Three-terminal transport through a quantum dot in the Kondo regime: Conductance, dephasing, and current-current correlations

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by  David Sanchez

Three-terminal transport through a quantum dot in the Kondo regime: Conductance,
dephasing and current-current correlations
David Sa´nchez and Rosa Lo´pez
D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland
(Dated: February 2, 2008)
5
0 We investigate the nonequilibrium transport properties of a three-terminal quantum dot in the
0 stronglyinteractinglimit. Atlowtemperatures,aKondoresonancearisesfromtheantiferromagnetic
2 coupling between the localized electron in the quantumdot and theconduction electrons in source
n and drain leads. It is known that the local density of states is accessible through the differential
a conductance measured at the (weakly coupled) third lead. Here, we consider the multiterminal
J current-currentcorrelations (shotnoiseandcrosscorrelations measured attwodifferentterminals).
7 We discuss the dependence of the current correlations on a number of external parameters: bias
1 voltage, magnetic field and magnetization of the leads. When the Kondo resonance is split by
fixing the voltage bias between two leads, the shot noise shows a nontrivial dependence on the

] voltage applied to the third lead. We show that the cross correlations of the current are more
ll sensitive than the conductance to the appearance of an external magnetic field. When the leads
a are ferromagnetic and their magnetizations point along opposite directions, we find a reduction of
h
the cross correlations. Moreover, we report on the effect of dephasing in the Kondo state for a
-
s two-terminal geometry when thethird lead plays therole of a fictitious voltage probe.
e
m PACSnumbers: 72.15.Qm,72.70.+m,73.63.Kv
.
t
a
I. INTRODUCTION expected to suffer from dephasing, leading to a decrease
m
in the conductance. This issue has recently attracted a
- lot of attention4,5,6. In this work, we mimic in a phe-
d The Kondo effect represents a distinguished example
n nomenological way the effect of dephasing on the trans-
of strong many-body correlations in condensed matter
o port properties of a two-terminal quantum dot in the
physics.1Overthelastfifteenyears,muchefforthasbeen
c Kondo regime by introducing a fictitious voltage probe.
[ madeinunderstandingthe implicationsofthe Kondoef-
fect on the scattering properties of phase-coherent con- Now,intheabsenceofdephasing,thebuildingblockof
2
ductors. Indeed, the electric transport through a quan- theKondoresonanceisanarrowpeak(ofwidth∼kBTK)
v
5 tum dot connected to two terminals becomes highly cor- around EF in the local density of states (LDOS) of
8 relatedwhenthe temperature T is loweredbelow achar- the dot. However, full quantum-dot spectroscopy of
4 acteristic energy scale given by kBTK.2 At equilibrium, the LDOS cannot be accomplished with a two-terminal
3 the Kondo temperature T depends on the parameters transport setup. In particular, one cannot gain experi-
K
0 ofthesystem,i.e.,thecouplingofthedottotheexternal mentalaccesstothepredictedlargevoltageinducedsplit-
4 leads due to tunneling, the dot onsite repulsion (charg- ting of the LDOS when eVsd > kBTK.7,8,9,10 A way to
0
ingenergy)andthepositionoftheresonantlevelrelative circumvent this problem is by attaching a third lead,
t/ to the Fermi energy E . All of them can be tuned in a as demonstrated independently by Sun and Guo11 and
a F
m controlled way.3 Lebanon and Schiller.12 In subsequent laboratory work,
De Franceschi et al.13 observed a split Kondo resonance
Inaquantumdotwithasufficientlylargechargingen-
-
by employing a slightly modified technique—one of the
d ergy(U ≫k T)andasingleenergylevelwellbelowE ,
B F
n the dynamics of the quasilocalized electron becomes al- leadswasreplacedbyanarrowwiredrivenoutofequilib-
o rium where left and right moving carriers have different
mostfrozen. Therefore,a quantumdot canbe viewedas
c electrochemical potentials.
an artificial realization (at the nanoscale) of a magnetic
:
v impurity with spin S = 1/2. At very low temperatures Motivated in part by the works cited in the preceding
i
X (T < TK), charge fluctuations in the dot are suppressed paragraph, we are concerned in this paper as well with
andtherearisesaneffectiveantiferromagneticinteraction the nonequilibrium Kondo physics and the fluctuations
r
a between the electrons of the reservoir and the S = 1/2 of the current through a quantum dot attached to three
localized moment. Remarkably, the measured conduc- leads. As is well known,the investigationof the current-
tancereachesthemaximumvalueforaquantumchannel current correlations in mesoscopic conductors has been
(2e2/h) and the dot appears to be perfectly transparent a fruitful area of research.14 Nevertheless, there are still
when a small voltage eV is applied between the source very scarce applications to strongly correlated systems
sd
and the drain contacts. Nevertheless, the coherent cor- as the shot noise is a purely nonequilibrium property,
related motion of the delocalized electrons forming the and thus more difficult to treat. Hershfield15 calculates
Kondo cloudcan be disturbed when either the bias volt- the zero-frequency shot noise using perturbation theory
age or the temperature are of the orderof T . In such a in the charging energy (valid when the Kondo correla-
K
case,the many-bodywavefunctionofthe Kondostate is tions are not large; e.g., at T > T ). Yamaguchi and
K
2
Kawamura16 choose the tunneling part of the Hamilto- rity. We shownextthatthesignofthe crosscorrelations
nian as the perturbing parameter. Ding and Ng17 study ofthecurrentisnegative,asexpectedfromthefermionic
the frequency dependence of the noise by means of the character of the Kondo correlations at very low temper-
equation-of-motion method (also reliable for T > T ). ature. Moreover, we discuss the effect of bias voltage,
K
Meir and Golub18 perform an exhaustive study of the external magnetic fields, and spin-polarized tunneling in
influence of bias voltage in the shot noise of a quantum the cross correlations. We finish this section with an in-
dot in the Kondo regime. Dong and Lei19 discuss the vestigationoftheeffectofspinpolarizedtransportinthe
effect on the shot noise of both external magnetic fields shot noise. Finally, Sec. IV contains our conclusions.
