Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag

*Brahim Ait Hammou, Abdelhamid El Kaaouachi, Abdellatif El Oujdi, Adil Echchelh, Said Dlimi, Chi-Te Liang and Jamal Hemine*

## **Abstract**

In this work, we model the dielectric functions of gold (Au) and silver (Ag) which are typically used in photonics and plasmonics. The modeling has been performed on Au and Ag in bulk and in nanometric states. The dielectric function is presented as a complex number with a real part and an imaginary part. First, we will model the experimental measurements of the dielectric constant as a function of the pulsation ω by appropriate mathematical functions in an explicit way. In the second part we will highlight the contributions to the dielectric constant value due to intraband and interband electronic transitions. In the last part of this work we model the dielectric constant of these metals in the nanometric state using several complex theoretical models such as the Drude Lorentz theory, the Drude two-point critical model, and the Drude three-point critical model. We shall comment on which model fits the experimental dielectric function best over a range of pulsation.

**Keywords:** Modeling bulk and nanometric dielectric function, noble metals Au and Ag, interband transitions, intraband transitions, UV and IR pulsations

### **1. Introduction**

All the intrinsic effects corresponding to the process of light-matter interaction are contained in the dielectric function *ε ω*ð Þ**.** In the case of an isotropic material, the optical response is described by following equation

$$
\varepsilon(a) = \varepsilon\_1(a) + i\varepsilon\_2(a) \tag{1}
$$

where *ε ω*ð Þis generally a complex scalar value which depends upon the pulsation ω of the field. If the medium has an anisotropy, this magnitude is in the form of a tensor. It is often convenient to describe the optical response in an equivalent way from the complex refractive index *n*~ ¼ *n* þ *iκ* as *n* denotes the refractive index describing the phase speed of the wave and *κ* denotes the extinction index describing the absorption of the wave during propagation in the material. These two indices are directly related to the dielectric constant of the material. In fact, the real and imaginary parts of the dielectric function are deduced from the relation:

$$e\_1 = n^2 - \kappa^2 \tag{2}$$

$$
\varepsilon\_2 = 2n\kappa \tag{3}
$$

**Figure 2.**

**Table 1.**

**Table 2.**

**Table 3.**

**Table 4.**

**Table 5.**

**69**

*[Ref. [13])© The Optical Society.*

*Values of the model parameters.*

*Values of the model parameters.*

*Values of the model parameters.*

**Parameter** *ε***<sup>3</sup>**

*Values of the model parameters.*

*Values of the model parameters.*

**Parameter** *ε***<sup>4</sup>**

**Parameter** *ε***<sup>2</sup>**

*<sup>R</sup>***<sup>0</sup>** *A*<sup>0</sup>

*<sup>R</sup>***<sup>0</sup>** *A*<sup>00</sup>

**<sup>1</sup>** *t*<sup>0</sup>

**1(rad/s)** *A*<sup>0</sup>

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

**<sup>1</sup>** *t*<sup>00</sup>

**Parameter** *ε***<sup>1</sup>**

**Parameter** *ε***<sup>0</sup>**

*Real part( ) and imaginary part ( ) of the dielectric function of bulk Au([adapted] with permission from*

Value 10499.713 �1300.876 3.813 � <sup>10</sup><sup>14</sup> �9163.320 1.217 � <sup>10</sup><sup>14</sup> 0.9999

**<sup>2</sup>** *t*<sup>0</sup>

Value 0.305 83.392 1.091 � <sup>10</sup><sup>15</sup> 86.023 1.091 � 1015 86.639 1.091 � <sup>10</sup><sup>15</sup> 0,99101

**<sup>1</sup> (rad/s)** *A*<sup>00</sup>

Value 5.281 �0.006 �9.588 � <sup>10</sup><sup>14</sup> �0.006 �9.576 � <sup>10</sup><sup>14</sup> 0.99671

Value 0.868 0.735 �9.816 � <sup>10</sup><sup>14</sup> 3.483 � 1014 0.96592

Value 2.539 7.140 � 1015 �2.399 6.978 � <sup>10</sup><sup>14</sup> 3.783 � <sup>10</sup><sup>24</sup> 0.99613

*<sup>R</sup>***<sup>0</sup>** *A***<sup>1</sup>** *t***1(rad/s)** *A***<sup>2</sup>** *t***2(rad/s) R-Square (COD)**

**<sup>2</sup>** *t*<sup>00</sup>

*<sup>R</sup>***<sup>0</sup>** *A ω<sup>C</sup>* **(rad/s)** *B* **(rad/s) R-Square (COD)**

*<sup>R</sup>***<sup>0</sup>** *ωC*<sup>0</sup> **(rad/s)** *H ω***<sup>1</sup> (rad/s)** *ω***<sup>2</sup> (rad/s) R-Square (COD)**

**2(rad/s)** *A***<sup>3</sup>** *t***3(rad/s) R-Square**

**(COD)**

**<sup>2</sup> (rad/s) R-Square (COD)**

Several important physical quantities can be deduced from the complex refractive index *n*~ and dielectric function *ε ω*ð Þsuch as the reflectivity coefficient *R* and the attenuation coefficient *α*. In the fields of photonics, and plasmonics, researchers use the dielectric function in their calculations and investigations [1–8].

## **2. Modeling of the bulk experimental dielectric function**

Here we try to model the experimental dielectric function *<sup>ε</sup>exp*ð Þ *<sup>ω</sup>* of noble metals (Ag and Au) on a wide pulse interval ω with adequate mathematical functions. We consider the measured values of the dielectric function reported by Dold and Mecke [9], Winsemius *et al.* [10], Leveque *et al.* [11], and Thèye *et al.* [12] respectively**.** These measurements are summarized by Rakiç *et al.* [13] in **Figure 1** (for Ag), and in **Figure 2** (for Au).

#### **2.1 Modeling the experimental dielectric function of bulk Ag**

In this part, we will model the real and imaginary parts of the experimental dielectric function of Ag (**Figure 1**). For this we will divide the values of the pulsation ω into several intervals in order to allow us to determine the best fit to suitable mathematical functions over a certain interval. All the results will be presented in **Tables 1**–**8**.

*2.1.1 Modeling the real part of the bulk dielectric function of Silver ε exp <sup>R</sup>*�*Ag* ð Þ *<sup>ω</sup>*

For 1, 899 � <sup>10</sup><sup>14</sup> *rad=s*<sup>≤</sup> *<sup>ω</sup>*≤1, 275 � <sup>10</sup><sup>15</sup> *rad=<sup>s</sup>*

$$\varepsilon\_{R\_{-\lg}}^{\exp}(w) = \varepsilon\_{R\_0}^0 + A\_1 \left( \mathbf{1} - e^{-w/t\_1} \right) + A\_2 \left( \mathbf{1} - e^{-w/t\_2} \right) \tag{4}$$

#### **Figure 1.**

*Real part ( ) and imaginary part ( ) of the dielectric function of bulk Ag ([adapted] with permission from [Ref. [13])© The Optical Society.*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

#### **Figure 2.**

*Real part( ) and imaginary part ( ) of the dielectric function of bulk Au([adapted] with permission from [Ref. [13])© The Optical Society.*


#### **Table 1.**

*Values of the model parameters.*


#### **Table 2.**

*Values of the model parameters.*


#### **Table 3.**

*Values of the model parameters.*


#### **Table 4.**

*Values of the model parameters.*


**Table 5.**

*Values of the model parameters.*


*ε exp*

8 ><

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

>:

*ε exp*

*ε exp*

> *ε exp*

*ε exp*

*ε exp*

**71**

*<sup>R</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>a</sup>*<sup>0</sup>

*ε exp*

*<sup>I</sup>*�*Ag* ð Þ¼ *<sup>ω</sup> <sup>B</sup>*<sup>0</sup>

results are listed in **Tables 9**–**15**.

*ε exp*

<sup>1</sup> þ *B*<sup>0</sup>

*<sup>I</sup>*�*Ag* ð Þ¼ *<sup>ω</sup> <sup>ε</sup>*<sup>1</sup>

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*<sup>I</sup>*�*Ag* ð Þ¼ *<sup>ω</sup> <sup>ε</sup>*<sup>1</sup>

<sup>2</sup> � *B*<sup>0</sup> 1 � � *p*

**2.2 Modeling the experimental dielectric function of bulk Au**

*2.2.1 Modeling the real part ofthe bulk dielectric function of Gold ε*

For 3*:*<sup>319</sup> � <sup>10</sup><sup>14</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>1*:*<sup>515</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

For 1*:*<sup>515</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>3</sup>*:*<sup>345</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

For 3*:*<sup>345</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>4*:*<sup>271</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

For 4*:*<sup>271</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>7</sup>*:*<sup>560</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

For 3*:*<sup>047</sup> � <sup>10</sup><sup>14</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>1</sup>*:*<sup>511</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

For 1*:*<sup>511</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>4</sup>*:*<sup>265</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

*ε exp*

<sup>2</sup> � *a*<sup>0</sup> 1 � � *q*

<sup>1</sup> þ *a*<sup>0</sup>

For 4*:*<sup>252</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>7*:*<sup>453</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

*<sup>I</sup>*<sup>0</sup> *for*ð Þ *ω* <*TD*

*<sup>I</sup>*<sup>0</sup> <sup>þ</sup> *<sup>B</sup>*<sup>0</sup> <sup>1</sup> � *<sup>e</sup>*�ð Þ *<sup>ω</sup>*�*TD*

<sup>1</sup> <sup>þ</sup> <sup>10</sup>ð Þ *Lnω*01�*<sup>ω</sup> <sup>h</sup>*<sup>1</sup>

In this section, we will model the real and imaginary parts of the experimental dielectric function of bulk Au (**Figure 2**). For this we will proceed in the same way as for bulk Ag by dividing the values of the pulsation ω into various intervals. All the

*<sup>R</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>a</sup>*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*2*ω*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*3*ω*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*4*ω*<sup>4</sup> <sup>þ</sup> *<sup>a</sup>*5*ω*<sup>5</sup> <sup>þ</sup> *<sup>a</sup>*6*ω*<sup>6</sup> <sup>þ</sup> *<sup>a</sup>*7*ω*<sup>7</sup> <sup>þ</sup> *<sup>a</sup>*8*ω*<sup>8</sup> <sup>þ</sup> *<sup>a</sup>*9*ω*<sup>9</sup>

<sup>1</sup> <sup>þ</sup> <sup>10</sup> *Lnω*<sup>0</sup> ð Þ <sup>01</sup>�*<sup>ω</sup> <sup>g</sup>*<sup>1</sup>

*<sup>R</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>c</sup>*<sup>0</sup> <sup>þ</sup> *<sup>c</sup>*1*<sup>ω</sup>* <sup>þ</sup> *<sup>c</sup>*2*ω*<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*3*ω*<sup>3</sup> <sup>þ</sup> *<sup>c</sup>*4*ω*<sup>4</sup> <sup>þ</sup> *<sup>c</sup>*5*ω*<sup>5</sup> <sup>þ</sup> *<sup>c</sup>*6*ω*<sup>6</sup> <sup>þ</sup> *<sup>c</sup>*7*ω*<sup>7</sup> <sup>þ</sup> *<sup>c</sup>*8*ω*<sup>8</sup> <sup>þ</sup> *<sup>c</sup>*9*ω*<sup>9</sup>

*<sup>I</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>d</sup>*<sup>0</sup> <sup>þ</sup> *<sup>d</sup>*1*<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*2*ω*<sup>2</sup> <sup>þ</sup> *<sup>d</sup>*3*ω*<sup>3</sup> <sup>þ</sup> *<sup>d</sup>*4*ω*<sup>4</sup> <sup>þ</sup> *<sup>d</sup>*5*ω*<sup>5</sup> <sup>þ</sup> *<sup>d</sup>*6*ω*<sup>6</sup> <sup>þ</sup> *<sup>d</sup>*7*ω*<sup>7</sup> <sup>þ</sup> *<sup>d</sup>*8*ω*<sup>8</sup> <sup>þ</sup> *<sup>d</sup>*9*ω*<sup>9</sup>

*2.2.2 Modeling the imaginary part ofthe bulk dielectric function of Gold ε*

þ

*<sup>R</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>b</sup>*<sup>0</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*2*ω*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*3*ω*<sup>3</sup> <sup>þ</sup> *<sup>b</sup>*4*ω*<sup>4</sup> <sup>þ</sup> *<sup>b</sup>*5*ω*<sup>5</sup> (14)

*<sup>R</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>f</sup>* <sup>0</sup> <sup>þ</sup> *<sup>f</sup>* <sup>1</sup>*<sup>ω</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ω*<sup>2</sup> <sup>þ</sup> *<sup>f</sup>* <sup>3</sup>*ω*<sup>3</sup> <sup>þ</sup> *<sup>f</sup>* <sup>4</sup>*ω*<sup>4</sup> <sup>þ</sup> *<sup>f</sup>* <sup>5</sup>*ω*<sup>5</sup> (17)

1 � *q* <sup>1</sup> <sup>þ</sup> <sup>10</sup> *Lnω*<sup>0</sup> ð Þ <sup>02</sup>�*<sup>ω</sup> <sup>g</sup>*<sup>2</sup>

� � (13)

*τ* � � *for*ð Þ *<sup>ω</sup>*≥*TD*

þ

1 � *p* <sup>1</sup> <sup>þ</sup> <sup>10</sup>ð Þ *Lnω*02�*<sup>ω</sup> <sup>h</sup>*<sup>2</sup> � � (11)

> *exp <sup>R</sup>*�*Au* ð Þ *<sup>ω</sup>*

(10)

(12)

(15)

(16)

*exp <sup>I</sup>*�*Au* ð Þ *<sup>ω</sup>*

#### **Table 6.**

*Values of the model parameters.*


#### **Table 7.**

*Values of the model parameters.*


#### **Table 8.**

*Values of the model parameters.*

For 1, 275 � <sup>10</sup><sup>15</sup> *rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>4, 873 � <sup>10</sup><sup>15</sup> *rad=<sup>s</sup> ε exp <sup>R</sup>*�*Ag* ð Þ¼ *<sup>ω</sup> <sup>A</sup>*<sup>0</sup> 1*e* �*ω=t* 0 <sup>1</sup> þ *A*<sup>0</sup> 2*e* �*ω=t* 0 <sup>2</sup> þ *A*3*e* �*ω=t*<sup>3</sup> <sup>þ</sup> *<sup>ε</sup>*<sup>1</sup> *<sup>R</sup>*<sup>0</sup> (5)

For 4*:*<sup>873</sup> � <sup>10</sup><sup>15</sup> *rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>5*:*<sup>650</sup> � <sup>10</sup><sup>15</sup> *rad=<sup>s</sup>*

$$
\varepsilon\_{R\_{-\text{Ag}}}^{\text{exp}}(o) = A\_1^{\prime \prime} e^{-o/t\_1^{\prime \prime}} + A\_2^{\prime \prime} e^{-o/t\_2^{\prime \prime}} + \varepsilon\_{R\_0}^2 \tag{6}
$$

For 5*:*<sup>650</sup> � <sup>10</sup><sup>15</sup> *rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>6*:*<sup>198</sup> � <sup>10</sup><sup>15</sup> *rad=<sup>s</sup>*

$$
\varepsilon\_{R\_{-\text{Ag}}}^{\text{exp}}(\alpha) = \varepsilon\_{R\_0}^3 + A \sin \left( \pi \frac{\alpha - \alpha\_C}{B} \right) \tag{7}
$$

For 6*:*<sup>198</sup> � 1015 *rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>7*:*<sup>605</sup> � <sup>10</sup><sup>15</sup> *rad=<sup>s</sup>*

$$\begin{cases} \begin{aligned} \varepsilon\_{R\_{-A\_{\xi}}}^{\exp}(o) &= \varepsilon\_{R\_{0}}^{4} + He^{-\frac{1}{2}\left(\frac{a - a\_{C'}}{\alpha\_{1}}\right)^{2}} for(o < a\_{C'})\\ \varepsilon\_{R\_{-A\_{\xi}}}^{\exp}(o) &= \varepsilon\_{R\_{0}}^{4} + He^{-\frac{1}{2}\left(\frac{a - a\_{C'}}{\alpha\_{2}}\right)^{2}} for(o \ge a\_{C'}) \end{aligned} \end{cases} \tag{8}$$

*2.1.2 Modeling the imaginary part of the bulk dielectric function of Silver ε exp <sup>I</sup>*�*Ag* ð Þ *<sup>ω</sup>*

For 1, 900 � <sup>10</sup><sup>14</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>1, 340 � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

$$\varepsilon\_{I\_{-\mathcal{A}\_{\rm g}}}^{\rm exp}(\omega) = \varepsilon\_{I\_0}^0 + B\_1 \mathbf{e}^{-(\alpha - \alpha \mathbf{e})/V\_1} + B\_2 \mathbf{e}^{-(\alpha - \alpha \mathbf{e})/V\_2} + B\_3 \mathbf{e}^{-(\alpha - \alpha \mathbf{e})/V\_3} \tag{9}$$

For 1*:*<sup>340</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>4*:*<sup>251</sup> � 1015*rad=<sup>s</sup>*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

$$\begin{cases} \varepsilon\_{I\_{-\&\xi}}^{\exp}(o) = \varepsilon\_{I\_0}^1 for(o < TD) \\\\ \varepsilon\_{I\_{-\&\xi}}^{\exp}(o) = \varepsilon\_{I\_0}^1 + B' \left(1 - e^{-\frac{(o-TD)}{\mathfrak{r}}}\right) for(o \ge TD) \end{cases} \tag{10}$$

For 4*:*<sup>252</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>7*:*<sup>453</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

$$\varepsilon\_{I\text{\\_}\text{\\_}}^{\text{exp}}(\boldsymbol{\omega}) = \mathbf{B}\_1^\prime + \left(\mathbf{B}\_2^\prime - \mathbf{B}\_1^\prime\right) \left[ \frac{\mathbf{p}}{\mathbf{1} + \mathbf{1}\mathbf{0}^{(Lna\_{01} - \boldsymbol{\omega})\mathbf{b}\_1}} + \frac{\mathbf{1} - \mathbf{p}}{\mathbf{1} + \mathbf{1}\mathbf{0}^{(Lna\_{02} - \boldsymbol{\omega})\mathbf{b}\_2}} \right] \tag{11}$$

#### **2.2 Modeling the experimental dielectric function of bulk Au**

In this section, we will model the real and imaginary parts of the experimental dielectric function of bulk Au (**Figure 2**). For this we will proceed in the same way as for bulk Ag by dividing the values of the pulsation ω into various intervals. All the results are listed in **Tables 9**–**15**.

*2.2.1 Modeling the real part ofthe bulk dielectric function of Gold ε exp <sup>R</sup>*�*Au* ð Þ *<sup>ω</sup>*

For 3*:*<sup>319</sup> � <sup>10</sup><sup>14</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>1*:*<sup>515</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

$$\varepsilon\_{R\_{-4u}}^{\text{exp}}(\boldsymbol{\alpha}) = \boldsymbol{a}\_0 + \boldsymbol{a}\_1 \boldsymbol{\alpha} + \boldsymbol{a}\_2 \boldsymbol{\alpha}^2 + \boldsymbol{a}\_3 \boldsymbol{\alpha}^3 + \boldsymbol{a}\_4 \boldsymbol{\alpha}^4 + \boldsymbol{a}\_5 \boldsymbol{\alpha}^5 + \boldsymbol{a}\_6 \boldsymbol{\alpha}^6 + \boldsymbol{a}\_7 \boldsymbol{\alpha}^7 + \boldsymbol{a}\_8 \boldsymbol{\alpha}^8 + \boldsymbol{a}\_9 \boldsymbol{\alpha}^9 \tag{12}$$

For 1*:*<sup>515</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>3</sup>*:*<sup>345</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

$$\varepsilon\_{R\text{\\_}a}^{\text{exp}}(\boldsymbol{\omega}) = \boldsymbol{a}\_1^\prime + \left(\boldsymbol{a}\_2^\prime - \boldsymbol{a}\_1^\prime\right) \left[ \frac{q}{\mathbf{1} + \mathbf{1}\mathbf{0}^{(Lna\prime\_{\Omega} - a)\mathbf{g}\_1}} + \frac{\mathbf{1} - q}{\mathbf{1} + \mathbf{1}\mathbf{0}^{(Lna\prime\_{\Omega} - a)\mathbf{g}\_2}} \right] \tag{13}$$

For 3*:*<sup>345</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup>4*:*<sup>271</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

$$\varepsilon\_{R\_{-4u}}^{\varepsilon xp}(a) = b\_0 + b\_1 a + b\_2 a^2 + b\_3 a^3 + b\_4 a^4 + b\_5 a^5 \tag{14}$$

For 4*:*<sup>271</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>* <sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>7</sup>*:*<sup>560</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup>*

$$\varepsilon\_{R\_{-4u}}^{\exp}(a\boldsymbol{\rho}) = \varepsilon\_0 + \mathfrak{c}\_1 a \boldsymbol{\rho} + \mathfrak{c}\_2 a \boldsymbol{\rho}^2 + \mathfrak{c}\_3 a \boldsymbol{\rho}^3 + \mathfrak{c}\_4 a \boldsymbol{\rho}^4 + \mathfrak{c}\_5 a \boldsymbol{\rho}^5 + \mathfrak{c}\_6 a \boldsymbol{\rho}^6 + \mathfrak{c}\_7 a \boldsymbol{\rho}^8 + \mathfrak{c}\_8 a \boldsymbol{\rho}^9 \tag{15}$$

*2.2.2 Modeling the imaginary part ofthe bulk dielectric function of Gold ε exp <sup>I</sup>*�*Au* ð Þ *<sup>ω</sup>*

$$\begin{aligned} \text{For } 3.047 \times 10^{14} rad/s \le \alpha \le 1.511 \times 10^{15} rad/s\\ \varepsilon\_{I\_{-3u}}^{\text{exp}}(\boldsymbol{\alpha}) = d\_0 + d\_1 \boldsymbol{\alpha} + d\_2 \boldsymbol{\alpha}^2 + d\_3 \boldsymbol{\alpha}^3 + d\_4 \boldsymbol{\alpha}^4 + d\_5 \boldsymbol{\alpha}^5 + d\_6 \boldsymbol{\alpha}^6 + d\_7 \boldsymbol{\alpha}^7 + d\_8 \boldsymbol{\alpha}^8 + d\_9 \boldsymbol{\alpha}^9 \end{aligned} \tag{16}$$

