Quantum Hall Bogoliubov Interferometer Vadim Khrapai Osipyan Institute of Solid State Physics Russian Academy of Sciences 142432 Chernogolovka Russian Federation and

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Quantum Hall Bogoliubov Interferometer

Vadim Khrapai

Osipyan Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russian Federation and

National Research University Higher School of Economics,

20 Myasnitskaya Street, 101000 Moscow, Russian Federation

A quantum Hall interferometer containing a grounded superconducting terminal is proposed. This

geometry allows to control the Andreev and normal scattering amplitudes of sub-gap Bogoliubov

quasiparticles with the Aharonov-Bohm phase, as well as with the constrictions deﬁning the inter-

ferometer loop. The conductance matrix of such a three-terminal NSN interference device exhibits

a much richer behavior as compared to its two-terminal Fabry-P´erot counterpart, which is illus-

trated by non-trivial behavior of non-local charge and heat responses. A single edge version of the

interferometer enables full on-demand control of the electron-hole superposition, including resonant

enhancement of arbitrary small Andreev reﬂection probability up to 1, and can be used as a building

block in future more complex interference setups.

Chiral one-dimensional transport of quasiparticles

along the boundary of a gapped two-dimensional electron

system (2DES) is a fundamental aspect of the quantum

Hall eﬀect [1,2]. A combination of phase-coherent bal-

listic propagation over large distances together with a

controllable backscattering by gate-deﬁned constrictions

result in a plethora of quasiparticle interference phenom-

ena in quantum Hall edge channels [3]. Matching these

unique capabilities with a superconducting proximity ef-

fect may greatly advance the research in semiconductor-

superconductor hybrids.

A semi-classical transport of Bogoliubov quasiparticles

by skipping orbits along 2DES-superconductor interface

has been realized in Refs. [4–6]. In this low magnetic

ﬁeld range the Andreev reﬂection process, which is con-

strainted by momentum conservation, is allowed for scat-

tering between diﬀerent edge modes or in the presence of

disorder [7–9]. More recently, the experiments extended

towards the physics of chiral Andreev edge states in the

quantum Hall regime [10–13]. Observations of a small

but ﬁnite cross-Andreev signals in non-local conductance

measurements evidence the Andreev reﬂection within a

single chiral edge mode in graphene [14,15], that may re-

sult from pecularities of a valley spectrum at the edge [16]

or, most likely, from strong disorder scattering inherent

to real superconductors [17]. Quantum interference ef-

fects play an important role in quasiparticle transport

along the 2DES-superconductor interface [9,14,18,19].

Two-particle interference eﬀects were discussed in devices

combining quantum Hall edge channels and superconduc-

tivity [20,21].

In this work, I propose a quantum Hall Bogoliubov in-

terferometer, that is a modiﬁcation of a Fabry-P´erot type

interferometer [22,23] with a superconducting terminal

inside. This geometry enables a ﬁne tuning of the local

and non-local normal and Andreev scattering amplitudes

by means of the Aharonov-Bohm (AB) phase and con-

strictions deﬁning the interferometer loop. The proposed

interferometer represents a versatile three-terminal NSN

FIG. 1. Sketch of the Bogoliubov interferometer. A light-grey

rectangle depicts a mesa of the 2DES, with two normal ter-

minals one the sides and a superconducting terminal in the

middle, marked, respectively, by N1,N2and S. The edge

channel forms a loop of the interferometer with the help of

two constrictions, its chirality is shown by arrows. Magniﬁed

views of the constrictions 1 and 2 are given on the left and

right hand sides. The transmission matrices used in the calcu-

lation are shown nearby the corresponding scattering regions.

device that enables full control over the non-local charge

and heat quasiparticle transport, including a resonant

enhancement of an arbitrarily small Andreev reﬂection

probability up to 1.

A sketch of the Bogoliubov interferometer is depicted

in Fig. 1a. The light-grey rectangle represents a mesa

of the 2DES, which is divided in three regions separated

by two gate-deﬁned constrictions (schematized by pairs

of dark-grey vertical rectangles). The inner region is a

Fabry-P´erot-like interferometer for the chiral edge chan-

nels, an essential novel part of which is the grounded

superconducting terminal S (dark-red). The outer re-

gions on either side of the interferometer contain normal

terminals N1and N2, which are assumed to be ideally

coupled to the chiral edge channels. The edge channels

propagating downstream the normal terminals contain

normal quasiparticles and are shown in black. Passing

the S-terminal, quasiparticles experience both Andreev

and normal scattering and become coherent superposi-

arXiv:2210.05525v5 [cond-mat.mes-hall] 7 Jun 2023

tions of the electron-like and hole-like excitations (Bogoli-

ubov quasiparticles), as illustrated by the blue-red color.

The whole structure is placed in a quantizing perpen-

dicular magnetic ﬁeld corresponding to the ﬁlling factor

ν= 2 (the lowest spin-degenerate Landau level ﬁlled).

The phase Φ is tuned by the AB ﬂux through the inter-

ferometer.

Following Ref. [17], below I assume spin-degeneracy

of the chiral edge states and neglect possible edge re-

construction. Hence, all scattering matrices are spin-

degenerate and the spin index is suppressed for brevity.

