#  First-principles study of electron transport in ScN

Sai Mu  Materials Department, University of California, Santa Barbara,
California, 93106-5050, USA  Andrew J. E. Rowberg  Materials Department,
University of California, Santa Barbara, California, 93106-5050, USA  Joshua
Leveillee  Oden Institute for Computational Engineering and Sciences, The
University of Texas at Austin, Austin, Texas 78712, USA  Department of
Physics, The University of Texas at Austin, Austin, Texas 78712, USA
Feliciano Giustino  Oden Institute for Computational Engineering and Sciences,
The University of Texas at Austin, Austin, Texas 78712, USA  Department of
Physics, The University of Texas at Austin, Austin, Texas 78712, USA  Chris G.
Van de Walle  Materials Department, University of California, Santa Barbara,
California, 93106-5050, USA

###### Abstract

We investigate the conduction-band structure and electron mobility in rocksalt
ScN based on density functional theory. The first-principles band structure
allows us to obtain band velocities and effective masses as a function of
energy. Electron-phonon scattering is assessed by explicitly computing the
$q$-dependent electron-phonon matrix elements, with the inclusion of the long-
range electrostatic interaction. The influence of free-carrier screening on
the electron transport is assessed using the random phase approximation. We
find a notable enhancement of electron mobility when the carrier concentration
exceeds 1020 cm-3. We calculate the room-temperature electron mobility in ScN
to be 587 cm2/Vs at low carrier concentrations. When the carrier concentration
is increased, the electron mobility starts to decrease significantly around
$n=10^{19}$ cm-3, and drops to 240 cm2/Vs at $n=10^{21}$ cm-3. We also explore
the influence of strain in (111)- and (100)-oriented ScN films. For (111)
films, we find that a 1.0% compressive epitaxial strain increases the in-plane
mobility by 72 cm2/Vs and the out-of-plane mobility by 50 cm2/Vs. For (100)
films, a 1.0% compressive epitaxial strain increases the out-of-plane mobility
by as much as 172 cm2/Vs, but has a weak impact on the in-plane mobility. Our
study sheds light on electron transport in ScN at different electron
concentrations and shows how strain engineering could increase the electron
mobility.

##  I Introduction

ScN is a semiconducting transition-metal nitride that has received significant
attention due to the demonstrated synthesis of high-quality thin films Oshima
_et al._ (2014); Ohgaki  _et al._ (2013); Saha  _et al._ (2013) and their
potential applications in thermoelectric Zide  _et al._ (2005); Saha  _et al._
(2012); Kerdsongpanya  _et al._ (2017) and electronic Biswas and Saha (2019)
devices. The material exhibits high mobility and carrier concentrations Al-
Atabi  _et al._ (2020); Deng  _et al._ (2015) and ambipolar dopability Nayak
_et al._ (2019). Crystallizing in the rocksalt structure, ScN can be
integrated with several technologically important III-nitride semiconductors,
leading to reduced dislocation density with the use of a GaN epilayer Johnston
_et al._ (2009), and giant interfacial carrier densities in ScN/GaN
heterostructures Adamski  _et al._ (2019).

The electronic structure of ScN has historically been a topic of debate Biswas
and Saha (2019). Early reports suggested that it might be a semimetal Neckel
_et al._ (1975); Monnier  _et al._ (1985), with the conduction-band minimum
(CBM) at X lower in energy than the valence-band maximum (VBM) at $\Gamma$,
but it is now understood that ScN is an indirect-band gap semiconductor.
Experimental values for the fundamental band gap have been reported ranging
from 0.9 to 1.3 eV Al-Brithen  _et al._ (2004); Gall  _et al._ (2001a); Saha
_et al._ (2013); Deng  _et al._ (2015), while computational studies have
produced values between 0.8 and 1.1 eV Saha  _et al._ (2017); Qteish  _et al._
(2006); Stampfl  _et al._ (2001); Lambrecht (2000). A more rigorous study of
the electronic structure is still lacking, particularly relating to how band
curvature, which governs velocities and effective masses, changes with carrier
energy.

A high carrier mobility is desirable for many of ScN’s applications. In
thermoelectric energy conversion devices, ScN’s high mobility contributes to a
higher figure of merit Kerdsongpanya  _et al._ (2017); Burmistrova  _et al._
(2013). Similarly, ScN integration with GaN-based electronic or optoelectronic
devices is made attractive by the high electron mobility Moram  _et al._
(2010); Johnston  _et al._ (2009). It is thus crucial to understand the
factors that impact the mobility, as well as how it varies as a function of
carrier concentration. It is possible to achieve $n$-type doping through the
use of ON or FN donors Kumagai  _et al._ (2018); Cetnar  _et al._ (2018).
Reported measurements of room-temperature electron mobility in ScN cover a
wide range, from 1 to 284 cm2/Vs Oshima  _et al._ (2014); Saha  _et al._
(2017, 2013); Dismukes  _et al._ (1972); Burmistrova  _et al._ (2013); Ohgaki
_et al._ (2013); Biswas and Saha (2019). The highest reported value, 284
cm2/Vs, was obtained at a carrier concentration of 3.7$\times$ 1018 cm-3 in
ScN films grown on sapphire by hydride vapor phase epitaxy Oshima  _et al._
(2014).

Accurate values for effective mass, as well as knowledge of how these masses
change with carrier concentration, are essential for understanding mobility. A
wide range of electron effective masses has been reported, with experimental
measurements for transport effective masses ranging between 0.1 and 0.4
$m_{0}$ Harbeke  _et al._ (1972); Deng  _et al._ (2015), and results from
computations ranging from 0.18 to 0.35 $m_{0}$ Qteish  _et al._ (2006); Deng
_et al._ (2015). These numbers reflect effective masses at the CBM; at high
carrier concentrations, the masses can change significantly.

In this paper we report the results of first-principles calculations based on
density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham
(1965) and density functional perturbation theory (DFPT) to obtain a
comprehensive picture of the electronic structure and electron transport in
ScN. We use a hybrid functional Heyd  _et al._ (2003, 2006) combined with
Wannier interpolation Pizzi  _et al._ (2020) to obtain an accurate description
of the band structure. This allows us to determine band velocities and
effective masses as a function of electron energy, and also to locate
inflection points. Our approach provides accurate values that take the
significant degree of anharmonicity of the electronic band structure into
account. We also use the Wannier-interpolated density of states (DOS) and
Fermi-Dirac statistics to determine the carrier concentrations for holes and
electrons as a function of Fermi level.

We also perform detailed calculations of the electron-phonon (el-ph)-limited
electron mobility. At room temperature, the electron mobility is limited
primarily by interactions with the longitudinal optical phonon mode that
generates a long-range electric field. Properly treating this long-range
interaction is crucial. We also include the effects of free-carrier screening
on electron mobility, by screening the el-ph matrix elements using the
Lindhard dielectric function calculated within the random phase approximation
(RPA) for the homogeneous electron gas Lindhard (1954); Hedin (1965). We
present a detailed analysis of scattering rates and contributions from
different phonon modes, and analyze the dependence of mobility on carrier
concentrations.

Additionally, we study the impact of strain on mobility. Thin films of ScN
have been grown on MgO Al-Brithen and Smith (2000); Saha  _et al._ (2017);
Ohgaki  _et al._ (2013); Burmistrova  _et al._ (2013), sapphire Dismukes  _et
al._ (1972); Oshima  _et al._ (2014), and Si Moram  _et al._ (2006)
substrates. Since the lattice mismatch is large, the resulting films are
likely relaxed, but residual strain may be present Saha  _et al._ (2017);
Moram  _et al._ (2006). A smaller lattice mismatch occurs for (111)-oriented
ScN grown on the $c$ plane of wurzite GaN ($+$0.2%) and AlN ($-$2.3%), making
pseudomorphic growth plausible. Since strain is likely to be present in ScN
films and is known to affect the electronic structure and carrier mobility of
semiconductors Poncé  _et al._ (2019), we will explore its impact on electron
transport.

