ARTICLE
OPEN
Effective mass and Fermi surface complexity factor from ab
initio band structure calculations
Zachary M. Gibbs
1
, Francesco Ricci
2
, Guodong Li
3
, Hong Zhu
4
, Kristin Persson
5
, Gerbrand Ceder
4,5
, Geoffroy Hautier
2
,
Anubhav Jain
5
and G. Jeffrey Snyder
3
The effective mass is a convenient descriptor of the electronic band structure used to characterize the density of states and electron
transport based on a free electron model. While effective mass is an excellent first-order descriptor in real systems, the exact value
can have several definitions, each of which describe a different aspect of electron transport. Here we use Boltzmann transport
calculations applied to ab initio band structures to extract a density-of-states effective mass from the Seebeck Coefficient and an
inertial mass from the electrical conductivity to characterize the band structure irrespective of the exact scattering mechanism. We
identify a Fermi Surface Complexity Factor: N
*
v
K
*
from the ratio of these two masses, which in simple cases depends on the number
of Fermi surface pockets ðN
*
v
Þ and their anisotropy K
*
, both of which are beneficial to high thermoelectric performance as
exemplified by the high values found in PbTe. The Fermi Surface Complexity factor can be used in high-throughput search of
promising thermoelectric materials.
npj Computational Materials (2017) 3:8 ; doi:10.1038/s41524-017-0013-3
INTRODUCTION
The calculation of electronic band structures using density
functional theory (DFT) is now so routine that it is becoming faster
to compute certain physical properties than make samples and
measure them—inspiring the materials genome initiative efforts
worldwide. Ab initio calculations are important from a materials’
design perspective in that they provide insight into the underlying
electronic states that give rise to experimentally measurable
properties. Dielectric, optical and transport properties such as
electrical conductivity, Hall effect, and thermoelectric power
(Seebeck effect) require knowledge not only of the electronic
structure readily available from ab initio calculations, but may also
require an assumption about the scattering. Using a constant
relaxation time approximation, very precise predictions of transport
properties that depend on fine details of the band structure can be
made, however, the electrical conductivity predicted for instance
can be greatly misleading because the relaxation time is
approximated, often to an arbitrary constant. A recent study
performed by the authors demonstrated that while Seebeck
coefficient was reproduced fairly well across a variety of compounds
(provided that the band gap was not severely underestimated), the
experimental values on conductivities can be highly inaccurate
using a constant relaxation time.
1
While some scattering mechan-
isms can now be calculated using ab initio methods, they are far
from routine and require special algorithms. The goal of this study is
to extract transport information from band structure calculations
that does not depend on any scattering assumption.
In the common free-electron approximation lexicon of electro-
nic and optical properties we typically describe charge carriers as
having an effective mass m
*
and a relaxation or scattering time τ.
Combining with the density of carriers n, the electrical
conductivity σ = ne
2
τ/m
*
and drift mobility μ
d
= eτ/m* can also
be expressed. This description, while not exact, has proven
immensely helpful in the understanding and engineering of
electronic materials that have profoundly changed our civilization.
This representation already separates transport into electronic
structure (through the m*) and scattering (τ) terms as well as
allowing the flexibility of varying n (through doping, for example).
It is, therefore, natural to expect that knowledge of electronic
structure should reveal the appropriate effective mass for various
values of n but not necessarily the scattering-dependent transport
properties (such as σ and μ
d
) until the scattering is known.
The free-electron description has been integral to semiconduc-
tor physics despite many examples of profound deviations. In the
free-electron model the electronic structure is represented by a
single, isotropic Fermi surface described by particle with mass and
charge of an electron. A free-electron like description is only
helpful to describe real semiconductors if we allow multiple (N
v
)
free-electron like pockets to describe the Fermi surface and that
each pocket may be anisotropic (described by anisotropy term K)
with effective mass m
*
that differs from free-electron mass.
Because of the multiple pockets the total density of electronic
states will be N
v
times that of a single pocket.
In thermoelectrics, for example, these and other material
parameters are helpful to predict the promise of a material for
use in a thermoelectric device.
