Anomalous Josephson eect in planar noncentrosymmetric superconducting devices

2025-04-27 0 0 942.75KB 11 页 10玖币

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Anomalous Josephson eﬀect in planar noncentrosymmetric superconducting devices

Jaglul Hasan*,1Konstantin N. Nesterov*,1, 2 Songci Li,1Manuel Houzet,3Julia S. Meyer,3and Alex Levchenko1

1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

2Bleximo Corporation, Berkeley, California 94710, USA

3Universit´e Grenoble Alpes, CEA, Grenoble INP, IRIG, Pheliqs, 38000 Grenoble, France

(Dated: November 19, 2022)

In two-dimensional electron systems with broken inversion and time-reversal symmetries, a

Josephson junction reveals an anomalous response: the supercurrent is nonzero even at zero

phase diﬀerence between two superconductors. We consider details of this peculiar phenomenon

in the planar double-barrier conﬁgurations of hybrid circuits, where the noncentrosymmetric

normal region is described in terms of the paradigmatic Rashba model of spin-orbit coupling. We

analyze this anomalous Josephson eﬀect by means of both the Ginzburg-Landau formalism and the

microscopic Green’s functions approach in the clean limit. The magnitude of the critical current is

calculated for an arbitrary in-plane magnetic ﬁeld orientation, and anomalous phase shifts in the

Josephson current-phase relation are determined in terms of the parameters of the model in several

limiting cases.

Spin-Coherent Phenomena in Semiconductors: Special Issue in Honor of Emmanuel I. Rashba.

I. INTRODUCTION

In Josephson junctions (JJ) of conventional s-wave su-

perconductors, the supercurrent-phase relation j(φ) is

expected to obey rather general properties that depend

neither on the junction’s geometry nor on the scattering

processes taking place in the junction region, in other

words, they apply to tunnel, ballistic, and diﬀusive junc-

tions [1]. (i) The ﬁrst basic property follows from the

2πperiodicity of the superconducting order parameter,

which implies that j(φ) = j(φ+ 2π). (ii) The second

property reﬂects the fact that changing the direction of

the phase gradient applied across the junction reverses

the direction of the superﬂow, j(φ) = −j(−φ), and there-

fore the supercurrent-phase relation is an odd function.

(iii) The current must vanish at all integer phases modulo

2π, namely, j(2πn) = 0 for n∈Z. This condition states

an obvious thermodynamic requirement that a ﬁnite su-

percurrent is induced only by a nonzero phase gradient,

so it vanishes for φ= 0, and then by virtue of period-

icity must vanish at other phases multiple of 2π. (iv)

The combination of the ﬁrst two properties dictates that

j(πn) = 0 for n∈Z; therefore it is suﬃcient to con-

sider j(φ) only in the interval 0 < φ < π. Addition-

ally, it should be noted that in general, symmetries of

the full Hamiltonian describing a Josephson junction, or

their absence, can be related to the particular features in

the pattern of the supercurrent-phase relation.

The anomalous Josephson eﬀect (AJE), where the

above-formulated properties of the supercurrent-phase

relation are altered, can be realized in superconduc-

tors with broken time-reversal symmetry, leading to

spontaneous currents. There are two kinds of systems

*J.H. and K.N.N. contributed equally to this work.

where these eﬀects have been discussed: (i) JJs between

magnetic superconductors with unconventional pair-

ing symmetry [2–5]; (ii) superconductor-ferromagnet-

superconductor (SFS) junctions, and their more complex

hybrids with additional noncollinear ferromagnetic layers

and insulating barriers [6–11]. In particular, in the orig-

inal work of Geshkenbein and Larkin [2] devoted to JJs

based on heavy-fermion superconductors, the following

current-phase relation was predicted:

j(φ) = j1sin φ+j2cos φ=jcsin(φ+φ0),(1)

where jc=pj2

1+j2

2is the critical current, and φ0=

arctan(j2/j1) is the anomalous phase shift, whose mi-

croscopic form depends on the system under consider-

ation and speciﬁc model assumptions. In general, the

current-phase relation is not simply sinusoidal. Indeed,

the contribution of higher-order harmonics may be non-

negligible, which is often the case at temperatures much

lower than the critical. Therefore the generalized form of

Eq. (1) can be presented as the Fourier series,

j(φ) = X

n≥1

[j1nsin(nφ) + j2ncos(nφ)] ,(2)

and contributions with j2nare typically present as long

as time-reversal symmetry is broken.

A diﬀerent kind of AJE was proposed later in Refs.

[12, 13]; see also important preceding works [14, 15]. The

key insight of those works is that current-phase relation

of the type Eq. (2) can be realized even in junctions

of conventional superconductors when the normal layer

between them is a noncentrosymmetric metal, i.e., with

broken inversion symmetry. As a guiding example, calcu-

lations were presented for a weak link with Rashba-type

spin-orbit coupling [16], and Eq. (1) was derived mi-

croscopically in the quasi-one-dimensional geometry. To

arXiv:2210.01037v2 [cond-mat.supr-con] 19 Nov 2022

separate this anomalous Josephson eﬀect from that in un-

conventional JJs, the term “φ0-junction” was introduced

[12].

