# Significant reduction in semiconductor interface resistance via interfacial
atomic mixing

Qichen Song  Department of Mechanical Engineering, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139  Jiawei Zhou  Department of
Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139  Gang Chen<EMAIL_ADDRESS>Department of Mechanical
Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139

###### Abstract

The contact resistance between two dissimilar semiconductors is determined by
the carrier transmission through their interface. Despite the ubiquitous
presence of interfaces, quantitative simulation of charge transport across
such interfaces is difficult, limiting the understanding of interfacial charge
transport. This work employs Green’s functions to study the charge transport
across representative Si/Ge interfaces. For perfect interfaces, it is found
that the transmittance is small and the contact resistance is high, not only
because the mismatch of carrier pockets makes it hard to meet the momentum
conservation requirement, but also because of the incompatible symmetries of
the Bloch wave functions of the two sides. In contrast, atomic mixing at the
interface increases the carrier transmittance as the interface roughness opens
many nonspecular transmission channels, which greatly reduces the contact
resistance compared with the perfect interface. Specifically, we show that
disordered interfaces with certain symmetries create more nonspecular
transmission. The insights from our study will benefit the future design of
high-performance heterostructures with low contact resistance.

Nonspecular electron transmission, interfacial transport, disordered interface

##  I Introduction

The importance of interfaces in advanced semiconductor devices has been
clearly pointed out by Herbert Kroemer with his famous statement, “the
interface is the device” [1]. Semiconductor heterostructures play essential
roles in vertical-cavity surface-emitting lasers [2, 3, 4], heterostructure
bipolar transistors [5, 6, 7], quantum cascade lasers [8], quantum well
infrared photodetectors [9], thermionic microcoolers [10, 11], spin qubit
devices [12, 13], thermoelectric power generators [14, 15, 16, 17], etc.
However, the interfaces in heterostructures strongly scatter electrons and
cause the contact resistance [18, 19, 20, 21]. The interface scattering
probabilities are not only determined by the intrinsic properties of bulk
materials, but by the non-intrinsic properties such as the interface
structures. Specifically, the interface roughness due to atomic mixing [22],
as a common type of interface disorder, alters the contact resistance. In
order to design proper interface structures that minimize interfacial
resistance, it is crucial to understand how the atomic mixing affects electron
scattering at interfaces.

The nonequilibrium Green’s function (NEGF) is often used to describe the
structure-dependent charge transport [23]. Many works using NEGF combined with
Landauer formula for conductance are conducted to study the transport across
molecular junctions [24, 25, 26, 27], nanotransistors [28, 29], grain
boundaries in two-dimensional materials [30], metal-semiconductor interfaces
[31, 32], metal-metal interfaces in magnetic multilayers [33, 34, 35, 36] and
semiconductor interfaces [37, 38]. In particular, Bellotti et al. investigated
the carrier transport through semiconductor interfaces in the presence of
positional and compositional disorders using NEGF and found that the disorder
significantly impedes the coherent propagation of carriers through multiple
interfaces [37]. Tibaldi et al. performed a large-scale NEGF calculation of
the carrier transport in a realistic tunnel junctions for vertical-cavity
surface-emitting lasers and achieved a good agreement with experimental
$I$-$V$ curve. However, the interface roughness in the transverse directions
is neglected in these works, as the computational cost of NEGF increases
dramatically with the cross-section areas of the interface. Aside from NEGF
calculation, Daryoosh et al. used a simple effective mass model to study the
carrier transport through barriers in metal-based superlattices and found that
the nonspecular (diffuse) scattering can dramatically increase the
thermoelectric figure of merit $zT$ [17]. Los studied how the transmission
probability varies with the average fluctuations of potential energies due to
interface disorders under the effective mass approximation [39]. However, the
effective mass approximation adopted in these works can poorly describe
practical semiconductors with band pockets not at zone center. Due to the
multi-valley nature [15] of the band structures of semiconductors, new physics
shall emerge for electron interfacial transport.

Figure 1: (a) The partitioning for Green’s function calculation with repeated
unit cell in the lead region being numbered with 0, 1 …. (b) The transverse
momentum conservation law and how the transverse momenta in the unit-cell
representation and the supercell representation are related. (c) Specular
transmission process through a perfect interface where the transverse momentum
is conserved, $\mathbf{k}_{i,\parallel}=\mathbf{k}_{f,\parallel}$. (d)
Nonspecular transmission processes through a disordered interface, where
$\mathbf{k}_{i,\parallel}=\mathbf{k}_{f,\parallel}+m\mathbf{G}_{x}+n\mathbf{G}_{y}$,
allows the scattering between valleys with different $\mathbf{k}_{\parallel}$.

In this work, we apply the mode-resolved Green’s function formalism with
tight-binding Hamiltonian to study charge transport across perfect and rough
interfaces due to atomic mixing. In particular, we take advantage of the
transverse translational symmetry to reduce the computational cost of surface
Green’s function. The tight-binding Hamiltonian makes sure the multiple
carrier pockets in the Brillouin zone are properly described. We vary the
degree of disorders in transverse directions and perpendicular direction and
study the specular and nonspecular interface scattering processes with mode
resolution. Moreover, we unveil the roles of disorders and symmetries in
assisting nonspecular transmission. We show that over one order of magnitude
of reduction of the specific contact resistance can be achieved by the
interfacial atomic mixing.

##  II Methodology

In the Green’s function calculation, we first divide the system into three
regions, two semi-infinite lead regions and a device region, as depicted in
Fig. 1 (a). The repeated cells along $z$ direction in the lead region are
indexed by 0, 1, … and the period length is $a_{z,\alpha}$ with $\alpha=L,R$.
The whole supercell structure is periodic along directions parallel to the
interface. Inside the lead region, there are $N_{\mathrm{uc},x}\times
N_{\mathrm{uc},y}$ identical unit cells along the transverse directions. The
transverse lattice vector for the supercell is
$\mathbf{R}_{\mathrm{sc},\beta}=N_{\mathrm{uc},\beta}\mathbf{R}_{\mathrm{uc},\beta}$
with $\beta=x,y$, where $\mathbf{R}_{\mathrm{uc},\beta}$ is the transverse
lattice vector for the unit cell. As a result, the transverse momentum in the
supercell representation can be uniquely unfolded in a momentum defined in the
unit-cell representation, as elucidated in Fig. 1 (b). The unfolded momentum
can be expressed by,

$\mathbf{k}_{\mathrm{uc},\parallel}=\mathbf{k}_{\mathrm{sc},\parallel}+a\mathbf{G}_{\mathrm{sc},x}+b\mathbf{G}_{\mathrm{sc},y}$
(1)

where $\mathbf{G}_{\mathrm{sc},x}$ and $\mathbf{G}_{\mathrm{sc},y}$ are the
transverse reciprocal lattice vectorsa, and $a$ and $b$ are integers to be
determined. Finding the correct pair of $a$ and $b$ is known as an unfolding
problem and we use the unfolding scheme by Popescu and Zunger [40] to resolve
the correct $\mathbf{k}_{\mathrm{uc},\parallel}$.

