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
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
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-
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
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)
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
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
gr(t,t′) = −iθ(t − t′)
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
ga(t,t′) = iθ(t′− t)
g<(t,t′) = i
g>(t,t′) = −i
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) = [Σr
t(ω) = −i2ΓN
?f(ω − µN)0
0f(ω + µN)
?1 − f(ω − µN)
t(ω) = i2ΓN
01 − f(ω + µN)
(1 − f(ω)). (18)
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.
1,2,3) is a Pauli matrix in Nambu space.
√ω2−∆2θ(|ω| − ∆) +
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=
i√∆2−ω2θ(∆ − |ω|) and f(x) =
The full lesser and greater ones are calculated from the
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,
2nd(ω) = −U2
G<,>(ω) = −Gr(ω)
(ω) + Σ<,>
1st= U?N? with
2nd(ω) = −U2
2πΠ<(ω + ω1)σ2
2πΠ>(ω + ω1)σ2
22(ω − ω1)
21(ω − ω1)
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,
ω − ω1± iη
2nd(ω1) − Σ>
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
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,
dot + HN+ HTN,
↓+ 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-
energies at the QD can be exactly obtained as,
atm(ω) = U2χ((ω + iη)I − ∆)−1,
atm(ω) = U2χ((ω − iη)I − ∆)−1,
atm(ω) = −i2ηU2χ((ω + iη)I − ∆)−1
×((I − σ3)/2 + ?N?)
×((ω − iη)I − ∆)−1,
atm(ω) = i2ηU2χ((ω + iη)I − ∆)−1
×((I + σ3)/2 − ?N?)
×((ω − iη)I − ∆)−1,
χ ≡ ?nd?(1 − ?nd?) − |?d↓d↑?|2,
?ǫd+ U(1 − ?nd?)∆d− U?d↓d↑?
−ǫd− U(1 − ?nd?)
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-
H = Hdot+ HN+ HS+ HTN+ HTS,
Hdot = (ǫd+ U?nd?)
↓+ H.c.), (35)
where ?nd? and ?d↓d↑? are the effective parameters repre-
senting the energy shifts of the one-particle Green’s func-
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,
2nd(ω) → U2χ0((ω + iη)I − ∆0)−1,
2nd(ω) → U2χ0((ω − iη)I − ∆0)−1,
2nd(ω) → −i2ηU2χ0((ω + iη)I − ∆0)−1
×((I − σ3)/2 + ?N?0)
×((ω − iη)I − ∆0)−1,
2nd(ω) → i2ηU2χ0((ω + iη)I − ∆0)−1
×((I + σ3)/2 − ?N?0)
×((ω − iη)I − ∆0)−1,
χ0 ≡ ?nd?0(1 − ?nd?0) − |?d↓d↑?0|2,
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
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,
2nd(ω) = A
2nd(ω) = A[Σa
U(ω) = U?N? +U2χ
I + O(1
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,
I + O(1
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
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,
1 − ?nd? − ?nd?
−?d↓d↑?∗− ?d↓d↑?∗−1 + ?nd? + ?nd?
−?d↓d↑? − ?d↓d↑?
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,
Multiplying the modified self-
We now confirm that the above self-energy reproduces
the atomic-limit form.
energy (48) on the left and right by the inverse matrices
U2χ((I − σ3)/2 + ?N?0).
2nd(ω), we take the superconducting
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,
U2χ((I − σ3)/2 + ?N?).
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,
2nd(ω) → Σ<
atm(ω) (∆ → ∞,ΓN→ 0). (51)
We also define the modified greater self-energy as,
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,
We indeed confirm that the modified self-energies satisfy
D.Current conservation and consistency of the
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
??2f(ω − µN)Gr
. Since we assume that the system
?ˆIN? = −4eΓN
11(ω) + G<
?ˆIS? = −4eΓS
11(ω) + β∗(ω)G<
12(ω) + β∗(ω)G<
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,
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,
U?d↓d↑? = U?d↓d↑? + [?Σr
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
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
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-
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
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 0.2 0.4 0.6 0.8 1
eV / ∆
eV / ∆
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
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
where?ΓNand?ΓSare the renormalized couplings defined
?ΓN = zΓN,
z = (1 +ΓS
?ΓS = z(ΓS+ [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
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
U / ΓN
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
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 5 10 15 20
U / 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
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
width?ΓN. The change from a single resonance to the
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
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
= 1, the strong
0 0.1 0.20.30.4 0.5
ΓS / U
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
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.
