2. Computer model

The phonon BTE can be defined as [7]:

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model http://dx.doi.org/10.5772/intechopen.75595 17

$$\frac{\partial f(r,v,t)}{\partial t} + v \nabla f(r,v,t) = -\frac{f - f\_0}{\tau\_R} \tag{2}$$

where f is the distribution function, f <sup>0</sup> is the equilibrium distribution function, ν is the group velocity, and τ<sup>R</sup> is the relaxation time related to resistive collisions written as

$$
\pi\_R = \frac{3 \times k}{\mathbb{C} \times \mathbb{v}^2} \tag{3}
$$

where κ is the thermal conductivity written as

(13) nm for the current (2018) and less than (6) nm for long-term (2026) [4]. The Fourier's law has generally used to predict the diffusive heat conduction [5]. In nanoscale the characteristic time and size of nanodevice was smaller than the mean free path (MFP). The classical heat

where q is the heat flux, ∇T is the temperature gradient and κ is the bulk thermal conductivity. Actually, the BTE is an effective method to study the non-continuous temperature and heat flux in nanosystems [6, 7]. Many transport models have been derived from the BTE used to investigate the thermal transport in solid interface [8], nano-transistor [9–12] and carbon

Alvarez et al. [14] have studied the nonlocal effect in nanoscale devices. They inquired the heat transport in ballistic regime. They found that the thermal conductivity bank on the Knudsen

Nano-heat transport includes both temporal aspects and spatial aspects. More elaborated model have been developed to describe the nanoheat conduction [15]. The phonon hydrodynamic model [16–19], single- phase-lag model [2] and dual-lag phase model [1] have been applied in modeling thermal transport in nanostructures. Nasri et al. [2, 3] have been investigated the heat transfer in many architecture of nano-MOSFET. They found that the Tri-gate SOI-MOSFET with a wideness of 20 nm is more thermally stable than the device having a length of 10 nm. To study the nature of collision, it is found that the temperature jump boundary condition was an accurate approach to explore the heat transport in interfaces [2]. The ballistic-diffusive equation (BDE) was used to explain the temperature dependence in nano-structure [20, 21]. Humian et al. [22] proposed the BDE to evaluate the heat transport in two-dimensional domain. They have been used the finite element analysis to validate the BDE model. Yang et al. [23] solved the BDE model to access the heat transfer in two-dimensional conventional MOSFET. In this work, we have been developed the BDE model to address the phonon transport in nanodevice. Due to the miniaturization the thermal conductivity, reduce by scattering [24, 25]. The scattering mechanism induced to the use of the ETC [26–29]. We have proposed a theoretical approach, which describe the nature of phonon collision with boundary. The specularity parameter defined as the probability of reflection at boundary [25, 30]. We include this parameter in the ETC to portend the rise of the temperature in nanostructure. To validate our results, the proposed model is tested with results obtained by Yang et al. [23] and a previous work [2, 9]. The proposed (ETC) will be compared with the results obtained by McGaughey et al. [31]. To compute the proposed BDE model depended with the temperature jump boundary condition, we have used the FEM. This method is a useful procedure to model the

q ¼ �κ∇T (1)

conduction based on local equilibrium lead to the linear equation

nanotubes [13].

thermal properties of nanodevice [2].

The phonon BTE can be defined as [7]:

2. Computer model

number.