We consider the elastic interface scattering limit, where the energy $E$ of
the incident electron is conserved. In addition, the in-plane translational
symmetry of the supercell dictates that the transverse momentum
$\mathbf{k}_{\mathrm{sc},\parallel}$ must be conserved during an interface
scattering event. When the device region contains a perfect interface with the
same in-plane periodicity as the lead region, the transverse momentum
$\mathbf{k}_{\mathrm{uc},\parallel}$ is conserved. However, when the device
region consists of a rough interface, $\mathbf{k}_{\mathrm{uc},\parallel}$ is
not always conserved. This is because the interface roughness breaks the
internal transverse translational symmetry within the supercell and
$\mathbf{k}_{\mathrm{sc},\parallel}$ can be unfolded into different
$\mathbf{k}_{\mathrm{uc},\parallel}$ for the incident state and the
transmitted state. As illustrated in Fig. 1 (c), for a perfect interface,
$\mathbf{k}_{\mathrm{uc},\parallel}$ is conserved, and we denote this type of
scattering process the specular transmission. For a rough interface shown in
Fig. 1 (d), $\mathbf{k}_{\mathrm{uc},\parallel}$ can be either conserved or
nonconserved. Particularly, we denote the scattering process with nonconserved
$\mathbf{k}_{\mathrm{uc},\parallel}$ the nonspecular transmission.

We define the transmission probability matrix from the left side
$T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ as the ratio between the normal
current of the transmitted state $j$ to the incident state $i$. Formally, we
can express the specular and nonspecular transmission probability matrix with,

$\begin{cases}T_{\mathrm{s},ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})=T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel}),\;\text{when}\;\mathbf{k}_{\mathrm{uc},\parallel,j}=\mathbf{k}_{\mathrm{uc},\parallel,i}\\\
T_{\mathrm{ns},ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})=T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel}),\;\text{when}\;\mathbf{k}_{\mathrm{uc},\parallel,j}\neq\mathbf{k}_{\mathrm{uc},\parallel,i}\end{cases}$
(2)

The elements of the transmission probability matrix from the left side is
given by,

$T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})=|t_{RL,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})|^{2}$
(3)

where the transmission matrix
$t_{RL,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ is related to the Green’s
function via the following relation [41]:

$t_{RL}(E,\mathbf{k}_{\mathrm{sc},\parallel})=i\sqrt{V^{r}_{R}}[U^{r}_{R}]^{-1}G_{N+1,0}[U^{a\dagger}_{L}]^{-1}\sqrt{V^{a}_{L}}$
(4)

The formal definitions and detailed calculations of the velocity matrices
$V_{R/L}^{r/a}$, eigenvector matrices $U_{R/L}^{r/a}$, and Green’s function
$G_{N+1,0}$ can be found in Appendix A. Note that the calculation of the
velocity matrices and eigenvector matrices require the surface Green’s
function $g^{a/r}_{L/R}(E,\mathbf{k}_{\mathrm{sc},\parallel})$. We apply the
Fourier transform to the Hamiltonian to obtain the block-diagonal surface
Green’s function. Then, we apply the inverse Fourier transform to obtain the
surface Green’s function. These procedures allow us to invert small matrix
multiple times rather than directly inverting the large matrix, which greatly
boosts the computational efficiency. The detailed implementation can be found
in our prior work on studying diffuse phonon scattering by rough interfaces
[42].

Figure 2: (a) The band structure for bulk Si, Ge along high-symmetry lines in
the first Brillouin zone. Left inset: the three-dimensional first Brillouin
zone of Si (Ge) and its projection on the (001) plane. Right inset: the atomic
structure for Si (Ge) unit cell along [001] direction. (b) The band structures
for Ge and tensile-strained Ge (2 %). (c) The band structures for Si and Ge
slabs along high-symmetry lines in the surface Brillouin zone. The slab
contains 108 atom layers (27$a$ in thickness with $a$ the lattice constant).

##  III Band structures

We study the [001] Si/Ge interface as it is a classical semiconductor
interface used in a wide range of applications such as quantum information
storage [13], strained field-effect transistors [43, 44] and thermoelectrics
[45, 15]. To start with, we examine the bands structures for Si and Ge
individually. We use $sp^{3}d^{5}s^{*}$ Slater-Koster tight-binding model [46,
47] to construct the Hamiltonian, with hopping integral parametrizations from
Niquet et al [48]. More specifically, the hopping integral varies with the
bond length according to a power law,

$V_{\alpha\beta\gamma}(L)=V_{0,\alpha\beta\gamma}\left(\frac{L}{L_{0}}\right)^{\chi_{\alpha\beta\gamma}}$
(5)

where $\alpha$ and $\beta$ refer to the orbital types, $\gamma$ is the type of
bond, $L$ is the bond length, $L_{0}$ is the unstrained bond length,
$V_{0,\alpha\beta\gamma}$ is the hopping integral for unstrained bond, and
$\chi_{\alpha\beta\gamma}$ is the power-law exponent. The band structures
using this set of hopping integrals have shown an excellent agreement with GW
calculations with various strain ratios [48]. The bulk Si and Ge have
mismatched lattice constants with $a_{\mathrm{Si}}=5.431$ Å and
$a_{\mathrm{Ge}}=5.658$ Å. Correspondingly, the unstrained bond lengths for Si
and Ge are 2.352 and 2.450 Å , respectively. For simplicity, we study the
lattice-matched interface and we assume the Si-Si and Ge-Ge bond lengths are
the same, $L=$ 2.398 Å , which is relaxed Si-Ge bond length found by Niquet et
al [48]. Furthermore, we rescale the $L_{0}$ for Si-Si bond and Ge-Ge bond to
be $L_{0}=$ 2.398 Å to ensure that the Si’s and Ge’s band structures are the
same with their unstrained bulk band structures [49, 50]. In our calculation,
the spin-orbital coupling is not included.

First, we compare the band structures $E_{n}(\mathbf{k})$ of bulk Si and Ge
along high-symmetry paths and examine the distributions of electron/hole
pockets in the first Brillouin zone, as depicted in Fig. 2 (a), which clearly
shows that the conduction band pockets for Si and Ge are distributed very
differently, whereas their valence band pockets are quite similar. In
particular, the highest valence bands for Si and Ge are both at $\Gamma$
point. In contrast, the conduction band edge for Si is close to the X point
along the $\Gamma$X path (in the following, we denote this point $\Delta$),
while the conduction band edge for Ge is at the L point. In addition, there
are six pockets for Si’s lowest conduction band at the $\Delta$ point, while
there are four pockets (or eight half-pockets) for Ge at the L point. The
second-lowest conduction band for Si is at the X point. The second- and third-
lowest conduction bands for Ge are at the $\Gamma$ point and $\Delta$ point,
respectively.

Next, we look into the symmetry properties of the Bloch wave functions in
order to develop an understanding of how symmetry affects the transmission. In
the bra-ket notation, the transmission matrix element is directly proportional
to the Green’s function matrix element,

$t_{ji}\propto\bra{u_{j}}\hat{G}\ket{u_{i}}$ (6)

where $\hat{G}=\left(EI-\hat{H}\right)^{-1}$ is the Green’s function operator
and $\hat{H}$ is the Hamiltonian operator [51, 52]. It is easy to show that
$\hat{G}$ inherits all symmetries of $\hat{H}$ [53]. For $\ket{u_{j}}$ and
$\ket{u_{i}}$ with certain type of symmetries, the transmission matrix element
$t_{ji}$ is guaranteed to vanish according to group theory [54]. Thus, it is
essential to identify the symmetries of Bloch wave functions of the two sides.

