In Fig. 16(b) we show the comparison between calculated and measured
mobilities, in a logarithmic scale. In all cases the experimental data lay
below the predicted phonon-limited Hall mobility. The difference can be as
severe as 100% in some cases. The calculated drift mobility is found to be
closer to the experimental data, but we have to emphasize that most
experimental data reported in this figure are for the Hall mobility, as
indicated in Table 8. The SERTA drift mobility appears to be closer to
experiments, and in some cases below the experimental data (for example in
diamond, AlSb and GaAs). This behavior is probably due to a cancellation of
errors, and should not be taken as an endorsement of the SERTA method.
Whenever possible we recommend to perform complete BTE calculations. We note
that, in calculations of phonon-limited carrier mobilities, it is not
meaningful to validate computational methods by direct comparison with
experiments, since experimental data incorporate additional scattering
mechanisms.

###  IV.6 Corrections to band structures and matrix elements from experimental
data

In this section we investigate the role of many-body renormalization of the
band structures and electron-phonon matrix elements on the mobility. Since it
is not currently possible to perform complete many-body calculations of the
BTE, we estimate these effects by using the experimental effective masses and
by rescaling the electron-phonon matrix elements via the experimental
dielectric permittivities.

To this aim, starting from the experimental masses reported in Table 5, for
the electrons we determine the conductivity effective mass defined as the
harmonic mean of the transverse and longitudinal effective masses:

$\frac{3}{m_{\rm cond}^{*,\rm
el}}=\frac{1}{m_{\parallel}^{*}}+\frac{2}{m_{\perp}^{*}}.$ (54)

For the holes we average the band contribution with the following weighted
average formula which assumes band parabolicity Ricci et al. (2017):

$m_{\rm cond}^{*,\rm h}=\frac{(m_{\rm hh}^{*})^{5/2}+(m_{\rm
lh}^{*})^{5/2}}{(m_{\rm hh}^{*})^{3/2}+(m_{\rm lh}^{*})^{3/2}}.$ (55)

Using these experimentally-derived quantities, we rescale our calculated
electron and hole Hall mobilities using:

$\mu^{\rm mass}=\mu(m_{\rm cond}^{*,\rm DFT}/m_{\rm cond}^{*,\rm exp}).$ (56)

The relative change in mobility obtained by this procedure is reported in
Table 6. In all materials considered here this rescaling decreases the hole
mobility, with the exception of SiC and BN. For the electron mobility, in all
cases but diamond the mobility increases upon rescaling.

| Hall mobility | Effective mass | Screening | Both
---|---|---|---|---
| cm2/Vs | rescaling | rescaling | rescaling
C-e | 1705 | -16% | -9% | -24%
C-h | 2467 | -28% | -9% | -35%
Si-e | 1770 | +2% | -14% | -13%
Si-h | 574 | -47% | -14% | -54%
GaAs-e | 17860 | - | -42% | -42%
GaAs-h | 1068 | -32% | -42% | -63%
SiC-e | 2257 | +3% | -11% | -8%
SiC-h | 111 | +17% | -11% | +4%
AlP-e | 698 | - | -14% | -14%
AlP-h | 92 | - | -14% | -14%
GaP-e | 442 | +56% | -22% | +21%
GaP-h | 353 | -32% | -22% | -47%
BN-e | 956 | +11% | -3% | +7%
BN-h | 281 | +57% | -3% | +52%
AlAs-e | 640 | +20% | -22% | +16%
AlAs-h | 357 | -45% | -22% | -47%
AlSb-e | 431 | +76% | -10% | +59%
AlSb-h | 1000 | -38% | -10% | -44%
SrO-e | 65 | - | -14% | -14%
SrO-h | 5 | - | -14% | -14%
Table 6:  Effect of rescaling the Hall mobilities obtained via the BTE using
experimental effective masses and dielectric constants.  Figure 17:
Comparison between theoretical and experimental mobility with experiment.
Black triangles indicate the impact of rescaling the matrix elements while
blue crosses indicate additional effective masses rescaling. The mean absolute
error (MAE) and mean absolute relative error (MARE) are indicated as inset in
the bottom figure and do not include the electron mobility of GaAs.

Along the same line, we also proceed to renormalize the electron-phonon matrix
elements by using the experimental dielectric screening. This strategy allows
us to capture effects similar to the many-body renormalization of the
electron-phonon coupling Antonius et al. (2014); Li et al. (2019a). Since the
mobility contains the square moduli of the matrix elements, we rescale the
mobility using:

$\mu^{\rm screen}=\mu(\epsilon^{\rm DFT}/\epsilon^{\rm exp})^{2}.$ (57)

In this operation we employ the dielectric constants reported in Table 5.

In Fig. 17 we show the impact of rescaling the matrix elements (black
triangles) on the mobility together with the rescaling of the matrix elements
and effective masses (blue crosses). For all cases we also report the mean
absolute error (MAE) and the mean absolute relative error (MARE) between the
computed and experimental mobilities.

Our best first-principles BTE Hall mobilities have a large absolute error
around 1000 cm2/Vs due to the strong overestimation of the high electron
mobility of GaAs and with a MARE of 64 %. If we remove GaAs from the dataset,
which is the compound with the highest mobility and hence it weighs
disproportionately in the calculation of the average, we obtain a maximum
absolute error of 335 cm2/Vs, and a MARE of 62%. When we proceed to
renormalize the electron-phonon matrix elements using the experimental
dielectric constant, the results improve drastically with a MAE below 205
cm2/Vs and MARE of 35 %. However, this improvement is partially cancelled when
we also use the experimental effective masses, leading to a MARE of 55 %.

