Section 2 Applications

(FDL); 24-26 September 2013;

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[39] Zhang L, Ge N, Yang JJ, Li Z, Williams RS, Chen Y. Low voltage twostate-variable memristor model of vacancy-drift resistive switches. Applied Physics A: Materials Science &

[40] Oblea AS, Timilsina A, Moore D, Campbell KA. In: 2010 International Joint Conference on Neural Networks

[41] Miao F, Strachan JP, Yang JJ, Zhang MX, Goldfarb I, Torrezan AC, et al. Anatomy of a nanoscale conduction channel reveals the mechanism of a high performance memristor. Advanced Materials. 2011;**23**(47):5633-5640

[42] Miller KJ. [Ph.D. thesis]. Iowa State

switching memory. Nano Letters. 2008;

[45] Jeong DS, Schroeder H, Waser R. Coexistence of bipolar and unipolar resistive switching behaviors in a Pt∕TiO2∕Pt stack. Electrochemical and Solid-State Letters. 2007;**10**(8):G51-G53

[46] Sun X, Li G, Zhang XA, Ding L, Zhang W. Coexistence of the bipolar and unipolar resistive switching behaviours in Au/SrTiO3/Pt cells.

[43] Jo SH, Lu W. CMOS compatible nanoscale nonvolatile resistance

[44] Strachan JP, Torrezan AC, Medeiros-Ribeiro G, Williams RS. Measuring the switching dynamics and energy efficiency of tantalum oxide memristors. Nanotechnology. 2011;

Processing. 2015;**119**(1):1-9

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[38] Mbarek K, Ouaja Rziga F, Ghedira S,

*Memristors - Circuits and Applications of Memristor Devices*

Journal of Physics D: Applied Physics.

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[48] Chang T, Jo SH, Lu W. Short-term

[49] Lee SB, Chang SH, Yoo HK, Yoon MJ, Yang SM, Kang BS. Reversible changes between bipolar and unipolar resistance-switching phenomena in a Pt/

SrTiO x/Pt cell. Current Applied Physics. 2012;**12**(6):1515-1517

[50] Schindler C, Thermadam SCP, Waser R, Kozicki MN. Bipolar and unipolar resistive switching in Cu-Doped SiO2. IEEE Transactions on Electron Devices. 2007;**54**(10):

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2011;**44**(12):125404

Conference on Control, Automation and Diagnosis (ICCAD); Hammamet, Tunisia; 2017. pp. 054-059

pp. 1636-9874

**Chapter 4**

**Abstract**

circuits.

analog processors

**1. Introduction**

for multiple-solution mazes.

**45**

Memristive Grid for Maze Solving

Memcomputing represents a novel form of neuro-oriented signal processing that uses the memristor as a key element. In this chapter, a memristive grid is developed in order to achieve the specific task of solving mazes. This is done by resorting to the dynamic behavior of the memristance in order to find the shortest path that determines trajectory from entrance to exit. The structure of the maze is mapped onto the memristive grid, which is formed by memristors that are defined by fully analytical charge-controlled functions. The dependance on the electric charge permits to analyze the variation of the branch memristance of the grid as a function of time. As a result of the dynamic behavior of the developed memristor model, the shortest path is formed by those memristive branches exhibiting the fastest

memristance change. Special attention is given to achieve a realistic implementation

memristors and CMOS circuitry. HSPICE is used in combination with MATLAB to establish the simulation flow of the memristive grid. Besides, the memristor model is recast in VERILOG-A, a high-level hardware description language for analog

For thousands of years, mazes have intrigued the human mind [1]. The labyrinths have been used in research with laboratory animals, in order to study their ability to recognize their environment [2–4]. In the 1990s, artificial intelligence of robots was studied by examining their ability to traverse unfamiliar mazes [5–7]. Maze exploration algorithms are closely related to graph theory and have been used

There are several algorithms for maze solving in the literature, they can be classified in two very well-defined groups: the algorithms used by a traveler in the maze without knowledge of a general view of the maze, and the algorithms used for a program that can have a whole view the whole maze. Some examples of the first ones are the wall follower, random mouse, pledge algorithm [10], and Trémaux's algorithm [11]. In the second group, shortest path algorithms are most useful, because they can find the solution not only for a simple connected maze, but also

In this chapter, we put a main idea into practice, namely that the topology of a maze can be mapped onto a memristive grid. By exploiting the analog computations

memristive grids have demonstrated their ability for computing shortest paths in a

performed by solving Kirchoft's Current Laws (KCL) in a parallel manner,

of the fuses of the grid, which are formed by an anti-series connection of

**Keywords:** memristive grids, symbolic memristor modeling, maze-solving,

in both mathematics and computer science [8, 9].

*Arturo Sarmiento-Reyes and Yojanes Rodríguez Velásquez*

#### **Chapter 4**

## Memristive Grid for Maze Solving

*Arturo Sarmiento-Reyes and Yojanes Rodríguez Velásquez*

#### **Abstract**

Memcomputing represents a novel form of neuro-oriented signal processing that uses the memristor as a key element. In this chapter, a memristive grid is developed in order to achieve the specific task of solving mazes. This is done by resorting to the dynamic behavior of the memristance in order to find the shortest path that determines trajectory from entrance to exit. The structure of the maze is mapped onto the memristive grid, which is formed by memristors that are defined by fully analytical charge-controlled functions. The dependance on the electric charge permits to analyze the variation of the branch memristance of the grid as a function of time. As a result of the dynamic behavior of the developed memristor model, the shortest path is formed by those memristive branches exhibiting the fastest memristance change. Special attention is given to achieve a realistic implementation of the fuses of the grid, which are formed by an anti-series connection of memristors and CMOS circuitry. HSPICE is used in combination with MATLAB to establish the simulation flow of the memristive grid. Besides, the memristor model is recast in VERILOG-A, a high-level hardware description language for analog circuits.

**Keywords:** memristive grids, symbolic memristor modeling, maze-solving, analog processors

#### **1. Introduction**

For thousands of years, mazes have intrigued the human mind [1]. The labyrinths have been used in research with laboratory animals, in order to study their ability to recognize their environment [2–4]. In the 1990s, artificial intelligence of robots was studied by examining their ability to traverse unfamiliar mazes [5–7]. Maze exploration algorithms are closely related to graph theory and have been used in both mathematics and computer science [8, 9].

There are several algorithms for maze solving in the literature, they can be classified in two very well-defined groups: the algorithms used by a traveler in the maze without knowledge of a general view of the maze, and the algorithms used for a program that can have a whole view the whole maze. Some examples of the first ones are the wall follower, random mouse, pledge algorithm [10], and Trémaux's algorithm [11]. In the second group, shortest path algorithms are most useful, because they can find the solution not only for a simple connected maze, but also for multiple-solution mazes.

In this chapter, we put a main idea into practice, namely that the topology of a maze can be mapped onto a memristive grid. By exploiting the analog computations performed by solving Kirchoft's Current Laws (KCL) in a parallel manner, memristive grids have demonstrated their ability for computing shortest paths in a

describes the nonlinear drift mechanism, with a homotopy perturbation method

In order to obtain a charge-dependent memristance model, the nonlinear drift

where *η* defines the direction of the drift and it can be �1. Besides, *f <sup>w</sup>* is the window function used to define the nonlinear and bounded behavior of the state

**Figure 2** shows the resulting window plots for various values of *k*. In addition, *κ*

*<sup>κ</sup>* <sup>¼</sup> *<sup>μ</sup>Ron*

The main goal is to obtain a solution to Eq. (1) in the form of an analytical expression x(q). Once, this is done, this solution is substituted into the coupled

where *M t*ð Þ is the total memristance. Besides, *Ron* and *Roff* are the on-state and

In order to obtain an analytical solution to Eq. (1), we resort to the methodology reported in [24, 29], which is based on the homotopy perturbation method (HPM). HPM finds *x q*ð Þ for a given order of the homotopy method as well as the integer value of exponent of the window function (*k*). Furthermore, it should be also

As a result, the sign of the charge as well as the direction of the drift (*η*) allows us to introduce two operators that are used to simplify the final expressions for the solution. These operators are denoted as Λ and Θ. **Table 1** shows how they are

resistor equivalent of **Figure 3** which is expressed as [17]:

pointed out that the charge may be positive or negative.

defined depending on the signs of the charge and *η*.

where *μ*, R*on*, and Δ are the mobility, the ON-state resistance, and the dimension

*dq* <sup>¼</sup> *ηκ<sup>f</sup> <sup>w</sup>*ð Þ *x q*ð Þ (1)

*<sup>f</sup> <sup>w</sup>* <sup>¼</sup> <sup>1</sup> � ð Þ <sup>2</sup>*<sup>x</sup>* � <sup>1</sup> <sup>2</sup>*<sup>k</sup>* (2)

*M t*ðÞ¼ *Ronx q*ð Þþ *Roff* ½ � 1 � *x q*ð Þ (4)

<sup>Δ</sup><sup>2</sup> (3)

that yields an analytical expression for the memristance [24–27].

differential equation is expressed in terms of the electric charge:

variable *x q*ð Þ, and it is given as [28]:

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

the off-state resistances respectively.

is given as:

of the device.

**Figure 2.**

**47**

*Window function for different values of* k*.*

*dx q*ð Þ

**Figure 1.** *Basic circuit elements.*

given maze, levering on the dynamic adjustment of their intrinsic memristance [12, 13].

The parallel solution of KCL introduces a resemblance of the memristive grid as an analog processor [14] in counterposition to a digital approach in which the processing can also be done in parallel way, but the overhead in additional conversion circuitry is too high.

Two important milestones appear in the history of the memristor. The first one in 1971 when professor Leon O. Chua introduced the memristor as the fourth basic circuit element in his seminal paper [15]. It established that the memristor completes the number of possible relationships between the four fundamental circuit variables: current, voltage, magnetic flux, and electric charge—as depicted in **Figure 1**. Later, an extension to memristive systems was published in [16].

The second milestone occurred in 2008, when a team at Hewlett-Packard Laboratories fabricated a device whose behavior exhibited the memristance phenomenon [17]. Since the advent of the memristor as an actual device, research and technological development in several areas related to memristive applications have been increased.

In the field of signal processing, the memristor has special preponderance in neuro-computing and artificial neural networks because it allows new architectures and processing paradigms with important features based on biological neuronal systems [18–22]. In summary, a novel form of neuro-computing is on scene, namely memcomputing [23].

Memristive grids represent a family of neuro-computing systems that are able of achieving in a very flexible way several tasks for analog applications. In the next paragraphs, we present a specially tailored memristive grid that is focused on solving mazes.

The rest of the manuscript is organized as follows: in Section 2, the developed models are recast in a set of fully analytical expressions for the memristance, which are given as charge-controlled functions that are further used in this application. The components of the memristive grid are introduced in Section 3. The maze-solving procedure is introduced in Section 4 by explaining the simulation flow of the memristive grid. Subsequently, several mazes are solved in order to illustrate the operation of the memristive grid in Section 6. Finally, a series of conclusions is drawn.

#### **2. Development of a charge-controlled memristor model**

In this section, a charge-controlled memristor model is introduced. The model has been developed by solving the ordinary differential equation (ODE) that

describes the nonlinear drift mechanism, with a homotopy perturbation method that yields an analytical expression for the memristance [24–27].

In order to obtain a charge-dependent memristance model, the nonlinear drift differential equation is expressed in terms of the electric charge:

$$\frac{d\varkappa(q)}{dq} = \eta \mathfrak{x} \mathfrak{f}\_w(\varkappa(q)) \tag{1}$$

where *η* defines the direction of the drift and it can be �1. Besides, *f <sup>w</sup>* is the window function used to define the nonlinear and bounded behavior of the state variable *x q*ð Þ, and it is given as [28]:

$$f\_w = \mathbf{1} - \left(\mathbf{2x} - \mathbf{1}\right)^{2k} \tag{2}$$

**Figure 2** shows the resulting window plots for various values of *k*. In addition, *κ* is given as:

$$
\kappa = \frac{\mu R\_{on}}{\Delta^2} \tag{3}
$$

where *μ*, R*on*, and Δ are the mobility, the ON-state resistance, and the dimension of the device.

The main goal is to obtain a solution to Eq. (1) in the form of an analytical expression x(q). Once, this is done, this solution is substituted into the coupled resistor equivalent of **Figure 3** which is expressed as [17]:

$$M(t) = R\_{on} \varkappa(q) + R\_{q\overline{f}}[\mathbf{1} - \varkappa(q)] \tag{4}$$

where *M t*ð Þ is the total memristance. Besides, *Ron* and *Roff* are the on-state and the off-state resistances respectively.

In order to obtain an analytical solution to Eq. (1), we resort to the methodology reported in [24, 29], which is based on the homotopy perturbation method (HPM). HPM finds *x q*ð Þ for a given order of the homotopy method as well as the integer value of exponent of the window function (*k*). Furthermore, it should be also pointed out that the charge may be positive or negative.

As a result, the sign of the charge as well as the direction of the drift (*η*) allows us to introduce two operators that are used to simplify the final expressions for the solution. These operators are denoted as Λ and Θ. **Table 1** shows how they are defined depending on the signs of the charge and *η*.

**Figure 2.** *Window function for different values of* k*.*

given maze, levering on the dynamic adjustment of their intrinsic memristance

*v q*

**Resistor**

*Memristors - Circuits and Applications of Memristor Devices*

**Capacitor**

**Memristor**

*i* φ

**Inductor**

an analog processor [14] in counterposition to a digital approach in which the processing can also be done in parallel way, but the overhead in additional conver-

The parallel solution of KCL introduces a resemblance of the memristive grid as

Two important milestones appear in the history of the memristor. The first one in 1971 when professor Leon O. Chua introduced the memristor as the fourth basic circuit element in his seminal paper [15]. It established that the memristor completes the number of possible relationships between the four fundamental circuit variables: current, voltage, magnetic flux, and electric charge—as depicted in **Figure 1**. Later, an extension to memristive systems was published in [16].

The second milestone occurred in 2008, when a team at Hewlett-Packard Laboratories fabricated a device whose behavior exhibited the memristance phenomenon [17]. Since the advent of the memristor as an actual device, research and technological development in several areas related to memristive applications have

In the field of signal processing, the memristor has special preponderance in neuro-computing and artificial neural networks because it allows new architectures and processing paradigms with important features based on biological neuronal systems [18–22]. In summary, a novel form of neuro-computing is on scene, namely

Memristive grids represent a family of neuro-computing systems that are able of achieving in a very flexible way several tasks for analog applications. In the next paragraphs, we present a specially tailored memristive grid that is focused on

The rest of the manuscript is organized as follows: in Section 2, the developed models are recast in a set of fully analytical expressions for the memristance, which are given as charge-controlled functions that are further used in this application. The components of the memristive grid are introduced in Section 3. The maze-solving procedure is introduced in Section 4 by explaining the simulation flow of the memristive grid. Subsequently, several mazes are solved in order to illustrate the operation of the memristive grid in Section 6. Finally, a series of conclusions is drawn.

In this section, a charge-controlled memristor model is introduced. The model

has been developed by solving the ordinary differential equation (ODE) that

**2. Development of a charge-controlled memristor model**

[12, 13].

**Figure 1.**

*Basic circuit elements.*

sion circuitry is too high.

been increased.

solving mazes.

**46**

memcomputing [23].

In a similar way, an expression for the memristance for order-3 and *k* ¼ 1 can be

It can be noticed that HPM produces nested expressions of the memristance, that is to say, a given memristance of a given order is expressed as function of the

<sup>8</sup>Λ*κ<sup>q</sup>* � 3 2*X*<sup>2</sup>

<sup>P</sup>1*e*8Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>2*e*16Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>3*e*24Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>4*e*32Λ*κ<sup>q</sup>*

<sup>þ</sup>P5*e*40Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>6*e*48Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>7*e*56Λ*κ<sup>q</sup>*

<sup>0</sup>P9*e*

<sup>0</sup> � 63*X*<sup>0</sup> þ 19

0*e* <sup>40</sup>Λ*κ<sup>q</sup>* <sup>þ</sup>

<sup>16</sup>Λ*κ<sup>q</sup>* � *<sup>X</sup>*<sup>3</sup>

24 5 *e* <sup>0</sup>P10*e*

<sup>48</sup>Λ*κ<sup>q</sup>* � <sup>8</sup> 9 *e* 56Λ*κq*

24Λ*κq*

106 45

*<sup>X</sup>*<sup>0</sup> � <sup>184</sup> 9

(12)

8Λ*κq*

<sup>8</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>2</sup>*X*<sup>3</sup>

<sup>32</sup>Λ*κ<sup>q</sup>* � <sup>13</sup>*X*<sup>5</sup>

<sup>0</sup> <sup>þ</sup> <sup>75</sup>*X*<sup>2</sup>

<sup>0</sup> � 65*X*<sup>0</sup> þ 13

<sup>0</sup> <sup>þ</sup> <sup>5</sup>*X*<sup>2</sup> <sup>0</sup> � <sup>4</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>0</sup> � 630*X*<sup>0</sup> þ 405

<sup>0</sup> þ 26*X*<sup>0</sup> � 10

<sup>0</sup> � <sup>32</sup>*X*<sup>3</sup>

<sup>0</sup> � <sup>2</sup>*X*<sup>3</sup>

<sup>0</sup> � <sup>2</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � �*<sup>e</sup>*

<sup>0</sup> � <sup>8</sup> 3 *X*<sup>0</sup> þ 2 3

� �*e*<sup>32</sup>Λ*κ<sup>q</sup>* � <sup>1</sup>

16Λ*κq*

1 A

(10)

(11)

*MO*3*K*<sup>1</sup> <sup>¼</sup> *MO*2*K*<sup>1</sup> <sup>þ</sup> *Rd* <sup>Θ</sup>ð Þ *Xo* � <sup>1</sup> <sup>4</sup> �*e*<sup>Λ</sup>4*κ<sup>q</sup>* � <sup>3</sup>*e*<sup>Λ</sup>8*κ<sup>q</sup>* � <sup>3</sup>*e*<sup>Λ</sup>12*κ<sup>q</sup>* � *<sup>e</sup>*<sup>Λ</sup>16*κ<sup>q</sup>* � � <sup>h</sup>

<sup>0</sup> �*e*<sup>Λ</sup>16*κ<sup>q</sup>* � <sup>3</sup>*e*<sup>Λ</sup>12*κ<sup>q</sup>* � <sup>3</sup>*e*<sup>Λ</sup>8*κ<sup>q</sup>* � *<sup>e</sup>*<sup>Λ</sup>4*κ<sup>q</sup>* � �ð Þ <sup>1</sup> � <sup>Θ</sup> �

For order-1 and *k* ¼ 2, the memristance is given as follows:

<sup>0</sup> � <sup>3</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � �*e*<sup>24</sup>Λ*κ<sup>q</sup>* � <sup>4</sup>

<sup>0</sup> � <sup>6</sup>*X*<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>9</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>3</sup> � �*<sup>e</sup>*

<sup>0</sup>*e*<sup>16</sup>Λ*κ<sup>q</sup>* � <sup>2</sup>*X*<sup>3</sup>

Θ

þ 8 9 *X*4 <sup>0</sup>P11*e*

<sup>0</sup> � <sup>22</sup>*X*<sup>2</sup>

<sup>0</sup> <sup>þ</sup> <sup>36</sup>*X*<sup>4</sup>

<sup>0</sup> <sup>þ</sup> <sup>130</sup>*X*<sup>2</sup>

<sup>0</sup> <sup>þ</sup> <sup>495</sup>*X*<sup>2</sup>

<sup>0</sup> þ 18*X*<sup>0</sup> þ 9

<sup>0</sup> þ 9*X*<sup>0</sup> þ 18

0 @

> � 1 45 *X*3 <sup>0</sup>P8*e*

<sup>0</sup> � <sup>2</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>2</sup> � �*<sup>e</sup>*

<sup>0</sup>*e*<sup>24</sup>Λ*κ<sup>q</sup>* <sup>þ</sup>

In a similar way, the memristance for order-2 and *k* ¼ 2 is given:

3 *X*4 <sup>0</sup> � <sup>8</sup> 3 *X*3 <sup>0</sup> <sup>þ</sup> <sup>4</sup>*X*<sup>2</sup>

2 3 *X*4 0*e* 32Λ*κq*

obtained:

*MO*1*K*<sup>2</sup> ¼ *Rd*Θ

<sup>þ</sup>*X*<sup>4</sup>

memristance of lower orders.

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

þ *Rd*

*MO*2*K*<sup>2</sup> ¼ *MO*1*K*<sup>2</sup> þ *Rd*

<sup>P</sup><sup>1</sup> ¼ � <sup>128</sup> 45 *X*6 0 þ 128 15 *X*5 <sup>0</sup> � <sup>89</sup> 9 *X*4 0 þ 50 9 *X*3 0 þ 11 3 *X*2 <sup>0</sup> � <sup>226</sup> 45 *X*<sup>0</sup> þ

<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>*X*<sup>4</sup>

<sup>P</sup><sup>3</sup> <sup>¼</sup> <sup>8</sup>*X*<sup>6</sup>

<sup>P</sup><sup>4</sup> ¼ � <sup>16</sup> 9 *X*6 0 þ 16 3 *X*5 <sup>0</sup> � <sup>488</sup> 9 *X*4 0 þ 896 9 *X*3 <sup>0</sup> � <sup>400</sup> 3 *X*2 0 þ 760 9

<sup>P</sup><sup>5</sup> <sup>¼</sup> <sup>65</sup>*X*<sup>4</sup>

<sup>P</sup><sup>6</sup> <sup>¼</sup> <sup>2</sup>*X*<sup>2</sup>

<sup>P</sup><sup>8</sup> <sup>¼</sup> <sup>40</sup>*X*<sup>4</sup>

<sup>P</sup><sup>9</sup> <sup>¼</sup> <sup>2</sup>*X*<sup>2</sup>

<sup>P</sup><sup>10</sup> <sup>¼</sup> <sup>4</sup>*X*<sup>3</sup>

<sup>P</sup><sup>11</sup> <sup>¼</sup> <sup>2</sup>*X*<sup>3</sup>

**49**

<sup>P</sup><sup>7</sup> <sup>¼</sup> <sup>56</sup> 9 *X*6 <sup>0</sup> � <sup>56</sup> 3 *X*5 0 þ 280 9 *X*4 <sup>0</sup> � <sup>280</sup> 9 *X*3 0 þ 56 3 *X*2 <sup>0</sup> � <sup>56</sup> 9 *X*<sup>0</sup> þ 8 9

<sup>0</sup> � <sup>8</sup>*X*<sup>3</sup>

<sup>0</sup> � <sup>24</sup>*X*<sup>5</sup>

<sup>0</sup> � <sup>130</sup>*X*<sup>3</sup>

<sup>0</sup> � <sup>2</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � � *<sup>X</sup>*<sup>4</sup>

<sup>0</sup> � <sup>204</sup>*X*<sup>3</sup>

<sup>0</sup> � 6*X*<sup>0</sup> þ 9

<sup>0</sup> � <sup>12</sup>*X*<sup>2</sup>

<sup>0</sup> � <sup>6</sup>*X*<sup>2</sup>

<sup>þ</sup>2 3*X*<sup>2</sup>

� 1 3 *X*<sup>0</sup> 2*X*<sup>3</sup>

<sup>þ</sup>3*X*<sup>2</sup>

4 <sup>3</sup> *<sup>X</sup>*<sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>1</sup> � � *<sup>X</sup>*<sup>2</sup>

#### **Figure 3.** *Coupled series equivalent of the memristor.*


#### **Table 1.**

*Operators for the signs of η and q.*

As a matter of an example, the expression of *x q*ð Þ for order-1 with *k* ¼ 1 is given as: *XO*1*K*1ð Þ¼ *<sup>q</sup>* <sup>Θ</sup> <sup>1</sup> <sup>þ</sup> ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> <sup>2</sup> *e* <sup>8</sup>Λ*κ<sup>q</sup>* � ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>2</sup> *<sup>e</sup>* 4Λ*κq* h i <sup>þ</sup> ð Þ <sup>1</sup> � <sup>Θ</sup> *<sup>X</sup>*<sup>0</sup> �*X*0*<sup>e</sup>* <sup>8</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> ð Þ *<sup>X</sup>*<sup>0</sup> <sup>þ</sup> <sup>1</sup> *<sup>e</sup>* <sup>4</sup>Λ*κ<sup>q</sup>* � � (5)

After substituting Eq. (5) in Eq. (4), it results in the memristance expression:

$$\begin{split} M\_{\text{O1K1}} &= \Theta \Big[ R\_d (\text{Xo} - \mathbf{1}) \Big[ (\mathbf{X}\_0 - \mathbf{2}) e^{\Lambda 4 \kappa q} - (\mathbf{X} \mathbf{o} - \mathbf{1}) e^{\Lambda 8 \kappa q} \Big] + R\_{\text{ON}} \Big] \\ &+ (\mathbf{1} - \Theta) \Big[ R\_d \mathbf{X}\_0 \Big[ \mathbf{X}\_0 e^{\Lambda 8 \kappa q} - (\mathbf{X}\_0 + \mathbf{1}) e^{\Lambda 4 \kappa q} \Big] + R\_{\text{off}} \Big] \end{split} \tag{6}$$

where the variable *Rd* is given as:

$$R\_d = R\_{q\overline{f}} - R\_{on} \tag{7}$$

For order-2 and *k* ¼ 1, the solution to Eq. (1) is given as:

$$X\_{O2K1}(q) = \Theta \left[ 1 + (X\text{o} - 1) \left( X\_0^2 - 3X\text{o} + 3 \right) e^{4\Lambda aq} - (X\text{o} - 1)^2 (2X\text{o} - 3) e^{8\Lambda aq} + (X\text{o} - 1)^3 e^{12\Lambda aq} \right] \tag{8}$$

$$+ (1 - \Theta) \left[ X\_0 (X\_0^2 + X\text{o} + 1) e^{4\Lambda aq} - X\_0^2 (2X\text{o} + 1) e^{8\Lambda aq} + X\_0^3 e^{12\Lambda aq} \right] \tag{8}$$

Again, after substituting the expression above in Eq. (4) and after some reductions, it is possible to obtain the memristance for order-2 and *k* ¼ 1 as:

$$\begin{split} M\_{O2K1} &= M\_{O1K1} + Rd \Big[ \Theta (\mathbf{X}o - \mathbf{1})^3 (-e^{\Lambda 4cq} - 2e^{\Lambda 8cq} - e^{\Lambda 12cq}) \\ &+ X\_0^3 (-e^{\Lambda 12cq} - 2e^{\Lambda 8cq} - e^{\Lambda 4cq}) (\mathbf{1} - \Theta) \Big] \end{split} \tag{9}$$

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

In a similar way, an expression for the memristance for order-3 and *k* ¼ 1 can be obtained:

$$\begin{split} M\_{O3K1} &= M\_{O2K1} + Rd \Big[ \Theta(\mathbf{X}o - \mathbf{1})^4 \left( -e^{\Lambda 4aq} - \mathbf{3} e^{\Lambda 8aq} - \mathbf{3} e^{\Lambda 12aq} - e^{\Lambda 16aq} \right) \\ &+ X\_0^4 \Big( -e^{\Lambda 16aq} - \mathbf{3} e^{\Lambda 12aq} - \mathbf{3} e^{\Lambda 8aq} - e^{\Lambda 4aq} \Big) (\mathbf{1} - \Theta) \Big] \end{split} \tag{10}$$

It can be noticed that HPM produces nested expressions of the memristance, that is to say, a given memristance of a given order is expressed as function of the memristance of lower orders.

For order-1 and *k* ¼ 2, the memristance is given as follows:

$$M\_{\rm OK2} = R\_d \Theta \left[ \begin{aligned} &\frac{4}{3} \left( X\_0^2 + 1 \right) \left( X\_0^2 - 2X\_0 + 2 \right) e^{3\Lambda \alpha q} - 3 \left( 2X\_0^2 - 2X\_0 + 1 \right) e^{16\Lambda \alpha q} \\ &+ 2 \left( 3X\_0^2 - 3X\_0 + 1 \right) e^{2\Lambda \lambda \alpha q} - \left( \frac{4}{3} X\_0^4 - \frac{8}{3} X\_0^3 + 4 X\_0^2 - \frac{8}{3} X\_0 + \frac{2}{3} \right) e^{2\Lambda \alpha q} - 1 \right] \\ &+ R\_d \left[ \begin{array}{c} & - \frac{1}{3} X\_0 \left( 2X\_0^3 - 6X\_0^2 + 9X\_0 + 3 \right) e^{8\Lambda \alpha q} \\ &+ 3X\_0^2 e^{16\Lambda \alpha q} - 2X\_0^3 e^{24\Lambda \alpha q} + \frac{2}{3} X\_0^4 e^{32\Lambda \alpha q} \end{array} \right] + R\_{\rm off} \end{aligned} \tag{11}$$

In a similar way, the memristance for order-2 and *k* ¼ 2 is given:

*MO*2*K*<sup>2</sup> ¼ *MO*1*K*<sup>2</sup> þ *Rd* Θ <sup>P</sup>1*e*8Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>2*e*16Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>3*e*24Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>4*e*32Λ*κ<sup>q</sup>* <sup>þ</sup>P5*e*40Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>6*e*48Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>P</sup>7*e*56Λ*κ<sup>q</sup>* 0 @ 1 A � 1 45 *X*3 <sup>0</sup>P8*e* <sup>8</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> <sup>2</sup>*X*<sup>3</sup> <sup>0</sup>P9*e* <sup>16</sup>Λ*κ<sup>q</sup>* � *<sup>X</sup>*<sup>3</sup> <sup>0</sup>P10*e* 24Λ*κq* þ 8 9 *X*4 <sup>0</sup>P11*e* <sup>32</sup>Λ*κ<sup>q</sup>* � <sup>13</sup>*X*<sup>5</sup> 0*e* <sup>40</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> 24 5 *e* <sup>48</sup>Λ*κ<sup>q</sup>* � <sup>8</sup> 9 *e* 56Λ*κq* 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 <sup>P</sup><sup>1</sup> ¼ � <sup>128</sup> 45 *X*6 0 þ 128 15 *X*5 <sup>0</sup> � <sup>89</sup> 9 *X*4 0 þ 50 9 *X*3 0 þ 11 3 *X*2 <sup>0</sup> � <sup>226</sup> 45 *X*<sup>0</sup> þ 106 45 <sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>*X*<sup>4</sup> <sup>0</sup> � <sup>8</sup>*X*<sup>3</sup> <sup>0</sup> � <sup>22</sup>*X*<sup>2</sup> <sup>0</sup> þ 26*X*<sup>0</sup> � 10 <sup>P</sup><sup>3</sup> <sup>¼</sup> <sup>8</sup>*X*<sup>6</sup> <sup>0</sup> � <sup>24</sup>*X*<sup>5</sup> <sup>0</sup> <sup>þ</sup> <sup>36</sup>*X*<sup>4</sup> <sup>0</sup> � <sup>32</sup>*X*<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>75</sup>*X*<sup>2</sup> <sup>0</sup> � 63*X*<sup>0</sup> þ 19 <sup>P</sup><sup>4</sup> ¼ � <sup>16</sup> 9 *X*6 0 þ 16 3 *X*5 <sup>0</sup> � <sup>488</sup> 9 *X*4 0 þ 896 9 *X*3 <sup>0</sup> � <sup>400</sup> 3 *X*2 0 þ 760 9 *<sup>X</sup>*<sup>0</sup> � <sup>184</sup> 9 <sup>P</sup><sup>5</sup> <sup>¼</sup> <sup>65</sup>*X*<sup>4</sup> <sup>0</sup> � <sup>130</sup>*X*<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>130</sup>*X*<sup>2</sup> <sup>0</sup> � 65*X*<sup>0</sup> þ 13 <sup>P</sup><sup>6</sup> <sup>¼</sup> <sup>2</sup>*X*<sup>2</sup> <sup>0</sup> � <sup>2</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � � *<sup>X</sup>*<sup>4</sup> <sup>0</sup> � <sup>2</sup>*X*<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>5</sup>*X*<sup>2</sup> <sup>0</sup> � <sup>4</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � � <sup>P</sup><sup>7</sup> <sup>¼</sup> <sup>56</sup> 9 *X*6 <sup>0</sup> � <sup>56</sup> 3 *X*5 0 þ 280 9 *X*4 <sup>0</sup> � <sup>280</sup> 9 *X*3 0 þ 56 3 *X*2 <sup>0</sup> � <sup>56</sup> 9 *X*<sup>0</sup> þ 8 9 <sup>P</sup><sup>8</sup> <sup>¼</sup> <sup>40</sup>*X*<sup>4</sup> <sup>0</sup> � <sup>204</sup>*X*<sup>3</sup> <sup>0</sup> <sup>þ</sup> <sup>495</sup>*X*<sup>2</sup> <sup>0</sup> � 630*X*<sup>0</sup> þ 405 <sup>P</sup><sup>9</sup> <sup>¼</sup> <sup>2</sup>*X*<sup>2</sup> <sup>0</sup> � 6*X*<sup>0</sup> þ 9 <sup>P</sup><sup>10</sup> <sup>¼</sup> <sup>4</sup>*X*<sup>3</sup> <sup>0</sup> � <sup>12</sup>*X*<sup>2</sup> <sup>0</sup> þ 18*X*<sup>0</sup> þ 9 <sup>P</sup><sup>11</sup> <sup>¼</sup> <sup>2</sup>*X*<sup>3</sup> <sup>0</sup> � <sup>6</sup>*X*<sup>2</sup> <sup>0</sup> þ 9*X*<sup>0</sup> þ 18 (12)

As a matter of an example, the expression of *x q*ð Þ for order-1 with *k* ¼ 1 is given as:

Θ ¼ 1

Θ ¼ 0

**ON R (x) OFF R (x)**

**doped** undoped

α =

**R RON OFF**

*q*≥0 *q*<0

After substituting Eq. (5) in Eq. (4), it results in the memristance expression:

� �

<sup>Λ</sup>8*κ<sup>q</sup>* � ð Þ *<sup>X</sup>*<sup>0</sup> <sup>þ</sup> <sup>1</sup> *<sup>e</sup>* <sup>Λ</sup>4*κ<sup>q</sup>* � � <sup>þ</sup> *Roff*

<sup>4</sup>Λ*κ<sup>q</sup>* � ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> <sup>2</sup>

<sup>12</sup>Λ*κ<sup>q</sup>* � �

<sup>4</sup>Λ*κ<sup>q</sup>* � *<sup>X</sup>*<sup>2</sup>

Again, after substituting the expression above in Eq. (4) and after some reduc-

*MO*2*K*<sup>1</sup> <sup>¼</sup> *MO*1*K*<sup>1</sup> <sup>þ</sup> *Rd* <sup>Θ</sup>ð Þ *Xo* � <sup>1</sup> <sup>3</sup> �*e*<sup>Λ</sup>4*κ<sup>q</sup>* � <sup>2</sup>*e*<sup>Λ</sup>8*κ<sup>q</sup>* � *<sup>e</sup>*<sup>Λ</sup>12*κ<sup>q</sup>* ð Þ

<sup>0</sup> �*e*<sup>Λ</sup>12*κ<sup>q</sup>* � <sup>2</sup>*e*<sup>Λ</sup>8*κ<sup>q</sup>* � *<sup>e</sup>*<sup>Λ</sup>4*κ<sup>q</sup>* ð Þð Þ <sup>1</sup> � <sup>Θ</sup> �

h i

<sup>0</sup>ð Þ 2*X*<sup>0</sup> þ 1 *e*

4Λ*κq*

<sup>Λ</sup>4*κ<sup>q</sup>* � ð Þ *Xo* � <sup>1</sup> *<sup>e</sup>* <sup>Λ</sup>8*κ<sup>q</sup>* � � <sup>þ</sup> *RON*

� � (6)

*Rd* ¼ *Roff* � *Ron* (7)

ð Þ 2*X*<sup>0</sup> � 3 *e*

<sup>8</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> *<sup>X</sup>*<sup>3</sup> 0*e*

þ ð Þ 1 � Θ *X*<sup>0</sup> �*X*0*e*

<sup>8</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> ð Þ *<sup>X</sup>*<sup>0</sup> <sup>þ</sup> <sup>1</sup> *<sup>e</sup>* <sup>4</sup>Λ*κ<sup>q</sup>* � �

Λ ¼ 1 Θ ¼ 0

Λ ¼ 1 Θ ¼ 1

<sup>8</sup>Λ*κ<sup>q</sup>* <sup>þ</sup> ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> <sup>3</sup>

*e* 12Λ*κq*

(8)

(9)

(5)

<sup>8</sup>Λ*κ<sup>q</sup>* � ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>2</sup> *<sup>e</sup>*

h i

*XO*1*K*1ð Þ¼ *<sup>q</sup>* <sup>Θ</sup> <sup>1</sup> <sup>þ</sup> ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> <sup>2</sup>

*Operators for the signs of η and q.*

*Coupled series equivalent of the memristor.*

**Figure 3.**

**Table 1.**

**48**

*e*

*η*<sup>þ</sup> Λ ¼ �1

*Memristors - Circuits and Applications of Memristor Devices*

*η*� Λ ¼ �1

*MO*1*K*<sup>1</sup> ¼ Θ *Rd*ð Þ *Xo* � 1 ð Þ *X*<sup>0</sup> � 2 *e*

where the variable *Rd* is given as:

<sup>þ</sup> ð Þ <sup>1</sup> � <sup>Θ</sup> *<sup>X</sup>*<sup>0</sup> *<sup>X</sup>*<sup>2</sup>

<sup>þ</sup> *<sup>X</sup>*<sup>3</sup>

*XO*2*K*1ð Þ¼ *<sup>q</sup>* <sup>Θ</sup> <sup>1</sup> <sup>þ</sup> ð Þ *<sup>X</sup>*<sup>0</sup> � <sup>1</sup> *<sup>X</sup>*<sup>2</sup>

þ ð Þ 1 � Θ *RdX*<sup>0</sup> *X*0*e*

For order-2 and *k* ¼ 1, the solution to Eq. (1) is given as:

<sup>0</sup> � <sup>3</sup>*X*<sup>0</sup> <sup>þ</sup> <sup>3</sup> � �*<sup>e</sup>*

tions, it is possible to obtain the memristance for order-2 and *k* ¼ 1 as:

h

<sup>0</sup> <sup>þ</sup> *<sup>X</sup>*<sup>0</sup> <sup>þ</sup> <sup>1</sup> � �*<sup>e</sup>*

Eqs. (6), (9)–(12) are indeed the analytical expressions that constitute memristor models. References [29, 30] contain a proper characterization of the resulting models.

the blocked paths are represented by memristive fuses in red, on the contrary, the paths that can be followed are represented by memristive fuses in white. It clearly results that the walls should be given by memristive fuses with the switch in the OFF-state (high-resistance), while the open paths are constituted by memristive

On the top of this, the memristive grid can be straightforwardly adapted to other kinds of mazes. Mazes with multiple entrances are represented with multiple input voltages. Similarly, mazes with multiple outputs are given by setting multiple

A close look of the solution path in **Figure 6a** can lead us to a graph-theoretical explanation on how the memristive grid solves the maze, because the open ways in the maze can be regarded as an unweighted graph where the solution path is subgraph. The solution path can be found by using a *breath-first-search* (BFS) algorithm in order to traverse the graph which yields indeed the shortest-path

The application of BFS is illustrated by determining the shortest path between nodes 3 and 6 of the graph from **Figure 7a**. Here, node 3 can be regarded as the input (*i*) and node 6 as the output (*o*). The algorithm starts by selecting the initial node (3). From this, a first level of coloring is achieved by selecting the neighboring nodes (2, 4, 5). This procedure is repeated until all nodes have been visited. For this

graph, it suffices with 2 levels. The shortest path is defined by the sequence

*Mapping the maze onto the memristive grid. (a) Maze, (b) Grid and (c) Merging the maze and the grid.*

fuses with the switch in the ON-state (low-resistance).

because we deal with an unweighted graph [31].

