**Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory**

Vasily Kushpil *Nuclear Physics Institute of Academy Science of the Czech Republic Czech Republic* 

### **1. Introduction**

The feedback theory is a quite advantageous tool in the process of description of complex systems' behaviour or the procedure of complex processes. We can characterize the complexity of the system from its structural properties when there is a large amount of bounds between many elements of system. We can also characterize the complexity of the system as its functional complexity when the current state of system is defined as a result of many self-consistent states originating during the evolution of the system. It can be said in most general words that the feedback theory (FBT) describes the behaviour of a system or a process when the current status is defined as a result of achieving the self-consistency of the main system parameters. The feedback theory is most fruitfully applied to the description of those complex systems for which, due to their statistical character it is impossible to construct the full physical-mathematical model with a simple solution, but for which it is easy to select statistical parameters of the system and define their connections with the system properties that are of our interest. This approach allows applying FBT to describe the evolution of complex systems in chemistry, physics, biology, sociology and economics [1].

In this paper, we discuss the avalanche process in semiconductor avalanche photo detectors (APD) proceeding from three different points of view on this process. Then, we sum up the conclusions made in relation to the avalanche process in connection with avalanche photo detectors. We present APD as a converter of photo radiation to electric current with further current amplification. As a result, we can interpret the APD operation from various points that will allow us, finally, both to make several interesting conclusions on their practical application and analyze a possibility to choose optimal parameters in APD manufacturing.

To describe efficiently the processes in the frames of the feedback theory, it is necessary to fetch out the main physical parameters that determine the process procedure and establish quantitative ratios between the character of the process procedure and the change of these parameters. Further, we discuss the main characteristic parameter – the multiplication factor of the avalanche photo detectors M as an applied biasing voltage function (of the material parameters and topology) and attempt to describe the conditions for the achievement of selfconsistency with the choice of the feedback coefficient function. It should be mentioned that at present FBT is most widely used in the control theory [2], i.e. in the description of the

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 209

Ko>>1 the gain is defined mainly with the feedback coefficient K1/ that allows a desired

It should be stressed that the transfer function (1.1) in radio technology is given in frequency representation [3]. Speaking more accurately, it is regarded as a result of Laplace transformations on the complex plane. In this case the transfer function is equivalent to the

 H(s)=Y(s)/X(s)=L[y(t)]/L[x(t)] (2.2) The frequency representation for radio technical amplifying systems is more informative and suitable for their designing and study. We will discuss here only the time representation of the system coefficient of gain (amplifying) for avalanche detectors. The reason is the suitability and simplicity of this representation for the description of the avalanche photo detectors' operation and sufficiently simple procedure of measuring of the APD pulse characteristic. The time representation of the system gain is easily obtained if the APD pulse characteristic is known. In this stage, we do not consider the transfer function for APD, not to make the description and comprehension of the idea to describe the process of avalanche

Let us consider an idealized model of avalanche multiplication of electrons in a homogeneous semiconductor with the following suppositions: electron multiplication takes place in a limited region with the length La with constant electron field voltage Ea=const in this region. The probability of multiplication along the Lf electron free flight is P<1. See

If there are N0 electrons at the beginning of multiplication in the output of this region there will be N>>N0 electrons, as a result of multiplication. It can be expressed quantitatively in

N=N0\*(1+P+P2+P3+...+Pn) where n=La/Lf (3.1)

N=N0(1-P)-1 (3.2)

 Nx=N0\*(1+(P-ß)+(P-ß)2+(P-ß)3+...+(P-ß)n)=N0\*(1-(P-ß))-1 (3.3) In this expression ß<1 and it describes a probability of electron multiplication suppression

 M=(Nx/N0)=M0/(1+ß\*M0), where M0=(1-P)-1 (3.4) Therefore, we see that with this simple model the physical sense of the feedback coefficient is obvious. It describes a probability of electron multiplication suppression in its free flight.

coefficient of gain of the system be set by the choice of only one parameter .

frequency characteristic of the chain if active sources are absent.

multiplication in the FBT frames too complicated.

The multiplication factor is defined as M=N/N0

For n>>1 this expression can be written in the form

along its free flight. From expressions (3.1) and (3.3) we get

Formally, let us re-write (2.1) in the form

Figure 3.1.

the form (3.1).

**3. Probability model of the avalanche multiplication** 

behaviour of the systems for which it is necessary to achieve the desired character of behaviour. It can be usually attained by the choice of the corresponding system feedback function if the system transfer function without feedback is known. From this point of view, the application of FBT should allow us to approach the designing of avalanche photo detectors from the taken characteristics. We will obtain an opportunity to build the necessary APD transfer function and define the necessary feedback function with it. We have to learn to connect the topology and technology with the parameters of the established Feedback Function (FBF) and then, proceeding from the given FBF, we will be able to obtain topological and technological parameters. The practical application of the described physical-technological algorithm is not discussed in this paper. Our task is to show how it is possible to come from different APD models to their generalized description using FBT.

#### **2. Feedback theory basics**

Let us regard the main notions of the feedback theory in radio technique. Hereinafter, we will have to describe an avalanche photodiode as an amplifying device with feedback and present it in the form of the corresponding equivalent scheme.

In radio technique the behaviour of a system with feedback is described by the function of a special type – the system function (in a complex form, as the transfer function H(s)). Let us discuss a simple scheme of the signal amplifier shown in Fig.2.1 to whose input a signal is given that depends on time x(t), and the signal y(t) is received in the output.

Fig. 2.1. Feedback in amplifier.

The system coefficient of gain function is defined as a ration K(t)=y(t)/x(t). If a part if the signal from the system output is transmitted to the input it is supposed that the feedback is active in the system and the feedback coefficient is defined as (t)=((x'(t)-x(t))/y(t)), where x'(t) – is the signal in the system input in the action of the feedback in the system. At (t)>0 the feedback is positive; at (t)<0 the feedback is negative; at (t)=0 there is no feedback in the system. The system gain is written in the system with acting feedback in the form (2.1)

$$\mathbf{K} = \mathbf{K}\_0 / \left( 1 \text{-} \beta^\* \mathbf{K}\_0 \right) \tag{2.1}$$

where Ko is gain of the system without the feedback. Negative feedback (NFB) is most often used in electronic systems as in this case the system stability increases. Besides, providing

behaviour of the systems for which it is necessary to achieve the desired character of behaviour. It can be usually attained by the choice of the corresponding system feedback function if the system transfer function without feedback is known. From this point of view, the application of FBT should allow us to approach the designing of avalanche photo detectors from the taken characteristics. We will obtain an opportunity to build the necessary APD transfer function and define the necessary feedback function with it. We have to learn to connect the topology and technology with the parameters of the established Feedback Function (FBF) and then, proceeding from the given FBF, we will be able to obtain topological and technological parameters. The practical application of the described physical-technological algorithm is not discussed in this paper. Our task is to show how it is possible to come from different APD models to their generalized description using FBT.

Let us regard the main notions of the feedback theory in radio technique. Hereinafter, we will have to describe an avalanche photodiode as an amplifying device with feedback and

In radio technique the behaviour of a system with feedback is described by the function of a special type – the system function (in a complex form, as the transfer function H(s)). Let us discuss a simple scheme of the signal amplifier shown in Fig.2.1 to whose input a signal is

The system coefficient of gain function is defined as a ration K(t)=y(t)/x(t). If a part if the signal from the system output is transmitted to the input it is supposed that the feedback is active in the system and the feedback coefficient is defined as (t)=((x'(t)-x(t))/y(t)), where x'(t) – is the signal in the system input in the action of the feedback in the system. At (t)>0 the feedback is positive; at (t)<0 the feedback is negative; at (t)=0 there is no feedback in the system. The system gain is written in the system with acting feedback in the form (2.1)

 K=Ko/(1-\*Ko) (2.1) where Ko is gain of the system without the feedback. Negative feedback (NFB) is most often used in electronic systems as in this case the system stability increases. Besides, providing

present it in the form of the corresponding equivalent scheme.

given that depends on time x(t), and the signal y(t) is received in the output.

**2. Feedback theory basics** 

Fig. 2.1. Feedback in amplifier.

Ko>>1 the gain is defined mainly with the feedback coefficient K1/ that allows a desired coefficient of gain of the system be set by the choice of only one parameter .

It should be stressed that the transfer function (1.1) in radio technology is given in frequency representation [3]. Speaking more accurately, it is regarded as a result of Laplace transformations on the complex plane. In this case the transfer function is equivalent to the frequency characteristic of the chain if active sources are absent.

$$\mathbf{H}(\mathbf{s}) = \mathbf{Y}(\mathbf{s}) / \mathbf{X}(\mathbf{s}) = \mathbf{L} \left[ \mathbf{y}(\mathbf{t}) \right] / \mathbf{L} \left[ \mathbf{x}(\mathbf{t}) \right] \tag{2.2}$$

The frequency representation for radio technical amplifying systems is more informative and suitable for their designing and study. We will discuss here only the time representation of the system coefficient of gain (amplifying) for avalanche detectors. The reason is the suitability and simplicity of this representation for the description of the avalanche photo detectors' operation and sufficiently simple procedure of measuring of the APD pulse characteristic. The time representation of the system gain is easily obtained if the APD pulse characteristic is known. In this stage, we do not consider the transfer function for APD, not to make the description and comprehension of the idea to describe the process of avalanche multiplication in the FBT frames too complicated.

### **3. Probability model of the avalanche multiplication**

Let us consider an idealized model of avalanche multiplication of electrons in a homogeneous semiconductor with the following suppositions: electron multiplication takes place in a limited region with the length La with constant electron field voltage Ea=const in this region. The probability of multiplication along the Lf electron free flight is P<1. See Figure 3.1.

If there are N0 electrons at the beginning of multiplication in the output of this region there will be N>>N0 electrons, as a result of multiplication. It can be expressed quantitatively in the form (3.1).

The multiplication factor is defined as M=N/N0

$$\text{N=N\_0\*(1+P+P2+P3+...+Pn)}\text{ where }\text{n=L\_a/L}\tag{3.1}$$

For n>>1 this expression can be written in the form

$$\mathbf{N} = \mathbf{N}\_0 (\mathbf{1} \cdot \mathbf{P})^{\mathbf{1}} \tag{3.2}$$

Formally, let us re-write (2.1) in the form

$$\mathbf{N}\_{\mathbf{x}} = \mathbf{N}\_{0}\mathbf{\*}\mathbf{(1} + \mathbf{(P-\mathcal{S})} + \mathbf{(P-\mathcal{S})}\mathbf{2} + \mathbf{(P-\mathcal{S})}\mathbf{3} + \dots + \mathbf{(P-\mathcal{S})}\mathbf{n}\tag{3.3}$$

In this expression ß<1 and it describes a probability of electron multiplication suppression along its free flight. From expressions (3.1) and (3.3) we get

$$\mathbf{M} = (\mathbf{N}\_{\mathbf{x}}/\mathbf{N}\_{0}) \mathbf{=} \mathbf{M}\_{0}/(1 + \mathcal{S}^{\*}\mathbf{M}\_{0}), \text{ where } \qquad \mathbf{M}\_{0} = (1 \cdot \mathbf{P})^{\perp} \tag{3.4}$$

Therefore, we see that with this simple model the physical sense of the feedback coefficient is obvious. It describes a probability of electron multiplication suppression in its free flight.