and particle-hole symmetry breaking. Avishai et al.20
calculate the Fano factor when the leads are s-wave su-
perconductors whereas the case of p-wave superconduc- II. MODEL
tivity is treated by Aono et al.21 The authors22 examine
thebehavioroftheFanofactoratzerotemperaturewhen We model the electric transportthrough the quantum
theformationoftheKondoresonancecompeteswiththe dotusingthe AndersonHamiltonianinthe limit oflarge
presence of ferromagnetic leads and spin-flip processes. onsite Coulomb interaction U → ∞. This way we ne-
Lo´pez et al.23 make use of the two-impurity Anderson glect double occupancy in the dot and the Hamiltonian
Hamiltonian to address the shot noise in double quan- is written in terms of the slave-bosonlanguage:32
tum dot systems. To the best of our knowledge, a study
of the current fluctuations in a multiprobe Kondo impu- H= ε c† c + ε f†f
kασ kασ kασ 0σ σ σ
rity is still missing. This is the gap we want to fill here. Xkασ Xσ
In mesoscopic conductors, Bu¨ttiker24 shows that the + (V c† b†f +H.c.)
kα kασ σ
sign of the current cross correlations depends on the X
kασ
statistics of the carriers. It is positive (negative) for
+λ(b†b+ f†f −1), (1)
bosons(fermions)duetostatisticalbunching(antibunch- σ σ
X
σ
ing). This statement is based on a series of assump-
tions (e.g., zero-impedance external circuits, spin inde- wherec† (c ) is the creation(annihilation)operator
pendent transport, normal thermal leads). Positive cor- kασ kασ
describing an electronic state k with spin σ ={↑,↓}and
relations can be found if these conditions are not met
energy dispersion ε in the lead α = {1,2,3}, ε is
kασ 0σ
(see Ref. 25 for references on this subject). Here, we
the (spin-dependent) energy level in the dot and V is
kα
just mention a few studies based on structures involv-
the coupling matrix element. The original dot second-
ing a quantum dot. Bagret and Nazarov26 consider a
quantization operators have been replaced in Eq. (1) by
Coulomb-blockaded quantum dot attached to paramag-
a combinationof the pseudofermion operatorf and the
σ
netic leads whereas the ferromagnetic case and the spin-
boson field b. Hopping off the dot is described by the
blockade case are treated by Cottet et al.27 B¨orlin et processc† b†f : wheneveranelectronisannihilatedby
al.28 and Samuelsson and Bu¨ttiker29 examine the cross kασ σ
f , an empty state in the dot is created by b† and then
correlations of a chaotic dot in the presence of a super- σ
c† generatesanelectronwithspin σ in the conduction
conducting lead. In the spin dependent case, Sa´nchez et kασ
al.30 findthatthesignofthecrosscorrelationsisaffected bandofcontactα. The bosonoperatorb (b†)maybe re-
gardedasaprojectionoperatorontothevacuum(empty)
by Andreev cross reflections. In the context of quantum
stateofthequantumdot. Tomakesurethatastatewith
computation, measuring current cross correlations have
been shown31 to yield a indirect identification of the ex- double occupancyis nevercreated,a constraintwith La-
grange multiplier λ is added to the Hamiltonian.
istence of streams of entangled particles. Therefore, the
The currentoperatorIˆ thatyieldsthe electronicflow
crosscorrelationsareavaluabletoolincharacterizingthe α
from lead α is given by
electron transport in phase-coherent conductors.
