Presentation given at MRS Fall Meeting, Boston MA (Dec 2022)

1. Efficient methods for accurately calculating
thermoelectric properties – electronic and
thermal transport
Anubhav Jain
Lawrence Berkeley National Laboratory
MRS Fall meeting, Nov 2022
Slides (already) posted to hackingmaterials.lbl.gov

2. There are many efforts to identify new thermoelectric
materials through calculations
2
Urban, J. J.; Menon, A. K.; Tian, Z.; Jain, A.;
Hippalgaonkar, K. New Horizons in
Thermoelectric Materials: Correlated
Electrons, Organic Transport, Machine
Learning, and More. Journal of Applied
Physics 2019, 125 (18), 180902.

3. Such screening studies are typically limited to
using approximate models for estimating zT
3
Electron mobility Thermal conductivity Figure of merit
1. Constant / uniform relaxation
time approximation
2. Semi-empirical models[1]
1. Glassy limit thermal
conductivity models[2]
2. Semi-empirical models[1]
1. Combining previous
models for electron and
thermal conductivity
(optimize for doping, T)
2. Descriptors that implicitly
optimize[1] for doping, T
!=constant
[1] Yan, J.; Gorai, P.; Ortiz, B.; Miller, S.; Barnett, S. A.; Mason, T.; Stevanović, V.; Toberer, E. S. Material
Descriptors for Predicting Thermoelectric Performance. Energy Environ. Sci. 2015, 8 (3), 983–994
[2] D. G. Cahill, S. K. Watson, R. O. Pohl, Phys. Rev. B 1992, 46, 6133-6140 ; D. G. Cahill, R. O. Pohl, Ann. Rev.
Phys. Chem. 1988, 39, 93-121.
3 fitted parameters (A0, B, s)
Band effective mass based on
DOS effective mass and valley
degeneracy
2 fitted parameters (A1, A2)
Later extended to include
coordination number effects

4. For example, we previously calculated a large
amount of transport data under cRTA
4
~50,000 crystal
structures and
band structures
from Materials
Project are used as
a source
F. Ricci, et al., An ab initio electronic transport
database for inorganic materials, Sci. Data. 4
(2017) 170085.
We compute
electronic transport
properties with
BoltzTraP and
atomate
About 300GB of
electronic transport
data is generated. All
data is available free
for download
https://contribs.materialsproject.org/projects/carrier_transport/
All data is available
free for download via
Materials Project or
direct download from
journal article.

5. We used this data to identify materials with
decent figure-of-merit, but no breakthroughs
5
• Calculations:
trigonal p-
TmAgTe2 could
have power
factor up to 8
mW/mK2
• requires 1020/cm3
carriers
experiment
computation
• Calculations: p-YCuTe2 could
only reach PF of 0.4
mW/mK2
• SOC inhibits PF
• if thermal conductivity is low
(e.g., 0.4, we get zT ~1)
• Expt: zT ~0.75 – not too far
from calculation limit
• carrier concentration of 1019
• Decent performance, but
unlikely to be improved with
further optimization
• Expt: p-zT only 0.35 despite
very low thermal
conductivity (~0.25 W/mK)
• Limitation: carrier
concentration (~1017/cm3)
• likely limited by TmAg
defects, as determined by
followup calculations
• Later, we achieved zT ~ 0.47
using Zn-doping
TmAgTe2
YCuTe2
Collaborations w/Jeff Snyder & M.A. White

6. How to bring the field forward?
• Approximate and semi-empirical techniques used in high-throughput
screening studies typically suffer from two issues
• The first (and obvious) one is accuracy
• The second one is information content
• In addition to being less accurate, approximate techniques give you much less
information than full theoretical techniques
• Can we retain accuracy and information content, while minimizing
computational cost and retaining automation for future studies?
6

7. Outline
• Electronic properties using the AMSET method
• Phonon and thermal properties by efficient data fitting
7

8. The Boltzmann transport equation
determines carrier transport properties
8
group velocity (easy)
lifetime (hard)

9. τ
...
DFPT
AMSET
∝ DOS–1 / semi-
empirical
constant lifetime
A model to explicitly calculate
scattering rates while remaining
computationally efficient
Aim: accuracy comparable to EPW at
1/100th – 1/1000th computational cost
AND with rich information content
AMSET is a new framework for calculating
transport properties including e- lifetimes

