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Electro-thermal simulation based on coupled Boltzmann
transport equations for electrons and phonons
T. Nghiem, Jérôme Saint-Martin, P. Dollfus
To cite this version:
T. Nghiem, Jérôme Saint-Martin, P. Dollfus. Electro-thermal simulation based on coupled Boltzmann
transport equations for electrons and phonons. Journal of Computational Electronics, Springer Verlag,
2016, 15 (1), pp.3 - 15. �10.1007/s10825-015-0773-2�. �hal-01906694�
1
Electro-thermal simulation based on coupled
Boltzmann transport equations for electrons
and phonons
T.T. Trang Nghiêm
1,2
, J. Saint-Martin
1
and P. Dollfus
1
1
Institute of Fundamental Electronics, UMR 8622, CNRS-University of Paris-Sud,
Orsay, France
2
The center for Thermal Sciences of Lyon, UMR 5008, CNRS–INSA–University of
Lyon 1, Villeurbanne, France
Phone: +33 1 69 15 72 83
Fax: +33 1 69 15 40 20
E-mail: jerome.saint-martin@u-psud.fr
Abstract: To study the thermal effect in nano-transistors, a simulator based on the self-consistent
solution of the Boltzmann transport equations (BTEs) for both electrons and phonons has been
developed. It has been used here to investigate the self-heating effect in a 20-nm long double gate
MOSFET. In this model, a Monte Carlo (MC) solver for electrons has been coupled with a direct
solver for the phonon transport. This method is particularly efficient to provide a deep insight on the
out-of-equilibrium thermal dissipation occurring at the manometer scale when the length of the
devices is smaller than the mean free path of both charge and thermal carriers. This approach allows
us to evaluate accurately the phonon emission and absorption spectra in both real and energy spaces.
Keywords: phonon, electron, Boltzmann, Transport Equation, Silicon, MOS,
DGMOS, transistors, self-heating.
1. Introduction
Heat conduction/dissipation and self-heating effects are taking an increasing place
in the design of solid-state devices and circuits. At the simulation level, different
methods have now reached a high degree of maturity to describe accurately the
electronic transport under uniform lattice temperature. They include in particular
the Monte Carlo (MC) method to solve the Boltzmann or Wigner transport equation
and the non-equilibrium Green's function (NEGF) formalism. With different levels
of approximation, these techniques are now able all derails of the material band
structure [1] [2] [3], multisubband transport [4] [5] [6], quantum transport [7] [8]
and scattering effects [9] [10] [11] [12]. However, introducing accurate description
2
of heat conduction and its coupling with charge transport in device simulators able
to consider devices of realistic size is still an issue, in spite of recent efforts.
In crystalline materials heat is mainly carried by phonons, the pseudo-particle
associated with the lattice vibrations. The phonon mean free path (MFP) is usually
limited by interaction with other phonons, impurities and interfaces [13]. In silicon,
the mean value is estimated to be typically 300 nm at room temperature [14]. On
distance scales larger than the MFP, the phonon system remains close to thermal
equilibrium and may be well described by the classical Fourier heat equation.
However, in modern electronic devices the length of the active region is in the order
of a few tens of nanometers and in the presence of a perturbation, phonon scattering
events are too rare for the system to recover local thermodynamic equilibrium.
Thus, in such devices the use of a macroscopic description of thermal transport as
the Fourier heat equation is questionable. In this case, the phonon Boltzmann
transport equation (pBTE), which has the ability to correctly describe both
equilibrium and non-equilibrium phenomena, is much more relevant. Simplified
equations have been derived from the pBTE and different methods for solving them
have been developed with different levels of approximation. It is worth mentioning
the phonon radiative transfer equation, the ballistic-diffusive equation [15, 16], the
discrete ordinate method (DOM) [17, 18] and the lattice Boltzmann method [19-
21]. However, to accurately solve the pBTE, the stochastic Monte Carlo approach
has been shown to be a powerful technique that can manage the details of the
collision processes at the microscopic scale [13, 22-26] [27]. Additionally, it can be
used in complex geometry devices. To solve the coupled electron and phonon
transport equations on an equal footing within the Boltzmann approach suitable for
full device simulation, it is very tempting to connect a thermal solver with an
electron MC (eMC) code and to introduce a local dependence of the electron
transport (scattering rates) on the state of phonon system. Different methods have
been developed. In a simple approach, the eMC simulation has been coupled with
a solver of the Fourier heat equation to study the size effect on the thermoelectric
properties of III-V heterostructures [28]. Then a split-flux model of phonon
transport has been self-consistently coupled with eMC simulation of Si FETs [29].