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 211

(GAPD) [4], the probability interpretation of multiplication coefficient has quite a definite meaning. Namely, as GAPD operates in the region where the field intensity is critical and P~1, the avalanche suppressing should be defined with a certain internal process whose probability should be practically constant to provide for low noise of the detector. The cell structure of GAPD is the simplest and most natural way to develop them, as the cell dimensions and the intensity of the electric field in it define the maximal amplifying. The smaller is the dimension of the cell the better are noise properties of GAPD due to electron fluctuations decrease in it. Further, we will continue the discussion of this subject in the

Let us discuss a classical one-dimensional system of continuity equations for P-N transition

Fig. 4.1. Avalanche multiplications of electrons in high electrical field region of P-N junction.

*dJ dJn <sup>p</sup> J J nn pp dx dx* 

The solution of this system will be interesting for us in the form of multiplication factor

1 1 exp '

 

*n np*

*dx dx*

  (4.1)

(4.2)

phase of more accurate mathematical calculations.

shown in Fig. 4.1 [5]

at boundary conditions

**4. Physical model of the avalanche multiplication** 

Jn(0)=Jno Jp(0)=J-Jpo

J(x)=Jn(x)+Jp(x)=const

Mn=Jn/Jno for the case of purely electronic injection.

*M n*

Jn(Z)=J-Jpo Jp(Wdepl)=Jpo

It is to imagine a situation when to account for the multiplication probability of not only electrons but also holes, we upgrade the expression (3.3) so as P=Pe+Ph and ß= ße+ßh , while Pe, Ph are multiplicity probabilities for electrons and holes in their free flight length. In this case Nx=Ne+Nh.

Fig. 3.1. Illustration of Probability model for avalanche process.

This simple model presupposes independence of multiplication processes of electrons and holes. It is not easy to implement it in practice.

$$\mathbf{N\_{x}=N\_{e}+N\_{h}=N\_{0e}\*(1\text{-}(P\_{e}\text{-}\theta\_{e}))^{\cdot 1}+N\_{0h}\*(1\text{-}(P\_{h}\text{-}\theta\_{h}))^{\cdot 1}\tag{3.5}$$

We would like to note that it is the first illustration of moving from a certain definite (in this case, simple probability) model for avalanche photo-detectors to further interpretation of this model in the frames of FBT.

We shall discuss in more detail some not very obvious statements of the suggested model. First, we suppose that all electron multiplication events are independent as a whole. The second supposition comes from the sum row notion in the form of the algebraic expression (3.2). This expression has uncertainty at P=1. It leads to a natural restriction P<1 and shows that the multiplication probability can be as close to the entity but cannot be equal to it. Both statements are not significant for the main conclusion – the result of the mathematical description of the avalanche generation process can be presented in the form of a formula for the coefficient of gain of the amplifier with feedback after some identical transformations, and, consequently, we can apply it for the analysis of the operation of FBT avalanche photo-detectors. To illustrate some advantages of the interpretation, we consider (3.4) in the case M0>>1. Then M~1/ß and, consequently, the multiplication factor is definitely determined by the feedback coefficient ß. In other words, the avalanche suppression processes in APD define the multiplication factor. As such requirements correspond to the operation principle by convention Geiger Mode Avalanche Photo Diode (GAPD) [4], the probability interpretation of multiplication coefficient has quite a definite meaning. Namely, as GAPD operates in the region where the field intensity is critical and P~1, the avalanche suppressing should be defined with a certain internal process whose probability should be practically constant to provide for low noise of the detector. The cell structure of GAPD is the simplest and most natural way to develop them, as the cell dimensions and the intensity of the electric field in it define the maximal amplifying. The smaller is the dimension of the cell the better are noise properties of GAPD due to electron fluctuations decrease in it. Further, we will continue the discussion of this subject in the phase of more accurate mathematical calculations.

### **4. Physical model of the avalanche multiplication**

Let us discuss a classical one-dimensional system of continuity equations for P-N transition shown in Fig. 4.1 [5]

Fig. 4.1. Avalanche multiplications of electrons in high electrical field region of P-N junction.

$$\frac{d\mathbf{J}\_m}{d\mathbf{x}} = \frac{d\mathbf{J}\_p}{d\mathbf{x}} = \alpha\_n \cdot \mathbf{J}\_n + \alpha\_p \cdot \mathbf{J}\_p \tag{4.1}$$

at boundary conditions

210 Photodetectors

It is to imagine a situation when to account for the multiplication probability of not only electrons but also holes, we upgrade the expression (3.3) so as P=Pe+Ph and ß= ße+ßh , while Pe, Ph are multiplicity probabilities for electrons and holes in their free flight length. In this

This simple model presupposes independence of multiplication processes of electrons and

 Nx=Ne+Nh.=N0e\*(1-(Pe-ße))-1 + N0h\*(1-(Ph-ßh))-1 (3.5) We would like to note that it is the first illustration of moving from a certain definite (in this case, simple probability) model for avalanche photo-detectors to further interpretation of

We shall discuss in more detail some not very obvious statements of the suggested model. First, we suppose that all electron multiplication events are independent as a whole. The second supposition comes from the sum row notion in the form of the algebraic expression (3.2). This expression has uncertainty at P=1. It leads to a natural restriction P<1 and shows that the multiplication probability can be as close to the entity but cannot be equal to it. Both statements are not significant for the main conclusion – the result of the mathematical description of the avalanche generation process can be presented in the form of a formula for the coefficient of gain of the amplifier with feedback after some identical transformations, and, consequently, we can apply it for the analysis of the operation of FBT avalanche photo-detectors. To illustrate some advantages of the interpretation, we consider (3.4) in the case M0>>1. Then M~1/ß and, consequently, the multiplication factor is definitely determined by the feedback coefficient ß. In other words, the avalanche suppression processes in APD define the multiplication factor. As such requirements correspond to the operation principle by convention Geiger Mode Avalanche Photo Diode

Fig. 3.1. Illustration of Probability model for avalanche process.

holes. It is not easy to implement it in practice.

this model in the frames of FBT.

case Nx=Ne+Nh.

$$\mathbf{J}^{\mathrm{u}}(\mathbf{Z}) = \mathbf{J}^{\mathrm{L}}\mathbf{J}^{\mathrm{loc}} \qquad \qquad \mathbf{J}^{\mathrm{b}}(\mathbf{W}^{\mathrm{depl}}) = \mathbf{J}^{\mathrm{b}\alpha}$$

$$\mathbf{J}^{\mathrm{u}}(\mathbf{0}) = \mathbf{J}^{\mathrm{u}\alpha} \qquad \qquad \mathbf{J}^{\mathrm{b}}(\mathbf{0}) = \mathbf{J}^{\mathrm{d}\alpha}$$

The solution of this system will be interesting for us in the form of multiplication factor Mn=Jn/Jno for the case of purely electronic injection.

$$M\_m = \frac{1}{1 - \left[ \left( a\_m \cdot \exp \left( -\mathfrak{f} \left( a\_m - a\_p \right) \text{dx} \right) \right) \text{dx} \right]} \tag{4.2}$$

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 213

Let us consider the character of possible behaviour of the feedback coefficient in the region of avalanche multiplication. We differentiate (4.6) in coordinate. As a result we get three

Case A(x)>B(x) dβn/dx – the rate of hole multiplication in the direction of differentiation

Case A(x)=B(x) dβn/dx – the rate of hole multiplication in the direction of differentiation is

Case A(x)<B(x) dβn/dx – the rate of hole multiplication in the direction of differentiation

Thus, having done a formally identical transformation of the initial equation, we get an opportunity to mark three characteristic cases for positive FB. The method discussed above will be further applied and allow us to make a clear and generalizing classification of

There two reasons that show that there is no need to analyze deeper the obtained expressions. Firstly, our main aim is an acquaintance with a possibility itself to interpret an avalanche process on various levels of abstraction and for various models. Secondly, further studies will have sense if we use a specific APD topology. There exists a simpler model of FBT for avalanche detectors to make general description of some common properties of APD. This model allows one to use a more illustrative and simpler approach, from the point

It is well-known in APD applications that in practice the use of the physical model to describe a specific APD will be too complex and far from informative. That is why various empiric or semi-empiric models are used. Miller suggested the first model of this type in 1955 [6]. It turns out that the new approach based on the identical transformation of the

In the first variant of formula (5.1) the influence on avalanche amplification was not considered of the APD internal base and the contact resistance (the equivalent APD scheme

> 1 *no n*

Later, the formula was modified and presented in the form (5.2) for equivalent APD scheme

1

*M*

1

*br*

*V V*

(5.1)

(5.2)

1

*V iR V*

*br*

*fb*

*n n*

main formula of this model allows one to obtain sufficiently interesting results.

*M*

variants of behaviour.

gradually decreases.

gradually increases.

various APD types.

of practical application of APD.

**5. Miller's formula** 

is in Fig. 5.1а).

shown in Fig. 5.1в.

constant.

if Mno is written in the form

$$M\_{no} = \frac{1}{1 - \mathbb{I}\left(\alpha\_n \cdot \exp\left(-\mathbb{I}\left(\alpha\_n\right) \text{dx}\right)\right)}\tag{4.3}$$

It is possible to represent (4.2) in the following form

$$\boldsymbol{M}\_{\boldsymbol{m}} = \frac{\boldsymbol{M}\_{\boldsymbol{m}o}}{1 + \boldsymbol{\beta}\_{\boldsymbol{m}} \cdot \boldsymbol{M}\_{\boldsymbol{m}o}} \tag{4.4}$$

In this expression the feedback coefficient is in the form

$$\mathcal{B}\_m = \left\{ \boldsymbol{\alpha}\_n \cdot \boldsymbol{e} \, \right. \left. \left. \left( 1 - \boldsymbol{e} \right)^{-\beta(x)} \boldsymbol{d} \boldsymbol{x} \right) \tag{4.5}$$

where

$$A(\mathbf{x}) = \int a\_n d\mathbf{x} \quad \text{and} \quad B(\mathbf{x}) = \int a\_p d\mathbf{x}$$

We can easily obtain the expression for the avalanche multiplication factor in the case of purely "holes" injection, simply changing n to p in formula (4.4) and vice versa. This formal transformation, as it may seem, of a well-known expression makes it possible for us to regard electron and hole multiplication from the FBT point of view.

