Replicas of the Fano resonances induced by phonons in a subgap Andreev tunneling

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by  J. Baranski

Replicas of the Fano resonances induced by phonons in a subgap Andreev tunneling
J. Baran´ski and T. Doman´ski
Institute of Physics, M. Curie Sk lodowska University, 20-031 Lublin, Poland
(Dated: January 17, 2013)
3
We study influence of the phonon modes on a subgap spectrum and Andreev conductance for
1
the double quantum dot vertically coupled between a metallic and superconducting lead. For the
0
monochromatic phonon reservoir we obtain replicas of the interferometric Fano-type structures
2
appearing simultaneously in the particle and hole channels. We furthermore confront the induced

n on-dotpairingwiththeelectroncorrelationsandinvestigatehowthephononmodesaffectthezero-
a bias signature of theKondoeffect in Andreevconductance.
J

6 PACSnumbers: 73.63.Kv;73.23.Hk;74.45.+c;74.50.+r
1

] I. INTRODUCTION
l
e phonon
- bath
r Electron transport through nano-size transistors con-
t
s taining the quantum dots, molecules and/or nanowires
.
t is determined by the available energy levels (tunable
a
m by external gate voltage) and strongly depends on the QD
2
Coulomb interactions [1]. Discretization of the energy N S
- t
d levelsisresponsibleforoscillationsofthedifferentialcon-
n ductanceuponvaryingthegatevoltage,whereasthecor- QD
o relation effects lead to the Coulomb blockade and can Γ 1 Γ
N S
c induce (at low temperatures) the Kondo resonance en-
[ hancingthezero-biasconductanceto aunitaryvalue 2he2 V = µ - µ
1 [2,3]. Besidespromisingperspectivesfortheapplications N S
v in modern electronics/spintronics the nanoscopic struc-
4
tures represent also valuable testing grounds for probing FIG.1: (coloronline)Schematicviewofthedoublequantum
7
the many-body effects. Magnetic, superconducting or dotcoupledinT-shapeconfigurationbetweenthemetallic(N)
6
othertypesoforderingsabsorbedfromtheexternalleads andsuperconducting(S)electrodeswhereanexternalphonon
3
. can be confronted with the on-dot electron correlations bath affects theside-attached quantumdot (QD2).
1
in a fully controllable manner.
0
3 In this regard, especially interesting are the hetero-
1 junctions where the quantum dots (QDs) are in con- freedom. Transport through this T-shape double quan-
v: tact with the superconducting (S) electrodes. Nonequi- tum dot (DQD) occurs predominantly via the central
i librium charge transport can occur there either via the quantum dot (QD1), whereas electron leakage to/from
X
usual single particle tunneling (upon breaking the elec- the side-coupledquantum dot (QD ) brings in the inter-
2
r tron pairs) or by activating the anomalous (Andreev or ferenceeffects. Inthe caseofbothnormal(N)electrodes
a
Josephson)channels. Theresultingcurrentsaresensitive and assuming a weak interdot coupling it has been ar-
toacompetitionbetweentheinducedon-dotpairingand gued[17,18]thatthe differentialconductanceshouldre-
the Coulomb repulsion. In such context there have been vealthe asymmetric Fano-type lineshapes. This fact has
experimentally explored the signatures of π-junction [4], been indeed observed experimentally [19].
Josephson effect [5], superconducting quantum interfer-
Recentlywehaveexploredtheinterferometricpatterns
ence [6], quantum entanglement by splitting the Copper
fortheDQDcaseplacedbetweenthemetallicandsuper-
pairs [7], multiple Andreev scattering [8], and interplay
conducting electrodes [20]. In such heterostructures the
oftheon-dotpairingwiththeKondoeffect[9–11]. These
interferometric lineshapes appear simultaneously at neg-
and similar related activities have been discussed theo-
ative and positive energies because of the mixed particle
retically by various groups [12–16].
and hole degrees in the effective quantum dot spectrum
Since in practical realizations the nanoscopic objects [21]. These effects manifest themselves in the subgap
are never entirely separated from an environment (e.g. Andreev conductance [10]. Stability of interferometric
a given substrate or external photon/phonon quanta) Fano structures on a dephasing by external fermionic
therefore transport properties are also affected by the bath have been analyzed in Refs [22, 23]. Here we ex-
interference effects. A convenientprototype for studying tend such study addressing the role of bosonic bath in
such phenomena is the tunneling setup shown in figure the setup displayed in Fig. 1. We argue that monochro-
1, where the central quantum dot is coupled to the side- matic phonon bath induces a number of the Fano-type
attachedquantumdotandeventuallytootherdegreesof replicas depending on the adiabadicity ratio λ/ω .
