# Directional electron–filtering at a superconductor–semiconductor interface

Daniel Breunig  Institut für Theoretische Physik und Astrophysik, Universität
Würzburg, Am Hubland, D-97074 Würzburg, Germany  Song-Bo Zhang  Institut für
Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074
Würzburg, Germany  Björn Trauzettel  Institut für Theoretische Physik und
Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Würzburg-Dresden Cluster of Excellence ct.qmat, Germany  T. M. Klapwijk  Kavli
Institute of NanoScience, Faculty of Applied Sciences, Delft University of
Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands  Physikalisches
Institut, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Würzburg-Dresden Cluster of Excellence ct.qmat, Germany

###### Abstract

We evaluate the microscopically relevant parameters for electrical transport
of hybrid superconductor–semiconductor interfaces. In contrast to the commonly
used geometrically constricted metallic systems, we focus on materials with
dissimilar electronic properties like low–carrier density semiconductors
combined with superconductors, without imposing geometric confinement. We find
an intrinsic mode–selectivity, a directional momentum-filter, due to the
differences in electronic band structure, which creates a separation of
electron reservoirs each at the opposite sides of the semiconductor, while at
the same time selecting modes propagating almost perpendicular to the
interface. The electronic separation coexists with a transport current
dominated by Andreev reflection and low elastic back-scattering, both
dependent on the gate-controllable electronic properties of the semiconductor.

##  I Introduction

The recent interest in the interaction of topological materials and
conventional superconductors raises questions about the electronic properties
of their interfaces. In general, in the analysis of quantum transport
properties of one-dimensional edge channels, a distinction is made between
two-point and four-point measurements [1, 2, 3]. A Josephson-junction is
intrinsically a 2-terminal device in which the same contact is used for source
or drain, as well as for the voltage terminals. Although quantum-transport
analysis works very well for constriction-type metallic Josephson-junctions
consisting of one and the same material [4, 5], an application to Josephson-
junctions consisting of dissimilar materials with wide planar interfaces
(Fig.1) is often implicitly assumed, but not obviously justified. We will
demonstrate that hybrid Josephson-junctions deserve an explicit analysis,
based on the different electronic properties [6, 7, 8, 9]. The useful feature
of semiconductors, that the electronic properties can be changed by changing
the Fermi level with a gate, also affects the electronic transport-properties
of the interface. These interface properties play a crucial role in mediating
the macroscopic phase coherence between the two superconductors across the
semiconductor, not only quantitatively, but also conceptually. This aspect
becomes particularly urgent when studying the voltage-carrying state of a
ballistic hybrid Josephson-junction, but it is also important for the zero-
voltage state because of its effect on the boundary conditions for the Andreev
bound states [10].

Fig. 1: Sketch of a two-dimensional interface-junction considered in this
work. A semiconductor (SM, blue) is brought in contact to a normal metal (N,
gray) or an s-wave superconductor (S, gray). We assume two interfaces in
$x$–direction, while the $y$–direction remains translational invariant.

Our analysis starts with the experimental observation of a so-called excess
current in Josephson-junctions (like Fig. 1) based on HgTe-heterostructures
using superconductors like Al, Nb, and MoRe [11, 12, 13]. The devices consist
of large cross-sectional areas of a few micron by 10 to 80 nanometer.
Different arrangements are possible such as planar electrodes on the active
surface of the HgTe-layer, in which it is assumed that superconductivity is
induced by the proximity-effect. More recently, mesa-structures are used with
superconducting contacts made on the sides of the thin HgTe-layer [14] (like
also used for graphene [15] and, very recently, for InSb quantum wells [16]).
In all cases a characteristic excess current is observed for applied voltages
higher than the superconducting energy-gap. This feature is since Blonder et
al. [17] understood as due to enhanced charge transport by Andreev-reflection
for a range of electron-energies, which match the energy-gap. The principle
can be understood by studying the quantum-mechanics of a single interface
between a superconductor and a normal material. In order to make these
interface-properties observable in a conductance experiment Blonder et al.
have chosen a suitable geometry, compatible with the experiments. It positions
this transmissive interface in an orifice in an otherwise opaque screen, which
separates the electron systems of the two materials. This separation allows
the assignment of different chemical potentials to the two reservoirs [18]
with the occupation in one reservoir characterized by $f_{0}(E)$ and the other
one by $f_{0}(E+eV)$ with $f_{0}$ the Fermi-Dirac distribution. For a given
applied voltage, the conductance through the orifice is calculated as the
difference between the flux of right-movers and left-movers, like it has
become customary in studying quantum point contacts. In this framework, an
orifice has resistance, even in the absence of backscattering, known as the
Sharvin-resistance. It is assumed that the diameter of the orifice is smaller
than the elastic mean free path, which is feasible for metallic point contacts
and also for tunnel-junctions with a pinhole, but it is definitely not in
agreement with the dimensions used in the HgTe-samples [11, 12, 13, 14]. In
these electronic transport experiments an excess current is found, which
indicates very weak elastic back-scattering, i.e. a high transmission-
coefficient, which scales with the area. The conceptual problem is therefore
to understand the experimental observation of a conductance of a geometrically
large planar interface with excellent Andreev-reflection properties, including
a low value of elastic back-scattering, while missing in the experimental
setup an obvious mechanism through which the highly transmissive interface
could also provide a separation of the two electronic reservoirs, each at a
different chemical potential determined by the applied voltage.