Let us study some limit cases for formula (4.5).

Firstly, the trivial case ( ) 1 *B x e* means that there is no feedback or ( ) 0 *B x dx <sup>p</sup>* - there is no multiplication of the hole component. On the other hand, the presence of the holes multiplication process is a factor that means the presence of positive feedback. This fact easily shows the case ( ) 1 *B x e* when we get from formula (4.5)

$$\mathcal{J}\_{\mathfrak{n}} = -[\![\alpha\_{\mathfrak{n}} \cdot \!e^{-}] \!(A(\mathfrak{x}) - B(\mathfrak{x})) ]\_{d\mathfrak{x}} \tag{4.6}$$

In the obtained expression, the minus mark indicates the feedback type – namely positive feedback, and the ratio of two components of the exponent shows the FB depth.

Concerning the restrictions on the transfer from representation (4.2) to representation (4.4), they are connected with the necessity to follow the implementation of requirement

$$[(a\_p \exp(-\lceil a\_n \rceil) \text{dx})\text{x}\text{'}\text{1}\tag{4.7}$$

For FB coefficient the following condition should be implemented

$$
\beta\_n \cdot M\_n > 1 \qquad \text{or}$$

$$
0 \le \|\alpha\_\eta \cdot e^{-(A(\chi) - B(\chi))}\|\_{d\chi} < \frac{1}{M\_{n\eta}}.
$$

Let us consider the character of possible behaviour of the feedback coefficient in the region of avalanche multiplication. We differentiate (4.6) in coordinate. As a result we get three variants of behaviour.

Case A(x)>B(x) dβn/dx – the rate of hole multiplication in the direction of differentiation gradually decreases.

Case A(x)=B(x) dβn/dx – the rate of hole multiplication in the direction of differentiation is constant.

Case A(x)<B(x) dβn/dx – the rate of hole multiplication in the direction of differentiation gradually increases.

Thus, having done a formally identical transformation of the initial equation, we get an opportunity to mark three characteristic cases for positive FB. The method discussed above will be further applied and allow us to make a clear and generalizing classification of various APD types.

There two reasons that show that there is no need to analyze deeper the obtained expressions. Firstly, our main aim is an acquaintance with a possibility itself to interpret an avalanche process on various levels of abstraction and for various models. Secondly, further studies will have sense if we use a specific APD topology. There exists a simpler model of FBT for avalanche detectors to make general description of some common properties of APD. This model allows one to use a more illustrative and simpler approach, from the point of practical application of APD.

#### **5. Miller's formula**

212 Photodetectors

1

*<sup>n</sup> e e*

It is possible to represent (4.2) in the following form

In this expression the feedback coefficient is in the form

() *A x dx*

Let us study some limit cases for formula (4.5).

Firstly, the trivial case ( ) 1 *B x*

easily shows the case ( ) 1 *B x*

regard electron and hole multiplication from the FBT point of view.

 1 1 exp ' *Mno dx dx n n*

> *n no*

 ( ) ( ) <sup>1</sup> *A x B x <sup>n</sup> dx*

*<sup>n</sup>* and ( ) *B x dx*

*e* means that there is no feedback or ( ) 0 *B x dx*

( ( ) ( )) *Ax Bx e dx*

We can easily obtain the expression for the avalanche multiplication factor in the case of purely "holes" injection, simply changing n to p in formula (4.4) and vice versa. This formal transformation, as it may seem, of a well-known expression makes it possible for us to

no multiplication of the hole component. On the other hand, the presence of the holes multiplication process is a factor that means the presence of positive feedback. This fact

In the obtained expression, the minus mark indicates the feedback type – namely positive

Concerning the restrictions on the transfer from representation (4.2) to representation (4.4),

1

*n n M* or

<sup>1</sup> ( ( ) ( )) <sup>0</sup> *Ax Bx <sup>e</sup> dx n M*

*e* when we get from formula (4.5)

*n n*

 

feedback, and the ratio of two components of the exponent shows the FB depth.

they are connected with the necessity to follow the implementation of requirement

*p n* 

( exp( ) ) ' 1 *dx dx*

For FB coefficient the following condition should be implemented

(4.4)

*<sup>p</sup>* - there is

(4.5)

(4.6)

*no*

(4.7)

*<sup>p</sup>*

*M M no n M*

  (4.3)

if Mno is written in the form

where

It is well-known in APD applications that in practice the use of the physical model to describe a specific APD will be too complex and far from informative. That is why various empiric or semi-empiric models are used. Miller suggested the first model of this type in 1955 [6]. It turns out that the new approach based on the identical transformation of the main formula of this model allows one to obtain sufficiently interesting results.

In the first variant of formula (5.1) the influence on avalanche amplification was not considered of the APD internal base and the contact resistance (the equivalent APD scheme is in Fig. 5.1а).

$$M\_{no} = \frac{1}{1 - \left(\frac{V}{V\_{br}}\right)^n} \tag{5.1}$$

Later, the formula was modified and presented in the form (5.2) for equivalent APD scheme shown in Fig. 5.1в.

$$M\_n = \frac{1}{1 - \left(\frac{V - i \cdot R\_{fb}}{V\_{br}}\right)^n} \tag{5.2}$$

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 215

In the first case external chains of APD connection define the rate of the avalanche multiplication suppression; in the second case, the influence of the reactive component on the rate of the avalanche multiplication suppression is strong. Due to this, the suggested model allows one to describe accurately the operation of APD with internal local negative FB. The FB locality implies that the process of the avalanche suppressing is much quicker than the changes in the parameters of external chains of APD power supply. Thus, we can choose parameters Cfb and Rfb in such a way that we can accurately describe the APD behaviour with negative local FB. A step further is more complicated as it is necessary to connect the obtained distributed parameters with topological and technological parameters

Fig. 5.2. Metal - Resistive Layer - Silicon (MRS APD) cell. The diameter of current filament

In the discussed example (Figure 5.2) we can proceed from the fact that we have already known the parameters of the resistive layer (resistivity) ρSiC and area N+ region SN. The value Cfb is more significant for the description of the dynamic mode, while for the static

To describe the dynamic mode, the value Cfb is more significant, while for the static mode

cross section area of the avalanche in stationary mode (avalanche current is constant).

*fb SiC*

*<sup>d</sup> <sup>R</sup> <sup>S</sup>* 

*fb SiC <sup>d</sup> <sup>R</sup> <sup>S</sup>* 

*o*

1 *SiC* where S0 determines the

where S1 determines the

must be change on boundary n-SiC – p-Si. (SiC/p-Si)~10

Rfb is more significant. We can expect here that

one Rfb is more important, so we can expect that *SiC*

here *fb fb*

of the APD under study.

*br br V iR V V*

Fig. 5.1. The evolution of equivalent circuit in Miller's formula.

It turns out that the Miller formula can be transformed to the form of expression (2.0). Let us do transformation for formula (5.3) (the equivalent APD scheme is suggested in [7] and shown in Fig. 5.1с).

$$M\_n = \frac{1}{1 - \left(\frac{V - V\_{fl}}{V\_{br}}\right)^n} \tag{5.3}$$

In the case under consideration the transformation has sense as the model exactly describes the APD operation with an internal FB. In (5.4) we show the feedback coefficient in the general form to indicate that the feedback value is defined by the change of the voltage Vfb. On the other hand, in this expression a significant FB difference is clearly seen if the FB resistance consists of purely active component and if it contains the reactive component [8].

$$M(V) = \frac{M\_o(V)}{1 - \beta \cdot M\_o(V)}\tag{5.4}$$

Fig. 5.1. The evolution of equivalent circuit in Miller's formula.

( ) ( ) 1 ()

It turns out that the Miller formula can be transformed to the form of expression (2.0). Let us do transformation for formula (5.3) (the equivalent APD scheme is suggested in [7] and

1

*M*

1

*V V V*

 

> *o o*

*M V M V* 

*fb br*

(5.3)

*<sup>M</sup> <sup>V</sup>* (5.4)

*n n*

In the case under consideration the transformation has sense as the model exactly describes the APD operation with an internal FB. In (5.4) we show the feedback coefficient in the general form to indicate that the feedback value is defined by the change of the voltage Vfb. On the other hand, in this expression a significant FB difference is clearly seen if the FB resistance consists of purely active component and if it contains the reactive

shown in Fig. 5.1с).

component [8].

In the first case external chains of APD connection define the rate of the avalanche multiplication suppression; in the second case, the influence of the reactive component on the rate of the avalanche multiplication suppression is strong. Due to this, the suggested model allows one to describe accurately the operation of APD with internal local negative FB. The FB locality implies that the process of the avalanche suppressing is much quicker than the changes in the parameters of external chains of APD power supply. Thus, we can choose parameters Cfb and Rfb in such a way that we can accurately describe the APD behaviour with negative local FB. A step further is more complicated as it is necessary to connect the obtained distributed parameters with topological and technological parameters of the APD under study.

Fig. 5.2. Metal - Resistive Layer - Silicon (MRS APD) cell. The diameter of current filament must be change on boundary n-SiC – p-Si. (SiC/p-Si)~10

In the discussed example (Figure 5.2) we can proceed from the fact that we have already known the parameters of the resistive layer (resistivity) ρSiC and area N+ region SN. The value Cfb is more significant for the description of the dynamic mode, while for the static one Rfb is more important, so we can expect that *SiC fb SiC o <sup>d</sup> <sup>R</sup> <sup>S</sup>* where S0 determines the cross section area of the avalanche in stationary mode (avalanche current is constant).

To describe the dynamic mode, the value Cfb is more significant, while for the static mode Rfb is more significant. We can expect here that 1 *SiC fb SiC <sup>d</sup> <sup>R</sup> <sup>S</sup>* where S1 determines the

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 217

*V VV V*

We study the character of behaviour of the FB coefficient depending on the voltage changes V.

the derivative on V from (4.4) will be in the form (5.9)

*dV*

*dM Gef*

If we define the avalanche gain efficiency coefficient in the form

characteristic cases of FB demonstration in APD in Fig. 5.3.

