Effect of magnetic pair breaking on Andreev bound states and resonant supercurrent in quantum dot Josephson junctions

Effect of magnetic pair breaking on Andreev bound states and resonant supercurrent in quantum dot Josephson junctions

Grygoriy Tkachov^1,2 and Klaus Richter¹

1Institute for Theoretical Physics, Regensburg University, 93040 Regensburg, Germany

2Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 共Received 22 December 2006; published 24 April 2007兲

We propose a model for resonant Josephson tunneling through quantum dots that accounts for Cooper pair-breaking processes in the superconducting leads caused by a magnetic field or spin-flip scattering. The pair-breaking effect on the critical supercurrentI_cand the Josephson current-phase relationI共␸兲is largely due to the modification of the spectrum of Andreev bound states below the reduced共Abrikosov-Gorkov兲quasipar- ticle gap. For a quantum dot formed in a quasi-one-dimensional channel, bothI_candI共␸兲can show a signifi- cant magnetic-field dependence induced by pair breaking despite the suppression of the orbital magnetic-field effect in the channel. This case is relevant to recent experiments on quantum dot Josephson junctions in carbon nanotubes. Pair-breaking processes are taken into account via the relation between the Andreev scattering matrix and the quasiclassical Green functions of the superconductors in the Usadel limit.

DOI:10.1103/PhysRevB.75.134517 PACS number共s兲: 74.50.⫹r, 73.63.⫺b

I. INTRODUCTION

Since its discovery, the Josephson effect¹has been studied for a variety of superconducting weak links.^2–4The research has recently entered another phase with the experimental re- alization of quantum dot weak links, exploiting electronic properties of finite-length carbon nanotubes coupled to su- perconducting leads.^5–7 In particular, since its theoretical prediction,^8–10resonant Josephson tunneling through discrete electronic states has been observed in carbon nanotube quan- tum dots.⁶As demonstrated in Refs.6and7, the quantum dot junctions exhibit transistorlike functionalities, e.g., a periodic modulation of the critical current with a gate voltage tuning successive energy levels in the dot on- and off-resonances with the Fermi energy in the leads. This property has already been implemented in a recently proposed carbon nanotube superconducting quantum interference device¹¹with possible applications in the field of molecular magnetism.

Motivated by the experiments on resonant Josephson tun- neling, in this paper, we investigate theoretically how robust it is with respect to pair-breaking perturbations in the super- conducting leads. Cooper pair breaking can be induced by a number of factors, e.g., by paramagnetic impurities,¹²an ex- ternal magnetic field,¹³or by structural inhomogeneities pro- ducing spatial fluctuations of the superconducting coupling constant.¹⁴ It can cause a drastic distortion of the Bardeen- Cooper-Schrieffer共BCS兲superconducting state, which mani- fests itself in the smearing of the BCS density of states lead- ing to gapless superconductivity.^12,13

While the pair-breaking effect on bulk superconductivity is now well understood, its implications for quantum super- conducting transport have been studied to a much lesser ex- tent共see, e.g., Refs.4and15and16兲which, to our knowl- edge, does not cover Josephson tunneling through quantum dots. On the other hand, in low-dimensional systems, pair- breaking effects may be observable in a common experimen- tal situation when, for instance, a carbon nanotube weak link is subject to a magnetic field. Since the orbital field effect in the quasi-one-dimensional channel is strongly suppressed,

pair breaking in the superconducting leads can be the main source of the magnetic-field dependence of the Josephson current. This situation is addressed in our work.