0
2
Inthenextsectionweintroducethemicroscopicmodel Gi(τ1,τ2) = −iTˆτhΨˆi(τ1)Ψˆ†i(τ2) defined in a representa-
adnisdcubsrsietflhyesfpoercmifaylcahsapreaccttserciostniccerennienrggythscealaedso.pWteedaalspo- ItnionthoefetqhueilNibarmiubmu csponindoirtsioΨnˆs†i (≡for(dˆµ†i↑,dˆ=i↓)µ, Ψˆ)is≡uch(Ψˆm†i)a†-.
N S
proximations. InthesectionIIIweanalyzespectroscopic
trix Green’s function depends only on time difference
fingerprints of the phonon modes for the case of uncor-
τ τ . ThecorrespondingFouriertransformcanbe then
1 2
rected quantum dots. Finally, in section IV, we address −
expressed by the Dyson equation
the correlation effects (Coulomb blockade and Kondo
physics)duetotheon-dotrepulsionbetweentheopposite G−i 1(ω)=g−i 1(ω)−Σ0i(ω)−ΣUi (ω), (4)
spin electrons. Appendix A provides phenomenological
arguments for the Fano-type interferometric patters of where the bare propagators gi(ω) of uncorrelated quan-
the double quantum dot structures. tum dots are given by
ω ε 0
g−i 1(ω)= −0 i ω+ε . (5)
II. MICROSCOPIC MODEL i
(cid:18) (cid:19)
First part of the selfenergy Σ0(ω) comes from a com-
ThedoublequantumdotheterojunctionshowninFig. i
binedeffectofthe interdotcoupling,hybridizationofthe
1 can be described by the Anderson-type Hamiltonian
central QD with the external leads (3) and the phonon
1
Hˆ =Hˆ +Hˆ +Hˆ (1) bathcontributionactingonQD . TheothertermΣU(ω)
leads DQD T 2 i
appearingin(4)accountsforthemany-bodyeffectsorig-
where Hˆ =Hˆ +Hˆ denote the normal N and su-
leads N S inating from the on-dot Coulomb repulsion U .
perconducting S charge reservoirs, HˆDQD refers to both Let us begin by first specifying the selfeneirgy Σ0(ω)
quantum dots (together with the phonon bath) and the 1
for the uncorrelated central quantum dot. The usual di-
last term HˆT describes the hybridization to external agrammatic approachyields
leads. We treat the conducting lead as a Fermi gas
HˆN = k,σξkNcˆ†kσNcˆkσN and we assume that isotropic Σ01(ω)= Vkβgβ(k,ω)Vk∗β +tG2(ω)t∗, (6)
superconductor is described by the BCS Hamiltonian k,β
THˆhSe=oPpPekra,σtoξrksSccˆˆ(k†k†σσ)βScˆckoσrSre−sp∆oPndkt(ocˆ†ka↑nSncˆi†−hkil↓aSt+ioncˆ−(kc↓rSeacˆkti↑oSn)). twhheerleeadgsβ.(kIn,ωp)Xardteicnuoltaer,thwee mhaavterifxorGtrheeenn’osrmfuanlclteioands of
of the itinerant electrons with spin σ = , and energies
ξpkoβte=ntεikaβls−µµββ.are measuredwith respect↑t↓o the chemical gN(k,ω)= ω−01ξkN 01 (7)
The double quantum dot along with the phonon bath (cid:18) ω+ξkN (cid:19)
is described by following local part and for the superconducting electrode
HˆDQD = εidˆ†iσdˆiσ +t dˆ†1σdˆ2σ+H.c. (2) g (k,ω)= ωu2kEk + ω+vk2Ek −ωukEvkk + ωu+kvEkk (8)
+ Ui dˆX†iσ,diˆi dˆ†i dˆi +ω0Xaˆσ†aˆ(cid:16)+λ dˆ†2σdˆ2(cid:17)σ(aˆ†+aˆ), S −ω−u−kEvkk + ωu+kvEkk ω−+u2kEk + ω−vk2Ek !