3!5!6, which is shown in red in **Figure 7b**.

instances of the ground node.

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

**3.1 An algorithmic view**

**Figure 6.**

**51**

#### **3. Implementing the memristive grid**

A memristive grid is a rectangular array of memristive branches, as shown in **Figure 4**. Herein, the memristive branches have been denoted as *bricked* circuit elements called memristive fuses. In addition, a memristive fuse is composed of a series connection of two memristors in anti-series and a switching device [14].

The switch is used to define the structure of the labyrinth, if the switch is in the ON-state, then the way is free, while if the switch is in the OFF-state then a wall is encountered. **Figure 5** shows the equivalent of the memristive fuse.

In order to illustrate the use of the memristive grid in describing a maze, the maze of **Figure 6a** is used. The entrance of the maze is marked by the green arrow and the output is marked by a red arrow, and the walls are shown in red. The maze is mapped onto the memristive grid as shown in **Figure 6b** by denoting the entrance of the maze as a voltage source, while the output of the maze is given by the ground node. For sake of clarity, both figures are merged into **Figure 6c**, where

**Figure 4.** *Description of the memristive grid.*

**Figure 5.** *Configuration of the memristive fuse for maze solving.*

#### *Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

the blocked paths are represented by memristive fuses in red, on the contrary, the paths that can be followed are represented by memristive fuses in white. It clearly results that the walls should be given by memristive fuses with the switch in the OFF-state (high-resistance), while the open paths are constituted by memristive fuses with the switch in the ON-state (low-resistance).

On the top of this, the memristive grid can be straightforwardly adapted to other kinds of mazes. Mazes with multiple entrances are represented with multiple input voltages. Similarly, mazes with multiple outputs are given by setting multiple instances of the ground node.

#### **3.1 An algorithmic view**

Eqs. (6), (9)–(12) are indeed the analytical expressions that constitute memristor models. References [29, 30] contain a proper characterization of the

A memristive grid is a rectangular array of memristive branches, as shown in **Figure 4**. Herein, the memristive branches have been denoted as *bricked* circuit elements called memristive fuses. In addition, a memristive fuse is composed of a series connection of two memristors in anti-series and a switching device [14].

The switch is used to define the structure of the labyrinth, if the switch is in the ON-state, then the way is free, while if the switch is in the OFF-state then a wall is

In order to illustrate the use of the memristive grid in describing a maze, the maze of **Figure 6a** is used. The entrance of the maze is marked by the green arrow and the output is marked by a red arrow, and the walls are shown in red. The maze is mapped onto the memristive grid as shown in **Figure 6b** by denoting the entrance of the maze as a voltage source, while the output of the maze is given by the ground node. For sake of clarity, both figures are merged into **Figure 6c**, where

encountered. **Figure 5** shows the equivalent of the memristive fuse.

resulting models.

**Figure 4.**

**Figure 5.**

**50**

*Description of the memristive grid.*

*Configuration of the memristive fuse for maze solving.*

**3. Implementing the memristive grid**

*Memristors - Circuits and Applications of Memristor Devices*

A close look of the solution path in **Figure 6a** can lead us to a graph-theoretical explanation on how the memristive grid solves the maze, because the open ways in the maze can be regarded as an unweighted graph where the solution path is subgraph. The solution path can be found by using a *breath-first-search* (BFS) algorithm in order to traverse the graph which yields indeed the shortest-path because we deal with an unweighted graph [31].

The application of BFS is illustrated by determining the shortest path between nodes 3 and 6 of the graph from **Figure 7a**. Here, node 3 can be regarded as the input (*i*) and node 6 as the output (*o*). The algorithm starts by selecting the initial node (3). From this, a first level of coloring is achieved by selecting the neighboring nodes (2, 4, 5). This procedure is repeated until all nodes have been visited. For this graph, it suffices with 2 levels. The shortest path is defined by the sequence 3!5!6, which is shown in red in **Figure 7b**.

**Figure 6.** *Mapping the maze onto the memristive grid. (a) Maze, (b) Grid and (c) Merging the maze and the grid.*

It clearly results that the overall performance of the grid in solving mazes is based on the model of the memristors that form the fuses. Even though the models are recast in fully symbolic form—which represent a great advantage, numeric values should be assigned to the parameters of the model, as given in **Table 2**. Since variations of the model parameters may appear, it is important to notice that the anti-series connection alleviates the possible effects of those variations. Specific sensitivity analysis on the parameter variations of the charge-controlled models are

In the memristive fuse, an ideal switch can be used in the process of finding the solution, however, with the aim to have a more realistic switch, a transmission gate is used instead. The transmission gate is a switch in CMOS technology, it consists of an NMOS transistor and a PMOS transistor connected in parallel, as in **Figure 9a**. Both devices in combination can fully transmit any signal value between *Vdd* (the supply voltage of the transistors) and ground. In order to switch, each transistor requires a complementary control input. Therefore, it is necessary to add an inverter connected between the control input and the PMOS gate [30, 32].

If the control input is *Vdd* then the switch is closed, and as a result, the transmission gate can pass the input signal to output because it exhibits a low-resistance. On the contrary, if the control input is grounded, then the switch is opened and the

In order to simulate the transmission gate of the memristive fuse, a CMOS 180 nm technology is used. The parameters of the two complementary transistors are shown in **Table 3**. The equivalent resistance of the transmission gate both states

The resistance values are extracted making a sweep of the input voltage and measure the equivalent average resistance of the transistors in the ON-state

CMOS TG *W μm L μm* PMOS 1.44 0.18 NMOS 0.48 0.18

given in [30].

**Figure 9.**

**Table 3.**

**53**

*3.2.1 Switch implementation*

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

transmission gate presents a high-resistance.

*Transmission gate. (a) Configuration and (b) symbol.*

*Transmission gate: transistor parameters.*

as a function of the input voltage is shown in **Figure 10**.

**Figure 7.** *BFS algorithm to obtain the shortest path. (a) A graph and (b) The BFS algorithm.*

As a result, by representing the graph with the memristive grid, it allows us to define ways for the current to flow through the open paths by gradually changing the equivalent memristance of the fuse. Besides, what is more relevant, since the current is given as the time-derivative of the charge, then the solution of the maze is always given by the shortest path to ground which represents the path with the fastest changing memristance.

#### **3.2 Technical specifications of the memristive fuse**

The memristive fuse from **Figure 5** contains a pair of memristors in anti-series connection. Such a memristor connection produces an *M*-*q* characteristic that is composed of the overlapping of the *M*-*q* curves of the memristor expressions for *η*� and *η*þ. **Figure 8a** shows the *M*-*q* characteristics for the model of order-1, *k* ¼ 5 and **Figure 8b** shows the schematic curve with the values of *Roff* and *Rinit*. Physical parameters of the memristor model are given by the nominal values of the HP memristor. A summary of the specs for the memristor model is given in **Table 2**.

**Figure 8.**

*Memristance-charge characteristic of the anti-series connection. (a) MO*1*K*<sup>5</sup> *and (b) M-q.*


**Table 2.**

*Memristor parameters of the anti-series connection.*

#### *Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

It clearly results that the overall performance of the grid in solving mazes is based on the model of the memristors that form the fuses. Even though the models are recast in fully symbolic form—which represent a great advantage, numeric values should be assigned to the parameters of the model, as given in **Table 2**. Since variations of the model parameters may appear, it is important to notice that the anti-series connection alleviates the possible effects of those variations. Specific sensitivity analysis on the parameter variations of the charge-controlled models are given in [30].

#### *3.2.1 Switch implementation*

As a result, by representing the graph with the memristive grid, it allows us to define ways for the current to flow through the open paths by gradually changing the equivalent memristance of the fuse. Besides, what is more relevant, since the current is given as the time-derivative of the charge, then the solution of the maze is always given by the shortest path to ground which represents the path with the

The memristive fuse from **Figure 5** contains a pair of memristors in anti-series connection. Such a memristor connection produces an *M*-*q* characteristic that is composed of the overlapping of the *M*-*q* curves of the memristor expressions for *η*� and *η*þ. **Figure 8a** shows the *M*-*q* characteristics for the model of order-1, *k* ¼ 5 and **Figure 8b** shows the schematic curve with the values of *Roff* and *Rinit*. Physical parameters of the memristor model are given by the nominal values of the HP memristor. A summary of the specs for the memristor model is given in **Table 2**.

fastest changing memristance.

**Figure 7.**

**Figure 8.**

**Table 2.**

**52**

*μ<sup>v</sup> <sup>m</sup>*<sup>2</sup> *Vs*

*Memristor parameters of the anti-series connection.*

**3.2 Technical specifications of the memristive fuse**

*Memristance-charge characteristic of the anti-series connection. (a) MO*1*K*<sup>5</sup> *and (b) M-q.*

h i <sup>Δ</sup> ½ � *nm Ron* ½ � <sup>Ω</sup> *Roff* ½ � <sup>Ω</sup> *Rinit* ½ � <sup>Ω</sup> *<sup>k</sup>* Order

<sup>1</sup> � <sup>10</sup>�<sup>14</sup> <sup>10</sup> <sup>100</sup> <sup>16</sup> � <sup>10</sup><sup>3</sup> <sup>1</sup> � 103 5 1

*BFS algorithm to obtain the shortest path. (a) A graph and (b) The BFS algorithm.*

*Memristors - Circuits and Applications of Memristor Devices*

In the memristive fuse, an ideal switch can be used in the process of finding the solution, however, with the aim to have a more realistic switch, a transmission gate is used instead. The transmission gate is a switch in CMOS technology, it consists of an NMOS transistor and a PMOS transistor connected in parallel, as in **Figure 9a**. Both devices in combination can fully transmit any signal value between *Vdd* (the supply voltage of the transistors) and ground. In order to switch, each transistor requires a complementary control input. Therefore, it is necessary to add an inverter connected between the control input and the PMOS gate [30, 32].

If the control input is *Vdd* then the switch is closed, and as a result, the transmission gate can pass the input signal to output because it exhibits a low-resistance. On the contrary, if the control input is grounded, then the switch is opened and the transmission gate presents a high-resistance.

In order to simulate the transmission gate of the memristive fuse, a CMOS 180 nm technology is used. The parameters of the two complementary transistors are shown in **Table 3**. The equivalent resistance of the transmission gate both states as a function of the input voltage is shown in **Figure 10**.

The resistance values are extracted making a sweep of the input voltage and measure the equivalent average resistance of the transistors in the ON-state

#### **Figure 9.**

*Transmission gate. (a) Configuration and (b) symbol.*


#### **Table 3.**

*Transmission gate: transistor parameters.*

#### **Figure 10.**

*Resistance characteristic of the transmission gate for both states. (a) ON-state and (b) OFF-state.*


#### **Table 4.**

*Selected values for R TG on and R TG off .*

(switch closed, **Figure 10a**) and OFF-state (switch opened, **Figure 10b**). **Table 4** shows the selected values for *R TG on* and *R TG off* .

In addition, it can be noticed that the initial value of the ON-state resistance is given as:

$$R\_{TG\_{\text{init}}} = R\_{TG\_{\text{on}}}|\_{Vin=0} = \textbf{1.266} \tag{13}$$

The simulation flow is shown in **Figure 11** and is described as follows:

grid and an input file for HSPICE is generated. The inputs in the maze are

tion results are saved in a .tr0 output file.

Eq. (14):

**55**

**Figure 11.** *Simulation flow.*

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

**5. Mazes under-test**

• Mazes with a single solution

• Mazes with multiple solutions

**Maze generator:** The first stage in the solution process is to generate the maze by using a script in MATLAB that generates the maze and it is shown as a plot. The walls of the maze are shown in green color in the resulting plot. From this graphical description, the maze can be automatically mapped onto the memristive

represented by input voltage sources of 1 *V* and the exits are connected to ground. **Electric simulation:** The netlist obtained by the maze generator is simulated with HSPICE. Here, a transient analysis for 20 *s* is carried out, this time is enough to find the solutions of the mazes under-test, however, the exact time when the solutions are found depends on the maze dimensions (grid). The transient simula-

**Graphic display of the results:** In order to visualize the results, a script in MATLAB imports the output simulation signals obtained with HSPICE. The resistance dynamic change (Δ*R t*ð Þ) is calculated at each simulation time and then the paths of the maze are represented by a graph, where the color in each branch indicates the level of Δ*R t*ð Þ at a given time. For sake of readiness, we have selected

During the transient simulation, the equivalent resistance of the fuses is obtained at every instant *t*. It clearly results that Δ*R* is obtained by calculating the difference between the measured resistance and the minimum resistance from

Consequently, the fuses that first reach the highest Δ*R* define indeed the solution path of the maze. In mazes with multiple solutions, fuses that belong to the shortest path reach high values of Δ*R* more fastly. As time lapses, other solution paths are revealed reaching high values of Δ*R*. For a given time, all fuses within the

In order to prove the behavior of the memristive grid in maze solving, this

Δ*R t*ðÞ¼ *R t*ðÞ� *Rfuseinit* (16)

maxð Þ¼ Δ*R t*ð Þ *R t*ðÞ� *Rfusemax* (17)

white for the minimum change and black for the maximum change.

solution paths reach the maximum Δ*R*, which is given by

section presents several cases that have been ordered as follows:

As a result of the specifications above, a couple of parameters are of special interest, namely, the initial resistance and the maximum resistance of the memristive fuses. At the start, the fuses present an initial resistance which is given as the sum of the initial resistance of the memristors in the anti-series connection plus initial resistance of the ON-state of the transmission gate:

$$R\_{fus\_{init}} = \mathcal{Z}R\_{init} + R\_{TG\_{init}} \tag{14}$$

which is 3.266 kΩ.

Moreover, the maximum resistance of the fuse is given as:

$$R\_{\text{fuse}\_{\text{max}}} = R\_{\text{gf}\_1} + R\_{on\_2} + R\_{TG\_{av}} \tag{15}$$

It is worthy to notice that the maximum fuse resistance does not contain *Roff* of both memristor, but *Roff* of one memristor and *Ron* of the other memristor due to the anti-series connection.

#### **4. Simulation flow**

Since the solution path for a given maze is obtained by determining the path where the fastest change in resistance occurs, the core of the solution process involves a transient analysis. We have chosen to achieve the electrical simulation of the memristive grid by using HSPICE. Both memristors of the fuse are defined as nonlinear resistors in the input netlist.

#### *Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

**Figure 11.** *Simulation flow.*

(switch closed, **Figure 10a**) and OFF-state (switch opened, **Figure 10b**). **Table 4**

*Resistance characteristic of the transmission gate for both states. (a) ON-state and (b) OFF-state.*

*R TG on* Ω *R TG off* Ω <sup>2</sup>*:*<sup>504</sup> � 103 <sup>10</sup>*:*<sup>854</sup> � <sup>10</sup><sup>9</sup>

As a result of the specifications above, a couple of parameters are of special

memristive fuses. At the start, the fuses present an initial resistance which is given as the sum of the initial resistance of the memristors in the anti-series connection

It is worthy to notice that the maximum fuse resistance does not contain *Roff* of both memristor, but *Roff* of one memristor and *Ron* of the other memristor due to the

Since the solution path for a given maze is obtained by determining the path where the fastest change in resistance occurs, the core of the solution process involves a transient analysis. We have chosen to achieve the electrical simulation of the memristive grid by using HSPICE. Both memristors of the fuse are defined as

*R TG init* ¼ *R TG on* j

plus initial resistance of the ON-state of the transmission gate:

Moreover, the maximum resistance of the fuse is given as:

interest, namely, the initial resistance and the maximum resistance of the

In addition, it can be noticed that the initial value of the ON-state resistance is

*Vin*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>266</sup> (13)

*Rfuseinit* ¼ 2*Rinit* þ *R TG init* (14)

*Rfusemax* ¼ *Roff* <sup>1</sup> þ *Ron*<sup>2</sup> þ *R TG on* (15)

shows the selected values for *R TG on* and *R TG off* .

*Memristors - Circuits and Applications of Memristor Devices*

given as:

**Table 4.**

*Selected values for R TG on and R TG off .*

**Figure 10.**

which is 3.266 kΩ.

anti-series connection.

**4. Simulation flow**

**54**

nonlinear resistors in the input netlist.

The simulation flow is shown in **Figure 11** and is described as follows:

**Maze generator:** The first stage in the solution process is to generate the maze by using a script in MATLAB that generates the maze and it is shown as a plot. The walls of the maze are shown in green color in the resulting plot. From this graphical description, the maze can be automatically mapped onto the memristive grid and an input file for HSPICE is generated. The inputs in the maze are represented by input voltage sources of 1 *V* and the exits are connected to ground.

**Electric simulation:** The netlist obtained by the maze generator is simulated with HSPICE. Here, a transient analysis for 20 *s* is carried out, this time is enough to find the solutions of the mazes under-test, however, the exact time when the solutions are found depends on the maze dimensions (grid). The transient simulation results are saved in a .tr0 output file.

**Graphic display of the results:** In order to visualize the results, a script in MATLAB imports the output simulation signals obtained with HSPICE. The resistance dynamic change (Δ*R t*ð Þ) is calculated at each simulation time and then the paths of the maze are represented by a graph, where the color in each branch indicates the level of Δ*R t*ð Þ at a given time. For sake of readiness, we have selected white for the minimum change and black for the maximum change.

During the transient simulation, the equivalent resistance of the fuses is obtained at every instant *t*. It clearly results that Δ*R* is obtained by calculating the difference between the measured resistance and the minimum resistance from Eq. (14):

$$
\Delta R(t) = R(t) - R\_{\text{fuse}\_{\text{init}}} \tag{16}
$$

Consequently, the fuses that first reach the highest Δ*R* define indeed the solution path of the maze. In mazes with multiple solutions, fuses that belong to the shortest path reach high values of Δ*R* more fastly. As time lapses, other solution paths are revealed reaching high values of Δ*R*. For a given time, all fuses within the solution paths reach the maximum Δ*R*, which is given by

$$\max(\Delta R(t)) = R(t) - R\_{fuse\_{\max}}\tag{17}$$

#### **5. Mazes under-test**

In order to prove the behavior of the memristive grid in maze solving, this section presents several cases that have been ordered as follows:


#### **5.1 Single-solution mazes**

The first set to be solved consists of three mazes with a single-entrance and single-output and the solution is given by a unique path.

### *5.1.1 The* 5 � 5 *maze*

The first maze, from **Figure 12a**, is treated in full with the aim of highlighting the details of the solution procedure. The first stage of the procedure yields the memristive grid associated to the mapping of the maze, which is shown in **Figure 12b**. The resulting netlist of the memristive grid is then simulated in a transient analysis with HSPICE.

It can be noticed that there are 24 memristive fuses in the open paths of the maze. The electrical simulation is applied in order to measure the instantaneous resistance of the fuses. On the one hand, **Figure 13a** shows the transient behavior of the resistance of those fuses for the first <sup>1</sup> 5 s. It can be noticed that all fuses start with the same resistance at *t* ¼ 0, namely *Rfuseinit* . As a result, at *t* ¼ 0, Δ*R* ¼ 0 for all fuses and the maze is not walked yet and the output display shows the open paths in white color, as shown in **Figure 13b**.

As time lapses, at *t* ¼ 0*:*197s, only the fuses belonging to the solution path exhibit significant changes in their resistance. Here, the blue lines correspond to fuses outside the solution path, while the red lines correspond to fuses that belong to solution path. These changes are represented in the output display of **Figure 13c** for the same time in yellow. The solution path can already be distinguished.

On the other hand, **Figure 14a** shows *R t*ð Þ of the memristive fuses for 0<*t*<20s. The red lines show that the fuses belonging to the solution path reached a maximum, while the blue lines remain in low levels of resistance, i.e., they belong to paths that finish in dead-ends.

fuses having a value of Δ*R*≈8*:*0*k*Ω. At *t* ¼ 3*:*7886s, the fuses of the solution path

*Transient analysis of the maze in Figure 12 for small values of t. (a) R t*ð Þ *of the fuses for* 0<*t* <0*:*197*s,*

In summary, it can be concluded that the memristive grid achieves the solution of the maze in a parallel processing by calculating the resistance of the fuses simultaneously. The progress of the solution procedure can be regarded as tracking the dynamic behavior of Δ*R*, which directly points out the solution path of the maze. On top of this, the output display allows us to visualize this procedure with the help

The memristive grid has also been applied to single-solution mazes that have larger sizes. The first maze is of 10 � 10 dimension and it is depicted in **Figure 15a**

The second case is a 15 � 15 maze, which is shown altogether with its solution in

The second set to be solved consists of three mazes that have solutions with

show Δ*R* ¼ 15*:*0*k*Ω.

*(b) t* ¼ 0 s*, and (c) t* ¼ 0*:*197 s*.*

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

**Figure 13.**

of a color scale.

showing these mazes.

**5.2 Multiple solutions mazes**

**Figure 16**.

multiple paths.

**57**

*5.1.2 The* 10 � 10 *and* 15 � 15 *mazes*

Within this time-window, two snapshots of the output display have been taken at *t* ¼ 1*:*3929s and *t* ¼ 3*:*7886s—as depicted in the plots of **Figure 14b** and **c**, respectively. In the first display, the solution path is already highlighted in red with

**Figure 12.**

*Mapping the* 5 � 5 *single-solution maze onto the memristive grid. (a) A* 5 � 5 *maze and (b) associated memristive grid.*

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

• Maze with two inputs and two outputs

*Memristors - Circuits and Applications of Memristor Devices*

• A 3D maze

*5.1.1 The* 5 � 5 *maze*

**5.1 Single-solution mazes**

transient analysis with HSPICE.

of the resistance of those fuses for the first <sup>1</sup>

in white color, as shown in **Figure 13b**.

paths that finish in dead-ends.

**Figure 12.**

**56**

*memristive grid.*

• An octogonal maze with three inputs and a single output

single-output and the solution is given by a unique path.

The first set to be solved consists of three mazes with a single-entrance and

The first maze, from **Figure 12a**, is treated in full with the aim of highlighting the details of the solution procedure. The first stage of the procedure yields the memristive grid associated to the mapping of the maze, which is shown in **Figure 12b**. The resulting netlist of the memristive grid is then simulated in a

It can be noticed that there are 24 memristive fuses in the open paths of the maze. The electrical simulation is applied in order to measure the instantaneous resistance of the fuses. On the one hand, **Figure 13a** shows the transient behavior

5

On the other hand, **Figure 14a** shows *R t*ð Þ of the memristive fuses for 0<*t*<20s. The red lines show that the fuses belonging to the solution path reached a maximum, while the blue lines remain in low levels of resistance, i.e., they belong to

Within this time-window, two snapshots of the output display have been taken

at *t* ¼ 1*:*3929s and *t* ¼ 3*:*7886s—as depicted in the plots of **Figure 14b** and **c**, respectively. In the first display, the solution path is already highlighted in red with

*Mapping the* 5 � 5 *single-solution maze onto the memristive grid. (a) A* 5 � 5 *maze and (b) associated*

with the same resistance at *t* ¼ 0, namely *Rfuseinit* . As a result, at *t* ¼ 0, Δ*R* ¼ 0 for all fuses and the maze is not walked yet and the output display shows the open paths

As time lapses, at *t* ¼ 0*:*197s, only the fuses belonging to the solution path exhibit significant changes in their resistance. Here, the blue lines correspond to fuses outside the solution path, while the red lines correspond to fuses that belong to solution path. These changes are represented in the output display of **Figure 13c** for the same time in yellow. The solution path can already be distinguished.

s. It can be noticed that all fuses start

**Figure 13.** *Transient analysis of the maze in Figure 12 for small values of t. (a) R t*ð Þ *of the fuses for* 0<*t* <0*:*197*s, (b) t* ¼ 0 s*, and (c) t* ¼ 0*:*197 s*.*

fuses having a value of Δ*R*≈8*:*0*k*Ω. At *t* ¼ 3*:*7886s, the fuses of the solution path show Δ*R* ¼ 15*:*0*k*Ω.

In summary, it can be concluded that the memristive grid achieves the solution of the maze in a parallel processing by calculating the resistance of the fuses simultaneously. The progress of the solution procedure can be regarded as tracking the dynamic behavior of Δ*R*, which directly points out the solution path of the maze. On top of this, the output display allows us to visualize this procedure with the help of a color scale.

### *5.1.2 The* 10 � 10 *and* 15 � 15 *mazes*

The memristive grid has also been applied to single-solution mazes that have larger sizes. The first maze is of 10 � 10 dimension and it is depicted in **Figure 15a** showing these mazes.

The second case is a 15 � 15 maze, which is shown altogether with its solution in **Figure 16**.

#### **5.2 Multiple solutions mazes**

The second set to be solved consists of three mazes that have solutions with multiple paths.

#### **Figure 14.**

*Transient analysis of the maze in Figure 12 for larger values of t. (a) R t*ð Þ *of the fuses for* 0<*t* <20*s, (b) t* ¼ 1*:*3929 s*, and (c) t* ¼ 3*:*7886 s*.*

Furthermore, the red lines show a steepest behavior which is a result that the red lines are associated to the fuses belonging to the solution with the shortest path. Besides, the blue lines are associated to fuses for the second solution path.

which denoted by the darkest yellow tones in **Figure 18c**. After a while, at

other solution, which can be compared by using the color bar.

*t* ¼ 20s—as shown in **Figure 19**.

**Figure 16.**

**Figure 17.**

**59**

*A* 5 � 5 *double-solution maze.*

15 � 15 *maze and solution at t* ¼ 3*:*7886*s.*

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

*5.2.2 Other mazes with multiple solutions*

exit, but there are four possible solution paths.

It can be observed that all paths start from *Rfuseinit* when *t* ¼ 0, i.e., the maze has not yet been walked—as given in the display of **Figure 18b**. After 0.2204 s, both solutions paths are already distinguishable, but the shortest path exhibits higher Δ*R*,

*t* ¼ 0*:*638, the solution given by the shortest path is perfectly differentiable from the

In this paragraph, two case studies are presented. The first one is the maze shown in **Figure 20a**, which is a 10 � 10 maze that has a single entrance and a single

A snapshot at 1.901 s has been taken—see **Figure 20b**. The four solution paths are visible in different colors. The shortest path is shown in red exhibiting the highest Δ*R* at the time of evaluation. On the opposite, the solution with the longest path is given in pale yellow. This example shows the usefulness of the color palette

After a larger sweep of time, the resistances of the fuses for both solutions have coalesced into an asymptotic level, which is the maximum value of the resistance at

**Figure 15.** 10 � 10 *maze and solution at t* ¼ 1*:*3929*s.*

### *5.2.1 The* 5 � 5 *maze with multiple solutions*

This maze is shown in **Figure 17**. It is a very simple example that is explained to some extent in order to illustrate the procedure for finding the paths that constitute the solutions.

After carrying out the transient simulation, the resistance of the memristive fuses is obtained. **Figure 18a** shows *R t*ð Þ for 0 <*t*<0*:*65s. Herein, the attention is focused only on the resistance of the fuses belonging to the solution paths.

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

**Figure 16.** 15 � 15 *maze and solution at t* ¼ 3*:*7886*s.*

**Figure 17.** *A* 5 � 5 *double-solution maze.*

Furthermore, the red lines show a steepest behavior which is a result that the red lines are associated to the fuses belonging to the solution with the shortest path. Besides, the blue lines are associated to fuses for the second solution path.

It can be observed that all paths start from *Rfuseinit* when *t* ¼ 0, i.e., the maze has not yet been walked—as given in the display of **Figure 18b**. After 0.2204 s, both solutions paths are already distinguishable, but the shortest path exhibits higher Δ*R*, which denoted by the darkest yellow tones in **Figure 18c**. After a while, at *t* ¼ 0*:*638, the solution given by the shortest path is perfectly differentiable from the other solution, which can be compared by using the color bar.

After a larger sweep of time, the resistances of the fuses for both solutions have coalesced into an asymptotic level, which is the maximum value of the resistance at *t* ¼ 20s—as shown in **Figure 19**.

#### *5.2.2 Other mazes with multiple solutions*

In this paragraph, two case studies are presented. The first one is the maze shown in **Figure 20a**, which is a 10 � 10 maze that has a single entrance and a single exit, but there are four possible solution paths.

A snapshot at 1.901 s has been taken—see **Figure 20b**. The four solution paths are visible in different colors. The shortest path is shown in red exhibiting the highest Δ*R* at the time of evaluation. On the opposite, the solution with the longest path is given in pale yellow. This example shows the usefulness of the color palette

*5.2.1 The* 5 � 5 *maze with multiple solutions*

10 � 10 *maze and solution at t* ¼ 1*:*3929*s.*

the solutions.

**58**

**Figure 15.**

**Figure 14.**

*(b) t* ¼ 1*:*3929 s*, and (c) t* ¼ 3*:*7886 s*.*

*Memristors - Circuits and Applications of Memristor Devices*

This maze is shown in **Figure 17**. It is a very simple example that is explained to some extent in order to illustrate the procedure for finding the paths that constitute

After carrying out the transient simulation, the resistance of the memristive fuses is obtained. **Figure 18a** shows *R t*ð Þ for 0 <*t*<0*:*65s. Herein, the attention is focused only on the resistance of the fuses belonging to the solution paths.

*Transient analysis of the maze in Figure 12 for larger values of t. (a) R t*ð Þ *of the fuses for* 0<*t* <20*s,*

#### **Figure 18.**

*Progress of the solution search for small t for the maze in Figure 17. (a) R t*ð Þ *for* 0<*t* <0*:*65*s, (b) t* ¼ 0*s, (c) t* ¼ 0*:*2204*s, and (d) t* ¼ 0*:*638 *s.*

**5.3 Octogonal maze**

**Figure 20.**

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

**Figure 21.**

**5.4 A 3D maze**

**Figure 22.**

**61**

*Octogonal maze and solution.*

A nonrectangular maze is given in **Figure 22a**, which is an octogonal concentric maze with three entrances and with the output goal in the center of the maze. The three entrances are denoted as **S**, **E**, and **NW**. Entrances **E** and **NW** cannot reach the solution, while entrance **S** does. Given the impossibility of the output display for dealing with nonrectangular mazes, the octogonal maze is converted into an isomorphic view that is given in **Figure 22b** that shows the solution path in red.

*Multiple-solution maze with two entrances and one exits. (a) Maze, (b) t* ¼ 1*:*0276*s, and (c) t* ¼ 8*:*0716*s.*

*Multiple-solution maze with one entrance and one exit. (a) Maze, (b) t* ¼ 1*:*901*s, and (c) t* ¼ 20*s.*

In order to illustrate that the memristive grid is able to deal with a threedimensional maze, a three-layer maze is solved. For sake of readiness, **Figure 23** shows the maze in separated levels in a puzzle-fashion. The ball on the top-layer

#### **Figure 19.** *Transient analysis for larger values of t. (a) R t*ð Þ *and (b) display at t* ¼ 20*s.*

of the output display on its full extent, because all possible solution paths are visible and it gives more insight on the progress of the solution procedure. After a long time, all the solutions reach the same resistance value as shown in **Figure 20c**.

The second example of this paragraph is a maze with two entrances and two exits that is shown in **Figure 21a**. We show in **Figure 21b** a snapshot taken at 1.0276 s. At this point, the memristive grid has been able to find both shortest paths for the solutions between the entrances and the outputs. After a while, at *t* ¼ 8*:*0716s, the output display shows the connection between both paths—as given in **Figure 21c**.

