Renormalization effects in interacting quantum dots coupled to superconducting leads

2 MB·English

by  David Futterer

Renormalization effects in interacting quantum dots coupled to superconducting leads
David Futterer,1 Jacek Swiebodzinski,1 Michele Governale,2 and Ju¨rgen K¨onig1
1Theoretische Physik, Universit¨at Duisburg-Essen and CENIDE, 47048 Duisburg, Germany
2School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,
Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
(Dated: October 9, 2012)
We study subgap transport through an interacting quantum dot tunnel coupled to one normal
and two superconducting leads. To check the reliability of an approximation of an infinitely-large
gap∆inthesuperconductingleadsandweaktunnelcouplingtothenormallead,weperforma1/∆
expansion, and we analyze next-to-leading order corrections in the tunnel coupling to the normal
2 lead. Furthermore, we propose a resummation approach to calculate the Andreev bound states for
1 finite∆. Theresultsaresubstantiallymoreaccuratethanthoseobtainedbymean-fieldtreatments
0 and favorably compare with numerical exact results.
2
t PACSnumbers: 74.45.+c,73.63.Kv
c
O
I. INTRODUCTION perconductingleadshasbeenintroducedinRef.31. This
8
method has been applied in the limit of an infinitely-

] Thetunnelcouplingtosuperconductingreservoirscan large superconducting gap, ∆ , to study a va-
→ ∞
ll induce superconducting correlations in quantum dots.1,2 riety of transport phenomena in different quantum-
a Interesting linear- and nonlinear transport effects aris- dot setups,36–43 including, for example, pure spin-
h ing from Andreev reflection3,4 can be studied in hy- current generation,38 shot-noise suppression,41 Cooper-
-
s brid normal-superconducting systems comprising quan- pair splitting,39,40 and time dependent driving.42
e
tum dots in the presence of strong Coulomb interac- Thecalculationsintheinfinite-gaplimitneglectquasi-
m
tion. During the past years, there has been tremen- particle contributions to transport. They, furthermore,
t. dous experimental progress in coupling quantum dots approximatetheABSenergiessincethelatterareafunc-
a
to superconductors.5–22 Prominent examples include the tion of the superconducting gap. In the experiments,
m
observationof0 π transitionsinaJosephsonjunction17 the superconducting gap ∆ is often of the same order of
- and splitting of−Cooper pairs.18–20 magnitudeasotherenergyscalessuchaschargingenergy
d
n An exact theoretical treatment of the transport prop- or tunnel-coupling strength. For this reason, it is quite
o erties of such devices requires to account for Coulomb natural to investigate the quality of the approximation
c interaction, superconducting correlations, and non- introduced by considering the infinite-gap limit. This is
[ equilibrium at the same time. To circumvent this chal- the main goal of the present paper. Furthermore, we an-

1 lenge, various approximation schemes have been pro- alyze the effect of next-to-lowest order corrections in the
v posed. In the case of vanishing Coulomb interaction, tunnel coupling to the normal leads.
7 an exact solution can be obtained within a scattering A finite superconducting gap in the leads affects the
6 approach.23 As a first step beyond, Coulomb interaction AndreevandJosephsontransportthroughquantumdots
2 has been treated perturbatively,24–29 which, however, is intwoways. First,itmodifiestheABSs. Thisaffectsthe
2
not justified for the large charging energies that are typ- positionofbiasorgatevoltagesatwhichanewtransport
.
0 ical for small quantum dots. An alternative possibility channelisopened. Second,thevalueofthecurrentaway
1
is to allow for strong Coulomb interaction but perform fromthesethresholdvoltagesischanged. Toaddressthe
2
a perturbation expansion in the strength of the tunnel effect of a finite ∆, we start by performing a systematic
1
coupling between dot and leads.30,31 However, in the expansion in 1/∆ around ∆ . Exemplarily, we will
:
v limitofaninfinitely-largesuperconductinggap,anexact focusonaquantumdottunne→lco∞upledtoonenormaland
i
X treatment of both the Coulomb interaction and the tun- twosuperconductingreservoirsandcomparethecurrents
nel coupling between quantum dot and superconducting obtainedbythe1/∆expansionwiththe∆ limit.31
ar leads is possible.31–35 In this case, quasiparticle tunnel- We find that away from the threshold volta→ges∞, both the
ing is completely suppressed and transport from and to AndreevandtheJosephsoncurrentsarehardlymodified,
the superconducting leads is fully sustained by Cooper i.e.,the∆ limitprovidesquiteanaccuratedescrip-
→∞
pairs. The formation of Andreev-bound states (ABSs) tion of subgap transport. The 1/∆ expansion, further-
in the quantum dot indicates that superconducting cor- more,indicatesarenormalizationoftheABSenergies. It
relations are induced via the proximity to superconduct- can, however, only predict the direction towards which
ing leads. The dependence of the ABS energies on the the energies are shifted. To determine the position of
quantum dot’s level position is directly reflected in the the ABS, we go beyond the 1/∆ expansion and perform
transport spectrum of the system. a partial resummation of diagrams. The results within
A diagrammatic real-time approach to transport this approach are considerably more accurate than those
through quantum dots coupled both to normal and su- obtained by a simple Hartree-Fock treatment and com-
2
where (cid:15) is the energy of the spin-degenerate single-
N particlelevel, andU theon-siteCoulombrepulsion. The
dot’sannihilation(creation)operatorsofspinσaregiven
ΓN by d(σ†), and nσ =d†σdσ is the corresponding number op-
erator. Tunneling between dot and leads is described by
ΓL ΓR the tunneling Hamiltonian
L QD R
(cid:88)(cid:16) (cid:17)
Htunn,η =Vη c†ηkσdσ+H.c. , (3)
kσ
FIG.1: (Coloronline)Setup: aquantumdotistunnelcoupled
to one normal and two superconducting leads. withthespinandmomentumindependenttunnelmatrix
elements V . We define the tunnel-coupling strengths as
η
Γ =2π V 2ρ , where ρ is the density of states in lead
η η η η
parefavorablywiththenumericalrenormalizationgroup η that w|e a|ssume to be independent of spin and energy.
