Andreev processes in mesoscopic multi-terminal graphene Josephson junctions Fan Zhang1Asmaul Smitha Rashid2Mostafa Tanhayi Ahari3Wei Zhang2 Krishnan Mekkanamkulam Ananthanarayanan2Run Xiao1George J. de

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Andreev processes in mesoscopic multi-terminal graphene Josephson junctions

Fan Zhang,1Asmaul Smitha Rashid,2Mostafa Tanhayi Ahari,3Wei Zhang,2

Krishnan Mekkanamkulam Ananthanarayanan,2Run Xiao,1George J. de

Coster,4Matthew J. Gilbert,5, 3 Nitin Samarth,1, ∗and Morteza Kayyalha2, †

1Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA

2Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA

3Materials Research Laboratory, The Grainger College of Engineering,

University of Illinois, Urbana-Champaign, IL 61801, USA

4DEVCOM Army Research Laboratory, 2800 Powder Mill Rd, Adelphi, MD, 20783, USA

5Department of Electrical Engineering, University of Illinois, Urbana-Champaign, IL 61801, USA

(Dated: October 25, 2022)

There is growing interest in using multi-terminal Josephson junctions (MTJJs) as a platform

to artiﬁcially emulate topological phases and to investigate complex superconducting mechanisms

such as quartet and multiplet Cooper pairings. Current experimental signatures in MTJJs have

led to conﬂicting interpretations of the salient features. In this work, we report a collaborative

experimental and theoretical investigation of graphene-based four-terminal Josephson junctions. We

observe resonant features in the diﬀerential resistance maps that resemble those ascribed to multiplet

Cooper pairings. To understand these features, we model our junctions using a circuit network of

coupled two-terminal resistively and capacitively shunted junctions (RCSJs). Under appropriate

bias current, the model predicts that supercurrent ﬂow between two terminals in a four-terminal

geometry may be represented as a sinusoidal function of a weighted sum of the superconducting

phases. We ﬁnd that the resonant features generated by the RCSJ model are insensitive to the

diﬀusive or ballistic form of the current-phase relation and junction transparency. Our study suggests

that diﬀerential resistance measurements alone are insuﬃcient to conclusively distinguish resonant

Andreev reﬂection processes from semi-classical circuit-network eﬀects.

I. INTRODUCTION

The Josephson eﬀect is a centerpiece of many quantum

device applications, including superconducting quan-

tum interference devices (SQUIDs) and superconducting

qubits [1,2]. Increasing the number of superconducting

terminals beyond the typical two terminals in Joseph-

son junctions (JJs) leads to non-local coupling of super-

conducting order parameters through a common scatter-

ing region. This non-local coupling has been predicted

to lead to quartet Cooper pairings [3] and macroscopic

multi-channel eﬀects such as phase drag and magnetic

ﬂux transfer [4–6]; it may also be used to form a supercon-

ducting phase qubit [7]. More recently, multi-terminal

Josephson junctions (MTJJs) have been proposed as a

platform to emulate topological phases in artiﬁcial di-

mensions [8–21]. In MTJJs with nsuperconducting ter-

minals, the energy of the Andreev bound states is a func-

tion of (n−1) independent phases. In this context, the

phase diﬀerences between superconducting terminals are

treated as quasi-momenta of a crystal forming a Brillouin

zone in (n−1) dimensions. The resultant band structure

may display topological properties such as Weyl singu-

larities [10,11]. While providing strong motivation for

∗Corresponding author: nsamarth@psu.edu

†Corresponding author: mzk463@psu.edu

studying the multi-terminal Josephson eﬀect, the explo-

ration of MTJJs as a platform for engineering artiﬁcial

topological systems is still nascent.

MTJJs have been experimentally explored in various

materials platforms including graphene/MoRe [22–24]

and InAs/Al [25–29]. These experiments focused on the

gate and magnetic ﬁeld dependence of the supercurrent

ﬂow between adjacent and nonadjacent superconducting

terminals [27]. Additionally, these experiments studied

the non-trivial geometric response of the critical current

contour (CCC), a generic characteristic deﬁning the re-

gion in which all the superconducting terminals are at

zero voltage [26]. Recent studies have also reported signa-

tures of quartet pairings [23,25,30], arising from crossed

Andreev reﬂection processes. However, studies of three-

terminal Josephson junctions [24] and a network of tunnel

junctions [31] argue that the resonant features may also

arise from circuit-network eﬀects.

In this paper, we use a coordinated experimental and

theoretical approach to critically understand and exam-

ine the transport manifestations of Andreev processes

in four-terminal JJs fabricated in hBN-encapsulated

graphene heterostructures. We use device geometries

wherein the superconducting terminals are connected via

a common scattering region such that no individual two-

terminal JJ is formed between the adjacent superconduc-

tors (Fig. 1). Hence, our device geometry allows us to

better explore phenomena that are due to non-local cou-

arXiv:2210.04408v3 [cond-mat.supr-con] 24 Oct 2022

pling of superconducting order parameters. We model

our junctions using a circuit network of coupled resis-

tively and capacitively shunted junctions (RCSJs). We

show that the semi-classical RCSJ model reproduces the

observed resonant features, which are similar in nature to

those predicted for multiplet Cooper pairings [23,25,30].

To elucidate the underlying mechanisms giving rise to

these features, we calculate the contribution of the quasi-

particle current which reveals the pair current contribu-

tion to the total current. We further consider materials-

speciﬁc properties such as the Fermi surface geometry of

the normal material and junction transparency by incor-

porating the relevant current-phase relation (CPR) into

the RCSJ model. We show that while the Fermi sur-

face geometry is primarily responsible for the shape of

the CCC [26], the resonant features outside of the CCC

are robust to changes in the Fermi surface and junc-

tion transparency. Finally, our study demonstrates that

circuit-network eﬀects, predicted by the RCSJ model,

lead to macroscopic signatures in diﬀerential resistance

maps that are identical to those ascribed to distinctly

quantum processes such as multiplet pairings.