Fig. 5.3. Classification of APD using model of FB.

that corresponds to the APD operation in the unstable mode;

0 2 *ef* 0 *dM <sup>d</sup> G M dV dV*

( ) *n n br n br*

<sup>2</sup> (1 )

0

(5.10)

0

Kef < 1 – with the growth of voltage the feedback increases slower than the avalanche gain

*M dV*

*d*

1 *ef*

*dV <sup>K</sup> dM*

If we define the FB efficiency parameter in the form (5.10) it is possible to outline three

(5.8)

*<sup>M</sup>* (5.9)

dimension of the spreading area of the filament of the current on the boundary NSiC-PSi, proceeding from the term of achieving the current stationary when M(t)=const, and

$$\mathbf{C}\_{\mathcal{f}^{\text{b}}} = \frac{\boldsymbol{\mathcal{E}}\_{o} \cdot \boldsymbol{\mathcal{E}}\_{\text{Sic}} \cdot \mathbf{S}\_{1}}{d\_{\text{Sic}}}$$

where S1 defines the maximum dimension of the avalanche current from which the avalanche suppression starts.

We can define a characteristic constant τfb (5.5) that we will call the time of feedback of the process of avalanche multiplication suppression. This value is determined by the APD technology and topology. It should be noted that the constant does not depend on the resistive layer thickness, but does depend on the spreading area cross-section and the thickness of the avalanche current area cross-section.

$$
\sigma\_{fb} = \varepsilon\_o \cdot \varepsilon\_{SiC} \cdot \rho\_{SiC} \cdot \frac{S\_1}{S\_0} = \tau\_{SiC} \cdot \frac{S\_1}{S\_0} \tag{5.5}
$$

It seems obvious that for local negative FB it is necessary to implement the condition τfb ≥ τav , namely, the time of FB suppression must be a little longer than the time of the avalanche generation τav . If we consider the model of the abrupt P-N junction and presuppose that the avalanche region Wav ~0.1 of the thickness of the depletion region Wdepl and the velocity of the electron movement in the multiplication region Vsat we get an evaluation in the form (5.6).

$$
\varepsilon\_o \cdot \varepsilon\_{\text{SiC}} \cdot \rho\_{\text{SiC}} \cdot \frac{S\_1}{S\_0} \ge \frac{W\_{av}}{V\_{sat}} \tag{5.6}
$$

As Vbr is depend on the silicon parameters [9], and Vsat=const, we get the evaluation (for the n-SiC/p-Si structure) of the minimal concentration of acceptors in p-Si in the form (5.7).

$$\mathcal{W}\_{av} \le V\_{sat} \cdot \tau\_{SiC} \frac{S\_1^2}{S\_0^2} \tag{5.7}$$

*SiC o SiC SiC* — Maxwell time for n-SiC.

This implies (for simplicity we suppose that S1=S0) that at ρSiC=156 [ohm/cm] the acceptors concentration in silicon must be lower Np < 2\*1015 1/cm3. But at the same time the concentration cannot be too low. Otherwise FB will not have time to suppress locally the avalanche production process. If we take into account that an agreement of this kind is necessary for each new value ρSiC and, besides, that in the general case S1≠S0 depends on the impurity concentration and conditions on the border n-SiC/p-Si, it becomes clear why the suggested FB mechanism was implemented by only one research group [10].

The presented derivations are one more example of FBT application for the description of APD operation. In the Chapter 6 that considers examples of application of the models based on FBT ideas we discuss the model of the n-SiC/p-Si structure in more detail.

Let us regard the general analysis of formula (5.4)[11]. We mark that in the first approximation we thought that n=1. If we take the general case n>1, the feedback coefficient must be presented in the form (5.8)

dimension of the spreading area of the filament of the current on the boundary NSiC-PSi, proceeding from the term of achieving the current stationary when M(t)=const, and

where S1 defines the maximum dimension of the avalanche current from which the

We can define a characteristic constant τfb (5.5) that we will call the time of feedback of the process of avalanche multiplication suppression. This value is determined by the APD technology and topology. It should be noted that the constant does not depend on the resistive layer thickness, but does depend on the spreading area cross-section and the

*fb o SiC SiC SiC*

It seems obvious that for local negative FB it is necessary to implement the condition τfb ≥ τav , namely, the time of FB suppression must be a little longer than the time of the avalanche generation τav . If we consider the model of the abrupt P-N junction and presuppose that the avalanche region Wav ~0.1 of the thickness of the depletion region Wdepl and the velocity of the electron movement in the multiplication region Vsat we get an evaluation in the form (5.6).

> 1 0 *av o SiC SiC*

*S W S V*

 

 

As Vbr is depend on the silicon parameters [9], and Vsat=const, we get the evaluation (for the n-SiC/p-Si structure) of the minimal concentration of acceptors in p-Si in the form (5.7).

> *av sat SiC <sup>S</sup> W V*

This implies (for simplicity we suppose that S1=S0) that at ρSiC=156 [ohm/cm] the acceptors concentration in silicon must be lower Np < 2\*1015 1/cm3. But at the same time the concentration cannot be too low. Otherwise FB will not have time to suppress locally the avalanche production process. If we take into account that an agreement of this kind is necessary for each new value ρSiC and, besides, that in the general case S1≠S0 depends on the impurity concentration and conditions on the border n-SiC/p-Si, it becomes clear why the

The presented derivations are one more example of FBT application for the description of APD operation. In the Chapter 6 that considers examples of application of the models based

Let us regard the general analysis of formula (5.4)[11]. We mark that in the first approximation we thought that n=1. If we take the general case n>1, the feedback coefficient

  1 1 0 0

*sat*

*S*

(5.5)

(5.6)

(5.7)

*S S S S*

 

*o SiC* 1

*<sup>S</sup> <sup>C</sup> d* 

*SiC*

*SiC o SiC SiC*

 

must be presented in the form (5.8)

 

avalanche suppression starts.

thickness of the avalanche current area cross-section.

 

— Maxwell time for n-SiC.

suggested FB mechanism was implemented by only one research group [10].

on FBT ideas we discuss the model of the n-SiC/p-Si structure in more detail.

*fb*

$$\beta = \frac{V'' - (V - V\_{br})''}{V\_{br}''} \tag{5.8}$$

We study the character of behaviour of the FB coefficient depending on the voltage changes V. If we define the avalanche gain efficiency coefficient in the form

$$G\_{\varepsilon f} = \frac{dM\_0}{dV} - M\_0^2 \cdot \frac{d\beta}{dV} \quad \text{the derivative on V from (4.4) will be in the form (5.9)}$$

$$\frac{dM}{dV} = \frac{G\_{ef}}{(1 - \beta \cdot M)^2} \tag{5.9}$$

If we define the FB efficiency parameter in the form (5.10) it is possible to outline three characteristic cases of FB demonstration in APD in Fig. 5.3.

Fig. 5.3. Classification of APD using model of FB.

$$K\_{ef} = \frac{\frac{d\beta}{dV}}{\frac{1}{M\_0} \cdot \frac{dM\_0}{dV}}\tag{5.10}$$

Kef < 1 – with the growth of voltage the feedback increases slower than the avalanche gain that corresponds to the APD operation in the unstable mode;

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 219

second is the dimension and structure of the GAPD cell. Let us discuss a GAPD cell as it is

The electric field in the cell is close or more to the critical one; therefore, the probability of the avalanche generation of a secondary carrier P~1. Moreover, there exists a certain threshold value of photo-generated electrons No necessary for the start of the avalanche process. Thus, to suppress the avalanche in cell from thermal noises for the thermogenerated carriers NT the NT < No inequation should be implemented. We can evaluate NT supposing that the noise generates only by the detector dark current. This supposition will allow us evaluate theoretically the dependence of the size of cell on the concentration of the doping applied for the APD material. It is obvious that the higher is the doping concentration the lower is the maximal cell volume ∆Ū at the same cell area. At backward bias voltage Vbias, the dark current Jd and equals to NT will determine the concentration of minor carriers generated per time unit in the depletion region. For APD with a long base, presupposing that the dark current is defined by the volume component from [12] it follows

Let us consider to be definite a cell of the S area in the form of a abrupt P-N

*bulk sat*

*depl d br*

*<sup>U</sup> U SW N V*

(6.1.1)

*i*

*N* 

<sup>2</sup> (,) *bulk sat*

junction on silicon of the N type with the donor concentration Nd and depletion region size

2 *i*

*N U <sup>N</sup> U*

*T*

shown in Fig. 6.1.

Fig. 6.1.1. Structure of GAPD for model.

Wdepl(Nd,Vbr) at break-down voltage Vbr.

*i bulk sat*

*N U <sup>N</sup> U* or

2 *i*

*eN U <sup>J</sup>* 

*bulk*

Consequently 0 <sup>2</sup>

*d*

Kef = 1 – with the growth of voltage the feedback increases as quickly as the avalanche gain that corresponds to the APD operation in the self stabilization mode;

Kef > 1 – with the growth of voltage the feedback increases quicker than the avalanche gain that corresponds to APD operation in the self-suppressing mode.

A classification of this type allows us to define the character of the FB behavior. For some particular cases <sup>0</sup> 0 *d* 1 *dM dV M dV* for example we can state:

<sup>0</sup> *<sup>d</sup> dV* FB coefficient monotonously grows with the voltage increase; thus, theoretically, there must exist voltage when, if higher than its value, relative changes of the avalanche gain <sup>0</sup> 0 1 *dM <sup>M</sup> dV* are lower than the FB introduced suppressing. Initially, the condition 0 0 *d* 1 *dM dV M dV* is implemented. <sup>0</sup> *<sup>d</sup> dV* The FB coefficient is constant with the voltage growth; therefore, theoretically there

must be voltage for which relative changes in avalanche gain <sup>0</sup> 0 1 *dM <sup>M</sup> dV* correspond to FB introduced suppressing. Initially, the condition <sup>0</sup> *d* 1 *dM* is implemented.

0

*dV M dV*

<sup>0</sup> *<sup>d</sup> dV* The FB coefficient monotonously decreases with the voltage growth; thus, theoretically there must exist voltage when, if high than its value, relative changes of the avalanche gain <sup>0</sup> 0 1 *dM <sup>M</sup> dV* become higher than the FB introduced suppressing. Initially, the condition <sup>0</sup> 0 *d* 1 *dM dV M dV* is implemented.

All the above stated, despite the fact that only cases of quite particular character were considered, can further assist in sorting out correctly the main mechanism of FB for each specific case. And this will be the first and basic step how to connect experimental data with the APD physics, topology and technology.