2–4
For typical semiconductor
transport where the electrons are scattered by acoustic phonons
(deformation potential scattering)
5
the thermoelectric quality
factor B, given by:
B ¼
2k
2
B
ħ
3π
N
v
m
*
c
C
l
κ
L
Ξ
2
T
ð1Þ
Received: 30 October 2016 Revised: 13 January 2017 Accepted: 25 January 2017
1
California Institute of Technology, Division of Che mistry and Chemical Engineering, Pasadena, CA, USA;
2
Institute of Condensed Matter and Nanosciences (IMCN), Université
catholique de Louvain, Chemin des étoiles 8, bte L7.03.01, Louvain-la-Neuve, Belgium;
3
Department of Materials Science and Engineering, Northwestern University, 2220 Campus
Drive, Evanston, IL, USA;
4
Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Massachusetts, MA, USA and
5
Lawrence Berkeley National Lab, 1 Cyclotron Road, Berkeley, CA, USA
Correspondence: G Jeffrey Snyder (jeff.snyder@northwestern.edu)
www.nature.com/npjcompumats
Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences
determines the maximum zT (materials efficiency) when the
semiconductor is optimally doped, where κ
L
is the lattice thermal
conductivity, Ξ is the deformation potential, C
l
is the average
longitudinal elastic modulus, m
*
c
is the conductivity effective mass,
and T is temperature.
Two materials parameters, valley degeneracy N
v
and conduc-
tivity effective mass m
*
c
appear in the quality factor that concern
the static band structure and should be accessible by DFT. A small
conductivity effective mass m
*
c
should lead to higher mobility,
6
and have led to the reinvestigation of structures with very light
bands such as SnTe.
7
Valley degeneracy (N
v
) has been shown to
be critical for achieving high zT.
8–12
Many of the best thermo-
electric materials are known to have, recently have been found to
have, or are being engineered to have high valley degeneracy,
including: the lead and tin chalcogenides,
5, 10 , 13
diamond like
copper selenides,
14, 15
Skutterudites,
16
Mg
2
Si,
11, 17
Half Heuslers,
18
and Zintl phases.
19
While isolated pockets of Fermi surfaces can
improve the quality factor, Fermi surface pockets connected by
threads in the lead chalcogenides
20
and the complex Fermi
surfaces of bismuth telluride
21
have also been suggested to be
beneficial for zT.
Here, we show that DFT calculations can be used to compute
two types of effective masses characteristic of the electronic
structure independent of the scattering mechanism. An inertial
mass m
*
c
from the electrical conductivity and a density of states
(DOS) mass m
*
S
from the Seebeck coefficient and carrier
concentration. Then we introduce the ratio of these two masses
as the Fermi Surface complexity factor ðN
*
v
K
*
Þ¼ðm
*
S
=m
*
c
Þ
3=2
,
which we recognize as related to valley degeneracy N
v
and carrier
pocket anisotropy K.
RESULTS
Boltztrap conductivity mass ðm
*
c
Þ
The conductivity effective mass is computed directly from the
Boltztrap calculation for electrical conductivity, ðm
*
c
Þ
1
¼ σ=ne
2
τ
using the constant relaxation time approximation (CRTA). This
definition has been used to conduct a high-throughput search for
low hole and electron low effective mass (high mobility)
transparent conductive oxides
22–24
and also for analyzing specific
thermoelectric materials.
25, 26
Specifically using the notation of
Madsen and Singh
27
the effective mass tensor can be computed
from
Eq. (12) of ref.
27
ðm
*
c
αβ
ðT; μÞÞ
1
¼
σ
αβ
ðT; μÞ
e
2
τ
´
1
nðT; μÞ
ð2Þ
Where e is the electron charge and τ is the user-specified constant
relaxation time. When multiple bands contribute to conduction,
m
*
c
is a weighted average over all contributing bands.
The net charge carrier concentration n (or doping concentra-
tion) is measured relative to a band edge and is a positive quantity
whether electrons or holes are dominant charge carriers. In
BoltzTraP, N
val
, the number of valence electrons/cell required
to place the Fermi level in the band gap to make the material
“undoped” at 0 K is set by the user. Thus
N
val
V
¼
R
gðϵÞf
μ¼undoped
ðT ¼ 0 K; ϵÞdϵ where V is the volume of the unit
cell (cm
3
), gðϵÞ is the density of states (states/volume/Energy), and
f
μ¼undoped
ðT; ϵÞ is the Fermi-Dirac distribution function where the
electron chemical potential (μ or E
F
) is that of the undoped
material. The doping concentration is then n ¼j
N
val
V
R
gðϵÞf
μ
ðT; ϵÞdϵj using the notation of Madsen and Singh,
27
which
is directly computed from the N in the BoltzTrap output as n(T; μ)
=(|N(T; μ)|)/(V).
