Phase-dependent charge and heat current in thermally biased short Josephson junctions formed at helical edge states Paramita DuttaID1

2025-05-02 0 0 2.4MB 12 页 10玖币

侵权投诉

Phase-dependent charge and heat current in thermally biased short Josephson

junctions formed at helical edge states

Paramita Dutta ID 1

1Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India∗

(Dated: August 14, 2023)

We explore the phase-dependent charge and heat current in the short Josephson junctions with

two normal metal regions attached at opposite ends, formed at helical edge states of two-dimensional

topological insulators (TIs). For all ﬁnite phases, an asymmetry appears around the zero energy

in the transmission spectra except for ϕ=nϕ0, where nis a half-integer and ϕ0(= 2π) is the

ﬂux quantum. The phase-induced asymmetry plays a key role in inducing charge and heat current

through the thermally biased junction. However, the current amplitudes are sensitive to the size of

the junction. We show that in the short Josephson junction when subject to a temperature gradient,

the charge current shows an odd-symmetry in phase. It indicates that the phase-tunable asymmetry

around the zero-energy is not suﬃcient to induce a dissipative thermoelectric current in the junction.

This is in contrast to the behavior of long Josephson junction as shown in the literature. The phase-

tunable heat currents are obtained with amplitudes set by the phase diﬀerence, base temperature,

and system size.

I. INTRODUCTION

Study of thermal gradient-induced current in super-

conductors and superconducting junctions has been re-

juvenated in recent years, breaking the concepts of poor

thermal current in superconductors [1–8]. The thermal

current in ordinary superconductors were expected to be

low or even vanishing, primarily because of the super-

conducting gap in the density of states. The symmetry

in the energy spectrum is responsible for low charge cur-

rent in the linear regime [9]. On top of that, thermal

bias-induced charge current interferes with the super-

ﬂow, and this causes the separation of the two currents

tricky. For these reasons, conventional Bardeen-Cooper-

Schrieﬀer (BCS) superconductors were not considered as

active thermoelectric materials for several years [10]. Un-

conventional superconductors were also studied in few

works to enhance the thermal current i.e., the non-

dissiptaive charge current [11,12].

Recently, some eﬀorts have been put to enhance ther-

mal charge current, within the linear regime, in su-

perconducting junctions instead of bare superconduc-

tors by breaking the spin-symmetry using ferromagnetic

elements [1], forming ferromagnet/superconductor [2–8,

13] or anti-ferromagnet/superconductor hybrid struc-

tures [14]. Research in this direction has been boosted af-

ter the experimental veriﬁcation in 2016 [2], where an ex-

cellent agreement with the theoretical prediction [1] was

conﬁrmed. Very recently. it has been predicted that

a nonlinear thermal current can ﬂow in the presence of

spontaneously broken particle-hole symmetry [15].

Search for ways to control the thermal currents in

superconductor junctions is continued. In recent work,

Kalenkov et al. have shown that depending on the topol-

ogy and the temperature gradient it is possible to gen-

∗paramita@prl.res.in

erate a large phase-coherent charge current in a Joseph-

son junction (JJ) [16]. In JJ, one can avoid using exter-

nal elements like non-magnetic or magnetic impurity [17],

or any engineering like creating vacancy [18], which has

been utilized to enhance thermal current in other junc-

tions. In fact, non-trivial thermal bias-induced voltage

can be achieved just by tuning the superconducting phase

of JJ [15,16,19–21]. In another work, phase-tunable

thermal-bias induced charge current is shown in a bal-

listic junction [22]. Hence, the phase-tunability makes JJ

more powerful compared to other superconducting junc-

tions.

To generate the phase-tunable thermal current in JJ,

topological materials have also been considered in very

few works [20–25] as the combination of global topology

and local superconducting order has been established to

host exotic transport properties in the literature [26–30].

