1. Introduction

Considerable exploration focused on the fast progress of nanodevice. In recent years, the thermal analysis is important to inquire the phonon transport in nano-materials. The study of the heat conduction in nano-electronics lead to compare the thermal stability of nanotransistors [1–3]. The increase of the heat dissipation has been owned by the miniaturization and the reduction of the thermal conductivity [3]. The smaller channel device was assumed to

© 2016 The Author(s). Licensee InTech. 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 eproduction in any medium, provided the original work is properly cited. © 2018 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.

(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 conduction based on local equilibrium lead to the linear equation

$$
\eta = -\kappa \nabla T \tag{1}
$$

∂f rð Þ ; v; t

velocity, and τ<sup>R</sup> is the relaxation time related to resistive collisions written as

where f is the distribution function, f

f <sup>m</sup> and ballistic term f <sup>b</sup> [20]:

where κ is the thermal conductivity written as

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

series to the first order of Eq. (7), we obtain [7]:

defined as [20].

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

∂f <sup>b</sup>ð Þ r; v; t

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

∂f <sup>m</sup>ð Þ r; v; t

qmð Þ¼ t;r

∂qmð Þ r; t

τR

We use the energy conservation equation to eliminate qm

ð

ε

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>v</sup>∇f rð Þ¼� ; <sup>v</sup>; <sup>t</sup> <sup>f</sup> � <sup>f</sup> <sup>0</sup>

<sup>τ</sup><sup>R</sup> <sup>¼</sup> <sup>3</sup> � <sup>k</sup>

<sup>κ</sup> <sup>¼</sup> <sup>C</sup> � <sup>v</sup> � <sup>Λ</sup>

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

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>v</sup>∇<sup>f</sup> <sup>b</sup>ð Þ¼� <sup>r</sup>; <sup>v</sup>; <sup>t</sup> <sup>f</sup> <sup>b</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>v</sup>∇<sup>f</sup> <sup>m</sup>ð Þ¼� <sup>r</sup>; <sup>v</sup>; <sup>t</sup> <sup>f</sup> <sup>m</sup> � <sup>f</sup> <sup>0</sup>

where ε is the kinetic energy and Dð Þε is the density of states. By the development in Taylor

�∇:q rð Þþ ; <sup>t</sup> <sup>q</sup>\_<sup>h</sup> <sup>¼</sup> <sup>∂</sup>u rð Þ ; <sup>t</sup>

where q\_<sup>h</sup> is the volumetric heat generation, q is the heat flux and u is the internal energy

τR

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model

<sup>0</sup> is the equilibrium distribution function, ν is the group

<sup>C</sup> � <sup>v</sup><sup>2</sup> (3)

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

<sup>3</sup> (4)

f ¼ f <sup>m</sup> þ f <sup>b</sup> (5)

v rð Þ ; t f <sup>m</sup>ð Þ r; ε; t εDð Þε dε (8)

<sup>∂</sup><sup>t</sup> <sup>þ</sup> qmð Þ¼� <sup>r</sup>; <sup>t</sup> <sup>κ</sup>∇Tmð Þ <sup>r</sup>; <sup>t</sup> (9)

<sup>∂</sup><sup>t</sup> (10)

τR

τR

(2)

17

(6)

(7)

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 nanotubes [13].

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 number.

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 thermal properties of nanodevice [2].
