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arXiv:1101.5239v1 [cond-mat.mes-hall] 27 Jan 2011

Interplay of Kondo and superconducting correlations in the nonequilibrium Andreev

transport through a quantum dot

Yasuhiro Yamada,1Yoichi Tanaka,2and Norio Kawakami1

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan

2Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

(Dated: January 28, 2011)

Using the modified perturbation theory, we theoretically study the nonequilibrium Andreev trans-

port through a quantum dot coupled to normal and superconducting leads (N-QD-S), which is

strongly influenced by the Kondo and superconducting correlations. From the numerical calcula-

tion, we find that the renormalized couplings between the leads and the dot in the equilibrium

states characterize the peak formation in the nonequilibrium differential conductance. In particular,

in the Kondo regime, the enhancement of the Andreev transport via a Kondo resonance occurs in

the differential conductance at a finite bias voltage, leading to an anomalous peak whose position

is given by the renormalized parameters. In addition to the peak, we show that the energy levels

of the Andreev bound states give rise to other peaks in the differential conductance in the strongly

correlated N-QD-S system. All these features of the nonequilibrium transport are consistent with

those in the recent experimental results [R. S. Deacon et al., Phys. Rev. Lett. 104, 076805 (2010);

Phys. Rev. B 81, 12308 (2010)]. We also find that the interplay of the Kondo and superconducting

correlations induces an intriguing pinning effect of the Andreev resonances to the Fermi level and

its counter position.

PACS numbers: 73.63.Kv, 74.45.+c, 72.15.Qm, 73.23.-b

I. INTRODUCTION

Electron transport through nanofabrications has at-

tracted much attention in the studies of fundamental

quantum physics as well as potential future devices. In

particular, a quantum dot (QD), which has discrete en-

ergy levels where the electrons are correlated, provides an

ideal arena to study the local Coulomb interaction effect

on the transport1–3. The magnetic doublet states with

spin 1/2 are stabilized at an isolated QD with an odd

number of electrons and the strong Coulomb interaction,

which results in the Coulomb blockade for the transport

through the QD coupled to leads. At sufficiently low

temperatures, however, the local moment of the doublet

states is screened by the electrons of the leads owing to

the Kondo effect, and thus the Kondo singlet is stabi-

lized, resulting in an anomalous enhancement of the zero

bias conductance4–9.

If we replace the leads by s-wave superconductors, a

different situation arises; the doublet is not screened due

to the lack of low-lying energy states of the leads. Even

in the system, a singlet state can be stabilized due to

the superconducting proximity effect, and the system

thus shows a transition between the doublet and the sin-

glet10–17. Away from the transition point, one of the two

states becomes the ground state and the other an ex-

cited state which is localized at the QD, i.e., the Andreev

bound state. In this system, however, it is difficult to di-

rectly observe the Andreev bound states via transport

measurements because of a supercurrent and a multiple

Andreev reflection process18–28.

Recently, Deacon et. al. have observed the Andreev

bound states experimentally not in the above system

but in the system with a QD coupled to normal and

superconducting leads (N-QD-S) where an Andreev re-

flection dominates the transport29,30.

system, however, the doublet states should be replaced

by the Kondo singlet state owing to the screening by

the electrons of the normal lead (N-lead), leading to a

crossover between the Kondo singlet to the supercon-

ducting singlet. A lot of studies have thus far focused

on how the competition between the Kondo and super-

conducting correlations affects the Andreev transport ex-

perimentally29–33and theoretically34–48. Indeed, Kondo-

type anomalous phenomena have been observed in the

measurement of zero bias conductance in the recent ex-

periment30.

Experimentally, characteristics of the Andreev bound

states emerge under nonequilibrium steady-state condi-

tions where a finite bias voltage is applied to the N-lead.

Some theoretical studies have dealt with the nonequilib-

rium transport properties in an N-QD-S system with em-

phasis on the influence of the Kondo effect34–36,39–42,45–47

and also on the Andreev bound states48. However, the

coexistence of the phenomena related to the Kondo ef-

fect and the Andreev bound states in the experiments

indicates the necessity of further theoretical studies; it is

needed for the comprehensive understanding of the trans-

port to include the Andreev bound state as well as the

interplay between the Kondo and superconducting corre-

lations into the theory.

In this paper, we study the nonequilibrium Andreev

transport, by taking into account the above different as-

pects of the N-QD-S system in a unified way. To this end,

we employ the modified second order perturbation theory

(MPT) used previously by Cuevas et al.38and extend it

to the nonequilibrium steady-state conditions. The MPT

was originally formulated in the equilibrium Anderson

In the N-QD-S

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2

model49, then has been used in several different systems,

e.g., a quantum dot coupled to normal leads50–54and as a

impurity solver for the dynamical mean-field theory55,56.

Furthermore, we exploit the exact solution of the QD-

S system with an infinitely large superconducting gap,

which still has the essence of the Andreev bound states,

to improve the perturbation theory. By systematically

examining the nonequilibrium transport properties in a

wide variety of the system parameters, we demonstrate

that the theoretical results obtained in this paper are

qualitatively in agreement of the recent experiments.29,30

We note that a part of the present results was briefly re-

ported in ref. 47.

This paper is organized as follows. In Sec. II, the

model Hamiltonian is introduced and we formulate the

modified second-order perturbation theory in Keldysh-

Nambu space of the Green’s function. Section III, we

assess the validity of our method in the equilibrium case

and define the renormalized parameters which character-

ize the electron transport in the nonequilibrium states.

The results of nonequilibrium transport are shown in Sec.

IV. We also analyze the superconducting pair amplitude

and the local density of states at the QD in the nonequi-

librium states. The correspondence between the theo-

retical and experimental results is also discussed in this

section. A summary is given in Sec. V.

II. MODEL AND METHOD

A. Model Hamiltonian and Keldysh Green’s

function in Nambu space

In order to describe the electron transport in the N-

QD-S system, we use a single level QD coupled to a nor-

mal metal and a superconductor, which is applicable for

the system with large level spacing of the QD,

H = HQD+ HN+ HS+ HTN+ HTS, (1)

where

HQD= ǫd

?

(ǫN

σ

ndσ+ Und↑nd↓, (2)

HN=

?

?

?

?

kσ

k− µN)c†

kσckσ,(3)

HS=

qσ

(ǫS

q− µS)a†

qσaqσ+

?

q

(∆Sa†

q↓a†

−q↑+H.c.),(4)

HTN=

kσ

(tNc†

kσdσ+ H.c.),(5)

HTS=

qσ

(tSa†

qσdσ+ H.c.). (6)

Here, d†

has an energy level ǫd and the Coulomb interaction U.

Here, ndσ≡ d†

ator of an electron with spin σ and wave vector k (q) in

σcreates an electron with spin σ at the QD which

σdσ. c†

kσ(a†

qσ) denotes the creation oper-

the normal (superconducting) lead. The superconduct-

ing lead is assumed to be described by the BCS Hamilto-

nian with a superconducting gap ∆S= ∆exp(iθS). The

QD is coupled to the normal and superconducting leads

labeled by α = N,S with hybridization tα.

In order to define nonequilibrium steady states of

the N-QD-S system with a bias voltage V , we treat

the Coulomb interaction U as a perturbation.

non-interacting problem can be solved exactly with the

Keldysh Green’s function technique in Nambu space,

from which we can define the chemical potentials of the

normal and superconducting leads as µN = eV and

µS= 0, respectively.

In the noninteracting case, several different Green’s

functions in Nambu space at the QD are defined as

?

?

?

?

The

gr(t,t′) = −iθ(t − t′)

?[d↑(t),d†

?[d†

?[d↑(t),d†

?[d†

↑(t′)]?0 ?[d↑(t),d↓(t′)]?0

↓(t),d†

↑(t′)]?0 ?[d↑(t),d↓(t′)]?0

↓(t),d†

↑(t′)d↑(t)?0 ?d↓(t′)d↑(t)?0

?d†

?d↑(t)d†

?d†

where grand gadenote the retarded and advanced

Green’s functions, which are also used in the equilibrium

case, and g<and g>represent the lesser and greater

Green’s functions. We consider a sufficiently wide band

of electrons in the leads, in which the coupling strength

ΓN(S)(ω) ≡ π|tN(S)|?

dom in the two leads, we obtain the Fourier transformed

Green’s functions,

↑(t′)]?0 ?[d†

↓(t),d↓(t′)]?0

?

,(7)

ga(t,t′) = iθ(t′− t)

↑(t′)]?0 ?[d†

↓(t),d↓(t′)]?0

?

?

?