Particularly, junctions involving two-dimensional (2D)

topological insulators (TI) [31] have drawn great atten-

tion because of its potential to inﬂuence scattering pro-

cesses [32–34] and most importantly to host Majorana

fermions [31,32,35–38]. The one-dimensional (1D) he-

lical edge states make 2D TIs [39] more eﬀective by pre-

venting all the backscatterings but admitting only two

processes: (i) Andreev reﬂections and (ii) electron trans-

missions through the junction [28,40]. Also, there are

recent predictions for the detection of topological bound

states via thermal current in some junctions including

JJ [22,41].

Notably, the charge current consists of dissipative and

non-dissipative parts. The dissipative charge current

describes the conventional thermoelectricity while, the

traditional non-dissipative Josephson current is also un-

avoidable in the same junction. The previous studies

involve JJ where the superconductors act as leads with

various widths of the middle normal regions. The eﬀect

of ﬁnite sized superconductors is yet to explore. Moti-

vated by this, we study the charge and heat current in

arXiv:2210.00936v2 [cond-mat.supr-con] 11 Aug 2023

thermally biased short JJ (sJJ) when it is formed at the

1D helical edges of 2D TI in proximity to ordinary su-

perconductor. Within the short JJ, an extremely short

normal region is sandwiched between two ﬁnite size su-

perconductors. The phase-tunable topological Andreev

bound states (ABS) formed in such normal metal/short

Josephson junction/normal metal (N-sJJ-N) junction at

the edge of 2D TI help generate charge current under

voltage bias condition as seen in Ref. [40] which further

motivates to search for the phase-tunable thermal cur-

rent in the same junction. We show that the appearance

of the asymmetry around the zero energy in the trans-

mission spectra plays a key role here. The charge and

heat currents generated by the thermal gradient are tun-

able by the phase of the junction with their amplitudes

being sensitive to the lengths of the ﬁnite-sized supercon-

ductors. Remarkably, the charge current generated in the

junction is entirely non-dissipative. The phase-tunability

is not suﬃcient to produce a dissipative charge current

when the junction is short, which is in contrast to the pre-

vious results in long JJ. We demonstrate that the charge

and heat current can be optimized when the supercon-

ductors’ lengths are of the order of coherence length. Our

work thus predicts topological Josephson junction, where

the heat and dissipative charge current is smoothly con-

trollable by the phase of the junction, can behave in a

way completely diﬀerent from the long JJ and thus help

in choosing proper system size for thermal current.

II. MODEL AND HAMILTONIAN

x= 0 x=LSx= 2LS

LSLS

Δϕ

T+ΔT/2 T−ΔT/2

FIG. 1. N-sJJ-N junction with two ends maintained at two

diﬀerent temperatures. The whole junction is placed at the

1D helical edge states shown by the red and blue lines with

counterpropagating channels for up and down spin marked by

blue and red arrow, respectively.

We consider a sJJ where two ﬁnite size superconduc-

tors, each having length LS, are coupled via a tiny insu-

lator region. We take this insulator region as tiny just to

simplify the calculation. A ﬁnite width of the insulator

region will not aﬀect our results qualitatively. The junc-

tion is formed at the edge of a 2D TI and attached to two

normal metal regions on opposite sides to form N-sJJ-N

set-up. The superconductivity is proximity induced by

using a traditional BCS superconductor as presented in

Fig. 1. The lengths of the two superconductors are set

exactly equal to each other (denoted by LS) for simplic-

ity. A small diﬀerence between them will not aﬀect our

results qualitatively. The phase diﬀerence between the

two superconductors of the sJJ can be tuned by external

magnetic ﬂux ϕ. We describe each part of the N-sJJ-N

junction by Bogoliubov-de Gennes (BdG) Hamiltonian in

the basis Ψ(x) = ψ↑(x), ψ↓(x), ψ†

↓(x),−ψ†

↑(x)as [28],

HBdG =H˜

∆

∆†−H†,(1)

where the normal part Hamiltonian is given by

H=−ivF∂xσz−µσ0.(2)

