3. Boundary and initial condition

When the ballistic transport appear the temperature jump at boundary occur and cause the reduction of the thermal conductivity [27, 32]. Ben Aissa et al. [12] have been explored a nanoheat conduction in cylindrical surrounding-gate (SG) MOSFET. They used the DPL with the temperature jump applied in the interface oxide-semiconductor defined as

$$
\Delta T\_{jump} = -d \times \text{Kn} \times L \times \nabla T \tag{22}
$$

where d is an adjustable coefficient, the ETC defined as [12]:

$$d = \frac{R \times \kappa\_{\text{eff}}}{Kn \times L\_c} \tag{23}$$

where R is the thermal boundary resistance. The proposed ETC given by Ben Aissa et al. [12, 32] is written as:

$$\kappa\_{\sharp\sharp} = \kappa / (1 + (4 \times \text{Kn})) \tag{24}$$

Hua et al. [33, 34] discussed the temperature jump in nanofilms. They have studied the phonon transport in interfaces. They derived a boundary temperature jump defined as

Table 1. First-order temperature jump condition.

$$T - T\_W = -d \times \Lambda \times \frac{\partial T}{\partial \mathbf{x}} \tag{25}$$

where TW is the temperature jump at the wall.

Yang et al. [35] explained the impact of the temperature jump in the continuum flow and slip flow. They have investigated the heat transfer in nanofluids. At the wall, the temperature jump lead to the following expression by Gad-el-Hak [35]

$$T\_S - T\_W = \frac{2\beta}{\beta + 1} \frac{2 - \sigma\_T}{\sigma\_T} \frac{\Lambda}{\Pr} \frac{dT}{dy} \bigg|\_{\text{wall}} \tag{26}$$

where TS is the system temperature and TW is the wall temperature, σ<sup>T</sup> is the thermal accommodation coefficient, β is the ratio of specific heats, and Pr is the gas Prandtl number. Singh et al. [36] noted that in ideal monoatomic gas (Kn < 0:1), the temperature jump at the solid interface rewritten as

$$T\_S - T\_W = \frac{1.25 \times \Lambda}{\text{Pr}} \frac{dT}{dy} \bigg|\_{\text{Wall}} \tag{27}$$

To solve the BDE model coupled with the temperature jump at boundary we use the FEM [4].

Si <sup>3000</sup> <sup>150</sup> 1.5 � 106 <sup>100</sup> <sup>10</sup> SiO2 <sup>5900</sup> 1.4 1.75 � 106 0.4 0.04

) C (J m�<sup>3</sup> k�<sup>1</sup>

where ½ � B , ½ � B<sup>1</sup> and ½ � D is a matrix valued, f g Tt , f g Ttt and f gT represent the nodal temperature,

<sup>þ</sup> <sup>2</sup> ½ � <sup>B</sup> Δt <sup>2</sup> þ ½ � D

The materials used in our simulation are Silicon [2] and Silicon dioxide [2] and their thermal

The reduction of the thermal conductivity have a strong dependence with the Knudsen number. We take account the specularity parameter and Knudsen number because we studied the

is the nodal temperature at the time tp.

½ � B f g Tt þ ½ � B<sup>1</sup> f g Ttt � ½ � D f gT ¼ f g m (28)

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model

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� Tp�<sup>1</sup> ½ � B

Δt 2 

þ f g m <sup>p</sup> (29)

) Λ (nm) Kn

The finite-element approximation used in the BDE model can be defined as:

) K (Wm�<sup>1</sup> K�<sup>1</sup>

f g m is the matrix vector.

Symbol V (m s�<sup>1</sup>

The discretization of Eq. (25) leads to

Δt

Figure 1. Schematic geometries of the MOSFET transistor.

Table 2. Thermal properties of Silicon and Silicon dioxide.

<sup>þ</sup> ½ � <sup>B</sup> Δt 2 ¼ Tp ½ � B Δt

Tpþ<sup>1</sup> ½ � B

where Δt is the time step and Tp

properties are illustrated in Table 2.

5. Results and discussion

mechanism of boundary scattering.

Due to the utility of the temperature jump, the following expressions are summarized in Table 1.