In this work, we consider electron transport through ie
Iˆ = [Nˆ ,H], (2)
a strongly interacting quantum dot attached to three α ~ α
leads. Section II explains the theoretical framework
(slave-boson mean-field theory) we use to compute the where Nˆα = kσc†kασckασ. The general form of the
conductance and the current-current correlations. We power spectrumPof the current fluctuations reads33
show that the expressions for the cross correlations may
be inferred from scattering theory applied to a Breit- S (ω)=2 dτeiωτh{δIˆ (τ),δIˆ (0)}i
Wigner resonance with renormalized parameters. Sec- αβ Z α β
tion III is devoted to the results. First, we assume that
=2 dτeiωτ h{Iˆ (τ),Iˆ (0)}i−hIˆ ihIˆ i , (3)
the third lead is a fictitious voltage probe and investi- Z h α β α β i
gate the effect of dephasing with increasing coupling to
the probe. Then, we consider that lead as a real probe δIˆ = Iˆ −I describing the fluctuations of the current
α α α
and relate the differential conductance measured at one away from its average value I = hIˆ i. We are inter-
α α
electrode with the LDOS of the artificial Kondo impu- ested in the zero-frequency limit of S (ω). Since the
αβ
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 3
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
In evaluating the above Green functions we need the
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)ck(cid:0)(cid:1),3,(cid:0)(cid:1)σ (cid:0)(cid:1)(cid:0)(cid:1) S coupling strength given by Γ (ǫ) = π |V |2δ(ǫ −
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)µ(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 23 ασ k kα
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)3(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) εkασ). In the wide band limit, one neglePcts the energy
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) dependence of Γ and the hybridization width is taken as
Γ = Γ (E ) for −D ≤ ε ≤ D (D is the high-energy
ασ ασ F
Γ cutoff). We notice that in the presence of Kondo cor-
3
relations the lifetime broadening becomes renormalized
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Γ Γ (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)Γ(cid:0)(cid:1)ασ(cid:0)(cid:1)→ Γ˜ασ = π k|V˜kα|2δ(ǫ−εkασ) and the bare level
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)c(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) 1 dσ 2 (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)c(cid:0)(cid:0)(cid:1)(cid:1)k,2,(cid:0)(cid:0)(cid:1)(cid:1)σ(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)εf(cid:0)(cid:0)(cid:1)(cid:1)u0lσl(cid:0)(cid:0)(cid:1)(cid:1)iesxpshreifstseidontooPfε˜0thσe=Foεu0σrie+r-tλr.anWsfoermcaendnloeswsegrivGerethene
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)k,(cid:0)(cid:1)1,σ(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)µ(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
µ 2 function:
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)1(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Γ˜ f (ǫ)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) G< (ǫ)=2i α ασ α , (6)
fσ,fσ (ǫP−ε˜ )2+Γ˜2
0σ σ
where Γ˜ = Γ˜ is the total hybridization width
σ α ασ
FIG. 1: The system under consideration. The central island per spin and fPα(ǫ) = θ(µα−ǫ) is the Fermi function at
is a resonant level coupled to three leads. The level may be zerotemperatureofleadαwithelectrochemicalpotential
shiftedthroughacapacitativecouplingtoagate. Inthelimit µ = E +eV . On the other hand, G< (ω) can be
of a vanishingly small capacitance, double occupancy in the α F α fσ,kασ
cast in terms of G< (ω) with the help of the equation
dot is forbidden and Kondo effect can arise. The current– fσ,fσ
of motion of the operators and then applying the ana-
current cross correlations are measured between leads 2 and
3. lytical continuation rules in a complex time contour.34
Therefore, we obtain a closed system of two nonlinear
equations[Eqs.(4)and(5)] with unknowns|b|2 and λto
energy scale kBTK in typical experiments is of the or- be found self-consistently.
der of 100 mK, the frequencies should be ω . 2.4 GHz.
FromtheprecedentargumentsandEq.(2)wecaneas-
Moreover,weshallworkatT =0(seebelow)sothatthe ilyestablishanexpressionfortheexpectationvalueofthe
current will fluctuate due to quantum fluctuations only
electric current:
(we disregardthermal fluctuations).
e
I = dǫT˜σ (ǫ)[f (ǫ)−f (ǫ)], (7)
α h Z αβ α β
X
βσ
A. Mean-field approximation
which has exactly the same transparent form as the
Themean-fieldsolutionoftheHamiltonian(1)consists Landauer-Bu¨ttikerformula36 inthetwochannel(oneper
ofconsideringthe effectofthebosoninanaveragedway, spin)caseappliedto adouble-barrierresonant-tunneling
replacing the true operator b(t) by its expectation value system:
hb(t)i. Within this approximation the Hamiltonian de-
4Γ˜ Γ˜
scribes noninteracting quasiparticles with renormalized T˜σ (ǫ)= ασ βσ , (8)
couplings: Vkα |b| → V˜kα. The theory is then suitable αβ (ǫ−ε˜0σ)2+Γ˜2σ
for studying thpe Fermi-liquid fixed point of the Kondo
whichhasasimpleBreit-Wignerlineshape. Forthesame
problem (i.e., at T ≪ T ) in which the averaged occu-
K
reason,the quasiparticledensityofstatesisaLorentzian
pationinthe dotis always1. The dominantfluctuations
functioncenteredaroundtheFermilevel(theAbrikosov-
in the system are those associated to spin.
Suhl resonance). This result is expected since we are
The stationary state of the boson field is determined
dealingwithaFermiliquidbutwestressthatthephysics
fromthe t→∞limit of its equationofmotionusing the
it containsshould not be confusedwith a noninteracting
Keldysh technique for systems out of equilibrium:34,35
quantum dot since:
V˜ G< (t,t)=−iλ|b|2, (4) (i)T˜dependsimplicitlyon|b|2andλ,anditmustthen
X kα fσ,kασ beself-consistentlycalculatedforeachsetofparameters:
kασ
contact voltages {V }, magnetic field ∆ = ε −ε ,
γ Z 0↑ 0↓
where G< (t,t) = ihc† (t)f (t)i is the lead-dot gate voltage ε (V ), and lead magnetization.
fσ,kασ kασ σ 0 g
lesser Green function. Next, we take into account the (ii) T˜ is renormalized by Kondo correlations (as the
constraint: bare Γ and ε are),
0
G< (t,t)=i(1−|b|2), (5) (iii)T˜ hasanontrivialdependenceonthebiasvoltage.
fσ,fσ All these features give rise to a number of effects
X
σ
that are not seen in a noninteracting resonant-tunneling
G< (t,t) = ihf†(t)f (t)i being the dot lesser Green diode. There are many instances: regions of negative
fσ,fσ σ σ
function. It yields the nonequilibrium distribution func- differential conductance in the current–voltage charac-
tion in the dot. teristics of a double quantum dot,37 a crossover from
4
Kondophysicstoanantiferromagneticsingletinthetwo- be calculated in the next order. This is valid as long as
impurity problem,23 an anomalous sign of the zero-bias werestrictourselvestotheFermi-liquidfixedpointofthe
magnetoresistance,22etc. Below,weshalldiscussanother Kondoproblem. Wearenotawareofreal1/N correction
example without counterpart in a noninteracting Breit- calculations of shot noise. Although Meir and Golub18
Wigner resonance: When the Kondo peak splits due to perform a noncrossing approximation (NCA), they just
a large bias voltage. substitute the NCA propagators into Eq. (9), with the
limitations exposed above.
