# Effect of Electron-Phonon and Electron-Impurity Scattering on Electronic
Transport Properties of Silicon/Germanium Superlattices

Manoj Settipalli  Ann and H.J. Smead Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, CO 80303, USA  Vitaly S Proshchenko
Ann and H.J. Smead Aerospace Engineering Sciences, University of Colorado
Boulder, Boulder, CO 80303, USA  Sanghamitra Neogi
<EMAIL_ADDRESS>Ann and H.J. Smead Aerospace Engineering
Sciences, University of Colorado Boulder, Boulder, CO 80303, USA

###### Abstract

Semiconductor superlattices have been extensively investigated for
thermoelectric applications, to explore the effects of compositions, interface
structures, and lattice strain environments on the reduction of thermal
conductivity, and improvement of efficiency. Most studies assumed that
electronic properties of superlattices remain unaffected compared to their
bulk counterparts. However, recent studies demonstrated that electronic
properties of silicon (Si)/germanium (Ge) superlattices show significant
variations depending on compositions and growth substrates. These studies used
a constant relaxation time approximation, and ignored the effects of electron
scattering processes. Here, we consider electron scattering with phonons and
ionized impurities, and report first-principles calculations of electronic
transport properties including the scattering rates. We investigate two
classes of Si/Ge short-period superlattices: superlattices with varied
compositions grown on identical substrates and with identical compositions but
grown on different substrates. We illustrate the relationship between the
energy bands of the superlattices and the electron-phonon relaxation times. We
model the electron-ionized impurity interaction potentials by explicitly
accounting for the in-plane and the cross-plane structural anisotropy of the
configurations. Our analysis reveals that the inclusion of electron-phonon and
electron-impurity scattering processes can lead to $~{}\sim{1.56}$-fold
improved peak power-factor of superlattices, compared to that of bulk Si. We
observe that superlattices can also display dramatically reduced power-factors
for certain lattice strain environments. Such reduction could cancel out
potential thermoelectric efficiency improvements due to reduced thermal
conductivities. Our study provides insight to predict variation of electronic
properties due to changes in lattice strain environments, essential for
designing superlattices with optimized electronic properties.

##  I Introduction

Semiconductor nanostructures, in particular silicon (Si)/germanium (Ge) based
materials have risen to prominence over the past few decades owing to their
applications in key technologies, including electronics [1, 2, 3, 4],
optoelectronics [5, 6], thermoelectrics [7, 8, 9, 10], spintronics [11, 12],
and quantum applications [13]. Nanostructuring approaches have opened
remarkable possibilities to design novel Si/Ge based materials, with tailored
electronic and thermal properties, to enable these applications. A
thermoelectric material operates at its peak efficiency when the
thermoelectric figure of merit, ZT$=\text{S}^{2}\sigma\text{T}/\kappa$, is
maximized. Here, S is the Seebeck coefficient or thermopower, $\sigma$ is the
electrical conductivity, $\kappa$ is the electronic and the lattice thermal
conductivity, and T is the absolute temperature. Over the past two decades,
several nanostructuring strategies have been proposed to reduce the thermal
conductivity of Si based thermoelectric materials, and thus, improve ZT. Some
examples include two-dimensional silicene [14], Si-Ge nanomeshes [15],
layered-graded Si1-xGex/Si superlattices (SL) [16], and Si membranes with and
without surface nanostructures [17, 18, 19, 20, 21, 22, 23]. Si/Ge based SLs,
in particular, have been extensively studied to maximize the reduction of
thermal conductivity by varying layer compositions [24, 25, 26], interface
structures [27, 28, 29], and lattice strain environments [27, 28, 29].
Although most studies focused on identifying strategies to reduce $\kappa$,
there has been an increasing interest in improving the electronic power-factor
(PF), S${}^{2}\sigma$. The combination of reduced thermal conductivity and
improved PF promises to greatly improve the thermoelectric efficiency of a
nanostructured material.

Prediction of electronic properties of materials, using first-principles
density functional theory (DFT)-based approaches, has seen rapid progress in
recent years. However, it still remains a challenge to predict electronic
transport properties of nanostructured materials using these approaches. The
structural variability introduced by non-uniform strain environments,
dislocations and other growth dependent parameters is often too difficult to
represent in first-principles models due to large computational expenses,
making the prediction unreliable. As a consequence, only a few first-
principles studies analyzed the electronic structure [30, 31, 32, 33] or
transport properties [34, 35, 36, 37, 38, 39] of even planar Si/Ge SLs.
Electronic transport properties of nonpolar semiconductors are significantly
affected by electron scattering due to lattice vibrations, ionized impurities,
neutral impurities, dislocations, vacancies, and interstitials [40, 41, 42].
The effect of various scattering mechanisms on electronic and thermal
properties of Si nanostructures has been acknowledged in previous studies [43,
44, 45]. However, most of these studies used carrier concentration-dependent
electron relaxation times [46, 19, 34, 47], parametrized by the experimental
bulk Si mobility data [48, 49]. The effects of electron-phonon scattering on
thermal conductivities and the resultant ZT of bulk [46] and nanocrystalline
[44] Si, Si nanostructures [43], SiGe alloys [44, 45], and SiGe compounds
[50], have been investigated with first-principles approaches. In comparison,
the role of the different scattering processes on electronic transport
properties of Si/Ge based materials have received little attention [51]. In
particular, the effects of different electron scattering processes on the
electronic transport coefficients of Si/Ge based superlattices, such as the
Seebeck coefficients, the electronic resistivities or the power-factors, are
not established.

In this article, we illustrate the effects of electron-phonon (EPS) and
electron-ionized impurity scattering (IMS) processes on the electronic
transport properties of n-type [001]-Si/Ge based superlattices. We investigate
two classes of eight-atom Si/Ge SLs with diverse lattice strain environments:
SLs with varied layer compositions grown on same substrate and SLs with
identical compositions but grown on different substrates. The role of lattice
strain and interface roughness on electronic properties of p-type
Ge/Si0.5Ge0.5 [52, 53] and n-type SiGe/Si SLs [8, 54] has been highlighted by
experimental studies. In our recent first-principles studies [36, 37, 38, 39],
we showed that the cross-plane electronic transport properties of n-type
[001]-Si/Ge SLs can be tuned by varying the SL layer compositions, periods,
and growth substrates. We noted that such nanostructuring strategies
positively impact the PF in the high-doping regime. However, the past studies
primarily employed the constant relaxation time approximation (CRTA) within
the BTE framework. Here, we calculate the electronic structure properties of
the Si/Ge SLs by performing DFT calculations, as implemented in the Quantum
Espresso package [55]. We compute the EPS and IMS rates using a perturbative
approach that follows Fermi’s golden rule. The electron-phonon matrix elements
are computed using the electron-phonon Wannier (EPW) package based on
maximally localized Wannier functions [56, 57]. This approach provides
accurate interpolation of the matrix elements from coarse grids to dense grids
[58], and thus reduces computational expenses. We calculate the cross-plane
electronic transport properties (S, $\rho$, PF) of the Si/Ge SLs by
incorporating the energy dependent electron scattering rates within the semi-
classical Boltzmann transport equation (BTE) framework. Our study establishes
the role of the strain-modulated electronic density of states, group
velocities and energy-dependent scattering rates on the electronic transport
properties. We highlight the role of different scattering processes, and how
the predicted properties change when different electron scattering rate
approximations, such as constant relaxation time and acoustic deformation
potentials scattering approximations, are used. The physical insights
demonstrated here not only are valuable to predict band movement and electron
relaxation in complex heterostructures, but essential to develop strain
engineering approaches to optimize electronic properties.

##  II Methods

Figure 1 shows the representative configurations of the model systems we
investigate in this work: eight-atom Si${}_{\text{n}}$Ge${}_{\text{m}}$ SL
unit cells composed of n and m monolayers of Si and Ge, respectively
($\text{n}+\text{m}=4$). The configurations include atomically ordered
electronic well (Si) and barrier (Ge) layers (Fig. 1(e)), similar to other
short-period SLs investigated in previous studies [59, 60]. We consider
pristine SL models that are free of dislocations and other defects.

Figure 1: Representative configurations investigated: (a) Eight-atom
conventional unit cell of bulk Si. Eight-atom unit cells of superlattices: (b)
Si1Ge3, (c) Si2Ge2, and (d) Si3Ge1. (e) Supercell of Si2Ge2 superlattice with
periodic replicas along the [001] direction, shown with light shade.

We investigate the electronic transport properties of SL models with carrier
concentration between $10^{17}$ and $10^{21}\ \text{cm}^{-3}$ at 300 K. In
this doping and temperature range, EPS and IMS processes are the dominant
mechanisms causing electron relaxation [61, 62, 63].

###  II.1 Electronic Transport Coefficients

We calculate the Seebeck coefficient (S), and the resistivity ($\rho$) of
Si/Ge SLs following the BTE framework. We evaluate the integrals [64, 65]

$\mathcal{L}_{\parallel,\perp}^{(n)}(\text{E}_{\text{F}},\text{T})=\int
d\text{E}\>\Xi_{\parallel,\perp}(\text{E})(\text{E}-\text{E}_{\text{F}})^{n}\left(-\frac{\partial
f(\text{E,T})}{\partial\text{E}}\right),$ (1)

over the respective SL first Brillouin (BZ) zone. Here, $\parallel$ and
$\perp$ refer to the in-plane and the cross-plane directions of the SLs,
respectively. The $\mathcal{L}$-integrands depend on the transport
distribution function (TDF), $\Xi(\text{E})$, the n${}^{\text{th}}$ power of
energy difference between energy, E, and the Fermi level, E${}_{\text{F}}$,
and the derivative of Fermi-Dirac distribution function $f$ at temperature T.
The TDF is proportional to an energy dependent relaxation time,
$\tau(\text{E})$, and an area integral, determined by the density of states
(DOS) $\left(\propto\oint_{\text{E}_{\bf k}=\text{E}}\frac{d\mathcal{A}}{|{\bf
v_{k}}|}\right)$ weighted by the group velocity squared, ${\bf v}_{{\bf
k},(\parallel,\perp)}^{2}$, and given by:

$\Xi_{\parallel,\perp}(\text{E})=\frac{\tau(\text{E})}{\hslash(2\pi)^{3}}\oint_{\text{E}_{\bf
k}=\text{E}}\frac{d\mathcal{A}}{|{\bf v_{k}}|}({\bf v}_{{\bf
k},(\parallel,\perp)})^{2}.$ (2)

We focus on the cross-plane ($\perp$) transport properties, and drop the
subscripts ($\parallel$,$\perp$) in the subsequent discussions. The cross-
plane transport coefficients are determined from the $\mathcal{L}$-integrals,
and are given by:

$\displaystyle\sigma$ $\displaystyle=\mathcal{L}^{0},\quad\rho=\sigma^{-1},$
(3) S $\displaystyle=\frac{1}{e\text{T}}\mathcal{L}^{1}/\mathcal{L}^{0},$ (4)

where $e$ is electron charge. We compute $\tau(\text{E})$’s using first-
principles approach and include them to obtain the transport coefficients,
following the BTE framework as implemented in the BoltzTrap code [65]. We
calculate the room temperature normal (diffusive) transport coefficients [66]
of eight-atom Si/Ge SLs at several carrier concentrations of interest.

###  II.2 Energy Dependent Electron Relaxation Times

We provide a brief summary of the current understanding of the EPS and the IMS
processes, and discuss our approach to calculate the electron relaxation times
in the SLs. We encourage the interested reader to consult the extensive
literature for further study [57, 67, 61]. We ignore the effects of electron-
electron interactions, since past studies showed that these mechanisms do not
alter the bulk Si mobility predictions by more than a few percent, at any
doping level [62, 68].

####  II.2.1 Electron-Phonon Scattering

We compute the EPS rates employing different approximations, and discuss the
effect of these approximations on the predicted rates and the electronic
transport properties of the SLs. To obtain the ab initio EPS rates, we
evaluate the first order electron-phonon matrix elements,

$\displaystyle
g^{\nu}_{mn}(\textbf{k},\textbf{q})=\langle\psi_{m\textbf{k}+\textbf{q}}|\partial_{\textbf{q}\nu}U|\psi_{n\textbf{k}}\rangle.$
(5)

Here, $\psi_{n\textbf{k}}$ is the electronic wave function of the Kohn-Sham
state with eigenenergy $\text{E}_{n\textbf{k}}$ corresponding to band $n$ and
wavevector k, $\omega_{\textbf{q}\nu}$ is the frequency of the phonon mode
with wavevector q and branch $\nu$, and $U$ is the Kohn-Sham self-consistent
electron-ion potential. We compute the electron and phonon eigenstates using
DFT and the electron-phonon matrix elements,
$g^{\nu}_{mn}(\textbf{k},\textbf{q})$, using DFPT [67, 57], on a coarse grid
that samples the respective reciprocal spaces. We then obtain interpolated
electron and phonon eigenstates and $g^{\nu}_{mn}(\textbf{k},\textbf{q})$, on
a dense grid, using maximally localized Wannier functions as implemented in
the EPW package, which provides accurate interpolation of electron-phonon
matrix elements from coarse grids to dense grids [58]. The Wannier90 package
within EPW, which performs these interpolations, was demonstrated to
successfully predict the electron-phonon coupling and superconducting
properties of several solid state systems [67, 57]. This approach, in
conjunction with exact solution of linearized BTE, has shown to accurately
predict bulk Si’s resistivities and Seebeck coefficients matching measured
data [69]. Using the dense grid electron-phonon matrix elements,
$g^{\nu}_{mn}(\textbf{k},\textbf{q})$, we obtain the electron self energy
($\Sigma$):

$\displaystyle\Sigma_{n\textbf{k}}$
$\displaystyle=\frac{1}{\hslash}\sum_{m\nu}\int\frac{d\textbf{q}}{\Omega_{BZ}}|g^{\nu}_{mn}(\textbf{k},\textbf{q})|^{2}$
$\displaystyle\times\bigg{\\{}\left[\frac{n(\omega_{\textbf{q}\nu},\text{T})+f(\text{E}_{m\textbf{k}+\textbf{q}}/\hslash,\text{T})}{\text{E}_{n\textbf{k}}/\hslash-\text{E}_{m\textbf{k}+\textbf{q}}/\hslash-\omega_{\textbf{q}\nu}+i\eta}\right]$
$\displaystyle+\left[\frac{n(\omega_{\textbf{q}\nu},\text{T})+1-f(\text{E}_{m\textbf{k}+\textbf{q}}/\hslash,\text{T})}{\text{E}_{n\textbf{k}}/\hslash-\text{E}_{m\textbf{k}+\textbf{q}}/\hslash+\omega_{\textbf{q}\nu}+i\eta}\right]\bigg{\\}}.$
(6)