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-
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 = ∞).
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
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
√2(|0? − | ↑↓?),
√2(|0? + | ↑↓?).
Note that the singly occupied states are degenerate (zero
energy), and |S1? and |S2? have the different energies,
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+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
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.
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
-1 -0.5 0 0.5 1
eV / ∆
1 2 5 10
ΓS / ΓN
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-
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
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.
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
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
-2 -1.5 -1-0.5 0 0.5 1 1.5 2
ω / ∆
0 0.20.40.6 0.81
eV / ∆
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
-1-0.5 0 0.5 1
eV / ΓN
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
-1 -0.5 0 0.5 1
eV / ∆
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-
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-
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
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
∼)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
0.05 0.1 0.15 0.2 0.25
ΓS / U
eV / ∆
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
-2-1.5-1-0.5 0 0.5 1 1.5 2
ω / ∆
0 0.20.4 0.60.81
eV / ∆
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-
0.05 0.1 0.15 0.2 0.25
ΓS / U
eV / ∆
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,
∼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
0.2 0.25 0.3 0.35
ΓS / U
eV / ∆
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 =
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<
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
FIG. 18: (Color online) (a) Mean-field approximation results
of the differential conductance: ΓN/U = 0.05, ǫd/U = −0.5,
∆/U = 0.025 and kBT/ΓN = 0.01. The energy of the An-
dreev bound states, ωb/∆, as a function of ǫd/U is denoted
by diamonds. The black line indicates the boundary of the
magnetic solution. (b) MPT results of the differential con-
ductance. The parameters are same as those in (a).
lation deduced from the MPT calculation.
We also calculate the differential conductance, dI/dV ,
with the mean-field approximation. Figure 18(a) shows
the false-color scale representation of dI/dV as functions
of eV/∆ and the logarithm of ΓS/U. The black solid
line indicates the magnetic boundary, and the diamond
denotes the energy of the Andreev bound states. In the
region where the magnetic solution exists, the ridge posi-
tion of dI/dV features a curve similar to the energy of the
Andreev bound states. On the other hand, in the large
ΓS region where the non-magnetic solution only exists,
the ridge is located around the gap edge and coincides
with the Andreev bound states. As a result, the mean-
field solutions still capture the essential nature of the
Andreev bound states, except around the boundary line.
At the boundary, the conductance changes in a discon-
tinuous manner, implying that the physics related to the
Andreev bound states can be qualitatively understood
without quantum fluctuations48.
On the other hand, quantum fluctuations induce the
intriguing phenomena a la Kondo, which cannot be de-
scribed at the mean-field level in the region where the
magnetic and superconducting correlations compete with
each other.Figure 18(b) shows the MPT results of
dI/dV . In addition to the good correspondence between
the ridge of dI/dV and the energy of the Andreev bound
states, the anomalous enhancement occurs around the
transition point of the ground state in the QD-S system,
which is caused by the Kondo-assisted Andreev reflec-
tion. The quantum fluctuation effects are significant at a
finite bias voltage in the vicinity of the transition point;
The Kondo-assisted Andreev reflection, as well as the
pinning effect of the Andreev resonances discussed in the
previous section, is seen only at a finite bias voltage and
smeared away from the transition point. Applying the
bias voltage resembles increasing temperature concern-
ing the enhanced weight of the excited states.