For 1*:*<sup>511</sup> � <sup>10</sup><sup>15</sup>*rad=s*<sup>≤</sup> *<sup>ω</sup>* <sup>≤</sup> <sup>4</sup>*:*<sup>265</sup> � <sup>10</sup><sup>15</sup>*rad=<sup>s</sup> ε exp <sup>R</sup>*�*Au* ð Þ¼ *<sup>ω</sup> <sup>f</sup>* <sup>0</sup> <sup>þ</sup> *<sup>f</sup>* <sup>1</sup>*<sup>ω</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ω*<sup>2</sup> <sup>þ</sup> *<sup>f</sup>* <sup>3</sup>*ω*<sup>3</sup> <sup>þ</sup> *<sup>f</sup>* <sup>4</sup>*ω*<sup>4</sup> <sup>þ</sup> *<sup>f</sup>* <sup>5</sup>*ω*<sup>5</sup> (17)


**Parameter**

**73**

**Value**

**9873.897**

*d*<sup>5</sup>

2.212 1069

> **Table 13.**

*Values of the model parameters.*

**Parameter**

Value **Table 14.** *Values of the model parameters.*

115.516

1.891 1013

1.299 1028

4.538 1044

7.892 1060

5.361 1076

0.9973

*f* **0**

*f* **1**

**(rad/s)1**

*f* **2**

**(rad/s)2**

*f* **3**

**(rad/s)3**

*f* **4**

**(rad/s)4**

*f* **5**

**(rad/s)5**

**R-Square (COD)**

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

**Parameter**

**Value**

**53432.059**

*j*<sup>5</sup>

2.324 1072

> **Table 15.**

*Values of the model parameters.*

(rad/s)5

*j*<sup>6</sup>

2.918 1088

(rad/s)6

*j*<sup>7</sup>

2.316 10104

(rad/s)7

*j*<sup>8</sup>

1.056 10120

(rad/s)8

*j*<sup>9</sup>

2.111 10137

(rad/s)9

0.9974

**1.046 1013**

**8.722 1026**

**4.106 1041**

**1.210 1056**

*j***<sup>0</sup>**

*j***<sup>1</sup>**

**(rad/s)1**

*j***<sup>2</sup>**

**(rad/s)2**

*j***<sup>3</sup>**

**(rad/s)3**

*j***<sup>4</sup>**

**(rad/s)4**

**R-Square (COD)**

(rad/s)5

*d*6

1.598 1084

(rad/s)6

*d*<sup>7</sup>

7.276 10100

(rad/s)7

*d*<sup>8</sup>

1.895 10115

(rad/s)8

*d*<sup>9</sup>

2.152 10131

(rad/s)9

0.9999

**9.845 1011**

**4.479 1025**

**1.188 1039**

**2.002 1054**

*d***0**

*d***<sup>1</sup>**

**(rad/s)1**

*d***2**

**(rad/s)2**

*d***<sup>3</sup>**

**(rad/s)3**

*d***4**

**(rad/s)4**

**R-Square (COD)**

 *parameters.*


*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

> **Table 15.** *Values of the model parameters.*

$$\begin{aligned} \text{For } 4.265 \times 10^{15} \text{ rad/s} & \le \alpha \le 7.560 \times 10^{15} \text{ rad/s} \\\\ \varepsilon\_{I\_{-4u}}^{\text{exp}}(\boldsymbol{\omega}) &= j\_0 + j\_1 \boldsymbol{\omega} + j\_2 \boldsymbol{\omega}^2 + j\_3 \boldsymbol{\omega}^3 + j\_4 \boldsymbol{\omega}^4 + j\_5 \boldsymbol{\omega}^5 + j\_6 \boldsymbol{\omega}^6 + j\_7 \boldsymbol{\omega}^7 + j\_8 \boldsymbol{\omega}^8 + j\_9 \boldsymbol{\omega}^9 \end{aligned} \tag{18}$$

**3. Highlighting the contribution of interband and intraband transitions**

In metals, there aretwo types of contribution in the dielectric function, namely

The first term corresponds to the intraband component of the dielectric function. It is referred to the optical transitions of a free electron from the conduction band to a higher energy level of the same band. The second term corresponds to the interband component of the dielectric constant. It is referred to optical transitions between the valence bands (mainly d band and s-p conduction bands). Due to Pauli's exclusion principle, an electron from a valence band can only be excited to the conduction band. There is therefore an energy threshold *EIB* for interband transitions which is placed in the visible band for Au and near the UV region for Ag. This component is often overlooked in the infrared range (this is valid only for alkali metals) where the optical response is dominated by intraband absorption.

*ε ω*ð Þ¼ *<sup>ε</sup><sup>D</sup>*ð Þþ *<sup>ω</sup> <sup>ε</sup>iB*ð Þ *<sup>ω</sup>* (19)

contribution of interband transitions denoted *<sup>ε</sup>iB*ð Þ *<sup>ω</sup>* and that of the intraband transitions denoted *<sup>ε</sup><sup>D</sup>*ð Þ *<sup>ω</sup>* . The dielectric function can be written as the sum of two

This type of transitions dominates the optical response beyond *EIB*.

We note that *<sup>γ</sup>* <sup>¼</sup> *<sup>τ</sup>*�<sup>1</sup> and *<sup>τ</sup>* is the elastic diffusion time

well-known free-electronorDrude Model [14]:

Where *ε*<sup>0</sup> is dielectric constant in vacuum.

The intraband part *<sup>ε</sup><sup>D</sup>*ð Þ *<sup>ω</sup>* of the dielectric function is described by the

*<sup>ε</sup><sup>D</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

*<sup>τ</sup>* <sup>¼</sup> *<sup>l</sup>*<sup>0</sup> *v f*

In the Drude model, there appears a pulsation called the plasmon frequency of a

The electronic structure of bulk noble metals such as Ag and Au, the respective values of the conduction electron density *nc*, the effective mass *meff* , the Fermi speed *v <sup>f</sup>* conduction electrons and the mean free path of electron *l*0, are listed in the

The intraband dielectric function described by the Drude model [14] as denoted

ffiffiffiffiffiffiffiffiffiffiffiffi *nce*<sup>2</sup> *<sup>ε</sup>*0*meff* <sup>s</sup>

where *γ* is the collision rate (probability of collision per unit of time).

where *l*<sup>0</sup> is the mean free path of electrons and *v <sup>f</sup>* is the Fermi speed.

*ω<sup>p</sup>* ¼

**3.1 Contribution of intraband transitions to dielectric function**

*p ω ω*ð Þ þ *iγ*

(20)

(21)

(22)

**in the expression of dielectric function**

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

terms

εω = ε1ω + iε2ω.

bulk metal given by

following **Table 16** [15]:

*<sup>ε</sup><sup>D</sup>*ð Þ *<sup>ω</sup>* can be written as

**75**

All the results of these different models for both bulk Ag and bulk Au are plotted in the following **Figures 3** and **4**.

In both cases, the experimental results can be well fitted by the models of experimental dielectric function in both its real and imaginary parts with mathematical functions with high accuracy.

#### **Figure 3.**

*Real ( ) and imaginary ( ) parts of the experimental dielectric function of bulkAg (Ref. [13]). Solid colored curves represent, by pulse intervals, the different mathematical models used in the modeling.*

#### **Figure 4.**

*Real ( ) and imaginary ( ) parts of the experimental dielectric function of bulkAu (Ref. [13]). Solid colored curves represent, by pulse intervals, the different mathematical models used in the modeling.*

## **3. Highlighting the contribution of interband and intraband transitions in the expression of dielectric function**

In metals, there aretwo types of contribution in the dielectric function, namely contribution of interband transitions denoted *<sup>ε</sup>iB*ð Þ *<sup>ω</sup>* and that of the intraband transitions denoted *<sup>ε</sup><sup>D</sup>*ð Þ *<sup>ω</sup>* . The dielectric function can be written as the sum of two terms

$$
\varepsilon(a) = \varepsilon^{D}(a) + \varepsilon^{i\mathcal{B}}(a) \tag{19}
$$

The first term corresponds to the intraband component of the dielectric function. It is referred to the optical transitions of a free electron from the conduction band to a higher energy level of the same band. The second term corresponds to the interband component of the dielectric constant. It is referred to optical transitions between the valence bands (mainly d band and s-p conduction bands). Due to Pauli's exclusion principle, an electron from a valence band can only be excited to the conduction band. There is therefore an energy threshold *EIB* for interband transitions which is placed in the visible band for Au and near the UV region for Ag. This component is often overlooked in the infrared range (this is valid only for alkali metals) where the optical response is dominated by intraband absorption. This type of transitions dominates the optical response beyond *EIB*.

The intraband part *<sup>ε</sup><sup>D</sup>*ð Þ *<sup>ω</sup>* of the dielectric function is described by the well-known free-electronorDrude Model [14]:

$$\varepsilon^{D}(\boldsymbol{\alpha}) = \mathbf{1} - \frac{\boldsymbol{\alpha}\_p^2}{\boldsymbol{\alpha}(\boldsymbol{\alpha} + i\boldsymbol{\gamma})} \tag{20}$$

where *γ* is the collision rate (probability of collision per unit of time). εω = ε1ω + iε2ω.

We note that *<sup>γ</sup>* <sup>¼</sup> *<sup>τ</sup>*�<sup>1</sup> and *<sup>τ</sup>* is the elastic diffusion time

$$
\pi = \frac{l\_0}{v\_f} \tag{21}
$$

where *l*<sup>0</sup> is the mean free path of electrons and *v <sup>f</sup>* is the Fermi speed.

In the Drude model, there appears a pulsation called the plasmon frequency of a bulk metal given by

$$
\rho\_p = \sqrt{\frac{n\_c e^2}{\varepsilon\_0 m\_{\text{eff}}}} \tag{22}
$$

Where *ε*<sup>0</sup> is dielectric constant in vacuum.

The electronic structure of bulk noble metals such as Ag and Au, the respective values of the conduction electron density *nc*, the effective mass *meff* , the Fermi speed *v <sup>f</sup>* conduction electrons and the mean free path of electron *l*0, are listed in the following **Table 16** [15]:

#### **3.1 Contribution of intraband transitions to dielectric function**

The intraband dielectric function described by the Drude model [14] as denoted *<sup>ε</sup><sup>D</sup>*ð Þ *<sup>ω</sup>* can be written as


#### **Table 16.**

*Electronic properties of Ag and Au.*

$$\varepsilon^{D}(\boldsymbol{\alpha}) = \varepsilon\_{R}^{D}(\boldsymbol{\alpha}) + i\varepsilon\_{I}^{D}(\boldsymbol{\alpha}) \tag{23}$$

The real and imaginary parts of the relative dielectric function (intraband) are written as follows

$$\varepsilon\_R^D(\boldsymbol{\alpha}) = 1 - \frac{\boldsymbol{\alpha}\_p^2}{\boldsymbol{\alpha}^2 + \boldsymbol{\gamma}^2} \tag{24}$$

$$
\epsilon\_I^D(w) = \frac{o\_p^2 \chi}{o(o^2 + \chi^2)}\tag{25}
$$

Usually, for noble metals *ω* ≫ *γ* in the near UVrange and up to the near IR, we can write

$$\varepsilon\_R^D(\alpha) = \mathbf{1} - \frac{\alpha\_p^2}{\alpha^2} \tag{26}$$

$$
\epsilon\_I^D(\alpha) = \frac{\alpha\_p^2 \gamma}{\alpha^3} \tag{27}
$$

The following **Table 17** shows the values of plasma frequencies and the collision rate of noble metals (Gold, Silver):

The results of the calculations of the contribution of intraband effects to dielectric function are represented in their real and imaginary parts for bulk Ag (**Figure 5**) and for bulk Au (**Figure 6**).

For the noble metals (Gold, Silver), we observed that the real and imaginary parts decrease withincreasing pulsation. The further away from the pulsations corresponding to IR radiation and the closer we get to the pulsations corresponding to UV radiation, these values decrease.

#### **3.2 Contribution of interband transitions to dielectric function**

The interband dielectric function denoted *<sup>ε</sup>iB*ð Þ *<sup>ω</sup>* is described by the following term


*<sup>ε</sup>iB*ð Þ¼ *<sup>ω</sup> <sup>ε</sup>iB <sup>R</sup>* ð Þþ *<sup>ω</sup> <sup>i</sup>εiB <sup>I</sup>* ð Þ *ω* (28) The real and imaginary parts of the interband dielectric function are written

*exp <sup>R</sup>* ð Þ� *<sup>ω</sup> <sup>ε</sup><sup>D</sup>*

*exp <sup>I</sup>* ð Þ� *<sup>ω</sup> <sup>ε</sup><sup>D</sup>*

*exp <sup>R</sup>* ð Þþ *ω iε*

*exp*

*<sup>I</sup>* ð Þ *ω* are the experimental values, real and

*<sup>R</sup>* ð Þ *ω* (29)

*<sup>I</sup>* ð Þ *ω* (30)

*<sup>I</sup>* ð Þ *ω* (31)

εiB <sup>R</sup> ð Þ¼ ω *ε*

*Real ( ) imaginary part ( ) of intraband dielectric function of bulk Au.*

*Real ( ) imaginary part ( ) of the intraband dielectric function of bulk Ag.*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

εiB *<sup>I</sup>* ð Þ¼ ω *ε*

<sup>ε</sup>*exp*ð Þ¼ <sup>ω</sup> *<sup>ε</sup>*

*exp*

imaginary parts of the complex dielectric function, respectively.

respectively as follows:

where

**77**

**Figure 6.**

**Figure 5.**

Here <sup>ε</sup>*exp*ð Þ <sup>ω</sup> , *<sup>ε</sup>*

*exp*

*<sup>R</sup>* ð Þ *ω* , *and ε*

**Table 17.**

*Plasma frequency and collision rate values for Ag and Au.*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

#### **Figure 5.**

*Real ( ) imaginary part ( ) of the intraband dielectric function of bulk Ag.*

#### **Figure 6.**

*Real ( ) imaginary part ( ) of intraband dielectric function of bulk Au.*

The real and imaginary parts of the interband dielectric function are written respectively as follows:

$$
\varepsilon\_{\rm R}^{\rm iB}(o) = \varepsilon\_{\rm R}^{\rm exp}(o) - \varepsilon\_{\rm R}^{D}(o) \tag{29}
$$

$$
\varepsilon\_I^{\rm iB}(\alpha) = \varepsilon\_I^{\rm exp}(\alpha) - \varepsilon\_I^D(\alpha) \tag{30}
$$

where

$$
\varepsilon^{\exp}(\alpha) = \varepsilon\_R^{\exp}(\alpha) + i \varepsilon\_I^{\exp}(\alpha) \tag{31}
$$

Here <sup>ε</sup>*exp*ð Þ <sup>ω</sup> , *<sup>ε</sup> exp <sup>R</sup>* ð Þ *ω* , *and ε exp <sup>I</sup>* ð Þ *ω* are the experimental values, real and imaginary parts of the complex dielectric function, respectively.

The results of the calculations of the contribution of interband transitions to real and imaginary parts of dielectric function are represented respectively for bulk Ag (**Figure 7**) and for bulk Au (**Figure 8**).

We note that for Gold, the real and imaginary parts of the contribution of interband transitions decrease by increasing the values of the pulses in the IR domain to the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>5</sup> � 1015rad*=*s with a real part almost higher than the imaginary part. For pulsation values in the UV range above the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>5</sup> � 1015rad*=*s, with

In **Figures 9** and **10** (for Ag) and **Figures 11** and **12** (for Au), we have presented experimental values, the contributions of intraband and interband transitions to the

Concerning the real part, we note that, the real part due to the intraband transi-

domain corresponding to IR radiation and up to the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup> � 1015rad*=*s. In this range of pulsations, we can conclude that the participation in the dielectric function

*Real parts of the dielectric function of Ag: experimental values ( ) ([Ref. [13]), intraband transitions ( ),*

*Imaginary parts of the dielectric function of Ag: experimental values ( ) ([Ref. [13]), intraband transitions*

*<sup>R</sup>* ð Þ *ω* is in very good agreement with the experimental values in the full

a slight variation the imaginary part becomes superior to the real part.

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

real and imaginary parts of the dielectric function.

tions noted *ε<sup>D</sup>*

**Figure 9.**

**Figure 10.**

**79**

*( ), interband transitions ( ).*

*interband transitions ( ).*

As shown in **Figure 7**, the real part of the contribution of interband effects to the dielectric function of bulk Ag decreases with increasing pulsation in the IR radiation domain then is still almost constant with small variations from *<sup>ω</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>15rad*=*<sup>s</sup> and this until the end of the UV radiation range. Concerning the imaginary part of this contribution, it is almost less important than the real part. It decreases with increasing pulsation in the range corresponding to IR radiation, then varies very little to the value of the pulsation *<sup>ω</sup>* <sup>¼</sup> <sup>6</sup> � <sup>10</sup>15rad*=*s in the range of UV radiation, in this area increase from 0.4 to 1.4 and finally remains constant for the rest of the UV pulses. The imaginary part becomes superior to the real part for UV pulses higher than the value *<sup>ω</sup>* <sup>¼</sup> <sup>6</sup> � <sup>10</sup>15rad*=*s.

**Figure 7.** *Real ( ) and imaginary part ( ) of the interband dielectric function of bulk Ag.*

**Figure 8.** *Real ( ) and imaginary part ( ) of the interband dielectric function of bulk Au.*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

We note that for Gold, the real and imaginary parts of the contribution of interband transitions decrease by increasing the values of the pulses in the IR domain to the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>5</sup> � 1015rad*=*s with a real part almost higher than the imaginary part. For pulsation values in the UV range above the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup>*:*<sup>5</sup> � 1015rad*=*s, with a slight variation the imaginary part becomes superior to the real part.

In **Figures 9** and **10** (for Ag) and **Figures 11** and **12** (for Au), we have presented experimental values, the contributions of intraband and interband transitions to the real and imaginary parts of the dielectric function.

Concerning the real part, we note that, the real part due to the intraband transitions noted *ε<sup>D</sup> <sup>R</sup>* ð Þ *ω* is in very good agreement with the experimental values in the full domain corresponding to IR radiation and up to the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup> � 1015rad*=*s. In this range of pulsations, we can conclude that the participation in the dielectric function

**Figure 9.**

*Real parts of the dielectric function of Ag: experimental values ( ) ([Ref. [13]), intraband transitions ( ), interband transitions ( ).*

#### **Figure 10.**

*Imaginary parts of the dielectric function of Ag: experimental values ( ) ([Ref. [13]), intraband transitions ( ), interband transitions ( ).*

corresponding to IR radiation and up to the value *<sup>ω</sup>* <sup>¼</sup> <sup>2</sup> � <sup>10</sup>15rad*=*s. In this range of pulsations, we can conclude that the participation in the dielectric function due to interband transitions is negligible compared to that of intraband transitions. For *<sup>ω</sup>*<sup>&</sup>gt; <sup>2</sup> � <sup>10</sup>15rad*=*s and up to the UV radiation domain, the two contributions due to intraband and interband transitions differ significantly from the experimental measurements. These two contributions are very similar in the field of UV for

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

Concerning the imaginary part, as for Silver, we see that the imaginary part due

In this section, we study the dielectric functions of nanometric Ag and nanometric

The collective excitation of nanoparticle conduction electrons gives them new

For the study of resonant nanostructures, it is important to have a good description of the permittivity of the metal in a large frequency band. For this purpose, the validity band of the Drude Model is often extended by adding Lorentzian terms [16]

> þ<sup>X</sup> 2

The dielectric function described by the Drude Lorentz model is written as follows

*DL*ð Þþ *<sup>ω</sup> <sup>i</sup>εim*

*l*¼1

*Ω*2

*fl Ω*2 *l*

*<sup>l</sup>* � *<sup>ω</sup>*<sup>2</sup> � *<sup>i</sup>*Γ*l<sup>ω</sup>* (32)

*DL*ð Þ *ω* (33)

*D ω*<sup>2</sup> þ *iγω*

*<sup>ε</sup>DL*ð Þ¼ *<sup>ω</sup> <sup>ε</sup><sup>R</sup>*

• Plasmons guided along a metallic film of nanometric cross-section

• Surface plasmons located in a metallic particle of nanometric size

Au. They are composed of a few tens to several thousand atoms. Their very small characteristic dimensions, in the nanometer range (i.e. well under optical wavelengths), give rise to extraordinary electronic and optical properties that cannot be observed in bulk materials. These properties are clearly influenced by the size, form of the nanoparticle and the nature of the host environment. We consider the measured values of dielectric function used in the previous paragraphs, and try to model those using theoretical models for nanometals such as the Drude Lorentz (DL) model, the Drude two-point critical model DCP and the Drude three-point critical model DCP3. The optical properties of metallic nanoparticles are dominated by the collective oscillation of conduction electrons induced by interaction with electromagnetic

experimental part in the whole field of pulsations from the IR domain to the end of the UV domain. This indicates that the imaginary part due to intraband transitions

is negligible compared to the imaginary part due to interband transitions.

**4. Modeling the dielectric functions of nanometri cAg and**

optical properties; we consider the following two effects:

*<sup>ε</sup>DL*ð Þ¼ *<sup>ω</sup> <sup>ε</sup>*<sup>∞</sup> � *<sup>ω</sup>*<sup>2</sup>

*<sup>I</sup>* ð Þ ω is practically confused with the imaginary

*<sup>ω</sup>*<sup>≥</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>15rad*=*s.

**nanometric Au**

radiation (IR, UV).

(1D confinement).

(0D confinement).