Here $\Omega_{BZ}$ is the BZ volume, $n(\omega_{\textbf{q}\nu},\text{T})$ and
$f(\text{E}_{m\textbf{k}+\textbf{q}}/\hslash,\text{T})$ are the Bose-Einstein
and Fermi-Dirac occupation functions for phonons and electrons at temperature
T, respectively. The real and imaginary parts of self energy [57, 70] are
given by $\Sigma_{n\textbf{k}}^{\prime}$ and
$\Sigma_{n\textbf{k}}^{\prime\prime}$, respectively:
$\Sigma_{n\textbf{k}}=\Sigma_{n\textbf{k}}^{\prime}+i\Sigma_{n\textbf{k}}^{\prime\prime}$.
We obtain the electron-phonon relaxation times ($\tau_{\text{ep-EPW}}$) from
the imaginary parts of the self energy:

$\displaystyle\frac{1}{\tau_{\text{ep-
EPW},n\textbf{k}}}=\frac{2|\Sigma_{n\textbf{k}}^{\prime\prime}|}{\hslash}.$
(7)

The occupation functions in Eq. 6 introduce the effect of temperature on
electron relaxation times. However, we assume that the lattice vibrations are
in thermal equilibrium during the scattering process. This assumption is not
strictly valid when a temperature gradient is present across a material. The
temperature gradient could establish an additional electrical current in the
same direction as the heat flow, resulting in a phonon drag contribution to
the Seebeck effect that boosts the diffusive part. The effect of such coupled
electron-phonon dynamics on the thermoelectric properties of silicon recently
received renewed attention [66, 71, 69]. Several phonon engineering strategies
have been proposed to improve S with nanostructuring [69, 72], however,
contradictory results have been reported [73, 74]. To accurately account for
the effect of phonon engineering approaches on electronic transport
properties, one needs to solve the coupled BTE’s for electrons and phonons,
which makes it especially challenging for SL nanostructures, where phonons are
strongly affected and might deviate from bulk character. We believe that our
predicted trends of thermoelectric properties of SLs with diverse lattice
strain environment will persist after considering this effect. We leave the
discussion of phonon drag contributions to the SL thermoelectric properties
for future study. We also did not include the direct effect of doping on EPS
rates, and assumed that electrons are in their equilibrium energy states.
Previous ab initio study of SiGe compound showed that electron relaxation
times are relatively unaffected by change in carrier concentration, as opposed
to phonon relaxation times which can be significantly reduced by
electron–phonon coupling at high carrier concentrations [50].

We compare the ab initio EPW predicted EPS rates with the rates obtained
considering acoustic deformation potential scattering in the elastic limit. In
this limit, the EPS rates in non-polar semiconductors are assumed to generally
follow the DOS [75]. We refer to this concept as DOS scattering. We exploit
the proportionality to predict the relaxation times within the DOS scattering
approximation as $\tau_{\text{ep-DOS}}$: $1/\tau_{\text{ep-
DOS}}(\text{E})=\text{K}_{\text{el-ph}}\times\text{DOS}(\text{E})$. We apply
the DOS scattering approximation near the conduction band edge, the region of
interest for transport properties of n-doped SLs. We obtain the scaling
constants, K${}_{\text{el-ph}}$, by aligning DOS(E) with the EPS-EPW rates.
The proportionality constant, K${}_{\text{el-ph}}$, is related to the
deformation potential squared. Recently, deformation potential based concepts
have been utilized to develop computationally efficient approaches (AMSET) to
predict EPS rates [76]. In bulk systems like Si (Ge), electron transport
properties are mainly determined by the $\Delta$ (L) valley electrons, and the
deformation potentials can successfully describe the properties of the states
within these valleys. Subsequently, the DOS scattering approximation has been
exploited to accurately predict the electron and hole mobilities of bulk
silicon [77, 78]. In an earlier study, we computed thermopower (S) of Si/Ge
SLs employing DOS scattering approximation (see Fig. S2, in Ref. 37), and
compared with S obtained using CRTA. We noted that the trends of S obtained
with the two approximations match quite well, although the values differ from
each other. However, we acknowledged that a detailed analysis of the validity
of the DOS scattering approximation would be highly beneficial since this
approach could significantly expedite the computation of EPS rates. In this
article, we discuss the applicability of DOS approximation to predict energy-
dependent EPS rates and resulting electronic transport properties of Si/Ge
SLs. We demonstrate that it is not straightforward to apply such
approximation, especially since the value of K${}_{\text{el-ph}}$ is highly
dependent on the fitting procedure. We include figures demonstrating the
variability of K${}_{\text{el-ph}}$ values and the effect on the predicted
electronic transport properties in the Supplementary Information (SI). We
conclude that further work is required to substitute the expensive EPS-EPW
approach with the DOS scattering approximation approach.

####  II.2.2 Electron-Ionized Impurity Scattering

In addition to electron-phonon coupling, electron relaxation in Si/Ge SLs is
strongly affected by scattering due to ionized impurities, which are
ubiquitous in doped semiconductors [79]. We employ a modified Brooks-Herring
(BH) formulation to model the electron-ionized impurity scattering mechanisms.
In the BH model, the ionized impurities are assumed to be static, randomly
spaced objects, and a spherically symmetric Yukawa potential is used to model
the screening effect on electrons. This model provides an improved
representation of real systems by explicitly including a long-range Coulomb
tail for a screened ionized impurity, which was missing in the bare Coulomb
potential form of the Conwell and Weisskopf’s model [80]. The impurities are
assumed to not have any internal excitations, so the electron scatters from
them elastically. The impurity concentration is assumed to be dilute,
therefore, interference between successive scatterings is neglected. We
implement this simplifying assumption to reduce the complexity of the problem,
however, we acknowledge that this assumption may not be strictly valid at
higher carrier concentrations. The electron states are Bloch waves except for
occasional scattering from isolated impurities. The impurity causes the
electron in state k and band $n$ to elastically scatter to state
$(\textbf{k}+\textbf{q}-\textbf{G})$ and band $m$, which has the same energy.
Here, G is a reciprocal lattice vector. We use Fermi’s golden rule to
calculate the electron relaxation time due to impurity scattering as [61, 62]:

$\displaystyle\frac{1}{\tau_{im,n\textbf{k}}}$
$\displaystyle=\frac{2\pi\text{n}_{\text{i}}}{\hslash}\frac{V^{2}}{(2\pi)^{3}}\sum_{\textbf{G}}\sum_{m}\int
d\textbf{q}\>H^{2}_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$
$\displaystyle\times\delta(\text{E}_{n\textbf{k}}-\text{E}_{m\textbf{k}+\textbf{q}-\textbf{G}})(1-\cos{\theta}).$
(8)

Here, $\text{n}_{\text{i}}$ is the density of scattering centers. We assume
that all dopants in our models are ionized at $\text{T}=300$K. Therefore, the
density of scattering centers ($\text{n}_{\text{i}}$) is same as the carrier
concentration ($\text{n}_{\text{e}}$). $V$ is the volume of the SL
configurations, $\theta$ is the angle between electron group velocities in
state k and band $n$, $\textbf{v}_{n\textbf{k}}$, and in state
$\textbf{k}^{\prime}=\textbf{k}+\textbf{q}-\textbf{G}$ and band $m$,
$\textbf{v}_{\textbf{k}+\textbf{q}-\textbf{G}}$, before and after scattering,
respectively. The $\textbf{G}=0$ terms correspond to normal scattering and the
$\textbf{G}\neq 0$ to umklapp scattering [81], respectively.

The matrix element, $H_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$,
represents the perturbation term of the electron-ionized impurity interaction
Hamiltonian. We model the perturbation using an anisotropic impurity
scattering potential, $\phi(\textbf{r})$. We derive the anisotropic potential,
$\phi(\textbf{r})$, by solving the screened Poisson-Boltzmann equation, given
by

$\displaystyle\nabla\cdot(\mbox{\boldmath$\varepsilon$}\nabla\
-\lambda_{D}^{-2})\phi(\textbf{r})=\frac{-e\delta^{3}(\textbf{r})}{\varepsilon_{0}},\hskip
14.45377pt\mbox{\boldmath$\varepsilon$}=\begin{bmatrix}\varepsilon_{\parallel}&0&0\\\
0&\varepsilon_{\parallel}&0\\\ 0&0&\varepsilon_{\perp}\end{bmatrix}.$ (9)

Here, $\lambda_{D}$ is the modified Debye screen length,
$\lambda_{D}=\sqrt{\varepsilon_{0}k_{B}\text{T}/e^{2}\text{n}_{\text{i}}}$,
$k_{B}$ is the Boltzmann constant and $\varepsilon_{0}$ is the dielectric
permittivity of vacuum. We consider an anisotropic dielectric tensor,
$\varepsilon$, consisting of different in-plane ($x-y$)
($\varepsilon_{\parallel}$) and cross-plane ($z$) ($\varepsilon_{\perp}$)
dielectric constants, to explicitly account for the in-plane and cross-plane
structural anisotropy of the SL configurations, respectively. The in-plane SL
structural symmetry further imposes,
$\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_{\parallel}$. We solve for
$\phi(\textbf{r})$ from Eq. 9, by taking Fourier transform and performing a
transformation of variables from (r and k) to ($\textbf{r}^{\prime}$ and
$\textbf{k}^{\prime}$), where
$(r_{x}^{\prime},r_{y}^{\prime},r_{z}^{\prime})=(r_{x}/\sqrt{\varepsilon_{\parallel}},r_{y}/\sqrt{\varepsilon_{\parallel}},r_{z}/\sqrt{\varepsilon_{\perp}})$
and
$(k_{x}^{\prime},k_{y}^{\prime},k_{z}^{\prime})=(k_{x}\sqrt{\varepsilon_{\parallel}},k_{y}\sqrt{\varepsilon_{\parallel}},k_{z}\sqrt{\varepsilon_{\perp}})$.
The solution, in the form of a Yukawa potential, is given by

$\displaystyle\phi(\textbf{r})=\frac{e}{4\pi
r^{\prime}\varepsilon_{0}\sqrt{\varepsilon_{\parallel}^{2}\varepsilon_{\perp}}}e^{-r^{\prime}/\lambda_{D}},$
(10)

where
$r^{\prime}=|\textbf{r}^{\prime}|=\sqrt{(r_{x}^{2}+r_{y}^{2})/\varepsilon_{\parallel}+r_{z}^{2}/\varepsilon_{\perp}}$.
We obtain the dielectric constants $\varepsilon_{\parallel}$ and
$\varepsilon_{\perp}$ using first-principles calculations, as described in the
next subsection, and compute $\phi(\textbf{r})$ numerically for the subsequent
analysis. We use the anisotropic potential, $\phi(\textbf{r})$, to obtain the
perturbation matrix elements,
$H_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$, from

$\displaystyle H_{n\textbf{k},m\textbf{k}+\textbf{q-G}}$
$\displaystyle=\frac{1}{V}\int
d\textbf{r}\psi^{*}_{m\textbf{k}+\textbf{q}-\textbf{G}}(\textbf{r})\phi(\textbf{r})\psi_{n\textbf{k}}(\textbf{r}),$
(11)

where the electronic wave functions $\psi_{n\textbf{k}}$ are given by,
$\psi_{n\textbf{k}}=1/\sqrt{V}u_{n\textbf{k}}(\textbf{r})e^{i\textbf{k}\cdot\textbf{r}}$.
$u_{n\textbf{k}}(\textbf{r})$’s are the periodic parts of the Bloch states.
Inserting the expression of $\phi(\textbf{r})$ from Eq. 10 into Eq. 11 and
performing the Fourier integral over the crystal volume, we obtain

$\displaystyle H_{n\textbf{k},m\textbf{k}+\textbf{q-G}}$
$\displaystyle=\frac{e^{2}\lambda_{D}^{2}}{V\varepsilon_{0}(1+|\textbf{q}^{\prime}|^{2}\lambda_{D}^{2})}\zeta_{n\textbf{k},\textbf{k}+\textbf{q}-\textbf{G}},$
(12) $\displaystyle\text{with}\
\zeta_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$ $\displaystyle=\int
d\textbf{r}u^{*}_{m\textbf{k}+\textbf{q}-\textbf{G}}(\textbf{r})u_{n\textbf{k}}(\textbf{r})e^{i\textbf{G}\cdot\textbf{r}},$
(13)

where
$\textbf{q}^{\prime}=(\sqrt{\varepsilon_{\parallel}}\textbf{i}+\sqrt{\varepsilon_{\parallel}}\textbf{j}+\sqrt{\varepsilon_{\perp}}\textbf{k})\cdot\textbf{q}$.
Here, $\zeta_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$ is the overlap
integral, which we assume to be unity [69, 82]. We compute the IMS rates using
the matrix elements, $H_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$,
following Eq. 8.

Below, we include a brief discussion about two of the approximations we made
to calculate the IMS rates. First is the approximation that considers the
overlap integral to be unity. We performed rigorous evaluations of the overlap
integrals using the Wannier functions and calculated the resultant scattering
rates and the electronic transport coefficients. We noted that the full
considerations of $\zeta_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$ do
not significantly affect the electronic transport properties at most carrier
concentrations of interest. However, the computational cost increases by
$\sim~{}3-4$-fold compared to the cases that assumed
$\zeta_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}=1$, for different
carrier concentrations. The predicted S is considerably less affected due to
the inclusion of the overlap factors compared to $\rho$. For example, the two
S predictions stay within 5% for Si1Ge3 for all $n_{e}$ of interest and
maximum deviation of S is about 15% above $n_{e}=10^{20}$ cm-3 (Si), and
$7.5\times 10^{20}$ cm-3 (Si2Ge2). However, such incorporation results in
nonuniform variations of predicted resistivities in the high carrier
concentration regime. We did not observe a definite trend of the deviations of
SL transport coefficients by including the overlap factors. We acknowledge
that this aspect needs a thorough investigation, however, leave it for future
work and focus on the discussion of the trends of transport properties of SL
systems. Additionally, we only include contributions from the $G=0$ normal
scattering terms and ignore the $G\neq 0$ umklapp terms, due to the fact that
$H_{n\textbf{k},m\textbf{k}+\textbf{q}-\textbf{G}}$ decreases with increasing
$|\textbf{q}^{\prime}|$, as can be seen from Eq. 12. A number of umklapp terms
could contribute in principle, however, normal scattering terms are expected
to play a dominant role to determine the EMS rates [81]. We acknowledge that
umklapp terms may be comparable to the normal terms for SL systems, however,
further studies are required to establish the relative importance of the two
terms. We do not include these terms to keep the computational expenses at
minimum.

