Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
Effects of Quantum Corrections and Isotope
Scattering on Silicon Thermal Properties
Javier V. Goicochea
IBM Research GmbH, Zurich Research Laboratory, 8803 Rüschlikon, Switzerland
Marcela Madrid
Pittsburgh Supercomputing Center, Pittsburgh, PA 15213, USA
Cristina Amon
Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S 1A4, Canada
Abstract- A quantum correction procedure is proposed to
correct silicon thermal properties estimated with molecular
dynamics (MD). The procedure considers the energy
quantization per mode basis and the anharmonic nature of the
potential energy function (including the thermal expansion of
the crystal) and is applied to reported thermal properties of
silicon estimated with MD in ref. [11], such as temperature,
specific heat and thermal conductivity. The procedure
facilitates the use of these properties as input to faster
numerical methods, such as those based on the Boltzmann
transport equation under the single relaxation time
approximation. In addition, the effect of isotope scattering is
included in reported values of phonon-phonon relaxation times.
The effects of the correction procedure and the scattering with
isotopes are analyzed in terms of the change of phonon specific
heat, mean free path and thermal conductivity. We have found
that the application of quantum corrections yields a significant
reduction in the contribution of high-frequency modes to the
overall thermal conductivity. This contribution is further
reduced by the inclusion of isotope scattering. At 220 K, the
total contribution of optical modes reduces from 12.3 % (before
quantum corrections) to 5.8 %; and to 2 % when the isotope
scattering is also considered. The quantum corrections and the
inclusion of isotope scattering are found to bring the estimated
thermal conductivity into close agreement with experimental
values. The relative contributions of the acoustic and optical
modes after quantum corrections agrees very well with recently
reported ab initio results.
I. INTRODUCTION
Several alternatives have been proposed to link MD with
other continuum or sub-continuum models [1, 2]. Most
approaches divide the macroscopic domain in regions where
different numerical methods are employed. The different
levels of interaction have been typically described using MD
and finite element (FE) methods [1, 2]. In some cases,
quantum-mechanics models are concurrently used to
estimate unknown parameters required by MD. The link
between these methods is achieved by defining a special
interaction region (also called hand-shake region), where the
FE-mesh overlaps with the atoms defined in MD. Usually, a
special Hamiltonian is used to describe such interaction [2,
3]. Problems like crack propagation, dynamic of fractures,
nano-indentation and dislocation generation have been
addressed by these methods. Conversely, neither of these is
suitable for describing the sub-continuum heat transport in
semiconductors. The common simplification of the
interatomic potential and the assumption of long wave
behavior (both required for solving the FE method), changes
the scattering mechanisms of interacting heat carriers and
affects the description of the heat transport in
semiconductors. In addition, the assumption of long wave
behavior neglects the existence of optical modes, which are
important in describing sub-continuum heat transfer under
self-heating conditions [4-9].
For semiconductor materials, the sub-continuum heat
transport has been studied by means of the Boltzmann
transport equation (BTE) for phonons. Phonons are
quantized lattice vibrations, subject to different scattering
mechanisms that affect how the heat is transported in the
crystalline structure [10]. The complexity of the scattering
mechanisms has led to the development of phenomenological
models whose thermal predictions depend on thermal
properties difficult to estimate or measure in advance, such
as phonon relaxation times and dispersion relations. To
avoid this, MD simulations and the normal mode
decomposition have been recently used to estimate all
thermal properties of silicon required as input to the BTE,
including phonon relaxation times, dispersion relations,
group velocities and specific heat, at 300 and 1000 K [11].
However to use these quantities, quantum corrections (QCs)
are necessary since both methods interpret the physics of
phonons differently.
The main idea of QCs is to map the results obtained
classically (using MD) onto their quantum analogs at the
same energy level. In a quantum system, the phonon
occupation number is a function of the temperature and
frequency, and the energy is quantized in units of h ω . In a
classical, the modes are equally excited regardless of the
temperature, have an energy expectation value of about and
are continuous functions of the instantaneous position and
momentum of the particles in the system [12]. These
©EDA Publishing/THERMINIC 2009 197
ISBN: 978-2-35500-010-2