The influence of pair breaking on the Josephson current cannot, in general, be accounted for by mere suppression of the order parameter in the superconducting leads. As was pointed out in Ref.15, it is a more subtle effect involving the modification of the spectrum of current-carrying states in the junction, in particular, the subgap states usually referred to as Andreev bound states共ABSs兲.^10,17We illustrate this idea for quantum dot junctions in the simple model of a short super- conducting constriction with a scattering region containing a single Breit-Wigner resonance near the Fermi energy. The Josephson current is calculated using the normal-state scat- tering matrix of the system and the Andreev reflection matrix.^9,10 Unlike Refs.9 and10, we focus on dirty super- conductors for which the Andreev matrix can be quite gen- erally expressed in terms of the quasiclassical Green functions,¹⁸ allowing us to treat pair breaking in the super- conducting leads nonperturbatively. Although we account for all energies 共below and above the Abrikosov-Gorkov gap

⌬g兲, it turns out that the behavior of the Josephson current can be well understood in terms of a pair-breaking-induced modification of the ABS, which depends sensitively on the relation between the Breit-Wigner resonance width⌫and the superconducting pairing energy⌬. Both the critical supercur- rent and the Josephson current-phase relation are analyzed under experimentally realizable conditions.

II. MODEL AND FORMALISM

We consider a junction between two superconductors S₁ and S₂ adiabatically narrowing into quasi-one-dimensional ballistic wires S₁

⬘

^{and S}2

⬘

coupled to a normal conductor N 共Fig.1兲. The transformation from the superconducting elec- tron spectrum to the normal-metal one is assumed to take place at the boundariesS₁S₁

⬘

^andS₂

⬘

^S2, implying the pairing potential of the form²⌬共x兲=⌬e^i␸1 forx⬍−L/ 2, ⌬共x兲= 0 for 兩x兩艋L/ 2, and⌬共x兲=⌬e^i␸2forx⬎L/ 2 with the order param-

1098-0121/2007/75共13兲/134517共5兲 134517-1 ©2007 The American Physical Society

eter phase difference ␸⬅␸2−␸1 and the junction length L ⰆបvF/⌬共vF is the Fermi velocity inS_1,2兲.

The Josephson coupling can be interpreted in terms of the Andreev process,¹⁹whereby an electron is retroreflected as a Fermi-sea hole from one of the superconductors with the subsequent hole-to-electron conversion in the other one.

Such an Andreev reflection circle facilitates a Cooper pair transfer betweenS₁andS₂. Normal backscattering from dis- ordered superconducting bulk into a single-channel junction is suppressed due to the smallness of the junction width com- pared to the elastic mean free path ᐉ. The N region in the middle of the junction is thus supposed to be the only source of normal scattering. In such type of weak links, the Joseph- son current is conveniently described by the scattering ma- trix expression of Refs.10and20that can be written at finite temperatureT as the following sum over the Matsubara fre- quencies␻n=共2n+ 1兲␲^kBT 共Ref.20兲:

I= −2e ប2k_BT ⳵

⳵␸

兺

n=0

⬁

ln Det关1ˆ−sˆ_A共E兲sˆ_N共E兲兴E=i␻n. 共1兲 Here, sˆN共E兲 is a 4⫻4 unitary matrix relating the incident electron and hole waves on theNregion to the outgoing ones 共Fig.1兲. It is diagonal in the electron-hole space,

sˆ_N=

冋

^{see0共E兲 shh0共E兲}

册

^{, see共E兲=}

冋

^rt21¹¹共^共E兲E兲 r^t12₂₂^共E兲共E兲

册

^.

The matrix see共E兲 describes electron scattering in terms of the reflection and transmission amplitudes,rjk共E兲 andtjk共E兲, for a transition fromS_k

⬘

^toSj

⬘

共j,k= 1 , 2兲. The hole scattering matrix is related to the electron one byshh共E兲=s_ee^*共−E兲. The Andreev scattering matrix sˆA共E兲 is off-diagonal in the electron-hole space,

sˆA=

冋

^{she0共E兲 seh0共E兲}

册

^{, 共2兲}

where the 2⫻2 matrices she共E兲 and seh共E兲 govern the electron-to-hole and hole-to-electron scatterings of the super- conductors. Equation共1兲 is valid for all energies as long as normal scattering from the superconductors is absent.^10,20