To describe the symmetry properties of Bloch wave functions in Si and Ge, the
Bouckaert-Smoluchowski-Wigner (BSW) [55] notation is adopted in Fig. 2 (a),
which marks the irreducible representations for the Bloch wave function. The
different irreducible representations of the same group (labeled by the same
Greek letter with different subscripts) are orthogonal to each other. The
character tables for different groups can be found in group theory textbooks
[54] and online databases [56]. They describe how the Bloch wave function
transforms under different symmetry operations. For instance, the states of
the lowest conduction band of Si at the $\Delta$ point transform as
$\Delta_{1}$ representation under the symmetry operations of the $C_{4v}$
group. On the other hand, the states of the second-lowest conduction band in
Ge at the $\Delta$ point transform as $\Delta_{2^{\prime}}$ representation.
Without the loss of generality, we consider the $\Delta$ points along the z
axis [(0,0,1) axis]. In this case, one of the $C_{4v}$ group elements is the
symmetry operation $\hat{S}=\\{C_{4}|\bm{\tau}_{d}\\}$ with
$\bm{\tau}_{d}=\frac{1}{4}(a,a,a)$, which first rotates the Bloch wave
function by 90∘ with respect to the z axis and then applies the translation
operator by $\bm{\tau}_{d}$. When applying $\hat{S}$ to a state $\ket{u_{i}}$
of $\Delta_{1}$ symmetry, we have $\hat{S}\ket{u_{i}}=1\cdot
e^{ik_{z}a/4}\ket{u_{i}}$, where $k_{z}$ is the wave vector’s z component. The
phase factor $e^{ik_{z}a/4}$ appears because the space group of Si (Ge)
structure is nonsymmorphic. In comparison, when applying the same operator
$\hat{S}$ to a state $\ket{u_{i}}$ of $\Delta_{2^{\prime}}$ symmetry, we have
$\hat{S}\ket{u_{i}}=-1\cdot e^{ik_{z}a/4}\ket{u_{i}}$. Intuitively, one can
regard $\Delta_{1}$ as “even” and $\Delta_{2^{\prime}}$ as “odd” in a more
generalized way. If the incident and transmitted states are not compatible,
they will never interact. Hence, knowing the symmetry properties of wave
functions (i.e., their irreducible representations) will be useful in the
later analysis of the transmission probabilities.

Figure 3: The surface density of states
$\mathrm{SDOS}(E,\mathbf{k}_{\mathrm{uc},\parallel})$ for (a) Si, (b) Ge, and
(c) sGe. The logarithm of the surface density of state is indicated by colors
and the spin degree of freedom 2 is not multiplied. The transmission spectra
$T(E,\mathbf{k}_{\mathrm{uc},\parallel})$ along high-symmetry lines in the
surface Brillouin zone through (d) a perfect Si/Ge [001] interface and (e) a
perfect Si/sGe [001] interface. The color indicates the value of
$T(E,\mathbf{k}_{\mathrm{uc},\parallel})$. (f) Left panel: the transmission
function
$\Theta(E)=\frac{1}{N_{\mathbf{k}_{\mathrm{uc},\parallel}}}\sum_{\mathbf{k}_{\mathrm{uc},\parallel}}T(E,\mathbf{k}_{\mathrm{uc},\parallel})$
for a perfect Si/Ge interface and a perfect Si/sGe interface
($N_{\mathbf{k}_{\mathrm{uc},\parallel}}=40\times 40$ is used). Right panel:
the transmittance
$\tau_{\alpha}(E)=\frac{\Theta(E)}{\Theta_{\mathrm{bulk},\alpha}(E)}$ from Si
and Ge side, where $\Theta_{\mathrm{bulk},\alpha}(E)$ is the transmission
function for bulk $\alpha$ material.

Moreover, the strain effect can change the relative positions for different
valleys in the reciprocal space. We find that the strain generally has a
smaller impact on Si compared with Ge, thus, we only consider the case of
applying strain to Ge. Since we have already assumed the Si and Ge have the
same bond lengths $L=$ 2.398 Å, we change the equilibrium $L_{0}$ of Ge from
2.398 Å to 2.343 Å while keeping $L$ unchanged. The corresponding hopping
integrals $V_{\alpha\beta\gamma}$ defined in Eq. 5 are altered and the Ge band
structure is accordingly changed. Equivalently, we have applied a tensile
strain of 2 % to Ge. In Fig. 2 (b), we compare the band structures of strained
Ge (sGe for short) with the relaxed Ge. The elongated bond pushes the second-
lowest conduction band downwards and makes it the lowest conduction band. It
also shifts the third-lowest band further upwards. Meanwhile, the strained Ge-
Ge bond also makes the valence band upwards, thus causing a smaller band gap.
The features of the sGe band structure are consistent with other works [13,
57].

When forming an interface, the translational symmetry is broken along the
direction normal to the interface, and the band structures are now projected
to the two-dimensional surface Brillouin zone, as depicted in the left inset
of Fig. 2 (c). We conduct a slab calculation to study the projected band
structure. The slab is periodic along the x and y directions and finite in the
z direction. The unit cell for Si (Ge) slab along [001] direction contains
four atoms, as shown in the right inset of Fig. 2 (a). Note that the atomic
structure of Si (Ge) has mirror symmetries with respect to (110) and
(1$\bar{1}$0) planes. In Fig. 2 (c), we find that the highest valence bands of
Si and Ge are both projected to the $\bar{\Gamma}$ point. Two out of the six
lowest conduction band pockets of Si are projected to the $\bar{\Gamma}$
point, two pockets are projected to a point between the $\bar{\Gamma}$ and
$\mathrm{\bar{X}}$ points, and the remaining two pockets are projected to a
point between the $\bar{\Gamma}$ and $\mathrm{\bar{Y}}$ points. As for Ge, two
of the four lowest conduction band pockets at L points are projected to the
$\mathrm{\bar{X}}$ point, and the remaining two are projected to the
$\mathrm{\bar{Y}}$ point. Since we use a slab to compute the projected band
structure, we observe the surface states [58] for Si and Ge in the band gap.
They each have two degenerate surface states within the x-y plane, one for the
top surface, one for the bottom surface. However, in the direction normal to
the interface (z direction), these surface states are localized, thus do not
contribute to the interfacial transport.

