of 6
Piezoelectric Electron-Phonon Interaction from
Ab Initio
Dynamical Quadrupoles:
Impact on Charge Transport in Wurtzite GaN
Vatsal A. Jhalani ,
1
,*
Jin-Jian Zhou ,
1
,*
Jinsoo Park ,
1
Cyrus E. Dreyer,
2,3
and Marco Bernardi
1
,
1
Department of Applied Physics and Materials Science, California Institute of Technology,
Pasadena, California 91125, USA
2
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA
3
Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA
(Received 20 February 2020; accepted 27 August 2020; published 21 September 2020)
First-principles calculations of
e
-ph interactions are becoming a pillar of electronic structure theory.
However, the current approach is incomplete. The piezoelectric (PE)
e
-ph interaction, a long-range
scattering mechanism due to acoustic phonons in noncentrosymmetric polar materials, is not accurately
described at present. Current calculations include short-range
e
-ph interactions (obtained by interpolation)
and the dipolelike Frölich long-range coupling in polar materials, but lack important quadrupole effects for
acoustic modes and PE materials. Here we derive and compute the long-range
e
-ph interaction due to
dynamical quadrupoles, and apply this framework to investigate
e
-ph interactions and the carrier mobility
in the PE material wurtzite GaN. We show that the quadrupole contribution is essential to obtain accurate
e
-ph matrix elements for acoustic modes and to compute PE scattering. Our work resolves the outstanding
problem of correctly computing
e
-ph interactions for acoustic modes from first principles, and enables
studies of
e
-ph coupling and charge transport in PE materials.
DOI:
10.1103/PhysRevLett.125.136602
When atoms move due to lattice vibrations, the potential
seen by an electron quasiparticle changes due to both short-
range and long-range forces. Early theories of such
electron-phonon (
e
-ph) interactions focused on simplified
models tailored to specific materials
[1]
. For example, the
deformation potential, first formulated for metals, quanti-
fies the
e
-ph interactions with acoustic phonons in the long-
wavelength limit
[2]
. In ionic and polar covalent materials,
in which atoms can be thought of as carrying a net charge,
Fröhlich identified a dipolelike long-range
e
-ph interaction
due to longitudinal optical (LO) phonons
[3]
. Also impor-
tant is the piezoelectric (PE)
e
-ph interaction, which arises
from the strain induced by acoustic phonons in polar
materials lacking inversion symmetry
[4,5]
. Its conven-
tional formulation by Mahan expresses the
e
-ph coupling in
terms of the macroscopic PE constants of the material
[5]
.
Vogl unified these
e
-ph interactions
[6]
, showing that the
dipole, PE, and deformation-potential contributions origi-
nate from a multipole expansion
[7]
of the
e
-ph potential,
and analyzed its behavior in the long-wavelength limit
(phonon wave vector
q
0
). The Fröhlich dipole term
diverges as
1
=q
in this limit, and it is proportional to the
sum of atomic dipoles: if each atom
κ
is associated with a
Born charge tensor,
Z
κ
, contracting this tensor with the
phonon eigenvector
e
ν
q
gives the atomic contributions to
the dipole field,
Z
κ
e
ν
q
. Next in the multipole expansion is
the quadrupole field generated by the atomic motions,
which approaches a constant value as
q
0
. If each atom
is associated with a dynamical quadrupole tensor,
Q
κ
, the
atomic quadrupole contributions can be written as
Q
κ
e
ν
q
.
Both the dipole and the quadrupole terms contribute to
the PE
e
-ph interaction
[6]
. The multipole expansion also
gives octupole and higher terms, which vanish at
q
0
and
can be grouped together into a short-range deformation
potential.
