Waiting time distributions in Quantum spin hall based heterostructures F. Schulz1D. Chevallier2and M. Albert3 1Department of Physics University of Basel Klingelbergstrasse 82 CH-4056 Basel Switzerland

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Waiting time distributions in Quantum spin hall based heterostructures

F. Schulz,1D. Chevallier,2and M. Albert3

1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

2Bleximo Corp., 701 Heinz Ave, Berkeley, CA 94710, USA

3Universit´e Cˆote d’Azur, CNRS, Institut de Physique de Nice, 06560 Valbonne, France

(Dated: October 5, 2022)

For the distinction of the Andreev bound states and Majorana bound states, we study the wait-

ing time distributions (WTDs) for heterostructures, based on one dimensional edge states of a two

dimensional topological insulators (TI) in combination with an proximitized s-wave superconductor

(SC) and an applied magnetic ﬁeld. We show for the time reversal symmetric (TRS) situation of a

Josephson junction details of the WTD. This includes diﬀerent transport processes, diﬀerent num-

bers of Andreev bound states and the phase diﬀerence of the SC. We further consider a Zeeman ﬁeld

in the normal part of the junctions revealing novel features in the WTD along the phase transition

between trivial bound states and Majorana bound states. We ﬁnally discuss clear signatures to

discriminate between them.

The quest of Majorana fermionic states in condensed-

matter physics, have been the subject of a strong in-

terest during the past decades, in particular because of

their exotic properties, such as non-Abelian statistics,

that open the way to using them for quantum computa-

tion [1,2]. Among many proposals to create them, het-

erostructures based on two dimensional TIs are one in-

teresting lead of research. Previous investigations on the

topologically non-trivial bulk band inversion in TIs result

in quasi one-dimensional TRS protected counter propa-

gating edge states [3–5]. These spin polarized states in

combination with an s-wave superconductor (SC) and a

magnetic ﬁeld can be used to engineer Majorana Bound

States (MBSs) within diﬀerent geometrical setups [6–9].

Induced superconductivity in the edge states of TIs has

experimentally been realized in quantum well setups [10–

16], or alternatively on thin-layer TIs [17–21]. The com-

bination with a magnetic ﬁeld for such systems results

in a zero energy mode in the conductance, potentially

allowing the characterization of a MBS [22]. The re-

alization of topological superconductivity has also been

realized on semiconducting nanowires [23–25]. However,

usual conductance measurements to identify ﬁngerprints

of MBS for such topological setups remain elusive, such

that we are motivated for alternative measurements. One

possibility would be to resort to the Waiting Time Distri-

butions (WTDs) [26–38], namely the distribution of time

delay between the detection of two consecutive charge

carriers, which has been shown to reveal traces of MBS

in such non-trivial systems [36–38].

Necessary techniques for the counting of single parti-

cles have come very precise, allowing a novel measure-

ment for the processes of speciﬁc scattering events [39–

48]. The resulting WTD in turn provides statistics in-

cluding signatures that may manifest a clear distinction

between topologically trivial and non trivial states within

the system. WTDs in semiconducting hybrid systems

have recently been studied, providing distinct features

of a one dimensional p-wave SC in comparison to an

Figure 1. (a) Schematic of the setup. The scattering area,

containing two superconducting (blue) regions enclosing a fer-

romagnet (yellow) are connected to source and drain elec-

trodes. Further indicated are the possible scattering states

of an incoming spin up electron, which may be reﬂected as

electron/hole or be transmitted as electron/hole. (b) Energy

spectra with the eﬀective paring amplitudes in the scattering

regions.

s-wave SC [36]. Further theoretical investigations have

shown results for entangled electrons on a SC interface

for MBSs [37,38].

In this paper we study WTDs in materials based on

topological hybrid junctions as presented on Fig. 1. We

combine the WTD as a statistical characterization tool to

detect several transport eﬀects that appear within such

topological junctions. Starting with a simple application

on almost perfectly Andreev reﬂected electrons, we con-

tinue with more complicated situations, such as the in-

clusion of several transmitting resonances. Finally, by in-

cluding a magnetic ﬁeld, we create topological protected

MBSs, which we measure and discuss along their distinct

signatures in the WTD along the phase transition.

The article is setup as follows, in Sec. Iwe introduce

arXiv:2210.01080v2 [cond-mat.mes-hall] 4 Oct 2022

the model that is used for the underlying N-S-F-S-N junc-

tion. In Sec. II we recap the substance of WTDs. The re-

sults are presented in Sec. III, where we elaborate WTDs

for speciﬁc types of hybrid quantum conductors, closely

related to the Fu and Kane structure [7]. We study in

details the WTDs for both, ABS and MBS and identify

signatures to distinguish them. Conclusion and outlook

are given in Sec. IV and several technical details are avail-

able in Appendices.