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

**Figure 20.**

*Multiple-solution maze with one entrance and one exit. (a) Maze, (b) t* ¼ 1*:*901*s, and (c) t* ¼ 20*s.*

**Figure 21.**

*Multiple-solution maze with two entrances and one exits. (a) Maze, (b) t* ¼ 1*:*0276*s, and (c) t* ¼ 8*:*0716*s.*

#### **5.3 Octogonal maze**

A nonrectangular maze is given in **Figure 22a**, which is an octogonal concentric maze with three entrances and with the output goal in the center of the maze. The three entrances are denoted as **S**, **E**, and **NW**. Entrances **E** and **NW** cannot reach the solution, while entrance **S** does. Given the impossibility of the output display for dealing with nonrectangular mazes, the octogonal maze is converted into an isomorphic view that is given in **Figure 22b** that shows the solution path in red.

#### **5.4 A 3D maze**

In order to illustrate that the memristive grid is able to deal with a threedimensional maze, a three-layer maze is solved. For sake of readiness, **Figure 23** shows the maze in separated levels in a puzzle-fashion. The ball on the top-layer

**Figure 22.** *Octogonal maze and solution.*

of the output display on its full extent, because all possible solution paths are visible and it gives more insight on the progress of the solution procedure. After a long time, all the solutions reach the same resistance value as shown in **Figure 20c**. The second example of this paragraph is a maze with two entrances and two exits that is shown in **Figure 21a**. We show in **Figure 21b** a snapshot taken at 1.0276 s. At this point, the memristive grid has been able to find both shortest paths

*Progress of the solution search for small t for the maze in Figure 17. (a) R t*ð Þ *for* 0<*t* <0*:*65*s, (b) t* ¼ 0*s, (c)*

*t* ¼ 8*:*0716s, the output display shows the connection between both paths—as given

for the solutions between the entrances and the outputs. After a while, at

*Transient analysis for larger values of t. (a) R t*ð Þ *and (b) display at t* ¼ 20*s.*

*Memristors - Circuits and Applications of Memristor Devices*

in **Figure 21c**.

**60**

**Figure 19.**

**Figure 18.**

*t* ¼ 0*:*2204*s, and (d) t* ¼ 0*:*638 *s.*

indicates the starting point of the maze, while the ball in the low-layer points out to the output of the maze. The layers communicate each other with holes that are

The memristive grid that describes this maze counts 16 nodes per layer which yields a total of 48 nodes. Every layer possesses 24 branches (the external walls do not count) plus four inter-layer branches, i.e., 78 memristive fuses to describe the

Finally, the output display given in **Figure 24** shows the progress of the solution

In the following, the code for the memristor model as used within HSPICE is

\*———————————————————————————————————————————————————————

.PARAM Pol1='-(256/45)\*POW(Xo,10)+32\*POW(Xo,9)-(576/7)\*POW(Xo,8)+128\*POW

.PARAM Pol2='(256/45)\*POW(Xo,10)-(224/9)\*POW(Xo,9)+(352/7)\*POW(Xo,8)-(1280/

+(672/5)\*POW(Xo,6)+(504/5)\*POW(Xo,5)-56\*POW(Xo,4)+24\*POW(Xo,3)-9\*POW

+(736/15)\*POW(Xo,6)-(136/5)\*POW(Xo,5)+(32/3)\*POW(Xo,4)-(8/3)\*POW(Xo,3)+

.PARAM Pol5='-56\*POW(Xo,4)+224\*POW(Xo,3)-336\*POW(Xo,2)+224\*Xo-56' .PARAM Pol6='-(504/5)\*POW(Xo,5)+504\*POW(Xo,4)-1008\*POW(Xo,3)+1008\*POW

.PARAM Pol7='-(672/5)\*POW(Xo,6)+(4032/5)\*POW(Xo,5)-2016\*POW(Xo,4)

.PARAM Pol9='-(576/7)\*POW(Xo,8)+(4608/7)\*POW(Xo,7)-2304\*POW(Xo,6)

+5760\*POW(Xo,4)+4608\*POW(Xo,3)-2304\*POW(Xo,2)+(4608/7)\*Xo-576/7'

+4032\*POW(Xo,4)-2688\*POW(Xo,3)+1152\*POW(Xo,2)-288\*Xo+32'

.PARAM Pol8='-128\*POW(Xo,7)+896\*POW(Xo,6)-2688\*POW(Xo,5)+4480\*POW(Xo,4)-

.PARAM Pol10='-32\*POW(Xo,9)+288\*POW(Xo,8)-1152\*POW(Xo,7)+2688\*POW(Xo,6)-

.PARAM Pol11='-(256/45)\*POW(Xo,10)+(512/9)\*POW(Xo,9)-256\*POW(Xo,8)+(2048/

depicted as circles on the floor and disks on the roof of them.

maze.

procedure.

**6. Code for the model**

*Memristive Grid for Maze Solving*

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

\*INAOE, summer 2018 \*Yojanes Rodríguez .LIB MemModels

\*HPMQ Joglekar k=5 O1 .SUBCKT HPMQK5O1N N+ N-

.PARAM Xo=0.99 .PARAM mu=10f .PARAM eta=-1 .PARAM Roff=16e3 .PARAM Ron=100 .PARAM Delta=10n

(Xo,7)-

(Xo,2)-Xo'

21)\*POW(Xo,7)+

(Xo,2)-504\*Xo+504/5'

+2688\*POW(Xo,3)-

+4608\*POW(Xo,5)-

4032\*POW(Xo,5)+

**63**

given.\*Charge-controlled models

.PARAM kappa='Ron\*mu/(POW(Delta,2))'

+POW(Xo,2)-(1441/315)\*Xo+1126/315' .PARAM Pol3='-9\*POW(Xo,2)+18\*Xo-9'

+2016\*POW(Xo,2)+(4032/5)\*Xo-(672/5)'

+4480\*POW(Xo,3)+2688\*POW(Xo,2)-896\*Xo+128'

.PARAM Pol4='-24\*POW(Xo,3)+72\*POW(Xo,2)-72\*Xo+24'

**Figure 24.** *Solutions of the 3D-maze.*

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

indicates the starting point of the maze, while the ball in the low-layer points out to the output of the maze. The layers communicate each other with holes that are depicted as circles on the floor and disks on the roof of them.

The memristive grid that describes this maze counts 16 nodes per layer which yields a total of 48 nodes. Every layer possesses 24 branches (the external walls do not count) plus four inter-layer branches, i.e., 78 memristive fuses to describe the maze.

Finally, the output display given in **Figure 24** shows the progress of the solution procedure.

### **6. Code for the model**

**Figure 23.** *A 3D-maze.*

*Memristors - Circuits and Applications of Memristor Devices*

**Figure 24.**

**62**

*Solutions of the 3D-maze.*

```
In the following, the code for the memristor model as used within HSPICE is
given.*Charge-controlled models
*INAOE, summer 2018
*Yojanes Rodríguez
.LIB MemModels
*———————————————————————————————————————————————————————
*HPMQ Joglekar k=5 O1
.SUBCKT HPMQK5O1N N+ N-
.PARAM Xo=0.99
.PARAM mu=10f
.PARAM eta=-1
.PARAM Roff=16e3
.PARAM Ron=100
.PARAM Delta=10n
.PARAM kappa='Ron*mu/(POW(Delta,2))'
.PARAM Pol1='-(256/45)*POW(Xo,10)+32*POW(Xo,9)-(576/7)*POW(Xo,8)+128*POW
(Xo,7)-
+(672/5)*POW(Xo,6)+(504/5)*POW(Xo,5)-56*POW(Xo,4)+24*POW(Xo,3)-9*POW
(Xo,2)-Xo'
.PARAM Pol2='(256/45)*POW(Xo,10)-(224/9)*POW(Xo,9)+(352/7)*POW(Xo,8)-(1280/
21)*POW(Xo,7)+
+(736/15)*POW(Xo,6)-(136/5)*POW(Xo,5)+(32/3)*POW(Xo,4)-(8/3)*POW(Xo,3)+
+POW(Xo,2)-(1441/315)*Xo+1126/315'
.PARAM Pol3='-9*POW(Xo,2)+18*Xo-9'
.PARAM Pol4='-24*POW(Xo,3)+72*POW(Xo,2)-72*Xo+24'
.PARAM Pol5='-56*POW(Xo,4)+224*POW(Xo,3)-336*POW(Xo,2)+224*Xo-56'
.PARAM Pol6='-(504/5)*POW(Xo,5)+504*POW(Xo,4)-1008*POW(Xo,3)+1008*POW
(Xo,2)-504*Xo+504/5'
.PARAM Pol7='-(672/5)*POW(Xo,6)+(4032/5)*POW(Xo,5)-2016*POW(Xo,4)
+2688*POW(Xo,3)-
+2016*POW(Xo,2)+(4032/5)*Xo-(672/5)'
.PARAM Pol8='-128*POW(Xo,7)+896*POW(Xo,6)-2688*POW(Xo,5)+4480*POW(Xo,4)-
+4480*POW(Xo,3)+2688*POW(Xo,2)-896*Xo+128'
.PARAM Pol9='-(576/7)*POW(Xo,8)+(4608/7)*POW(Xo,7)-2304*POW(Xo,6)
+4608*POW(Xo,5)-
+5760*POW(Xo,4)+4608*POW(Xo,3)-2304*POW(Xo,2)+(4608/7)*Xo-576/7'
.PARAM Pol10='-32*POW(Xo,9)+288*POW(Xo,8)-1152*POW(Xo,7)+2688*POW(Xo,6)-
4032*POW(Xo,5)+
+4032*POW(Xo,4)-2688*POW(Xo,3)+1152*POW(Xo,2)-288*Xo+32'
.PARAM Pol11='-(256/45)*POW(Xo,10)+(512/9)*POW(Xo,9)-256*POW(Xo,8)+(2048/
```

```
3)*POW(Xo,7)-
+(3584/3)*POW(Xo,6)+(7168/5)*POW(Xo,5)-(3584/3)*POW(Xo,4)+
+(2048/3)*POW(Xo,3)-256*POW(Xo,2)+(512/9)*Xo-256/45'
*Integrator
Ecur Ni 0 VOL = 'I(Rmem)'
R Ni Na 1k
C Na No 1m
Eop No GND GND Na 1Meg
Echarge charge GND No GND -1
Rmem N+ N- R='(V(charge)>0)?(+(256/45)*POW(Xo,10)*exp(-200*kappa*V
(charge))-
+32*POW(Xo,9)*exp(-180*kappa*V(charge))+(576/7)*POW(Xo,8)*exp(-160*kappa*V
(charge))-
+128*POW(Xo,7)*exp(-140*kappa*V(charge))+(672/5)*POW(Xo,6)*exp(-
120*kappa*V(charge))-
+(504/5)*POW(Xo,5)*exp(-100*kappa*V(charge))+56*POW(Xo,4)*exp(-80*kappa*V
(charge))-
+24*POW(Xo,3)*exp(-60*kappa*V(charge))+9*POW(Xo,2)*exp(-40*kappa*V
(charge))+
+(Pol1)*exp(-20*kappa*V(charge)))*(Roff-Ron)+Roff:((Pol2)*exp(20*kappa*V
(charge))+
+(Pol3)*exp(40*kappa*V(charge))+(Pol4)*exp(60*kappa*V(charge))+
+(Pol5)*exp(80*kappa*V(charge))+(Pol6)*exp(100*kappa*V(charge))+
+(Pol7)*exp(120*kappa*V(charge))+(Pol8)*exp(140*kappa*V(charge))+
+(Pol9)*exp(160*kappa*V(charge))+(Pol10)*exp(180*kappa*V(charge))+
+(Pol11)*exp(200*kappa*V(charge)))*(Roff-Ron)+Ron'
.ENDS
*———————————————————————————————————————————————————————
```
maze. The solution is found by sensing the variations of the resistance of the fuses that belong to the path, which implies that the memristive grid achieves the shortest

Finally, the maze grid has proven its reliability in solving mazes with different levels of complexity. A series of examples has been analyzed: single-solution mazes,

path algorithm.

*Memristive Grid for Maze Solving*

**Author details**

**65**

San Andrés Cholula, Puebla, Mexico

provided the original work is properly cited.

Arturo Sarmiento-Reyes\* and Yojanes Rodríguez Velásquez

\*Address all correspondence to: jarocho@inaoep.mx

Electronics Department, National Institute for Astrophysics, Optics and Electronics,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

multiple-solution mazes, and a 3D maze.

*DOI: http://dx.doi.org/10.5772/intechopen.84678*

```
.ENDL MemModels
```
#### **7. Conclusions**

A specially tailored memristive grid has been used as an analog processor for solving mazes. The memristives branches of the grid (fuses) are formed by an antiseries connection of two memristors and a switch. On one side, we have introduced a family of symbolic models for the memristor that are defined by charge-controlled functions. The fact that the models are charge-controlled allows us to monitor the velocity of the variation of the equivalent memristance of the fuses by carrying out a transient analysis with HSPICE. It is worth to mention that the model has been recast in VERILOG-A. On the other side, with the aim of producing a more realistic scenario, the switches are implemented by a transmission gate in CMOS technology. In this form, the resulting grid is in fact a hybrid CMOS-Memristor circuit.

The simulation flow-work is formed by an input stage developed in MATLAB, the electric simulation in HSPICE and the output stage again in MATLAB. The input stage is responsible for mapping the structure of the maze onto the memristive grid. The outcome of this stage is an input file with the netlist of the grid. The intermediate stage executes the transient simulation. The output stage is used to display the variation of the resistance of the fuses and it literally draws the solution path of the

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

3)\*POW(Xo,7)-

\*Integrator

R Ni Na 1k C Na No 1m

(charge))-

(charge))-

(charge))-

(charge))+

(charge))+

.ENDS

**64**

.ENDL MemModels

**7. Conclusions**

Ecur Ni 0 VOL = 'I(Rmem)'

Eop No GND GND Na 1Meg

120\*kappa\*V(charge))-

Echarge charge GND No GND -1

+(3584/3)\*POW(Xo,6)+(7168/5)\*POW(Xo,5)-(3584/3)\*POW(Xo,4)+ +(2048/3)\*POW(Xo,3)-256\*POW(Xo,2)+(512/9)\*Xo-256/45'

*Memristors - Circuits and Applications of Memristor Devices*

Rmem N+ N- R='(V(charge)>0)?(+(256/45)\*POW(Xo,10)\*exp(-200\*kappa\*V

+128\*POW(Xo,7)\*exp(-140\*kappa\*V(charge))+(672/5)\*POW(Xo,6)\*exp(-

+24\*POW(Xo,3)\*exp(-60\*kappa\*V(charge))+9\*POW(Xo,2)\*exp(-40\*kappa\*V

+(Pol3)\*exp(40\*kappa\*V(charge))+(Pol4)\*exp(60\*kappa\*V(charge))+ +(Pol5)\*exp(80\*kappa\*V(charge))+(Pol6)\*exp(100\*kappa\*V(charge))+ +(Pol7)\*exp(120\*kappa\*V(charge))+(Pol8)\*exp(140\*kappa\*V(charge))+ +(Pol9)\*exp(160\*kappa\*V(charge))+(Pol10)\*exp(180\*kappa\*V(charge))+

+(Pol11)\*exp(200\*kappa\*V(charge)))\*(Roff-Ron)+Ron'

+(Pol1)\*exp(-20\*kappa\*V(charge)))\*(Roff-Ron)+Roff:((Pol2)\*exp(20\*kappa\*V

\*———————————————————————————————————————————————————————

A specially tailored memristive grid has been used as an analog processor for solving mazes. The memristives branches of the grid (fuses) are formed by an antiseries connection of two memristors and a switch. On one side, we have introduced a family of symbolic models for the memristor that are defined by charge-controlled functions. The fact that the models are charge-controlled allows us to monitor the velocity of the variation of the equivalent memristance of the fuses by carrying out a transient analysis with HSPICE. It is worth to mention that the model has been recast in VERILOG-A. On the other side, with the aim of producing a more realistic scenario, the switches are implemented by a transmission gate in CMOS technology.

In this form, the resulting grid is in fact a hybrid CMOS-Memristor circuit.

The simulation flow-work is formed by an input stage developed in MATLAB, the electric simulation in HSPICE and the output stage again in MATLAB. The input stage is responsible for mapping the structure of the maze onto the memristive grid. The outcome of this stage is an input file with the netlist of the grid. The intermediate stage executes the transient simulation. The output stage is used to display the variation of the resistance of the fuses and it literally draws the solution path of the

+32\*POW(Xo,9)\*exp(-180\*kappa\*V(charge))+(576/7)\*POW(Xo,8)\*exp(-160\*kappa\*V

+(504/5)\*POW(Xo,5)\*exp(-100\*kappa\*V(charge))+56\*POW(Xo,4)\*exp(-80\*kappa\*V

maze. The solution is found by sensing the variations of the resistance of the fuses that belong to the path, which implies that the memristive grid achieves the shortest path algorithm.

Finally, the maze grid has proven its reliability in solving mazes with different levels of complexity. A series of examples has been analyzed: single-solution mazes, multiple-solution mazes, and a 3D maze.

### **Author details**

Arturo Sarmiento-Reyes\* and Yojanes Rodríguez Velásquez Electronics Department, National Institute for Astrophysics, Optics and Electronics, San Andrés Cholula, Puebla, Mexico

\*Address all correspondence to: jarocho@inaoep.mx

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

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[2] Barnes CA. Memory deficits associated with senescence: A neurophysiological and behavioral study in the rat. Journal of Comparative and Physiological Psychology. 1979;**93**(1):74

[3] Olton DS, Samuelson RJ. Remembrance of places passed: Spatial memory in rats. Journal of Experimental Psychology: Animal Behavior Processes. 1976;**2**(2):97

[4] Morris R. Developments of a watermaze procedure for studying spatial learning in the rat. Journal of Neuroscience Methods. 1984;**11**(1): 47-60

[5] Dracopoulos DC. Robot path planning for maze navigation. In: The 1998 IEEE International Joint Conference on Neural Networks Proceedings, IEEE World Congress on Computational Intelligence; Vol. 3. IEEE; 1998. pp. 2081-2085

[6] Lumelsky VJ. A comparative study on the path length performance of maze-searching and robot motion planning algorithms. IEEE Transactions on Robotics and Automation. 1991;**7**(1): 57-66

[7] Werbos PJ, Pang X. Generalized maze navigation: SRN critics solve what feedforward or Hebbian nets cannot. In: IEEE International Conference on Systems, Man, and Cybernetics; Vol. 3. IEEE; 1996. pp. 1764-1769

[8] Milková E, Slaby A. Graph algorithms in mutual contexts. In: 7th WSEAS International Conference Proceedings on Mathematics and

Computers in Science and Engineering; Vol. 1. World Scientific and Engineering Academy and Society; 2008. pp. 721-726 Prodromakis T. Integration of nanoscale memristor synapses in neuromorphic

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[27] He J-H. Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation. 2004;**156**(2):527-539

[28] Joglekar YN, Wolf SJ. The elusive memristor: Properties of basic electrical circuits. European Journal of Physics.

[29] Sarmiento-Reyes A, Velásquez YR. Chapter 5: Charge-controlled memristor grid for edge detection. In: Ciufudean C, editor. Advances in Memristor Neural Networks. London, United Kingdom: InTechOpen; 2018. pp. 91-113

[30] Velásquez YAR. Development of an analytical model for a charge-controlled memristor and its applications [Master's thesis]. Puebla, Mexico: National Institute for Astrophysics, Optics and

[31] Recski A. Matroid Theory and Its Applications. Berlin, Germany:

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Electronics (INAOE); 2017

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Nanotechnology. 2013;**24**(38):384010

[19] Vittoz EA. Future of analog in the

International Symposium on Circuits and Systems. IEEE; 1990. pp. 1372-1375

[20] Chua L. Memristor, Hodgkin-Huxley, and edge of chaos. In: Adamatzky A, Chua LO. Memristor Networks. Basel, Switzerland: Springer;

[21] Jo SH, Chang T, Ebong I, Bhadviya BB, Mazumder P, Lu W. Nanoscale memristor device as synapse in neuromorphic systems. Nano Letters.

[22] Naous R, Al-Shedivat M, Salama KN. Stochasticity modeling in memristors. IEEE Transactions on Nanotechnology. 2016;**15**(1):15-28

Memcomputing: A computing paradigm to store and process information on the same physical platform. In: 2014 International Workshop on

Computational Electronics (IWCE).

[24] Sarmiento-Reyes A, Hernández-

Hernández-Mejía C, Diaz Arango GU. A fully symbolic homotopy-based memristor model for applications to circuit simulation. Analog Integrated Circuits and Signal Processing. 2015;

[25] He J-H. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering.

[26] Vazquez-Leal H. Generalized homotopy method for solving nonlinear differential equations. Computational

Martínez L, Vázquez-Leal H,

[23] Pershin YV, Di Ventra M.

computing architectures.

*Memristive Grid for Maze Solving*

VLSI environment. In: IEEE

2014. pp. 67-94

2010;**10**(4):1297-1301

IEEE; 2014. pp. 1-2

**85**(1):65-80

**67**

1999;**178**(3):257-262

[9] Bondy JA, Murty USR. Graph Theory with Applications. Vol. 290. London: Macmillan; 1976

[10] Abelson H, DiSessa AA. Turtle Geometry: The Computer as a Medium for Exploring Mathematics. Cambridge, Massachusetts: MIT Press; 1986

[11] Müller H. A one-symbol printing automaton escaping from every labyrinth. Computing. 1977;**19**(2): 95-110

[12] Vourkas I, Stathis D, Sirakoulis G. Massively parallel analog computing: Ariadnes thread was made of memristors. IEEE Transactions on Emerging Topics in Computing. 2015; **6**(1):145-155

[13] Ye Z,Wu SHM, Prodromakis T. Computing shortest paths in 2D and 3D memristive networks. In: Adamatzky A, Chua LO. Memristor Networks. Basel, Switzerland: Springer; 2014. pp. 537-552

[14] Pershin YV, Di Ventra M. Solving mazes with memristors: A massively parallel approach. Physical Review E. 2011;**84**(4):046703

[15] Chua L. Memristor—The missing circuit element. IEEE Transactions on Circuit Theory. 1971;**18**(5):507-519

[16] Chua LO, Kang S-M. Memristive devices and systems. Proceedings of the IEEE. 1976;**64**(2):209-223

[17] Strukov DB, Snider GS, Stewart DR, Williams RS. The missing memristor found. Nature. 2008;**453**(7191):80-83

[18] Indiveri G, Linares-Barranco B, Legenstein R, Deligeorgis G,

*Memristive Grid for Maze Solving DOI: http://dx.doi.org/10.5772/intechopen.84678*

Prodromakis T. Integration of nanoscale memristor synapses in neuromorphic computing architectures. Nanotechnology. 2013;**24**(38):384010

**References**

2000

[1] Kern H, Saward J, Schons M, Clay AH, Thomson SB, Velder KA. Through the Labyrinth: Designs and Meanings Over 5,000 Years. Art and Design Series. New York: Prestel Publishing;

*Memristors - Circuits and Applications of Memristor Devices*

Computers in Science and Engineering; Vol. 1. World Scientific and Engineering Academy and Society; 2008. pp. 721-726

[9] Bondy JA, Murty USR. Graph Theory with Applications. Vol. 290. London:

[10] Abelson H, DiSessa AA. Turtle Geometry: The Computer as a Medium for Exploring Mathematics. Cambridge,

Massachusetts: MIT Press; 1986

[11] Müller H. A one-symbol printing automaton escaping from every labyrinth. Computing. 1977;**19**(2):

[12] Vourkas I, Stathis D, Sirakoulis G. Massively parallel analog computing:

[13] Ye Z,Wu SHM, Prodromakis T. Computing shortest paths in 2D and 3D memristive networks. In: Adamatzky A, Chua LO. Memristor Networks. Basel, Switzerland: Springer; 2014. pp. 537-552

[14] Pershin YV, Di Ventra M. Solving mazes with memristors: A massively parallel approach. Physical Review E.

[15] Chua L. Memristor—The missing circuit element. IEEE Transactions on Circuit Theory. 1971;**18**(5):507-519

[16] Chua LO, Kang S-M. Memristive devices and systems. Proceedings of the

[17] Strukov DB, Snider GS, Stewart DR, Williams RS. The missing memristor found. Nature. 2008;**453**(7191):80-83

[18] Indiveri G, Linares-Barranco B, Legenstein R, Deligeorgis G,

IEEE. 1976;**64**(2):209-223

Ariadnes thread was made of memristors. IEEE Transactions on Emerging Topics in Computing. 2015;

Macmillan; 1976

95-110

**6**(1):145-155

2011;**84**(4):046703

[2] Barnes CA. Memory deficits associated with senescence: A

[3] Olton DS, Samuelson RJ.

1976;**2**(2):97

47-60

57-66

**66**

neurophysiological and behavioral study in the rat. Journal of Comparative and Physiological Psychology. 1979;**93**(1):74

Remembrance of places passed: Spatial memory in rats. Journal of Experimental Psychology: Animal Behavior Processes.

[4] Morris R. Developments of a watermaze procedure for studying spatial learning in the rat. Journal of Neuroscience Methods. 1984;**11**(1):

[5] Dracopoulos DC. Robot path planning for maze navigation. In: The

1998 IEEE International Joint Conference on Neural Networks Proceedings, IEEE World Congress on Computational Intelligence; Vol. 3.

IEEE; 1998. pp. 2081-2085

[6] Lumelsky VJ. A comparative study on the path length performance of maze-searching and robot motion planning algorithms. IEEE Transactions on Robotics and Automation. 1991;**7**(1):

[7] Werbos PJ, Pang X. Generalized maze navigation: SRN critics solve what feedforward or Hebbian nets cannot. In: IEEE International Conference on Systems, Man, and Cybernetics; Vol. 3.

IEEE; 1996. pp. 1764-1769

[8] Milková E, Slaby A. Graph

algorithms in mutual contexts. In: 7th WSEAS International Conference Proceedings on Mathematics and

[19] Vittoz EA. Future of analog in the VLSI environment. In: IEEE International Symposium on Circuits and Systems. IEEE; 1990. pp. 1372-1375

[20] Chua L. Memristor, Hodgkin-Huxley, and edge of chaos. In: Adamatzky A, Chua LO. Memristor Networks. Basel, Switzerland: Springer; 2014. pp. 67-94

[21] Jo SH, Chang T, Ebong I, Bhadviya BB, Mazumder P, Lu W. Nanoscale memristor device as synapse in neuromorphic systems. Nano Letters. 2010;**10**(4):1297-1301

[22] Naous R, Al-Shedivat M, Salama KN. Stochasticity modeling in memristors. IEEE Transactions on Nanotechnology. 2016;**15**(1):15-28

[23] Pershin YV, Di Ventra M. Memcomputing: A computing paradigm to store and process information on the same physical platform. In: 2014 International Workshop on Computational Electronics (IWCE). IEEE; 2014. pp. 1-2

[24] Sarmiento-Reyes A, Hernández-Martínez L, Vázquez-Leal H, Hernández-Mejía C, Diaz Arango GU. A fully symbolic homotopy-based memristor model for applications to circuit simulation. Analog Integrated Circuits and Signal Processing. 2015; **85**(1):65-80

[25] He J-H. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering. 1999;**178**(3):257-262

[26] Vazquez-Leal H. Generalized homotopy method for solving nonlinear differential equations. Computational

and Applied Mathematics. 2014;**33**(1): 275-288

[27] He J-H. Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation. 2004;**156**(2):527-539

[28] Joglekar YN, Wolf SJ. The elusive memristor: Properties of basic electrical circuits. European Journal of Physics. 2009;**30**(4):661

[29] Sarmiento-Reyes A, Velásquez YR. Chapter 5: Charge-controlled memristor grid for edge detection. In: Ciufudean C, editor. Advances in Memristor Neural Networks. London, United Kingdom: InTechOpen; 2018. pp. 91-113

[30] Velásquez YAR. Development of an analytical model for a charge-controlled memristor and its applications [Master's thesis]. Puebla, Mexico: National Institute for Astrophysics, Optics and Electronics (INAOE); 2017

[31] Recski A. Matroid Theory and Its Applications. Berlin, Germany: Spring-Verlag; 1989

[32] Hodges DA, Jackson HG. Analysis and Design of Digital Integrated Circuits. Boston, USA: McGraw-Hill; 1988

**Chapter 5**

**Abstract**

**1. Introduction**

nano-structures [2].

**69**

Mathematical Analysis of

In this chapter we present mathematical study of memristor systems. More precisely, we apply local activity theory in order to determine the edge of chaos regime in reaction-diffusion memristor cellular nanoscale networks (RD-MCNN) and in memristor hysteresis CNN (M-HCNN). First we give an overview of mathematical models of memristors, CNN and complexity. Then we consider the above mentioned two models and we develop constructive algorithm for determination of edge of chaos in them. Based on these algorithms numerical simulations are provided. Two applications of M-HCNN model in image processing are presented.

**Keywords:** memristor, cellular nanoscale networks, reaction-diffusion systems,

Memristors form an important emerging technology for memory and neuromorphic computing applications (**Figure 1**). Chua has developed the fundamentals of the memistor framework nearly 40 years ago [1]. Since then, the industry has been engaged in the search for novel materials and technologies of these

Mathematical models of the complex dynamics which can be exhibited by nanodevices is presented in [3]. General and simple models are very important in the investigations of nonlinear dynamics in memristors [4]. Such models of memristorbased circuits are presented in [5, 6]. In order to develop novel hybrid [7] hardware architectures combining memory storage and data processing in the same physical location and at the same time [8] to explain the behavior of biological systems [9] new accurate mathematical models need to be introduced (**Figure 2**). Although several physical models [10–13] have been derived in order to study phenomena characterizing these nano-devices, a circuit theoretic-based mathematical treatment allowing the development of memristor circuits is still restricted to few cases. Most of these mathematical models have been studied in [14]. The merit for the first model of a titanium dioxide-based nano-structure may be ascribed to Williams [15]. This model is simple not specifying boundary conditions in the state equation. Literature was later enriched with a number of more complex models taking into account nonlinear effects on ionic transport and defining behavior at boundaries. The model proposed by Joglekar and Wolf [16] may allow for single-valued stateflux characteristics only, under any sign-varying periodic input with zero mean. By contrast, only multi-valued state-flux characteristics may be reproduced by Biolek's

Memristor CNN

*Angela Slavova and Ronald Tetzlaff*

hysteresis, edge of chaos, image processing

#### **Chapter 5**

## Mathematical Analysis of Memristor CNN

*Angela Slavova and Ronald Tetzlaff*

#### **Abstract**

In this chapter we present mathematical study of memristor systems. More precisely, we apply local activity theory in order to determine the edge of chaos regime in reaction-diffusion memristor cellular nanoscale networks (RD-MCNN) and in memristor hysteresis CNN (M-HCNN). First we give an overview of mathematical models of memristors, CNN and complexity. Then we consider the above mentioned two models and we develop constructive algorithm for determination of edge of chaos in them. Based on these algorithms numerical simulations are provided. Two applications of M-HCNN model in image processing are presented.

**Keywords:** memristor, cellular nanoscale networks, reaction-diffusion systems, hysteresis, edge of chaos, image processing

#### **1. Introduction**

Memristors form an important emerging technology for memory and neuromorphic computing applications (**Figure 1**). Chua has developed the fundamentals of the memistor framework nearly 40 years ago [1]. Since then, the industry has been engaged in the search for novel materials and technologies of these nano-structures [2].

Mathematical models of the complex dynamics which can be exhibited by nanodevices is presented in [3]. General and simple models are very important in the investigations of nonlinear dynamics in memristors [4]. Such models of memristorbased circuits are presented in [5, 6]. In order to develop novel hybrid [7] hardware architectures combining memory storage and data processing in the same physical location and at the same time [8] to explain the behavior of biological systems [9] new accurate mathematical models need to be introduced (**Figure 2**). Although several physical models [10–13] have been derived in order to study phenomena characterizing these nano-devices, a circuit theoretic-based mathematical treatment allowing the development of memristor circuits is still restricted to few cases. Most of these mathematical models have been studied in [14]. The merit for the first model of a titanium dioxide-based nano-structure may be ascribed to Williams [15]. This model is simple not specifying boundary conditions in the state equation. Literature was later enriched with a number of more complex models taking into account nonlinear effects on ionic transport and defining behavior at boundaries. The model proposed by Joglekar and Wolf [16] may allow for single-valued stateflux characteristics only, under any sign-varying periodic input with zero mean. By contrast, only multi-valued state-flux characteristics may be reproduced by Biolek's

algorithms are flexible because they can be implemented with an appropriate design. Local and non-local learning algorithms can be implemented in a straightforward way. The disadvantage is that they cannot have high connectivity and

tant in the development of CNN-based computational methods.

Many scientists have struggled to uncover the elusive origin of "complexity" and

its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, non-equilibrium phenomena, etc. [22, 30–32]. In his works, Schrödinger [31] defines the exchange of energy as a necessary condition for complexity of open systems. Prigogine [32] states the instability of the homogeneous systems as a new principle of nature, whereas Turing finds the origin of morphogenesis in symmetry breaking. In [31] Smale considers a reaction-diffusion system

Cellular Nonlinear/Nanoscale Networks (CNN) have been introduced in 1988 by Chua and Yang [20, 21] as a new class of information processing systems which show important potential applications (**Figure 3**). By endowing a single CNN cell with local analog and logic memory, some communication circuitry and further units, the CNN Universal Machine (CNN-UM) has been invented by Roska [22–24]. Analog CNN-UM chip hardware implementations have been developed some time ago. A CNN-UM chip represents a parallel computer with stored programmability allowing real-time processing of multivariate data. CNN have very promising applications in image processing and pattern recognition [22, 25–27]. Although, recently realized systems, e.g., the EyeRis 1.3 system, the MIPA4k, and SCAMP-5, are characterized as sensor-processor systems for high speed vision by reaching frames rates more than 20 kHz, their low resolution (e.g., 176 144 pixel in the EyeRis 1.3 system) limits the applicability to certain problems in practice only. Since the cell size cannot be decreased considerably in conventional CMOS technology, nano-elements will play an important role in future CNN-UM chip realizations. Especially, memristors [28] which are considered for synaptic connections in first realizations [29], will play an important role for the realization of future CNN-UM sensor-processor systems by taking their rich dynamical behavior into account. However, a deep mathematical treatment of CNN with memristors, briefly called memristor CNN in the following, has not been provided so far. Especially, the derivation of methods allowing the determination of the parameter space of a memristor CNN showing emergent complex behavior, is being essentially impor-

internal dynamics.

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

**Figure 3.** *CNN architecture.*

**71**

#### **Figure 1.** *Crossbar architecture and crossbar elements of a memristor.*

**Figure 2.** *Four basic circuit elements.*

model [17] under the same type of excitation. A comparison of the dynamic behavior of titanium dioxide memristor circuits assuming different boundary models including the BCM model is given in [18]. The results underline the sensitivity of these nonlinear circuits to the modeling accuracy. In the following, models have been proposed also for memristors based on other materials, e.g., for a tantalum oxide element [13, 19] and for a niobium dioxide memristors [12]. Nevertheless, recent investigations [14] uncover numerical problems occurring in the numerical solution of the strongly nonlinear differential-algebraic equations or in a SPICE simulation [8] of these memristors.

Computing with memory is one of the main properties of neural networks, in which the performance is within localized memory storage. Therefore, they can provide high density analogue storage and can be integrated locally with computing elements which is the main advantage of the network learning algorithms. The memristive array works as a conventional memory, i.e., the weigh values can be calculated outside the array and can be programmed to the correct addresses. The

#### *Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

algorithms are flexible because they can be implemented with an appropriate design. Local and non-local learning algorithms can be implemented in a straightforward way. The disadvantage is that they cannot have high connectivity and internal dynamics.