The calculations are performed at zero energy, that is for

quasiparticles at the chemical potential of the S termi-

nal, well inside the bulk and superconductor gaps. The

wave-function of a Bogoliubov quasiparticle is a two-

component vector (aeah)T, where ae,ahare the am-

plitudes of the electron-like and hole-like components,

respectively. The propagation around the interferome-

ter is described by 2 ×2 matrix that takes the sum of

the amplitudes of all possible trajectories. For exam-

ple, the transmission matrix corresponding to the en-

trance via constriction 1 and a single full-turn around

the interferometer, is expressed as ˆ

M0ˆ

T1, where ˆ

M0=

R′

1ˆ

Cˆ

R2ˆ

Bˆ

Sˆ

A. Using the scattering amplitudes I express

A= diag(eiϕA, e−iϕA) (and similar for ˆ

B, ˆ

C) and ˆ

Ri=

diag(ri, r∗

i); ˆ

Ti= diag(ti, t∗

i) (and similar for ˆ

R′

i,ˆ

T′

i).

Matrices ˆ

A,ˆ

Band ˆ

Cdescribe a free propagation along

the edge that results only in a phase accumulation. Ma-

trices ˆ

Riand ˆ

Tidescribe a reﬂection and transmission of

a quasiparticle incident on the i-th constriction from the

lower edge. ˆ

R′

iand ˆ

T′

idescribe the same processes for a

quasiparticle incident from the upper edge (cf. Fig. 1).

Ri,ˆ

Ti,ˆ

R′

iand ˆ

T′

iare 2 ×2 blocks of the total 4 ×4 scat-

tering matrix of the i-th constriction and are constrained

by its unitarity [24]. Therefore, |ri|=|r′

i|=√Ri,

|ti|=|t′

i|=√Ti,r∗

it′

i=−r′

it∗

i, where Tiand Riare,

respectively, the transmission and reﬂection probabilities

and Ti+Ri= 1. Finally, the only non-diagonal matrix ˆ

describes a propagation along the 2DES-superconductor

boundary. Its non-diagonal elements generate rotations

in the electron-hole basis owing to the Andreev scatter-

ing:

S=tee teh

the thh;thh =t∗

ee;teh =−t∗

he.

A detailed microscopic analysis of the matrix ˆ

Swas

recently performed in Ref. [17]. In the following I neglect

possible quasiparticle loss in the superconductor, so that

|tee|2+|teh|2= 1. The full scattering amplitudes are

contained in the blocks i→j

i→j

≡



see

ji seh

she

ji shh



,

where i, j ∈1,2 label the N terminals and sαβ

ji represents

a scattering amplitude of a quasiparticle of the type β

from terminal Nito a quasiparticle of the type αin the

terminal Nj(α, β ∈e, h), see Ref. [25]. For example:

1→2

=ˆ

T2ˆ

Bˆ

Sˆ

A(1 −ˆ

M0)−1ˆ

T1,

where (1 −ˆ

M0)−1= (1 + ˆ

M0+ˆ

2+. . .). Without the

loss of generality, the phases accumulated by a quasipar-

ticle before entering the interferometer and after leaving

it are assumed zero, so that the corresponding evolution

is given by identity matrices.

The transmission coeﬃcients, deﬁned as Tαβ

ij =|sαβ

ij |2,

are used to calculate the conductance matrix Gij ≡

∂Ii/∂Vj. Here, Iiis the electric current ﬂowing in the

device through terminal Niand Vjis the voltage bias on

terminal Nj.Gij =G0δij −Tee

ij +Teh

ij [25], where δij

is the Kronecker delta symbol and G0= 2e2/h (here 2

stands for the spin degeneracy):

G=G0T1T2

D



1−1

−1 1



+G02|teh|2

D2



1R2T1T2R1R2

T1T2T2

2R1



, D ≡1 + R1R2−2pR1R2|tee|cos Φ (1)

where Φ = ϕA+ϕB+ϕC+ arg(r′

1r2tee), is the phase ac-

cumulated by an electron during one full turn around the

interferometer. The increment of the AB phase is related

to change of a magnetic ﬁeld δB by δΦ=2πδBA/Φ0,

where Ais the area enclosed by the interferometer and

Φ0=h/e is the ﬂux quantum.

Equation (1) shows how the Andreev amplitude teh

impacts the conductance matrix of the Bogoliubov in-

terferometer. For teh = 0 the current in the supercon-

ductor vanishes and the usual result for a Fabry-P´erot

interferometer is recovered. In this case, the current con-

servation constraints ˆ

Gso that all conductances have the

same absolute value, see the ﬁrst term in Eq. (1). In the

presence of Andreev scattering the S terminal comes into

play and the Bogoliubov interferometer becomes a versa-

tile three-terminal NSN device. This can be observed via

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QuantumHallBogoliubovInterferometerVadimKhrapaiOsipyanInstituteofSolidStatePhysics,RussianAcademyofSciences,142432Chernogolovka,RussianFederationandNationalResearchUniversityHigherSchoolofEconomics,20MyasnitskayaStreet,101000Moscow,RussianFederationAquantumHallinterferometercontainingagroundedsuperc...

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Quantum Hall Bogoliubov Interferometer Vadim Khrapai Osipyan Institute of Solid State Physics Russian Academy of Sciences 142432 Chernogolovka Russian Federation and

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