B.Nonequeliburium transport for generic (ǫd, V )
cases: comparison with experiments
In the remainder of the section, we discuss the differ-
ential conductance as functions of the energy level at the
QD, ǫd, and the bias voltage, V . Note that ǫdcan be eas-
ily controlled by the gate voltage in actual experiments.
We compare the analysis done here with the recent ex-
periments qualitatively in good agreement.29,30
1.Transport via Andreev bound states
Let us first focus on the simple cases where the Kondo-
assisted Andreev transport is not so important, for which
we can highlight the crossover behavior in the transport
due to the Andreev resonances.
Figure 19 is the false color-scale plot of dI/dV as a
function of ǫdfor several values of ΓS. We also show the
ǫd dependence of the energy of Andreev bound states,
which is calculated for ΓN= T = 0 with using the NRG
method68. In Fig. 19(a) and (b), the system is in the S-
lead dominant coupling regime. It is clearly seen that the
value of the bias voltage where dI/dV takes a peak value
coincides approximately with the energy of the Andreev
bound states. The peak ridges correspond to Peak C
defined in the previous subsection. For ΓS/U = 0.16, the
system shows a crossover around ǫd/U = −0.5 and the
conductance has the maximum value, 4e2/h (center of
the figure 19(c)). Note that the energy of Andreev bound
states does not touch the zero axis since ΓS/U = 0.16
characterizing the crossover is slightly different from the
exact transition point. However, the peak positions still
FIG. 19: (Color online) False color-scale representation of the differential conductance as functions of V and ǫdfor ΓN/U = 0.1,
∆/U = 0.05 and kBT/ΓN = 0.01: (a) ΓS/U = 0.5, (b) ΓS/U = 0.25, (c) ΓS/U = 0.16, (d) ΓS/U = 0.1, (e) ΓS/U = 0.05. The
energies of the Andreev bound states, ωb/∆, are also plotted.
approximately correspond to the energy of the Andreev
With decreasing ΓS, the system enters the N-lead dom-
inant coupling regime as shown in Fig. 19(d). In the
case of ΓN = T = 0 with the other parameters same as
in (d), the magnetic doublet becomes the ground state
around ǫd/U = −0.5, and a transition occurs, where
the softening of the Andreev bound states occurs around
ǫd/U ≃ −0.5 ± 0.32. After the transition, the Andreev
bound states move toward the gap edges. For finite ΓN,
the magnetic doublet is changed to the Kondo singlet,
which is realized around the center of the figure. Re-
garding dI/dV , the peaks corresponding to the Andreev
bound states are not so clearly seen as in the S-lead dom-
inant coupling regime. This is because ΓNis not small in
comparison with ΓS. The peaks around the symmetric
point are especially indistinct because there are two kind
of peaks of dI/dV : Peak A originating from the Kondo-
assisted Andreev reflection and Peak B corresponding to
the Andreev bound states. With further decreasing ΓS,
the bare value of the coupling of the S-lead, ΓS, becomes
smaller than that of the N-lead, ΓN. For ΓN> ΓS, these
peaks almost disappear in the subgap voltage. However,
there is a remnant of the Kondo-assisted Andreev reflec-
tion, which makes a dip at V = 0 in Fig. 19(e). For
ΓS → 0, the remnant is also diminished and there are
only peaks at the gap edges as in the noninteracting N-
lead dominant coupling regime.
As a result, both for ΓN/ΓS→ 0 and ∞, the subgap
conductance, except at the gap edges, is completely sup-
pressed for any values of ǫd. A wide variety of patterns of
the differential conductance result from the competition
between the Coulomb interaction and the superconduct-
ing correlations at the QD.
It is to be noted here that the pronounced gap-edge
peaks in the cases of ΓN ≪ ΓS and ΓN ≫ ΓS, and the
prominent peaks corresponding to the Andreev bound
states in the S-lead dominant regime are qualitatively in
agreement with the recent experimental results29,30. In
the experiment29,30, the softening of the Andreev bound
states in the N-lead dominant coupling regime is also
observed in the distinctive peaks in the dI/dV measure-
ment. However, we do not find such a distinctly sep-
arated peak in the case ΓN ≥ ΓS. We will discuss the
visibility of the peaks separately below.
