Two-electron photoemission spectroscopy in Topological Superconductors Ka Ho Wong1 Ameya Patwardhan23 Peter Abbamonte23 Fahad Mahmood23 and Dirk K. Morr1 1Department of Physics University of Illinois at Chicago Chicago IL 60607 USA

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Two-electron photoemission spectroscopy in Topological Superconductors

Ka Ho Wong1, Ameya Patwardhan2,3, Peter Abbamonte2,3, Fahad Mahmood2,3, and Dirk K. Morr1

1Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA

2Department of Physics, University of Illinois at Urbana-Champaign,Urbana,IL 61801, USA and

3Materials Research Laboratory, University of Illinois at Urbana-Champaign,Urbana,IL 61801, USA

We demonstrate that the photo-electron counting rate, P(2), measured in two electron coincidence

spectroscopy (2e-ARPES) experiments, provides unprecedented insight into the nature of topological

superconductivity. In particular, we show that the spin dependence of P(2) allows one to detect

superconducting spin-triplet correlations that are induced in a topological superconductor even in the

absence of an associated triplet superconducting order parameter. This ability to detect spin-triplet

correlations allows one to distinguish between two recently proposed scenarios for the microscopic

origin of topological superconductivity in FeSe0.45Te0.55. Finally, we show that P(2) exhibits a

characteristic intensity maximum that can be employed to detect topological phase transitions.

I. INTRODUCTION

Topological superconductors harbor Majorana zero

modes (MZMs) whose non-Abelian statistics in combi-

nation with their topologically protection against disor-

der and decoherence eﬀects provide an exciting platform

for the realization of topological quantum computing [1].

However, the experimental observation and identiﬁcation

of MZMs in a variety of superconducting systems [2–11]

met signiﬁcant challenges due to the system’s small su-

perconducting gaps, which are often only of the order

of a few hundred µeV . The recent report of topolog-

ical superconductivity in the iron-based superconductor

FeSe0.45Te0.55[12–21], possessing a signiﬁcantly larger su-

perconducting gap of a few meV, might therefore provide

a more suitable platform for the unambiguous identiﬁ-

cation of MZMs, and the realization of topology based

devices and topological quantum computing.

The origin of topological surface superconductivity in

FeSe0.45Te0.55 was initially proposed to arise from band-

inversion [12, 22–24] – rendering FeSe0.45Te0.55 a 3D

topological insulator – and the gaping of the ensuing sur-

face Dirac cone by proximity induced superconductivity

(we refer to this as the 3DTI+mechanism). However,

the recent experimental observation of ferromagnetism

on the surface of FeSe1−xTex[13–15, 25] has shed doubts

on this interpretation, as topological superconductivity

arising from the 3DTI+mechanism, being protected by

a time-reversal symmetry, is destroyed already for rather

weak surface ferromagnetism [26, 27]. A competing sce-

nario was therefore proposed [28, 29] in which the very

ferromagnetism observed experimentally in combination

with the two-dimensional nature of superconductivity in

FeSe0.45Te0.55 and a Rashba spin-orbit interaction on

the surface induced by the broken inversion symmetry,

gives rise topological surface superconductivity (we refer

to this as the 2DTSC mechanism). Clearly, further ex-

periments are required to distinguish between these two

proposed scenarios.

In this article, we demonstrate that the photo-electron

counting rate [30], P(2), measured in two electron co-

incidence spectroscopy (2e-ARPES) experiments, can

provide unprecedented insight into the nature of topo-

logical superconducting phases, and thus identify the

microscopic origin of topological superconductivity in

FeSe0.45Te0.55. In 2e-ARPES experiments, the absorp-

tion of a single photon leads to the emission of two co-

incident photo-electrons. As previously shown [31], the

energy dependence of P(2) cannot only reveal the total

center of mass momentum of a Cooper pair, but also its

spin state. As a result, 2e-ARPES experiments can iden-

tify superconducting spin-triplet correlations which are

induced within the 2DTSC mechanism in FeSe1−xTex,

but are all but absent in the 3DTI+mechanism. In ad-

dition, we show that 2eARPES experiments can identify

topological phase transitions which coincide with a max-

imum in P(2) for photo-electrons with equal spin. These

results open a new venue to distinguish between proposed

mechanisms for the emergence of topological supercon-

ductivity in FeSe1−xTex.

II. THEORETICAL MODEL

In the following, we consider the 2eARPES photo-

electron counting rate for two diﬀerent types of topolog-

ical superconductors: (i) a two-dimensional topological

superconductor with broken time reversal symmetry, as

described by the 2DTSC mechanism, and (ii) a topologi-

cal superconductor on the surface of a three-dimensional

topological insulator, arising from the proximity coupling

of its surface Dirac cone to an s-wave superconductor, as

described by the 3DTI+mechanism. The former system

is described by the Hamiltonian [28, 32]

HSC =X

k"ξkc†

k,σck,σ + ∆0c†

k,↑c†

−k,↓+ck,↓c−k,↑

+ 2αX

δ,σ,σ0

sin(k·δ

δ)c†

k,σ(δ

δ×σ

σ)z

σσ0ck,σ0

−JS X

σ,σ0

c†

k,σσz

σσ0ck,σ0#.(1)

arXiv:2210.11738v1 [cond-mat.supr-con] 21 Oct 2022

Here, c†

k,σ creates an electron with momentum kand spin

σand ξk=−2t(cos kx+ cos ky)−µis the tight binding

dispersion with −tbeing the nearest-neighbor hopping

amplitude, and µbeing the chemical potential. More-

over, ∆0is the s-wave superconducting order parame-

ter, αis the Rashba spin-orbit interaction, with δ

δbeing

the unit vector connecting nearest neighbor sites, and J

is the magnetic exchange coupling between the ordered

moments of magnitude Sand the conduction electrons.