**4.1 Drude Lorentz (DL) model**

depending in the following form

**81**

to the interband transitions noted εiB

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

*Real parts of the dielectric function of Au: experimental values ( ) ([Ref. [13]), intraband transitions ( ), interband transitions ( ).*

#### **Figure 12.**

*Imaginary parts of the dielectric function of Au: experimental values ( ) ([Ref. [13]), intraband transitions ( ),interband transitions ( ).*

due to inter-band transitions is negligible compared to that of intra-band transitions. For *<sup>ω</sup>*<sup>&</sup>gt; <sup>4</sup>*:*<sup>5</sup> � 1015*rad=<sup>s</sup>* in the range of UV radiation, the two contributions due to intraband and interband transitions are equivalent but differ from the experimental measurements.

Concerning the imaginary part, we note that the imaginary part due to the interband transitions noted εiB *<sup>I</sup>* ð Þ ω is practically confused with the imaginary experimental part in the whole field of pulsations from the IR domain to the end of the UV domain. This indicates that the imaginary part due to intraband transitions is negligible compared to the imaginary part due to interband transitions.

For Gold, we note that, the real part due to the intraband transitions noted *ε<sup>D</sup> <sup>R</sup>* ð Þ *ω* is in very good agreement with the real experimental values in the whole range

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

corresponding to IR radiation and up to the value *<sup>ω</sup>* <sup>¼</sup> <sup>2</sup> � <sup>10</sup>15rad*=*s. In this range of pulsations, we can conclude that the participation in the dielectric function due to interband transitions is negligible compared to that of intraband transitions. For *<sup>ω</sup>*<sup>&</sup>gt; <sup>2</sup> � <sup>10</sup>15rad*=*s and up to the UV radiation domain, the two contributions due to intraband and interband transitions differ significantly from the experimental measurements. These two contributions are very similar in the field of UV for *<sup>ω</sup>*<sup>≥</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>15rad*=*s.

Concerning the imaginary part, as for Silver, we see that the imaginary part due to the interband transitions noted εiB *<sup>I</sup>* ð Þ ω is practically confused with the imaginary experimental part in the whole field of pulsations from the IR domain to the end of the UV domain. This indicates that the imaginary part due to intraband transitions is negligible compared to the imaginary part due to interband transitions.

## **4. Modeling the dielectric functions of nanometri cAg and nanometric Au**

In this section, we study the dielectric functions of nanometric Ag and nanometric Au. They are composed of a few tens to several thousand atoms. Their very small characteristic dimensions, in the nanometer range (i.e. well under optical wavelengths), give rise to extraordinary electronic and optical properties that cannot be observed in bulk materials. These properties are clearly influenced by the size, form of the nanoparticle and the nature of the host environment. We consider the measured values of dielectric function used in the previous paragraphs, and try to model those using theoretical models for nanometals such as the Drude Lorentz (DL) model, the Drude two-point critical model DCP and the Drude three-point critical model DCP3.

The optical properties of metallic nanoparticles are dominated by the collective oscillation of conduction electrons induced by interaction with electromagnetic radiation (IR, UV).

The collective excitation of nanoparticle conduction electrons gives them new optical properties; we consider the following two effects:


#### **4.1 Drude Lorentz (DL) model**

For the study of resonant nanostructures, it is important to have a good description of the permittivity of the metal in a large frequency band. For this purpose, the validity band of the Drude Model is often extended by adding Lorentzian terms [16] depending in the following form

$$\varepsilon\_{\rm DL}(\boldsymbol{\alpha}) = \varepsilon\_{\infty} - \frac{\alpha\_{\rm D}^2}{\alpha^2 + i\gamma\alpha} + \sum\_{l=1}^2 \frac{f\_l \Omega\_l^2}{\Omega\_l^2 - \alpha^2 - i\Gamma\_l \alpha} \tag{32}$$

The dielectric function described by the Drude Lorentz model is written as follows

$$
\varepsilon\_{\rm DL}(o) = \varepsilon\_{\rm DL}^{\mathbb{R}}(o) + i \varepsilon\_{\rm DL}^{\rm im}(o) \tag{33}
$$

where:

The real part of the dielectric function according to the DL model

$$\varepsilon\_{\rm DL}^{\rm R}(\boldsymbol{\alpha}) = \varepsilon\_{\rm os} - \frac{\boldsymbol{\alpha}\boldsymbol{\alpha}^{2}}{\boldsymbol{\alpha}^{2} + \boldsymbol{\eta}^{2}} + \frac{f\_{1}\mathfrak{Q}\_{1}^{2}\big(\varOmega\_{1}^{2} - \boldsymbol{\alpha}^{2}\big)}{\left(\varOmega\_{1}^{2} - \boldsymbol{\alpha}^{2}\right)^{2} + \left(\varGamma\_{1}\boldsymbol{\alpha}\right)^{2}} + \frac{f\_{2}\mathfrak{Q}\_{2}^{2}\big(\varOmega\_{2}^{2} - \boldsymbol{\alpha}^{2}\big)}{\left(\varOmega\_{2}^{2} - \boldsymbol{\alpha}^{2}\right)^{2} + \left(\varGamma\_{2}\boldsymbol{\alpha}\right)^{2}}\tag{34}$$

The imaginary part of the dielectric function according to the DL model:

$$\epsilon\_{DL}^{\rm im}(\boldsymbol{\omega}) = \frac{\gamma \boldsymbol{\omega}\_{\rm D}}{\boldsymbol{\omega}^3 + \gamma^2 \boldsymbol{\omega}} + \frac{f\_1 \Omega\_1^2 \Gamma\_1 \boldsymbol{\omega}}{\left(\varOmega\_1^2 - \varOmega^2\right)^2 + \left(\Gamma\_1 \boldsymbol{\omega}\right)^2} + \frac{f\_2 \Omega\_2^2 \Gamma\_2 \boldsymbol{\omega}}{\left(\varOmega\_2^2 - \varOmega^2\right)^2 + \left(\Gamma\_2 \boldsymbol{\omega}\right)^2} \tag{35}$$

The studies of Vial and Laroche [16] on the permittivity of Auand Ag metals used in their model with the parameters are listed in **Table 18**.

The results of modeling the experimental dielectric function in its real and imaginary parts using the DL model are shown in **Figures 13** and **14** for Au(in **Figures 15** and **16** for Ag).

#### **4.2 Drude model with two critical points DCP**

In order to describe the metal in the largest possible range of pulsations, another formula describing the two-point critical Drude model (DCP) [16] will appear in this paragraph.

The dielectric function of Au and Ag is can be expressed from as [17]:

$$\varepsilon\_{\rm DCP}(\boldsymbol{\alpha}) = \varepsilon\_{\rm os} - \frac{\alpha\_{\rm D}^2}{\alpha^2 + i\gamma\alpha} + \sum\_{l=1}^2 A\_l \Omega\_l \left[ \frac{e^{i\phi}}{\Omega\_l - \boldsymbol{\alpha} - i\boldsymbol{\Gamma}\_l} + \frac{e^{-i\phi}}{\Omega\_l + \boldsymbol{\alpha} + i\boldsymbol{\Gamma}\_l} \right] \tag{36}$$

The dielectric function described by the Drude two-critical-point model is written as follows

$$\varepsilon\_{\rm DCP}(o) = \varepsilon\_{\rm DCP}^{\rm R}(o) + i \varepsilon\_{\rm DCP}^{\rm im}(o) \tag{37}$$

*εR*

**Figure 14.**

**Figure 13.**

*εim*

**83**

*DCP*ð Þ¼ *<sup>ω</sup> <sup>ε</sup>*<sup>∞</sup> � *<sup>ω</sup><sup>D</sup>*

*DCP*ð Þ¼ *<sup>ω</sup> γω<sup>D</sup>*

þ<sup>X</sup> 2

*the DCP model ( ) and the DCP3model ( ).*

*l*¼1

*( ), the DCPmodel ( ), and the DCP3 model ( ).*

2 *ω*<sup>3</sup> þ *γ*<sup>2</sup>*ω* þ<sup>X</sup> 2

*AlΩ<sup>l</sup>*

*l*¼1

2 *ω*<sup>2</sup> þ *γ*<sup>2</sup>

> ð Þ *Ω<sup>l</sup>* � *ω* cos *ϕ<sup>l</sup>* � Γ*<sup>l</sup>* sin *ϕ<sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* � *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup><sup>2</sup>

*Imaginary part of the dielectric function of nanometric Ag:experimental values ( ) ([Ref. [13]), the DL model*

*Real part of the dielectric function of nanometric Ag: experimental values ( ) ([Ref. [13]), the DL model ( ),*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

The imaginary part of the dielectric function according to the DCP model:

ð Þ *Ω<sup>l</sup>* � *ω* sin *ϕ<sup>l</sup>* þ Γ*<sup>l</sup>* cos *ϕ<sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* � *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup><sup>2</sup>

*l*

*l*

" #

" #

<sup>þ</sup> ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* cos *<sup>ϕ</sup><sup>l</sup>* � <sup>Γ</sup>*<sup>l</sup>* sin *<sup>ϕ</sup><sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup>*<sup>l</sup>*

<sup>þ</sup> ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* sin *<sup>ϕ</sup><sup>l</sup>* � <sup>Γ</sup>*<sup>l</sup>* cos *<sup>ϕ</sup><sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup><sup>2</sup>

2

*l*

(38)

(39)

*AlΩ<sup>l</sup>*

where

The real part of the dielectric function according to the DCP model:


#### **Table 18.**

*Optimized parameters of the Drude Lorentz model for noble metals (Au and Ag).*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

#### **Figure 13.**

*Real part of the dielectric function of nanometric Ag: experimental values ( ) ([Ref. [13]), the DL model ( ), the DCP model ( ) and the DCP3model ( ).*

#### **Figure 14.**

*Imaginary part of the dielectric function of nanometric Ag:experimental values ( ) ([Ref. [13]), the DL model ( ), the DCPmodel ( ), and the DCP3 model ( ).*

$$\begin{split} \varepsilon\_{\text{DCP}}^{R}(\boldsymbol{\omega}) &= \varepsilon\_{\text{ox}} - \frac{\boldsymbol{\alpha}\_{\text{D}}}{\boldsymbol{\alpha}^{2} + \boldsymbol{\eta}^{2}} \\ &+ \sum\_{l=1}^{2} \boldsymbol{A}\_{l} \boldsymbol{\Omega}\_{l} \bigg[ \frac{\left(\boldsymbol{\Omega}\_{l} - \boldsymbol{\omega}\right) \cos \phi\_{l} - \boldsymbol{\Gamma}\_{l} \sin \phi\_{l}}{\left(\boldsymbol{\Omega}\_{l} - \boldsymbol{\omega}\right)^{2} + \boldsymbol{\Gamma}\_{l}^{2}} + \frac{\left(\boldsymbol{\Omega}\_{l} + \boldsymbol{\omega}\right) \cos \phi\_{l} - \boldsymbol{\Gamma}\_{l} \sin \phi\_{l}}{\left(\boldsymbol{\Omega}\_{l} + \boldsymbol{\omega}\right)^{2} + \boldsymbol{\Gamma}\_{l}^{2}} \end{split} \tag{38}$$

The imaginary part of the dielectric function according to the DCP model:

$$\begin{split} \epsilon\_{\text{DCP}}^{\text{im}}(\boldsymbol{\alpha}) &= \frac{\gamma \boldsymbol{\alpha}\_{\text{D}}^{\text{2}}}{\boldsymbol{\alpha}^{3} + \boldsymbol{\gamma}^{2} \boldsymbol{\alpha}} \\ &+ \sum\_{l=1}^{2} A\_{l} \boldsymbol{\Omega}\_{l} \left[ \frac{\left(\boldsymbol{\Omega}\_{l} - \boldsymbol{\alpha}\right) \sin \phi\_{l} + \boldsymbol{\Gamma}\_{l} \cos \phi\_{l}}{\left(\boldsymbol{\Omega}\_{l} - \boldsymbol{\alpha}\right)^{2} + \boldsymbol{\Gamma}\_{l}^{2}} + \frac{\left(\boldsymbol{\Omega}\_{l} + \boldsymbol{\alpha}\right) \sin \phi\_{l} - \boldsymbol{\Gamma}\_{l} \cos \phi\_{l}}{\left(\boldsymbol{\Omega}\_{l} + \boldsymbol{\alpha}\right)^{2} + \boldsymbol{\Gamma}\_{l}^{2}} \right] \end{split} \tag{39}$$

**Figure 15.**

*Real part of the dielectric function of nanometric Au: experimental values ( ) ([Ref. [13]), the DLmodel ( ), the DCP model ( ), and the DCP3 model ( ).*

*<sup>ε</sup>DCP*3ð Þ¼ *<sup>ω</sup> <sup>ε</sup>*<sup>∞</sup> <sup>þ</sup> *<sup>σ</sup>*

*ε***<sup>∞</sup>** *ω<sup>D</sup>*

*A*<sup>2</sup> *ϕ*<sup>2</sup>

*From Appl. Phys.B 93, 140 (2008).*

**(rad/s)**

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

(rad)

written as follows:

**Table 19.**

*(Gold, Silver).*

Where:

*DCP*3ð Þ¼ *<sup>ω</sup> <sup>ε</sup>*<sup>∞</sup> <sup>þ</sup><sup>X</sup>

*=ε*0 *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> 3

pulsation zone in the IR domain.

3

*AlΩ<sup>l</sup>*

*AlΩ<sup>l</sup>*

The parameters of this model are given in **Table 20**.

**Figures 13** and **14** for Au (in **Figures 15** and **16** for Ag).

*l*¼1

*l*¼1

*εR*

*εim*

**85**

*DCP*3ð Þ¼ *<sup>ω</sup>* �*<sup>σ</sup>*

*=ε*0 *<sup>i</sup><sup>ω</sup>* <sup>þ</sup><sup>X</sup> 3

*l*¼1

*<sup>ε</sup>DCP*3ð Þ¼ *<sup>ω</sup> <sup>ε</sup><sup>R</sup>*

The real part of the dielectric function in the DCP3 model:

The imaginary part of the dielectric function in the DCP3 model:

*AlΩ<sup>l</sup>*

The dielectric function described by the Drude two-critical-point model is

*Optimized parameters of the Drude two-point critical point model DCP, dielectric function for noble metals*

ð Þ *Ω<sup>l</sup>* � *ω* cos *ϕ<sup>l</sup>* � Γ*<sup>l</sup>* sin *ϕ<sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* � *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup><sup>2</sup>

ð Þ *Ω<sup>l</sup>* � *ω* sin *ϕ<sup>l</sup>* þ Γ*<sup>l</sup>* cos *ϕ<sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* � *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup>*<sup>l</sup>*

The results of modeling the real and imaginary parts of experimental dielectric

Concerning the real part of the dielectric function of nanometric Ag; the model that is very much in agreement with the experiment up to the value of the pulsation *<sup>ω</sup>*<sup>≈</sup> <sup>6</sup> � <sup>10</sup><sup>15</sup> rad*=*s, placed in the UV region, is the DCP model. The other models are also valid up to the value of *<sup>ω</sup>*≈<sup>3</sup> � <sup>10</sup><sup>15</sup> rad*=*s. This interval covers the whole

For the imaginary part of nanometric Ag, we find that for pulsations located in the IR domain and less than *<sup>ω</sup>*<1*:*<sup>5</sup> � <sup>10</sup>15rad*=*s, the most appropriate model with experience in this range is the DCP3 model. For pulsation values 1*:*5 �

<sup>10</sup>15rad*=*s<*ω*<sup>&</sup>lt; <sup>6</sup> � <sup>10</sup>15rad*=*s, the most suitable model to the measured values of the

function using the Drude three-point critical point model DCP3 are shown in

*eiϕ Ω<sup>l</sup>* � *ω* � *i*Γ*<sup>l</sup>*

**γ (rad/s)** *A***<sup>1</sup>** *ϕ***<sup>1</sup>**

Γ2 (rad/s)

Au 1.1431 1.3202 � <sup>10</sup><sup>16</sup> 1.0805 � <sup>10</sup><sup>14</sup> 0.26698 �1.2371 3.8711 � <sup>10</sup><sup>15</sup> Ag 15.833 1.3861 � <sup>10</sup><sup>16</sup> 4.5841 � <sup>10</sup><sup>13</sup> 1.0171 �0.93935 6.6327 � <sup>10</sup><sup>15</sup>

> *Ω*2 (rad/s)

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

Au 3.0834 �1.0968 4.1684 � 1015 2.3555 � 1015 4.4642 � <sup>10</sup><sup>14</sup> Ag 15.797 1.8087 9.2726 � 1017 2.3716 � 1017 1.6666 � 1015

*DCP*3ð Þþ *<sup>ω</sup> <sup>i</sup>εim*

*l*

þ

" #

" #

*e*�*i<sup>ϕ</sup> Ω<sup>l</sup>* þ *ω* þ *i*Γ*<sup>l</sup>* � � (40)

**(rad)**

Γ1 (rad/s)

*Ω***1 (rad/s)**

*DCP*3ð Þ *ω* (41)

<sup>þ</sup> ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* cos *<sup>ϕ</sup><sup>l</sup>* � <sup>Γ</sup>*<sup>l</sup>* sin *<sup>ϕ</sup><sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup>*<sup>l</sup>*

<sup>2</sup> � ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* sin *<sup>ϕ</sup><sup>l</sup>* <sup>þ</sup> *<sup>Ω</sup><sup>l</sup>* cos *<sup>ϕ</sup><sup>l</sup>* ð Þ *<sup>Ω</sup><sup>l</sup>* <sup>þ</sup> *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>Γ</sup>*<sup>l</sup>*

2

2

(43)

(42)

**Figure 16.**

*Imaginary part of dielectric of nanometric Au: experimental values ( ) ([Ref. [13]), the DL model ( ), the DCP model ( ), and theDCP3 model ( ).*

The work of Alexandre Vial's [17] on the permittivity of the noble metals (Gold, Silver) made model with the parameters are listed in **Table 19**.

The results of modeling the experimental dielectric function in its real and imaginary parts using the Drude two-point DCP critical point model are shown in **Figures 13** and **14** for Au (in **Figures 15** and **16** for Ag).

#### **4.3 Drude model with three critical points DCP3**

The DCP3 model describes the response of the dielectric function in a wider pulsation band, it should be noted that the DCP3 model gives a very good description of the dielectric function of noble metals; it is expressed by the relation [18]:


*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

#### **Table 19.**

*Optimized parameters of the Drude two-point critical point model DCP, dielectric function for noble metals (Gold, Silver).*

$$\varepsilon\_{\rm DCP3}(\boldsymbol{\omega}) = \varepsilon\_{\rm os} + \frac{\sigma/\varepsilon\_{\rm to}}{i\alpha} + \sum\_{l=1}^{3} A\_l \Omega\_l \left[ \frac{e^{i\phi}}{\Omega\_l - \omega - i\Gamma\_l} + \frac{e^{-i\phi}}{\Omega\_l + \omega + i\Gamma\_l} \right] \tag{40}$$

The dielectric function described by the Drude two-critical-point model is written as follows:

$$
\varepsilon\_{\rm DCP3}(o) = \varepsilon\_{\rm DCP3}^{\mathbb{R}}(o) + i \varepsilon\_{\rm DCP3}^{im}(o) \tag{41}
$$

Where:

The real part of the dielectric function in the DCP3 model:

$$\varepsilon\_{\rm DCP3}^{R}(\boldsymbol{\omega}) = \varepsilon\_{\rm os} + \sum\_{l=1}^{3} A\_{l} \Omega\_{l} \left[ \frac{(\varOmega\_{l} - \boldsymbol{\omega}) \cos \phi\_{l} - \varGamma\_{l} \sin \phi\_{l}}{(\varOmega\_{l} - \boldsymbol{\omega})^{2} + \varGamma\_{l}^{2}} + \frac{(\varOmega\_{l} + \boldsymbol{\omega}) \cos \phi\_{l} - \varGamma\_{l} \sin \phi\_{l}}{(\varOmega\_{l} + \boldsymbol{\omega})^{2} + \varGamma\_{l}^{2}} \right] \tag{42}$$

The imaginary part of the dielectric function in the DCP3 model:

$$\varepsilon\_{\rm DCP3}^{\rm im}(\boldsymbol{\alpha}) = \frac{-\sigma\_{\boldsymbol{\varepsilon}\_{0}}}{\boldsymbol{\alpha}} + \sum\_{l=1}^{3} A\_{l} \Omega\_{l} \left[ \frac{(\varOmega\_{l} - \boldsymbol{\alpha})\sin\phi\_{l} + \varGamma\_{l}\cos\phi\_{l}}{(\varOmega\_{l} - \boldsymbol{\alpha})^{2} + \varGamma\_{l}^{2}} - \frac{(\varOmega\_{l} + \boldsymbol{\alpha})\sin\phi\_{l} + \varOmega\_{l}\cos\phi\_{l}}{(\varOmega\_{l} + \boldsymbol{\alpha})^{2} + \varGamma\_{l}^{2}} \right] \tag{43}$$

The parameters of this model are given in **Table 20**.

The results of modeling the real and imaginary parts of experimental dielectric function using the Drude three-point critical point model DCP3 are shown in **Figures 13** and **14** for Au (in **Figures 15** and **16** for Ag).

Concerning the real part of the dielectric function of nanometric Ag; the model that is very much in agreement with the experiment up to the value of the pulsation *<sup>ω</sup>*<sup>≈</sup> <sup>6</sup> � <sup>10</sup><sup>15</sup> rad*=*s, placed in the UV region, is the DCP model. The other models are also valid up to the value of *<sup>ω</sup>*≈<sup>3</sup> � <sup>10</sup><sup>15</sup> rad*=*s. This interval covers the whole pulsation zone in the IR domain.

For the imaginary part of nanometric Ag, we find that for pulsations located in the IR domain and less than *<sup>ω</sup>*<1*:*<sup>5</sup> � <sup>10</sup>15rad*=*s, the most appropriate model with experience in this range is the DCP3 model. For pulsation values 1*:*5 � <sup>10</sup>15rad*=*s<*ω*<sup>&</sup>lt; <sup>6</sup> � <sup>10</sup>15rad*=*s, the most suitable model to the measured values of the



dielectric function is still the DCP3 model. For

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

*ω* ≈ 5

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

range corresponding to IR radiation and up to the value

� <sup>10</sup>15rad

*ω* ≥ 2 *:* 5

compared to that of intraband transitions. For

better than the DCP model.

the value of the pulsation

*<sup>ω</sup>*<sup>≈</sup> <sup>5</sup> � <sup>10</sup>15rad*=*s.

