Fraunhofer pattern in the presence of Majorana zero modes

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F. Dominguez,1E. G. Novik,2and P. Recher1, 3

1Institut für Mathematische Physik, Technische Universität Braunschweig, 38106 Braunschweig, Germany

2Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany

3Laboratory for Emerging Nanometrology, 38106 Braunschweig, Germany

(Dated: January 17, 2024)

We propose a new platform to detect signatures of the presence of Majorana bound states (MBSs)

in the Fraunhofer pattern of Josephson junctions featuring quantum spin Hall edge states on the

normal part and Majorana bound states at the NS interfaces. We use a tight-binding model to

demonstrate a drastic change in the periodicity of the Fraunhofer pattern when comparing trivial

and non-trivial regimes. We explain these results in terms of the presence of additional parallel-spin

electron-hole reﬂections, which due to the spin-momentum locking, occur as cross Andreev reﬂec-

tions, accumulating a diﬀerent magnetic ﬂux and yielding a change in the Fraunhofer periodicity.

We show that this detection scheme exhibits some advantages compared to previous ones as it is

robust against disorder, ﬁnite temperature and works in equilibrium. Furthermore, we introduce

a scattering model that captures the main results of the microscopic calculations with MBSs and

extend our discussion to the main diﬀerences found using accidental zero energy ABSs.

I. INTRODUCTION

In condensed matter, Majorana fermion

quasiparticles1, i.e. γ†=γexhibit unconventional

properties for the charge, which is neutral, and

the occupation number, which is not well-deﬁned

γ†γ=γγ†= 1.2–4These exotic quasiparticles emerge as

zero energy excitations in topological superconductors,

among which, p-wave superconductors are the most

studied platforms due to the possibility of engineering

them by proximitizing semiconductor systems with

strong spin-orbit interactions.5–7Furthermore, Ma-

jorana zero modes exhibit a fractional nature, and

therefore, they always appear in pairs as the result

of delocalizing the information of a single fermion

c= (γ1+iγ2)/2onto the boundaries of the system.

For example, in one-dimensional (1D) systems a pair

of zero-dimensional bound states becomes localized at

the boundaries of the topological superconductor4,6,7,

whereas in two-dimensions, they emerge as an even

number of chiral vortices.8Apart from their intrinsic

interest, there are practical applications due to their in-

dividual charge neutrality, which protects them from the

local coupling to environmental charge ﬂuctuations, and

more importantly, due to the possibility of performing

computational operations by the adiabatic exchange of

Majorana bound states, also known as braiding, which

leads to an adiabatic state change within a degenerate

ground state manifold3. For further information and

references there is a collection of reviews that cover

diﬀerent aspects of these exotic particles, see for example

Refs. 9–15

Signatures of MBSs are found in several transport ex-

periments. Two of the most studied experiments, the

zero bias conductance in a NS junction8,16,17 and the frac-

tional Josephson eﬀect4in the Josephson junction, report

deviations from the theoretical predictions. In the case

of the zero bias conductance, the signature consists of a

quantized value G= 2e2/h at zero temperature. How-

ever, most of the experimental realizations have shown

a substantially suppressed value18–21, and only in one

of them the conductance is consistently close to 2e2/h,

exhibiting deviations below and above22. There are dif-

ferent explanations that can justify such deviations, some

of them are compatible with a topological ground state,

like eﬀects of a ﬁnite temperature, or a ﬁnite coupling to

the opposite MBS, while other explanations are compat-

ible with a trivial ground state, like the scattering with

quasi-Majorana bound states23–25 or coupling to trivial

zero-energy Andreev bound states (ZEABSs)26,27. The

situation is similar for the fractional Josephson eﬀect,

which can be probed by means of the Shapiro experiment,

where odd Shapiro steps vanish4,5,28–32 or the Josephson

radiation33,34. Experimentally however, in most cases

only few odd steps are suppressed when the driving fre-

quency is low enough35–38, and only one contribution re-

ports the lack of the ﬁrst four odd steps39. In this occa-

sion, the signal can also be explained in terms of a topo-

logical state that coexists with trivial ones31,40,41, how-

ever, it is also possible that the behavior can be explained

in terms of non-adiabatic transitions between Andreev

bound states30–32,42–46. Therefore, the need for detection

schemes that are more susceptible to the triplet supercon-

ducting nature of the MBSs, such as the measurement

of triplet correlations by coupling the topological super-

conductor to a spin-dependent current in a 3-terminal

setup47,48 or the spin susceptibility of a Josephson junc-

tion are of utmost importance49.