(NRG) data of Mart´ın-Rodero and Levy Yeyati.44
Finally, wepresent(forthe∆ limit)resultsfrom
→∞
a systematic perturbation expansion to next-to-lowest
B. Diagrammatic Technique
order in the tunnel coupling to the normal leads. In
this way, we go beyond the regime of weak tunnel cou-
1. Effective Dot Hamiltonian
plingbutdonotcoverKondocorrelations.15,16,24–27,45–57
We find that the next-to-leading correction leads to a
renormalization of the gate-voltage position at which a Themainideaofthediagrammaticreal-timetechnique
0 π transition occurs in the Josephson current. This is to integrate out the leads’ degrees of freedom to arrive
re−normalization also affects the Andreev current and the at a reduced density matrix for the quantum dot. As
average quantum-dot charge. a basis for the reduced density matrix one may use the
eigenstates of the decoupled quantum dot, H . This
dot
is a convenient choice for weak tunnel couplings to the
II. MODEL AND METHOD superconductors.30 In the opposite limit of strong cou-
pling to the superconductor, a resummation of an infi-
A. Hamiltonian nite number of diagrams is required. This resummation
can be carried out exactly in the ∆ limit.31 For
→ ∞
a systematic 1/∆-expansion, however, it is more conve-
Wefocusonathree-terminalsetupcomposedofanor-
nient to work in the eigenbasis of H = H H ,
mal conductor and two superconductors tunnel coupled eff dot − p
to an interacting single-level quantum dot, see Fig. 1. with the pairing Hamiltonian Hp =χ∗d d +χd†d† and
T(cid:80)he system’s Hamiltonian is given by H = Hdot + χ= 12(cid:80)η=L,RΓηeiΦη. Theremainingpa↓rt↑ofthe↑H↓amil-
H +H , with η L,R,N . We assume the tonian
η η tunn,η ∈ { }
normal lead to be a reservoir of noninteracting electrons
(cid:88)
andmodelthesuperconductorswiththemean-fieldBCS H H =H + H +H , (4)
eff p η tunn,η
−
Hamiltonian η
Hη =(cid:88)(cid:15)ηkc†ηkσcηkσ−δη,S(cid:88)(cid:0)∆eiΦηcη−k↓cηk↑+H.c.(cid:1), is treated diagrammatically as a perturbation. Thereby,
kσ k H has been chosen such that in the ∆ limit the
p
(1) → ∞
diagrammatic contributions stemming from H exactly
p
with (cid:15)ηk being the single-particle energies and c(η†k)σ are cancel those from the tunneling to the superconducting
theannihilation(creation)operatorsforelectronsinlead leads, i.e., H describes the hybrid system of quantum
eff
η with momentum k and spin σ. Here, ∆ is the modulus dot and superconducting leads in the limit of an infinite
of the superconducting pair potential, which we assume superconducting gap.
to be the same in both superconductors, and Φ is the The eigenstates of H are , , and =
η eff
mcoeraresuspreonadllinogccpuhrarisneginenleeargdieηs=wiSthwrietshpeSct∈to{Lth,Re}F.erWmei √12(cid:104)∓e−iΦχ/2(cid:113)1∓ 2(cid:15)δA |0(cid:105)+eiΦχ/|↑2(cid:105)(cid:113)1|↓±(cid:105)2(cid:15)δA |d(cid:105)(cid:105)|,±(cid:105)with
levelofthesuperconductors,i.e. µL,R =0. Furthermore, eigenenergies (cid:15), (cid:15), and(cid:113)E± = 2δ ±(cid:15)A, respectively. Here,
we assume all occurring excitation energies of the dot to Φ = arg(χ), (cid:15) = (δ)2+ χ2, and δ = 2(cid:15) + U is
lie inside the superconducting gap of the leads so that χ A 2 | |
thedetuningbetweentheenergiesforemptyanddoubly-
only subgap transport takes place.
occupied quantum dot. The superconducting proximity
The Anderson Hamiltonian describing the single-level
effect is indicated by the fact that 0 and d are no
quantum dot reads | (cid:105) | (cid:105)
longer eigenstates but appear as linear combinations in
H =(cid:88)(cid:15)d d +Un n , (2) . The mixing between 0 and d becomes largest
dot †σ σ |±(cid:105) | (cid:105) | (cid:105)
↑ ↓ around zero detuning, δ 0.
σ ∼
3
The excitation energies associated with Heff are given with Wχχχχ(cid:48)(cid:48)1Iη being the generalized current rates. For a
bythedifferencesoftheeigenenergiesofstateswitheven 2
systematic perturbation expansion in powers of Γ and
N
and odd dot occupation numbers, 1/∆oneneedstoexpandP,W,andWIη inEqs.(6)and
(9).