**5. Conclusion**

denoted

*ω* > 4 *:* 5

noted

**87**

� <sup>10</sup>15rad

up to the value

*=*

*ω* ¼ 2 *:* 5

similar in the field of UV for

values (**Figure 15**). From the value of

*ω* > 6

� <sup>10</sup>15rad

the two models DCP and DCP3 are very close to the experiment. The DCP3 model is

In the case of the real part of the dielectric function of nanometric Au, the DL, DCP, and DCP3 models are allin very good agreement with the experiment up to

radiation region. Beyond this value, the three models deviate from the measured

In this work, we modeled the dielectric function of noble metals (silver and gold) in their bulk and nanometric states. Initially, we modeled the measured dielectric functions of these two metals using explicit mathematical functions and the results are in very good agreement with the experiment. Moreover, we have decomposed these measured values of the dielectric functions; in their real and imaginary parts; into several intervals according to the pulsations that sweep the domains corresponding to IR and UV radiation via the intermediate values. The obtained results are very conclusive, and depending on the pulsation domain studied, it is possible to use the corresponding mathematical function in simulations and calculations. Then, we highlighted the importance of the contributions of intraband and interband transitions in dielectric function for both Ag and Au. For Ag, we note that the imaginary part of the dielectric function due to interband transitions

<sup>ε</sup>iB*<sup>I</sup>* ð Þ <sup>ω</sup> is almost the same with the imaginary experimental part in the whole field of pulsations from the IR domain to the end of the UV domain. This indicates that the imaginary part due to intraband transitions is negligible compared to the imaginary part due to interband transitions. Concerning the real part of the dielectric function, we note that, the real part due to the intraband transitions noted *<sup>ε</sup>DR* ð Þ *<sup>ω</sup>* is in very good agreement with the real experimental values in the entire

range of pulsation, we can conclude that the contribution tothe dielectric function due to interband transitions is negligible compared to that of intraband transitions. In the case of Au, we note that the real part of the dielectric function due for

intraband and interband transitions are equivalent but they differ from the experiment. to the intraband transitions denoted as *<sup>ε</sup>DR* ð Þ *<sup>ω</sup>* is in very good agreement with the real experimental values over the whole range corresponding to IR radiation and

the participation in the dielectric function due to interband transitions is negligible

radiation domain, the two contributions due to intraband and interband transitions differ significantly from the experimental results. The two contributions are very

� <sup>10</sup>15rad

<sup>ε</sup>iB*<sup>I</sup>* ð Þ <sup>ω</sup> is practically the same with the imaginary experimental part in the

and as for Ag, we note that the imaginary part due to the interband transitions

*ω* ¼ 4

*=*s. In this range of pulsation, we can conclude that

� <sup>10</sup>15rad

*=*

*=*s. Concerning the imaginary part,

*s* and in the field of UV radiation, the two contributions due to

*ω* > 2 *:* 5 � <sup>10</sup>15rad

*=*s. In this

*s* and up to the UV

*ω* ≈ 6 *:*75

DCP3 agree well the experimental values.As shown in **Figure 16**, concerning the imaginary part of the dielectric function of nanometricAu, we note that the DCP model is in excellent agreement with the experiment over the whole domain of pulsation values including values corresponding to both IR and UV radiation.

� 1015rad

The DL model is also close to the experimental values up to the value

� 1015rad

*=*s; which is the beginning of the UV

*=*s in the UV range,

*=*s two models DCP and

*Optimized*

 *parameters*

 *of Drude at three critical points DCP3 of the dielectric function of noble metals (Au and Ag).*

*Magnetic Skyrmions*

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

dielectric function is still the DCP3 model. For *<sup>ω</sup>*<sup>&</sup>gt; <sup>6</sup> � 1015rad*=*s in the UV range, the two models DCP and DCP3 are very close to the experiment. The DCP3 model is better than the DCP model.

In the case of the real part of the dielectric function of nanometric Au, the DL, DCP, and DCP3 models are allin very good agreement with the experiment up to the value of the pulsation *<sup>ω</sup>*≈<sup>5</sup> � 1015rad*=*s; which is the beginning of the UV radiation region. Beyond this value, the three models deviate from the measured values (**Figure 15**). From the value of *<sup>ω</sup>* <sup>≈</sup>6*:*<sup>75</sup> � <sup>10</sup>15rad*=*s two models DCP and DCP3 agree well the experimental values.As shown in **Figure 16**, concerning the imaginary part of the dielectric function of nanometricAu, we note that the DCP model is in excellent agreement with the experiment over the whole domain of pulsation values including values corresponding to both IR and UV radiation. The DL model is also close to the experimental values up to the value *<sup>ω</sup>*<sup>≈</sup> <sup>5</sup> � <sup>10</sup>15rad*=*s.

## **5. Conclusion**

In this work, we modeled the dielectric function of noble metals (silver and gold) in their bulk and nanometric states. Initially, we modeled the measured dielectric functions of these two metals using explicit mathematical functions and the results are in very good agreement with the experiment. Moreover, we have decomposed these measured values of the dielectric functions; in their real and imaginary parts; into several intervals according to the pulsations that sweep the domains corresponding to IR and UV radiation via the intermediate values. The obtained results are very conclusive, and depending on the pulsation domain studied, it is possible to use the corresponding mathematical function in simulations and calculations. Then, we highlighted the importance of the contributions of intraband and interband transitions in dielectric function for both Ag and Au. For Ag, we note that the imaginary part of the dielectric function due to interband transitions denoted εiB *<sup>I</sup>* ð Þ ω is almost the same with the imaginary experimental part in the whole field of pulsations from the IR domain to the end of the UV domain. This indicates that the imaginary part due to intraband transitions is negligible compared to the imaginary part due to interband transitions. Concerning the real part of the dielectric function, we note that, the real part due to the intraband transitions noted *εD <sup>R</sup>* ð Þ *ω* is in very good agreement with the real experimental values in the entire range corresponding to IR radiation and up to the value *<sup>ω</sup>* <sup>¼</sup> <sup>4</sup> � <sup>10</sup>15rad*=*s. In this range of pulsation, we can conclude that the contribution tothe dielectric function due to interband transitions is negligible compared to that of intraband transitions.

In the case of Au, we note that the real part of the dielectric function due for *<sup>ω</sup>*<sup>&</sup>gt; <sup>4</sup>*:*<sup>5</sup> � <sup>10</sup>15rad*=<sup>s</sup>* and in the field of UV radiation, the two contributions due to intraband and interband transitions are equivalent but they differ from the experiment. to the intraband transitions denoted as *ε<sup>D</sup> <sup>R</sup>* ð Þ *ω* is in very good agreement with the real experimental values over the whole range corresponding to IR radiation and up to the value *<sup>ω</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>15rad*=*s. In this range of pulsation, we can conclude that the participation in the dielectric function due to interband transitions is negligible compared to that of intraband transitions. For *<sup>ω</sup>*>2*:*<sup>5</sup> � <sup>10</sup>15rad*=<sup>s</sup>* and up to the UV radiation domain, the two contributions due to intraband and interband transitions differ significantly from the experimental results. The two contributions are very similar in the field of UV for *<sup>ω</sup>*≥2*:*<sup>5</sup> � <sup>10</sup>15rad*=*s. Concerning the imaginary part, and as for Ag, we note that the imaginary part due to the interband transitions noted εiB *<sup>I</sup>* ð Þ ω is practically the same with the imaginary experimental part in the

whole field of pulsations from the IR domain to the end of the UV domain. This indicates that the imaginary part due to intraband transitions is negligible compared to the imaginary part due to interband transitions.

In the last part of this paper, we have modeled the dielectric functions of Ag and Au, using theoretical models that deal with nanometric systems such as the Drude Lorenz model, the Drude two-point critical model, and the Drude three-point critical model.In the case of nanometic Ag, the real part of the dielectric function model agrees well with the experiment up to the value of the pulsation *ω*≈ 6 1015rad*=*s, which is located in the UV radiation region, is the DCP model. The other models are also valid up to the value of *<sup>ω</sup>*<sup>≈</sup> <sup>3</sup> <sup>10</sup>15rad*=*s. This interval covers the entire pulsation zone located in the IR domain. For the imaginary part of Ag, we find that for pulsation located in the IR domain and less than *ω*< 1*:*5x1015rad*=*s, the most appropriate model is the DCP3 model. For pulsation values 1*:*5 <sup>10</sup>15rad*=*s<*ω*<sup>&</sup>lt; <sup>6</sup> <sup>10</sup>15rad*=*s, model that deviates the least from the measured values of the dielectric function is still the DCP3 model. For *<sup>ω</sup>* <sup>&</sup>gt;<sup>6</sup> <sup>10</sup>15rad*=*s in the UV domain, the two models DCP and DCP3 are very close to the experience with a better approach using the DCP3 model.

For nanometric Au, concerning the real part of the dielectric function, the three models DL, DCP, and DCP3 are all in very good agreement with the experiment up to the value of the pulsation *<sup>ω</sup>*≈<sup>5</sup> <sup>10</sup>15rad*=*s; which is the beginning of the UV radiation region. Beyond this value, the three models deviate from the experiment. From the value of *<sup>ω</sup>*<sup>≈</sup> <sup>6</sup>*:*<sup>75</sup> <sup>10</sup>15rad*=*s the two models DCP and DCP3 meet the experimental values. Concerning the imaginary part, we note that the DCP model is in very good agreement with the experiment on the whole range of pulsation values including values corresponding to both IR and UV radiation. The DL model is also very close to the experimental values up to the value *<sup>ω</sup>* <sup>≈</sup><sup>5</sup> <sup>10</sup><sup>15</sup> rad*=*s.

## **Acknowledgements**

We are grateful to Professor Uwe Thumm who hosted us for three months in his James R. Macdonald laboratory at the Kansas State University in the USA, and who offered us an opportunity to collaborate on this subject, as part of the Fulbright Grant Merit Award.

**Author details**

Adil Echchelh<sup>2</sup>

Kenitra, Morocco

**89**

Brahim Ait Hammou<sup>1</sup>

, Said Dlimi<sup>3</sup>

and Climate Group, BP, Agadir, Morocco

provided the original work is properly cited.

, Abdelhamid El Kaaouachi<sup>1</sup>

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

*DOI: http://dx.doi.org/10.5772/intechopen.96123*

, Chi-Te Liang<sup>4</sup> and Jamal Hemine<sup>2</sup>

1 Faculty of Sciences of Agadir, Materials and Physicochemistry of the Atmosphere

2 Faculty of Sciences Ibn Tofail, Laboratory of Energetic Engineering and Materials,

© 2021 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,

3 Faculty of Sciences of Agadir, Physics Department, BP, Agadir, Morocco

4 Department of Physics, National Taiwan University, Taipei, Taiwan

\*Address all correspondence to: kaaouachi21@yahoo.fr

\*, Abdellatif El Oujdi<sup>2</sup>

,

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

## **Author details**

Brahim Ait Hammou<sup>1</sup> , Abdelhamid El Kaaouachi<sup>1</sup> \*, Abdellatif El Oujdi<sup>2</sup> , Adil Echchelh<sup>2</sup> , Said Dlimi<sup>3</sup> , Chi-Te Liang<sup>4</sup> and Jamal Hemine<sup>2</sup>

1 Faculty of Sciences of Agadir, Materials and Physicochemistry of the Atmosphere and Climate Group, BP, Agadir, Morocco

2 Faculty of Sciences Ibn Tofail, Laboratory of Energetic Engineering and Materials, Kenitra, Morocco

3 Faculty of Sciences of Agadir, Physics Department, BP, Agadir, Morocco

4 Department of Physics, National Taiwan University, Taipei, Taiwan

\*Address all correspondence to: kaaouachi21@yahoo.fr

© 2021 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] M. J. Ambrosio and U. Thumm, "Electronic structure effects in spatiotemporally resolved photoemission interferograms of copper surfaces," Phys. Rev. A **96**, 051403(R) (2017).

[3] S. R. Leone, C. W. McCurdy, J. Burgdörfer, L. S. Cederbaum, Z. Chang, N. Dudovich, J. Feist, C. H. Greene, M. Ivanov, R. Kienberger, U. Keller, M. F. Kling, Z.-H. Loh, T. Pfeifer, A. N. Pfeiffer, R. Santra, K. Schafer, A. Stolow, U. Thumm, and M. J. J. Vrakking,"What will it take to observe processes in 'real time'?," Nat. Photon. **8**, 162-166 (2014).

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[5] Yanzeng Li, Margaret Kocherga, Serang Park, Marc Lata, Micheal McLamb, Glenn Boreman, Thomas A. Schmedake, and Tino Hofmann, "Optical dielectric function of Si(2,6-bis (benzimidazol-2<sup>0</sup> -yl)pyridine)2 determined by spectroscopic ellipsometry," Opt. Mater. Express 9, 3469–3475 (2019).

[6] M. J. Ambrosio and U. Thumm, "Comparative time-resolved photoemission from the Cu(100) and Cu(111)," surfaces Phys. Rev. A 94, 063424 (2016).

[7] N. Manrique and H. Riascos, "Estimation of Dielectric Constant and Thickness of Copper Thin Films Using Surface Plasmon Resonance," in *Latin America Optics and Photonics Conference*, OSA Technical Digest (Optical Society of America, 2018), paper Th4A.16.

[8] F. Roth, C. Lupulescu, E. Darlatt, A. Gottwald, and W. Eberhardt,"Angle resolved Photoemission from Cu single crystals; Known Facts and a few Surprises about the Photoemission Process," J. Electron Spectrosc. **208**, 2-10 (2016).

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*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag*

[17] A. Vial and T. Laroche, "Comparison of gold and silver dispersion laws suitable for FDTD simulations," Appl. Phys*.*B **93**, 139-149

[18] J.Y. Lu and Y.H. Chang, "Implementation of an efficient dielectric function into the finite difference time domain method for simulating the coupling localized surface plasmons of nanostructures," Superlattices and Microstructures **47**,

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[10] P. Winsemius, H. P. Langkeek, and F. F. van Kampen,"Structure dependence of the optical properties of Cu, Ag and Au," Physica**79B**, 529–546 (1975).

[11] G. Leveque, C. G. Olson, and D. W. Lynch, "Reflectance spectra and dielectric functions of Ag in the region of interband transitions," Phys. Rev. B **27,** 4654–4660 (1983).

[12] M. L. Thèye, "Investigation of the optical properties of Au by means of thin semitransparent films," Phys. Rev. B **2**, 3060-3078 (1970).

[13] Aleksandar D. Rakić, Aleksandra B. Djurišić, Jovan M. Elazar, and Marian L. Majewski,"Optical properties of metallic films for vertical-cavity optoelectronic devices," Applied Optics **37**, 5271-5284 (1998).

[14] C. Kittel, "Introduction to solid states," Wiley, New York, (1971).

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[16] A.Vial and T. Laroche,"Description of dispersion proprieties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,"

*Modeling the Bulk and Nanometric Dielectric Functions of Au and Ag DOI: http://dx.doi.org/10.5772/intechopen.96123*

J. Phys. D. Appl. Phys. **40**, 7152-7158 (2007).

[17] A. Vial and T. Laroche, "Comparison of gold and silver dispersion laws suitable for FDTD simulations," Appl. Phys*.*B **93**, 139-149 (2008).

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**93**

**1. Introduction**

**Chapter 5**

**Abstract**

Magnetic Properties of Heusler

In this chapter, results of our recent investigations on the structural, microstructural and magnetic properties of Cu-based Heusler alloys and MFe2O4 (M = Mn, Fe, Co, Ni, Cu, Zn) nanostructures will be discussed. The chapter is divided into two parts, the first part describes growth and different characterizations of Heusler alloys while in the second part magnetic properties of nano-ferrites are discussed. The Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10 at %) alloys have been synthesized in the form of ribbons. The alloys with x ≤ 8 show the formation of Heusler single phase of the Cu2MnAl structure. Further increase of Ga content gives rise to the formation of γ-Cu9Al4 type phase together with Cu2MnAl Heusler phase. The alloys are ferromagnetically ordered and the saturation magnetization (Ms) decreases slightly with increasing Ga concentration. Annealing of the ribbons significantly changes the magnetic properties of Cu50Mn25Al25-xGax alloys. The splitting in the zero field cooled (ZFC) and field cooled (FC) magnetization curves at low temperature has been observed for alloys. Another important class of material is Nanoferrites. The structural and magnetization behaviour of spinel MFe2O4 nanoferrites are quite different from that of bulk ferrites. X-ray diffraction study revealed spinel structure of MFe2O4 nanoparticles. The observed ferromagnetic behaviour of MFe2O4 depends on the nanostructural shape as well as ferrite inversion degree. The magnetic interactions in Ce doped CoFe2O4 are antiferromagnetic that was confirmed by zero field/field cooling measurements at 100 Oe. Log R (Ω) response measurement of MgFe2O4 thin film

was taken for 10–90% relative humidity (% RH) change at 300 K.

netic compounds of iron oxides Fe2O3 and FeO with general formula: + +++

**Keywords:** Heusler alloy, nanoferrites, magnetization, microstructure, sensor

Heusler alloys were discovered in 1903 when Heusler reported that the addition of sp. elements (Al, In, Sb, Sn or Bi) turn Cu-Mn alloy into a ferromagnetic material even though the alloy contains none of the ferromagnetic elements [1, 2]. Heusler reported that these alloys have stoichiometric composition X2YZ and exhibit an ordered *L*21 crystal structure with space group *Fm*3*m* [3, 4]. Ferrites are ferrimag-

−− ∗ 3 233 *Fe Fe Fe Fe O* 1 1 1.67 0.33 4 *y y yy* (\*indicates vacancies), which can be partly changed by other transition metal (TM) oxides [5]. The ferrites can be classified according to their crystalline structures: hexagonal (MFe12O19), garnet (M3Fe5O12) and spinel (MFe2O4), where M = TM = Mn, Fe, Co, Ni, Cu, Zn. Both Heusler alloys and

Alloys and Nanoferrites

*Devinder Singh and Kuldeep Chand Verma*

## **Chapter 5**

## Magnetic Properties of Heusler Alloys and Nanoferrites

*Devinder Singh and Kuldeep Chand Verma*

## **Abstract**

In this chapter, results of our recent investigations on the structural, microstructural and magnetic properties of Cu-based Heusler alloys and MFe2O4 (M = Mn, Fe, Co, Ni, Cu, Zn) nanostructures will be discussed. The chapter is divided into two parts, the first part describes growth and different characterizations of Heusler alloys while in the second part magnetic properties of nano-ferrites are discussed. The Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10 at %) alloys have been synthesized in the form of ribbons. The alloys with x ≤ 8 show the formation of Heusler single phase of the Cu2MnAl structure. Further increase of Ga content gives rise to the formation of γ-Cu9Al4 type phase together with Cu2MnAl Heusler phase. The alloys are ferromagnetically ordered and the saturation magnetization (Ms) decreases slightly with increasing Ga concentration. Annealing of the ribbons significantly changes the magnetic properties of Cu50Mn25Al25-xGax alloys. The splitting in the zero field cooled (ZFC) and field cooled (FC) magnetization curves at low temperature has been observed for alloys. Another important class of material is Nanoferrites. The structural and magnetization behaviour of spinel MFe2O4 nanoferrites are quite different from that of bulk ferrites. X-ray diffraction study revealed spinel structure of MFe2O4 nanoparticles. The observed ferromagnetic behaviour of MFe2O4 depends on the nanostructural shape as well as ferrite inversion degree. The magnetic interactions in Ce doped CoFe2O4 are antiferromagnetic that was confirmed by zero field/field cooling measurements at 100 Oe. Log R (Ω) response measurement of MgFe2O4 thin film was taken for 10–90% relative humidity (% RH) change at 300 K.

**Keywords:** Heusler alloy, nanoferrites, magnetization, microstructure, sensor

## **1. Introduction**

Heusler alloys were discovered in 1903 when Heusler reported that the addition of sp. elements (Al, In, Sb, Sn or Bi) turn Cu-Mn alloy into a ferromagnetic material even though the alloy contains none of the ferromagnetic elements [1, 2]. Heusler reported that these alloys have stoichiometric composition X2YZ and exhibit an ordered *L*21 crystal structure with space group *Fm*3*m* [3, 4]. Ferrites are ferrimagnetic compounds of iron oxides Fe2O3 and FeO with general formula: + +++ −− ∗ 3 233 *Fe Fe Fe Fe O* 1 1 1.67 0.33 4 *y y yy* (\*indicates vacancies), which can be partly changed by other transition metal (TM) oxides [5]. The ferrites can be classified according to their crystalline structures: hexagonal (MFe12O19), garnet (M3Fe5O12) and spinel (MFe2O4), where M = TM = Mn, Fe, Co, Ni, Cu, Zn. Both Heusler alloys and

nanoferrites are important magnetic materials owing to their promise to qualify for many potential applications.

## **1.1 Heusler alloys**

Heusler alloys have attracted avid attention due to their various properties suitable for technological applications such as high-density magnetic recording or magneto-optics applications [6–8]. It has been shown that the ordered combination of two binary B2 compounds XY and YZ leads to the formation of Heusler structure. Both these compounds may have CsCl type crystal structure e.g. CoAl and CoMn yield Co2MnAl [9]. Thus the possible formation of new Heusler alloys will depend upon the ability of compounds to form B2 structure. It has also been observed to leave one of the four sublattices unoccupied (C1b structure). The L21 compounds are known as full-Heusler alloys while the latter compounds are often called half- or semi-Heusler alloys (**Figure 1**). Most of the Heusler alloys are saturate in weak magnetic field and ordered ferromagnetically. There are various parameters which are found to be very important in determining the magnetic properties [6–9, 11–13]; these include crystal structure, composition and heat treatment. The magnetic shape memory effect as well as other properties arises due to magneto-crystalline coupling in Heusler alloys [6–9, 11]. These made Heusler alloys to possess very interesting magnetic properties. A series of interesting diverse magnetic phenomenon like itinerant and localized magnetism, antiferromagnetism, helimagnetism, Pauli paramagnetism or heavy-fermionic behaviour can be studied in the same family of Heusler alloys [3, 4, 6, 7]. At low temperatures several Heusler alloys, e.g. Ni2MnGa, Co2NbSn etc., undergo a martensitic transition from a highly symmetric cubic austenitic to a low symmetry martensitic phase. Unlike atomic order–disorder transitions a martensitic transition is caused by non-diffusional

#### **Figure 1.**

*Representation of various structures of semi- and full-Heusler alloys. The lattice is consisted to have four interpenetrating f.c.c. lattices in all cases. It should be noted that if all atoms are identical, the lattice would simply become b.c.c [10].*

**95**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

studied in Heusler alloys [24–30].