We average the k-dependent scattering rates for each type of scattering,
$I=\text{ep},\>\text{im}$, to obtain the energy-dependent electron relaxation
times. We numerically evaluate $\tau_{I}(\text{E})$ from

$\displaystyle\frac{1}{\tau_{\text{ep-EPW}}(\text{E})}$
$\displaystyle=\frac{1}{\text{DOS}(\text{E})}\sum_{n}\int_{\text{BZ}}\frac{d\textbf{k}}{\Omega_{\text{FBZ}}}\frac{1}{\tau_{ep-
EPW,n\textbf{k}}}\delta(\text{E}-\text{E}_{n\textbf{k}}),$
$\displaystyle\frac{1}{\tau_{\text{im}}(\text{E})}$
$\displaystyle=\frac{1}{\text{DOS}(\text{E})}\sum_{n}\int_{\text{BZ}}\frac{d\textbf{k}}{\Omega_{\text{FBZ}}}\frac{1}{\tau_{im,n\textbf{k}}}\delta(\text{E}-\text{E}_{n\textbf{k}}),$
(14)

by averaging $1/\tau_{I,n\textbf{k}}$, corresponding to
$\text{E}_{n\textbf{k}}$ in the energy window $\text{E}\pm\Delta$ with
$\Delta=5\times 10^{-4}$ Ry. Here $\Omega_{\text{FBZ}}$ is the FBZ volume. We
provide the relevant numerical details in the following Computational Details
section. This numerical averaging procedure has been demonstrated to
successfully predict the energy-dependent relaxation times in several studies
[82], including superconducting cuprates that have highly anisotropic cell-
axes and conduction properties [82, 83, 84, 85, 65]. We calculate the total
electron relaxation times due to electron-phonon (ep) and electron-ionized
impurity (im) scattering processes, using Matthiessen’s rule [82]:

$\displaystyle\frac{1}{\tau(\text{E})}$
$\displaystyle=\frac{1}{\tau_{\text{ep-
EPW}}(\text{E})}+\frac{1}{\tau_{\text{im}}(\text{E})},\ \ \text{ and}$ (15)
$\displaystyle\frac{1}{\tau(\text{E})}$
$\displaystyle=\frac{1}{\tau_{\text{ep-
DOS}}(\text{E})}+\frac{1}{\tau_{\text{im}}(\text{E})}.$ (16)

###  II.3 Computational Details

####  II.3.1 Energy Dependent Electron Relaxation Times

We calculate the electronic structure properties of eight-atom Si/Ge SLs by
performing self-consistent field (SCF) and non self-consistent field (NSCF)
DFT calculations, as implemented in the plane-waves code Quantum Espresso (QE)
[55]. We use non-relativistic norm-conserving pseudopotentials for Si and Ge
where the valence electrons ($3s^{2}3p^{2}$) are treated with the Perdrew-
Zunger parametrization of the local-density approximation (LDA) of the
exchange–correlation energy functional [86]. Similar numerical approach has
been followed in previous DFT studies that computed electron scattering rates
in Si [57, 69, 43]. We use cutoff energy of 45 Ry to expand the Kohn-Sham
orbitals in terms of a plane wave basis set for all calculations. A
convergence threshold of $10^{-9}$ Ry is chosen for self-consistency. We
ignore the effects of spin-orbit coupling on the energy bands in our
calculations. The LDA approximation is known to under-predict bulk Si and Ge
bandgaps. We compute the band gaps of the Si/Ge SLs separately using the Heyd-
Scuseria-Ernzerhof (HSE06) hybrid functionals [87]. We use a scissors operator
to correct the LDA band gaps using the HSE06 predictions, shown in Table 1. We
provide the details regarding geometry optimization and HSE06 band gap
calculations in SI.

We perform NSCF calculations to obtain electronic states and DFPT calculations
to obtain phonon modes. We use a coarse grid sampling of the respective
Brillouin zones to compute electronic and phonon states and electron-phonon
matrix elements, $g^{\nu}_{mn}(\textbf{k},\textbf{q})$. We use Wannier
functions to interpolate $g^{\nu}_{mn}(\textbf{k},\textbf{q})$ on a fine grid
using the EPW code Poncé  _et al._ [57]. We compute the EPS rates at T$=$ 300
K using the interpolated $g^{\nu}_{mn}(\textbf{k},\textbf{q})$, by following
the steps outlined in Eqs. 5$-$7 of the previous subsection. We first
calculate the EPS rates in two-atom primitive unit cell of undoped bulk Si and
compare with literature data [57]. This step establishes confidence in our
numerical approach to produce results consistent with published data. We
perform SCF and NSCF calculations to compute the electronic states and DFPT
calculations to compute the phonon modes of bulk Si. We calculate the
electronic states, phonon modes, and electron-phonon matrix elements on a
$6\times 6\times 6$ coarse Monkhorst-pack grid [88]. We then obtain
interpolated electron-phonon matrix elements on a fine grid of
$30,000/150,000$ randomly selected k/q-points, using the EPW code [57]. We
choose these coarse and fine grid sizes to compare our results with literature
data, obtained with similar coarse (fine) $6\times 6\times 6$/$6\times 6\times
6$ (30,000/150,000) k/q-points grids [57]. We average $\tau_{n\textbf{k}}$
over a $\sim 30,000$ k-mesh to obtain energy dependent electron relaxation
times due to phonons, $\tau_{\text{ep-EPW}}(\text{E})$. We show the resulting
EPW-EPS rates in two-atom bulk Si models in Fig. 2 (grey circles). We use a
Gaussian broadening of $10$ meV, and the HSE06 predicted bulk Si band gap of
1.08 eV to adjust the EPS-EPW rates. The green circles in Fig. 2 represent the
reported EPW predicted scattering rates in two-atom bulk Si models, extracted
from Ref. 57. Note that the literature data are adjusted to display the Si
band gap, 1.08 eV, used for this article. The red line corresponds to EPS
rates obtained with the DOS scattering approximation, $\tau_{\text{ep-DOS}}$:
$1/\tau_{\text{ep-DOS}}(\text{E})=\text{K}_{\text{el-
ph}}\times\text{DOS}(\text{E})$, with $\text{K}_{\text{el-ph}}=4.2$. We use a
dense $\sim 200,000$ k-mesh for NSCF calculations and an energy step of
$0.001$ Ry to obtain DOS(E). The DOS has the units of number of
states/Ryd/spin/unit cell, henceforth referred to as DOS.u.. The comparison
between the EPS-EPW and EPS-DOS rates shows that the DOS approximation is
quite effective to describe EPS near the conduction band edge. However, the
EPS-DOS predictions deviate as we move away from the edge.

Figure 2: Electron-phonon scattering rates in (i) two-atom primitive cell
(grey) and (ii) eight-atom conventional cell (blue) of bulk Si. (iii)
Reference data (green) is extracted from Ref. 57. DOS scattering rates in
eight-atom conventional cell are shown with red line. We obtain identical EPS-
DOS rates for two-atom primitive cell, confirming that DOS scattering rates
are independent of the unit cell sizes for bulk Si.

To find the appropriate coarse and fine grid sizes for the SLs, we perform
extensive tests to compute EPS rates in a similar size eight-atom bulk Si
conventional unit cell, and compare these results with the reference two-atom
Si EPS rates [57]. The use of grid sizes, similar to the above mentioned
2-atom calculations, to obtain the EPS rates in several eight-atom Si/Ge SLs,
demands unfeasible computational expenses. Additionally, such sampling might
not be necessary for the respective eight-atom BZs, which is four times
smaller than the primitive cell BZ. Figure 2 shows the comparison between EPS
rates in eight-atom bulk Si conventional unit cell (blue), two-atom primitive
cell, and the reference data (green). The eight-atom EPS rates, obtained with
coarse (fine) $6\times 6\times 6$/$6\times 6\times 6$ (15,000/50,000)
k/q-points grids match well with the two-atom results and the reference data,
obtained with coarse (fine) $6\times 6\times 6$/$6\times 6\times 6$
(30,000/150,000) k/q-points grids. We average $\tau_{n\textbf{k}}$ over a
$\sim 15,000$ k-mesh for eight-atom bulk Si, to obtain energy dependent EPS
rates. We then systematically vary the coarse and the fine BZ sampling grids
to obtain converged results for the eight-atom EPS rates. This aspect is
particularly important since BZ sampling significantly impacts the computed
transport properties. The importance of grid size based convergence tests has
been acknowledged in multiple studies [89]. We find that EPS rates obtained
with $6\times 6\times 6$/$3\times 3\times 3$ (15,000/50,000) coarse (fine)
k/q-point grids match well with $6\times 6\times 6$/$6\times 6\times 6$
(15,000/50,000) coarse (fine) k/q-point grid results.

Such dense sampling proves to be expensive for calculating EPS rates in SLs
with a diverse lattice strain environments. We check further to find an
appropriate BZ sampling of SLs that produces converged EPS rates, and also
helps to keep the computational costs manageable. We compute EPS rates in
Si2Ge2 SL, assumed to be grown on Si, with (coarse/fine) $4\times 4\times
4$/$4\times 4\times 4$ (10,000/30,000) k/q-point grids. The rates match well
with the coarse (fine) $6\times 6\times 6$/$3\times 3\times 3$ (15,000/50,000)
k/q-points grid results for Si2Ge2 SL on Si. We use the coarse (fine) $4\times
4\times 4$/$4\times 4\times 4$ (10,000/30,000) BZ sampling to calculate all
the Si/Ge SL EPS rates presented in this article. We average
$\tau_{n\textbf{k}}$ over a $\sim 10,000$ k-mesh to obtain energy dependent
electron-phonon relaxation times in eight-atom SLs. We report the results from
our extensive convergence tests in SI for the benefit of the readers
interested in reproducing our results. In order to compute IMS relaxation
times, $\tau_{\text{im}}(\text{E})$, we perform NSCF calculations using a
dense $\sim 200,000$ k-mesh, and obtain dielectric constant tensors using DFPT
calculations as implemented in QE [55]. We implement the random phase
approximation including local field effects (LRPA) to obtain the dielectric
constants. We list the LRPA calculated anisotropic dielectric constants and
the HSE06 predicted band gaps of the Si/Ge SLs used in our calculations in
Table 1. We find that the LRPA predicted dielectric constant of bulk Si
(12.66) is greater than that of the measured value of 11.9 [90] by $\sim
6.4\%$, however, matches well with previous GW+BSE result (12.7) [91]. We
combine $\tau_{\text{ep-EPW}}(\text{E})$ and $\tau_{\text{im}}(\text{E})$ to
calculate $\tau(E)$ using Matthiessen’s rule, (Eq. 16) as described in the
previous subsection.

Table 1: In-plane and cross-plane diagonal components of the anisotropic dielectric tensors and HSE06 computed band gaps of the Si/Ge SLs. System | Substrate | $\varepsilon_{\parallel}$ | $\varepsilon_{\perp}$ | Band gap (HSE06)
---|---|---|---|---
Si1Ge3 | Si | 14.38 | 14.87 | 0.49 eV
Si2Ge2 | Si | 13.66 | 13.95 | 0.55 eV
Si3Ge1 | Si | 12.87 | 13.05 | 0.69 eV
Si2Ge2 | Si0.5Ge0.5 | 14.11 | 13.83 | 0.80 eV
Si2Ge2 | Ge | 14.95 | 13.99 | 0.63 eV
Figure 3: Comparison of predicted (a) Seebeck coefficient (S) and (b)
resistivity ($\rho$) of bulk Si and literature data. The dashed red and black
lines represent EPS-EPW and EPS-DOS results, respectively. The solid red and
black lines represent predictions made considering EPS-EPW+IMS and EPS-
DOS+IMS, respectively. The solid green line represent results obtained using
the CRTA approximation. The difference between the lines show how S and $\rho$
predictions vary for different scattering approximations. For CRTA
calculations, $\tau$ is extracted from the measured bulk Si mobility [48]. The
black triangles represent measured data for diffusive S, extracted from Refs.
92, 93. The blue +’s (EPS-EPW) and blue circles (EPS-EPW+IMS) represent past
first-principles study predictions of diffusive S, for comparison. These
values are extracted from Ref. 69. The measured data for $\rho$ are extracted
from Ref. 94 (inverted black triangles).

###  II.4 Electronic Transport Coefficients

We calculate the electronic transport coefficients at several carrier
concentrations of interest, by incorporating the energy dependent relaxation
times, $\tau(\text{E})$. We consider different scattering approximations to
obtain $\tau$, and compare the resulting transport properties. We obtain TDFs
using the BoltzTrap code which considers $\tau(\text{E})=1$ for all energies.
With our in-house code, we incorporate $\tau(\text{E})$ with the TDFs, and
compute S and $\rho$ following steps outlined in Eqs. 1-4. We use an energy
step of 0.001 Ry to compute the BTE integrals. We use the HSE06 corrected band
gap for bulk Si (1.08 eV) for both S and $\rho$ calculations, as opposed to
the LDA predicted band gap used in past research [69]. Figure 3 shows the
comparison of the predicted (a) Seebeck coefficient (semi-log plot) and (b)
the resistivity (log-log plot) of eight-atom bulk Si with literature results.
The Seebeck coefficients predicted with $\tau_{\text{ep-EPW}}$ and
$\tau_{\text{ep-DOS}}$ are shown with red and black dashed lines in Fig. 3(a),
respectively. The match between the two results suggests that the DOS
scattering approximation works well to describe the bulk Si Seebeck
coefficients. The blue pluses are previous first-principles EPS-EPW results
extracted from Ref. 69. The data represent diffusive S predicted using a
k-dependent scattering approach and exact BTE formulation. We find that our S
predictions made considering an energy-dependent electron relaxation time,
$\tau(\text{E})$, compare well with exact BTE based predictions. The solid red
and black lines represent predictions made considering total electron
relaxation times due to EPS-EPW+IMS and EPS-DOS+IMS, respectively. The
corresponding literature results, extracted from Ref.69, are shown with blue
circles. The close match between the two sets of results can be noted. Past
studies reported that the diffusive part of S does not depend strongly on the
nature of the scattering mechanisms and switching off phonon or impurity
scattering affects the S result by less than 8% [69]. We also note that the
predicted S results, including the IMS rates (solid lines), do not show
considerable deviations from the results with only EPS rates included (dashed
lines). The maximum deviation of 18% between the two results can be observed
at n${}_{\text{e}}\sim 10^{19}$ cm-3. The black triangles represent measured
data for diffusive contributions to S, and are extracted from Refs. 92, 93. We
find that our results compare well with the measured data, especially in the
low doping regime. At high doping, for n${}_{\text{e}}$>$10^{17}$cm-3, our
results are slightly higher than the experimental data. The discrepancy
between theoretical predictions and measured data was attributed to possible
electron-plasmon interactions [69]. The solid green line represent results
obtained using $\tau$ within the CRTA approximation. We calculate the TDFs and
consider doping-dependent relaxation times, $\tau(\text{n}{{}_{\text{e}}})$,
to calculate S. We use the $\tau(\text{n}{{}_{\text{e}}})$ of unstrained bulk
Si, parametrized by Ref. [34] using the values first reported in Ref. [48]. We
find that CRTA predicted S is higher than the EPS-EPW+IMS results by atmost
10%, indicating the relative independence of S on scattering mechanisms.
Similar observation can be made by noting the match between Seebeck
coefficients predicted using EPW approach and the DOS approximation.