The current-current correlations can be deduced ei-
B. Current-current correlations ther using Eq. (9) or using the scattering approach for
the multiterminal case (see Ref. 24). The latter formal-
We consider now the current fluctuations given by ism amounts to replacing the bare parameters by the
Eq. (3) at zero frequency S (0). To simplify the no- renormalizedones23. We consider the illustrative case of
αβ
tationwe introduceG (ω)=G (ω) as the dot Green having different electrochemical potentials in two leads,
0 fσ,fσ
function. After lengthy algebra, we have µα 6= µβ (e.g., α = 2 and β = 3) at zero temperature.
We find
4e2
S (0)= dǫΓ˜ Γ˜ [G<G>−GaG>f 2e2
αβ h Z α β 0 0 0 0 α S (0)=− dǫTr(s† s s† s )(f −f )(f −f ),
23 h Z 2γ 2δ 3δ 3γ γ a δ b
+G<Ga(1−f )−G<Gr(1−f )+GrG>f −GaGaf (1−f ) Xγ,δ
0 0 β 0 0 α 0 0 β 0 0 α β
(11)
δ
−Gr0Gr0fβ(1−fα)−iπαΓ˜βα(G<0(1−fβ)−G>0fα)]. (9) Bwrheeirte-Wsαigβniesrtrheesorneannocrem:alized scattering amplitude of a
Thisformula(orvariationsofit)hasbeenalreadyem-
ployed in the literature. Wei et al.38 prove it using the 2i Γ˜ασΓ˜βσ
Fisher-Lee-Baranger-Stone relation39 to write the scat- sσαβ(ǫ)=δαβ − ǫ−qε˜ +iΓ˜ . (12)
0σ σ
tering matrix elements in terms of the retarded Green
function of the dot, Gr. Dong and Lei19 and Lo´pez et In Eq. (11) the trace Tr(...) is over spin indices. The
0
al.22,23 find it in Kondo problems within a slave-boson Fermifunctionsf andf arearbitrary.24 Choosingf =
a b a
mean-field framework. Actually, in Ref. 23 it is shown f =f , we obtain
b 3
that the shot noise in a two-terminal geometry reads
S ∼ T˜(1−T˜), i.e., the well known result for the par- 2e2
tition noise but with renormalized transmissions. Souza S23(0)=− h Z dǫ{T˜1σ2T˜1σ3[f1−f3]2
et al.40 calculatethenoiseofanultrasmallmagnetictun- Xσ
nel junction by means of Eq. (9) within a Hartree-Fock +R˜σ T˜σ[f −f ]2+2T˜σT˜σ[f −f ][f −f ]}, (13)
22 32 2 3 12 23 1 3 2 3
framework. In general,we cansaythat Eq.(9) is consis-
tent within mean-field theories. However, some caution whereR2σ2 is thereflectionprobability(ingeneralRαα =
is needed if one wishes to go beyond a mean-field level. 1 − βTαβ). Notice that generally one cannot write
In deriving Eq. (9), one needs to apply Wick theorem, the mPultilead current–current correlations only in terms
which is valid only for noninteracting (quasi)-particles. of transmission probabilities as in Eq. (13). This was
More specifically, one finds terms that read: firstly pointed out by Bu¨ttiker,44 suggesting the appear-
ance of exchange effects in noise measurements. Here,
hc†kασ(t)fσ(t)c†kβσ′(0)fσ′(0)i= since we are dealing with a (renormalized) Breit-Wigner
resonance, exchange corrections due to phase differences
hc†kασ(t)fσ(t)ihc†kβσ′(0)fσ′(0)i do not play any role.
+hc†kασ(t)fσ′(0)ihfσ(t)c†kβσ′(0)i. (10) noFisoer,cio.em.,ptlheteencuesrsr,enwte-cguivrreennotwcotrhreelfaotrimonuslamfoearstuhreedshaott
The first term in the left-hand side corresponds to dis- thesamelead(e.g.,lead1). Followingthewayofreason-
connected diagrams that cancel out the term hIˆ ihIˆ i of ing that led to Eq. (13) we obtain
α β
Eq. (3) whereas the second term contributes to Eq. (9).