These results indicate that improving the accuracy of the band structures and
electron-phonon matrix elements, for example by using hybrid functionals Yin
et al. (2013) or many-body perturbation theory Antonius et al. (2014); Li et
al. (2019a) will be important to achieve agreement with experiment. Whether
these improvements will lead to close agreement with experiment, or else a
more powerful theory beyond the BTE formalism will be needed in the future,
remains an important open question.

Finally, we note that regardless of the level of approximation or re-scaling
applied, the electron mobility of 3C-SiC and AlAs, and the hole mobility of
GaP are significantly higher than experiment. This suggests that higher
mobilities could be achieved in these materials, paving the way for improved
electronic devices.

##  V Conclusion

In this work we have carried out an extensive investigation of the carrier
transport properties of 10 simple semiconductors including diamond, silicon,
GaAs, 3C-SiC, AlP, GaP, c-BN, AlAs, AlSb, and SrO. Exploiting a precise
interpolation approach and an efficient procedure for numerical convergence,
we have been able to compute from first-principles transport coefficients with
high accuracy. This key step has allowed us to directly quantify the impact of
a wide range of approximations often used in transport calculations. In
addition, we have been able to assess the role of a number of physical factors
such as the contribution of long-range dipole and quadrupole interactions, the
SOC, effective masses and lattice parameters as well as the effect of
temperature and magnetic field. One of the most interesting result of this
extensive study is that our most accurate Hall mobility calculations seem to
systematically overestimate the experimental data. In a few cases the
overestimation is as large as a factor of two. This is a quite promising
outcome as in our calculation we only included electron-phonon scattering and
our result should be regarded as an upper limit to the Hall mobility
achievable in highly pure samples, suggesting the possibility of further
improvement in the transport properties of semiconductors of technological
relevance. Here it is also important to stress that our work has clearly shown
that the Hall factor exhibits significant variations across materials and as a
function of temperature, ranging between 0.7 and 2. This is a clear indication
that the common practice of comparing computed drift mobilities with
experimental Hall mobilities should be done with particular care. Finally, it
is important to stress that in our analysis we have also showed that part of
the difference between theory and experiment may be connected with the
inaccuracy in the DFT effective masses and electron-phonon matrix elements,
but further work using explicit many-body perturbation theory calculations
will be needed to confirm this point.

Our work demonstrates that first-principles calculations of carrier transport
are making considerable progress. We believe we are reaching a point where
detailed comparison with high-quality experimental data on pure samples will
be increasingly important. For example, we highlight the need for new accurate
measurements of the carrier mobility of SiC, AlAs, and GaP, and we hope to
reignite interest in the experimental community to generate modern transport
datasets for theorists to compare with. This manuscript describes a
comprehensive protocol to perform such delicate calculations, and as a
blueprint for reporting first-principles transport data in a way that is
accessible, reliable, and reproducible. In this spirit, we encourage the
adoption of the FAIR Wilkinson et al. (2016) data and software principles in
the study first-principles transport.

###### Acknowledgements.

The authors would like to thank C. Verdi, X. Gonze, Junfeng Qiao, Hyungjun
Lee, Massimiliano Stengel, Miquel Royo, for useful discussions. Computer time
was provided by the PRACE-17 and PRACE-21 resources MareNostrum at BSC-CNS,
and the Texas Advanced Computing Center (TACC) at the University of Texas at
Austin. S.P. acknowledges support from the European Unions Horizon 2020
Research and Innovation Programme, under the Marie Skłodowska-Curie Grant
Agreement SELPH2D No. 839217. F.M. and N.B. acknowledge the Cirrus UK National
Tier-2 HPC Service at EPCC. E.R.M. acknowledges support from the National
Science Foundation (Award No. OAC-1740263). N.M. acknowledges support from the
Swiss National Science Foundation and the NCCR MARVEL. F.G.’s contribution to
this work was supported as part of the Computational Materials Sciences
Program funded by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, under Award DE-SC0020129.

##  Appendix A Optimal Wigner-Seitz construction

The interpolation of the Hamiltonian, dynamical matrix, and electron-phonon
matrix elements from real space to reciprocal space via Eqs. (II.3),(22) and
(23) is usually performed by using a radial cutoff on the direct lattice
vectors Mostofi et al. (2014); Poncé et al. (2016). While this is certainly
the simplest possible choice, it is not the most efficient, especially in the
case of unit cells with high aspect ratios.

In order to improve the accuracy of the interpolation, it is convenient to
select the direct lattice vectors so that the distance between two Wannier
centers or the distance between two atoms are within a cutoff radius. For
example, in the interpolation of the Hamiltonian we construct a shifted
Wigner-Seitz cell of lattice vectors $\mathbf{R}_{p}$ with the requirement
that this cell be centered on one of the two Wannier functions, say
$\mathbf{r}_{n}$. The radial cutoff is then imposed with respect to the
distance $|\mathbf{R}_{p}+\mathbf{r}_{m}-\mathbf{r}_{n}|$, where
$\mathbf{r}_{m}$ denotes the center of the second Wannier function.

In practice, we determine the set of Wigner-Seitz vectors to describes the
electronic Hamiltonian:

$H_{mn}(\mathbf{R}_{p})=(1/N_{p})\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\mathbf{k}\in(i_{p}\times
j_{p}\times
o_{p})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!U_{mm^{\prime}\mathbf{k}}^{\dagger}H_{m^{\prime}n^{\prime}\mathbf{k}}U_{n^{\prime}n\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{R}_{p}},$
(58)

such that $\mathbf{R}_{p}+\mathbf{r}_{m}-\mathbf{r}_{n}\in(i_{p}\times
j_{p}\times o_{p})$ where $\mathbf{r}_{m}$ is the origin vector pointing to
the position of the $m$th Wannier center. A similar approach was recently
implemented in the Wannier90 software for the interpolation of the Hamiltonian
Pizzi et al. (2020).