We also calculate the (ǫd, V ) dependence of |?d↓d↑?| as
shown in Fig. 20. For ΓS/U = 0.5, it is seen that |?d↓d↑?|
is large around the center of the figure and decreases
with increasing ǫd since the dominant SC-singlet state,
which consists of the superposition of doubly-occupied
and empty states, is weakened in the empty or doubly-
occupied region. For finite V , |?d↓d↑?| decreases rapidly
around the voltage corresponding to the energy of the
Andreev bound states of a doublet character. For small
ΓS, the peak of |?d↓d↑?| is divided into two which are
located around the voltage corresponding to the energy
of the Andreev bound states of a SC-singlet character.
In spite that the electron and hole components of An-
FIG. 20: (Color online) False color-scale representation of |?d↓d↑?| as functions of V and ǫd for ΓN/U = 0.1, ∆/U = 0.05 and
kBT/ΓN = 0.01: (a) ΓS/U = 0.5, (b) ΓS/U = 0.25, (c) ΓS/U = 0.16, (d) ΓS/U = 0.1, (e) ΓS/U = 0.05. The energies of the
Andreev bound states, ωb/∆, as a function of ǫd/U are also plotted.
dreev bound states have the same energy, there is asym-
metry in dI/dV and |?d↓d↑?| as a function of V . For
instance, |?d↓d↑?| is strongly suppressed in the right top,
in comparison with the one in the right bottom in Fig.
20. Moreover, the asymmetric feature becomes promi-
nent with decreasing ΓS. In order to explain how the
asymmetry emerges, let us consider the ΓS → 0 limit.
Figure 21 shows the schematic phase diagram of the N-
QD system with strong U.
dominant for ǫd/U = −0.5 and V = 0, and is weakened
with increasing |ǫd+ 0.5|. Note that V just shifts the
chemical potential of the N-lead, so that only one of the
two parameters, eV and ǫd, becomes relevant; the sys-
tem stays in the same state along µN = ǫd line in the
figure. For −0.5 + ∆/U < ǫd/U, therefore, the Kondo
correlation is enhanced if we fix ǫdand increase V . This
enhancement of the Kondo correlation would occur in the
N-QD-S system with small ΓS, giving rise to the asym-
The Kondo correlation is
2.How to observe Kondo-assisted Andreev transport
The subgap peak structure in the differential conduc-
tance in the N-lead dominant coupling regime is a bit
more complicated than that in the S-lead dominant cou-
pling regime because both of the Kondo-assisted Andreev
reflection and the Andreev bound states could affect the
FIG. 21: (Color online) Schematic phase diagram of the N-
QD-S system for ΓS = 0 and large U.
conductance profile as seen in Fig. 19. Therefore how we
can observe the conductance peaks in the N-lead domi-
nant coupling regime depends sensitively on the ratio of
the bare coupling strengths. We find that if the N-lead
dominant system is in the condition ΓN/ΓS< 1, that is
if?ΓN/?ΓS > 1 and ΓN/ΓS < 1, the conductance peaks
results below in the case satisfying this specific condition.
Figure 22 shows the differential conductance for several
become prominent in the subgap voltage. We show some
-1-0.5 0 0.5 1
eV / ∆
FIG. 22: (Color online) Differential conductance as a function
of V for several values of (ΓS, ∆): ΓN/U = 0.05, ǫd/U =
−0.5, and kBT/ΓN = 0.01.
values of (Γ, ∆) and ǫd/U = −0.5. Note that the system
is in the N-lead dominant coupling regime for any choices
of (Γ, ∆) (see also the phase diagram of Fig. 7). The
two peaks in the vicinity of V = 0 are related to the
Kondo-assisted Andreev reflection and the other two at
eV/∆ ≃ ±0.5 originate from the Andreev bound states.