The ﬁrst term of Eq. (2) is the kinetic energy term fol-

lowing the linear dispersion relation of the 1D metal-

lic edge states of 2D TI. The second term includes the

chemical potential µ. The Pauli matrices σi, act in spin

space and ψ†

σ(x) (ψσ(x)) is the creation (annihilation)

operator for an electron with spin σ∈ {↑,↓} at position

x. The oﬀ-diagonal matrices of Eq. (1) are responsible

for the proximity-induced superconductivity described by

the pair potential as: ˜

∆(x) = ∆(x)iσy. We set it as:

∆(x)=∆lfor the left (0 < x < LS) and ∆(x)=∆reiϕ for

the right superconductor (LS<x<2LS) to have a ﬁnite

phase diﬀerence in our sJJ, otherwise ∆(x)=0 in all nor-

mal regions. We set the Fermi velocity vF=1 and ∆0=1

so that for the symmetric junction where ∆L= ∆R= ∆,

the superconducting coherence length is ξ=ℏvF/∆. We

show all the results for µN= 0 (for normal regions) and

µS= 2 (for superconducting regions) but our results are

insensitive to the chemical potential qualitatively.

The temperature dependence of the superconduct-

ing gap is taken following the relation ∆(T) =

∆0Tanh(1.74pTc/T −1) where Tis the system temper-

ature. For the symmetric junction, we take ∆L= ∆R

and show all the results for T/Tc= 0.3. On the other

hand, for the asymmetric junction, we consider ∆L̸=∆R.

To understand the eﬀect of the gap asymmetry on the

transport properties clearly, we maximize the diﬀerence

between two gaps (∆L−∆R). To model the asymmetry,

we take ∆L= ∆(0.7) i.e., reduced superconducting gap

corresponding to T/Tc= 0.7 and ∆R=∆0.

III. THEORETICAL FORMALISM

We consider a temperature gradient across the junc-

tion without any bias voltage. The temperatures of the

two leads are maintained at T+ ∆T /2 and T−∆T /2 (as

shown in Fig. 1) to set the temperature diﬀerence across

the junction as ∆T. Note that, Tis scaled by the super-

conducting transition temperature Tc. The applied tem-

perature gradient acts in two ways: (i) it tunes the gaps

in the density of states of the two superconductors of the

sJJ, and (ii) it also aﬀects the quasiparticles’ occupation

factors in the junction [42]. Consequently, there appear

two diﬀerent types of currents: charge current and heat

current. The variation in superconducting gaps aﬀects

the usual Josephson current, which is non-dissipative. It

can be expressed in terms of the variation of the gap as:

δIc=Pl(∂Ic/∂∆l)δ∆lconsidering the contribution by

each lead lconnected to the superconductor with gap

∆l[42]. On the other hand, the occupation factor af-

fects the charge current induced by the thermal gradient,

which is dissipative. The dissipative and non-dissipative

parts of the charge current can be separated by reversing

the sign of the superconducting phase ϕ. The dissipative

part is even in phase ϕi.e., I(ϕ) = I(−ϕ), whereas the

non-dissipative component is odd in ϕ:I(ϕ)= −I(ϕ) [42].

Charge current: To evaluate the charge current in-

duced by the temperature gradient ∆T, we employ the

Landauer transport theory. It can be written as the

diﬀerence between the currents ﬂowing in the oppo-

site directions (coming from opposite leads) as [5,42]

Ic=Ic

L−Ic

Rwhere

l=2e

hZ∞

dω ie

l(ω)−ih

l(ω)f(ω/Tl),(3)

where eis the electronic charge, his the Planck’s con-

stant, ωis the incoming electron energy, and fis the

Fermi distribution function. Here, lstands for L or R

to represent the left or right normal metal leads, respec-

tively, and ie(h)

ldenotes the contributions by the elec-

trons (holes) in l-th lead accordingly. Now, the heat cur-

rent should be zero for ∆T= 0 following the second law

of thermodynamics. Assuming this, the net current due

to the temperature gradient can be calculated in terms

of only one lead as

Ic=2e

h∆TZ∞

dω ie

L(ω)−ih

L(ω)∂f (ω/Tl)