Therefore,theparticularHamiltonianhastobecastfirst 2e2
inaquadraticform. ZhuandBalatsky41incorrectlystate S11(0)= h Z dǫ{T˜1σ2R˜1σ1[f1−f2]2
X
that Eq. (9) takes into account the many-body effects. σ
Also,itisnotclearhowthisformulaisinferredwithinthe +T˜1σ3R˜1σ1[f1−f3]2+T˜1σ2T˜1σ3[f2−f3]2}. (14)
equation-of-motion method employed by Lu¨ and Liu.42
In our case, the mean-field approximationis known to
be the leading term in a 1/N expansion,43 where N =2 III. RESULTS
is the spin degeneracy. Therefore, we neglect the fluctu-
ationsofboththebosonfield(δb=0)andtherenormal- In the following, we present results obtained by self-
ization of the resonant level (δλ = 0),19,32 which could consistently solving Eqs. (4) and (5) for each bias volt-
5
age. The rest of parameters is changed in the next sub- Γ3=0
Γ=Γ
sections. Throughout this work, we have checked that 1 3 1 1
Γ=5Γ
current conservation (I1+I2+I3 =0) is fulfilled.45 (a) Γ3=10Γ1 (b)
Tunneling effects are incorporated at all orders since 3 1 h)
at equilibriukmBTthK0e=KoΓ˜n=doDteemxppe(−raπtu|εr0e|/is2Γfo)u,nd to be(15) 2units of 2e/h) 0.5 0.52n units of 2e/
wΓh=ich i3αs=c1lΓeaαrliystahentoontpaelrhtyubrbriadtiizvaetiroensublrto.adIneniEnqg.. (T1h5e) G (in 11 (0) (i1
referPence energy will be always set at EF = 0 and the G1
energy cutoff is D = 100Γ. The bare level is ε = −6Γ,
0
deep below EF to ensure a pure Kondo regime. 0 0
-2 -1 0 1 2 0 2 4 6 8 10
0 Γ /Γ
V /T 3 1
sd K
A. Dephasing
FIG. 2: (a) Differential conductance G11 versus bias voltage
V1 as afunction of thebare couplingΓ3 to thevoltage probe
Before turning to the determination of current cross (reservoir 3) for Γ1 = Γ2 and ε0 = −6Γ. (b) Linear conduc-
correlators,webrieflydiscusswithanapplicationtheca- tance G11(0) showing thereduction of the peak in (a) versus
pabilities of three-terminal setups to illustrate some dif- the coupling to the voltage probe. The dots are numerical
ficult aspects of the physics of the two-terminal Kondo results where the line corresponds to an analytical formula
effect. As mentioned in the Introduction, we investigate (see text).
the action of a fictitious voltage probe46 (say, lead 3) in
order to simulate decoherence effects on the formation
of the Kondo resonance between leads 1 and 2.47 These Eqs. (7) and (16):
contacts play the role of source and drain, respectively.
The voltage probe model46 describes decoherence since e 4Γ˜1Γ˜2
I = dǫA (ǫ)[f (ǫ)−f (ǫ)], (17)
an electron that is absorbed into the probe looses its co- ~Γ˜ +Γ˜ Z 0 1 2
1 2
herence. The exiting electron is replaced by an electron
(with an unrelated phase) injected by the probe. where A0(ε) = −ImGr0(ε)/π is the LDOS in the dot.
Equation (17) has the form of a formula for a two-
At low temperatures the principal source of dephas-
ing is due to quasi-elastic scattering.48 We consider then terminal current50 with Gr0(ε) = [ε−ε˜0 +i(Γ˜1 +Γ˜2 +
a voltage probe that preserves energy.49 The current Γ˜3)]−1. It is straightforward to show that a nonzero Γ3
throughthevoltageprobeiszeroateveryenergyǫ. Thus, leads to deviations of Eq. (17) from the unitary limit.
fromEq.(7)the distributionfunctionatthe probe reads In Fig. 2(b) we plot the linear conductance G =
G(V =0)asafunctionofΓ /Γ fromthe resultsfound
sd 3 1
numerically. At zero bias we can find from Eq. (17) an
T13(ǫ)f1(ǫ)+T23(ǫ)f2(ǫ) analytical expression for the reduction of the peak:
f (ǫ)= . (16)
3
T (ǫ)+T (ǫ)
13 23
2e2 2
G = . (18)
We have to insert this result into Eqs. (4) and (5) and h 2+Γ /Γ
3 1
solve self-consistently for the hybridization couplings Γ˜
andtheresonancelevelε˜ inthepresenceofquasi-elastic It is shown in Fig. 2(b) (full line). In the limit of
0
scatteringforeachvalueoftheappliedbiasvoltage. Then Γ3/Γ1 ≪ 1 a similar expression for the reduction of the
wecomputenumericallythedifferentialconductanceG= peak was found by Kaminski et al.,4 the source of deco-
dI/dV , where I =I =−I and V =V −V . herence being an ac voltage applied to the dot level.
sd 1 2 sd 1 2
Figure2(a)showsGfordifferentvaluesofthecoupling
to the probe (we set Γ = Γ ) . For Γ = 0 we obtain
2 1 3
the well known zero-bias anomaly, which arises from the B. Multiterminal conductance
formation of the Kondo resonance at V = 0. As Γ
sd 3
increasesweobserveaquenchingoftheKondopeak. The From now on, we consider lead 3 as a real electrode
degree of the conductance suppression depends on the with tunable voltage V . We set V = V = 0 and vary
3 3 2
coupling to the probe. At each bias, µ (which has to the tunneling coupling Γ . The self-consistent results of
3 3
be self-consistently calculated)adjusts itself to fulfill the Eqs. (4) and (5) are inserted in Eqs. (7) to calculate the
conditionofzeronetcurrentateachenergyǫ. Hence,Γ differential conductance through lead 1: G =dI /dV .