##  II Fermi surface mismatch

Fig. 2: (a) 3D and (b) top view illustration of the FSM. At any energy,
electron states lie on circles with different radii $\kappa_{l}$ and
$\kappa_{r}$ in $k$–space. Only a fraction of the states (blue sectors) can
contribute to transmission, the rest (red sectors) is reflected. $\theta_{c}$
represents the critical angle of incidence.

The key concept in this work is Fermi surface mismatch (FSM), which we
illustrate for an NN’–junction in Fig. 2. Assuming different Fermi-momenta,
for instance, due to different effective masses at the left ($L$) and right
($R$) part of the interfaces, the available electron states lie on circles in
momentum space with generally different radii. For clarity, we focus here on
the two–dimensional (2D) case and isotropic Fermi–surfaces. Because of the
different size of the Fermi-surfaces, there are modes in $L$ with no
corresponding modes in $R$, i.e. electrons with an angle of incidence larger
than a critical angle $\theta_{c}$ cannot be transmitted across the interface,
but are reflected back. Consequently, they do not contribute to the
conductance across the junction, in contrast to the modes with an angle of
incidence $|\theta|<\theta_{c}$ (while possibly being limited by an additional
scattering potential). In many real systems, a quasi-2D semiconductor is
coupled to 3D metallic reservoirs. This implies that many modes of the 3D
metals (corresponding to a mode index that parameterizes the direction
perpendicular to the quasi–2D system) are not transmitted into the
semiconductor. If two regions are weakly coupled in space, it is possible to
define separate thermal equilibria for each of them. We argue here that FSM is
a feasible mechanism to decouple (and therefore only weakly couple) two
regions in space with a larger Fermi surface by one region in space with a
smaller Fermi surface, cf. Fig. 1.

FSM is not a new concept [19, 20, 21, 22]. Our main message, however, is its
effect on the observable conductance and, more importantly, its meaning for
the proper definition of local thermal equilibria and thus the applicability
of contemporary transport theories. We demonstrate below that in analogy to
the well-known Landauer conductance in a geometrically confined geometry of
$G={2e^{2}}/h\sum|t_{n,m}|^{2}$, the system has a conductance per unit length
given by

$\displaystyle
G_{0e}=\frac{2e^{2}}{h}\kappa_{l}\int\limits_{-\theta_{c}}^{\theta_{c}}\mathrm{d}\theta_{l}\cos\theta_{l}\;T\left(\theta_{l}\right)$
(1)

with $\theta$ the angle of incidence of electrons approaching the interface,
$\theta_{c}$ a critical angle determined by the electronic mismatch between
the reservoirs and the semiconductor, and $T\left(\theta_{l}\right)$ the
transmission-coefficient for the incident wave 333The factor $2$ in Eq. (1)
takes spin degeneracy into account.. The conductance is tunable by changing
the Fermi-level of the semiconductor, which changes $\theta_{c}$ and
$T\left(\theta_{l}\right)$. The system can be viewed as providing a
confinement in momentum space rather than in real space as anticipated by
Büttiker [7] and Landauer [8], assuming the absence of any limiting
contribution to the conductance by elastic or inelastic scattering. It is
analogous to the geometric Sharvin–resistance. In Eq. (1), it is labeled
$G_{0e}$ to indicate its source in the electronic mismatch between the two
material-systems, and could be called an electronic Sharvin–resistance.

We analyze the quantum-mechanical properties of a single interface from the
metallic superconductor into the semiconductor. Based on this analysis we will
argue that, to make the interface properties observable in a conductance
experiment [11, 12, 13], one needs two interfaces. Since the excess current is
a property observable at voltages higher than the superconducting energy gap,
the crucially relevant energy-conditions are then equivalent to those of one
interface, the semiconductor and the superconductor. The experimental data at
lower voltages will require, in addition to the quantum-mechanical interface
properties, the determination of a non-equilibrium distribution function for
the occupation of the states in the semiconductor in the spirit of the SINIS-
model discussed by Octavio et al. [24, 25], but including the directionality.

##  III Model

For the S contact, we assume intrinsic s–wave pairing and use the
Bogoliubov–de Gennes (BdG) formalism [26]. Introducing the Nambu spinor
$(\hat{c},\;\hat{c}^{\dagger})^{T}$, where $\hat{c}^{\dagger}$ is the electron
creation operator, we define the BdG Hamiltonian as

$\displaystyle\mathcal{H}_{BdG}(x)=[\mathcal{H}_{0}(x)+H\delta(x)]\tau_{z}+\Delta(x)\tau_{x},$
(2)

where $\tau_{x,z}$ are Pauli matrices and $\mathcal{H}_{0}$ the normal–state
Hamiltonian

$\displaystyle\mathcal{H}_{0}(x)$
$\displaystyle=\hat{k}_{x}\frac{\hbar^{2}}{2m(x)}\hat{k}_{x}+\frac{\hbar^{2}k_{y}^{2}}{2m(x)}-\mu(x)$
(3)

with $\Delta(x)$ the position–dependent order parameter in the S region and
$H$ a localized repulsive potential to model an interface with a conventional
elastic scattering potential. The form of Eq. (3) guarantees that the
Hamiltonian is hermitian and well defined [27, 28], since the momentum
operator in $x$–direction, $\hbar\hat{k}_{x}=-i\hbar\partial_{x}$, does not
commute with a position–dependent effective mass $m(x)$. Here, $\hbar$ is the
reduced Planck constant, $\mu(x)$ the electrochemical potential, related to
the carrier density, and $k_{y}$ is the $y$–component of the wave vector,
which parameterizes the transverse modes [29]. For simplicity, we assume
quadratic dispersion relations in all parts. However, all our findings only
depend quantitatively but not qualitatively on this choice.