In 2010, Sadi anf Kelsall proposed to couple the 2D heat equation with 2D eMC
simulation to describe self-heating effects in SOI transistors thanks to the position-
dependence of the temperature and thermal conductivity [30]. A similar model has
3
been developed to analyse the thermal effects in quantum cascade lasers [31]. To
make it possible to capture out-of-equilibrium thermal phenomena, the Vasileska
and Goodnick group solved the energy balance equations of thermal transport. By
coupling this approach with eMC simulation, it has been possible to describe the
optical phonon bottleneck in ultra-short transistors and the resulting current
degradation through an analytic formulation of thermal conductivity in thin Si films
[32] [33]. Kamakura et al. have implemented a MC method to solve the BTE for
both electrons and phonons for 1D Si diodes with simplified phonon scattering rates
but this approach has not been extended to transistors yet [34]. Recently, Ni et al.
have used the phonon generation spectrum extracted from eMC simulation as input
for a pBTE solver with anisotropic relaxation times and Brillouin zone to evaluate
the hotspot temperature in a MOSFET [35]. This detailed description requires large
computational resources [36].
In this paper, we introduce a computationally efficient approach to solving
deterministically the steady-state 1D pBTE within the relaxation time
approximation (RTA). This phonon transport solver has been self-consistently
coupled to an eMC device simulation to study the electro-thermal effects in
nanoelectronic devices. This model provides deep insight into the out-of-
equilibrium phonon effects in small devices.
This paper is organized as follows. Sections 2 and 3 are dedicated to the
presentation of the main features of the thermal simulator based on the direct
solution of the pBTE, and on the phonon scattering mechanisms included in the
model for silicon, respectively. In Section 4 this model is used to explore the
different regimes of phonon transport in silicon bars of different lengths, from
diffusive to ballistic limits. Next, in Section 5 we explain how this thermal transport
model has been coupled to an electron Monte Carlo simulation code to build an
electro-thermal device simulator. This simulator has been used to investigate the
self-heating effects in an ultra-short Double Gate MOS field effect transistor (DG-
MOSFET). The results presented in section 6 put forward the effect of out-of-
equilibrium phonon distribution resulting from high phonon generation rate at the
drain-end of the channel.
4
2. Thermal simulator
2.1. Phonon dispersion
The phonon dispersion in silicon is composed of six phonon modes, i.e. two
transverse acoustic (TA), one longitudinal acoustic (LA) acoustic, two transverse
optical (TO) and one longitudinal optical (LO) modes. In this work, the TA modes
on one side and the TO modes on the other side have been considered to be
degenerate, which is exact along the main crystallographic directions. For each
mode s the dispersion relation has been assumed to follow an analytic and quadratic
expression of the form
2
0, ,
s s g s s
q v q a q
(1)
where q is the modulus of the phonon wave vector. The parameters
0,s
,
,gs
v
and
s
a
are taken from Ref. [37] where they have been optimized to fit the actual
dispersions along the direction [100]. Accordingly, the Brillouin zone was assumed
to be isotropic in this model, i.e with a spherical symmetry. The maximum value of
the wave vector norm
max
qa
is the radius of the Brillouin zone where a is the
lattice parameter.
2.2. Thermal transport equation
Since the out-of-equilibrium character of phonon transport may be significant in
nano-devices, the use of the Boltzmann transport formalism to study the heat
diffusion is particularly relevant. In contrast to the case of charged particles, the
trajectories of phonons are not modified by any external driving forces. Hence, the
corresponding drift term is absent in the Boltzmann transport equation for phonons
(pBTE), the steady-state form of which can be written for each mode s as
, s s
. . , ,
gr
col
s
l
N
v q N r q G r q
t
, (2)
where
q
is the phonon wave vector,
r
is the position vector,
,gs
v
is the group
velocity,
,
s
N r q
is the phonon distribution (i.e. the number of phonons with a
wave vector 𝑞 in the range
/2q dq
),
coll
N
t
is the scattering term and
,
s
G r q
is the generation term.
