of 5
PHYSICAL REVIEW B
108
, 165202 (2023)
Charge transport in BAs and the role of two-phonon scattering
Iretomiwa Esho
and Austin J. Minnich
*
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
(Received 9 May 2023; revised 11 August 2023; accepted 12 September 2023; published 17 October 2023)
The semiconductor BAs has drawn significant interest due to experimental reports of simultaneous high
thermal conductivity and ambipolar charge mobility. The
ab initio
prediction of high electron and hole mobility
assumed the dominance of charge carrier scattering by one phonon. Recently, higher-order electron-phonon
scattering processes in polar and nonpolar semiconductors have been reported to have a non-negligible impact
on charge transport properties, suggesting they may play a role in BAs as well. Here, we report an
ab initio
study of two-phonon electron and hole scattering processes in BAs. We find that inclusion of these higher-order
processes reduces the computed room-temperature electron and hole mobility in BAs by around 40% from the
one-phonon value, resulting in an underestimate of experimental values by a similar percentage. We suggest
an experimental approach to test these predictions using luminescence spectroscopy that is applicable to the
defective samples which are presently available.
DOI:
10.1103/PhysRevB.108.165202
I. INTRODUCTION
BAs is a semiconductor of substantial recent interest begin-
ning from the
ab initio
prediction of high thermal conductivity
comparable to that of diamond [
1
] owing in part to the high
optical phonon energy (
80 meV) which inhibits phonon
scattering. The prediction of high optical phonon energy was
initially confirmed by inelastic x-ray scattering [
2
], but reports
of the thermal conductivity values were significantly lower
than the predictions due to scattering by As vacancies [
3
].
Theoretically, four-phonon processes were found to make a
non-negligible contribution to phonon scattering, yielding a
lower thermal conductivity compared with the original pre-
dictions [
4
]. After improvements in synthesis resulting in
higher-quality samples, the high thermal conductivity was
confirmed experimentally [
5
7
] and was found to be in quan-
titative agreement with predictions including four-phonon
scattering.
BAs has also been predicted to exhibit simultaneous
high electron and hole mobilities, with computed room-
temperature values of 1400 and 2110 cm
2
V
1
s
1
, respec-
tively [
8
,
9
]. However, initial experiments that estimated the
mobility from conductivity and thermoelectric measurements
and a single parabolic band model yielded a lower hole mobil-
ity of 400 cm
2
V
1
s
1
[
10
]; recent direct Hall measurements
yielded
500 cm
2
V
1
s
1
on bulk samples [
11
]. The lower
values obtained experimentally have been attributed to scat-
tering by charged impurities in the defective samples which
could be synthesized. Recent experiments have circumvented
the need for high-quality macroscopic samples by measur-
ing the ambipolar diffusivity of photoexcited carriers in a
local region of the sample using transient grating experi-
ments [
11
] or transient reflectivity microscopy [
12
]. Using
*
aminnich@caltech.edu
the Einstein relation to convert the measured diffusivity into
a mobility, these experiments obtained ambipolar mobilities
of 1500
1600 cm
2
V
1
s
1
at some locations on the sample.
These values are in good agreement with those calculated
from first principles [
8
].
Most first-principles studies of the electron-phonon in-
teractions employ the lowest level of perturbation theory
involving one electron and one phonon (1ph) [
13
,
14
], and
this level of theory was also used for BAs [
8
,
9
]. Given
the contribution of higher-order phonon processes to thermal
transport in BAs [
4
7
], it is of interest to consider the role of
higher-order processes in charge transport. Although evidence
for the contribution of multiphonon processes to electron-
phonon scattering has been previously reported [
15
17
], only
recently have first-principles studies included the contribu-
tion of higher-order scattering processes, such as that of an
electron with two phonons (2ph) in the electron-phonon in-
teraction [
18
20
]. In GaAs at room temperature, the 2ph
scattering rates were predicted to be on the order of the
1ph rates [
18
], resulting in an
40% reduction in the com-
puted mobility at 300 K. Good quantitative agreement with
experimental mobility was obtained only considering this
correction. Corrections to the high-field transport properties
of GaAs of a similar magnitude were also found [
19
]. For
nonpolar semiconductors, Hatanpää
et al.
reported improved
agreement of the warm electron coefficient in Si over temper-
atures from 190 to 310 K with the inclusion of 2ph processes
[
20
]. These studies suggest that inclusion of 2ph processes
for electron-phonon scattering may be necessary to accurately
predict the charge transport properties of semiconductors.
