Reconciling phase shift in Josephson junction experiments with even-parity superconductivity in Sr 2RuO 4 Austin W. Lindquist1and Hae-Young Kee1 2

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Reconciling πphase shift in Josephson junction experiments with even-parity

superconductivity in Sr2RuO4

Austin W. Lindquist1and Hae-Young Kee1, 2, ∗

1Department of Physics and Center for Quantum Materials,

University of Toronto, 60 St. George St., Toronto, Ontario, M5S 1A7, Canada

2Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada

The superconducting state of Sr2RuO4was once thought to be a leading candidate for p-wave

superconductivity. A constant Knight shift below the transition temperature provided evidence

for spin-triplet pairing, and a πphase shift observed in Josephson junction tunneling experiments

suggested odd-parity pairing, both of which are described by p-wave states. However, with recent

experiments observing a signiﬁcant decrease in the Knight shift below the transition temperature,

signifying a spin-singlet state, the odd-parity results are left to be reconciled. In this work, we

show that an even-parity pseudospin-singlet state originating from interorbital pairing via spin-

orbit coupling can explain what has been assumed to be evidence for an odd-parity state. In the

presence of small mirror symmetry breaking, interorbital pairing is uniquely capable of displaying

odd-parity characteristics required to explain these experimental results. Further, we discuss how

these experiments may be used to diﬀerentiate the proposed pairing states of Sr2RuO4.

I. INTRODUCTION

The puzzle of superconductivity in Sr2RuO4(SRO) has

been a longstanding problem with many seemingly contra-

dictory experimental results [1–4]. Once a leading candi-

date for p+ip-wave spin-triplet superconductivity [5,6], re-

cent experiments show a drop in the Knight shift below the

superconducting transition temperature, potentially ruling

this state out [7–9]. Other notable experiments suggest a

two component order parameter [10–12] which breaks time-

reversal symmetry [13–16] and features gap nodes [17–19].

Attempts to explain these results have led to the recent

proposals of various multicomponent, even-parity pairing

states [20–28].

The proposed even-parity states are capable of explain-

ing many experimental results, however, little progress has

been made in explaining the experimental data supporting

odd-parity superconductivity [29–32]. Primarily, phase-

sensitive Josephson junction experiments observe a πphase

shift of the superconducting order parameter under inver-

sion [29]. Previous studies have shown these results to be

consistent with odd-parity pairing [33,34], whereas con-

ventional even-parity spin-singlet states have remained in

contradiction with these observations. Conventional even-

parity superconductors with inversion symmetry breaking

have been shown to display both even- and odd-parity

character in non-centrosymmetric superconductors [35,36],

however, this eﬀect would be much smaller in SRO.

In this work, we study even-parity intraband pseudospin-

singlet superconductivity, evolved from interorbital spin-

triplet pairing via spin-orbit coupling (SOC) in the pres-

ence of small mirror symmetry breaking hoppings as a route

to reconcile these remaining contradictions. The mirror

symmetry breaking hopping term we introduce occurs near

surfaces, interfaces, or strain as sketched in the experimen-

tal setup shown in Fig. 1a in Sec. II. We then explain the

setup of the Josephson junction calculations in Sec. III and

show the current-phase relations for conventional even- and

∗hy.kee@utoronto.ca

SRO (C)

s-wave (A) s-wave (A)

I (B) I (B)

(a)

(b)

Figure 1. (a) Schematic of the Josephson junction setup show-

ing the s-wave superconductor, insulator (I), and SRO regions,

denoted A,B, and Cin the main-text equations, labeled here in

parentheses. In experiment SRO and the s-wave superconduc-

tor are separated in the z-direction by SiO, and no tunneling

occurs at this interface, signiﬁed by the ﬁlled black region here.

(b) One example of interorbital hopping which is only allowed

where mirror symmetry is broken. When a z-mirror plane ex-

ists, the orbital overlap is 0, but when broken, the overlap is

ﬁnite.

odd-parity pairing states. Finally, in Sec. IV we present in-

terorbital pairing and show that signatures of pairing states

which feature gap nodes in the tunneling direction may

match those expected of an odd-parity state. This behav-

ior is made possible by the multiorbital nature of SRO and

the intraband pseudospin-singlet pairing evolved from in-

terorbital spin-triplet pairing.

arXiv:2210.00554v2 [cond-mat.supr-con] 11 Jan 2023

II. MICROSCOPIC HAMILTONIAN

The Josephson junction consists of a conventional single

band s-wave superconducting region, a normal insulator re-

gion, and the superconducting SRO region. A schematic of

the regions is shown in Fig. 1(a), and they are denoted A,

B, and C, respectively, in the equations below. Within re-

gions Aand Bwe use the single band kinetic Hamiltonian,

written here for region A,

HA=X

k,iy,δy,σ

ξA(k, δy)c†

A,k,iy,σcA,k,iy+δy,σ (1)

where ξA(k, δy) is the electron dispersion in region Abe-

tween slabs at positions iyand iy+δy, and c†

A,k,iy,σ cre-

ates an electron in slab iyof region Awith momentum

k= (kx, kz) and spin σ. The form of all dispersion terms

as well as the values of the hopping parameters are given

in Appendix A. The superconductivity in region Ais de-

scribed by,

HSC

A=X

k,iy

eiφA∆Ac†

A,k,iy,↑c†

A,−k,iy,↓+ h.c., (2)

where ∆Ais the s-wave order parameter, and φAis the

superconducting phase. Hopping between regions Aand

Bis taken to have the same parameters as hopping within

either of the regions.

