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, Steele Laboratory,
California Institute of Technology, Pasadena, California 91125, USA.
2
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800
3
Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010
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 non-centrosymmetric polar materials,
is not accurately described at present. Current calculations include short-range
e
-ph interactions
(obtained by interpolation) and the dipole-like Fr ̈ohlich 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 inves-
tigate
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.
When atoms move due to lattice vibrations, the po-
tential 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 sim-
plified models tailored to specific materials [1]. In met-
als, the so-called “deformation potential” quantifies the
short-range
e
-ph interactions with acoustic phonons [2].
In ionic and polar covalent materials, in which atoms can
be thought of as carrying a net charge, Fr ̈ohlich identified
a dipole-like long-range
e
-ph interaction due to longitudi-
nal optical (LO) phonons [3]. In polar materials lacking
inversion symmetry, the piezoelectric (PE)
e
-ph interac-
tion, due to strain induced by acoustic phonons, is also
important [4, 5]. Its conventional 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
originate from a multipole expansion [7] of the
e
-ph po-
tential, and analyzed its behavior in the long-wavelength
limit (phonon wave vector
q
→
0). The Fr ̈ohlich dipole
term diverges as 1
/q
in this limit, and is proportional
to the sum of atomic dipoles: If each atom
κ
is asso-
ciated 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 contri-
butions can be written as
Q
κ
e
ν
q
. Both the dipole and
the quadrupole terms contribute to the PE
e
-ph interac-
tion [6]. The multipole expansion also gives octopole 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 cal-
culations of
e
-ph interactions from first principles. In
turn, the
e
-ph interactions can be used in the Boltz-
mann transport equation (BTE) framework to predict
electron scattering processes and charge transport [10–
21]. The recent development of the
ab initio
Fr ̈ohlich
e
-ph interaction [22, 23] has been a first step toward im-
plementing 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 materi-
als, is sizable in nonpolar materials, and is particularly
important in PE materials such as wurtzite crystals or
titanates.
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 includ-
ing 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 er-
rors resulting from including only the Fr ̈ohlich term in
GaN [24], which greatly overestimates the acoustic mode
e
-ph interactions [24, 25]. Our companion paper applies
this framework to silicon and the PE material PbTiO
3
.
The
e
-ph quadrupole contribution is sizable in both cases,
and essential to correctly compute the
e
-ph matrix ele-
ments. Taken together, the quadrupole interaction com-
pletes the theory and enables accurate
ab initio
e
-ph cal-
culations in all materials and for all phonon modes.
arXiv:2002.08351v1 [cond-mat.mtrl-sci] 19 Feb 2020
2
The key ingredient in first-principles
e
-ph calculations
are the
e
-ph matrix elements,
g
mnν
(
k
,
q
), which en-
code the probability amplitude for scattering from an
initial Bloch state
|
n
k
〉
with band index
n
and crystal
momentum
k
into a final state
|
m
k
+
q
〉
by emitting
or absorbing a phonon with branch index
ν
and wave
vector
q
[17, 26]. Following Vogl [6], we separate the
long-range dipole and quadrupole contributions from the
short-range part:
g
mnν
(
k
,
q
) =
g
dipole
mnν
(
k
,
q
) +
g
quad
mnν
(
k
,
q
) +
g
S
mnν
(
k
,
q
) (1)
where
g
dipole
is the
ab initio
Fr ̈ohlich interaction writ-
ten in terms of Born effective charges [22, 23],
g
quad
is
the quadrupole interaction written in terms of dynam-
ical quadrupoles, and
g
S
collects octopole and higher-
order short-range terms.
For polar acoustic modes,
g
dipole
+
g
quad
is a generalized replacement for the phe-
nomenological PE
e
-ph coupling expressed in terms of
the PE constants [4, 5]. We derive the first-principles
quadrupole
e
-ph interaction
g
quad
mnν
(
k
,
q
) by superimpos-
ing two oppositely oriented dipole moments, as discussed
in detail in our companion paper, and obtain:
g
quad
mnν
(
k
,
q
) =
e
2
Ω
ε
0
∑
κ
(
~
2
ω
ν
q
M
κ
)
1
2
∑
G
6
=
−
q
1
2
(q
α
+ G
α
)(
Q
αγ
κ,β
e
(
κ
)
ν
q
,β
)(q
γ
+ G
γ
)
(q
α
+ G
α
)
αγ
(q
γ
+ G
γ
)
〈
m
k
+
q
|
e
i
(
q
+
G
)
·
(
r
−
τ
κ
)
|
n
k
〉
,
(2)
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 in-
dex
κ
,
G
are reciprocal lattice vectors,
e
(
κ
)
ν
q
is the phonon
eigenvector projected on atom
κ
, and
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 [27–
29]. Here, they are computed by symmetrizing the first-
order-in-
q
polarization response [28, 30]:
Q
αγ
κ,β
=
i
Ω
∂
P
q
α,κβ
∂q
γ
∣
∣
∣
∣
∣
q
=0
+
∂
P
q
γ,κβ
∂q
α
∣
∣
∣
∣
∣
q
=0
(3)
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 solu-
tion [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 [26]. The mobility is then
obtained as the energy integral [17]:
μ
αβ
(
T
) =
e
n
c
Ω
N
k
∫
d
E
(
−
∂f
∂E
)
∑
n
k
F
α
n
k
(
T
)
v
β
n
k
δ
(
E
−
ε
n
k
)
(4)
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
(
T
)
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 [31].