, (8)

g<(t,t′) = i

?d†

↑(t′)d†

↓(t)?0 ?d↓(t′)d†

↑(t′)?0 ?d↑(t)d↓(t′)?0

↓(t)d†

↓(t)?0

, (9)

g>(t,t′) = −i

↑(t′)?0 ?d†

↓(t)d↓(t′)?0

,(10)

k(q)δ(ω − ǫN(S)

k(q)) becomes a con-

stant ΓN(S). Integrating out the electron degrees of free-

gr(ω) = ((ω + iη)I − ǫdσ3− Σr

ga(ω) = ((ω − iη)I − ǫdσ3− Σa

g<(ω) = −gr(ω)Σ<

g>(ω) = −gr(ω)Σ>

t(ω))−1,

t(ω))−1,

(11)

(12)

(13)

(14)

t(ω)ga(ω),

t(ω)ga(ω),

where

Σr

t(ω) =

?−i(ΓN+ ΓSβ(ω))

t(ω) = [Σr

iΓSβ(ω)∆S

−i(ΓN+ ΓSβ(ω))

ω

iΓSβ(ω)∆∗

S

ω

?

, (15)

Σa

t(ω)]†,(16)

Σ<

t(ω) = −i2ΓN

?f(ω − µN)0

0f(ω + µN)

−∆S

−∆∗

?

−i2ΓSRe[β(ω)]

?1 − f(ω − µN)

?

1

ω

S

ω

1

?

f(ω),(17)

Σ>

t(ω) = i2ΓN

0

01 − f(ω + µN)

−∆S

1

?

+i2ΓSRe[β(ω)]

?

1

ω

−∆∗

S

ω

?

(1 − f(ω)). (18)

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3

FIG. 1: Second order self-energy diagrams we consider in this

paper. The solid line indicates the propagator and the dashed

line the Coulomb interaction.

Here,

1,2,3) is a Pauli matrix in Nambu space.

|ω|

√ω2−∆2θ(|ω| − ∆) +

[ex/kBT+ 1]−1.

If we obtain the self-energies due to the Coulomb inter-

action, the full retarded and advanced Green’s functions

are determined from the Dyson equation,

η is a positive infinitesimal and σi (i=

β(ω) =

ω

i√∆2−ω2θ(∆ − |ω|) and f(x) =

Gr,a(ω) =

?

[gr,a(ω)]−1− Σr,a

U(ω)

?−1

. (19)

The full lesser and greater ones are calculated from the

relation,

?

Here we examine the self-energy ΣU, using the pertur-

bation theory in the Keldysh Green’s function formal-

ism. The first order contributions to the retarded and

advanced self-energies are Σr,a

?

where ?nd? denotes the expectation value of the electron

number at the QD per spin. There is no first-order con-

tribution to the lesser and greater self-energies because

the Coulomb interaction at the QD takes place without

delay. The second order skeleton diagrams are depicted

in Fig. 1. The corresponding contributions to the lesser

and greater self-energies are obtained from the equations,

?

Σ>

2nd(ω) = −U2

G<,>(ω) = −Gr(ω)

Σ<,>

t

(ω) + Σ<,>

U

(ω)

?

Ga(ω). (20)

1st= U?N? with

?nd?

?d↓d↑?∗−?nd?

?N? ≡

?d↓d↑?

?

,(21)

Σ<

2nd(ω) = −U2

dω1

2πΠ<(ω + ω1)σ2

dω1

2πΠ>(ω + ω1)σ2

?g>(ω1)?Tσ2,(22)

?g<(ω1)?Tσ2, (23)

?

where

Π<,>(ω) =

?

dω1

2π

?

g<,>

11(ω1)g<,>

22(ω − ω1)

−g<,>

12(ω1)g<,>

21(ω − ω1)

?

, (24)

where gij denotes the (i,j) component of g. Using the

above self-energies, we calculate the second order contri-

butions to the retarded and advanced self-energies,

Σr,a

2nd(ω) =

i

2π

?

dω1

ω − ω1± iη

?Σ<

2nd(ω1) − Σ>

2nd(ω1)?.

(25)

Although the second order self-energies are believed

to give reasonable results for the nonequilibrium trans-

port through a strongly interacting QD coupled to two

normal leads (N-QD-N system) at least for the particle-

hole symmetric case57,58, this technique is not directly

applicable to the N-QD-S system because of the lack of

symmetry of the leads. Main difficulty in our N-QD-S

system comes from the fact that the simple second or-

der self-energies do not correctly give the formula in the

”atomic limit” where the QD and the leads are discon-

nected. Indeed, the qualitatively correct description of

the Kondo effect in a particle-hole symmetric N-QD-N

system with the second order self-energy is ensured by

the fact that the corresponding formula becomes exact

not only in the weak-U but also in the atomic limit55. In

the N-QD-S system, furthermore, the superconducting

correlations at the QD, which come from the supercon-

ducting proximity effects, must be taken into account, so

that we have to introduce a suitable ”atomic limit” to

study the strong-U regime. For this purpose, we here

make use of the exact solution of the QD-S system with

an infinitely large superconducting gap.

Below, we describe how to construct the modified self-

energies,?Σ2nd, which reproduce the correct results in

above second-order perturbation. The self-energy due to

the Coulomb interaction up to second order thus reads

ΣU= Σ1st+?Σ2nd.

the atomic limit as well as the weak-U limit within the

B.Superconducting atomic limit

In the limit of ∆ → ∞, the quasiparticle degree of

freedom in the superconducting lead is decoupled from

the QD. Therefore, the Hamiltonian is simplified as,

H∆inf= H∆inf

dot + HN+ HTN,

?

(26)

H∆inf

dot

= ǫd

σ

ndσ+ (∆dd†

↑d†

↓+ H.c.) + Und↑nd↓, (27)

where ∆d ≡ ΓSexp(iθS). In this case, ΓS corresponds

to the effective superconducting gap at the QD owing to

the proximity effect.

In the limit of ΓN→ 0, the QD is decoupled from the

normal lead and the effective Hamiltonian(26) becomes a

one-site problem with the superconducting paring poten-

tial and the Coulomb interaction. Hereafter, we call the

limit of (∆ → ∞,ΓN/U → 0) ”superconducting atomic

limit”17. In the superconducting atomic limit, the self-

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4

energies at the QD can be exactly obtained as,

Σr

atm(ω) = U2χ((ω + iη)I − ∆)−1,

Σa

atm(ω) = U2χ((ω − iη)I − ∆)−1,

Σ<

atm(ω) = −i2ηU2χ((ω + iη)I − ∆)−1

×((I − σ3)/2 + ?N?)

×((ω − iη)I − ∆)−1,

Σ>

atm(ω) = i2ηU2χ((ω + iη)I − ∆)−1

×((I + σ3)/2 − ?N?)

×((ω − iη)I − ∆)−1,

(28)

(29)

(30)

(31)

where

χ ≡ ?nd?(1 − ?nd?) − |?d↓d↑?|2,

∆ ≡

∆∗

(32)

?ǫd+ U(1 − ?nd?)∆d− U?d↓d↑?

−ǫd− U(1 − ?nd?)

d− U?d↓d↑?∗

?

.(33)

Note that we omit the first order contributions of U in

the atomic-limit self-energies.

Next, we consider the second order self-energies, fol-

lowing the formula derived in the previous section. We

assume that one-particle Green’s functions in the second

order diagrams (Fig. 1) are dressed with energy shifts,

which are determined by the following one-body Hamil-

tonian,

H = Hdot+ HN+ HS+ HTN+ HTS,

?

+(U?d↓d↑?d†

(34)

Hdot = (ǫd+ U?nd?)

σ

↑d†

d†

σdσ

↓+ H.c.), (35)

where ?nd? and ?d↓d↑? are the effective parameters repre-

senting the energy shifts of the one-particle Green’s func-

tion.

In the superconducting atomic limit, the above one-

particle Green’s functions behave like a δ function, so

that the second order self-energies can be evaluated as,

Σr

2nd(ω) → U2χ0((ω + iη)I − ∆0)−1,

Σa

2nd(ω) → U2χ0((ω − iη)I − ∆0)−1,

Σ<

2nd(ω) → −i2ηU2χ0((ω + iη)I − ∆0)−1

×((I − σ3)/2 + ?N?0)

×((ω − iη)I − ∆0)−1,

Σ>

2nd(ω) → i2ηU2χ0((ω + iη)I − ∆0)−1

×((I + σ3)/2 − ?N?0)

×((ω − iη)I − ∆0)−1,

(36)

(37)

(38)

(39)

where

χ0 ≡ ?nd?0(1 − ?nd?0) − |?d↓d↑?0|2,

∆0 ≡

∆∗

?

(40)

?

ǫd+ U?nd?

d+ U?d↓d↑?∗

?nd?0

?d↓d↑?∗

∆d+ U?d↓d↑?

−ǫd− U?nd?

?

?