The left and right domain, which are N and S, respectively, are described by
$\Delta(x)=\Delta_{0}\Theta(x)$, with $\Theta\left(\cdot\right)$ the Heaviside
function. The effective masses are assumed to be constant in each part, but
not equal, $m(x)=m_{l}\Theta(-x)+m_{r}\Theta(x)$. Furthermore, we assume a
global electrochemical potential, $\mu(x)=\mu$, which is significantly larger
than the excitation energy $E$ and the order parameter $\Delta_{0}$, i.e.
$\mu\gg E,\Delta_{0}$ 111For simplicity, we assume a step–wise variation of
parameters across the junction, which should be justified if the Fermi wave
lengths of the two regions (on the two sides of the junction) differ
substantially.. Under these assumptions (for detailed derivations, see App. A
and B), the eigenstates can be written as

$\displaystyle\psi^{\pm}_{e}(x)=\begin{pmatrix}1\\\ 0\end{pmatrix}e^{\pm
ik_{l}x},$ $\displaystyle\psi^{\pm}_{h}(x)=\begin{pmatrix}0\\\
1\end{pmatrix}e^{\mp ik_{l}x},$ (4)

in $L$ and

$\displaystyle\psi^{\pm}_{eq}(x)=\begin{pmatrix}u\\\ v\end{pmatrix}e^{\pm
ik_{r}x},$ $\displaystyle\psi^{\pm}_{hq}(x)=\begin{pmatrix}v\\\
u\end{pmatrix}e^{\mp ik_{r}x},$ (5)

in $R$, where the superscript $(\pm)$ indicates the group velocity with
respect to the $x$–axis. The subscript distinguishes electrons ($e$), holes
($h$), electron–like ($eq$), and hole–like ($hq$) quasiparticles. Defining
$\theta_{l/r}=\arcsin\left(k_{y}/\kappa_{l/r}\right)$, the wave numbers and
the associated group velocities are given by
$k_{l/r}=\kappa_{l/r}\cos\theta_{l/r}$ and

$\displaystyle v_{l}=\frac{\hbar k_{l}}{m_{l}},$ $\displaystyle
v_{r}=\frac{\hbar
k_{r}}{m_{r}}\left(\left|u\right|^{2}-\left|v\right|^{2}\right),$ (6)

respectively. Here, $\kappa_{l/r}=\sqrt{2m_{l/r}\mu}/\hbar$ are the Fermi wave
numbers (cf. Fig. 2) and $u^{2}=1-v^{2}=\left(1+\Omega/E\right)/2$ with
$\Omega=\sqrt{E^{2}-\Delta_{0}^{2}}$ the superconducting coherence factors.
Using scattering theory, we calculate the probabilities for Andreev [31] ($A$)
and normal ($B$) reflection and thus the transmission coefficient for each
incident electron as

$\displaystyle T(E,\theta_{l})=1+A(E,\theta_{l})-B(E,\theta_{l}),$ (7)

which is energy and mode–dependent.

##  IV Angle-dependent transmission

We start our analysis with an NN’ junction, by setting $\Delta_{0}=0$
everywhere, while maintaining FSM. Several results and conclusions found in NS
systems can be inferred from the normal state transmission, providing a
convenient theoretical foundation for the subsequent analysis. To quantify
FSM, we define the ratio of the Fermi wave numbers as
$r\equiv\kappa_{r}/\kappa_{l}=\sin\theta_{c}$, with $0<r\leq 1$. The results
for the complementary regime, $r>1$, are essentially the same, corresponding
to the mirror image of Fig. 2(b).

Fig. 3: Fraction of the propagating modes, available for transmission, as a
function of FSM. The inset is drawn for a relatively small mismatch of $r=0.5$
for illustrative reasons.

Figure 3 illustrates the propagating modes as a function of $r$, which we
define as the fraction of incident electrons from $L$ that have an angle of
incidence smaller than the critical angle $\theta_{c}$. This fraction is given
by $\theta_{c}/\pi=\arcsin(r)/\pi$. The other modes are reflected with unit
probability due to the absence of corresponding modes in $R$. In a homogeneous
setup ($r=1$), all modes can pass through the interface, but this fraction
decreases gradually if $r$ is reduced from unity. After $r\approx 0.5$, the
number of available modes decays nearly linearly and vanishes as $r\to 0$. For
a practical superconductor-semiconductor system, $r$ reaches easily a value in
the order of $0.01$, rendering the critical angle to a small cone.

