Eects of spinorbit coupling and in-plane Zeeman elds on the critical current in two-dimensional hole gas SNS junctions Jonas Lidal and Jeroen Danon

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Eﬀects of spin–orbit coupling and in-plane Zeeman ﬁelds on the critical current

in two-dimensional hole gas SNS junctions

Jonas Lidal and Jeroen Danon

Center for Quantum Spintronics, Department of Physics,

Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

(Dated: February 3, 2023)

Superconductor–semiconductor hybrid devices are currently attracting much attention, fueled by

the fact that strong spin–orbit interaction in combination with induced superconductivity can lead

to exotic physics with potential applications in fault-tolerant quantum computation. The detailed

nature of the spin dynamics in such systems is, however, often strongly dependent on device details

and hard to access in experiment. In this paper we theoretically investigate a superconductor–

normal–superconductor junction based on a two-dimensional hole gas with additional Rashba spin

orbit–coupling, and we focus on the dependence of the critical current on the direction and magnitude

of an applied in-plane magnetic ﬁeld. We present a simple model, which allows us to systematically

investigate diﬀerent parameter regimes and obtain both numerical results and analytical expressions

for all limiting cases. Our results could serve as a tool for extracting more information about the

detailed spin physics in a two-dimensional hole gas based on a measured pattern of critical currents.

I. INTRODUCTION

Hybrid devices made of superconductors and semicon-

ductors have gained much interest in recent years due to

their rich and complex behavior. Spin–orbit coupling in

combination with superconducting correlations induced

via the proximity eﬀect can give rise to exotic spin physics

inside the semiconductor, which could be exploited to en-

gineer topological superconductivity [1–7]. Since such

topological superconductors are expected to host low-

energy Majorana modes that obey non-Abelian anyonic

statistics, they could provide a platform for implement-

ing fault-tolerant quantum computation with topologi-

cally protected qubit operations [8–10].

Arguably the simplest hybrid device one can cre-

ate using superconducting and normal elements is the

superconductor–normal–superconductor (SNS) junction,

which ﬁnds applications in a wide range of directions,

including superconducting qubits [11–15] and electronic

and magnetic measuring devices [16–20]. In addition to

being an essential component of superconducting circuits,

an SNS junction can also be used for studying the un-

derlying properties of the constituent elements of the hy-

brid structure. For the case of a semiconducting normal

region, an SNS setup allows to probe details of the spin–

orbit interaction in the semiconductor and its interplay

with the Zeeman eﬀect [21–23], as well as to study phase

transitions into and out of topological phases [24–26].

One quantity that encodes several details of the under-

lying physics of the system is the critical current through

the SNS junction, i.e., the maximal supercurrent the

junction can support. By applying a magnetic ﬁeld per-

pendicular to a two-dimensional junction, information

about the current density distribution can be extracted

from the measured critical current [27]. For a uniform

current distribution, the critical current as a function

of the out-of-plane magnetic ﬁeld emerges as a so-called

Fraunhofer pattern, which reﬂects the ﬂux enclosed by

the junction. A deviation from a Fraunhofer pattern

is a sign of a non-uniform current distribution and the

pattern of critical current can be directly related to the

actual current distribution proﬁle in the junction [28–34].

The ﬁeld-dependent behavior of an SNS junction is

heavily inﬂuenced by the properties of the normal part,

and junctions based on a wide range of materials have

been explored in the past [35–38]. In this paper we focus

on SNS junctions comprised of a two-dimensional hole

gas (2DHG) contacted by two conventional superconduc-

tors. Our choice is motivated by the recent surge in in-

terest for lower-dimensional quantum devices hosted in

2DHGs [39–46], which was sparked by their interesting

properties including strong inherent and tunable spin–

orbit interaction [47–51] and highly anisotropic and tun-

able g-tensors [52–55], all caused by the underlying p-

type orbital structure of the valence band states [56].

Additionally, germanium-based hole gases have recently

shown great promise for straightforward integration with

superconducting elements [57–60].