In recent years we received compelling experimental

veriﬁcation of these anomalous Josephson phenomena in

various heterojunctions [17–25]. These devices represent

a diverse class of systems that diﬀer from each other by

material components, dimensionality, quality of contacts,

and purity of interlayers between superconducting banks,

thus reﬂecting the prevalence and robustness of the afore-

mentioned eﬀects. Theoretical studies address a broad

spectrum of questions related but not limited to (i) types

of spin-orbit interaction, including spin-active interfaces;

(ii) eﬀects of impurities; and (iii) electronic band struc-

ture, in particular topological properties. There are a

number of notable theoretical contributions to this topic,

and we can highlight studies of the AJE in quantum dots

[26, 27], semiconducting nanowires [28–34], and topolog-

ical [35–42] and nontopological systems [43–57] that in-

volve a combination of unconventional superconductors,

topological surface or edge states, and ferromagnets.

The continuous improvement in the quality of mate-

rials, where the electron mean free path is comparable

to or even may exceed the dimensions of the junction,

call for the investigation of AJE in the clean limit, which

thus far has received very limited theoretical attention.

This task is accomplished in the present work, and the

rest of the paper is organized as follows. In Sec. II we

apply Ginzburg-Landau (GL) phenomenology to address

the anomalous Josephson eﬀect in the two-dimensional

electron gas (2DEG) with Rashba-type spin-orbit cou-

pling. Even though the GL formalism has its limitations,

it gives us an opportunity to fully analytically investi-

gate the ﬁeld dependence of the critical current and the

phase shift in the two-dimensional geometry. The prin-

cipal results of this work are presented in Sec. III, where

we develop a microscopic theory of the AJE based on

the Gor’kov equations for two complementary junction

geometries. This analysis expands previous considera-

tions of the AJE that exploited semiclassical approxima-

tions for ballistic (Eilenberger limit) and diﬀusive (Usadel

limit) systems. In Sec. IV we provide summary of our

ﬁndings in comparison to earlier related works.

II. GINZBURG-LANDAU FORMALISM

To elucidate the unusual properties of the AJE, we

start with the simple Ginzburg-Landau (GL) model be-

fore delving into the microscopic calculation. The geom-

etry that we consider is depicted schematically in Fig. 1

in which a normal region of two-dimensional electron gas

(2DEG) of length Land width dis ﬂanked by two con-

ventional s-wave superconducting banks. Let us consider

the situation where the time reversal symmetry (TRS) in

the system is broken by an in-plane magnetic ﬁeld hand

the space inversion symmetry in the normal region is bro-

ken by the presence of a Rashba-type spin-orbit coupling

FIG. 1. Geometry of a planar SINIS Josephson junction with

Rashba-type 2DEG as the normal region. The in-plane mag-

netic ﬁeld his directed arbitrarily in the x−yplane and the

vector nis along the zaxis. The total length of the normal

region is Land its total width is d. The complex supercon-

ducting order parameter in the leads is |ψ0|e±iφ/2so that φ

is the total phase diﬀerence across the junction.

(SOC) term [16] given by α[σ×p]·n. Here σis the Pauli

spin matrix-vector, pis the particle momentum, nis the

unit vector along the direction of the asymmetric poten-

tial gradient, and the parameter αdenotes the strength

of the spin-orbit interaction, which has units of velocity.

In the presence of SOC, the GL free energy Ω was

derived by Edelstein [58] (see also related works Refs.

[59–62]):

Ω (ψ, ψ∗) = Zdr"a|ψ|2+b

2|ψ|4+1

4m|∂ψ|2

−[n×h]· {ψ(∂ψ)∗+ψ∗(∂ψ)}+h2

8π#,

(3)

where ψ(r) is the spatially inhomogeneous superconduct-

ing order parameter, h=∇×Awhere Ais the vector

potential, and ∂=−i∇−2eAis the gauge invariant

derivative. Here and in what follows, we work in the units

~=kB=c= 1. This form of the GL functional applies

to both clean and disordered superconductors. The dif-

ference is in the dependence of the expansion coeﬃcients

a, b, , and also the gradient term, on the strength of

spin-orbit α, critical temperature Tcof a superconductor,

and elastic scattering time τinduced by disorder poten-

tial. The conventional part of the GL functional, namely,

the ﬁrst three terms in Eq. (3), weakly depends on the

SOC. In contrast, the coeﬃcient =α

vFpFfd(αpF

Tc, Tcτ),

with vFand pFbeing the Fermi velocity and the Fermi

momentum, respectively, depends sensitively on α. The

asymptotic form of the function fdis established for two-

and three-dimensional superconductors in various limit-

ing cases, see Refs. [58, 61, 62] for details. Both pa-

rameters αpF/Tcand Tcτcan be of the order of unity in

materials.