The paper is organized as follows. Section II describes the computational
methodology and theoretical background. The main results are summarized in
Sec. III, including the electronic (Sec. III.1) and phonon band structure
(Sec. III.3) of bulk ScN, electron concentrations as a function of Fermi level
(Sec. III.2), electron-phonon coupling matrix elements (Sec. III.4), electron
scattering rates and their phonon-mode decomposed contributions (Sec. III.5),
room-temperature electron mobility at various carrier concentrations (Sec.
III.6), and the effect of screening (Sec. III.7). The strain dependence of
electron mobility at low carrier concentrations is investigated in Sec.
III.8). Section IV concludes the paper.

##  II Methodology

###  II.1 Computational Details

We conduct first-principles calculations based on DFT Hohenberg and Kohn
(1964); Kohn and Sham (1965) and the Heyd, Scuseria, and Ernzerhof (HSE) Heyd
_et al._ (2003, 2006) screened hybrid functional with 25% mixing of exact
exchange, as implemented in the Vienna Ab initio Simulation Package (VASP)
Kresse and Hafner (1993); Kresse and Furthmüller (1996). We use an energy
cutoff of 520 eV for the plane-wave basis set, and the core electrons are
described with projector-augmented-wave potentials Blöchl (1994); Kresse and
Joubert (1999) with the Sc $4s^{2}$ $3d^{1}$ and N $2s^{2}$ $2p^{3}$ electrons
treated as valence. We integrate over the Brillouin zone (BZ) using a 10
$\times$ 10 $\times$ 10 $k$-point grid. The HSE calculated equilibrium lattice
parameter is 4.48 Å, very close to the experimental value of 4.50 Å Travaglini
_et al._ (1986). To enable dense sampling of the conduction band, we use
Wannier interpolation based on the Wannier90 code Pizzi  _et al._ (2020) with
more than 2000 k-points sampled along each high-symmetry path in the BZ. To
assess the effect of strain in (111)- and (100)-oriented ScN films, we
constrain the in-plane lattice parameters and relaxed the out-of-plane lattice
parameter.

The electronic structure calculations preceding phonon and electron mobility
calculations are performed using the Quantum ESPRESSO software package
Giannozzi  _et al._ (2009, 2017). We therefore perform calculations using the
local density approximation (LDA) Perdew and Wang (1992). In Sec. III.1 we
demonstrate, by comparing with the HSE calculations, that the relevant parts
of the band structure are accurately reproduced with this approach. The LDA
calculations are performed with norm-conserving pseudopotentials Troullier and
Martins (1991) and the same valence electron configurations as specified
above. The lattice parameter is set to the experimental value of 4.50 Å
Travaglini  _et al._ (1986). We used a plane-wave cutoff of 72 Ry and a 24
$\times$ 24 $\times$ 24 $k$-point grid to integrate over the BZ.

The phonon band structure is calculated using DFPT Baroni  _et al._ (2001)
with a threshold of 10-14 Ry for self-consistency and using a coarse 6
$\times$ 6 $\times$ 6 $q$-point grid. A non-analytical correction Gonze  _et
al._ (1994) to the phonon bands has been included to capture the split in
frequencies between longitudinal and transverse optical modes. Based on the
lattice vibrational spectra and electronic wavefunctions, we calculate el-ph
coupling using the EPW code Poncé  _et al._ (2016). The el-ph matrix elements
are first calculated on a coarse grid of 12 $\times$ 12 $\times$ 12 $k$\- and
6 $\times$ 6 $\times$ 6 $q$-point grids, and then Wannier interpolation is
performed at denser $k$\- and $q$-point grids. To calculate the scattering
rates and mobility, we use 120 $\times$ 120 $\times$ 120 $k$\- and $q$-point
grids. We have confirmed that our calculated mobility changes by less than 1%
when the grid size is increased to 140 $\times$ 140 $\times$ 140\. The
Perturbo code Zhou  _et al._ (2021) was also employed for transport
calculations, yielding consistent results for the mobility.

###  II.2 Electron-Phonon Matrix Elements

ScN has two atoms in its primitive unit cell, each of which carries an
isotropic Born effective charge tensor. The atomic motion associated with a
longitudinal optical (LO) phonon in the long-wavelength limit
($\mathbf{q}\rightarrow 0$) generates a long-range electric field that couples
strongly to carriers Ziman (2001). This is known as the Fröhlich interaction
Fröhlich (1954). As a consequence, LO-phonon scattering is the major
scattering mechanism at finite temperatures, when LO phonon modes are
thermally excited.

In the Fröhlich model Fröhlich (1954), the el-ph (LO) coupling matrix
$g_{\mathbf{q}}$ is proportional to $1/\mathbf{q}$, diverging when
$\mathbf{q}\rightarrow 0$. This singularity requires special numerical
treatment in calculations for the el-ph interaction. We have adopted the
approach of Verdi _et al._ Verdi and Giustino (2015) to calculate the el-ph
matrix elements by separating the short-range and long-range contributions to
the coupling strength. The el-ph matrix element
$g_{mn,\nu}(\mathbf{k},\mathbf{q})$ that quantifies the scattering process
between Kohn-Sham states ($n$,$\mathbf{k}$) and ($m$,$\mathbf{k+q}$),
interacting with a phonon with wavevector $\mathbf{q}$ and band index $\nu$,
can be written as:

$g_{mn,\nu}(\mathbf{k},\mathbf{q})=g^{S}_{mn,\nu}(\mathbf{k},\mathbf{q})+g^{L}_{mn,\nu}(\mathbf{k},\mathbf{q}),$
(1)

$g^{S}_{mn,\nu}$ is the short-range contribution, which is obtained from
Wannier interpolation on a dense grid Verdi and Giustino (2015).
$g^{L}_{mn,\nu}(\mathbf{k},\mathbf{q})$ is the long-range contribution, which
is computed analytically from

$\begin{split}&g^{L}_{mn,\nu}(\mathbf{k},\mathbf{q})=\\\
&i\frac{e^{2}}{\Omega\epsilon_{0}}\sum_{j}\sqrt{\frac{\hbar}{2M_{j}\omega_{\mathbf{q},\nu}}}\sum_{\mathbf{G}\mathbf{q}}\frac{(\mathbf{q}+\mathbf{G})\cdot\mathbf{Z}^{*}_{j}\cdot
e_{j,\nu}(\mathbf{q})}{(\mathbf{q}+\mathbf{G})\cdot\epsilon^{\infty}\cdot{(\mathbf{G}+\mathbf{q})}}\\\
&\times\left\langle\psi_{m,\mathbf{k}+\mathbf{q}}\left|e^{i(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}}\right|\psi_{n,\mathbf{k}}\right\rangle.\end{split}$
(2)

$\Omega$ is the volume of the cell, $\epsilon_{0}$ the permittivity of vacuum,
$\mathbf{G}$ is the reciprocal lattice vector, and $\mathbf{Z}^{*}_{j}$ is the
Born effective charge tensor of atom $j$ that has a mass $M_{j}$.
$\mathbf{e}_{j,\nu}(\mathbf{q})$ are the components of the eigenvector of the
dynamical matrix for the phonon with band index $\nu$ and wavevector
$\mathbf{q}$ for atom $j$. $\epsilon^{\infty}$ is the macroscopic high-
frequency dielectric tensor.
$\left\langle\psi_{m,\mathbf{k}+\mathbf{q}}\left|e^{i(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}}\right|\psi_{n,\mathbf{k}}\right\rangle$
is the overlap matrix element between the initial and the final state and is
evaluated in the limit of $\mathbf{q}+\mathbf{G}\rightarrow 0$.