Cellular Nonlinear/Nanoscale Networks (CNN) have been introduced in 1988 by Chua and Yang [20, 21] as a new class of information processing systems which show important potential applications (**Figure 3**). By endowing a single CNN cell with local analog and logic memory, some communication circuitry and further units, the CNN Universal Machine (CNN-UM) has been invented by Roska [22–24]. Analog CNN-UM chip hardware implementations have been developed some time ago. A CNN-UM chip represents a parallel computer with stored programmability allowing real-time processing of multivariate data. CNN have very promising applications in image processing and pattern recognition [22, 25–27]. Although, recently realized systems, e.g., the EyeRis 1.3 system, the MIPA4k, and SCAMP-5, are characterized as sensor-processor systems for high speed vision by reaching frames rates more than 20 kHz, their low resolution (e.g., 176 144 pixel in the EyeRis 1.3 system) limits the applicability to certain problems in practice only. Since the cell size cannot be decreased considerably in conventional CMOS technology, nano-elements will play an important role in future CNN-UM chip realizations. Especially, memristors [28] which are considered for synaptic connections in first realizations [29], will play an important role for the realization of future CNN-UM sensor-processor systems by taking their rich dynamical behavior into account. However, a deep mathematical treatment of CNN with memristors, briefly called memristor CNN in the following, has not been provided so far. Especially, the derivation of methods allowing the determination of the parameter space of a memristor CNN showing emergent complex behavior, is being essentially important in the development of CNN-based computational methods.

Many scientists have struggled to uncover the elusive origin of "complexity" and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, non-equilibrium phenomena, etc. [22, 30–32]. In his works, Schrödinger [31] defines the exchange of energy as a necessary condition for complexity of open systems. Prigogine [32] states the instability of the homogeneous systems as a new principle of nature, whereas Turing finds the origin of morphogenesis in symmetry breaking. In [31] Smale considers a reaction-diffusion system

**Figure 3.** *CNN architecture.*

model [17] under the same type of excitation. A comparison of the dynamic behavior of titanium dioxide memristor circuits assuming different boundary models including the BCM model is given in [18]. The results underline the sensitivity of these nonlinear circuits to the modeling accuracy. In the following, models have been proposed also for memristors based on other materials, e.g., for a tantalum oxide element [13, 19] and for a niobium dioxide memristors [12]. Nevertheless, recent investigations [14] uncover numerical problems occurring in the numerical solution of the strongly nonlinear differential-algebraic equations or in a SPICE

Computing with memory is one of the main properties of neural networks, in which the performance is within localized memory storage. Therefore, they can provide high density analogue storage and can be integrated locally with computing elements which is the main advantage of the network learning algorithms. The memristive array works as a conventional memory, i.e., the weigh values can be calculated outside the array and can be programmed to the correct addresses. The

simulation [8] of these memristors.

**Figure 1.**

**Figure 2.**

**70**

*Four basic circuit elements.*

*Crossbar architecture and crossbar elements of a memristor.*

*Memristors - Circuits and Applications of Memristor Devices*

which has properties such that it makes the Turing interacting system to oscillate. Recent laboratory observations suggesting that chaotic regimes may in fact represent the ground state of central nervous system point to the intriguing possibility of exploiting and controlling chaos for future scientific and engineering applications.

**2.1 Definition of local activity for reaction-diffusion equations**

are locally active at some cell equilibrium points [33].

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

*<sup>∂</sup>u x*ð Þ

*<sup>∂</sup>v x*ð Þ

Let us discretize first equation of (1) in space:

Then we obtain the resulting point dynamics

*gu w<sup>l</sup> j uj*�<sup>1</sup> � *uj*

*duj dt* ¼

*dvj*

*dt* <sup>¼</sup> *<sup>f</sup> <sup>v</sup>*ð Þ*: ,*

*dt* <sup>¼</sup> *gu uj*�<sup>1</sup> � *uj*

gradient is represented by linear resistors.

*d uj*ð Þ*t*

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *gu*∇<sup>2</sup>

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *gv*∇<sup>2</sup>

larger energy signal.

and *gu uj*þ<sup>1</sup> � *uj*

memristors:

(**Figure 4**).

**73**

We call reaction-diffusion CNN equations locally active if and only if their cells

In [34] principle of local activity is explained with the assumption of zero energy at the zero time. Therefore we can say that the cell is acting as a source of small signal energy and it is able to give rise to an initially very small input signal to a

From mathematical point of view the signal should be very small in order to take the linear terms of Taylor series expansion of the cell model. In this way we can derive explicit analytical inequalities for the cell to be locally active at some equilibrium points in which the Taylor series expansion is computed. In other words, we can say that complex behavior of cells arises from infinitesimal small perturbations.

*u x*ð Þþ *f <sup>u</sup>*ð Þ *u x*ð Þ*; v x*ð Þ

(1)

(4)

*u x*ð Þþ *f <sup>v</sup>*ð Þ *u x*ð Þ*; v x*ð Þ

<sup>Δ</sup>*x*<sup>2</sup> <sup>þ</sup> *<sup>f</sup> <sup>u</sup>*ð Þ*: ,* (2)

*dt* <sup>¼</sup> *i,* (3)

where *gu, <sup>v</sup>* are the diffusion coefficients, *f u, <sup>v</sup>*ð Þ*:* state for the reaction model. In (1) *u*(*x*) and *v*(*x*) are represented by voltages on the RD hardware, and the

<sup>þ</sup> *gu uj*þ<sup>1</sup> � *uj*

where *j* is the spatial index, *Δx* is the discrete step in space, terms *gu uj*�<sup>1</sup> � *uj*

 represent respectively current flowing into the *j*th node from ( *j* � 1)th and ( *j* + 1)th nodes via two resistors whose conductance is represented by *gu*. We consider the memristor model [9], in which the resistors are replaced with

*dw*

*j uj*þ<sup>1</sup> � *uj* <sup>Δ</sup>*x*<sup>2</sup> <sup>þ</sup> *<sup>f</sup> <sup>u</sup>*ð Þ*: ,*

where the voltage across the memristor is *v*, the current of the memristor is *i*, the nominal internal state of the memristor corresponding to the charge flow is *w*, and

*i* ¼ *gu*ð Þ *w v,*

the monotonically non decreasing function is *gu*ð Þ *w* when *w* is increasing. We shall replace resistors for diffusion in analog RD LSIs with memristors

<sup>þ</sup> *gu <sup>w</sup><sup>r</sup>*

where *gu*ð Þ*:* is the monotonically increasing function defined by:

In this section we shall consider reaction-diffusion system [35, 36]:

Among other things, a mathematical proof is given in [22, 30, 33, 34] showing that none of the complexity related jargons cited above can explain emergent complex behavior in reaction-diffusion system without introducing local activity. The theory of local activity offers a constructive method for determination of complexity. We shall propose algorithm for hysteresis CNN which defines the domain of the cell parameters in which the system is capable of exhibiting complexity. The main advantage of local activity theory is that the complex behavior of reaction-diffusion system can be explained in a rigorous way by explicit mathematical formulas determining a small subset of locally active parameters' region called edge of chaos. Cell kinetic equations which are locally active can exhibit limit cycles or chaos if the cells are uncoupled from each other by letting all diffusion coefficients to be zero. In this case complex spatiotemporal phenomena arise, such as spatiotemporal chaos or scroll waves.

In particular, constructive and explicit mathematical inequalities can be obtained for identifying that region in the CNN parameter space. By restricting the cell parameter space to the local activity domain, a major reduction in the computing time required by the parameter search algorithms is achieved [33, 34].

In the next sections we shall present two mathematical models of memristor CNN. First one is reaction-diffusion memristor CNN, and the second one is memristor hysteresis CNN. We shall derive algorithm for determination of edge of chaos regime in these models based on local activity theory [33].

#### **2. Reaction-diffusion memristor CNN (RD-MCNN)**

Nonlinear reaction-diffusion types of equations are widely used to describe phenomena in different fields. We shall determine for reaction-diffusion CNN the domain of the cell parameters in which the cells are locally active and therefore they can exhibit complex behavior. Edge of chaos (EC) is associated with a region of parameter space in which complex phenomena and thus information processing can appear.

In this section the principle of local activity will be applied in studying complex behavior of reaction-diffusion CNN with memristor synapses (RD-MCNN). Semiconductor reaction-diffusion (RD) large scale circuits (LSI) implementing RD dynamics, called reaction-diffusion chips, are mostly designed by digital, analog, or mixed-signal complementary-metal-oxide-semiconductor (CMOS) circuits of CNN and cellular automata (CA).

In our model each cell will be arranged on a two-dimensional square grid and will be connected to adjacent cells through coupling devices that mimic 2-D spatial diffusion and transmit the cell's state to its neighboring cells, as in conventional CNN.

We shall consider a discrete medium of identical cells which interact locally and therefore the homogeneous medium exhibits a non-homogeneous static or spatiotemporal patterns under homogeneous initial and boundary conditions. The theory of local activity will be formulated mathematically and implemented in circuit models. We shall start with reaction-diffusion CNN equations as a special class of spatially extended dynamical systems and we shall define the principle of local activity which will not be based on observations but on rigorous mathematical analysis.

which has properties such that it makes the Turing interacting system to oscillate. Recent laboratory observations suggesting that chaotic regimes may in fact represent the ground state of central nervous system point to the intriguing possibility of exploiting and controlling chaos for future scientific and engineering applications. Among other things, a mathematical proof is given in [22, 30, 33, 34] showing

*Memristors - Circuits and Applications of Memristor Devices*

that none of the complexity related jargons cited above can explain emergent complex behavior in reaction-diffusion system without introducing local activity. The theory of local activity offers a constructive method for determination of complexity. We shall propose algorithm for hysteresis CNN which defines the domain of the cell parameters in which the system is capable of exhibiting complexity. The main advantage of local activity theory is that the complex behavior of reaction-diffusion system can be explained in a rigorous way by explicit mathematical formulas determining a small subset of locally active parameters' region called edge of chaos. Cell kinetic equations which are locally active can exhibit limit cycles or chaos if the cells are uncoupled from each other by letting all diffusion coefficients to be zero. In this case complex spatiotemporal phenomena arise, such as

In particular, constructive and explicit mathematical inequalities can be obtained for identifying that region in the CNN parameter space. By restricting the cell parameter space to the local activity domain, a major reduction in the comput-

In the next sections we shall present two mathematical models of memristor CNN. First one is reaction-diffusion memristor CNN, and the second one is memristor hysteresis CNN. We shall derive algorithm for determination of edge of

Nonlinear reaction-diffusion types of equations are widely used to describe phenomena in different fields. We shall determine for reaction-diffusion CNN the domain of the cell parameters in which the cells are locally active and therefore they can exhibit complex behavior. Edge of chaos (EC) is associated with a region of parameter space in which complex phenomena and thus information processing can

In this section the principle of local activity will be applied in studying complex behavior of reaction-diffusion CNN with memristor synapses (RD-MCNN). Semiconductor reaction-diffusion (RD) large scale circuits (LSI) implementing RD dynamics, called reaction-diffusion chips, are mostly designed by digital, analog, or mixed-signal complementary-metal-oxide-semiconductor (CMOS) circuits of CNN

In our model each cell will be arranged on a two-dimensional square grid and will

be connected to adjacent cells through coupling devices that mimic 2-D spatial diffusion and transmit the cell's state to its neighboring cells, as in conventional CNN. We shall consider a discrete medium of identical cells which interact locally and therefore the homogeneous medium exhibits a non-homogeneous static or spatiotemporal patterns under homogeneous initial and boundary conditions. The theory of local activity will be formulated mathematically and implemented in circuit models. We shall start with reaction-diffusion CNN equations as a special class of spatially extended dynamical systems and we shall define the principle of local activity which

will not be based on observations but on rigorous mathematical analysis.

ing time required by the parameter search algorithms is achieved [33, 34].

chaos regime in these models based on local activity theory [33].

**2. Reaction-diffusion memristor CNN (RD-MCNN)**

spatiotemporal chaos or scroll waves.

appear.

**72**

and cellular automata (CA).

#### **2.1 Definition of local activity for reaction-diffusion equations**

We call reaction-diffusion CNN equations locally active if and only if their cells are locally active at some cell equilibrium points [33].

In [34] principle of local activity is explained with the assumption of zero energy at the zero time. Therefore we can say that the cell is acting as a source of small signal energy and it is able to give rise to an initially very small input signal to a larger energy signal.

From mathematical point of view the signal should be very small in order to take the linear terms of Taylor series expansion of the cell model. In this way we can derive explicit analytical inequalities for the cell to be locally active at some equilibrium points in which the Taylor series expansion is computed. In other words, we can say that complex behavior of cells arises from infinitesimal small perturbations.

In this section we shall consider reaction-diffusion system [35, 36]:

$$\begin{aligned} \frac{\partial u(\mathbf{x})}{\partial t} &= \mathbf{g}\_u \nabla^2 u(\mathbf{x}) + f\_u(u(\mathbf{x}), v(\mathbf{x})) \\ \frac{\partial v(\mathbf{x})}{\partial t} &= \mathbf{g}\_v \nabla^2 u(\mathbf{x}) + f\_v(u(\mathbf{x}), v(\mathbf{x})) \end{aligned} \tag{1}$$

where *gu, <sup>v</sup>* are the diffusion coefficients, *f u, <sup>v</sup>*ð Þ*:* state for the reaction model. In (1) *u*(*x*) and *v*(*x*) are represented by voltages on the RD hardware, and the gradient is represented by linear resistors.

Let us discretize first equation of (1) in space:

$$\frac{d\,u\_{j}(t)}{dt} = \frac{\mathbf{g}\_{u}\left(u\_{j-1} - u\_{j}\right) + \mathbf{g}\_{u}\left(u\_{j+1} - u\_{j}\right)}{\Delta\mathbf{x}^{2}} + f\_{u}(.),\tag{2}$$

where *j* is the spatial index, *Δx* is the discrete step in space, terms *gu uj*�<sup>1</sup> � *uj* and *gu uj*þ<sup>1</sup> � *uj* represent respectively current flowing into the *j*th node from ( *j* � 1)th and ( *j* + 1)th nodes via two resistors whose conductance is represented by *gu*.

We consider the memristor model [9], in which the resistors are replaced with memristors:

$$\dot{u} = \mathbf{g}\_u(w)v,\\ \frac{dw}{dt} = \dot{\mathbf{t}},\tag{3}$$

where the voltage across the memristor is *v*, the current of the memristor is *i*, the nominal internal state of the memristor corresponding to the charge flow is *w*, and the monotonically non decreasing function is *gu*ð Þ *w* when *w* is increasing.

We shall replace resistors for diffusion in analog RD LSIs with memristors (**Figure 4**).

Then we obtain the resulting point dynamics

$$\begin{cases} du\_j = \mathcal{g}\_u \left( w\_j^l \right) \left( u\_{j-1} - u\_j \right) + \mathcal{g}\_u \left( w\_j^r \right) \left( u\_{j+1} - u\_j \right) \\ \qquad \Delta x^2 \\ \frac{dv\_j}{dt} = f\_v(.), \end{cases} \tag{4}$$

where *gu*ð Þ*:* is the monotonically increasing function defined by:

**Figure 4.** *Circuit realization.*

$$\lg\_u\left(w\_j^{l,r}\right) = \lg\_{min} + \left(\mathbf{g}\_{max} - \mathbf{g}\_{min}\right) \frac{\mathbf{1}}{\mathbf{1} + e^{-\beta w\_j^{l,r}}},\tag{5}$$

*∂u*

*∂v ∂t*

*f* <sup>1</sup>ð Þ¼� *u; v*

8 >>>><

>>>>:

*f*(*u*) is monotonically non decreasing function

*f u*ð Þ¼

carbon oxide and to the temperature.

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

*duij*

*dvij*

Nagumo CNN model:

*duij*

*dvij*

**75**

*f* <sup>2</sup>ð Þ¼ *u; v f u*ð Þ� *v,*

where

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>1</sup>ð Þþ *<sup>u</sup>; <sup>v</sup> <sup>d</sup>*1∇<sup>2</sup>

1 *ε*

<sup>¼</sup> *<sup>f</sup>* <sup>2</sup>ð Þþ *<sup>u</sup>; <sup>v</sup> <sup>d</sup>*2∇<sup>2</sup>

*u,*

(7)

(8)

(9)

(11)

<sup>3</sup> *; v* <sup>∗</sup> 3 � �.

*,* k ¼ 1*,* 2 (10)

*v,*

*a* � �*,*

<sup>3</sup> <sup>≤</sup> *<sup>u</sup>*<sup>≤</sup> <sup>1</sup>

1 3

*u u*ð Þ � <sup>1</sup> *<sup>u</sup>* � *<sup>b</sup>* <sup>þ</sup> *<sup>v</sup>*

0*,* 0≤ *u*≤

1 *u* . 1

The parameters *ε*, *a*, *b* are physical parameters, related to pressures of oxide,

*dt* <sup>¼</sup> *<sup>f</sup>* <sup>1</sup> *uij; vij* � � <sup>þ</sup> *<sup>d</sup>*<sup>1</sup> *<sup>w</sup>l,r* � � *ui*�1*<sup>j</sup>* <sup>þ</sup> *ui*þ1*<sup>j</sup>* <sup>þ</sup> *uij*�<sup>1</sup> <sup>þ</sup> *uij*þ<sup>1</sup> � <sup>4</sup>*uij* � �

*dt* <sup>¼</sup> *<sup>f</sup>* <sup>2</sup> *uij; vij* � � <sup>þ</sup> *<sup>d</sup>*<sup>2</sup> *<sup>w</sup>l,r* � � *vi*�1*<sup>j</sup>* <sup>þ</sup> *vi*þ1*<sup>j</sup>* <sup>þ</sup> *vij*�<sup>1</sup> <sup>þ</sup> *vij*þ<sup>1</sup> � <sup>4</sup>*vij* � �*,*

where *dk wl,r* � � denotes the monotonically increasing function defined as

We shall apply the constructive algorithm for determining edge of chaos (EC)

• We map memristive FitzHugh-Nagumo system into the associated FitzHugh-

• We find the equilibrium points of (11). According to the theory of dynamical

*<sup>f</sup>* <sup>1</sup> *<sup>u</sup>*<sup>∗</sup> *; <sup>v</sup>* <sup>∗</sup> ð Þ¼ <sup>0</sup>

*dt* <sup>¼</sup> *<sup>f</sup>* <sup>1</sup> *uij; vij* � � <sup>þ</sup> *<sup>d</sup>*1ð Þ *<sup>w</sup> ui*�1*<sup>j</sup>* <sup>þ</sup> *ui*þ1*<sup>j</sup>* <sup>þ</sup> *uij*�<sup>1</sup> <sup>þ</sup> *uij*þ<sup>1</sup> � <sup>4</sup>*uij* � �

*dt* <sup>¼</sup> *<sup>f</sup>* <sup>2</sup> *uij; vij* � � <sup>þ</sup> *<sup>d</sup>*2ð Þ *<sup>w</sup> vi*�1*<sup>j</sup>* <sup>þ</sup> *vi*þ1*<sup>j</sup>* <sup>þ</sup> *vij*�<sup>1</sup> <sup>þ</sup> *vij*þ<sup>1</sup> � <sup>4</sup>*vij* � �*,*

*dk <sup>w</sup>l,r* � � <sup>¼</sup> *dmin* <sup>þ</sup> ð Þ *dmax* � *dmin :* <sup>1</sup>

region for the memristive FitzHugh-Nagumo system (9) and (10).

systems equilibrium points *u*<sup>∗</sup> *, v* <sup>∗</sup> are these for which:

System (12) may have one, two or three real roots *u*<sup>∗</sup>

In general, these roots are functions of the cell parameters *a*, *b*, *ε.*

Then the memristive dynamics is defined as in (6).

We discretize spatially system (7) and the resulting point dynamics are given as:

<sup>4</sup> *,* <sup>1</sup>

1 � *e*

�*βwl,r k*

*<sup>f</sup>* <sup>2</sup> *<sup>u</sup>*<sup>∗</sup> *; <sup>v</sup>* <sup>∗</sup> ð Þ¼ <sup>0</sup> (12)

<sup>1</sup> *; v* <sup>∗</sup> 1 � �*, u*<sup>∗</sup>

<sup>2</sup> *; v* <sup>∗</sup> 2 � �*, u*<sup>∗</sup>

<sup>1</sup> � <sup>27</sup>*u u*ð Þ � <sup>1</sup> <sup>2</sup>

where *β* is the gain, *gmin* and *gmax* are the minimum and maximum coupling strengths, respectively, and *wl,r <sup>j</sup>* denotes the variables for determining the coupling strength (*l*—leftward, *r*—rightward).

We introduce the following memristive dynamics for *wl,r j* :

$$
\pi \frac{d w\_j^{l,r}}{dt} = \mathbf{g}\_u \left( w\_j^{l,r} \right) . \eta\_1. (u\_{j-1} - u\_j), \tag{6}
$$

where the right-hand side represents the current of the memristors in (2), *η*<sup>1</sup> denotes the polarity coefficient—*η*<sup>1</sup> = + 1: *w<sup>l</sup> j* , *<sup>η</sup>*<sup>1</sup> <sup>=</sup> �1: *<sup>w</sup><sup>r</sup> j* .

In the next we shall study RD-MCNN model of FitzHugh-Nagumo system.

Simplification of Hodgkin-Huxley model (**Figure 5**) can be given by FitzHugh-Nagumo system consisting of two coupled partial differential equations with two diffusion coefficients. In generally it describes the electrical potential across cell membrane when the flow of ionic channels is changed. It also can be presented as the model of electrical waves of the heart.

In this section we shall present the following FitzHugh-Nagumo system with two diffusion terms:

**Figure 5.** *Memristive model.*

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

$$\begin{aligned} \frac{\partial u}{\partial t} &= f\_1(u, v) + d\_1 \nabla^2 u, \\ \frac{\partial v}{\partial t} &= f\_2(u, v) + d\_2 \nabla^2 v, \end{aligned} \tag{7}$$

where

*gu <sup>w</sup>l,r j* 

*Memristors - Circuits and Applications of Memristor Devices*

*τ dwl,r j*

denotes the polarity coefficient—*η*<sup>1</sup> = + 1: *w<sup>l</sup>*

the model of electrical waves of the heart.

two diffusion terms:

**Figure 5.** *Memristive model.*

**74**

strengths, respectively, and *wl,r*

**Figure 4.** *Circuit realization.*

strength (*l*—leftward, *r*—rightward).

¼ *gmin* þ *gmax* � *gmin*

where *β* is the gain, *gmin* and *gmax* are the minimum and maximum coupling

*j* 

where the right-hand side represents the current of the memristors in (2), *η*<sup>1</sup>

In the next we shall study RD-MCNN model of FitzHugh-Nagumo system. Simplification of Hodgkin-Huxley model (**Figure 5**) can be given by FitzHugh-Nagumo system consisting of two coupled partial differential equations with two diffusion coefficients. In generally it describes the electrical potential across cell membrane when the flow of ionic channels is changed. It also can be presented as

In this section we shall present the following FitzHugh-Nagumo system with

*j*

We introduce the following memristive dynamics for *wl,r*

*dt* <sup>¼</sup> *gu <sup>w</sup>l,r*

1

*:η*1*: uj*�<sup>1</sup> � *uj*

, *<sup>η</sup>*<sup>1</sup> <sup>=</sup> �1: *<sup>w</sup><sup>r</sup>*

1 þ *e*

*<sup>j</sup>* denotes the variables for determining the coupling

*j* .

�*βwl,r j*

*j* :

*,* (6)

*,* (5)

$$\begin{aligned} f\_1(u,v) &= -\frac{1}{\varepsilon} u(u-1) \left( u - \frac{b+v}{a} \right), \\ f\_2(u,v) &= f(u) - v, \end{aligned} \tag{8}$$

*f*(*u*) is monotonically non decreasing function

$$f(u) = \begin{cases} 0, & 0 \le u \le \frac{1}{3} \\ 1 - \frac{27u(u-1)^2}{4}, & \frac{1}{3} \le u \le 1 \\ 1 & u \ge 1 \end{cases}$$

The parameters *ε*, *a*, *b* are physical parameters, related to pressures of oxide, carbon oxide and to the temperature.

We discretize spatially system (7) and the resulting point dynamics are given as:

$$\begin{aligned} \frac{du\_{\vec{\eta}}}{dt} &= f\_1(u\_{\vec{\eta}}, v\_{\vec{\eta}}) + d\_1(w^{l,r}) \left( u\_{i-1\vec{\eta}} + u\_{i+1\vec{\eta}} + u\_{\vec{\eta}-1} + u\_{\vec{\eta}+1} - 4u\_{\vec{\eta}} \right) \\ \frac{dv\_{\vec{\eta}}}{dt} &= f\_2(u\_{\vec{\eta}}, v\_{\vec{\eta}}) + d\_2(w^{l,r}) \left( v\_{i-1\vec{\eta}} + v\_{i+1\vec{\eta}} + v\_{\vec{\eta}-1} + v\_{\vec{\eta}+1} - 4v\_{\vec{\eta}} \right), \end{aligned} \tag{9}$$

where *dk wl,r* � � denotes the monotonically increasing function defined as

$$d\_k(w^{l,r}) = d\_{\min} + (d\_{\max} - d\_{\min}) \cdot \frac{1}{1 - e^{-\beta w\_k^{l,r}}}, \mathbf{k} = \mathbf{1}, \mathbf{2} \tag{10}$$

Then the memristive dynamics is defined as in (6).

We shall apply the constructive algorithm for determining edge of chaos (EC) region for the memristive FitzHugh-Nagumo system (9) and (10).

• We map memristive FitzHugh-Nagumo system into the associated FitzHugh-Nagumo CNN model:

$$\begin{aligned} \frac{d u\_{\vec{\eta}}}{dt} &= f\_1(u\_{\vec{\eta}}, v\_{\vec{\eta}}) + d\_1(w) \left( u\_{i-1\vec{\eta}} + u\_{i+1\vec{\eta}} + u\_{\vec{\eta}-1} + u\_{\vec{\eta}+1} - 4u\_{\vec{\eta}} \right) \\ \frac{d v\_{\vec{\eta}}}{dt} &= f\_2 \left( u\_{\vec{\eta}}, v\_{\vec{\eta}} \right) + d\_2(w) \left( v\_{i-1\vec{\eta}} + v\_{i+1\vec{\eta}} + v\_{\vec{\eta}-1} + v\_{\vec{\eta}+1} - 4v\_{\vec{\eta}} \right), \end{aligned} \tag{11}$$

• We find the equilibrium points of (11). According to the theory of dynamical systems equilibrium points *u*<sup>∗</sup> *, v* <sup>∗</sup> are these for which:

$$\begin{aligned} f\_1(\boldsymbol{u}^\*, \boldsymbol{v}^\*) &= \mathbf{0} \\ f\_2(\boldsymbol{u}^\*, \boldsymbol{v}^\*) &= \mathbf{0} \end{aligned} \tag{12}$$

System (12) may have one, two or three real roots *u*<sup>∗</sup> <sup>1</sup> *; v* <sup>∗</sup> 1 � �*, u*<sup>∗</sup> <sup>2</sup> *; v* <sup>∗</sup> 2 � �*, u*<sup>∗</sup> <sup>3</sup> *; v* <sup>∗</sup> 3 � �. In general, these roots are functions of the cell parameters *a*, *b*, *ε.*


$$LAR\left(E\_k^\*\right) : a\_{22} \ge 0,\\
\text{or } 4a\_{11}a\_{22} \le \left(a\_{12} + a\_{21}\right)^2$$

• Additional stability condition in this case is:

$$Tr(E\_k^\*) \le \mathbf{0} \text{ and } D(E\_k^\*) > \mathbf{0}$$

It can be shown that this is the only region which corresponds to locally asymptotically stable equilibrium points of our model.

> **Figure 6.** *Simulations.*

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

**Figure 7.**

**77**

*Pattern formation: (a) spatial pattern formation and (b) clockwise spiral wave patterns.*


Then we check the conditions for the local activity and stability of the equilibrium points. The result is that only *E*1*, E*<sup>2</sup> satisfy these conditions.

By the above presented algorithm we can prove the following theorem:

**Theorem 1.** *We say that MCNN model of FitzHugh-Nagumo system (7) and (8) operates in EC region if and only if the following conditions for the cell parameters are satisfied:*

$$a \ll 1, \frac{a-b-1}{a} \le 1.$$

*In other words there is at least one equilibrium point which is in SLAR E*<sup>∗</sup> *k :*

In the simulations of the above algorithm we can see the cell parameter projection on the (*T*, Δ, *a*22)-plane (**Figure 6**). We have red subregion in which we have three equilibrium points of our model and at least one is both stable and locally active; blue subregion in which we have either one or three equilibrium points and every equilibrium point is unstable; green subregion in which there is only one equilibrium point and it is both stable and locally active. By definition, red and green subregions in **Figure 6a** together constitute the edge of chaos. In **Figure 6b** we can see the plot of edge of chaos regime in the parameter (*a*, *b*, *ε*) plane.

Through extensive numerical simulations we obtain that non uniform spatial patterns are generated in our CNN model with memristor synapses depending on initial conditions—see **Figure 7**.

Through the above numerical simulations, the following things were demonstrated: (a) excitable waves propagating on the memristor can modulate the memristor conductance which depends on the memristor's polarity; (b) change of memristor conductance can modulate the velocity of the excitable wave propagation, and it is inversely proportional to the time constant of the model; (c) the model under consideration generates nonuniform spatial patterns which process depends on the initial condition of FitzHugh-Nagumo system (7) and (8), memristor polarity and stimulation.

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

• We calculate the four cell coefficients *a*11*, a*12*, a*21*, a*<sup>22</sup> of the Jacobian matrix of

: *<sup>a</sup>*<sup>22</sup> . <sup>0</sup>*,* or 4*a*11*a*<sup>22</sup> , ð Þ *<sup>a</sup>*<sup>12</sup> <sup>þ</sup> *<sup>a</sup>*<sup>21</sup>

It can be shown that this is the only region which corresponds to locally asymp-

Then we check the conditions for the local activity and stability of the equilib-

**Theorem 1.** *We say that MCNN model of FitzHugh-Nagumo system (7) and (8) operates in EC region if and only if the following conditions for the cell parameters are*

> *a* � *b* � 1 *a*

In the simulations of the above algorithm we can see the cell parameter projection on the (*T*, Δ, *a*22)-plane (**Figure 6**). We have red subregion in which we have three equilibrium points of our model and at least one is both stable and locally active; blue subregion in which we have either one or three equilibrium points and

one equilibrium point and it is both stable and locally active. By definition, red and green subregions in **Figure 6a** together constitute the edge of chaos. In **Figure 6b** we can see the plot of edge of chaos regime in the parameter (*a*, *b*, *ε*) plane.

Through extensive numerical simulations we obtain that non uniform spatial patterns are generated in our CNN model with memristor synapses depending on

Through the above numerical simulations, the following things were demon-

strated: (a) excitable waves propagating on the memristor can modulate the memristor conductance which depends on the memristor's polarity; (b) change of memristor conductance can modulate the velocity of the excitable wave propagation, and it is inversely proportional to the time constant of the model; (c) the model under consideration generates nonuniform spatial patterns which process depends on the initial condition of FitzHugh-Nagumo system (7) and (8),

, 1*:*

, 0 and *D E*<sup>∗</sup>

*k*

• We define locally active region for each equilibrium point *E*<sup>∗</sup>

*<sup>k</sup> , k* ¼ 1*,* 2*,* 3.

*k*

*k* :

2

*k :* of the Jacobian

*k :*

) and the determinant *D E*<sup>∗</sup>

*k* . 0

(12) about each system equilibrium point *E*<sup>∗</sup>

*Memristors - Circuits and Applications of Memristor Devices*

matrix of (12) for each equilibrium point.

*LAR E*<sup>∗</sup> *k*

totically stable equilibrium points of our model.

*<sup>E</sup>*<sup>1</sup> <sup>¼</sup> ð Þ <sup>0</sup>*;* <sup>0</sup> *, E*<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup>*;* <sup>1</sup> *, E*<sup>3</sup> <sup>¼</sup> *<sup>b</sup>*þ<sup>1</sup>

initial conditions—see **Figure 7**.

memristor polarity and stimulation.

*satisfied:*

**76**

• Additional stability condition in this case is:

*Tr E*<sup>∗</sup> *k*

• We define the stable and locally active region *SLAR E*<sup>∗</sup>

• In our particular case, we have three equilibrium points

rium points. The result is that only *E*1*, E*<sup>2</sup> satisfy these conditions.

*ε* ≪ 1*,*

*<sup>a</sup> ;* <sup>1</sup> .

By the above presented algorithm we can prove the following theorem:

*In other words there is at least one equilibrium point which is in SLAR E*<sup>∗</sup>

every equilibrium point is unstable; green subregion in which there is only

• Then we calculate the trace *Tr E*<sup>∗</sup>

#### **3. Hysteresis CNN with memristor synapses**

In this section we shall present mathematical study of hysteresis CNN (HCNN) with memristor synapses. In our model the cells are of first order and they have hysteresis switches. It is known from the literature [21–26, 35, 37–41] that such models have many applications because they operate in two modes—bi-stable multi-vibrator mode and relaxation oscillator mode. We shall consider HCNN working in second one. When CNN operates in the relaxation oscillator mode then various patterns and nonlinear waves can be generated. Associative (static) and dynamic memories functions can be derived from the hysteresis CNN [35, 37, 41].

Let us consider hysteresis CNN with memristor synapses, which we shall call memristor hysteresis CNN (M-HCNN):

$$\frac{du\_{\vec{\eta}}}{dt} = -m\left(u\_{\vec{\eta}}\right) + \sum\_{k\_l, l \in N\_{\vec{\eta}}} \left(a\_{k-i, l-j} f(u\_{kl})\right) \qquad \mathbf{1} \le i, j \le \mathbf{N},\tag{13}$$

where *uij* denotes the state of the cell, the output *yij* <sup>¼</sup> *f uij* � � is dynamic hysteresis function defined by:

$$f(u(t)) = \begin{cases} \mathbf{1}, \text{for } u(t) > -\mathbf{1}, f(u(t\_{-})) = \mathbf{1} \\ -\mathbf{1}, \text{for } u(t) = -\mathbf{1} \\ -\mathbf{1}, \text{for } u(t) \le \mathbf{1}, f(u(t\_{-})) = -\mathbf{1} \\ \mathbf{1}, \text{for } u(t) = \mathbf{1}, \end{cases} \tag{14}$$

*<sup>t</sup>*� <sup>¼</sup> lim*<sup>ε</sup>*!<sup>0</sup> ð Þ *<sup>t</sup>* � *<sup>ε</sup> , <sup>ε</sup>* . 0, *<sup>m</sup>*ð Þ*:* is defined as *m uij* � � <sup>¼</sup> *uij M t*ð Þ in which by *M t*ð Þ we denote the memristance. When we insert memristor [9] in HCNN model we expect to obtain better resolution in static and dynamic images [41]. We introduce a memristor in HCNN by replacing the original linear resistor. In this way it can exhibit nonlinear current-voltage characteristic with locally negative differential resistance. The main advantage of our memristor HCNN (M-HCNN) is the versatility and compactness due to the nonvolatile and programmable synapse circuits. In the circuit realization of M-HCNN the output function is not complex.