In Ref.10, the Andreev matrix关Eq.共2兲兴was obtained by matching the solutions of the Bogolubov–de Gennes equa- tions in the wires S_1,2

⬘

to the corresponding solutions in impurity-free leads. Gorkov’s Green function formalism in

combination with the quasiclassical theory²¹ allows one to generalize the results of Ref. 10 to dirty leads with a short mean free path ᐉⰆបv_F/⌬. In the latter case, the matrices she共E兲andseh共E兲can be expressed in terms of the quasiclas- sical Green functions of the superconductors as follows:¹⁸

seh=

冤

^{g1f共1E共E兲0兲+ 1 g2f共E兲20共E兲+ 1}

冥

^,

she=

冤

^{g−1共E兲f1†0共E兲+ 1 g−2共E兲f02†共E兲+ 1}

冥

^.

Here, g_1,2 and f_1,2 共f_1,2^† 兲 are, respectively, the normal and anomalous retarded Green functions inS_1,2. These matrices are diagonal in the electrode space due to a local character of Andreev reflection in our geometry.

Neglecting the influence of the narrow weak link on the bulk superconductivity, we can use the Green functions of the uncoupled superconductors S_1,2 described by the position-independent Usadel equation,²¹

冋

^{E␶ˆ3+⌬ˆj+2iប␶pb␶ˆ3gˆj␶ˆ3,gˆj}

册

^{= 0, 共3兲}

with the normalization conditiongˆ_j²=␶^ˆ0for the matrix Green function,

gˆj=

冋

^{gfj†j −fgjj}

册

^{, ⌬ˆj=}

冋

^{−⌬e0−i␸j ⌬e0i␸j}

册

^{, j= 1,2.}

Here,␶^ˆ0and␶^ˆ3are the unity and Pauli matrices, respectively, and关¯,¯兴denotes a commutator. Equation共3兲accounts for a finite pair-breaking rate␶pb

−1 whose microscopic expression depends on the nature of the pair-breaking mechanism. For instance, for thin superconducting films in a parallel mag- netic field, ␶pb

−1=共v_Fᐉ/ 18兲共␲^dB/⌽0兲² 共Ref. 13兲, where d is the film thickness and⌽0is the flux quantum. For paramag- netic impurities,␶pbcoincides with the spin-flip time.¹²In the case of the spatial fluctuations of the superconducting cou- pling,␶pb

−1is proportional to the variance of the fluctuations.¹⁴ From Eq.共3兲, one obtains the Green functions

gj= u

冑

u²− 1=ue^−i␸jfj, f^†_j= −e^−2i␸jfj, 共4兲

E

⌬=u

冉

^{1 −}

^冑

^{1 −␨ u2}

冊

^{, ␨=␶pbប⌬, 共5兲}

where, following Refs.12and13, we introduce a dimension- less pair-breaking parameter␨. The matricesseh andshecan be expressed using Eq.共4兲as follows:

s_eh=␣

冋

^{ei0␸1 e0i␸2}

册

^{, she=␣}

冋

^{e−i0␸1 e−i0␸2}

册

^{, 共6兲}

FIG. 1. Scheme of a superconducting constriction with a normal scattering regionN. The arrows indicate the electrons共e兲and holes 共h兲 incident on and outgoing fromN.

␣^=u−

冑

u²− 1. 共7兲 We note that pair breaking modifies the energy dependence of the Andreev reflection amplitude␣ according to the non- BCS Green functions关Eqs. 共4兲 and共5兲兴. A few words con- cerning the applicability of this result are due here.

First of all, there is no restriction on energy E; e.g., for

␨艋1, Eqs.共6兲 and共7兲 are valid both below and above the reduced 共Abrikosov-Gorkov兲 quasiparticle gap

⌬g=⌬共1 −␨^2/3兲^3/2. In particular, for 兩E兩艋⌬g one can show thatu is real and兩u兩艋共1 −␨^2/3兲^1/2⬍1共Ref. 13兲, correspond- ing to perfect Andreev reflection with␣^{= exp}关−iarccos共u兲兴. Since in the Usadel limitᐉⰆvF␶pb, normal scattering from the superconductors is suppressed due to the smallness of the junction width also in the presence of pair breaking. The absence of normal transmission at兩E兩艋⌬gis consistent with the Abrikosov-Gorkov approach assuming no impurity states inside the gap and the validity of the Born approximation.^12,13For兩E兩艌⌬g, the relevant solution of Eq.