( ) { } + + ++ − − − 2 3 23 2

**1.2 Nanoferrites**

cooperative movement of the atoms in the crystal. Heusler alloys exhibit magnetic shape memory effect (MSM), when they are magnetic in the martensitic phase [14–16]. This occurs especially in those cases when the Y constituent is Mn, but other transition elements are also possible. When these alloys are applied in the

In recent years, Ni-Mn-X (X = In, Sn, Sb) Heusler alloys have attracted much interest because of their magnetic shape memory effect and magnetic field-induced martensitic transformation [17–19]. In recent years, the interest in Heulser compounds has been increased due to the evolving field of spin-electronics [7]. These compounds have also been studied as potential spin-injector materials [6, 8, 20]. In the alloys X2YZ, a value close to 4μB is usually observed if the magnetic moment is carried by Mn atoms. These compounds are considered as ideal model systems to study the effects of atomic disorder and changes in the electron concentration on the magnetic properties. These compounds have localized magnetic properties though they are metals. Extensive magnetic and other measurements have been performed on quaternary Heusler alloys in order to understand the role of 3d (X) and sp. (Z) atoms on magnetic properties [21]. It has been shown that both the magnetic moment formation and the type of the magnetic order get influenced by the sp. electron concentration which is very important in establishing the magnetic properties. During the past few years, the family of ferromagnetic Heusler alloy systems has been extensively studied due to their main advantages in comparison to half-metallic systems e.g. structural similarity with the binary semiconductors and predicted perfect spin polarization at the Fermi level as well as high Curie temperature [22, 23]. The effect of alloying addition on the magnetic properties has been

A series of spinel-structured ferrites, MFe2O4, where M = TM = Mn, Fe, Co, Ni, Cu, Zn were reported for novel data storage, recording devices, microwave technology and biomedical applications [31–35]. The spinel structure has general formula AB2O4 which had octahedral (B) and tetrahedral (A) sites. The spinel is normal, if M2+ occupies only the A sites; the spinel is inverse, if it occupies only the B sites. When Mn2+occupies both A and B sites, the MFe2O4 has formed a mixed spinel structure while the other metal ferrites have an inverse spinel structure [3]. The magnetic spins at B sites align in parallel in direction to the applied magnetic field, whereas those present at A sites align antiparallel. Spinel ferrite has general formula: ( ) + +− 2 32 *A BO* 2 4 ; where <sup>2</sup><sup>+</sup> *<sup>A</sup>* and <sup>3</sup><sup>+</sup> *<sup>B</sup>*2 are the divalent and trivalent cations occupying tetrahedral (A) and octahedral [B] sites. **Figure 2** shows the inverse spinel structure of NiFe2O4 [36]. The inverse spinel has general formula:

1 2 <sup>4</sup> *M Fe M Fe O xx x x* ; the round and square brackets denote (A) and [B] sites respectively, whereas *x* represents inversion degree. Also for normal spinel AB2O4, the A2+cations occupies 1*/*8 of the *fcc* tetrahedral sites (Td) while the B3+occupy 16 of the 32 available octahedral sites (Oh). An inverse spinel structure with Ni2+ at octahedral sites (labelled as Ni(Oh)) and Fe3+ equally distributed between octahedral (Fe(Oh))and tetrahedral sites (Fe(Td)) of the O2− fcc cell have been adopted for NiFe2O4 (**Figure 2**). The complete structure crystallizes into a cubic system <sup>7</sup> *Oh* with space group 227. The oxygen atoms occupy the 32e positions, Fe(Td) atoms occupy the 8a ones while the Ni(Oh) and Fe(Oh) atoms are distributed on the 16d positions, using Wyckoffnotations [36]. The efficacy of the material depends on its microstructural properties that are sensitive to mode of preparation.

Moreover, CoFe2O4 is the most versatile hard ferrite with mixed cubic spinel

martensitic state, an external magnetic field can induce large strains.

#### *Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

*Magnetic Skyrmions*

**1.1 Heusler alloys**

many potential applications.

nanoferrites are important magnetic materials owing to their promise to qualify for

Heusler alloys have attracted avid attention due to their various properties suitable for technological applications such as high-density magnetic recording or magneto-optics applications [6–8]. It has been shown that the ordered combination of two binary B2 compounds XY and YZ leads to the formation of Heusler structure. Both these compounds may have CsCl type crystal structure e.g. CoAl and CoMn yield Co2MnAl [9]. Thus the possible formation of new Heusler alloys will depend upon the ability of compounds to form B2 structure. It has also been observed to leave one of the four sublattices unoccupied (C1b structure). The L21 compounds are known as full-Heusler alloys while the latter compounds are often called half- or semi-Heusler alloys (**Figure 1**). Most of the Heusler alloys are saturate in weak magnetic field and ordered ferromagnetically. There are various parameters which are found to be very important in determining the magnetic properties [6–9, 11–13]; these include crystal structure, composition and heat treatment. The magnetic shape memory effect as well as other properties arises due to magneto-crystalline coupling in Heusler alloys [6–9, 11]. These made Heusler alloys to possess very interesting magnetic properties. A series of interesting diverse magnetic phenomenon like itinerant and localized magnetism, antiferromagnetism, helimagnetism, Pauli paramagnetism or heavy-fermionic behaviour can be studied in the same family of Heusler alloys [3, 4, 6, 7]. At low temperatures several Heusler alloys, e.g. Ni2MnGa, Co2NbSn etc., undergo a martensitic transition from a highly symmetric cubic austenitic to a low symmetry martensitic phase. Unlike atomic order–disorder transitions a martensitic transition is caused by non-diffusional

*Representation of various structures of semi- and full-Heusler alloys. The lattice is consisted to have four interpenetrating f.c.c. lattices in all cases. It should be noted that if all atoms are identical, the lattice would* 

**94**

**Figure 1.**

*simply become b.c.c [10].*

cooperative movement of the atoms in the crystal. Heusler alloys exhibit magnetic shape memory effect (MSM), when they are magnetic in the martensitic phase [14–16]. This occurs especially in those cases when the Y constituent is Mn, but other transition elements are also possible. When these alloys are applied in the martensitic state, an external magnetic field can induce large strains.

In recent years, Ni-Mn-X (X = In, Sn, Sb) Heusler alloys have attracted much interest because of their magnetic shape memory effect and magnetic field-induced martensitic transformation [17–19]. In recent years, the interest in Heulser compounds has been increased due to the evolving field of spin-electronics [7]. These compounds have also been studied as potential spin-injector materials [6, 8, 20]. In the alloys X2YZ, a value close to 4μB is usually observed if the magnetic moment is carried by Mn atoms. These compounds are considered as ideal model systems to study the effects of atomic disorder and changes in the electron concentration on the magnetic properties. These compounds have localized magnetic properties though they are metals. Extensive magnetic and other measurements have been performed on quaternary Heusler alloys in order to understand the role of 3d (X) and sp. (Z) atoms on magnetic properties [21]. It has been shown that both the magnetic moment formation and the type of the magnetic order get influenced by the sp. electron concentration which is very important in establishing the magnetic properties. During the past few years, the family of ferromagnetic Heusler alloy systems has been extensively studied due to their main advantages in comparison to half-metallic systems e.g. structural similarity with the binary semiconductors and predicted perfect spin polarization at the Fermi level as well as high Curie temperature [22, 23]. The effect of alloying addition on the magnetic properties has been studied in Heusler alloys [24–30].

## **1.2 Nanoferrites**

A series of spinel-structured ferrites, MFe2O4, where M = TM = Mn, Fe, Co, Ni, Cu, Zn were reported for novel data storage, recording devices, microwave technology and biomedical applications [31–35]. The spinel structure has general formula AB2O4 which had octahedral (B) and tetrahedral (A) sites. The spinel is normal, if M2+ occupies only the A sites; the spinel is inverse, if it occupies only the B sites. When Mn2+occupies both A and B sites, the MFe2O4 has formed a mixed spinel structure while the other metal ferrites have an inverse spinel structure [3]. The magnetic spins at B sites align in parallel in direction to the applied magnetic field, whereas those present at A sites align antiparallel. Spinel ferrite has general formula: ( ) + +− 2 32 *A BO* 2 4 ; where <sup>2</sup><sup>+</sup> *<sup>A</sup>* and <sup>3</sup><sup>+</sup> *<sup>B</sup>*2 are the divalent and trivalent cations occupying tetrahedral (A) and octahedral [B] sites. **Figure 2** shows the inverse spinel structure of NiFe2O4 [36]. The inverse spinel has general formula: ( ) { } + + ++ − − − 2 3 23 2 1 2 <sup>4</sup> *M Fe M Fe O xx x x* ; the round and square brackets denote (A) and [B] sites respectively, whereas *x* represents inversion degree. Also for normal spinel AB2O4, the A2+cations occupies 1*/*8 of the *fcc* tetrahedral sites (Td) while the B3+occupy 16 of the 32 available octahedral sites (Oh). An inverse spinel structure with Ni2+ at octahedral sites (labelled as Ni(Oh)) and Fe3+ equally distributed between octahedral (Fe(Oh))and tetrahedral sites (Fe(Td)) of the O2− fcc cell have been adopted for NiFe2O4 (**Figure 2**). The complete structure crystallizes into a cubic system <sup>7</sup> *Oh* with space group 227. The oxygen atoms occupy the 32e positions, Fe(Td) atoms occupy the 8a ones while the Ni(Oh) and Fe(Oh) atoms are distributed on the 16d positions, using Wyckoffnotations [36]. The efficacy of the material depends on its microstructural properties that are sensitive to mode of preparation. Moreover, CoFe2O4 is the most versatile hard ferrite with mixed cubic spinel

#### **Figure 2.**

*Inverse spinel unit cell of NiFe2O4 Oxygen atoms are adjustable to complete the Ni(Oh) and Fe(Oh). Ni atoms (blue) Fe (green) and O atoms (red) [36].*

structure having Fd3m space group and exhibits high coercivity ~5400 Oe, high magneto-crystalline anisotropy and moderate saturation magnetization [37, 38].

Spintronics is concerned with highly spin-polarized materials to enhance tunnelling magnetoresistance of magnetic tunnel junctions which are active members of magnetic random access memory elements. For the optimal operation of spintronics, the highly spin-polarized materials used for increasing spin-polarization of currents injected into semiconductors [38]. There is a way to achieve high spin-polarization by employing fully spin-polarized ferromagnetic metals, such as half-metals. Another way is to exploit the band structure features of tunnel barrier materials, *i.e.*, MgO, and filtering electronic wave functions. Spinel NiFe2O4, CoFe2O4, and MnFe2O4 are also used as such spin filters. Spin-dependent gap should result in spin-dependent barrier for tunnelling of electrons through the insulator, giving rise to spin filtering. In particular, a spin filtering efficiency of up to 22% is reported for NiFe2O4 barrier [39]. The resistive switching performance characteristics of a Pt/NiFe2O4/Pt structure such as low operating voltage, high device yield, long retention time (up to 105 s), and good endurance (up to 2.2 × 104 cycles) can be used to demonstrate the opportunity of spinel ferrites in non-volatile memory devices [33]. Since the resistive switching memory cell has capacitor-like metal/ insulator/metal configuration, which can be switched reversibly between two different resistance states, i.e*.*, high-resistance state (HRS) and low-resistance state (LRS).

**Figure 3(a)** shows the current–voltage (I-V) characteristics of the Pt/NiFe2O4/Pt devices during repetitive switching cycles. Unipolar resistive switching characteristics were clearly observed in both the forward- and backward-bias sweeping processes. The drastic decrease in the current has been observed on increasing the forward voltage to its critical value of 0.6–1.0 V. This indicates the switching of the Pt/NiFe2O4/ Pt device from the LRS to the HRS. The device was switched from HRS to LRS, when soft breakdown occurred on increasing the voltage in the range of 1.8–2.2 V. NiFe2O4 has the inverse spinel structure which has Fe-O bonds that are stronger than Ni-O bonds leads to oxygen vacancies formation [40]. **Figure 3(b)** shows magnetic reduction effect in changing the valence of Fe3+ and Ni2+ions [41]. The oxygen vacancies and the reduction of cations may cause the decrease in magnetization and the increase

**97**

**Figure 3.**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

in the electrical conductivity. Due to the annihilation of oxygen vacancies (driven by the thermal effect) and the change in the valence of cations (due to redox effect in the reset process) has led to rupture of filaments. The process of formation and rupture of the conducting filaments in NiFe2O4 films has been described in **Figure 3(c)** and **(d)**.

*(a) I-V characteristics with switching cycles of Pt/NiFe2O4/Pt devices (b) Magnetic hysteresis at 300 K for HRS* 

*and LRS (c & d) Conducting filaments mechanism in LRS and HRS [33].*

**2. Structure/microstructure and magnetic properties of Heusler alloys**

**2.1 Microstructural and structural features**

The Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10) alloys with a thickness of ∼40 to 50 μm and lengths of ∼1 to 2 cm has been synthesized using melt spinning technique [24]. The substitution of Ga in place of Al is of special interest as it does not significantly change the lattice constant and the valence electron ratio (e/a) of the investigated alloy system [42–49]. The effect of heat treatment on the magnetic and phase transformation behaviour of Cu50Mn25Al25-xGax ribbons will be discussed. The study is focussed on the structural/microstructural changes with substitution of Ga and their correlation with magnetic properties of Cu50Mn25Al25-xGax alloys.

The effect of Ga substitution on the stability of Heusler phase has been investigated. The XRD patterns of melt-spun Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10) alloys are shown in **Figure 4(a)**. A single Heusler phase of the Cu2MnAl structure (space group: *Fm*3*m*, *a* = 5.949 Ǻ) was observed for the alloys with x = 0, 2, 4 and 8, while the alloy with x = 10 reveals some diffraction peaks corresponding to γ-Cu9Al4 type phase (Space group: *P*43*m*, *a* = 8.702 Ǻ). Thus it can be said that the Heusler phase was also observed in the Ga substituted alloys and is stable up to x = 8. In addition to the formation of Heusler phase, the precipitation of additional crystalline phase of γ-Cu9Al4 type has been observed for the alloys with x > 8. For the alloys from x = 0 to x = 8, a slight increase in the lattice parameter was observed. This is

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

#### **Figure 3.**

*Magnetic Skyrmions*

structure having Fd3m space group and exhibits high coercivity ~5400 Oe, high magneto-crystalline anisotropy and moderate saturation magnetization [37, 38]. Spintronics is concerned with highly spin-polarized materials to enhance tunnelling magnetoresistance of magnetic tunnel junctions which are active members of magnetic random access memory elements. For the optimal operation of spintronics, the highly spin-polarized materials used for increasing spin-polarization of currents injected into semiconductors [38]. There is a way to achieve high spin-polarization by employing fully spin-polarized ferromagnetic metals, such as half-metals. Another way is to exploit the band structure features of tunnel barrier materials, *i.e.*, MgO, and filtering electronic wave functions. Spinel NiFe2O4, CoFe2O4, and MnFe2O4 are also used as such spin filters. Spin-dependent gap should result in spin-dependent barrier for tunnelling of electrons through the insulator, giving rise to spin filtering. In particular, a spin filtering efficiency of up to 22% is reported for NiFe2O4 barrier [39]. The resistive switching performance characteristics of a Pt/NiFe2O4/Pt structure such as low operating voltage, high device yield,

*Inverse spinel unit cell of NiFe2O4 Oxygen atoms are adjustable to complete the Ni(Oh) and Fe(Oh). Ni atoms* 

s), and good endurance (up to 2.2 × 104

be used to demonstrate the opportunity of spinel ferrites in non-volatile memory devices [33]. Since the resistive switching memory cell has capacitor-like metal/ insulator/metal configuration, which can be switched reversibly between two different resistance states, i.e*.*, high-resistance state (HRS) and low-resistance

**Figure 3(a)** shows the current–voltage (I-V) characteristics of the Pt/NiFe2O4/Pt devices during repetitive switching cycles. Unipolar resistive switching characteristics were clearly observed in both the forward- and backward-bias sweeping processes. The drastic decrease in the current has been observed on increasing the forward voltage to its critical value of 0.6–1.0 V. This indicates the switching of the Pt/NiFe2O4/ Pt device from the LRS to the HRS. The device was switched from HRS to LRS, when soft breakdown occurred on increasing the voltage in the range of 1.8–2.2 V. NiFe2O4 has the inverse spinel structure which has Fe-O bonds that are stronger than Ni-O bonds leads to oxygen vacancies formation [40]. **Figure 3(b)** shows magnetic reduction effect in changing the valence of Fe3+ and Ni2+ions [41]. The oxygen vacancies and the reduction of cations may cause the decrease in magnetization and the increase

cycles) can

**96**

long retention time (up to 105

*(blue) Fe (green) and O atoms (red) [36].*

state (LRS).

**Figure 2.**

*(a) I-V characteristics with switching cycles of Pt/NiFe2O4/Pt devices (b) Magnetic hysteresis at 300 K for HRS and LRS (c & d) Conducting filaments mechanism in LRS and HRS [33].*

in the electrical conductivity. Due to the annihilation of oxygen vacancies (driven by the thermal effect) and the change in the valence of cations (due to redox effect in the reset process) has led to rupture of filaments. The process of formation and rupture of the conducting filaments in NiFe2O4 films has been described in **Figure 3(c)** and **(d)**.

## **2. Structure/microstructure and magnetic properties of Heusler alloys**

The Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10) alloys with a thickness of ∼40 to 50 μm and lengths of ∼1 to 2 cm has been synthesized using melt spinning technique [24]. The substitution of Ga in place of Al is of special interest as it does not significantly change the lattice constant and the valence electron ratio (e/a) of the investigated alloy system [42–49]. The effect of heat treatment on the magnetic and phase transformation behaviour of Cu50Mn25Al25-xGax ribbons will be discussed. The study is focussed on the structural/microstructural changes with substitution of Ga and their correlation with magnetic properties of Cu50Mn25Al25-xGax alloys.

#### **2.1 Microstructural and structural features**

The effect of Ga substitution on the stability of Heusler phase has been investigated. The XRD patterns of melt-spun Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10) alloys are shown in **Figure 4(a)**. A single Heusler phase of the Cu2MnAl structure (space group: *Fm*3*m*, *a* = 5.949 Ǻ) was observed for the alloys with x = 0, 2, 4 and 8, while the alloy with x = 10 reveals some diffraction peaks corresponding to γ-Cu9Al4 type phase (Space group: *P*43*m*, *a* = 8.702 Ǻ). Thus it can be said that the Heusler phase was also observed in the Ga substituted alloys and is stable up to x = 8. In addition to the formation of Heusler phase, the precipitation of additional crystalline phase of γ-Cu9Al4 type has been observed for the alloys with x > 8. For the alloys from x = 0 to x = 8, a slight increase in the lattice parameter was observed. This is

**Figure 4.**

*(a) XRD patterns of Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10) melt-spun ribbons (b) XRD patterns of Cu50Mn25Al25-xGax ribbons annealed at 903 K for 30 h (Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

expected due to nearly equal atomic size of Ga and Al. **Table 1** shows the variation of lattice parameter with Ga content. There is no significant increase in the lattice parameter for x = 10 (**Table 1**). The formation of extra phases for concentrations x > 8 may be due to the excess of Ga atoms which no longer replace the corresponding number of Al atoms in the crystal structure. Thus, the lattice constant of the Heusler phase remains almost constant. It is therefore can be concluded that the Ga atoms can substitute for Al atoms in the structure only within the content range of 0 ≤ x ≤ 8. Hence, x = 8 is the critical Ga concentration (xc), beyond which the alloys are having mixed phases.

The phase transformation behaviour of Cu50Mn25Al25-xGax alloys on heat treatment has been investigated. The alloys with x = 0 and x = 8 are annealed at 903 K for 30 hrs and there XRD patterns are shown in **Figure 4(b)**. The XRD pattern for x = 0 shows the presence of Cu2MnAl, β-Mn and γ-Cu9Al4 phases. The decomposition reaction of the Cu2MnAl phase during the annealing process leads to the appearance of these two phases. Based on the previous studies done on the decomposition process of Cu-Mn-Al alloys, it has been shown that the Cu2MnAl phase is metastable and can decompose into β-Mn and γ-Cu9Al4 phases at annealing temperatures from 800 to 900 K [50, 51]. This decomposition then follows the reaction (Cu2MnAl → β-Mn + γ-Cu9Al4). However, the x = 8 alloy shows different decomposition behaviour. The Cu2MnAl phase decomposes into only γ-Cu9Al4 on annealing. There are no traces of β-Mn phase here. In addition to this, Cu2MnAl precipitated as a majority phase for the alloy with x = 0 while for the alloy with x = 8, γ-Cu9Al4 precipitated as a majority phase.


#### **Table 1.**

*The lattice constants, saturation magnetization (Ms) at 5 K and 300 K for melt-spun and annealed ribbons of Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys (Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

**99**

**Figure 5.**

*permission from Elsevier, Copyright 2012, Elsevier).*

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

paramagnetic in nature [50].