Xi ↑ ↑ ↓ ↓ Xσ with quasiparticle energy Ek= ξk2S +∆2 and the BCS
w(chreearetiowne)uospeetrhaetosrtsandˆd(†a)rdfornoetlaectitornonfsorattheeacahnnqiuhailnattuiomn coefficients u2k,vk2 = 21 1± ξEkkSp, ukvk = 2E∆k. In the
i wide band limit approxhimationiwe assume the constant
dot QD . Their energy levels are denoted by ε and
Ui referi=to1,2the on-dot Coulomb potentials. Since wie are thryebartidΓizataisonaccoounpvleinngiesnΓtβun=it2fπor ekne|Vrgkiβe|s2.δW(ωe−tξhkeβn)afonrd-
interested in the Fano-type interference we focus on the N P
mally have
electron transport only via the central quantum dot
HˆT = Vkβ dˆ†1σcˆkσβ +H.c. . (3) |VkN|2 gβ(k,ω) = −iΓ2N 10 01 (9)
β=XN,SXk,σ (cid:16) (cid:17) Xk (cid:18) (cid:19)
Twhhiesnseilteucattrioonntucannnebliengedxtireencdtelydintovomlveosrebogtehnetrhaelqcuaasnes- |VkS|2 gS(k,ω) = −iΓ2Sγ(ω) ∆1 ∆ω1 (10)
k (cid:18) ω (cid:19)
tum dots. For clarity reasons we postpone such analysis X
for the future studies. where [12]
γ(ω)= √ω|2ω|∆2 for |ω|>∆, (11)
A. Outline of the formalism ω− for ω <∆.
( i√∆2 ω2 | |
−
Energy spectrum and transport properties of the sys- Deep in a subgap regime (i.e. for ω ∆) only the off-
| | ≪
tem(1)canbeinferredfromthematrixGreen’sfunction diagonal terms of (10) survive, approaching the static
3
value Γ /2. From the physical point of view a magni-
S
−
tude Γ /2 ∆ canbeinterpretedason-dotpairing
S d1
gap in|−duced|in≡the QD . Such situation has been stud- ρ (ω)
1 d2 ε~
ied in the literature by a number of authors [24] apply- 150 2
ingvariousmethodstoaccountforthecorrelationeffects 150
ΣU(ω). ρd2(ω)
1
100 100 ~ε +U
2 2
B. Influence of the phonon modes 50
50 U
2
0
Generally speaking, the phonon modes have both the -1.22-1.2-1.18
quantitative and qualitative influence on the electron 0 ω / ΓN
transport through nanodevices [25]. In particular they
-5 -4 -3 -2 -1 0 1 2 3 ω / Γ
can be responsible for such effects as: appearance of the N
multiple side-peaks, polaronic shift in the energy levels,
FIG. 2: (color online) Spectral function of the side-attached
lowering of the on-dot potential U (even to the nega-
i
quantum dot obtained at low temperature for the model pa-
tivevaluespromotingthepairhopping[26]),suppression
rameters ε2 =−1ΓN, U2 =5ΓN, ω0 =0.2ΓN, g =1, ε1 =0,
of the hybridization couplings Γ and often serve as a
β ∆=10ΓN assuming a weak interdot coupling t=0.2ΓN.
source of the decoherence. These and related subjects
have been so far studied by many groups, mainly con-
sidering the single quantum dots coupled to the normal
In figure 2 we illustrate the characteristic spectrum
leads [27]. Here we would like to focus on the different
of the side-attached quantum dot (see section IV.A for
situation (Fig.1) considering the phononmodes coupled
technical details). We notice two groups of the narrow
to the side-attachedquantumdot. This is reminiscentof
peaks. The lowerone startsfromthe energyε˜ =ε ∆
2 p
the setup discussed by M. Bu¨ttiker [28] except that the (where ∆ = λ2/ω is the polaronic shift) followed−by a
p 0
fermion reservoir is here replaced by the phonon bath.
number of equidistant phonon peaks spaced by ω . The
In analogy to (6) we express the selfenergy Σ0(ω) of 0
2 upper phonon branch is separated by U and it mani-
2
QD by the following contributions
2 fests the charging effect [25]. For the coupling g = 1 we
observe only about five phonon peaks but in the antia-
Σ02(ω)=tG1(ω)t∗+Σp2h(ω), (12) diabatic regime (g 1) their number considerably in-
≫
where tG (ω)t originates from the interdot hybridiza- creases. Such tendency is shownin section III discussing
1 ∗
tionandthesecondtermisinducedbythephononreser- the spectrum of QD1. Let us also stress that the in-
voir. Since QD is assumed to be weakly coupled with terference peaks have a rather tiny but yet finite width
2
the central dot we approximate the selfenergy Σp2h(ω) ∝t2/ΓN [20].