We show the bulk Si resistivity predicted using $\tau_{\text{ep-EPW}}$ and
$\tau_{\text{ep-DOS}}$ with red and black dashed lines in Fig. 3(b),
respectively. The $\tau_{\text{ep-DOS}}$ resistivity follows the predicted
$\tau_{\text{ep-EPW}}$, with the values slightly higher in the log scale. This
result indicates that it may be reasonable to use the DOS approximation
($\tau_{\text{ep-DOS}}$) to predict the trends of resistivities at low carrier
concentrations. This observation matches with our conclusion from a previous
study as well [37]. The inverted black triangles in Fig. 3(b) represent
measured bulk Si resistivities reported in Ref. 94. The resistivity obtained
with $\tau_{\text{ep-EPW}}$, closely matches with the measured data [94] for
n${}_{\text{e}}<10^{17}$ cm-3. At high doping, for
n${}_{\text{e}}$>$10^{17}$cm-3, both the $\tau_{\text{ep-EPW}}$ and
$\tau_{\text{ep-DOS}}$ results are lower than the experimental data. The
consideration of IMS processes significantly influences the resistivity
predictions at higher carrier concentrations and improves the match with the
measured data. The solid red and black lines represent $\rho$ computed
considering electron relaxation due to EPS-EPW+IMS and EPS-DOS+IMS,
respectively. The increase of carrier concentration leads to higher electron
scattering rate due to increased electron-impurity scattering. The inclusion
of IMS processes does not influence the predictions at lower concentrations,
as expected. The CRTA predicted $\rho$ (green line) closely matches the
EPW+IMS predicted $\rho$ at lower concentrations, however, is higher than the
EPW+IMS predictions at higher concentrations, highlighting the strong effect
of different scattering mechanisms on $\rho$, especially in the high doping
regime. The results shown in Fig. 3 establishes that it is essential to
consider the electron relaxation times due to both EPS and IMS processes for
accurate prediction of the Seebeck coefficient (diffusive contribution) and
the resistivity of bulk Si at 300 K, especially at the high doping regime.

##  III Results and Discussion

We now turn to the main focus of this study, which is to demonstrate the
effect of the scattering processes on the electronic transport properties of
Si/Ge SLs, with diverse lattice strain environments. The strain environment in
fabricated Si/Ge heterostructures is highly nonuniform: $\sim 3-4\%$ strain
values have been reported in Si/Ge nanowire heterostructures with
compositionally abrupt interfaces, grown via the VLS process [95]. The lattice
strain environment in SLs is strongly dependent on growth substrates [38] and
layer compositions [37]. We illustrated in our previous publications that the
layer compositions and external substrate induced strains prompt non-uniform
monolayer separations, and modulate the Seebeck coefficients of Si/Ge SLs,
within the CRTA [37, 38, 36, 39]. Here, we highlight the effect of EPS and IMS
processes on the Seebeck coefficients, resistivities, and power-factors of
these Si/Ge SLs with diverse strain environment. We also discuss how these
predictions compare with the DOS scattering approximation and CRTA (SI Fig. 9)
results. We compare the transport properties with the corresponding bulk Si
properties for a better appreciation of the modulation of electronic transport
in the SLs.

###  III.1 Effect of EPS on Electronic Transport Properties

In the SLs with varied compositions and substrate induced strain environments,
the energy bands shift due to lattice strain [37, 36, 38]. Here, we illustrate
the effect of these band energy shifts on the EPS processes and the resulting
cross-plane transport properties of the SLs. Alongside, we discuss the
applicability of the DOS scattering approximation by comparing the computed
transport properties with the EPW predictions.

Figure 4: Electronic properties of Si/Ge SLs with varied compositions: (ii)
Si1Ge3, (iii) Si2Ge2, and (iv) Si3Ge1 SLs, grown on a fixed Si substrate. (a)
Electron-phonon scattering rates predicted by the EPW approach (circles) and
the DOS scattering approximation (dashed lines with symbols)
($1/\tau_{\text{ep-DOS}}(\text{E})=\text{K}_{\text{el-
ph}}\times\text{DOS}(\text{E})$). The proportionality constants,
$\text{K}_{\text{el-ph}}$, are shown in respective legends. (b) Colormap of
electronic band structures of (i) bulk Si and (ii) Si1Ge3, (iii) Si2Ge2, and
(iv) Si3Ge1 SLs, weighted by the imaginary part of electron self energy. (c)
Seebeck coefficients, (d) electronic resistivities and (e) power-factors of
Si/Ge SLs incorporating EPS-EPW (solid) and EPS-DOS rates (dashed lines with
circles).

####  III.1.1 Superlattices with Varied Compositions

We investigate three SLs with different compositions, namely Si1Ge3, Si2Ge2,
and Si3Ge1 SLs. The model SLs are assumed to be grown on a Si substrate.
Figure 1 shows the representative configurations of the Si/Ge SLs investigated
in the present study. We define the in-plane strain in the monolayers,
$\Epsilon_{\text{Si}\parallel}$, with respect to bulk Si, as
$\Epsilon_{\text{Si}\parallel}=(a_{\parallel}/a_{\text{Si}}-1)$. Here,
$a_{\parallel}$ is the effective in-plane lattice constant of the SLs and
$a_{\text{Si}}=5.40\;\text{\AA}$ is the LDA predicted bulk Si lattice
constant. Using our definition, the in-plane lattice strain in the Si region
of the SLs measures to $\Epsilon_{\text{Si}\parallel}=0\%$. We show the EPW
predicted EPS rates in (i) bulk Si (grey circles), (ii) Si1Ge3 (red circles),
(iii) Si2Ge2 (green circles) and (iv) Si3Ge1 (blue circles), in Fig. 4(a). We
aligned the CBM of the different SLs to compare electron relaxation due to
phonons in different systems. The E $=0$ eV point in the y-axis refers to the
common conduction band minima (CBM) of the SLs. We show the EPS rates for
$0\leq\text{E}\leq 0.5$ eV, because the Fermi levels of the three SLs, with
n-type doping of $\sim 10^{17}-10^{21}\ \text{cm}^{-3}$, are on the range of
$\sim-0.14-0.45$ eV at 300 K. In order to establish a qualitative
understanding of the EPS rates, we discuss the relationship between the
imaginary part of the electron-phonon self energies, Im$\Sigma_{n\textbf{k}}$
and the SL band structures along symmetry directions. Although, the EPS rates
are calculated from Im$\Sigma_{n\textbf{k}}$, by sampling the entire SL BZ,
comparison of the electronic properties along SL symmetry directions reveals
physical insights regarding electron relaxation in diverse SLs.

Figure 4(b) shows the SL band structures in the X$-\Gamma-$Z symmetry
directions, superimposed with Im$\Sigma_{n\textbf{k}}$. The panels in Fig.
4(b) show the band structures of (i) bulk Si and (ii) Si1Ge3, (iii) Si2Ge2 and
(iv) Si3Ge1 SLs, respectively, along the in-plane X$-\Gamma$ and cross-plane
$\Gamma-$Z SL symmetry directions. The bulk Si bands are contributed from
$\Delta$ valleys (panel (i)). The SL bands, shown in panels ((ii)-(iv)), are
formed due to the combined effect of zone folding of bulk bands (panel (i))
dictated by structural symmetry, and periodic potential perturbation [38, 30,
96, 31, 97]. The varied compositions of these SLs result in diverse lattice
strain environment, that regulates contributions from the valleys and modifies
the SL bands [37, 36]. As we increase the number of Si monolayers in the SLs,
$L_{\text{Si}}=1\rightarrow 3$ (Si1Ge3 $\rightarrow$ Si2Ge2 $\rightarrow$
Si3Ge1), for a fixed $L\ (L=4)$ and fixed $a_{\parallel}=a_{\text{Si}}$, the
average cross-plane strain ($\propto(1-L_{Si}/L$)) decreases. The cross-plane
strain is high in (ii) Si1Ge3 SL and decreases for (iii) Si2Ge2 and (iv)
Si3Ge1 SLs. The reduced cross-plane strain induces a shift of type 2
conduction minibands, contributed by $\Delta_{\perp}$ valleys, to low energies
in the $\Gamma-$Z direction. The minima of the type 2 $\Gamma-$Z conduction
bands shift from (ii) 0.45 eV to (iii) 0.35 eV to (iv) 0.28 eV. The interplay
between strain environment and confinement is more complex for the low energy
type 1 and type 4 conduction bands in the $\Gamma-$X direction, formed by
$\Delta_{\parallel}$ valleys. The intersections of type 1 bands with the
$\Gamma$ point remain mostly same, however, the dispersive natures are
significantly different. The colors in the band structures plot (Fig. 4(b))
represent the corresponding imaginary part of the self-energies,
Im$\Sigma_{n\textbf{k}}$, with values as indicated by the color bar at the
side panel. The self-energies are small in the entire low energy conduction
zone of interest E $=0-0.5$ eV, due to the absence of states in the bandgap
region and also, low DOS in this energy range. However, color variations of
the SL bands between panels ((ii)(green)$\rightarrow$(iv)(blue)) can be noted,
when compared to the bulk bands of panel (i), that indicates change of the
self-energies. The electron self-energies are directly influenced by the
energy shifts and the dispersive nature of the SL bands, as we illustrate in
the following discussion. The decrease of Im$\Sigma_{n\textbf{k}}$ of type 2
SL bands ((ii) >(iii) >(iv)) is due to the energy shift of the type 2
conduction minibands in the $\Gamma-$Z direction, induced by the changes of
the cross-plane lattice strain environment. The energy shift reduces the
availability of lower energy states and thus, lowers the probabilities of
phonon-assisted transitions, resulting in smaller electron self-energies
(green $\rightarrow$ blue). Similarly, the changes of self-energies associated
with the type 4 bands ((ii), (iv)>(iii)), are connected to the nature of type
1 bands. The Im$\Sigma_{n\textbf{k}}$ associated with the type 1 bands remain
small in all three SLs due to the absence of available lower energy states.
However, the dispersive nature of these bands changes significantly in
different SLs. The altered type 1 band induces changes of the electron self-
energies in other SL bands, as shown in the band structures of the three
panels ((ii)-(iv)). The increase of available density of states increases
probabilities for phonon-assisted transitions, resulting in higher
Im$\Sigma_{n\textbf{k}}$ for type 4 bands in panels (ii) and (iv).

We illustrate below that the EPS rates are strongly connected to the nature of
the SL bands and the self-energies, by highlighting the relationship between
the resutls shown in panels Fig. 4(a) and Fig. 4(b). The EPS rates in the
three SLs are shown in Fig. 4(a). The EPS rates in all three SLs (red, green,
blue) are lower than bulk Si (grey) for E $\gtrsim$ 0.2 eV due to low
Im$\Sigma_{n\textbf{k}}$. At this energy range, Im$\Sigma_{n\textbf{k}}$
values in bulk Si are on the range $\sim$ 15-20 meV (red-black), higher than
the values in the SLs (green-yellow). Comparing the EPS rates in (ii) Si1Ge3
SLs (red) and (iii) Si2Ge2 SLs (green), one can note that they match till E
<0.1 eV. Above this energy range, the rates in (ii) Si1Ge3 SLs (red) increase
at a faster rate. This increase could be attributed to the higher self-
energies of type 4 bands of (ii) Si1Ge3 SLs, compared to those in (iii) Si2Ge2
SLs, in this energy range. As we discussed previously, the higher self-energy
of type 4 bands of (ii) Si1Ge3 SLs is influenced by the nature of type 1
bands. The lowest EPS rates are observed for (iv) Si3Ge1 SL at most energies
(blue circles) due to the small electron self-energies of (iv) Si3Ge1 SL
bands. However, the results match the rates in (ii) Si1Ge3 SLs (red) at E
$\gtrsim$ 0.4 eV, due to the increase of self-energies of electrons,
especially in type 4 bands (Fig. 4(b)-(iv)). At E $\gtrsim$ 0.4 eV, the
electrons in type 4 bands also participate in the scattering of process. The
above discussion illustrates that the EPS rates are controlled by the nature
of the SL bands. Consequently, the DOS scattering approximation could be used
to describe the electron relaxation processes reasonably well. The EPS-DOS
rates ($1/\tau_{\text{ep-DOS}}$), are shown in Fig. 4(a) with (i) black solid
line (bulk Si) and black dashed lines with (ii) triangles (Si1Ge3), (iii) +’s
(Si2Ge2) and (iv) circles (Si3Ge1), respectively. As can be seen from Fig.
4(a), the EPS-DOS rates follow the EPW predictions closely. The
proportionality constants, $\text{K}_{\text{el-ph}}$, are given by (ii) 6.8,
(iii) 6.0 and (iv) 7.8, respectively. The slight variations of
$\text{K}_{\text{el-ph}}$ are connected to the varying DOS in the three SLs.
The electronic DOS is low in (iv) Si3Ge1 compared to the other SLs for the
energy range of interest. However, the EPS rates in the three SLs are mostly
similar. The lower DOS and similar EPS rates result in higher K${}_{\text{el-
ph}}$ for (iv) Si3Ge1 ($1/\tau_{ep}(\text{E})=\text{K}_{\text{el-
ph}}\uparrow\times\text{DOS}(\text{E})\downarrow$). We like to point out that
the determination of the scaling constants , K${}_{\text{el-ph}}$, is highly
dependent on the fitting procedure. As we show in Fig. 4 of the SI document,
different values of the scaling constants could be obtained for relaxed Si2Ge2
SL, depending on the numerical procedure. The variations of K${}_{\text{el-
ph}}$ do not change the resulting S but affect $\rho$ (SI Fig. 5). This
discussion highlights the importance of exercising caution while using the DOS
scattering rates to compute electronic transport properties.