In this work, we investigate signatures of the presence

of MBSs at the NS interfaces that arise in the Fraun-

hofer pattern of a planar Josephson junction featuring

quantum spin Hall edge states on the normal part, see

Fig. 1. Here, the spin-momentum locking of the helical

edge states forces local and crossed Andreev reﬂections

(LAR and CAR) to take diﬀerent spin-symmetries, that

is, rhe

¯ss for the LAR and rhe

ss for the CAR. Note that in

trivial junctions, the CAR is zero for a homogeneous or

eﬀectively linear in momentum spin-orbit coupling. In

arXiv:2210.02065v3 [cond-mat.mes-hall] 16 Jan 2024

FIG. 1. [Panel (a)] Schematic representation of the planar

quantum spin Hall Josephson junction. Here, pink and white

areas represent superconducting and normal parts subjected

to a magnetic ﬂux Φ. [Panel (b)] Possible Andreev bound

states present in the setup. Due to spin-momentum locking,

the crossed (local) Andreev bound states hold parallel (an-

tiparallel) spin. Solid and dashed arrows represent electron-

and hole-like helical edge states, respectively.

contrast, in the presence of a MBS, LAR and CAR are

of the same order. Interestingly, the presence of a mag-

netic ﬂux can unravel both contributions since electrons

accumulate a diﬀerent magnetic ﬂux when performing a

LAR or a CAR process50,51. In this way, we investigate

the critical current of a Josephson junction as a func-

tion of an applied magnetic ﬂux, also known as Fraun-

hofer pattern, expecting to observe an abrupt change on

the proﬁle when passing from the trivial to the topolog-

ical regime. Since this detection scheme is performed in

equilibrium, it removes the complications of parity con-

servation required in Fu and Kane’s proposal5and the

possibility of non-adiabatic transitions that can appear

in the Shapiro experiment or the Josephson radiation.

Furthermore, using a scattering model we demonstrate

that the change in Fraunhofer pattern proﬁle is absent

in the presence of accidental zero energy Andreev bound

states if the direction of the Zeeman ﬁeld is parallel to

the helicity operator of the quantum spin Hall states.

In the same spirit but with a diﬀerent mechanism,

other proposals suggest to use quantum Hall edge states

to observe a change in the Fraunhofer pattern proﬁle

when MBSs are present52. Alternatively, one can study

the change of periodicity in the conductance of a NSN

junction interferometer fabricated on a 2D-topological

insulator53. A recent experiment shows an abrupt π

shift in the Fraunhofer pattern of a Josephson junction

in a SQUID loop conﬁguration, formed by two parallel

Al/InAs Josephson junctions54. There, the phase shift

is attributed to a topological phase transition produced

by an external in-plane Zeeman ﬁeld, but the mechanism

for this transition is not further discussed.

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

troduce the proximitized BHZ model and the topological

phase studied in this work. Then, in Sec. III we explore

the Fraunhofer pattern of the thin superconducting junc-

tion where to expect a change in the Fraunhofer pattern

periodicity. Finally, in Sec. IV we introduce an eﬀective

scattering model that captures the essential ingredients of

the microscopic model and compute the resulting Fraun-

hofer pattern in the presence of MBSs and accidental zero

energy crossings. Further information is given in several

appendices.

II. MICROSCOPIC MODEL AND TRANSPORT

FORMALISM

In this section, we introduce the proximitized

Bernevig-Hughes-Zhang (BHZ) model55 and the trans-

port formalism used to calculate the critical current of

the Josephson junction.