Here, we report an
ab initio
study of the role of two-phonon
scattering of electrons and holes on the charge transport
properties of BAs. We find that the two-phonon rates may
be as large as
50% of the one-phonon rates, leading to a
marked reduction in the calculated ambipolar mobility from
1420 to 810 cm
2
V
1
s
1
at room temperature and a 35–50%
2469-9950/2023/108(16)/165202(5)
165202-1
©2023 American Physical Society
IRETOMIWA ESHO AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
108
, 165202 (2023)
correction to the carrier mobility over temperatures from 150
to 350 K. The experimental origin of the discrepancy could
arise from the superdiffusion of hot carriers shortly after
photoexcitation, an effect which has been observed using
scanning ultrafast electron microscopy, leading to an over-
estimate of the ambipolar diffusivity. On the theory side, an
underestimate of the predicted value is possible owing to
cancellation between the iterated and direct contributions to
2ph scattering, the latter of which is neglected here. To test
our predictions given the defective samples presently avail-
able, we suggest an experimental approach based on direct
measurements of hot-carrier lifetimes using the broadening of
photoluminescence spectra.
II. COMPUTATIONAL METHODS
We computed the mobility of electrons and holes in BAs
using established methods based on density functional the-
ory (DFT) and density functional perturbation theory (DFPT)
[
14
,
21
23
]. Briefly, we obtained the electronic structure and
electron-phonon matrix elements using
QUANTUM ESPRESSO
[
24
] with a relaxed lattice constant of 4
.
819 Å, a coarse
12
×
12
×
12
k
grid, and a plane-wave cutoff of 80 Ry. A
fully relativistic ultrasoft potential with the Perdew-Burke-
Ernzerhof (PBE) exchange-correlation functional was used.
For the DFPT calculations, we employed a 6
×
6
×
6 phonon
q
grid. The band structure and electron-phonon matrix el-
ements were interpolated onto fine 160
3
and 80
3
k
and
q
grids, respectively, using
PERTURBO
[
25
]. Increasing the grid
density to 200
3
and 100
3
for the
k
and
q
grids, respectively,
changed the mobility by 2%. The Fermi level was chosen
so as to obtain a carrier concentration of 10
15
cm
3
at all
temperatures. The energy window was set to 200 meV above
(below) the band extremum for electrons (holes). Increasing
the energy window to 250 meV changed the mobility by 0.6%.
We explicitly constructed the collision matrix and solved the
Boltzmann transport equation using numerical linear algebra,
from which transport properties were calculated. Details of
this approach are given elsewhere [
19
21
]. The contributions
of the next-to-leading-order electron-phonon scattering (2ph)
processes originally derived in Ref. [
18
] were computed fol-
lowing the implementation used in Ref. [
19
]. The 2ph rates
were iterated five times. Increasing the number of iterations to
6 changed the mobility by 2.7%.
III. RESULTS
The calculated scattering rates for electrons and holes are
shown in Figs.
1(a)
and
1(b)
. The trend of the 1ph scattering
rates agrees with that reported previously [
8
]; quantitative
differences are due to the differing exchange-correlation func-
tional or pseudopotential necessitated by the use of
PERTURBO
in this paper. We observe the characteristic sharp increase
in the scattering rate for electrons and holes near ̄
h
ω
LO
80 meV as longitudinal optical (LO) phonon emission starts
to dominate the electron-phonon interaction. The 2ph rates
largely follow the same trend and are on the order of the 1ph
rates, consistent with previously published 2ph calculations
forGaAs[
18
,
19
] and Si [
20
]. At 300 K, the 2ph rates are
around 50% of the 1ph rates. Prior works have examined
FIG. 1. Scattering rates vs energy for (a) electrons and (b) holes
in BAs including 1ph (circles) and 2ph processes (triangles) at 300 K.