3 11 1 1
isaphenomenologicalparameterthatincludesdephasing Figure3(a)showsG asafunctionofV . AtΓ =0the
11 1 3
processes present in the quantum dot. To see this, we conductance at V = 0 achieves the unitary limit as in
1
can write down the current through, say, lead 1, using the two-terminal case. With increasing the coupling to
6
1 Γ =0 1 0/h)K 0-10 -5 V1/0TK0 5 10 -10 -5 V10/Γ 5 100 Γ/h)
3 T 2
2(in units of 2e/h)0.05 ΓΓ33==ΓΓ11/2 000...4680T/TKK 2S (in units of 4e23----0000....4231 (a) (b)----0000....4321 S (in units of 4e23
1
G1 0.2 -0.3 -0.3
3 3
2 2
γ γ
-0.4 -0.4
(a) (b)
-0.5 0 (c) (d)
-4 -2 0 2 40 3 6 9 -0.5 -0.5
0 0 -10 -5 0 5 10 -10 -5 0 5 10
V /T V /T 0 V /Γ
1 K 1 K V /T 1
1 K
FIG. 3: (a) Differential conductance G11 versus bias voltage
V1 for ε0 = −6Γ (TK0 = 8×10−3Γ). (b) Dependence of the FIG. 4: (a) Current-current cross correlation measured in
Kondotemperature on V1. leads 2 and 3, S23(0), as a function of the bias voltage in
theinjectinglead,V1. Kondocorrelations involveanincrease
of S23(0) for voltages larger than 2TK. (b) Same as (a) for
a noninteracting quantum dot with a resonant level exactly
third lead, G (0) decreases. For equal tunnel couplings
11 at EF. (c) and (d) correspond to the Fano factor γ23 as
(Γ =Γ =Γ =Γ/3), G (0) does not reach1 (in units
1 2 3 11 a function of voltage for the interacting and noninteracting
of 2e2/h) but instead G11(0) = 8/9, in agreement with case respectively.
Ref. 51. This is an immediate consequence of having
three leads with identical couplings. Interestingly, the
KondotemperatureofFig.3(b)doesnotvanishabruptly Inparticular,thecurrent–voltagecharacteristicsshowsa
forV =2T0,asknowninthetwo-terminalcase(seethe region of negative differential conductance in the Kondo
1 K
case Γ = 0). This is an important result as it implies case[seeFig.3(a)]whereasitreachesaconstantvalueat
3
that Kondo correlations survive at large voltages. The large voltages for an noninteracting quantum dot.
effect is reminiscent of the situation found by Aguado Toavoideffectsduetomoderatebiases,inwhatfollows
and Langreth37 in tunnel-coupled double quantum dots, we shall concentrate on a normalized S23. We define the
though the physical origin is clearly distinct. Fano factor of S23 as
S
23
γ = . (19)
23
2e |I ||I |
C. Sign of the current cross-correlations. 2 3
p
Comparison with a noninteracting quantum dot
Ifthescatteringregionwereasimplebarrieroftransmis-
sion T, γ would be simply −1. This number changes
23
We now focus on the current-current correlations of when the system under consideration is a quantum dot.
the current for V = V = 0 and equal couplings Γ = In Figs. 4(a) and (b), we plot S for the Kondo and
3 2 1 23
Γ =Γ =Γ/3. Later,weshallallowfornonzerovoltage the noninteracting case, respectively. Their correspond-
2 3
differences between leads 2 and 3. In Fig. 4(a), we show ing Fano factors are shownin Figs. 4(c) and (d). We see
the cross correlator S (0) obtained from Eq. (13). As that γ has a minimum at V =0. Analytically, we find
23 23 1
expected, S is zero for V = 0 and negative elsewhere. γ (0) = −4/9≃ −0.44, which is in excellent agreement
23 1 23
This reflects the fermionic nature of the quasiparticles. with the numerical result. Likewise, we can assess the
For comparison, we plot in Fig. 4(b) the corresponding limit of γ at very high voltages (V ≫ T0). We get
23 1 K
S for a noninteracting resonant double-barrier struc- γ = −2/9 ≃ −0.22. As observed, both curves tend to
23 23
ture with the level at E (of course, for ε = −6Γ the this value,thoughfor anoninteractingquantumdotitis
F 0
spectrum S is always very small as the transmission more quickly due to the independence of Γ on the bias
23
is). Inthis case,the physics isgovernedby the barecou- voltage.
pling Γ.52 On the contrary, in the Kondo problem the
dominating energy scale is T . Qualitatively, Fig. 4(a)
K
and 4(b) look the same until V ∼ 2T . The cross cor- D. Effect of the nonequilibrium splitting on the
1 K
relatorin the Kondo case increaseswith voltagewhile in current-current correlations
the noninteracting case S saturates at large voltages.
23
It is easy to show that the saturation value is given by Now we turn to an exciting case. Consider the follow-
−8π/81 ≃ −0.31 (in units of 4e2Γ/h). The reason for ing bias configuration: V =−V 6=0 and determine the
2 3
the increase of S (0) in Fig. 4(a) is that T is voltage differentialconductanceG asafunctionofV . Thecase
23 K 11 1
dependentunlikethebareΓ,eveninthewide-bandlimit. V =−V = 0 has been treated before. However, due to
2 3
7
0 0.8
(a) (b)
0.9 ∆V=0 0/h)K-0.25 0.60/h)K
2of 2e/h) 0.6 ∆∆∆VVV===T23 KTT0K00 2units of 4eT -0.5 ∆∆∆∆VVVV====0 0 TTKK00 0.42units of 4eT
n units 0.3 ∆V=4 TKK0 S (in 23-0.75 ∆∆∆∆VVVV====2323 T TTTKKKK0000 0.2 S (in 11
(i ∆∆VV==44 TT 00
1 KK
G1 0
-1 0
-10 -5 0 5 10 -10 -5 0 5 10
V/T 0 V/T 0
1 K 1 K
-0.3
-6 -4 -2 0 2 4 6
0 FIG. 6: (a) Cross correlations of the current measured be-
V /T
1 K tween leads 2 and 3 for the case treated in Fig. 5. (b) Same
as (a) for the shot noise in lead 1.