Lastly, we study the density of states for the projected band. The density of
states for the projected band structures at the given energy $E$ and
transverse momentum $\mathbf{k}_{\mathrm{uc},\parallel}$ is obtained by taking
the imaginary part of retarded surface Green’s function given by Eq. 17 for
the lead,

$\mathrm{SDOS}(E,\mathbf{k}_{\mathrm{uc},\parallel})=-\frac{1}{\pi}\mathrm{Im}g^{r}_{\alpha}(E,\mathbf{k}_{\mathrm{uc},\parallel})$
(7)

where $g^{r}_{\alpha}(E,\mathbf{k}_{\mathrm{uc},\parallel})$ is the retarded
surface Green’s function for $\alpha$ lead with $\alpha=\mathrm{Si,Ge}$. From
the density of states shown in Fig. 3 (a), where we use color to indicate
$\mathrm{ln}\left[\mathrm{SDOS}(E,\mathbf{k}_{\mathrm{uc},\parallel})\right]$,
we identify the localized states in the band gap, the continuum spectrum of
propagating conduction band electrons and the resonant states inside the
continuum spectrum.

Figure 4: The ensemble-averaged total transmission function $\Theta(E)$ and
nonspecular transmission function $\Theta_{\mathrm{ns}}(E)$ through rough
Si/Ge interfaces for (a) electrons and (b) holes in solid lines compared with
the transmission function for a perfect interface in dashed line. 21
configurations of $2\times 2$, 8 ml disordered interfaces are used for
ensemble average. (c), (d) The transmission function for Si/sGe interfaces.
(e), (f) The ensemble-averaged mode-resolved specular and nonspecular
scattering probabilities at $E$ = 1.77 eV for rough Si/sGe interfaces defined
by Eq. 8 as a function of in-plane momentum
$\mathbf{k}_{\mathrm{uc},\parallel}$. The calculation uses a 40 $\times$ 40
$\mathbf{k}_{\mathrm{sc},\parallel}$-point mesh. Equivalently, it corresponds
to an 80 $\times$ 80 $\mathbf{k}_{\mathrm{uc},\parallel}$ mesh.

##  IV Transmission through a perfect interface

We first study the electron transmission through a perfect interface. There
are several relevant physical quantities, and we want to clarify their
definitions here to avoid confusion.
$t_{RL,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ is the transmission matrix,
which is used to compute the transmission probability matrix.
$T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ is the transmission probability
matrix, which described mode-to-mode transition probability, normalized by
normal incident current. $T_{i}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ refers
the transmission probability (transmittance) across the interface for an
incident electron $i$. The transmission spectrum
$T(E,\mathbf{k}_{\mathrm{sc},\parallel})$ is the number of transmission
channels including all subbands that have same $E$ and
$\mathbf{k}_{\mathrm{sc},\parallel}$. The transmission function (we use
transmission in short in figures) $\Theta(E)$ describes the total number of
transmission channels at the given energy $E$ and is the sum of all
transmission channels with different $\mathbf{k}_{\mathrm{sc},\parallel}$.
Note that $t_{RL,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$,
$T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$,
$T_{i}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ depend on which side incident
electron is from, whereas $T(E,\mathbf{k}_{\mathrm{sc},\parallel})$,
$\Theta(E)$ are independent of the side of incidence.

The transmission probability can be computed by summing the transmission
probability matrix over all possible final states. The specular part and
nonspecular part of the transmission probability read as,

$\begin{cases}T_{\mathrm{s},i}(E,\mathbf{k}_{\mathrm{sc},\parallel})=\sum_{j}T_{\mathrm{s},ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})\\\
T_{\mathrm{ns},i}(E,\mathbf{k}_{\mathrm{sc},\parallel})=\sum_{j}T_{\mathrm{ns},ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})\end{cases}$
(8)

The transmission function is the measure of conductance channels and can be
expressed by, $\Theta(E)=\Theta_{\mathrm{s}}(E)+\Theta_{\mathrm{ns}}(E)$,
where the specular and nonspecular transmission functions are defined by,

$\Theta_{\mathrm{s}}(E)=\frac{1}{N_{\mathbf{k}_{\mathrm{sc},\parallel}}}\sum_{i,\mathbf{k}_{\mathrm{sc},\parallel}}T_{\mathrm{s},i}(E,\mathbf{k}_{\mathrm{sc},\parallel})$
(9)
$\Theta_{\mathrm{ns}}(E)=\frac{1}{N_{\mathbf{k}_{\mathrm{sc},\parallel}}}\sum_{i,\mathbf{k}_{\mathrm{sc},\parallel}}T_{\mathrm{ns},i}(E,\mathbf{k}_{\mathrm{sc},\parallel})$
(10)

For the case of perfect interface, all the transmission processes are
specular, hence we have
$T_{\mathrm{s},ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})=T_{ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$.
In addition, for the perfect interface, we only need to construct a unit cell
as the supercell such that the in-plane momenta in the unit-cell
representation and the supercell representation are the same,
$\mathbf{k}_{\mathrm{uc},\parallel}=\mathbf{k}_{\mathrm{sc},\parallel}$.

The transmission spectrum is attained by
$T(E,\mathbf{k}_{\mathrm{uc},\parallel})=\sum_{i}T_{i}(E,\mathbf{k}_{\mathrm{uc},\parallel})$,
where we sum over all subbands with the same $E$ and
$\mathbf{k}_{\mathrm{uc},\parallel}$. In Figs. 3 (d) and (e), we show the
transmission spectra $T(E,\mathbf{k}_{\mathrm{uc},\parallel})$ through the
Si/Ge and Si/sGe interfaces. Comparing with the surface density of states
through examining the Figs. 3 (a)-(d), we see that the transmission is non-
zero only when the surface density of states for Si and sGe overlap. This is
due to the energy and momentum conservation requirement. For example, the Ge’s
lowest conduction band at $\mathrm{\bar{X}}$ and $\mathrm{\bar{Y}}$ does not
have any corresponding states in Si, thus cannot contribute to transmission.
Most of the overlapped states are the valleys at $\mathrm{\bar{\Gamma}}$ and
along the $\mathrm{\bar{\Gamma}}\mathrm{\bar{M}}$ path, which corresponds to
the lowest conduction band in Si, and second- and third-lowest conduction
bands in Ge.

Because of the mismatch of conduction band valleys of Si and Ge, a large
“transport gap” of 1.65 eV emerges at the $\mathrm{\bar{\Gamma}}$ point. For a
Si/sGe interface, the transmission spectra for holes change slightly from a
Si/Ge interface. The transport gap is 1.91 eV, which is even larger due to
fewer energy and momentum matched conduction bands. From the energy-resolved
transmission and transmittance in Fig. 3 (f), we also find that strain has
much smaller impact on the hole transmission than the electron transmission.
This is because the valence bands stay at the $\mathrm{\bar{\Gamma}}$ point
even with strain, while the strain changes the position of conduction bands in
reciprocal space more profoundly.