The conductivity mass tensor (m
*
c
) should be a good
representation of the band structure relevant to electrical
conductivity independent of the scattering mechanism. This is a
more relevant output from Boltztrap than conductivity (σ) as CRTA
is typically a poor approximation for scattering in real thermo-
electric materials. Also while σ changes rapidly in a semiconductor
with differing Fermi level (or n), m
*
c
should be constant to a first-
order approximation as demonstrated in Fig.
1 making it a more
robust descriptor of the BoltzTraP calculation result. Like
conductivity, the drift mobility μ
d
requires knowledge of the
scattering through τ. Thus, while mobility could be easily
computed from BoltzTraP:
μ
d
αβ
ðT; μÞ¼
σ
αβ
ðT; μÞ
e
´
1
nðT; μÞ
ð3Þ
the uncertainty of τ makes effective mass m
*
c
, not mobility μ
d
, the
parameter that most directly represents the electronic structure.
Note that the effective mass m
*
c
used here differs from the
commonly used definition involving the second derivative of a
particular band dispersion: 1=m
*
αβ
¼ð1=ħ
2
Þð∂
2
E=∂k
α
∂k
β
Þ either
evaluated at a particular k-point (energy) or evaluated at the band
edge. In general, particularly for complex band structures, these
varying definitions can give different values for effective mass. It is
mathematically equivalent (by integration by parts
28
) to use the
second derivative mass, but one must remember that this second
derivative mass must be averaged over the entire band of filled
states at 0 K. Thus using the second derivative, the effective mass
m
*
c
αβ
ðT; μÞ requires an integration over all states weighted by the
Fermi function.
23
While m
*
c
is a good measure of the average
inertial effective mass of electrons in a system, it does not indicate
much about the density of electronic states in a system that has
multiple bands.
Boltztrap Seebeck mass (m
*
S
)
The effective density of electronic states can be estimated through
the Seebeck coefficient characterized by the DOS effective mass:
m
*
S
.
26
This can be performed in analogy to the Pisarenko plot
(S vs n), which is commonly used to estimate an effective mass m
*
S
from experimental data.
29, 30
Treating the BoltzTraP transport
results as if it were experimental data, we solve for an effective
reduced chemical potential η
eff
that yields the computed Seebeck
coefficient. For each BoltzTraP calculated pair of S(T; μ) (using 1/3
the trace of the S
ij
(T; μ) tensor of Eq 16 in Madsen and Singh
27
)
and n(T; μ) we solve for a m
*
S
(T; μ). First, from S(T; μ)wefind the
reduced chemical potential η
eff
that would give that same
Seebeck coefficient for a single parabolic band:
SðT; μÞ¼
k
B
q
ð2 þ λÞ
ð1 þ λÞ
F
1þλ
ðη
eff
Þ
F
λ
ðη
eff
Þ
η
eff
ð4Þ
Where λ is the scattering exponent, q is −e for electrons and + e
for holes, and F
1+λ
are the Fermi functions given by:
F
j
ðηÞ¼
Z
1
0
ϵ
j
dϵ
1 þ e
ϵη
ð5Þ
We adjust the common scattering assumption, which is usually by
acoustic phonons in experiments, to coincide with the CRTA used
in BoltzTraP (i.e., set λ = 1/2). The reduced chemical potentials
(η
eff
=(μ − E
B
)/(k
B
T)) are relative to the band edge where E
B
is the
energy of the (valence or conduction) band edge and the sign of η
is always such that η
eff
> 0 has chemical potential (μ in Madsen
and Singh;
27
E
F
in BoltzTraP output) in the band.
