Proximity-induced equilibrium supercurrent and perfect superconducting diode effect due to band asymmetry Pavan Hosur

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Proximity-induced equilibrium supercurrent and perfect superconducting diode

eﬀect due to band asymmetry

Pavan Hosur

Department of Physics and Texas Center for Superconductivity,

University of Houston, Houston, Texas 77004

Daniel Palacios

Jan and Dan Duncan Neurological Research Institute at Texas Children’s Hospital, Houston, TX, 77030 and

Graduate Program of Quantitative and Computational Biosciences,

Baylor College of Medicine, Houston, TX, 77030

(Dated:)

We theoretically investigate the consequences of proximity-induced conventional superconductiv-

ity in metals that break time-reversal and inversion symmetries through their energy dispersion.

We discover behaviors impossible in an isolated superconductor such as an equilibrium supercurrent

that apparently violates a no-go theorem and, at suitable topological defects, non-conservation of

electric charge reminiscent of the chiral anomaly. The equilibrium supercurrent is expected to be

trainable by a helical electromagnetic ﬁeld in the normal state. Remarkably, if the band asymmetry

exceeds the critical current of the parent superconductor in appropriate units, we predict a perfect

superconducting diode eﬀect with diode coeﬃcient unity. We propose toroidal metals such as UNi4B

and metals with directional scalar spin chiral order as potential platforms.

I. INTRODUCTION

Nonreciprocal phenomena in superconductors

(SCs) have a long history. Among diode-like sys-

tems, early examples included ampliﬁcation of the

luminescence of light-emitting diodes when the diode

was attached to a SC [1, 2]. More recently, the

asymmetry in the current-voltage characteristics of

non-centrosymmetric metals under a magnetic ﬁeld

was seen to be enhanced if the metal turned su-

perconducting [3–5]. Recent theoretical and exper-

imental breakthroughs in the theory and realiza-

tion of superconducting and Josephson diodes [3–

26], which carry immense technological potential by

avoiding the enormous heating losses of semicon-

ductor diodes, have driven fervent activity in the

ﬁeld. These diodes are characterized by unequal

critical supercurrents in opposite directions, result-

ing in Ohmic and dissipationless transport, respec-

tively, for current magnitudes between the two crit-

ical currents. Such diode eﬀects are intimately con-

nected to the exotic Fulde-Ferrell superconductiv-

ity, deﬁned by ﬁnite momentum Cooper pairs in the

ground state [7, 13, 14, 18, 23, 27]. Another exotic

non-reciprocal phenomenon entails the existence of

spontaneous supercurrents in a preferred direction

through Josephson junctions [26, 28–52] and SCs

with spin-orbit coupling in proximity to magnetism

[53–58]. While details vary, all the above approaches

rely crucially on one principle: broken Tand Isym-

metries. Violation of these symmetries results in

other peculiar phenomena, such as unusual vortex

dynamics in non-centrosymmetric SCs [59–62].

In this work, we revisit the problem of non-

reciprocity in superconducting systems and explore

it in a minimal scenario. Speciﬁcally, we consider

metals with an asymmetric dispersion εk̸=ε−kand

no Berry phases, proximity-couple them to a con-

ventional, s-wave SC and focus on a uniform sys-

tem without any Josephson junctions. We dub met-

als with εk̸=ε−kband asymmetric metals (BAMs)

and refer to BAMs that acquire conventional super-

conductivity as band asymmetric superconductors

(BASCs). Since kis inequivalent to −kin BAM,

intrinsic pairing tendencies in them, if any, are ex-

pected to be towards exotic Fulde-Ferrell supercon-

ductivity built from ﬁnite momentum Cooper pairs

[63]. On the other hand, band asymmetry elimi-

nates a Cooper instability at weak interactions, so a

more practical route to superconductivity in BAMs

may be extrinsic. We show that even this minimal

setup leads to strange behaviors impossible in iso-

lated SCs, namely, (i) an apparent violation of a ba-

sic no-go theorem due to an equilibrium current den-

sity (Sec. II), (ii) a perfect superconducting diode

eﬀect (SDE) without ﬁne-tuning (Sec. III), and (iii)

topological defects that violate charge conservation

(Sec. IV). We conclude by mentioning suitable ex-

perimental platforms (Sec. V).

II. EQUILIBRIUM SUPERCURRENT

We ﬁrst derive the equilibrium current Ieq in

a one-dimensional (1D) BAM. Generalization to

higher dimensions is straightforward. While sponta-

neous supercurrents have been studied [26, 28–58],

their signiﬁcance with respect to basic quantum me-

arXiv:2210.09346v2 [cond-mat.supr-con] 31 Aug 2023

Figure 1. Depositing a BAM wire on a conventional

SC will generate an equilibrium current Ieq , a SDE in

general, and a perfect SDE with unit diode coeﬃcient

if the band asymmetry exceeds a threshold determined

by the critical Cooper pair momentum of the parent SC.