,(41)

?N?0 ≡

?d↓d↑?0

0−?nd?0

.(42)

Here, ?nd?0 and ?d↓d↑?0 are the expectation values of

the particle number per spin and the superconducting

correlation at the QD under the one-body Hamiltonian

(34).

We find that in the superconducting atomic limit, the

second-order self-energies have the functional forms simi-

lar to the exact ones, except that they have different con-

stants; (χ0, ∆0, ?N?0) and (χ, ∆, ?N?). Exploiting this

fact, we construct the modified second order self-energies

in the following.

C.Modified second order perturbation theory

First, we formulate the modified self-energies for the

retarded and advanced sectors. For the sake of clarity, we

follow the procedure of the modified perturbation theory

(MPT) in the N-QD-N system by Kajueter and Kotliar55.

In this procedure, the modified self-energies are assumed

to have the following functional forms,

where A and B should be determined for the self-energies

to reproduce the exact ones in the high-energy limit as

well as the superconducting atomic limit.

high-energy limit of the self-energies can be calculated

from the continued-fraction expansion of the correspond-

ing Green’s functions59,60,

?Σr

?Σa

2nd(ω) = A

?

?

[Σr

2nd(ω)]−1− B

?−1

?−1

2nd(ω) = A[Σa

2nd(ω)]−1− B

(43)

The exact

Σr,a

U(ω) = U?N? +U2χ

ω

I + O(1

ω2).(44)

The first term coincides with the first order self-energy

in U. On the other hand, in the limit of ω → ∞, the

modified self-energies are expanded as,

?Σr,a

2nd(ω) =AU2χ0

ω

I + O(1

ω2). (45)

The coefficient A in eq. (45) is determined from the con-

dition that the leading terms of the modified self-energies

are identical to the corresponding ones in the exact self-

energies in eq. (44). Accordingly, we set A as χ/χ0.

We next determine the matrix B from the condition

that the modified self-energies give the correct values in

the superconducting atomic limit. In the limit, the mod-

ified retarded and advanced self-energies become

?Σr,a

2nd(ω) → U2χ((ω ± iη)I − ∆0− U2χ0B)−1.

In order to eliminate the difference between the r.h.s of

the above equation and that of eqs. (28) and (29), we set

B as follows,

?

(46)

B =

1

Uχ0

1 − ?nd? − ?nd?

−?d↓d↑?∗− ?d↓d↑?∗−1 + ?nd? + ?nd?

−?d↓d↑? − ?d↓d↑?

?

(47)

.

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5

Note that in the limit of ∆ → 0 or ΓS → 0, the su-

perconducting correlations at the QD vanishes and the

off-diagonal terms in eq. (43) become zero.

more, A = ?nd?(1 − ?nd?)/(?nd?0(1 − ?nd?0)) and the

non-diagonal terms of B vanish. As a result, the modi-

fied self-energies of eq.(43) just coincide with those in the

previous studies55. Therefore, we believe that the mod-

ified self-energies obtained here are proper extensions of

those used in the N-QD-N system.

We have to calculate the modified lesser and greater

self-energies in order to obtain the transport properties.

By generalizing the strategy used previously55, we define

the modified lesser self-energy as,

Further-

?Σ<

2nd(ω) =

1

A

?Σr

2nd(ω)[Σr

2nd(ω)]−1Σ<

2nd(ω)

×[Σa

2nd(ω)]−1?Σa

Multiplying the modified self-

2nd(ω)(48)

We now confirm that the above self-energy reproduces

the atomic-limit form.

energy (48) on the left and right by the inverse matrices

of?Σr

??Σr

U2χ((I − σ3)/2 + ?N?0).

2nd(ω) and?Σa

2nd(ω), we take the superconducting

atomic limit,

2nd(ω)

?−1?Σ<

2nd(ω)

??Σa

2nd(ω)

?−1

→−2iη

(49)

In the limit, the above matrix does not depend on ω. In

a similar way, we multiply the atomic-limit lesser self-

energy (48) on the left and right by the same matrices.

The resulting matrix also becomes a constant in the limit,

??Σr

U2χ((I − σ3)/2 + ?N?).

2nd(ω)

?−1

Σ<

atm(ω)

??Σa

2nd(ω)

?−1

→−2iη

(50)

Therefore, the difference between eq. (49) and eq. (50)

can be ignored in the limit of η → 0, and eq. (48) repro-

duces the atomic-limit form indeed,

?Σ<

2nd(ω) → Σ<

atm(ω) (∆ → ∞,ΓN→ 0).(51)

We also define the modified greater self-energy as,

?Σ>

2nd(ω) =

1

A

?Σr

2nd(ω)[Σr

2nd(ω)]−1Σ>

2nd(ω)

×[Σa

2nd(ω)]−1?Σa

2nd(ω), (52)

which gives an appropriate form in the limit.

So far, we have formulated the modified retarded, ad-

vanced, lesser and greater self-energies. However, these

four self-energies are not independent, but have to satisfy

the following equality,

?Σ<

2nd(ω) −?Σ>

2nd(ω) =?Σr

2nd(ω) −?Σa

2nd(ω). (53)

We indeed confirm that the modified self-energies satisfy

the equality.

D. Current conservation and consistency of the

energy shifts

Using the resulting full Green’s function, we calculate

the current though the N-QD-S system.

flowing in the normal (N) and superconducting (S) leads

can be calculated from the time evolution of the particle

number operatorsˆ NN,Sin each lead:ˆIN(t) = −edˆ NN(t)

andˆIS(t) = −edˆ NS(t)

is in a nonequilibrium steady state, the expectation val-

ues of these operators are time-independent, which are

given by

??2f(ω − µN)Gr

?

?

?

The current

dt

dt

. Since we assume that the system

?ˆIN? = −4eΓN

h

Im

11(ω) + G<

11(ω)?dω,

(54)

?ˆIS? = −4eΓS

h

Im dω

×

2˜ ρS(ω)f(ω)Gr

11(ω) + β∗(ω)G<

11(ω)

−∆S

ω

2˜ ρS(ω)f(ω)Gr

12(ω) + β∗(ω)G<

12(ω)

??

, (55)

where ˜ ρS(ω) ≡ Re[β(ω)] and Gij denotes the (i,j) com-

ponent of G. We define the current I in this system as

I = ?ˆIN? = −?ˆIS?. However, it is known that the current

calculated by the second order perturbation theory may

not be conserved in some quantum dot systems except

in a special condition57,58,61. In the N-QD-S system, the

simple application of the second order self-energy usually

breaks the current conservation rule, i.e. ?ˆIN?+?ˆIS? ?= 0.

In our modified second order perturbation theory, the

problem in the current still exists. In order to resolve this

difficulty within our framework, we introduce the source

term λ coupled to the current operator and add the term

into the one-body Hamiltonian (34),

λ(IN+ IS). (56)

The effective parameters ?nd?, ?d↓d↑? and λ are de-

termined by the following consistency conditions on the

energy shifts and the current conservation,

where [?Σr

mined in a self-consistent manner.

Here, we check the U → 0 limit of the modified self-

energies. In the small-U limit,?Σr,a

from the one-particle Green’s functions dressed with the

U?nd? = U?nd? + Re[?Σr

?ˆIN? + ?ˆIS? = 0,

2nd(µN)]11

U?d↓d↑? = U?d↓d↑? + [?Σr

2nd(µS)]12

(57)

2nd]ijdenotes the (i,j) component of the modi-

fied retarded self-energy. ?nd? and ?d↓d↑? are also deter-

2ndbecomes the sim-

2ndwhich is calculatedple second order self-energy Σr,a

Page 6

6

mean-field energy shift because the consistency condi-

tions in eq (57) are reduced to ?nd? = ?nd? = ?nd?0

and ?d↓d↑? = ?d↓d↑? = ?d↓d↑?0.

also reduced to Σ<,>

2ndevaluated with using the mean-field

Green’s functions.

Note that the above modified self-energies are applica-

ble to impurity systems with or without superconducting

correlation. We will show below that the above method

works very well except for some special cases with large

bias voltage where we cannot find the convergent parame-

ters ?nd?, ?d↓d↑? and λ. In this paper, we mainly focus on

the reasonable parameter region where the bias voltage

is not so large. We demonstrate that a variety of intrigu-

ing phenomena emerge due to the interplay between the

superconducting correlation and the Kondo effect, some

of which indeed reproduce the experimental results qual-

itatively well.

Therefore,?Σ<,>

2ndis

III. LINEAR-RESPONSE CONDUCTANCE

AND PHASE DIAGRAM

In this section, we study the transport properties in

the linear-response regime and check the validity of our

approximation for the electron transport. In addition,

the renormalized couplings of tunneling are introduced,

which clearly specify various regimes appearing in the

nonequilibrium electron transport addressed in the next

section.