#### **6. Examples of FBT application for APD**

#### **6.1 Logistic model of front of avalanche in GAPD**

In this chapter we consider model that describes the process of avalanche generating in GAPD. The peculiarity of the model is in the fact that we will not solve the Fundamental Equation System for semiconductors and will not actually use the Poisson equation. The dynamics of an avalanche process in GAPD is determined by two important characteristics of this detector. The first is the difference of the bias voltage and breakdown voltage and the second is the dimension and structure of the GAPD cell. Let us discuss a GAPD cell as it is shown in Fig. 6.1.

Fig. 6.1.1. Structure of GAPD for model.

218 Photodetectors

Kef = 1 – with the growth of voltage the feedback increases as quickly as the avalanche gain

Kef > 1 – with the growth of voltage the feedback increases quicker than the avalanche gain

A classification of this type allows us to define the character of the FB behavior. For some

FB coefficient monotonously grows with the voltage increase; thus, theoretically,

The FB coefficient is constant with the voltage growth; therefore, theoretically there

The FB coefficient monotonously decreases with the voltage growth; thus,

theoretically there must exist voltage when, if high than its value, relative changes of the

All the above stated, despite the fact that only cases of quite particular character were considered, can further assist in sorting out correctly the main mechanism of FB for each specific case. And this will be the first and basic step how to connect experimental data with

In this chapter we consider model that describes the process of avalanche generating in GAPD. The peculiarity of the model is in the fact that we will not solve the Fundamental Equation System for semiconductors and will not actually use the Poisson equation. The dynamics of an avalanche process in GAPD is determined by two important characteristics of this detector. The first is the difference of the bias voltage and breakdown voltage and the

0 *d* 1 *dM dV M dV*

*<sup>M</sup> dV* become higher than the FB introduced suppressing. Initially, the

0 1 *dM M dV*

is implemented.

correspond to FB

*<sup>M</sup> dV* are lower than the FB introduced suppressing. Initially, the condition

there must exist voltage when, if higher than its value, relative changes of the avalanche

that corresponds to the APD operation in the self stabilization mode;

for example we can state:

must be voltage for which relative changes in avalanche gain <sup>0</sup>

introduced suppressing. Initially, the condition <sup>0</sup>

is implemented.

that corresponds to APD operation in the self-suppressing mode.

particular cases <sup>0</sup>

0

avalanche gain <sup>0</sup>

condition <sup>0</sup>

0 1 *dM*

0 *d* 1 *dM dV M dV*

the APD physics, topology and technology.

**6. Examples of FBT application for APD** 

**6.1 Logistic model of front of avalanche in GAPD** 

is implemented.

<sup>0</sup> *<sup>d</sup> dV* 

<sup>0</sup> *<sup>d</sup> dV* 

<sup>0</sup> *<sup>d</sup> dV* 

gain <sup>0</sup> 0 1 *dM*

0 *d* 1 *dM dV M dV* 0 *d* 1 *dM dV M dV*

> The electric field in the cell is close or more to the critical one; therefore, the probability of the avalanche generation of a secondary carrier P~1. Moreover, there exists a certain threshold value of photo-generated electrons No necessary for the start of the avalanche process. Thus, to suppress the avalanche in cell from thermal noises for the thermogenerated carriers NT the NT < No inequation should be implemented. We can evaluate NT supposing that the noise generates only by the detector dark current. This supposition will allow us evaluate theoretically the dependence of the size of cell on the concentration of the doping applied for the APD material. It is obvious that the higher is the doping concentration the lower is the maximal cell volume ∆Ū at the same cell area. At backward bias voltage Vbias, the dark current Jd and equals to NT will determine the concentration of minor carriers generated per time unit in the depletion region. For APD with a long base, presupposing that the dark current is defined by the volume component from [12] it follows 2 *i d bulk eN U <sup>J</sup>* Let us consider to be definite a cell of the S area in the form of a abrupt P-N

> junction on silicon of the N type with the donor concentration Nd and depletion region size Wdepl(Nd,Vbr) at break-down voltage Vbr.

$$N\_T \approx \frac{N\_i \cdot \Delta U}{2 \cdot \tau\_{bulk} \cdot U\_{sat}} \tag{6.1.1}$$

Consequently 0 <sup>2</sup> *i bulk sat N U <sup>N</sup> U* or <sup>2</sup> (,) *bulk sat depl d br i <sup>U</sup> U SW N V N* 

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 221

The model given above describes the exponential change of the FB factor from the maximal

to zero that corresponds to the maximal current through GAPD.

In Fig. 6.1.2 we show the results of calculations of the avalanche front line for different values of No. The lower are the threshold values the faster augment the avalanche front. It corresponds to the idea that higher probability of the charge carrier ionization in APD

Fig. 6.1.2. Simulation of raising current in avalanche versus time by using Logistic model

Paper [11] suggests a model where the feedback factor was given in the form *fb*

Proceeding from the supposition that in GAPD the voltage should change by the value V-Vbr with the avalanche suppressing due to the voltage drop on the amplifying cell to

V-Vbr=Vfb. Let the FB factor be equal to β=Vfb/Vbr. We consider the charge gain in the GAPD cell whose capacity is Co. The cell discharge charge is Qo=Co\*Vfb. If the threshold number of electrons No, the avalanche gain coefficient of the charge Mx~Qo/e\*No = Co\*Vfb/e\*No. On

suppress the avalanche, we think it is the very similar FB voltage

*br V V* .

value max

0 <sup>1</sup> (1 ) *<sup>M</sup>*

**6.2 GAPD Analysis. FBT interpretation** 

corresponds to the fast front of the avalanche process.

It should be noted that, if the threshold value No is large, the avalanche probability is small and the front time increases. Big lifetime of the minor charge carriers in SCR (Space-Charge Region) makes it possible to produce GAPD cells of a larger area. After we have evaluated the maximal cell area let us consider the following model.

Let the maximal number of electrons in the cell available for multiplication be equal to N. At the moment of time t after the start of the avalanche multiplication process there are already x electrons; therefore, the number of electrons potentially available for multiplication will be (N-x). If the length of the avalanche gain region is Lav and the velocity of electron in the electric field Usat=const, (the time of front of avalanche in the order Tav~Lav/Usat). The differential equation for the described situation is given in the form (6.1.2).

$$\frac{d\mathbf{x}}{dt} = \mathbf{k} \cdot \mathbf{x} \cdot (\mathbf{N} - \mathbf{x})\tag{6.1.2}$$

with the initial requirement 0 *xt N* ( 0)

The solution of this equation is well known as a logistic equation [13].

$$\exp(t) = \frac{N}{(1 - a \cdot \exp(-k \cdot N \cdot t))} \tag{6.1.3}$$

where <sup>0</sup> 0 *N N a N* 

The constant 1/k must have time dimensionality, so we input a certain constant of the time of the avalanche front and define it on the basis of the dimension analysis. 0 0 0 <sup>1</sup> *av av sat N LN <sup>T</sup> k NV N* (at No=1 τo=Tav/N has a simple physical interpretation – it is the

time of one multiplication act, and if the threshold number No grows the time of one multiplication act becomes longer). Now we write (5.1.3) in the form

$$\mathbf{x}(t) = \frac{N}{(1 - a \cdot \exp(-\frac{t}{\tau\_0}))} \tag{6.1.4}$$

To transfer to the interpretation of the solution from the FBT point of view, we introduce new variables. M(t) = x(t)/No – the avalanche gain coefficient at the moment of time t, Mo=N/No - maximal avalanche gain in GAPD.

We write (6.1.4) in the form

$$M(t) = \frac{M\_0}{(1 + \beta(t) \cdot M\_0)}\tag{6.1.5}$$

where 0 0 <sup>1</sup> ( ) (1 ) exp( ) *<sup>t</sup> <sup>t</sup> M* 

It should be noted that, if the threshold value No is large, the avalanche probability is small and the front time increases. Big lifetime of the minor charge carriers in SCR (Space-Charge Region) makes it possible to produce GAPD cells of a larger area. After we have evaluated

Let the maximal number of electrons in the cell available for multiplication be equal to N. At the moment of time t after the start of the avalanche multiplication process there are already x electrons; therefore, the number of electrons potentially available for multiplication will be (N-x). If the length of the avalanche gain region is Lav and the velocity of electron in the electric field Usat=const, (the time of front of avalanche in the order Tav~Lav/Usat). The

( ) *dx kx N x*

( ) (1 exp( ))

The constant 1/k must have time dimensionality, so we input a certain constant of the time of the avalanche front and define it on the basis of the dimension analysis.

(at No=1 τo=Tav/N has a simple physical interpretation – it is the

time of one multiplication act, and if the threshold number No grows the time of one

*<sup>N</sup> x t <sup>t</sup> a*

(1 exp( ))

0

( ) (1 ( ) ) *<sup>M</sup> M t*

0

*t M* (6.1.5)

To transfer to the interpretation of the solution from the FBT point of view, we introduce new variables. M(t) = x(t)/No – the avalanche gain coefficient at the moment of time t,

0

*<sup>N</sup> x t*

*dt* (6.1.2)

*a kNt* (6.1.3)

(6.1.4)

differential equation for the described situation is given in the form (6.1.2).

The solution of this equation is well known as a logistic equation [13].

multiplication act becomes longer). Now we write (5.1.3) in the form

( )

the maximal cell area let us consider the following model.

with the initial requirement 0 *xt N* ( 0)

0 0

*N LN <sup>T</sup> k NV N*

*sat*

Mo=N/No - maximal avalanche gain in GAPD.

0 0

<sup>1</sup> ( ) (1 ) exp( ) *<sup>t</sup> <sup>t</sup> M*

where <sup>0</sup>

*a*

0

where

0 *N N*

*N* 

<sup>1</sup> *av av*

We write (6.1.4) in the form

The model given above describes the exponential change of the FB factor from the maximal value max 0 <sup>1</sup> (1 ) *<sup>M</sup>* to zero that corresponds to the maximal current through GAPD.

In Fig. 6.1.2 we show the results of calculations of the avalanche front line for different values of No. The lower are the threshold values the faster augment the avalanche front. It corresponds to the idea that higher probability of the charge carrier ionization in APD corresponds to the fast front of the avalanche process.

Fig. 6.1.2. Simulation of raising current in avalanche versus time by using Logistic model

### **6.2 GAPD Analysis. FBT interpretation**

Paper [11] suggests a model where the feedback factor was given in the form *fb br V V* .