Figure 5: Electronic transport properties of (i) bulk Si and Si2Ge2 SLs with
in-plane strain (ii) 0%, (iii) 1.85%, and (iv) 3.98%, induced by growth
substrates. The proportionality constants, $\text{K}_{\text{el-ph}}$, are
shown in respective legends. (a) Electron-phonon scattering rates predicted by
the EPW approach (circles) and the DOS scattering approximation (dashed lines
with symbols). (b) Colormap of electronic band structures of (i) bulk Si and
Si2Ge2 SLs with (i) 0%, (ii) 1.85%, and (iii) 3.98% in-plane strains, weighted
by the imaginary part of electron self energy. (c) Seebeck coefficients, (d)
electronic resistivities and (e) power-factors of strained Si/Ge SLs
incorporating EPS-EPW (solid) and EPS-DOS rates (dashed lines with circles).

The electronic transport coefficients: Seebeck coefficients (Eq. 4),
resistivities (Eq. 3), and power-factors are obtained from the
$\mathcal{L}$-integrals (Eq. 1). The strain-modulated DOS, electron group
velocities and electron scattering rates affect the energy-dependence of the
TDFs, as depicted in Eq. 2. The overlap of the energy dependent TDFs with the
Fermi window functions
$\left((\text{E}-\text{E}_{\text{F}})^{n}\left(-\frac{\partial
f}{\partial\text{E}}\right)\right)$ results in modulated
$\mathcal{L}$-integrals, and the electronic transport coefficients. In Fig.
4(c) and (d), we show the S and $\rho$ of Si1Ge3 (red), Si2Ge2 (green), and
Si3Ge1 SLs (blue), respectively, computed incorporating $\tau_{\text{ep-EPW}}$
(solid lines) and $\tau_{\text{ep-DOS}}$ (dashed lines with circles). The S of
the Si1Ge3 and the Si2Ge2 SLs follow the bulk Si values closely. The similar
Seebeck coefficients of (ii) and (iii) SLs can be explained by noting that
these two systems share similar DOS, resulting in similar K${}_{\text{el-ph}}$
values. The DOS in (iv) Si3Ge1 SLs is somewhat lower, resulting in higher
K${}_{\text{el-ph}}$. The S of Si3Ge1 deviates from the bulk Si values and
displays reduced values at most carrier concentrations, n${}_{\text{e}}$. We
note that S of Si3Ge1 SL shows a non-monotonic behavior with increase of
n${}_{\text{e}}$. The Seebeck value is low compared to all other cases for
n${}_{\text{e}}\lesssim 10^{20}$cm-3 ($\text{E}_{\text{F}}\sim$ 2.5 eV) and
matches them at high n${}_{\text{e}}$. The behavior can be explained by
discussing the band structure shown in Fig. 4(b)(iv). The lattice strain in
Si3Ge1 SL increases the energy separation between the type 1 and type 4
$\Delta_{\parallel}$ bands. The gap between energy levels decreases the
overlap between TDF and the Fermi window, causing S to reduce at
n${}_{\text{e}}<2\times 10^{20}$cm-3 ($\text{E}_{\text{F}}$ <0.25 eV).
However, when the Fermi level ($\text{E}_{\text{F}}$) is aligned close to the
edge of the $\Delta_{\perp}$ minibands (type 2), S increases at high doping
regime.

The effect of varied scattering rates or inversely the electron relaxation
times is more pronounced on the resistivities of the SLs. The low EPS rates in
Si3Ge1 SLs indicate high relaxation times and consequentially, low
resistivities. The $\rho$ of Si3Ge1 SLs is lower than the bulk values at most
carrier concentrations, $\text{n}_{\text{e}}\lesssim\times 10^{20}$ cm-3. The
$\rho$ of Si1Ge3 and Si2Ge2 SLs remain close to the bulk values due to similar
EPS rates. The resistivities obtained with EPS-DOS rates deviate slightly from
the EPW predictions. The difference between the electronic transport property
predictions from the two approaches become more apparent when S and $\rho$ are
combined together to calculate PF ($\text{S}^{2}/\rho$). The variations of S
and $\rho$ lead to considerable modulations of PF, especially at high
$\text{n}_{\text{e}}$. The PF of Si1Ge3 SL is $\sim 1.4$-fold larger than bulk
Si at $\text{n}_{\text{e}}\sim 2.7\times 10^{20}$ cm-3, while the PF of Si3Ge1
PF remains lower than that of bulk Si for all $\text{n}_{\text{e}}$. The
lowest PF ($\text{S}^{2}/\rho$) of Si3Ge1 SL could be attributed to the strong
reduction of S at most carrier concentrations. Overall, we note that the peak
PFs of the chosen Si/Ge SLs are not higher than that of bulk Si. We like to
point out that most studies of thermoelectric properties of phonon engineered
systems assume that electronic properties remain unaffected. Here, we show
that compositions of SLs modulate the SL bands and their EPS processes. The
modulations lead to strong reduction of PFs at most n${}_{\text{e}}$ compared
to bulk Si. We like to point out that one needs to exercise caution while
performing the numerical evaluation of the $\mathcal{L}$-integrals. In Si Fig.
6 and Fig. 7, we show how different alignments between EPS rates and DOS near
the CB edge can affect the integrals and impact both S and $\rho$ (SI Fig. 7).

####  III.1.2 Superlattices with Lattice Strain Induced by Substrates

To model SLs with substrate induced strain environments, we consider Si2Ge2
SLs grown on Si, Si0.5Ge0.5, and Ge substrates, respectively. Due to the
lattice mismatch, the substrates induce in-plane strains of
$\Epsilon_{\text{Si}\parallel}=$ $0\%$, $1.85\%$, and $3.98\%$, in the Si
region of the SL. In Fig. 5(a), we show the EPS rates in substrate strained
Si2Ge2 SLs, computed with two different approaches. The EPS-EPW rates,
$1/\tau_{\text{ep-EPW}}$, are shown with (i) grey (bulk Si), (ii) red (0%),
(iii) green (1.85%) and (iv) blue (3.98%) circles. The EPS-DOS rates are shown
with (i) red solid line (bulk Si) and black dashed lines with (ii) triangles
(0%), (iii) +’s (1.85%) and (iv) circles (3.98%), respectively. Similar to the
previous case, E $=0$ eV point in the y-axis refers to the common CBM of the
strained SLs. We choose a similar x-axis range because the Fermi levels of the
three strained SLs with n-type doping of $\sim 10^{17}-10^{21}\
\text{cm}^{-3}$ are on the range $\sim-0.13-0.44$ at 300 K. In order to
develop a qualitative understanding of the trends of the EPS rates in the
strained SLs, we discuss the variations of the imaginary part of the electron-
phonon self energies, Im$\Sigma_{n\textbf{k}}$, in relation to the SL band
structures along different symmetry directions. As we have demonstrated for
the SLs with varied compositions grown on fixed substrates, the comparison of
the electronic properties along different symmetry directions provides
physical understanding regarding the EPS rates in SLs.

Figure 5(b) shows the SL band structures along the in-plane X$-\Gamma$ and
cross-plane $\Gamma-$Z symmetry directions, superimposed with
Im$\Sigma_{n\textbf{k}}$. The panels in Fig. 5(b) shows the band structures of
(i) bulk Si and Si2Ge2 SLs with in-plane strain of (ii) 0%, (iii) 1.85% and
(iv) 3.98%, respectively. The SL bands shown in panels ((ii)-(iv)) are formed
due to the combined effect of zone folding of bulk bands, and periodic
potential perturbation. The substrate induced lattice strain induces valley
splittings and regulates contributions from the split valleys to modify the SL
bands [38, 36]. With the increase of $\Epsilon_{\text{Si}\parallel}$
((ii)$\rightarrow$(iv)), the low energy conduction bands (type 1, type 4) in
the $\Gamma-$X direction, formed by the $\Delta_{\parallel}$ valleys, move
upward in energy. The strain-induced band shifts can be particularly noted by
considering the upward shift of the conduction bands and downward shift of
valence bands together. It is good to keep in mind that we aligned the CBM of
the different SLs to E $=0$ eV in these plots. The intersection of the lowest
$\Gamma-$X conduction band (type 1), with the $\Gamma$ point shifts upward
from (ii) 0.01, (iii) 0.01 to (iv) 0.27. On the other hand, the low energy
conduction minibands (type 2) in the $\Gamma-$Z direction, contributed by the
$\Delta_{\perp}$ valleys, shift downward with the increase of in-plane strain.
The intersection of these bands with the $\Gamma$ point moves from (ii) 0.46
to (iii) 0.20 (iv) to 0.17 eV. The other set of parabolic conduction bands
(type 3) along $\Gamma-$Z direction, formed due to the overlap of
$\Delta_{\parallel}$ valleys at the $\Gamma$ point, shifts upward in energy
together with the type 1 bands. It can be noted that the CBM shifts from the
$\Delta_{\parallel}$ bands (type 1) to the $\Delta_{\perp}$ bands (type 2),
along cross-plane $\Gamma-Z$ direction, with the increase of
$\epsilon_{\text{Si}\parallel}$. Similar energy shifts of the strain-split
$\Delta$ valley bands have been observed in other short-period SLs (e.g.,
Si4Ge4) [38].

The colors of the band structures plot (Fig. 5(b)) represent the corresponding
imaginary part of the self-energies. The self-energy values are as indicated
by the color bar at the side panel. The self-energies are small in the entire
low energy conduction zone of interest $0-0.5$ eV, due to the absence of
states in the bandgap region. The low self-energies are also due to the small
density of CB states in this region, however, the density varies for different
strain cases leading to variations of the self-energies. We illustrate in the
following that the changes of the self-energies are related to the strain
induced energy shifts of the SL bands. As discussed above, the energy of the
type 2 $\Delta_{\perp}$ valley band lowers ((ii) $\rightarrow$ (iv)) with the
increase of in-plane strain, along the cross-plane $\Gamma-Z$ direction. The
self-energy of the shifted $\Delta_{\perp}$ band decreases with increase of
strain (blue-green (ii) $\rightarrow$ blue (iv)), due to the reduced
availability of lower energy states. Additionally, the energy gap between the
type 2 band and other bands (3, 4) increases due to the energy shift, as can
be noted from panel (iv) of Fig. 5(b). The increase of the energy gap lowers
the probabilities of phonon-assisted energy transitions. As a result, the
self-energies remain low till E $\lesssim$ 0.3 eV. For E >0.3 eV, the self-
energy of the bands of the (iv) 3.98% strained SL increases due to increased
probabilities for phonon-assisted transitions. We show the colormap of band
structures and Im$\Sigma_{n\textbf{k}}$ of the region $-2$ to $2$ eV, in SI
Fig. 1, for a broader perspective.

These variations of the self-energies with E directly influence the EPS rates
in 3.98% strained Si2Ge2 SL, as can be seen from Fig. 5(a) (blue circles). The
EPS rates are low till (E $\sim$ 0.3 eV) and increase sharply afterwards.
Similarly, one can relate the EPS rates in the Si2Ge2 SLs with (ii) 0% (red)
and (iii) 1.85 % (green) in-plane strain environment, to the nature of the
bands of the two strained SLs. As can be seen from panels (ii) and (iii) of
Fig. 5(b), the band structures of type 1, 3, and 4 bands are quite similar
near the CB edge. As a consequence, the EPS rates in these two SLs match for E
$\lesssim$ 0.1 eV. For E $\gtrsim$ 0.1 eV, the EPS rates for case (iii)
(green) are higher than those for case (ii). The increased EPS rates for case
(iii) SL are due to the following reasons. The scattering rates of electrons
in the type 2 $\Delta_{\perp}$ valley band contribute to the EPS rates of case
(iii) SL at approximately, E $\sim$ 0.1 eV. The type 2 band of the (iii) 1.85%
strained SL is shifted to lower energy due to increased strain. Additionally,
the small energy gap between the bands increases probabilities for phonon-
assisted transitions, resulting in higher EPS rates for case (iii) (green).
The EPS rates for case (ii) SL (red) increases around E $\gtrsim 0.4$ eV once
electrons in type 2 band participate in the scattering process. The DOS
scattering rates follow the EPW predictions closely, with proportionality
constants, $\text{K}_{\text{el-ph}}$, given by (ii) 6.0, (iii) 6.6 and (iv)
6.8, respectively. Overall, we note from Fig. 4(a) and Fig. 5(a) that the DOS
scattering approximation can describe the trends of the EPS rates of short-
period superlattices. However, it is essential to determine
$\text{K}_{\text{el-ph}}$ accurately to predict electronic transport
properties. $\text{K}_{\text{el-ph}}$ is determined by aligning DOS data with
the EPS-EPW rates. As we have discussed previously, $\text{K}_{\text{el-ph}}$
is highly dependent on the numerical fitting procedure (SI Fig. 4-5).
Therefore, it is difficult to solely rely on this approach to predict the
transport properties without the knowledge of higher accuracy EPS rates (Fig.
4(e) and Fig. 5(e)).