Density functional theory (DFT)
[8]
and density func-
tional perturbation theory (DFPT)
[9]
have enabled calcu-
lations of
e
-ph interactions from first principles. In turn, the
e
-ph interactions can be used in the Boltzmann transport
equation (BTE) framework to predict electron scattering
processes and charge transport
[10
22]
. The
e
-ph inter-
actions computed with DFPT include all the contributions
from the multipole expansion. Yet, DFPT is too costly to
use directly on the ultrafine Brillouin zone grids needed to
compute
e
-ph relaxation times and charge transport using
the BTE. The current approach relies on Fourier interpo-
lation techniques to capture the short-range
e
-ph inter-
actions
[23
26]
while adding the dipole Fröhlich term in
reciprocal space for polar materials. Deriving and com-
puting the Fröhlich interaction
[27,28]
has been a first step
toward implementing Vogl
s modern
e
-ph theory in first-
principles calculations and correctly capturing the long-
range
e
-ph contributions. However, a key piece is still
missing in the
ab initio
framework: the quadrupole
e
-ph
interaction, which critically corrects the dipole term in
polar materials, is sizable also in nonpolar materials, and is
particularly important in PE materials such as wurtzite
crystals or titanates.
PHYSICAL REVIEW LETTERS
125,
136602 (2020)
0031-9007
=
20
=
125(13)
=
136602(6)
136602-1
© 2020 American Physical Society
In this Letter, we derive the
ab initio
quadrupole
e
-ph
interaction and compute it for a PE material, wurtzite
gallium nitride (GaN), using dynamical quadrupoles
computed from first principles. We show that including
the quadrupole term provides accurate
e
-ph matrix
elements and is essential to obtaining the correct acoustic
phonon contribution to carrier scattering. We compute
the electron and hole mobilities in GaN including the
quadrupole interaction, obtaining results in agreement with
experiment. Our analysis highlights the large errors result-
ing from including only the Fröhlich term in GaN
[29]
,
which greatly overestimates the acoustic mode
e
-ph inter-
actions
[29,30]
. Our companion paper
[31]
applies this
framework to silicon and the PE material PbTiO
3
. The
quadrupole
e
-ph interaction, which critically corrects the
dipole term in polar materials, is sizable in nonpolar
materials, where it is the leading long-range
e
-ph inter-
action, and it is essential to correctly compute the
e
-ph
matrix elements. Taken together, the quadrupole interaction
completes the theory and enables accurate
ab initio
e
-ph
calculations in all materials and for all phonon modes.
The key ingredient in first-principles
e
-ph calculations
are the
e
-ph matrix elements,
g
mn
ν
ð
k
;
q
Þ
, which encode the
probability amplitude for scattering from an initial Bloch
state
j
n
k
i
with band index
n
and crystal momentum
k
into
a final state
j
m
k
þ
q
i
by emitting or absorbing a phonon
with branch index
ν
and wave vector
q
[17,32]
. Following
Vogl
[6]
, we separate the long-range dipole and quadrupole
contributions from the short-range part:
g
mn
ν
ð
k
;
q
Þ¼
g
dipole
mn
ν
ð
k
;
q
Þþ
g
quad
mn
ν
ð
k
;
q
Þþ
g
S
mn
ν
ð
k
;
q
Þð
1
Þ
where
g
dipole
is the
ab initio
Fröhlich interaction written in
terms of Born effective charges
[27,28]
,
g
quad
is the
quadrupole interaction written in terms of dynamical
quadrupoles, and
g
S
collects octupole and higher-order
short-range terms. For polar acoustic modes,
g
dipole
þ
g
quad
is a generalized replacement for the phenomenological PE
e
-ph coupling expressed in terms of the PE constants
[4,5]
.