Similarly, in the case of the dynamical matrix we consider pairs of atoms
$\boldsymbol{\tau}_{\kappa}$ and $\boldsymbol{\tau}_{\kappa^{\prime}}$. For
each such pair, we construct a Wigner-Seitz cell centered at
$\boldsymbol{\tau}_{\kappa}$, and we impose a truncation based on the distance
$|\mathbf{R}_{p^{\prime}}+\boldsymbol{\tau}_{\kappa^{\prime}}-\boldsymbol{\tau}_{\kappa}|$.
For the optimal choice of a Wigner-Seitz vector for a discretized Brillouin
zone grid $i_{p^{\prime}}\times j_{p^{\prime}}\times o_{p^{\prime}}$ , the
short-range part of the dynamical matrix takes the form Gonze and Lee (1997):

$D_{\kappa\alpha,\mu}^{\mathcal{S}}(\mathbf{R}_{p^{\prime}})=\\\
(1/N_{p^{\prime}})\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\mathbf{q}\in(i_{p^{\prime}}\times
j_{p^{\prime}}\times
o_{p^{\prime}})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e^{-i\mathbf{q}\cdot\mathbf{R}_{p^{\prime}}}e_{\kappa\alpha,\mathbf{q}\nu}^{\dagger}D_{\nu,\kappa^{\prime}\beta}^{\mathcal{S}}(\mathbf{q})e_{\kappa^{\prime}\beta,\mathbf{q}\mu},$
(59)

for all direct lattice vectors $\mathbf{R}_{p^{\prime}}$ such that
$\mathbf{R}_{p^{\prime}}+\boldsymbol{\tau}_{\kappa^{\prime}}-\boldsymbol{\tau}_{\kappa}\in(i_{p^{\prime}}\times
j_{p^{\prime}}\times o_{p^{\prime}})$ and 0 otherwise. In Eq. (59) we have
made a rigid shift to the central primitive cell such that it only depends on
one lattice vector $\mathbf{R_{p^{\prime}}}$. As a result one needs to store a
set of atom-dependent Wigner-Seitz cells. In first-principles software such as
Quantum ESPRESSO Giannozzi et al. (2017), a Fourier-transform grid is
constructed with weights centered on pairs of atoms and zeros elsewhere. In
EPW, we slightly optimize the computational cost of the interpolation by only
retaining the union of nonzero elements between all the Wigner-Seitz cells.

Figure 18:  Simplified schematic (2 dimensional 2$\times$2 k/q grid) of the
electron-phonon matrix elements where $\mathbf{r}_{n}$ and $\mathbf{r}_{m}$
indicate the position of the electronic Wannier centres, and
$\boldsymbol{\tau}_{\kappa}$ is the position of the atom $\kappa$ within a
primitive cell. The square lattice indicates the unit cells of the crystal and
$\mathbf{R}_{p}$ and $\mathbf{R}_{p^{\prime\prime}}$ are the lattice vectors
for the electronic and phononic grids, respectively. When performing the
electron or phonon Fourier transform, we use the direct lattice point $p$ or
$p^{\prime\prime}$, respectively, if the trial point
$\mathbf{t}=\mathbf{R}_{p}+\mathbf{r}_{m}-r_{n}$ or
$\mathbf{t}^{\prime\prime}=\mathbf{R}_{p^{\prime\prime}}+\boldsymbol{\tau}_{\kappa}-\mathbf{r}_{n}$
is closer to $\mathbf{0}$ than any other supercell center $\mathbf{S}$. The
red square denotes the central supercell.

In the case of the electron-phonon matrix elements, we exploit the same ideas
as above, this time for both the Wannier functions and the atomic positions.
The Wigner-Seitz cells for electrons and phonons are chosen so that both the
$|\mathbf{R}_{p}+\mathbf{r}_{m}-\mathbf{r}_{n}|$ and
$|\mathbf{R}_{p^{\prime\prime}}+\boldsymbol{\tau}_{\kappa}-\mathbf{r}_{n}|$
distances are minimized:

$g_{nm\kappa\alpha}(\mathbf{R}_{p},\mathbf{R}_{p^{\prime\prime}})=\frac{1}{N_{p}N_{p^{\prime\prime}}}\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\mathbf{k}\in(i_{p}\times
j_{p}\times o_{p})\\\ \mathbf{q}\in(i_{p^{\prime\prime}}\times
j_{p^{\prime\prime}}\times
o_{p^{\prime\prime}})\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e^{-i(\mathbf{k}\cdot\mathbf{R}_{p}+\mathbf{q}\cdot\mathbf{R}_{p^{\prime\prime}})}\\!\\!\sum_{m^{\prime}n^{\prime}\nu}\\\
\times\sqrt{\frac{2M_{\kappa}\omega_{\mathbf{q}\nu}}{\hbar}}e_{\kappa\alpha,\mathbf{q}\nu}^{\dagger}U_{mm^{\prime}\mathbf{k+q}}^{\dagger}g_{m^{\prime}n^{\prime}\nu}(\mathbf{k,q})U_{n^{\prime}n\mathbf{k}},$
(60)