16 Green Electronics

$$\mathbf{x} = \frac{\mathbf{C} \times \mathbf{v} \times \boldsymbol{\Lambda}}{\mathbf{3}} \tag{4}$$

where Λ is the mean free path defined as Λ ¼ vτ<sup>R</sup> and C is the volumetric heat capacity [20]. The ballistic-diffusive approximation is to divide the distribution function into a diffusive term f <sup>m</sup> and ballistic term f <sup>b</sup> [20]:

$$f = f\_m + f\_b \tag{5}$$

where f <sup>b</sup> arise from the boundary scattering [20], defined as

$$\frac{\partial f\_b(r, v, t)}{\partial t} + v \nabla f\_b(r, v, t) = -\frac{f\_b}{\tau\_R} \tag{6}$$

The second part grouped into f <sup>m</sup>. The basic equation for f <sup>m</sup> is defined as:

$$\frac{\partial f\_m(r, v, t)}{\partial t} + v \nabla f\_m(r, v, t) = -\frac{f\_m - f\_0}{\tau\_R} \tag{7}$$

We can calculate the diffusive flux [7, 20].

$$\eta\_m(t, r) = \int\_{\varepsilon} v(r, t) f\_m(r, \varepsilon, t) \varepsilon \mathcal{D}(\varepsilon) d\varepsilon \tag{8}$$

where ε is the kinetic energy and Dð Þε is the density of states. By the development in Taylor series to the first order of Eq. (7), we obtain [7]:

$$
\pi\_R \frac{\partial q\_m(r,t)}{\partial t} + q\_m(r,t) = -\kappa \nabla T\_m(r,t) \tag{9}
$$

We use the energy conservation equation to eliminate qm

$$-\nabla.q(r,t) + \dot{q}\_h = \frac{\partial u(r,t)}{\partial t} \tag{10}$$

where q\_<sup>h</sup> is the volumetric heat generation, q is the heat flux and u is the internal energy defined as [20].

$$\begin{aligned} q(t,r) &= q\_b(t,r) + q\_m(t,r) \\ \mathbf{u}(t,r) &= \boldsymbol{u}\_b(t,r) + \boldsymbol{u}\_m(t,r) \end{aligned} \tag{11}$$

For high values of Kn, where the regime ballistic is dominant, the thermal conductivity reduced by scattering (reflection at boundary) [24, 25], Eq. (17) predict that the ETC behaves

CW

∇qbð Þ r; t <sup>C</sup> <sup>þ</sup> <sup>q</sup>\_<sup>h</sup>

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model

ΔTJump ¼ �d � Kn � L � ∇T (22)

κeff ¼ κ=ð Þ 1 þ ð Þ 4 � Kn (24)

<sup>C</sup> ∇∇Tmð Þ� <sup>r</sup>; <sup>t</sup>

where τ<sup>b</sup> is the relaxation time related to the phonon scattering at boundary [7] defined as:

<sup>¼</sup> <sup>1</sup> � <sup>p</sup> 1 þ p 

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

> <sup>d</sup> <sup>¼</sup> <sup>R</sup> � <sup>κ</sup>eff Kn � Lc

where R is the thermal boundary resistance. The proposed ETC given by Ben Aissa et al.

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

� ν L

<sup>τ</sup><sup>b</sup> <sup>¼</sup> <sup>3</sup> � <sup>κ</sup>eff

<sup>2</sup>Kn (18)

http://dx.doi.org/10.5772/intechopen.75595

C þ τb C ∂q\_ h

<sup>C</sup> � <sup>ν</sup><sup>2</sup> (20)

(21)

<sup>∂</sup><sup>t</sup> (19)

19

(23)

κeffð Þ¼ Kn κ

∂Tmð Þ r; t <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup>eff

Substituting Eq. (18) into Eq. (20), we obtain the Ziman formula [24]

1 τb

temperature jump applied in the interface oxide-semiconductor defined as

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

[19]:

where <sup>1</sup>

[12, 32] is written as:

In our case, the BDE model rewritten as:

τb ∂2 Tmð Þ r; t ∂t <sup>2</sup> þ

<sup>τ</sup><sup>b</sup> is the collision rate.

3. Boundary and initial condition

We can rewritten two temperature Tb and Tm such that [20, 21].

$$\frac{\partial u}{\partial t} = \mathcal{C} \frac{\partial T}{\partial t} = \frac{\partial u\_m}{\partial t} + \frac{\partial u\_b}{\partial t} = \mathcal{C} \left(\frac{\partial T\_m}{\partial t} + \frac{\partial T\_b}{\partial t}\right) \tag{12}$$

where T ¼ Tm þ Tb

Tm is the temperature to the diffusive part and Tb arise from the ballistic parts.