We derive the first-principles quadrupole
e
-ph interaction
g
quad
mn
ν
ð
k
;
q
Þ
by superimposing two oppositely oriented
dipole moments, as discussed in detail in our companion
paper
[31]
, and obtain:
g
quad
mn
ν
ð
k
;
q
Þ¼
e
2
Ω
ε
0
X
κ

2
ω
ν
q
M
κ

1
=
2
X
G
≠−
q
1
2
ð
q
α
þ
G
α
Þð
Q
αγ
κ
;
β
e
ð
κ
Þ
ν
q
;
β
Þð
q
γ
þ
G
γ
Þ
ð
q
α
þ
G
α
Þ
ε
αγ
ð
q
γ
þ
G
γ
Þ
h
m
k
þ
q
j
e
i
ð
q
þ
G
Þ
·
ð
r
τ
κ
Þ
j
n
k
i
;
ð
2
Þ
where
e
is the electron charge,
Ω
is the unit cell volume,
M
κ
and
τ
κ
are the mass and position of the atom with index
κ
,
G
are reciprocal lattice vectors,
e
ð
κ
Þ
ν
q
is the phonon eigenvector
projected on atom
κ
, and
ε
is the dielectric tensor of the
material. Summation over the Cartesian indices
α
,
β
,
γ
is
implied.
The dynamical quadrupoles
Q
κ
entering Eq.
(2)
are third-
rank tensors defined as the second order term in the long-
wavelength expansion of the cell-integrated charge-density
response to a monochromatic displacement
[33
35]
.
Here, they are computed by symmetrizing the first-order-
in-
q
polarization response
[34,36]
:
Q
αγ
κ
;
β
¼
i
Ω

̄
P
q
α
;
κβ
q
γ




q
¼
0
þ
̄
P
q
γ
;
κβ
q
α




q
¼
0

.
ð
3
Þ
With the full first-principles
e
-ph matrix elements in
Eq.
(1)
in hand, we compute the phonon-limited mobility at
temperature
T
using the BTE in both the relaxation time
approximation (RTA) and with a full iterative solution
[17]
.
Briefly, we compute the
e
-ph scattering rates (and their
inverse, the relaxation times
τ
n
k
), from the lowest order
e
-ph self-energy
[32]
. The mobility is then obtained as the
energy integral
[17]
:
μ
αβ
ð
T
Þ¼
e
n
c
Ω
N
k
Z
dE

f
E

X
n
k
F
α
n
k
ð
T
Þ
v
β
n
k
δ
ð
E
ε
n
k
Þ
;
ð
4
Þ
where
n
c
is the carrier concentration,
f
is the Fermi-Dirac
distribution,
N
k
is the number of
k
points, and
v
n
k
and
ε
n
k
are electron band velocities and energies, respectively. The
F
α
n
k
term is obtained as
τ
n
k
ð
T
Þ
v
α
n
k
in the RTA or by solving
the BTE iteratively.
We carry out
ab initio
calculations on wurtzite GaN with
relaxed lattice parameters, using the same settings as in our
recent work
[37]
. The ground state properties and electronic
wave functions are computed using DFT in the generalized
gradient approximation
[38]
with the
QUANTUM ESPRESSO
code
[39,40]
. We include spin-orbit coupling by employing
fully relativistic norm-conserving pseudopotentials
[41,42]
(generated with
PSEUDO DOJO
[43]
) and correct the DFT
band structure using
GW
results. We use DFPT
[9]
to
compute phonon frequencies and eigenvectors, and obtain
the
e
-ph matrix elements
g
nm
ν
ð
k
;
q
Þ
on coarse
8
×
8
×
8
k
-point and
q
-point grids
[44]
using our
PERTURBO
code
[17]
. We obtain the dynamical quadrupoles
Q
κ
by computing
̄
P
q
α
;
κβ
in Eq.
(3)
with the methodology
in Ref.
[36]
as implemented in the
ABINIT
code
[45]
.
PHYSICAL REVIEW LETTERS
125,
136602 (2020)
136602-2
All efforts were made to maintain consistency with the DFT
and DFPT calculations and the results were validated
against DFPT clamped-ion PE constants (see the
Supplemental Material
[46]
). We subtract the long-range
terms
g
dipole
þ
g
quad
, interpolate the
e
-ph matrix elements
using Wannier functions
[48]
to fine
k
- and
q
-point grids,
and then add back the long-range terms. The resulting
matrix elements are employed to compute relaxation times
and mobilities
[46]
with the
PERTURBO
code
[17]
.