adopting the local solution. The selfenergy Σph(ω) can
2
be determined by means of the Lang-Firsov canonical
C. Subgap transport
transformation which effectively gives [29]
ω−ε2−1Σp2h(ω) =Xl ω−A0ε˜(2l)(l) ω+Aε0˜(2l)(l) ! (13) afrtoeCmdhoathnrgelyemtberyatantlhslpiecoAlretnaiddnraeareesvutbmhgeenacphcaornengivsiemmrt.eedE|elVienc|tt<orot∆hnesisCcgooemonpienergr-
pairs in superconductor simultaneously reflecting holes
withthe quasiparticleenergiesε˜ (l)=ε ∆ +lω ,the
corresponding polaronic shift ∆2 =λ2/ω2−anpd tem0pera- back to the normal lead. Such anomalous current IA(V)
p 0 can be expressed by the Landauer-type formula [30]
ture dependent spectral weights (l) [25, 29]
A
2e
A(l) = e−g√1+2Nph(l) enω0/kBT (14) IA(V)= h dωTA(ω)[f(ω−eV,T)−f(ω+eV,T)](,16)
Z
I 2g2 N (l)[1+N (l)] .
× l ph ph wheref(ω,T)=1/ eω/kBT +1 istheFermi-Diracfunc-
(cid:18) q (cid:19) tion. Andreev transmittance T (ω) depends on the off-
A
Asusually,weintroduceadimensionlessadiabadicitypa- diagonal part of the(cid:2)Green’s fun(cid:3)ction G (ω) via [30]
1
rEaimnsetteeirngdi=strωλib202u,tNiopnh(aln)d=I(cid:2)ledωe0n/koBteT t−he1(cid:3)m−1odisifitehdeBBeosssee-l TA(ω)=Γ2N|G1,12(ω)|2. (17)
functions. In particular, for the ground state the equa-
The transmittance (17) can be regarded as a qualitative
tion (14) simplifies to
measure of the proximity induced on-dot pairing. Under
gl optimal conditions it approaches unity when ω is close
lim (l)=e−g θ(l). (15) to the quasiparticle energies ε2+(Γ /2)2. The An-
T 0A l! ± 1 S
→
p
4
dreevtransmittanceT (ω)isalsosensitivetootherstruc-
A
tures, for instance originating from the interferometric
ρ (ω)
effects [20, 22, 23]. d1
InwhatfollwosweshallstudythedifferentialAndreev 0.5 g = 0.1 ρd1(ω)
conductance ω0 = 1 ΓN 0.04
0.4 t = 0.2 Γ 0.03
N
∂I (V) 0.02
G (V)= A (18) 0.3 0.01
A
∂V 0
0.2 0.8 0.9 1
exploringitsdependenceonthephononmodes. Westart ω / ΓN
this analysis assuming that both quantum dots uncorre- 0.1
latedandwenextextendit(insectionIV)byconsidering
the correlation effects. As a general remark, let us no- 0
tice that the even function T ( ω)=T (ω) implies the -3 -2 -1 0 1 2 ω / Γ
A − A N
symmetric Andreev conductance GA(−V)=GA(V) re- ρd1(ω)
gardless of any particular features due to interference, 0.5 g = 1 ρd1(ω)
phonons, correlations or whatever. Physically this is ω = 0.2 Γ 0.08
0 N
caused by the fact that particle and hole degrees of free- 0.4 t = 0.2 Γ 0.06
dom participate equally in the Andreev scattering. N 0.04
0.3 0.02
0
0.2 Γ 0.8 1 1.2 1.4
III. UNCORRELATED QUANTUM DOTS N Γ ω / ΓN
S
0.1
Upon neglecting the correlationselfenergies ΣU(ω)=0
i
one has to solve the following coupled equations 0
-3 -2 -1 0 1 2 ω / Γ
N
1 iΓ Γ ρ (ω)
G−11(ω) = g−11(ω)−|t|2G2(ω)+ 2 ΓSN iΓSN (19) d 10.5 g = 10 ρd1(ω)
(cid:18) (cid:19)
G−21(ω) = g−21(ω)−|t|2G1(ω)−Σp2h(ω). (20) 0.4 ωt =0 =0. 20 .Γ01N ΓN 000...000345
0.02
We havecomputednumericallythe matrix Green’sfunc- 0.3 0.01
tions Gi(ω) for a mesh of energy points appropriate for 0
themodelparameters(mainlydependentonω ,gandt). 0.2 0.95 1 1.05
0 ω / ΓN
Practically already about ten iterations proved to yield
a fairly convergent solution. 0.1
Since the proximity induced on-dot pairing predomi-
0
nantly affects the energy region around µ we first con-
S -3 -2 -1 0 1 2 ω / Γ
sider the instructive case ε =0, ε =ε . In figure 3 we N
1 2 1
6
present the equilibrium spectrum ρ (ω) of the central
d1
quantum dot obtained for three representative coupling FIG. 3: (color online) The interferometric Fano-type line-
constants g =λ/ω corresponding to the adiabatic limit shapes appearing at T = 0 in the spectral function ρd1(ω)
0 of QD1. Numerical calculations have been done for the un-
g 1 (upper panel), the antiadiabatic regime g 1
≪ ≫ correlated quantum dots Ui =0 using the model parameters
(bottompanel)andtheintermediatecase(middlepanel).