where the lattice vectors are obtained such that
$\mathbf{R}_{p^{\prime\prime}}+\boldsymbol{\tau}_{\kappa}-\mathbf{r}_{n}\in(i_{p^{\prime\prime}}\times
j_{p^{\prime\prime}}\times o_{p^{\prime\prime}})$ and
$\mathbf{R}_{p}+\mathbf{r}_{m}-\mathbf{r}_{n}\in(i_{p}\times j_{p}\times
o_{p})$. We note that the set of lattice vectors
$\mathbf{R}_{p^{\prime\prime}}$ used for the interpolation of electron-phonon
matrix elements can in general be different from the set employed for the
interpolation of the dynamical matrix. This procedure used to determine the
optimal set of vectors is sketched in Fig. 18. The set of direct lattice
vectors $\mathbf{R}_{p}$ and $\mathbf{R}_{p^{\prime\prime}}$ are selected such
that the trial point
$\mathbf{t}=\mathbf{R}_{p}+\mathbf{r}_{m}-\mathbf{r}_{n}$, constructed from
the distance between two Wannier centers, or
$\mathbf{t}^{{}^{\prime\prime}}=\mathbf{R}_{p^{\prime\prime}}+\boldsymbol{\tau}_{\kappa}-\mathbf{r}_{n}$
constructed from a Wannier center and an atomic position in the central cell
and any other cell, is closer to the supercell center $\mathbf{0}$ than any
other supercell centers $\mathbf{S}$. In the case of a trial point at equal
distance between $\mathbf{0}$ and $\mathbf{S}$ (i.e., on the surface of the
supercell Wigner-Seitz cell) a weight proportional to the number of
degeneracies is given. The optimal Wigner-Seitz construction is activated with
the new use_ws input variable in EPW. As a sanity check, we verified that the
mobility of 3C-SiC calculated on a 8$\times$8$\times$8 k-point and
4$\times$4$\times$4 q-point grid was the same as the one obtained from a
2$\times$1$\times$1 supercell with a 4$\times$8$\times$8 k-point and
2$\times$4$\times$4 q-point grid.

Figure 19:  Acoustic phonon dispersions of 3C-SiC near the zone center,
interpolated from a coarse 4$\times$4$\times$4 $\mathbf{k}$-point and
$\mathbf{q}$-point grid using a $\Gamma$-centered fixed Wigner-Seitz cell with
dipole-dipole correction (dash gray line), an optimal Wigner-Seitz cell with
dipole-dipole correction (red line), and an optimal Wigner-Seitz cell with
dipole-dipole, dipole-quadrupole and quadrupole-quadrupole corrections (blue
line). The black dots are reference datapoints obtained via direct DFPT
calculations.  Figure 20:  Deformation potential of 3C-SiC without spin-orbit
coupling of the first electronic band ($m=n=1$) on a coarse
3$\times$3$\times$3 $\mathbf{k}$-point and $\mathbf{q}$-point grid including
dipole and quadrupole corrections. The Wigner-Seitz cell is either
$\Gamma$-centered or constructed using
$\mathbf{R}_{p}+\mathbf{r}_{m}-\mathbf{r}_{n}\in(i_{p}\times j_{p}\times
o_{p})$ for the electrons and
$\mathbf{R}_{p^{\prime\prime}}+\boldsymbol{\tau}_{\kappa}-\mathbf{r}_{n}\in(i_{p^{\prime\prime}}\times
j_{p^{\prime\prime}}\times o_{p^{\prime\prime}})$ for the phonons,
respectively. The black dots are reference datapoints obtained via direct DFPT
calculations.

In Fig. 19 we report the calculated phonon dispersions of the three acoustic
modes of 3C-SiC. The dashed gray line represents the interpolation from a
coarse $\mathbf{q}$-point grid using a $\Gamma$-centered fixed Wigner-Seitz
cell with dipole-dipole correction. The red line is for an optimal Wigner-
Seitz cell with dipole-dipole correction. The blue line is for an optimal
Wigner-Seitz cell with dipole-dipole, dipole-quadrupole and quadrupole-
quadrupole corrections. We can see that the $\Gamma$-centered Wigner-Seitz
cell construction yields spurious soft phonon modes along the $\Gamma-L$
direction. The improved Wigner-Seitz construction eliminates this artifact.

We also show in Fig. 20 a comparison between the deformation potential of
3C-SiC computed with a $\Gamma$-centered Wigner-Seitz cell or with the
improved construction. The new construction produces results closer to those
obtained directly with DFPT everywhere. We note that the results shown in Fig.
20 are for an unconverged small coarse grid of 3$\times$3$\times$3 for both
$\mathbf{k}$\- and $\mathbf{q}$-points. When using denser coarse grids, these
difference rapidly disappear (at least for the present case of cubic
semiconductors). Given that we use much denser coarse grids in this work, for
simplicity we employed the basic $\Gamma$-centered construction in all
calculations reported in this work.