What is intriguing is that at the $\mathrm{\bar{\Gamma}}$ point, Si and sGe
have overlapped conduction band pockets, yet the transmission
$T(E,\mathbf{k}_{\mathrm{uc},\parallel})$ is still almost zero. This implies
that there are other factors other than energy and momentum conservation which
limits the transmission. We found out that the zero transmission originates
from the different symmetries of the wave functions. In three-dimensional
Brillouin zone, the lowest conduction band of sGe is at the $\Gamma$ point
with $\Gamma_{2^{\prime}}$ symmetry. Under the symmetry operation
$\hat{S}=\\{C_{4}|\bm{\tau}_{d}\\}$ mentioned above, it transforms as
$\hat{S}\ket{u_{R,\Gamma}}=-\ket{u_{R,\Gamma}}$. In comparison, for the lowest
conduction band of Si at the $\Delta$ point, it satisfies
$\hat{S}\ket{u_{L,\Delta}}=e^{ik_{L,z}a/4}\ket{u_{L,\Delta}}$. The Hamiltonian
for a perfect Si/sGe interface should always have “even” symmetry
representation $\Delta_{1}$. Hence, it follows that
$\hat{S}\hat{H}=e^{ik_{L,z}a/4}\hat{H}$ and
$\hat{S}\hat{G}=e^{-ik_{L,z}a/4}\hat{G}$. As a result, the transmission matrix
element should satisfy the condition,
$t_{ji}\propto\bra{u_{R,\Gamma}}\hat{G}\ket{u_{L,\Delta}}=\bra{\hat{S}u_{R,\Gamma}}\hat{S}\hat{G}\ket{\hat{S}u_{L,\Delta}}=-\bra{u_{R,\Gamma}}\hat{G}\ket{u_{L,\Delta}}$.
Consequently, we obtain that $T_{ji}=|t_{ji}|^{2}=0$. Similarly, for the
electrons at the $\Delta$ point with $\mathbf{k}=(0,0,k_{R,z})$, they have
$\Delta_{2^{\prime}}$ symmetry and transform as
$\hat{S}\ket{u_{R,\Delta}}=-e^{ik_{R,z}a/4}\ket{u_{R,\Delta}}$. Resultantly,
we have
$t_{ji}\propto\bra{u_{R,\Delta}}\hat{G}\ket{u_{L,\Delta}}=\bra{\hat{S}u_{R,\Delta}}\hat{S}\hat{G}\ket{\hat{S}u_{L,\Delta}}=-\bra{u_{R,\Delta}}\hat{G}\ket{u_{L,\Delta}}$
and, correspondingly, $T_{ji}=0$. In short, the transmission at $\bar{\Gamma}$
is exactly zero, dictated by symmetry.

##  V Transmission through rough interfaces

We add interface disorders in the form of atomic mixing. In particular, we
randomly swap pairs of Si and Ge atoms that have the same distance to the
interface. We use a larger supercell with in-plane periodicity to describe the
rough interface. To mimic an actual rough Si/Ge interface observed in
experiments [22], we make sure that the further away from the interface, the
fewer or equal number of atom pairs are swapped. In the following, we define
two measures of the degree of interface disorders along the interface normal
and along the transverse directions.

The first measure is the number of atom layers that are involved in atomic
mixing. If there are two layers of Si and two layers of Ge atoms that are
involved in atomic intermixing, the number of atoms that are swapped per layer
follows a pattern of 1|2|2|1. We label such interface structure by 4 ml, in
short for four mixing layers. A larger ml number corresponds to the larger
degree of disorders in the cross-plane direction. The atomic number density of
Si across the rough interface with different mixing layers can be found in
Fig. A1 in the Appendix.

The second measure is the size of the transverse supercell. For example, when
we construct a 2$\times$2 transverse supercell with 4 ml structure, there are
two out of four atoms for the Si atom layer closest to the interface and one
out of four atoms for the Si atom layer secondly closest to the interface
involved in atomic mixing. When we use a larger transverse supercell
(3$\times$3 or 4$\times$4), we let the number of swapped atoms unchanged. The
larger transverse supercell we use, the smaller degree of disorders along the
transverse directions. For a given ml number and a given supercell size, we
generate 21 random configurations and compute the ensemble average of the
transmission and reflection probability matrix elements.

Figure 5: The specific contact resistance for (a) electrons and (b) holes for
rough Si/Ge interfaces in solid lines and for the perfect Si/Ge interface in
dashed lines at various temperatures. The specific contact resistance for (c)
electrons and (d) holes for rough Si/sGe interfaces in solid lines and the
perfect Si/sGe interface in dashed lines at various temperatures.

We found that the total transmissions for electrons and holes are both
enhanced for rough Si/Ge and Si/sGe interfaces compared with the corresponding
perfect interfaces, as shown in Figs. 4 (a)-(d). Particularly, the total
transmission function for a perfect Si/sGe interface with energy $E$ ranging
from 1.5 to 1.9 eV is zero, whereas the transmission function for the
corresponding rough interface is largely enhanced by nonspecular scattering
processes. For the hole transmission, we notice that the nonspecular part is
much smaller than the specular part and the enhancement in the total
transmission is not significant, although the transmission for both electrons
and holes is enhanced by the rough interface compared with the perfect
interface. This is because holes are close to $\mathbf{k}=0$ and have long
wavelengths. A more detailed discussion on the enhancement of hole
transmission can be found in Appendix C. In Fig. A2 and Fig. A3, we have shown
the dependence of transmission function on the degree of disorders along
transverse and longitudinal directions. It turns out that the smaller
transverse supercell dimensions and large mixing layer numbers are in favor of
the nonspecular transmission. The smaller transverse supercell provides a
large $\mathbf{G}_{\parallel}$, which allows the transition between valleys
with large momentum mismatch. The larger degree of disorders along the
perpendicular direction can lower the lateral symmetry to a greater extent and
provides more channels that are previously forbidden by symmetry. Moreover,
the effective thickness $\delta$ of the interface roughness along the
perpendicular direction increases with increasing ml number. The interface
roughness preferably couples with carriers with
$|k_{z}|\sim\frac{2\pi}{\delta}$. The enhancement of transmission will be
promoted if the corresponding valley satisfies
$|k_{z}|\sim\frac{2\pi}{\delta}$.

In Fig. 4 (e), we plot the mode-resolved specular transmission and reflection
probabilities at $E$ = 1.77 eV as a function of their unfolded momentum
$\mathbf{k}_{\mathrm{uc},\parallel}$. We find that the overlapped valleys for
Si and sGe at the $\bar{\Gamma}$ point lead to small specular transmission
probability. This is because the atomic mixing at the interface breaks the
symmetry of Hamiltonian $\hat{H}$ and the above-mentioned symmetry-forbidden
transmission at the $\bar{\Gamma}$ point is now allowed. In Fig. 4 (f), we
show the nonspecular transmission and reflection probabilities. The majority
of nonspecular transmission processes are found to be starting from the
$\bar{\Gamma}$ and $\bar{\mathrm{M}}$ points in Si to the $\bar{\mathrm{X}}$
and $\bar{\mathrm{Y}}$ points in sGe. These processes correspond to the
transition between the lowest conduction band of Si at the $\Delta$ point and
the lowest conduction band of Ge at the $\mathrm{L}$ point in the three-
dimensional Brillouin zone. Si’s conduction band at the $\Delta$ point and
Ge’s conduction band at the $L$ point are both far from the $\Gamma$ point and
the conduction electrons have small wavelengths. The characteristic length of
disorders has to be small to contribute to the nonspecular interface
scattering. Thus, smaller transverse supercell dimensions, i.e., atomic-scale
disordered structures, are in favor of more nonspecular transmission channels.