Recent work highlighted the importance of dynamical quadrupoles in
calculations of carrier mobilities Brunin  _et al._ (2020); Jhalani  _et al._
(2020). In this work we do not include quadrupole corrections because ScN
crystallizes in the $Fm\bar{3}m$ space group, hence quadrupole tensors vanish
by symmetry (see for example the similar case of SrO in Ponce  _et al._
(2021)).

In doped semiconductors, the el-ph interaction is screened by the free charge
carriers. We take this effect into account by screening the el-ph matrix
elements using the Lindhard dielectric function, i.e.,
$g^{\mathrm{screen}}_{mn,\nu}(\mathbf{q})=g_{mn,\nu}(\mathbf{q})/\epsilon(\mathbf{q})$.
The Lindhard function is evaluated at the phonon frequency $\omega_{{\bf
q}\nu}$, and corresponds to the RPA screening of the homogeneous electron gas
Lindhard (1954); Hedin (1965). The screened el-ph matrix elements are then
used in the calculation of the electron mobility Caruso and Giustino (2016);
Verdi  _et al._ (2017).

###  II.3 Electron Transport and Carrier Mobility

Electron mobility is calculated as

$\mu_{\alpha\beta}=-\frac{e}{\Omega
n_{e}}\sum_{n\in\mathrm{CB}}\int\frac{d^{3}\mathbf{k}}{\Omega_{BZ}}v_{n\mathbf{k},\alpha}\partial_{E_{\beta}}f_{n\mathbf{k}},$
(3)

where $\Omega$ is the volume of the unit cell, $n_{e}$ is the electron
concentration, $\Omega_{BZ}$ is the volume of the BZ, $f_{n,\mathbf{k}}$ is
the Fermi-Dirac distribution function, and
$v_{n\mathbf{k},\alpha}=\hbar^{-1}\partial\epsilon_{n\mathbf{k}}/{\partial
k_{\alpha}}$ is the band velocity for the single-particle electron eigenvalue
$\epsilon_{n\mathbf{k}}$. One can define the band- and $k$-dependent el-ph
relaxation time ($\tau_{n\mathbf{k}}$) that goes into the integral on the
right side of Eq. (3). The scattering rate ($\tau^{-1}_{n\mathbf{k}}$, defined
as the inverse of the relaxation time) can be directly calculated from the
imaginary part of the el-ph self-energy Im$\Sigma_{nk}$ using
$\tau^{-1}_{n\mathbf{k}}=(2/\hbar)\mathrm{Im}\Sigma_{nk}$, which is expressed
as

$\begin{split}\tau^{-1}_{n\mathbf{k}}&=\frac{2\pi}{\hbar}\sum_{\mathbf{q}\nu,m}|g_{mn,\nu}(\mathbf{k},\mathbf{q})|^{2}\\\
&\times\\{(n_{\mathbf{q},\nu}+f_{\mathbf{k}+\mathbf{q},m})\delta(\epsilon_{\mathbf{k}+\mathbf{q},m}-\epsilon_{\mathbf{k},n}-\hbar\omega_{\mathbf{q},\nu})\\\
&+(1+n_{\mathbf{q},\nu}-f_{\mathbf{k}+\mathbf{q},m})\delta(\epsilon_{\mathbf{k}+\mathbf{q},m}-\epsilon_{\mathbf{k},n}+\hbar\omega_{\mathbf{q},\nu})\\},\end{split}$
(4)

where $\omega_{\mathbf{q},\nu}$ is the frequency and $n_{\mathbf{q},\nu}$ the
occupation (using Bose-Einstein statistics) of the phonon mode $\nu$ at
wavevector $\mathbf{q}$. The two Dirac delta functions indicate the energy
conservation conditions for el-ph scattering events, with the former
corresponding to a phonon absorption process and the latter to a phonon
emission process. The scattering rate is temperature dependent. While the el-
ph matrix is calculated at $T$=0, the temperature dependence of the scattering
rate arises from the electron and phonon occupations at finite temperatures.
The electron mobility is obtained from an iterative solution of the Boltzmann
transport equation Poncé  _et al._ (2018).

##  III Results and Discussion

###  III.1 Details of Conduction-Band Structure

The band structure of rocksalt ScN as calculated with HSE is displayed in Fig.
1. The VBM is located at the $\Gamma$ point and the CBM at the X point; there
are six equivalent X points on the BZ boundary. The valence band has O $2p$
character, while the conduction band is comprised of Sc $3d$ states. The
lowest three conduction bands belong to the unoccupied $t_{2g}$ orbitals,
which arise from crystal-field splitting of Sc $3d$ orbitals in the octahedral
ligand field. We find an indirect band gap of 0.79 eV from $\Gamma$ to X, and
optical direct gaps of 1.91 eV at X and 3.58 eV at $\Gamma$. These values are
in reasonable agreement with the range of computational Saha  _et al._ (2017);
Qteish  _et al._ (2006); Stampfl  _et al._ (2001); Lambrecht (2000) and
experimental Al-Brithen  _et al._ (2004); Gall  _et al._ (2001a); Saha  _et
al._ (2013); Deng  _et al._ (2015) results that have been previously reported.
Arguably the most detailed measurements were reported by Deng _et al._ Deng
_et al._ (2015) who performed optical measurements that yielded direct gaps of
2.02 eV at X and 3.75 eV at $\Gamma$. Using scanning tunneling spectroscopy,
Al-Brithen _et al._ Al-Brithen  _et al._ (2004) obtained an indirect band gap
of 0.9$\pm$0.1 eV. Our results are also in good agreement with the hybrid
functional calculations of Deng _et al._ Deng  _et al._ (2015) and with GW
calculations Qteish  _et al._ (2006).

Figure 1: The band structure of rocksalt ScN. Results from HSE calculations
(black solid lines) and Wannier90 interpolation (blue dashed lines) are
overlaid. Indirect and direct band gaps are indicated.

In Fig. 1 we overlay the Wannier-interpolated bands onto the band structure
calculated with HSE, demonstrating that the Wannier fit is well converged.
Figure 1 shows very different CB dispersion along the path connecting the X
point (0, 0.5, 0.5) with $\Gamma$ (0, 0, 0) versus the path connecting X with
W (0.25, 0.75, 0.5). These high-symmetry pathways are perpendicular and are
generally referred to as the longitudinal (X$\rightarrow\\!\Gamma$) and
transverse (X$\rightarrow$W) directions. The secondary conduction-band valleys
are considerably higher in energy than those at X; specifically, the K valley
is 2.52 eV higher than the CBM at X, while the $\Gamma$ valley is 2.78 eV
higher. Therefore, electrons will only occupy the X valleys over a wide range
of temperatures and carrier concentrations, and scattering to other valleys
will not be a concern.

In this paper we focus on electron transport; however, in the course of
investigating the HSE band structure, we have similarly analyzed the valence
bands; we include our results for the valence bands in the Supplemental
Material (SM) SM .

In Fig. 2 we plot the band velocities and effective masses as a function of
energy above the CBM. We determine the band velocities by taking the first
derivative with respect to k on our Wannier-interpolated HSE band structure
using the Wannier90 band interpolation tool Pizzi  _et al._ (2020). The band
velocity is defined as

${v}_{\bf k}=\frac{1}{\hbar}\frac{\partial\epsilon_{\bf k}}{\partial{\bf
k}}\,.$ (5)

We then perform finite-differences calculations on the first derivative to
determine the second derivative with respect to k, which is related to the
effective mass tensor as:

$\left[\frac{1}{m^{*}_{\mathbf{k}}}\right]_{ij}=\frac{1}{\hbar^{2}}\frac{\partial^{2}\epsilon_{\mathbf{k}}}{\partial
k_{i}\partial k_{j}}\,.$ (6)

In this way, we obtain accurate results for the evolution of effective mass as
we move away from the CBM.

Figure 2: (a) Band velocity and (b) effective mass as a function of energy
above the CBM for the lowest-lying conduction band along X$\rightarrow$W and
X$\rightarrow\\!\Gamma$ directions. The vertical lines in (b) correspond to
inflection points in the band structure, where the effective mass diverges.

