# Induced superconducting pairing in integer quantum Hall edge states

Mehdi Hatefipour1 Joseph J. Cuozzo2 Jesse Kanter1 William M. Strickland1
Christopher R. Allemang3 Tzu-Ming Lu3,4 Enrico Rossi2 Javad Shabani1
<EMAIL_ADDRESS>1Center for Quantum Phenomena, Department of Physics, New
York University, NY 10003, USA
2 Department of Physics, William & Mary, Williamsburg, Virginia 23187, USA
3 Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
4 Center for Integrated Nanotechnologies, Sandia National Laboratories,
Albuquerque, New Mexico, 87123, USA

###### Abstract

Indium Arsenide (InAs) near surface quantum wells (QWs) are promising for the
fabrication of semiconductor-superconductor heterostructures given that they
allow for a strong hybridization between the two-dimensional states in the
quantum well and the ones in the superconductor. In this work we present
results for InAs QWs in the quantum Hall regime placed in proximity of
superconducting NbTiN. We observe a negative downstream resistance with a
corresponding reduction of Hall (upstream) resistance, consistent with a very
high Andreev conversion. We analyze the experimental data using the Landauer-
Büttiker formalism, generalized to allow for Andreev reflection processes. We
attribute the high efficiency of Andreev conversion in our devices to the
large transparency of the InAs/NbTiN interface and the consequent strong
hybridization of the QH edge modes with the states in the superconductor.

Anyons with non-Abelian statistics are of great fundamental interest Moore and
Read (1991) and can be used to realize topologically protected, and therefore
intrinsically fault-tolerant qubits Kitaev (2003); Das Sarma et al. (2005);
Nayak et al. (2008). Non-Abelian anyons are expected to be realized in few
fractional quantum Hall (QH) states Read and Green (2000); Cheng (2012);
Alicea and Fendley (2016); Vaezi (2013, 2014) such as the QH states with
filling factor $\nu=\frac{5}{2}$ Willett et al. (1987); Miller et al. (2007);
Radu et al. (2008), and, possibly, $\nu=\frac{12}{5}$ Zhu et al. (2015).
However, so far, no unambiguous experimental confirmation exists of the
presence of non-Abelian anyons in such QH states.

An alternative route to realize non-Abelian anyons relies on inducing
superconducting pairing between counter-propagating edge modes of QH states
that, intrinsically, support only Abelian anyons Qi et al. (2010); Lindner et
al. (2012); Clarke et al. (2014); Mong et al. (2014). These theoretical
proposals build on an earlier proposal for creating Majorana zero modes, the
anyons with the simplest non-Abelian statistics, using 1D modes at the edge of
a 2D topological insulator (TI) in contact with a superconductor (SC) Fu and
Kane (2008). In contrast to TIs, in two-dimensional electron gases (2DEGs) in
the QH regime, by varying filling factor $\nu$, states can be realized with a
variety of topological orders. This allows access to more exotic edge states
needed for engineering anyons with richer non-Abelian statistics. Key in all
these theoretical proposals is the ability to induce superconducting pairing,
via the proximity effect, between the QH edge modes.

The strength of the superconducting correlations that can be induced in a QH-
SC heterojunction can be evaluated by obtaining the amplitude of the Andreev
reflection of QH edge modes. The early search for Andreev reflection in QH-SC
systems focused on InAs and InGaAs semiconductor magneto-resistance
oscillations at relatively low magnetic fields Nitta et al. (1994) followed
later by reports of induced superconductivity in QH states Wan et al. (2015).
More recently there have been reports of observation of induced
superconductivity Rickhaus et al. (2012); Park et al. (2017), cross Andreev
conversion Önder Gül et al. (2021); Lee et al. (2017), edge state mediated
supercurrent Amet et al. (2016), and interference of chiral Andreev edge
states Zhao et al. (2020); Hoppe et al. (2000); Kurilovich et al. (2022);
Manesco et al. (2021) in graphene. To make further progress, It is essential
to reliably demonstrate the ability to induce robust superconducting
correlations into the edge modes of a QH state.

In this work we show that in high quality InAs/NbTiN heterostructures, very
strong superconducting correlations can be induced in the edge modes of
integer QH states realized in the InAs-based quantum wells (QWs). Such
correlations appear to be robust, showing no oscillations as a function of
doping, for gate voltages within the QH plateaus. We analyze the experimental
data in conjunction with a microscopic model to extract the details of the
processes determining the transport properties of the QH-SC interface.