If we rescale $H\to Z\sqrt{\frac{\kappa_{l}\kappa_{r}}{m_{l}m_{r}}}\hbar^{2}$
and define $\tilde{v}=v_{r}/v_{l}$, we obtain the transmission coefficient for
$E\ll\mu$ as

$\displaystyle T(E,\theta_{l})\approx
T(\theta_{l})=\frac{4\tilde{v}\;\Theta(\theta_{c}-\left|\theta_{l}\right|)}{\frac{4Z^{2}\tilde{v}}{\cos\theta_{l}\cos\theta_{r}}+\left(1+\tilde{v}\right)^{2}}.$
(8)

The step function expresses the mode selectivity, which is essentially
determined by the FSM ratio $r$, and suppresses any transport across the
junction for $\left|\theta_{l}\right|>\theta_{c}$.

For perpendicular incidence, Eq. (8) reduces to

$\displaystyle
T_{0}=T\left(\theta_{l}=0\right)=\frac{4r}{4Z^{2}r+\left(1+r\right)^{2}},$ (9)

and we have $T_{0}<1$ for all $0<r<1$, even in the absence of an elastic
scattering potential ($Z=0$). Eq. (9) describes the 1D limit, where FSM leads
to non–perfect transmission due to the mismatch of the group velocities. Since
it reduces the conductance in 1D junctions, FSM was earlier interpreted as an
effective barrier, which further increased the repulsive potential at the
interface [6]. The implications of FSM are, however, richer in higher
dimensions, allowing electrons with
$\left|\theta_{l}\right|<\theta_{c}=\arcsin\left(\kappa_{r}/\kappa_{l}\right)$
to contribute strongly to the transmission, as illustrated in Fig. 4.

Fig. 4: Transmission as a function of $\theta_{l}$ for different values of $Z$
and $r$. The homogenous system with an elastic scattering potential (red) is
compared to those with different degrees of electronic mismatch and $Z=0$
(blue, black). We choose $\mu=10^{6}\,\Delta_{0}$.

For the homogeneous case ($r=1$) and a finite $Z$ (red curve), the zero–mode
conductance deviates from unity since the interface causes electron
reflection. It decreases further for finite angles of incidence until it
vanishes at $\theta_{l}=\pm\pi/2$.

This behavior is distinctively different in a system featuring FSM for a clean
interface (blue and black curves). First of all, we observe a sharp cut–off at
$\theta_{l}=\pm\theta_{c}$, which is due to the absence of states in $R$ at
larger angles of incidence. Moreover, the transmission coefficient increases
from the value for perpendicular incidence, $T_{0}$, when $\theta_{l}$
increases, and becomes unity at

$\displaystyle\left|\theta_{l}\right|=\theta_{m}\equiv\arcsin\left(\frac{r}{\sqrt{1+r^{2}}}\right)<\theta_{c}.$
(10)

This behavior can be attributed to the group velocities $v_{l}$ and $v_{r}$,
which are, in general, different from each other if $r\neq 1$. At
$\theta_{l}=\pm\theta_{m}$, however, they coincide, allowing for perfect
electron transmission across the interface. For larger angles of incidence,
the transmission coefficient quickly decreases and vanishes for
$\left|\theta_{l}\right|\geq\theta_{c}$. Note that, for large FSM,
$\theta_{c}$ approaches zero and nearly coincides with $\theta_{m}$, which
yields sharp peaks at $\theta_{l}=\pm\theta_{m}\approx\pm\theta_{c}$ (black
dashed line).

In a case, in which both FSM and an interface barrier are present, a
competition between the effects of the two quantities $r$ and $Z$ occurs. The
dominant parameter can be identified by comparing $Z$ to a critical barrier
strength defined by

$\displaystyle Z_{\rm
crit}\equiv\frac{\left|1-r^{2}\right|}{2\sqrt{r(1+r^{2})}}.$ (11)

For $Z<Z_{\rm crit}$, we observe the same pattern as in a system with only
FSM, while the transmission $T$ decreases monotonously from its zero–mode
value if the elastic scattering potential at the interface dominates,
$Z>Z_{\rm crit}$, see Fig. 5.

Fig. 5: Transmission as a function of $\theta_{l}$ for the NS-case (solid) and
compared to the NN’ case (dashed). We compare systems that are dominated (i)
by FSM (blue), (ii) by the barrier (red) and (iii) those in the intermediate
regime (gray). We choose $\mu=10^{6}\,\Delta_{0}$ and $r=0.6$.

Note that, even though FSM dominates for $Z<Z_{\rm crit}$, the maximum at
$\theta_{l}=\pm\theta_{m}$ is no longer at unity. A finite $Z$ promotes
reflection at the interface, even if $v_{l}=v_{r}$. The barrier–dominated
transport, $Z>Z_{\rm crit}$, is affected in a similar manner by $r$, as the
latter restricts the conductance to be finite only for
$\left|\theta_{l}\right|<\theta_{c}$, contrary to the homogeneous system where
all incident modes, $\left|\theta_{l}\right|<\pi/2$, have a certain
probability to be transferred across the junction. Notably, FSM and interface
barrier affect the transport rather differently.