The band velocity along the X$\rightarrow$W (transverse) direction is clearly
greater, by more than a factor of two, than along the X$\rightarrow\\!\Gamma$
(longitudinal) direction [Fig. 2(a)]. This correlates with the significantly
lower effective mass along X$\rightarrow$W [Fig. 2(b)]. We can also locate the
position of inflection points, i.e., energies for which the slope of the band
velocity goes to zero, or, correspondingly, where the effective mass diverges.
The inflection point along X$\rightarrow\\!\Gamma$ is approximately 1.6 eV
above the CBM, while along X$\rightarrow$W it is almost 2.5 eV above the CBM.
If carriers are present at these inflection points, either by doping or by
injection, they could be used in high-frequency oscillators Krömer (1958);
Gribnikov  _et al._ (2001); Rowberg  _et al._ (2020).

Our calculated values for the longitudinal and transverse effective masses at
the CBM are listed in Table 1. The values agree well with those derived from
previous computational studies Qteish  _et al._ (2006); Deng  _et al._ (2015).
We additionally calculate the transport effective mass
($m_{\mathrm{tr}}$)—sometimes also referred to as conductivity effective
mass—and density-of-states effective mass ($m_{\mathrm{DOS}}$). These are
defined as

${m_{\mathrm{tr}}}=3\bigg{(}\frac{1}{m_{l}}+\frac{2}{m_{t}}\bigg{)}^{-1}$ (7)

and

${m_{\mathrm{DOS}}}={(m_{l}m^{2}_{t})}^{1/3}\,.$ (8)

The values as a function of energy above the CBM are shown in Fig. 3. As
carriers are added into the conduction band, both masses increase, making
clear that it is not warranted to assume that the masses retain the value they
have at the CBM.

Figure 3: Density-of-states and transport effective masses, shown as a
function of electron energy above the CBM.  Figure 4: Electron concentrations
at $T$=300 K, shown as a function of the Fermi level position.

The values for $m_{\mathrm{tr}}$ and $m_{\mathrm{DOS}}$ at the CBM are
included in Table 1. Our value for $m_{\mathrm{tr}}$=0.24 $m_{0}$ is smaller
than that reported by Deng $et$ $al.$ Deng  _et al._ (2015), who extracted
$m_{\mathrm{tr}}$=0.40$\pm$0.02 $m_{0}$ based on a fit of the plasma frequency
that did not take the variation of $m_{\mathrm{tr}}$ with carrier
concentration into account. However, our calculated values are greater than
those measured by Harbeke $et$ $al.$, who obtained effective masses of 0.1–0.2
$m_{0}$ using infrared reflectivity Harbeke  _et al._ (1972).

For the density-of-states mass, we note that our definition does not include a
factor $M_{c}^{2/3}$, where $M_{c}$ is the number of equivalent band minima
($M_{c}=3$ for the CBM in ScN) Van Zeghbroeck (2011). In this manner, we are
following the definition used by Deng $et$ $al.$ in Ref. Deng  _et al._ ,
2015, with whom our density-of-states effective mass agrees well. They
reported that $m_{\mathrm{DOS}}$=0.33$\pm$0.02 $m_{0}$ based on an analysis of
the Burstein-Moss shift in optical absorption data Deng  _et al._ (2015).

Table 1: Calculated electron effective masses $m_{l}$, $m_{t}$,
$m_{\mathrm{tr}}$ and $m_{\mathrm{DOS}}$ (in units of $m_{0}$) using HSE and
LDA band structures. Reported experimental values (from Ref. Deng  _et al._ ,
2015) are listed for comparison.

| $m_{l}$ | $m_{t}$ | $m_{\mathrm{tr}}$ | $m_{\mathrm{DOS}}$
---|---|---|---|---
HSE | 1.43 | 0.17 | 0.24 | 0.34
LDA | 1.67 | 0.19 | 0.27 | 0.39
Expta |  |  | 0.40$\pm$0.02 | 0.33$\pm$0.02

aRef. Deng  _et al._ , 2015

While the electronic structure computed with HSE is in good agreement with
previous experimental and theoretical results, we cannot use HSE for the
calculation of mobility. Indeed, we need the Quantum ESPRESSO code Giannozzi
_et al._ (2009, 2017) for the calculation of electron-phonon matrix elements,
but the DFPT implementation in this code is not compatible with HSE. We
therefore perform these calculations consistently with LDA, and demonstrate
that the conduction-band structure that is relevant for electron mobility
agrees well with the HSE band structure. This agreement indicates that the
band velocities and effective masses are reasonably accurate in LDA. The
underestimation of the band gap in LDA likely leads to a small amount of
overscreening of the electron-phonon matrix elements, therefore the calculated
mobility should be understood as a theoretical upper bound to the measured
mobility Ponce  _et al._ (2021). As a quantitative measure of the agreement
between LDA and HSE results in the vicinity of the CBM, we compare electron
effective masses calculated with LDA with those calculated with HSE, as listed
in Table 1. The similarity in electron effective masses when comparing LDA
with HSE assures us that using the LDA band structure is sufficient to yield
reliable results for electron mobility.

###  III.2 Electron Concentrations

Using our Wannier-interpolated band structure and the corresponding density of
states, we evaluate the relationship between carrier concentrations and Fermi
level. Electron concentrations $n$ are determined by integrating the density
of states as a function of energy from the CBM Van Zeghbroeck (2011):

$n=\int_{E_{CBM}}^{\infty}g_{C}(E)f(E)dE\,.$ (9)

Here, $g_{C}(E)$ is the density of states in the conduction band as a function
of energy, and $f(E)$ is the Fermi-Dirac occupation function, given by:

$f(E)=\frac{1}{1+\exp(\frac{E-E_{F}}{k_{B}T})}\,,$ (10)

where $k_{B}$ is Boltzmann’s constant and $E_{F}$ is the Fermi energy. In Fig.
4, we plot $n$ at room temperature as a function of Fermi level, obtained
through numerical integration. A similar analysis for holes in the valence
band is included in the Supplemental Material SM .

Deng _et al._ Deng  _et al._ (2015) reported optical band gaps as a function
of increasing electron concentration starting at $1.0\times 10^{20}$ cm-3. The
observed an increase in optical band gap, caused by a Burstein-Moss shift, by
0.37 eV when increasing the carrier concentration from $1.0\times 10^{20}$
cm-3 to $1.0\times 10^{21}$ cm-3. If we assume a flat valence band around the
X point, as Deng _et al._ did when fitting their data, we can compare this
increase with the expected shift in Fermi level shown in Fig. 4. We find that
increasing $n$ from $1.0\times 10^{20}$ cm-3 to $1.0\times 10^{21}$ cm-3
corresponds to a shift of 0.33 eV, in very good agreement with the results of
Deng _et al._ Deng  _et al._ (2015). We also obtain excellent agreement with
the results of Gall _et al._ Gall  _et al._ (2001a), who estimated a Burstein-
Moss shift of 0.25 eV for a carrier concentration of $5.0\times 10^{20}$ cm-3.
Our results indicate that the same concentration will increase the position of
the Fermi level by 0.23 eV.