Proceeding from the supposition that in GAPD the voltage should change by the value V-Vbr with the avalanche suppressing due to the voltage drop on the amplifying cell to suppress the avalanche, we think it is the very similar FB voltage

V-Vbr=Vfb. Let the FB factor be equal to β=Vfb/Vbr. We consider the charge gain in the GAPD cell whose capacity is Co. The cell discharge charge is Qo=Co\*Vfb. If the threshold number of electrons No, the avalanche gain coefficient of the charge Mx~Qo/e\*No = Co\*Vfb/e\*No. On

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 223

avalanche multiplication is Wav~0.1\*Wdepl(Vbr), and Ro – is the dimension of the region of

*S R Q CNV V <sup>V</sup> <sup>V</sup>*

To calculate the maximally possible charge of the avalanche gain let us suppose that the electric field in the region of the avalanche gain Wav is constant E=const and the ionization coefficient α(E) is known. We think that an avalanche appears as a micro-plasma current filament of a constant and independent of material thickness and S0/S1=1 (see chapter 6). Then, the maximally achievable charge of the avalanche multiplication can be calculated

> 01 0 1 ( ) ( ) *<sup>c</sup> av av av <sup>E</sup> Q eN E S W eN S W <sup>E</sup>*

In the expression αo, Ec the model parameters and e- is the electron charge. Let us suppose that the field intensity E is equal to the maximal for the given PN junction at voltage V=Vbr.

*eN W N V E E*

In Fig. (6.2.1) and (6.2.2) calculation results are presented in the given formulas (6.2.3) and (6.2.5) for No=1, Ro=2,3,5,7,10,15 um, S1=1мкм2 and ∆V=0.1,0.25,0.5, 1.0, 2.0, 3.0, 5.0 V. Although the dimension of the current filament is not taken into account in calculations of the dependence that shows the maximal charge change of the avalanche gain Qav(Nd,Wav(Nd,Vbr)) achievable in multiplying, while it is easy to make the corresponding correction, the main thing is to connect it rightly with real GAPD topology and technology. The observed decrease of the charge at high Nd concentrations is explained by the effect of

In Fig. (6.2.1) the red line shows how the maximal charge Qav(Nd,Wav(Nd,Vbr)) achievable in multiplying changes if the donor concentration in the base silicon changes in the limits from 1012 cm-3 to 1016 cm-3, and the straight lines show how the available charge in the cell changes Qmax(Nd,Wav(Nd,Vbr,Ro)) at various values of the avalanche region radius Ro and ∆V=1. The region where the feedback mechanism operates to restrict the avalanche at the expense of the limitation of the charge available for gain in the cell is defined by the

0

0

0 1 ( ) (,) (,) *Si c av av d br d d d br <sup>E</sup> Q eN S W NV eN W N V* 

 

(,) ( ) <sup>2</sup>

0 00

 

*depl br Si br*

*Si Si*

 

(,) *d d d br Si*

(6.2.5)

*W V V*

0

 

*e N*

  2

(6.2.3)

(6.2.4)

*d*

avalanche current filament. The maximal cell charge is:

*d br*

*d*

*e N* 

max

max 0

As a result, we obtain expression (6.2.5)

where <sup>0</sup> <sup>2</sup> ( , ) 0.1 *Si br*

*<sup>V</sup> W NV*

*av d br*

decrease Wav(Nd,Vbr).

inequation Qav≥ Qmax.

from equation (6.2.4).

the other hand, we can define the voltage of a separate gain at V=Vbr when β~1, so the choice *fb br V V* will be justified if Mo is chosen in the corresponding way. From the expression (6.2.1) we obtain the value Mo.

$$M = \frac{\mathbf{C}\_0 \cdot V\_{fb}}{e \cdot N} = \frac{M\_0}{1 + \frac{V\_{fb}}{V\_{br}} \cdot M\_0} \tag{6.2.1}$$

The result of the solution is given by way of the charge gain coefficient.

$$M\_0 = \frac{\mathbf{C}\_0 \cdot V\_{\beta\hbar}}{e \cdot N\_0 - \mathbf{C}\_0 \cdot V\_{\beta\hbar} \cdot \frac{V\_{\beta\hbar}}{V\_{br}}} = \frac{Q\_0}{Q\_i + \frac{V\_{\beta\hbar}}{V\_{br}} \cdot Q\_0} \tag{6.2.2}$$

In this expression Qo is an intensified charge from the cell and Qi is the initial charge that switches the avalanche in GAPD. Thus, we have managed to present the FB factor in the desired form and, at the same time, keep the general kind of the expression for the avalanche gain in the form 1 *o o <sup>M</sup> <sup>M</sup> <sup>M</sup>* .

Let us discuss now the feedback model when the avalanche restrictions happen due to limitation of the charge available for the avalanche multiplication in the cell. As the main parameter is the type and concentration of GAPD material doping, we will try to connect our abstract model with silicon parameters. The maximal charge available for multiplication in the cell is defined by the cell and SCR dimensions and equals to Qmax=Co\*Vfb~Na 1/2 (we suppose that the basic material is silicon of the N type with donor concentration Nd). The possible charge of the avalanche gain Qav is defined by the electric field intensity in the gain area, Qav=e\*No\*M~Nam where m>1.2 than the length of the avalanche region.

As the change of the field intensity per length unit in low-ohm material is higher than in high-ohm material, the rate of the avalanche gain change in the low-ohm material in the SCR avalanche part is higher and the avalanche generates in smaller space. It leads to its greater instability and imposes strict requirements to the time of FB establishment, as it was indicated above (formula (5.7)). In this case the FB mechanism due to the limitation of the charge available for gain in the cell is more preferable.

There should exist conditions when FB due to charge quantity limitation in the cell is possible and this condition will be Qav≥ Qmax. In carrying out this inequation, the maximal charge is defined by the cell geometry and not by electric field intensity in it. Meeting this requirement will allow one to develop GAPD with high field intensity without fear of destroying the cell with high currents due to local instability of the avalanche process.

Let us study a simple model shown in Fig. 6.1 the abrupt P-N junction on the N-type silicon. If Nd>>Na the dimension Wdepl of SCR at the bias voltage equal to breakdown voltage is known and the maximal charge accumulated in the cell at the excess of breakdown voltage by ∆V=V-Vbr is defined from equation (6.2.3). Let us suppose that the region of the avalanche multiplication is Wav~0.1\*Wdepl(Vbr), and Ro – is the dimension of the region of avalanche current filament. The maximal cell charge is:

$$Q\_{\rm max} = C\_0 (N\_{d'} V\_{br}) \cdot \Delta V \approx \frac{\varepsilon\_0 \cdot \varepsilon\_{Si} \cdot S}{W\_{dept} (V\_{br})} \cdot \Delta V = \frac{\varepsilon\_0 \cdot \varepsilon\_{Si} \cdot \pi \cdot R\_0^2}{\sqrt{\frac{2 \cdot \varepsilon\_0 \cdot \varepsilon\_{Si} \cdot V\_{br}}{e \cdot N\_d}}} \cdot \Delta V \tag{6.2.3}$$

To calculate the maximally possible charge of the avalanche gain let us suppose that the electric field in the region of the avalanche gain Wav is constant E=const and the ionization coefficient α(E) is known. We think that an avalanche appears as a micro-plasma current filament of a constant and independent of material thickness and S0/S1=1 (see chapter 6). Then, the maximally achievable charge of the avalanche multiplication can be calculated from equation (6.2.4).

$$Q\_{av} \approx e \cdot N\_0 \cdot \alpha(E) \cdot S\_1 \cdot \mathcal{W}\_{av} = e \cdot N\_0 \cdot \alpha \cdot (-\frac{E\_c}{E}) \cdot S\_1 \cdot \mathcal{W}\_{av} \tag{6.2.4}$$

In the expression αo, Ec the model parameters and e- is the electron charge. Let us suppose that the field intensity E is equal to the maximal for the given PN junction at voltage V=Vbr.

$$E = E\_{\text{max}} = \frac{\boldsymbol{\varepsilon} \cdot \mathbf{N}\_d \cdot \mathcal{W}\_d (\mathbf{N}\_{d'} V\_{br})}{\varepsilon\_0 \cdot \varepsilon\_{Si}}$$

As a result, we obtain expression (6.2.5)

222 Photodetectors

the other hand, we can define the voltage of a separate gain at V=Vbr when β~1, so the

0 0

*fb*

*fb*

*C V <sup>Q</sup> <sup>M</sup> V V*

*C V <sup>M</sup> <sup>M</sup> e N V*

The result of the solution is given by way of the charge gain coefficient.

*o o*

area, Qav=e\*No\*M~Nam where m>1.2 than the length of the avalanche region.

*<sup>M</sup> <sup>M</sup> <sup>M</sup>* .

charge available for gain in the cell is more preferable.

0

will be justified if Mo is chosen in the corresponding way. From the

<sup>0</sup> 1

*V*

0 0

0 0 0

*fb i*

*eN C V Q Q V V*

In this expression Qo is an intensified charge from the cell and Qi is the initial charge that switches the avalanche in GAPD. Thus, we have managed to present the FB factor in the desired form and, at the same time, keep the general kind of the expression for the

Let us discuss now the feedback model when the avalanche restrictions happen due to limitation of the charge available for the avalanche multiplication in the cell. As the main parameter is the type and concentration of GAPD material doping, we will try to connect our abstract model with silicon parameters. The maximal charge available for multiplication in the cell is defined by the cell and SCR dimensions and equals to Qmax=Co\*Vfb~Na1/2 (we suppose that the basic material is silicon of the N type with donor concentration Nd). The possible charge of the avalanche gain Qav is defined by the electric field intensity in the gain

As the change of the field intensity per length unit in low-ohm material is higher than in high-ohm material, the rate of the avalanche gain change in the low-ohm material in the SCR avalanche part is higher and the avalanche generates in smaller space. It leads to its greater instability and imposes strict requirements to the time of FB establishment, as it was indicated above (formula (5.7)). In this case the FB mechanism due to the limitation of the

There should exist conditions when FB due to charge quantity limitation in the cell is possible and this condition will be Qav≥ Qmax. In carrying out this inequation, the maximal charge is defined by the cell geometry and not by electric field intensity in it. Meeting this requirement will allow one to develop GAPD with high field intensity without fear of destroying the cell with high currents due to local instability of the avalanche process.