The eﬀective spin–orbit interaction and Zeeman cou-

pling that together can give rise to its useful properties

depend strongly on many details of the 2DHG, includ-

ing its exact out-of-plane conﬁning potential, the carrier

density, strain, and the local electrostatic landscape. For

this reason it is not always straightforward to access the

relevant underlying spin–orbit and g-tensor parameters

in experiment for a given system. Here, we theoretically

study the dependence of the critical current through a

2DHG-based SNS junction on the direction and magni-

tude of an applied in-plane magnetic ﬁeld. We show how

to derive an elegant expression for the ﬁeld-dependent

critical current in a semi-classical limit (where the sys-

tem is large compared to its Fermi wave length), which al-

lows for straightforward numerical evaluation of the cur-

rent. Assuming that we can describe the dynamics of the

holes in the normal region with a simple 4 ×4 Luttinger

Hamiltonian and that the carrier density is low enough

that only the lowest (heavy-hole) subband is occupied, we

identify several diﬀerent parameter regimes where diﬀer-

arXiv:2210.13266v2 [cond-mat.mes-hall] 2 Feb 2023

FIG. 1. Schematic of the SNS junction: two identical conven-

tional superconductors Sland Sr, connected by a 2D hole

gas. The junction has a length of Land a width of W,

as indicated. An example of a diagram contributing to the

Cooper-pair propagator is drawn in the normal region, where

the electron(hole) propagator is depicted with a solid(dashed)

line.

ent spin-mixing mechanisms could be dominating and we

calculate the ﬁeld-dependent critical current in all these

regimes. We are able to connect each mechanism to clear

qualitative features in the pattern of critical current that

emerges and we present analytical expressions for the

current in most limiting cases. Our results could thus

help distinguishing the dominating spin-mixing process

at play in an experiment, and as such give insight in

the strength and nature of the underlying spin–orbit and

Zeeman couplings in the system.

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

we will introduce the setup we consider and the model

we use to describe it. We outline our method of calcu-

lating the critical current through the junction and ex-

plain how we tailor it to the situation where all trans-

port in the normal region is carried by the heavy holes.

In Sec. III we present both our numerical and analytical

results, systematically going through the diﬀerent param-

eter regimes that could be reached. Finally, in Sec. IV

we present a short conclusion.

II. MODEL

Fig. 1shows a cartoon of the system we consider:

A 2DHG is contacted from two sides by two identical

conventional superconductors to create an SNS junction,

where we assume the coupling between the superconduc-

tors and the normal region to be weak. We deﬁne the

length Land width Wof the junction as indicated in

the Figure and choose the coordinate system such that

the average ﬂow of supercurrent is in the x-direction and

the out-of-plane direction is denoted by z. We assume

the clean junction limit, as the width of experimentally

viable devices is typically of the order 100 nm to 1 µm

[59,61,62], while the mean free path of, e.g., Ge 2DHGs

has been measured to be up to ∼6µm [58,59].

We will ﬁrst introduce the method we chose for cal-

culating the supercurrent through the junction. In the

ground state, the current is given by

I(φ) = 2e

∂F

∂φ ,(1)

where Fis the free energy of the junction and φis the

diﬀerence in phase between the two superconductors.

We describe the coupling between the hole gas and the

superconducting leads with a tunneling Hamiltonian

Ht=X

σZdy tlˆ

ψ†

σ(0, y)ˆ

Ψσ,L(0, y)

+trˆ

ψ†

σ(W, y)ˆ

Ψσ,R(W, y) + H.c.,

(2)

where ˆ

ψ†

σ(r) is the creation operator for an electron with

spin σat position rin the normal region, and ˆ

Ψ†

σ,L(R)(r)

for an electron with spin σat position rin the left(right)

superconductor. The lines x= 0, W deﬁne the interfaces

between the superconductors and the normal region.

We assume the coupling amplitudes tl,r to be small

enough to justify a perturbative treatment of Ht. Weak

coupling can result from, e.g., interfacial disorder, but

could also be a consequence of the diﬀerence in underly-

ing orbital structure of the electronic wave functions in

the superconductors’ conduction band and the semicon-

ductor’s valence band. The leading-order correction to F

that depends on φis second order in the self energy due

to the proximity of the superconductors, or fourth order

in the coupling Hamiltonian Ht[63],

F(4) =−1

4!βZβ

dτ1...4hˆ

TτHt(τ1)Ht(τ2)Ht(τ3)Ht(τ4)i,

(3)

where β= 1/T is the inverse temperature, ˆ

Tτis the

(imaginary) time-ordering operator, and ~=kB= 1.

