Non-Equilibrium Josephson and Andreev Current through Interacting
  Quantum Dots
7 Non-Equilibrium Josephson and Andreev Current
through Interacting Quantum Dots
Marco G. Pala1, Michele Governale2, Jürgen König2
1 IMEP-MINATEC (UMR CNRS/INPG/UJF 5130), 38016 Grenoble, France
2 Institut für Theoretische Physik III, Ruhr-Universität Bochum, 44780 Bochum,
Germany
E-mail: pala@minatec.inpg.fr, michele@tp3.ruhr-uni-bochum.de,
koenig@tp3.ruhr-uni-bochum.de
Abstract. We present a theory of transport through interacting quantum dots
coupled to normal and superconducting leads in the limit of weak tunnel coupling.
A Josephson current between two superconducting leads, carried by first-order tunnel
processes, can be established by non-equilibrium proximity effect. Both Andreev and
Josephson current is suppressed for bias voltages below a threshold set by the Coulomb
charging energy. A π-transition of the supercurrent can be driven by tuning gate or
bias voltages.
PACS numbers: 74.45.+c,73.23.Hk,73.63.Kv,73.21.La
Submitted to: New J. Phys.
http://arxiv.org/abs/0704.0204v2
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 2
1. Introduction
Non-equilibrium transport through superconducting systems attracted much interest
since the demonstration of a Superconductor-Normal-Superconductor (SNS) transistor
[1]. In such a device, supercurrent suppression and its sign reversal (π-transition)
are achieved by driving the quasi-particle distribution out of equilibrium by means of
applied voltages [2, 3, 4, 5]. Another interesting issue in mesoscopic physics is transport
through quantum dots attached to superconducting leads. For DC transport through
quantum dots coupled to a normal and a superconducting lead, subgap transport is due
to Andreev reflection [6, 7, 8, 9, 10, 11]. Also transport between two superconductors
through a quantum dot has been studied extensively. The limit of a non-interacting
dot has been investigated in [12]. Several authors considered the regime of weak tunnel
coupling where the electrons forming a Cooper pair tunnel one by one via virtual states
[13, 14, 15]. The Kondo regime was also addressed [13, 16, 17, 18, 19]. Multiple
Andreev reflection through localized levels was investigated in [20, 21]. Numerical
approaches based on the non-crossing approximation [22], the numerical renormalization
group [23] and Monte Carlo [24] have also been used. The authors of [25] compare
different approximation schemes, such as mean field and second-order perturbation in
the Coulomb interaction. In double-dot systems the Josephson current has been shown
to depend on the spin state of the double dot [26]. Experimentally, the supercurrent
through a quantum dot has been measured through dots realized in carbon nanotubes
[27] and in indium arsenide nanowires [28].
In this Letter we study the transport properties of a system composed of an
interacting single-level quantum dot between two equilibrium superconductors where
a third, normal lead is used to drive the dot out of equilibrium. A Josephson coupling
in SNS heterostructures can be mediated by proximity-induced superconducting
correlations in the normal region. In case of a single-level quantum dot, superconducting
correlations are indicated by the correlator 〈d↓(0)d↑(t)〉, where dσ is the annihilation
operator of the dot level with spin σ. To obtain a large pair amplitude, i.e. the equal-
time correlator 〈d↓d↑〉, at least two conditions need to be fulfilled: (i) the states of an
empty and a doubly-occupied dot should be nearly energetically degenerate and (ii) the
overall probability of occupying the dot with an even number of electrons should be
finite. For a non-interacting quantum dot, i.e. vanishing charging energy U for double
occupancy, this can be achieved by tuning the level position ǫ in resonance with the
Fermi energy of the leads, ǫ = 0 [12]. In this case, the Josephson current can be viewed
as transfers of Cooper pairs between dot and leads and the expression of the current
starts in first order in the tunnel coupling strength Γ.
The presence of a large charging energy U ≫ kBT,Γ destroys this mechanism since
the degeneracy condition 2ǫ+U ≈ 0 is incompatible with a finite equilibrium probability
to occupy the dot with an even number of electrons. Nevertheless, a Josephson current
can be established by higher-order tunnelling processes (see, for example, [13, 14, 15]),
associated with a finite superconducting correlator 〈d↓(0)d↑(t)〉 at different times. The
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 3
amplitude of the Josephson coupling is, however, reduced by a factor Γ/∆, i.e., the
current starts only in second order in Γ, and the virtual generation of quasiparticles
in the leads suppresses the Josephson current for large superconducting gaps ∆. In
particular, it vanishes for ∆ → ∞.