By using the same reasoning of Eq. (9), Eq. (6) becomes [20].

$$
\tau\_R \frac{\partial^2 T\_b(r, t)}{\partial t^2} + \mathbb{C} \frac{\partial T\_b}{\partial t}(r, t) = -\tau\_R \times \frac{\partial \left(\nabla. q\_b(r, t)\right)}{\partial t} \tag{13}
$$

Substituting Eqs. (9) and (13) into Eq. (10) we obtain the ballistic-diffusive-equation [20, 23].

$$
\tau\_R \frac{\partial^2 T\_m(r, t)}{\partial t^2} + \frac{\partial T\_m(r, t)}{\partial t} = \frac{\kappa}{\mathbb{C}} \nabla \nabla T\_m(r, t) - \frac{\nabla q\_m(r, t)}{\mathbb{C}} + \frac{\dot{q}\_h}{\mathbb{C}} + \frac{\tau\_R}{\mathbb{C}} \frac{\partial \dot{q}\_h}{\partial t} \tag{14}
$$

The conventional Fourier heat conduction equation cannot predict the heat transport in nanostructure. Hua et al. [26] studied the ETC in nanostructure. They derived a model for the ETC based on the phonon BTE written as [26]:

$$\kappa\_{\sharp\overline{\mathcal{Y}}} = \kappa / (\mathbf{1} + \boldsymbol{\alpha} \times \mathbf{K} \mathbf{n}) \tag{15}$$

where Kn <sup>¼</sup> <sup>Λ</sup> <sup>L</sup> is the Knudsen number, L is the length of nanofilms and α is a coefficient depend with the geometries.

For Kn ¼ 0, Eq. (15) becomes κeff ¼ κ (diffusive regime).

For Kn > 1, the thermal conductivity reduce due to the ballistic transport. Using the Fourier's law the ETC defined as [26]:

$$\kappa\_{\text{eff}} = \frac{q \times L}{\Delta T} \tag{16}$$

where L is length of the nanostructure q is the heat flux and ΔT is the temperature difference. Kaiser et al. [28] proposed a non-Fourier heat conduction at the nanoscale. They recently derived an analytic expression for the ETC. In addition, they proved the impact of the temperature jump in nanostructure. In this work, we propose a theoretical model for the ETC [19], defined as

$$k\_{\rm eff}(\text{Kn}) = \kappa \left[ 1 - \frac{2\text{Kn} \times \tanh(1/2\text{Kn})}{1 + \text{C}\_{\text{W}} \times \tanh(1/2\text{Kn})} \right] \tag{17}$$

where CW ¼ 2 � 1þp 1�p is a constant related to the properties of the walls [19] and <sup>p</sup> is the specularity parameter [30].

For high values of Kn, where the regime ballistic is dominant, the thermal conductivity reduced by scattering (reflection at boundary) [24, 25], Eq. (17) predict that the ETC behaves [19]:

$$\kappa\_{\text{eff}}(\text{Kn}) = \kappa \left( \frac{\text{C}\_{\text{W}}}{2 \text{Kn}} \right) \tag{18}$$

In our case, the BDE model rewritten as:

q tð Þ¼ ;r qbð Þþ t;r qmð Þ t;r uð Þ¼ t;r ubð Þþ t;r umð Þ t;r

∂ub

<sup>∂</sup><sup>t</sup> ð Þ¼� <sup>r</sup>; <sup>t</sup> <sup>τ</sup><sup>R</sup> �

<sup>C</sup> ∇∇Tmð Þ� <sup>r</sup>; <sup>t</sup>

<sup>L</sup> is the Knudsen number, L is the length of nanofilms and α is a coefficient depend