Figure 5(c) and (d) show the S and $\rho$ of Si2Ge2 SLs with in-plane strain
of $\Epsilon_{\text{Si}\parallel}=$ $0\%$ (red), $1.85\%$ (green), and
$3.98\%$ (blue), respectively, computed incorporating $\tau_{\text{ep-EPW}}$
(solid lines) and $\tau_{\text{ep-DOS}}$ (dashed lines with circles). We note
that S of the 3.98% strained Si2Ge2 SL (blue) shows a non-monotonic behavior
with increase of n${}_{\text{e}}$. The Seebeck value peaks at high
n${}_{\text{e}}$ preceded by values lower than those of bulk Si. Such results
are uncommon since the Seebeck coefficients of doped semiconductors and metals
are expected to decrease monotonically with increase of carrier concentration,
in agreement with the Pisarenko relation. The S modulations are directly
related to the nature of bands in the strained SLs, as can be illustrated by
discussing the band structure shown in panels of Fig. 5(b). Comparing the band
structures shown in panels (iii) and (iv), it can be observed that the energy
gap between type 2 ($\Delta_{\perp}$) and type 1, 3, 4 ($\Delta_{\parallel}$)
bands increases due to the substrate-strain induced shift of SL energy bands.
The gap between energy levels results in low DOS near CB edge and decreased
overlap between TDF and the Fermi window (Eq. 1). As a result, S is reduced at
n${}_{\text{e}}<2.6\times 10^{20}$cm-3 ($\text{E}_{\text{F}}$ <0.25 eV)
compared to that of bulk Si. However, when the Fermi level
($\text{E}_{\text{F}}$) is aligned close to the edge of the
$\Delta_{\parallel}$ minibands (type 1, 3, 4), the higher DOS leads to a
S-peak at high $\text{n}_{\text{e}}$. Such substrate strain modulated non-
monotonic Seebeck coefficients have been observed in other short-period SLs
[38, 98, 37, 36, 39].

The $\rho$ of Si2Ge2 SLs with 3.98% in-plane strain also displays nonmonotonic
behavior, due to the nature of the $\Delta_{\parallel}$ and $\Delta_{\perp}$
bands in the strained SL (Fig. 5(d)). The resistivities of the SLs are greater
compared to that of bulk Si at most n${}_{\text{e}}$. $\rho$ of 3.98% strained
SL (iv, blue) are higher than those of (ii) (red) and (iii) (green) SLs at all
n${}_{\text{e}}$ of interest, with a noticeable increase at
n${}_{\text{e}}\sim 2.2\times 10^{20}$cm-3 ($\text{E}_{\text{F}}$ $\sim$ 0.25
eV). The cross-plane electrical resistivity $\rho$ can be expressed as,
$\rho\propto
1/\int\tau(\text{E})v_{z}^{2}(\text{E})\text{DOS}(\text{E})\left(\partial
f/d\text{E}\right)d\text{E}$ (Eq. 1-3). As illustrated previously, the DOS
scattering approximation ($1/\tau_{\text{ep-
DOS}}(\text{E})=\text{K}_{\text{el-ph}}\times\text{DOS}(\text{E})$) could
describe the EPS rates in strained Si2Ge2 SLs reasonably well. As shown in
Fig. 4 and Fig. 5, S and $\rho$ predicted using $\tau_{\text{ep-DOS}}$(E)
(dashed lines with circles) follow the $\tau_{\text{ep-EPW}}$(E) (solid lines)
predictions closely, for carefully chosen values of $\text{K}_{\text{el-ph}}$.
Inserting this approximation, $\rho$ can be simplified to, $\rho\propto 1/\int
v_{z}^{2}(\text{E})\left(\partial f/d\text{E}\right)d\text{E}$. This
expression indicates that electron group velocities play a significant role in
determining $\rho$. We show a comparison between $v_{z}^{2}(\text{E})$ in the
strained Si2Ge2 SLs in SI Fig. 8. The results show that $v_{z}^{2}(\text{E})$
in the 3.98% strained SL, lies below those of the other two SLs for all
energies of interest, resulting in high resistivities. Between $0\leq
E\lesssim 0.14$ eV, $v_{z}^{2}(\text{E})$ increases fast with E, causing
decrease of $\rho$ with n${}_{\text{e}}$ up to n${}_{\text{e}}\sim 9.5\times
10^{19}$cm-3. For 0.14 <E <0.3 eV, $v_{z}^{2}(\text{E})$ remains comparatively
‘flat’ causing the ‘bump’ in the resistivity. $v_{z}^{2}(\text{E})$ increases
again for E >0.3 eV. The product of Fermi window and $v_{z}^{2}(\text{E})$,
$v_{z}^{2}(\text{E})\left(\partial f/d\text{E}\right)$, encompasses the fast
increase of $v_{z}^{2}(\text{E})$ and results in the change of trend, followed
by a steady decrease of $\rho$. The resistivity trend starts to change near E
$\sim 0.22$ eV (n${}_{\text{e}}\sim 1.97\times 10^{20}$cm-3).

Figure 6: (i) Seebeck coefficients, (ii) electronic resistivities and (iii)
power-factors of SLs with diverse strain environments: (a) Si/Ge SLs with
varied compositions, grown on identical substrates, (b) Si2Ge2 SLs grown on
different substrates. Solid lines represent predictions made by considering
electron scattering due to phonons and ionized impurities. Dashed lines
display results obtained by considering electron scattering due to phonons
only, for comparison.

The non-monotonic S and $\rho$ lead to significant modulations in the power-
factors, as shown in Fig. 5(e). S and $\rho$ predicted using $\tau_{\text{ep-
DOS}}$(E) (dashed lines with circles) match the $\tau_{\text{ep-EPW}}$(E)
(solid lines) predictions, however, the two PF predictions show considerable
differences. We note that the PFs for all strained Si2Ge2 SLs are low compared
to bulk Si, for n${}_{\text{e}}<9.1\times 10^{19}$cm-3, with the values
decreasing with increase of substrate induced strain. However, some of the
strained SLs show an increased PF at high doping regime, n${}_{\text{e}}$
>$10^{20}$cm-3. This follows from the band structure physics of Si/Ge SLs
grown on substrates inducing high strains. The strain induces shift of
$\Delta_{\parallel}$ valley minibands to higher energies, resulting in closely
spaced minibands at high n${}_{\text{e}}$ or high $\text{E}_{\text{F}}$. The
dense miniband structures, the resultant electronic group velocities and
scattering rates directly result in a$~{}\sim$1.8 fold improvement of PF over
bulk Si at n${}_{\text{e}}\sim 2.7\times 10^{20}$cm-3, for Si2Ge2 SL with
$\Epsilon_{\text{Si}\parallel}=0\%$ (red, grown on Si substrate). However, the
peak PFs of the SLs are not improved compared to bulk Si at most carrier
concentrations, to prove beneficial for thermoelectric applications.

###  III.2 Effect of IMS on Electronic Transport Properties

In addition to electron scattering due to phonons, electronic transport
properties of heavily doped semiconductor superlattices are strongly
influenced by scattering due to impurities, which has hardly been investigated
using first-principles approaches. As outlined in the Methods section, we
obtain the electron relaxation times due to the IMS processes,
$\tau_{\text{im}}$ and combine with those due to the EPS processes,
$\tau_{\text{ep-EPW}}$ using Matthiessen’s rule, to calculate the total
$\tau$(E). In Fig. 6, we show the electronic transport properties of (a) SLs
of varied compositions, grown on Si substrates, and (b) Si2Ge2 SLs grown on
different substrates, respectively. The solid lines represent predictions
obtained by considering the total energy-dependent electron relaxation time,
$\tau$(E). The dashed lines represent predictions obtained by considering the
electron-phonon relaxation time, $\tau_{\text{ep-EPW}}$(E), only. The dashed
lines display the same results we have shown in Fig. 4 and 5, obtained using
the EPS-EPW rates. We include both the predictions in the same figure to
compare the effect of different scattering mechanisms on the electronic
transport properties of the SLs.

Figure 6(a-i) show the Seebeck coefficients of SLs of varied compositions,
grown on Si substrates. The comparison between the solid lines and the dashed
lines exhibit the change of S results, due to different scattering processes.
This is unlike the results reported for bulk Si, that the diffusive part of S
does not depend strongly on the nature of the scattering mechanisms [69]. A
maximum 1.77-fold increase of S can be observed for Si3Ge1 SL (blue) when
considering both EPS and IMS processes, compared to EPS only results. However,
the trend of S for the different SLs remains preserved. The Seebeck
coefficients of Si1Ge3 (red) and Si2Ge2 SLs (green) remain close to S of bulk
Si (black) at most $\text{n}_{\text{e}}$, while S of Si3Ge1 SL (blue) is
reduced at all $\text{n}_{\text{e}}$ (Fig. 6(a-i)). Figure 6(b-i) shows the
resistivities of SLs obtained by considering electron scattering only due to
phonons (dashed lines) compared to those obtained by considering both EPS and
IMS processes (solid lines). Similar to S, the resistivities of Si3Ge1 SL
(blue) are reduced, while Si1Ge3 (red) and Si2Ge2 SLs (green) values are
closer to bulk Si values (black) at most n${}_{\text{e}}$. The consideration
of the IMS processes increases the resistivities at almost all carrier
concentrations, as expected. The effect is more pronounced, as shown in the
log scale. A maximum 4.3-fold increase of $\rho$ is observed for the Si3Ge1 SL
(blue), when considering both EPS and IMS processes, compared to EPS only
results. Figure 6(a-iii) shows the comparison between power-factors of SLs
obtained by considering only EPS processes (dashed lines) and both EPS and IMS
processes (solid lines). The consideration of IMS processes significantly
increase resistivities and reduce PFs for all systems. However, the variations
of S and $\rho$ lead to a peak PF of 4.60 for Si1Ge3 SL (red) at
n${}_{\text{e}}\sim 3.7\times 10^{19}$cm-3, which is $\sim 1.56$-fold (or
$56\%$) greater than the bulk Si peak PF of $2.95$ at n${}_{\text{e}}\sim
2.4\times 10^{19}$cm-3. The maximum increase of PF over bulk Si value can be
noted for n${}_{\text{e}}\sim 7.4\times 10^{19}$cm-3. The PF of Si1Ge3 SL is
$\sim 1.62$-times the bulk Si PF value at this carrier concentration.

We make similar observations regarding the role of IMS processes on the
electronic transport properties of Si2Ge2 SLs, grown on substrates that induce
different in-plane strains. The S results shown with solid lines in Fig.
6(b-i) are modified from the results shown with dashed lines, that only
considers electron relaxation due to the EPS processes. A maximum 1.4-fold
increase of EPS+IMS S results is observed for Si2Ge2 SL on Si substrate (red).
Figure 6(b-ii), shows the resistivities obtained by considering EPS only
(dashed lines) compared to those obtained by considering both EPS and IMS
processes (solid lines). Similar to the SLs with varied compositions, we note
that the consideration of the IMS processes increases the resistivities at
almost all carrier concentrations. A maximum 3.7-fold increase of $\rho$ is
observed for the Si2Ge2 SL on Si substrate (red). Consequentially, the PFs of
these SLs, as shown with solid lines in Fig. 6(b-iii), are significantly
different from the EPS only results (dashed lines). The comparison between the
solid and the dashed lines shows that the incorporation of the combined
$\tau$(E) results in reduced PFs compared to the EPS only results, for all
systems. However, we find that the PF of the SL grown on Si substrate (red) is
higher than that of bulk Si for n${}_{\text{e}}\sim 7.6\times
10^{18}-5.2\times 10^{20}$ cm-3. A maximum $\sim 1.9$-fold improvement is
observed at n${}_{\text{e}}\sim 2.3\times 10^{20}$cm-3. Peak to peak
comparisons between PFs of different systems show that peak PF of the SL grown
on Si substrate is $3.67$ at n${}_{\text{e}}\sim 5.5\times 10^{19}$cm-3. This
value is $\sim 1.24$-fold (or $24\%$) greater than the bulk Si peak PF of
$2.95$ at n${}_{\text{e}}\sim 2.4\times 10^{19}$cm-3.

Overall, we observe that the consideration of both the scattering processes
decreases PFs of bulk Si and Si/Ge SLs, compared to the results, obtained by
only including EPS processes. However, the PFs of the SLs are comparatively
less affected than bulk Si, when the impurity scattering mechanisms are
included. This observation can be explained through a qualitative discussion
of electron mean free paths (MFPs) in bulk Si and Si/Ge heterostructures, and
the effect of IMS processes on the MFPs. First-principles studies have
reported that the electron MFPs can go up to about 60 nm in bulk silicon at
300 K at low carrier concentration, $10^{16}\ \text{cm}^{-3}$. The MFPs in Si
strongly depend on carrier concentrations, and decrease monotonically with
increase of carrier concentration, due to the electron-impurity scattering
processes. The MFPs reduce to about 20 nm in bulk Si with carrier
concentration of $10^{19}\ \text{cm}^{-3}$. In this doping regime, the
dominant contribution to the electrical conductivity comes from electrons with
MFPs less than 10 nm [99]. Such studies about electron MFPs in Si/Ge
heterostructures are limited, however, the profiles of the chosen short-period
SLs introduce potential perturbations at length scales smaller than the
electron MFPs contributing to electron transport [100]. Therefore,
contribution from electrons with lower MFPs can be expected to dominate, which
are not considerably affected by the IMS processes. This argument could be
used to explain the observation that the influence of IMS processes on the
electronic transport properties of short-period SLs is less severe compared to
bulk Si.

##  IV Summary and Discussion

Here we demonstrate the variation of electronic properties of semiconductor
SLs due to changes in the lattice strain environments. In particular, we
consider electron relaxation in diverse SLs due to scattering with phonons and
ionized impurities, and report first-principles calculations of electronic
transport properties including the energy-dependent relaxation times. Previous
first-principles studies did not include the effect of energy dependent
electron relaxation times on electronic transport properties of SLs [34, 37,
38, 39]. We also discuss how these predictions change when different
scattering approximations, such as constant relaxation time and DOS scattering
approximations, are used. We investigate two classes of short-period Si/Ge SLs
with diverse lattice strain environments: SLs with varied layer compositions
grown on identical substrates and SLs with identical compositions but grown on
different substrates. We calculate the electronic structure properties of the
eight-atom Si/Ge SLs by performing DFT calculations, as implemented in the
Quantum Espresso package [55]. We consider electron relaxation due to
scattering with phonons and ionized impurities, and compute both the EPS and
IMS rates using a perturbative approach following Fermi’s golden rule. We
compute electron-phonon matrix elements using the EPW package based on
maximally localized Wannier functions [56, 57], which provide accurate
interpolation of the matrix elements from coarse grids to dense grids [58] and
allows to reduce computational expenses. We perform extensive tests to
determine the coarse and fine grid sizes that results in converged electronic
properties. We illustrate the physical mechanisms that influence EPS processes
and determine the electronic properties of the SLs. Our analysis shows that
the energy bands are highly sensitive to the lattice strain environments of
the SLs with diverse compositions or SLs grown on different substrates. The
layer compositions and substrate induced strains result in non-uniform
separations between SL monolayers. The non-uniform layer separations modify
the contributions from $\Delta_{\perp}$ and $\Delta_{\parallel}$ valleys, and
induce energy shift of the SL bands. We illustrate that these energy shifts
strongly influence the electron-phonon self-energies. We obtain the EPS rates
from the imaginary part of the electron-phonon self-energies. Our analysis
establishes a direct relationship between the energy bands and the EPS rates
in the SLs with diverse lattice strain environments.