Then, using the effective reduced chemical potential η
eff
,we
find the effective mass m
*
S
(T; μ) that would give the n(T;μ)
calculated from Boltztrap.
nðT; μÞ¼
1
2π
2
2m
*
S
ðT; μÞk
B
T
ħ
2
3
2
F
1=2
ðη
eff
Þ
ð6Þ
It should be noted that m
*
S
is a scalar quantity and not a tensor
unlike m
*
c
αβ
. Although the Seebeck coefficient S
ij
(T;μ) is a tensor, for
Effective mass and Fermi surface complexity factor
ZM Gibbs et al
2
npj Computational Materials (2017) 8 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences
parabolic bands of the same sign, and for τ that may depend on
energy but not direction,
26
the Seebeck only depends on the
reduced chemical potential η
eff
and the scattering exponent λ
both of which are scalars. Considering that the density of states
gðϵÞ, is also a scalar quantity it is appropriate that a density of
states effective mass such as m
*
S
should also be a scalar quantity.
This single parabolic band Seebeck coefficient likely represents
the thermopower from the configurational entropy of the
electrons in the available states represented by gðϵÞ, which would
support our interpretation of m
*
S
as an appropriate measure of the
density of states effective mass.
The density of states effective mass is a convenient single
metric to describe the density of states. Like m
*
c
, m
*
S
remains
relatively constant with doping while S , σ, g, n change, and,
therefore, m
*
S
is a better descriptor of the density of states for
semiconductors than any single value of g (recall that gðϵÞ¼
ð8 π
ffiffiffi
2
p
=h
3
Þm
*3=2
ffiffiffi
ϵ
p
is a typical description for g of a semicon-
ductor). Even though there may be multiple or nonparabolic
bands, a single effective mass is the best first order characteriza-
tion of a semiconductor and is accurate within the uncertainty of
measurements on new materials. Particularly for use in thermo-
electrics, m
*
S
is anticipated to be more relevant than an effective
mass from a direct comparison to the calculated density of
states (g)
2
because it most directly relates to the Seebeck
coefficient both in theory and experiment.
Effective valley degeneracy ðN
*
v
Þ
A Fermi surface consisting of N
v
identical, isolated surfaces will
have total number of electronic states that is N
v
times the number
of states in each isolated surface. If each isolated surface can be
described with a density of states mass m*
b
(recall gðϵÞ/m
*
3=2
b
) for
each band or pocket then the total density of states mass, here
computed using the Seebeck coefficient, is m
*
S
¼ N
2=3
v
m
*
b
. Valley
degeneracy then manifests itself by increasing the density of
states effective mass relative to the single valley effective mass
(m*
b
), which should be related to the inertial mass m
*
c
. Symmetry
imposed valley degeneracy results from band extrema that exist at
low symmetry points in a high symmetry crystal or orbital
degeneracy of the constituent atoms. In order to maximize N
v
, the
band extrema should be off high symmetry points (such as
gamma), which is influenced by the symmetry of the most
relevant atomic orbitals.
31
When multiple bands contribute to conduction but they are not
exactly degenerate (band extrema not at the same energy), we
consider this an increased effective valley degeneracy. One can
approximate N
*
v
by counting the total number of charge carrier
Fermi surfaces (as in done for calculating N
v
in Figs. 1 and 2,or
within a particular energy window.
2
We think of effective valley
degeneracy, N
*
v
, as describing the effective number carrier pockets
contributing to conduction, where the contribution may only be
partial if that pocket is displaced by more than ~k
B
T from the
chemical potential (μ).
Effective anisotropy factor (K
*
)
Only in the simplest cases can Fermi surfaces can be described as
spherical pockets; many materials contain more complicated
Fermi surfaces. The next level of complexity involves ellipsoidally
shaped pockets where the anisotropy parameter, K ¼ m
*
k
=m
*
?
,
quantifies the degree of anisotropy.
32
Many material systems have
been shown to display K different from unity including: Si/Ge,
33, 34
IV–VI materials,
32, 35, 36
III–V materials,
37
and others.
38
The
conductivity effective mass of such systems is calculated from
the harmonic average along each direction:
m
*
c
¼ 3ðm
*
1
k
þ 2m
*
1
?
Þ
1
, which determines the carrier mobility
ðμ ¼ eτ=m
*
c
Þ. The conductivity mass m
*
c
is, in general, different
from the single valley density of states mass m*
b
(geometric
average: m
*
b
¼ðm
*
k
m
*
2
?