We can define the specular and nonspecular reflection probabilities similarly
to the transmission. By examining the specular and nonspecular reflection
probabilities, we find that the newly emerged nonspecular reflection channels
are accompanied by the removal of the specular transmission channels at the
same $\mathbf{k}_{\mathrm{uc},\parallel}$. Although the increasing nonspecular
reflection probability is detrimental for interfacial transport, there are
overall more nonspecular transmission channels than the nonspecular reflection
channels, thus, the total transmission is still enhanced.

With the knowledge of the transmission function, we proceed to compute the
contact resistance. The Landauer-Büttiker formalism is used to compute the
two-probe conductance,

$G_{12}=-\frac{2e^{2}}{h}\int dE\Theta(E)\frac{\partial f}{\partial E}$ (11)

where $h$ is the Plank constant, $f=\frac{1}{e^{(E-\mu)/k_{B}T}+1}$ is the
Fermi-Dirac distribution function, and the factor 2 describes the spin degree
of freedom. The subscripts 1 and 2 refer to the left and right sides. The
four-probe conductance can be computed by [59, 51],

$G_{4}=\frac{1}{G^{-1}_{12}-\frac{1}{2}\left(G^{-1}_{11}+G^{-1}_{22}\right)}$
(12)

where $G_{11}$ and $G_{22}$ are the two-probe conductance for bulk material
$1$ and $2$, respectively. In practical calculations of the conductance for a
bulk material, we let the two leads and devices all consist of the same
materials. Then, the specific contact resistance is defined by,

$\rho_{c}=\frac{A}{G_{4}}$ (13)

where $A$ is the cross-section area.

Figure 6: (a) The disordered structure with both $\sigma_{x}$ and $\sigma_{y}$
symmetry. (b) The disordered structure without $\sigma_{x}$ yet with
$\sigma_{y}$ symmetry. (c) The disordered structure without $\sigma_{x}$ and
$\sigma_{y}$ symmetry. $\sigma_{x}$ and $\sigma_{y}$ refer to the mirror
symmetries along the x and y directions. In (a)-(c), the open circles are the
Si atoms and the filled circles refer to the swapped Ge atoms. The larger
circles represent the atoms closer to the interface. We only plot the Si side
here, and on the Ge side the swapped Si atoms have the same transverse
positions with the swapped Ge atoms on the Si side. (a)-(c) All correspond to
$4\times 4$, 2-ml structures. (d) The projected bulk band structures of Si and
sGe along the $\bar{\mathrm{\Gamma}}\bar{\mathrm{X}}$ in the surface Brillouin
zone sorted by their symmetries. (e) The nonspecular transmission function
$\Theta_{\mathrm{ns}}(E)$ for the three disordered interface structures
plotted in (a)-(c).

In Fig. 5, we observe over an order of magnitude reduction in the contact
resistance for conduction electrons through the rough Si/sGe interface at
various temperatures compared with the corresponding perfect interface. For
the rough Si/Ge interface, a much smaller reduction in electron contact
resistance is found. This is because the perfect Si/Ge interface does not have
the symmetry-forbidden transmission for low-energy conduction electrons as the
perfect Si/sGe interface does. As a result, for Si/Ge interface, the lowered
symmetry due to interface roughness does not benefit as much as the Si/sGe
interface. For valence bands, the hole contact resistances for Si/Ge and
Si/sGe interfaces are only slightly reduced by the interface disorders, as
most holes from two sides have compatible momenta and symmetries.

Last but not least, we want to examine how the symmetry of the disordered
interface changes the nonspecular transmission. The nonspecular transmission
probability can be analyzed using perturbation theory [60, 42] and we argue
that the nonspecular transmission probability is proportional to the
scattering matrix element, $t_{ns,ji}\propto\bra{u_{j}}\Delta H\ket{u_{i}}$,
where the perturbed potential is the difference between the potential energy
for disordered interface and perfect interface, $\Delta
H=H_{\mathrm{rough}}-H_{\mathrm{perfect}}$. For different disordered interface
structures, the symmetry of $\Delta H$ can be different.

In Fig. 6, we show the nonspecular transmission for three representative
disordered interface configurations. In Fig. 6 (d), we have plotted the
projected band structures of Si and sGe, sorted according to the symmetries of
Bloch wave functions under mirror operation. The conduction band for Si and
sGe are both even under the $\sigma_{x}$ operation, thus, it is preferred to
have $\Delta H$ with even symmetry as well such that
$t_{\mathrm{ns},ji}\propto\bra{u_{j}}\Delta
H\ket{u_{i}}=\bra{\sigma_{x}u_{j}}\sigma_{x}\Delta
H\ket{\sigma_{x}u_{i}}=\bra{u_{j}}\Delta H\ket{u_{i}}$ and
$t_{\mathrm{ns},ji}$ is not forbidden by symmetry. As for the case with no
mirror symmetries along $x$ or $y$ directions, the symmetry of the whole
system is lowered and the symmetry analysis for $t_{\mathrm{ns},ji}$ does not
work. Although there are some nonspecular transmission channels for the case
with no mirror symmetry, the nonspecular transmission still favors the
disordered structures with compatible symmetries with the initial and final
states than those without.

In general, our symmetry analysis for wave functions and disordered structures
can be applied to study other interfaces between semiconductors with
mismatched band valleys. The extent of the contact resistance reduction
depends on the specific materials on two sides of the interface and can only
be known from Green’s function calculations. However, the computational cost
of Green’s function calculation increases rapidly with number of atoms. On top
of that, when the material is polar, the band edge profile near the interface
(especially for complex oxide interfaces [61]) can vary significantly over a
long distance, thus the Poisson equation has to be solved using a large
supercell. Due to these challenges, we only study Si/Ge interfaces in this
work.

##  VI Conclusion

We have studied the charge transport through a [001] Si/Ge interface. The
transmission though a perfect interface must be specular. The electron
transmission through the Si/Ge interface is very low due to momentum-
mismatched band structures. The incompatible symmetries of the electron states
at different pockets also forbid the transmission, leading to a high contact
resistance. However, with atomic mixing at the interface, the symmetry is
lowered and the previously forbidden transmission is allowed. In addition, the
nonspecular transmission connecting electron pockets with different transverse
momentum is enabled by those interface disorders. As a result, the specific
contact resistance is reduced by over an order of magnitude.

##  VII Acknowledgements

We thank G.D. Mahan for helpful discussions. This work is partially supported
by the MRSEC Program of the National Science Foundation under Award No.
DMR-1419807. G.C. gratefully acknowledges MIT support.