Let us study a simple model shown in Fig. 6.1 the abrupt P-N junction on the N-type silicon. If Nd>>Na the dimension Wdepl of SCR at the bias voltage equal to breakdown voltage is known and the maximal charge accumulated in the cell at the excess of breakdown voltage by ∆V=V-Vbr is defined from equation (6.2.3). Let us suppose that the region of the

*M*

*fb fb*

*br br*

(6.2.1)

(6.2.2)

*fb br*

choice *fb*

*br V V* 

avalanche gain in the form 1

expression (6.2.1) we obtain the value Mo.

$$Q\_{av} = e \cdot N\_0 \cdot \alpha \cdot (-\frac{\varepsilon\_0 \cdot \varepsilon\_{Si} \cdot E\_c}{e \cdot N\_d \cdot W\_d (N\_{d'} \cdot V\_{br})}) \cdot S\_1 \cdot W\_{av} (N\_{d'} \cdot V\_{br}) \tag{6.2.5}$$

where <sup>0</sup> <sup>2</sup> ( , ) 0.1 *Si br av d br d <sup>V</sup> W NV e N* 

In Fig. (6.2.1) and (6.2.2) calculation results are presented in the given formulas (6.2.3) and (6.2.5) for No=1, Ro=2,3,5,7,10,15 um, S1=1мкм2 and ∆V=0.1,0.25,0.5, 1.0, 2.0, 3.0, 5.0 V. Although the dimension of the current filament is not taken into account in calculations of the dependence that shows the maximal charge change of the avalanche gain Qav(Nd,Wav(Nd,Vbr)) achievable in multiplying, while it is easy to make the corresponding correction, the main thing is to connect it rightly with real GAPD topology and technology. The observed decrease of the charge at high Nd concentrations is explained by the effect of decrease Wav(Nd,Vbr).

In Fig. (6.2.1) the red line shows how the maximal charge Qav(Nd,Wav(Nd,Vbr)) achievable in multiplying changes if the donor concentration in the base silicon changes in the limits from 1012 cm-3 to 1016 cm-3, and the straight lines show how the available charge in the cell changes Qmax(Nd,Wav(Nd,Vbr,Ro)) at various values of the avalanche region radius Ro and ∆V=1. The region where the feedback mechanism operates to restrict the avalanche at the expense of the limitation of the charge available for gain in the cell is defined by the inequation Qav≥ Qmax.

studies.

and concentration donor in cell.

establishment.

**6.3 APD equivalent scheme – SPICE model** 

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 225

to calculate the GAPD construction from the point that the emerged avalanche will discharge the cell to the full extent and switch off due to the carriers' deficit, with no unstable unlimited by anything avalanche process. In practice, the discussed above mechanisms of avalanche quenching are implemented simultaneously, both in the form of decreasing the voltage in the cell and achieving depletion of free multiplication carriers in the avalanche region. To answer this question it is necessary to conduct more elaborate

Fig. 6.2.2. The selection optimal over voltage V=(Vbias -Vbr) for GAPD. The red line is the maximal avalanche charge can be the limit for over voltage for given GAPD with fixed size

A model to shape an avalanche front in APD on the basis of n-SiC/p-Si P-N junction is suggested for application in paper [11]. The form of the FB factor (its time dependence during the front shaping) was obtained on the basis of experimental data, with an initial supposition that the n-SiC layer properties determine the characteristic time of FB

The full equivalent schemes of MRS APD and Micro Channel Avalanche Photo Diode (MCAPD) are shown in Fig. 6.3.1. We are interested in one of its elements – a nonlinear

Fig. 6.2.1. The selection optimal size of cell for GAPD. The concentration of donors in cell must be selected near cross point of red line of maximum avalanche charge and the lines of different radius of cell.

In Fig. (6.2.2) the red line shows how the maximal charge Qav(Nd,Wav(Nd,Vbr)) changes if it is attained in multiplying and the donor concentration in the base silicon changes in the limits from 1012 cm-3 to 1016 cm-3, while the straight lines show how the charge available in the cell changes Qmax(Nd,Wav(Nd,Vbr,Ro)) at various voltages ∆V if Ro=5. The region of application of the feedback mechanism to restrict the avalanche at the expense of the limitation of the charge available for gain in the cell is defined by the inequation Qav≥ Qmax.

Let us discuss an example of application of the obtained dependences. If we choose silicon with Nd=1016 cm-3 at the expected ∆V=1 the size Ro of the charge region available for multiplication should be less than 3 μm (for example, 2 μm) and the mode is implemented when the maximal avalanche gain is restricted by the cell geometry and not by the intensity of the cell electric field. At ∆V=2 the size Ro of the charge region available for multiplication should be less than 2 μm. In another Figure we see that if we choose silicon with Nd=1016 cm-3 at the chosen Ro=5 the excess voltage ∆V for the charge multiplication should be less than 0.5 (for example, 0.25); if Ro=2 we can choose ∆V=2 but not larger. The discussed examples show that the implementation of the restriction mode of the charge in the cell is quite complicated due to the complex interconnections of many parameters. It is necessary

Fig. 6.2.1. The selection optimal size of cell for GAPD. The concentration of donors in cell must be selected near cross point of red line of maximum avalanche charge and the lines of

In Fig. (6.2.2) the red line shows how the maximal charge Qav(Nd,Wav(Nd,Vbr)) changes if it is attained in multiplying and the donor concentration in the base silicon changes in the limits from 1012 cm-3 to 1016 cm-3, while the straight lines show how the charge available in the cell changes Qmax(Nd,Wav(Nd,Vbr,Ro)) at various voltages ∆V if Ro=5. The region of application of the feedback mechanism to restrict the avalanche at the expense of the limitation of the charge available for gain in the cell is defined by the inequation Qav≥

Let us discuss an example of application of the obtained dependences. If we choose silicon with Nd=1016 cm-3 at the expected ∆V=1 the size Ro of the charge region available for multiplication should be less than 3 μm (for example, 2 μm) and the mode is implemented when the maximal avalanche gain is restricted by the cell geometry and not by the intensity of the cell electric field. At ∆V=2 the size Ro of the charge region available for multiplication should be less than 2 μm. In another Figure we see that if we choose silicon with Nd=1016 cm-3 at the chosen Ro=5 the excess voltage ∆V for the charge multiplication should be less than 0.5 (for example, 0.25); if Ro=2 we can choose ∆V=2 but not larger. The discussed examples show that the implementation of the restriction mode of the charge in the cell is quite complicated due to the complex interconnections of many parameters. It is necessary

different radius of cell.

Qmax.

to calculate the GAPD construction from the point that the emerged avalanche will discharge the cell to the full extent and switch off due to the carriers' deficit, with no unstable unlimited by anything avalanche process. In practice, the discussed above mechanisms of avalanche quenching are implemented simultaneously, both in the form of decreasing the voltage in the cell and achieving depletion of free multiplication carriers in the avalanche region. To answer this question it is necessary to conduct more elaborate studies.

Fig. 6.2.2. The selection optimal over voltage V=(Vbias -Vbr) for GAPD. The red line is the maximal avalanche charge can be the limit for over voltage for given GAPD with fixed size and concentration donor in cell.

#### **6.3 APD equivalent scheme – SPICE model**

A model to shape an avalanche front in APD on the basis of n-SiC/p-Si P-N junction is suggested for application in paper [11]. The form of the FB factor (its time dependence during the front shaping) was obtained on the basis of experimental data, with an initial supposition that the n-SiC layer properties determine the characteristic time of FB establishment.

The full equivalent schemes of MRS APD and Micro Channel Avalanche Photo Diode (MCAPD) are shown in Fig. 6.3.1. We are interested in one of its elements – a nonlinear

and [15].

given in Fig. 6.3.2 and 6.3.3.

Fig. 6.3.2. Simulation result for MRS APD [11], [14].

Table 6.1. (Parameters for simuliation)

MRS Rfb=1K Cfb=1200pF R=1K

MP APD Cfb=5pF Dfb(CJO)=10pF Dfb(BV)=100mV Dfb(IBV)=20nA

Model

Model

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 227

For test of method were defined of FB factors for APD of two types described in papers [14]

The dependences that correspond to the definition are presented in Table 6.1.1 as fragments of the SPICE code and the simulation result and its comparison with experimental data is

source of current I(t) (APD) controlled by current (Rfb, Dfb, Cfb). The source amplifies the photocurrent Io(t) M times and by a certain time dependence of the FB factor β(t) it correctly describes the shaping of the APD front.

Let us consider the method of definition of time dependence of the FB factor to be applied in the further suggested model. The method can be applied to linear APD, but not to GAPD.

Let the APD under study be under the effect of a rectangular light pulse that generates current Io(t) in APD. An intensified current pulse I(t) can be given in the form

$$I(t) = I\_0(t) \cdot \frac{M\_0}{1 + \beta(t) \cdot M\_0} \tag{6.3.1}$$

Fig. 6.3.1. Two different model were investigated for analysis by SPICE. APD - is source of current that controlled by voltage. Rfb - linear element of FB can be used for MRS APD [14], the Dfb is not linear element of FB and can be used for MC APD [15].

We suppose that Mo=const does not depend on time Mo≠Mo(t) and only β=β(t).

Then it is easy to obtain the dependence for the FB factor from (5.3.1)

$$\frac{I\_0(t)}{I(t)} = \beta(t) + \frac{1}{M\_0} \tag{6.3.2}$$

As we suppose that Mo-1=const any time dependence of the FB factor is obviously reflected by equation (6.3.2).

source of current I(t) (APD) controlled by current (Rfb, Dfb, Cfb). The source amplifies the photocurrent Io(t) M times and by a certain time dependence of the FB factor β(t) it correctly

Let us consider the method of definition of time dependence of the FB factor to be applied in the further suggested model. The method can be applied to linear APD, but not to GAPD.

Let the APD under study be under the effect of a rectangular light pulse that generates

0

0

*t M* (6.3.1)

current Io(t) in APD. An intensified current pulse I(t) can be given in the form

0

() () 1 () *<sup>M</sup> It I t*

Fig. 6.3.1. Two different model were investigated for analysis by SPICE. APD - is source of current that controlled by voltage. Rfb - linear element of FB can be used for MRS APD [14],

> ( ) <sup>1</sup> ( ) ( ) *I t <sup>t</sup> It M*

As we suppose that Mo-1=const any time dependence of the FB factor is obviously reflected

0

(6.3.2)

the Dfb is not linear element of FB and can be used for MC APD [15].

Then it is easy to obtain the dependence for the FB factor from (5.3.1)

by equation (6.3.2).

We suppose that Mo=const does not depend on time Mo≠Mo(t) and only β=β(t).

0

describes the shaping of the APD front.

For test of method were defined of FB factors for APD of two types described in papers [14] and [15].

The dependences that correspond to the definition are presented in Table 6.1.1 as fragments of the SPICE code and the simulation result and its comparison with experimental data is given in Fig. 6.3.2 and 6.3.3.

Fig. 6.3.2. Simulation result for MRS APD [11], [14].