The main purpose of the present paper is to propose a new mechanism that
circumvents the above-stated hindrance to achieve a finite pair amplitude in an
interacting quantum dot, and, thus, restores a Josephson current carried by first-order
tunnel processes that survives in the limit ∆ → ∞. For this aim, we attach a third,
normal, lead to the dot that drives the latter out of equilibrium by applying a bias
voltage, so that condition of occupying the dot with an even number of electrons is
fulfilled even for 2ǫ+ U ≈ 0.
We relate the current flowing into the superconductors to the nonequilibrium
Green’s functions of the dot. In the limit of a large superconducting gap, ∆ → ∞,
the current is only related to the pair amplitude. The latter is calculated by means
of a kinetic equation derived from a systematic perturbation expansion within real-
time diagrammatic technique that is suitable for dealing with both strong Coulomb
interaction and nonequilibrium at the same time.
2. Model
We consider a single-level quantum dot connected to two superconducting and one
normal lead via tunnel junctions, see figure 1. The total Hamiltonian is given by H =
ΓΓS S
Figure 1. Setup: a single-level quantum dot is connected by tunnel junctions to one
normal and two superconducting leads with tunnelling rates ΓN and ΓSL,R , respectively.
η=N,SL,SR
(Hη +Htunn,η). The quantum dot is described by the Anderson model
σdσ +Un↑n↓, where nσ = d
σdσ is the number operator for spin σ =↑, ↓, ǫ is
the energy level, and U is the charging energy for double occupation. The leads, labeled
by η = N, SL, SR, are modeled by Hη =
kσ ǫkc
ηkσcηkσ−
η−k↓ +H.c.
, where
∆η is the superconducting order parameter (∆N = 0). The tunnelling Hamiltonians are
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 4
Htunn,η = Vη
ηkσdσ +H.c.
. Here, Vη are the spin- and wavevector-independent
tunnel matrix elements, and cηkσ(c
ηkσ) and dσ(d
σ) represent the annihilation (creation)
operators for the leads and dot, respectively. The tunnel-coupling strengths are
characterized by Γη = 2π|Vη|
k δ(ω − ǫk).
3. Current formula
We start with deriving a general formula for the charge current in lead η by using
the approach of Ref. [29] generalized to superconducting leads. Similar formulae
that relate the charge current to the Green’s function of the dot in the presence of
superconducting leads have been derived in previous works, in particular for equilibrium
situations, see e.g. Refs. [16, 22]. The formula derived below is quite general, as
it allows for arbitrary bias and gate voltages, temperatures, and superconducting
order parameters for a quantum dot coupled to an arbitrary number of normal and
superconducting leads. For this, it is convenient to use the operators ψηk = (cηk↑, c
η−k↓)
and φ = (d↑, d
T in Nambu formalism. The current from lead η is expressed as
Jη = e 〈dNη/dt〉 = i(e/h̄)〈[H,Nη]〉 = i(e/h̄)〈[Htunn,η, Nη]〉 ‡, with Nη =
ηkτ3ψηk,
where τ1, τ2, τ3 indicate the Pauli matrices in Nambu space and e > 0 the electron
charge. Evaluating the commutator leads to
Jη = −
Re {Tr [τ3VηG
D,ηk(ω)]} , (1)
with Vη = Diag(Vη,−V
η ) and the lead–dot lesser Green’s functions (G
D,ηk(ω))m,n
that are the Fourier transforms of i〈ψ
ηkn(0)φm(t)〉. In the following, we assume the
tunnelling matrix elements Vη to be real (any phase of Vη can be gauged away by
substituting ∆η → ∆η exp(−2i arg Vη)). The Green’s function G
D,ηk is related to
the full dot Green’s functions and the lead Green’s functions by a Dyson equation in
Keldysh formalism: G<D,ηk(ω) = G
R(ω)V†ηg
ηk(ω)+G
<(ω)V†ηg
ηk(ω), where G
R(<)(ω) is
the retarded (lesser) dot Green’s function, and and g
ηk (ω) the lead advanced (lesser)
Green’s function. Using this relation and assuming energy-independent tunnel rates Γη,
we obtain for the current Jη = J1η + J2η with
J1η =
ΓηDη(ω)Im
ω − µη
2GR(ω)fη(ω) +G
, (2)
J2η =
ΓηSη(ω)Re
, (3)
where ∆η =
∆∗η 0
, and fη(ω) = [1 + exp(ω − µη)/(kBT )]
−1 is the Fermi
function, with T being the temperature and kB the Boltzmann constant. The dot
Green’s functions (G<D(ω))m,n and
GRD(ω)
are defined as the Fourier transforms
‡ Note that [Hη, Nη] 6= 0 but 〈[Hη, Nη]〉 = 0 for η = SL,R.