##  Appendix B Additional figures and tables

Figure 21:  Conduction band edge of silicon – including spin-orbit coupling (SOC) – where the zero energy has been set at the X high-symmetry point and computed using DFT (black line). (a) Electronic bands obtained using Wannier interpolation with $\mathbf{k}$-point grids ranging from $10^{3}$ to $16^{3}$ and constructed from an initial set of four $sp^{3}$ projections on each Si atom. Both the valence and conduction bands have been Wannierized, totalling 16 Wannier functions due to SOC. (b) Electronic bands obtained using Wannier interpolation with $\mathbf{k}$-point grids ranging from $10^{3}$ to $16^{3}$ and constructed from an initial set of one $s$ and five $d$ projections on one Si atom only. Only the conduction bands have been Wannierized, totalling 12 Wannier functions due to SOC.  Figure 22:  Effective quadrupole term $\tilde{Q}_{\kappa\alpha}^{\beta\gamma}$ obtained from the least-mean square fit minimization of direct DFPT calculations for 3C-SiC. In these calculations we used the LDA exchange and correlation functional and without nonlinear core correction, so that we can compare with Ref. Brunin et al., 2020b. The two non-equivalent values of the quadrupole tensor are obtained by using the phonon frequencies, Born-effective charges, dielectric constant, eigendisplacement vectors and wavefunction overlaps from explicit DFPT calculations, and by optimizing the quadrupole tensor using Eq. (40). We repeat the fitting for several sets of $\mathbf{q}$-points on spherical surfaces of varying radii $R$. The radii are in units of $2\pi/a$.  Compounds | Coarse | Temp. | Wind. | Fine | Hole mobility | Hall factor | Wind. | Fine | Electron mobility | Hall factor
---|---|---|---|---|---|---|---|---|---|---
| k | q | (K) | eV | grids | SERTA | BTE | SERTA | BTE | eV | grid | SERTA | BTE | SERTA | BTE
Diamond |  83 |  43 | 100 | 0.40 |  1003 | 6694.2 | 6700.1 | 0.721 | 0.749 | 0.40 |  803 | 11016 | 10675 | 0.656 | 0.662
|  |  | 300 |  |  | 1048.5 | 1051.3 | 0.755 | 0.792 |  |  | 1713.4 | 1746.3 | 0.809 | 0.795
| 83 | 43 | 100 | 0.30 | 1003 | 6694.4 | 6700.4 | 0.721 | 0.745 | 0.30 | 803 | 11016 | 10676 | 0.656 | 0.662
|  |  | 300 |  |  | 1049.6 | 1052.4 | 0.755 | 0.793 |  |  | 1714.2 | 1747.0 | 0.809 | 0.795
|  83 |  43 | 100 | 0.20 |  1003 | 6703.6 | 6709.5 | 0.720 | 0.748 | 0.20 |  803 | 11035 | 10693 | 0.655 | 0.661
|  |  | 300 |  |  | 1084.2 | 1087.7 | 0.752 | 0.790 |  |  | 1778.2 | 1817.1 | 0.804 | 0.792
Silicon |  83 |  43 | 100 | 0.30 |  603 | 7157.9 | 7053.0 | 0.129 | 0.135 | 0.30 |  603 | 12592 | 13137 | 0.638 | 0.593
|  |  | 300 |  |  | 691.43 | 708.94 | 0.292 | 0.335 |  |  | 1331.0 | 1358.9 | 0.939 | 0.900
| 83 | 43 | 100 | 0.20 | 603 | 7175.5 | 7070.4 | 0.129 | 0.135 | 0.20 | 603 | 12618 | 13163 | 0.637 | 0.592
|  |  | 300 |  |  | 695.19 | 712.82 | 0.337 | 0.295 |  |  | 1334.1 | 1362.2 | 0.937 | 0.898
|  83 |  43 | 100 | 0.15 |  603 | 7236.4 | 7130.4 | 0.128 | 0.134 | 0.15 |  603 | 12740 | 13290 | 0.631 | 0.586
|  |  | 300 |  |  | 710.44 | 728.95 | 0.297 | 0.339 |  |  | 1352.5 | 1381.2 | 0.927 | 0.889
GaAs |  83 |  43 | 100 | 0.30 |  1003 | 3118.7 | 3160.3 | 0.518 | 0.535 | 0.30 |  2003 | 529560 | 660550 | 0.711 | 0.801
|  |  | 300 |  |  | 326.35 | 432.33 | 0.932 | 0.907 |  |  | 23875 | 45296 | 1.060 | 0.971
| 83 | 43 | 100 | 0.20 | 1003 | 3125.2 | 3166.8 | 0.517 | 0.534 | 0.20 | 2003 | 531150 | 662530 | 0.709 | 0.799
|  |  | 300 |  |  | 326.78 | 433.04 | 0.933 | 0.908 |  |  | 23925 | 45253 | 1.059 | 0.973
|  83 |  43 | 100 | 0.15 |  1003 | 3156.8 | 3198.9 | 0.511 | 0.529 | 0.15 |  2003 | 537830 | 670860 | 0.701 | 0.789
|  |  | 300 |  |  | 329.13 | 436.55 | 0.953 | 0.936 |  |  | 24216 | 45262 | 1.054 | 0.981
3C-SiC |  83 |  43 | 100 | 0.30 |  603 | 274.85 | 308.89 | 0.457 | 0.470 | 0.30 | 603 | 3418.8 | 3123.1 | 0.964 | 1.111
|  |  | 300 |  |  | 82.11 | 91.82 | 0.532 | 0.583 |  |  | 875.61 | 923.19 | 0.727 | 0.780
|  83 |  43 | 100 | 0.20 | 603 | 275.03 | 309.10 | 0.457 | 0.469 | 0.20 |  603 | 3423.1 | 3127.1 | 0.963 | 1.109