FIG. 5: Differential conductance G11 versus bias voltage V1
fordifferentvaluesofthevoltagedifference∆V ≡|V2−V3|&
2TK0. next consider the shot noise in lead 1 S , which is an
11
even function of the applied V .
1
InFig.6(b)weplottheresultsofEq.(14). Weobserve
thefactthatthebosonfieldnevervanishes,wecanstudy
that S at V = 0 is nonvanishing with increasing ∆V,
thesituation∆V ≡|V −V |&2T0. Asremarkedinthe 11 1
2 3 K causing a divergence of the Fano factor. This is not re-
Introduction,it hasbeen argued11,12 andexperimentally
latedtotheKondophysicsbutwiththefactthatthelead
observed13 that in a three-lead geometry the splitting of
1 at V =0 acts as a voltage probe with zero impedance
the Kondo resonance due to voltageis visible, unlike the 1
sincethenetcurrentflowingthroughitiszero. Including
two-terminalcase. Moreover,in Refs. [12,51] it has been
the fluctuations of the potentials would probably cancel
noticed that the conductance G is not sensitive to the
11 out the divergence. A consequence of Kondo physics is
strength of the coupling to the third lead, showing al-
thattheminimumatV =0turnsintoamaximum. This
ways a two-peak structure. Of course, only when the 1
occurs when the splitting in G is sharply formed [see
third lead is weakly coupled to the dot G is a measure 11
11 Fig.5].
of the LDOS. But since we are interested in the trans-
portpropertiesofthesystem,ourchoiceofequalcoupling
constantsdoesnotaffecttheresultsfortheconductances
E. Spin dependent transport and current cross
and the current-current correlations.
correlations
In Fig. 5 we plot the behavior of the differential con-
ductance G . At ∆V = 0 we obtain the zero-bias
11
So far we have assumed spin-independent transport.
anomaly of Fig. 3(a). As ∆V increases, G is split
11
at V ∼ T0. Both splitting peaks lie at V ∼ V and Let us go back to the bias configuration of Secs. IIIB
2 K 1 3
andIIIC (V =V =0)andfocusonthe spin-dependent
V ∼ V , i.e., when a pair of electrochemical potentials 2 3
1 2
transport properties. It is customary in the theoretical
are aligned. It is also at those points where the Kondo
studies of spintronic transport to take into account the
temperature is larger. We emphasize that this effect has
no similitude in the electronic transport through a non- influence of external magnetic fields and ferromagnetic
electrodes, among other parameters.53 Firstly, we shall
interacting quantum dot. Still, a mean-field theory of
change the external Zeeman field and then enable the
the Kondo effect as presented here is able to capture
presence of spin-polarized tunneling.
this physics. At the same time that the splitting in G
11
develops, the height of the peaks decreases, suppressing
the zero-biasanomaly,althoughnotsostronglyasinthe
experiment13 duetotheabsenceofinelasticscatteringin 1. Magnetic field
this case.
We now use Eq. (13) to calculate the cross correla- We assume that the leads are paramagnetic and that
tions between leads 2 and 3. The results are presented the magnetic field is applied only to the dot, resulting in
in Fig. 6(a). The dependence of S23 on voltage is rather a Zeeman gap of the bare resonant level: ∆Z = ε0↑ −
asymmetric, hindering the observation of a clear indica- ε . It is well known that, as a consequence, the Kondo
0↓
tion due to the voltage induced splitting. The asymme- resonance is split when ∆ ∼T0.7
Z K
try is caused by the third term of the right-hand side Figure 7(a)shows the differential conductanceG for
11
of Eq. (13), which is not symmetric under the operation different values of the Zeeman field. The conductance is
V → −V when ∆V > 0. That is the reason why we split and quenched with increasing ∆ , as expected. In
1 1 Z
8
1 -0.1 -0.1
∆∆==00 pp==00
0.8 Zz
22e/h) 0.6 ∆∆∆∆zZZ====0T0T..K5500TTKK00 -0.2 -0.2 pppp====0000....3636 -0.2
units of 0.4 ∆∆zzZ==11..K55TTKK00 -0.3γ23 γ23-0.3 -0.3γ23
n 0.2
(i
G 11 0 -0.4 -0.4 -0.4
-0.2 (a) (b) (a) (b)
-5 0 5 -10 -5 0 5 10-0.5 -0.5 -5 0 5 -5 0 5 -0.5
V1/TK0 V1/TK0 V1/TK0 V1/TK0
FIG.7: (a)DifferentialconductanceG11 versusV11 asafunc- FIG. 8: Fano factor of the cross correlator, γ23 vs V1/TK0 for
tion of the Zeeman term ∆Z for V2 = V3 = 0. (b) Same as different lead magnetization when V2 = V3 = 0. (a) Parallel
(a) for the Fanofactor of thecross correlator, γ23. alignment between themagnetizations of the leads with spin
polarizations: p1 = p2 = p3 = p . (b) Antiparallel case with
p1=−p2 =−p3=p.