A. Zero bias conductance and the renormalized

couplings in the equilibrium states

Let us first consider the zero bias conductance obtained

in two different ways within the same framework of mod-

ified perturbation theory (MPT) to confirm the consis-

tency of our approximation. Here, we concentrate on the

symmetric coupling case, ΓN/ΓS= 1, with particle-hole

symmetry, ǫd/U = −0.5, in the equilibrium state.

We first obtain the zero bias conductance by directly

differentiating the current by the bias voltage. In this

case, we have to calculate the lesser and greater self-

energies in order to obtain the current from eqs. (54)

and (55). The current-voltage (IV) characteristics thus

obtained for several values of U, are shown in Fig 2(a).

In addition to the suppression of the current, we can see

the enhanced nonlinear behavior. In order to observe the

nonlinearity in more detail, we show the conductance,

I/V , near the zero bias voltage in Fig 2(b). In this fig-

ure, the conductance curve for U/ΓN = 0 is almost flat

near the zero bias voltage, implying that the linear re-

sponse theory can be safely applied in this finite voltage

region. For U/ΓN = 5, the conductance is suppressed,

yet keeps the flat structure. With further increase in

U, however, the conductance shows a convex curve; the

linear response regime is restricted to the very tiny volt-

age region, i.e., |eV |/∆<

∼0.01 for U/ΓN = 20. There-

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

eV / ∆

I (e/h)

U/ΓN=0

U/ΓN=5

U/ΓN=10

U/ΓN=15

U/ΓN=20

1

2

3

4

0 0.1

eV / ∆

0.2

I/V (e2/h)

(a)(b)

FIG. 2: (Color online) (a) Current-voltage characteristics for

several values of U: ΓS/ΓN = 1, ǫd/U = −0.5, ∆/ΓN = 0.5

and kBT/ΓN = 0.005. (b) Conductance as a function of the

bias voltage. The parameters used are the same as in (a).

fore, theoretical studies only on the zero bias conductance

are not enough to understand the transport properties in

the actual experiments in the strong Coulomb interaction

regime.

In the case of ǫd/U = −0.5, we have an alternative

expression for the zero bias conductance at absolute zero

in terms of the renormalized couplings as38,44

dI

dV

???

V =0=16e2

h

??ΓS/?ΓN

1 +

?2

?

??ΓS/?ΓN

?2?2,(58)

where?ΓNand?ΓSare the renormalized couplings defined

?ΓN = zΓN,

z = (1 +ΓS

by

(59)

?ΓS = z(ΓS+ [Σr

U(0)]12),(60)

|∆|−d[Σr

U(ω)]11

dω

???

ω=0)−1.(61)

Here, [Σr

matrix of the retarded self-energy at the QD. It is worth-

while to note that?ΓNand?ΓScan be calculated only from

states and there is no need to calculate the lesser and

greater self-energies.

Figure 3(a) shows the zero bias conductance obtained

in the above-mentioned two different ways. The values of

the conductance obtained from the differentiation (trian-

gle) well coincide with those obtained from Eq.(58) with

the renormalized parameters (circle). This fact confirms

the consistency of our MPT treatment at least around

the zero bias voltage. The consistency is assured by the

effective parameter λ introduced for the current conser-

vation in the MPT framework. As pointed out in the

previous section, a simple second order perturbation may

U]ijdenotes the (i,j) component of the Nambu

the retarded (or advanced) self-energy in the equilibrium

Page 7

7

0

1

2

3

4

Conductance (e2/h)

renormalized parameters

differentiation

0

0.2

0.4

0.6

0.8

05 1015 20

U / ΓN

(a)

(b)

Γ∼

Γ∼

N/∆

S/∆

FIG. 3: (Color online) (a) Zero bias conductance as a function

of U obtained from the two different methods for ΓS/ΓN = 1,

ǫd/U = −0.5, ∆/ΓN = 0.5, kBT/ΓN = 0, which are respec-

tively denoted by circles (scheme of renormalized parameters)

and triangles (scheme of direct differentiation). (b) Renormal-

ized couplings?ΓN and?ΓS. The parameters used are the same

as in (a).

break the current conservation law; there are two differ-

ent definitions of the current, IN and IS. If the current

conservation law is violated, at least one of the values of

the zero bias conductance calculated from IN and IS is

not consistent with the one obtained with the renormal-

ized parameters. Therefore, the confirmation done here

is important to obtain the sensible results for the current

at finite bias voltage.

In Fig. 3(a), the zero bias conductance decreases with

increasing U, which is due to the suppression of the An-

dreev reflection by the Coulomb interaction. The sup-

pression of the Andreev reflection between the QD and

the S-lead and also the single-electron tunneling between

the QD and the N-lead are seen in the Coulomb inter-

action dependence of the renormalized couplings (Fig.

3(b)). With increasing U,?ΓS decreases more rapidly

proximately decoupled into two parts in the low energy

region: S-lead and QD-N systems. Therefore, the Kondo

singlet state becomes dominant in the ground state of

the QD for large U, leading to the suppression of the

Andreev reflection.

We next discuss the zero bias conductance in the

large ∆ region, in comparison with the results obtained

with the numerical renormalization group (NRG) calcu-

lation44. Figure 4 shows the zero bias conductance as

a function of U for several values of ∆. We note that

similar calculations have been done by Cuevas et al.38

Let us first look at the case of infinitely large gap, where

the closed and open circles denote the conductance ob-

tained with MPT and NRG calculations. Although our

approach is based on the perturbation expansion in U,

the MPT results reproduce the NRG results in both weak

than?ΓN, indicating that the entire N-QD-S system is ap-

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20

Conductance (e2/h)

U / ΓN

∆/ΓN=1

∆/ΓN=2

∆/ΓN=10

∆/ΓN=∞

NRG ∆/ΓN=∞

FIG. 4: (Color online) Zero bias conductance as a function

of U for several values of ∆. The closed and open symbols

denote the results of MPT and NRG. The other parameters

are ΓS/ΓN = 1, ǫd/U = −0.5 and kBT/ΓN = 0.

and strong U regions since in the MPT framework the

effective parameters ?nd? and ?d↓d↑? are self-consistently

determined to reproduce the atomic limit correctly. Only

in the intermediate region around U/ΓN = 6, we see

some discrepancies between the two results (less than 0.3

e2/h).

Let us now discuss how the zero bias conductance de-

pends on the superconducting gap ∆ in Fig. 4. With

decreasing ∆, the conductance for finite Coulomb inter-

action is enhanced. It should be noted that the renor-

malization factor z is determined not only by ∆ and ΓS

but also by the retarded self-energy due to the Coulomb

interaction, as seen in eq. (61); for finite ∆ and ΓS, z is

smaller than unity even in the noninteracting case. For

small ∆, the renormalization by the Coulomb interac-

tion is weak, leading to the enhancement of the zero bias

conductance, as compared with the case of ∆ = ∞.

We now look at the local density of states (LDOS) at

the QD with and without the Coulomb interaction U.

The LDOS in the noninteracting case is shown in Fig.

5(a). For ΓS = 0, there is a broad resonance around

the Fermi energy due to the decoupling of the QD from

the S-lead, which means that the QD is in the mixed

valence regime. For small ΓS, the weight of the LDOS

is suppressed at ω = ±∆ since at the same energies, the

divergence of DOS of S-lead occurs. With further increas-

ing ΓS, the LDOS develops a pseudo gap at the Fermi

energy and a double-peak structure appears inside the

gap owing to the superconducting proximity effect; the

superconducting (SC) singlet state becomes dominant at

the QD. Since the two resonances inside the gap are re-

duced to the Andreev bound states for ΓN/ΓS = 0, we

here refer to them as the Andreev resonances. The An-

dreev resonances are located at ω ≃ ±?ΓSwith the same

Andreev resonances clearly characterizes the crossover in

the dominant couplings at the QD, which occurs around

ΓS/ΓN= 1.

width?ΓN. The change from a single resonance to the

Page 8

8

FIG. 5: (Color online) (a) LDOS for several values of ΓS:

U = ǫd = 0, ∆/ΓN = 0.5 and kBT = 0.

several values of U: ΓS/ΓN = 1, ǫd/U = −0.5, ∆/ΓN = 0.5

and kBT = 0. The inset is the enlarged picture in the region

around the Fermi energy.

(b) LDOS for

In Fig.

regime) is shown for several choices of U.

increase in U, the Andreev resonances are merged into

a single resonance, indicating that the superconducting

correlations are reduced by the strong Coulomb interac-

tion and the Kondo correlations are enhanced instead;

the Kondo singlet state dominates the SC singlet state

at the QD. The broad peaks corresponding to the charge

excitations are also observed at ω ≃ U/2 for large U.