We model the electron-ionized impurity interaction potential with a long-range
Coulomb tail that represents a screened ionized impurity. Past first-
principles studies demonstrated that incorporation of such long-range Coulomb
tail is essential to account for the effect of ionized dopants on electronic
transport properties of bulk Si [99]. We show that the EPS processes determine
the resistivity of bulk Si at low n${}_{\text{e}}$, while the consideration of
IMS processes improves the resistivity predictions at high n${}_{\text{e}}$,
matching measured data [94]. Our study substantiates the observation that the
IMS processes play a significant role in determining electronic resistivities
of bulk Si at high carrier concentrations [69, 57]. On the other hand, IMS
processes do not significantly affect the diffusive components of the Seebeck
coefficients of bulk Si. We further discuss the effect of IMS processes on
electron transport properties of Si/Ge SLs, which has not been explored using
first-principles approach. We explicitly account for the in-plane and the
cross-plane structural anisotropy of the SL configurations, to model the
electron-ionized impurity interaction potentials. We obtain anisotropic
dielectric tensors from first-principles calculations and use them to compute
the electron-ionized impurity perturbation matrix elements. Past studies of
IMS rates in bulk Si used isotropic dielectric tensor, and implemented the
Brook-Herring’s scattering equation [69, 61]. We illustrate here a modified
Brooks-Herring formulation that explicitly considers the effects of
anisotropic potentials. However, we note that the % dielectric anisotropy
(defined here as,
$|(\epsilon_{\perp}-\epsilon_{\parallel})|/\epsilon_{\parallel}\times 100$),
of the chosen SLs stays within 6.5% and can be considered weak. We report here
the values obtained by including the anisotropic dielectric tensor
($\epsilon_{\perp}$ and $\epsilon_{\parallel}$), defined in Eq. 9 with
$\epsilon_{\perp}$ and $\epsilon_{\parallel}$ shown in Tab. 1. Separately, we
compute the electronic transport properties using the isotropic BH formalism
with an average dielectric constant,
$((\epsilon_{\perp}+2\epsilon_{\parallel})/3$). We find that the two sets of
predictions do not vary significantly. We do not include the calculations here
for brevity. Anisotropic impurity scattering potentials have not been employed
in past studies to model scattering rates and resultant SL electronic
transport properties. We anticipate that our calculations considering the
anisotropic dielectric tensors will encourage future studies of electron-
impurity scattering rates in diverse anisotropic systems, going beyond bulk
materials.

We demonstrate that the electronic transport coefficients of short-period
Si/Ge SLs show significant variations when energy dependent relaxation times
are taken into account. We find that the inclusion of the EPS processes
negatively impacts the power-factor of the SLs at most $\text{n}_{\text{e}}$,
in comparison with the bulk Si values. Interestingly, when IMS processes are
included along with EPS, significant changes are exhibited by the transport
properties. We find that Si2Ge2 and Si1Ge3 SLs grown on Si substrate show
$~{}\sim{1.62}$-fold and $~{}\sim{1.9}$-fold improvement over bulk Si at
$\text{n}_{\text{e}}\sim 7.4\times 10^{19}\text{cm}^{-3}$ and
$\text{n}_{\text{e}}\sim 2.3\times 10^{20}\text{cm}^{-3}$, respectively. Their
overall peak PFs are $~{}\sim{1.56}$-fold and $~{}\sim{1.24}$-fold greater
than the bulk Si peak PF. We observe that the PFs can be drastically reduced
in SLs with certain lattice strain environments. We illustrate that the
electronic and therefore, thermoelectric properties can be tuned by varying
the compositions and substrate induced strains of Si/Ge SLs. Although this
concept has been previously discussed using analytical and first-principles
based CRTA-BTE approaches [101, 36, 38, 37, 102], the energy dependent
scattering effects were not considered. Nanostructured Si/Ge SLs are being
extensively studied to reduce the thermal conductivity of Si based
thermoelectric materials. Additionally, significant reduction of thermal
conductivity of Si has been reported at the high doping regime, owing to
phonon scattering due to electrons [46]. The power-factor improvements at high
$\text{n}_{\text{e}}$ predicted by our study, could provide promising avenues
for thermoelectric applications, when combined with low thermal
conductivities. We hope that our results will encourage experimental studies
to validate this promising physical phenomenon, illustrated by taking into
account the effect of different energy dependent scattering processes. Our
results establish the role of the strain-modulated DOS, electron group
velocities and energy-dependent electron scattering rates on the electronic
transport properties. The physical insights demonstrated in this analysis not
only are valuable to predict band movement and electron relaxation in complex
heterostructures, but essential to develop strain engineering approaches to
optimize electronic properties. For example, understanding of the interactions
of electrons with real and quasiparticles is crucial to optimize the
electronic properties of thermoelectric and other technology enabling
materials.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We are indebted to Samuel Poncé for useful discussions and assistance with
implementing EPW code for electron-phonon scattering rate calculations. We
gratefully acknowledge funding from the Defense Advanced Research Projects
Agency (Defense Sciences Office) [Agreement No.: HR0011-16-2-0043]. This work
used the Extreme Science and Engineering Discovery Environment (XSEDE), which
is supported by National Science Foundation grant number ACI-1548562. We are
grateful for the computing resources provided by the Department of Defense
(DoD) High Performance Computing Modernization Program (HPCMP) at the Air
Force Research Laboratory DoD Supercomputing Research Center (AFRL-DSRC). We
acknowledge the computing resources provided the RMACC Summit supercomputer,
which is supported by the National Science Foundation (awards ACI-1532235 and
ACI-1532236), the University of Colorado Boulder, and Colorado State
University. The Summit supercomputer is a joint effort of the University of
Colorado Boulder and Colorado State University.