U
EA∆,→γ(cid:48)∞,γ =γ(cid:48) 2 +γ(cid:15)A, (5) We refer to the current flowing out of the normal
lead as the Andreev current and to the difference of the
withγ ,γ = . TheyarenothingbuttheAndreevbound currents between left and right superconductor as the
(cid:48)
state (ABS)±energies64 in the ∆ limit. Josephson current. In the limit ∆ , the Andreev
→∞ currentandtheJosephsoncurrentar→edi∞rectlyconnected
to the isospin via31
2. Generalized Master Equation
2e Φ
J∆ = Γ I∆ sin , (10)
jos→∞ ¯h S x→∞ 2
The system’s dynamics is determined by a general-
4e Φ
iwzehdichmiasstoebrtaeqinueadtiboynifnotretghreatriendguocuedttdheenlseitayd’msadtergirxeρesreodf, Ja∆n→d∞ = − ¯h ΓSIy∆→∞cos 2. (11)
freedom. Its matrix elements Pχχ21 = (cid:104)χ1|ρred|χ2(cid:105) obey, For the 1/∆ corrections to the currents, however, one
in the stationary limit, the generalized master equation
needs to use the more general Eq. (9).
i(E E )Pχ1 = (cid:88) Wχ1χ(cid:48)1Pχ(cid:48)1, (6)
χ1 − χ2 χ2 χ2χ(cid:48)2 χ(cid:48)2
χ(cid:48)1χ(cid:48)2 4. Diagrammatic Rules
whereWχ1,χ(cid:48)1 aregeneralizedratesthatcanbecomputed
χ2,χ(cid:48)2 Starting from Refs. 31 and 37 we derive the rules for
in a diagrammatic way, see Sec. IIB4. As explained in calculating the generalized rates and the generalized
the previous section, we work in the basis , ,+, . current rates in the basis of the effective Hamiltonian.
{↑ ↓ −}
This has the consequence that the off-diagonal matrix The diagrammatic expansion is performed with respect
elementsontherighthandsideofEq.(6)vanishtolowest to the tunneling parts of the Hamiltonian and, in
order in ΓN and 1/∆ as long as (cid:15)A > ΓN. Only their addition, HP. In the following, we assume the two
next-order correction remains finite. ∼ superconductors being at zero chemical potential, i.e.
Superconducting correlations in the quantum dot are µ = µ = 0 and a symmetric tunneling coupling,
L R
associatedwithcoherentsuperpositionsofanemptyand Γ =Γ =Γ .
L R S
adoubly-occupieddot. Thismotivatesthefollowingdef-
inition of an isospin,30,31
(1) Draw all topologically different diagrams with
fixed ordering of the vertices with respect to (real) time.
Pd+P0 Pd P0 P P
I = 0 d, I =i 0 − d, I = d− 0 , (7) The vertices may be bullets (representing tunneling) or
x y z
2 2 2 crosses (for H ). The bullets are connected in pairs by
p
tunneling lines carrying energy ω . The tunneling lines
formulatedinthebasis , ,0,d (thetransformationto i
{↑ ↓ } can be normal or anomalous. For each anomalous line,
thebasis , ,+, isstraightforward). Afinitevalueof
{↑ ↓ −} choose the direction (forward or backward with respect
the isospin components I and/or I indicates the pres-
x y
to the Keldysh contour) arbitrarily.
ence of superconducting proximity effect.
(2) For each vertical cut between two vertices, assign a
It turns out that the generalized master equations for
factor1/(∆E+i0+),where∆E isthedifferencebetween
theisospincomponentsdonotcoupletotheprobabilities
the left-going and the right-going energies, including the
ofthequantumdottobesinglyoccupiedwitheitherspin.
energies of the dot states E and the tunneling lines ω .
They can be written in the form of a Bloch-like equation χ i
(3) For each tunneling line arising from the normal lead
0= dI =A R I+I B, (8) assign a factor 21πΓNfN±(ωN) and for each tunneling
line arising from a superconducting lead assign a factor
dt − · ×
where the terms with A, R, and B describe the genera- 21πΓS√|ω|ηω|η2|−∆2Θ(|ωη|−∆)f±(ωi), where fN+(ω) =
tion, relaxation, and coherent rotation of the isospin. f+(ω µN)=f(ω µN)= 1+exp[(ω µN)/kBT] −1
− − { − }
and fN−(ω) = f−(ω−µN) = 1−f(ω−µN). The upper
(lower) sign applies for lines going backward (forward)
3. Current formulae with respect to the Keldysh contour. For anomalous
lines, multiply an additional factor sign(ω ) ∆ .
The current in lead η is given by Moreover, assign a factor eiΦχe−iΦη for±an outigo|ωini|g
Jη =−¯he χ(cid:88)χ(cid:48)χ(cid:48) Wχχχχ(cid:48)2(cid:48)1IηPχχ(cid:48)2(cid:48)1, (9) a(Φ4nχ)d=eE−aaricΦghχ(χebi)Φuaηlnledftorχva=enrt12eiΓnxcSo(cid:80)cmoinηn∈ngSeceatinsΦoηm.aaloduost lisntea,tewhαere,
1 2 | (cid:105)
4
with α +, , to another dot state σ , with (a) ￿ (b) ￿
∈ { −} | (cid:105) + +
σ , , and to a tunneling line describing a ↑ ↑
tunn∈elin{g↑ ↓e}lectron of spin σ(cid:48) (cid:113)∈ {↑,↓}. For each ωL ωN ωR
bullet vertex assign a factor 1−SαSσSσ(cid:48)2(cid:15)δA/√2, − E− ↓ E+ ↑ − E− ✕ E+ ↑
where Sα=+ = Sσ= = Sσ(cid:48)= = 1 and
Sα= =Sσ= =Sσ(cid:48)= = ↑1. ↑ FIG. 2: Example diagrams contributing to the rate W↑↑−+.
(5) −Each c↓ross ver↓tex−connects a state α with
| (cid:105)
α +, to a state α with α +, .
(cid:48) (cid:48) (cid:90) (cid:90)
∈ { −} | (cid:105) ∈ { −} dω dω
F(or1)Seac(cid:113)h1crosδs2vχerteaxndwiftohr αα == αα(cid:48) aassssiiggnn aa ffaaccttoorr −i 2πL 2πNΓNΓSei(ΦL−Φχ)fN+(ωN)f−(ωL)
− α − 4(cid:15)2A| | (cid:54) (cid:48) 1 1 1
δ χ.