10. primary input: uniform k-mesh band structure calculation
Step 1: Band structure (probably DFT)

11. Fourier interpolation of eigenvalues and group velocities
(there is some custom resampling to get even more accurate integrations)
Step 2: Interpolation of band structure to
dense mesh

12. lifetimes calculated using scattering equations that
depend on first-principles inputs
Step 3: Use band structure and scattering
equations to determine scattering rates

13. calculate mobility, conductivity, Seebeck & thermal conductivity
Step 4: Transport properties by solving BTE

14. Acoustic deformation potential (ad)
deformation potential, elastic constant
Ionized impurity (ii)
dielectric constant
Piezoelectric (pi)
dielectric constant, piezoelectric coefficient
Polar optical phonon (po)
dielectric constant, polar phonon frequency
Scattering rates determined by DFT inputs
*note: scalar equations shown, these are generalized to tensor forms

15. • All first principles inputs
• Nothing fit or tuned
to experimental data
• Rich information content
• scattering
mechanisms
• E & k-dependent
scattering rates
• Short runtime
• ~60 mins of DFT
calculations on 64
cores
• Plus ~40 mins of time
to run AMSET
AMSET predicts switch from
impurity to polar phonon scattering in GaN
Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.;
Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering
Rates from First Principles. Nat Commun 2021, 12 (1), 2222.

16. • Anisotropy is
also captured by
AMSET
• Allows for
analyzing non-
cubic systems
and getting
direction-
dependent
properties
AMSET can also calculate the anisotropic
transport properties of realistic materials
Crystal structure image from:
Pletikosić, Ivo et al. (2017). Band structure of a IV-
VI black phosphorus analogue, the thermoelectric
SnSe. Physical Review Letters. 120.
Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.;
Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering
Rates from First Principles. Nat Commun 2021, 12 (1), 2222.
SnSe

17. AMSET shows close agreement to experiment for the
mobility and Seebeck coefficient across many materials
Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.;
Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering
Rates from First Principles. Nat Commun 2021, 12 (1), 2222.

18. Timing for calculations are very reasonable and
easily within reach of most groups
Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.;
Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering
Rates from First Principles. Nat Commun 2021, 12 (1), 2222.
Total calculation times are
~500X faster than
DFPT+Wannier
The AMSET portion of
the calculation scales well
with system size

19. Docs: https://hackingmaterials.lbl.gov/amset/
Support: https://matsci.org/c/amset
Paper:
installation
pip install amset
usage
amset run --static-dielectric 10 ...
Can be controlled through the
command line or python interface,
integration with atomate2 for
automatic workflows
Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.; Persson, K. A.; Jain, A. Efficient Calculation
of Carrier Scattering Rates from First Principles. Nat Commun 2021, 12 (1), 2222.
AMSET is an open source python package that you can
run today, and has already been used in many
downstream studies

20. Outline
• Electronic properties using the AMSET method
• Phonon and thermal properties by efficient data fitting
20

21. Calculating thermal properties of materials
• The vibrational thermal properties of materials are determined by
phonon behavior
• In lattice dynamics, we typically tailor expand the phonon interactions by
atomic displacements:
And differentiate to solve for the interatomic force constants
21

22. After obtaining the force constants to various
orders, one can calculate materials properties
22
Harmonic term (Φ2)
Second order:
phonon dispersion, phonon DOS,
free energy, heat capacity (Cv), entropy
Anharmonic terms (Φ3, Φ4)
Φ
2
(
h
a
r
m
o
n
i
c
)
Fourth order:
thermal expansion, more accurate
thermal conductivity & free energies
Third order:
Gruneisen parameter, thermal
conductivity
More Thermal Properties
Higher Physical Accuracy
Computational Feasibility
4 th
order
of IFC
3 rd
order of
IFC
2 nd
order of IFC
…
φ
4
Φ3
(anharm
onic)
…

23. The problem – obtaining force constants can
require many DFT calculations
23
To obtain 2nd order IFCs
To obtain 3rd order IFCs
2 displacements in a supercell
(# of supercells needed: 1000s-10000s)
…
1 displacement in a supercell
(Usually <5 supercells needed)
Finite-displacement method IFCs extracted from HiPhive
To obtain any order of IFCs (2nd, 3rd,…) in one shot
…
displace each atom in a supercell
(Only need 5~10 supercells in total!)
• Traditionally, one performs systematic
displacements, each of which only has a
few atom movements and solves only a
small portion of the IFC matrix
• For higher-order terms, the IFC matrix
contains many distinct terms and many
calculations are needed
• Primitive cells with reduced symmetry and
many atoms can easily require 1000 or
more calculations
• The scaling goes something like: O(Nn)
where N is the number of sites and n is the
order of IFC you want. Not scalable!