ε1=0, ε2=1ΓN, t=0.2ΓN, ΓS =5ΓN and ∆=10ΓN.
OntopoftwoLorentzianpeakscenteredatthequasipar-
ticleenergies Γ /2weclearlyseeformationoftheFano
S
±
resonances. Theyappearatenergiesε˜ +lω andattheir
2 0
intheprecedingsection. Againwenoticethebroadmax-
mirror reflections (because of the particle - hole mixing
imacenteredatthe subgapquasiparticleenergies Γ /2
[21]). Number of these phonon features depends on the S
±
accompaniedbyanumberoftheFano-typeresonancesat
adiabadicityparameterg. Fortheadiabaticregimethere
(ε˜ +lω ). SpectroscopicmeasurementsoftheAndreev
appear only a few phonon features whereas in the oppo- 2 0
±
conductance would thus be able to detect such phonon
site antiadiabatic limit there is a whole bunch of such
induced interferometric features.
narrow structures. In the latter case they seem to have
an irregularstructure, but after closer inspection we can The subgap quasiparticle lorentzian peaks (often re-
clearly see the Fano-type shapes (see the inset). ferred as the bound Andreev states) depend on the en-
Phonon driven replicas of the Fano lineshapes appear ergy level ε1. In the case of single quantum dot (i.e.
also in the Andreev transmittance (see Fig. 4). In a dis- for vanishing t) the spectral function ρd1(ω) consists of
tinction to the spectral function ρ (ω) the resonances two lorentzians at E (where E = ε2+(Γ /2)2)
d1 ± 1 1 1 S
showupinasymmetrizedwayduetoreasonsmentioned broadened by Γ . Their spectral weights are given by
N
p
5
T (ω)
A ρd1(ω)
1 g = 1 0.6
0.8 ω0 = 0.2 ΓN 0.4
t = 0.2 ΓN 0.2
0.6
0
0.4
1
0.2 2
0 ε1 / ΓN 3
h ] 1 -3 -2 -1 Γ 0S 1 2 ω / ΓN GA(V) [ 44e 25/h ] -6 -4 -2 0 ω / Γ 2N 4 6
2/ 1
e 0.8
4
V) [ 0.6 ΓN -ε~2 ε~2 0.5
G(A 0.4 0 0
0.2 1
0 2
-3 -2 -1 0 1 2 eV / ΓN ε1 / ΓN 3
FpaIGne.l)4a:nd(cothloerdoiffnelirneen)tiTalhseubAgnadpreceovndturactnasnmciettGanAc(eV)(u(pbpoetr- 4 5 -6 -4 -2 0 eV / Γ2N 4 6
tompanel)obtainedfortheintermediatecouplinglimitg=1
using thesame parameters as in figure 3.
FIG. 5: (color online) Dependence of the spectral function
ρd1(ω) and the Andreev conductance GA(V) on ε1 (tunable
by the gate voltage). Calculations have been done for the
same parameters as in figure4.
the BCS factors 1(1 ε /E ). This fact has some im-
2 ± 1 1
portance also for the interferometric features. Figure 5
showsthespectrumρd1(ω)forseveralvaluesofε1. When A. Influence of U2
the energy ε moves away from the Fermi level (by ap-
1
plying the gate voltage) we observe a gradual redistri-
We start by considering the effects of finite U , ne-
butionofthe quasiparticlespectralweightsaccompanied 2
glectingcorrelationsonthecentralquantumdot(U =0).
with suppression of the Fano resonances, especially at 1
Sincetheside-attachedquantumdotisweaklyhybridized
(ε˜ +lω ). The Andreev transmittance T (ω) and dif-
−feren2tialc0onductanceG (V)areevenfunctiAonstherefore with QD1 therefore the indirect influence of external
A leads on QD should be rather meaningless. For this
such particle-hole redistribution is not pronounced, nev- 2
reason we impose a diagonal structure of the selfenergy
ertheless suppression of the phonon induced Fano line-
shapes is well noticeable. Σdiag(ω) 0
ΣU(ω) i,↑ , (21)
i ≃ 0 − Σdi,iag(−ω) ∗ !