## 4. Structure to model and numerical method

The architecture used in this present work is the two-dimensional conventional MOSFET. The proposed structure shown in Figure 1. The substrate is compound by Silicon (Si). The Si-MOSFETs thickness used in this model is 50 nm. The channel length is Lc = 10 nm. In order to compare our results with similar works, the reference temperature is T0 = 300 K and the maximal power generation is q\_ <sup>h</sup> <sup>¼</sup> 1019w=m<sup>3</sup> [23]. The right and left boundaries are assumed to be adiabatic. The temperature jump boundary condition is applied in the interface (Si-SiO2). In this side the phenomena of collision phonon-wall is more frequent. The MFP used in this proposed work is Λ ¼ 100 nm [23]. Using Eq. (17), we found that the thermal conductivity reaches 18 Wm�<sup>1</sup> k�<sup>1</sup> . In this case, the adjustable coefficient d attains 0.09 for R = 0.503 K m2 W�<sup>1</sup> [37].

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model http://dx.doi.org/10.5772/intechopen.75595 21

Figure 1. Schematic geometries of the MOSFET transistor.

T � TW ¼ �d � Λ �

Yang et al. [35] explained the impact of the temperature jump in the continuum flow and slip flow. They have investigated the heat transfer in nanofluids. At the wall, the temperature jump

> 2 � σ<sup>T</sup> σT

Λ Pr dT dy Wall

dT dy Wall

<sup>h</sup> <sup>¼</sup> 1019w=m<sup>3</sup> [23]. The right and left boundaries are assumed

β þ 1

TS � TW <sup>¼</sup> <sup>1</sup>:<sup>25</sup> � <sup>Λ</sup>

where TS is the system temperature and TW is the wall temperature, σ<sup>T</sup> is the thermal accommodation coefficient, β is the ratio of specific heats, and Pr is the gas Prandtl number. Singh et al. [36] noted that in ideal monoatomic gas (Kn < 0:1), the temperature jump at the solid

Pr

Due to the utility of the temperature jump, the following expressions are summarized in

The architecture used in this present work is the two-dimensional conventional MOSFET. The proposed structure shown in Figure 1. The substrate is compound by Silicon (Si). The Si-MOSFETs thickness used in this model is 50 nm. The channel length is Lc = 10 nm. In order to compare our results with similar works, the reference temperature is T0 = 300 K and the

to be adiabatic. The temperature jump boundary condition is applied in the interface (Si-SiO2). In this side the phenomena of collision phonon-wall is more frequent. The MFP used in this proposed work is Λ ¼ 100 nm [23]. Using Eq. (17), we found that the thermal conductivity reaches 18

. In this case, the adjustable coefficient d attains 0.09 for R = 0.503 K m2 W�<sup>1</sup> [37].

TS � TW <sup>¼</sup> <sup>2</sup><sup>β</sup>

Pr For Kn < 0:1

where TW is the temperature jump at the wall.

Present work 0.09 For Kn ¼ 10

interface rewritten as

maximal power generation is q\_

Table 1.

Wm�<sup>1</sup> k�<sup>1</sup>

lead to the following expression by Gad-el-Hak [35]

Work Adjustable coefficient d Ben Aissa et al. [12] 0.05 For Kn ¼ 3:33 Hua and Cao [33] 0.66 For Kn < 5

Sing et al. [36] <sup>1</sup>:<sup>25</sup>

20 Green Electronics

Table 1. First-order temperature jump condition.

4. Structure to model and numerical method

∂T

<sup>∂</sup><sup>x</sup> (25)

(26)

(27)


Table 2. Thermal properties of Silicon and Silicon dioxide.

To solve the BDE model coupled with the temperature jump at boundary we use the FEM [4]. The finite-element approximation used in the BDE model can be defined as:

$$[B]\{T\_t\} + [B\_1]\{T\_H\} - [D]\{T\} = \{m\} \tag{28}$$

where ½ � B , ½ � B<sup>1</sup> and ½ � D is a matrix valued, f g Tt , f g Ttt and f gT represent the nodal temperature, f g m is the matrix vector.

The discretization of Eq. (25) leads to

$$\left\{{T\_{p+1}\}\left(\frac{[\mathcal{B}]}{\Delta t} + \frac{[\mathcal{B}]}{\Delta t^2}\right) = \left\{{T\_p\}\left(\frac{[\mathcal{B}]}{\Delta t} + 2\frac{[\mathcal{B}]}{\Delta t^2} + [\mathcal{D}]\right) - \left\{{T\_{p-1}}\right\}\left(\frac{[\mathcal{B}]}{\Delta t^2}\right) + \{m\}\_p\right\} \tag{29}$$

where Δt is the time step and Tp is the nodal temperature at the time tp.

The materials used in our simulation are Silicon [2] and Silicon dioxide [2] and their thermal properties are illustrated in Table 2.