A. Tight-binding Hamiltonian

The proximitized Bogoliubov de Gennes (BdG) BHZ

Hamiltonian is given by H= (1/2) RdxdyΨ†HΨwith

H=He−EFˆ

∆(y)

∆(y)∗EF− Hh,(1)

with the Fermi energy EFand Hh=T HeT−1is the

Hamiltonian for holes, which is obtained from the one

for electrons (He) by performing a time-reversal trans-

formation T=isyσ0C, with Cbeing the complex con-

jugate operator. Here, ˆ

∆(y) = ∆0(y) exp[iφ(y)]14×4

with the pairing potential ∆0(y), which is ﬁnite and con-

stant for |y|> Ln/2and 0 otherwise, with φ(y) = ϕ/2

[φ(y) = −ϕ/2] for y > Ln/2(y < −Ln/2), see Fig. 1.

This Hamiltonian is written in the electron-hole basis

Ψ(x, y) = (ψe, ψh)t, with

ψe(x, y)=(cE,↑, cH,↑, cE,↓, cH,↓)t

(x,y),(2)

ψh(x, y) = (c†

E,↓, c†

H,↓,−c†

E,↑,−c†

H,↑)t

(x,y),(3)

where c(†)

a,σ is the annihilation (creation) operator of an

electron with orbital a=E, H and spin ↑/↓at position

(x, y). The Pauli matrices σiand si, span the orbital and

spin degrees of freedom.

He=H0+HR+HD+HZis the BHZ Hamiltonian,

where

H0=A(ˆ

kxσxsz−ˆ

kyσy) + ξ(ˆ

k) + M(ˆ

k)σz,(4)

HR=ασ0+σz

2(ˆ

kysx−ˆ

kxsy),(5)

HD=δ0σysy+δe

σ0+σz

2(ˆ

kxsx−ˆ

kysy)(6)

+δh

σ0−σz

2(ˆ

kxsx+ˆ

kysy),

HZ=Be

⊥sz(σ0+σz)/2 + Bh

⊥sz(σ0−σz)/2(7)

+B∥sθσ0,

with ξ(ˆ

k) = C−Dˆ

k2,M(ˆ

k) = M−Bˆ

k2and

A, B, C, D, M, are band structure parameters. Here,

FIG. 2. Panel (a), energy dispersion of the (normal) over-

lapped quantum spin Hall edge states in the thin limit, with

(CN= 0). Inset: zoom within the gap opened by the over-

lap between the quantum spin Hall edge states. Panel (b),

local density of states as a function of the energy ωand Zee-

man energy B⊥. Panel (c), BdG band spectrum inversion as

a function of the Zeeman energy B⊥, similar to the energy

dispersion of the Rashba wire in Refs. 6and 7. The width of

the sample is W= 144 nm, the critical ﬁeld Bc≈1.3meV and

CS=−9.2meV.

2|M|gives approximately the energy band gap of the sys-

tem and its sign determines the topological character of

the Hamiltonian: M > 0(M < 0), sets the Hamiltonian

in the trivial (topological) insulating regime. Its topolog-

ical character expresses through the emergence of helical

quantum spin Hall edge states. In the presence of time-

reversal symmetry, these quantum spin Hall edge states

propagate without backscattering even in the presence of

disorder55.

HRis the Rashba spin-orbit coupling with αbeing the

Rashba spin-orbit constant, which can be tuned by an

external electric ﬁeld56.HDaccounts for the bulk inver-

sion asymmetry (BIA) contribution with δ0, δe, δh,are

bulk inversion asymmetry parameters, which are speciﬁc

to the material under consideration. The eﬀects of an

external magnetic ﬁeld Henter via the Peierls substitu-

tion, discussed below, and also via the Zeeman Hamilto-

nian HZ, with in-plane H∥and out of plane H⊥com-

ponents. The corresponding Zeeman energies are given

by B∥=g∥µB|H∥|/2and Be/h

⊥=ge/h

⊥µB|H⊥|/2, with

the parallel g∥≈2and perpendicular electron/hole band

g-factor ge/h

⊥≈22.7(−1.21) and the Bohr magneton µB.

Further, we have introduced the in-plane spin-operator

sθ= cos θsx+ sin θsywith θthe in-plane polar angle.