The ground state prop-
erties and electronic wave functions are computed us-
ing DFT in the generalized gradient approximation [32]
with the
Quantum ESPRESSO
code [33, 34]. We in-
clude spin-orbit coupling by employing fully relativis-
tic norm-conserving pseudopotentials [35, 36] (generated
with Pseudo Dojo [37]) and correct the DFT band struc-
ture using
GW
results. We use DFPT [9] to compute
phonon frequencies and eigenvectors, and obtain the
e
-ph
matrix elements
g
nmν
(
k
,
q
) on coarse 8
×
8
×
8
k
-point
and
q
-point grids [38] using our
Perturbo
code [17].
We obtain the dynamical quadrupoles
Q
κ
by comput-
ing
P
q
α,κβ
in Eq. (3) with the methodology in Ref. [30]
as implemented in the
ABINIT
code [39], and validate
the results against DFPT clamped-ion PE constants (see
Supplemental Material [40]). We subtract the long-range
terms
g
dipole
+
g
quad
, interpolate the
e
-ph matrix elements
using Wannier functions [41] to fine
k
- and
q
-point grids,
and then add back the long-range terms. The result-
ing matrix elements are employed to compute relaxation
times and mobilities [40] 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 interpolation. 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. Follow-
ing Ref. [22], 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
(
q
) =
~
−
1
√
2
ω
ν
q
M
uc
∑
mn
|
g
mnν
(
k
= Γ
,
q
)
|
2
/N
b
,
(5)
with
M
uc
the unit cell mass and
N
b
the number of bands.
We compute
D
ν
tot
(
q
) with various approximations to an-
alyze the role of the quadrupole
e
-ph interactions.
3
FIG. 1.
Mode-resolved
e
-ph coupling strength,
D
ν
tot
(
q
),
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
(
q
) data computed with
e
-ph matrix elements from
DFPT (black circles) is used as a benchmark, and compared
with Wannier interpolation plus Fr ̈ohlich interaction (blue) or
plus Fr ̈ohlich and quadrupole interactions (orange).
In Fig. 1, we use
D
ν
tot
(
q
) obtained from direct DFPT
calculations of the matrix elements as a benchmark,
and compare calculations that include only the long-
range Fr ̈ohlich dipole interaction and both the dipole and
quadrupole interactions. The short-range
e
-ph interac-
tions are included in both cases as a result of the Wannier
interpolation. Including the quadrupole term dramati-
cally improves the accuracy of the
e
-ph matrix elements
for the longitudinal acoustic (LA) and transverse acous-
tic (TA) modes at small
q
(near Γ in Fig. 1). The dis-
crepancy between the dipole-only calculation and DFPT
is completely 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, can-
cel each other out since they are nearly equal and oppo-
site for acoustic modes in GaN.
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
(0)
ν
q
+
iq
e
(1)
ν
q
[6], one finds two PE contributions
of order
q
0
for
q
→
0 [6]. One is from the Born charges,
Z
κ
(
iq
e
(1)
ν
q
), and is a dipole-like interaction generated by
atoms with a net charge experiencing different displace-
ments due to an acoustic mode. The other is from the
dynamical quadrupoles,
Q
κ
e
(0)
ν
q
, and is associated with a
clamped-ion electronic polarization [42]. As a result, the
ab initio
Fr ̈ohlich term includes only part of the PE
e
-ph
interaction, so the dipole-only scheme fails in GaN be-
cause it neglects the all-important quadrupole electronic
contribution. We also implemented and tested Mahan’s
phenomenological PE coupling [5],
̃
g
PE
ν
(
q
) = 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
αγ
q
γ
(6)
where
e
α,βγ
are the PE constants of GaN [40]. In the
q
→
0 limit, this model includes both ionic-motion and
electronic effects [6], and is a less computationally de-
manding alternative that does not require computing the
dynamical quadrupoles. We find that the
e
-ph coupling
D
ν
tot
(
q
) obtained from the Mahan model improves over
the dipole-only scheme [40], but exhibits discrepancies
with direct DFPT calculations at finite
q
, and overall is
inadequate 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
= 1
/τ
n
k
)
computed at 300 K. We focus on the energy range of
interest for charge transport near 300 K, namely an en-
ergy window within
∼
100 meV of the band edges. Since
these energies are below the LO phonon emission thresh-
old (90 meV in GaN), LO scattering is suppressed and
FIG. 2. Electron-phonon scattering rates at 300 K. We com-
pare calculations including the long-range Fr ̈ohlich interaction
only (top) with results including the Fr ̈ohlich and quadrupole
interactions (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
meV of the band edges. The gray shading represents the en-
ergies 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.