The computed 2ph rates for electrons and holes are around 50% of
the 1ph rates. Computed total 2ph (triangles), 1e1a (squares), 2a (di-
amonds), and 2e (circles) scattering rates vs energy for (c) electrons
and (d) holes show the subprocesses that comprise the total 2ph rates.
Below 150 meV, the 1e1a processes have the largest contribution to
the 2ph rates at 300 K.
the influence of the exchange-correlation functional on charge
carrier mobilities, finding variations on the order of
10–15%
in Si [
26
] and BAs [
8
]. Although this uncertainty may influ-
ence the predicted absolute mobility values, we expect the
relative contribution of 2ph processes compared with 1ph
processes to be insensitive to the choice of functional.
The 2ph processes exhibit several different subtypes be-
cause the two phonons involved in scattering can each be
emitted or absorbed. Following Ref. [
18
], processes where
a phonon is emitted and another absorbed are denoted 1e1a,
and processes where two phonons are sequentially emitted or
absorbed are 2e and 2a, respectively. The individual subpro-
cesses contributing to the total 2ph rate are shown in Figs.
1(c)
and
1(d)
for electrons and holes, respectively. Below ̄
h
ω
LO
80 meV, 1e1a processes are dominant. Note that the total
1e1a rate includes processes where a phonon is first emitted
and another absorbed, and processes where a phonon is first
absorbed and another is subsequently emitted. Two-phonon
emission (2e) processes are comparatively weak in this re-
gion since LO phonon emission is prohibited until the energy
threshold of 2 ̄
h
ω
LO
. Two-phonon absorption (2a) processes
are generally weak throughout the energy range studied, ex-
cept at sufficiently low energies where emission and therefore
1e1a events become increasingly unlikely such that 2a rates
are comparable to 1e1a rates. Between ̄
h
ω
LO
80 meV and
2 ̄
h
ω
LO
, the 1e1a and 2e rates increase as LO phonon emission
starts to dominate the electron-phonon scattering processes,
a feature observed in polar semiconductors [
27
]. Beyond
2 ̄
h
ω
LO
, carriers are energetic enough to emit two LO phonons,
165202-2
CHARGE TRANSPORT IN BAS AND THE ROLE ...
PHYSICAL REVIEW B
108
, 165202 (2023)
FIG. 2. (a) Electron and (b) hole mobility in BAs vs temperature
at the 1ph (dashed line) and (1
+
2)ph (solid line) level of theory.
For holes, the correction to the mobility at room temperature from
including 2ph processes is
37%, while for electrons this correction
is
43%, demonstrating the significant contribution of 2ph processes
to the mobility at room temperature.
and 2e processes have the largest contribution to the total
2ph scattering rate. This energy dependence of the individual
2ph subprocesses in BAs is consistent with those reported for
GaAs and Si [
18
20
].
We next examine the effect of 2ph processes on the
electron and hole mobility. The computed 1ph and (1
+
2)ph
mobility versus temperature is shown in Figs.
2(a)
and
2(b)
for electrons and holes, respectively. With only 1ph processes,
we obtain room-temperature electron mobility
μ
e
=
1066 cm
2
V
1
s
1
and hole mobility
μ
h
=
2000 cm
2
V
1
s
1
,
in quantitative agreement with previous 1ph predictions that
employ the same PBE exchange-correlation functional (see
Supplemental Material of Ref. [
8
] for calculations using the
same functional as in this paper). With the inclusion of 2ph
processes,
μ
e
and
μ
h
decrease to 600 and 1240 cm
2
V
1
s
1
,
respectively, corresponding to a 43 and 37% reduction at
room temperature. Over the temperature range from 150 to
350 K, this correction ranges from 36% at 350 K to 41% at
150 K for holes, and from 44% at 350 K to 46% at 150 K for
electrons. These corrections to the electron mobility are of a
comparable magnitude to those obtained for GaAs (
45%)
[
18
,
19
], but slightly higher than those for Si (
35%) [
20
].