↓
h i
and here i=2. We next approximate the diagonal terms
of (21) by the atomic limit solution
IV. CORRELATION EFFECTS 1 1 n2,σ¯ n2,σ¯
= − + (22)
ω−ε2−Σd2,iσag(ω) ω−ε2 ω−ε2−U2
Inthis sectionweaddressqualitativeeffects causedby where ¯ = , ¯ = . It has been pointed out [31] that
the Coulomb repulsion between the opposite spin elec- ↓ ↑ ↑ ↓
theselfenergydefinedinequation(22)coincideswiththe
trons. Roughly speaking, we expect some possible sig- second order perturbation formula
natures of the charging effect (Coulomb blockade) and
eventualhallmarksof the Kondophysics. Since the elec- (U )2n (1 n )
Σdiag(ω) = U n + 2 2,σ¯ − 2,σ¯ (23)
tron transport occurs in our setup via QD1 we suspect 2,σ 2 2,σ¯ ω ε U (1 n )
2 2 2,σ¯
thatpredominantlytheCoulombpotentialU canhavea − − −
1
significantrole. Forcompletenessweshallhoweverstudy and it can be generalized into more sophisticated treat-
the influence of correlations on both quantum dots. mentsintheschemeofiterativeperturbativetheory[32].
6
ρ (ω)
d1 -~ε
2
0.3
ρ (ω)
~ε +U d1 ρd1(ω)
2 2
0.2 -~ε2-U2 ~ε2 0.15 0.05 Kondo peak
0.1
0.1
0.05
0 0
-5 -4 -3 -2 -1 0 1 2 3 ω / ΓN 0 ω / ΓN 1
2/h ] 0 .18 UU12 == 05 ΓN 0 -15 -10 -5 0 5 10 ω / ΓN
e
4
[ FIG. 7: (color online) Spectral function ρd1(ω) of the cen-
V) 0.6 t1r0a−l3qΓuNa,ntεu1m=d−ot2ΓobNt,aiUn2ed=in15thΓeNK, oΓnSdo=re4gΓiNm,etfo=r k0B.2TΓN=,
(A 0.4 ω0 = 0.2ΓN, g = 1, ∆ ≫ ΓN. The vertical arrows indicate
G
positions of the subgap Andreev states at ±E1 and at their
0.2 Coulomb satellites. We can notice that the Kondo peak at
ω=0 is distinct from thephonon induced Fano resonances.
0
-5 -4 -3 -2 -1 0 1 2 eV / Γ 5
N
ertiesthanabovediscussedU . Thecentralquantumdot
2
FIG. 6: (color online) Spectralfunction ρd1(ω) of thecentral isdirectlycoupledtobothexternalleadsthereforeonone
quantum dot (top panel) and the differential Andreev con- handbyitexperiencestheinducedon-dotpairing(dueto
ductance (bottom panel) obtained for ε1 = 0, ε2 = −1ΓN, ΓS) and, the other hand, the Kondo effect (due to ΓN).
t = 0.2ΓN, g = 1, ω0 = 0.2ΓN, ∆ = 10ΓN assuming the These phenomena are known to be antagonistic. Their
Coulomb potential U2 = 5ΓN. The corresponding ρd2(ω) is nontrivial competition in a context of the quantum dots
shown in Fig. 2.
has been discussed theoretically by many groups using
various techniques (see Ref. [13] for a survey).
To recover basic qualitative features we shall follow
For the weakinterdotcoupling t we expect howeverthat
here our previous studies [34] which proved to yield sat-
corrections to (22,23) are not crucial. We skip here the
isfactory results for the single quantum dot on interface
higherordersuperexchangemechanismleadindtotheex-
betweenthemetallicandsuperconductingleads[10]. We
otic Kondo effect [33] which is beyond the scope of our choosethe correlationselfenergyΣU(ω)in the form(21)
present study. 2
and determine its diagonal parts by the equation of mo-
The top panel of figure 6 shows the spectrum ρ (ω)
d1 tion approach [35]. Formally we use
obtained at low temperature for U = 5Γ , U = 0. As
2 N 1
far as the side attached quantum dot spectrum is con- Σdiag(ω)=U [n Σ (ω)] (24)
cerned it reveals a bunch of phonon peaks formed near 1,σ 1 1,σ¯− 1
U [n Σ (ω)][Σ (ω)+U (1 n )]
the energy ε˜ and another group of states around the 1 1,σ¯ 1 3 1 1,σ¯
2 + − − ,
Coulomb satellite ε˜2+U2 (see figure 2). These phonon ω−ε1−Σ0(ω)−[Σ3(ω)+U1(1−n1,σ¯)]
signaturesappearinρ (ω)astheFano-typeresonances.