2(cid:15)A| | × ωN +E (cid:15)+i0+ωN +ωL+i0+ωN +E+ (cid:15)+i0+
(6) Assign an overall prefactor i. −− −
− ∆ (cid:18) δ (cid:19)2
Furthermore, assign a factor 1 for each sign(ω ) Θ(ω ∆) 1 .
(a) vertex on the lower propa−gator; × L (cid:112)ωL2 −∆2 | L|− − 2(cid:15)A
(b) crossing of tunneling lines;
Thesecondexampleisadiagram,seeFig.2(b),contain-
(c) outgoing (incoming) bullet vertex ending (starting)
ing a line associated with the right superconductor and
in the state + or ;
| (cid:105) |↑(cid:105) a cross vertex and it yields the following expression
(d) outgoing (incoming) anomalous tunneling line, in
which the earlier (later) tunnel vertex with respect to (cid:90)
dω 1
the Keldysh contour is connected to a tunneling spin-up i 2πRΓSei(ΦR−Φχ)f−(ωR)E ω (cid:15)+i0+
electron. −− R−
1 ∆
(S7u)mFoorveeracahlltduinangrealimngs.line, integrate over the energy ωi. × E+−ωR−(cid:15)+i0+sign(ωR)(cid:112)ωR2 −∆2
(cid:115)
The generalized current rates Wχχχχ(cid:48)2(cid:48)1Iη are evalu- × Θ(|ωR|−∆)δ2|(cid:15)χ| 1− 4δ(cid:15)22 .
ated in the following way: A A
(8) Multiply each diagram that contributes to the
χχ(cid:48)
corresponding generalized rate W 1 and where the
χχ(cid:48) III. FINITE-GAP EFFECTS
2
rightmost line is associated with lead η with a factor:
(a) for the rightmost line being a normal line: 1 if the
A. 1/∆ Expansion
line is going from the lower to the upper propagator, -1
if it is going from the upper to the lower propagator,
We assume symmetric tunnel couplings to the super-
and 0 otherwise;
conducting leads, Γ = Γ = Γ . Furthermore, we
(b) for the rightmost line being an anomalous line: 1 L R S
choose a gauge such that Φ = Φ = Φ/2. For both
for incoming lines within the upper and outgoing lines L − R
the Josephson and the Andreev current we calculate the
within the lower propagator, 1 for outgoing lines
− value for the ∆ limit and the correction to first
within the upper and incoming lines within the lower → ∞
order in 1/∆. The tunnel coupling to the normal lead,
propagator, and 0 otherwise.
Γ isaccountedforinlowestnon-vanishingorder,i.e.,to
N
zeroth order for the Josephson current and to first order
for the Andreev current.
Figures 3 and 4 show the Josephson current and the
Andreev current in the limit of ∆ as a function
→ ∞
of the level position (cid:15) and the chemical potential of the
normal lead µ . This is the limit considered in Ref. 31
N
and is used as a reference for the present discussion. A
We want to remark that all terms arising from rule cutthroughthedensityplotsatafixedlevelposition(cid:15)=
(5) are exactly cancelled by the terms arising from the 0.4U is shown in Fig. 5 (solid blue line). The currents
−
superconducting tunneling lines in the infinite-gap limit. displayastep-likebehaviorwiththepositionofthesteps
ThisreflectsthefactthattheeffectiveHamiltoniancom- reflecting the ABS energies. This is compared with the
pletely describes the hybrid system of quantum dot and currents expanded up to first order in 1/∆, dashed red
superconducting leads in the infinite-gap limit. To illus- lines in Fig. 5. From this comparison, we can draw two
trate how the graphical representation is translated into conclusions.
analytical expressions we show two exemplary diagrams First,awayfromtheABS,thesolidanddashedcurves
here. The first example is a diagram containing one line almost coincide, i.e., the 1/∆ correction to the current
associated with the normal lead and one line associated is small. This leads to the conclusion that the ∆
→ ∞
withtheleftsuperconductor, seeFig.2(a), anditreads calculations provide a useful and quantitatively accurate
5
(cid:209)J (cid:144)(cid:72)e(cid:71) (cid:76) (cid:209)J (cid:144)(cid:72)e(cid:71) (cid:76)
jos S and S
1.5
1.5
0.38 0.005
1.0 1.0
0.5 0.5
U U
(cid:144)N 0.0 0 (cid:144)N 0.0 0
Μ Μ
(cid:45)0.5 (cid:45)0.5
(cid:45)1.0 (cid:45)1.0
(cid:45)0.38 (cid:45)0.005
(cid:45)1.5 (cid:45)1.5
(cid:45)1.5 (cid:45)1.0 (cid:45)0.5 0.0 0.5 (cid:45)1.5 (cid:45)1.0 (cid:45)0.5 0.0 0.5
Ε(cid:144)U Ε(cid:144)U
FIG. 3: (Color online) Density plot of the Josephson current FIG. 4: (Color online) Density plot of the Andreev current
in the limit of ∆ → ∞ as a function of the level position (cid:15) in the limit of ∆ → ∞ as a function of the level position (cid:15)
andthechemicalpotentialofthenormalleadµ . Theother andthechemicalpotentialofthenormalleadµ . Theother
N N
parameters are Γ = 0.2U, Γ = 0.001U, Φ = π/2, and parameters are Γ = 0.2U, Γ = 0.001U, Φ = π/2, and
S N S N
k T =0.01U. k T =0.01U.