### 5. Results and discussion

The reduction of the thermal conductivity have a strong dependence with the Knudsen number. We take account the specularity parameter and Knudsen number because we studied the mechanism of boundary scattering.

Figure 2. Effective thermal conductivity vs. Knudsen number.

In Figure 2 we shows the impact of the thermal conductivity which depend on the specularity parameter. In this case, we use Eq. (17) to inquire the ETC. It is obvious that the thermal conductivity reduce when the Knudsen number increase. In similarity to the analytical model proposed by Hua and Cao et al. [32], it is found that the thermal conductivity reaches 62% of the bulk value for Kn = 1.

Figure 3 plot the ETC for various the length of nanofilms. For low thin films (L = 10 nm) the thermal conductivity attains 10–20% of the bulk value. The ballistic transport involve the rapid increase of the ETC. For p = 0.25 we shows the same shape obtained by Ma [29]. For thin films (L > 1000 nm), p = 0.25 is a good approximation.

The advantage of our proposed model is the capture of the increase of the temperature better than the other transport model (DPL, SPL and classical BDE). We associated the BDE model with the temperature jump. The obtained results are presented along the centerline (Lx/2, Y = 0) at the time t = 30 ps. In the ballistic regime (Kn = 10) we use Eq. (18). For high Knudsen number the heat transport influenced by the mechanism of scattering related to the boundary. This type of collision was examined by Guo et al. [38] they deduced a discrete-ordinate-method (DOM) derived by the Callaway's model [38]. They found acceptable results to determinate the ETC in a rectangular graphene ribbon. The Callaway's model based on a simple boundary scattering.

Figure 4 illustrate the comparison of the peak temperature rise in the nano-transistor at t = 30 ps. The classical BDE, BTE, DPL, SPL, Fourier law and our proposed model reaches respectively 318.7, 327, 320.5, 318.9, 305 and 322.7 K. The new BDE model capture the increase of the temperature near the BTE. For low thin-film (Lc = 10 nm) one can see that the temperature attaints the maximal at short time. The saturation of the temperature varied to 12–15 ps for all model transport. The classical Fourier law cannot predict the temperature profile due the nature of phonon thermal transport [39].

The temperature peak rise in the Y-direction at the centerline of the nanodevice is demonstrated in Figure 5. The decrease of the temperature is owned to the reduction of the thermal conductivity. Our present model has the same form with the classical BDE model. The difference appear in low temperature due to the collision rate, which depend on the specularity.

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Figure 4. Comparison of the peak temperature at the centerline.

Figure 3. Effective thermal conductivity in Silicon thin film at room temperature.

Figure 6 illustrate the 2D distribution of the temperature at t = 30 ps. In a short time, the temperature increase in the left and right side of the channel region. This side known as the heat zone of the nanodevice. A self-heating processes appear due to nature of the phonon Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model http://dx.doi.org/10.5772/intechopen.75595 23

Figure 3. Effective thermal conductivity in Silicon thin film at room temperature.

In Figure 2 we shows the impact of the thermal conductivity which depend on the specularity parameter. In this case, we use Eq. (17) to inquire the ETC. It is obvious that the thermal conductivity reduce when the Knudsen number increase. In similarity to the analytical model proposed by Hua and Cao et al. [32], it is found that the thermal conductivity reaches 62% of

Figure 3 plot the ETC for various the length of nanofilms. For low thin films (L = 10 nm) the thermal conductivity attains 10–20% of the bulk value. The ballistic transport involve the rapid increase of the ETC. For p = 0.25 we shows the same shape obtained by Ma [29]. For thin films

The advantage of our proposed model is the capture of the increase of the temperature better than the other transport model (DPL, SPL and classical BDE). We associated the BDE model with the temperature jump. The obtained results are presented along the centerline (Lx/2, Y = 0) at the time t = 30 ps. In the ballistic regime (Kn = 10) we use Eq. (18). For high Knudsen number the heat transport influenced by the mechanism of scattering related to the boundary. This type of collision was examined by Guo et al. [38] they deduced a discrete-ordinate-method (DOM) derived by the Callaway's model [38]. They found acceptable results to determinate the ETC in a rectangular graphene ribbon. The Callaway's model based on a simple boundary scattering. Figure 4 illustrate the comparison of the peak temperature rise in the nano-transistor at t = 30 ps. The classical BDE, BTE, DPL, SPL, Fourier law and our proposed model reaches respectively 318.7, 327, 320.5, 318.9, 305 and 322.7 K. The new BDE model capture the increase of the temperature near the BTE. For low thin-film (Lc = 10 nm) one can see that the temperature attaints the maximal at short time. The saturation of the temperature varied to 12–15 ps for all model transport. The classical Fourier law cannot predict the temperature profile due the

the bulk value for Kn = 1.