The U dependence of the LDOS in Fig. 5(b) is consis-

tent with the preceding MPT calculations by Cuevas et

al.38, though they did not address its relationship to the

Kondo and SC singlet states.

In the particle-hole symmetric case (ǫd/U = −0.5)

with symmetric couplings ΓS/ΓN

Coulomb interaction favors the Kondo singlet ground

state, as discussed above. If we change the ratio ΓS/ΓN,

however, the SC singlet state can be dominant in the

ground state in the strong Coulomb interaction regime.

Such examples are shown in Fig. 6, where the zero-bias

conductance and the corresponding renormalized cou-

plings are plotted as a function of ΓS for ΓN/U = 0.05.

For ΓS/U = ΓN/U = 0.05, the N-lead is strongly coupled

to the QD;?ΓN>?ΓS. As ΓSincreases,?ΓSincreases more

5(b), the LDOS at ΓS/ΓN = 1 (crossover

With the

= 1, the strong

0

1

2

3

4

Conductance (e2/h)

0

0.2

0.4

0.6

0.8

0 0.10.20.30.4 0.5

ΓS / U

(a)

(b)

Γ∼

Γ∼

N/∆

S/∆

FIG. 6: (Color online) (a) Zero bias conductance as a function

of ΓS for ΓN/U = 0.05, ǫd/U = −0.5, ∆/U = 0.025 and

kBT/ΓN = 0. (b) The renormalized parameters as a function

of ΓS.

rapidly than?ΓN, although both of them are enhanced be-

the superconducting proximity effects. We can indeed

see that the crossover in the dominant couplings occurs

around ΓS/U ≃ 0.1. Further increase in ΓS leads to

the enhancement of?ΓSand the suppression of?ΓN, driv-

where the SC singlet is dominant at the QD.

Since the zero bias conductance has the maximum

value for?ΓS/?ΓN = 1 (see eq. (58)), the conductance

Fig. 6(a). Away from the crossover regime, the conduc-

tance decreases both in the N-lead and S-lead dominant

coupling regimes.For any finite values of ΓN/U and

∆/U, the crossover in the dominant couplings occurs at

a finite ΓS/U. We will see in the next section that the

renormalized quantities?ΓNand?ΓSalso characterize the

cause the Coulomb interaction effects are suppressed by

ing the system into the S-lead dominant coupling regime

shows a peak structure around ΓS/U ≃ 0.1 as shown in

differential conductance even at a finite bias voltage.

B. Phase diagram

We summarize the results for the equilibrium N-QD-

S system in the phase diagram specified in terms of the

dominant couplings. Figure 7(a) shows the phase dia-

gram of the particle-hole symmetric N-QD-S system as

functions of logarithms of ΓN/U and ΓS/U. In this fig-

ure, the solid lines, which are determined by?ΓN =?ΓS,

gion divided by the crossover line, the coupling between

the QD and the N-lead (S-lead) is dominant,?ΓN >?ΓS

First, let us look at the case of ∆ = ∞ in Fig.

7(a).For large ΓN/U, the crossover line approaches

ΓN/ΓS = 1 denoted by the dotted line because the ef-

fects of the Coulomb interaction become weak there and

characterize the crossover behavior. In the left (right) re-

(?ΓN<?ΓS).

Page 9

9

FIG. 7: (Color online) (a) Phase diagram for the particle-hole

symmetric N-QD-S system in equilibrium conditions.

dominant state for finite ΓN and the ground state for ΓN = 0

at the QD are denoted in italic. (b) Phase diagram for the

particle-hole symmetric QD-S system for ΓN = 0. The first

order transition points of the ground state are denoted by the

diamond (∆/U = 0.025) and the square (∆/U = ∞).

The

then?ΓN/?ΓS ≃ ΓN/ΓS (see eqs. (59) and (60)). In the

coupled only to the N-lead and the Coulomb interaction

is week, so that the QD is in a mixed-valent singlet state.

With increasing ΓS, the proximity effects are enhanced

and a pseudogap is formed in the LDOS at the QD, thus

leading to the SC dominant state at the QD. On the

other hand, in the small ΓN/U region (πΓN/U < 1), the

crossoverline considerably deviates from the noninteract-

ing one (ΓN/ΓS= 1) due to the renormalization effects

by the Coulomb interaction. For πΓN/U < 1, the mixed

valence state is gradually changed into the Kondo singlet

state, so that the crossover from the Kondo singlet state

to the SC singlet state occurs as ΓS/U increases for small

ΓN/U.

The crossover line terminates at ΓS/U = 0.5 in the

limit of ΓN/U → 0 as shown in Fig. 7(b), where the

crossover is changed to a doublet-singlet transition. This

is because the N-QD-S system is completely divided into

two parts at ΓN = 0, the N-lead and the QD-S system.

This doublet-singlet transition is easily seen in the case

of ∆ = ∞, where the effective Hamiltonian of the QD-S

system is simplified since the coupling between the Bo-

goliubov quasiparticles and the QD vanishes; It has a

single level with the superconducting pairing potential

characterized by the hybridization ΓS as already noted

in eq. (27). The resulting effective Hamiltonian can be

diagonalized by the Bogoliubov transformation, leading

to four eigenstates: two singly occupied states with spin

region of large ΓN/U but small ΓS/U, the QD is strongly

1/2, | ↑? and | ↓?, and two states with total spin 0 consist-

ing of a linear combination of the doubly occupied and

empty states. In the particle-hole symmetric case, the

spin-0 singlet states are given by

|S1? =

1

√2(|0? − | ↑↓?),

1

√2(|0? + | ↑↓?).

(62)

|S2? =

(63)

Note that the singly occupied states are degenerate (zero

energy), and |S1? and |S2? have the different energies,

ES1=U

dates for the ground state are the magnetic doublet state,

|σ?, and the SC singlet state, |S1?. Either of these two

states can be the ground state depending on the param-

eters, and a first order transition occurs at ΓS/U = 0.5,

which is denoted by the black square on the ΓN= 0 line

in Fig. 7(b); for ΓS/U < 0.5 (ΓS/U > 0.5), the ground

state is the doublet state (singlet state). If ΓN is small

but has a finite value, the local moment of the doublet

ground state is screened by the electrons in the N-lead

and the Kondo singlet state becomes the ground state.

Therefore, the characteristic behavior in the crossover

line for small ΓN/U reflects a remnant of the doublet-

singlet transition at ΓN= 0.

When the superconducting gap ∆ becomes finite, the

system is not so much simplified because of the existence

of the coupling between the quasiparticles in the S-lead

and the QD even for ΓN= 0. Therefore, the competition

between the Kondo effect and the superconducting prox-

imity effect becomes important. Even in this case, there

is still a doublet-singlet transition, which is confirmed by

several authors in the problem of a magnetic impurity

in superconductors62–68and the 0 − π transition of the

QD-Josephson junctions10–17. The transition point shifts

toward lower ΓSwith decreasing ∆. We denote the tran-

sition point for ∆/U = 0.025, which is obtained with the

NRG calculation, by the diamond in Fig. 7(b).

Summarizing, the ground state of our system is always

in the singlet phase for finite ΓN, where three different-

type singlet regions are smoothly connected to each other

via crossover behaviors. Only for ΓN= 0, there exists a

transition between the singlet and doublet states.

2−ΓSand ES2=U

2+ΓS. Therefore, the candi-

C. Andreev bound states

Here some comments are in order on the nature of ex-

cited states. We start with the ΓN = 0 and ∆ = ∞

case. Since there are only four discrete eigenstates at

the QD in this case as discussed above, the excited

states are localized at the QD. For ΓS/U > 0.5, |σ? be-

comes the first excited state with the energy ωb= |ES1|.

In contrast, for ΓS/U < 0.5, |S1? becomes the ex-

cited state with the energy ωb. The other singlet state,

|S2?, is always the second excited state with the energy

ωb2= ES2−min(ES1,0) which is larger than ωb. The one

Page 10

10

particle excitation from the ground state to the excited

states localized at the QD is observed as sharp peaks

in the LDOS. These sharp peaks correspond to the An-

dreev bound states. Therefore, there may be four An-

dreev bound states in the LDOS when the ground state

is a magnetic doublet with energy ±ωband ±ωb2. On the

other hand, when |S1? becomes the ground state, there

are only two peaks with energy ±ωbbecause there is no

one particle excitation from |S1? to |S2?.

In the finite ∆ case, there still exist the Andreev bound

states inside the gap which correspond to |σ? or |S1? for

any values of ∆ though the energy of the bound states

ωbcannot be obtained easily. Moreover, the second ex-

cited state corresponding to |S2? may be outside of the

gap and be absorbed into the continuum energy spectrum

for small ∆17. In that case, there are only two Andreev

bound states at the QD even though the magnetic dou-

blet state is the ground state.