B B
description of sub-gap transport also for finite values of culations are exact but numerically demanding.44,58–61
the superconducting gap. Instead, we propose a resummation approach that is
Second, we find peaks at the positions of the ABSs in more accurate than mean field but less computationally
the1/∆correctiontothecurrent. Whilethepeaksthem- heavy than NRG.
selves are artefacts of the 1/∆ expansion, they carry im- In the following, we consider the limit of weak tunnel
portant physical information about the renormalization couplingtothenormallead,i.e.,wediscussthespectrum
of the ABSs as compared to their ∆ value. This to zeroth order in ΓN. Furthermore, we choose Φ = 0
can be understood in the following w→ay.∞The currents without lossof generality (toconsider the case ofa finite
varystepwiseatthepositionoftheABSs. Thelatterare Φ one has to replace ΓS by ΓScos(Φ/2); to model the
renormalized due to finite ∆. Therefore, a systematic case of a single superconducting lead, ΓS needs to be
expansion in the small parameter with which the renor- replaced by ΓS/2).
malization scales gives rise to the occurrence of peaks at
the original position of the steps. The sign of the peaks
relative to the sign of the steps indicates the direction of 1. Exact Green’s function for the noninteracting dot
the renormalization. The ABS is renormalized towards
(awayfrom)zeroifstepandpeakhavethesame(theop- For a noninteracting dot, U = 0, the exact Green’s
posite)sign. IntheexamplesplottedinFig.5thecurves function for spin σ is given, in Nambu space, by
indicatearenormalizationofallfourABSstowardszero.
(cid:16) (cid:17) 1
Gˆr = ω I hˆ (U =0) Σ − , (12)
2 2 σ S
· × − −
B. Renormalization of Andreev Bound States withI beingtheidentitymatrixinndimensions. The
n n
matrix×
The1/∆expansionindicatesthattheABSsarerenor- (cid:18) (cid:19)
(cid:15) 0
malized as compared to the ∆ limit. It even pre- hˆ (U =0)= σ (13)
→ ∞ σ 0 (cid:15)
dicts the sign of the renormalization. It does, however, − −σ
not provide a possibility to calculate the ABSs at finite
accounts for the single-particle energies, and the self-
∆. For this task, alternative approaches are required. In
energy due to tunneling reads
the limit of vanishing Coulomb interaction, the problem
can be solved exactly.23,52 A mean-field treatment of the (cid:32) (cid:33)
ω ∆
Coulomb interaction provides easily an estimate of the Σˆ =Γ −√∆2 ω2 √∆2 ω2 . (14)
ABSs.21,44 Numerical renormalization group (NRG) cal- S S ∆− −ω
√∆2 ω2 −√∆2 ω2
− −
6
(cid:72)a(cid:76) 2. Hartree-Fock approximation
0.5
(cid:76)S 0.25 A Hartree-Fock (HF) treatment of the Coulomb in-
(cid:71)
e teraction has been used in Ref. 21 to fit the experi-
(cid:144)(cid:72)s 0. mental data. This HF approach corresponds to replace
o
Jj (cid:45)0.25 hˆσ(U =0) in Eq. (12) by
(cid:209)
(cid:18) (cid:19)
(cid:45)0.5 (cid:15)˜ 0
hˆ (U)= σ , (15)
(cid:68)(cid:174)(cid:165) σ 0 (cid:15)˜
(cid:72)b(cid:76) − −σ
0.004 1(cid:144)(cid:68)
where(cid:15)˜ =(cid:15)˜ +U. ForU =0,theexactGreen’sfunction
(cid:76)S 0.002 Eq. (12↓) is r↑ecovered. Furthermore, it reproduces the
(cid:71)
e exact ABSs in the ∆ limit.
(cid:144)(cid:72)d 0. An improved versio→n∞is self-consistent HF, in which
n
Ja the dot occupations are calculated self-consistently.44 In
(cid:45)0.002
(cid:209) the following, however, we use the simpler version of HF
(cid:45)0.004 for comparison with the results from the resummation
approach.
(cid:45)1.5 (cid:45)1.0 (cid:45)0.5 0.0 0.5 1.0 1.5
Μ (cid:144)U
N
3. Resummation Approach
FIG. 5: (Color online) Plot of (a) the Josephson current and
(b) the Andreev current as a function the chemical potential
of the normal lead µ . The solid (blue) curve shows the An important feature of the diagrammatic technique
N
∆ → ∞ limit, the dashed (red) curve includes the first 1/∆ presented in this paper is that Coulomb interaction can
correction. Theotherparametersare(cid:15)=−0.4U,ΓS =0.2U, be included beyond the mean-field level. The downside
Γ =0.001U, ∆=5U , Φ=π/2, and k T =0.01U.
N B is that for an exact solution one has to sum up infinitely
many diagrams. A great simplification is achieved by
working in the basis , ,+, . As discussed above,
{↑ ↓ −}
in the ∆ limit, all diagrams that contain super-
1.0 → ∞
conducting lines are cancelled by cross vertices. This
(cid:68)(cid:76)(cid:144)0 0.5 (cid:68)ex(cid:174)a(cid:165)ct mneelalninsetshoantlayl.lcontributingdiagramscontainnormaltun-
(cid:61) At finite ∆, the situation is different. Now, both su-
U 0.0 perconducting lines and cross vertices dress the original
(cid:72)
Γ
Γ', normal tunnel lines. We define the propagator Π(ω) as
A,(cid:45)0.5 thediagrampartbetweentwoverticesthatareconnected
E
byanormaltunnellinewithenergyω runningfromright
(cid:45)1.0 to left. This propagator obeys a Dyson equation
(cid:45)2 (cid:45)1 0 1 2
Ε(cid:144)(cid:68) Π(ω)=Π(0)(ω)+Π(ω)Σ(ω)Π(0)(ω), (16)
where Π(0)(ω) is the free propagator (without supercon-
FIG. 6: (Color online) Plot of the U = 0 Andreev-bound
states for a tunnel-coupling strength to the superconducting ductinglinesandcrossvertices)andΣ(ω)theselfenergy.
lead of Γ = 0.25∆ as a function of the level position (cid:15). All In the ∆ limit, the self energy is exactly zero.