These four peaks become sharper and more prominent
from the top to bottom lines. In particular, the Kondo-
like peaks approach the zero bias voltage since the peaks
are located at eV ≃ ±?ΓN and?ΓN decreases from top to
ΓN, only a single peak around V = 0, instead of the
two peaks at finite bias voltages, may be observed in real
We next look at the ǫddependence of the N-QD-S sys-
tem for ΓN/U = 0.05, ΓS/U = 0.1 and ∆/U = 0.075
(Fig. 23).In this case,?ΓN >?ΓS is satisfied at the
tion to the condition of ΓN/ΓS< 1, which leads to the
crossoverin the dominant couplings may occur away from
the symmetric point as shown Fig. 23(a). The crossover
points are approximately coincide with the doublet-
singlet transition points denoted by arrows for ΓN = 0.
In the particle-hole symmetric case,?ΓN/ΓN≃ 0.1 char-
the Kondo-type peaks in dI/dV . For kBT <?ΓN, both of
bound states at the QD contribute to the nonlinear elec-
tron transport. Figure 23(b) shows the ǫddependence of
dI/dV at kBT/ΓN = 0.01. Regarding the central two
peaks related to the Kondo effect, one of the peaks in-
creases its height away from ǫd/U = −0.5, whereas the
other disappears. We note that the curve of dI/dV ends
at certain voltages for ǫd/U = −0.4, 0.3, 0.2 and −0.1
since we cannot get the convergent solution in the frame-
work of MPT. Therefore, the movement of the one of
the two peaks owing to the Andreev bound states is not
clear in this figure, unfortunately. Nevertheless, we can
bottom. Therefore, we conclude that for U ≫ ΓS,∆ ≫
particle-hole symmetric point (ǫd/U = −0.5) in addi-
acterizes the position and the temperature dependence of
the Kondo-assisted Andreev reflection and the Andreev
see that the other peak moves to lower bias voltages and
merges into the single peak with the prominent peak re-
lated to the Kondo effect. With further increasing ǫd, the
merged peak moves toward the gap edge and there is no
peak in the subgap voltage.
A quantitative comparison of the peak positions with
the energy of the Andreev bound state, ωb/∆, is shown
in Fig. 23(c). Softening of the Andreev bound states
clearly emerges in the peak structure of the differen-
tial conductance. The softening reflects the transition
of the ground states in the case that ΓN = 0.
Andreev reflection ridges around V
ridges are separated around ǫd/U − 0.5. In the case of
∼−0.75 or −0.25>
are not observed since the system is away from the Kondo
regime, and there appear only the gap-edge peaks related
to the Andreev bound states. The dashed lines in the fig-
ure denote the conditions where the level of the QD and
the chemical potential of the N-lead satisfy the equations;
µN = ǫd+ U and µN = ǫd. In spite that the Andreev
bound states have the symmetric energy spectrum, the
large asymmetry is found in dI/dV as a function of V ,
except for ǫd/U ?= −0.5. The asymmetry reflects the fact
that the Kondo correlations appear differently depending
on the sign of the bias voltage, as discussed in Fig. 21.
It is remarkable that the overall features of the differ-
ential conductance are consistent with those in the re-
cent experiment30, including the observation of the fin-
gerprints of two kinds of peaks; Kondo-type peaks at
eV/∆ ≃ 0 and the peaks due to the Andreev bound
states. There still seems to be a small discrepancy be-
tween the theory and the experiment30. In our theory,
the Kondo-type ridges are separated at the particle-hole
symmetric point, while a single Kondo ridge is observed,
instead of the two ridges, in the experiment. We believe
that the single ridge is a consequence of the special condi-
tion, U ≫ ΓS,∆ ≫ ΓN, used in the experiment. In this
case, the two Kondo-type peaks for ǫd/U = −0.5 would
be located near V = 0 and overlap with each other, as
discussed in Fig. 22. Hence, the two Kondo ridges could
be also connected at the particle-hole symmetric point in
such a special condition.