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 5
of i〈φ†n(0)φm(t)〉 and −iθ(t)〈{φm(t), φ
n(0)}〉, respectively. The two weighting functions
Dη(ω) and Sη(ω) are given by
Dη(ω) =
|ω − µη|
(ω − µη)2 − |∆η|2
θ(|ω − µη| − |∆η|)
Sη(ω) =
|∆η|2 − (ω − µη)2
θ(|∆η| − |ω − µη|).
The terms J1η and J2η involve excitation energies ω above and below the superconducting
gap, respectively. For η = N, only the part of J1η that involves normal (diagonal)
components of the Green’s functions contributes, and the current reduces to the result
presented in [29]. For superconducting leads, this part describes quasiparticle transport
that is independent of the superconducting phase difference. The other part of J1η
involves anomalous (off-diagonal) components of the Green’s functions and is, in general,
phase dependent. The contribution to the Josephson current stemming from this term is
the dominant one in the regime considered in [13, 14, 15]. The excitation energies above
the gap are only accessible either for transport voltages exceeding the gap or by including
higher-order tunnelling, involving virtual states with quasiparticles in the leads, and,
therefore, J1η vanishes for large |∆η|. In this case J2η, that involves only anomalous
Green’s functions with excitation energies below the gap, dominates transport. It is, in
general, phase dependent, and describes both Josephson as well as Andreev tunnelling.
In the following we consider the limit |∆η| → ∞, where the current simplifies to
Γη|〈d↓d↑〉| sin(Ψ− Φη) , (4)
with Φη being the phase of ∆η and 〈d↓d↑〉 = |〈d↓d↑〉| exp(iΨ) the pair amplitude of the
dot that has to be determined in the presence of Coulomb interaction, coupling to all
(normal and superconducting) leads and in non-equilibrium due to finite bias voltage.
We now consider a symmetric three-terminal setup with ΓSL = ΓSR = ΓS,
∆SL = |∆| exp(iΦ/2) and ∆SR = |∆| exp(−iΦ/2), and µSL = µSR = 0. The quantities
of interest are the the current that flows between the two superconductors (Josephson
current) Jjos = (JSL − JSR)/2 and the current in the normal lead (Andreev current)
Jand = JN = −(JSL + JSR).
Furthermore, we focus on the limit of weak tunnel coupling, ΓS < kBT . In this
regime, an Josephson current through the dot in equilibrium would be suppressed even
in the absence of Coulomb interaction, U = 0, since the influence of the superconductors
on the quantum-dot spectrum could not be resolved for the resonance condition ǫ ≈ 0.
This can, e.g., be seen in the exactly-solvable limit of U = 0 together with ΓN = 0,
where the Josephson current is Jjos = (e/2h̄)Γ
S sin(Φ) [f(−ǫA(Φ))− f(ǫA(Φ))] /ǫA(Φ)
with ǫA(Φ) =
ǫ2 + Γ2S cos
2(Φ/2). This provides an additional motivation to look for a
non-equilibrium mechanism to proximize the quantum dot.
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 6
4. Kinetic equations for quantum-dot degrees of freedom
The Hilbert space of the dot is four dimensional: the dot can be empty, singly
occupied with spin up or down, or doubly occupied, denoted by |χ〉 ∈ {|0〉, | ↑〉,
| ↓〉, |D〉 ≡ d
↓|0〉}, with energies E0, E↑ = E↓, ED. For convenience we define the
detuning as δ = ED −E0 = 2ǫ+ U . The dot dynamics is fully described by its reduced
density matrix ρD, with matrix elements P
≡ (ρD)χ2χ1 . The dot pair amplitude 〈d↓d↑〉
is given by the off-diagonal matrix element P 0D. The time evolution of the reduced
density matrix is described by the kinetic equations
P χ1χ2 (t) +
(Eχ1 −Eχ2)P
(t) =
(t, t′)P
(t′). (5)
We define the generalized transition rates byW
−∞ dt
(t, t′), which are the
only quantities to be evaluated in the stationary limit. Together with the normalization
condition
χ Pχ = 1, (5) determines the matrix elements of ρD. Furthermore, in (5)
we retain only linear terms in the tunnel strengths Γη and the detuning δ. Hence, we
calculate the rates W
to the lowest (first) order in Γη for δ = 0. This is justified in
the transport regime ΓS,ΓN, δ < kBT .