## References

  * Thompson  _et al._ [2004] S. E. Thompson, M. Armstrong, C. Auth, M. Alavi, M. Buehler, R. Chau, S. Cea, T. Ghani, G. Glass, T. Hoffman, _et al._ , A 90-nm logic technology featuring strained-silicon, IEEE Transactions on Electron Devices 51, 1790 (2004).
  * Meyerson [1994] B. S. Meyerson, High-speed silicon-germanium electronics, Scientific American 270, 62 (1994).
  * Nissim and Rosencher [2012] Y. Nissim and E. Rosencher,  _Heterostructures on silicon: one step further with silicon_ , Vol. 160 (Springer Science & Business Media, 2012).
  * Paul [2004] D. J. Paul, Si/SiGe heterostructures: from material and physics to devices and circuits, Semicond. Sci. Technol. 19, R75 (2004).
  * Koester  _et al._ [2006] S. J. Koester, J. D. Schaub, G. Dehlinger, and J. O. Chu, Germanium-on-soi infrared detectors for integrated photonic applications, IEEE Journal of selected topics in quantum electronics 12, 1489 (2006).
  * Liu  _et al._ [2010] J. Liu, X. Sun, R. Camacho-Aguilera, L. C. Kimerling, and J. Michel, Ge-on-si laser operating at room temperature, Optics letters 35, 679 (2010).
  * Alam and Ramakrishna [2013] H. Alam and S. Ramakrishna, A review on the enhancement of figure of merit from bulk to nano-thermoelectric materials, Nano Energy 2, 190 (2013).
  * Taniguchi  _et al._ [2020] T. Taniguchi, T. Ishibe, N. Naruse, Y. Mera, M. M. Alam, K. Sawano, and Y. Nakamura, High thermoelectric power factor realization in si-rich sige/si superlattices by super-controlled interfaces, ACS Applied Materials & Interfaces 12, 25428 (2020).
  * Chen  _et al._ [2003] G. Chen, M. Dresselhaus, G. Dresselhaus, J.-P. Fleurial, and T. Caillat, Recent developments in thermoelectric materials, International Materials Reviews 48, 45 (2003).
  * Dresselhaus  _et al._ [2007] M. S. Dresselhaus, G. Chen, M. Y. Tang, R. Yang, H. Lee, D. Wang, Z. Ren, J.-P. Fleurial, and P. Gogna, New directions for low-dimensional thermoelectric materials, Adv. Mater. 19, 1043 (2007).
  * Awschalom  _et al._ [2013] D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu, and J. R. Petta, Quantum spintronics: engineering and manipulating atom-like spins in semiconductors, Science 339, 1174 (2013).
  * Jansen [2012] R. Jansen, Silicon spintronics, Nature materials 11, 400 (2012).
  * Zwanenburg  _et al._ [2013] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Silicon quantum electronics, Reviews of modern physics 85, 961 (2013).
  * Zhang  _et al._ [2014] X. Zhang, H. Xie, M. Hu, H. Bao, S. Yue, G. Qin, and G. Su, Thermal conductivity of silicene calculated using an optimized stillinger-weber potential, Physical Review B 89, 054310 (2014).
  * Perez-Taborda  _et al._ [2016] J. A. Perez-Taborda, M. M. Rojo, J. Maiz, N. Neophytou, and M. Martin-Gonzalez, Ultra-low thermal conductivities in large-area si-ge nanomeshes for thermoelectric applications, Scientific reports 6, 32778 (2016).
  * Reparaz  _et al._ [2012] J. Reparaz, I. C. Marcus, A. R. Goñi, M. Garriga, and M. Alonso, Influence of alloy inhomogeneities on the determination by raman scattering of composition and strain in si1–x ge x/si (001) layers, Journal of Applied Physics 112, 023512 (2012).
  * Neogi  _et al._ [2015] S. Neogi, J. S. Reparaz, L. F. C. Pereira, B. Graczykowski, M. R. Wagner, M. Sledzinska, A. Shchepetov, M. Prunnila, J. Ahopelto, C. M. Sotomayor-Torres, _et al._ , Tuning thermal transport in ultrathin silicon membranes by surface nanoscale engineering, ACS nano 9, 3820 (2015).
  * Neogi and Donadio [2015] S. Neogi and D. Donadio, Thermal transport in free-standing silicon membranes: influence of dimensional reduction and surface nanostructures, The European Physical Journal B 88, 73 (2015).
  * Mangold  _et al._ [2016] C. Mangold, S. Neogi, and D. Donadio, Optimal thickness of silicon membranes to achieve maximum thermoelectric efficiency: A first principles study, Applied Physics Letters 109, 053902 (2016).
  * Xiong  _et al._ [2017] S. Xiong, D. Selli, S. Neogi, and D. Donadio, Native surface oxide turns alloyed silicon membranes into nanophononic metamaterials with ultralow thermal conductivity, Physical Review B 95, 180301 (2017).
  * Neogi and Donadio [2020] S. Neogi and D. Donadio, Anisotropic in-plane phonon transport in silicon membranes guided by nanoscale surface resonators, Physical Review Applied 14, 024004 (2020).
  * Honarvar and Hussein [2018] H. Honarvar and M. I. Hussein, Two orders of magnitude reduction in silicon membrane thermal conductivity by resonance hybridizations, Physical Review B 97, 195413 (2018).
  * Davis and Hussein [2014] B. L. Davis and M. I. Hussein, Nanophononic metamaterial: Thermal conductivity reduction by local resonance, Phys. Rev. Lett. 112, 055505 (2014).
  * Hahn  _et al._ [2016] K. R. Hahn, S. Cecchi, and L. Colombo, Effect of asymmetric concentration profile on thermal conductivity in ge/sige superlattices, Applied Physics Letters 108, 203102 (2016).
  * Ferrando-Villalba  _et al._ [2015] P. Ferrando-Villalba, A. F. Lopeandía, F. X. Alvarez, B. Paul, C. de Tomás, M. I. Alonso, M. Garriga, A. R. Goñi, J. Santiso, G. Garcia, _et al._ , Tailoring thermal conductivity by engineering compositional gradients in si 1- x ge x superlattices, Nano Research 8, 2833 (2015).
  * Zhang  _et al._ [2017] H. Zhang, H. Han, S. Xiong, H. Wang, S. Volz, and Y. Ni, Impeded thermal transport in composition graded sige nanowires, Applied Physics Letters 111, 121907 (2017).
  * Chen  _et al._ [2013] P. Chen, N. Katcho, J. Feser, W. Li, M. Glaser, O. Schmidt, D. G. Cahill, N. Mingo, and A. Rastelli, Role of surface-segregation-driven intermixing on the thermal transport through planar si/ge superlattices, Physical review letters 111, 115901 (2013).
  * Samvedi and Tomar [2009] V. Samvedi and V. Tomar, The role of interface thermal boundary resistance in the overall thermal conductivity of si–ge multilayered structures, Nanotechnology 20, 365701 (2009).
  * Li and Yang [2012] X. Li and R. Yang, Effect of lattice mismatch on phonon transmission and interface thermal conductance across dissimilar material interfaces, Physical Review B 86, 054305 (2012).
  * Satpathy  _et al._ [1988] S. Satpathy, R. M. Martin, and C. G. Van de Walle, Electronic properties of the (100)(si)/(ge) strained-layer superlattices, Physical Review B 38, 13237 (1988).
  * Hybertsen and Schlüter [1987] M. S. Hybertsen and M. Schlüter, Theory of optical transitions in si/ge (001) strained-layer superlattices, Physical Review B 36, 9683 (1987).
  * Ciraci and Batra [1988] S. Ciraci and I. P. Batra, Strained si/ge superlattices: Structural stability, growth, and electronic properties, Physical Review B 38, 1835 (1988).
  * Bernard and Zunger [1991] J. E. Bernard and A. Zunger, Strain energy and stability of si-ge compounds, alloys, and superlattices, Physical Review B 44, 1663 (1991).
  * Hinsche  _et al._ [2012] N. Hinsche, I. Mertig, and P. Zahn, Thermoelectric transport in strained si and si/ge heterostructures, Journal of Physics: Condensed Matter 24, 275501 (2012).
  * Shi  _et al._ [2012] L. Shi, J. Jiang, G. Zhang, and B. Li, High thermoelectric figure of merit in silicon-germanium superlattice structured nanowires, Applied Physics Letters 101, 233114 (2012).
  * Settipalli and Neogi [2020] M. Settipalli and S. Neogi, Theoretical prediction of enhanced thermopower in n-doped si/ge superlattices using effective mass approximation, Journal of Electronic Materials , 1 (2020).
  * Proshchenko  _et al._ [2019a] V. S. Proshchenko, M. Settipalli, and S. Neogi, Optimization of seebeck coefficients of strain-symmetrized semiconductor heterostructures, Applied Physics Letters 115, 211602 (2019a).
  * Proshchenko  _et al._ [2021] V. S. Proshchenko, M. Settipalli, A. K. Pimachev, and S. Neogi, Role of substrate strain to tune energy bands–seebeck relationship in semiconductor heterostructures, Journal of Applied Physics 129, 025301 (2021).
  * Pimachev and Neogi [2021] A. K. Pimachev and S. Neogi, First-principles prediction of electronic transport in fabricated semiconductor heterostructures via physics-aware machine learning, npj Computational Materials 7, 1 (2021).
  * Dekker [1981] A. J. Dekker, Nonpolar semiconductors, in  _Solid State Physics_ (Springer, 1981) pp. 319–347.
  * Nag [2012] B. R. Nag,  _Electron transport in compound semiconductors_ , Vol. 11 (Springer Science & Business Media, 2012).
  * Gupta  _et al._ [2020] R. Gupta, N. Kumar, P. Kaur, and C. Bera, Theoretical model for predicting thermoelectric properties of tin chalcogenides, Physical Chemistry Chemical Physics 22, 18989 (2020).
  * Fu  _et al._ [2017] B. Fu, G. Tang, and Y. Li, Electron–phonon scattering effect on the lattice thermal conductivity of silicon nanostructures, Physical Chemistry Chemical Physics 19, 28517 (2017).
  * Yang and Minnich [2017] L. Yang and A. J. Minnich, Thermal transport in nanocrystalline si and sige by ab initio based monte carlo simulation, Scientific reports 7, 44254 (2017).
  * Xu  _et al._ [2019] Q. Xu, J. Zhou, T.-H. Liu, and G. Chen, Effect of electron-phonon interaction on lattice thermal conductivity of sige alloys, Applied Physics Letters 115, 023903 (2019).
  * Liao  _et al._ [2015] B. Liao, B. Qiu, J. Zhou, S. Huberman, K. Esfarjani, and G. Chen, Significant reduction of lattice thermal conductivity by the electron-phonon interaction in silicon with high carrier concentrations: A first-principles study, Physical review letters 114, 115901 (2015).
  * Proshchenko  _et al._ [2019b] V. S. Proshchenko, P. P. Dholabhai, T. C. Sterling, and S. Neogi, Heat and charge transport in bulk semiconductors with interstitial defects, Physical Review B 99, 014207 (2019b).
  * Jacoboni  _et al._ [1977] C. Jacoboni, C. Canali, G. Ottaviani, and A. A. Quaranta, A review of some charge transport properties of silicon, Solid-State Electronics 20, 77 (1977).
  * Baccarani and Ostoja [1975] G. Baccarani and P. Ostoja, Electron mobility empirically related to the phosphorus concentration in silicon, Solid-State Electronics 18, 579 (1975).
  * Fan  _et al._ [2018] D. Fan, H. Liu, L. Cheng, J. Liang, and P. Jiang, A first-principles study of the effects of electron–phonon coupling on the thermoelectric properties: a case study of the sige compound, Journal of Materials Chemistry A 6, 12125 (2018).
  * Murphy-Armando and Fahy [2008] F. Murphy-Armando and S. Fahy, First-principles calculation of carrier-phonon scattering in n-type si 1- x ge x alloys, Physical Review B 78, 035202 (2008).
  * Ferre Llin  _et al._ [2013] L. Ferre Llin, A. Samarelli, S. Cecchi, T. Etzelstorfer, E. Müller Gubler, D. Chrastina, G. Isella, J. Stangl, J. Weaver, P. Dobson, _et al._ , The cross-plane thermoelectric properties of p-ge/si0. 5ge0. 5 superlattices, Applied physics letters 103, 143507 (2013).
  * Samarelli  _et al._ [2013] A. Samarelli, L. Ferre Llin, S. Cecchi, J. Frigerio, T. Etzelstorfer, E. Müller, Y. Zhang, J. Watling, D. Chrastina, G. Isella, _et al._ , The thermoelectric properties of ge/sige modulation doped superlattices, Journal of applied physics 113, 233704 (2013).
  * Koga  _et al._ [2000] T. Koga, S. Cronin, M. Dresselhaus, J. Liu, and K. Wang, Experimental proof-of-principle investigation of enhanced z 3d t in (001) oriented si/ge superlattices, Applied Physics Letters 77, 1490 (2000).
  * 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._ , Quantum espresso: a modular and open-source software project for quantum simulations of materials, Journal of physics: Condensed matter 21, 395502 (2009).
  * Giustino  _et al._ [2007] F. Giustino, M. L. Cohen, and S. G. Louie, Electron-phonon interaction using wannier functions, Physical Review B 76, 165108 (2007).
  * Poncé  _et al._ [2016] S. Poncé, E. R. Margine, C. Verdi, and F. Giustino, Epw: Electron–phonon coupling, transport and superconducting properties using maximally localized wannier functions, Computer Physics Communications 209, 116 (2016).
  * Park  _et al._ [2014] C.-H. Park, N. Bonini, T. Sohier, G. Samsonidze, B. Kozinsky, M. Calandra, F. Mauri, and N. Marzari, Electron–phonon interactions and the intrinsic electrical resistivity of graphene, Nano letters 14, 1113 (2014).
  * Zhang  _et al._ [2013] L. Zhang, J.-W. Luo, A. Saraiva, B. Koiller, and A. Zunger, Genetic design of enhanced valley splitting towards a spin qubit in silicon, Nature communications 4, 1 (2013).
  * Peter and Cardona [1996] Y. Peter and M. Cardona,  _Fundamentals of semiconductors: physics and materials properties_ (Springer Science & Business Media, Berlin, Heidelberg, 1996) p. Chapter 9.
  * Li [2012] S. S. Li,  _Semiconductor physical electronics_ (Springer Science & Business Media, 2012).
  * Restrepo  _et al._ [2009] O. Restrepo, K. Varga, and S. Pantelides, First-principles calculations of electron mobilities in silicon: Phonon and coulomb scattering, Applied Physics Letters 94, 212103 (2009).
  * Lundstrom [2009] M. Lundstrom,  _Fundamentals of carrier transport_ (Cambridge university press, 2009).
  * Ashcroft and Mermin [1976] N. W. Ashcroft and N. D. Mermin,  _Solid State Physics_ (Saunders, Philadelphia, 1976).
  * Madsen and Singh [2006] G. K. Madsen and D. J. Singh, Boltztrap. a code for calculating band-structure dependent quantities, Computer Physics Communications 175, 67 (2006).
  * Zhou  _et al._ [2015] J. Zhou, B. Liao, B. Qiu, S. Huberman, K. Esfarjani, M. S. Dresselhaus, and G. Chen, Ab initio optimization of phonon drag effect for lower-temperature thermoelectric energy conversion, Proceedings of the National Academy of Sciences 112, 14777 (2015).
  * Noffsinger  _et al._ [2010] J. Noffsinger, F. Giustino, B. D. Malone, C.-H. Park, S. G. Louie, and M. L. Cohen, Epw: A program for calculating the electron–phonon coupling using maximally localized wannier functions, Computer Physics Communications 181, 2140 (2010).
  * Meyer and Bartoli [1987] J. Meyer and F. Bartoli, Ionized-impurity scattering in the strong-screening limit, Physical Review B 36, 5989 (1987).
  * Fiorentini and Bonini [2016] M. Fiorentini and N. Bonini, Thermoelectric coefficients of n-doped silicon from first principles via the solution of the boltzmann transport equation, Physical Review B 94, 085204 (2016).
  * Giustino [2017] F. Giustino, Electron-phonon interactions from first principles, Reviews of Modern Physics 89, 015003 (2017).
  * Mahan  _et al._ [2014] G. Mahan, L. Lindsay, and D. Broido, The seebeck coefficient and phonon drag in silicon, Journal of Applied Physics 116, 245102 (2014).
  * Nadtochiy  _et al._ [2019] A. Nadtochiy, V. Kuryliuk, V. Strelchuk, O. Korotchenkov, P.-W. Li, and S.-W. Lee, Enhancing the seebeck effect in ge/si through the combination of interfacial design features, Scientific reports 9, 1 (2019).
  * Boukai  _et al._ [2008] A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard Iii, and J. R. Heath, Silicon nanowires as efficient thermoelectric materials, nature 451, 168 (2008).
  * Sadhu  _et al._ [2015] J. Sadhu, H. Tian, J. Ma, B. Azeredo, J. Kim, K. Balasundaram, C. Zhang, X. Li, P. Ferreira, and S. Sinha, Quenched phonon drag in silicon nanowires reveals significant effect in the bulk at room temperature, Nano letters 15, 3159 (2015).
  * Fischetti [1991] M. V. Fischetti, Monte carlo simulation of transport in technologically significant semiconductors of the diamond and zinc-blende structures. i. homogeneous transport, IEEE transactions on electron devices 38, 634 (1991).
  * Ganose  _et al._ [2021] A. M. Ganose, J. Park, A. Faghaninia, R. Woods-Robinson, K. A. Persson, and A. Jain, Efficient calculation of carrier scattering rates from first principles, Nature communications 12, 1 (2021).
  * Witkoske  _et al._ [2017] E. Witkoske, X. Wang, M. Lundstrom, V. Askarpour, and J. Maassen, Thermoelectric band engineering: The role of carrier scattering, Journal of Applied Physics 122, 175102 (2017).
  * Yu  _et al._ [2008] D. Yu, Y. Zhang, and F. Liu, First-principles study of electronic properties of biaxially strained silicon: Effects on charge carrier mobility, Physical Review B 78, 245204 (2008).
  * Chattopadhyay and Queisser [1981] D. Chattopadhyay and H. Queisser, Electron scattering by ionized impurities in semiconductors, Reviews of Modern Physics 53, 745 (1981).
  * Conwell and Weisskopf [1950] E. Conwell and V. Weisskopf, Theory of impurity scattering in semiconductors, Physical review 77, 388 (1950).
  * Jacoboni and Reggiani [1983] C. Jacoboni and L. Reggiani, The monte carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Reviews of modern Physics 55, 645 (1983).
  * Deng  _et al._ [2020] T. Deng, G. Wu, M. B. Sullivan, Z. M. Wong, K. Hippalgaonkar, J.-S. Wang, and S.-W. Yang, Epic star: a reliable and efficient approach for phonon-and impurity-limited charge transport calculations, npj Computational Materials 6, 1 (2020).
  * Schulz  _et al._ [1992] W. W. Schulz, P. B. Allen, and N. Trivedi, Hall coefficient of cubic metals, Physical Review B 45, 10886 (1992).
  * Samsonidze and Kozinsky [2018] G. Samsonidze and B. Kozinsky, Accelerated screening of thermoelectric materials by first-principles computations of electron–phonon scattering, Advanced Energy Materials 8, 1800246 (2018).
  * Allen  _et al._ [1988] P. B. Allen, W. E. Pickett, and H. Krakauer, Anisotropic normal-state transport properties predicted and analyzed for high-t c oxide superconductors, Physical Review B 37, 7482 (1988).
  * Perdew and Zunger [1981] J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Physical Review B 23, 5048 (1981).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened coulomb potential, Journal of chemical physics 118, 8207 (2003).
  * Monkhorst and Pack [1976] H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Physical review B 13, 5188 (1976).
  * Poncé  _et al._ [2018] S. Poncé, E. R. Margine, and F. Giustino, Towards predictive many-body calculations of phonon-limited carrier mobilities in semiconductors, Physical Review B 97, 121201 (2018).
  * Dunlap Jr and Watters [1953] W. Dunlap Jr and R. Watters, Direct measurement of the dielectric constants of silicon and germanium, Physical Review 92, 1396 (1953).
  * Zheng  _et al._ [2017] F. Zheng, J. Tao, and A. M. Rappe, Frequency-dependent dielectric function of semiconductors with application to physisorption, Physical Review B 95, 035203 (2017).
  * Geballe and Hull [1955] T. Geballe and G. Hull, Seebeck effect in silicon, Physical Review 98, 940 (1955).
  * Herring [1954] C. Herring, Theory of the thermoelectric power of semiconductors, Physical Review 96, 1163 (1954).
  * Mousty  _et al._ [1974] F. Mousty, P. Ostoja, and L. Passari, Relationship between resistivity and phosphorus concentration in silicon, Journal of Applied Physics 45, 4576 (1974).
  * Wen  _et al._ [2015] C.-Y. Wen, M. C. Reuter, D. Su, E. A. Stach, and F. M. Ross, Strain and stability of ultrathin Ge layers in Si/Ge/Si axial heterojunction nanowires, Nano letters 15, 1654 (2015).
  * Van de Walle and Martin [1986] C. G. Van de Walle and R. M. Martin, Theoretical calculations of heterojunction discontinuities in the si/ge system, Physical Review B 34, 5621 (1986).
  * Tserbak  _et al._ [1993] C. Tserbak, H. Polatoglou, and G. Theodorou, Unified approach to the electronic structure of strained si/ge superlattices, Physical Review B 47, 7104 (1993).
  * Bahk  _et al._ [2012] J.-H. Bahk, R. B. Sadeghian, Z. Bian, and A. Shakouri, Seebeck enhancement through miniband conduction in iii–v semiconductor superlattices at low temperatures, Journal of electronic materials 41, 1498 (2012).
  * Qiu  _et al._ [2015] B. Qiu, Z. Tian, A. Vallabhaneni, B. Liao, J. M. Mendoza, O. D. Restrepo, X. Ruan, and G. Chen, First-principles simulation of electron mean-free-path spectra and thermoelectric properties in silicon, EPL (Europhysics Letters) 109, 57006 (2015).
  * Tsu and Esaki [1973] R. Tsu and L. Esaki, Tunneling in a finite superlattice, Applied Physics Letters 22, 562 (1973).
  * Hinsche  _et al._ [2011] N. F. Hinsche, I. Mertig, and P. Zahn, Effect of strain on the thermoelectric properties of silicon: an ab initio study, Journal of Physics: Condensed Matter 23, 295502 (2011).
  * Koga  _et al._ [1999] T. Koga, X. Sun, S. Cronin, and M. Dresselhaus, Carrier pocket engineering applied to “strained” si/ge superlattices to design useful thermoelectric materials, Applied Physics Letters 75, 2438 (1999).