The normal state Hamiltonian of region Cis,

HC=X

k,iy,a,σ

ξa

C(k, δy)ca†

C,iy,k,σca

C,iy+δy,k,σ

k,iy,a6=b,σ

ξab

C(k, δy)ca†

C,k,iy,σcb

C,k,iy+δy,σ + h.c. + HSOC,

(3)

which includes intraorbital and interorbital dispersions,

ξa

C(k, δy) and ξab

C(k, δy), respectively, where aand bare

the orbital indices representing the yz,xz, and xy orbitals,

as well as SOC terms. Finally, the Hamiltonian describ-

ing hopping between regions Band C, where the interface

occurs between iy= 1 and 2, has the form,

Hint =X

k,a,σ

ξa

int(k)ca†

C,k,2,σcB,k,1,σ + h.c., (4)

which features orbital dependence in the SRO region, as

denoted by ξa

int(k).

We also consider the eﬀects of mirror symmetry break-

ing in the z-direction (out of plane direction). This eﬀect

is largest near the surface normal to the z-direction, but

imperfections of the interface leading to a broken mirror

plane in the z-direction have previously been proposed to

occur [37], where it was shown that the experimental re-

sults may be explained by a dxz +idyz-wave state if the

tunneling directions tilt out of the xy plane. Additionally,

the growth of Au0.5In0.5directly onto SRO to create the

junction may cause strain in SRO. Any deformations of

the lattice that this leads to may further contribute to bro-

ken mirror symmetry throughout the sample. Dislocations

may also contribute to this mirror symmetry breaking, and

have been found to occur near interfaces [38]. The lack of

mirror symmetry in the z-direction means that hopping be-

tween the xy and xz(yz) orbitals is allowed to be ﬁnite in

the y(x) direction. An example of such hopping is shown

in Fig. 1(b) for the xy to xz interorbital hopping. These

hoppings have the form:

hISB

k=−α[2isin kx(cyz†

k,iy,σcxy

k,iyσ−cxy†

k,iy,σcyz

k,iy,σ)

−(cxz†

C,k,iy,σcxy

C,k,iy+1,σ −cxy†

C,k,iy,σcxz

C,k,iy+1,σ + h.c.)],(5)

where αrepresents the hopping integral, which depends on

the strength of the mirror symmetry breaking, and the use

of δy= 1 here represents nearest-neighbor hopping between

slabs.

In the next section, we describe the setup for the Joseph-

son tunneling calculations and apply it with conventional

s- and p-wave superconducting states in the SRO region.

Then, we consider interorbital superconductivity and show

how the current-phase relation (CPR) is aﬀected by the

mirror symmetry breaking terms introduced here, showing

that they may behave like the conventional s-wave state,

or potentially more like the p-wave state, depending on the

nodal structure as well as the strength of the mirror sym-

metry breaking.

III. JOSEPHSON CALCULATIONS

To calculate the Josephson CPR, we use the lattice

Green’s function method presented in Ref. 34, which con-

siders only p-wave pairing to explain experimental results.

Semi-inﬁnite Green’s functions are obtained for the s-wave

and SRO regions using the recursive Green’s function ap-

proach [39]. A single layer of the normal insulator is added

on the surfaces of both of these regions by the Dyson equa-

tions,

0(k, iωl)=(iωl−ˆu0(k)−ˆ

t0,−1ˆ

−1(k, iωl)ˆ

t−1,0)−1,(6)

1(k, iωl)=(iωl−ˆu1(k)−ˆ

t1,2ˆ

2(k, iωl)ˆ

t2,1)−1,(7)

where the interface is in the xz plane. Here, ˆ

n(k, iωl) is

the Green’s function of layer nin region m, ˆun(k) is the

part of the Hamiltonian of layer n, and ˆ

tn,n+1 is the part

of the Hamiltonian featuring hopping the the y-direction,

representing hopping between layers nand n+ 1. The left

and right systems are combined using the two equations,

G00(k, iωl) = {[ˆ

0(k, iωl)]−1−ˆ

t01 ˆ

1(k, iωl)ˆ

t10}−1,(8)

G11(k, iωl) = {[ˆ

1(k, iωl)]−1−ˆ

t10 ˆ

0(k, iωl)ˆ

t01}−1.(9)

These are then used to obtain the nonlocal Green’s func-

tions

G01(k, iωl) = ˆ

0(k, iωl)ˆ

t01 ˆ

G11(k, iωl),(10)

G10(k, iωl) = ˆ

1(k, iωl)ˆ

t10 ˆ

G00(k, iωl).(11)

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ReconcilingphaseshiftinJosephsonjunctionexperimentswitheven-paritysuperconductivityinSr2RuO4AustinW.Lindquist1andHae-YoungKee1,2,1DepartmentofPhysicsandCenterforQuantumMaterials,UniversityofToronto,60St.GeorgeSt.,Toronto,Ontario,M5S1A7,Canada2CanadianInstituteforAdvancedResearch,Toronto,Ontario,M5...

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Reconciling phase shift in Josephson junction experiments with even-parity superconductivity in Sr 2RuO 4 Austin W. Lindquist1and Hae-Young Kee1 2

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