##  Appendix A Mode-resolved Green’s function formalism

The mode-resolved Green’s function formalism to compute the transmission and
reflection probability matrix is developed by Khomyakov et al.. We present a
brief introduction to the formalism as follows for completeness. We first
construct the Hamiltonian for the structure shown in Fig. 1 (a). For a given
$\mathbf{k}_{\mathrm{sc},\parallel}$, the Hamiltonian writes,

$H(\mathbf{k}_{\mathrm{sc},\parallel})=\begin{pmatrix}\ddots&&&&\\\
&H^{L}_{11}&H^{L}_{01}&&&\\\ &H^{L}_{10}&H^{L}_{00}&H_{LD}&&\\\
&&H_{DL}&H_{D}&H_{DR}&&\\\ &&&H_{RD}&H^{R}_{00}&H^{R}_{01}&\\\
&&&&H^{R}_{10}&H^{R}_{11}&\\\ &&&&&&\ddots\end{pmatrix}$ (14)

with the matrix blocks corresponding to different cells of the supercell along
the interface normal as well as the interactions between neighboring cells. In
the semi-infinite lead region, we have $H^{L}_{nn}=H^{L}_{00}$ and
$H^{R}_{nn}=H^{R}_{00}$, where $n$ denote the $n$th repeated supercell cell in
the lead region as denoted in Fig. 1 (a). $H_{D}$ is the Hamiltonian
corresponding to the device region. $H_{LD/DL}$ and $H_{RD/DR}$ describe
interactions between the lead and the device region.

The Green’s function matrix is defined by,

$\left[(E\pm
i\eta)I-H(\mathbf{k}_{\mathrm{sc},\parallel})\right]G^{r/a}(E,\mathbf{k}_{\mathrm{sc},\parallel})=I$
(15)

where $I$ is the identity matrix, $\eta$ is an infinitesimal positive real
number, and the superscripts $r$ and $a$ denote retarded and advanced Green’s
function, depending on the sign in front of $i\eta$. The retarded Green’s
function at given energy $E$ in the block-matrix form is explicitly expressed
by,

$G^{r}(E,\mathbf{k}_{\mathrm{sc},\parallel})=\begin{pmatrix}\ddots&&&&&&\\\
&G^{L}_{11}&G^{L}_{01}&&&\udots\\\ &G^{L}_{10}&G^{L}_{00}&G_{LD}&G_{0,N+1}&\\\
&&G_{DL}&G_{D}&G_{DR}&&\\\ &&G_{N+1,0}&G_{RD}&G^{R}_{00}&G^{R}_{01}&\\\
&\udots&&&G^{R}_{10}&G^{R}_{11}&\\\ &&&&&&\ddots\end{pmatrix}$ (16)

In particular, the matrix blocks $G^{L}_{00}$ and $G^{R}_{00}$ are useful
physical quantities to compute the transmission matrix, called the surface
Green’s function for the left and right lead which satisfies,

$g^{r}_{L/R}(E,\mathbf{k}_{\mathrm{sc},\parallel})=G^{L/R}_{00}=[(E+i\eta)I-H^{L/R}_{00}-\Sigma^{r}_{L/R}]^{-1}$
(17)

where the self-energies of the left and right leads are
$\Sigma^{r}_{L}=H^{L}_{10}g^{r}_{L}H^{L}_{01}$ and
$\Sigma^{r}_{R}=H^{R}_{01}g^{r}_{R}H^{R}_{10}$. In this work, the surface
Green’s functions are iteratively solved using the decimation technique [62].

The matrix block $G_{N+1,0}$ describes the response at $N+1$ th cell ($0$ th
cell in the right lead) cell given the perturbation at $0$ th cell in the left
lead. We use Dyson’s equation to compute the matrix $G_{N+1,0}$. To obtain
$G_{N+1,0}$, we need to evaluate the matrix block $G_{D}$ for the device
region, which contains a large amount of atoms for the case of rough
interfaces. The efficient computation for device Green’s function $G_{D}$ is
thus challenging. To overcome this difficulty, the device Green’s function is
calculated using the recursive technique [63, 64] and the detail for our
implementation of the recursive Green’s function can be found in our prior
work [42].

To compute the transmission mentioned in Eq. 4, we need to compute the
eigenvector matrices and velocity matrices. We outline how to compute these
matrices in the following. For a given transverse momentum
$\mathbf{k}_{\mathrm{sc},\parallel}$ and energy $E$, there are multiple
subbands in the lead region with different perpendicular momenta $k_{z}$’s.
What is more, the lead is semi-infinite, which supports both propagating (real
$k_{z}$) and evanescent (imaginary $k_{z}$) states. We need to resolve the
perpendicular momentum $k_{z}$ and its corresponding velocity $v_{z}$ to
compute the ratio of scattered current to the incident current to obtain the
transmission and reflection probability matrix.

We first introduce an auxiliary matrix for the right lead,

$F^{r}_{R}=g^{r}_{R}H^{R}_{10}$ (18)

and compute its eigenvalue $\Lambda^{r}_{R}$ and eigenvector $U^{r}_{R}$ via,

$F^{r}_{R}U^{r}_{R}=\Lambda^{r}_{R}U^{r}_{R}$ (19)

It has been pointed out by Khomyakov et al. [41] that the eigenvalue
$\Lambda^{r}_{R,i}$ stores the phase information of the electron and the
eigenvector matrix $U^{r}_{R,i}$ contains the Bloch wave functions for state
$i$. If $|\Lambda^{r}_{R,i}|\neq 1$, it corresponds to an evanescent state. If
$|\Lambda^{r}_{R,i}|=1$, it corresponds to a propagating state. We can extract
the perpendicular momentum by
$k_{R,i}=\frac{1}{a_{R}}\mathrm{log}\Lambda^{r}_{R,i}$. Similarly, for the
left lead, we define the auxiliary matrix, and its eigenvalues and
eigenvectors write,

$F^{a}_{L}=g^{a}_{L}H^{L}_{01}$ (20)
$F^{a}_{L}U^{a}_{L}=\Lambda^{a}_{L}U^{a}_{L}$ (21)

where $g^{a}_{L}=(g^{r}_{L})^{\dagger}$ is the advanced surface Green’s
function for the left lead.

The velocity along the transport direction (perpendicular to interface)
$v_{z}$ can be described by the velocity matrix,

$V^{a}_{L}=-U^{a\dagger}_{L}\Gamma^{a}_{L}U^{a}_{L}$ (22)
$V^{r}_{R}=U^{r\dagger}_{R}\Gamma^{r}_{R}U^{r}_{R}.$ (23)

where $\Gamma=i(\Sigma-\Sigma^{\dagger})$. The diagonal elements of these
matrices correspond to the group velocities along z direction of different
states.

The reflection probability matrix from $\alpha$ side
$R_{\alpha\alpha,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})$ is similarly
defined by,

$R_{\alpha\alpha,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})=|r_{\alpha\alpha,ji}(E,\mathbf{k}_{\mathrm{sc},\parallel})|^{2}$
(24)

Specifically, the reflection matrices from the left and right sides are,

$r_{LL}(E,\mathbf{k}_{\mathrm{sc},\parallel})=i\sqrt{V^{r}_{L}}[U^{r}_{L}]^{-1}\left(G_{0,0}-Q^{-1}_{L}\right)[U^{a\dagger}_{L}]^{-1}\sqrt{V^{a}_{L}}$
(25)
$r_{RR}(E,\mathbf{k}_{\mathrm{sc},\parallel})=i\sqrt{V^{r}_{R}}[U^{r}_{R}]^{-1}\left(G_{N+1,N+1}-Q^{-1}_{R}\right)[U^{a\dagger}_{R}]^{-1}\sqrt{V^{a}_{R}}$
(26)

where
$Q^{-1}_{L}=(E+i\eta)I-H^{L}_{00}-H^{L}_{10}g^{r}_{L}H^{L}_{01}-H^{L}_{01}g^{r}_{L^{\prime}}H^{L}_{10}$
and
$Q^{-1}_{R}=(E+i\eta)I-H^{R}_{00}-H^{R}_{01}g^{r}_{R}H^{L}_{10}-H^{R}_{10}g^{r}_{R^{\prime}}H^{R}_{01}$
are the retarded Green’s functions for bulk materials.
$g^{r}_{\alpha^{\prime}},\;\alpha=L,R$, are the retarded surface Green’s
function similar to Eq. 17, except that they describe the semi-infinite lead
of the same material extending to infinity in the opposite direction given by,