共5兲is complex and has positive Imurelated to the density of states of the superconductor.¹³Equations共5兲and共7兲are thus the generalization of the known result ␣⁼共E/⌬0兲

−

冑

^共E/⌬0兲²− 1共Ref.22兲for transparent point contact, where

⌬0⬅兩⌬兩_␨=0 is the BCS gap. It is convenient to measure all energies in units of ⌬0 for which Eqs. 共4兲–共7兲 should be complemented with the self-consistency equation for ⌬. At T= 0, the case we are eventually interested in, this equation can be written as^12,13

ln共␨0/␨兲= −␲␨^/4, ␨艋1, 共8兲

ln共␨0/␨兲=

冑

␨^{2− 1/}共2␨兲− ln共␨⁺

冑

␨^{2− 1}兲

−共␨/2兲arctan共1/

冑

_␨²− 1兲, ␨艌1, 共9兲 with␨ being now a function of a new pair-breaking param- eter ␨0=ប/共␶pb⌬0兲 ranging from zero to the critical value

␨0= 0.5 at which␨⁼⬁and⌬= 0.^12,13

Inserting Eqs.共2兲and共6兲forsˆ_A共E兲into Eq.共1兲and taking the limitT→0, we obtain the Josephson current for an arbi- trarysˆN共E兲as

I= −4e h

冕

0

⬁

d␻ ⳵

⳵␸^{ln兵1 +}␣^4Detsee共E兲Dets_ee^*共−E兲

−␣²关r11共E兲r₁₁^* 共−E兲+r₂₂共E兲r₂₂^*共−E兲+e^−i␸t₂₁共E兲t₁₂^*共−E兲 +e^i␸t₁₂共E兲t21* 共−E兲兴其E=i␻. 共10兲

III. ANDREEV BOUND STATES IN A RESONANT JUNCTION

Let us assume that the N region is a small quantum dot and electrons can only tunnel via one of its levels characterized by its position Er with respect to the Fermi level and broadening⌫. For the simplest Breit-Wigner scat- tering matrix with r₁₁=r₂₂=共E−E_r兲/共E−E_r+i⌫兲 and t₁₂=t₂₁=⌫/i共E−Er+i⌫兲, Eq.共10兲reads

I= −共2e/h兲Tsin␸

冕

0

⬁

d␻

再

^u2

^冋

^R+T

^冉

^{1 +}

^冑

^{1 −⌫/⌬u2−␨}

^冊

²

^册

− 1 +Tsin²

冉

^␸2

冊 冎

^{E=i−1␻, 共11兲}

where T= 1 −R=⌫²/共Er

2+⌫²兲 is the Breit-Wigner transmis- sion probability at the Fermi level. The parameter⌫/⌬ ac- counts for the energy dependence of the resonant supercon- ducting tunneling. In Eq.共11兲, the integrand has, in general, poles given by the equation

u²

冋

^R+T

^冉

^{1 +}

^冑

^{1 −⌫/⌬u2−␨}

^冊

²

册

^{= 1 −Tsin2}

^冉

^␸2

^冊

^{. 共12兲}

Along with Eq. 共5兲, they determine the energies of the ABSs localized below the Abrikosov-Gorkov gap

⌬g=⌬共1 −␨^2/3兲^3/2. It is instructive to understand how the pair breaking modifies the ABS spectrum since this is reflected on both the current-phase relationI共␸兲 and the critical current Ic⬅maxI共␸兲.

We start our analysis with an analytically accessible case of an infinitely broad resonant level, ⌫/⌬→⬁, where Eq.