A schematic representation of the 2DTSC is shown in

Fig. 1(a). This topological superconductor is in the topo-

logical class D, and its topological invariant is the Chern

number [32], C. Its topological phase diagram in terms

of Cis shown in Fig. 1(b) in the (µ, JS)-plane.

The Hamiltonian of the 3DTI system is given by [33]

H3D=−tX

r,j=1,2,3Ψ†

r+ˆej

Γ1−iΓj+1

2Ψr+H.c.

+mX

Ψ†

rΓ1Ψr(2)

with spinor

Ψ†

r=c†

r,1,↑, c†

r,2,↑, c†

r,1,↓, c†

r,2,↓,(3)

where cr,a,σ annihilates an electron with spin σin orbital

a= 1,2 at site r, and Γ(0,1,2,3,4) = (1⊗1,1⊗sz,−σy⊗

sx, σx⊗sx,−1⊗sy) with σiand si(i=x, y, z) being

Pauli matrices. Within this model, a topological super-

conducting phase emerges on the surface of the 3DTI due

to proximity coupling to a superconductor, and the en-

suing opening of a gap in the 3DTI’s surface Dirac cone.

The proximity induced superconductivity is described by

the Hamiltonian

H∆= ∆0X

r,a=1,2

c†

r,a,↑c†

r,a,↓+H.c. , (4)

where ∆0is the induced superconducting order parame-

ter with s-wave symmetry. We note that to demonstrate

the qualitative diﬀerent form of P(2) in the 2DTSC and

3DTI+systems (see below), it is suﬃcient to consider for

simplicity an s-wave symmetry of the superconducting

order parameter, rather than the s±-symmetry found in

FeSe0.45Te0.55. A more material speciﬁc calculation of

P(2) that also takes into account the multi-band struc-

ture of FeSe0.45Te0.55 [28], is reserved for a future study.

In 2eARPES spectroscopy, there are two distinct pro-

cesses in which the absorption of a single photon leads to

the ejection of a correlated pair of electrons [30, 31, 34–

40], giving rise to the photo-electron counting rate P(2).

In the ﬁrst one, the absorption of a photon results in the

excitation of a valence band electron into a free photo-

electron state, which subsequently ejects a second va-

lence electron via an electron energy-loss (EELS)-like

scattering event, mediated by the Coulomb interaction.

In the second process, the photon ﬁrst excites a photo-

electron from a core-level (rather than the conduction

FIG. 1. (a) Schematic representation of the 2DTSC with

local magnetic moments interaction with conduction electrons

on the surface of an s-wave superconductor. (b) Topological

phase diagram of the 2DTSC in terms of the Chern number

Cin the (µ, JS)-plane.

band). The resulting core hole is then ﬁlled by a valence

electron, leading to the emission of a second valence elec-

tron through an Auger process. As previously shown [31],

both processes lead to a very similar energy, momentum

and spin dependence of P(2). However, the use of lower

photon energy, laser based XUV sources will not allow

2e-ARPES experiments to directly probe core states. Be-

low, we will therefore consider P(2) as arising from the

ﬁrst process only, which is described by the Hamiltonian

Hscat =X

k,q,σ,ν

γν(q)d†

k+q,σck,σ aq,ν +a†

−q,ν 

k,p,q,α,β

V(q)d†

k+q,αd†

p−q,β dp,β ck,α +h.c. (5)

Here, γν(q) is the eﬀective electron-photon dipole in-

teraction, d†

k,σ(ck,σ) creates (destroys) a photo-electron

(conduction electron) with momentum kand spin σ,

and V(q) = V0/q2+κ2is the Fourier transform of

the (screened) Coulomb interaction, with κ−1being the

screening length. We note that in the 3DTI, possessing

two orbitals per site, the c-electron operators in Eq.(5)

acquire an orbital index. Since the qualitative nature of

P(2) does not depend on κ[31], we take for concrete-

ness κ−1= 10a0for the results shown below. Moreover,

as the photon momentum is much smaller than typical

fermionic momenta, we set it equal to zero, such that

γν(q) = γ0is simply a momentum-independent constant.

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Two-electronphotoemissionspectroscopyinTopologicalSuperconductorsKaHoWong1,AmeyaPatwardhan2;3,PeterAbbamonte2;3,FahadMahmood2;3,andDirkK.Morr11DepartmentofPhysics,UniversityofIllinoisatChicago,Chicago,IL60607,USA2DepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,Urbana,IL61801,USAand3Materi...

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Two-electron photoemission spectroscopy in Topological Superconductors Ka Ho Wong1 Ameya Patwardhan23 Peter Abbamonte23 Fahad Mahmood23 and Dirk K. Morr1 1Department of Physics University of Illinois at Chicago Chicago IL 60607 USA.pdf

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Two-electron photoemission spectroscopy in Topological Superconductors Ka Ho Wong1 Ameya Patwardhan23 Peter Abbamonte23 Fahad Mahmood23 and Dirk K. Morr1 1Department of Physics University of Illinois at Chicago Chicago IL 60607 USA

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