###  III.3 Phonon Band Structure of ScN

Our calculated phonon band structure for ScN is shown in Fig. 5. With two
atoms in the primitive cell, ScN possesses six phonon modes in total: one
longitudinal acoustic (LA) mode, two transverse acoustic (TA) modes, one
longitudinal optical (LO) mode, and two transverse optical (TO) modes. Our
calculated phonon frequencies at $\Gamma$ and X are listed in Table 2. The
calculated phonon frequencies generally agree with those from a previous LDA
calculation Paudel and Lambrecht (2009), except that
$\omega_{\mathrm{TO}}(\Gamma)$ in the present study is 11.9 meV lower. This
discrepancy may be attributed to the different pseudopotentials and codes
being used. Experimentally measured phonon frequencies Uchiyama  _et al._
(2018); Travaglini  _et al._ (1986); Gall  _et al._ (2001b) are also listed in
Table 2. While our calculated phonon frequencies at the X point are in
reasonable agreement with experiment Uchiyama  _et al._ (2018); Travaglini
_et al._ (1986); Gall  _et al._ (2001b), the optical phonon frequencies at the
$\Gamma$ point are underestimated. However, we note that the values obtained
from different experiments [inelastic X-ray scattering (IXS) Uchiyama  _et
al._ (2018) versus Raman spectroscopy Travaglini  _et al._ (1986); Gall  _et
al._ (2001b)] also differ substantially. Our calculated LO-TO splitting
[$\omega_{\mathrm{LO}}(\Gamma)-\omega_{\mathrm{TO}}(\Gamma)=42.3$ meV] is
consistent with the IXS measurement (42.6 meV). This value arises from the
nonanalytical correction at $\mathbf{q}\rightarrow 0$ and requires both the
high-frequency dielectric constant ($\epsilon^{\infty}$) and the diagonal term
of the isotropic Born effective charge tensor, which we will call $Z^{*}$. Our
calculations yield $\epsilon^{\infty}$=9.8 and $Z^{*}$(Sc)=$-Z^{*}$(N)=3.8
$e$. These values are comparable to those calculated with GGA+$U$ in Ref. Saha
_et al._ , 2010: $\epsilon^{\infty}$=12.9 and $Z^{*}$(Sc)=4.4 $e$. The origin
of the differences between results calculated at different levels of theory is
well-known and has previously been discussed Dal Corso  _et al._ (1994). In
our own HSE calculations we found $\epsilon^{\infty}$=8.4 and $Z^{*}$(Sc)=4.1
$e$, in general agreement with the LDA-calculated values.

Figure 5:  Calculated phonon band structure along high-symmetry lines in the
BZ. Frequencies for the LO and TO phonon modes at the $\Gamma$ point are
indicated.  Table 2: Calculated LO, TO, LA, TA phonon frequencies (in meV) for
ScN at the $\Gamma$ and X points. Results from a previous calculation Paudel
and Lambrecht (2009) are listed for comparison. Experimental results from
inelastic X-ray scattering (IXS) Uchiyama  _et al._ (2018) and Raman
spectroscopy Travaglini  _et al._ (1986); Gall  _et al._ (2001b) are also
listed.

| $\Gamma$ | X
---|---|---
| $\omega_{\mathrm{LO}}$ | $\omega_{\mathrm{TO}}$ | $\omega_{\mathrm{LO}}$ | $\omega_{\mathrm{TO}}$ | $\omega_{\mathrm{LA}}$ | $\omega_{\mathrm{TA}}$
This work | 75.8 | 33.4 | 69.3 | 50.7 | 42.8 | 32.8
Prev. calc.a | 78.4 | 45.3 | 70.5 | 51.5 | 43.9 | 33.1
IXSb | 85.1 | 42.5 | 67.5 | 45.7 | 31.8 | 25.1
Ramanc | 84.3 | 52.1 | – | 52.0 | 44.0 | 37.0

aRef. Paudel and Lambrecht, 2009; bRef. Uchiyama  _et al._ , 2018; cRef.
Travaglini  _et al._ , 1986; Gall  _et al._ , 2001b

###  III.4 Electron-Phonon Matrix Elements

In Fig. 6, we present our results for el-ph matrix elements
$|g_{mn,\nu}(\mathbf{k},\mathbf{q})|$ between two electron states in the
lowest conduction band ($m=n=5$, $k=$X) and an optical phonon branch $\nu$. We
consider the intraband electron interaction with either the polar LO mode or
the nonpolar TO mode, and the resulting $|g_{mn,\nu}(\mathbf{k},\mathbf{q})|)$
is plotted as a function of phonon wavevector $\mathbf{q}$ along the high-
symmetry paths of the BZ. The el-ph matrix elements for the polar LO mode are
clearly much larger than those for the TO mode. For the electron-LO phonon
interaction, a strong $1/\mathbf{q}$ dependence is observed for small
$\mathbf{q}$, due to the long-range contribution
$|g^{L}_{mn,\nu}(\mathbf{k},\mathbf{q})|$ [see Eq. (2)]. The importance of
this long-range contribution is highlighted by a comparison with the
calculated values for the short-range contribution
$|g^{S}_{mn,\nu}(\mathbf{k},\mathbf{q})|$, shown as the dashed red line in
Fig. 6. Regarding acoustic modes, we found that the el-ph matrix elements for
the LA mode are comparable in magnitude to those for the TO mode, while those
for the TA mode are negligibly small.

Figure 6:  Magnitude of el-ph matrix elements
$|g_{mn,\nu}(\mathbf{k},\mathbf{q})|$ for electrons in the lowest CB ($m=n=5$,
$\mathbf{k}$=X) and for the highest phonon branch (LO mode, $\nu=6$, red solid
line) and the second-highest phonon branch (TO mode, $\nu=5$, black solid
line). The red dashed line shows calculated values for the short-range
contribution $|g^{S}_{mn,\nu}(\mathbf{k},\mathbf{q})|$.

###  III.5 Scattering Rate

Figure 7:  (a) Relaxation time of electrons ($\tau$, fs). (b) Mode-resolved
imaginary part of the electron self-energy (lm$\Sigma$, meV), which is
proportional to the scattering rate $\tau^{-1}$ with a proportionality factor
$(\hbar/2)$. Contributions from the dominant LO, TO, and LA modes are shown,
and the characteristic frequencies $\omega_{\mathrm{LO}}$ and
$\omega_{\mathrm{TO}}$ are indicated.

Using the calculated el-ph matrix elements, we evaluate the room-temperature
relaxation times [Fig. 7(a)] for electrons in the conduction band [see Eq.
(4)]. The results are shown for energies up to 0.4 eV above the CBM. As the
satellite valleys (K and $\Gamma$ valleys) are much higher in energy than the
X valley (see Fig. 1), only small-$\mathbf{q}$ intravalley electron scattering
within the X valley is possible. The relaxation time decreases slightly over
the energy range 0–76 meV, ranging from 104 to 114 fs. Above 76 meV, the
relaxation time exhibits a sharp decrease, reaching a plateau around 7–8 fs at
higher electron energies.

To gain more insight into el-ph scattering, we plot the mode-resolved el-ph
scattering rates in Fig. 7(b). As expected, and consistent with the mode-
resolved el-ph matrix elements shown in Fig. 6, scattering with LO phonons
dominates. The total scattering rate mainly follows the trend of LO
scattering, which exhibits a threshold frequency of
$\omega_{\mathrm{LO}}(\Gamma)=76$ meV. At energies less than
$\omega_{\mathrm{LO}}$ from the CBM, the phase space for LO phonon emission
vanishes, leaving LO phonon absorption as the sole source of scattering
events. At energies above $\omega_{\mathrm{LO}}$, both LO phonon emission and
absorption are possible, sharply increasing the scattering rate.

Compared to LO phonon scattering, TO phonon scattering is much weaker. TO
phonon scattering also has a threshold frequency (at $\omega_{\mathrm{TO}}$)
below which phonon emission is suppressed and the scattering rate is lower.
Threshold frequencies in scattering rates are commonly observed in inelastic
electron–optical phonon scattering processes Lundstrom (2009).

Scattering with acoustic phonons, on the other hand, is nearly elastic, and
its rate increases with carrier energy without any threshold frequency
Lundstrom (2009). Overall, the LA phonon scattering rates are comparable to
those of TO phonons at high carrier energies, but they are noticeably higher
at low carrier energies.