共12兲 reduces to u²= 1 −Tsin²共␸/ 2兲, yielding the ABS ener- gies ±E共␸兲 共see Eq.共5兲兲,

E共␸兲=⌬

冑

^{1 −}Tsin²共␸/2兲

冋

^{1 −}

^冑

^{T兩sin共␨␸/2兲兩}

册

^{. 共13兲}

Requiring E共␸兲艋⌬g, we find that the ABSs exist in the phase interval where sin²共␸^{/ 2}兲艌␨^2/3/T and only if␨^2/3艋T. The numerical solution of Eqs. 共5兲, 共8兲, and 共12兲 confirms that the interval of the existence of ABSs gradually shrinks from 0艋␸艋2␲ to a narrower one with increasing pair breaking 共see Fig. 2兲. Outside this interval the Josephson current is carried by the continuum states 共E艌⌬g兲 alone, which is automatically accounted for by Eq.共11兲. An equa- tion of the same form as Eq.共13兲was derived earlier for a nonresonant system and by a different method.¹⁵

By contrast, the ABS spectrum for a narrow resonant level turns out to be much less sensitive to pair breaking. Indeed, under condition⌫/⌬Ⰶ1 −␨^{, Eqs.}共5兲and共12兲reproduce the known result, E共␸兲=

冑

E_r²+⌫²

冑

^{1 −Tsin2共}␸/ 2兲 共Refs. 4 and 9兲. In particular, forEr→0, the ABSs exist within the reso- nance widthE共␸兲⬍⌫and are separated from the continuum FIG. 2. 共Color online兲Phase dependence of the Andreev bound state for a broad resonant level with⌫= 15⌬0close to the Fermi energy 共E_r= 0.1⌫兲; dashed line shows the normalized gap for a given value of the pair-breaking parameter␨0.

by a gap⌬g−⌫. Solving Eqs.共5兲,共8兲, and共12兲numerically, we find that until this gap closes at a certain value of␨0, the ABS spectrum remains virtually intact共see Fig.3兲. For big- ger ␨0, the spectrum gets modified in a way similar to the previous case共cf., third panels in Figs.2and3兲. In the case of a very narrow resonance, the characteristic value of␨0 is

⬇0.45, corresponding to ␨⬇1, i.e., to the onset of gapless superconductivity.^12,13

IV. CRITICAL CURRENT AND CURRENT-PHASE RELATION: RESULTS AND DISCUSSION

For numerical evaluation of the Josephson current 关Eq.

共11兲兴, we first setE=i␻^{in Eq.}共5兲and then make the trans- formationu→i␯^{, yielding}␻^/⌬=␯共1 −␨^/

冑

1 +␯²兲. Using this relation, in Eq. 共11兲 we change the integration over ␯ ^with the Jacobiand␻^/d␯⁼⌬关1 −␨^/共1 +␯²兲^3/2兴,

I=共2e⌬/h兲Tsin␸

冕

_␯^⬁₀^d␯

冉

^{1 −}共1 +^␨␯²兲^3/2

冊

⫻

再

^␯2

^冋

^R+T

^冉

^{1 +}

^冑

^{1 +⌫/⌬␯2−␨}

^冊

²

^册

^{+ 1 −Tsin2}

^冉

^␸2

^冊 冎

^−1.

共14兲 Positiveness of␻^{in Eq.}共11兲enforces the choice of the lower integration limit:␯0= 0 for ␨艋1 and␯0=

冑

_␨²− 1 for␨艌1.