22 Green Electronics

(L > 1000 nm), p = 0.25 is a good approximation.

Figure 2. Effective thermal conductivity vs. Knudsen number.

nature of phonon thermal transport [39].

Figure 4. Comparison of the peak temperature at the centerline.

The temperature peak rise in the Y-direction at the centerline of the nanodevice is demonstrated in Figure 5. The decrease of the temperature is owned to the reduction of the thermal conductivity. Our present model has the same form with the classical BDE model. The difference appear in low temperature due to the collision rate, which depend on the specularity.

Figure 6 illustrate the 2D distribution of the temperature at t = 30 ps. In a short time, the temperature increase in the left and right side of the channel region. This side known as the heat zone of the nanodevice. A self-heating processes appear due to nature of the phonon

Figure 5. Peak temperature rise versus Y-axis at the centerline of the MOSFET at t = 10 ps.

collision which is characterized by a frequent scattering at boundary. The augmentation of the temperature cause an important dissipation of energy, which affect the environment. In the last years, organic electronic was made to reduce the energy consumption and the thermal resis-

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model

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Figure 7 shows the comparison of the heat flux in the Y-direction at the centerline of nano-MOSFET. Using the BDE model for p = 0.18, we obtain the same shape and amplitude given by the BTE. The temperature jump at boundary is a good argument to predict the non-Fourier heat transfer [41]. The increase of the heat flux is caused by two raison: the reduction of the thermal conductivity and the length of nanostructure. To reduce the heat dissipation in nanoelectronic materials, it is necessary that the current densities was minimized [9]. A preferment devise is characterized by minimal power consumption. In a technological concept, the graphene is an excellent material which described by high thermal conductivity and low

In this chapter, we report a nano-heat conduction based on the BDE model. The temperature jump is good proof to study the thermal properties of nano-materials. Our proposed model is efficient approach for the non-Fourier heat conduction. In addition, our obtained results agree with other transport model. In nanostructure the reduction of the thermal conductivity and phonon collision mechanism. Our study explain the distribution of the temperature in 10 nm

tance between materials [40].

Figure 7. Comparison of the heat flux in the Y direction at t = 10 ps.

temperature rise [25, 42].

6. Conclusions

Figure 6. A 2D temperature distribution for p = 0.18 at t =30 ps.

Figure 7. Comparison of the heat flux in the Y direction at t = 10 ps.

collision which is characterized by a frequent scattering at boundary. The augmentation of the temperature cause an important dissipation of energy, which affect the environment. In the last years, organic electronic was made to reduce the energy consumption and the thermal resistance between materials [40].

Figure 7 shows the comparison of the heat flux in the Y-direction at the centerline of nano-MOSFET. Using the BDE model for p = 0.18, we obtain the same shape and amplitude given by the BTE. The temperature jump at boundary is a good argument to predict the non-Fourier heat transfer [41]. The increase of the heat flux is caused by two raison: the reduction of the thermal conductivity and the length of nanostructure. To reduce the heat dissipation in nanoelectronic materials, it is necessary that the current densities was minimized [9]. A preferment devise is characterized by minimal power consumption. In a technological concept, the graphene is an excellent material which described by high thermal conductivity and low temperature rise [25, 42].

### 6. Conclusions

Figure 5. Peak temperature rise versus Y-axis at the centerline of the MOSFET at t = 10 ps.

24 Green Electronics

Figure 6. A 2D temperature distribution for p = 0.18 at t =30 ps.

In this chapter, we report a nano-heat conduction based on the BDE model. The temperature jump is good proof to study the thermal properties of nano-materials. Our proposed model is efficient approach for the non-Fourier heat conduction. In addition, our obtained results agree with other transport model. In nanostructure the reduction of the thermal conductivity and phonon collision mechanism. Our study explain the distribution of the temperature in 10 nm MOSFET. The maximal temperature is located in the interface oxide-semiconductor. To reduce the effect of the thermal transport in nano-electronic materials it is obvious that we should replace Si based materials by organic technologies (carbon based). Green electronics are involved in recent integrated circuits, solar cell and high-speed processors. In addition, green materials has a wide biocompatibility and a safe impact on the environment [43].

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