###  III.6 Carrier Mobility

With our calculated scattering rates, we now compute the room-temperature
electron mobility of ScN using Eq. (3). Since the energy scale for phonons (up
to about 76 meV, Fig. 5) is much smaller than for electrons, and because the
factor $-\frac{\partial f_{n\mathbf{k}}}{\partial\epsilon_{n\mathbf{k}}}$ is
nonzero only over a narrow range near the Fermi level, only states with
energies close to the Fermi level are involved in scattering. To compute the
room-temperature mobility we focus on an energy window of 0.8 eV centered at
the Fermi level. We considered electron concentrations ranging from $10^{16}$
cm-3 to $10^{21}$ cm-3. The Fermi level that corresponds to a particular
carrier concentration has been labeled in Fig. 8. At $n=10^{16}$ cm-3, the
Fermi level is 0.19 eV below the CBM, while at $n=10^{19}$ cm-3 it is
positioned 0.01 eV below the CBM. At our highest carrier concentration
considered, $n=10^{21}$ cm-3, the Fermi level is 0.38 eV above the CBM.

Figure 8:  Position of the Fermi level for different electron concentrations
at room temperature. The zero of energy is set at the CBM.

In Fig. 9, we plot the room-temperature electron mobility as a function of the
electron concentration. When $n<10^{18}$ cm-3, the electron mobility is
relatively insensitive to $n$. The electron mobility then drops quickly for
higher When $n$ increases above $10^{18}$ cm-3, the mobility starts dropping
quickly, from 578 cm2/Vs at $n=10^{18}$ cm-3 to 165 cm2/Vs at $n=10^{21}$
cm-3.

Figure 9:  Calculated room-temperature mobilities (in cm2/Vs) for ScN as a
function of electron concentration. The calculated mobilities with and without
the screening effect are depicted using red and black dots, respectively.
Experimental values (denoted by stars) are from Oshima et al. Oshima  _et al._
(2014), Dismukes et al. Dismukes  _et al._ (1972), Burmistrova et al.
Burmistrova  _et al._ (2013), Ohgaki et al. Ohgaki  _et al._ (2013), Saha et
al. Saha  _et al._ (2013), Cetnar et al. Cetnar  _et al._ (2018) and Al-Atabi
et al. Al-Atabi  _et al._ (2020). Figure 10:  Room-temperature scattering
rates as a function of carrier energy at different carrier concentrations. The
zero of energy is set at the CBM. The Fermi-level position is labeled with
green dashed lines.

To gain more insight into these results, we plot the total scattering rates
versus electron energy for six different electron concentrations in Fig. 10.
For $n=10^{16}$, $10^{17}$, and $10^{18}$ cm-3 the Fermi level is well below
the CBM and the shape and magnitude of the scattering rate are similar. At
$n=10^{19}$ cm-3 the Fermi level approaches the CBM and the scattering rate
near the CBM starts to rise, because more electronic states participate in
electron scattering [see the first term in Eq. (4)]. When the Fermi level is
well above the CBM, for $n=10^{20}$ cm-3, the scattering rates for low-energy
carriers near the CBM are greatly increased. The rates saturate at $n=10^{21}$
cm-3.

For large carrier concentrations, there is always a dip in the scattering rate
at the Fermi level. We have verified from mode-resolved contributions to the
scattering rates that this dip is caused by electron–LO phonon scattering. The
dip in the scattering rate around the Fermi energy originates from the fact
that electron scattering by optical phonons is forbidden below the TO/LO
energy (see also Ref. Krishnaswamy  _et al._ , 2017. The reduced scattering
rate at the Fermi level is also consistent with the fact that electrons near
the Fermi surface of a metal are scattered less compared to electrons far from
the Fermi surface. We note that only the scattering rates near the Fermi level
(within $\sim$0.2 eV of $E_{F}$, at room temperature) affect the electron
mobility. The scattering rate near the Fermi level gradually increases when
the carrier concentration is increased above $n=10^{19}$ cm-3 [Figs. 7(b) and
10]. This trend stems from the increasing electron density of states and the
associated phase space for scattering as the Fermi level moves up inside the
conduction band, thus explaining the reduction in electron mobility.

###  III.7 Screening Effect on Electron Transport

Inclusion of free-carrier screening reduces the el-ph matrix elements and
thereby increases the relaxation time and enhances the mobilities, as shown by
the results in Fig. 9. To identify the carrier concentration at which
screening is expected to play an notable role Caruso and Giustino (2016);
Verdi  _et al._ (2017), we estimate the plasma frequency $\omega_{\mathrm{P}}$
of free electrons and compare it to the frequency of LO phonons, which
dominate the electron-phonon scattering. When
$\omega_{\mathrm{P}}<\omega_{\mathrm{LO}}$, free-carrier screening is
ineffective Verdi  _et al._ (2017); we expect it to start affecting the
electron-phonon interaction when $\omega_{\mathrm{P}}>\omega_{\mathrm{LO}}$.
We calculate $\omega_{\mathrm{P}}$ as a function of carrier concentration $n$
based on the standard expression Lundstrom (2009)

$\omega_{\mathrm{P}}=\sqrt{\frac{ne^{2}}{m^{*}\epsilon_{0}\epsilon^{\infty}}}\,.$
(11)

$\omega_{\mathrm{P}}$ is 48 meV at $n=10^{19}$ cm-3, and increases to 150 meV
at $n=10^{20}$ cm-3; comparing these values with the LO phonon frequency
($\omega_{\mathrm{LO}}=76$ meV), we expect free-carrier screening to become
important around $n=10^{20}$ cm-3. At $n$=$10^{21}$ cm-3 we find
$\omega_{\mathrm{P}}$=330 meV and el-ph coupling is notably suppressed,
resulting in a signficant mobility enhancement (Fig. 9)—from 165 cm2/Vs to 240
cm2/Vs, an increase by 45%.

Figure 9 also contains values from various experiments Dismukes  _et al._
(1972); Oshima  _et al._ (2014); Burmistrova  _et al._ (2013); Ohgaki  _et
al._ (2013); Saha  _et al._ (2013). Our calculated mobility values are
systematically higher. This is to be expected, because we only include
electron-phonon scattering; our values thus constitute an upper bound on the
mobility. Other scattering mechanisms, such as ionized impurity scattering,
are present in actual samples. We also note that our calculations are for the
drift mobility, while some of the experimental data Dismukes  _et al._ (1972);
Ohgaki  _et al._ (2013); Cetnar  _et al._ (2018) are for the Hall mobility.
Recent benchmarks on standard semiconductors indicate that the calculated
electron Hall mobility is typically lower than the drift mobility Ponce  _et
al._ (2021).

###  III.8 Strain Effects on Mobility

Strain engineering is an established strategy for tailoring carrier mobilities
in semiconductors. Different substrates will lead to different strains, either
as residual strains in a partially relaxed layer, or as a result of
pseudomorphic growth of thin films. We investigated application of in-plane
strain (denoted by $\varepsilon_{\parallel}$) in a (111) plane or in a (100)
plane. The former may occur for epitaxial growth on AlN or GaN, while the
latter would occur as residual strain when ScN is grown on Si or MgO.

Figure 11:  Calculated change in room-temperature mobility ($\delta\mu_{e}$,
left axis) and relaxed out-of-plane strain ($\varepsilon_{\perp}$, right axis)
under different in-plane strains ($\varepsilon_{\parallel}$) in (a)
(111)-oriented ScN films and (b) (100)-oriented ScN films.
$\delta\mu^{\parallel}_{e}$ denotes the variation of in-plane mobility while
$\delta\mu^{\perp}_{e}$ denotes the variation of out-of-plane mobility.