S → ∞
energies are normalized to ∆. The solid (blue) line shows For an approximative treatment of the case of finite ∆,
the exact result, the dashed (black) line the bound states we now include all self energy parts that contain one su-
computed by means of the ∆ → ∞ expression, which in the perconducting tunnel line or one cross vertex. Examples
non-interacting case is simply ±(cid:15)A. are shown in Fig. 7.
Since the Hilbert space is four dimensional, the di-
mension of the corresponding Liouville space and, thus,
of the matrices appearing in Eq. (16) is, in general, 16.
The ABSs, i.e. the excitation energies, are found as the However, as a consequence of spin and charge conserva-
real part of the poles of the Green’s function’s determi- tion while tunneling, the Dyson equation decouples into
nant. An example is shown in Fig. 6. In the noninter- blocks of 4 4 matrices. The presence of the normal
×
acting case, there are only two ABSs. Their energies for tunneling line running from right to left which carries a
finite ∆ are reduced as compared to the ∆ limit. definite charge and spin puts constraints on the possible
→ ∞
Inparticular,thecurvatureoftheenergyasafunctionof states at the beginning and end of the upper (χ ) and
1
the bare level position (cid:15) changes for large (cid:15), such that lower (χ ) contour. In the following, we choose, without
2
| |
the ABSs lie within the window [ ∆,∆]. loss of generality, (χ ,χ ) ( , ),( ,+),( , ),(+,
1 2
− ∈ { − ↑ ↓ ↓ − ↑
7
(b) (cid:72)a(cid:76)
(a) 1.0
￿ E
− − − − −
ω 0.5
(cid:68) Resum
(cid:144)
ω ΓΓ', 0.0 NRG
A, HF
↑ ￿ ↑ ↑ E ↑ E
− (cid:45)0.5
FIG. 7: Examples for diagrams that contribute to the self-
energy. (cid:45)1.0
(cid:45)2 (cid:45)1 0 1
Ε(cid:144)(cid:68)
) , which is the relevant case when the normal tunneling (cid:72)b(cid:76)
} 1.0
line carries spin . Then,
↓
(cid:16) (cid:17) 1 0.5
Π(0)(ω) − =
(cid:68) Resum
(cid:144)
ω·I4×4−diag(EA,+,−,EA,−,−,EA,−,+,EA,+,+()17) ΓΓ', 0.0 NRG
A, HF
for the inverse free propagator, and the self energy E
(cid:45)0.5
 
W−−(ω) W−+↓(ω) W−↓(ω) W−+(ω)
 ↑↑ ↑ ↑− ↑↑  (cid:45)1.0
 
W+↓−(ω) W+↓↓+(ω) W+↓↓ (ω) W+↓+(ω) (cid:45)2 (cid:45)1 Ε(cid:144)(cid:68) 0 1
Σ(ω)= ↑ − ↑ 
 
W↓−(ω) W↓↓+(ω) W↓↓ (ω) W↓+(ω) (cid:72)c(cid:76)1.0
 −↑ − −− −↑ 
 
W+−(ω) W++↓(ω) W+↓(ω) W++(ω), 0.5
↑↑ ↑ ↑− ↑↑ (18) (cid:68) Resum
iaslgeifvt-egnoibnygtehxetegrennaelralilnizeedwirtahteesneWrgχχy21χχω21(cid:48)(cid:48).(ω) that contain (cid:144)ΓΓ', 0.0 NRG
A, HF
We solve Eq. (16) for the full propagator to obtain E
(cid:45)0.5
Π(ω)=(cid:20)(cid:16)Π(0)(ω)(cid:17)−1 Σ(ω)(cid:21)−1. (19)
− (cid:45)1.0
(cid:45)6 (cid:45)4 (cid:45)2 0 2
Ε(cid:144)(cid:68)
Theexcitationenergiesoftheproximizeddotareprobed
by the external energy ω and they are given by the real
poles of the full propagator. As mentioned above, we FIG.8: (Coloronline)PlotoftheAndreev-boundstatesfrom
approximate the self-energy Σ(ω) by including only di- theresummationapproach,NRG62,andtheHartree-Fockap-
agrams that contain a single superconducting tunneling proximationasafunctionofthelevelposition(cid:15)for(a)U =∆
line or one cross vertex, in addition to the external line and ΓS =0.5∆, (b) U =∆ and ΓS =0.2∆, and (c) U =4∆
of energy ω, see Fig. 7. and ΓS =0.2∆.
Although the resummation scheme does not define a
controlled approximation, its results are with a few lim-
itations quantitatively accurate, as we discuss in the fol- or four solutions. The ABSs are renormalized as com-
lowing. Itisclearthat,byconstruction,the∆ limit pared to the ∆ solution. The sign of the renomal-
→∞ →∞
is reproduced exactly. Also for the noninteracting case, izationagreeswiththepredictionofthe1/∆expansionof
U =0, the resummation approach yields the exact solu- Sec.IIIA.Forcomparison,wealsoshowtheresultsfrom
tion. As an artifact, however, the resummation scheme HF as dashed (black) lines. While qualitatively similar,
produces for U = 0 two extra, unphysical, solutions in there are substantial quantitative deviations. First, HF
addition to the two correct ones shown in Fig. 6. seemstounderestimatetherenormalizationoftheABSs.
The most interesting regime, however, is the case of Second,thepositionofthecrossingpointsoftheABSen-
finite U and finite ∆. Results obtained from the resum- ergies in the resummation scheme is shifted as compared
mationapproachfordifferentvaluesofΓ andU arepre- to the HF and the ∆ result, see Fig. 8(c).