In the evaluation of (3) we focus on the fully connected

diagrams only, since those are the ones that can probe

the phase diﬀerence between the two superconductors.

Anticipating that we will make a semi-classical approx-

imation later, assuming that the dimensions of the junc-

tion are much larger than the Fermi wave length λF, we

will take the Andreev reﬂection at the NS interface to

be local and energy-independent. After applying Wick’s

theorem to the correlator in (3) this allows us to simplify

the correction to

F(4) =−λlλrZZ dy dy0

×Re nei[ϕl(y)−ϕr(y0)]C(W, y0; 0, y)o,

(4)

where λl,r =πt2

l,rνeﬀ parameterize the strength of

the coupling to the superconducting leads, with νeﬀ

the local eﬀective one-dimensional tunneling density of

states of the superconductors (giving the λ’s dimensions

energy×meters). The phase diﬀerence

ϕl(y)−ϕr(y0) = φ+π(y+y0)BzW

Φ0

,(5)

with Φ0=h/2ethe ﬂux quantum, depends on the two

y-coordinates in such a way that it captures the coupling

to an out-of-plane magnetic ﬁeld [64], due to the ﬂux

Φ = BzW L penetrating the junction. We assume that

the magnetic ﬁeld Bzis small enough that it does not

signiﬁcantly aﬀect the trajectories of the charges [65].

We used the function

C(r0;r) = T

Tr¯

G(r0,r;iωk)σy¯

G(r0,r;−iωk)Tσy,

(6)

were the ¯

Gare 2×2 matrices in spin space,

G(r0,r;iωk) = G↑↑(r0,r;iωk)G↑↓(r0,r;iωk)

G↓↑(r0,r;iωk)G↓↓(r0,r;iωk),(7)

with Gσ0σ(r0,r;iωk) = −Rβ

0dτ eiωkτhˆ

Tτˆ

ψσ0(r0, τ)ˆ

ψ†

σ(r,0)i

the thermal Green function at Matsubara frequency ωk=

(2k+1)πT for (spin-dependent) electronic propagation in

the normal region. The correlation function C(W, y0; 0, y)

as used in (4) can thus be interpreted as the probability

amplitude for a Cooper pair to cross the junction, from

the point (0, y) to the point (W, y0), as illustrated by the

simple diagram shown in Fig. 1. In writing Eq. (4) we

further assumed the pairing in the superconductors to

be conventional s-type, described by pairing terms like

H(S)

pair =−Pkn∆0ˆ

Ψ†

k↑ˆ

Ψ†

−k↓+ ∆∗

0ˆ

Ψ−k↓ˆ

Ψk↑o. Within all

approximations made, other details of the dynamics in-

side the superconductors will only aﬀect the magnitude

of the two coupling parameters λl,r.

We assume that the carriers in the normal region can

be described by a 4×4 Luttinger Hamiltonian [56],

H0=1

2m0





P+Q0 0 M

0P+Q M∗0

0M P −Q0

M∗0 0 P−Q





,(8)

where

P=γ1k2+hk2

zi,(9a)

Q=γ2k2−2hk2

zi,(9b)

M=−1

2√3(γ2+γ3)k2

−+ (γ2−γ3)k2

+,(9c)

using k±=kx±ikyand k2=k2

x+k2

y, and with m0being

the bare electron mass and γ1,2,3the three dimension-

less material-speciﬁc Luttinger parameters. This Hamil-

tonian is written in the basis of the angular-momentum

states {|+3/2i,|−3/2i,|+1/2i,|−1/2i} with total angu-

lar momentum 3/2 and it includes an extra minus sign,

i.e., it describes the dynamics from a hole perspective.

The z-coordinate (along which the holes are strongly con-

ﬁned) has already been integrated out, hk2

zi ∼ 1/d2, with

dthe transverse conﬁnement length, and we neglect the

eﬀects of strain for simplicity.