d1
where
Due to the absorbed superconducting order we can no-
tice effectively four groups of such Fano-type structures
1 1
tnieoanrsb.yTthheeAennedrrgeieevsεt˜r2a,nε˜s2m+iUtt2aanncedTaAt(tωh)eiirsmsyirmromrerterflizeecd- Σν(ω)=Xk |VkN|2(cid:20)ω−ξkN + ω−U1−2ε1+ξkN(cid:21)
versionsofwhatisshowninfigure6thereforethe result- f(ξkN) for ν =1 (25)
ing differential conductance is even function of applied × 1 for ν =3
voltage V (see the bottom panel in Fig. 2). (cid:26)
and as usually Σ0(ω)= k VkN 2/(ω ξkN)= iΓN/2.
| | − −
Let us remark that upon neglecting the terms Σ (ω)
1
P
B. Influence of U1 and Σ3(ω) the selfenergy (24) nearly coincides with the
second order perturbation formula (23)
Correlations originating from the Coulomb repulsion
lim Σdiag(ω)= (26)
U have a totally different effect on the transport prop- 1,σ
1 Σ1,Σ3 0
→
7
locatednearbytheKondopeak. Thezero-biasfeatureit-
selfisalsoquite sensitiveto the asymmetryratioΓ /Γ
S N
-practicallyitisvisibleonlywhenbothcouplingsΓ are
β
GA(V) [ 4e2/h ] comparable [34].
0.05
0.04
0.03
0.5 V. CONCLUSIONS
0.02
0.01 1
0
-15 1.5 We have studied influence of the phonon modes on
-10
-5 0 2 ε2 / ΓN the spectral and on the Andreev transportin the double
5
eV / ΓN 10 15 2.5 quantumdotverticallycoupledbetweenthemetallicand
superconducting electrodes. Our studies focused on the
weakinterdotcoupling t assuming that phononsdirectly
FIG. 8: (color online) The differential Andreev conductance affect only the side-attached quantum dot. Under such
GA(V) for the set of parameters discussed in figure 7 and circumstancesanexternalphononbathleadstosomein-
several values of ε2. terferometric effects, reminiscent of the dephasing setup
introduced by Bu¨ttiker [28].
[U ]2n (1 n ) InparticularwefindanumberoftheequidistantFano-
U n + 1 1,σ¯ − 1,σ¯ type patterns manifested both in the effective spectrum
1 1,σ¯
ω ε U (1 n ) Σ (ω)
1 1 1,σ¯ 0 andinthesubgaptransportproperties. Theselineshapes
− − − −
appear at (ε˜ +lω ) (where ω is the phonon energy, l
2 0 0
exceptofΣ (ω)presentinthenumeratorof(27). Thisin- ±
0 is an integer number and ε˜ = ε λ2/ω describes the
2 2 0
dicatesthatequation(24)isabletocapturethecharging −
QD energyshiftedbyapolaronicterm). Theycanbere-
2
effect (Coulomb blockade). The additional terms Σ (ω)
ν gardedasreplicasoftheinitialinterferometricstructures
provide corrections which are important in the Kondo
in absence of the phonon bath formed at ε [20, 23].
2
regime, i.e. for ε < 0 < ε +U at temperatures below ±
1 1 1 Electron correlations U on the quantum dots can in-
T =0.29 U Γ /2exp πε1(ε1+U1) . Under these con- i
K 1 N ΓNU1 duceadditionalCoulombsatellitesoftheseFanofeatures.
ditions atpµN there formhs the narroiw Kondo resonance We have investigated in some detail how the correla-
of a width scaled by k T . In our present method such tion effects get along with the Fano interference tak-
B K
Kondo resonance is only qualitatively reproduced [34]. ing into account the induced on-dot pairing. We notice
Its structure in the low energy regime ω k T must thatphononfeaturesaresensitivetothesubgapAndreev
B K
| |≤
beinferredfromtherenormalizationgrouporothermore states (dependent on U ) and to the Kondo effect. The
i
sophisticated treatments. latter one is important whenever the phonon lineshapes
Let us now point out the main properties character- are induced in a vicinity of the Kondo peak. We thus
istic for the Kondo regime. The equilibrium spectrum expectthatquantuminterferencewoulddestructivelyaf-
of QD illustrated in figure 7 consists of four Andreev fecttheKondophysicsbypartlysuppressingitszero-bias
1
bound states (indicated by the vertical arrows) centered hallmark [10].