BAs exhibits several distinct features compared with other
polar semiconductors such as GaAs. In GaAs and other polar
materials, LO phonons make the overwhelming contribution
to electron-phonon scattering [
27
]. In BAs, carrier scatter-
ing relevant to mobility is instead primarily due to acoustic
phonons owing to the high optical phonon energy (80 versus
35 meV in GaAs) that limits scattering by LO phonon emis-
sion as well as the decreased LO phonon absorption scattering
from decreased thermal population [
8
]. Additionally, in GaAs,
intervalley processes have a negligible effect on low-field
charge transport because of the

-
L
energy separation of
300 meV, but scattering processes in BAs are more similar to
those in Si in that they involve intervalley transfers mediated
by zone-edge wave vector phonons. Our calculations reveal
that intervalley processes account for 43% of (1
+
2)ph scat-
tering in BAs at 300 K and 20% at 150 K. The decrease with
decreasing temperature occurs due to the reduced population
of the zone-edge phonons required for intervalley scattering.
As a comparison, intervalley processes account for 61% of
(1
+
2)ph scattering in Si at 300 K and 25% at 150 K.
IV. DISCUSSION
We consider our calculated mobility values in the context
of recent optical experiments on BAs that reported an ambipo-
lar carrier mobility [
11
,
12
]. At the 1ph level of theory, we
predict a high ambipolar mobility
μ
a
=
2
μ
e
μ
h
/
(
μ
e
+
μ
h
)of
1420 cm
2
V
1
s
1
at 300 K using 1ph theory, consistent with
a prior computed value of 1570 cm
2
V
1
s
1
with the PBE
exchange-correlation functional [
8
] and in agreement with
recent experimental reports [
11
,
12
]. Including 2ph processes
reduces
μ
a
to 810 cm
2
V
1
s
1
, a 43% reduction. Considering
the (1
+
2)ph mobility value, the apparent agreement between
theory and experiment is substantially degraded, with the ex-
periment now overestimating the theory.
This discrepancy could arise from several factors. First,
the quantity that was measured in the optical experiments
of Refs. [
11
,
12
] was the ambipolar diffusion coefficient of
photoexcited charge carriers, from which the mobility was
obtained through the Einstein relation. In Refs. [
11
,
12
], the
photoexcitation wavelength for determination of the ambipo-
lar diffusion coefficient was chosen to be around the available
estimates of the band-gap energy (
2eV[
28
31
]). If the
photon energy exceeds the band-gap energy, the photoex-
cited carriers will have energy in excess of thermal energies,
potentially causing the extracted transport properties to dif-
fer from their linear response values. This hot-carrier effect
was observed in both Ref. [
11
] and Ref. [
12
]asalarger
measured electronic diffusivity for pump wavelengths

500
nm. Evidence for the absence of the hot-carrier effect for the
final reported diffusivity values was presented, for example,
in Fig. 1 D of Ref. [
11
], as the plateau of the measured
electronic decay rate with increasing wavelength. On the other
hand, scanning ultrafast electron microscopy (SUEM) studies
have reported observations of superdiffusion of photoexcited
carriers in semiconductors persisting over hundreds of pi-
coseconds [
32
34
]. This phenomenon has been attributed to
the additional contribution to carrier diffusion of a pressure
gradient in the nondegenerate hot-carrier gas after photoex-
citation [
32
]. In Refs. [
11
,
12
], the diffusivity was extracted
from the electronic decay curve over timescales from tens to
hundreds of picoseconds, conceivably leading to an extracted
diffusivity that was influenced by superdiffusion.