The in-plane strain $\varepsilon_{\parallel}$ is accompanied by an out-of-
plane lattice relaxation, described by $\varepsilon_{\perp}$, along either the
[111] direction or the [100] direction. For the [111]-oriented films, our HSE
calculations show that 1.0% compressive (tensile) epitaxial strain yields
0.44% tensile (compressive) out-of-plane strain $\varepsilon_{\perp}$ [Fig.
11(a)]. For the [100]-oriented films, 1.0% compressive (tensile) epitaxial
strain yields 0.56% tensile (compressive) out-of-plane strain [Fig. 11(b)].
Based on the strain relationships in Fig. 11, we obtain a Poisson’s ratio of
0.180 for [111]–oriented films and 0.219 for [100]-oriented films. Both values
compare well with the experimental measurements: 0.188 for [111] Moram  _et
al._ (2006) and 0.201 for [100] Gall  _et al._ (1999).

We evaluated the change in the room-temperature electron mobility at
$n=10^{16}$ cm-3 as a function of in-plane strain. Figure 11(a) shows our
results for (111) films. Epitaxial strain in the (111) plane does not lift the
degeneracy of the CBM at the X point; it only alters the band curvature of the
conduction band and hence the electron effective mass and band velocity. We
see that compressive strain increases the mobility. This can be attributed to
an increase in the dispersion of the conduction band. The mobility along the
out-of-plane direction ($\mu^{\perp}_{e}$) follows the same trend as the in-
plane mobility, but with a lower rate of change. Tensile strain leads to a
change of the same magnitude in the opposite direction; the behavior is close
to linear over this range of strains. A linear fitting shows that a 1.0%
compressive epitaxial strain increases the in-plane mobility by 72 cm2/Vs and
the out-of-plane mobility by 50 cm2/Vs.

The results for (100) films are displayed in Fig. 11(a). Epitaxial strain in
the (100) plane splits the threefold-degenerate conduction band at the X point
(not including spin degeneracy) into a singly-degenerate and a doubly-
degenerate band. Under compressive in-plane strain, the singly-degenerate band
is higher in energy. The splitting between the singly-degenerate and doubly-
degenerate band extrema is 36 meV per 1% epitaxial strain. The out-of-plane
mobility ($\mu^{\perp}_{e}$) displays a strong dependence on strain: a 1%
compressive in-plane strain raises the mobility by 172 cm2/Vs. The in-plane
mobility ($\mu^{\parallel}_{e}$) is not very sensitive to strain, and exhibits
a notable nonlinearity. This can be attributed to the lifting of the
degeneracy at the X points, which impacts electron occupation and scattering.

##  IV Conclusion

In summary, we reported an in-depth first-principles study of the conduction-
band structure, electron-phonon interaction and room-temperature electron
transport in ScN. We determined band velocities and effective masses as a
function of carrier concentration. Electron mobilities were calculated as a
function of carrier concentrations ranging from $10^{16}$ to $10^{21}$ cm-3.
Effects of free-carrier screening were included; we found that screening
significantly enhances electron mobility when the carrier concentration
exceeds 1020 cm-3. We calculated a room-temperature electron mobility of 587
cm2/Vs at $n$=1016 cm-3, decreasing to 240 cm2/Vs at $n$=1021 cm-3. We also
showed that strain can strongly impact the electron mobility. In
(111)-oriented ScN films a 1.0% compressive strain increases the in-plane
mobility by 72 cm2/Vs and the out-of-plane mobility by 50 cm2/Vs. The increase
in out-of-plane mobility is even greater in (100)-oriented ScN films: a 1.0%
compressive strain increases the out-of-plane mobility by 172 cm2; the impact
of (100) strain on the in-plane mobility is quite weak. These results
demonstrate how pseudomorphic strain can potentially be used to enhance the
electron mobility. In light of the critical importance of high mobility for
ScN’s applications, including in thermoelectric and (opto)electronic devices,
our results will help support its further technological development.

###### Acknowledgements.

We are grateful to John Cetnar, David Look, Mark Turiansky, Nicholas Adamski,
Karthik Krishnaswamy, Wennie Wang, and Youngho Kang for valuable discussions,
to Jinjian Zhou for assistance with the Perturbo code, and to Samuel Poncé for
useful discussions about the EPW code. This work was supported by the Air
Force Office of Scientific Research under award number FA9550-18-1-0237.
A.J.E.R. was also supported by the National Science Foundation (NSF) Graduate
Research Fellowship Program under Grant No. 1650114. Any opinions, findings,
and conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the NSF. J.L. and F.G.
acknowledge support from the Computational Materials Sciences Program funded
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
under Award DE-SC0020129. We acknowledge computational resources provided by
the Extreme Science and Engineering Discovery Environment (XSEDE), which is
supported by NSF Grant No. ACI-1548562, and by the DOD High Performance
Computing Modernization Program at the AFRL DSRC and ERDC DSRC under Project
No. AFOSR46403464. Use was also made of computational facilities purchased
with funds from NSF (CNS-1725797) and administered by the Center for
Scientific Computing (CSC). The CSC is supported by the California NanoSystems
Institute and the Materials Research Science and Engineering Center (NSF
DMR-1720256) at UC Santa Barbara.