S →∞
sented in Fig. 8 as solid (blue) curves. Depending on the Areliablequalitycheckoftheproposedapproximation,
valuesoftheparametersofthemodel, wefindeithertwo however, is only given by the comparison with the full
8
NRGresults.62Wefindaremarkablygoodagreementbe-
(cid:72)a(cid:76) 2
tweenNRGdata,dotted(red)linesinFig.8,andresum- J(cid:72)0(cid:76)(cid:61)0
jos
mation approach. The ABS spectrum is quantitatively
J(cid:72)0(cid:76)(cid:43)J(cid:72)1(cid:76)(cid:61)0
reproduced in almost the entire parameter space. The 1 jos jos
main difference is that the resummation scheme some-
U
times yields four solutions when NRG only predicts two.
(cid:144) 0
Furthermore, sometimes the resummation approach pro- N
Μ
duces solutions with finite imaginary part. Since the
ABSsintheabsenceofcouplingtoanormalreservoirare (cid:45)1
sharpduetothegapinthequasiparticlespectrum,these
solutions are discarded. This gives rise to the small gaps (cid:45)2
inthesolid(blue)lineinFig.8(a)and(b). Bothfeatures (cid:45)0.55 (cid:45)0.50 (cid:45)0.45
Ε(cid:144)U
(extra pair of solutions and poles with finite imaginary
part) are artifacts of the approximation employed.
(cid:72)b(cid:76) 2
(cid:182) J(cid:72)1(cid:76)(cid:61)0
Ε and
IV. BEYOND WEAK COUPLING TO THE 1 (cid:182)Ε(cid:72)Ja(cid:72)1n(cid:76)d(cid:43)Ja(cid:72)2n(cid:76)d(cid:76)(cid:61)0
NORMAL LEAD
U
(cid:144) 0
N
In the previous section we have discussed how a finite Μ
superconducting gap in the leads renormalizes the ABSs
(cid:45)1
of a quantum dot weakly tunnel-coupled to a normal
lead. Now, we want to address the influence of the tun-
nelcouplingtothenormalleadbeyondtheweak-coupling (cid:45)2
(cid:45)0.55 (cid:45)0.50 (cid:45)0.45
limit. For this, we go back to ∆ but include next- Ε(cid:144)U
→∞
to-leading order corrections in Γ . To distinguish the
N
contributions of different order in ΓN, we introduce in FIG. 9: (Color online) Plot of (a) the position of the 0−
the following an index (n) for n-th order. We study the π transition of the Josephson current and (b) the extreme
Josephson and the Andreev current as well as the aver- values of the Andreev current. The chosen parameters are
age charge of the quantum dot. Within a perturbation k T/U = 0.05, Γ /U = 0.05, Γ /U = 0.2, and Φ = π/2.
B N S
expansion, they start to zeroth, first, and zeroth order, The thin black lines show the position of the ABS.
respectively,i.e.,thenext-to-leadingordercorrectionsare
of first, second, and first order, respectively. To evaluate
them we have to include diagrams with two tunneling ized,Fig.9(a). Theoriginofthisrenormalizationcanbe
lines from the normal lead. This increases the number understood in terms of an effective, tunnel-coupling in-
of diagrams considerably and we generate and evaluate ducedfieldthatactsontheisospinintroducedinEq.(7).
them by means of a computer code. A finite bias voltage leads to the generation of the z-
It is natural for a systematic perturbation expansion component of the isospin. The tunnel coupling to the
in a small parameter that higher-order corrections are superconductors leads to an effective field along the x-
quantitatively small. Therefore, we focus, in the follow- direction. Thiscausestheisospintorotateandacquirea
ing,onqualitativefeaturesthatappearintheregionclose finitey-component. AccordingtoEq.(10),however,afi-
to the point of zero detuning δ =2(cid:15)+U =0 when going nitex-componentoftheisospinisneededforaJosephson
beyond the weak-coupling limit. Around zero detuning, current. This is accomplished with an additional finite
the weights of the states 0 and d entering the coher- z-component of the effective field. To lowest order, this
| (cid:105) | (cid:105)
ent superpositions become equal, indicating a large component is given by the detuning, B(0) =δ, which ex-
|±(cid:105) z
proximizationofthequantumdot. Forbiasvoltageslarge plains the position of the 0 π transition in Fig. 3. The
enough such that the quantum dot has a finite probabil- next-order correction, B(1),−renormalizes this position.
z
itytobeeitheremptyordoublyoccupied,Josephsonand
Weremarkthatthisrenormalizationofthe0 π tran-
Andreev currents set in. −
sitionhasalreadybeenpredictedinRef.30,inwhichthe
In the regime of large bias voltage, the Josephson cur-
tunnel couplings to both the superconductors and the
rent changes sign as function of the quantum dot’s level
normalleadsaswellasthedetuningδ havebeensimulta-
position (cid:15). This indicates a transition from a 0- to a π-
neously treated as small parameters, i.e., the systematic
junction behavior. In the limit ∆ and weak tunnel
coupling to the normal lead,31 this→0∞π transition hap- perturbation expansion included already both Bz(0) and
pensexactlyatzerodetuning,(cid:15)= U−/2,andthenbends Bz(1). On the other hand, the formation of ABSs could
when the bias voltage approaches−the highest ABS, see notbetreatedinthatexpansion. InRef.31,ontheother
Fig. 3. When including the next-order correction in ΓN, hand,noconstraintswereputonΓS andδ,i.e.,theABSs
the position of this 0 π transition becomes renormal- could be described, but the renormalization B(1) did not
z
−
9
Q(cid:72)0(cid:76)(cid:61)(cid:45)e or equivalently Iz = 0, for both zeroth and first or-
2 der. (We do not plot the position in the Coulomb-
Q(cid:72)0(cid:76)(cid:43)Q(cid:72)1(cid:76)(cid:61)(cid:45)e
blockade regime.) As discussed above, the position of
Q(0)/( e) = 1 is at zero detuning in the large-bias
1 −
regime and splits in the intermediate-bias regime, with
U Q(0)/( e) < 1 between and Q(0)/( e) > 1 outside the
(cid:144)N 0 two lin−es. When including the next-−order correction, the
Μ situationchanges. Now,thelineof[Q(0)+Q(1)]/( e)=1
−
(cid:45)1 is split even for the large-bias regime, opening a region
with [Q(0)+Q(1)]/( e) < 1 in between. To understand
−
this, we analyze I at zero detuning, δ = 0. Solving
(cid:45)2 z
(cid:45)0.55 (cid:45)0.50 (cid:45)0.45 the generalized master equation31,63 immediately yields
Ε(cid:144)U
I(0)(δ =0) = 0 and I(1)(δ =0) = A(1)/B(0), where
z z y x
−
B(0) =2χ and
FIG. 10: (Color online) Plot of the average dot charge. The x | |
chosenparametersarekBT/U =0.05,ΓN/U =0.05,ΓS/U = 1 (cid:104) (cid:105)
0.2, and Φ=π/2. A(y1) = 4 Im W+−−−(1)+W+−++(1)+W−+−−(1)+W−+++(1) ,
(cid:115)
(cid:20) (cid:18) (cid:19)(cid:21)
ΓN δ2 (cid:88) 1 EA,γ(cid:48),γ µN
= 1 Re ψ +i − .