**2.2 Magnetic properties**

Further TEM characterization studies of melt-spun as well as annealed samples were carried out. TEM microstructure and its corresponding diffraction pattern of Cu50Mn25Al25 and Cu50Mn25Al17Ga8 melt-spun alloys are shown in **Figure 5** (on the left). The formation of nano-meter sized grains in the range of 100 to 200 nm of the alloys with x = 0 and x = 8 have been observed (*c.f.* **Figure 5(a)**-**(d)**). The substitution of Ga in Cu50Mn25Al25-xGax alloy system does not results in any significant microstructural variation. The confirmation of Cu2MnAl Heusler phase has been done based on the analysis of selected area diffraction pattern (SADP) of the alloys. The TEM microstructure and diffraction pattern for the annealed ribbon of x = 8 are shown in **Figure 5** (on the right). The SADP analysis reveals two crystalline phases; Cu2MnAl and γ-Cu9Al4. This is consistent with the result obtained by XRD analysis which also shows the formation of Cu2MnAl and γ-Cu9Al4 phases. The large grains belong to Cu2MnAl phase while the small ones identified as γ-Cu9Al4 phase. As one might know, Cu2MnAl phase is ferromagnetic while γ-Cu9Al4 phase is

The temperatures of 5 K and 300 K were used to measure the magnetization curves (M-H curves) of Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys. Based on the results presented in **Figure 6(a)** and **(b)**, It can be said that the magnetization is saturated in a magnetic field of about 1500 to 2500 Oe, indicating that ribbons are fairly homogenous ferromagnets [51]. The soft ferromagnetic behaviour (at 5 K and 300 K) were observed for the Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys. **Table 1** gives the list of the saturation magnetization (Ms) derived from the M-H curves for all the compositions. The Ms for the composition with x = 0 at 5 K is ~95 emu/g and at 300 K is ~85 emu/g respectively. These are very close to the reported magnetization values for this alloy [52]. It can be seen from **Table 1** that Ms decreases with increasing Ga content. This decrease is very small from x = 0 (~ 95 emu/g) to x = 8 (~ 83 emu/g) at 5 K. However, for x > 8, the Ms decreases from ~83 emu/g (for x = 8) to ~20 emu/g (for x = 10). The large decrease in the Ms for x = 10 may be explained on the basis that the Cu2MnAl Heusler phase which is responsible for the ferromagnetism is stable only up to x = 8. The decomposition of Heusler phase into γ-Cu9Al4 type crystalline phase for the alloys with x > 8, is responsible for the significant decrease in the Ms for x = 10. The Ms values in the range 95–83 emu/g (at 5 K) has

*(left) TEM microstructures and the corresponding selected area diffraction patterns of the melt-spun alloys with x = 0 (a and b) and x = 8 (c and d) showing the formation of Cu2MnAl Heusler phase. (Right) TEM microstructures and the corresponding selected area diffraction patterns of the annealed alloy of x = 8 showing the existence of (a and b) Cu2MnAl and (c and d) γ-Cu9Al4 phases (Reprinted from reference [24] with kind* 

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

Further TEM characterization studies of melt-spun as well as annealed samples were carried out. TEM microstructure and its corresponding diffraction pattern of Cu50Mn25Al25 and Cu50Mn25Al17Ga8 melt-spun alloys are shown in **Figure 5** (on the left). The formation of nano-meter sized grains in the range of 100 to 200 nm of the alloys with x = 0 and x = 8 have been observed (*c.f.* **Figure 5(a)**-**(d)**). The substitution of Ga in Cu50Mn25Al25-xGax alloy system does not results in any significant microstructural variation. The confirmation of Cu2MnAl Heusler phase has been done based on the analysis of selected area diffraction pattern (SADP) of the alloys. The TEM microstructure and diffraction pattern for the annealed ribbon of x = 8 are shown in **Figure 5** (on the right). The SADP analysis reveals two crystalline phases; Cu2MnAl and γ-Cu9Al4. This is consistent with the result obtained by XRD analysis which also shows the formation of Cu2MnAl and γ-Cu9Al4 phases. The large grains belong to Cu2MnAl phase while the small ones identified as γ-Cu9Al4 phase. As one might know, Cu2MnAl phase is ferromagnetic while γ-Cu9Al4 phase is paramagnetic in nature [50].

## **2.2 Magnetic properties**

*Magnetic Skyrmions*

**Figure 4.**

are having mixed phases.

*from Elsevier, Copyright 2012, Elsevier).*

x = 8, γ-Cu9Al4 precipitated as a majority phase.

**Lattice parameter a (**Ǻ**)**

expected due to nearly equal atomic size of Ga and Al. **Table 1** shows the variation of lattice parameter with Ga content. There is no significant increase in the lattice parameter for x = 10 (**Table 1**). The formation of extra phases for concentrations x > 8 may be due to the excess of Ga atoms which no longer replace the corresponding number of Al atoms in the crystal structure. Thus, the lattice constant of the Heusler phase remains almost constant. It is therefore can be concluded that the Ga atoms can substitute for Al atoms in the structure only within the content range of 0 ≤ x ≤ 8. Hence, x = 8 is the critical Ga concentration (xc), beyond which the alloys

*(a) XRD patterns of Cu50Mn25Al25-xGax (x = 0, 2, 4, 8 and 10) melt-spun ribbons (b) XRD patterns of Cu50Mn25Al25-xGax ribbons annealed at 903 K for 30 h (Reprinted from reference [24] with kind permission* 

The phase transformation behaviour of Cu50Mn25Al25-xGax alloys on heat treatment has been investigated. The alloys with x = 0 and x = 8 are annealed at 903 K for 30 hrs and there XRD patterns are shown in **Figure 4(b)**. The XRD pattern for x = 0 shows the presence of Cu2MnAl, β-Mn and γ-Cu9Al4 phases. The decomposition reaction of the Cu2MnAl phase during the annealing process leads to the appearance of these two phases. Based on the previous studies done on the decomposition process of Cu-Mn-Al alloys, it has been shown that the Cu2MnAl phase is metastable and can decompose into β-Mn and γ-Cu9Al4 phases at annealing temperatures from 800 to 900 K [50, 51]. This decomposition then follows the reaction (Cu2MnAl → β-Mn + γ-Cu9Al4). However, the x = 8 alloy shows different decomposition behaviour. The Cu2MnAl phase decomposes into only γ-Cu9Al4 on annealing. There are no traces of β-Mn phase here. In addition to this, Cu2MnAl precipitated as a majority phase for the alloy with x = 0 while for the alloy with

**x (at %) As-synthesized ribbons Annealed ribbons**

 5.947 95 85 53 45 5.960 88 79 — — 5.986 83 73 5 3 5.981 20 16 — —

*The lattice constants, saturation magnetization (Ms) at 5 K and 300 K for melt-spun and annealed ribbons of Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys (Reprinted from reference [24] with kind permission from* 

**Ms (emu/g) at 300 K**

**Ms (emu/g) at 5 K**

**Ms (emu/g) at 300 K**

**Ms (emu/g) at 5 K**

**98**

**Table 1.**

*Elsevier, Copyright 2012, Elsevier).*

The temperatures of 5 K and 300 K were used to measure the magnetization curves (M-H curves) of Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys. Based on the results presented in **Figure 6(a)** and **(b)**, It can be said that the magnetization is saturated in a magnetic field of about 1500 to 2500 Oe, indicating that ribbons are fairly homogenous ferromagnets [51]. The soft ferromagnetic behaviour (at 5 K and 300 K) were observed for the Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys. **Table 1** gives the list of the saturation magnetization (Ms) derived from the M-H curves for all the compositions. The Ms for the composition with x = 0 at 5 K is ~95 emu/g and at 300 K is ~85 emu/g respectively. These are very close to the reported magnetization values for this alloy [52]. It can be seen from **Table 1** that Ms decreases with increasing Ga content. This decrease is very small from x = 0 (~ 95 emu/g) to x = 8 (~ 83 emu/g) at 5 K. However, for x > 8, the Ms decreases from ~83 emu/g (for x = 8) to ~20 emu/g (for x = 10). The large decrease in the Ms for x = 10 may be explained on the basis that the Cu2MnAl Heusler phase which is responsible for the ferromagnetism is stable only up to x = 8. The decomposition of Heusler phase into γ-Cu9Al4 type crystalline phase for the alloys with x > 8, is responsible for the significant decrease in the Ms for x = 10. The Ms values in the range 95–83 emu/g (at 5 K) has

#### **Figure 5.**

*(left) TEM microstructures and the corresponding selected area diffraction patterns of the melt-spun alloys with x = 0 (a and b) and x = 8 (c and d) showing the formation of Cu2MnAl Heusler phase. (Right) TEM microstructures and the corresponding selected area diffraction patterns of the annealed alloy of x = 8 showing the existence of (a and b) Cu2MnAl and (c and d) γ-Cu9Al4 phases (Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

#### **Figure 6.**

*Magnetization curves at 5 K (a) and 300 K (b) for melt-spun ribbons of Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys. Magnetization curves (c) and (d) shows a small increase in M with H for the alloy with x = 10 (Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

been observed for the alloys with x = 0, 4 and 8 while the Ms value of 20 emu/g (at 5 K) has been observed for x = 10. Thus there is very slight variation in the Ms for x = 0, 4 and 8. The slight decrease in the Ms for x = 4 and x = 8 as compared to x = 0 may be due to several reasons. In the present alloy system, Ga is a non-magnetic element. The concentration of Al is only varied while the concentration of Cu and Mn remains fixed in Cu50Mn25Al25-xGax alloy system. It has been reported that Mn-Mn coupling is responsible for the magnetic properties of Cu50Mn25Al25 alloy [52, 53]. As discussed earlier, the substitution of Ga may increase the Mn-Mn distance in the lattice. This is due to sight increase in the lattice constant (see **Table 1**). Thus, the decrease in the magnetization has been observed due to increase in the lattice constant which reduces the ferromagnetic coupling of Cu50Mn25Al25-xGax alloy system. Such type of observation has also been reported in other alloy systems [14, 26, 29, 30]. Also, it is well known that Ga3+ is normally substituted in isovalent state vis-a-vis that of Al3+ [54–59]. However, it is worth mentioning here that Ga1+ may possess monovalent state. It has been confirmed Al, Ga mainly provide conduction electrons for the exchange interaction and the Mn atoms show a localized moment in this material [3]. It may be said by considering the varied concentration ratio of Al and Ga that the concentration of conduction electron plays a very important role in determining the magnetic properties. Thus, the magnetic properties may be affected by the presence of mixed valence states of Ga. The concentration of Ga and the disordered occupation of Mn atoms may also be important factors which affects the magnetic properties. Therefore, the conduction electron concentration is believed to be critical in stabilizing the Heusler structure [3]. In the present case,

**101**

**Figure 7.**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

ferromagnetic phase in the alloy with x = 10.

the concentration of Mn is fixed i.e. 25 at. %. Thus, the ferromagnetism behaviour of this material may be related to the concentration of Mn, atomic sites and situation of conduction electrons provided by Cu, Al and Ga. Further, the M-H curves for the melt spun alloy with x = 10 at 5 K and 300 K respectively are shown in **Figure 6(c)** and **(d)**. A close observation of the curves indicates a small increase in M with H superimposed on the flat saturated ferromagnetic M-H curve. This may be due to the formation of a paramagnetic phase (γ-Cu9Al4) in addition to the majority Cu2MnAl

The effect of heat treatment on the magnetization has also been studied. On annealing, the magnetic properties of the alloy have changed. The hysteresis loops of the annealed ribbons (for x = 0 and x = 8) at 5 K and 300 K are shown in **Figure 7**. The Ms of the annealed ribbons are found to be lower than their respective as-synthesized ribbons (**Table 1**). The Ms ~ 53 emu/g has been found for annealed ribbon of x = 0 from the magnetization curve (M-H) at 5 K, which is lower than that of the value found for the as-synthesized ribbons of x = 0 *i.e.* Ms ~ 95 emu/g. The decomposition of Cu2MnAl ferromagnetic phase into β-Mn and γ-Cu9Al4 paramagnetic phases may decrease the magnetization [50]. However, along with these phases, Cu2MnAl Heusler phase is still a majority phase for the alloy with x = 0. The drastic decrease in the magnetization value at 5 K from ~83 emu/g to ~5 emu/g has been observed for x = 8. This is attributed to the different decomposition behaviour of x = 0 and x = 8 alloys. The M-H curves of the annealed ribbon for x = 8 (at 5 K and 300 K) are shown in **Figure 7(c)** and **(d)** (c & d). The increase in M with increase in the applied field is

*Magnetization curves at 5 K (a) and 300 K (b) for annealed ribbons of Cu50Mn25Al25-xGax (x = 0 and 8) alloys. Magnetization curves (c) and (d) shows an increase in M with H for the alloy with x = 8 (Reprinted* 

*from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

*Magnetic Skyrmions*

been observed for the alloys with x = 0, 4 and 8 while the Ms value of 20 emu/g (at 5 K) has been observed for x = 10. Thus there is very slight variation in the Ms for x = 0, 4 and 8. The slight decrease in the Ms for x = 4 and x = 8 as compared to x = 0 may be due to several reasons. In the present alloy system, Ga is a non-magnetic element. The concentration of Al is only varied while the concentration of Cu and Mn remains fixed in Cu50Mn25Al25-xGax alloy system. It has been reported that Mn-Mn coupling is responsible for the magnetic properties of Cu50Mn25Al25 alloy [52, 53]. As discussed earlier, the substitution of Ga may increase the Mn-Mn distance in the lattice. This is due to sight increase in the lattice constant (see **Table 1**). Thus, the decrease in the magnetization has been observed due to increase in the lattice constant which reduces the ferromagnetic coupling of Cu50Mn25Al25-xGax alloy system. Such type of observation has also been reported in other alloy systems [14, 26, 29, 30]. Also, it is well known that Ga3+ is normally substituted in isovalent state vis-a-vis that of Al3+ [54–59]. However, it is worth mentioning here that Ga1+ may possess monovalent state. It has been confirmed Al, Ga mainly provide conduction electrons for the exchange interaction and the Mn atoms show a localized moment in this material [3]. It may be said by considering the varied concentration ratio of Al and Ga that the concentration of conduction electron plays a very important role in determining the magnetic properties. Thus, the magnetic properties may be affected by the presence of mixed valence states of Ga. The concentration of Ga and the disordered occupation of Mn atoms may also be important factors which affects the magnetic properties. Therefore, the conduction electron concentration is believed to be critical in stabilizing the Heusler structure [3]. In the present case,

*(Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

*Magnetization curves at 5 K (a) and 300 K (b) for melt-spun ribbons of Cu50Mn25Al25-xGax (x = 0, 4, 8 and 10) alloys. Magnetization curves (c) and (d) shows a small increase in M with H for the alloy with x = 10* 

**100**

**Figure 6.**

the concentration of Mn is fixed i.e. 25 at. %. Thus, the ferromagnetism behaviour of this material may be related to the concentration of Mn, atomic sites and situation of conduction electrons provided by Cu, Al and Ga. Further, the M-H curves for the melt spun alloy with x = 10 at 5 K and 300 K respectively are shown in **Figure 6(c)** and **(d)**. A close observation of the curves indicates a small increase in M with H superimposed on the flat saturated ferromagnetic M-H curve. This may be due to the formation of a paramagnetic phase (γ-Cu9Al4) in addition to the majority Cu2MnAl ferromagnetic phase in the alloy with x = 10.

The effect of heat treatment on the magnetization has also been studied. On annealing, the magnetic properties of the alloy have changed. The hysteresis loops of the annealed ribbons (for x = 0 and x = 8) at 5 K and 300 K are shown in **Figure 7**. The Ms of the annealed ribbons are found to be lower than their respective as-synthesized ribbons (**Table 1**). The Ms ~ 53 emu/g has been found for annealed ribbon of x = 0 from the magnetization curve (M-H) at 5 K, which is lower than that of the value found for the as-synthesized ribbons of x = 0 *i.e.* Ms ~ 95 emu/g. The decomposition of Cu2MnAl ferromagnetic phase into β-Mn and γ-Cu9Al4 paramagnetic phases may decrease the magnetization [50]. However, along with these phases, Cu2MnAl Heusler phase is still a majority phase for the alloy with x = 0. The drastic decrease in the magnetization value at 5 K from ~83 emu/g to ~5 emu/g has been observed for x = 8. This is attributed to the different decomposition behaviour of x = 0 and x = 8 alloys. The M-H curves of the annealed ribbon for x = 8 (at 5 K and 300 K) are shown in **Figure 7(c)** and **(d)** (c & d). The increase in M with increase in the applied field is

#### **Figure 7.**

*Magnetization curves at 5 K (a) and 300 K (b) for annealed ribbons of Cu50Mn25Al25-xGax (x = 0 and 8) alloys. Magnetization curves (c) and (d) shows an increase in M with H for the alloy with x = 8 (Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

clearly evident. This may be due to the precipitation of paramagnetic phase (γ-Cu9Al4 type) as a majority phase for the alloy with x = 8.

The temperature dependence of magnetization curves (M-T curves) for the melt-spun and annealed ribbons under a field of 500 Oe has been investigated. **Figure 8(a-d)** shows the M-T curves of the melt-spun Cu50Mn25Al25-xGax (x = 0, 4, 8 & 10) alloys. The alloys with x = 4 and 8 have the same type of M-T curves as the parent alloy Cu50Mn25Al25 (x = 0) shows the characteristics of a ferromagnet. With increase in Ga concentration, the saturation magnetization M(T) decreases. At low temperature in the M-T curves for x = 0, 4 and 8, it follows the expression M(T) = M(0)(1-ATn ) with n = 3/2 [52]. The data for T < 100 K as a function of T3/2 is shown in **Figure 8(a)–(c)**. The T3/2 dependence of the magnetization at this low T range has been shown by the solid line in the inset graph which is a linear fit to the data. The magnetization measurements of Cu50Mn25Al25-xGax (x = 0, 4, 8 & 10) alloys in the zero field cooled (ZFC) and field cooled (FC) modes are shown in **Figure 8**. The magnetization measurements in ZFC and FC modes can give an estimate of magnetic ordering temperature in a system with long range magnetic order or blocking/freezing temperature in the case of medium range order [60].

The splitting in the ZFC and FC magnetization curves has been observed for the alloys with x = 0, 4, 8 and 10 (**Figure 8**). The alloys with x = 0, 4, 8 and 10 shows the splitting at ~145 K, ~125 K, ~25 K and ~ 225 K respectively. The splitting temperature found to be decreases from x = 0 to x = 8 and then increases for x = 10. The reason for this may be attributed to the fact that the alloys up to x = 8 exhibit single phase while the alloy with x = 10 is biphasic in nature. Thus, the synthesis acquired anisotropy

#### **Figure 8.**

*Temperature dependence of magnetization for melt-spun ribbons of Cu50Mn25Al25-xGax alloys with x = 0 (a), x = 4 (b), x = 8 (c) and x = 10 (d) in a field of 500 Oe in the ZFC and FC modes. The data for T < 100 K as a function of T3/2 is shown in the inset graphs of (a), (b) and (c) (Reprinted from reference [24] with kind permission from Elsevier, Copyright 2012, Elsevier).*

**103**

**Figure 9.**

*Elsevier).*

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

51.231 and 55.85o

of the Cu50Mn25Al25-xGax (x = 0–10) melt spun alloys causes the splitting in the ZFC and FC curves. The synthesis of ribbons using the melt spinning process involves the sudden cooling of molten alloy when it falls on to the rotating Cu-wheel. The sudden and fast cooling of the molten alloy arises from the very high speed (~ 4000 rpm) of the rotating wheel. It is well known that the side of the ribbon which is in direct contact with the wheel has higher cooling rate than the one which is not in contact [61]. Thus, the variation in cooling rate along the thickness of the ribbon result in some microstructural changes between contact side (CS) and non-contact side

(NCS) [61]. This has led to synthesis acquired anisotropy in the ribbons.

**3. Structure/microstructure and magnetic properties of nanoferrites**

The nanoferrites of NiFe2O4, CoFe2O4 and MnFe2O4 thin films were prepared by a MOD method [62]. **Figure 9(a)** shows the XRD pattern of NiFe2O4, CoFe2O4 and MnFe2O4 thin films. The ferrite peaks at 2θ = 30.481, 34.991, 37.481, 42.581, 48.021,

and (511) reflections of spinel structure. A small amount of α-Fe2O3 phase had also been formed as an impurity phase. The lattice constants are calculated from the XRD data *i.e. a*(Å) ~8.161, 8.312 and 8.425, respectively for NiFe2O4, CoFe2O4 and MnFe2O4. These calculated values of lattice constants are closer to bulk NiFe2O4

*(a) XRD pattern of NiFe2O4, CoFe2O4 and MnFe2O4 thin films. AFM images (b) NiFe2O4 (c) CoFe2O4 (d) MnFe2O4 thin films (Reprinted from reference [62] with kind permission from Elsevier, Copyright 2011,* 

respectively attributed to (220), (311), (222), (400), (331), (422)

**3.1 XRD and AFM analysis of NiFe2O4, CoFe2O4 and MnFe2O4 thin films**

#### *Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

*Magnetic Skyrmions*

M(T) = M(0)(1-ATn

type) as a majority phase for the alloy with x = 8.

clearly evident. This may be due to the precipitation of paramagnetic phase (γ-Cu9Al4

The temperature dependence of magnetization curves (M-T curves) for the melt-spun and annealed ribbons under a field of 500 Oe has been investigated. **Figure 8(a-d)** shows the M-T curves of the melt-spun Cu50Mn25Al25-xGax (x = 0, 4, 8 & 10) alloys. The alloys with x = 4 and 8 have the same type of M-T curves as the parent alloy Cu50Mn25Al25 (x = 0) shows the characteristics of a ferromagnet. With increase in Ga concentration, the saturation magnetization M(T) decreases. At low temperature in the M-T curves for x = 0, 4 and 8, it follows the expression

is shown in **Figure 8(a)–(c)**. The T3/2 dependence of the magnetization at this low T range has been shown by the solid line in the inset graph which is a linear fit to the data. The magnetization measurements of Cu50Mn25Al25-xGax (x = 0, 4, 8 & 10) alloys in the zero field cooled (ZFC) and field cooled (FC) modes are shown in **Figure 8**. The magnetization measurements in ZFC and FC modes can give an estimate of magnetic ordering temperature in a system with long range magnetic order or blocking/freezing temperature in the case of medium range order [60]. The splitting in the ZFC and FC magnetization curves has been observed for the alloys with x = 0, 4, 8 and 10 (**Figure 8**). The alloys with x = 0, 4, 8 and 10 shows the splitting at ~145 K, ~125 K, ~25 K and ~ 225 K respectively. The splitting temperature found to be decreases from x = 0 to x = 8 and then increases for x = 10. The reason for this may be attributed to the fact that the alloys up to x = 8 exhibit single phase while the alloy with x = 10 is biphasic in nature. Thus, the synthesis acquired anisotropy

*Temperature dependence of magnetization for melt-spun ribbons of Cu50Mn25Al25-xGax alloys with x = 0 (a), x = 4 (b), x = 8 (c) and x = 10 (d) in a field of 500 Oe in the ZFC and FC modes. The data for T < 100 K as a function of T3/2 is shown in the inset graphs of (a), (b) and (c) (Reprinted from reference [24] with kind* 

) with n = 3/2 [52]. The data for T < 100 K as a function of T3/2

**102**

**Figure 8.**

*permission from Elsevier, Copyright 2012, Elsevier).*

of the Cu50Mn25Al25-xGax (x = 0–10) melt spun alloys causes the splitting in the ZFC and FC curves. The synthesis of ribbons using the melt spinning process involves the sudden cooling of molten alloy when it falls on to the rotating Cu-wheel. The sudden and fast cooling of the molten alloy arises from the very high speed (~ 4000 rpm) of the rotating wheel. It is well known that the side of the ribbon which is in direct contact with the wheel has higher cooling rate than the one which is not in contact [61]. Thus, the variation in cooling rate along the thickness of the ribbon result in some microstructural changes between contact side (CS) and non-contact side (NCS) [61]. This has led to synthesis acquired anisotropy in the ribbons.