Þ
1=3
); they are equal only for spherical
pockets (K ¼ m
*
k
=m
*
?
¼ 1). For non-ellipsoidally shaped Fermi
surfaces, we can define the effective anisotropy parameter K
*
in
terms of the effective masses:
26
K
*
¼
m
*
b
m
*
c
3=2
¼
ð2K þ 1Þ
3=2
3
3=2
K
ð7Þ
Fig. 1 Effective masses and complexity factor at 300 K from BoltzTraP for some relatively simple structures CdTe (a–e) and AlAs (f–j). a, f
Computed electronic band structure, b, g Conductivity mass, m*
c
, versus Fermi level and Seebeck DOS mass, m*
S
, as functions of the Fermi level
across the valence and conduction bands. c, h Fermi surface complexity fac tor N
*
v
K
*
and true valley degeneracy N
v.
d, i Primary conduction
Fermi surface (0.03 eV above the conduction band edge (CBM) in CdTe and 0.1 eV above the band edge for AlAs), and e, j Valence band Fermi
surface (0.05 eV below the valence band edge (VBM) for both CdTe and AlAs). The valence band Fermi surfaces are colored differently for the
three degenerate bands
Effective mass and Fermi surface complexity factor
ZM Gibbs et al
3
Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2017) 8
Fermi surface complexity factor ðN
*
v
K
*
Þ
Combining the effects of valley degeneracy N
*
v
and anisotropy K
*
we define the Fermi surface complexity factor, ðN
*
v
K
*
Þ in terms of
the effective masses m
*
c
and m
*
S
we derive from DFT calculations:
ðN
*
v
K
*
Þ¼
m
*
S
m
*
c
3=2
ð8Þ
Although it is not clear how to define the single valley effective
mass (m*
b
) in general, for simple cases of multiple, degenerate,
ellipsoidal Fermi surface pockets we concluded N
*
v
¼ðm
*
S
=m
*
b
Þ
3=2
and K
*
¼ðm
*
b
=m
*
c
Þ
3=2
. So while we do not to calculate or even
clearly define N
*
v
or K
*
individually, we can clearly define ðN
*
v
K
*
Þ
through the equation above. Because m
*
c
αβ
is, in general, a tensor
while m
*
S
is a scalar, ðN
*
v
K
*
Þ
αβ
is most generally a second rank
tensor like m
*
c
αβ
. For simplicity in this work, only isotropic
compounds are demonstrated below so that only scalar values
are needed.
The Fermi surface complexity factor N
*
v
K
*
can be a good
indicator of the effective valley degeneracy N
*
v
. Compared to the
valley degeneracy N
*
v
, which can easily be three or more, the
anisotropy factor K
*
, in some cases, may only lead to small
deviations from unity, resulting in a complexity factor N
*
v
K
*
≈ N
*
v
.
For ellipsoidal Fermi pockets a factor of two difference in the
directional effective mass, m
*
k
=m
*
?
¼ 2, results only in K
*
= 1.08
while a highly anisotropic m
*
k
=m
*
?
¼ 12 is required to get just
K
*
=2.
The Fermi surface complexity factor N
*
v
K
*
may also be a good
indicator of the suitability of a material for thermoelectrics as
noted by Parker et al.
26
who also considered a ratio similar to
m
*
S
/m
*
c
. Complex Fermi surfaces have been shown to be beneficial
for zT. Complexity in the form of high valley degeneracy N
v
(multiple carrier pockets in the Fermi surface contributing to
conduction) such as found in PbTe
39
is widely recognized to be
beneficial to thermoelectric performance as seen in the quality
factor. However, other complexities such as narrow threads of
Fermi surface has also been identified to improve thermoelectric
transport.
20, 36, 40, 41
The advantage of anisotropy factor K
*
may be most apparent
when the scattering is also complex. Separating the scattering
time τ
0
from the quality factor
42
] results in a quality factor strongly
dependent on N
*
v
K
*
:
B ¼
k
2
B
Tðk
B
TÞ
3=2
3π
2
ħ
3
τ
0
κ
L
N
*
v
K
*
ðm
*
c
Þ
1=2
ð9Þ
In these cases where the Fermi surface has additional
complexity beneficial to thermoelectrics, N
*
v
K
*
may be an
auspicious metric for thermoelectric materials.