On the theory side, a possible cause of an underestimate
for the computed mobility is the cancellation of the two
contributions to electron-phonon scattering at second order.
Electron-2ph processes arise from the 1ph term, correspond-
ing to the first derivative of the interatomic potential with
respect to lattice displacements taken to second order in
perturbation theory, or a direct 2ph term involving the simul-
taneous interaction of an electron with two phonons with a
strength given by second-order derivative of the interatomic
potential [
35
,
36
]. These two terms exhibit a nontrivial interac-
tion owing to a cancellation in the long-wavelength acoustic
phonon limit which arises from translational invariance of the
crystal [
37
]. In this paper and other recent
ab initio
studies
of 2ph scattering, only the first term is included, and thus
neglect of the second term will lead to an overestimate of 2ph
165202-3
IRETOMIWA ESHO AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
108
, 165202 (2023)
scattering rate. This cancellation has long complicated the
study of 2ph scattering in semiconductors [
36
,
38
]. A recent
study of 2ph scattering in Si suggested that the correction
could be on the order of 10–20% in that material [
20
]. It
is possible that this effect could lead to an underestimate of
the computed mobility in BAs; further study is needed to
investigate this hypothesis.
Absent higher-quality samples, verifying the prediction
of the role of 2ph scattering using transport measurements
is challenging due to the contribution of extrinsic defect
scattering. We suggest an alternative approach based on
continuous-wave luminescence spectroscopy which allows
the lifetimes of electronic states away from the band minimum
to be determined [
39
]. These states are less influenced by
impurity scattering compared with those near the band edge.
While the contribution of these higher-energy states to car-
rier mobility is negligible, the contribution of 2ph processes
to the total scattering rate is largely independent of energy,
as shown in Fig.
1
. Therefore evidence of the influence of
2ph scattering on mobility can be obtained by comparing the
photoluminescence linewidths of these higher energy states
with theory. In these experiments, hot electrons excited by
a continuous-wave laser emit photons by recombination, and
the spectrum of the emitted light exhibits a broadening that
is determined by the lifetime of the state. We may predict
the difference in broadening at the 1ph and (1
+
2)ph levels
of theory in BAs using the same
ab initio
theory employed
for transport calculations. In Fig.
3
, we plot the predicted
full width at half maximum (FWHM) of the luminescence
peak, 2

=
τ
1
, versus energy for electrons. At 0.4 eV above
the conduction band minimum (CBM), we predict 2

13
and 21 meV for 1ph and (1
+
2)ph, respectively. This 8-meV
difference is almost an order of magnitude higher than the
experimental uncertainty reported in Ref. [
39
] and thus should
be discernible.
V. SUMMARY
In summary, we have reported
ab initio
calculations of
ambipolar mobility in BAs considering 2ph electron-phonon
processes. We find that the inclusion of these processes re-
duces the predicted electron and hole mobility by 43 and 37%
FIG. 3. Calculated broadening vs energy for electrons due to
electron-phonon scattering at 77 K and carrier concentration of
10
15
cm
3
. The difference in broadening between 1ph theory (dashed
line) and (1
+
2)ph theory (solid line) is expected to be distin-
guishable considering prior reports of experimental uncertainties of
1meV[
39
].
at room temperature, respectively, lowering the ambipolar mo-
bility by 43% and underestimating experimental reports by a
similar amount. We hypothesize that the discrepancy between
our results and recent optical experiments could in part arise
from the superdiffusion of hot carriers, or an underestimation
of the calculated mobility owing to cancellations at second
order of perturbation theory. We have suggested an experi-
mental approach based on hot-electron luminescence to test
these predictions.
ACKNOWLEDGMENTS
I.E. was supported by a National Science Foundation Grad-
uate Research Fellowship under Grant No. DGE-1745301.
A.J.M. was supported by AFOSR under Grant No. FA9550-
19-1-0321. The authors thank Benjamin Hatanpää for useful
discussions and providing data on intervalley scattering in Si.
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