contribute to lowest order in ΓN. Only the present cal- 4π − 4(cid:15)2A γ,γ(cid:48)= 2 2πkBT
culation,witharbitraryΓS andδ andnext-to-leadingor- ±
der on ΓN enables us to address both the ABSs and the Here Wχ1χ(cid:48)1(1) are generalized rates in first order in Γ
renormalization of the position of the 0 π transition. χ2χ(cid:48)2 N
− andψ(x)isthedigammafunction. Itis,thus,theisospin
While the Josephson current changes sign at small de-
tuning,theAndreevcurrentbecomesextremalasafunc- generation term A(y1), describing combined Andreev pro-
tion of (cid:15), see Fig. 4. In Fig. 9 (b) we show the position cesses that involve both the normal and the supercon-
of this extremum in the (cid:15)-µ -plane. (We do not show ducting leads, which reduce Q/( e) below 1.
N −
the position of the extremum in the Coulomb-blockade
regime). To lowest order and for large bias voltage, the
extremumisatzerodetuning. Thenext-ordercorrection, V. CONCLUSIONS
however, leads to a renormalization of this position, for
thesamereasonastherenormalizationofthe0 π tran- The theoretical description of sub-gap (Josephson and
−
sition in the Josephson current. Andreev) transport through quantum dots with strong
Finally, we address the average quantum dot charge Coulomb interaction coupled to normal and supercon-
Q, which is related to the z-component of the isospin ducting leads simplifies substantially in the limit of
via Q = e(1+2I ). Again, we focus on the region of infinitely-large superconducting gap ∆ and weak tunnel
z
−
small detuning where the proximity effect is most pro- coupling Γ to the normal lead. For experimental de-
N
nounced. IntheCoulomb-blockaderegime(atsmallbias vices with finite ∆ and larger Γ these calculations may
N
voltage), the dot is preferably singly occupied, Q = e. stillbeusedasanapproximation. Theassessmentofthe
−
A larger bias voltage has the tendency to inject a sec- quality of this approximation to describe sub-gap trans-
ond electron into the quantum dot. Superconducting port has been the focus of this paper. In particular we
correlations, however, mix the states of empty and dou- came to the following conclusions. The position of the
ble occupation and, thus, reduce the average number of ABSs strongly depend on ∆, i.e., a ∆ approxima-
→ ∞
electrons. For zero detuning δ and to zeroth order in tion is insufficient. These ABSs define threshold volt-
Γ ,thechargeQremains eevenforlargebiasvoltage. ages in the current-voltage characteristics at which new
N
−
Onlywhenincreasing δ thenumberofchargesincreases transport channels open. The numerical values for the
| |
above 1. In the intermediate-bias regime for finite de- Josephson and Andreev current between these threshold
tuningvaluestheaveragechargetendstowards0or 2e, voltages,however,seemtobenicelyapproximatedbythe
−
depending on the sign of the detuning. This effect can ∆ limit.
→∞
easily be understood by looking at the eigenstates : Similarly,wefindthatthenext-to-leadingordercorrec-
|±(cid:105)
intheintermediate-biasregimeforpositivevaluesofµ , tioninthetunnelcouplingΓ tothenormalleadmainly
N N
i.e. E < µ < E , the state can already yields small quantitative corrections. But there are also
A,+, N A,+,+
be excited−while state + is still inacces|s−ib(cid:105)le. From the qualitative differences: the position of the 0 π transi-
| (cid:105) −
definition of state it follows that a positive (nega- tion in the Josephson current and the peak positions in
|−(cid:105)
tive) detuning causes the weight of the zero component the Andreev current are shifted.
to dominate over (be dominated by) the double compo- In conclusion, we find that the ∆ and weak-
→ ∞
nent resulting in an decrease (increase) of the average couplingcalculationsprovideaverygoodapproximation
number of electrons. for sub-gap transport with the limitation that the posi-
In Fig. 10 we show the position of Q/( e) = 1, tionsoftheABSsforfinite∆arenotproperlydescribed.
−
10
To address the latter, we proposed a resummation ap- KO 1987/5 is gratefully acknowledged. We are grateful
proachthatsubstantiallyimprovesovermean-fieldtreat- to A. Mart´ın-Rodero and A. Levy Yeyati for fruitful dis-
mentsandfavorablycompareswithmoreelaborateNRG cussions and for sending us their NRG data. We would
calculations. like to thank A. Hucht for helpful discussions.
Acknowledgments
D.F. appreciates the hospitality of The Victoria Uni-
versity of Wellington. Financial support from DFG via
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