DISCUSSION
II–VI and III–V Materials—CdTe and AlAs
As a first example, the conduction band of CdTe is centered
directly at Γ (N
v
= 1) and has a spherical Fermi surface (as shown in
Fig. 2 Band structure, m*
c
and m*
S
versus Fermi level, and N
*
v
K
*
at 300 K versus Fermi level, and valence and conduction band Fermi surfaces for
PbTe (a–e), PbSe (f–j), and PbS (k–o). The valence and conduction band edges are shown as a dashed line. Fermi surfaces are drawn at 0.1 eV
above the conduction band edge (CBM) for PbTe, PbSe, and PbTe, respectively, or 0.13, 0.3, and 0.4 eV below the valence band edge (VBM) for
PbTe, PbSe, and PbS, respectively
Effective mass and Fermi surface complexity factor
ZM Gibbs et al
4
npj Computational Materials (2017) 8 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences
Fig. 1d). This simple band structure shows the effective masses
(m
*
S
and m
*
c
) as computed from calculated Boltztrap data (shown
in Fig.
1 as described earlier) are constant with varying doping
(Fermi level) and roughly consistent with experimental and
traditionally calculated values of m*.
43
The Fermi surface complex-
ity factor ðN
*
v
K
*
Þ also shown in Fig.
1c is very close to 1 for the
conduction band, consistent with a single (N
*
v
= 1) spherical pocket
(K
*
= 1). The computed N
*
v
K
*
is approximately 1 over a wide range
of Fermi levels even as the Fermi level is moved deeper into the Γ
band. A slight decrease in value and increase in noise is observed
as the Fermi level moves farther into the band presumably due to
sensitivity of m
*
S
to the small variations in Seebeck at these Fermi
levels.
Next, consider the slightly more complicated conduction band
of AlAs, which has multiple carrier pockets at different points in
the Brillouin zone (Fig.
1i), as the Fermi level is moved into the
conduction band. The AlAs Fermi surface for the primary
conduction band at X has multiple pockets (N
v
= 3) (Fig. 1i). We
calculate ðN
*
v
K
*
Þ¼3:5 near the conduction band edge consistent
with our expectation of N
v
~ 3 and K* close to 1.
One might expect that N
*
v
should make a stepwise transition
from one value of N
v
to another as the Fermi level approaches and
enters the additional band. This behavior is observed in Fig.
1h
where N
*
v
K
*
is compared to a hypothetical N
v
just by considering
the symmetry of the extrema alone (green line in Fig.
1g, N
v
ðE
F
Þ¼
P
N
v;i
HðE
F
E
i
Þ where H is the Heaviside step function and E
i
is
the energy of the i
th
band extrema).
2
As the Fermi level moves into
the conduction band, we reach the Γ
c
(N
v
= 1, 0.28 eV above X
c
)
and L
c
(N
v
= 4, 0.51 eV above X
c
) bands where the total N
v
increases to 4 and 8, respectively. The Fermi surface complexity
factor ðN
*
v
K
*
Þ increases steadily from the band edge resulting in a
value of 3.5 and 6.1 at the Γ
c
and L
c
band edge energies,
respectively. The thermoelectric Fermi surface complexity factor
ðN
*
v
K
*
Þ mirrors the true N
v
both qualitatively and quantitatively—
consistent with an anisotropy component, K
*
, that is not far from
unity. The
Supplementary material includes m
*
S
, m
*
c
, N
v
, ðN
*
v
K
*
Þ for
the other III–V materials (Table
S1).
Enhancement in K
*
appears in the valence band of both CdTe
and AlAs, which consist of three degenerate bands: Γ
1,2,3v
, with
different effective masses (light hole, heavy hole, and split-off
band) due to p-orbital degeneracy.
31
As a result, the Fermi surface
—even though it is centered at the Γ point—will have a non-trivial
topology (as described by Mecholsky et al.
41
for silicon), which
appears to result in a larger K
*
component to the Fermi surface
complexity factor and a N
*
v
K
*
that exceeds the expected
degeneracy of N
v
= 3 for Γ
1,2,3v
. For the valence band, ðN
*
v
K
*
Þ=6
and 9 for CdTe and AlAs, respectively. Mecholsky et al.