4
FIG. 3.
Electron (top) and hole (bottom) mobilities in
wurtzite GaN, computed in the [1000] plane. We show our
computed results obtained with the RTA (solid lines) and it-
erative BTE (dashed lines), both with the long-range Fr ̈ohlich
interaction only (blue) and with dipole plus quadrupole inter-
actions (orange). Experimental results from Refs. [43–46] for
electrons and Refs. [47–49] for holes are shown for comparison.
dominated by thermally activated LO phonon absorp-
tion processes. On the other hand, there is a large phase
space for acoustic phonon scattering, especially for inter-
band processes in the valence band. For electrons, includ-
ing 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 negligi-
ble contribution (Fig. 2). This result is due to the can-
cellation of the dipole and quadrupole terms for acoustic
phonons discussed above. As expected, the LO contri-
bution becomes dominant for electrons in the conduction
band, where small-
q
intravalley scattering is controlled
by the Fr ̈ohlich interaction 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 quadrupole term, our calcula-
tion 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 Fr ̈ohlich interaction is included to
less than 50% when including the quadrupole term. The
large acoustic phonon scattering for holes found in re-
cent calculations [24] is partially due to the fact that the
quadrupole term was missing. The possibility of increas-
ing hole mobility by engineering lower acoustic scattering
based on these investigations [25] clearly does not prop-
erly take into account PE scattering (and PE charges
induced by straining GaN), and should be revisited.
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, com-
puted both using Wannier interpolation plus the Fr ̈ohlich
interaction and with our improved scheme including
the quadrupole term. Experimental mobility measure-
ments [43–49] are also given for comparison. Compared
to calculations that include only the dipole Fr ̈ohlich in-
teraction, including the quadrupole term removes the ar-
tificial overestimation of acoustic phonon scattering, thus
increasing the computed mobility and correcting its tem-
perature dependence, especially at lower temperatures,
where acoustic scattering is dominant.
We find good agreement between our computed elec-
tron and hole mobilities and experimental results, es-
pecially when comparing with the highest mobilities
measured in samples with low doping concentrations
(
∼
10
15
cm
−
3
for n-type [43] and
∼
10
16
−
10
17
cm
−
3
for p-
type [47, 50] GaN). In these high purity samples, charge
transport is governed by phonon scattering in the tem-
perature range we investigate, 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 [51].
For holes, in which acoustic scattering is significant, the
temperature dependence of the mobility is improved af-
ter including the quadrupole interaction, as shown in the
upper panel of Fig. 3. The exponent
n
in the temper-
ature dependence of the mobility,
μ
∼
T
−
n
, is
n
= 2
.
3
after including the quadrupole term, the same value as
the exponent obtained by fitting the experimental data
(for comparison,
n
= 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 correctly predict
the temperature trend, as we found recently for electrons
in GaAs [52], which similar to GaN has dominant LO
phonon scattering in the conduction band.
Our improved scheme increases the computed mobili-
ties, placing them slightly above the experimental values.
This trends is physically correct
−
our computed mobil-
ities are an upper bound as experimental samples may
exhibit additional scattering from defects and interfaces.
In addition, a slight mobility overestimation is expected
in polar materials because including two-phonon scatter-
ing processes would lower the mobility and bring it closer
to experiment, as we have recently shown in GaAs [52].
5
In summary, we have presented a framework for com-
puting 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 acous-
tic phonon interactions and paves the way to studies of
charge transport in PE materials.
V.J. thanks the Resnick Sustainability Institute at Cal-
tech 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. CAREER-1750613 for theory devel-
opment and ACI-1642443 for code development. J.-J.Z.
acknowledges 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.
−
While writing this manuscript, we be-
came aware of a related preprint by another group [53].
Their article analyzes how the quadrupole term improves
e
-ph matrix element interpolation, while ours focuses
more broadly on the physics of
e
-ph interactions, acoustic
phonons, and piezoelectric materials.
∗
Corresponding author: bmarco@caltech.edu
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