Figure 1: (a) Schematic of gated NbTiN/InAs hybrid device structure (only the
portion which is defined by the rectangle in Fig. 1(b)). (b) Device pin-out
configuration. Contacts 1,2,3,5, and 6 are normal; contacts 4, 4′, 4′′ are
superconducting. Contacts 1 and 4′′ are used as the source and drain,
respectively. Contacts which are not labeled had electrical connection issue
during the experiment. (c) Andreev conversion via CAES interference along the
QH-SC interface (top) and a supporting tight binding calculation of the
difference between the electron and hole LDOS (LDOSe-h) (bottom). (d) Measured
$R_{D}^{2L}$ as a function of $V_{g}$ and $B$ in a dirty interface device
(device B). (e) Measured $R_{D}^{2L}$ as a function of $V_{g}$ and $B$ in the
cleaned interface device (device A). IQHSs are labeled from complementary
$R_{xy}$ data. The dashed line shows the position of the cut shown in Fig. 2
(a).  Figure 2:  $R_{U}-R_{D}$ and $R_{xy}$ shown as a function of $V_{g}$.
All traces taken at $B=11$T (Device A). IQHSs are labeled and markers are
shown for the states used in Fig. 3(c).

Figure 1(a) shows a cross sectional schematic of the fabricated device used in
this work. The QW is formed by a 4 nm layer of In0.81Ga0.19As layer, a 7 nm
layer of InAs, and a 10 nm top layer of In0.81Ga0.19As. The QW is grown on
InxAl1-xAs buffer where the indium content is step-graded from $x=$ 0.52 to
0.81. A delta-doped Si layer with electron doping $n\sim 1\times 10^{12}$ cm-2
is placed 6 nm below the QW. This epitaxial structure has been used in
previous studies on mesoscopic superconductivity Kjaergaard et al. (2016);
Bøttcher et al. (2018); Mayer et al. (2019, 2020), in the development of
tunable qubits Casparis et al. (2019), and in studies aimed at realizing and
detecting topological superconducting states Suominen et al. (2017); Fornieri
et al. (2019); Dartiailh et al. (2021).

A Hall bar, Fig. 1(b), is fabricated by electron beam lithography. In order to
study the 2DEG/SC interface, a 90 nm thick layer of NbTiN was sputtered as the
superconducting contacts with a $150\mu m$-wide interface after performing wet
etch surface cleaning (Device A). We also fabricated a similar device with
intentional no surface cleaning step before NbTiN sputtering (Device B). A
metallic top gate is created by depositing a layer of Al oxide followed by an
Al layer to control the QW electron density Shabani et al. (2016). The
mobility of the QW is determined to be $\mu\sim 12,000$ $cm^{2}/V.s$ at $n\sim
8.51\times 10^{11}$cm-2 corresponding to an electron mean free path of
$l_{e}\sim 180$ nm. All data reported here were taken at $T\sim 30$ mK. We
have provided more information on transport properties of the sample in the
SI. We note that while we focus mainly on one device (Device A) in the main
text, we have studied a few other similar devices which their data have been
shown in SI.

When the sample is placed in a magnetic field, in the classical picture,
electrons and holes will alternate their skipping orbits across the interface
of the superconductor and 2DEG Chtchelkatchev and Burmistrov (2007). In the
full quantum-mechanical analysis the electron and hole edge states hybridize
due to the proximity of the SC and form a coherent chiral Andreev edge state
(CAES) extended along the QH-SC interface Hoppe et al. (2000); van Ostaay et
al. (2011); Khaymovich et al. (2010). A schematic of CAES propagation along
the QH-SC interface is shown in Fig. 1 (c). In this picture, if more holes
than electrons reach the normal lead downstream from the superconducting
electrode (lead 5), then a negative potential difference
$(V_{5}-V_{4}^{\prime})$ develops. In Fig. 1 (c) we also show the the local
density of states of a CAES obtained with a tight binding (TB) calculation
performed using the python package Kwant Groth et al. (2014). In the TB model
the presence of the magnetic field is taken into account via a Peierls phase,
and the superconductivity of the QW proximitized by NbTiN via a mean field an
s-wave pairing term of strength $\tilde{\Delta}$. The details of the TB model
can be found in the SI.

Figures 1(d) and (e) show the results for the downstream resistance,
$R_{D}^{2L}$, measured between the voltage contacts 5 and $4^{{}^{\prime}}$,
as a function of gate voltage $V_{g}$ and magnetic field $B$. Hall resistance
data measured between contacts 2 and 5 allow us to determine the filling
factor of the different regions of Fig. 1 (d), (e). Figure 2(a) shows the
horizontal cut at $B=11$ T of Fig. 1(e), $R_{D}^{4L}$, and the corresponding
longitudinal resistance $R_{xx}$. From the $R_{xx}$ measurements we see that
we have well developed integer QH states (IQHS). From Figs. 1(e) and 2(a) we
clearly observe that $R_{D}$ is negative for IQHS, a fact that strongly
suggests the presence of Andreev processes at the QH-SC interface for these
IQHS. We notice the importance of a clean InAs/NbTiN interface by comparing
the magnitude of negative resistance in Figs. 1 (d) and (e). The clean
interface on device A has been achieved by etching the surface of defined
NbTiN pattern area by buffered oxide etchant (BOE) for 2 seconds immediately
followed by loading into sputtering tool’s load lock in order to minimize the
time for the native oxide growth at the interface. On the other hand, for
device B and all the other devices mentioned in SI, this cleaning step has
been skipped and NbTiN sputtered on the defined region after its exposure to
air. For the rest of the paper, we focus only on device A results. The
upstream resistance $R_{U}^{2L}$ (measured between contacts 3 and 4) exhibits
plateaus in correspondence to the $R_{xy}$ plateaus in magnetic field but with
resistance values lower than $R_{xy}$. Moreover, $R_{U}^{2L}-R_{D}^{2L}$
recovers the quantized Hall value, $R_{xy}$, as shown in Fig. 2 (c). Note that
this difference does not necessarily match the $R_{xy}$ data outside the QH
regime.

These results can be understood within the Landauer-Büttiker (LB) theory,
generalized to allow for the presence of a superconducting lead Datta et al.
(1996); Beconcini et al. (2018).

We start with the six-terminal setup shown in Fig. 1(b) (see also the SI). We
assume the terminal 1, 2, 3, 5, 6 to be ideal metallic leads, and contact 4 to
be a superconducting lead. We first consider the limit in which no normal
reflection or transmission processes take place at the superconducting lead.
Let $I_{i}$, $V_{i}$, the currents and voltages, respectively, at the
terminals $i=(1,2,3,4,5,6)$. Without loss of generality, we can set $V_{4}=0$.
We can use the charge conservation equation $\sum_{i}I_{i}=0$ to express
$I_{4}$ in terms of the currents at the other leads. With these considerations
the LB equations reduce to the following system of linear equations:

$\begin{pmatrix}I_{1}\\\ I_{2}\\\ I_{3}\\\ I_{5}\\\
I_{6}\end{pmatrix}\\!\\!=\\!\\!\frac{\nu}{R_{H}}\\!\begin{pmatrix}1&0&0&0&-1\\\
-1&1&0&0&0\\\ 0&-1&1&0&0\\\ 0&0&2A-1&1&0\\\
0&0&0&-1&1\end{pmatrix}\\!\\!\begin{pmatrix}V_{1}\\\ V_{2}\\\ V_{3}\\\
V_{5}\\\ V_{6}\end{pmatrix}$ (1)

where $\nu$ is the number of edge states, $R_{H}$ is the Hall resistance, and
$A$ is the average probability, per edge mode, of Andreev reflection.
Considering that no current flows into leads 2, 3, 5, 6, so that
$I_{2}=I_{3}=I_{5}=I_{6}=0$, and $V_{1}=V_{2}=V_{3}$, $V_{5}=V_{6}$, and
setting $I\equiv I_{1}$, it is straightforward to solve Eq. (1) to obtain

$\displaystyle R_{U}^{2L}=\frac{V_{2}}{I}=\frac{R_{H}}{\nu}\frac{1}{2A}$ (2)
$\displaystyle
R_{D}^{2L}=\frac{V_{5}}{I}=\frac{R_{H}}{\nu}\left(\frac{1}{2A}-1\right)$ (3)

Figure 3: (a) $R_{D}^{2L}$ vs $1/{\nu}$ for different IQHSs and values of $B$,
and linear fits corresponding to each magnetic field. (b) $A$ obtained from
the slope of linear fits to $R_{D}$ and $R_{U}$ data vs $1/{\nu}$ with their
corresponding error bars. (c) $(R_{U}^{2L}-R_{D}^{2L})/R^{\nu}_{xy}$ for
different $\nu$s and values of $B$.

Figure 3 (a) shows the scaling of $R_{D}$ with respect to $1/\nu$ for
different values of $B$. From the slope of the fits to the experimental data
shown in Fig. 3 (a) we obtain the value of $A_{\rm exp}=55\%$, independent, to
very good approximation, on the value of $V_{g}$ within the QH plateaus.
Figure 3 (c) shows the consistency of the measured values of $R_{D}^{2L}$ and
$R_{U}^{2L}$ with the LB predictions by plotting the ratio
$(R_{U}^{2L}-R_{D}^{2L})/R_{xy}$ as function of $B$ that, according to Eqs.
(2-3), is expected to be equal to 1.

To understand qualitatively how such values of $A$ can arise, we consider an
effective 1D Bogoliubov de Gennes Hamiltonian, $H_{BdG}$, for the 1D chiral
edge modes: $H_{BdG}=\int dx~{}\psi^{\dagger}(x)\mathcal{H}(x)\psi(x)$ where
$\psi(x)=(c_{x\uparrow},~{}c^{\dagger}_{x\downarrow})$ is the spinor formed by
the annihilation (creation) operator for a fermion at position $x$ and spin up
(down) and
$\mathcal{H}(x)=v_{d}(-i\partial_{x})\tau_{0}-v_{d}k_{F}\tau_{z}+\tilde{\Delta}\tau_{x}$.
Here, and in the remainder, we set $\hbar=1$. In the equation for $H(x)$,
$v_{d}$ is the drift velocity of the edge modes, $\tau_{i}$ are the Pauli
matrices Nambu space, $k_{F}$ is the edge modes Fermi wave vector (measured
with respect to the QH-SC interface)

and $\tilde{\Delta}$ is the superconducting pairing induced via the proximity
effect by the superconducting lead. Using the expression for $H$ we can obtain
the transfer matrix $M$ relating $\psi(x)$ at the two ends of the length
$L_{sc}$ of the QH-SC interface (see SI) van Ostaay et al. (2011); Gao et al.
(2020), and then the expression for the electron-hole conversion probability

$A=\frac{\sin^{2}(\delta\phi)}{[1+(v_{d}k_{F}/\tilde{\Delta})^{2}]}.$ (4)

In Eq. (4), $\delta\phi$ is the difference of the phases accumulated by the
electron-like and hole-like edge modes along the length of the QH-SC
interface. Let $k_{F}^{(e)}$, $k_{F}^{(h)}$ be the Fermi wave vector of the
electron-like and hole-like edge modes, and $\delta
k\equiv|k_{F}^{(e)}-k_{F}^{(h)}|=[\tilde{\Delta}^{2}+(v_{d}k_{F})^{2}]^{1/2}/v_{d}$.
We then write $\delta\phi=L_{sc}\delta k$.

Considering that $L_{sc}=150\mu{\rm m}$ is quite large any small change of
$\delta k$, induced for example by changes in $V_{g}$, should result in a
significant change of $A$ and therefore of $R_{U}$ and $R_{D}$. However, in
the experiment, within the QH plateaus, $R_{U}$ and $R_{D}$ do not show any
oscillation as a function of $V_{g}$. It is natural to conclude that this
might be due to scattering processes leading to a dephasing of the electron-
like and hole-like modes along the QH-SC interface Kurilovich et al. (2022).
In this case the effective $A_{\rm eff}$ can be obtained by averaging over
$L_{sc}$ on the right hand side of Eq. (4) to obtain $A_{\rm eff}=\langle
A\rangle=(1/2){[1+(v_{d}k_{F}/\tilde{\Delta})^{2}]}$. Considering that
$(v_{d}k_{F}/\tilde{\Delta})^{2}>0$ we see that in this case we cannot recover
the value of $A$ extracted from the experimental results, $A=0.55>0.5$.

To explain the large value of $A$, accompanied by the lack of oscillations as
a function of $V_{g}$, we are led to two possibilities. The first possibility
is that $\delta_{k}$ does not change appreciably as a function of $V_{g}$.
From Eq. (4), considering that $0<\sin^{2}(\delta\phi)<1$, we can see that to
have $A=0.55$ we must have $v_{d}k_{F}/\tilde{\Delta}<0.9$. In the limit when
$\delta\phi$ is such that $\sin^{2}(\delta\phi)\approx 0.55$, we must have
$v_{d}k_{F}/\tilde{\Delta}\ll 1$. In this limit we can write $\delta
k\approx\tilde{\Delta}/v_{d}$. Considering that to good approximation,
$\tilde{\Delta}$ and $v_{d}$ are independent of $V_{g}$, we recover the
observed values of $R_{U}$ and $R_{D}$, with no oscillations, in the QH
plateaus. Notice that the condition $v_{d}k_{F}/\tilde{\Delta}\ll 1$ is
equivalent to the condition $\delta k\xi\approx 1$, where $\xi\equiv
v_{d}/\Delta$ can be interpreted as the superconducting coherence length of
the edge modes in proximity of the SC. The other possibility is that dephasing
processes are accompanied by a finite probability of single electron tunneling
into the superconductor and breaking of particle-hole (p-h) symmetry. This
would allow to have a situation in which electron-like states are more likely
than h-like states to tunnel into the superconductor and therefore contribute
less to the downstream current explaining a negative $R_{D}$ even when $A\leq
1/2$. If we denote by $T$ the probability, per edge mode, of an electron-like
state to tunnel in the SC, in Eqs. (2), (3) we would replace $2A$ with $2A+T$.
In this case from the measurements of $R_{U}$ and $R_{D}$ we recover $2A+T$.
Assuming $\langle A\rangle=1/2$ would imply $T=0.2$. The smallest value of
$\langle A\rangle$, consistent with particle conservation, is $15\%$ to which
it would correspond $T=0.8$, a value that implies a very strong breaking of
particle-hole symmetry at the QH-SC interface. It is difficult to distinguish
between these two possibilities given that we cannot measure separately the
quasiparticle and supercurrent contributions to the charge current flowing
from the QH region into the superconducting lead.
In conclusion, we have fabricated a quantum Hall-superconductor (QH-SC)
epitaxial heterostructure based on InAs and NbTiN and characterized the
transport properties of its QH edge modes propagating along a superconducting
interface. We have observed negative values for the downstream resistance
$R_{D}$ between a normal lead and the superconducting lead and a corresponding
suppression of the upstream resistance $R_{U}$ such that in the QH plateaus
the difference $R_{U}-R_{D}$ is equal to the Hall resistance $R_{H}$. The
negative values of $R_{D}$ are an unambiguous sign that at the QH-SC interface
there is a very large electron-hole conversion probability, $A$. Using a
Landauer-Büttiker analysis we were able to explain the relation between
$R_{D}$ and $R_{U}$ and express both resistances in terms of a single
effective probability for Andreev reflections at the QH-SC interface. Our
analysis led us to the conclusion that either the edge modes propagate along
the QH-SC interface with negligible dephasing resulting in an electron-hole
conversion close to 55%, or, if dephasing processes dominate, that a strong
breaking of particle-hole symmetry at the QH-SC interface must occur.

Even the lower bounds estimates for $A$ that we extract from our measurements
are remarkable, larger than any published results for QH-SC devices. This
shows that in our InAs devices very strong superconducting correlations can be
induced into the QH edge modes, an essential prerequisite to use QH-SC
heterojunctions to realize non-Abelian anyons and topologically protected
qubits and quantum gates based on such unusual quantum states.

We thank Dr. Shashank Misra for his insightful comment and feedback. The NYU
team acknowledge partial support from DARPA grant no. DP18AP900007 and ONR
grant no. N00014-21-1-2450. JJC and ER acknowledge support from ARO Grant No.
W911NF-18-1-0290. JJC also acknowledges support from the Graduate Research
Fellowship awarded by the Virginia Space Grant Consortium (VSGC). Sandia
National Laboratories is a multimission laboratory managed and operated by
National Technology and Engineering Solutions of Sandia LLC, a wholly owned
subsidiary of Honeywell International Inc. for the U.S. DOE’s National Nuclear
Security Administration under contract DE-NA0003525. This work was performed,
in part, at the Center for Integrated Nanotechnologies, a U.S. DOE, Office of
BESs, user facility. This paper describes objective technical results and
analysis. Any subjective views or opinions that might be expressed in the
paper do not necessarily represent the views of the U.S. DOE or the United
States Government.

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