at ε2+(Γ /2)2 and (ε +U )2+(Γ /2)2. The Itwouldbeofinterestforthefuturestudiestocheckif
± 1 S ± 1 1 S
fact that ε is located aside the superconducting en- the presently discussed effects are still preserved when
1
p p
ergy gap causes asymmetry of the quasipartice spectral the interdot coupling t is comparable to the external
weights. Besides these broad lorentzians we additionally hybridization Γ . We suspect, that the Fano-type pat-
β
notice the narrow peak at the Fermi level signifying the terns shall evolve into some new qualities typical for the
Kondo effect. Such Kondo peak is considerably reduced complex molecular structures. Furthermore, the role of
in comparison to the normal case Γ = 0 because of a Coulomb interaction U might prove to be more influen-
S 2
competition with the on-dot pairing [34]. On top of this tial via the higher order exchange integrations inducing
picture we recognize the phonon degrees of freedom ap- someexotickindsoftheindirectKondoeffect[33]. These
pearing as the Fano-type resonances at (ε˜ +lω ). nontrivial issues deserve further studies eventually using
2 0
±
The above listed effects are also detectable in the dif- some complementary methods.
ferential Andreev conductance (see Fig. 8). G (V) has
A
local maxima at voltages corresponding to the energies
of the subgap bound states. Furthermore, similarly to
Acknowledgments
ourpreviousstudies [34], wenotice thatthe Kondopeak
leads enhances the zero-biasAndreev conductance. This
property has been indeed observed experimentally [10]. WeacknowledgethediscussionswithK.I.Wysokin´ski,
InthepresentsituationweadditionallyobservetheFano- B.R.Bul kaandS.Andergassen. Theprojectissupported
type resonances. They destructively affect the zero-bias bytheNationalCenterofScienceunderthegrantNN202
enhancementwheneverthephononfeatureshappentobe 263138.
8
Appendix A: Fano-type interference: Similar reasoning can be applied to the T-shape dou-
phenomenological arguments ble quantum dot system shown in Fig. 1. For simplic-
ity let as neglect the phonon bath and assume that
Here we would like to explain in a simple way why both electrodes are normal conductors. In the case
theinterferometricFanostructuresshowupinthetrans- of weak interdot coupling t2 Γ2 the side-attached
≪ β
port properties of DQD system. In general, the Fano- dot QD2 plays the role of ”resonant” channel with its
type lineshapes [36] emerge whenever the localized (res- transmission amplitude t (ω) = √G t/2 , where
r rω ε2+it/2
onant) electron waves interfere with a continuum (or G = 2e2. On the other hand the oth−er central dot
with sufficiently broad electron states). Following [37] r h
QD provides a relatively broad background t (ω) =
1 d
let us consider the very instructive example in which
the ”direct” transmission channel td = √Gdeiφd (here 2he2(Γ4NΓ+NΓΓSS)2 ω−ε(1Γ+Ni(+ΓΓNS+)/Γ2S)/2. For energies ω ∼ ε2
Gd denotes its conductance and φd stands for an ar- tqhe latter amplitude is nearly constant td(ω) √Gdeiφd
bitrary phase) is combined with the transmission am- with √G = 2e2 4ΓNΓS 1 ≃ . Un-
opnliatundt”e lterv(eωl)ε=. √FGrormω−gε(erΓ+nLei+(rΓaΓLRl+)c/Γo2Rn)s/i2deorfataionnosth[e3r5]”rtehse- der suchdcondiqtionhs(tΓhNe+ΓreSs)u2l(cid:12)(cid:12)t2in(εg1−cεo2n)/d(uΓNct+aΓnSc)e+iG(cid:12)(cid:12)(ω) =
r t +t (ω)2 indeed reduces to(cid:12) the Fano structur(cid:12)e (A1).
corresponding conductance of such ”resonant” level is | d r |
G = 2e2 4ΓLΓR . Effectively these two channels yield Some more specific microscopic arguments in support
r h (ΓL+ΓR)2 for the Fano-type interference of the strongly correlated
the following asymmetric structure quantum dots have been discussed at length e.g. by
Maruyama [17] and by Zˇitko [18].
ω˜+q 2
G(ω)= t +t (ω)2 =G | | , (A1)
| d r | d ω˜2+1
where ω˜ = (ω−εr)/(21Γ) and q = i+e−iφd GGdr is the
characteristic asymmetry factor. q
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ρ ω
( )
d1
0.5 ε / Γ = 1 0.03
2 N ρ (ω)
ω Γ d1
/ = 0.5
0 N 0.02
0.4 Γ
g / = 0.5
N
0.01
0.3
0
0.2 1.2 1.3
ω / Γ
N
0.1
0
-4 -2 0 2 ω Γ 4
/
N