The conventional Fourier heat conduction equation cannot predict the heat transport in nanostructure. Hua et al. [26] studied the ETC in nanostructure. They derived a model for the ETC

For Kn > 1, the thermal conductivity reduce due to the ballistic transport. Using the Fourier's

<sup>κ</sup>eff <sup>¼</sup> <sup>q</sup> � <sup>L</sup>

where L is length of the nanostructure q is the heat flux and ΔT is the temperature difference. Kaiser et al. [28] proposed a non-Fourier heat conduction at the nanoscale. They recently derived an analytic expression for the ETC. In addition, they proved the impact of the temperature jump in nanostructure. In this work, we propose a theoretical model for the ETC [19], defined as

keffð Þ¼ Kn <sup>κ</sup> <sup>1</sup> � <sup>2</sup>Kn � tanh 1ð Þ <sup>=</sup>2Kn

1 þ CW � tanh 1ð Þ =2Kn 

is a constant related to the properties of the walls [19] and p is the

Substituting Eqs. (9) and (13) into Eq. (10) we obtain the ballistic-diffusive-equation [20, 23].

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>C</sup> <sup>∂</sup>Tm

∂t þ

∂Tb ∂t

<sup>∂</sup> <sup>∇</sup>:qbð Þ <sup>r</sup>; <sup>t</sup>

κeff ¼ κ=ð Þ 1 þ α � Kn (15)

C þ τR C ∂q\_<sup>h</sup>

<sup>Δ</sup><sup>T</sup> (16)

∇qmð Þ r; t <sup>C</sup> <sup>þ</sup> <sup>q</sup>\_<sup>h</sup>

<sup>∂</sup><sup>t</sup> (13)

<sup>∂</sup><sup>t</sup> (14)

(17)

We can rewritten two temperature Tb and Tm such that [20, 21].

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup>um ∂t þ

Tm is the temperature to the diffusive part and Tb arise from the ballistic parts.

<sup>2</sup> <sup>þ</sup> <sup>C</sup> <sup>∂</sup>Tb

∂Tmð Þ r; t <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>κ</sup>

∂u <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>C</sup> <sup>∂</sup><sup>T</sup>

By using the same reasoning of Eq. (9), Eq. (6) becomes [20].

τR ∂2 Tbð Þ r; t ∂t

τR ∂2 Tmð Þ r; t ∂t <sup>2</sup> þ

based on the phonon BTE written as [26]:

For Kn ¼ 0, Eq. (15) becomes κeff ¼ κ (diffusive regime).

where T ¼ Tm þ Tb

18 Green Electronics

where Kn <sup>¼</sup> <sup>Λ</sup>

with the geometries.

where CW ¼ 2 �

specularity parameter [30].

1þp 1�p 

law the ETC defined as [26]:

(11)

(12)

$$
\tau\_b \frac{\partial^2 T\_m(r, t)}{\partial t^2} + \frac{\partial T\_m(r, t)}{\partial t} = \frac{\kappa\_{\ell\ell}}{C} \nabla \nabla T\_m(r, t) - \frac{\nabla q\_b(r, t)}{C} + \frac{\dot{q}\_h}{C} + \frac{\tau\_b}{C} \frac{\partial \dot{q}\_h}{\partial t} \tag{19}
$$

where τ<sup>b</sup> is the relaxation time related to the phonon scattering at boundary [7] defined as:

$$
\pi\_b = \frac{\mathbf{3} \times \mathbf{x}\_{\text{eff}}}{\mathbf{C} \times \nu^2} \tag{20}
$$

Substituting Eq. (18) into Eq. (20), we obtain the Ziman formula [24]

$$\frac{1}{\tau\_b} = \left(\frac{1-p}{1+p}\right) \times \left(\frac{\nu}{L}\right) \tag{21}$$

where <sup>1</sup> <sup>τ</sup><sup>b</sup> is the collision rate.