As shown in Fig. 5, in which the metal is in the superconducting state, the
angle-dependent features are also present, but the transmission is enhanced
due to Andreev-reflection. If $\Delta_{0}$ is finite, the transmission
coefficient at the Fermi energy becomes

$\displaystyle
T(E=0,\theta_{l})=\frac{8\tilde{v}^{2}\;\Theta(\theta_{c}-\left|\theta_{l}\right|)}{\left(\frac{4Z^{2}\tilde{v}}{\cos\theta_{l}\cos\theta_{r}}+1+\tilde{v}^{2}\right)^{2}}.$
(12)

In comparing this expression to Eq. (8), we find two essential differences:
First, the maximum value of the conductance is twice that of the NN’ setup,
which is due to the additional hole channel in NS systems. Secondly, the
barrier strength $Z$ enters the denominator in fourth order, while it is only
second order in Eq. (8), which is a well-known characteristic of Andreev-
reflection [17, 32]. Therefore, the conductance is more sensitive to a
scattering potential at the interface. Transmission across the junction in the
normal state becomes thus more likely than Andreev reflection for
$\Delta_{0}\neq 0$ and finite $Z$. We illustrate this behavior in Fig. 5 for
$Z=2$. All this is a well-known subgap feature. For $E\gg\Delta_{0}$, however,
the $Z$ sensitivity of the conductance is comparable in NN’ and NS junctions.

##  V Conductance

The given analysis of the selectivity of the interface is applicable from a
metal into the semiconductor. A similar selectivity does not occur in the
opposite direction, from a semiconductor to a metal. Therefore it is not
possible to use the semiconductor and the metal as two separate equilibrium
reservoirs to calculate the conductance. Instead we can consider a low
carrier-density semiconductor sandwiched between two metal reservoirs, of
which one can be superconducting. In analogy to the analysis of Octavio et al.
[24, 25], we distinguish inside the semiconductor, where we assume no
energy–relaxation, right-moving and left-moving populations, $f_{\rightarrow}$
and $f_{\leftarrow}$, see App. C. The equilibrium reservoirs are the two
metallic electrodes attached to the semiconductor, which in the usual way take
care of energy relaxation [5]. In the commonly used geometric confinement,
there are, for electrons in the reservoirs, closed channels with $T=0$ and
open channels with $T\approx 1$. These open channels are characterized by the
filling factors according to the Fermi-Dirac distribution of the reservoirs
and by their directionality.

We proceed by calculating the current-voltage characteristic of the device
shown in Fig. 1. However, since we are interested in the energy-dependent
properties of the conductance we take one of the electrodes as a normal metal,
to avoid the complex dynamics of the Josephson-effect. This normal metal and
the superconductor serve as equilibrium reservoirs in the spirit of Landauer-
Büttiker-theory, whereas the semiconductor serves as the conductor in the
central region. With this point of view, the metal-semiconductor-
superconductor device will carry a current given by

$\displaystyle
I\left(V\right)=\frac{1}{e\,R_{N}}\int\limits_{-\theta_{c}}^{\theta_{c}}\mathrm{d}\theta_{l}\cos\theta_{l}\int\limits_{-\infty}^{\infty}\mathrm{d}E\left[f_{0}(E-eV)-f_{0}(E)\right]T(E,\theta_{l}),$
(13)

where $R_{N}$ is the normal–state resistance and $T(E,\theta_{l})$ is defined
by Eq. (7). For a symmetric arrangement like a superconductor-semiconductor-
superconductor device, above the low voltage-range, the excess current-voltage
characteristic will be given by Eq. (13) multiplied by 2. For illustrative
purposes, we provide explicit figures of $\partial I/\partial V$ and $I(V)$ in
App. C.

##  VI Conclusion

We have shown that, in the absence of other scattering processes, the
conductance of a metal-semiconductor-metal device is limited by Fermi surface
mismatch, leading to a directional selection of propagating modes in momentum
space. We have applied this understanding to the electronic transport of
superconductor-semiconductor-superconductor samples and argued that the
observed excess current at applied voltages above the energy-gap can be
understood based on this point of view. In addition, it has implications for
our understanding of the proximity-effect under driven conditions [33]. The
tunability of the Fermi surface mismatch by a gate-voltage attached to the
semiconductor provides a rich opportunity for further experimental and
theoretical work.

###### Acknowledgements.

We acknowledge inspiring discussions with the members of the experimental
group of Martin Stehno and Laurens Molenkamp, which led to this study. The
work was supported by the DFG (SPP1666 and SFB1170 “ToCoTronics”), the
Würzburg–Dresden Cluster of Excellence ct.qmat, EXC2147, project-id 390858490,
and the Elitenetzwerk Bayern Graduate School on “Topological Insulators”.