Let us consider M-HCNN model working in a relaxation oscillator mode described by

$$\frac{du\_{\vec{\eta}}}{dt} = -m\left(u\_{\vec{\eta}}\right) - 2\left.h\left(u\_{\vec{\eta}}\right) + bf\left(u\_{\vec{\eta}}\right), \mathbf{1} \le i, j \le N. \tag{15}$$

cell parameters in which the cells are locally active. We shall develop constructive and explicit mathematical inequalities for identifying the region in the M-HCNN model (15) parameter space where complexity phenomena may emerge. By restricting the cell parameter space to the local activity domain we can achieve a major reduction in the computing time required by the parameter search algorithms. This will allow to determine and control chaos which will be useful for the

We shall develop constructive algorithm for studying the dynamics of our M-

to discretize the M-HCNN model (15). Then in relaxation mode the dynamics of an isolated cell when there are no control and threshold parameters can

*dt* ¼ �*m uij* � � � <sup>2</sup>*h uij* � � <sup>þ</sup> *bf uij* � � <sup>¼</sup> *F uij* � �*:* (16)

010 1 �4 1 010 1

CA in order

0

B@

future scientific and engineering applications [34, 41, 42].

(1) We chose the Laplace template of the following type

HCNN model (15) based on [33]:

*duij*

be written:

**79**

**Figure 8.**

**Figure 9.**

*Relaxation oscillator defined by (15).*

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

*Spiral waves in HCNN model (15).*

Below is the picture of relaxation oscillator under consideration (**Figure 8**). M-HCNN model (15) generates patterns close to the bifurcation point *b = 3.* Computer simulations of (15) when we use the Laplace template 010 1 �4 1 010 0 B@ 1 CA

show the generation of spiral waves for b = 3 (see **Figure 9**):

#### **3.1 Determination of edge of chaos domain in M-HCNN model**

We shall apply theory of local activity [33, 34] in order to study the dynamics of M-HCNN model (15). The theory which will be presented below offers both constructive analytical and numerical method for obtaining local activity of M-HCNN. It is known [35, 41] that the cells of HCNN can exhibit complexity in the domain of *Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

**3. Hysteresis CNN with memristor synapses**

*Memristors - Circuits and Applications of Memristor Devices*

memristor hysteresis CNN (M-HCNN):

*dt* ¼ �*m uij*

*f ut* ð Þ¼ ð Þ

*duij*

*dt* ¼ �*m uij*

� � <sup>þ</sup> <sup>∑</sup>

*k,l* ∈ *Nij*

where *uij* denotes the state of the cell, the output *yij* ¼ *f uij*

8 >>><

>>>:

*t*� ¼ lim*<sup>ε</sup>*!<sup>0</sup> ð Þ *t* � *ε , ε* . 0, *m*ð Þ*:* is defined as *m uij*

*duij*

esis function defined by:

described by

**78**

In this section we shall present mathematical study of hysteresis CNN (HCNN) with memristor synapses. In our model the cells are of first order and they have hysteresis switches. It is known from the literature [21–26, 35, 37–41] that such models have many applications because they operate in two modes—bi-stable multi-vibrator mode and relaxation oscillator mode. We shall consider HCNN working in second one. When CNN operates in the relaxation oscillator mode then various patterns and nonlinear waves can be generated. Associative (static) and dynamic memories functions can be derived from the hysteresis CNN [35, 37, 41]. Let us consider hysteresis CNN with memristor synapses, which we shall call

*ak*�*i,l*�*<sup>j</sup> f u*ð Þ *kl*

1*,*for *u t*ð Þ . � 1*,f u t* ð Þ¼ ð Þ � 1

�1*,*for *u t*ð Þ , 1*,f u t* ð Þ¼� ð Þ � 1

� � <sup>¼</sup> *uij*

�1*,*for *u t*ðÞ¼�1

denote the memristance. When we insert memristor [9] in HCNN model we expect to obtain better resolution in static and dynamic images [41]. We introduce a memristor in HCNN by replacing the original linear resistor. In this way it can exhibit nonlinear current-voltage characteristic with locally negative differential resistance. The main advantage of our memristor HCNN (M-HCNN) is the versatility and compactness due to the nonvolatile and programmable synapse circuits. In

1*,*for *u t*ðÞ¼ 1*,*

the circuit realization of M-HCNN the output function is not complex.

� � � <sup>2</sup> *h uij*

Computer simulations of (15) when we use the Laplace template

**3.1 Determination of edge of chaos domain in M-HCNN model**

show the generation of spiral waves for b = 3 (see **Figure 9**):

Let us consider M-HCNN model working in a relaxation oscillator mode

Below is the picture of relaxation oscillator under consideration (**Figure 8**). M-HCNN model (15) generates patterns close to the bifurcation point *b = 3.*

We shall apply theory of local activity [33, 34] in order to study the dynamics of M-HCNN model (15). The theory which will be presented below offers both constructive analytical and numerical method for obtaining local activity of M-HCNN. It is known [35, 41] that the cells of HCNN can exhibit complexity in the domain of

� � <sup>þ</sup> *bf uij*

� � 1≤ *i, j*≤ N*,* (13)

� � is dynamic hyster-

*M t*ð Þ in which by *M t*ð Þ we

� �*,* 1≤ *i, j*≤ *N:* (15)

0

B@

010 1 �4 1 010 1

CA

(14)

**Figure 8.** *Relaxation oscillator defined by (15).*

#### **Figure 9.** *Spiral waves in HCNN model (15).*

cell parameters in which the cells are locally active. We shall develop constructive and explicit mathematical inequalities for identifying the region in the M-HCNN model (15) parameter space where complexity phenomena may emerge. By restricting the cell parameter space to the local activity domain we can achieve a major reduction in the computing time required by the parameter search algorithms. This will allow to determine and control chaos which will be useful for the future scientific and engineering applications [34, 41, 42].

We shall develop constructive algorithm for studying the dynamics of our M-HCNN model (15) based on [33]:

$$\text{(1) We chose the Laplace template of the following type } \begin{pmatrix} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \end{pmatrix} \text{ in order.}$$

to discretize the M-HCNN model (15). Then in relaxation mode the dynamics of an isolated cell when there are no control and threshold parameters can be written:

$$\frac{du\_{\vec{\eta}}}{dt} = -m\left(u\_{\vec{\eta}}\right) - 2h\left(u\_{\vec{\eta}}\right) + b\mathcal{f}\left(u\_{\vec{\eta}}\right) = F\left(u\_{\vec{\eta}}\right). \tag{16}$$


**Remark.** In order to provide physical implementation it is important to have appropriate circuit model for which we can use the results from the classical circuit theory. In order to obtain locally active cells it is sufficient the cell to act as a source of small signal in at least one equilibrium point. In this way the cell injects a net small signal average power into the passive resistive grids.

Let us now define stable and locally active region for the M-HCNN model (16). **Definition 1**. *We say that the cell is both stable and locally active region at the equilibrium point Ek for M-HCNN model (16) if*

> *a*<sup>22</sup> . 0 *or* 4*a*11*a*<sup>22</sup> , ð Þ *a*<sup>12</sup> þ *a*<sup>21</sup> <sup>2</sup> *and Tr E*ð Þ*<sup>k</sup>* , 0 *and* Δð Þ *Ek* . 0*:*

*This region in the parameter space is called SLAR E*ð Þ*<sup>k</sup> .*

(5) We shall define the EC region our M-HCNN model (16). According to [33, 34] it is such region in the cell parameter space where we can expect emergence of complex phenomena and information processing.

*tn* ¼ *n*Δ*t, n* ¼ 1*,* 2*,* …

We shall start with the application of our M-HCNN model for edge extraction.

It is known that for feature extraction we firstly extract the edges of the image, which contain most of the information for the image shape. In the example provide on **Figure 11** we show the original image—(a), and then the results which we obtain simulating M-HCNN model—(b) and standard CNN model—(c). It can be seen that the results from M-HCNN (16) model and CNN model are very similar.

The simulations are presented below (see **Figure 11**):

*Example of edge extraction: (a) original image, (b) M-HCNN, and (c) standard CNN.*

*Edge of chaos domain for M-HCNN model (16).*

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

**Figure 10.**

**Figure 11.**

**81**

**Definition 2**. *For M-HCNN model (16) edge of chaos region is such that there exists at least one equilibrium point both locally active and stable.*

Based on the above algorithm we can prove the following theorem:

**Theorem 2**. *We say that M-HCNN model (16) is working in edge of chaos regime if and only if the following conditions are satisfied:* �*1 < b < 3. In other words there is at least one equilibrium point which is locally active and stable.*

We obtain the following edge of chaos domain for our M-HCNN model (16) through computer simulation:

The location of 16 cell parameter points arbitrarily chosen within the locally active domain. We have locally active and stable (or edge of chaos) in red; locally active and unstable in green; locally passive in blue (**Figure 10**).

#### **3.2 Simulation results and some applications**

In this section we shall consider two applications of M-HCNN model (16) in image processing. First one is for edge extraction and the second one is for noise removal. In our simulations we use programing environment MATLAB and we use a forward Euler algorithm with a time step size *Δt* = 0.01. The dynamic hysteresis function *h*(*x*) can be programmed applying this algorithm is:

$$h(u(t\_n)) = \begin{cases} \mathbf{1}, & \text{for } u(t\_n) > -\mathbf{1}, \quad h(u(t\_{n-1})) = \mathbf{1} \\ -\mathbf{1}, & \text{for } u(t\_n) = -\mathbf{1}, \\ -\mathbf{1}, & \text{for } u(t\_n) < \mathbf{1}, \\ \mathbf{1}, & \text{for } u(t\_n) = \mathbf{1}, \end{cases} \qquad h(u(t\_{n-1})) = -\mathbf{1}$$

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

(2) We can find the equilibrium points *Ek* of (16), which satisfy the equation

(4) We denote the trace of the Jacobian matrix at equilibrium point by *Tr E*ð Þ*<sup>k</sup>*

**Remark.** In order to provide physical implementation it is important to have appropriate circuit model for which we can use the results from the classical circuit theory. In order to obtain locally active cells it is sufficient the cell to act as a source of small signal in at least one equilibrium point. In this way the cell injects a net

Let us now define stable and locally active region for the M-HCNN model (16). **Definition 1**. *We say that the cell is both stable and locally active region at the*

<sup>2</sup> *and*

*a*<sup>22</sup> . 0 *or* 4*a*11*a*<sup>22</sup> , ð Þ *a*<sup>12</sup> þ *a*<sup>21</sup>

*Tr E*ð Þ*<sup>k</sup>* , 0 *and* Δð Þ *Ek* . 0*:*

(5) We shall define the EC region our M-HCNN model (16). According to [33, 34] it is such region in the cell parameter space where we can expect

**Definition 2**. *For M-HCNN model (16) edge of chaos region is such that there exists*

**Theorem 2**. *We say that M-HCNN model (16) is working in edge of chaos regime if and only if the following conditions are satisfied:* �*1 < b < 3. In other words there is at*

We obtain the following edge of chaos domain for our M-HCNN model (16)

The location of 16 cell parameter points arbitrarily chosen within the locally active domain. We have locally active and stable (or edge of chaos) in red; locally

In this section we shall consider two applications of M-HCNN model (16) in image processing. First one is for edge extraction and the second one is for noise removal. In our simulations we use programing environment MATLAB and we use a forward Euler algorithm with a time step size *Δt* = 0.01. The dynamic hysteresis

�1*, for u t*ð Þ¼� *<sup>n</sup>* 1*,*

1*, for u t*ð Þ¼ *<sup>n</sup>* 1*,*

1*, for u t*ð Þ*<sup>n</sup>* . � 1*, hut* ð Þ¼ ð Þ *<sup>n</sup>*�<sup>1</sup> 1

�1*, for u t*ð Þ*<sup>n</sup>* , 1*, hut* ð Þ¼� ð Þ *<sup>n</sup>*�<sup>1</sup> 1

emergence of complex phenomena and information processing.

Based on the above algorithm we can prove the following theorem:

(3) We calculate the cell coefficients *a*11ð Þ *Ek , a*12ð Þ *Ek , a*21ð Þ *Ek , a*22ð Þ *Ek* ,

� � <sup>¼</sup> 0. In general, this system may have four real roots as functions of

*F uij*

the cell parameters.

and determinant by Δð Þ *Ek* .

*Memristors - Circuits and Applications of Memristor Devices*

small signal average power into the passive resistive grids.

*This region in the parameter space is called SLAR E*ð Þ*<sup>k</sup> .*

*at least one equilibrium point both locally active and stable.*

*least one equilibrium point which is locally active and stable.*

**3.2 Simulation results and some applications**

*hut* ð Þ¼ ð Þ*<sup>n</sup>*

**80**

active and unstable in green; locally passive in blue (**Figure 10**).

function *h*(*x*) can be programmed applying this algorithm is:

8 >>><

>>>:

through computer simulation:

*equilibrium point Ek for M-HCNN model (16) if*

*k* ¼ 1*,* 2*,* 3*,* 4.

**Figure 10.** *Edge of chaos domain for M-HCNN model (16).*

$$t\_n = n\Delta t, n = 1, 2, \dots$$

We shall start with the application of our M-HCNN model for edge extraction. The simulations are presented below (see **Figure 11**):

It is known that for feature extraction we firstly extract the edges of the image, which contain most of the information for the image shape. In the example provide on **Figure 11** we show the original image—(a), and then the results which we obtain simulating M-HCNN model—(b) and standard CNN model—(c). It can be seen that the results from M-HCNN (16) model and CNN model are very similar.

**Figure 11.** *Example of edge extraction: (a) original image, (b) M-HCNN, and (c) standard CNN.*

architecture, the CNN Universal Machine has been invented [24, 30]. The unique combined property of memristors [44] to store the information for a very long time after the power is switched off may allow the development and circuit implemen-

We develop algorithm for determination of EC domain of the cell parameter space for M-HCNN model (16). Two applications are presented—for edge detection and noise removal. The conclusions of the simulation results are that the image does not change when we vary the memristor weights which is possible because of the binary quantization of the output. The speed of the numerical simulations of our M-HCNN model could be enlarged due to the need of more iterations of the algorithm in order to obtain stable solutions. But the quality of the image does not change.

Acknowledgements are due to the project TE 257/25-1 between DFG and Bul-

First author acknowledges as well the provided access to the e-infrastructure of the Centre for Advanced Computing and Data Processing, with the financial support by the Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014–2020) and co-financed by the European Union through the European structural and Investment funds.

1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Institute of Fundamentals of Electrical Engineering, Technical University of

tation of memcomputing paradigms.

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

**Acknowledgements**

**Author details**

Angela Slavova<sup>1</sup>

Dresden, Dresden, Germany

provided the original work is properly cited.

Bulgaria

**83**

\* and Ronald Tetzlaff <sup>2</sup>

\*Address all correspondence to: angela.slavova@gmail.com

garian Academy of Sciences.

**Figure 12.**

*Simulation of noise removal by M-HCNN model and by standard CNN model: (a) noise, (b) M-HCNN, and (c) CNN.*

Another application which we shall present is for noise removal. The results of our simulations are given on the figure below (see **Figure 12**):

The applications of CNN show that the linear image processing can be compared to spatial convolution with infinite impulse response kernels [21, 41, 42]. When taking the image by a camera from the real world there is a possibility it to be polluted with some noise. That is why noise removal is very important for CNN applications such as AI devices, IoT. In our case, the simulations of M-HCNN model (16) shown in **Figure 12** present very good processing performance of noise removal similar to the simulations of standard CNN.

#### **4. Conclusions and discussion**

In this chapter, we stated the local activity theory for reaction-diffusion equations and hysteresis systems. However it can be generalized to other systems. In particular, the developed constructive procedure is applicable to any system whose cells and couplings are described by deterministic mathematical models. The crux of the problem is to derive testable necessary and sufficient conditions which guarantee that the system has a unique steady state solution at *t* ! ∞. A homogeneous non conservative medium cannot exhibit complexity unless the cells, or the coupling network is locally active.

In the second part of the chapter we focus our attention on HCNN model which has memristor synapses. The concept of CNN is based on some aspects of neurobiology and is adapted to integrated circuits. CNN are defined as spatial arrangements of locally coupled dynamical systems, cells. The CNN dynamics is determined by a dynamic law of an isolated cell, by the coupling laws between the cell and by boundary and initial conditions. The dynamic law and the coupling laws of a cell are often combined and described by a nonlinear ordinary differential- or difference equation (ODE), the state equation of a cell. Thus a CNN is given by a system of coupled ODEs with a very compact representation in the case of translation invariant state equations. Despite of having a compact representation CNN can show very complex dynamics like chaotic behavior, self-organization, pattern formation or nonlinear oscillation and wave propagation. Analog CNN chip hardware implementations have been developed [23]. The future of CNN implementation is in nano-structure computer architecture. CNN not only represent a new paradigm for complexity but also establish novel approaches to information processing by nonlinear complex systems. CNN have very impressive and promising applications in image processing and pattern recognition [22, 43]. After the introduction of the CNN paradigm, CNN Technology got a boost when the analogic cellular computer

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

architecture, the CNN Universal Machine has been invented [24, 30]. The unique combined property of memristors [44] to store the information for a very long time after the power is switched off may allow the development and circuit implementation of memcomputing paradigms.

We develop algorithm for determination of EC domain of the cell parameter space for M-HCNN model (16). Two applications are presented—for edge detection and noise removal. The conclusions of the simulation results are that the image does not change when we vary the memristor weights which is possible because of the binary quantization of the output. The speed of the numerical simulations of our M-HCNN model could be enlarged due to the need of more iterations of the algorithm in order to obtain stable solutions. But the quality of the image does not change.

#### **Acknowledgements**

Another application which we shall present is for noise removal. The results of

*Simulation of noise removal by M-HCNN model and by standard CNN model: (a) noise, (b) M-HCNN, and*

The applications of CNN show that the linear image processing can be compared to spatial convolution with infinite impulse response kernels [21, 41, 42]. When taking the image by a camera from the real world there is a possibility it to be polluted with some noise. That is why noise removal is very important for CNN applications such as AI devices, IoT. In our case, the simulations of M-HCNN model (16) shown in **Figure 12** present very good processing performance of noise

In this chapter, we stated the local activity theory for reaction-diffusion equations and hysteresis systems. However it can be generalized to other systems. In particular, the developed constructive procedure is applicable to any system whose cells and couplings are described by deterministic mathematical models. The crux of the problem is to derive testable necessary and sufficient conditions which guarantee that the system has a unique steady state solution at *t* ! ∞. A homogeneous non conservative medium cannot exhibit complexity unless the cells, or the

In the second part of the chapter we focus our attention on HCNN model which has memristor synapses. The concept of CNN is based on some aspects of neurobiology and is adapted to integrated circuits. CNN are defined as spatial arrangements of locally coupled dynamical systems, cells. The CNN dynamics is determined by a dynamic law of an isolated cell, by the coupling laws between the cell and by boundary and initial conditions. The dynamic law and the coupling laws of a cell are often combined and described by a nonlinear ordinary differential- or difference equation (ODE), the state equation of a cell. Thus a CNN is given by a system of coupled ODEs with a very compact representation in the case of translation invariant state equations. Despite of having a compact representation CNN can show very complex dynamics like chaotic behavior, self-organization, pattern formation or

nonlinear oscillation and wave propagation. Analog CNN chip hardware

implementations have been developed [23]. The future of CNN implementation is in nano-structure computer architecture. CNN not only represent a new paradigm for complexity but also establish novel approaches to information processing by nonlinear complex systems. CNN have very impressive and promising applications in image processing and pattern recognition [22, 43]. After the introduction of the CNN paradigm, CNN Technology got a boost when the analogic cellular computer

our simulations are given on the figure below (see **Figure 12**):

removal similar to the simulations of standard CNN.

*Memristors - Circuits and Applications of Memristor Devices*

**4. Conclusions and discussion**

**Figure 12.**

*(c) CNN.*

**82**

coupling network is locally active.

Acknowledgements are due to the project TE 257/25-1 between DFG and Bulgarian Academy of Sciences.

First author acknowledges as well the provided access to the e-infrastructure of the Centre for Advanced Computing and Data Processing, with the financial support by the Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014–2020) and co-financed by the European Union through the European structural and Investment funds.

#### **Author details**

Angela Slavova<sup>1</sup> \* and Ronald Tetzlaff <sup>2</sup>

1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

2 Institute of Fundamentals of Electrical Engineering, Technical University of Dresden, Dresden, Germany

\*Address all correspondence to: angela.slavova@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

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[2] Mikolajick T. From basic switching mechanism to device applications. Workshop on Memristor Science & Technology; Dresden; 2014

[3] Corinto F, Ascoli A. A boundary condition-based approach to the modeling of memristor nanostructures. IEEE Transactions on Circuits and Systems I. 2012;**59**(11):2713-2726

[4] Corinto F, Ascoli A, Gilli M. Nonlinear dynamics of memristor oscillators. IEEE Transactions on Circuits and Systems. 2010;**58**(6): 1323-1336

[5] Talukdar A, Radwan AG, Salama KN. Nonlinear dynamics of memristor based 3rd order oscillatory system. Microelectronics Journal. 2012;**43**(3): 169-175

[6] Ascoli A, Corinto F, Tetzlaff R. Generalized boundary condition memristor model. International Journal of Circuit Theory and Applications. 2016;**44**(1):60-84

[7] Strukov DB, Stewart DR, Borghetti JL, Li X, Pickett M, Ribeiro GM, et al. Hybrid CMOS/memristor circuits. In: Proceedings of IEEE International Symposium on Circuits and Systems. 2010:1967-1970. Inspec Accesion Number:114622876

[8] Tetzlaff R. Memristor and Memristive Systems. Springer. 2014

[9] Chua LO. The fourth element. Proceedings of the IEEE. 2012;**100**(6): 1920-1927

[10] Kvatinski S, Friedman EG, Kolodny A, Weiser UC. TEAM: Threshold

adaptive memristor model. IEEE Transactions on Circuits and Systems I. 2013;**60**(1):211-221

[19] Torrezan AC, Strachan JP,

[21] Chua LO, Yang L. CNN: Applications. IEEE Transactions on Circuits and Systems. 1988;**35**:1273-1299

Medeiros-Ribeiro G, Williams RS. Subnanosecond switching of a tantalum oxide memristor. Nanotechnology. 2011;**22**(48):485203(1)-485203(7)

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

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[31] Smale S. A mathematical model of two cells via Turing's equation. Lectures in Applied Mathematics. 1974;**6**:15-26

Becoming: Time and Complexity in the Physical Sciences. San Francisco: W.H.

[33] Chua LO. Local activity is the origin of complexity. International Journal of Bifurcation and Chaos. 2005;**15**(11):

[34] Mainzer K, Chua LO. Local Activity Principle: The Cause of Complexity and Symmetry Breaking. London: Imperial

[30] Dogaru R. Universality and Emergent Computation in Cellular Neural Networks. Singapore: World

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Scientific; 2003

Freeman; 1980

3435-3456

College Press; 2013

Publishers; 2003

[35] Slavova A. Cellular Neural Networks: Dynamics and Modelling. Mathematical Modelling: Theory and Applications 16. Kluwer Academic

[36] Slavova A, Rashkova V. Edge of chaos in reaction-diffusion CNN models. In: Mathematical Analysis, Differential Equations and Their Applications. Academic Publishing House "M.Drinov"; 2011. pp. 207-216

[37] Slavova A. Chaotic behavior of hysteresis cellular nonlinear networks and its control. In: Proc. NDES; Dresden. Germany; 2010. pp. 74-77

[38] Slavova A. Applications of some mathematical methods in the analysis of cellular neural networks. Journal of

Computational and Applied Mathematics. 2000;**114**:387-404

[20] Chua LO, Yang L. Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems. 1988;**35**:1257-1272

[22] Chua LO, Roska T. Cellular Neural Networks and Visual Computing: Foundations and Applications. Cambridge University Press; 2001

[23] Roska T. In: Vandewalle J, editor. Cellular Neural Networks. Wiley, John & Sons, Incorporated Pub; 1994

[24] Roska T, Rodríguez-Vázquez A. Toward the Visual Microprocessor-VLSI Design and the Use of Cellular Neural Network (CNN) Universal Machine Computers. London: J. Wiley; 2001

[25] Chua LO. CNN: A Paradigm for Complexity. World Scientific Pub Co;

[26] Manganaro G, Arena P, Fortuna L. Cellular Neural Networks: Chaos, Complexity and VLSI Processing.

[27] Hanggi M, Moschytz G. Cellular Neural Networks: Analysis, Design and Optimization. Kluwer Academic

[28] Ascoli A, Slesazeck S, Mähne H, Tetzlaff R, Mikolajick T. Nonlinear dynamics of a locally-active memristor. IEEE Transactions on Circuits and Systems. 2015;**62**(4):1165-1174. DOI:

Springer Verlag; 1999

Publishers Pub; 2000

10.1109/TCSI.2015.2413152

[29] Ascoli A, Lanza V, Corinto F, Tetzlaff R. Synchronization conditions

1998

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[11] Pickett MD, Strukov DB, Borghetti JL, Yang JJ, Snider GS, Stewart RD, et al. Switching dynamics in titanium dioxide memristive devices. Journal of Applied Physics. 2009;**106**(7):074508

[12] Pickett MD, Williams RS. Sub-100 fJ and sub-nanosecond thermally driven threshold switching in niobium oxide crosspoint nanodevices. Nanotechnology. 2012;**23**:215202

[13] Strachan JP, Torrezan AC, Miao F, Pickett MD, Yang JJ, Yi W, et al. State dynamics and modeling of tantalum oxide memristors. IEEE Transactions on Electron Devices. 2013;**60**(7):2194-2202

[14] Ascoli A, Tetzlaff R, Biolek Z, Kolka Z, Biolkovà V, Biolek D. The art of finding accurate memristor model solutions. IEEE Journal on Emerging and Selected Topics in Circuits and Systems. 2015. DOI: 10.1109/ JETCAS.2015.2426493

[15] Strukov DB, Snider GS, Stewart DR, Williams RS. The missing memristor found. Nature Letters. 2008;**453**. DOI: 10.1038/nature06932

[16] Joglekar YN, Wolf ST. The elusive memristive element: Properties of basic electrical circuits. European Journal of Physics. 2009;**30**:661-675

[17] Biolek Z, Biolek D, Biolkovà V. Spice model of memristor with nonlinear dopant drift. Radioengineering. 2009; **18**(2):210-214

[18] Ascoli A, Corinto F, Senger V, Tetzlaff R. Memristor model comparison. IEEE Circuits and Systems Magazine. 2013:89-105. DOI: 10.1109/ MCAS.2013.2256272

*Mathematical Analysis of Memristor CNN DOI: http://dx.doi.org/10.5772/intechopen.86446*

[19] Torrezan AC, Strachan JP, Medeiros-Ribeiro G, Williams RS. Subnanosecond switching of a tantalum oxide memristor. Nanotechnology. 2011;**22**(48):485203(1)-485203(7)

**References**

[1] Chua LO. Memristor: The missing circuit element. IEEE Transactions on Circuits Theory. 1971;**18**(5):507-519

*Memristors - Circuits and Applications of Memristor Devices*

adaptive memristor model. IEEE Transactions on Circuits and Systems I.

Physics. 2009;**106**(7):074508

crosspoint nanodevices.

Nanotechnology. 2012;**23**:215202

[13] Strachan JP, Torrezan AC, Miao F, Pickett MD, Yang JJ, Yi W, et al. State dynamics and modeling of tantalum oxide memristors. IEEE Transactions on Electron Devices. 2013;**60**(7):2194-2202

[14] Ascoli A, Tetzlaff R, Biolek Z, Kolka Z, Biolkovà V, Biolek D. The art of finding accurate memristor model solutions. IEEE Journal on Emerging and Selected Topics in Circuits and Systems. 2015. DOI: 10.1109/ JETCAS.2015.2426493

[15] Strukov DB, Snider GS, Stewart DR, Williams RS. The missing memristor found. Nature Letters. 2008;**453**. DOI:

[16] Joglekar YN, Wolf ST. The elusive memristive element: Properties of basic electrical circuits. European Journal of

[17] Biolek Z, Biolek D, Biolkovà V. Spice model of memristor with nonlinear dopant drift. Radioengineering. 2009;

[18] Ascoli A, Corinto F, Senger V, Tetzlaff R. Memristor model

comparison. IEEE Circuits and Systems Magazine. 2013:89-105. DOI: 10.1109/

10.1038/nature06932

Physics. 2009;**30**:661-675

**18**(2):210-214

MCAS.2013.2256272

[11] Pickett MD, Strukov DB, Borghetti JL, Yang JJ, Snider GS, Stewart RD, et al. Switching dynamics in titanium dioxide memristive devices. Journal of Applied

[12] Pickett MD, Williams RS. Sub-100 fJ and sub-nanosecond thermally driven threshold switching in niobium oxide

2013;**60**(1):211-221

[2] Mikolajick T. From basic switching mechanism to device applications. Workshop on Memristor Science &

[3] Corinto F, Ascoli A. A boundary condition-based approach to the modeling of memristor nanostructures. IEEE Transactions on Circuits and Systems I. 2012;**59**(11):2713-2726

[4] Corinto F, Ascoli A, Gilli M. Nonlinear dynamics of memristor oscillators. IEEE Transactions on Circuits and Systems. 2010;**58**(6):

3rd order oscillatory system.

[5] Talukdar A, Radwan AG, Salama KN. Nonlinear dynamics of memristor based

Microelectronics Journal. 2012;**43**(3):

[6] Ascoli A, Corinto F, Tetzlaff R. Generalized boundary condition memristor model. International Journal of Circuit Theory and Applications.

[7] Strukov DB, Stewart DR, Borghetti JL, Li X, Pickett M, Ribeiro GM, et al. Hybrid CMOS/memristor circuits. In: Proceedings of IEEE International Symposium on Circuits and Systems. 2010:1967-1970. Inspec Accesion

1323-1336

169-175

2016;**44**(1):60-84

Number:114622876

1920-1927

**84**

[8] Tetzlaff R. Memristor and Memristive Systems. Springer. 2014

[9] Chua LO. The fourth element. Proceedings of the IEEE. 2012;**100**(6):

[10] Kvatinski S, Friedman EG, Kolodny A, Weiser UC. TEAM: Threshold

Technology; Dresden; 2014

[20] Chua LO, Yang L. Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems. 1988;**35**:1257-1272

[21] Chua LO, Yang L. CNN: Applications. IEEE Transactions on Circuits and Systems. 1988;**35**:1273-1299

[22] Chua LO, Roska T. Cellular Neural Networks and Visual Computing: Foundations and Applications. Cambridge University Press; 2001

[23] Roska T. In: Vandewalle J, editor. Cellular Neural Networks. Wiley, John & Sons, Incorporated Pub; 1994

[24] Roska T, Rodríguez-Vázquez A. Toward the Visual Microprocessor-VLSI Design and the Use of Cellular Neural Network (CNN) Universal Machine Computers. London: J. Wiley; 2001

[25] Chua LO. CNN: A Paradigm for Complexity. World Scientific Pub Co; 1998

[26] Manganaro G, Arena P, Fortuna L. Cellular Neural Networks: Chaos, Complexity and VLSI Processing. Springer Verlag; 1999

[27] Hanggi M, Moschytz G. Cellular Neural Networks: Analysis, Design and Optimization. Kluwer Academic Publishers Pub; 2000

[28] Ascoli A, Slesazeck S, Mähne H, Tetzlaff R, Mikolajick T. Nonlinear dynamics of a locally-active memristor. IEEE Transactions on Circuits and Systems. 2015;**62**(4):1165-1174. DOI: 10.1109/TCSI.2015.2413152

[29] Ascoli A, Lanza V, Corinto F, Tetzlaff R. Synchronization conditions in simple memristor neural networks. Journal of the Franklin Institute. 2015. DOI: 10.1016/j.jfranklin.2015.06.003

[30] Dogaru R. Universality and Emergent Computation in Cellular Neural Networks. Singapore: World Scientific; 2003

[31] Smale S. A mathematical model of two cells via Turing's equation. Lectures in Applied Mathematics. 1974;**6**:15-26

[32] Prigogine I. From Being to Becoming: Time and Complexity in the Physical Sciences. San Francisco: W.H. Freeman; 1980

[33] Chua LO. Local activity is the origin of complexity. International Journal of Bifurcation and Chaos. 2005;**15**(11): 3435-3456

[34] Mainzer K, Chua LO. Local Activity Principle: The Cause of Complexity and Symmetry Breaking. London: Imperial College Press; 2013

[35] Slavova A. Cellular Neural Networks: Dynamics and Modelling. Mathematical Modelling: Theory and Applications 16. Kluwer Academic Publishers; 2003

[36] Slavova A, Rashkova V. Edge of chaos in reaction-diffusion CNN models. In: Mathematical Analysis, Differential Equations and Their Applications. Academic Publishing House "M.Drinov"; 2011. pp. 207-216

[37] Slavova A. Chaotic behavior of hysteresis cellular nonlinear networks and its control. In: Proc. NDES; Dresden. Germany; 2010. pp. 74-77

[38] Slavova A. Applications of some mathematical methods in the analysis of cellular neural networks. Journal of Computational and Applied Mathematics. 2000;**114**:387-404

[39] Slavova A, Tetzlaff R, Markova M. CNN computing of the interaction of fluxons. In: IEEE Proceedings of the General Assembly of URSI; Istanbul. 2011. p. 1-4

[40] Slavova A, Tetzlaff R. CNN computing of double sine-Gordon equation with physical applications. Comptes rendus de l'Academie bulgare des Sciences. 2014;**67**(1):21-28

[41] Itoh M, Chua LO. Star cellular neural networks for associative and dynamical memories. International Journal of Bifurcation and Chaos. 2004; **14**:1725-1772

[42] Mazumder P, Li S-R, Ebong IE. Tunneling-based cellular nonlinear network architectures for image processing. IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 2009;**17**(4):487-495

[43] Madan R. Chua's Circuit: A Paradigm for Chaos. Singapore: World Scientific; 1993

[44] Chua L. Everything you wish to know about memristors but afraid to ask. Radioengineering. 2015;**24**(2): 319-368

**87**

**Chapter 6**

**Abstract**

memristor development.

memristive behavior

**1. Introduction**

Memristor Behavior under Dark

and Violet Illumination in Thin

*Adolfo Henrique Nunes Melo, Raiane Sodre de Araujo,* 

*Eduardo Valença and Marcelo Andrade Macêdo*

Films of ZnO/ZnO-Al Multilayers

ZnO/ZnO-Al thin films were grown aiming the development of a memristor. Electrical voltage sweeps were imposed to induce dopant migration and to achieve several resistance states. A memristor behavior was observed, presenting adaptation to external electrical stimulus. Voltage sweeps occurred under the influence of violet light and in the dark, alternately, and the influence of the photon incidence on the current intensity was noticed. Throughout the alternating cycles between light and dark, less resistance was observed under illumination, but the migration of Al and O ions caused the formation of Al2O3 and ZnO oxides, resulting in a gradual increase in resistance. With constant voltage, the device presented continuous modification of resistance and sensitivity to the violet light with generation of free carriers. These results bring new opportunities for using memristors as violet light sensors as well as new insights for light-controlled

**Keywords:** memristor, ZnO, ZnO-Al, thin films, violet sensor, dark, illumination,

In 1971, Leon Chua predicted the existence of a fourth electronic passive element of two terminals, called a memristor (a union of the terms memory and resistance) [1]. A memristor is basically a resistor that has its resistance altered with external stimulation in a nonvolatile way. In other words, it maintains the state of resistance even if the stimulus is removed. In 1976, Chua and Kang determined that a wide class of devices and systems can be considered as memristives when they present time-dependent electrical resistance and also depend on application of electric voltage [2]. Memristor devices can be configured in nonvolatile memories, logic gates, and programmable connections having high-density integration or presenting complementary metal-oxide-semiconductor (CMOS) compatibility [3]. This CMOS compatibility makes memristors excellent candidates to go beyond Moore's law. In 2008, in Hewlett-Packard (HP) laboratories, thin films of titanium dioxide prepared with two terminals presented memristor characteristics [4]. Their basic structure was based on epitaxial growth of metal-insulator-metal (MIM) thin

#### **Chapter 6**

[39] Slavova A, Tetzlaff R, Markova M. CNN computing of the interaction of fluxons. In: IEEE Proceedings of the General Assembly of URSI; Istanbul.