The rates are evaluated by means of a real-time diagrammatic technique [30],
that we generalize to include superconducting leads. This technique provides a
convenient tool to perform a systematic perturbation expansion of the transport
properties in powers of the tunnel-coupling strength. In the following, we concentrate
on transport processes to first order in tunnelling (a generalization to higher orders is
straightforward). This includes the transfer of charges through the tunnelling barriers
as well as energy-renormalization terms that give rise to nontrivial dynamics of the
quantum-dot degrees of freedom.
We find for the (first-order) diagonal rates Wχ1χ2 ≡ W
the expressions Wσ0 =
ΓNfN(−U/2);W0σ = ΓN[1 − fN(−U/2)];WDσ = ΓNfN(U/2);WσD = ΓN[1 − fN(U/2)].
The N lead also contributes to the rates WDD00 = (W
∗ = −ΓN[1 + fN(−U/2) −
fN(U/2) + iB] where B =
U/2−µN
2πkBT
−U/2−µN
2πkBT
, with µN being
the chemical potential of the normal lead and ψ(z) the Digamma function. Notice that
B vanishes when µN = 0 or U = 0. The superconducting leads do not enter here
due to the gap in the quasi-particle density of states. These leads, though, contribute
to the off-diagonal rates W 000D = W
W 0D00
W 0DDD
= −WD000 = −W
− (W 00D0)
WDDD0
= −iΓS cos(Φ/2).
For an intuitive representation of the system dynamics we define, in analogy to [31],
a dot isospin by
PD0 + P
; Iy = i
PD0 − P
; Iz =
PD − P0
. (6)
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 7
From (5), we find that in the stationary limit the isospin dynamics can be separated
into three parts, 0 = dI/dt = (dI/dt)acc + (dI/dt)rel + (dI/dt)rot, with
[1− fN(−U/2)− fN(U/2)]êz (7)
= − ΓN[1 + fN(−U/2)− fN(U/2)]I (8)
= I×Beff (9)
where êz is the z-direction and Beff = {2ΓS cos(Φ/2), 0,−ΓNB − 2ǫ− U} is an effective
magnetic field in the isospin space. The accumulation term (7) builds up a finite isospin,
while the relaxation term (8) decreases it. Finally, (9) describes a rotation of the isospin
direction.
5. Non-equilibrium Josephson current
In the isospin language the current in the superconducting leads is
JSL,R =
ΓS [Iy cos(Φ/2)± Ix sin(Φ/2)] , (10)
where the upper(lower) sign refers to the left(right) superconducting lead. The Iy
component contributes to the Andreev current, while Ix is responsible for the Josephson
current. To obtain subgap transport, we first need to build up a finite isospin component
along the z-direction, i.e. we need a population imbalance between the empty and doubly
occupied dot [(this is generated by the accumulation term in (7)]; second, we need a
finite Beff which rotates the isospin so that it acquires an inplane component. In order
to have a finite Josephson current (Ix 6= 0), we need the z-component, −ΓNB − 2ǫ−U ,
of the effective magnetic field producing the rotation to be non zero.
The Josephson current and the Andreev current read
Jjos = −
[2ǫ+ U + ΓNB]ΓS sin(Φ)
|Beff |2 + Γ
N[1 + fN(−U/2)− fN(U/2)]
1− fN(−U/2)− fN(U/2)
1 + fN(−U/2)− fN(U/2)
Jand =
2ΓNΓS[1 + cos(Φ)]
|Beff |2 + Γ
N[1 + fN(−U/2)− fN(U/2)]
× [1− fN(−U/2)− fN(U/2)]. (12)
These results take into account only first-order tunnel processes, i.e. the rates W
are computed to first order in Γη. The factor [1 − fN(−U/2) − fN(U/2)] ensures that
no finite dot-pair amplitude can be established if the chemical potential of the normal
lead, µN, is inside the interval [−U/2, U/2] by at least kBT . In this situation both
the Josephson and the Andreev currents vanish. On the other hand, this factor takes
the value −1 if µN > U/2 and the value +1 if µN < −U/2. Hence, the sign of the
Josephson current can be reversed by the applied voltage (voltage driven π-transition).