##  Appendix A Model

The eigensystem of the full kernel Hamiltonian, written in the basis
$(\hat{c},\;\hat{c}^{\dagger})^{T}$,

$\displaystyle{\mathcal{H}}_{\mathrm{BdG}}(x)=\mathcal{H}_{0}\tau_{z}+\Delta(x)\tau_{x}$
(14)

with

$\displaystyle\mathcal{H}_{0}=\hat{k}_{x}\frac{\hbar^{2}}{2m(x)}\hat{k}_{x}+\frac{\hbar^{2}k_{y}^{2}}{2m(x)}-\mu(x)+H\delta(x),$
(15)

can be obtained by solving the Bogoliubov–de Gennes equation,

$\displaystyle{\mathcal{H}}_{\mathrm{BdG}}(x)\psi(x)=E\psi(x).$ (16)

As introduced in the main text, the position–dependence of the effective
masses as well as the superconducting order parameter and the electrochemical
potential are given by

$\displaystyle m(x)$ $\displaystyle=m_{l}\Theta(-x)+m_{r}\Theta(x),$ (17a)
$\displaystyle\Delta(x)$ $\displaystyle=\Delta_{0}\Theta(x),$ (17b)
$\displaystyle\mu(x)$ $\displaystyle=\mu,$ (17c)

which yields an interface at $x=0$ separating the left ($L$) from the right
($R$) domain. In the bulk regions, the eigenenergies are given by

$\displaystyle E\big{|}_{x\ll 0}$
$\displaystyle=\frac{\hbar^{2}k^{2}}{2m_{l}}-\mu,$ (18a) $\displaystyle
E\big{|}_{x\gg 0}$
$\displaystyle=\sqrt{\left(\frac{\hbar^{2}k^{2}}{2m_{r}}-\mu\right)^{2}+\Delta_{0}^{2}},$
(18b)

with $k^{2}=k_{x}^{2}+k_{y}^{2}$. Furthermore, the eigenstates read

$\displaystyle\psi^{\pm}_{e}(x)$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}e^{\pm ik_{e}x},$
$\displaystyle\psi^{\pm}_{h}(x)=\begin{pmatrix}0\\\ 1\end{pmatrix}e^{\mp
ik_{h}x},$ (19a) $\displaystyle\psi^{\pm}_{eq}(x)$
$\displaystyle=\begin{pmatrix}u\\\ v\end{pmatrix}e^{\pm ik_{eq}x},$
$\displaystyle\psi^{\pm}_{hq}(x)=\begin{pmatrix}v\\\ u\end{pmatrix}e^{\mp
ik_{hq}x},$ (19b)

where the subscripts and superscripts are explained in the main text and
$u^{2}=1-v^{2}=\left(1+\Omega/E\right)/2$ with
$\Omega=\sqrt{E^{2}-\Delta_{0}^{2}}$ are the superconducting coherence
factors. The wave numbers read

$\displaystyle k_{e/h}$
$\displaystyle=\kappa_{l}\sqrt{1\pm\frac{E}{\mu}-\left(\frac{k_{y}}{\kappa_{l}}\right)^{2}},$
(20a) $\displaystyle k_{eq/hq}$
$\displaystyle=\kappa_{r}\sqrt{1\pm\frac{\Omega}{\mu}-\left(\frac{k_{y}}{\kappa_{r}}\right)^{2}}$
(20b)

with $\kappa_{l/r}=\sqrt{2m_{l/r}\mu}/\hbar$, and we obtain the group
velocities according to the relation $v_{g}=\left(\partial k/\partial
E\right)^{-1}/\hbar$ as

$\displaystyle v_{e/h}=\frac{\hbar k_{e/h}}{m_{l}},$ $\displaystyle
v_{eq/hq}=\frac{\hbar
k_{eq/hq}}{m_{r}}\left(\left|u\right|^{2}-\left|v\right|^{2}\right).$ (21)

The simplified relations in the main text are derived under the assumption
$\mu\gg E,\Delta_{0}$. Then, we can drop the terms $E/\mu$ and $\Omega/\mu$ in
Eq. (20a) and obtain $k_{e}=k_{h}\equiv k_{l}$ and $k_{eq}=k_{hq}\equiv k_{r}$
as well as $v_{e}=v_{h}\equiv v_{l}$ and $v_{eq}=v_{hq}\equiv v_{r}$.

To determine the transport properties, we introduce the scattering state for
an electron approaching the interface from the left asymptotic domain as

$\displaystyle\phi(x)$
$\displaystyle=\begin{cases}\psi^{+}_{e}(x)+a\,\psi^{-}_{h}(x)+b\,\psi^{-}_{e}(x),&x<0\\\
c\,\psi^{+}_{eq}(x)+d\,\psi^{+}_{hq}(x),&x>0\end{cases}.$ (22)

The coefficients are related to Andreev ($a$) and normal ($b$) reflection as
well as transmission without ($c$) and with ($d$) branch crossing. The
corresponding probability amplitudes read

$\displaystyle A(E,k_{y})$
$\displaystyle=\frac{v_{h}}{v_{e}}\left|a_{1}\right|^{2},$ $\displaystyle
B(E,k_{y})=\left|b_{1}\right|^{2},$ (23a) $\displaystyle C(E,k_{y})$
$\displaystyle=\frac{v_{eq}}{v_{e}}\left|c_{1}\right|^{2},$ $\displaystyle
D(E,k_{y})=\frac{v_{hq}}{v_{e}}\left|d_{1}\right|^{2},$ (23b)

and we obtain the transmission coefficient for electrical current by means of
the relation

$\displaystyle T(E,k_{y})=1+A(E,k_{y})-B(E,k_{y}).$ (24)

The scattering coefficients $a,b,c,d$ are determined by matching the waves
appropriately at the interface, $x=0$, according to the conditions