(See text for details)

chanics has not been appreciated, which we do here.

In particular, we show how Ieq naively violates a the-

orem by Bloch that forbids current densities in the

thermodynamic limit in arbitrary systems of inter-

acting fermions [64–68], and then resolve the para-

dox.

We assume a single band with degeneracy g; for

spin-degenerate bands, g= 2. The Bloch Hamilto-

nian for such a BAM is

HBAM =ˆ

n=1

c†

kncknεk(1)

where ´k≡´dk

2π. Let us deposit the BAM wire

on a conventional, s-wave SC with zero Cooper pair

momentum, as sketched in Fig 1. The BAM will

develop conventional superconductivity too via the

proximity eﬀect. The Bogoliubov-deGennes Hamil-

tonian in the basis Ψk=cT

k,Tc†

kT−1Tis HBdG =

2PkΨ†

kH∆

k⊗IgΨkwhere

H∆

k=εk∆∗

∆0−ε−k(2)

and Igis a g×gidentity matrix. Ieq is given by

Ieq =ˆ

Tr jkfH∆

k−fH0

k (3)

where jk=e

2vk0

0−v−k⊗Igis the current

operator, f(X) = eX/T + 1−1and we have set

ℏ=kB= 1. We have explicitly subtracted a spuri-

ous current due to Hilbert space doubling that cap-

tures the current carried by the ﬁlled bands when

∆=0. This current vanishes in general lattice mod-

els and in continuum models with a symmetric dis-

persion. However, in an asymmetric continuum, it

is non-zero, regularization-dependent, and can even

diverge. For weak pairing, we ﬁnd (Appendix A)

Ieq ≈ge|∆0|2ˆ

v−k

(εk+ε−k)2tanh hεk

2Ti(4)

to leading order in ∆0.Ieq is generically non-zero

as long as εk̸=ε−k. To gain more insight into this

result, suppose the BAM has Fermi momenta Ki

and Fermi velocities vi. Linearizing the dispersion as

εKi+p≈vip,ε−Ki+p≈ε−Kiand assuming |ε−Ki| ≫

Λwhere Λis an energy cutoﬀ gives

Ieq ≈ge|∆0|2Λ2

2πX

v−Ki

|vi|ε3

−Ki

(5)

for T→0. If we assume Pi

Λ2v−Ki

|vi|ε3

−Ki∼10−9/eV,

which amounts to a 1 part-per-million band asym-

metry if Λand ε−Kiare each O(meV)and all Fermi

velocities are of the same order, then ∆0∼1K gives

a large Ieq ∼10mA which should be detectable via

the magnetic ﬁelds it creates.

The above current seems to contradict a seminal

theorem by Bloch, which states that the ground or

equilibrium state of a generic, interacting fermionic

system cannot carry a current density [64–68].

In particular, a recent reﬁnement of the theorem

showed that the current density along xis bounded

as |⟨Jx⟩| < O L−1

x, where Lxis the linear dimen-

sion in the xdirection [70]. Historically, Bloch’s the-

orem helped prove that persistent currents in iso-

lated superconducting and metallic [71, 72] rings

necessarily occur in excited states and are stabi-

lized by the quantization of magnetic ﬂux piercing

the ring. Thus, the persistent currents there have

a long lifetime that is limited only by the probabil-

ity of spontaneous or stimulated emission that re-

laxes them to the ground state. In contrast, BASCs

clearly carry a ground state current with a truly in-

ﬁnite lifetime, apparently evading Bloch’s theorem.

The spontaneous supercurrents described in Refs.

[53–58] are special cases of Ieq. However, Ieq dif-

fers fundamentally from spontaneous currents in T

and Ibreaking Josephson junctions that crucially

rely on the presence of a junction and decay expo-

nentially with junction thickness [26, 28–52] while

Ieq is independent of the length of the BAM wire.

The resolution to the paradox lies in the observa-

tion that Bloch’s theorem explicitly assumes charge

conservation whereas the BASC can freely exchange

pairs of electrons with the parent SC. Viewed dif-

ferently, the BAM-plus-SC system conserves charge,

obeys Bloch’s theorem and indeed has a vanishing

current density in the thermodynamic limit. How-

ever, the BASC alone can host a non-zero current

density, which physically corresponds to a surface

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Proximity-inducedequilibriumsupercurrentandperfectsuperconductingdiodeeffectduetobandasymmetryPavanHosurDepartmentofPhysicsandTexasCenterforSuperconductivity,UniversityofHouston,Houston,Texas77004DanielPalaciosJanandDanDuncanNeurologicalResearchInstituteatTexasChildren’sHospital,Houston,TX,77030andG...

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Proximity-induced equilibrium supercurrent and perfect superconducting diode effect due to band asymmetry Pavan Hosur

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