Since DFPT can accurately capture the long-range
e
-ph
interactions, it can be used as a benchmark for our approach
of including the dipole plus quadrupole terms after inter-
polation. Note that due to computational cost, DFPT
calculations cannot be carried out on the fine grids needed
to compute electrical transport, so the long-range terms
need to be added after interpolation. Following Ref.
[27]
,
we define a gauge-invariant
e
-ph coupling strength,
D
ν
tot
,
proportional to the absolute value of the
e
-ph matrix
elements:
D
ν
tot
ð
q
Þ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ω
ν
q
M
uc
X
mn
j
g
mn
ν
ð
k
¼
Γ
;
q
Þj
2
=N
b
r
;
ð
5
Þ
with
M
uc
the unit cell mass and
N
b
the number of bands.
We compute
D
ν
tot
ð
q
Þ
with various approximations to
analyze the role of the quadrupole
e
-ph interactions.
In Fig.
1
, we use
D
ν
tot
ð
q
Þ
obtained from direct DFPT
calculations of the matrix elements as a benchmark, and
compare calculations that include only the long-range
Fröhlich dipole interaction and both the dipole and quadru-
pole interactions. The short-range
e
-ph interactions are
included in both cases as a result of the Wannier inter-
polation. Including the quadrupole term dramatically
improves the accuracy of the
e
-ph matrix elements for
the longitudinal acoustic (LA) and transverse acoustic (TA)
modes at small
q
(near
Γ
in Fig.
1
). The discrepancy
between the dipole-only calculation and DFPT is com-
pletely corrected when including the quadrupole term,
which reproduces the DFPT benchmark exactly. While
the dipole-only scheme leads to large errors, the dipole and
quadrupole contributions, as can be seen, cancel each
other out since they are nearly equal and opposite for
acoustic modes in GaN. If the quadrupole correction is not
included, in principle the interpolation could be improved
at small
q
by using a denser
q
-point grid when computing
the DFPT matrix elements. However, due to the non-
analyticity of the quadrupole term, prohibitively dense
DFPT grids are required to approach the accuracy of the
quadrupole term
[25]
.
Both the dipole and quadrupole terms contribute to the
PE
e
-ph interaction from the LA and TA acoustic modes
[6]
. Expanding the phonon eigenvectors at
q
0
as
e
ν
q
e
ð
0
Þ
ν
q
þ
iq
e
ð
1
Þ
ν
q
[6]
, one finds two PE contributions of order
q
0
for
q
0
[6]
. One is from the Born charges,
Z
κ
ð
iq
e
ð
1
Þ
ν
q
Þ
,
and is a dipolelike interaction generated by atoms with a net
charge experiencing different displacements due to an
acoustic mode. The other is from the dynamical quadru-
poles,
Q
κ
e
ð
0
Þ
ν
q
, and is associated with a clamped-ion
electronic polarization
[49]
. As a result, the
ab initio
Fröhlich term includes only part of the PE
e
-ph interaction,
so the dipole-only scheme fails in GaN because it neglects
the all-important quadrupole electronic contribution. We
also implemented and tested Mahan
s phenomenological
PE coupling
[5]
,
̃
g
PE
ν
ð
q
Þ¼
4
π
e
2
4
πε
0

2
ω
ν
q
M
uc

1
=
2
q
α
e
α
;
βγ
e
ν
q
;
β
q
γ
q
α
ε
αγ
q
γ
;
ð
6
Þ
where
e
α
;
βγ
are the PE constants of GaN
[46]
. In the
q
0
limit, this model includes both ionic-motion and electronic
effects
[6]
, and is a less computationally demanding
alternative that does not require computing the dynamical
quadrupoles. We find that the
e
-ph coupling
D
ν
tot
ð
q
Þ
obtained from the Mahan model improves over the
dipole-only scheme
[46]
, but exhibits discrepancies with
direct DFPT calculations at finite
q
, and overall is inad-
equate for quantitative calculations.