We now assume that transverse conﬁnement is strong

enough to make the splitting δHL = 2γ2hk2

zi/m0the

largest energy scale involved, on the order of ∼100meV

in planar Ge [49,57,66], which allows us to focus on

the so-called heavy-hole (HH) subspace {|+3/2i,|−3/2i}

and treat the coupling to the light-hole (LH) states

{|+1/2i,|−1/2i} perturbatively. We will further assume

that the Andreev reﬂection at the interfaces with the su-

perconductors pairs hole states with opposite orbital and

spin angular momentum, such as |±3/2i. This allows us

to treat the low-energy HH subspace {|+3/2i,|−3/2i}

as an eﬀective spin-1/2 system that can host a supercur-

rent that can be described with the formalism presented

above [67].

Furthermore, we want to include the Zeeman eﬀect due

to an in-plane magnetic ﬁeld Bkas well as Rashba-type

spin–orbit coupling. We describe the in-plane Zeeman

eﬀect with the Hamiltonian

HZ=−2κ(B+J−+B−J+),(10)

where J±=Jx±iJyare the spin-3/2 raising and lowering

operators, B±=Bx±iBy, the hole g-factor is κ, and we

set µB= 1. The spin–orbit coupling, which can be due to

asymmetries in the conﬁning potential or to an externally

applied out-of-plane electric ﬁeld, is described with

HR=iαR(k+J−−k−J+),(11)

where αRcharacterizes the strength of the coupling.

We add these two ingredients to the projected two-

dimensional Luttinger Hamiltonian introduced above

and we make the so-called spherical approximation,

amounting to the assumption |γ2−γ3|  γ2+γ3, which

allows to drop the last term in (9c). For many commonly

used semiconductors, such as Ge, GaAs, InSb, and InAs

(but not for Si), this is a valid approximation [56]. Oth-

erwise we impose no constraints on the Luttinger param-

eters. Then the total Hamiltonian for the hole gas is

FIG. 2. A sketch of the spectrum of the Hamiltonian (12),

where we assumed the perturbations HZand HRto be small

enough to be neglected. We see the heavy-hole and light-hole

bands being split by the HH–LH splitting δHL and anticross

where they are mixed by the oﬀ-diagonal terms k2

±/2mx. The

blue shaded region around the Fermi energy, EF, indicates

the energy window within which all relevant dynamics are

assumed to happen, its width being of the order of |∆0|. Su-

perconducting pairing in the 2DHG is induced between holes

in the HH band with opposite spin and momentum, as illus-

trated in the Figure.

Htot =H0+HZ+HR

=





k2/2mH0−√3 (2κB−+iαRk−)−k2

−/2mx

0k2/2mH−k2

+/2mx√3 (−2κB++iαRk+)

√3 (−2κB++iαRk+)−k2

−/2mxδHL +k2/2mL−4κB−−4iαRk−

−k2

+/2mx−√3 (2κB−+iαRk−)−4κB++ 4iαRk+δHL +k2/2mL







,(12)

where we introduced the eﬀective HH and LH masses mH=m0/(γ1+γ2) and mL=m0/(γ1−γ2). We further

used mx= 2m0/√3(γ2+γ3), which governs the strength of the momentum-dependent HH-LH mixing. A scetch of

Hamiltonian (12) can bee seen in Fig. 2.

Our assumption that δHL is the largest energy scale involved allows us to treat the HH–LH coupling perturbatively.

We ﬁrst diagonalize the LH subspace in (12), after which we perform a Schrieﬀer-Wolﬀ transformation to decouple

the HH and LH subspaces. To second order in 1/δHL we ﬁnd the eﬀective HH Hamiltonian

HHH =k2/2mH−δ−1

HL(k2

−/mx)β1−4δ−2

HLβ2

1β2

−δ−1

HL(k2

+/mx)β∗

1−4δ−2

HL(β∗

1)2β∗

2k2/2mH,(13)

where we ignored the shift of the diagonal elements, and

we used β1=√3(2κB−+iαRk−) and β2=κB−+

iαRk−. We see that, depending on the magnitude of

δHL, the typical in-plane (Fermi) momentum kFof the

current-carrying holes, the strength of the spin–orbit cou-

pling αR, and the magnitude of the applied in-plane

magnetic ﬁeld, diﬀerent terms can dominate the eﬀec-

tive coupling of the two HH states. We used that in

the perturbative limit we consider here one always has

(δ−1

HLk2

±/mx)2δ−1

HLk2

±/mx, and we thus ignore the con-

tribution −4δ−2

HL(k2

−/mx)2(β∗

2σ++β2σ−) to Eq. (13).