24. The solution – perform non-systematic
displacements
• Instead of performing systematic
displacements, perform non-systematic
displacements in which many IFC terms are
“mixed up”
• Then, perform a best fit procedure to fit the
IFC matrix elements to the observed data
• Typically undetermined, so regularization is
important
• This method has been suggested by
several groups, for now we focus on the
implementation in the HiPhive code
(Erhart group, Chalmers University of
Technology)
24
IFCs extracted from HiPhive
To obtain any order of IFCs (2nd, 3rd,…) in one shot
…
displace each atom in a supercell
(Only need 5~10 supercells in total!)
Monte Carlo rattle penalizes displacements that lead to very small interatomic distances

25. HiPhive has been shown to give very good
accuracy with few calculations, but can require
parameter selection / tuning
25
HiPhive is itself not fully automatic. Things that need to be tuned include:
• Number of training structures
• Training structure supercell size and atom displacement strategy
• Interaction cutoff distance
• Method of regularization
Fransson, E.; Eriksson, F.; Erhart, P. Efficient
Construction of Linear Models in Materials
Modeling and Applications to Force
Constant Expansions. npj Comput Mater
2020, 6 (1), 135.

26. We have been testing parameters to
determine “high-throughput”
• Supercell size: 150 – 600 atoms, >= 18 Å in dimension
• Training structures: 3 – 15 (depending on lattice symmetry)
• Feature selection: Recursive feature elimination
• Cutoffs (Å) – initial cutoffs below, these are increased to hit convergence
26
Period 2nd order 3rd order 4th order
1 5.0 3.0 2.5
2 6.0 3.5 3.0
3 7.0 4.5 3.5
4 8.0 5.5 4.0
5 9.0 6.0 4.5
6 10.0 6.5 5.0
7 11.0 7.0 5.5

27. We’ve also been automating physical
considerations
Highly ionic compounds
require incorporation of
non-analytical
corrections
27
Dynamically unstable
compounds
Cubic SrTiO3 (Tc=105K)
100K
200K
Wave vector
Wave vector
Frequency
(THz)
Frequency
(THz)
Wave vector
vector
Frequency
(THz)
NaCl
no NAC
w/NAC

28. We are wrapping all of these into automated
workflows for high-throughput
28
VASP
DFT relaxation
VASP
(Large displaced)
Complete Φ
Imaginary
modes?
Stable Phonon
INPUT
Bulk modulus
ShengBTE
Boltzmann
Transport
• Free Energy
• Entropy
• Heat Capacity
• Gruneisen
• Thermal Expansion • Lattice Thermal
Conductivity
Yes
No
• Phase transition
• Thermoelectric
zT
Renormalization
at T > 0 K
Renormalized Φ
• Corrected
Free Energy
HiPhive
Harmonic Φ2
HiPhive
Anharmonic Φ3,
Φ4 etc
VASP
(Small displaced)

29. Testing and parameter selection is ongoing, but
we see major speedups compared to standard
methods
29
10x speedup
100x speedup
1000x speedup
All 5 testing systems show a 100-500x speedup based on DFT computations, due to
the very few supercells needed for HiPhive method.
Expecting fully automated workflows in 2023

30. Conclusions
• The community has already done a lot of great work with
approximate methods of electronic and thermal conductivity
• We are developing methods intended to be automatic and
also >100X faster, while also retaining accuracy and
information content
• Looking ahead, we hope such methods will also lead to
accurate databases of calculated electronic and thermal
properties in resources like The Materials Project
30

31. Acknowledgements
AMSET
• Alex Ganose
• Alireza Faghaninia
• Junsoo Park
31
Funding provided by:
• U.S. Department of Energy, Basic Energy Science, Early Career program
• U.S. Department of Energy, Basic Energy Science, Materials Project program
Slides (already) posted to hackingmaterials.lbl.gov
Phonons
• Zhuoying Zhu
• Junsoo Park
• Alex Ganose
Alex Ganose Alireza Faghaninia Junsoo Park Zhuoying Zhu