Guided by HgTe, we use diﬀerent in-plane and out of

plane g-factors57. While Be

∥=Bh

∥,Be

⊥̸=Bh

⊥because

of diﬀerent g-factors |gh

⊥/ge

⊥| ≪ 1, and thus, we assume

a negligible gh

⊥≈0. Nevertheless, similar results can be

found using ge

⊥=gh

⊥, as realized in InAs/GaSb wells58.

Following standard ﬁnite diﬀerence methods, we

discretize the Hamiltonian on a 2D lattice, turning

the continuum momenta ˆ

kx/yΨ(x, y) = −i∂x/yΨ(x, y),

into their discretized version, that is, ∂x/yΨ(x, y)→

2ax/y (Ψix/y+ax/y−Ψix/y−ax/y)and ∂2

x/yΨ(x, y)) →

x/y

(Ψix/y+ax/y−2Ψix/y+ Ψix/y−ax/y). In the normal

part of the junction |y|< Ln/2, we add a perpendic-

ular magnetic ﬁeld, whose orbital eﬀects enter via the

Peierls substitution ˆ

kx,y →ˆ

kx,y −(e/ℏ)Ax,y, with the

Landau gauge A=|H⊥|xey. Thus, ψ†

e,iyψe,iy±ay→

exp[±i(πΦp/Φ0)ix]ψ†

e,iyψe,iy±ay, with the ﬂux quanta

Φ0=h/2eand the magnetic ﬂux in a plaquette Φp=

A|H⊥|for |y|< Ln/2and 0 otherwise, and with A=

axaybeing the area of a single plaquet. Besides, we as-

sume a negligible Zeeman contribution generated by the

magnetic ﬁeld responsible for the magnetic ﬂux.

B. Topological superconductivity

There are diﬀerent ways to induce Majorana bound

states on the proximitized quantum spin Hall edges. The

most direct one, a proximitized helical edge state59, is

not relevant here because without conserving the parity,

its corresponding Fraunhofer pattern gives no diﬀerence

in the trivial and topological regimes. Here, we explore

a diﬀerent conﬁguration closer to the Rashba quasi 1D

well, where the proximitized BHZ model supports a chiral

Majorana edge mode for a Zeeman ﬁeld above the critical

ﬁeld B⊥> Bc60,61 in the NS junction. In addition, in the

quasi 1D limit a pair of proximitized quantum spin Hall

edge channels can exhibit a superconducting topological

phase when the width of the sample is smaller than the

localization length of the quantum spin Hall edge states

ξqsh, that is, W ≲ξqsh48. To understand this scenario,

we ﬁrst represent the non-proximitized spectrum of a thin

sample in Fig. 2(a), which develops a gap at the Dirac

point. In the vicinity of the gap, the resulting spectrum

exhibits the same structure as the Rashba wire, that is,

parabolic bands shifted in kby the spin-orbit coupling

∆k=±kso, see magniﬁcation in Fig. 2(a). Hence, it is

not surprising to expect an inversion of the proximitized

superconducting band produced by a Zeeman ﬁeld ap-

plied perpendicularly to the eﬀective spin-orbit ﬁeld di-

rection, see bottom panels in Fig. 2(c). As a result, two

Majorana bound states appear at the extremes of the su-

perconductor, which becomes visible imposing hard-wall

boundary conditions. See LDOS at the NS interface in

Fig.2(b)48. It is important to remark that even though

we need a ﬁnite overlap between opposite quantum spin

Hall edge states on the superconducting part, the quan-

tum spin Hall edge states placed on the normal part prop-

agate without scattering to the opposite edge. This is

possible by setting Cin the normal region, i.e. CN, in

such a way that the Dirac point is away from the Fermi

energy. Alternatively, one can use the same chemical po-

tential as in the superconducting region, i.e. CN=CS,

with a diﬀerent width WN>WSon the normal part

and superconducting parts. Although we use the former

conﬁguration (CN̸=CS), the latter (WN>WS)might

be easier to accomplish experimentally since it does not

involve a spatially dependent gate voltage control.

Experimentally, there are already two scenarios where

non-proximitized quantum spin Hall modes of opposite

edges overlap. Firstly, the addition of an electric ﬁeld

can push the edge modes towards the bulk62. Secondly,

materials like bismuthene on SiC develop topological line

defects through which the quantum spin Hall edges prop-

agating on opposite sides can overlap63.