Figure
2
shows the effect of using the more accurate
quadrupole scheme on the
e
-ph scattering rates (defined as
the inverse of the
e
-ph relaxation times,
Γ
n
k
¼
1
=
τ
n
k
)
computed at 300 K. We focus on the energy range of
interest for charge transport near 300 K, namely an energy
window within
100
meV of the band edges. Since these
energies are below the LO phonon emission threshold
FIG. 1. Mode-resolved
e
-ph coupling strength,
D
ν
tot
ð
q
Þ
,
computed using the two lowest conduction bands; the initial
state is fixed at
Γ
and
q
is varied along high-symmetry lines. The
D
ν
tot
ð
q
Þ
data computed with
e
-ph matrix elements from DFPT
(black circles) is used as a benchmark, and compared with
Wannier interpolation plus Fröhlich interaction (blue) or plus
Fröhlich and quadrupole interactions (orange).
PHYSICAL REVIEW LETTERS
125,
136602 (2020)
136602-3
(90 meV in GaN), LO scattering is suppressed and
dominated by thermally activated LO phonon absorption
processes. On the other hand, there is a large phase space
for acoustic phonon scattering, especially for interband
processes in the valence band. For electrons, including the
quadrupole term greatly suppresses the acoustic mode
contribution to the scattering rates, reducing it from nearly
half of the total scattering rate to a negligible contribution
(Fig.
2
). This result is due to the cancellation of the dipole
and quadrupole terms for acoustic phonons discussed
above. As expected, the LO contribution becomes domi-
nant for electrons in the conduction band, where small-
q
intravalley scattering is controlled by the Fröhlich inter-
action with 1/q behavior rather than by the PE dipole and
quadrupole terms, both of which have a
q
0
trend at small
q
.
Using correct
e
-ph matrix elements that include the quadru-
pole term, our calculation restores this physical intuition.
A similar but less pronounced trend is found for holes
in Fig.
2
, where at the peak of the mobility integrand
(
50
meV below the valence band edge) acoustic phonon
scattering is suppressed from 75% of the total scattering
rate when only the Fröhlich interaction is included to less
than 50% when including the quadrupole term. The effect is
less pronounced for holes due to the greater proportion of
larger-
q
interband scattering channels in the valence band,
which are less affected by the quadrupole interaction (see
Fig.
1
). Neglecting the quadrupole term in GaN may
partially account for the large acoustic phonon scattering
for holes found in recent calculations
[29,30]
.
Analysis of the temperature dependent mobility in GaN,
computed using Eq.
(4)
with both the RTA and iterative
BTE approaches, highlights the key role of the quadrupole
e
-ph interaction. Figure
3
shows the electron and hole
mobilities in the basal [1000] plane of GaN, computed both
using Wannier interpolation plus the Fröhlich interaction
and with our improved scheme including the quadrupole
term. Experimental mobility measurements
[50
56]
are
also given for comparison. Compared to calculations that
include only the dipole Fröhlich interaction, including the
quadrupole term removes the artificial overestimation of
acoustic phonon scattering, thus increasing the computed
mobility and correcting its temperature dependence, espe-
cially at lower temperatures, where acoustic scattering is
dominant. For completeness, we also compare the electron
mobility computed using
e
-ph matrix elements from
Mahan
s model [see Eq.
(6)
], and find that it underestimates
the
ab initio
quadrupole result by up to 20% at 200 K
and notably changes the temperature dependence of the
mobility (see the Supplemental Material
[46]
). The error
FIG. 2. Electron-phonon scattering rates at 300 K. We compare
calculations including the long-range Fröhlich interaction only
(top) with results including the Fröhlich and quadrupole inter-
actions (bottom). The short-range
e
-ph interactions are included
in both cases as a result of the interpolation. We plot the total
scattering rate (blue) as well as the contributions from the LO
(red) and acoustic (orange) modes for holes (left) and electrons
(right) as a function of carrier energy within 150 meVof the band
edges. The gray shading represents the energies at which carriers
contribute to the mobility, as given by the integrand in Eq.