$\displaystyle\lim\limits_{\varepsilon\to
0}\left(\phi_{1}(0+\varepsilon)-\phi_{1}(0-\varepsilon)\right)$
$\displaystyle=0,$ (25a) $\displaystyle\lim\limits_{\varepsilon\to
0}\left(\frac{\phi_{1}^{\prime}(0+\varepsilon)}{m_{r}}-\frac{\phi_{1}^{\prime}(0-\varepsilon)}{m_{l}}\right)$
$\displaystyle=\frac{2H}{\hbar^{2}}\phi_{1}(0).$ (25b)

##  Appendix B Transport properties

Starting with the NN’-setup ($\Delta_{0}=0\,\Rightarrow\,A=D=0$), convincing
ourselves that the wave numbers in $L$ can either be real or purely imaginary,
and evaluating Eqs. (25), this yields

$\displaystyle T(E,k_{y})$ $\displaystyle=1-B(E,k_{y})=C(E,k_{y})=$
$\displaystyle=\frac{4m_{l}m_{r}\hbar^{4}\mathrm{Re}(k_{e})\mathrm{Re}(k_{eq})}{4H^{2}m_{l}^{2}m_{r}^{2}+\hbar^{4}\left(m_{r}k_{e}+m_{l}k_{eq}\right)^{2}}.$
(26)

Note that for $\Delta_{0}=0$, $k_{eq}$ describes an electron, not a
quasiparticle. In order to obtain a finite conductance, both wave numbers (on
the left and the right hand side of the junction) need to be real, imposing
the condition
$\left(\left|k_{y}\right|\leq\kappa_{l}\right)\land\left(\left|k_{y}\right|\leq\kappa_{r}\right)$,
and we can simplify the transmission coefficient to

$\displaystyle
T(E,k_{y})=\frac{4\hbar^{2}v_{e}v_{eq}\Theta(\kappa_{l}-\left|k_{y}\right|)\Theta(\kappa_{r}-\left|k_{y}\right|)}{4H^{2}+\hbar^{2}\left(v_{e}+v_{eq}\right)^{2}}.$
(27)

Assuming $m_{l}>m_{r}$, a large electrochemical potential, $\mu\gg E$, and
rescaling $H\to Z\sqrt{(\kappa_{l}\kappa_{r})/(m_{l}m_{r})}\hbar^{2}$, we
obtain the NN’ transmission coefficient $T(E,k_{y})\approx T(k_{y})$ as stated
in Eq. (8) of the main text,

$\displaystyle
T(k_{y})=T(\theta_{l})=\frac{4\tilde{v}\;\Theta(\theta_{c}-\left|\theta_{l}\right|)}{\frac{4Z^{2}\tilde{v}}{\cos\theta_{l}\cos\theta_{r}}+\left(1+\tilde{v}\right)^{2}}.$
(28)

The NS–setup is generally more complicated, in particular due to the
additional hole channels. However, under the assumption $\mu\gg E,\Delta_{0}$,
we obtain the analytic expression

$\displaystyle
T(E,k_{y})=\begin{cases}\frac{8\tilde{v}^{2}\,\Theta(\kappa_{l}-\left|k_{y}\right|)\Theta(\kappa_{r}-\left|k_{y}\right|)}{\left(\frac{4Z^{2}\tilde{v}}{\cos\theta_{l}\cos\theta_{r}}+1+\tilde{v}^{2}\right)^{2}-\left(\frac{E}{\Delta_{0}}\right)^{2}\left(\frac{16Z^{4}\tilde{v}^{2}}{\cos^{2}\theta_{l}\cos^{2}\theta_{r}}+\frac{8Z^{2}\tilde{v}\left(1+\tilde{v}^{2}\right)}{\cos\theta_{l}\cos\theta_{r}}+\left(1-\tilde{v}^{2}\right)^{2}\right)},&E<\Delta\\\
\frac{4\tilde{v}\,\Theta(\kappa_{l}-\left|k_{y}\right|)\Theta(\kappa_{r}-\left|k_{y}\right|)}{\left(\frac{4Z^{2}\tilde{v}}{\cos\theta_{l}\cos\theta_{r}}+1+\tilde{v}^{2}\right)\left(u^{2}-v^{2}\right)+2\tilde{v}},&E>\Delta\end{cases}.$
(29)

This reduces to Eq. (12) in the main text for $E=0$ and to the normal–state
transmission, Eq. (8), for $E\gg\Delta_{0}$.

The competition between FSM and interface barrier is moderated by the quantity
$Z_{\mathrm{crit}}$, as we introduce it in the main text. For a strong
repulsive barrier, $Z>Z_{\mathrm{crit}}$, the conductance decreases
monotonously from its zero–mode value, while it increases from $T_{0}$ and
reaches a maximum at $\left|\theta_{l}\right|=\theta_{m}$ if
$Z<Z_{\mathrm{crit}}$. A rigorous, but tedious method to obtain an explicit
expression for this critical barrier strength is to demand $\theta_{m}$ to be
real. While this is always true for $Z=0$, a finite interface potential
$Z>Z_{\mathrm{crit}}$ shifts the position of the maxima into the complex
domain. Conveniently, a second order Taylor expansion of Eq. (27), for $\mu\gg
E,\Delta_{0}$ and $m_{r}=r^{2}m_{l}$, around $k_{y}=0$,