The free energy functional in Eq. (3) must be mini-

mized with respect to the order parameter ψ∗and the

vector potential Ato get the equilibrium GL equations.

Therefore, varying (3) with respect to ψ∗and Aand set-

ting that variation equal to zero for arbitrary variations

δψ∗and δA, the two GL equations are obtained in the

form

4m∂2ψ(r)−1

2mQ·∂ψ(r) + aψ(r)

+b|ψ(r)|2ψ(r)=0,

(4)

j=e

2m{ψ(∂ψ)∗+ψ∗(∂ψ)} − 4e|ψ|2(n×h)

−curl[n× {ψ(∂ψ)∗+ψ∗(∂ψ)}],

(5)

with the boundary condition,

[(∂−Q)ψ]·nb= 0.(6)

The unit vector nbis the normal vector at the system

boundary. The wave vector Q= 4m [n×h] represents

an emergent scale for this GL theory with Rashba SOC

and Zeeman ﬁeld. Its presence induces a spatially modu-

lated helical superconducting phase given by ψ∝eiQ·r.

We recall that the boundary conditions [Eq. (6)] on these

equations are obtained from the condition that the sur-

face integrals in the variation δΩ are zero. As a result

of this condition, the normal component of the supercur-

rent density (5) at the boundary of the superconductor

with vacuum is j·nb= 0.

This framework was used in the original work [12] to

derive the anomalous Josephson current in a quasi-one-

dimensional geometry with rigid boundary conditions.

Below we extend these results to a full two-dimensional

geometry with an arbitrarily oriented ﬁeld in the plane

of the 2DEG and also with more general boundary con-

ditions:

[(∂−Q)ψ]·nb=1

ilt

ψ. (7)

Here ltis the extrapolation length to the point outside the

boundary at which the order parameter ψwould vanish

if it maintained the slope it had at the surface [63]. The

value of ltdepends on the nature of the material to which

the interface is made, approaching zero for a magnetic

material or in the case of a high density of defects in the

interface (Dirichlet boundary condition), and inﬁnity for

an insulator or vacuum (Neumann boundary condition),

with normal metals lying in between.

To calculate the Josephson current in the geometry of

Fig. 1, we can neglect the orbital eﬀect of the ﬁeld. We

can also neglect the nonlinear term proportional to bin

Eq. (4). The solution for ψ(x, y) can be written as the

series

ψ=eiQr X

nAnex/ξn+Bne−x/ξncos[qn(y+d/2)]

(8)

where

qn=πn/d, ξ−1

n=p4ma −Q2+q2

n,(9)

d/ξ=2.5

d/ξ=5

d/ξ=10

d/ξ=20

d/ξ=50

-1.0 -0.5 0.5 1.0

Qyξ

0.2

0.4

0.6

0.8

1.0

jcQy

jc(0)

FIG. 2. Normalized critical current density, jc(Qy)/jc(0), is

plotted as a function of the Zeeman ﬁeld, Qy∝hx, for several

diﬀerent values of the sample width d, which is measured in

the units of the superconducting coherence length ξand for

a ﬁxed ratio L/ξ = 5.

with Q=qQ2

x+Q2

y= 4mqh2

x+h2

y= 4mh. The

expansion coeﬃcients Anand Bnare easily calculated

by using appropriate boundary conditions from Eq. (7)

for the two SN interfaces,

∂ψ

∂x −iQxψx=±L

=±1

lt|ψ0|e∓iφ

2,(10)

where, in the superconducting banks to the left and right

sides of the normal region, ψ(x < −L

2, y) = |ψ0|e+iφ

2and

ψ(x > L

2, y) = |ψ0|e−iφ

2. In these solution we require

4ma > Q2, because the smallest value of |n|= 0. This

is equivalent to the condition Qξ < 1, which means that

the length scale characterized by the inverse of the wave

vector Qmust be greater than the superconducting co-

herence length ξ=q1

4ma . Equation (5) can now be used

to ﬁnd the current density:

jx(y) = e

2mi ψ∗∂ψ

∂x −ψ∂ψ∗

∂x −eQx

m|ψ|2.(11)

Since the current density is inhomogeneous, we are in-

terested in a current density averaged over the sample

width d:

j(φ) = 1

+d/2

−d/2

jx(y)dy =e

2mi X

n∈Z2

ξn

(B∗

nAn−A∗

nBn).

(12)

This can be simpliﬁed to get the form of the anomalous

Josephson eﬀect in the φ0junction

j(φ) = jc(Qy) sin(φ+φ0).(13)

In this model, the critical current density jcand the

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AnomalousJosephsoneectinplanarnoncentrosymmetricsuperconductingdevicesJaglulHasan*,1KonstantinN.Nesterov*,1,2SongciLi,1ManuelHouzet,3JuliaS.Meyer,3andAlexLevchenko11DepartmentofPhysics,UniversityofWisconsin-Madison,Madison,Wisconsin53706,USA2BleximoCorporation,Berkeley,California94710,USA3Universit...

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Anomalous Josephson eect in planar noncentrosymmetric superconducting devices

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