(4)
.
The hole and electron energy zeros are the valence and
conduction band edges, respectively.
FIG. 3. Hole (top) and electron (bottom) mobilities in wurtzite
GaN, computed in the [1000] plane. We show our computed
results obtained with the RTA (solid lines) and iterative BTE
(dashed lines), both with the long-range Fröhlich interaction only
(blue) and with dipole plus quadrupole interactions (orange).
Experimental results from Refs.
[50
53]
for electrons and
Refs.
[54
56]
for holes are shown for comparison.
PHYSICAL REVIEW LETTERS
125,
136602 (2020)
136602-4
decreases at higher temperatures because the two
methods differ primarily for small-
q
acoustic scattering,
which becomes progressively less important at higher
temperatures.
We find good agreement between our computed electron
and hole mobilities and experimental results, especially
when comparing with the highest mobilities measured in
samples with low doping concentrations (
10
15
cm
3
for
n
-type
[50]
and
10
16
10
17
cm
3
for
p
-type
[54,57]
GaN).
In these high purity samples, charge transport is governed
by phonon scattering in the temperature range we investi-
gate, so these measurements are ideal for comparing with
our phonon-limited mobilities.
We focus on the iterative BTE results, whose accuracy is
superior to the RTA, also given for completeness
[58]
.For
holes, in which acoustic scattering is significant, the
temperature dependence of the mobility is improved after
including the quadrupole interaction, as shown in the upper
panel of Fig.
3
. The exponent
n
in the temperature
dependence of the mobility,
μ
T
n
,is
n
¼
2
.
3
after
including the quadrupole term, the same value as the
exponent obtained by fitting the experimental data (for
comparison,
n
¼
2
.
0
in the dipole-only calculation). The
temperature dependence of the electron mobility is only in
reasonable agreement with experiment. Including two-
phonon processes may be needed to further improve the
temperature trend
[59]
.
Our improved scheme increases the mobility values,
placing them slightly above the experimental values. This
trend is physically correct
our computed mobilities for an
ideal crystal are an upper bound to the experimental
mobilities. Other possible mechanisms not included in
our calculations, such as the two-phonon scattering proc-
esses
[59]
, are expected to lower the mobility and bring it in
greater agreement with experiment.
In summary, we have presented a framework for
computing the quadrupole
e
-ph interaction and including
it in
ab initio
e
-ph calculations. Our results show its crucial
contribution to acoustic phonon and PE scattering. Our
work enables accurate calculations of long-range acoustic
phonon interactions and paves the way to studies of charge
transport in PE materials.
V. J. thanks the Resnick Sustainability Institute at
Caltech for fellowship support. J. P. acknowledges support
by the Korea Foundation for Advanced Studies. This work
was supported by the National Science Foundation under
Grants No. DMR-1750613 for theory development and
No. ACI-1642443 for code development. J.-J. Z. acknowl-
edges partial support from the Joint Center for Artificial
Photosynthesis, a DOE Energy Innovation Hub, as follows:
the development of some computational methods employed
in this work was supported through the Office of Science
of the U.S. Department of Energy under Award No.
DE-SC0004993. C. E. D. acknowledges support from
the National Science Foundation under Grant No.
DMR-1918455. The Flatiron Institute is a division of the
Simons Foundation. This research used resources of the
National Energy Research Scientific Computing Center, a
DOE Office of Science User Facility supported by the
Office of Science of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231.
Note added.
Recently, we became aware of a related
work by another group that reaches similar conclusions
about the importance of the dynamical quadrupole term
to obtain an accurate physical description of
e
-ph inter-
actions
[25,26]
.
*
V. J. and J.-J. Z. contributed equally to this work.
Corresponding author.
bmarco@caltech.edu
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