I. MODEL

The heterostructure we consider consists of helical

counter propagating edge states of a Quantum spin hall

insulator, proximity coupled to a spatially restricted s-

wave superconductor and a magnetic ﬁeld [see Fig. 1

(a)]. Similar setups have been studied in the litera-

ture [6,7,49–51]. In the basis Ψ = (ψ↑, ψ↓, ψ†

↑, ψ†

↓), the

Hamiltonian of the system under consideration is the fol-

lowing

H=H0H∗

HSC −H∗

0,(1)

where H0=HTI +HZ. The Hamiltonian HTI =

vFkxσ1−µis the kinetic term of the edge states, with

Fermi-velocity vF(~= 1), momentum kx=−i∂xand

chemical potential µ, while HZ=σ1∆Z(x) is the Zee-

man ﬁeld pointing in x-direction. Furthermore, HSC =

−i∆(x)σ2includes the superconductor, which is assumed

to be grounded, and the Pauli matrices σ(τ) act in

spin (Nambu) space. The spatial arrangements of ∆

and ∆Zare chosen to have a ferromagnetic region en-

closed by two superconducting regions, such that ∆(x) =

∆[Θ(x)−Θ(x−L)]+∆eiφ[Θ(x−L−Lm)−Θ(x−2L−Lm)]

and ∆Z(x)=∆Z[Θ(x−L)−Θ(x−L−Lm)]. The pairing

mechanisms open a gap in the spectrum of the edge states

[see schematic in Fig. 1(b)], which is either at the Dirac-

point (DP) (∆Z) or at the Fermi level crossing (∆). In

addition we also allow the possibility to add a magnetic

ﬂux φto the system [see Fig. 1(a)].

We check the characterization of diﬀerent states ap-

pearing in the heterostructure, by ﬁrst presenting the

scattering features as a basis to identify distinct signa-

tures in the WTD. Whereby we use the seminal work of

Blonder, Tinkham and Klapwijk (BTK) to calculate the

corresponding scattering coeﬃcients [52]. For our inter-

ests we study two cases, ﬁrst, no Zeeman ﬁeld is applied

(trivial phase) and the middle region hosts freely propa-

gating states, and second, a non vanishing Zeeman ﬁeld

(topological phase above a certain treshold) couples the

two spin species and opens a gap in the middle region, re-

sulting in the emergence of MBSs on the interfaces of the

superconducting and magnetic regions [7]. We assume in

this work a rather long junction (Lm> vF/∆), such that

for ∆Z= 0 multiple bound states are well deﬁned within

the superconducting gap E < ∆ [see Fig. 2(a)]. Cal-

culating the wavefunction with the continuity conditions

on the interfaces at x= 0, L, L +Lmand x= 2L+Lm

results in the necessary transport coeﬃcients. With a

preserved TRS an incoming right moving electron from

the source is (∆Z= 0) protected from backscattering,

such that there are only two non vanishing processes.

Namely, the co-tunneling of an electron (te) form source

to drain and the local Andreev-reﬂection (rh) on the SC

interface at x= 0. The breaking of TRS in turn then

also allows normal electron reﬂection (re), including a

spin ﬂip, and the transmission of holes (th). While the

gap closing and reopening is usually used as an indica-

tor of the topological phase transition, we use only the

normal electron reﬂection signature, accompanied by its

probability at zero energy from zero to one. Details are

presented in App. A. Based on those energy dependent

coeﬃcients, we are highly interested in the corresponding

WTDs, which we introduce in the next section.

II. WAITING TIME DISTRIBUTION

For the sake of clarity, we recap in this section, the

formalism used for the calculation of WTDs [26–29] as

well as some well established results that will serve as

reference. The WTD W(τ) denotes the probability dis-

tribution for the time delay τbetween the detection of

two consecutive charge carriers. They can be electrons

or holes for instance. This quantities gives precise and

transparent informations about correlations in a trans-

port process. In general, it is customary to calculate it

from the Idle Time Probability (ITP) Π(τ), namely the

probability to detect zero particles during a time interval

τ. For stationary processes these two distributions are

connected by the following expression

W(τ) = hτi∂2Π(τ)

∂τ2. (2)

For non-interacting systems, the ITP is given by the de-

terminant formula [28]

Π(τ) = det1−Qτ, (3)

where Qτis a projector over the time window τwhose ex-

pression depends on the detection scheme (measurement

of consecutives electrons, holes, ...) and the scattering

matrix of the system. Explicit formulae are given be-

low for the processes of interest. In addition, the mean

waiting time is given by

hτi=−∂Π(τ)

∂τ 



−1

τ=0

.(4)

To continue, we specify a detection procedure for the

WTD. Thus, we calculate the ITP with the projection

of a single particle scattering state on an discrete

voltage window around the Fermi level. In the range

of the applied voltage V, the linear energy spectrum

is discretized into Ncompartments with wave vectors

k=n

~vF. A stationary process is then reached by the

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WaitingtimedistributionsinQuantumspinhallbasedheterostructuresF.Schulz,1D.Chevallier,2andM.Albert31DepartmentofPhysics,UniversityofBasel,Klingelbergstrasse82,CH-4056Basel,Switzerland2BleximoCorp.,701HeinzAve,Berkeley,CA94710,USA3UniversiteC^oted'Azur,CNRS,InstitutdePhysiquedeNice,06560Valbonne,Fran...

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Waiting time distributions in Quantum spin hall based heterostructures F. Schulz1D. Chevallier2and M. Albert3 1Department of Physics University of Basel Klingelbergstrasse 82 CH-4056 Basel Switzerland

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