$\displaystyle T(k_{y}\approx
0)=T_{0}+T_{2}\frac{k_{y}^{2}}{\kappa_{l}^{2}}+\mathcal{O}(k_{y}^{4}),$ (30)

with

$\displaystyle T_{0}$ $\displaystyle=\frac{4r}{4rZ^{2}+\left(1+r\right)^{2}},$
(31a) $\displaystyle T_{2}$
$\displaystyle=\frac{2\left(1-r^{2}\right)^{2}-8rZ^{2}\left(1+r^{2}\right)}{r\left(4rZ^{2}+\left(1+r\right)^{2}\right)},$
(31b)

yields exactly the same result as the more complicated method mentioned above.
By demanding $T_{2}$ to be positive, such that $T$ increases from its
zero–mode value $T_{0}$, we find

$\displaystyle
Z<Z_{\mathrm{crit}}=\frac{\left|1-r^{2}\right|}{2\sqrt{r\left(1+r^{2}\right)}}.$
(32)

This allows us to identify whether FSM or $Z$ dictates the mode–dependence of
the conductance.

##  Appendix C Derivation of the current

From the transmission, we directly obtain the currents in NN’ and NS systems,
and thus the excess current. To motivate Eq. (13) in the main text, we start
from the single–mode expression for the current density

$\displaystyle
J(k_{y})=2e\int\limits_{-\infty}^{\infty}N_{k_{y}}(E)v_{k_{y}}(E)\left[f_{k_{y},\rightarrow}(E)-f_{k_{y},\leftarrow}(E)\right]\mathrm{d}E$
(33)

with $N_{k_{y}}(E)$ and $v_{k_{y}}(E)$ the density of states and group
velocity for each 1D mode $k_{y}$ at energy $E$, respectively. We can further
simplify this equation by means of the relation
$N_{k_{y}}(E)=1/\left[2\pi\hbar v_{k_{y}}(E)\right]$ and obtain

$\displaystyle
J(k_{y})=\frac{2e}{h}\int\limits_{-\infty}^{\infty}\left[f_{k_{y},\rightarrow}(E)-f_{k_{y},\leftarrow}(E)\right]\mathrm{d}E.$
(34)

Assuming equilibrium reservoirs connected to $L$ and $R$, the distribution
functions are given by

$\displaystyle f_{k_{y},\rightarrow}(E)=$ $\displaystyle f_{0}(E-eV),$ (35a)
$\displaystyle f_{k_{y},\leftarrow}(E)=$ $\displaystyle
A(E,k_{y})\left[1-f_{k_{y},\rightarrow}(-E)\right]$
$\displaystyle+B(E,k_{y})f_{k_{y},\rightarrow}(E)$
$\displaystyle+\left[C(E,k_{y})+D(E,k_{y})\right]f_{0}(E)$ (35b)

and we obtain

$\displaystyle
J(k_{y})=\frac{2e}{h}\int\limits_{-\infty}^{\infty}\left[f_{0}(E-eV)-f_{0}(E)\right]T(E,k_{y})\mathrm{d}E.$
(36)

Considering FSM and the contributions from all modes, we obtain the current
for $\mu\gg E,\Delta_{0}$ according to Eq. (13) in the main text.

To conclude, we plot the differential conductance as well as the current
according to Eq. (13) for a particular FSM $r=0.2$ and a weak barrier
$Z=Z_{\mathrm{crit}}/4$ in Fig. 6.

Fig. 6: (a) $\partial I/\partial V$ as a function of the energy $E$ and (b)
current as a function of the bias $eV$ for $r=0.2$ and $Z=Z_{crit}/4$, summed
over all modes. The offset between $I(V)\big{|}_{\Delta_{0}>0}$ and
$I(V)\big{|}_{\Delta_{0}=0}$ is denoted the excess current $I_{exc}(V)$ and
reaches its asymptotic value for $eV\gg\Delta_{0}$.

While the normal–state transmission is constant, $\frac{\partial I}{\partial
V}\big{|}_{\Delta_{0}>0}$ features a resonance at $E=\Delta_{0}$ before it
converges to $\frac{\partial I}{\partial V}\big{|}_{\Delta_{0}=0}$ for
$E>\Delta_{0}$. This results in an offset between the currents
$I(V)\big{|}_{\Delta_{0}>0}$ and $I(V)\big{|}_{\Delta_{0}=0}$ at large biases,
while their slope is equal. This offset defines the excess current, which is,
more generally, given by

$\displaystyle
I_{\mathrm{exc}}(V)=I(V)\big{|}_{\Delta_{0}>0}-I(V)\big{|}_{\Delta_{0}=0}.$
(37)

As we can see, this excess current is finite even for a notable FSM, as long
as the interface barrier remains small, i.e., $Z<Z_{\mathrm{crit}}$.

These results have been obtained under the assumption of equilibrium
reservoirs on each side of the semiconductor–superconductor interface. In
practice, in the application to the experimental results such as for example
to the S-HgTe-S system,under bias conditions thermalisation in HgTe is
negligible. Therefore, a more detailed description, in the spirit of the
Octavio–Tinkham–Blonder–Klapwijk theory [24], is needed (in preparation).

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