|  |  | 300 |  |  | 82.76 | 92.56 | 0.534 | 0.584 |  |  | 893.84 | 943.62 | 0.731 | 0.784
|  83 |  43 | 100 | 0.15 |  603 | 276.16 | 310.37 | 0.455 | 0.467 | 0.15 |  603 | 3443.3 | 3145.5 | 0.958 | 1.103
|  |  | 300 |  |  | 87.12 | 97.76 | 0.544 | 0.592 |  |  | 1004.1 | 1053.6 | 0.726 | 0.784
AlP |  83 |  43 | 100 | 0.30 |  603 | 208.38 | 277.63 | 0.263 | 0.538 | 0.30 |  603 | 1036.02 | 1276.04 | 0.807 | 0.964
|  |  | 300 |  |  | 42.74 | 54.74 | 0.396 | 0.589 |  |  | 190.76 | 274.31 | 0.868 | 0.915
| 83 | 43 | 100 | 0.20 | 603 | 208.67 | 278.02 | 0.262 | 0.538 | 0.20 | 603 | 1037.23 | 1277.52 | 0.806 | 0.963
|  |  | 300 |  |  | 42.79 | 54.81 | 0.400 | 0.594 |  |  | 190.87 | 274.36 | 0.867 | 0.914
|  83 |  43 | 100 | 0.15 |  603 | 210.40 | 280.33 | 0.260 | 0.533 | 0.15 |  603 | 1042.66 | 1284.22 | 0.802 | 0.958
|  |  | 300 |  |  | 43.09 | 55.24 | 0.416 | 0.614 |  |  | 191.58 | 274.84 | 0.866 | 0.914
GaP |  83 |  43 | 100 | 0.30 |  603 | 846.71 | 937.41 | 0.257 | 0.414 | 0.30 |  603 | 2379.37 | 2237.48 | 0.619 | 0.722
|  |  | 300 |  |  | 136.15 | 171.62 | 0.412 | 0.546 |  |  | 225.58 | 275.75 | 0.892 | 0.909
| 83 | 43 | 100 | 0.20 | 603 | 848.06 | 938.90 | 0.257 | 0.414 | 0.20 | 603 | 2382.49 | 2240.41 | 0.618 | 0.721
|  |  | 300 |  |  | 136.30 | 171.88 | 0.419 | 0.555 |  |  | 225.76 | 276.00 | 0.891 | 0.909
|  83 |  43 | 100 | 0.15 |  603 | 853.61 | 945.04 | 0.255 | 0.411 | 0.15 |  603 | 2399.07 | 2256.00 | 0.614 | 0.716
|  |  | 300 |  |  | 137.25 | 173.31 | 0.436 | 0.575 |  |  | 226.73 | 277.23 | 0.889 | 0.906
c-BN |  83 |  43 | 100 | 0.40 |  603 | 985.49 | 1213.15 | 0.287 | 0.474 | 0.40 |  603 | 4856.13 | 4430.66 | 0.958 | 1.094
|  |  | 300 |  |  | 243.48 | 282.42 | 0.430 | 0.592 |  |  | 1255.37 | 1282.49 | 0.802 | 0.862
| 83 | 43 | 100 | 0.30 | 603 | 985.52 | 1213.18 | 0.287 | 0.474 | 0.30 | 603 | 4856.26 | 4430.78 | 0.958 | 1.094
|  |  | 300 |  |  | 243.62 | 282.59 | 0.430 | 0.592 |  |  | 1257.75 | 1285.23 | 0.803 | 0.863
|  83 |  43 | 100 | 0.20 |  603 | 986.82 | 1214.77 | 0.287 | 0.474 | 0.20 |  603 | 4863.11 | 4437.03 | 0.957 | 1.093
|  |  | 300 |  |  | 256.59 | 300.79 | 0.434 | 0.604 |  |  | 1427.53 | 1453.02 | 0.816 | 0.879
|  83 |  43 | 100 | 0.15 |  603 | 992.62 | 1221.85 | 0.287 | 0.472 | 0.15 |  603 | 4889.02 | 4460.67 | 0.953 | 1.088
|  |  | 300 |  |  | 272.69 | 324.29 | 0.436 | 0.614 |  |  | 1622.44 | 1631.41 | 0.798 | 0.867
AlAs |  83 |  43 | 100 | 0.30 |  603 | 547.44 | 665.68 | 0.372 | 0.613 | 0.30 |  603 | 1535.47 | 1595.03 | 0.674 | 0.865
|  |  | 300 |  |  | 88.59 | 118.90 | 0.710 | 0.835 |  |  | 212.47 | 280.23 | 0.864 | 0.912
| 83 | 43 | 100 | 0.20 | 603 | 548.32 | 666.75 | 0.372 | 0.613 | 0.20 | 603 | 1537.13 | 1596.76 | 0.673 | 0.864
|  |  | 300 |  |  | 88.71 | 119.11 | 0.711 | 0.836 |  |  | 212.57 | 280.32 | 0.863 | 0.911
|  83 |  43 | 100 | 0.15 |  603 | 553.67 | 673.25 | 0.368 | 0.607 | 0.15 |  603 | 1544.94 | 1604.87 | 0.670 | 0.860
|  |  | 300 |  |  | 89.35 | 120.13 | 0.721 | 0.852 |  |  | 212.99 | 280.73 | 0.862 | 0.909
|  83 |  43 | 100 | 0.10 |  603 | 575.50 | 699.82 | 0.355 | 0.584 | 0.10 |  603 | 1602.74 | 1666.23 | 0.647 | 0.828
|  |  | 300 |  |  | 93.27 | 125.11 | 0.779 | 0.943 |  |  | 225.04 | 291.55 | 0.863 | 0.928
AlSb |  83 |  43 | 100 | 0.30 |  603 | 1720.10 | 1785.65 | 0.346 | 0.515 | 0.30 |  603 | 1460.25 | 1394.00 | 0.645 | 0.703
|  |  | 300 |  |  | 304.05 | 383.03 | 0.828 | 0.849 |  |  | 196.86 | 221.20 | 0.835 | 0.842
| 83 | 43 | 100 | 0.20 | 603 | 1722.36 | 1787.99 | 0.345 | 0.515 | 0.20 | 603 | 1462.17 | 1395.84 | 0.644 | 0.702
|  |  | 300 |  |  | 304.61 | 383.90 | 0.837 | 0.861 |  |  | 196.95 | 221.33 | 0.836 | 0.842
|  83 |  43 | 100 | 0.15 |  603 | 1735.02 | 1801.14 | 0.343 | 0.511 | 0.15 |  603 | 1471.69 | 1404.93 | 0.640 | 0.698
|  |  | 300 |  |  | 308.14 | 388.85 | 0.892 | 0.934 |  |  | 197.38 | 221.81 | 0.839 | 0.847
SrO |  83 |  43 | 100 | 0.30 |  603 | 319.91 | 331.05 | 0.795 | 0.794 | 0.30 |  603 | 166.66 | 152.33 | 0.761 | 0.793
|  |  | 300 |  |  | 5.89 | 7.64 | 0.940 | 0.931 |  |  | 18.36 | 22.15 | 0.965 | 0.941
