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Electro-thermal simulation based on coupled Boltzmann transport equations for electrons and phonons

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Abstract

To study the thermal effect in nano-transistors, a simulator solving self-consistently the Boltzmann transport equations for both electrons and phonons has been developed. It has been used to investigate the self-heating effects in a 20 nm-long double-gate MOSFET (Fig. 1). A Monte Carlo solver for electrons is coupled with a direct solver for the steady-state phonon transport. The latter is based on the relaxation time approximation. This method is particularly efficient to provide a deep insight of the out-of-equilibrium thermal dissipation occurring at the nanometer scale when the device length is smaller than the mean free path of both charge and thermal carriers. It allows us to evaluate accurately the phonon emission and absorption spectra in both real and energy spaces.

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Acknowledgments

This work was partially supported by the French ANR through project NOE (12JS03-006-01).

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Authors and Affiliations

  1. Institute of Fundamental Electronics, CNRS, University of Paris-Sud, Université Paris Saclay, Orsay, France

    T. T. Trang Nghiêm, J. Saint-Martin & P. Dollfus

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  1. T. T. Trang Nghiêm
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Correspondence to J. Saint-Martin.

Appendix: Analytical temperature profile with non-uniform diffusivity

Appendix: Analytical temperature profile with non-uniform diffusivity

In steady-state, without heat generation, the diffusive heat equation can be reduced to

$$\begin{aligned} \nabla \left( {D_T \nabla T} \right) = 0, \end{aligned}$$
(13)

where \(D_{T}\) is the thermal diffusivity, defined from the thermal conductivity \(\kappa _T \) by \(D_{T}=\kappa _{T}/(\rho \times \hbox {c}_{s})\), where \(c_{s}\) is the specific heat and \(\rho \) the mass density.

According to experimental data, the temperature dependence of the thermal diffusivity \(D_{T}\) in silicon is assumed to follow

$$\begin{aligned} D_T = C\cdot T^{\alpha }, \end{aligned}$$
(14)

where C and \(\alpha \) are the fitting parameters.

Substituting (14) into (13) and noting that \(T^{\alpha }\nabla T = \frac{\nabla T^{\alpha +1}}{\alpha +1}\), a Laplace’s equation is derived for the variable \(U=T^{\alpha +1}\).

By integrating the above equation in its 1D form with boundary temperatures Th (at \(z=0\)) and Tc (at \(z=\hbox {L}\)), one obtains the temperature profile

$$\begin{aligned} T = \left( {\frac{z}{L}T_c^{\alpha +1} + \left( {1-\frac{z}{L}} \right) \;T_h^{\alpha +1} } \right) ^{1/\left( {\alpha +1} \right) } \end{aligned}$$
(15)

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Nghiêm, T.T.T., Saint-Martin, J. & Dollfus, P. Electro-thermal simulation based on coupled Boltzmann transport equations for electrons and phonons. J Comput Electron 15, 3–15 (2016). https://doi.org/10.1007/s10825-015-0773-2

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