Finally, some comment are in order for the tempera-
ture dependence. In Fig. 23(d), we present dI/dV at a
higher temperature kBTK/ΓN= 0.1, which is compara-
ble to?ΓN. The Kondo-type ridges are smeared with in-
bound states show little change. Therefore, at higher
temperatures, there would be only the crossover behav-
ior of the fingerprint of the Andreev bound states in the
∼−0.25, there are two Kondo-assisted
= 0.The two
∼ǫd/U, the Kondo-type ridges
creasing temperature, but the peaks due to the Andreev
In this paper, we have theoretically investigated the
nonequilibrium electron transport through a quantum
FIG. 23: (Color online) (a) Renormalized coupling strengths as a function of ǫd for ΓN/U = 0.05, ΓS/U = 0.1, ∆/U = 0.075
and T = V = 0. (b) Differential conductance as a function of the bias voltage for several values of ǫd/U and kBT/ΓN = 0.01.
From bottom to top, ǫd/U = −0.5, −0.4, −0.3, −0.2, −0.1, 0, 0.1, 0.2 and 0.3. The other parameters are the same as in (a).
(c) False color-scale representation of the differential conductance as a function of V and ǫd. The parameters are the same as
in (b). The energies of the Andreev bound states, ωb/∆, are denoted as diamonds. The bound states exist for ΓN = T = 0.
The dashed lines indicate the resonant conditions; µN = ǫd and µN = ǫd+ U. (d) At kBT/ΓN = 0.1 which is nearly equal to
that of?ΓN/ΓN, the central Kondo-enhanced Andreev ridges are suppressed.
dot coupled to the normal and superconducting leads
with particular emphasis on the interplay between the
Kondo effect and the superconducting correlations. For
this purpose, we have developed the modified second or-
der perturbation theory in Keldysh-Nambu formalism
under nonequilibrium steady-state conditions. We have
confirmed that this method is indeed efficient for analyz-
ing the nonequilibrium electron transport in the present
It has been shown that the renormalized couplings be-
tween the leads and the dot in the equilibrium states are
the key quantities that correctly describe nonequilibrium
transport properties. In particular, the enhancement of
the Andreev transport occurs via a Kondo resonance at
a finite bias voltage, giving rise to an anomalous peak
structure in the differential conductance, whose position
is determined solely by the above-mentioned renormal-
ized parameters. This peak formation is a remarkable
example of phenomena that are indeed caused by the
interplay between the Kondo and superconducting cor-
relations. A pinning effect of the Andreev resonances to
the Fermi level of the normal lead and its counter level
also evidences the interplay of the above two types of
correlations. Moreover, it has been shown that the en-
ergy levels of the Andreev bound states give rise to an
additional peak structure in the differential conductance
in the strongly correlated N-QD-S system.
We have demonstrated that the above characteristic
features of nonequilibrium differential conductance ob-
tained from our calculation are qualitatively in agreement
with those observed in the recent experiments29,30. Fi-
nally we note that there still exists a small discrepancy
between the theory and the experiments; the Kondo-type
ridges are separated in our theory (Fig. 23), while a sin-
gle Kondo ridge is observed experimentally.30We believe
that the single ridge is a consequence of a specific con-
dition employed in the experiments, U ≫ ΓS,∆ ≫ ΓN
and that the clearly-separated Kondo ridges predicted in
this paper will be observed if the proper conditions for
the system parameters are prepared experimentally.
We would like to thank A. Oguri, R. S. Deacon,
M. Imada and J. Bauer for fruitful discussions.
work was supported by KAKENHI (Nos.
20104010), the Grant-in-Aid for the Global COE Pro-
grams “The Next Generation of Physics, Spun from Uni-
versality and Emergence” from MEXT of Japan, and
JSPS through its FIRST ProgramD Y. Yamada is sup-
ported by JSPS Research Fellowships for Young Scien-
tists, and Y. Tanaka is supported by Special Postdoctoral
Researchers Program of RIKEN.
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