Set of parameters

Unless it is speciﬁed otherwise, we use the following

set of parameters: A= 373 meV nm, B=−857 meV

nm2,D=−682 meV nm2,M=−10.0meV. The pa-

rameters associated to spin-orbit coupling are α= 0.0,

δ0= 1.6meV, δe=−12.8meV nm, δh= 21.1meV nm,

T= 0.1K and ∆0= 0.15 meV. In the tight-binding cal-

culations, we use a discretization constant a≡ax=ay=

2.4nm. We set the Fermi energy EF= 0 and the chem-

ical potential on the normal and superconducting parts

is tuned by CNand CS, respectively.

In this work, we use the magnetic ﬁeld for two diﬀer-

ent purposes: a Zeeman ﬁeld to invert the superconduct-

ing gap and the magnetic ﬂux to study the Fraunhofer

pattern. Even though both ﬁelds are applied in the z-

direction, it is possible to study the critical current as

a function of both contributions independently because

of the diﬀerent order of magnitude of each contribution.

Here, the Zeeman ﬁeld is of the order of B⊥∼1T, while

the orbital ﬁeld is a few mT.

III. FRAUNHOFER PATTERN

The Fraunhofer pattern, i.e. the critical current as a

function of the magnetic ﬂux, contains important infor-

mation about the spatial distribution of the supercurrent

in a Josephson junction64. For example, when the nor-

mal part of the junction is dominated by bulk modes,

the critical current oscillates and decays for an increas-

ing magnetic ﬂux. Whereas, when the supercurrent is

carried along the edges (quantum spin Hall or quantum

Hall states), the critical current oscillates with a period

FIG. 3. Fraunhofer pattern for the thin superconducting

junction: Panel (a), critical current as a function of the

ﬂux and perpendicular Zeeman ﬁeld. The parameters of the

model are MS/N =−10 meV, W= 0.14 µm, Ln= 0.74 µm,

CS=−9.25 meV, CN= 2.0meV, ∆0= 0.15 meV. Panel (b),

Fraunhofer pattern for two diﬀerent line cuts B⊥= 0.0(triv-

ial regime) and B⊥= 3.5meV (topological regime).

(Φ/Φ0or 2Φ/Φ0) without decaying. Using standard equi-

librium Green’s function methods introduced in App. A,

we calculate the critical current of a Josephson junction

with quantum spin Hall edges on the normal part in the

absence and presence of Majorana bound states at the

NS interfaces.

In the 2D limit, where chiral Majorana modes propa-

gate at the boundary of the superconducting slabs, the

Fraunhofer pattern does not exhibit a qualitative change

when comparing the trivial and topological regimes, see

Fig. 11 in App. B 1. This is caused by the dominant

coupling between normal and proximitized quantum spin

Hall edge states, which gives rise to a supercurrent con-

tribution based on LAR, yielding negligible traces of the

presence of the chiral Majorana contribution. Therefore,

in order to observe a speciﬁc signature of the MBS in

the Fraunhofer pattern proﬁle, we need to reduce the su-

percurrent carried by local ABS so that we can enhance

the coupling between the quantum spin Hall edge states

and the central part of the superconducting slab, where

the MBSs are located. To this aim, we induce MBSs di-

rectly on the proximitized quantum spin Hall edge states

by considering a thin superconductor junction. Alterna-

tively, there is another way to increase the coupling be-

tween the helical modes and the MBSs, that involves a

positive mass M > 0in the superconducting leads, hav-

ing no helical edge states in the normal state. Since the

results are analogous to the thin superconducting junc-

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摘要：

FraunhoferpatterninthepresenceofMajoranazeromodesF.Dominguez,1E.G.Novik,2andP.Recher1,31InstitutfürMathematischePhysik,TechnischeUniversitätBraunschweig,38106Braunschweig,Germany2InstituteofTheoreticalPhysics,TechnischeUniversitätDresden,01062Dresden,Germany3LaboratoryforEmergingNanometrology,38106B...

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Fraunhofer pattern in the presence of Majorana zero modes

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