Table 6.1. (Parameters for simuliation)

Avalanche Process in Semiconductor Photo Detectors in the Context of the Feedback Theory 229

The FBT approach allows to hide information about real physics processes in different types of APD. We can exchange a physical model describing a complex behavior of the system by a simple universal model with predictable results. The basic steps are in definition of main statistical parameters and their connections with external conditions of APD. Practically in all cases it is possible to implement a formal transfer from the physical model of an avalanche generation to the description in FBT. The formal separation of processes into those of avalanche multiplication and avalanche suppression or restriction allows a simpler understanding of the physical origin of the phenomena. Based on FBT notions, it is possible to construct simple enough model to describe processes of avalanche gain. This model can be connected with important parameters of APD under study and allows to determine the

Application of the FBT approach to construct the SPICE model allows one to facilitate the procedure of definition of the equivalent APD circuit main parameters and correctly

[2] John Doyle, Bruce Francis, Allen Tannenbaum, (1990). Feedback Control Theory,

[3] William McC. Siebert, (1986). Circuits, signals, and systems, Published by MIT Press,

[4] K. Jradi, et.al., (2011). Computer-aided design (CAD) model for silicon avalanche Geiger

[6] S. L. Miller, (1955). Avalanche breakdown in germanium, Phys. Rev. 99, pp. 1234-1241 [7] D. Bisello, et al., (1995). Metal-Resistive layer-Silicon (MRS) avalanche detectors with

[8] F. Zappa, A. Lacaita, C. Samori, (1997). Characterization and Modeling of Metal-

[9] McKay, K. (1954). Avalanche Breakdown in Silicon, Physical Review 94:pp. 877-880 [10] Z. Sadygov et al., (1991). Proc. SPIE. Optical Memory and Neural Networks, pp. 158-168 [11] V. V. Kushpil, (2009). Application of simple negative feedback model for avalanche

mode systems design: Application to high sensitivity imaging systems, Nuclear Instruments and Methods in Physics Research Section A. Vol. 626-627, pp. 77-81

negative feedback, Nuclear Instruments and Methods in Physics Research Section

Resistance-Semiconductor Photodetectors, IEEE Trans. On Nucl. Sc., Vol. 44, No. 3

photodetectors investigation, Nuclear Instruments and Methods in Physics

[1] F.L. Lewis, (1993). Applied Optimal Control and Estimation, Prentice-Hall.

[5] S. M. Sze, (1981). Physics of Semiconductor Devices, New York: Wiley.

*((PI\*RADIUS\*RADIUS)-PI\*(RADIUS-*

**7. Conclusion** 

**8. References** 

*SKINDEPTHTR(RESISTIVITY,RELPERM,FREQ))\*\*2))\*DCRES} .FUNC SKINDEPTHTR(RESISTIVITY,RELPERM,FREQ) {503.3\*(SQRT(RESISTIVITY/(RELPERM\*FREQ)))}* 

*BG1 0 PINA I = {V(IN)\*POWER((40/(40-(V(0)-V(PINA)))),1)*

link of some technological parameters with APD properties.

McGraw-Hill in Cambridge, Mass, New York .

Research Section A, Vol. 610, Issue 1, pp.204-206 [12] (2003). Encyclopedia of Optical Engineering, pp. 128-130

describe the peculiarities of the front signal shaping.

Macmillan Publishing Co.

A, Vol. 360, Issue 1-2, pp.83-86

Fig. 6.3.3. Simulation result for MCAPD [15].

### **APD SPICE model:**

*.FUNC DPWR(D) {I(D)\*V(D)} .FUNC BPWR(Q) {IC(Q)\*VCE(Q)+IB(Q)\*VBE(Q)} .FUNC FPWR(M) {ID(M)\*VDS(M)} .FUNC HOTD(D,MAX) {IF((V(D)\*I(D)>MAX),1,0)} .FUNC HOTB(Q,MAX) {IF((VCE(Q)\*IC(Q)+IB(Q)\*VBE(Q)>MAX),1,0)} .FUNC HOTF(M,MAX) {IF((VDS(M)\*ID(M)>MAX),1,0)} .PARAM LOW3MIN={IMPORT(LOW3MIN.OUT,LOW3THRES)} .PARAM HIGH3MAX={IMPORT(HIGH3MAX.OUT,HIGH3THRES)} .PARAM LOWLVDS={IMPORT(LOWLVDS.OUT,LOWLIMIT)} .PARAM HILVDS={IMPORT(HILVDS.OUT,HILIMIT)} .PARAM LIMTLVDS={IMPORT(LIMTLVDS.OUT,LVDSLIMITS)} .FUNC SKINAC(DCRES,RESISTIVITY,RELPERM,RADIUS) {((PI\*RADIUS\*RADIUS)/ ((PI\*RADIUS\*RADIUS)-PI\*(RADIUS-SKINDEPTHAC(RESISTIVITY,RELPERM))\*\*2))\*DCRES} .FUNC SKINDEPTHAC(RESISTIVITY,RELPERM) {503.3\*(SQRT(RESISTIVITY/(RELPERM\*F)))} .FUNC SKINTR(DCRES,RESISTIVITY,RELPERM,RADIUS,FREQ) {((PI\*RADIUS\*RADIUS)/*  *((PI\*RADIUS\*RADIUS)-PI\*(RADIUS-SKINDEPTHTR(RESISTIVITY,RELPERM,FREQ))\*\*2))\*DCRES} .FUNC SKINDEPTHTR(RESISTIVITY,RELPERM,FREQ) {503.3\*(SQRT(RESISTIVITY/(RELPERM\*FREQ)))} BG1 0 PINA I = {V(IN)\*POWER((40/(40-(V(0)-V(PINA)))),1)*

### **7. Conclusion**

228 Photodetectors

Fig. 6.3.3. Simulation result for MCAPD [15].

*.FUNC BPWR(Q) {IC(Q)\*VCE(Q)+IB(Q)\*VBE(Q)}* 

*.FUNC HOTD(D,MAX) {IF((V(D)\*I(D)>MAX),1,0)}* 

*.FUNC HOTF(M,MAX) {IF((VDS(M)\*ID(M)>MAX),1,0)}* 

*.PARAM HILVDS={IMPORT(HILVDS.OUT,HILIMIT)}* 

*SKINDEPTHAC(RESISTIVITY,RELPERM))\*\*2))\*DCRES} .FUNC SKINDEPTHAC(RESISTIVITY,RELPERM) {503.3\*(SQRT(RESISTIVITY/(RELPERM\*F)))}* 

*.FUNC HOTB(Q,MAX) {IF((VCE(Q)\*IC(Q)+IB(Q)\*VBE(Q)>MAX),1,0)}* 

*.PARAM LOW3MIN={IMPORT(LOW3MIN.OUT,LOW3THRES)} .PARAM HIGH3MAX={IMPORT(HIGH3MAX.OUT,HIGH3THRES)} .PARAM LOWLVDS={IMPORT(LOWLVDS.OUT,LOWLIMIT)}* 

*.PARAM LIMTLVDS={IMPORT(LIMTLVDS.OUT,LVDSLIMITS)}* 

*.FUNC SKINAC(DCRES,RESISTIVITY,RELPERM,RADIUS) {((PI\*RADIUS\*RADIUS)/* 

*.FUNC SKINTR(DCRES,RESISTIVITY,RELPERM,RADIUS,FREQ) {((PI\*RADIUS\*RADIUS)/* 

**APD SPICE model:** 

*.FUNC DPWR(D) {I(D)\*V(D)}* 

*.FUNC FPWR(M) {ID(M)\*VDS(M)}* 

*((PI\*RADIUS\*RADIUS)-PI\*(RADIUS-*

The FBT approach allows to hide information about real physics processes in different types of APD. We can exchange a physical model describing a complex behavior of the system by a simple universal model with predictable results. The basic steps are in definition of main statistical parameters and their connections with external conditions of APD. Practically in all cases it is possible to implement a formal transfer from the physical model of an avalanche generation to the description in FBT. The formal separation of processes into those of avalanche multiplication and avalanche suppression or restriction allows a simpler understanding of the physical origin of the phenomena. Based on FBT notions, it is possible to construct simple enough model to describe processes of avalanche gain. This model can be connected with important parameters of APD under study and allows to determine the link of some technological parameters with APD properties.

Application of the FBT approach to construct the SPICE model allows one to facilitate the procedure of definition of the equivalent APD circuit main parameters and correctly describe the peculiarities of the front signal shaping.

### **8. References**


**11** 

Mikhail E. Belkin

*Russian Federation* 

**Multiscale Computer Aided Design of** 

**Microwave-Band P-I-N Photodetectors** 

*Moscow State Technical University of Radio-Engineering, Electronics and* 

Long wavelength InP-based p-i-n photodetectors (PD) are ubiquitous in modern optoelectronic circuits due to their inherent combination of ultra-high speed, high sensitivity in the most popular for modern telecom systems spectral range of 1.3-1.6 µm, and low bias voltages features that are impossible in principle for Si, GaAs or Ge counterparts. Typical material systems for the telecom spectral range are GaInAsP and GaInAs on InP substrate (Capasso et al., 1985). Now in the number of classical and present-day works (see, e.g. Bowers & Burrus, 1987; Beling & Campbell, 2009) is well-proved that compound semiconductor p-i-n photodetectors have the valid merits such as: high responsivity (up to 1 A/W), lowest dark current (below 10 fA), ultra-high bandwidth (up to 100 GHz and above), possibility of monolithic optoelectronic receiver module creation on common InP substrate. Through a wide time period this type of PD found the most intensive application inside a receiver of the long-haul digital fiber-optic systems. But recently the p-i-n PDs for the wavelength of 1.3 and 1.55 µm are advancing in various RF and microwave photonic apparatus, particularly, in microwave-band optoelectronic oscillators (X.S.Yao, 2002), frequency converters (J. Yao, 2009), as well in so called photonic antennas for the base stations of radio over fiber (RoF) systems (Sauer et al., 2007). While designing a receiving module for the above devices and systems the stage of circuit schematic development referred to active and passive, electronic and photonic components' modeling would be most complicated and labour-intensive. Based on microwave IC design practice in this case the modeling is realized as a description of electrical and optical features of the integrated functional elements by means of mathematical equations, equivalent circuits or tables.

At present time computer-aided design (CAD) exploitation for innovative high-tech production R&D acceleration is a common way (Minixhofer, 2006). This is especially important for microwave semiconductor component base with measurement equipment and experimental work's cost being considerably more expensive compare to that in lower frequency bands. Using modern CAD software the next two approaches may be relevant for this case: by means of so-called physical models and by means of equivalent circuit models. A physical modeling is generally the most accurate but at the same time the most complicated. It is executed through a computer simulation of dedicated physical processes

**1. Introduction** 

*Automation, Faculty of Electronics, Joint Research Laboratory* 

*"Microwave and Optoelectronic Devices", Moscow* 

