This document discusses calculating lattice vibrations in 2D materials using high-performance computing. It summarizes: 1) Using Density Functional Perturbation Theory as implemented in the Quantum Espresso software to calculate the lattice vibrations in graphene and transition metal dichalcogenides. 2) How Quantum Espresso uses MPI parallelization across images, k-points, orbitals, and plane waves to solve the eigenvalue problems arising in DFT and DFPT calculations efficiently. 3) The results obtained for phonon dispersions of graphene, MoS2, MoSe2, and WSe2 which provide insight for designing 2D devices with better heat management.

1. Calculation of lattice vibrations in 2D materials using HPC
Accelerated Quantum-Espresso
Cameron Foss
College of Engineering, University of Massachusetts Amherst
Nanotechnology Simulation and Theory Lab, Advisor: Zlatan Aksamija
Relaxation Calculation
References
[1] P. Giannozzi, Introduction to the calculation of phonons and of vibrational spectra. 2009, [2] P. Giannozzi, Quantum simulations of
materials using Quantum Espresso. J. Phys. Condens. Matter, 2009, [3] Eric Pop, Sanjiv Sinha, and Kenneth E. Goodson, Heat Generation
and Transport in Nanometer-Scale Transistors, IEEE Vol 94 No. 8 2006, [4] Breakthrough in Silicon Technology: Wafer-Level Strained Silicon
Technology Announced by SiGen, 2004, http://phys.org/news781.html, [5] Fengnian Xia , et al., Two-dimensional material nanophotonics,
Nature Photonics Vol. 8 2014, [6] Tania Roy, et al., Field-effect Transistors Built from All Two-Dimensional Material Components, ACS Nano,
2014 , [7] B. Radisavlijevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Single-Layer MoS2 transistors, Nature Nanotechnology, 2011
Abstract
Traditional downscaling of transistor length scales has reached fundamental limits with heat dissipation and physical size
constraints present at the atomic scale. With the advent of two-dimensional materials with graphene in 2004 the possibilities
of purely two-dimensional devices were realized, however heat dissipation continues to be a concern. Recent decades have
seen an increased interest in thermoelectrics and the need to fundamentally understand thermal currents at the nanoscale as
means to mitigating heat dissipation. Thermal transport can be calculated numerically via a phonon Boltzmann Transport
Equation, where a phonon is the fundamental quasi-particle of lattice vibrations. In this poster we present the lattice
vibrations calculated in graphene and various transition metal dichalcogenides calculated using Density Functional
Perturbation Theory as implemented in Quantum-Espresso1,2 (QE).
DFT + HPC
Quantum-Espresso primarily uses MPI
parallelization and uses the following hierarchy:
• World – group of all processes.
• Images – group of different self-consistent or
linear-response calculation.
• Pools – a subpartition of k-points for each
image.
• Bands – a subpartition of pools into groups
of KS orbitals
• PW – linear algebra operations on the PW
basis set are parallelized
• linear-algebra group – parallelization of
subspace diagonalization / iterative
orthonormalization
Motivation
Calculating lattice vibrations with DFT
SCF Energy Calculation
on Converged Structure
Phonon Calculation
Discrete Sampling of
the 1st Brillouin Zone
Physics and Numerical Approximations
• Modified KS system – DFT+ linear response
• Density Functional Perturbation Theory
𝑯𝚿 = E𝚿
• On band structure: energy states and orbitals
are the eigenvalues and eigenvectors
• On phonon dispersions: phonon frequency
and displacement vectors are the eigenpair.
Physics Approximations:
• Born-Oppenheimer Approx.
• Plane Wave Basis
• Pseudopotentials
Numerical Algorithms:
• In either DFT or DFPT we must solve an
eigenvalue problem
• Quantum-Espresso diagonalizes the
Hamiltonian through a conjugate gradient
(CG) iterative method
• LAPACK/BLAS are also used
Converged?
Yes
No
• Want to push physical limits to boost device
density and performance.
• Heat dissipation remains a major bottleneck
• Studying lattice vibrations in semiconductors gives
insight to designing devices with better heat
management
Results
• Runtimes over nprocs on XSEDE’s GORDON cluster.
• Only Parallelization over planes waves is used.
Graphene MoS2
MoSe2 WSe2
Core Ideas:
• Obtain the dispersion (or band structure) via
DFPT/DFT (Quantum-Espresso, VASP, ABINIT)
• Model transport via pBTE or Monte Carlo
• Build up to a device perspective
Basic
transistor
Strained Silicon
channel FINFETs
2D materials:
• 2D transistors and interconnects
• Increased packing density
• Vertical stacked vdW structures
• Transparent electronics
• Flexible electronics
Decreasing Dimensionality / Increasing Functionality
Phonons in Silicon under biaxial strain
Strained-Si and post-CMOS:
• SSi channels  higher e/h 𝜇
• Finfets  fully depleted channel
• Electron-phonon coupling
• Phononics/thermal cloaking
Solving Kohn-
Sham Equations
Solving modified
KS eqns.
Discretization of
Reciprocal Space
a
z
𝒁𝑻 =
𝑺 𝟐 𝝈𝑻
𝜿 𝒆𝒍 + 𝜿 𝒑𝒉
Modeling Transport via pBTE
• Phonon Boltzmann Transport Equation
𝐾 𝛼𝛽 𝑇
= 𝑘
𝑗 𝑞
ℏ𝜔𝑗 𝑞
𝑘𝑇
2
𝑒ℏ𝜔 𝑗 𝑞 /𝑘𝑇
𝑒ℏ𝜔 𝑗 𝑞 /𝑘𝑇
− 1
2 𝜏𝑗 𝑞 𝑣𝑗
𝛼
𝑞 𝑣𝑗
𝛽
( 𝑞)
where 𝑣𝑗 𝑞 =
𝜕𝜔𝑗 𝑞
𝜕 𝑞
• Scattering Rates
1
𝜏𝑗 𝑞
=
1
𝜏𝑗,𝑁 𝑞
+
1
𝜏𝑗,𝑈 𝑞
+
1
𝜏𝑗,𝐼 𝑞
+
1
𝜏𝑗,𝐵 𝑞
• Interfaces and boundaries
• Acoustic Mismatch vs Diffuse Mismatch Model
• Want to be able to tailor thermal conductivity to
particular purpose:
• TE generation/refrigeration
• to boost performanceSCF – self-consistent field
Intrinsic Extrinsic
𝝎(𝒒)
[3]
[3]
[4]
[5]
[6] [7]