| 83 | 43 | 100 | 0.20 | 603 | 321.22 | 332.41 | 0.791 | 0.791 | 0.20 | 603 | 166.93 | 152.58 | 0.760 | 0.792
|  |  | 300 |  |  | 5.93 | 7.68 | 0.935 | 0.929 |  |  | 18.36 | 22.09 | 0.964 | 0.940
|  83 |  43 | 100 | 0.15 |  603 | 327.94 | 339.35 | 0.775 | 0.774 | 0.15 |  603 | 167.90 | 153.46 | 0.756 | 0.787
|  |  | 300 |  |  | 6.59 | 8.37 | 1.028 | 0.972 |  |  | 18.41 | 22.01 | 0.965 | 0.944
Table 7:  Sensitivity of the calculated mobility and Hall factor to the size of the energy window in the BTE. The calculations have been performed using adaptive smearing. The numbers indicated in bold represent the smallest window such that the results are unaffected. The same energy window has been used throughout this work.  | Densest grids | k/q extrapolation |  | k/q extrapolation |
---|---|---|---|---|---
| Drift mobility | Drift mobility | Hall mobility | Exp. | Drift slope | Hall slope | Exp.
Materials | SERTA | BTE | SERTA | BTE | SERTA | BTE | mobility | SERTA | BTE | SERTA | BTE | slope
C-e | 1613 | 1671 | 1583 | 1634 | 1664 | 1705 |  1800 Haynes (2020) | -1.81 | -1.82 | -1.79 | -1.81 | -1.4† Nesladek et al. (2008)
|  |  |  |  |  |  | 1800 Madelung (2002) |  |  |  |  | -1.6† Gabrysch et al. (2011)
|  |  |  |  |  |  | 1802† Jansen et al. (2013) |  |  |  |  |  -1.88† Gabrysch et al. (2011)
|  |  |  |  |  |  | 1940† Gabrysch et al. (2011) |  |  |  |  |
|  |  |  |  |  |  | 2000 Vavilov and Konorova (1976) |  |  |  |  |
|  |  |  |  |  |  | 2750† Nesladek et al. (2008) |  |  |  |  |
C-h | 2256 | 2290 | 2364 | 2410 | 2409 | 2467 | 1550 Vavilov and Konorova (1976) | -1.97 | -1.99 | -1.88 | -1.89 | -1.72 Isberg et al. (2005)
|  |  |  |  |  |  |  2100 Madelung (2002) |  |  |  |  | -1.5 Madelung (2002)
|  |  |  |  |  |  | 2250† Nesladek et al. (2008) |  |  |  |  | -1.5† Nesladek et al. (2008)
|  |  |  |  |  |  | 2300† Gabrysch et al. (2011) |  |  |  |  |  -2.06† Gabrysch et al. (2011)
|  |  |  |  |  |  | 2534† Jansen et al. (2013) |  |  |  |  | -2.8 Madelung (2002)
Si-e | 1383 | 1429 | 1352 | 1383 | 1782 | 1770 |  1350† Madelung (2002) | -2.19 | -2.21 | -2.06 | -2.05 |  -2.31 Norton et al. (1973)
|  |  |  |  |  |  | 1350† Ludwig and Watters (1956) |  |  |  |  | -2.42 Madelung (2002)
|  |  |  |  |  |  | 1350† Jacoboni et al. (1977) |  |  |  |  |
|  |  |  |  |  |  | 1430† Norton et al. (1973) |  |  |  |  |
|  |  |  |  |  |  | 1450† Sze and Ng (2007) |  |  |  |  |
Si-h | 623 | 627 | 600 | 601 | 572 | 574 |  445† Jacoboni et al. (1977) | -2.55 | -2.53 | -3.02 | -2.93 | -2.44 Ludwig and Watters (1956)
|  |  |  |  |  |  | 480† Ludwig and Watters (1956) |  |  |  |  | -2.2 Madelung (2002)
|  |  |  |  |  |  | 480† Madelung (2002) |  |  |  |  |  -3.09 Logan and Peters (1960)
|  |  |  |  |  |  | 500† Sze and Ng (2007) |  |  |  |  |
GaAs-e | 8743 | 14799 | 8853 | 15099 | 12354 | 17860 | 7200 Madelung (2003) | -2.69 | -2.21 | -2.66 | -2.29 | -1.50 Stanley et al. (1991)
|  |  |  |  |  |  | 8000 Hicks and Manley (1969) |  |  |  |  | -2.12 Hicks and Manley (1969)
|  |  |  |  |  |  | 8865 Madelung (2003) |  |  |  |  |  -2.39 Lin et al. (1982)
|  |  |  |  |  |  | 8500 Rode and Knight (1971) |  |  |  |  |
|  |  |  |  |  |  |  9000 Madelung (2003) |  |  |  |  |
|  |  |  |  |  |  | 9750 Madelung (2003) |  |  |  |  |
GaAs-h | 392 | 500 | 389 | 500 | 988 | 1068 | 396 Hill (1970) | -2.45 | -2.22 | -2.40 | -2.31 | -2.23 Hill (1970)
|  |  |  |  |  |  | 450 Wenzel et al. (1998) |  |  |  |  |  -2.39 Kim et al. (1991)
|  |  |  |  |  |  | 450 Kim et al. (1991) |  |  |  |  |
|  |  |  |  |  |  | 450 Sotoodeh et al. (2000) |  |  |  |  |
|  |  |  |  |  |  | 460 Mears and Stradling (1971) |  |  |  |  |
|  |  |  |  |  |  |  500 Haynes (2020) |  |  |  |  |
SiC-e | 1885 | 2068 | 1803 | 1944 | 2204 | 2257 | 510 Madelung (2003) | -2.90 | -2.92 | -2.85 | -2.86 |  -1.50 Shinohara et al. (1988)
|  |  |  |  |  |  | 670 Kaplan et al. (1985) |  |  |  |  |
|  |  |  |  |  |  | 733 Shinohara et al. (1988) |  |  |  |  |
|  |  |  |  |  |  | 800 Mnatsakanov et al. (2001) |  |  |  |  |