In the recent experiments29,30, Deacon et al.

found that the Andreev bound states at the QD can be

detected in the nonequilibrium transport measurements

in an N-QD-S system. In the experiments, there are only

two kinds of peaks which correspond to two kinds of the

first excited bound states. This fact may be attributed

to the small superconducting gap prepared in the exper-

iments. This issue will be addressed in the next section

focusing on the nonequilibrium transport.

have

IV. NONEQUILIBRIUM ELECTRON

TRANSPORT

In this section, we study the nonequilibrium electron

transport in the N-QD-S system with a special focus on

the influence of the Kondo effect and the Andreev scatter-

ing on the nonlinear transport. In particular, we clarify

the origin of the characteristic structures in the conduc-

tance profile in comparison with the recent experiments.

Before elucidating the Coulomb interaction effects on

the nonequilibrium electron transport, it is instructive to

discuss the differential conductance, dI/dV , in the non-

interacting case. Here, we set ǫd= 0, where dI/dV has a

symmetric profile with respect to the V = 0 axis. Figure

8 shows the differential conductance for several values

of ΓS/ΓN as a function of the positive or negative bias

voltage V .

In the noninteracting case, the ratio of the bare cou-

plings directly determines the nature of the system. We

can see the dI/dV profiles characteristic in the N-lead

and S-lead dominant coupling regimes in Fig.

and (b), respectively. In the symmetric coupling case,

ΓS/ΓN= 1, the differential conductance has a maximum

value 4e2/h at V = 0. The zero bias conductance is sup-

pressed both for ΓS/ΓN < 1 and ΓS/ΓN > 1. In the

nonlinear regime, however, the differential conductance

in the two regimes has different characteristics.

In the N-lead dominant coupling regime as seen in Fig.

8(a), the differential conductance is suppressed in whole

8(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

-1-0.5 0 0.5 1

dI/dV (e2/h)

eV / ∆

(a)

ΓS/ΓN=0.2

ΓS/ΓN=0.5

ΓS/ΓN=1

(b)

(a)(b)

(c)

ΓS/ΓN=1

ΓS/ΓN=2

ΓS/ΓN=5

0

0.2

0.4

0.6

0.8

1

1 2 5 10

ΓS / ΓN

e|VC|/∆

Γ~

S/∆

FIG. 8: (Color online) (a), (b) Bias voltage dependence of the

differential conductance for several values of ΓS, U/ΓN = 0,

ǫd = 0, ∆/ΓN = 0.5 and kBT/ΓN = 0. Here, we only show

the results either for positive or negative V since dI/dV is

symmetric with respect to V = 0. (c) Peak position of dI/dV ,

VC, in comparison with?ΓS.

subgap voltages with decreasing ΓS because of the sup-

pression of the Andreev reflection. In the ΓS→ 0 limit,

the sharp peaks appear at the gap edges and the pro-

file of dI/dV becomes similar to the density of states

in the S-lead, indicating that the system in the limit

approaches the one with a NS tunnel junction69.

the other hand, in the S-lead dominant coupling regime

shown in Fig 8(b), the peak of zero bias conductance is

split into two, which then move toward the opposite gap

edges with keeping the unitary-limit value of 4e2/h when

ΓSincreases. In this case, the SC-singlet is dominant at

the QD in the equilibrium state, and the Andreev reso-

nances, which originate from the exited doublet state for

ΓN= 0, emerge in the LDOS at the QD as shown in Fig.

5(a). The positions of the resonances are approximately

given by ±?ΓS. We compare the voltage VC that gives

eVCmoves along the curve of?ΓSfor large ΓS/ΓN, which

enhancement of the transport through the Andreev res-

onances.

On

a peak in dI/dV with?ΓS in Fig. 8(c). It is seen that

confirms that the subgap peak in dI/dV results from the

A. Nonequilibrium transport for particle-hole

symmetric case: ǫd/U = −0.5

We now investigate the Coulomb interaction effects on

the nonequilibrium differential conductance at a finite

bias voltage. Since there are many relevant parameters

in the system, we will divide our discussions into two

cases. We first treat the simple case with a condition of

Page 11

11

FIG. 9: (Color online) (a) Bias voltage dependence of the

differential conductance for several values of U: ΓS/ΓN =

1, ǫd/U = −0.5, ∆/ΓN = 0.5 and kBT/ΓN = 0.01.

Plots of the peak position, VA, in the dI/dV curve near the

zero bias voltage and the renormalized coupling,?ΓN, which

is calculated for V = T = 0.

(b)

particle-hole symmetry, ǫd/U = −0.5. More generic cases

with arbitrary conditions for ǫdand U will be discussed

separately in the next subsection in comparison with the

experiments.

1. Coulomb interaction effects for ΓS/ΓN = 1

Let us start with a system with the symmetric cou-

plings for tunneling, ΓS/ΓN = 1, which may help us to

imagine what is essential in the nonequilibrium transport

in the interacting QD. Figure 9(a) shows the differential

conductance as a function of the bias voltage for several

values of the Coulomb interaction, U. According to the

analysis in the previous section (see Fig. 7), with increas-

ing U, the system enters the N-lead dominant coupling

regime where the Kondo singlet state becomes dominant.

In this Kondo regime, several peaks appear at subgap

voltages. We refer to the two sharp peaks near the zero

bias voltages as Peak A and the two broad peaks at higher

voltages as Peak B in Fig. 9(a). Although the heights

of the peaks are suppressed, both of Peak A and Peak B

become prominent for large U. Note that the U depen-

dence of the position of Peak A is different from that of

Peak B; Peak A approaches the zero bias voltage with

increasing U, whereas Peak B slightly shifts toward the

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1-0.5 0 0.5 1 1.5 2

ρ(ω)πΓN

ω / ∆

(a)

eV/∆=0

eV/∆=0.2

eV/∆=0.4

eV/∆=0.6

eV/∆=1

eV/∆=2

(b) (b)

-1

-0.5

0

0.5

1

0 0.2 0.40.60.81

eV / ∆

(a)

ω+/∆

ω-/∆

± µN/∆

FIG. 10: (Color online) (a) Local density of states at the QD

for several values of V : ΓS/ΓN = 1, U/ΓN = 20, ǫd/U =

−0.5, ∆/ΓN = 0.5 and kBT/ΓN = 0.01. (b) Peak position of

subgap resonances in (a) as a function of V .

gap edge. This fact implies that these two kinds of sub-

gap peaks in dI/dV have different origins. We will show

below that Peak A originates from the interplay between

the Kondo effect and the Andreev reflection at a finite

bias, while Peak B comes from the Andreev bound states

at the QD.

Let us focus on Peak A. Figure 9(b) shows the com-

parison of the position of Peak A, denoted as VA, and the

renormalized N-lead coupling?ΓN defined at V = T = 0.

system enters the Kondo regime with increasing U. Since

?ΓN is the characteristic energy scale of the Kondo ef-

resonance, Peak A is related to the Kondo effect. The

emergence of the Kondo effect is seen in the bias voltage

dependence of the LDOS at the QD shown in Fig. 10.

Although the LDOS for U = 0 is not changed by the

bias voltage, it is affected via the self-energy for finite U.

In particular, the LDOS in the Kondo regime substan-

tially changes its form under a finite bias voltage. Figure

10(a) shows the LDOS at the QD for U/ΓN = 20 and

ΓS/ΓN= 1. For V = 0, there is a sharp Kondo resonance

at the Fermi energy. With increasing V , the position of

the Kondo resonance follows the chemical potential of the

N-lead, µN = eV , suggesting that the Kondo screening

of the local moment is mainly caused by the normal lead.

It is seen that the value of eVAapproaches?ΓNwhen the

fect, which approximately gives the width of the Kondo

Page 12

12

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1

dI/dV (e2/h)

eV / ΓN

T/ΓN=0.005

T/ΓN=0.01

T/ΓN=0.05

T/ΓN=0.1

T/ΓN=0.25

FIG. 11: (Color online) Temperature dependence of the differ-

ential conductance for ΓS/ΓN = 1, U/ΓN = 20, ǫd/U = −0.5

and ∆/ΓN = 0.5.

A noticeable change in the LDOS at finite bias voltage

(Fig. 10(a)) is the appearance of the additional resonance

which is located at the counter position of the ordinary

Kondo resonance; the ordinary Kondo resonance has a

shoulder structure for eV/∆ = 0.2, which is changed into

an additional resonance for eV/∆ = 0.4. This consider-

ation naturally suggests that the additional resonance is

caused by the Andreev reflection through the ordinary

Kondo resonance (referred to as Kondo-assisted Andreev

reflection); an electron which comes from the N-lead

Fermi surface reaches the S-lead via the ordinary Kondo

resonance, and then it is converted as a hole via the An-

dreev reflection process. Since the electron has finite en-

ergy measured from the S-lead Fermi surface, eV , the re-

flected hole also has the same energy. This interpretation

clarifies why the position of the additional resonance is

located at the counter position of the Kondo resonance.