∂T .(4)

Note that, a ﬁnite amount of usual non-dissipative

Josephson current is always present in the system even

at ∆T= 0 for ∆ϕ̸= 0. The conservation of charge

current due to the condensate can be taken care by per-

forming fully self-consistent calculation [43,44]. Since,

we are only interested in the temperature driven part

(i.e., ∆T̸= 0), we calculate the current in terms of one

lead using Eq. (4). The lower limit of the integration in

Eq. (4) is to be replaced by the maximum among ∆Land

∆Rif TR

ee = 0 within the subgap energy.

Following the current conservation, the charge current

should be continuous and we can ﬁnd it out using the

BdG wavefunctions and ﬁnally express it in terms of the

transmission probabilities given by,

iη

L=TRL

ηη −TRL

η′η(5)

with η∈ {e,h}and Tl′l

ηη =|tl′l

ηη|2where Tl′l

η′η(tl′l

η′η) is

the probability (amplitude) of the transmission of η′type

particles from l′-th to l-th lead as η. In our case, Tl′l

η′η= 0

when l̸=l′for η̸=η′. The quasiparticles’ transmissions

take part in the dissipative part of the thermally induced

charge current. The expressions for the transmission am-

plitudes are mentioned in the Appendix A. From now on,

(a)

LS/ξ

0.2

0.5

1.5

-2-1 0 1 2

ω/Δ0

Tee

(b)

-2-1 0 1 2

ω/Δ0

Tee

FIG. 2. Transmission probability TR

ee as a function of ω/∆0

in symmetric junction for (a) ϕ=ϕ0/4 and (b) ϕ= 3ϕ0/4

with ϕ0= 2π.

we will use the notation TR

ηη in place of TRL

ηη throughout

the rest of the manuscript for simplicity.

Now, in the absence of any bias voltage, the linear

response of the non-dissipative charge current per unit

temperature diﬀerence is denoted as

L12 =Ic

∆T.(6)

Note that, this is not the conventional Seebeck current

as we explain in the next section. Reversing the phase

can help in separating the non-dissipative charge current

from the dissipative Seebeck current [42,45,46].

Heat current: To calculate the heat current, we follow

the similar prescription considering the contributions by

the individual leads as Iq=Iq

L−Iq

Rwhere

l=Z∞

ωdω ie

l(ω) + ih

l(ω)f(ω/Tl).(7)

Using the initial condition that the heat currents ﬂowing

in the opposite direction must cancel each other for ∆T=

0, we ﬁnally arrive at

Iq=2

h∆TZ∞

dω ie

L(ω) + ih

L(ω)∂f (ω/Tl)

∂T .(8)

where the contributions by the electron-like and hole-

like quasiparticles are give by Eq.(5). The heat current

per unit temperature diﬀerence is deﬁned as the thermal

conductance and it is given by

K=Iq

∆T.(9)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Phase-dependentchargeandheatcurrentinthermallybiasedshortJosephsonjunctionsformedathelicaledgestatesParamitaDuttaID11TheoreticalPhysicsDivision,PhysicalResearchLaboratory,Navrangpura,Ahmedabad-380009,India∗(Dated:August14,2023)Weexplorethephase-dependentchargeandheatcurrentintheshortJosephsonjunctio...

展开>> 收起<<

Phase-dependent charge and heat current in thermally biased short Josephson junctions formed at helical edge states Paramita DuttaID1.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Phase-dependent charge and heat current in thermally biased short Josephson junctions formed at helical edge states Paramita DuttaID1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: