BoltzTraP is a software tool that uses linearized Boltzmann transport theory to calculate electronic transport properties from first-principles band structures. It can calculate properties like electrical conductivity, Seebeck coefficient, and electronic thermal conductivity. The document discusses applications of BoltzTraP to analyze transport properties of metals and thermoelectric materials. Key applications highlighted include analyzing anisotropy, resistivity temperature dependence, and optimizing the electronic structure of materials for high thermoelectric performance.

1. BoltzTraP and Electronic Transport
David J. Singh
University of Missouri
Special Thanks to Georg K.H. Madsen
Today’s Menu:
Short introduction to Boltzmann transport and BoltzTraP
Application to Metals:
plasma frequency, anisotropy, resistivity analysis
Application to Thermoelectrics:
Seebeck coefficient
Transport effective mass
Electronic fitness function
Electronic thermal conductivity and Lorenz number

2. Band Structures and Electronic Transport
Have a band structure from calculations: What do we learn?
Orthorhombic NiSi
This is the dispersion of electronic
states, ε(k) in the Brillouin zone.
Generally, available as values on a
discrete grid of k points.
ε(k) is a periodic function in k.
The velocity is v(k)=∇kε(k)/ħ
Each state carries charge -e and
heat q=(ε(k)-µ) where µ is the
chemical potential.
Currents are sums over states
with occupations.

3. Linearized Boltzmann Theory in the
Relaxation Time Approximation
Basic Idea:
Fields push the distribution function from local equilibrium
(f0: Fermi function) due to their action on electrons, e.g.
electric field favors states with velocities that are towards lower
energy. Generically fields favor velocities “downhill”.
Scattering by itself works to bring the system back to local
equilibrium with a relaxation time.
Result is a steady state with currents.
Key point: Like specific heat, transport only involves states
near (a few kT) the Fermi level (EF).
Normally only very small part of the Brillouin zone is
active, especially at low T.

4. Formulas
Scalar Form:
Scalar Form:
Electrical Conductivity
Seebeck coefficient
V = S∆T
Closed circuit thermal
conductivity
Transport function:
Note the derivative of f0.

5. What BoltzTraP Does
• Smooth interpolation of the band energies (exactly reproduces
the values on the grid.1
• ε(k) = Σ wiΦi(k)
ΦI are real space stars: combinations of planewaves with the periodicity
of the reciprocal lattice and the symmetry of the zone.
Many more ΦI are used than the number of grid points  the flexibility
in choosing wi is used to make the interpolation smooth.
• Calculate velocities using the analytic gradients, ∇kΦi(k)
• Do k space integrals of the transport functions on a grid of
energies.
• Then to energy integrals with the temperature dependent Fermi
functions (fermiintegrals.F90)
These are done assuming that the relaxation time is constant.
Can modify fermiintegrals.F90 to put in other factors if desired, for
example τ = const./N(E) where N(E) is the density of states could be
used for acoustic phonon scattering.
Modified Shankland method: W.E. Pickett, H. Krakauer and P.B. Allen, Phys. Rev. B 38, 2721 (1988).

6. Constant Scattering Time Approximation
• Scattering depends on temperature, doping level, energy, and
sample details.
• The constant scattering time approximation (CSTA) consists in
neglecting the energy dependence of τ in doing the integrals.
• Does not require neglecting doping level, temperature or
other dependencies of τ. It puts as an energy independent
value that may depend on all the other parameters.
• This is how BoltzTraP without modification works.
Consequences:
(1) σ = (σ/τ)τ Can put in T dependent τ to analyze, e.g. ρ(T)
(2) S(T,n) can be directly calculated, since τ cancels in the ratio.
(3) The Lorenz number L(T,n)=κe/(σT) can be directly calculated.

7. Note on the Electronic Thermal Conductivity
TH TC
jQ
• Electrons carry both heat and charge.
Thermal gradient induces electron flow, i.e. charge current
and heat current. BoltzTraP calculates this case.
• In an open circuit case (common experimental condition for
thermal conductivity), no charge can flow, and instead there is a
Seebeck voltage that stops the charge flow. This also affects
thermal conductivity.
κe,open=κe - σS2T
This is usually a small correction, but can be important in
thermoelectrics. The open circuit Lorenz number is still a ratio that
can be calculated directly in the CSTA

8. Use in Metals
LaFeAsO (superconductor parent)
We have a calculated band structure  Fermi surface,
and we wish to characterize it and connect with
experiments.
(1) Transport Anisotropy: σαβ is a rank 2 tensor with the crystal symmetry. If we have
the CSTA with τ a single scalar number (often reasonable), the anisotropy of
(σ/τ)αβ from BoltzTraP gives us a prediction of the conductivity anisotropy.
Prediction for LaFeAsO was σxx/σzz~15, much lower than most cuprates high Tc
materials.
D.J. Singh and M.H. Du, Phys. Rev. Lett. 100, 237003 (2008).

9. Conductivity and Plasma Frequency in Metals
(2) Plasma Frequency (IR optical measurements): Related to Conductivity
Comparison of experimental and calculated plasma
frequencies yields a renormalization that can be
used to characterize fluctuations and correlations in
materials.
Easily extracted from BoltzTraP since
ħωp = ħ (σ/τ)1/2(1/ε0)1/2
Where vacuum permittivity, ε0=8.854x10-12 F/m
In anisotropic materials, ωp
2 is a rank 2 tensor
with the same properties as σ.
M. Qazilbash, J.J. Hamlin, R.E. Baumbach, L. Zhang, D.J. Singh,
M.B. Maple and D.N. Basov, Nature Physics 5, 647 (2009).

10. Thermopower in Metals
PtCoO2 (high conductivity oxide delafossite)
(3) Thermopower (Seebeck coefficient), S(T).
Typically very isotropic even in very
anisotropic semiconductors. Also high in
semiconductors low in metals.
A very useful probe of Fermiology.
K.P. Ong, D.J. Singh and P. Wu, Phys. Rev. Lett. 104, 176601 (2010).
Unlike semiconductors, metals can easily have open Fermi surfaces  highly
anisotropic Sαβ (note S couples two different quantities, so it is not necessarily
symmetric).
BoltzTraP prediction
(2010).
Experimental confirmation,
“Large thermopower anisotropy
in PdCoO2 thin films, P. Yadanov
et al., Phys. Rev. Mater. 3,
085403 (2019).

11. Analysis of Resistivity in Metals
(4) Temperature dependence of resistivity.
Resistivity, ρ(T) depends on band structure term (σ/τ), which is weakly temperature
dependent and τ, which we want to understand.
A. Pandey, et al., arXiv:2201.10325
P.B. Allen, “Empirical electron-phonon λ values from
resistivity of cubic metallic elements”, Phys. Rev. B 36,
2920 (1987) gave a formula for connecting electron-
phonon (or other boson) coupling to resistivity:
λtr = (ħωp
2)(ρ/T)FTH/(8π2kB)
where FTH is a function ~1 at high T.
 If there is experimental data for resistivity, we
can extract the slope dρ/dT at high T, and use the
calculated plasma frequency to get the coupling λtr
This can be compared to the specific heat enhancement, λγ where γexpt=(1+λγ)γbare and γbare is
the Sommerfeld value from the density of states.
It can also be compared with the λ inferred from superconducting properties.

12. Thermoelectric Materials
ZT = σS2T/κ
Performance is limited
by a materials specific
figure of merit, which is
a combination of
electrical and thermal
transport coefficients.

13. Thermoelectric Performance
Thermal Conductivity:
Electronic & Phononic
κ = κe + κl
Wiedemann-Franz
Relation:
κe = LσT ;L0=2.45x10-8
WΩ/K2
ZT = rS2/L [r = κe/κ]
Need high r (low κl) &
high S.
A Contraindicated Property of Matter
• High mobility, but low thermal conductivity.
• How can we scatter phonons but not
electron? Nanostructure, Rattling Ions,
Soft Modes … ?
• High conductivity and high thermopower.
• Special band structures, heavy light
band mixtures, dimensional cross-
over … ?
• Low κl (soft lattice) but high melting point.
• Mixture of hard and soft modes,
materials near lattice instability … ?
• Heavy carrier mass and controlled doping.
• Zintl chemistry, …?

14. Energy Bands in Semiconductors
E = p2/2m = ħ2k2/2m
Free electrons:
p
Semiconductors:
Only near EF (~kT) matters (Sommerfeld)
• Light mass  conductivity
• Heavy mass  thermopower
www.ioffe.ru
GaAs
EF

15. Importance of Doping
Ioffe (1957): Balance of Properties Scientific Optimization:
Then and Now
• Optimization of carrier
concentration is critical.
• Experimentally difficult: must
synthesize series of samples
with controlled doping.
1960’s: Models (e.g. parabolic
bands and scattering) fit to
experiment – interpolation /
extrapolation used to optimize.
1990’s: Predictive first band
structures for thermoelectrics.
Now: quantitative calculations
based on first principles and
Boltzmann transport theory.

16. PbTe
• Known since at least the 1950’s.
• Simple NaCl crystal structure
• Narrow gap (Eg~0.3 eV) semiconductor.
• Melting, Tm=1190 K.
• κ~1 W/mK @ 700 K.
• The basis of the highest performance
thermoelectric materials – e.g. Pb(Te,S)
nanostructured  ZT approaches 2.
What is so special?

17. PbTe Electronic Structure
Engel-Vosko GGA
p-type DOS
• Light mass non-parabolic band structure
near the band edges.
• Becomes much heavier away from the
band edge.
• NaCl structure thermoelectric known since at least the 1950’s.
• High performance thermoelectric materials – e.g. Pb(Te,S), nanostructured 
ZT approaches 2.

18. Origin of High DOS below 0.2 eV
• Hole pockets at the L-points become connected along ~001 (not
symmetry lines).
-0.20 eV -0.25 eV

19. T-Dependence at High p-Type Doping
Levels
Similar to PbTe:Tl experimental data  may be possible to get high ZT
without Tl. --- We also find behavior indicative of high ZT in p-type
PbSe especially at elevated T.
Expt.
Phys. Rev. B 81,
195217 (2010)

20. Experiment
Pei et al. (2011).
Key Points:
• BoltzTraP with first principles band structure and CSTA,
while approximate does not incorporate models or
experimental data in S(T)  a predictive method.

21. Examples of Band Complexities
Carrier Pocket Complexity
Bi2Te3
n-type
H. Shi, et al., Phys. Rev. Appl. 3, 014004 (2015).
X. Chen, D. Parker, D.J. Singh, Sci. Rep. 3, 3168 (2015).

22. Examples of Band Complexities
Multiple Carrier Pockets
H. Tamaki, H.K. Sato, T. Kanno, Adv. Mater. 28, 10182 (2016).
Mg3Sb2 n-type

23. Examples of Band Complexities
Band Convergence
W. Liu, et al., PRL 108, 166601 (2012).
Y. Pei, et al., Nature 473, 66 (2011).
Mg2Si1-xSnx

24. Examples of Band Complexities
Anisotropic Carrier Pockets
AgBiSe2
D. Parker, et al., Phys. Rev. Appl. 3, 064003 (2015).
p-type
n-type

25. Examples of Band Complexities
D.J. Singh, I.I. Mazin, PRB 56, 1650 (1997).
A.F. May, et al., PRB 79, 153101 (2009).
Heavy Band / Light Band
p-type skutterudite
La3-xTe4

26. How Good is a Band Structure?
Need a metric that quantifies:
Extent to which σ and S are decoupled by a
given band structure (T & doping dependent).
Also do not emphasize effective mass (not mass
dependent for a parabolic band).
Can use in conjunction with other screens, e.g.
Seebeck, dielectric constant, indicators of low κl

27. Electronic Fitness Function for Screening
76 compounds with exp.
lattice parameters: 42
HHs, 3FHs, 25 binary
semiconductors, and
several Zintls. About
half thermoelectrics
EFF:
t = (σ/τ)S2/N2/3
N = volumetric DOS.
Function of both temperature and
carrier density.
Need to screen for several different electronic structure features:
• Complex carrier pockets.
• Multiple valleys.
• Valley anisotropy.
• Reduced dimensionality.
• Heavy, light band mixtures.
• etc.
Do not screen for mass.
Easy to calculate from band
structure.