$g^{r}_{L^{\prime}}(E,\mathbf{k}_{\mathrm{sc},\parallel})=\left[(E+i\eta)I-H^{L}_{00}-\Sigma^{r}_{L^{\prime}}\right]^{-1}$
(27)
$g^{r}_{R^{\prime}}(E,\mathbf{k}_{\mathrm{sc},\parallel})=\left[(E+i\eta)I-H^{R}_{00}-\Sigma^{r}_{R^{\prime}}\right]^{-1}\\\
$ (28)

where the self-energies write,

$\Sigma^{r}_{L^{\prime}}=H^{L}_{01}g^{r}_{L^{\prime}}H^{L}_{10}$ (29)
$\Sigma^{r}_{R^{\prime}}=H^{R}_{10}g^{r}_{R^{\prime}}H^{R}_{01}$ (30)

The reflection matrix also depends on another two surface Green’s function
$g^{r}_{L}$ and $g^{a}_{R}$, as defined by Eq. 17. The auxiliary matrices,
eigenvalue matrices, and eigenvector matrices for these two surface Green’s
functions are,

$F_{L}^{r}=g^{r}_{L}H^{L}_{01}$ (31)
$F_{L}^{r}U_{L}^{r}=\Lambda_{L}^{r}U_{L}^{r}$ (32)
$F_{R}^{a}=g^{a}_{R}H^{R}_{10}$ (33)
$F_{R}^{a}U_{R}^{a}=\Lambda_{R}^{a}U_{R}^{a}$ (34)

The self-energies for these two surface Green’s functions are,

$\Sigma_{R}^{a}=H^{R}_{01}g^{r}_{R}H^{R}_{10}$ (35)
$\Sigma_{L}^{r}=H^{L}_{10}g^{a}_{L}H^{L}_{01}$ (36)

The corresponding broadening matrices are computed by
$\Gamma=i(\Sigma-\Sigma^{\dagger})$. The velocity matrices $V^{a}_{R}$ and
$V^{r}_{L}$ introduced in Eq. 25 and Eq. 26 are expressed by,

$V^{a}_{R}=-U^{a\dagger}_{R}\Gamma^{a}_{R}U^{a}_{R}$ (37)
$V^{r}_{L}=U^{r\dagger}_{L}\Gamma^{r}_{L}U^{r}_{L}$ (38)

##  Appendix B Interface atomic mixing

In Fig. A1, we present the ensemble-averaged atomic number density along z
direction for different ml and transverse supercell sizes. In Fig. A2 and Fig.
A3, we demonstrate the corresponding electron transmission for different
interface configurations. At fixed ml number, smaller transverse supercell
sizes give rise to higher nonspecular transmission function. At fixed
transverse supercell size, the higher ml numbers give rise to higher
nonspecular transmission function.

Figure A1: The atomic number density (the number of atoms of a given type per
unit volume) of Si at different atom layers of the interface. Figure A2: The
sensitivity of electron transmission through Si/Ge interfaces on the degree of
transverse and longitudinal disorders. In (a)-(c), we use a 2$\times$2
transverse supercell and change the ml number. In (d)-(f), we use 8 ml
structures and vary the in-plane supercell size. We use 20$\times$20,
15$\times$15, 10$\times$10 $\mathbf{k}_{\mathrm{sc},\parallel}$-point mesh for
2$\times$2, 3$\times$3, 4$\times$4 transverse supercells, respectively. Figure
A3: The sensitivity of electron transmission through Si/sGe interfaces on the
degree of transverse and longitudinal disorders. Other computational details
are the same with those for Fig. A2.

##  Appendix C The enhancement of hole transmission

From Fig. 4, we observe that the hole transmission functions for disordered
Si/Ge and Si/sGe interfaces are both slightly enhanced compared with
corresponding perfect interfaces. To investigate the origin of such
enhancement, we compare the mode-resolved hole scattering probabilities for
rough Si/sGe interfaces and the perfect Si/sGe interface in Fig. A4.

First, we notice that the specular part of the scattering probability is
generally much larger than the nonspecular part except for the reflectance
from the Si side. This is because the valence bands of Si and sGe are both at
the $\bar{\Gamma}$ point, which means that the momentum conservation is easier
to satisfy. Also, the symmetries for hole wave functions from two sides are
compatible, which puts no symmetry restriction on hole transmission and
reflection. Second, the nonspecular scattering probabilities (both
transmittance and reflectance) are promoted at the points with relatively
large $|\mathbf{k}_{\parallel}|$ on the Si and sGe side, such that those
points can differ by integer number of transverse reciprocal vectors
$\mathbf{G}_{\mathrm{sc},\parallel}$. Furthermore, compared with the perfect
interface, the specular transmittance for rough interfaces can be either
reduced or enhanced, depending on the transverse momentum. In contrast, the
specular reflectance for most holes from Si and sGe are reduced. Consequently,
the total hole transmission is slightly enhanced. However, such enhancement
might not be universal. The reasons for this are as follows.

For the perfect interface, there are already a considerable amount of
scattering channels (all of them are specular). When interfacial disorders are
introduced, the change in specular scattering probability $\Delta
P_{\mathrm{s},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})$ and the nonspecular
scattering probability
$P_{\mathrm{ns},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})$ are both small
perturbations compared in the specular scattering probability for the perfect
interface $P_{\mathrm{perfect},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})$.
The signs of these perturbations depend on both the wave functions of the
initial and final states. Specifically, it is difficult to predict how the
specular part of the scattering probability changes using symmetry analysis.
Eventually, the scattering probability for rough interfaces
$P_{\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})=P_{\mathrm{s},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})+P_{\mathrm{ns},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})$
can be either higher or lower than the scattering probability for the perfect
interface $P_{\mathrm{perfect},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})$.

Figure A4: The ensemble-averaged mode-resolved (a) specular and (b)
nonspecular scattering probabilities as a function of in-plane momentum
$\mathbf{k}_{\mathrm{uc},\parallel}$ at $E=-0.41$ eV. (c) The difference
between the ensemble-averaged specular scattering probability for disordered
interfaces and the specular scattering probability for the perfect interface,
$\Delta
P_{\mathrm{s},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})=P_{\mathrm{s},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})-P_{\mathrm{perfect},\alpha}(\mathbf{k}_{\mathrm{uc},\parallel})$,
where $P=T,R$ and $\alpha=\mathrm{Si,sGe}$, as a function of in-plane momentum
$\mathbf{k}_{\mathrm{uc},\parallel}$ at $E=-0.41$ eV.

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