## References

  * Oshima  _et al._ (2014) Y. Oshima, E. G. Víllora, and K. Shimamura, J. Appl. Phys. 115, 153508 (2014).
  * Ohgaki  _et al._ (2013) T. Ohgaki, K. Watanabe, Y. Adachi, I. Sakaguchi, S. Hishita, N. Ohashi, and H. Haneda, J. Appl. Phys. 114, 093704 (2013).
  * Saha  _et al._ (2013) B. Saha, G. Naik, V. P. Drachev, A. Boltasseva, E. E. Marinero, and T. D. Sands, J. Appl. Phys. 114, 063519 (2013).
  * Zide  _et al._ (2005) J. Zide, D. Klenov, S. Stemmer, A. Gossard, G. Zeng, J. Bowers, D. Vashaee, and A. Shakouri, Appl. Phys. Lett. 87, 112102 (2005).
  * Saha  _et al._ (2012) B. Saha, T. D. Sands, and U. V. Waghmare, J. Phys. Condens. Matter 24, 415303 (2012).
  * Kerdsongpanya  _et al._ (2017) S. Kerdsongpanya, O. Hellman, B. Sun, Y. K. Koh, J. Lu, N. Van Nong, S. I. Simak, B. Alling, and P. Eklund, Phys. Rev. B 96, 195417 (2017).
  * Biswas and Saha (2019) B. Biswas and B. Saha, Phys. Rev. Mater. 3, 020301 (2019).
  * Al-Atabi  _et al._ (2020) H. Al-Atabi, Q. Zheng, J. S. Cetnar, D. Look, D. G. Cahill, and J. H. Edgar, Appl. Phys. Lett. 116, 132103 (2020).
  * Deng  _et al._ (2015) R. Deng, B. Ozsdolay, P. Zheng, S. Khare, and D. Gall, Phys. Rev. B 91, 045104 (2015).
  * Nayak  _et al._ (2019) S. Nayak, M. Baral, M. Gupta, J. Singh, M. Garbrecht, T. Ganguli, S. Shivaprasad, and B. Saha, Phys. Rev. B 99, 161117 (2019).
  * Johnston  _et al._ (2009) C. Johnston, M. Moram, M. Kappers, and C. Humphreys, Appl. Phys. Lett. 94, 161109 (2009).
  * Adamski  _et al._ (2019) N. L. Adamski, C. E. Dreyer, and C. G. Van de Walle, Appl. Phys. Lett. 115, 232103 (2019).
  * Neckel  _et al._ (1975) A. Neckel, P. Rastl, R. Eibler, P. Weinberger, and K. Schwarz, J. Phys. C 9, 579 (1975).
  * Monnier  _et al._ (1985) R. Monnier, J. Rhyner, T. Rice, and D. Koelling, Phys. Rev. B 31, 5554 (1985).
  * Al-Brithen  _et al._ (2004) H. A. Al-Brithen, A. R. Smith, and D. Gall, Phys. Rev. B 70, 045303 (2004).
  * Gall  _et al._ (2001a) D. Gall, M. Städele, K. Järrendahl, I. Petrov, P. Desjardins, R. Haasch, T.-Y. Lee, and J. Greene, Phys. Rev. B 63, 125119 (2001a).
  * Saha  _et al._ (2017) B. Saha, M. Garbrecht, J. A. Perez-Taborda, M. H. Fawey, Y. R. Koh, A. Shakouri, M. Martin-Gonzalez, L. Hultman, and T. D. Sands, Appl. Phys. Lett. 110, 252104 (2017).
  * Qteish  _et al._ (2006) A. Qteish, P. Rinke, M. Scheffler, and J. Neugebauer, Phys. Rev. B 74, 245208 (2006).
  * Stampfl  _et al._ (2001) C. Stampfl, W. Mannstadt, R. Asahi, and A. J. Freeman, Phys. Rev. B 63, 155106 (2001).
  * Lambrecht (2000) W. R. Lambrecht, Phys. Rev. B 62, 13538 (2000).
  * Burmistrova  _et al._ (2013) P. V. Burmistrova, J. Maassen, T. Favaloro, B. Saha, S. Salamat, Y. Rui Koh, M. S. Lundstrom, A. Shakouri, and T. D. Sands, J. Appl. Phys. 113, 153704 (2013).
  * Moram  _et al._ (2010) M. Moram, M. Kappers, and C. Humphreys, Phys. Status Solidi C 7, 1778 (2010).
  * Kumagai  _et al._ (2018) Y. Kumagai, N. Tsunoda, and F. Oba, Phys. Rev. Appl. 9, 034019 (2018).
  * Cetnar  _et al._ (2018) J. S. Cetnar, A. N. Reed, S. C. Badescu, S. Vangala, H. A. Smith, and D. C. Look, Appl. Phys. Lett. 113, 192104 (2018).
  * Dismukes  _et al._ (1972) J. Dismukes, W. Yim, and V. Ban, J. Cryst. Growth 13, 365 (1972).
  * Harbeke  _et al._ (1972) G. Harbeke, E. Meier, and J. Dismukes, Optics Commun. 4, 335 (1972).
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Heyd  _et al._ (2003) J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).
  * Heyd  _et al._ (2006) G. Heyd, J. Scuseria, and M. Ernzerhof, J. Chem. Phys. 124, 219906 (2006).
  * Pizzi  _et al._ (2020) G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, L. Paulatto, S. Poncé’, T. Ponweiser, J. Qiao, F. Th ole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, and J. R. Yates, J. Phys. Condens. Matter 32, 165902 (2020).
  * Lindhard (1954) J. Lindhard, K. danske Vidensk. Selsk. Mat.-fys. Medd. 28 (1954).
  * Hedin (1965) L. Hedin, Phys. Rev. 139, A796 (1965).
  * Al-Brithen and Smith (2000) H. Al-Brithen and A. R. Smith, Appl. Phys. Lett. 77, 2485 (2000).
  * Moram  _et al._ (2006) M. A. Moram, Z. H. Barber, C. J. Humphreys, T. Joyce, and P. Chalker, J. Appl. Phys. 100, 023514 (2006).
  * Poncé  _et al._ (2019) S. Poncé, D. Jena, and F. Giustino, Phys. Rev. Lett. 123, 096602 (2019).
  * Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Travaglini  _et al._ (1986) G. Travaglini, F. Marabelli, R. Monnier, E. Kaldis, and P. Wachter, Phys. Rev. B 34, 3876 (1986).
  * Giannozzi  _et al._ (2009) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, _et al._ , J. Phys. Condens. Matter 21, 395502 (2009).
  * Giannozzi  _et al._ (2017) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, _et al._ , J. Phys. Condens. Matter 29, 465901 (2017).
  * Perdew and Wang (1992) J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).
  * Troullier and Martins (1991) N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991).
  * Baroni  _et al._ (2001) S. Baroni, S. De Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  * Gonze  _et al._ (1994) X. Gonze, J.-C. Charlier, D. Allan, and M. Teter, Phys. Rev. B 50, 13035 (1994).
  * Poncé  _et al._ (2016) S. Poncé, E. R. Margine, C. Verdi, and F. Giustino, Comput. Phys. Commun 209, 116 (2016).
  * Zhou  _et al._ (2021) J.-J. Zhou, J. Park, I.-T. Lu, I. Maliyov, X. Tong, and M. Bernardi, Comput. Phys. Commun 264, 107970 (2021).
  * Ziman (2001) J. M. Ziman,  _Electrons and phonons: the theory of transport phenomena in solids_ (Oxford university press, 2001).
  * Fröhlich (1954) H. Fröhlich, Adv. Phys. 3, 325 (1954).
  * Verdi and Giustino (2015) C. Verdi and F. Giustino, Phys. Rev. Lett. 115, 176401 (2015).
  * Brunin  _et al._ (2020) G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo, M. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese, and G. Hautier, Phys. Rev. Lett. 125, 136601 (2020).
  * Jhalani  _et al._ (2020) V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, and M. Bernardi, Phys. Rev. Lett. 125, 136602 (2020).
  * Ponce  _et al._ (2021) S. Ponce, F. Macheda, E. R. Margine, N. Marzari, N. Bonini, and F. Giustino, “First-principles predictions of hall and drift mobilities in semiconductors,” (2021), arXiv:2105.04192 [cond-mat.mtrl-sci] .
  * Caruso and Giustino (2016) F. Caruso and F. Giustino, Phys. Rev. B 94, 115208 (2016).
  * Verdi  _et al._ (2017) C. Verdi, F. Caruso, and F. Giustino, Nat. Commun. 8, 1 (2017).
  * Poncé  _et al._ (2018) S. Poncé, E. R. Margine, and F. Giustino, Phys. Rev. B 97, 121201 (2018).
  * (59) See Supplemental Material at [URL will be inserted by publisher] for a discussion of Wannier-interpolation to the HSE-calculated valence bands and corresponding results on band velocities, hole effective masses, and carrier concentrations as a function of energy below the VBM.
  * Krömer (1958) H. Krömer, Phys. Rev. 109, 1856 (1958).
  * Gribnikov  _et al._ (2001) Z. S. Gribnikov, R. R. Bashirov, and V. V. Mitin, IEEE J. Sel. Top. Quantum Electron. 7, 630 (2001).
  * Rowberg  _et al._ (2020) A. J. E. Rowberg, K. Krishnaswamy, and C. G. Van de Walle, Phys. Rev. B 102, 115201 (2020).
  * Van Zeghbroeck (2011) B. J. Van Zeghbroeck,  _Principles of Semiconductor Devices_ (Boulder: University of Colorado, 2011).
  * Paudel and Lambrecht (2009) T. R. Paudel and W. R. Lambrecht, Phys. Rev. B 79, 085205 (2009).
  * Uchiyama  _et al._ (2018) H. Uchiyama, Y. Oshima, R. Patterson, S. Iwamoto, J. Shiomi, and K. Shimamura, Phys. Rev. Lett. 120, 235901 (2018).
  * Gall  _et al._ (2001b) D. Gall, M. Stoehr, and J. Greene, Phys. Rev. B 64, 174302 (2001b).
  * Saha  _et al._ (2010) B. Saha, J. Acharya, T. D. Sands, and U. V. Waghmare, J. Appl. Phys. 107, 033715 (2010).
  * Dal Corso  _et al._ (1994) A. Dal Corso, S. Baroni, and R. Resta, Phys. Rev. B 49, 5323 (1994).
  * Lundstrom (2009) M. Lundstrom,  _Fundamentals of carrier transport_ (Cambridge university press, 2009).
  * Krishnaswamy  _et al._ (2017) K. Krishnaswamy, B. Himmetoglu, Y. Kang, A. Janotti, and C. G. Van de Walle, Phys. Rev. B 95, 205202 (2017).
  * Gall  _et al._ (1999) D. Gall, I. Petrov, P. Desjardins, and J. Greene, J. Appl. Phys. 86, 5524 (1999).