## **3. Structure/microstructure and magnetic properties of nanoferrites**

## **3.1 XRD and AFM analysis of NiFe2O4, CoFe2O4 and MnFe2O4 thin films**

The nanoferrites of NiFe2O4, CoFe2O4 and MnFe2O4 thin films were prepared by a MOD method [62]. **Figure 9(a)** shows the XRD pattern of NiFe2O4, CoFe2O4 and MnFe2O4 thin films. The ferrite peaks at 2θ = 30.481, 34.991, 37.481, 42.581, 48.021, 51.231 and 55.85o respectively attributed to (220), (311), (222), (400), (331), (422) and (511) reflections of spinel structure. A small amount of α-Fe2O3 phase had also been formed as an impurity phase. The lattice constants are calculated from the XRD data *i.e. a*(Å) ~8.161, 8.312 and 8.425, respectively for NiFe2O4, CoFe2O4 and MnFe2O4. These calculated values of lattice constants are closer to bulk NiFe2O4

#### **Figure 9.**

*(a) XRD pattern of NiFe2O4, CoFe2O4 and MnFe2O4 thin films. AFM images (b) NiFe2O4 (c) CoFe2O4 (d) MnFe2O4 thin films (Reprinted from reference [62] with kind permission from Elsevier, Copyright 2011, Elsevier).*

(8.17 Å), CoFe2O4 (8.32 Å) and MnFe2O4 (8.43 Å) [63, 64]. **Figure 9(b)-(d)** shows the AFM images respectively of NiFe2O4, CoFe2O4 and MnFe2O4 thin films. It exhibited homogeneous micro-structures with uniform size distribution of nano-grains. The thin film surfaces are smooth and crack-free. The grains exhibit a round shape with a small grain boundary region. The average size of grains AFM is 46, 61 and 75 nm and surface roughness is 2.5, 4 and 2 nm respectively for NiFe2O4, CoFe2O4 and MnFe2O4. The smaller nano-grains formation may attribute better stoichiometric ratio and low processing temperature of crystallization. Also the addition of PEG encapsulates the ferrites constituents into smaller groups during the heating process [65].

## **3.2 Magnetism of ferrite MFe2O4 nanoparticles**

## *3.2.1 Ferromagnetism of MFe2O4 [M = Mn, Co, Ni, Zn] with BaTiO3 nanocomposite*

The magnetic nanoparticles of multiferroic MFe2O4/BaTiO3 [M = Mn (MnFO/ BTO), Co (CFO/BTO), Ni (NFO/BTO) and Zn (ZFO/BTO)] thin films were fabricated by a MOD method [66]. The addition of ferrite MFe2O4 in perovskite BaTiO3 results in to lattice strain due to tetragonal distortion, expansion/contraction of MFO/BTO unit cell and lattice mismatch. The tetragonal BTO, spinel MFO phases and lattice strain effects are confirmed by XRD analysis. The average grain size for MnFO/BTO, CFO/BTO, NFO/BTO and ZFO/BTO is calculated from AFM images and is found to be 25, 102, 24 and 133 nm, respectively.

The ferromagnetic behaviour (magnetization versus applied magnetizing field (M-Hdc)) of MFO/ BTO thin films at room temperature is shown in **Figure 10(a)**. The values of remanent magnetization, Mr = 0.03, 3.75, 6.76 and 1.49 k J T−1 m−3 with coercivity, Hc = 0.013 × 105 , 0.079 × 105 , 0.167 × 105 and 0.135 × 105 Am−1 and saturation magnetization, Ms = 1.29, 20.25, 27.64 and 6.77 kJ T−1 m−3, respectively, measured for MnFO/ BTO, CFO/ BTO, NFO/ BTO and ZFO/ BTO. The magnetization values observed for MFO/BTO nanocomposite shown abrupt reduction than single phase MnFe2O4 (5.40 kJ T−1 m−3), ZnFe2O4 (230 kJ T−1 m−3), NiFe2O4 (50.60 kJ T−1 m−3) and CoFe2O4 (33.50 kJ T−1 m−3) [62, 67]. Due to non-magnetic

#### **Figure 10.**

*(a) Room temperature M-Hdc hysteresis of MFe2O4/BaTiO3 thin films (b & c) M-H hysteresis of Ce doped CoFe2O4 nanoparticles at room temperature. Inset shows SQUID measurement for ZFC/FC experiment (Reprinted from reference [66] with kind permission from Elsevier, Copyright 2017, Elsevier).*

**105**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

ferrite nanoparticles.

1.00 for NiFe2O4 nanoparticles.

*3.2.2 Ferromagnetic ordering in Ce doped CoFe2O4 nanoparticles*

ionic size): = − = × − + +× − ( )

It is reported that the core/shell nanoparticles of CoFe2-xCexO4 [*x* = 0.05 (CFCeO05), 0.1 (CFCeO10)] were prepared by a chemical combustion method [37]. XRD pattern results into spinel structure with cubic space group. From TEM images, the average particles size, D = 8 and 10 nm, respectively, for CFCeO05 and CFCeO10 sample. **Figure 10(b)** and **(c)** shows the M-H hysteresis with Ms = 42.54 and 10.41 emug−1 and Mr = 26.68 and 1.57, emug−1 with Hc = 1526 and 140 Oe, respectively, observed for CFCeO05 and CFCeO10. These values of magnetization are larger than nanostructured pure CoFe2O4 but smaller than bulk value (73 emug−1 at room temperature). This is due to higher surface energy and surface tension in the nanoparticles which changes cationic preferences of ferrite. It leads to an increase in degree of anti-site defects to cause more surface spin canted or disorder. Recently, Georgiadou *et al*. [78] suggested modification in cation occupancy in nanostructured CFO due to its inverse spinel structure. The theoretical expression for the net moment of MCe is given (Ce3+ ions occupy only the B sites for their large

> µ*M MM x x Ce B A* 5 1 3.8 5 *Ce* ; where MA and MB are the

BTO phase of nano-composite, the saturation magnetization of ferrite reduces. Since mixed perovskite BTO acts as a non-magnetic defect, which hinders the growth of the magnetic domains of spinel MFO and their movement under an external magnetic field [68]. Thus the non-magnetic elements weaken the A-B super exchange interaction which results in the increase of distance between the magnetic moments in A and B sites in the spinel structure [69]. The weaker A-B super exchange interaction of MFO by non-magnetic BTO phase is more affected by the thermal motion, resulting in the decrease of TC. Wang *et al.* [70] suggested that the decrease in the magnetization causes the relaxation process to increase, which might be related with oxygen vacancies redistribution. Bateer *et al.* [71] suggested that the decrease in particle size and the presence of a magnetic dead oranti-ferromagnetic layer on the surface results into reduction in saturation magnetization in

Moreover the origin of observed ferromagnetic behavior (**Figure 10**) of MFO ferrites nanocomposite must be related with two different mechanisms: ferrimagnetic coupling of Fe ionsat A-B sites in (M1-xFex)[MxFe2-x]O4 and the surface spin canting [72]. The structure of the ferrite becomes mixed spinel from its normal configuration when consider the cation inversion in which the shifting of M ions from A to B site and that of Fe ions from B to A-site changes the magnetic behavior of the nanocomposite [73]. The magnetic moments of Fe ions (at B-site) cancel out due to negative interaction among them and M ions have nothing to contribute. The Fe ions present at A-site are responsible for magnetization enhancement and contribute to net magnetic moment. Vamvakidis *et al.* [74] suggested that within the spinel structure, the reduction in inversion degree (~0.22) of MnFe2O4 (due to the partial oxidation of Mn2+ to Mn3+ ions) results into weaker super exchange interactions between tetrahedral and octahedral sites. It contributes to weak ferrimagnetism in MnFe2O4. Bullita *et al.* [75] reported that the cation distribution of ZnFe2O4 at the nanoscale level is contributed by partial inverted spinel structure which results in the increase of magnetization. Peddis *et al.* [76] shows the typical ferrimagnetic structure of inverse CoFe2O4 nanoparticles with an inversion degree of 0.74. These results in the better correlation between spin canting and cationic distribution to get competitively higher value of saturation magnetization. However, Carta *et al.* [77] reported the degree of inversion 0.20 for MnFe2O4 nanoparticles, 0.68 for CoFe2O4 nanoparticles and

*Magnetic Skyrmions*

*nanocomposite*

with coercivity, Hc = 0.013 × 105

(8.17 Å), CoFe2O4 (8.32 Å) and MnFe2O4 (8.43 Å) [63, 64]. **Figure 9(b)-(d)** shows the AFM images respectively of NiFe2O4, CoFe2O4 and MnFe2O4 thin films. It exhibited homogeneous micro-structures with uniform size distribution of nano-grains. The thin film surfaces are smooth and crack-free. The grains exhibit a round shape with a small grain boundary region. The average size of grains AFM is 46, 61 and 75 nm and surface roughness is 2.5, 4 and 2 nm respectively for NiFe2O4, CoFe2O4 and MnFe2O4. The smaller nano-grains formation may attribute better stoichiometric ratio and low processing temperature of crystallization. Also the addition of PEG encapsulates the

The magnetic nanoparticles of multiferroic MFe2O4/BaTiO3 [M = Mn (MnFO/ BTO), Co (CFO/BTO), Ni (NFO/BTO) and Zn (ZFO/BTO)] thin films were fabricated by a MOD method [66]. The addition of ferrite MFe2O4 in perovskite BaTiO3 results in to lattice strain due to tetragonal distortion, expansion/contraction of MFO/BTO unit cell and lattice mismatch. The tetragonal BTO, spinel MFO phases and lattice strain effects are confirmed by XRD analysis. The average grain size for MnFO/BTO, CFO/BTO, NFO/BTO and ZFO/BTO is calculated from AFM images

The ferromagnetic behaviour (magnetization versus applied magnetizing field (M-Hdc)) of MFO/ BTO thin films at room temperature is shown in **Figure 10(a)**. The values of remanent magnetization, Mr = 0.03, 3.75, 6.76 and 1.49 k J T−1 m−3

, 0.167 × 105

and 0.135 × 105

Am−1 and

, 0.079 × 105

saturation magnetization, Ms = 1.29, 20.25, 27.64 and 6.77 kJ T−1 m−3, respectively, measured for MnFO/ BTO, CFO/ BTO, NFO/ BTO and ZFO/ BTO. The magnetization values observed for MFO/BTO nanocomposite shown abrupt reduction than single phase MnFe2O4 (5.40 kJ T−1 m−3), ZnFe2O4 (230 kJ T−1 m−3), NiFe2O4 (50.60 kJ T−1 m−3) and CoFe2O4 (33.50 kJ T−1 m−3) [62, 67]. Due to non-magnetic

*(a) Room temperature M-Hdc hysteresis of MFe2O4/BaTiO3 thin films (b & c) M-H hysteresis of Ce doped CoFe2O4 nanoparticles at room temperature. Inset shows SQUID measurement for ZFC/FC experiment (Reprinted from reference [66] with kind permission from Elsevier, Copyright 2017, Elsevier).*

ferrites constituents into smaller groups during the heating process [65].

*3.2.1 Ferromagnetism of MFe2O4 [M = Mn, Co, Ni, Zn] with BaTiO3*

**3.2 Magnetism of ferrite MFe2O4 nanoparticles**

and is found to be 25, 102, 24 and 133 nm, respectively.

**104**

**Figure 10.**

BTO phase of nano-composite, the saturation magnetization of ferrite reduces. Since mixed perovskite BTO acts as a non-magnetic defect, which hinders the growth of the magnetic domains of spinel MFO and their movement under an external magnetic field [68]. Thus the non-magnetic elements weaken the A-B super exchange interaction which results in the increase of distance between the magnetic moments in A and B sites in the spinel structure [69]. The weaker A-B super exchange interaction of MFO by non-magnetic BTO phase is more affected by the thermal motion, resulting in the decrease of TC. Wang *et al.* [70] suggested that the decrease in the magnetization causes the relaxation process to increase, which might be related with oxygen vacancies redistribution. Bateer *et al.* [71] suggested that the decrease in particle size and the presence of a magnetic dead oranti-ferromagnetic layer on the surface results into reduction in saturation magnetization in ferrite nanoparticles.

Moreover the origin of observed ferromagnetic behavior (**Figure 10**) of MFO ferrites nanocomposite must be related with two different mechanisms: ferrimagnetic coupling of Fe ionsat A-B sites in (M1-xFex)[MxFe2-x]O4 and the surface spin canting [72]. The structure of the ferrite becomes mixed spinel from its normal configuration when consider the cation inversion in which the shifting of M ions from A to B site and that of Fe ions from B to A-site changes the magnetic behavior of the nanocomposite [73]. The magnetic moments of Fe ions (at B-site) cancel out due to negative interaction among them and M ions have nothing to contribute. The Fe ions present at A-site are responsible for magnetization enhancement and contribute to net magnetic moment. Vamvakidis *et al.* [74] suggested that within the spinel structure, the reduction in inversion degree (~0.22) of MnFe2O4 (due to the partial oxidation of Mn2+ to Mn3+ ions) results into weaker super exchange interactions between tetrahedral and octahedral sites. It contributes to weak ferrimagnetism in MnFe2O4. Bullita *et al.* [75] reported that the cation distribution of ZnFe2O4 at the nanoscale level is contributed by partial inverted spinel structure which results in the increase of magnetization. Peddis *et al.* [76] shows the typical ferrimagnetic structure of inverse CoFe2O4 nanoparticles with an inversion degree of 0.74. These results in the better correlation between spin canting and cationic distribution to get competitively higher value of saturation magnetization. However, Carta *et al.* [77] reported the degree of inversion 0.20 for MnFe2O4 nanoparticles, 0.68 for CoFe2O4 nanoparticles and 1.00 for NiFe2O4 nanoparticles.

## *3.2.2 Ferromagnetic ordering in Ce doped CoFe2O4 nanoparticles*

It is reported that the core/shell nanoparticles of CoFe2-xCexO4 [*x* = 0.05 (CFCeO05), 0.1 (CFCeO10)] were prepared by a chemical combustion method [37]. XRD pattern results into spinel structure with cubic space group. From TEM images, the average particles size, D = 8 and 10 nm, respectively, for CFCeO05 and CFCeO10 sample. **Figure 10(b)** and **(c)** shows the M-H hysteresis with Ms = 42.54 and 10.41 emug−1 and Mr = 26.68 and 1.57, emug−1 with Hc = 1526 and 140 Oe, respectively, observed for CFCeO05 and CFCeO10. These values of magnetization are larger than nanostructured pure CoFe2O4 but smaller than bulk value (73 emug−1 at room temperature). This is due to higher surface energy and surface tension in the nanoparticles which changes cationic preferences of ferrite. It leads to an increase in degree of anti-site defects to cause more surface spin canted or disorder.

Recently, Georgiadou *et al*. [78] suggested modification in cation occupancy in nanostructured CFO due to its inverse spinel structure. The theoretical expression for the net moment of MCe is given (Ce3+ ions occupy only the B sites for their large ionic size): = − = × − + +× − ( ) µ*M MM x x Ce B A* 5 1 3.8 5 *Ce* ; where MA and MB are the wof Fe3+cation is fixed to 5μB (spin only) and for octahedrally coordinated Co2+ cations is fixed to 3.8 which correspond to the Msat at 0 K of bulk CFO (95 emug−1) [79]. The net magnetic moments μCe cation is zero for the diamagnetic Ce4+ ions and non-zero for the paramagnetic Ce3+ ones. By replacing Fe3+ by Ce3+cation, the Ms is expected to vary as μCe by the sequential filling of electrons in the 4*f* shell. Unlikely, a clear deviation between the theoretically predicted magnetization by above equation and the experimental one (**Figure 10(b)** and **(c)**) is observed. This is ascribed into two reasons. One is the decrease in strong negative Fe3+-Fe3+ interaction that resulted from the doping of Ce because the spinel ferrimagnetic CFO is largely governed by the negative Fe3+-Fe3+ interaction (the spin coupling of the 3d electrons). The Ce3+-Fe3+ interaction (4*f*-3*d* coupling) as well as the Ce3+-Ce3+ one (indirect 4*f*-5*d*-4*f* coupling) exist this is very weak [80]. The other is the rearrangement of the Co2+ ion in the A and B sites resulted from the doping with RE Ce3+ ions. The migration of Co2+ ion into tetrahedral sublattice decreases the concentration of Fe3+ ion in A site, leading to enhance Ms. Also, the value of Hc is reduces with Ce3+ ion concentration in CFCeO10 and shows weak ferromagnetism. This is responsible due to variation in core/ shell formation and description of CFO lattice by vibrational modes [37]. The core/shell (CeO/CFO) system result into an increase in effective magnetic anisotropy caused by surface and interface exchange coupling effects. The huge difference in the coercivity value among CFCeO samples may ascribe to surface pinning that arises due to missing coordination of oxygen atoms and the shape effect of the spinel ferrite.

The origin of observed room temperature ferromagnetism of CFCeO samples is evaluated by the temperature dependent magnetization [M(T)] with field cooling (FC) and zero field cooling (ZFC) measurement (**Figure 10(b')** and **(c')** inset). The applied magnetic field is 100 Oe. These M-T measurements show that ZFC-FC curves of the CFCeO05 sample did not coincide with each other or slightly coincide around 300 K. It indicates that the nanoparticles are still magnetically blocked at around room temperature. However, CFCeO10 nanoparticles show blocking temperature of antiferromagnetism, spin glass etc. at about 91 K. This type of the magnetic response is due to different nano-core/ shell formation in CFCeO samples. The core/shell nanoparticles provides spin-phonon coupling in which a core of aligned spins is surrounded by a magnetically disordered shell [81].

## *3.2.3 Humidity response from MgFe2O4 thin films*

**Figure 11** shows the humidity response of MgFe2O4 thin films (measured in the range 10–90% RH at 25°C). The base resistance of thin film annealed at 400°C increased from 59 GΩ to 30 TΩ annealed at 800°C. There are many factors on which the resistance of ferrites depends such as porosity, vacancies and electron hopping between Fe2+ and Fe3+. In the present study it may be due to higher annealing temperature that increases the average pore size distribution which further creates more obstruction for the charge carrier's movement. It can be seen from **Figure 11** that with higher annealing temperature, the response of Log R (Ohm) approaches towards linearity with rising humidity 10–90% RH (relative humidity). Log R variation of the film annealed at 400°C was almost constant (up to 50% RH), after that the linear decrease was observed at high humidity value. This may be due to the presence of less pores available for adsorption and thus a very few water molecules are only able to chemisorbed in such pores. The slope of log R (with increasing humidity) increases for the film annealed at 600°C and it became almost linear for the film annealed at 800°C for the entire range of humidity which may be due to the increase of intra and inter pores.

**107**

**4. Conclusion**

**Figure 11.**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

*kind permission from Elsevier, Copyright 2011, Elsevier).*

In this chapter the recent progress in the development of Heusler alloys and nanoferrites are discussed. Formation of Heusler single phase of the Cu2MnAl structure has been found only for the Cu50Mn25Al25-xGax alloys with x ≤ 8. The size of the grains for the Heusler phase alloy is in the range 100–200 nm. Long term annealing of the alloys leads to the formation of β-Mn and γ-Cu9Al4 type phases. The saturation magnetization (Ms) decreases with increasing Ga concentration. The decomposition of the Cu2MnAl Heusler phase into β-Mn and γ-Cu9Al4 phases

*Log R vs. relative humidity response curve of MgFe2O4 thin films at 1 kHz (Reprinted from reference [82] with* 

The XRD pattern confirms the cubic phase of spinel ferrites in nanoferrites. The average grain's size for these ferrites is found to be less than 100 nm. Thus it can be said that the nanostructural formation of ferrite grains depends on the selection of chemical route and annealing temperature and the magnetic properties of nanoferrites are different from the bulk ferrites. The core/shell nanocomposites formation of Ce substituted CoFe2O4 hinders the superparamagnetism formed due to small magnetic nanoparticles results into long-range antiferromagnetic interactions. MgFe2O4 thin film annealed at 800°C has a linear log R (Ohm) response towards the

One of the authors (Devinder Singh) gratefully acknowledges the financial support by Department of Science and Technology (DST), New Delhi, India in the form of INSPIRE Faculty Award [IFA12-PH-39] and K.C. Verma is grateful to SERB Department of Science and Technology (DST), Govt. of India for the financial sup-

port under Fast-track Scheme for Young Scientist [SR/FTP/PS-180/2013].

during annealing leads to the decrease in the magnitude of Ms.

entire humidity range 10–90% RH.

**Acknowledgements**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

**Figure 11.**

*Magnetic Skyrmions*

and the shape effect of the spinel ferrite.