Note that the additional resonance discussed here was

previously realized by Sun et al.39, but was not discussed

in detail, in particular, about its physical relevance to

the transport properties. We will address this issue with

the use of the renormalized couplings, and demonstrate

that it indeed provides a source of the marked change in

nonequilibrium transport properties.

The positions of the Kondo and additional resonances,

ω+and ω−are shown in Fig. 10(b) as a function of the

bias voltage. For small V , ω+and ω−follow the dotted

lines which denote the position of the chemical poten-

tial of N-lead and its counter position, ±µN. Hence the

crossover of the LDOS from the single peak to the dou-

ble peaks occurs at eV ≃?ΓNwhere the distance between

comparing the results of the LDOS with dI/dV , we find

that the crossover voltage in the LDOS approximately

corresponds to the one giving Peak A in the differential

conductance. Summarizing all these results, we conclude

that Peak A in dI/dV originates from the Kondo-assisted

Andreev reflection at a finite bias voltage.

the two peaks is approximately given by their width. By

0

0.5

1

1.5

2

2.5

3

3.5

4

-1-0.5 0 0.5 1

dI/dV (e2/h)

eV / ∆

(a)(b)

ΓS/ΓN=0.5

ΓS/ΓN=1

ΓS/ΓN=1.5

ΓS/ΓN=2

ΓS/ΓN=2

ΓS/ΓN=3

ΓS/ΓN=4

ΓS/ΓN=7.5

FIG. 12: (Color online) Differential conductance as a function

of bias voltage for several values of ΓS: U/ΓN = 20, ǫd/U =

−0.5, ∆/ΓN = 0.5 and kBT/ΓN = 0.01.

In order to further confirm our interpretation for Peak

A in dI/dV , we calculate the temperature dependence

of dI/dV as shown in Fig. 11. With increasing tem-

perature, two peaks near zero bias voltage, which are

classified as Peak A, are smeared and absorbed into the

broad peaks of Peak B. The temperature dependence of

Peak A supports that it is due to the Kondo-assisted An-

dreev reflection. The characteristic temperature around

which Peak A is smeared coincides approximately with

?ΓN/ΓN ≃ 0.086. We note that the equilibrium quan-

dependence of Peak A in the nonlinear differential con-

ductance.

In contrast, we can see that Peak B in dI/dV is not

directly related to the Kondo effect according to its T-

dependence in Fig. 11; although the two peaks labeled

as Peak B decrease their heights with increasing T, the

broad peak structure still exists even at T/ΓN = 0.25

(higher than the Kondo temperature). We indeed find

that Peak B is solely controlled by the Andreev reflec-

tion, but not by the Kondo effect. Since the origin and

the physical implications of Peak B are naturally seen

in the asymmetric limit of ΓS/ΓN ≫ 1, we will discuss

the physical properties systematically in asymmetric cou-

plings below.

tity of?ΓN characterizes both the position and the T-

2.Coulomb interaction effects for ΓS/ΓN ?= 1

Here, we address how the asymmetry of couplings

(ΓS/ΓN ?= 1) alters the nonequilibrium transport prop-

erties. We start with the differential conductance dI/dV

shown in Fig. 12 for several values of ΓS in the case of

Page 13

13

strong interaction U/ΓN = 20. Here, we only show the

results either in positive or negative V since dI/dV is

symmetric with respect to V = 0 for ǫd/U = −0.5. The

crossover between the different regimes occurs around

ΓS/ΓN ≃ 2 where dI/dV has its maximum value 4e2/h

at zero bias voltage. The profiles of dI/dV in the N-lead

and S-lead dominant coupling regimes are shown in Fig.

12(a) and (b), respectively.

In Fig. 12(b), the dI/dV curves show properties analo-

gous to those in the noninteracting S-lead dominant cou-

pling case shown in Fig. 8(b); the peak of dI/dV moves

toward the gap edge with increasing ΓS. The peak val-

ues, however, depend on the Coulomb interaction and

become smaller than 4e2/h for large U. The suppression

of the peak is attributed to the inelastic scattering ow-

ing to the Coulomb interaction. However, in the extreme

limit of ΓS ≫ ΓN, U, the superconducting correlation

dominates the QD and thus the Coulomb interaction ef-

fects are reduced, leading to the suppression of the in-

elastic scattering. For ΓS/ΓN= 7.5, therefore, the peak

value increases again. For later discussions, we refer to

these peaks as Peak C. On the other hand, the results

of dI/dV in the N-lead dominant coupling regime(Fig.

12(a)) are a little bit complicated; the peak at zero bias

voltage splits with decreasing ΓS, forming a double-peak

structure both in the positive and negative half of the

subgap voltage regions. The origin of these two peaks is

the same as discussed above, so that we denote them as

Peak A and Peak B. With decreasing ΓS, the position

of Peak A hardly changes, while that of Peak B shifts

toward the gap edge. With further decreasing ΓS, both

of Peak A and Peak B reduce their heights. Therefore,

for ΓS→ 0, dI/dV in the gap is completely suppressed

as is the case for U = 0.

In order to elucidate the origin of the peak formation,

we plot the ΓSdependence of the positions of Peak A, B

and C, which are labeled as VA, VBand VC, in Fig. 13(a).

Here, we take U as the energy unit. For comparison, the

renormalized couplings,?ΓNand?ΓS, are also plotted. For

the system is in the N-lead (S-lead) dominant coupling

regime. It is seen that VAand VCapproach the values of

?ΓN and?ΓS in the limit of ΓS→ 0 and ∞, respectively.

ΓS/U ≃ 0.1, where the peak is located at V = 0 with

the unitary limit value, 4e2/h. At the crossover point,

the width of the zero bias peak is simply scaled by?Γ =

differential conductance calculated for several choices of

∆ and ΓSquickly decreases around |eV | =?Γ.

understood in terms of the Andreev bound states. The

open diamonds in Fig. 13(a) denote the energy ωb of

the Andreev bound states at the QD for ΓN = T = 0,

which is obtained with the NRG calculations66–68. As

mentioned in the previous section, the system shows a

transition between the magnetic doublet and SC singlet

ΓS/U<

∼(>

∼)0.1,?ΓN is larger (smaller) than?ΓS, namely,

The crossover between these two limits appears around

?ΓN =?ΓS as shown in Fig. 13(b). It is seen that the

The above properties in the conductance are clearly

FIG. 13: (Color online) (a) Semilog plot of the peak positions

of dI/dV in the N-QD-S system, VA and VB, as a function of

ΓS/U. The other parameters are ΓN/U = 0.05, ǫd/U = −0.5,

∆/U = 0.025 and kBT/ΓN = 0.01, which are the same as

in Fig. 12. For comparison, the renormalized couplings in

the N-QD-S system with V = T = 0,?ΓN and?ΓS, and the

energy of the Andreev bound states in the QD-S system with

T = 0, ωb, are also plotted. (b) Semilog plot of dI/dV at the

crossover point against V/?Γ for several choices of ∆ and ΓS.

The other parameters are the same as in (a). The value of?Γ

is 0.0551, 0.124, 0.157, 0.187 and 0.213 for each curve from

top to bottom. In the low voltage region with eV <?Γ, dI/dV

fall into a single curve. Inset shows the same data of dI/dV

as a function of eV/U.

states for ΓN = T = 0. The transition point is eval-

uated as ΓTP

S/U ≃ 0.129 from the condition ωb = 0.

For ΓS≤ (≥)ΓTP

ground state and the Andreev bound states originating

from the SC singlet (doublet) appear in the LDOS at the

QD (see also the discussion in Fig. 7). For finite ΓN,

the local moment of the doublet ground state is screened

by the electrons in the N-lead. Therefore, the transition

change into the crossover between the Kondo singlet and

the SC singlet. It is clearly seen in Fig. 13(a) that VB

is indeed related to the Andreev bound states since VB

approximately coincides with the energy of the Andreev

bound states. This is also the case for Peak C, and the

difference between them comes from whether the ground

state is the Kondo singlet (Peak B) or SC singlet (Peak

C). We therefore reveals the origin of Peak B and C;

when the energy corresponding to the Andreev bound

states is externally supplied by the applied bias voltage,

S, the doublet (SC singlet) becomes the

Page 14

14

0.05 0.1 0.15 0.2 0.25

ΓS / U

0

0.2

0.4

0.6

0.8

1

1.2

eV / ∆

0.005

0.01

0.015

0.02

0.025

FIG. 14: (Color online) False color-scale representation of

|?d↓d↑?| as a function of V and ΓS for ΓN/U = 0.05, ǫd/U =

−0.5, ∆/ΓN = 0.5 and kBT/ΓN = 0.01. The filled circles,

triangles and squares indicate the ΓS dependence of eVA/∆,

VB/∆ and VC/∆.

the weight of the excited state at the QD is increased, re-

sulting in the enhancement of the Andreev reflection. Al-

though the nontrivial correspondence between the peak

position of dI/dV and the energy of the Andreev bound

states has already been discovered by Deacon et al29,30

via the experimental studies, to our knowledge, this is the

first numerical calculation which systematically clarifies

the correspondence in both coupling regimes by taking

into account the Kondo effect.