*Memristors - Circuits and Applications of Memristor Devices*

[40] Slavova A, Tetzlaff R. CNN computing of double sine-Gordon equation with physical applications. Comptes rendus de l'Academie bulgare

des Sciences. 2014;**67**(1):21-28

[41] Itoh M, Chua LO. Star cellular neural networks for associative and dynamical memories. International Journal of Bifurcation and Chaos. 2004;

[42] Mazumder P, Li S-R, Ebong IE. Tunneling-based cellular nonlinear network architectures for image processing. IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[43] Madan R. Chua's Circuit: A Paradigm for Chaos. Singapore: World

[44] Chua L. Everything you wish to know about memristors but afraid to ask. Radioengineering. 2015;**24**(2):

2011. p. 1-4

**14**:1725-1772

2009;**17**(4):487-495

Scientific; 1993

319-368

**86**

## Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al Multilayers

*Adolfo Henrique Nunes Melo, Raiane Sodre de Araujo, Eduardo Valença and Marcelo Andrade Macêdo*

#### **Abstract**

ZnO/ZnO-Al thin films were grown aiming the development of a memristor. Electrical voltage sweeps were imposed to induce dopant migration and to achieve several resistance states. A memristor behavior was observed, presenting adaptation to external electrical stimulus. Voltage sweeps occurred under the influence of violet light and in the dark, alternately, and the influence of the photon incidence on the current intensity was noticed. Throughout the alternating cycles between light and dark, less resistance was observed under illumination, but the migration of Al and O ions caused the formation of Al2O3 and ZnO oxides, resulting in a gradual increase in resistance. With constant voltage, the device presented continuous modification of resistance and sensitivity to the violet light with generation of free carriers. These results bring new opportunities for using memristors as violet light sensors as well as new insights for light-controlled memristor development.

**Keywords:** memristor, ZnO, ZnO-Al, thin films, violet sensor, dark, illumination, memristive behavior

#### **1. Introduction**

In 1971, Leon Chua predicted the existence of a fourth electronic passive element of two terminals, called a memristor (a union of the terms memory and resistance) [1]. A memristor is basically a resistor that has its resistance altered with external stimulation in a nonvolatile way. In other words, it maintains the state of resistance even if the stimulus is removed. In 1976, Chua and Kang determined that a wide class of devices and systems can be considered as memristives when they present time-dependent electrical resistance and also depend on application of electric voltage [2]. Memristor devices can be configured in nonvolatile memories, logic gates, and programmable connections having high-density integration or presenting complementary metal-oxide-semiconductor (CMOS) compatibility [3]. This CMOS compatibility makes memristors excellent candidates to go beyond Moore's law.

In 2008, in Hewlett-Packard (HP) laboratories, thin films of titanium dioxide prepared with two terminals presented memristor characteristics [4]. Their basic structure was based on epitaxial growth of metal-insulator-metal (MIM) thin

films [5–7]. With the experimental development of memristors in HP laboratories, there has been increasing research in this area, and several materials have been constructed in the MIM structure for analysis of memristive behavior. There is an appreciable number of materials that have been applied in the design of the MIM structure, such as semiconductor insulation (III-V), including MgO, TiOx, ZrOx, HfOx, NbOx, AlOx, ZnOx, or rare-earth oxides containing Y, Ce, Sm, Gd, Eu, Nd, and perovskites (SrTiO3, Ba0.7Sr0.3TiO3) [8, 9]. In addition to such applications as nonvolatile memory, devices based on transparent memristors can be applied even in near-eye display technology that requires the construction of transparent memories, transparent switches, and optical sensors [10]. These devices can also act as synapse elements, creating neural computing machines resembling the behavior of the biological brain [7, 11, 12].

However, transparent conductive oxides (TCOs) have been widely studied because they are essential components in flat-panel monitors, solar cells, touch screens, light-emitting diodes (LEDs), ultraviolet (UV) detectors, and other optoelectronic devices [10, 13, 14]. In addition, new transparent devices are being developed in applications of neuromorphic circuits [4, 7, 15] and adaptive systems [16, 17] in which they are based on resistive commutation depending on the history of the electric voltage application. Among the various materials that have memristor characteristics, ZnO stands out for its low toxicity, low cost, wide resistive switching ratio (*Roff*/*Ron* ~1011), low power consumption, fast recording, and highdensity storage [5, 10, 18, 19]. For the construction of a transparent MIM system, an indium tin oxide (ITO) substrate can be used because of its optical properties (transmittance ~90%) and electrical properties (sheet resistance ~20 Ω/*sq*) [20]. Transparent thin films of ZnO have been extensively studied which acquires excellent optical and electrical properties when doped with such metals as Al [21], Ti [22], and Nb [13], but a multilayer insulator/insulator+metal system may be a good candidate for memristor characteristics, where the diffusion of metallic ions may favor the mechanism of adaptation to the external electrical stimulus. Some researchers have reported thin films of ZnO/ZnO:Al presenting optical transmission of > 80% and bandgap *Eg* = 3.32 *eV* [23] or ranging from 3.65 to 3.72 eV when grown under heat treatment (300–500°C) [24].

Bandgap studies have an important role for light detection (near the ultraviolet UV zone in the case of ZnO), biological and environmental research, and detection sensors, being a protagonist in several chemical processes, which makes its determination extremely important. Bandgap determination can favor the development of wavelength-sensitive circuits enabling the generation of electrical signals that can be measured. However, memristive systems require electric charge flux through the device, which causes variation in the internal electrical resistance. This can be controlled using light incidence in addition to the usual electrical voltage. Inspired by the biological processes, Chen et al. demonstrated a visual memory unit in which it was based on In2O3 resistive switching, where logic states (0) and (1) associated with the high-resistance state and low-resistance state, respectively, were achieved under dark and UV illumination conditions, where the existence of UV stimulation provides the possibility of light information being memorized and erased under voltage sweep and then records light patterns, such as butterfly or heart shaped, in arrangements of 10 × 10 pixels [25].

In this work, ITO/ZnO/ZnO-Al memristor devices were grown using magnetron sputtering. They were subjected to electrical voltage sweep to study homogeneous resistive switching behavior under illumination and dark ambient conditions. The analysis of optical transmission and absorption properties and bandgap determination are also presented.

**89**

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al…*

indium tin oxide (ITO) through physical deposition system RF/DC magnetron sputtering (AJA International). Three samples of ZnO thin films were grown on ITO using 100-W RF applied to a ceramic target of ZnO (99.9% purity, Macashew Technologies) as a function of the deposition time (40, 60, and 100 min). On these samples, a ZnO-Al film was grown by codeposition for 15 min, where an Al target (98.8%) was exposed to a 50 W DC source and simultaneously the ZnO target

ing pressure with Ar gas was 20 mTorr with continuous flux of 20 sccm. There was no supply of oxygen flow, and all depositions were without heating source. The samples in this case had an epitaxial architecture. For simplicity, coding was performed, where ZA1 refers to the sample ITO/ZnO(100 min)/ZnO-Al(15 min), ZA2 refers to ITO/ZnO(60 min)/ZnO-Al(15 min), and ZA3 to ITO/ZnO(40 min)/ ZnO-Al(15 min). The crystallinity of the samples was analyzed by X-ray diffraction (Bruker D8 Advance—CuKα radiation with *λ* = 0.154 nm). A concentration profile of chemicals per depth was obtained through the Rutherford backscattering spec-

data were obtained simultaneously at 120° and 170° scattering angles. This methodology was applied previously for the fabrication of these samples, and some results were previously published [26]. Transmission and optical absorption measurements were performed by UV-Vis spectrophotometry between 200 and 800 nm (Varian Cary 100 Scan UV-Vis spectrophotometer). All electrical measurements were performed using a voltage-current source (Keysight Agilent B2901) where, for the upper electrode, a Pt tip with ~200 μm in diameter was attached to a rod with micrometric displacement for a better approximation of the sample surface. The lower electrode (ITO) was grounded in all measurements. For realization of the measurements under illumination and dark conditions, a dark chamber was home built. A violet LED was coupled 1 cm away from the sample surface; this distance was suitably selected, aiming to provide a better homogeneity in the sample illumination where the Pt probe electrode would be acting. In addition, IR heating is minimized.

Previously published [26] X-ray diffraction analyses showed hexagonal crystalline phase formation with a wurtzite ZnO structure where the films grew preferably along the axis *c* perpendicular to the substrate in direction (002). No phases corresponding to Al were identified, indicating possible incorporation of Al3+ ions in place of Zn2+ without altering the structure. This agreed with the results of RBS, confirming the structure ITO/ZnO/ZnO-Al [26, 27]. The nonidentification of crystalline phases of Al may be relevant for the construction of a device in which the insertion of the metal ion may favor the electric conduction without reducing

The transmittance and absorbance spectra are shown in **Figure 1a** and **b**. The transmittance of the glass is given for reference only. The glass/ITO substrate presents an average transmittance, in the visible region, of 90%, while the ZA1, ZA2, and ZA3 films show transmittance of ~88, 80, and 79.8%, respectively, and have absorption bands at wavelengths of 350–650 nm. This indicates that the investigated thin films exhibit excellent optical properties in the visible and near-infrared region

and are semiconductors suitable for applications in electronic devices [28].

to 100 W RF. In all depositions, the base pressure was ~10−6

troscopy (RBS) technique carried out by bombardment of He+

**3. Structural and optical properties**

the transparency and enabling the memristive behavior.

Thin films of ZnO/ZnO-Al were deposited on substrates of 100-nm Asahi Glass

Torr, and the work-

(2.2 MeV). The RBS

*DOI: http://dx.doi.org/10.5772/intechopen.86557*

**2. Experimental details**

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al… DOI: http://dx.doi.org/10.5772/intechopen.86557*

#### **2. Experimental details**

*Memristors - Circuits and Applications of Memristor Devices*

of the biological brain [7, 11, 12].

grown under heat treatment (300–500°C) [24].

arrangements of 10 × 10 pixels [25].

tion are also presented.

films [5–7]. With the experimental development of memristors in HP laboratories, there has been increasing research in this area, and several materials have been constructed in the MIM structure for analysis of memristive behavior. There is an appreciable number of materials that have been applied in the design of the MIM structure, such as semiconductor insulation (III-V), including MgO, TiOx, ZrOx, HfOx, NbOx, AlOx, ZnOx, or rare-earth oxides containing Y, Ce, Sm, Gd, Eu, Nd, and perovskites (SrTiO3, Ba0.7Sr0.3TiO3) [8, 9]. In addition to such applications as nonvolatile memory, devices based on transparent memristors can be applied even in near-eye display technology that requires the construction of transparent memories, transparent switches, and optical sensors [10]. These devices can also act as synapse elements, creating neural computing machines resembling the behavior

However, transparent conductive oxides (TCOs) have been widely studied because they are essential components in flat-panel monitors, solar cells, touch screens, light-emitting diodes (LEDs), ultraviolet (UV) detectors, and other optoelectronic devices [10, 13, 14]. In addition, new transparent devices are being developed in applications of neuromorphic circuits [4, 7, 15] and adaptive systems [16, 17] in which they are based on resistive commutation depending on the history of the electric voltage application. Among the various materials that have memristor characteristics, ZnO stands out for its low toxicity, low cost, wide resistive switching ratio (*Roff*/*Ron* ~1011), low power consumption, fast recording, and highdensity storage [5, 10, 18, 19]. For the construction of a transparent MIM system, an indium tin oxide (ITO) substrate can be used because of its optical properties (transmittance ~90%) and electrical properties (sheet resistance ~20 Ω/*sq*) [20]. Transparent thin films of ZnO have been extensively studied which acquires excellent optical and electrical properties when doped with such metals as Al [21], Ti [22], and Nb [13], but a multilayer insulator/insulator+metal system may be a good candidate for memristor characteristics, where the diffusion of metallic ions may favor the mechanism of adaptation to the external electrical stimulus. Some researchers have reported thin films of ZnO/ZnO:Al presenting optical transmission of > 80% and bandgap *Eg* = 3.32 *eV* [23] or ranging from 3.65 to 3.72 eV when

Bandgap studies have an important role for light detection (near the ultraviolet UV zone in the case of ZnO), biological and environmental research, and detection sensors, being a protagonist in several chemical processes, which makes its determination extremely important. Bandgap determination can favor the development of wavelength-sensitive circuits enabling the generation of electrical signals that can be measured. However, memristive systems require electric charge flux through the device, which causes variation in the internal electrical resistance. This can be controlled using light incidence in addition to the usual electrical voltage. Inspired by the biological processes, Chen et al. demonstrated a visual memory unit in which it was based on In2O3 resistive switching, where logic states (0) and (1) associated with the high-resistance state and low-resistance state, respectively, were achieved under dark and UV illumination conditions, where the existence of UV stimulation provides the possibility of light information being memorized and erased under voltage sweep and then records light patterns, such as butterfly or heart shaped, in

In this work, ITO/ZnO/ZnO-Al memristor devices were grown using magnetron sputtering. They were subjected to electrical voltage sweep to study homogeneous resistive switching behavior under illumination and dark ambient conditions. The analysis of optical transmission and absorption properties and bandgap determina-

**88**

Thin films of ZnO/ZnO-Al were deposited on substrates of 100-nm Asahi Glass indium tin oxide (ITO) through physical deposition system RF/DC magnetron sputtering (AJA International). Three samples of ZnO thin films were grown on ITO using 100-W RF applied to a ceramic target of ZnO (99.9% purity, Macashew Technologies) as a function of the deposition time (40, 60, and 100 min). On these samples, a ZnO-Al film was grown by codeposition for 15 min, where an Al target (98.8%) was exposed to a 50 W DC source and simultaneously the ZnO target to 100 W RF. In all depositions, the base pressure was ~10−6 Torr, and the working pressure with Ar gas was 20 mTorr with continuous flux of 20 sccm. There was no supply of oxygen flow, and all depositions were without heating source. The samples in this case had an epitaxial architecture. For simplicity, coding was performed, where ZA1 refers to the sample ITO/ZnO(100 min)/ZnO-Al(15 min), ZA2 refers to ITO/ZnO(60 min)/ZnO-Al(15 min), and ZA3 to ITO/ZnO(40 min)/ ZnO-Al(15 min). The crystallinity of the samples was analyzed by X-ray diffraction (Bruker D8 Advance—CuKα radiation with *λ* = 0.154 nm). A concentration profile of chemicals per depth was obtained through the Rutherford backscattering spectroscopy (RBS) technique carried out by bombardment of He+ (2.2 MeV). The RBS data were obtained simultaneously at 120° and 170° scattering angles. This methodology was applied previously for the fabrication of these samples, and some results were previously published [26]. Transmission and optical absorption measurements were performed by UV-Vis spectrophotometry between 200 and 800 nm (Varian Cary 100 Scan UV-Vis spectrophotometer). All electrical measurements were performed using a voltage-current source (Keysight Agilent B2901) where, for the upper electrode, a Pt tip with ~200 μm in diameter was attached to a rod with micrometric displacement for a better approximation of the sample surface. The lower electrode (ITO) was grounded in all measurements. For realization of the measurements under illumination and dark conditions, a dark chamber was home built. A violet LED was coupled 1 cm away from the sample surface; this distance was suitably selected, aiming to provide a better homogeneity in the sample illumination where the Pt probe electrode would be acting. In addition, IR heating is minimized.

#### **3. Structural and optical properties**

Previously published [26] X-ray diffraction analyses showed hexagonal crystalline phase formation with a wurtzite ZnO structure where the films grew preferably along the axis *c* perpendicular to the substrate in direction (002). No phases corresponding to Al were identified, indicating possible incorporation of Al3+ ions in place of Zn2+ without altering the structure. This agreed with the results of RBS, confirming the structure ITO/ZnO/ZnO-Al [26, 27]. The nonidentification of crystalline phases of Al may be relevant for the construction of a device in which the insertion of the metal ion may favor the electric conduction without reducing the transparency and enabling the memristive behavior.

The transmittance and absorbance spectra are shown in **Figure 1a** and **b**. The transmittance of the glass is given for reference only. The glass/ITO substrate presents an average transmittance, in the visible region, of 90%, while the ZA1, ZA2, and ZA3 films show transmittance of ~88, 80, and 79.8%, respectively, and have absorption bands at wavelengths of 350–650 nm. This indicates that the investigated thin films exhibit excellent optical properties in the visible and near-infrared region and are semiconductors suitable for applications in electronic devices [28].

The values of the optical bandgap (*Eg*) were estimated using the Tauc relation in Eq. (1) [29]:

$$(ah\nu)^2 = A\left(h\nu - E\_{\mathfrak{g}}\right)^n\tag{1}$$

where *A* is a constant, *α* is the absorption coefficient, ν is the frequency of incident photons, *h* is the Planck constant, and *Eg* is the optical bandgap, which is associated with direct (*n* = 2) and direct (*n* = 1/2) transitions [28]. Adjustment was performed by linear extrapolation (αhν)<sup>2</sup> = 0, and the graph was plotted with relation to (*h*)<sup>2</sup> vs. *E*. The bandgap energy values of the samples are shown in **Figure 2** and **Table 1**. The bandgap energy obtained for the pure ITO was 3.75 eV, and the deposited films ZA1, ZA2, and ZA3 are in the range of 3.26, 3.23, and 3.19 eV, respectively. The bandgap reported in the literature for ZnO, Al-doped ZnO, and ITO is ~3.37, 3.28, and 4.2 eV, respectively [30–33]. It is important to note that in the ITO/ZnO/ZnO-Al thin films, an increase of the bandgap energy is observed with the increased thickness of the ZnO layer. This behavior depends on some process parameters such as crystallinity, grain size, and charge carrier density, which significantly affect bandgap energy [32, 34]. However, it is believed that for a greater thickness of the ZnO layer (Al poor region), the proportional number of defects that create traps between the valence band (VB) and the conduction band (CB) is smaller, resulting

#### **Figure 1.**

*(a) Transmittance and (b) absorbance spectra for ZA1, ZA2, ZA3 (ITO/glass substrate contributions were removed), ITO, and glass.*

**91**

**Figure 3.**

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al…*

**Thin film Time deposition ZnO layer (min) Thickness (nm) Bandgap (eV)** ITO — 100 3.77 ZA1 100 150 3.26 ZA2 60 110 3.23 ZA3 40 90 3.19

in a higher bandgap energy. However, for low ZnO thicknesses, the influence of the Al-rich region becomes relevant. Therefore, the number of energy levels between the VB and CB is higher, resulting in a lower average bandgap energy value. It is important to note that defects caused by the presence of Al metal forming an Al-rich ZnO (ZnO-Al) region, which may create energy levels within the bandgap, increase the availability of charge carriers over the sample when the photons impinge, thus

The memristor behavior as a function of the incidence violet light is presented in **Figure 3a**, where voltage sweeps between 0 and 2 V occurred five times and then five other sweeps between 0 and −5 V. The first five sweeps for the positive voltage polarity with the five negative polarity sweeps were performed under illumination of the violet LED (Illumination 1). Then, the same sweeping scheme was imposed on the sample; however, without ambient light (dark 1), the dark chamber was used to provide this situation. In total, the same sample was subjected to 10 alternating sweeps between illumination and dark. Observing specifically the first sweep under illumination, an adaptive-like response was found, in which the maximum value of the measured current intensity gradually increased, indicating that the electrical resistance of the sample decreases as the sweep occurs. This type of behavior is widely known in the scientific literature as the fingerprint of a memristor, where a

*DOI: http://dx.doi.org/10.5772/intechopen.86557*

*Optical properties of thin films of ITO/ZnO/AZO.*

reducing the electrical resistance.

**Table 1.**

**4. Memristor behavior under dark and illumination**

*(a) I–V characteristic curves of the ZA1 sample under continuous voltage scans* 

*of the first sweep cycles (0 – 2 – 0 V) × 5 under illumination and darkness.*

*(0 – 2 – 0 V) × 5 – (0 – (−2) – 0 V) × 5 under violet LED illumination and under darkness and (b) highlight* 

**Figure 2.** *Plots of (h) 2 vs. energy of all ZnO/ZnO-Al and ITO substrates.*

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al… DOI: http://dx.doi.org/10.5772/intechopen.86557*


**Table 1.**

*Memristors - Circuits and Applications of Memristor Devices*

(*h*)<sup>2</sup> = *A*(*h* − *Eg*)

Eq. (1) [29]:

relation to (*h*)<sup>2</sup>

The values of the optical bandgap (*Eg*) were estimated using the Tauc relation in

vs. *E*. The bandgap energy values of the samples are shown in

where *A* is a constant, *α* is the absorption coefficient, ν is the frequency of incident photons, *h* is the Planck constant, and *Eg* is the optical bandgap, which is associated with direct (*n* = 2) and direct (*n* = 1/2) transitions [28]. Adjustment was performed by linear extrapolation (αhν)<sup>2</sup> = 0, and the graph was plotted with

**Figure 2** and **Table 1**. The bandgap energy obtained for the pure ITO was 3.75 eV, and the deposited films ZA1, ZA2, and ZA3 are in the range of 3.26, 3.23, and 3.19 eV, respectively. The bandgap reported in the literature for ZnO, Al-doped ZnO, and ITO is ~3.37, 3.28, and 4.2 eV, respectively [30–33]. It is important to note that in the ITO/ZnO/ZnO-Al thin films, an increase of the bandgap energy is observed with the increased thickness of the ZnO layer. This behavior depends on some process parameters such as crystallinity, grain size, and charge carrier density, which significantly affect bandgap energy [32, 34]. However, it is believed that for a greater thickness of the ZnO layer (Al poor region), the proportional number of defects that create traps between the valence band (VB) and the conduction band (CB) is smaller, resulting

*(a) Transmittance and (b) absorbance spectra for ZA1, ZA2, ZA3 (ITO/glass substrate contributions were* 

*<sup>n</sup>* (1)

**90**

**Figure 2.** *Plots of (h)*

*2*

 *vs. energy of all ZnO/ZnO-Al and ITO substrates.*

**Figure 1.**

*removed), ITO, and glass.*

*Optical properties of thin films of ITO/ZnO/AZO.*

in a higher bandgap energy. However, for low ZnO thicknesses, the influence of the Al-rich region becomes relevant. Therefore, the number of energy levels between the VB and CB is higher, resulting in a lower average bandgap energy value. It is important to note that defects caused by the presence of Al metal forming an Al-rich ZnO (ZnO-Al) region, which may create energy levels within the bandgap, increase the availability of charge carriers over the sample when the photons impinge, thus reducing the electrical resistance.

#### **4. Memristor behavior under dark and illumination**

The memristor behavior as a function of the incidence violet light is presented in **Figure 3a**, where voltage sweeps between 0 and 2 V occurred five times and then five other sweeps between 0 and −5 V. The first five sweeps for the positive voltage polarity with the five negative polarity sweeps were performed under illumination of the violet LED (Illumination 1). Then, the same sweeping scheme was imposed on the sample; however, without ambient light (dark 1), the dark chamber was used to provide this situation. In total, the same sample was subjected to 10 alternating sweeps between illumination and dark. Observing specifically the first sweep under illumination, an adaptive-like response was found, in which the maximum value of the measured current intensity gradually increased, indicating that the electrical resistance of the sample decreases as the sweep occurs. This type of behavior is widely known in the scientific literature as the fingerprint of a memristor, where a

#### **Figure 3.**

*(a) I–V characteristic curves of the ZA1 sample under continuous voltage scans (0 – 2 – 0 V) × 5 – (0 – (−2) – 0 V) × 5 under violet LED illumination and under darkness and (b) highlight of the first sweep cycles (0 – 2 – 0 V) × 5 under illumination and darkness.*

device constructed in the form of metal-insulator (semiconductor)-metal presents electrical resistance dependent on the history of excitation by the application of an external electric field [1, 7, 26, 35].

The gradual conductivity increasing with the application of voltage is a desirable aspect related to memristors, as the memory effect associated to these devices is based on a change in the resistance state, usually a higher resistive state *ROFF* and lower resistive states, reaching a minimum resistance level at *RON* [5, 8, 36]. Computational logic states are, therefore, associated with these two values of electrical resistance (bit 0—*ROFF*; bit 1—*RON*). However, in this type of application, the memristors commonly present filamentary resistive switching mechanisms, where a conductive filament is formed by connecting one electrode to another [5, 6, 37]. The samples analyzed in this chapter are mechanisms based on a homogeneous resistive switching, in which electrical resistance states are gradually modified and controlled [26, 36, 38]. In this type of homogeneous resistive switching, it is possible to note an adaptive character to the applied electric voltage in which the state of resistance at a given instant depends on the entire history of the voltage sweep. Jo et al. presented a very interesting aspect of adaptation to the voltage in memristors of Si/Si + Ag thin films which behaved in a way similar to biological synaptic neurons, in other words, a neuromorphic behavior [7]. The samples worked in this chapter presented results of adaptation to voltage sweep very similar to Si/Si + Ag thin films; however, this work focused on the influence of violet LED light on memristive behaviors.

It is interesting to note that the sample showed a behavior of gradually increasing conductivity under illumination and darkness in all the sweeps. In addition, in the negative polarity, the reverse effect of decreasing conductivity was observed. These behaviors as memristors are explained in materials such as ZnO/ZnO-Ag, ZnO/ ZnO-Al, or WO3/Ag by ionic migrations through the insulating lattice [7, 26, 39]. As theoretically demonstrated by Strukov et al. [36], initially, a device with thickness D (distance between electrodes) presents maximum resistance *Roff*; however, the device may be constructed with a dopant-rich region that can continuously modulate the total resistance between *Roff* and *Ron* through ionic migrations. The control of the ionic migrations and, therefore, the resistance values reached by the device can be obtained with the application of electric voltage *V*(*t*). The boundary separating the dopant-rich region from the poor region moves as a function of the applied voltage, which can cause diffusion of the ions, and the normalized position (*w*(*t*)) of this boundary can have values assigned between 0 and 1, where 0 refers to the case where the resistance is maximal (*Roff*) and 1 to the minimum resistance (*Ron*). Previous work has already shown that this typical behavior of the current in a memristor can be characterized as Eq. (2): *<sup>I</sup>*(*t*) <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 1 *Ronw*(*t*) <sup>+</sup> *Roff*(1 <sup>−</sup> *<sup>w</sup>*(*t*))

$$I(t) = \frac{1}{R\_{ow}w(t) \star R\_{gf}(1 - w(t))}V(t) \tag{2}$$

**93**

**Figure 4.**

*Al2O3 oxide distribution after all sweeps.*

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al…*

The *I-V* characteristic curves under illumination present higher current values under the same voltage sweep than the dark response (see **Figure 3b**). This result indicates that the incidence of photons can significantly alter the number of free charge carriers in the CB, favoring the decrease of the electrical resistance. In this case, electrons trapped at energy levels between the VB and CB are excited due to the incidence of photons, increasing the electron population in the CB, which reduces the electrical resistance, a fact evidenced also in the bandgap values of these samples. However, as new voltage sweeps occur alternately between illumination and darkness (Illumination 1 → Dark 1 → Illumination 2 → Dark 2 → …→Dark 5), a gradual increase in resistance is observed (decrease of the current intensity generated by the same voltage scanning interval). When the first sweep is initiated, the ion diffusion

*Diffusion scheme of Al and oxygen dopant ions: (a) initially the Al ions are mostly in the Al-rich region, but, when the voltage sweep is initiated, Al and O ions migrate simultaneously in opposite directions; (b) after the first five sweep for positive polarity, a higher distribution of Al can be directed to the ZnO network, and oxygen ions can be allocated in the Al-rich region; (c) and (d) possible combinations of ions O with Al or Zn ions may prevent new migrations when the polarity is reversed, which indicates a formation of Al2O3 oxides in addition to the present ZnO; (e) ZnO and Al distribution scheme before voltage sweeps; and (f) scheme of ZnO and* 

*DOI: http://dx.doi.org/10.5772/intechopen.86557*

where a pinched hysteresis loop can be obtained [1, 7, 26, 36, 40].

Characteristic I-V curves for different voltage frequencies were published previously in [26]; when voltage excitation frequency is diminished, the step between one conduction state and another is increased. This behavior is typical of memristor and was theoretically predicted by Chua [1] which showed that as the excitation frequency tends to infinity, the area under the I-V curve tends to zero: the effect of adaptation to the excitation is decreased dramatically. On the other hand, when the frequency is reduced, the mechanism of adaptation is evidenced. The work presented in this chapter used a fixed frequency of 5 Hz (or 200 ms excitation period) in all sweeps and samples; on this frequency, an adaptive behavior was very well observed.

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al… DOI: http://dx.doi.org/10.5772/intechopen.86557*

The *I-V* characteristic curves under illumination present higher current values under the same voltage sweep than the dark response (see **Figure 3b**). This result indicates that the incidence of photons can significantly alter the number of free charge carriers in the CB, favoring the decrease of the electrical resistance. In this case, electrons trapped at energy levels between the VB and CB are excited due to the incidence of photons, increasing the electron population in the CB, which reduces the electrical resistance, a fact evidenced also in the bandgap values of these samples. However, as new voltage sweeps occur alternately between illumination and darkness (Illumination 1 → Dark 1 → Illumination 2 → Dark 2 → …→Dark 5), a gradual increase in resistance is observed (decrease of the current intensity generated by the same voltage scanning interval). When the first sweep is initiated, the ion diffusion

#### **Figure 4.**

*Memristors - Circuits and Applications of Memristor Devices*

external electric field [1, 7, 26, 35].

memristive behaviors.

be characterized as Eq. (2):

*<sup>I</sup>*(*t*) <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 1 *Ronw*(*t*) <sup>+</sup> *Roff*(1 <sup>−</sup> *<sup>w</sup>*(*t*))

where a pinched hysteresis loop can be obtained [1, 7, 26, 36, 40].

Characteristic I-V curves for different voltage frequencies were published previously in [26]; when voltage excitation frequency is diminished, the step between one conduction state and another is increased. This behavior is typical of memristor and was theoretically predicted by Chua [1] which showed that as the excitation frequency tends to infinity, the area under the I-V curve tends to zero: the effect of adaptation to the excitation is decreased dramatically. On the other hand, when the frequency is reduced, the mechanism of adaptation is evidenced. The work presented in this chapter used a fixed frequency of 5 Hz (or 200 ms excitation period) in all sweeps and samples; on this frequency, an adaptive behavior was very well observed.

device constructed in the form of metal-insulator (semiconductor)-metal presents electrical resistance dependent on the history of excitation by the application of an

The gradual conductivity increasing with the application of voltage is a desirable aspect related to memristors, as the memory effect associated to these devices is based on a change in the resistance state, usually a higher resistive state *ROFF* and lower resistive states, reaching a minimum resistance level at *RON* [5, 8, 36]. Computational logic states are, therefore, associated with these two values of electrical resistance (bit 0—*ROFF*; bit 1—*RON*). However, in this type of application, the memristors commonly present filamentary resistive switching mechanisms, where a conductive filament is formed by connecting one electrode to another [5, 6, 37]. The samples analyzed in this chapter are mechanisms based on a homogeneous resistive switching, in which electrical resistance states are gradually modified and controlled [26, 36, 38]. In this type of homogeneous resistive switching, it is possible to note an adaptive character to the applied electric voltage in which the state of resistance at a given instant depends on the entire history of the voltage sweep. Jo et al. presented a very interesting aspect of adaptation to the voltage in memristors of Si/Si + Ag thin films which behaved in a way similar to biological synaptic neurons, in other words, a neuromorphic behavior [7]. The samples worked in this chapter presented results of adaptation to voltage sweep very similar to Si/Si + Ag thin films; however, this work focused on the influence of violet LED light on

It is interesting to note that the sample showed a behavior of gradually increasing conductivity under illumination and darkness in all the sweeps. In addition, in the negative polarity, the reverse effect of decreasing conductivity was observed. These behaviors as memristors are explained in materials such as ZnO/ZnO-Ag, ZnO/ ZnO-Al, or WO3/Ag by ionic migrations through the insulating lattice [7, 26, 39]. As theoretically demonstrated by Strukov et al. [36], initially, a device with thickness D (distance between electrodes) presents maximum resistance *Roff*; however, the device may be constructed with a dopant-rich region that can continuously modulate the total resistance between *Roff* and *Ron* through ionic migrations. The control of the ionic migrations and, therefore, the resistance values reached by the device can be obtained with the application of electric voltage *V*(*t*). The boundary separating the dopant-rich region from the poor region moves as a function of the applied voltage, which can cause diffusion of the ions, and the normalized position (*w*(*t*)) of this boundary can have values assigned between 0 and 1, where 0 refers to the case where the resistance is maximal (*Roff*) and 1 to the minimum resistance (*Ron*). Previous work has already shown that this typical behavior of the current in a memristor can

*V*(*t*) (2)

**92**

*Diffusion scheme of Al and oxygen dopant ions: (a) initially the Al ions are mostly in the Al-rich region, but, when the voltage sweep is initiated, Al and O ions migrate simultaneously in opposite directions; (b) after the first five sweep for positive polarity, a higher distribution of Al can be directed to the ZnO network, and oxygen ions can be allocated in the Al-rich region; (c) and (d) possible combinations of ions O with Al or Zn ions may prevent new migrations when the polarity is reversed, which indicates a formation of Al2O3 oxides in addition to the present ZnO; (e) ZnO and Al distribution scheme before voltage sweeps; and (f) scheme of ZnO and Al2O3 oxide distribution after all sweeps.*

#### **Figure 5.**

*Current as a function of time for a fixed voltage of 1.5 V: the red arrow indicates the direction of growth of the characteristic resistive switching current, and the violet LED light was set to oscillate between on and off at intervals Δt = 100 s; the first five light exposures are emphasized in the curve (inset: first five violet light exposures in which the baseline was removed).*

process occurs, causing a distribution of Al3+ and O2<sup>−</sup> along the entire crystalline network of the sample in a way where simultaneous migrations of Al and O ions (or oxygen vacancies) are realized. This distribution of ion dopants to the ZnO network facilitates electronic conduction, which results in the gradual decrease of the resistance, a fact known as homogeneous resistive switching [26, 38]. However, after several sweeps for positive and negative polarity, the distribution of Al and O ions enables the formation of Al2O3 oxide in addition to the existing ZnO network, where this fact may result in the gradual increase of the resistance after each set of voltage sweeps. **Figure 4** illustrates this ion diffusion scheme throughout the sample. Similar results were observed for samples ZA2 and ZA3 (not shown in this work).

**Figure 5** shows a curve of the electric current intensity measured through the sample ZA1 as a function of the time for application of a constant voltage of 1.5 V. In each 100 s, the violet LED oscillated between on and off, where it was possible to perceive the sample response as a sensor of violet light by increasing the current intensity when illuminated for 100 s. This result is interesting because it demonstrates two simultaneous responses. The first is a homogeneous resistive switching response that indicates an adaptive process of the sample because there is no variation of the applied voltage modulus, and a gradual increase of current is observed. In addition, this memristorlike behavior is affected when the LED light is on. The first five illuminations in the sample are indicated in the inset of **Figure 5**, where the baseline has been removed. Considering that the incidence of photons in the sample can promote trapped charge carriers in the network to the CB, the amount of charge generated in the first five incidences of violet light was calculated knowing that *q* = ∫*Idt*. The calculated electric charges were, respectively, 320.4,420.0,242.1,and 650.9 *μC*.