We can now consider diﬀerent cases. Firstly, for a very

thin 2DHG we can assume that the term ∝δ−1

HL will

dominate, which leaves two qualitatively diﬀerent cou-

pling terms in HHH ,

H(1)

0,3=−i√3αR

mxδHL k3

−σ+−k3

+σ−,(14)

H(1)

1,2=−2√3κ

mxδHL B−k2

−σ++B+k2

+σ−,(15)

where the subscripts of Hrefer to the powers of Band k

appearing in the term, respectively, and the superscript

indicates the power of δ−1

HL. If the 2DHG is less thin,

then the term ∝δ−2

HL in (13) could also contribute, which

allows for four additional coupling terms,

H(2)

0,3= 12 iα3

δ2

HL k3

−σ+−k3

+σ−,(16)

H(2)

1,2= 60α2

Rκ

δ2

HL B−k2

−σ++B+k2

+σ−,(17)

H(2)

2,1=−96iαRκ2

δ2

HL B2

−k−σ+−B2

+k+σ−,(18)

H(2)

3,0=−48 κ3

δ2

HL B3

−σ++B3

+σ−.(19)

We now make the assumption that all relevant dynam-

ics happen on an energy scale very close to the Fermi level

EF. This allows us to linearize the kinetic energy in HHH

in kand to assume that the magnitude of the in-plane

momentum k≈kFin the oﬀ-diagonal terms. This leaves

us with a general 2×2 Hamiltonian eﬀectively describing

the HH subsystem (up to a constant)

HHH =vF(k−kF) + β(θ)·σ,(20)

where vF=kF/mHis the Fermi velocity and the ﬁeld β

includes the oﬀ-diagonal terms of HHH , depending only

on the angle θ, the in-plane direction of k. The vector

σ={σx, σy, σz}consists of the three Pauli matrices.

Following the approach of Ref. [64], we recognize that

HHH in (20) can be diagonalized in spin space and we de-

note the two k-dependent eigenspinors with |λki, where

λ=±. This allows to rewrite Eq. (20) as

HHH =X

λk=±k

kλPλk,(21)

in terms of the energies kλ=vF(k−kF) + λ|β(θ)|and

the projectors Pλk=|λkihλk|=1

2[1 + λˆ

β(θ)·σ], where

the dimensionless vector ˆ

β(θ) = β(θ)/|β(θ)|points along

the direction of the ﬁeld β(θ).

Assuming for simplicity translational invariance inside

the 2DHG, the correlation function C(r0;r) is only a func-

tion of the diﬀerence in coordinates and reduces at zero

temperature to (see the Supplementary Material for a

more detailed derivation)

C(r) = ZZ ∞

d d0

2(+0)Tr¯g(r,−)σy¯g(r,−0)Tσy

+ ¯g(r, )σy¯g(r, 0)Tσy,(22)

using the propagator

¯g(r, ) = 1

(2π)2X

λk=±kZdkeik·rδ(−kλ)Pλk.(23)

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摘要：

Eectsofspin{orbitcouplingandin-planeZeemaneldsonthecriticalcurrentintwo-dimensionalholegasSNSjunctionsJonasLidalandJeroenDanonCenterforQuantumSpintronics,DepartmentofPhysics,NorwegianUniversityofScienceandTechnology,NO-7491Trondheim,Norway(Dated:February3,2023)Superconductor{semiconductorhybriddev...

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Eects of spinorbit coupling and in-plane Zeeman elds on the critical current in two-dimensional hole gas SNS junctions Jonas Lidal and Jeroen Danon

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