|  |  |  |  |  |  | 890 Nelson et al. (1966) |  |  |  |  |
|  |  |  |  |  |  |  1000 Bhatnagar and Baliga (1993) |  |  |  |  |
SiC-h | 166 | 170 | 165 | 170 | 105 | 111 | 21 Madelung (2003) | -2.06 | -2.06 | -1.91 | -1.97 |  -1.75 Lebedev et al. (2008)
|  |  |  |  |  |  | 40 Yamanaka et al. (1987) |  |  |  |  |
|  |  |  |  |  |  |  100 Lebedev et al. (2008) |  |  |  |  |
AlP-e | 461 | 556 | 454 | 554 | 629 | 698 | 80 Madelung (2003) | -3.15 | -2.99 | -2.89 | -2.79 | -
AlP-h | 94 | 102 | 93 | 101 | 84 | 92 | - | -2.96 | -2.85 | -3.43 | -3.31 | -
GaP-e | 281 | 325 | 284 | 328 | 409 | 442 | 165 Craford et al. (1971) | -2.58 | -2.45 | -2.43 | -2.35 | -1.52 Kao and Eknoyan (1983)
|  |  |  |  |  |  |  300 Haynes (2020) |  |  |  |  |  -2.12 Craford et al. (1971)
GaP-h | 228 | 271 | 231 | 276 | 315 | 353 | 135 Madelung (2002) | -2.96 | -2.81 | -3.00 | -2.89 | -1.95 Kao and Eknoyan (1983)
|  |  |  |  |  |  |  150 Haynes (2020) |  |  |  |  |  -2.37 Casey et al. (1969)
BN-e | 573 | 1002 | 470 | 967 | 456 | 956 | - | -1.28 | -1.40 | -1.20 | -1.33 | -
BN-h | 196 | 321 | 170 | 319 | 124 | 281 | 500 Madelung (2002) | -1.31 | -1.49 | -1.18 | -1.51 |
AlAs-e | 409 | 507 | 405 | 501 | 578 | 640 | 161 Chand et al. (1984) | -3.14 | -2.94 | -2.84 | -2.71 | -
|  |  |  |  |  |  |  294 Madelung (2003) |  |  |  |  |
AlAs-h | 181 | 217 | 180 | 215 | 337 | 357 | 60 Sze and Ng (2007) | -3.08 | -2.86 | -2.79 | -2.73 | -
AlSb-e | 298 | 332 | 303 | 335 | 415 | 431 | 200 Madelung (2002) | -2.38 | -2.25 | -2.36 | -2.26 |  -1.46 Stirn and Becker (1966b)
|  |  |  |  |  |  |  300 Haynes (2020) |  |  |  |  |
AlSb-h | 491 | 598 | 490 | 600 | 918 | 1000 | 505 Madelung (2002) | -2.70 | -2.46 | -2.54 | -2.42 |  -1.95 Stirn and Becker (1966b)
|  |  |  |  |  |  |  550 Haynes (2020) |  |  |  |  |
SrO-e | 38 | 53 | 38 | 53 | 53 | 65 | - | -3.90 | -3.61 | -3.48 | -3.33 | -
SrO-h | 3.5 | 5.0 | 3.4 | 4.9 | 3.5 | 5.0 | - | -2.33 | -2.21 | -2.31 | -2.07 | -
Table 8:  Room temperature drift SERTA and BTE mobility (cm2/Vs) as well as
temperature exponent, as evaluated at the densest fine
$\mathbf{k}$/$\mathbf{q}$-point grids from Fig. 7, and drift SERTA and BTE
Hall mobility extrapolated to infinitely dense sampling. The experimental data
in bold are the reference values used in Figs. 16 and 17. The † symbol
indicates time-of-flight drift mobility experiments, while no symbols indicate
Hall measurements.  Figure 23:  Calculated hole drift mobility of c-BN for
various Brillouin zone grid sizes. The horizontal axis indicates the coarse
$\mathbf{q}$-point grid, and the lines correspond to different choices of the
coarse $\mathbf{k}$-point grid: same as the $\mathbf{q}$-point grid (blue),
twice as dense as the $\mathbf{q}$-point grid (green), and three times as
dense as the q-point grid (red). In all cases the fine grids were set to 603
points for both $\mathbf{k}$ and $\mathbf{q}$. We also show with dashed lines
the relative computational time needed in each case (normalized to the most
time-consuming calculation). These tests show that the results obtained by
using equal coarse grids for electrons and phonons are of the same accuracy as
those with denser electron grids, but much cheaper computationally. Therefore
we recommend to always use equal coarse grids.  Figure 24:  (a) Convergence of
the calculated Hall mobility of GaAs with the fine Brillouin zone grid size.
Both SERTA and BTE results are shown using the converged coarse grids from
Fig. 4. We consider several smearing parameters as well as the adaptive
smearing. (b) Calculated electron mobility of GaAs, extrapolated to infinitely
dense Brillouin zone sampling, for several smearing parameters. The
extrapolation of these results at zero smearing agrees with calculations using
adaptive smearing (stars).  Figure 25:  Calculated electron and hole
scattering rates as a function of energy at 150 K, 300 K, and 500 K. The
energy range displayed here encompasses the energy window used in the
calculations of carrier mobilities.

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