#### **5. Conclusions**

The ZnO/ZnO-Al thin films with memristor behavior showed a transparency of 88% in the visible region for a thickness of 150 nm, which makes it a relevant candidate in transparent electronics. The bandgap values were determined through the

**95**

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al…*

future research related to optical effects on memristor behaviors.

This study was financed in part by the Coordenação de Aperfeiçoamento de

optical absorption spectrum where the values are between 3.19 and 3.26 eV, similar to values in the literature for this type of material. The ZnO/ZnO-Al thin films' memristive behaviors were observed under the incidence of violet light and under darkness for cycles of voltage sweeps in which an adaptive character can be inferred. The incidence of light favored the increase of the number of carriers, but it did not impede the ion migration to form Al2O3 and ZnO oxides throughout the sample, a fact that gradually increased the resistance of the device. The memristor behavior was explained by the diffusion of Al ions that facilitated the electric conduction mechanism between illumination and dark conditions; however, for several voltage sweep cycles, the formation of oxides resulted in the reverse effect, increasing the resistance. Testing as a violet light sensor indicated the generation of electrical charges in the sample network while an adaptive behavior characteristic of the memristor occurred. In other words, two simultaneous phenomena were observed, in which the ZnO/ZnO-Al memristor was influenced by violet light, increasing the conductivity, at the time when it had homogeneous resistive switching due to the electric voltage. These results indicate a scientific advance in the area of resistive switching with the observation of ZnO/ZnO-Al memristive behavior dependent on the voltage application history and the ambient light conditions. In addition, new insight is provided for

*DOI: http://dx.doi.org/10.5772/intechopen.86557*

**Acknowledgements**

**Conflict of interest**

Pessoal de Nível Superior—Brasil (CAPES).

There is no conflict of interest.

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al… DOI: http://dx.doi.org/10.5772/intechopen.86557*

optical absorption spectrum where the values are between 3.19 and 3.26 eV, similar to values in the literature for this type of material. The ZnO/ZnO-Al thin films' memristive behaviors were observed under the incidence of violet light and under darkness for cycles of voltage sweeps in which an adaptive character can be inferred. The incidence of light favored the increase of the number of carriers, but it did not impede the ion migration to form Al2O3 and ZnO oxides throughout the sample, a fact that gradually increased the resistance of the device. The memristor behavior was explained by the diffusion of Al ions that facilitated the electric conduction mechanism between illumination and dark conditions; however, for several voltage sweep cycles, the formation of oxides resulted in the reverse effect, increasing the resistance. Testing as a violet light sensor indicated the generation of electrical charges in the sample network while an adaptive behavior characteristic of the memristor occurred. In other words, two simultaneous phenomena were observed, in which the ZnO/ZnO-Al memristor was influenced by violet light, increasing the conductivity, at the time when it had homogeneous resistive switching due to the electric voltage. These results indicate a scientific advance in the area of resistive switching with the observation of ZnO/ZnO-Al memristive behavior dependent on the voltage application history and the ambient light conditions. In addition, new insight is provided for future research related to optical effects on memristor behaviors.

#### **Acknowledgements**

*Memristors - Circuits and Applications of Memristor Devices*

process occurs, causing a distribution of Al3+ and O2<sup>−</sup> along the entire crystalline network of the sample in a way where simultaneous migrations of Al and O ions (or oxygen vacancies) are realized. This distribution of ion dopants to the ZnO network facilitates electronic conduction, which results in the gradual decrease of the resistance, a fact known as homogeneous resistive switching [26, 38]. However, after several sweeps for positive and negative polarity, the distribution of Al and O ions enables the formation of Al2O3 oxide in addition to the existing ZnO network, where this fact may result in the gradual increase of the resistance after each set of voltage sweeps. **Figure 4** illustrates this ion diffusion scheme throughout the sample. Similar

*Current as a function of time for a fixed voltage of 1.5 V: the red arrow indicates the direction of growth of the characteristic resistive switching current, and the violet LED light was set to oscillate between on and off at intervals Δt = 100 s; the first five light exposures are emphasized in the curve (inset: first five violet light* 

results were observed for samples ZA2 and ZA3 (not shown in this work).

charges were, respectively, 320.4,420.0,242.1,and 650.9 *μC*.

**Figure 5** shows a curve of the electric current intensity measured through the sample ZA1 as a function of the time for application of a constant voltage of 1.5 V. In each 100 s, the violet LED oscillated between on and off, where it was possible to perceive the sample response as a sensor of violet light by increasing the current intensity when illuminated for 100 s. This result is interesting because it demonstrates two simultaneous responses. The first is a homogeneous resistive switching response that indicates an adaptive process of the sample because there is no variation of the applied voltage modulus, and a gradual increase of current is observed. In addition, this memristorlike behavior is affected when the LED light is on. The first five illuminations in the sample are indicated in the inset of **Figure 5**, where the baseline has been removed. Considering that the incidence of photons in the sample can promote trapped charge carriers in the network to the CB, the amount of charge generated in the first five incidences of violet light was calculated knowing that *q* = ∫*Idt*. The calculated electric

The ZnO/ZnO-Al thin films with memristor behavior showed a transparency of 88% in the visible region for a thickness of 150 nm, which makes it a relevant candidate in transparent electronics. The bandgap values were determined through the

**94**

**5. Conclusions**

**Figure 5.**

*exposures in which the baseline was removed).*

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES).

#### **Conflict of interest**

There is no conflict of interest.

*Memristors - Circuits and Applications of Memristor Devices*

#### **Author details**

Adolfo Henrique Nunes Melo, Raiane Sodre de Araujo, Eduardo Valença and Marcelo Andrade Macêdo\* Department of Physics, Federal University of Sergipe, São Cristóvão, Sergipe, Brazil

\*Address all correspondence to: odecamm@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**97**

*Memristor Behavior under Dark and Violet Illumination in Thin Films of ZnO/ZnO-Al…*

Rapid Research Letters. 2007;**1**:86-88.

[10] Mundle R, Carvajal C, Pradhan AK. ZnO/Al:ZnO transparent resistive switching devices grown by atomic layer deposition for memristor applications. Langmuir. 2016;**32**:4983-4995. DOI: 10.1021/acs.langmuir.6b01014

[11] Serrano-Gotarredona T, Masquelier

T, Prodromakis T, Indiveri G, Linares-Barranco B. STDP and sTDP variations with memristors for spiking neuromorphic learning systems. Frontiers in Neuroscience. 2013;**7**:1-15.

DOI: 10.3389/fnins.2013.00002

[12] Thomas A. Memristor-based neural networks. Journal of Physics D: Applied Physics. 2013;**46**:093001. DOI: 10.1088/0022-3727/46/9/093001

[13] Melo AHN, Silva PB, Macedo MA. Structural, optical, and

AMR.975.238

electrical properties of ZnO/Nb/ZnO multilayer thin films. Advances in Materials Research. 2014;**975**:238-242. DOI: 10.4028/www.scientific.net/

[14] Snure M, Tiwari A. Structural, electrical, and optical characterizations of epitaxial Zn1−x GaxO films grown on sapphire (0001) substrate. Journal of Applied Physics. 2007;**101**:124912-124917.

[15] Yu S, Gao B, Fang Z, Yu H, Kang J, Wong H-SP. Stochastic learning in oxide binary synaptic device for neuromorphic computing. Frontiers in Neuroscience. 2013;**7**:186. DOI: 10.3389/

[16] Indiveri G, Linares-Barranco B, Hamilton TJ, van Schaik A, Etienne-Cummings R, Delbruck T, et al. Neuromorphic silicon neuron circuits. Frontiers in Neuroscience. 2011;**5**:1-23.

DOI: 10.3389/fnins.2011.00073

DOI: 10.1063/1.2749487

fnins.2013.00186

DOI: 10.1002/pssr.200701003

*DOI: http://dx.doi.org/10.5772/intechopen.86557*

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**Author details**

Marcelo Andrade Macêdo\*

Adolfo Henrique Nunes Melo, Raiane Sodre de Araujo, Eduardo Valença and

\*Address all correspondence to: odecamm@gmail.com

provided the original work is properly cited.

Department of Physics, Federal University of Sergipe, São Cristóvão, Sergipe, Brazil

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

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[22] Li H, Chen Q, Chen X, Mao Q, Xi J, Ji Z. Improvement of resistive switching in ZnO film by Ti doping. Thin Solid Films. 2013;**537**:279-284. DOI: 10.1016/j.

[23] Babu BJ, Velumani S, Asomoza R. An (ITO or AZO)/ZnO/Cu(In1-xGax) Se2 superstrate thin film solar cell structure prepared by spray pyrolysis.

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[32] Aydın H, Yakuphanoglu F, Aydın C. Al-doped ZnO as a multifunctional nanomaterial: Structural, morphological, optical and lowtemperature gas sensing properties. Journal of Alloys and Compounds. 2019;**773**:802-811

[33] Ou S-L, Liu H-R, Wang S-Y, Wuu D-S. Co-doped ZnO dilute magnetic semiconductor thin films by pulsed laser deposition: Excellent transmittance, low resistivity and high mobility. Journal of Alloys and Compounds. 2016;**663**:107-115. DOI: 10.1016/j.jallcom.2015.12.101

[34] Fallah HR, Ghasemi M, Hassanzadeh A. Influence of heat treatment on structural, electrical, impedance and optical properties of nanocrystalline ITO films grown on glass at room temperature prepared by electron beam evaporation. Physica E: Low-dimensional Systems and Nanostructures. 2007;**39**:69-74

[35] Kim H, Sah MP, Yang C, Roska T, Chua LO. Memristor bridge synapses. Proceedings of the IEEE. 2012;**100**:2061-2070. DOI: 10.1109/ JPROC.2011.2166749

[36] Strukov DB, Snider GS, Stewart DR, Williams RS. The missing memristor found. Nature. 2008;**453**:80-84. DOI: 10.1038/nature06932

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**101**

**1. Introduction**

**Chapter 7**

**Abstract**

Application of Probe

Characterization

(ReRAM) units based on the memristor devices.

nanocrystalline ZnO films, carbon nanotubes

Structures Formation and

Nanotechnologies for Memristor

*Vladimir A. Smirnov, Marina V. Il'ina, Vadim I. Avilov,* 

This chapter presents the results of experimental studies of the formation and investigation of the memristors by probe nanotechnologies. This chapter also perspectives and possibilities of application of local anodic oxidation and scratching probe nanolithography for the manufacture of memristors based on titanium oxide structures, nanocrystalline ZnO thin film, and vertically aligned carbon nanotubes. Memristive properties of vertically aligned carbon nanotubes, titanium oxide, and ZnO nanostructures were investigated by scanning probe microscopy methods. It is shown that nanocrystalline ZnO films manifest a stable memristor effect slightly dependent on its morphology. Titanium oxide nanoscale structures of different thicknesses obtained by local anodic oxidation demonstrate a memristive effect without the need to perform any additional electroforming operations. This experimentally confirmed the memristive switching of a two-electrode structure based on a vertically aligned carbon nanotube. These results can be used in the development of designs and technological processes of resistive random access memory

**Keywords:** nanotechnology, scanning probe microscopy, memristor, titanium oxide,

Reducing the elements of integrated circuits (ICs) leads to an increase in the speed of the processors and an increase in the amount of memory, but at the same time the bandwidth between them varies only slightly. This is referred as the von Neumann bottleneck and often limits the performance of the system [1]. One possible solution to this problem is the transition of computing systems to an architecture close to the structure of a biological brain, which is a set of elements of low power connected in parallel neurons interconnected via special channels synapses [1–4]. Processors built on this architecture have concurrent computing and will be able to surpass modern computers in tasks related to unstructured data classification, pattern recognition, as well as in applications with adaptable and self-learning

*Roman V. Tominov, Oleg I. Il'in and Oleg A. Ageev*

### **Chapter 7**

## Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization

*Vladimir A. Smirnov, Marina V. Il'ina, Vadim I. Avilov, Roman V. Tominov, Oleg I. Il'in and Oleg A. Ageev*

### **Abstract**

This chapter presents the results of experimental studies of the formation and investigation of the memristors by probe nanotechnologies. This chapter also perspectives and possibilities of application of local anodic oxidation and scratching probe nanolithography for the manufacture of memristors based on titanium oxide structures, nanocrystalline ZnO thin film, and vertically aligned carbon nanotubes. Memristive properties of vertically aligned carbon nanotubes, titanium oxide, and ZnO nanostructures were investigated by scanning probe microscopy methods. It is shown that nanocrystalline ZnO films manifest a stable memristor effect slightly dependent on its morphology. Titanium oxide nanoscale structures of different thicknesses obtained by local anodic oxidation demonstrate a memristive effect without the need to perform any additional electroforming operations. This experimentally confirmed the memristive switching of a two-electrode structure based on a vertically aligned carbon nanotube. These results can be used in the development of designs and technological processes of resistive random access memory (ReRAM) units based on the memristor devices.

**Keywords:** nanotechnology, scanning probe microscopy, memristor, titanium oxide, nanocrystalline ZnO films, carbon nanotubes

#### **1. Introduction**

Reducing the elements of integrated circuits (ICs) leads to an increase in the speed of the processors and an increase in the amount of memory, but at the same time the bandwidth between them varies only slightly. This is referred as the von Neumann bottleneck and often limits the performance of the system [1]. One possible solution to this problem is the transition of computing systems to an architecture close to the structure of a biological brain, which is a set of elements of low power connected in parallel neurons interconnected via special channels synapses [1–4]. Processors built on this architecture have concurrent computing and will be able to surpass modern computers in tasks related to unstructured data classification, pattern recognition, as well as in applications with adaptable and self-learning control systems. One of the possible ways of implementing such an architecture is to manufacture ICs based on neuromorphic structures, which are memory elements in the form of cells (neurons) interconnected by data buses (synapses) and are capable of changing their electrical resistance under the influence of an external electric field (the effect of resistive switch) [1–5]. Such structures can maintain cell resistance after the termination of the external electric field and are the basis of non-volatile resistive memory (ReRAM). The advantages of ReRAM include small size, high degree of integration, low power consumption, and high speed, which allows on its basis to realize the mass parallelism and low power calculations observed in the study of biological brain [6–8].

Currently, active theoretical and experimental studies of the effect of resistive switching in nanomaterials and nanostructures are underway to create elements of the ReRAM self-learning adaptive neuromorphic processor with high speed and low power consumption. An analysis of the publications showed that for the manufacture of ReRAM, films based on binary metal oxides (SiOx, TiO2, ZnO, HfOx, etc.) are promising from which amorphous titanium oxide and nanocrystalline zinc oxide can be distinguished, allowing for high response speed of the resistive switching process [8–10].

The creation of ReRAM elements for a neuromorphic processor is associated with the development and research of the formation processes of structures with a nanometer resolution. Existing lithographic methods of semiconductor technologies are approaching the limit of their resolution, characterized by a high degree of complexity and cost of equipment. Therefore, there is a need for research and development of new methods, the use of which will allow the creation of nanostructures of ReRAM elements, including at the prototyping stage.

One possible way of ReRAM elements prototyping is probe nanotechnologies usage, which is a combination of methods for nanostructures forming using a probe tip with visualization and process control in situ. Promising methods of probe nanotechnologies for ReRAM elements prototyping include local anodic oxidation (LAO) and scanning probe nanolithography (SPN) of atomic force microscope.

The method of local anodic oxidation is promising for the manufacture of oxide nanostructures (ONS) of titanium, which have reproducible memristor effect and do not require forming [11–14]. The advantages of the LAO method also include precision, the possibility of conducting research on electrochemical processes in local areas up to the size of several nanometers in situ diagnostics of the results of the formation of ONS on the substrate surface, the absence of additional technological operations for applying, exposing, and removing photoresist, as well as the relatively low cost of process equipment [15–20]. A variety of nanoimprint lithography is scratching probe nanolithography, which allows using the tip of an atomic force microscope probe to form profiled nanostructures in polymer films [21]. The simplicity of the method implementation allows using it in the development and study of promising design and technological solutions for prototyping ReRAM elements [22].

At the present stage of nanotechnology development, one of the most promising methods for surface diagnostics is scanning probe microscopy (SPM). The use of SPM methods allows the study of the local geometric, electrical, and mechanical properties of the sample surface [23–26].

Of interest for creating ReRAM with high cell density are memristor structures based on vertically aligned carbon nanotubes (VA CNTs) [27, 28]. The vertical orientation of the nanotubes provides a significant reduction in the memory cell area and the technology of producing VA CNTs based on the method of plasmaenhanced chemical vapor deposition (PECVD) allows localized growth of nanotubes in a process compatible with silicon technology [29, 30]. In addition, the use of VA CNTs as a storage element is expected to reduce the switching time to picoseconds [28, 31].

**103**

**Figure 1.**

*(c) current-voltage characteristics measuring scheme.*

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

A promising method of probe nanotechnologies for creation and characterization of memristor structures based on VA CNTs is the scanning tunneling microscopy (STM) [26]. This method allows you to create a controlled elastic deformation in VA CNT, the presence of which is a prerequisite for the occurrence of the memristor effect in VA CNT [32]. The mechanism of memristive switching of strained

This chapter describes application of scanning probe microscopy techniques for determination of resistive switching effects in vertically aligned carbon nanotubes, TiO2 nanostructures, and ZnO thin nanocrystalline films. Described techniques can be used for formation and nanodiagnostics of parameters of memristor structures

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

carbon nanotube is described in detail in [28].

**formed by local anodic oxidation**

1 × 1 μm and 1.1 nm thick (**Figure 1**).

operations.

for creation of metal oxide and CNT-based neuromorphic system.

**on titanium oxide by atomic force microscopy**

**2. Formation and investigation of memristor structures based** 

**2.1 Investigation of resistive switching of titanium oxide nanostructures** 

Then, using the AFM, the current-voltage (I-V) characteristics of the obtained structure were measured according to the scheme shown in **Figure 1c** with the application of a triangular-shaped voltage pulse (**Figure 2a** inset). It was shown that the titanium ONS obtained by the LAO exhibits a bipolar resistive switching effect (**Figure 2**) without additional doping and electroforming

The analysis showed that the current-voltage characteristic type corresponds to the switching mechanism due to potential barrier width modulation at the

*Titanium oxide nanostructures, formed by LAO: (a) AFM image; (b) profilogram along the line; and* 

The memristor effect study was carried out on titanium oxide nanostructures (ONS) using AFM spectroscopy using SPM Solver P47 Pro. For this purpose, local anodic oxidation (LAO) was carried out on the thin titanium film surface with a 20 nm thickness, and as a result, titanium ONS was formed with lateral dimensions *Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

A promising method of probe nanotechnologies for creation and characterization of memristor structures based on VA CNTs is the scanning tunneling microscopy (STM) [26]. This method allows you to create a controlled elastic deformation in VA CNT, the presence of which is a prerequisite for the occurrence of the memristor effect in VA CNT [32]. The mechanism of memristive switching of strained carbon nanotube is described in detail in [28].

This chapter describes application of scanning probe microscopy techniques for determination of resistive switching effects in vertically aligned carbon nanotubes, TiO2 nanostructures, and ZnO thin nanocrystalline films. Described techniques can be used for formation and nanodiagnostics of parameters of memristor structures for creation of metal oxide and CNT-based neuromorphic system.

#### **2. Formation and investigation of memristor structures based on titanium oxide by atomic force microscopy**

#### **2.1 Investigation of resistive switching of titanium oxide nanostructures formed by local anodic oxidation**

The memristor effect study was carried out on titanium oxide nanostructures (ONS) using AFM spectroscopy using SPM Solver P47 Pro. For this purpose, local anodic oxidation (LAO) was carried out on the thin titanium film surface with a 20 nm thickness, and as a result, titanium ONS was formed with lateral dimensions 1 × 1 μm and 1.1 nm thick (**Figure 1**).

Then, using the AFM, the current-voltage (I-V) characteristics of the obtained structure were measured according to the scheme shown in **Figure 1c** with the application of a triangular-shaped voltage pulse (**Figure 2a** inset). It was shown that the titanium ONS obtained by the LAO exhibits a bipolar resistive switching effect (**Figure 2**) without additional doping and electroforming operations.

The analysis showed that the current-voltage characteristic type corresponds to the switching mechanism due to potential barrier width modulation at the

#### **Figure 1.**

*Titanium oxide nanostructures, formed by LAO: (a) AFM image; (b) profilogram along the line; and (c) current-voltage characteristics measuring scheme.*

*Memristors - Circuits and Applications of Memristor Devices*

observed in the study of biological brain [6–8].

properties of the sample surface [23–26].

control systems. One of the possible ways of implementing such an architecture is to manufacture ICs based on neuromorphic structures, which are memory elements in the form of cells (neurons) interconnected by data buses (synapses) and are capable of changing their electrical resistance under the influence of an external electric field (the effect of resistive switch) [1–5]. Such structures can maintain cell resistance after the termination of the external electric field and are the basis of non-volatile resistive memory (ReRAM). The advantages of ReRAM include small size, high degree of integration, low power consumption, and high speed, which allows on its basis to realize the mass parallelism and low power calculations

Currently, active theoretical and experimental studies of the effect of resistive switching in nanomaterials and nanostructures are underway to create elements of the ReRAM self-learning adaptive neuromorphic processor with high speed and low power consumption. An analysis of the publications showed that for the manufacture of ReRAM, films based on binary metal oxides (SiOx, TiO2, ZnO, HfOx, etc.) are promising from which amorphous titanium oxide and nanocrystalline zinc oxide can be distinguished, allowing for high response speed of the resistive switching process [8–10]. The creation of ReRAM elements for a neuromorphic processor is associated with the development and research of the formation processes of structures with a nanometer resolution. Existing lithographic methods of semiconductor technologies are approaching the limit of their resolution, characterized by a high degree of complexity and cost of equipment. Therefore, there is a need for research and development of new methods, the use of which will allow the creation of nano-

One possible way of ReRAM elements prototyping is probe nanotechnologies usage, which is a combination of methods for nanostructures forming using a probe tip with visualization and process control in situ. Promising methods of probe nanotechnologies for ReRAM elements prototyping include local anodic oxidation (LAO) and scanning probe nanolithography (SPN) of atomic force microscope. The method of local anodic oxidation is promising for the manufacture of oxide nanostructures (ONS) of titanium, which have reproducible memristor effect and do not require forming [11–14]. The advantages of the LAO method also include precision, the possibility of conducting research on electrochemical processes in local areas up to the size of several nanometers in situ diagnostics of the results of the formation of ONS on the substrate surface, the absence of additional technological operations for applying, exposing, and removing photoresist, as well as the relatively low cost of process equipment [15–20]. A variety of nanoimprint lithography is scratching probe nanolithography, which allows using the tip of an atomic force microscope probe to form profiled nanostructures in polymer films [21]. The simplicity of the method implementation allows using it in the development and study of promising design and technological solutions for prototyping ReRAM elements [22]. At the present stage of nanotechnology development, one of the most promising methods for surface diagnostics is scanning probe microscopy (SPM). The use of SPM methods allows the study of the local geometric, electrical, and mechanical

Of interest for creating ReRAM with high cell density are memristor structures based on vertically aligned carbon nanotubes (VA CNTs) [27, 28]. The vertical orientation of the nanotubes provides a significant reduction in the memory cell area and the technology of producing VA CNTs based on the method of plasmaenhanced chemical vapor deposition (PECVD) allows localized growth of nanotubes in a process compatible with silicon technology [29, 30]. In addition, the use of VA CNTs as a storage element is expected to reduce the switching time to

structures of ReRAM elements, including at the prototyping stage.

**102**

picoseconds [28, 31].

**Figure 2.** *Current-voltage characteristic of titanium ONS, formed by the LAO.*

electrode/oxide interface described in [11, 12]. In this case, the potential barrier modulation occurs alternately at both the electrodes boundaries.

Initially, the structure is in high resistance state (HRS) (1.3 GΩ at 3.5 V) in the forward bias region (region I in **Figure 2**), while at a negative voltage applied to −5 V, the structure is in low resistance state (LRS) (0.27 GΩ at −3 V) (region II in **Figure 2**). Then, as the voltage rises in the reverse bias region from −6 to −10 V, the structure is switched, resulting in a HRS state (3.4 GΩ at −3 V) at the reverse bias (region III in **Figure 2**) and LRS (0.35 GΩ at 3.5 V) at up to 5 V direct bias (region IV in **Figure 2**). With an increase in the applied voltage from 6 to 10 V, the structure is switched to the initial state. In this case, the structure resistance ratio in the HRS state to the LRS state for positive voltages is 3.6, and for negative voltages is 12.6.

#### **2.2 Investigation of influence of AFM probe pressing force on the resistive switching in titanium oxide nanostructures**

For the experimental study, titanium ONS with 2 × 2 μm lateral dimensions and 6.8 nm thick was formed by LAO. Then, on its surface in the AFM contact mode was performed a spectroscopic measurement of dependence of the feedback circuit current on the cantilever beam bending. The spectrogram showed that with an increase in the feedback circuit current by 1.02 nA, the beam is bent by 20.8 nm. Since the cantilever beam stiffness is 2.5 N/m, it is possible to calculate the AFM probe pressing force to the ONS surface for a given feedback circuit current **Figure 3**.

Then, current-voltage characteristics were measured on the ONS surface in the ±10 V range with the AFM feedback circuit current values from 0.01 to 2 nA (**Figure 4**), after which the structure resistance values in the HRS and LRS states were measured at 3.5 V.

Obtained dependences analysis showed that an increase in the AFM probe clamping force to the surface from 0.51 to 102.8 nN leads to a decrease in the structure resistance in the HRS state from 1.12 × 1011 to 9.63 × 109 Ω and in the LRS state from 2.28 × 1010 to 1.38 × 109 Ω.

**105**

**Figure 3.**

**Figure 4.**

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

surface occurs. In further studies, 60 nN clamping force was used, ensuring reliable

Literature analysis showed that the top electrode material has a significant impact on the resistive switching of memristor structures. To study this effect, the current-voltage characteristics were measured in the mode of current AFM spectroscopy on the surface of oxide nanostructures formed by the LAO method; AFM probes with different conducting coating were used as the upper electrode. The

Obtained dependences analysis showed that the use of cantilevers with different coatings significantly affects the manifested memristor effect. So, when using a cantilever with a Pt coating, a symmetrical I-V characteristic was obtained (**Figure 5a**) with low current values, the structure resistance in the HRS state is 327 × 109 Ω, and in the LRS state is 22 × 109 Ω, while the resistance ratio in the high

When using a cantilever with a TiN coating, an asymmetric I-V characteristic was obtained (**Figure 5b**), while in the negative voltage region the memristor effect

**2.3 Investigation of influence of top electrode materials on the resistive** 

*Dependence of titanium ONS resistance in the HRS and LRS states on the pressing force.*

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

contact of the probe with the structure.

resistance to low resistance is 15.

**switching in titanium oxide nanostructures**

*Dependence of feedback circuit current from the cantilever beam bending.*

resulting characteristics are presented on (**Figure 5**).

This dependence can be explained by the fact that with a clamping force increase, an increase in the contact area between the AFM probe and the oxide *Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

**Figure 3.**

*Memristors - Circuits and Applications of Memristor Devices*

electrode/oxide interface described in [11, 12]. In this case, the potential barrier

**2.2 Investigation of influence of AFM probe pressing force on the resistive** 

For the experimental study, titanium ONS with 2 × 2 μm lateral dimensions and 6.8 nm thick was formed by LAO. Then, on its surface in the AFM contact mode was performed a spectroscopic measurement of dependence of the feedback circuit current on the cantilever beam bending. The spectrogram showed that with an increase in the feedback circuit current by 1.02 nA, the beam is bent by 20.8 nm. Since the cantilever beam stiffness is 2.5 N/m, it is possible to calculate the AFM probe pressing force to the ONS surface for a given feedback

Then, current-voltage characteristics were measured on the ONS surface in the ±10 V range with the AFM feedback circuit current values from 0.01 to 2 nA (**Figure 4**), after which the structure resistance values in the HRS and LRS states

Obtained dependences analysis showed that an increase in the AFM probe clamping force to the surface from 0.51 to 102.8 nN leads to a decrease in the structure resistance in the HRS state from 1.12 × 1011 to 9.63 × 109 Ω and in the LRS state

This dependence can be explained by the fact that with a clamping force increase, an increase in the contact area between the AFM probe and the oxide

Initially, the structure is in high resistance state (HRS) (1.3 GΩ at 3.5 V) in the forward bias region (region I in **Figure 2**), while at a negative voltage applied to −5 V, the structure is in low resistance state (LRS) (0.27 GΩ at −3 V) (region II in **Figure 2**). Then, as the voltage rises in the reverse bias region from −6 to −10 V, the structure is switched, resulting in a HRS state (3.4 GΩ at −3 V) at the reverse bias (region III in **Figure 2**) and LRS (0.35 GΩ at 3.5 V) at up to 5 V direct bias (region IV in **Figure 2**). With an increase in the applied voltage from 6 to 10 V, the structure is switched to the initial state. In this case, the structure resistance ratio in the HRS state to the LRS state for positive voltages is 3.6, and for negative

modulation occurs alternately at both the electrodes boundaries.

*Current-voltage characteristic of titanium ONS, formed by the LAO.*

**switching in titanium oxide nanostructures**

**104**

voltages is 12.6.

**Figure 2.**

circuit current **Figure 3**.

were measured at 3.5 V.

from 2.28 × 1010 to 1.38 × 109 Ω.

*Dependence of feedback circuit current from the cantilever beam bending.*

#### **Figure 4.**

*Dependence of titanium ONS resistance in the HRS and LRS states on the pressing force.*

surface occurs. In further studies, 60 nN clamping force was used, ensuring reliable contact of the probe with the structure.

#### **2.3 Investigation of influence of top electrode materials on the resistive switching in titanium oxide nanostructures**

Literature analysis showed that the top electrode material has a significant impact on the resistive switching of memristor structures. To study this effect, the current-voltage characteristics were measured in the mode of current AFM spectroscopy on the surface of oxide nanostructures formed by the LAO method; AFM probes with different conducting coating were used as the upper electrode. The resulting characteristics are presented on (**Figure 5**).

Obtained dependences analysis showed that the use of cantilevers with different coatings significantly affects the manifested memristor effect. So, when using a cantilever with a Pt coating, a symmetrical I-V characteristic was obtained (**Figure 5a**) with low current values, the structure resistance in the HRS state is 327 × 109 Ω, and in the LRS state is 22 × 109 Ω, while the resistance ratio in the high resistance to low resistance is 15.

When using a cantilever with a TiN coating, an asymmetric I-V characteristic was obtained (**Figure 5b**), while in the negative voltage region the memristor effect

**Figure 5.**

*Titanium ONS current-voltage characteristics, obtained using a cantilever coated by: (a) Pt; (b) TiN; and (c) carbon.*

is insignificant, the structure's resistance in the HRS state is 4.65 × 109 Ω, and in the LRS state −0.39 × 109 Ω, at the same time, resistance ratio in the high resistance to low resistance is 12. In the case of using a carbon-coated cantilever, an asymmetric I-V characteristic is also observed (**Figure 5c**), and there is no current through the ONS when applying voltages less than ±5 V in the case of a structure being in HRS and applying a voltage less than ±2 V in case of a structure being in LRS. The structure's resistance in the HRS state is 1.74 × 109 Ω, and in the LRS state, it is 0.3 × 109 Ω, and the resistance ratio in the high resistance to low resistance is 6.

Study results showed that the platinum-coated cantilever use as the top electrode is characterized by a largest resistance ratio in the HRS and LRS states.

#### **2.4 Investigation of influence of titanium oxide nanostructures geometric parameters on their resistive switching**

Another goal is to study titanium ONS thickness effect and applied voltage pulses on the memristor effect. For this, four ONSs were formed with lateral 2 × 2 μm dimensions and 1.6–3.6 nm height, which, based on the expression describing the oxide height and depth ratio presented in [3], corresponds to 3.6–8.2 nm thickness. The current-voltage characteristics were measured on these structures surface by applying ±2.4 voltage pulse (**Figure 6**).

Obtained expression analysis showed that with an increase in the ONS thickness, a decrease in the current corresponding to the LRS state and an increase in the resistance in this state are observed, to the extent that the memristor effect does not manifest itself when the oxide thickness is 8.2 nm.

The results allowed us to obtain voltage of the switching structure in the HRS state (Ures) and in the LRS state (Uset) dependence, as well as the corresponding

**107**

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

currents (Ires and Iset) on the titanium ONS thickness (**Figure 7**). In addition, the dependence of the structure resistance measured in a HRS and LRS state on the

*Titanium ONS current-voltage characteristics with a thickness: (a) 3.6 nm; (b) 5.4 nm; (c) 7.2 nm; and* 

**2.5 Investigation of influence of voltage pulses amplitude on the titanium** 

It is shown that increasing the thickness from 3.6 to 8.2 nm increases the resistance in the HRS state from 1.4 × 1011 to 8.8 × 1011 Ω, while the resistance in the LRS

To study the applied voltage pulses amplitude effect on the memristor effect, an titanium ONS with 2 × 2 μm lateral dimensions and 3.6 nm thickness was formed. Then, its current-voltage characteristic was measured in the voltage range from ±1

The analysis showed that when measuring the titanium ONS current-voltage characteristics with 3.6 nm thickness in the voltage range ± 1 V, the memristor effect is not observed, the structure shows the conductivity absence. When measuring the I-V characteristic in the ±2 V range, the structure also shows the conductivity absence, however, a small current surge in the I-V characteristic is already observed. When measuring the I-V characteristic in ±3 and ± 4 V range, this structure exhibits a memristor effect. It is shown that in the case of measuring the I-V characteristic in ±3 V range, the structure resistance in the HRS state is 39.2 × 109 Ω, and in the LRS

to 2.4 × 1011 Ω, while the structure resistance ratio in

ONS thickness measured at 1.5 V was obtained (**Figure 8**).

the high resistance to low resistance decreases from 79.4 to 3.6.

**oxide nanostructures resistive switching**

state increases from 1.765 × 109

to ±4 V (**Figure 9**).

**Figure 6.**

*(d) 8.2 nm.*

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

#### **Figure 6.**

*Memristors - Circuits and Applications of Memristor Devices*

is insignificant, the structure's resistance in the HRS state is 4.65 × 109 Ω, and in the LRS state −0.39 × 109 Ω, at the same time, resistance ratio in the high resistance to low resistance is 12. In the case of using a carbon-coated cantilever, an asymmetric I-V characteristic is also observed (**Figure 5c**), and there is no current through the ONS when applying voltages less than ±5 V in the case of a structure being in HRS and applying a voltage less than ±2 V in case of a structure being in LRS. The structure's resistance in the HRS state is 1.74 × 109 Ω, and in the LRS state, it is 0.3 × 109 Ω, and the resistance ratio in the high resistance to low resistance is 6.

*Titanium ONS current-voltage characteristics, obtained using a cantilever coated by: (a) Pt; (b) TiN; and* 

Study results showed that the platinum-coated cantilever use as the top electrode

is characterized by a largest resistance ratio in the HRS and LRS states.

**parameters on their resistive switching**

surface by applying ±2.4 voltage pulse (**Figure 6**).

manifest itself when the oxide thickness is 8.2 nm.