41
shows
that warped, non-ellipsoidal Fermi surfaces, which result from the
combination of light and heavy bands, significantly influence the
electronic transport in these systems altering the equivalent
effective masses.
IV–VI Materials—PbS–PbSe–PbTe
The lead chalcogenides (including PbTe, PbSe,
42, 44
PbS,
45
and
their alloys
46–54
) are known to be good thermoelectric materials
partly because of their complex electronic structure. The Fermi
surface complexity factor and effective masses were also
computed for these IV–VI compounds. For PbS and PbSe, the
conduction band at the L-point shows significant valley degen-
eracy of N
v
= 4 shows N
*
v
K
*
¼ 4 as expected for chemical potential
near the band edge. The primary valence band, also with N
v
=4
has a nearby secondary valence band along the Σ line, with its
own N
v
= 12, which can be seen by a rapidly rising N
*
v
K
*
that
approaches the simple sum of these valley degeneracies.
An exceptionally high N
*
v
K
*
is also found in p-type PbTe, above
that expected from N
*
v
of L and Σ bands indicating a significant
contribution from K
*
or a new band. In p-type PbTe the complex
Fermi surface characterized by threads which develop between
the L
-v
and Σ
−v
pockets, which have been concluded to contribute
to the high thermoelectric performance.
8, 55–59
This could be due
to the large surface area to volume ratio of the thread-like states,
which leads to an inherently large mobility and quality factor (and
corresponding large K*). Compared to PbS and PbSe, the bands in
PbTe are closer in energy (e.g., the L and Σ bands are computed to
be only ~ 0.12 eV apart) and these energies may change with
temperature. For example, in PbTe, the L and Σ bands are thought
to shift with temperature, eventually converging at
~ 700 K (ref.
56).
High-throughput computation
The vast electronic structure database constructed through the
Materials Project allows for large-scale screening of semiconduc-
tors for thermoelectrics, transparent conductors as well as other
applications. By combining DFT and BoltZtraP (using the CRTA)
thousands of compounds can be screened for effective mass
(m
*
c
, m
*
S
) and complexity factor N
*
v
K
*
(Supplementary Table
I).
Figure
3 shows the correlation between ðN
*
v
K
*
Þ and the
calculated maximum (Fermi level-dependent) power factor
(assuming constant τ =10
−14
s) for the large group of compounds
(~ 2300 isotropic compounds) at 600 K. We can see a good
correlation between the calculated Fermi surface complexity
factor and the maximum attainable power factor; this is expected
since the quality factor for constant relaxation time is expected to
scale according to Eqn.
9. Data regarding the maximum power
factor and N
*
v
K
*
is included in the
Supplementary material.
While experimental conductivity values are difficult to repro-
duce within the constant relaxation time,
1
it is remarkable that
known thermoelectric materials such as PbTe, GeTe, TiCoSb show
up with a high constant relaxation time power factor and a high
Fermi surface complexity factor. This indicates that, at least for
screening and ranking, the Fermi surface complexity factor is an
effective descriptor.
CONCLUSIONS
Both a density of states effective mass, m
*
S
and inertial effective
mass m
*
c
can be extracted even from complex DFT band structures
using the result of σ and S transport calculations, such as done in
BoltzTraP, even if the scattering mechanism is not known. Because
m
*
c
and m
*
S
are less influenced by τ and scattering mechanism than
σ and S, they are better descriptors of a band structure’s
contribution to transport than a CRTA value of σ and
S itself. We interpret the ratio of these two masses as a Fermi
Surface Complexity Factor ð N
*
v
K
*
Þ, which should be influenced by
effective valley degeneracy N
*
v
and anisotropy K
*
, both beneficial
to thermoelectric performance.
We have analyzed the maximum thermoelectric power factors
and for a large set compounds from the Materials Project to show
Fig. 3 Maximum power factor for ~ 2300 cubic compounds plotted
as a function of the Fermi surface complexity factor (evalulated at
the Fermi level which yields the maximum power factor) at T = 600 K
Effective mass and Fermi surface complexity factor
ZM Gibbs et al
5
Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2017) 8