The difference between VBand VCcan be more clearly

seen in the superconducting pairing correlation at the

QD, |?d↓d↑?|. Figure 14 shows the false color-scale repre-

sentation of |?d↓d↑?| as a function of ΓS and V . The

filled circles, triangles and squares on the representa-

tion indicate the ΓS dependence of eVA/∆, eVB/∆ and

eVC/∆ shown in Fig. 13. In the N-lead dominant cou-

pling regime, e.g., in the case of ΓS/U = 0.05, the su-

perconducting correlation at the QD is weak at V = 0

since the Kondo singlet is dominant. With increasing V ,

|?d↓d↑?| shows the peak at the bias voltage where Peak B

is located. This result clearly indicates the enhancement

of the weight of the SC singlet state at the finite bias

voltage. On the other hand, in the S-lead dominant cou-

pling regime, e.g., in the ΓS/U = 0.3 case, it is seen that

|?d↓d↑?| is relatively large at V = 0, and monotonically

decreases with increasing V . In particular, a rapid de-

crease of |?d↓d↑?| occurs around V = VC, implying that

the weight of the magnetic doublet state increases in the

system instead of the SC singlet state at V = VC. All

these features are consistent with the above interpreta-

tion of the peaks in the conductance.

It is instructive to consider the LDOS for finite V to see

the nature of the bound states. Since the V dependence

of the LDOS in the N-lead dominant coupling regime has

already been discussed in Fig. 10, here we focus on the

0

0.2

0.4

0.6

0.8

1

-2-1.5-1 -0.5 0 0.5 1 1.5 2

ρ(ω)πΓN

ω / ∆

(a)

(b)

eV/∆=0

eV/∆=0.2

eV/∆=0.4

eV/∆=0.6

eV/∆=1

eV/∆=2

0.5

0.55

0.6

0.65

0.7

00.20.40.60.81

eV / ∆

ω+/∆

-ω−/∆

µN/∆

FIG. 15: (Color online) (a) Local density of states at the

QD for several values of V for ΓS/U = 0.15, ΓN/U = 0.05,

ǫd/U = −0.5, ∆/ΓN = 0.025 and kBT/ΓN = 0.01.

S-lead dominant coupling regime. Figure 15(a) shows the

LDOS at the QD for ΓS/U = 0.15. In this S-lead dom-

inant coupling case, there are the Andreev resonances

corresponding to the excited doublet for ΓN = 0. The

double peaks of the LDOS in the equilibrium state are

located at ω/∆ ≃ ±?ΓS ≃ ±0.5. For eV/∆ = 0.2, the

tion is absent in the system. For eV/∆ > 0.4, however,

the bias voltage increases the distance between the two

resonances and smears them. In order to see the V de-

pendence of the resonances in detail, we plot the peak po-

sition of the resonances as a function of V in Fig. 15(b).

Here, ω+ and ω− denote the positions of the peaks for

positive and negative ω, respectively. With increasing

V , they move toward the opposite gap edges. In particu-

lar, the peaks become sensitive to the change of the bias

voltage around eV/∆ ≃ 0.3, which approximately cor-

responds to the value of eVC/∆. This feature indicates

that the peaks tend to follow the chemical potential of

N-lead and its counter position, µNand −µN, which can

be regarded as a kind of pinning effect of the Andreev

resonances. Note that the profiles of ω+ and −ω− ap-

proximately coincide with each other, implying that the

resonances keep their symmetric structure with respect

to the Fermi level of S-lead. Besides, in the N-lead dom-

inant coupling regime, the pinning of the resonance be-

comes more prominent as discussed in Fig. 10. The origin

resonances show little change as if the Coulomb interac-

Page 15

15

0.05 0.1 0.15 0.2 0.25

ΓS / U

0

0.2

0.4

0.6

0.8

1

1.2

eV / ∆

0

0.5

1

1.5

FIG. 16: (Color online) False color-scale representation of

dω+/dµN as functions of V and ΓS for ΓN/U = 0.05, ǫd/U =

−0.5, ∆/U = 0.025 and kBT/ΓN = 0.01. The filled circles,

triangles and squares indicate eVA/∆, VB/∆ and VC/∆.

of the pinning is attributed to the Kondo effect, and is

understood as follows: in the S-lead dominant coupling

case, the electron correlation is practically negligible in

the low energy and low voltage region, except for the

renormalization effects, since the SC singlet is dominant

at the QD. With increasing V , however, the weight of

the magnetic doublet state increases in the system near

V ≃ VC. Then the resulting doublet state is screened

by the electrons in the N-lead owing to the Kondo effect,

leading to the pinning of the resonances.

In order to further investigate the pinning of the res-

onances in the LDOS, we show the false color-scale rep-

resentation of dω+/dµN in Fig 16. Since dω+/dµN be-

comes large if the resonance of the LDOS in the pos-

itive ω region follows the chemical potential of the N-

lead, its value gives an estimate of how strong the Kondo

correlation is. In the N-lead dominant coupling regime,

ΓS/U<

∼0.1, the Kondo pinning effect is suppressed with

increasing V since the bias voltage destroys the Kondo

singlet state. In particular, dω+/dµNis rapidly decreases

when the value of the bias voltage approaches VB be-

cause the weight of the SC singlet is increased at the

voltage. In contrast, in the S-lead dominant coupling

regime, dω+/dµN increases with increasing V from zero

and takes a peak at a finite bias voltage where Peak

C appears in dI/dV . Therefore, it is intuitively under-

stood that the Kondo correlation is enhanced by the bias

voltage around V = VC. With further increase in V ,

the Kondo correlation is weakened again by the applied

bias voltage, which in turn leads to the suppression of

dω+/dµN. The different features in dω+/dµNat V = VB

and V = VCreflect the difference in the origin of the An-

dreev bound states. Therefore, all the results of dI/dV ,

|?d↓d↑?| and dω+/dµN are consistent with the scenario

that the weight of the excited states is enhanced when

No

magnetic

solution

0.2 0.25 0.3 0.35

ΓS / U

0

0.2

0.4

0.6

0.8

1

1.2

eV / ∆

0

0.05

0.1

0.15

0.2

0.25

0.3

FIG. 17: (Color online) False color-scale representation of

the magnetization at the QD as functions of V and ΓS for

ΓN/U = 0.05, ǫd/U = −0.5, ∆/U = 0.025 and kBT/ΓN =

0.01.

the strength of the bias voltage coincides with the energy

of the excited states.

3.Comparison with mean-field results

In order to clarify the role of electron correlations at a

finite bias voltage, it is instructive to compare the present

results with the mean-field approximation in U. In the

mean-field approximation, there are two types of the so-

lutions: the magnetic and nonmagnetic ones. The mag-

netic solution appears only in the strong Coulomb in-

teraction case, as known in the Anderson model70. Al-

though the magnetic solution does not describe the cor-

rect physics at zero temperature, we can infer the en-

hancement of the magnetic correlations at the QD via its

existence. Hence, the mean-field analysis highlights the

importance of the correlation effects, as discussed in the

Anderson model out of equilibrium71.

Figure 17 shows the local magnetization m = (?n↑? −

?n↓?)/2 at the QD as a function of V and ΓS. In the

case of 0.25<

∼ΓS/U<

magnetic solution exists only at a finite bias voltage; the

bias voltage induces the local moment at the QD. With

further increase in V , however, the magnetic solution dis-

appears again. Such a reentrant behavior is not found in

the nonequilibrium N-QD-N system for ǫd/U = −0.571.

Accordingly, we attribute the reentrant behavior to the

competition/cooperation of the magnetic and supercon-

ducting correlations at a finite bias voltage. The reen-

trant behavior of the magnetic boundary clearly indicates

that the magnetic correlation is enhanced by the bias

voltage. Of course, the magnetic state is an artifact of

the approximation and should be replaced by the Kondo

singlet state, but the reentrant behavior of the boundary

is consistent with the enhancement of the Kondo corre-

∼0.3, it is noteworthy that the