28. Electronic Fitness Function for Screening
EFF = (σ/τ)S2/N2/3
A simple and easy to
calculate function.
A useful screen for
electronic structures
that favor high ZT.
G. Xing, J. Sun, Y. Li, X. Fan, W. Zheng,
D.J. Singh, Phys. Rev. Mater. 1, 065405
(2017); erratum, ibid, 079901 (2017).
https://sites.google.com/view/david-singh-physics/software-transm

29. Calculating the EFF and Effective Mass
1. First principles electronic structure
2. Run BoltzTraP
3. Run transM (reads trace, condtens & outputtrans)
Outputs are doping level and temperature dependent
EFF and transport effective mass (useful for analyzing
semiconductor properties in general).
EFF = electronic fitness function for thermoelectrics
Transport Effective Mass = single parabolic band
effective mass that would give the same value of (σ/τ)
at the same carrier concentration as the actual value
from BoltzTraP
https://sites.google.com/view/david-singh-physics/software-transm

30. p-type Cubic GeTe
GeTe neutrons, Chattopadhyay, J. Phys. C
20, 143 (1987).
GeTe is rhombohedral at 600 K.
600 K
Cubic p-type GeTe is far
superior to other compounds at
600 K.

31. p-type Cubic GeTe
DOS
Cubic GeTe does not have
band convergence – it has
a very anisotropic L-point
pocket with a very light
transverse mass.
Xing et al., Phys. Rev. Mater. 1, 065405 (2017).
Xing et al., J. Appl. Phys. 123, 195105 (2018).

32. p-type Cubic GeTe
Z. Liu, J. Sun, J. Mao, H. Zhu, W. Ren, J. Zhou, Z. Wang, D.J. Singh, J. Sui, C.W. Chu and Z.
Ren, Phase-transition temperature suppression to achieve cubic GeTe and high thermoelectric
performance by Bi and Mn codoping, PNAS 115, 5332 (2018).

33. Electronic Thermal Conductivity
ZT = σS2T/κ, where κ is the thermal conductivity.
Typically written as a sum of electronic and lattice contributions
(ignore coupling of electronic and phonon transport):
κ = κl + κe (κl = lattice part, κe = electronic part)
Often desirable to analyze and “optimize” these separately from
experiment (electronic part will depend on doping for example)
Wiedemann-Franz Relation for the Electronic Part:
κe = LσT where L is the Lorenz number
Classical value of L is L0 = (π2/3)(kB/e)2 = 2.44x10-8 WΩK-2
Reasonable for metals at normal temperatures, but not
reliable for thermoelectrics in general.

34. Electrical Boundary Conditions
Electrons carry both heat and charge
 electrical currents also carry heat
 electrical boundary conditions can be important for the
electronic thermal conductivity
Formula (implemented in BoltzTraP):
is for the case where a current is induced by a temperature
gradient (closed circuit case).
Open circuit case (no electrical current, normally used for thermal
conductivity measurements):
Difference is large when S is large, i.e. low doped thermoelectrics.

35. Lorenz Number
The Lorenz number, L = κe/(σT) is a ratio
this is the case for both closed circuit and open circuit
boundary conditions (note κopen= κe - TσS2).
 The scattering time cancels in the CSTA and one can get a
direct calculations of L from BoltzTraP without adjustable
parameters.
BoltzTraP
n-type (dashed)
p-type (solid)
A. Putatunda, Materials Today Physics 8, 49 (2019).

36. Summary
• BoltzTraP is a tool for interpolating and analyzing
band structures.
• Produces transport related information using the
constant scattering time approximation (can modify in
fermiintegrals.F90).
• Predictive parameter free calculation of S(T).
• Metals: plasma frequency, anisotropy, resistivity
• Thermoelectrics: Seebeck coefficient, transport
effective mass, electronic fitness function, electronic
thermal conductivity and Lorenz number