Using Eqs.共8兲,共9兲, and共14兲, we are able to analyze the critical currentI_c⬅maxI共␸兲 in the whole range of the pair-

breaking parameter, 0艋␨0艋0.5共see Fig.4兲. In line with the discussed behavior of the Andreev bound states, for a narrow resonance, ⌫/⌬0Ⰶ1, the critical current starts to drop sig- nificantly only upon entering the gapless superconductivity regime, 0.45艋␨0艋0.5. On the other hand, for a broad reso- nance,⌫/⌬0Ⰷ1, the suppression ofIcis almost linear in the whole range. We note that in both cases the behavior of Ic

strongly deviates from that of the bulk order parameter共red curve兲 共Refs. 12 and 13兲 largely due to the pair-breaking effect on the ABS. In practice, theIc共␨0兲 dependence can be measured by applying a magnetic field 关the case where ␨0

=共B/B_*兲² and B_*=共⌽0/␲^d兲

冑

18⌬0/បv_Fᐉ兴 in an experiment similar to Ref.6 where a quantum dot, defined in a single- wall carbon nanotube, was strongly coupled to the leads with the ratio ⌫/⌬0⬇10. Carbon nanotube quantum dots with lower⌫/⌬0values are accessible experimentally, too.^5,11

We also found that the crossover between the gapped and gapless regimes is accompanied by a qualitative change in the shape of the Josephson current-phase relation I共␸兲, as demonstrated in Fig.5 for the on-resonance caseEr= 0 and

⌫=⌬0. The I共␸兲relation is anharmonic as long as the junc- tion with⌬g⫽0 supports the ABSs共black and blue curves兲.

The vanishing of the ABSs upon entering the gapless regime leads to a nearly sinusoidal current-phase relation 共red curve兲. A closely related effect is demonstrated in Fig. 6, showing the modification of the critical current resonance line shape with the increasing pair-breaking strength. In the absence of pair breaking, it is nonanalytic nearE_r= 0共black curves兲, reflecting the anharmonicI共␸兲due to the ABSs in a transparent channel.^9,10 On approaching the gapless regime,

FIG. 4. 共Color online兲 On-resonance critical current vs pair- breaking parameter␨0=ប/共␶pb⌬0兲for different⌫/⌬0. The behavior of the normalized order parameter共Refs.12and13兲is shown, for comparison, in red.

FIG. 5. 共Color online兲 On-resonance current-phase relation for different values of the pair-breaking parameter␨0and⌫=⌬0. FIG. 3. 共Color online兲Phase dependence of the Andreev bound

state for a narrow resonant level with⌫= 0.3⌬0andE_r= 0.1⌫.

FIG. 6.共Color online兲Critical current vs resonant level position:

␨0= 0共black兲,␨0= 0.25共blue兲, and␨0= 0.45共red兲.

this singularity is smeared out共red curves兲, which is accom- panied by the suppression of theIcamplitude. At finite tem- peratures TⰆ⌬/kB, the pair-breaking-induced smearing of the resonance peak will enhance the usual temperature effect.

In conclusion, we have proposed a model describing reso- nant Josephson tunneling through a quantum dot beyond the conventional BCS picture of the superconducting state in the leads. It allows for nonperturbative treatment of pair- breaking processes induced by a magnetic field or paramag- netic impurities in diffusive superconductors. We considered no Coulomb blockade effects, assuming small charging en- ergy in the dotECⰆ⌬0,⌫, which was, for instance, the case in the experiment of Ref.6. Our predictions, however, should be qualitatively correct also for weakly coupled dots with

⌫艋ECⰆ⌬0 at least as far as the dependence of the critical

supercurrent on the pair-breaking parameter is concerned.

Indeed, for a narrow resonance the Andreev bound states begin to respond to pair breaking only when the gap ⌬g

becomes sufficiently small 共see Fig. 3兲 so that for a finite ECⰆ⌬0, one can expect a sharp transition to the resistive state, too, similar to that shown in Fig.4for ⌫/⌬0Ⰶ1.

ACKNOWLEDGMENTS

We thank D. Averin, C. Bruder, P. Fulde, A. Golubov, M.

Hentschel, T. Novotny, V. Ryazanov, and C. Strunk for useful discussions. Financial support by the Deutsche Forschungs- gemeinschaft共GRK 638 at Regensburg University兲is grate- fully acknowledged.

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