The paper is organized as follows. In Sec. II, we dis-

cuss the details of device fabrication. In Sec. III, we de-

scribe the transport data for the asymmetric device. We

also theoretically analyze our junctions using an RCSJ

model. In Sec. IV, we discuss the transport results in

the symmetric device. We establish that multiple An-

dreev reﬂections (MARs) are responsible for the discrep-

ancies between the experimental and theoretical data. In

Secs. Vand VI, we consider the eﬀect of diﬀerent CPRs

ranging from ballistic to diﬀusive transport in the RCSJ

model and analyze the resultant diﬀerential resistance

maps. We conclude in Sec. VII.

II. DEVICE FABRICATION

We assemble hBN/graphene/hBN van der Waals het-

erostructures using a standard dry transfer technique,

followed by annealing in H2/Ar gas at 350 ◦C to re-

move polymer residues from the heterostructures [32,33].

The heterostructures are then patterned with electron

beam lithography, followed by dry etching (O2/CHF3),

to deﬁne the junction area. Another e-beam lithography

step is performed to deﬁne the contact patterns. Finally,

Ti(10 nm)/Al(100 nm) is evaporated to create supercon-

ducting edge contacts. Atomic force microscope (AFM)

images of our four-terminal JJs are shown in Fig. 1(a).

Devices A and B are four-terminal asymmetric and sym-

metric junctions, respectively. The channel length of de-

vice A is 0.8µm and 3 µm along I13 and I24 directions,

respectively. Device B has a circular geometry with a

diameter of 1.3µm. The Dirac point of device A and

B is at Vg=−4.5 V and Vg=−11.25 V, respectively.

In Fig. 1(a), we show the current directions used for the

transport measurements. We set I1=−I3and I2=−I4

in our experiments. We perform all of the measurements

FIG. 1. (a) The AFM image of device A. Inset shows the

AFM image of device B. The Josephson junctions are made

of hBN-graphene-hBN (outlined by white dashed line) edge

contacted with superconducting Al terminals. Arrows show

the directions of bias currents. (b) Color map of the diﬀer-

ential resistance (dV13 /dI1) versus applied dc current biases

I1=−I3and I2=−I4, as indicated in panel (a) of device

A. The diﬀerential resistance is measured using a lock-in am-

pliﬁer. Dashed red lines indicate superconducting branches

corresponding to Vjk = 0, where jand kare pairs of super-

conducting terminals as labeled in (b). Vertical lines in the

map along the I2direction are multiple Andreev reﬂections

corresponding to eV13 = 2∆/n with ∆ ≈169 µeV. All the

measurements are performed at Vg= 50 V and T= 12 mK.

at T= 12 mK, if not otherwise speciﬁed.

III. TRANSPORT IN THE ASYMMETRIC

DEVICE GEOMETRY

We now turn our attention to the transport char-

acteristics of the multi-terminal junctions in device A.

Fig. 1(b) shows a diﬀerential resistance dV13/dI1map

versus I1and I2measured at Vg= 50 V. The directions

indicated by the red dashed lines are branches along the

local minima of dV13/dI1and satisfy Vjk = 0 conditions

as labeled. These conditions correspond to supercurrent

ﬂow between terminals jand k. The vertical lines in

the map along the I2direction are signatures of MARs.

Along these lines, eV13 ≈2∆/n, where eis the electron

charge, ∆ is the superconducting gap, and nis an in-

teger. We calculate an induced superconducting gap of

∆≈169 µeV by ﬁtting eV13 to 2/n. The calculated gap is

consistent with the gap we obtain from the temperature-

dependent measurement. The coherence length based on

this gap size is ξ= ¯hvF/π∆≈1.2 µm, where ¯his the

reduced Planck constant and vF≈1×106m/s is the

Fermi velocity of graphene. The MAR signatures are

only observed along terminals 1 and 3, with distance ∼

0.8 µm< ξ, whereas no MAR is observed along terminals

2 and 4, with distance ∼3µm> ξ. This observation is

consistent with phase-coherent Andreev processes where

the superconducting phase keeps (loses) its coherence be-

fore being reﬂected multiple times between terminals 1

and 3 (2 and 4).

To more closely examine the ﬁne structure around the

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Andreevprocessesinmesoscopicmulti-terminalgrapheneJosephsonjunctionsFanZhang,1AsmaulSmithaRashid,2MostafaTanhayiAhari,3WeiZhang,2KrishnanMekkanamkulamAnanthanarayanan,2RunXiao,1GeorgeJ.deCoster,4MatthewJ.Gilbert,5,3NitinSamarth,1,andMortezaKayyalha2,y1DepartmentofPhysics,ThePennsylvaniaStateUnivers...

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Andreev processes in mesoscopic multi-terminal graphene Josephson junctions Fan Zhang1Asmaul Smitha Rashid2Mostafa Tanhayi Ahari3Wei Zhang2 Krishnan Mekkanamkulam Ananthanarayanan2Run Xiao1George J. de.pdf

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Andreev processes in mesoscopic multi-terminal graphene Josephson junctions Fan Zhang1Asmaul Smitha Rashid2Mostafa Tanhayi Ahari3Wei Zhang2 Krishnan Mekkanamkulam Ananthanarayanan2Run Xiao1George J. de

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