*3.2.3 Humidity response from MgFe2O4 thin films*

wof Fe3+cation is fixed to 5μB (spin only) and for octahedrally coordinated Co2+ cations is fixed to 3.8 which correspond to the Msat at 0 K of bulk CFO (95 emug−1) [79]. The net magnetic moments μCe cation is zero for the diamagnetic Ce4+ ions and non-zero for the paramagnetic Ce3+ ones. By replacing Fe3+ by Ce3+cation, the Ms is expected to vary as μCe by the sequential filling of electrons in the 4*f* shell. Unlikely, a clear deviation between the theoretically predicted magnetization by above equation and the experimental one (**Figure 10(b)** and **(c)**) is observed. This is ascribed into two reasons. One is the decrease in strong negative Fe3+-Fe3+ interaction that resulted from the doping of Ce because the spinel ferrimagnetic CFO is largely governed by the negative Fe3+-Fe3+ interaction (the spin coupling of the 3d electrons). The Ce3+-Fe3+ interaction (4*f*-3*d* coupling) as well as the Ce3+-Ce3+ one (indirect 4*f*-5*d*-4*f* coupling) exist this is very weak [80]. The other is the rearrangement of the Co2+ ion in the A and B sites resulted from the doping with RE Ce3+ ions. The migration of Co2+ ion into tetrahedral sublattice decreases the concentration of Fe3+ ion in A site, leading to enhance Ms. Also, the value of Hc is reduces with Ce3+ ion concentration in CFCeO10 and shows weak ferromagnetism. This is responsible due to variation in core/ shell formation and description of CFO lattice by vibrational modes [37]. The core/shell (CeO/CFO) system result into an increase in effective magnetic anisotropy caused by surface and interface exchange coupling effects. The huge difference in the coercivity value among CFCeO samples may ascribe to surface pinning that arises due to missing coordination of oxygen atoms

The origin of observed room temperature ferromagnetism of CFCeO samples

is evaluated by the temperature dependent magnetization [M(T)] with field cooling (FC) and zero field cooling (ZFC) measurement (**Figure 10(b')** and **(c')** inset). The applied magnetic field is 100 Oe. These M-T measurements show that ZFC-FC curves of the CFCeO05 sample did not coincide with each other or slightly coincide around 300 K. It indicates that the nanoparticles are still magnetically blocked at around room temperature. However, CFCeO10 nanoparticles show blocking temperature of antiferromagnetism, spin glass etc. at about 91 K. This type of the magnetic response is due to different nano-core/ shell formation in CFCeO samples. The core/shell nanoparticles provides spin-phonon coupling in which a core of aligned spins is surrounded by a magnetically disordered

**Figure 11** shows the humidity response of MgFe2O4 thin films (measured in the range 10–90% RH at 25°C). The base resistance of thin film annealed at 400°C increased from 59 GΩ to 30 TΩ annealed at 800°C. There are many factors on which the resistance of ferrites depends such as porosity, vacancies and electron hopping between Fe2+ and Fe3+. In the present study it may be due to higher annealing temperature that increases the average pore size distribution which further creates more obstruction for the charge carrier's movement. It can be seen from **Figure 11** that with higher annealing temperature, the response of Log R (Ohm) approaches towards linearity with rising humidity 10–90% RH (relative humidity). Log R variation of the film annealed at 400°C was almost constant (up to 50% RH), after that the linear decrease was observed at high humidity value. This may be due to the presence of less pores available for adsorption and thus a very few water molecules are only able to chemisorbed in such pores. The slope of log R (with increasing humidity) increases for the film annealed at 600°C and it became almost linear for the film annealed at 800°C for the entire range of humidity which may be due to the

**106**

increase of intra and inter pores.

shell [81].

*Log R vs. relative humidity response curve of MgFe2O4 thin films at 1 kHz (Reprinted from reference [82] with kind permission from Elsevier, Copyright 2011, Elsevier).*

### **4. Conclusion**

In this chapter the recent progress in the development of Heusler alloys and nanoferrites are discussed. Formation of Heusler single phase of the Cu2MnAl structure has been found only for the Cu50Mn25Al25-xGax alloys with x ≤ 8. The size of the grains for the Heusler phase alloy is in the range 100–200 nm. Long term annealing of the alloys leads to the formation of β-Mn and γ-Cu9Al4 type phases. The saturation magnetization (Ms) decreases with increasing Ga concentration. The decomposition of the Cu2MnAl Heusler phase into β-Mn and γ-Cu9Al4 phases during annealing leads to the decrease in the magnitude of Ms.

The XRD pattern confirms the cubic phase of spinel ferrites in nanoferrites. The average grain's size for these ferrites is found to be less than 100 nm. Thus it can be said that the nanostructural formation of ferrite grains depends on the selection of chemical route and annealing temperature and the magnetic properties of nanoferrites are different from the bulk ferrites. The core/shell nanocomposites formation of Ce substituted CoFe2O4 hinders the superparamagnetism formed due to small magnetic nanoparticles results into long-range antiferromagnetic interactions. MgFe2O4 thin film annealed at 800°C has a linear log R (Ohm) response towards the entire humidity range 10–90% RH.

### **Acknowledgements**

One of the authors (Devinder Singh) gratefully acknowledges the financial support by Department of Science and Technology (DST), New Delhi, India in the form of INSPIRE Faculty Award [IFA12-PH-39] and K.C. Verma is grateful to SERB Department of Science and Technology (DST), Govt. of India for the financial support under Fast-track Scheme for Young Scientist [SR/FTP/PS-180/2013].

*Magnetic Skyrmions*

## **Author details**

Devinder Singh1 \* and Kuldeep Chand Verma2

1 Amity School of Applied Sciences, Amity University, Lucknow Campus, Lucknow, UP, India

2 CSIR-Central Scientific Instrument Organisation, Chandigarh, India

\*Address all correspondence to: dsingh2@lko.amity.edu

© 2021 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.

**109**

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

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[11] Ouardi S, Shekhar C, Fecher GH, Kozina X, Stryganyuk G, Felser C, Ueda S, Kobayashi K. Electronic structure of Pt based topological Heusler alloys with C1b structure and "zero band gap". Appl Phys Lett

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[15] Kaul SN, Annie D' Santhoshini B, A bhyankar AC, Barquin LF, Henry P. Thermoelastic martensitic transformation in ferromagnetic Ni–Fe–Al alloys: Effect of site disorder. Appl Phys

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*Magnetic Skyrmions*

**108**

**Author details**

Devinder Singh1

Lucknow, UP, India

\* and Kuldeep Chand Verma2

\*Address all correspondence to: dsingh2@lko.amity.edu

provided the original work is properly cited.

1 Amity School of Applied Sciences, Amity University, Lucknow Campus,

© 2021 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 CSIR-Central Scientific Instrument Organisation, Chandigarh, India

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[55] Singh D, Singh D, Mandal RK, Srivastava ON, Tiwari RS. Glass forming ability, thermal stability and indentation characteristics of Ce75Al25−xGax metallic glasses. J Alloys & Compds 2014;**590**:15.

[56] Mandal RK, Tiwari RS, Singh D, Singh D. Influence of Ga substitution on the mechanical behavior of Zr69.5Al7.5 *x*GaxCu12Ni11 and Ce75Al25-*x*Gax metallic glass compositions. MRS Proceeding. 2015;1757. doi:10.1557/opl.2015.45.

[57] Yadav TP, Singh D, Shahi RR, Shaz MA, Tiwari RS, Srivastava ON. Formation of quasicrystalline phase in Al70-*x*GaxPd17Mn13 alloys. Phil Mag. 2011;**91**:2474.

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[59] Singh D, Singh D, Mandal RK, Srivastava ON, Tiwari RS. Effect of quenching rate on the microstructure and mechanical behavior of Ce75Al21Ga4 glassy alloy. Materials Characterization 2017;**134**:18.

[60] Konoplyuk SM, Kokorin VV, Kolomiets OV, Perekos AE, Nadutov VM. Magnetoresistance of Cu-Mn-Al melt-spun ribbons

containing the system of interacting ferromagnetic inclusions. J Magn Mag Materials 2011;**323**:763.

[61] Lalla NP, Tiwari RS, Srivastava ON. Transmission electron microscopic investigations of rapidly solidified Al-Mn-Ni quasicrystalline alloys. Phil Mag B 1991;**63**:629.

[62] Verma KC, Singh VP, Ram M, Shah J, Kotnala RK. Structural, microstructural and magnetic properties of NiFe2O4, CoFe2O4 and MnFe2O4 nano ferrite thin films. J Magn Magn Mater 2011;**323**:3271.

[63] Singh S, Singh M, Ralhan N K, Kotnala RK, Verma KC. Ferromagnetic and Dielectric Properties of Ni1 xZnxFe2O4 Nanoparticles Prepared via Chemical Combustion Route. Adv Sci Engg Med 2014; **6(6)**:688.

[64] Verma KC, Singh S, Tripathi SK, Kotnala RK. Multiferroic Ni0.6Zn0.4Fe2O4- BaTiO3 nanostructures: Magnetoelectric coupling, dielectric, and fluorescence. J Appl Phys 2014;**116**:124103.

[65] Verma KC, Sharma AK, Bhatt SS, Kotnala RK, Negi NS. Synthesis and characterization of nanostructured (Pb1-xSrx)TiO3 thin films by a modified chemical route. Phil Mag 2009;**89(27)**:2321.

[66] Verma KC, Singh D, Kumar S, Kotnala RK. Multiferroic effects in MFe2O4/BaTiO3 (M = Mn, Co, Ni, Zn) Nanocomposites. J Alloys Compds 2017;**709**:344.

[67] Sultan M, Singh R. FMR Studies on Nanocrystalline Zinc Ferrite Thin Films. J Phys: Conf Series 2010;**200**:072090.

[68] Verma KC, Kotnala RK. Nanostructural and lattice contributions to multiferroism in NiFe2O4/BaTiO3. Mater Chem Phys 2016;**174**:120.

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Chem C2015;**119**:8336.

1994;**6**:5707.

2011;**519**:6135.

Library, Amsterdam, 1961

Arvanitidid I, Kalogirou O, Samara CD. Unveiling the Physicochemical Features of CoFe2O4 Nanoparticles Synthesized via a Variant Hydrothermal Method: NMR Relaxometric Properties. J Phys

[79] Smith S, Wijn HPJ. Ferrites, Philips

[80] Rezlescu N, Rezlescu E, Pasnicu C, Craus ML. Effects of the rare-earth ions on some properties of a nickelzinc ferrite. J Phys: Condens Matter

[81] Sun QC, Birkel CS, Cao J, Tremel W, Musfeldt JL. Spectroscopic Signature of the Superparamagnetic Transition and Surface Spin Disorder in CoFe2O4 Nanoparticles.ACS Nano 2012;**6**:4876.

[82] Kotnala RK, Shah J, Mathpal MC, Verma KC, Singh S, Lovkush. Influence of annealing on humidity response of RF sputtered nanocrystalline MgFe2O4 thin films. Thin Solid Films

properties of NiFe2O4@BaTiO3 composites with well-matched interface. Sci Technol Adv Mater

[70] Wang P, Jin C, Zheng D, Li D, Gong J, Li P, Bai H, Strain and

Ferroelectric- Field Effects Co-mediated Magnetism in (011)-CoFe2O4/ Pb( Mg1/3Nb2/3)0.7Ti0.3O3 Multiferroic Heterostructures. ACS Appl Mater

[71] Bateer B *et al*. Synthesis, size and magnetic properties of controllable MnFe2O4 nanoparticles with versatile surface functionalities. Dalt Trans

[72] Singh JP *et al*. Observation of bulk like magnetic ordering below the blocking temperature in nanosized zinc ferrite. J Magn & Mag Mater 2012;**324**:

[74] Vamvakidis K *et al*. Reducing the inversion degree of MnFe2O4 nanoparticles through synthesis to enhance magnetization: evaluation of their 1 H NMR relaxation and heating efficiency. Dalton Trans

[73] Goya GF, Rechenberg HR. Magnetic properties of ZnFe2O4 synthesized by ball milling. J Magn & Mag Mater

[75] Bullita S *et al*. ZnFe2O4 nanoparticles dispersed in a highly porous silica aerogel matrix: a magnetic study. Phys Chem Chem Phys 2014;**16**:4843.

[76] Peddis D *et al*. Cationic distribution

nanoparticles. J Phys Condens Matter

[77] Carta D *et al*. A Structural and Magnetic Investigation of the Inversion Degree in Ferrite Nanocrystals MFe2O4 (M = Mn, Co, Ni). J Phys Chem C

and spin canting in CoFe2O4

2012;**13**:045001.

Interf 2016;**8**:24198.

2014;**43**:9885.

1999;**203**:141.

2014;**43**:12754.

2011;**23**:426004.

2009;**113**:8606.

2553.

*Magnetic Properties of Heusler Alloys and Nanoferrites DOI: http://dx.doi.org/10.5772/intechopen.95466*

properties of NiFe2O4@BaTiO3 composites with well-matched interface. Sci Technol Adv Mater 2012;**13**:045001.

*Magnetic Skyrmions*

2004;**278**:328.

Zhang X. Magnetism and transport properties of melt-spun ribbon Cu2MnAl Heusler alloy. J Magn Mag Materials

containing the system of interacting ferromagnetic inclusions. J Magn Mag

[61] Lalla NP, Tiwari RS, Srivastava ON. Transmission electron microscopic investigations of rapidly solidified Al-Mn-Ni quasicrystalline alloys. Phil

[62] Verma KC, Singh VP, Ram M, Shah J, Kotnala RK. Structural, microstructural and magnetic properties of NiFe2O4, CoFe2O4 and MnFe2O4 nano ferrite thin films. J Magn

[63] Singh S, Singh M, Ralhan N K, Kotnala RK, Verma KC. Ferromagnetic and Dielectric Properties of Ni1 xZnxFe2O4 Nanoparticles Prepared via Chemical Combustion Route. Adv Sci

[64] Verma KC, Singh S, Tripathi SK, Kotnala RK. Multiferroic Ni0.6Zn0.4Fe2O4- BaTiO3 nanostructures: Magnetoelectric coupling, dielectric, and fluorescence. J

[65] Verma KC, Sharma AK, Bhatt SS, Kotnala RK, Negi NS. Synthesis and characterization of nanostructured (Pb1-xSrx)TiO3 thin films by a modified chemical route. Phil Mag

[66] Verma KC, Singh D, Kumar S, Kotnala RK. Multiferroic effects in MFe2O4/BaTiO3 (M = Mn, Co, Ni, Zn) Nanocomposites. J Alloys Compds

[67] Sultan M, Singh R. FMR Studies on Nanocrystalline Zinc Ferrite Thin Films. J Phys: Conf Series 2010;**200**:072090.

Nanostructural and lattice contributions to multiferroism in NiFe2O4/BaTiO3. Mater Chem Phys 2016;**174**:120.

[69] Zhou JP, Lv L, Liu Q, Zhang YX, Liu P. Hydrothermal synthesis and

[68] Verma KC, Kotnala RK.

Magn Mater 2011;**323**:3271.

Engg Med 2014; **6(6)**:688.

Appl Phys 2014;**116**:124103.

2009;**89(27)**:2321.

2017;**709**:344.

Materials 2011;**323**:763.

Mag B 1991;**63**:629.

[53] Robinson J, McCormick P, Street R. Structure and properties of Cu2MnAl synthesized by mechanical alloying. J Phys: Condens Matter 1995;**7**:4259.

[54] Heinzig M, Jenks CJ, Hove MV, Fisher I, Canfield P, Thiel PA. Surface preparation and characterization of the icosahedral Al–Ga–Pd–Mn quasicrystal. J Alloys & Compds 2002;**338**:248.

[55] Singh D, Singh D, Mandal RK, Srivastava ON, Tiwari RS. Glass forming ability, thermal stability and indentation characteristics of Ce75Al25−xGax metallic glasses. J Alloys & Compds 2014;**590**:15.

[56] Mandal RK, Tiwari RS, Singh D, Singh D. Influence of Ga substitution on the mechanical behavior of Zr69.5Al7.5 *x*GaxCu12Ni11 and Ce75Al25-*x*Gax metallic glass compositions. MRS Proceeding. 2015;1757. doi:10.1557/opl.2015.45.

[57] Yadav TP, Singh D, Shahi RR, Shaz MA, Tiwari RS, Srivastava ON. Formation of quasicrystalline phase in Al70-*x*GaxPd17Mn13 alloys. Phil Mag.

[58] Singh D, Singh D, Mandal RK, Srivastava ON, Tiwari RS. Effect of annealing on the devitrification behavior and mechanical properties of rapidly quenched Ce-based glassy alloys.

J. Non-Cryst. Solids 2016;**445**:53.

[59] Singh D, Singh D, Mandal RK, Srivastava ON, Tiwari RS. Effect of quenching rate on the microstructure and mechanical behavior of Ce75Al21Ga4 glassy alloy. Materials Characterization

[60] Konoplyuk SM, Kokorin VV, Kolomiets OV, Perekos AE, Nadutov VM. Magnetoresistance of Cu-Mn-Al melt-spun ribbons

2011;**91**:2474.

2017;**134**:18.

**112**

[70] Wang P, Jin C, Zheng D, Li D, Gong J, Li P, Bai H, Strain and Ferroelectric- Field Effects Co-mediated Magnetism in (011)-CoFe2O4/ Pb( Mg1/3Nb2/3)0.7Ti0.3O3 Multiferroic Heterostructures. ACS Appl Mater Interf 2016;**8**:24198.

[71] Bateer B *et al*. Synthesis, size and magnetic properties of controllable MnFe2O4 nanoparticles with versatile surface functionalities. Dalt Trans 2014;**43**:9885.

[72] Singh JP *et al*. Observation of bulk like magnetic ordering below the blocking temperature in nanosized zinc ferrite. J Magn & Mag Mater 2012;**324**: 2553.

[73] Goya GF, Rechenberg HR. Magnetic properties of ZnFe2O4 synthesized by ball milling. J Magn & Mag Mater 1999;**203**:141.

[74] Vamvakidis K *et al*. Reducing the inversion degree of MnFe2O4 nanoparticles through synthesis to enhance magnetization: evaluation of their 1 H NMR relaxation and heating efficiency. Dalton Trans 2014;**43**:12754.

[75] Bullita S *et al*. ZnFe2O4 nanoparticles dispersed in a highly porous silica aerogel matrix: a magnetic study. Phys Chem Chem Phys 2014;**16**:4843.

[76] Peddis D *et al*. Cationic distribution and spin canting in CoFe2O4 nanoparticles. J Phys Condens Matter 2011;**23**:426004.

[77] Carta D *et al*. A Structural and Magnetic Investigation of the Inversion Degree in Ferrite Nanocrystals MFe2O4 (M = Mn, Co, Ni). J Phys Chem C 2009;**113**:8606.

[78] Georgiadou V, Tangoulis V, Arvanitidid I, Kalogirou O, Samara CD. Unveiling the Physicochemical Features of CoFe2O4 Nanoparticles Synthesized via a Variant Hydrothermal Method: NMR Relaxometric Properties. J Phys Chem C2015;**119**:8336.

[79] Smith S, Wijn HPJ. Ferrites, Philips Library, Amsterdam, 1961

[80] Rezlescu N, Rezlescu E, Pasnicu C, Craus ML. Effects of the rare-earth ions on some properties of a nickelzinc ferrite. J Phys: Condens Matter 1994;**6**:5707.

[81] Sun QC, Birkel CS, Cao J, Tremel W, Musfeldt JL. Spectroscopic Signature of the Superparamagnetic Transition and Surface Spin Disorder in CoFe2O4 Nanoparticles.ACS Nano 2012;**6**:4876.

[82] Kotnala RK, Shah J, Mathpal MC, Verma KC, Singh S, Lovkush. Influence of annealing on humidity response of RF sputtered nanocrystalline MgFe2O4 thin films. Thin Solid Films 2011;**519**:6135.

**Chapter 6**

**Abstract**

**1. Introduction**

**115**

Effect of M Substitution on

Magnetocaloric Properties of

R2Fe17-x Mx (R = Gd, Nd; M = Co,

The structure, magnetic and magnetocaloric properties of Nd2Fe17xCox (x = 0; 1; 2; 3, 4) and Gd2Fe17-xCux (x = 0, 0.5, 1 and 1.5) solid solutions have been studied. For this purpose, these samples were prepared by arc melting and subsequent annealing at 1073 K for a 7 days. Structural analysis by Rietveld method on X-ray diffraction (XRD) have determined that these alloys crystallize in the rhombohedral Th2Zn17-type structure (Space group R¯3 m) and the substitution of iron by nickel and copper leads to a decrease in the unit cell volume. The Curie temperature (TC) of the prepared samples depends on the nickel and copper content. Based on the Arrott plot, these analyses show that Nd2Fe17-xCox exhibits a second-order ferromagnetic to paramagnetic phase transition around the Curie temperature. These curves were also used to determine the magnetic entropy change ΔSMax and the relative cooling power. For an applied field of 1.5 T, ΔSMax increase from 3.35 J/kg. K for x = 0 to 5.83 J/kg. K for x = 2. In addition the RCP increases monotonously. This is due to an important temperature range for the magnetic phase transition, contributing to a large ΔSMax shape. Gd2Fe17-xCux solid solution has a reduction of the ferromagnetic phase transition temperature from 475 K (for x = 0) to 460 K (for x = 1.5) is due to the substitution of the magnetic element (Fe) by non-magnetic atoms (Cu). The magnetocaloric effect was determined in the vicinity of the Curie temperature TC. By increasing the Cu content, an increase in the values

Structural, Magnetic and

of magnetic entropy (ΔSMax) in a low applied field is observed.

**Keywords:** Rare-earth alloys and compounds, magnetization, magnetocaloric effect

During the last decades and until now, the production of cold has mainly been ensured by the technique of compression/expansion of a refrigerant. This process is developed and reliable, however it has a large number of disadvantages due to the use of toxic gases such as chlorofluorocarbon (CFC) or hydrochlorofluorocarbon (HCFC) which have proved to be very harmful to the environment (destruction of the ozone layers) and contribute to the greenhouse effect. Current environmental requirements and ecological standards limit conventional technologies. It is for this

Cu) Solid Solutions

*Mosbah Jemmali and Lotfi Bessais*

## **Chapter 6**