**2.4 Investigation of influence of titanium oxide nanostructures geometric** 

Another goal is to study titanium ONS thickness effect and applied voltage pulses on the memristor effect. For this, four ONSs were formed with lateral

2 × 2 μm dimensions and 1.6–3.6 nm height, which, based on the expression describing the oxide height and depth ratio presented in [3], corresponds to 3.6–8.2 nm thickness. The current-voltage characteristics were measured on these structures

Obtained expression analysis showed that with an increase in the ONS thickness, a decrease in the current corresponding to the LRS state and an increase in the resistance in this state are observed, to the extent that the memristor effect does not

The results allowed us to obtain voltage of the switching structure in the HRS state (Ures) and in the LRS state (Uset) dependence, as well as the corresponding

**106**

**Figure 5.**

*(c) carbon.*

*Titanium ONS current-voltage characteristics with a thickness: (a) 3.6 nm; (b) 5.4 nm; (c) 7.2 nm; and (d) 8.2 nm.*

currents (Ires and Iset) on the titanium ONS thickness (**Figure 7**). In addition, the dependence of the structure resistance measured in a HRS and LRS state on the ONS thickness measured at 1.5 V was obtained (**Figure 8**).

It is shown that increasing the thickness from 3.6 to 8.2 nm increases the resistance in the HRS state from 1.4 × 1011 to 8.8 × 1011 Ω, while the resistance in the LRS state increases from 1.765 × 109 to 2.4 × 1011 Ω, while the structure resistance ratio in the high resistance to low resistance decreases from 79.4 to 3.6.

#### **2.5 Investigation of influence of voltage pulses amplitude on the titanium oxide nanostructures resistive switching**

To study the applied voltage pulses amplitude effect on the memristor effect, an titanium ONS with 2 × 2 μm lateral dimensions and 3.6 nm thickness was formed. Then, its current-voltage characteristic was measured in the voltage range from ±1 to ±4 V (**Figure 9**).

The analysis showed that when measuring the titanium ONS current-voltage characteristics with 3.6 nm thickness in the voltage range ± 1 V, the memristor effect is not observed, the structure shows the conductivity absence. When measuring the I-V characteristic in the ±2 V range, the structure also shows the conductivity absence, however, a small current surge in the I-V characteristic is already observed. When measuring the I-V characteristic in ±3 and ± 4 V range, this structure exhibits a memristor effect. It is shown that in the case of measuring the I-V characteristic in ±3 V range, the structure resistance in the HRS state is 39.2 × 109 Ω, and in the LRS

state −1.4 × 109 Ω, the resistance ratio in the HRS and LRS states is 27.4. In the case of measuring the I-V characteristic in ±4 V range, the structure resistance the in the

#### **Figure 7.**

*Dependence of electrical parameters of titanium ONS on the oxide thickness: (a) voltage and (b) current.*

#### **Figure 8.**

*Dependence of titanium ONS resistance on the thickness.*

#### **Figure 9.**

*Current-voltage characteristics of titanium ONS in the different voltage range: (a) ±1 V; (b) ±2 V; (c) ±3 V; and (d) ±4 V.*

**109**

**Figure 10.**

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

HRS state is 1.08 × 109 Ω, and in the LRS state it is 0.14 × 109 Ω, the resistance ratio

Such a memristor effect dependence on the titanium ONS thickness and the applied voltage pulses amplitude is explained by the electric field intensity influence in the oxide on the oxygen vacancies transfer in the oxide volume between the electrodes and the titanium ONS switching between high-resistance and low-

**3. Formation and investigation of memristor structures based on nanocrystalline ZnO thin films by atomic force microscopy**

**3.1 Investigation of resistive switching of nanocrystalline ZnO thin films**

*ZnO film surface: (a) AFM-image; (b) AFM cross-sectional profile on (a); and (c) phase.*

A resistive switching effect in thin oxide films is attractive for manufacturing of neuromorphic system, which offers significant advantages over classical computers, such as an effective processing of data recognition. ZnO is the one of the promising materials, which is widely used in electronic element developments, sensors, and microsystem technology. Also, ZnO demonstrates resistive switching, which has just one phase and is compatible with semiconductor technology. To fabricate ZnO-based neuromorphic system, it is necessary to study resistive switching in ZnO films and today there are insufficient experimental results

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

in the HRS and LRS states is 7.5.

resistance states.

about it.

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

HRS state is 1.08 × 109 Ω, and in the LRS state it is 0.14 × 109 Ω, the resistance ratio in the HRS and LRS states is 7.5.

Such a memristor effect dependence on the titanium ONS thickness and the applied voltage pulses amplitude is explained by the electric field intensity influence in the oxide on the oxygen vacancies transfer in the oxide volume between the electrodes and the titanium ONS switching between high-resistance and lowresistance states.

#### **3. Formation and investigation of memristor structures based on nanocrystalline ZnO thin films by atomic force microscopy**

#### **3.1 Investigation of resistive switching of nanocrystalline ZnO thin films**

A resistive switching effect in thin oxide films is attractive for manufacturing of neuromorphic system, which offers significant advantages over classical computers, such as an effective processing of data recognition. ZnO is the one of the promising materials, which is widely used in electronic element developments, sensors, and microsystem technology. Also, ZnO demonstrates resistive switching, which has just one phase and is compatible with semiconductor technology. To fabricate ZnO-based neuromorphic system, it is necessary to study resistive switching in ZnO films and today there are insufficient experimental results about it.

**Figure 10.** *ZnO film surface: (a) AFM-image; (b) AFM cross-sectional profile on (a); and (c) phase.*

*Memristors - Circuits and Applications of Memristor Devices*

state −1.4 × 109 Ω, the resistance ratio in the HRS and LRS states is 27.4. In the case of measuring the I-V characteristic in ±4 V range, the structure resistance the in the

*Dependence of electrical parameters of titanium ONS on the oxide thickness: (a) voltage and (b) current.*

*Current-voltage characteristics of titanium ONS in the different voltage range: (a) ±1 V; (b) ±2 V; (c) ±3 V;* 

**108**

**Figure 9.**

**Figure 7.**

**Figure 8.**

*Dependence of titanium ONS resistance on the thickness.*

*and (d) ±4 V.*

To investigate resistive switching Al2O3/ZnO:In (42.1 ± 5.6 nm) as a wafer was used. ZnO thin films were grown using pulsed laser deposition under the following conditions: wafer temperature: 400°C, target-wafer distance: 50 mm, O2 pressure: 1 mTorr, pulse energy: 300 ml. To provide electrical contact to the bottom ZnO:In electrode, ZnO films were deposited through a mask.

Electrical properties of obtained ZnO films were measured by Ecopia HMS-3000 equipment (Ecopia Co., Republic of Korea). Obtained ZnO films had electron concentration 8.4 × 1019 cm<sup>−</sup><sup>3</sup> , electron mobility 12 cm2 /V∙s, and resistivity 5.2 × 10<sup>−</sup><sup>3</sup> Ω∙cm.

AFM-images of the ZnO film surface were obtained in semi-contact mode using scanning probe microscope Solver 47 Pro (NT-MDT, Russia). The AFMimage processing was performed using Image Analysis software. **Figure 10** shows experimental investigations of ZnO film morphology. It is shown that ZnO film surface has a granular structure (**Figure 10a** and **c**) with 1.53 ± 0.27 nm roughness (**Figure 10b**). The ZnO film thickness was measured by ZnO/ZnO:In stair scanning, and was equaled 32.3 ± 7.2 nm.

Electrical measurements were taken using nanolaboratory Ntegra with W2C probes. During the resistive switching investigation, ZnO:In film was grounded.

Current-voltage curves (CVC) were obtained from −3 to +3 V sweep for 15 cycles at the same point and for 15 cycles at different points on ZnO surface (**Figure 11a**). Based on the results obtained, resistance dependence on cycle number (uniformity test) and resistance dependence on number point were built (homogeneity test). It was shown RHRS and RLRS were equaled to 0.68 ± 0.07 GΩ and 0.11 ± 0.04 GΩ,

#### **Figure 11.**

*Investigation of resistive switching in ZnO film: (a) current-voltage characteristic; (b) uniformity; and (c) variability.*

**111**

**Figure 12.**

*profile on (a); and (c) time dependence of voltage.*

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

respectively at the same point on ZnO surface. At different points, RHRS and RLRS were equaled to 0.75 ± 0.13 GΩ and 0.12 ± 0.06 GΩ. RHRS/RLRS was equaled to 9.05 ± 5.65 at 0.7 V. Resistance dispersion during uniformity test was more than resistance dispersion during homogeneity test that can be explained by a granular structure of ZnO film. Time stability of resistive switching in ZnO was implemented using Ntegra in two stages. On the first stage, charge structure was formed on the ZnO surface at +5 V (**Figure 12a**). On the second stage, charged structure was scanned in Kelvin mode in the interval from 5 to 30 minutes with a step 5 minute (**Figure 12b** and **c**). It was shown that voltage decreased from 266 ± 17 to 68 ± 7 mV in 30 minutes

**3.2 Investigation of scratching probe nanolithography regimes for memristor** 

Resistive switching element manufacturing is associated with the development and research of the formation processes of structures with a nanometer resolution. Existing lithographic methods of semiconductor technologies are approaching the limit of their resolution, characterized by a high degree of complexity and cost of equipment. Therefore, there is a need for research and development of new methods, the use of which will allow the creation of nanostructures of resistive switching elements, including at the prototyping stage. A promising method for the formation of nanoscale structures that can be used to create resistive switching elements is nanoimprint lithography based on the use of special dies and films of polymeric materials. A type of nanoimprint lithography is scratching probe nanolithography

*Investigation of ZnO film surface charge: (a) Kelvin mode image of charged structure; (b) AFM cross-sectional* 

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

(**Figure 12a**).

**structure formation**

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

respectively at the same point on ZnO surface. At different points, RHRS and RLRS were equaled to 0.75 ± 0.13 GΩ and 0.12 ± 0.06 GΩ. RHRS/RLRS was equaled to 9.05 ± 5.65 at 0.7 V. Resistance dispersion during uniformity test was more than resistance dispersion during homogeneity test that can be explained by a granular structure of ZnO film.

Time stability of resistive switching in ZnO was implemented using Ntegra in two stages. On the first stage, charge structure was formed on the ZnO surface at +5 V (**Figure 12a**). On the second stage, charged structure was scanned in Kelvin mode in the interval from 5 to 30 minutes with a step 5 minute (**Figure 12b** and **c**). It was shown that voltage decreased from 266 ± 17 to 68 ± 7 mV in 30 minutes (**Figure 12a**).

#### **3.2 Investigation of scratching probe nanolithography regimes for memristor structure formation**

Resistive switching element manufacturing is associated with the development and research of the formation processes of structures with a nanometer resolution. Existing lithographic methods of semiconductor technologies are approaching the limit of their resolution, characterized by a high degree of complexity and cost of equipment. Therefore, there is a need for research and development of new methods, the use of which will allow the creation of nanostructures of resistive switching elements, including at the prototyping stage. A promising method for the formation of nanoscale structures that can be used to create resistive switching elements is nanoimprint lithography based on the use of special dies and films of polymeric materials. A type of nanoimprint lithography is scratching probe nanolithography

#### **Figure 12.**

*Memristors - Circuits and Applications of Memristor Devices*

electrode, ZnO films were deposited through a mask.

concentration 8.4 × 1019 cm<sup>−</sup><sup>3</sup>

and was equaled 32.3 ± 7.2 nm.

Ω∙cm.

To investigate resistive switching Al2O3/ZnO:In (42.1 ± 5.6 nm) as a wafer was used. ZnO thin films were grown using pulsed laser deposition under the following conditions: wafer temperature: 400°C, target-wafer distance: 50 mm, O2 pressure: 1 mTorr, pulse energy: 300 ml. To provide electrical contact to the bottom ZnO:In

Electrical properties of obtained ZnO films were measured by Ecopia HMS-3000 equipment (Ecopia Co., Republic of Korea). Obtained ZnO films had electron

, electron mobility 12 cm2

Electrical measurements were taken using nanolaboratory Ntegra with W2C probes.

Current-voltage curves (CVC) were obtained from −3 to +3 V sweep for 15 cycles at the same point and for 15 cycles at different points on ZnO surface (**Figure 11a**). Based on the results obtained, resistance dependence on cycle number (uniformity test) and resistance dependence on number point were built (homogeneity test). It was shown RHRS and RLRS were equaled to 0.68 ± 0.07 GΩ and 0.11 ± 0.04 GΩ,

*Investigation of resistive switching in ZnO film: (a) current-voltage characteristic; (b) uniformity; and (c)* 

During the resistive switching investigation, ZnO:In film was grounded.

AFM-images of the ZnO film surface were obtained in semi-contact mode using scanning probe microscope Solver 47 Pro (NT-MDT, Russia). The AFMimage processing was performed using Image Analysis software. **Figure 10** shows experimental investigations of ZnO film morphology. It is shown that ZnO film surface has a granular structure (**Figure 10a** and **c**) with 1.53 ± 0.27 nm roughness (**Figure 10b**). The ZnO film thickness was measured by ZnO/ZnO:In stair scanning,

/V∙s, and resistivity 5.2 × 10<sup>−</sup><sup>3</sup>

**110**

**Figure 11.**

*variability.*

*Investigation of ZnO film surface charge: (a) Kelvin mode image of charged structure; (b) AFM cross-sectional profile on (a); and (c) time dependence of voltage.*

**Figure 13.**

*Profiled nanostructures on photoresist surface: (a)AFM-image; and (b) AFM cross-sectional profile on (a).*

**Figure 14.**

*Investigation of scratching probe nanolithography regimes: (a) force of depth dependence; (b) velocity of depth dependence.*

(SPN), which allows using the tip of an atomic force microscope probe to form profiled nano-sized structures in polymer films. It was decided to use photoresist FP-383 as polymer film, because it is cheaper and has longer shelf life compared to other types of polymer films.

The solution of photoresist/thinner (FP-383/RPF383F) at volume ratio of 1:10 was transferred onto Si substrate using the centrifugal method at the rotation speed of a Laurell WS-400B-6NPP centrifuge at 5000 rpm. After the deposition of the film, the photoresist/thinner film was dried at a temperature of 90°С for 25 minutes. Thickness of the photoresist/thinner film was equaled to 32.1 ± 4.7 nm.

Scratching probe nanolithography on the photoresist/thinner film was performed using a Solver P47 Pro scanning probe microscope. Indentation was performed by applying an AFM probe to the surface of a FP-383 film with a fixed clamping force (the Set Point parameter in the AFM control program). Thus, arrays of the seven profiled lines-grooves were formed at different nanoindentation forces (**Figure 13**).

Analysis of the results obtained showed that nanoindentation force increase from 0.5 to 3.5 μN leads to nanostructure-groove depth increase from 2.7 ± 0.8 to 25.31 ± 2.11 nm (**Figure 14a**), tip velocity increase from 0.1 to 5 μm/s leads to nanostructure-groove depth decrease from 25.10 ± 1.2 to 8.87 ± 1.34 nm (**Figure 14b**).

**113**

**Figure 16.**

*sectional profile on (a); and (c) phase.*

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

The nonlinearity of the dependences obtained can be explained by the inhomogeneity of the viscoelastic properties of the FP-383 film. It should also be noted that the nature of the contact interaction between the probe and the film is complex and

*Profiled nanostructure on FP-383 film surface: (a) AFM-image; and (b) AFM cross-sectional profile on (a).*

*Resistive switching Al2O3/ZnO:In/ZnO/Ti/W2C memristor structure: (a) AFM-image; (b) AFM cross-*

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

is largely determined by the elastic forces.

**Figure 15.**

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

**Figure 15.** *Profiled nanostructure on FP-383 film surface: (a) AFM-image; and (b) AFM cross-sectional profile on (a).*

The nonlinearity of the dependences obtained can be explained by the inhomogeneity of the viscoelastic properties of the FP-383 film. It should also be noted that the nature of the contact interaction between the probe and the film is complex and is largely determined by the elastic forces.

**Figure 16.**

*Memristors - Circuits and Applications of Memristor Devices*

(SPN), which allows using the tip of an atomic force microscope probe to form profiled nano-sized structures in polymer films. It was decided to use photoresist FP-383 as polymer film, because it is cheaper and has longer shelf life compared to

*Investigation of scratching probe nanolithography regimes: (a) force of depth dependence; (b) velocity of depth* 

*Profiled nanostructures on photoresist surface: (a)AFM-image; and (b) AFM cross-sectional profile on (a).*

Scratching probe nanolithography on the photoresist/thinner film was performed using a Solver P47 Pro scanning probe microscope. Indentation was performed by applying an AFM probe to the surface of a FP-383 film with a fixed clamping force (the Set Point parameter in the AFM control program). Thus, arrays of the seven profiled lines-grooves were formed at different nanoindentation forces

Analysis of the results obtained showed that nanoindentation force increase from 0.5 to 3.5 μN leads to nanostructure-groove depth increase from 2.7 ± 0.8 to 25.31 ± 2.11 nm (**Figure 14a**), tip velocity increase from 0.1 to 5 μm/s leads to nanostructure-groove depth decrease from 25.10 ± 1.2 to 8.87 ± 1.34 nm

The solution of photoresist/thinner (FP-383/RPF383F) at volume ratio of 1:10 was transferred onto Si substrate using the centrifugal method at the rotation speed of a Laurell WS-400B-6NPP centrifuge at 5000 rpm. After the deposition of the film, the photoresist/thinner film was dried at a temperature of 90°С for 25 minutes. Thickness of the photoresist/thinner film was equaled to

**112**

other types of polymer films.

32.1 ± 4.7 nm.

**Figure 14.**

**Figure 13.**

*dependence.*

(**Figure 13**).

(**Figure 14b**).

*Resistive switching Al2O3/ZnO:In/ZnO/Ti/W2C memristor structure: (a) AFM-image; (b) AFM crosssectional profile on (a); and (c) phase.*

#### **3.3 Fabrication and investigation of resistive switching of memristor structures based on nanocrystalline ZnO thin films**

To fabricate memristor structure, photoresist FP-383 thin film with thickness 21.4 ± 3.1 nm was formed on Al2O3/ZnO:In/ZnO substrate. Then squared nanostructure-groove was formed on FP-383 film surface using scratching probe nanolithography at nanoindentation force 3.18 μN (**Figure 15**). Thin Ti film was deposited using BOC Edwards Auto 500 system. After that lift-off process was applied using dimethylformamide. AFM-image of the Al2O3/ZnO:In/ZnO/Ti resistive switching structure obtained is shown in **Figure 16**. Analysis of the result obtained showed that Ti film thickness was equaled to 4.1 ± 0.3 nm (**Figure 16b**). Ripped edges of Ti structures are result of lift-off process.

**Figure 17** shows current-voltage characteristic of Al2O3/ZnO:In/ZnO/Ti/W2C structure at −4 to +4 voltage sweep. It was shown that Al2O3/ZnO:In/ZnO/Ti/W2C structure has nonlinear, bipolar behavior when the electric potential gradient is the dominant parameter of resistive switching.

Investigation of resistive switching of Al2O3/ZnO:In/ZnO/Ti/W2C structure in the single point (uniformity test) shown that RHRS was 8.23 ± 1.93 GΩ and RLRS was 0.11 ± 0.06 GΩ (**Figure 17b**). At different points, RHRS and RLRS were equaled 7.65 ± 2.83 GΩ and 0.18 ± 0.11 GΩ, respectively (**Figure 17c**). It was shown, that RHRS/RLRS coefficient was equaled 135.31 ± 44.38 at 0.7 V.

In the end, it was shown that the use of Ti film allowed to increase RHRS/RLRS coefficient from 9.05 ± 5.65 to 135.31 ± 44.38 and to decrease the resistance dispersion of resistance switching (**Figures 11** and **17**). It can be explained by exception for the influence of air oxygen in Al2O3/ZnO:In/ZnO/Ti structure that significantly worsens the resistive switching.

#### **Figure 17.**

*Investigation of resistive switching in Al2O3/ZnO:In/ZnO/Ti/W2C structure: (a) current-voltage characteristic; (b) uniformity; and (c) homogeneity.*

**115**

**Figure 18.**

*(c) 6 V; and (d) 8 V.*

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

**4.1 Influence of voltage pulses amplitude on resistive switching of strained** 

. It should be

The dependence of the resistive switching of a vertically aligned carbon nanotube on the voltage pulse amplitude was studied using the STM spectroscopy using the probe nanolaboratory Ntegra. **Figure 18** shows the current-voltage characteristics of a strained carbon nanotube, obtained by applying a series of voltage sawtooth pulses with amplitude of 1–8 V, duration of 1 second, and tunnel gap of 1 nm. The diameter of a VA CNT of the investigated array was 95 ± 5 nm,

noted that the current-voltage characteristics are represented in the range from 0 to 50 nA, which relates to the peculiarities of the measuring system of the scanning

The measurement results showed that resistive switching of the VA CNT does not occur when the applied voltage amplitude is less than2 V (**Figure 18a**). This is due to the insufficient value of the external electric field for the formation of a low-resistance state in the nanotube [28]. The reproducible resistive switching was observed with a further increase in amplitude to 4 V and more (**Figure 18b–d**).

*CVCs of VA CNT upon application of a series of sawtooth voltage pulses with amplitude: (a) 2 V; (b) 4 V;* 

length 2.3 ± 0.2 μm, and density of nanotubes in the array was 18 μm<sup>−</sup><sup>2</sup>

**4. Investigation of resistive switching in vertically aligned carbon** 

**nanotubes using scanning tunnel microscopy**

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

**carbon nanotubes**

tunneling microscope.

#### **4. Investigation of resistive switching in vertically aligned carbon nanotubes using scanning tunnel microscopy**

#### **4.1 Influence of voltage pulses amplitude on resistive switching of strained carbon nanotubes**

The dependence of the resistive switching of a vertically aligned carbon nanotube on the voltage pulse amplitude was studied using the STM spectroscopy using the probe nanolaboratory Ntegra. **Figure 18** shows the current-voltage characteristics of a strained carbon nanotube, obtained by applying a series of voltage sawtooth pulses with amplitude of 1–8 V, duration of 1 second, and tunnel gap of 1 nm. The diameter of a VA CNT of the investigated array was 95 ± 5 nm, length 2.3 ± 0.2 μm, and density of nanotubes in the array was 18 μm<sup>−</sup><sup>2</sup> . It should be noted that the current-voltage characteristics are represented in the range from 0 to 50 nA, which relates to the peculiarities of the measuring system of the scanning tunneling microscope.

The measurement results showed that resistive switching of the VA CNT does not occur when the applied voltage amplitude is less than2 V (**Figure 18a**). This is due to the insufficient value of the external electric field for the formation of a low-resistance state in the nanotube [28]. The reproducible resistive switching was observed with a further increase in amplitude to 4 V and more (**Figure 18b–d**).

**Figure 18.** *CVCs of VA CNT upon application of a series of sawtooth voltage pulses with amplitude: (a) 2 V; (b) 4 V; (c) 6 V; and (d) 8 V.*

*Memristors - Circuits and Applications of Memristor Devices*

**based on nanocrystalline ZnO thin films**

structures are result of lift-off process.

dominant parameter of resistive switching.

worsens the resistive switching.

RHRS/RLRS coefficient was equaled 135.31 ± 44.38 at 0.7 V.

**3.3 Fabrication and investigation of resistive switching of memristor structures** 

To fabricate memristor structure, photoresist FP-383 thin film with thickness 21.4 ± 3.1 nm was formed on Al2O3/ZnO:In/ZnO substrate. Then squared nanostructure-groove was formed on FP-383 film surface using scratching probe nanolithography at nanoindentation force 3.18 μN (**Figure 15**). Thin Ti film was deposited using BOC Edwards Auto 500 system. After that lift-off process was applied using dimethylformamide. AFM-image of the Al2O3/ZnO:In/ZnO/Ti resistive switching structure obtained is shown in **Figure 16**. Analysis of the result obtained showed that Ti film thickness was equaled to 4.1 ± 0.3 nm (**Figure 16b**). Ripped edges of Ti

**Figure 17** shows current-voltage characteristic of Al2O3/ZnO:In/ZnO/Ti/W2C structure at −4 to +4 voltage sweep. It was shown that Al2O3/ZnO:In/ZnO/Ti/W2C structure has nonlinear, bipolar behavior when the electric potential gradient is the

Investigation of resistive switching of Al2O3/ZnO:In/ZnO/Ti/W2C structure in the single point (uniformity test) shown that RHRS was 8.23 ± 1.93 GΩ and RLRS was 0.11 ± 0.06 GΩ (**Figure 17b**). At different points, RHRS and RLRS were equaled 7.65 ± 2.83 GΩ and 0.18 ± 0.11 GΩ, respectively (**Figure 17c**). It was shown, that

In the end, it was shown that the use of Ti film allowed to increase RHRS/RLRS coefficient from 9.05 ± 5.65 to 135.31 ± 44.38 and to decrease the resistance dispersion of resistance switching (**Figures 11** and **17**). It can be explained by exception for the influence of air oxygen in Al2O3/ZnO:In/ZnO/Ti structure that significantly

*Investigation of resistive switching in Al2O3/ZnO:In/ZnO/Ti/W2C structure: (a) current-voltage characteristic;* 

**114**

**Figure 17.**

*(b) uniformity; and (c) homogeneity.*

The RHRS/RLRS ratio increased from 1 to 52 with an increase in the amplitude U from 1 to 8 V. This dependence is explained by the fact that the nanotube had the same resistance values of RHRS, while the resistance of its low resistance state decreased inversely to the increase in the applied voltage amplitude due to the compensation of the internal electric field arising during deformation of the VA CNT with an external electric field [28].

It should be noted that the values of RHRS and RLRS of the VA CNT are 2–3 times lower at U < 0 than at U > 0. It is due to the fact that associated with the occurrence of a piezoelectric charge, internal field of the nanotube [33, 34] is co-directed with an external electric field, and accordingly, reduces the resistance of the VA CNT when a negative voltage is applied and is oppositely directed and increases the total resistance of the VA CNT when a positive voltage is applied.

#### **4.2 Influence of deformation on resistive switching of strained carbon nanotubes**

The studies of the influence of deformation on resistive switching of a VA CNT were performed by the STM spectroscopy with a tunneling gap *d* = 0.2, 0.5, 1, and 2 nm. Controlled elastic deformation of VA CNT was formed on the basis of the previously developed technique [32] and was equal to the tunnel gap. The value *d* was determined on the basis of current-height characteristics and was controlled using the STM feedback system. **Figure 19** shows the experimental current-voltage characteristics obtained by applying voltage sawtooth pulses with amplitudes of 4 and 8 V.

Analysis of the obtained CVCs showed that at U = 4 V, the RLRS value initially decreased and increased again at ΔL = d = 2 nm (**Figure 19a**). This is due to the fact that the magnitude of the external electric field was not enough to compensate the internal electric field of the nanotube at a deformation of 2 nm. This effect disappeared as the voltage amplitude increased to U = 8 V due to an increase in the external electric field (**Figure 19b**). The values of RHRS and RLRS of the VA CNT decreased with increasing deformation (**Figure 19b**). The decrease in the of RHRS and RLRS of the VA CNT is due to the increase in the initial deformation ΔL ≈ d and the corresponding value of the piezoelectric charge, and is consistent with the mechanism of memristive switching of VA CNT [28].

It was also shown that the RHRS/RLRS ratio of the VA CNT does not depend on the deformation value and is determined by the value of the applied voltage: the RHRS/RLRS = 2–3 at U = 4 V (**Figure 19a**) and the RHRS/RLRS > 50 at U = 8 V (**Figure 19b**).

**Figure 19.**

*CVCs of VA CNT at various values of deformation ΔL ≈ d and at voltage sawtooth pulses amplitude: (a) 4 V; and (b) 8 V.*

**117**

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization*

Thus, application of scanning probe microscopy techniques for fabrication and determination of electrical parameters of memristor structures based on vertically aligned carbon nanotubes, titanium oxide nanostructures, and nanocrystalline ZnO thin films was presented. It is shown that titanium oxide nanostructures obtained by local anodic oxidation have a memristor effect without additional electroforming. The regularities of the manifestation of the memristor properties of oxide nanoscale structures of titanium are established, and the effect of the thickness of oxide nanoscale structures and the amplitude of applied voltage pulses on the displayed memristor effect in them is shown. It was found that the oxide nanoscale structures of titanium with a thickness of 1.6 nm have a resistance ratio in the high resistance to low resistance equal to 79.4. By using scratching probe nanolithography was made memristor structure based on nanocrystalline ZnO thin film obtained by pulsed laser deposition. The results can be used for micro- and nano-electronic elements manufacturing, as well as memristor structures, ReRAM elements using probe nanotechnologies, and for metal oxide and VA CNT-based neuromorphic system fabrication. The results of the study of resistive switching of vertical aligned carbon

nanotubes using scanning tunneling microscopy are presented.

Southern Federal University (project No. VnGr-07/2017-26).

The authors declare no conflict of interest.

The results were obtained using the equipment of Research and Education Center and the Center for collective use "Nanotechnologies" of Southern Federal

This work was financially supported by Russian Foundation for Basic Research (projects No. 16-29-14023 ofi\_m, 18-37-00299 mol\_a) and internal grant of the

Vladimir A. Smirnov\*, Marina V. Il'ina, Vadim I. Avilov, Roman V. Tominov,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Southern Federal University, Institute of Nanotechnologies, Electronics

and Electronic Equipment Engineering, Taganrog, Russia

\*Address all correspondence to: vasmirnov@sfedu.ru

provided the original work is properly cited.

*DOI: http://dx.doi.org/10.5772/intechopen.86555*

**5. Conclusion**

University.

**Acknowledgements**

**Conflict of interest**

**Author details**

Oleg I. Il'in and Oleg A. Ageev

*Application of Probe Nanotechnologies for Memristor Structures Formation and Characterization DOI: http://dx.doi.org/10.5772/intechopen.86555*

#### **5. Conclusion**

*Memristors - Circuits and Applications of Memristor Devices*

mechanism of memristive switching of VA CNT [28].

external electric field [28].

age is applied.

(**Figure 19b**).

The RHRS/RLRS ratio increased from 1 to 52 with an increase in the amplitude U from 1 to 8 V. This dependence is explained by the fact that the nanotube had the same resistance values of RHRS, while the resistance of its low resistance state decreased inversely to the increase in the applied voltage amplitude due to the compensation of the internal electric field arising during deformation of the VA CNT with an

It should be noted that the values of RHRS and RLRS of the VA CNT are 2–3 times lower at U < 0 than at U > 0. It is due to the fact that associated with the occurrence of a piezoelectric charge, internal field of the nanotube [33, 34] is co-directed with an external electric field, and accordingly, reduces the resistance of the VA CNT when a negative voltage is applied and is oppositely directed and increases the total resistance of the VA CNT when a positive volt-

**4.2 Influence of deformation on resistive switching of strained carbon nanotubes**

The studies of the influence of deformation on resistive switching of a VA CNT were performed by the STM spectroscopy with a tunneling gap *d* = 0.2, 0.5, 1, and 2 nm. Controlled elastic deformation of VA CNT was formed on the basis of the previously developed technique [32] and was equal to the tunnel gap. The value *d* was determined on the basis of current-height characteristics and was controlled using the STM feedback system. **Figure 19** shows the experimental current-voltage characteristics obtained by applying voltage sawtooth pulses with amplitudes of 4 and 8 V. Analysis of the obtained CVCs showed that at U = 4 V, the RLRS value initially decreased and increased again at ΔL = d = 2 nm (**Figure 19a**). This is due to the fact that the magnitude of the external electric field was not enough to compensate the internal electric field of the nanotube at a deformation of 2 nm. This effect disappeared as the voltage amplitude increased to U = 8 V due to an increase in the external electric field (**Figure 19b**). The values of RHRS and RLRS of the VA CNT decreased with increasing deformation (**Figure 19b**). The decrease in the of RHRS and RLRS of the VA CNT is due to the increase in the initial deformation ΔL ≈ d and the corresponding value of the piezoelectric charge, and is consistent with the

It was also shown that the RHRS/RLRS ratio of the VA CNT does not depend on the deformation value and is determined by the value of the applied voltage: the RHRS/RLRS = 2–3 at U = 4 V (**Figure 19a**) and the RHRS/RLRS > 50 at U = 8 V

*CVCs of VA CNT at various values of deformation ΔL ≈ d and at voltage sawtooth pulses amplitude: (a) 4 V;* 

**116**

**Figure 19.**

*and (b) 8 V.*

Thus, application of scanning probe microscopy techniques for fabrication and determination of electrical parameters of memristor structures based on vertically aligned carbon nanotubes, titanium oxide nanostructures, and nanocrystalline ZnO thin films was presented. It is shown that titanium oxide nanostructures obtained by local anodic oxidation have a memristor effect without additional electroforming. The regularities of the manifestation of the memristor properties of oxide nanoscale structures of titanium are established, and the effect of the thickness of oxide nanoscale structures and the amplitude of applied voltage pulses on the displayed memristor effect in them is shown. It was found that the oxide nanoscale structures of titanium with a thickness of 1.6 nm have a resistance ratio in the high resistance to low resistance equal to 79.4. By using scratching probe nanolithography was made memristor structure based on nanocrystalline ZnO thin film obtained by pulsed laser deposition. The results can be used for micro- and nano-electronic elements manufacturing, as well as memristor structures, ReRAM elements using probe nanotechnologies, and for metal oxide and VA CNT-based neuromorphic system fabrication. The results of the study of resistive switching of vertical aligned carbon nanotubes using scanning tunneling microscopy are presented.

The results were obtained using the equipment of Research and Education Center and the Center for collective use "Nanotechnologies" of Southern Federal University.

#### **Acknowledgements**

This work was financially supported by Russian Foundation for Basic Research (projects No. 16-29-14023 ofi\_m, 18-37-00299 mol\_a) and internal grant of the Southern Federal University (project No. VnGr-07/2017-26).

#### **Conflict of interest**

The authors declare no conflict of interest.

#### **Author details**

Vladimir A. Smirnov\*, Marina V. Il'ina, Vadim I. Avilov, Roman V. Tominov, Oleg I. Il'in and Oleg A. Ageev Southern Federal University, Institute of Nanotechnologies, Electronics and Electronic Equipment Engineering, Taganrog, Russia

\*Address all correspondence to: vasmirnov@sfedu.ru

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Series: Materials Science and

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S1995078018010111

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2014.06.002

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10.1063/1.3609065

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2018;**53**:8720-8746. DOI: 10.1007/

[3] Wang L, Zhang W, Chen Y, Cao Y, Li A, Wu D. Synaptic plasticity and learning behaviors mimicked in single inorganic synapses of Pt/HfOx/ZnOx/ TiN memristive system. Nanoscale Research Letters. 2017;**12**:65. DOI:

[4] Abbas H, Abbas Y, Truong S, Min K, Park M, Cho J, et al. A memristor crossbar array of titanium oxide for non-volatile memory and neuromorphic applications. Semiconductor Science and Technology. 2017;**32**:201732-065014.

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[8] Chang T, Chang K, Tsai T, Chu T, Sze S. Resistance random access memory.

Materials Today. 2016;**19**:254-264. DOI: 10.1016/j.mattod.2015.11.009

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*Memristors - Circuits and Applications of Memristor Devices*

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**120**

### *Edited by Alex James*

This Edited Volume *Memristors - Circuits and Applications of Memristor Devices* is a collection of reviewed and relevant research chapters, offering a comprehensive overview of recent developments in the field of Engineering. The book comprises single chapters authored by various researchers and edited by an expert active in the physical sciences, engineering, and technology research areas. All chapters are complete in itself but united under a common research study topic. This publication aims at providing a thorough overview of the latest research efforts by international authors on physical sciences, engineering, and technology,and open new possible research paths for further novel developments.

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Memristors - Circuits and Applications of Memristor Devices

Memristors

Circuits and Applications of Memristor Devices

*Edited by Alex James*