Fig. 7(b), we depict the Fano factor of the cross correla-
torγ . Itexhibits averyinterestingfeature. Due tothe
23
splitting of the Kondo peak, the minimum of the cross case. This means that p1 = p2 = p3 = p, where pα is
correlator at V = 0 becomes a local maximum, result- the spin polarization of lead α. Ferromagnetism in the
1
ing from the suppression of the Kondo effect. However, leads arises through spin-dependent densities of states
thischangeoccursbeforethesplittingoftheconductance νασ(ǫ) = kδ(ǫ−εkασ). Hence, the linewidths become
G11. Therefore, measuring the shot noise provides new spin depePndent: Γασ = (1±pα)Γα, where +(-) corre-
information in this case. The presence of the splitting sponds to up (down) spins. We prefer to restrict pα to
would be detected in an experiment more precisely by small values as strong magnetizations would require a
means of the shot noise. The underlying reason is that proper treatment of the reduction of the bandwidth D.
theformofEq.(13)differsfromthatofthecurrentwhich We observe that γ23 is rather insensitive to changes in
is basically proportional to T˜ alone, see Eq. (7). As a p in the same fashion as G11 is in the Fermi-liquid fixed
12
point22. Onlyatmoderatepolarizations(p=0.6)wesee
result,thewidthoftheG resonanceisabitlargerthan
11
thatthedipinγ getsnarrowerbecausetheKondotem-
theγ antiresonanceandtheformeristhenmorerobust 23
23
perature decreases as p increases56,57. In addition, γ is
thanthelatteragainsttheapplicationofmagneticfields. 23
alwaysnegativeincontrasttothe resultsobtainedinthe
Coulomb blockade regime where γ can take positive
23
values27. When the spin-flip scattering rate is smaller
than the tunneling rate, γ can be positive. However in
23
2. Ferromagnetic leads
the Kondo regime this condition is never met since the
rateofspinflipscattering∼1/T is alwaysmuchlonger
K
Therehasrecentlybeenconsiderabledebateaboutthe thanthe tunneling rate∼1/Γ. Figure8(b) isdevotedto
influence of ferromagnetic leads in the Kondo physics of the antiparallel case: p1 =−p2 =−p3 =p. Accordingly,
a quantum dot.54,55,56,57 In the preceding subsection, it γ23 is lifted with increasing lead polarization since the
was clear that an external magnetic field alters the real conductance peak decreases with increasing p (roughly,
part of the quantum-dot self-energy, breaking the spin with a factor 1−p2)22.
degeneracy. In the case of spin polarized tunneling, the
situation is more subtle.57 When the magnetic moments
of the contacts are aligned along the same direction, the IV. CONCLUSION
density of states of the localized electron undergoes a
splitting if particle-hole symmetry is broken.58 Recent In summary, we have investigated the Kondo temper-
transport experiments with C60 molecules and carbon ature,the differentialconductanceandcrosscorrelations
nanotubes have addressed this regime.59,60 However, in of the current when three leads are coupled to an arti-
our case the dot is in the strong coupling limit and the ficial Kondo impurity in the Fermi-liquid fixed point of
Kondo effect is pure in the sense that no charge fluctu- the infinite-U Anderson Hamiltonian (T ≪ T ). We
K
ations are allowed. Thus, no splitting is expected in the have performed a systematic study of the properties of
differential conductance. thecrosscorrelatorswhendcbias,Zeemansplittings,and
In Fig. 8(a), we show the cross correlator γ for dif- ferromagnetic leads influence the nonequilibrium trans-
23
ferent values of the lead magnetization in the parallel portthroughthe quantumdot. Ourmostrelevantresult
9
isthebehaviorofthe shotnoisewhentherearisesavolt- ferromagneticleads,bunchingeffectswillbeenhanced.63
age induced splitting in the quantum dot. Improvements of the model should go in the direction
In addition, we have studied the current of a two- of including fluctuations of the boson field and of the
terminal quantum dot attached to a voltage probe. We renormalized level. However, we do not expect large de-
have shown that increasing the coupling with the probe viations from the results reported here when T ≪ T .
K
inducesaquenchingoftheKondopeak. Despitethesim- These fluctuations will evidently become important as
plicity of this approach, it gives rise to results that are temperatureapproachesT . Experimentally,ourpredic-
K
inagreementwithmoresophisticatedmodels,4,61 though tionscanbetestedwithpresenttechnologysuchasGaAs
thepreciseprocessesresponsibleforthedecoherenceneed quantum dots13 or carbon-nanotube nanostructures.64
still to be derived from a microscopic model.
We have not exhausted all the possibilities that the
model offers and more complicated geometries with
appealing results can be envisaged. One could ad-
Acknowledgements
dress the situation with two injecting and two receiving
leads,whichcouldgiverisetoHanburyBrown-Twiss-like
effects.62 We expect that phase related exchange terms WegratefullyacknowledgeR.Aguado,M.Bu¨ttiker,S.
will arise especially at higher temperatures (T > T ), Pilgram and P. Samuelsson for helpful comments. This
K
whenthesingletstatebetweenthelocalizedspinandthe workwassupportedbytheEURTNunderContractNo.
conduction electrons is not yet well formed. We believe HPRN-CT-2000-00144,NanoscaleDynamics,andbythe
that in the presence of spin-polarized couplings due to Spanish MECD.
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