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 8
The considerations above establish the importance of the non-equilibrium voltage to
induce and control proximity effect in the interacting quantum dot. In figure 2 we show
in a density plot (a) Jjos and (b) Jand for Φ = π/2 as a function of the voltage µN and
the level position ǫ. Both the control of proximity effect by the chemical potential µN
and the voltage driven π-transition are clearly visible. If the detuning is too large,
|δ + ΓNB| >
Γ2N + 4Γ
S cos
2(Φ/2), it becomes difficult to build a superposition of
the states |0〉 and |D〉, which is necessary to establish proximity. As a consequence,
the Josephson and the Andreev current are algebraically suppressed by δ−1 and δ−2,
respectively. Figure 3 shows the Josephson current as a function of δ = 2ǫ + U . The
fact that the Josephson current is non zero for δ = 0 is due to the term ΓNB, i.e. of the
interaction induced contribution to the z-component of the effective field Beff acting on
the isospin. The term |B| has a maximum at µN = U/2, which causes this effect to be
more pronounced at the onset of transport. The fact that the value of the Josephson
current varies on a scale smaller than temperature indicates its nonequilibrium nature.
A π-transition of the Josephson current can also be achieved by changing the sign
of δ+ΓNB, as shown in figure 4 where Jjos is plotted as a function of the phase difference
Φ for different values of the level position. Notice that the current for δ = 0 (ǫ = −U/2)
is different from zero only due to the presence of the term ΓNB acting on the isospin.
6. Conclusion
In conclusion, we have studied non-equilibrium proximity effect in an interacting
single-level quantum dot weakly coupled to two superconducting and one normal lead.
We propose a new mechanism for a Josephson coupling between the leads that is
qualitatively different from earlier proposals based on higher-order tunnelling processes
via virtual states. Our proposal relies on generating a finite non-equilibrium pair
amplitude on the dot by applying a bias voltage between normal and superconducting
leads. The charging energy of the quantum dot defines a threshold bias voltage above
which the non-equilibrium proximity effect allows for a Josephson current carried
by first-order tunnelling processes, that is not suppressed in the limit of a large
superconducting gap. Both the magnitude and the sign of the Josephson current
are sensitive to the energy difference between empty and doubly-occupied dot. A π-
transition can be driven by either bias or gate voltage. In addition to defining a threshold
bias voltage, the charging energy induces many-body correlations that affect the dot’s
pair amplitude, visible in a bias-voltage-dependent shift of the π-transition as a function
of the gate voltage.
Acknowledgments
We would like to thank W. Belzig, R. Fazio, A. Shnirman, and A. Volkov for
useful discussions. M.G. and J.K. acknowledge the hospitality of Massey University,
Palmerston North, and of the CAS Oslo, respectively.
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots 9
Figure 2. Density plot of the a) Josephson and b) Andreev current, for fixed
superconducting-phase difference Φ = π/2, as a function of the dot-level position ǫ
and of the chemical potential of the normal lead µN. The symbols ± refer to the sign
of the current. The other parameters are ΓS = ΓN = 0.01U , and kBT = 0.05U .
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots10
-0.05 -0.025 0 0.025 0.05
/U=0.5
Figure 3. Josephson current, for fixed superconducting-phase difference Φ = π/2, as
a function of the detuning δ = ED − E0 = 2ǫ+ U for different values of the chemical
potential. The other parameters are ΓS = ΓN = 0.01U and kBT = 0.05U .
0 π/2 π 3π/2 2π
δ/U=0.01
δ/U=0
δ/U=-0.01
Figure 4. Josephson current as a function of the superconducting-phase difference
Φ for different values of the detuning. The other parameters are ΓS = ΓN = 0.01U ,
µN = U , and kBT = 0.05U .
Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots11
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http://arxiv.org/abs/cond-mat/0703264
	Introduction
	Model
	Current formula
	Kinetic equations for quantum-dot degrees of freedom
	Non-equilibrium Josephson current
	Conclusion
