Table Of ContentTransport in multilevel quantum dots: from the Kondo effect to the Coulomb
blockade regime
A. Levy Yeyati, F. Flores and A. Mart´ın-Rodero
Departamento de F´ısica Te´orica de la Materia Condensada C-V, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain
9 (February 1, 2008)
9
A new theoretical method is introduced to study coherent electron transport in an interacting
9
multilevel quantum dot. The method yields the correct behavior both in the limit of weak and
1
strongcouplingtotheleads,givingaunifieddescriptionofCoulombblockadeandtheKondoeffect.
n Resultsforthedensityofstatesandthetemperaturedependentconductanceforatwo-leveldotare
a
presented. The relevance of these results in connection to recent experiments on the Kondo effect
J in semiconducting quantumdots1,2 is discussed.
5
1
]
l
l The proper description of coherent electron trans- equilibrium Anderson models5,9. This type of approach
a
h port in the presence of strong electron-electron inter- hasbeenrecentlyrediscoveredandappliedtoanalyzethe
- actions has been one of the central issues in the field MotttransitionintheHubbardmodelforarbitraryband
s of mesoscopic systems3. Semiconducting quantum dots filling10,11.
e
m (QDs) provide an almost ideal system where the pre- For describing the multilevel QD we consider a model
dictions of the theory can be tested and new effects Hamiltonian which is a generalization of the single-level
.
at could be searched for. For instance, recent experi- Anderson model, H = Hdot + Hleads + HT, where
m mentshavedemonstratedthepossibilityofexploringthe H = ǫ dˆ† dˆ +U nˆ nˆ correspondstothe
Kondo effect, a prototypicalcorrelationeffect, using this dot m m m m l>m m l
d- technology1,2. uncouplePd QD (nˆm = dˆ†mPdˆm); Hleads = k∈L,Rǫkcˆ†kcˆk
n From the theoretical point of view, correlation effects tothe uncoupledleads,andHT = m,k∈LP,Rtm,kdˆ†mcˆk+
o have been mainly analyzed in this kind of systems by h.c.describesthecouplingbetweenthedotandtheleads.
c means of the simple Anderson model with a single spin- The labels m and l in H denote thPe different dot levels
[
degenerate level4,5. Within this model the Kondo ef- including spin quantum numbers.
1 fect arises due to fluctuations in the spin of an unpaired Our main objective is the determination of the dot
v electron6. So far,therehasbeenfew attempts to include retarded Green functions from which the different level
4
a multilevel spectrum for describing either single QDs chargesandthelinearconductancecanbeobtained. Ina
4
with quasi-degenerate levels or coupled QDs beyond a frequencyrepresentationtheycanbewrittenasG (ω)=
1 m
1 rate equation approach7. The actual situation in semi- ω−ǫHF −Σ (ω)−iΓ (ω)−iΓ (ω) −1, where
m m m,L m,R
0 conducting QDs should require the inclusion of a multi- ǫHF = ǫ + U n is the Hartree-Fock level (we
9 levelspectrumwhenevermanysingle-particlestateswith (cid:2)m m l6=m l (cid:3)
adopt the notation n =< nˆ >), iΓ (ω) =
9 asmalllevelseparationareinvolvedinthetransportpro- P l l m,L(R)
t/ cess. This could be the case in experiments measuring k∈L(R)t2m,k/(ω−ǫk+i0+) arethe hybridizationterms
a describing the coupling to the leads and Σ (ω) is a self-
the phase of the transmission amplitude through a QD m
m intheCoulombblockade(CB)regime8 andalsoinrecent ePnergy that takes into account electron-electron interac-
- experiments on the Kondo effect1,2. tions beyond the Hartree-Fock approximation. We shall
d The aimofthis letter is to introduceanew theoretical neglect the indirect coupling between dot states12 and
n
adopttheusualapproximationofΓ ,Γ beinginde-
o approach for describing correlation effects in multilevel m,L m,R
pendentofenergy. Inourapproach5,9, welookfor anin-
c QDs. The approach is constructed to yield the correct
: behaviorbothinthelimitofinfiniteandvanishingcharg- terpolativeself-energyyieldingthecorrectU/Γm →0,∞
v
limits.
i ing energy. This is achievedbyintroducing aninterpola-
X tive self-energy for the one-electron Green functions, an In the U/Γm → ∞ or “atomic” limit, Gm(ω) can be
r approach which has been successfully applied to several obtained using the equation of motion method13, which
a
interacting systems including the equilibrium and non- yields
G(at)(ω)= < l6=m(1−nˆl)> + <nˆl (s6=l)6=m(1−nˆs)> + ... + < l6=mnˆl > , (1)
m ω−ǫ +i0+ ω−ǫ −U +i0+ ω−ǫ −(M −1)U +i0+
Q m l6=m Q m m Q
X
where M denotes the total number of one-electron lev- the knowledge of many particle correlations < nˆ nˆ >,
1 2
els in H . In this expression all possible charge states < nˆ nˆ nˆ >, etc. However, for sufficiently large U fluc-
dot 1 2 3
of the dot give a contribution. Their evaluation requires tuations in the dot charge with respect to the mean
1
charge N by more than one electron become negligible Σ(at) = amU2(ω−ǫm+i0+)+bmU3 ,
and G(mat)(ω) is accurately given by the three poles ex- m (ω−ǫm+i0+)2+cmU(ω−ǫm+i0+)+dmU2
pression
(4)
Am
G(mat)(ω)≃ ω−ǫ −U(NN−1−1)+i0+ + wheream =(N −nm)[1−(N −nm)]+<nˆnˆ >m; cm =
m N−n −3N;d =<nˆnˆ > +3N2−1−(3N−1)(N−n )
AmN + AmN+1 , (2) andbmm =N2(1m−N)−(Nm−nm)dm. Ontheotherhanmd,
ω−ǫm−UN +i0+ ω−ǫm−U(N +1)+i0+ in the U/Γm → 0 limit Σm is given by the second order
expression9
whereN =Int[N]. Inordertoyieldtheexactfirstthree
mthoemfoelnlotawionfgEsqu.m(1r)utlheseweightfactorsAmN shouldsatisfy Σ(2)(ω)=U2 ∞ dǫ dǫ dǫ ρ˜m(ǫ1)ρ˜l(ǫ2)ρ˜l(ǫ3)
m 1 2 3 ω+ǫ −ǫ −ǫ +i0+
l6=mZ−∞ 2 1 3
Am +Am+Am =1 X
N−1 N N+1 [f f (1−f )−(1−f )(1−f )f ], (5)
1 3 2 1 3 2
(N −1)Am +NAm+(N +1)Am = n
N−1 N N+1 l
l6=m where fi = f(ǫi) denotes the Fermi distribution func-
X
(N −1)2Am +N2Am+(N +1)2Am = tion at the leads, and ρ˜m(ω) are effective densities of
N−1 N N+1 states given by πρ˜ (ω) = Γ /((ω −˜ǫ )2 +Γ2 ). The
m m m m
nl+<nˆnˆ >m, (3) effective levels ǫ˜m are introduced in order to fulfill the
l6=m Fermi-liquid relations associated with charge conserva-
X
tion (Friedel sum rule (FSR)14) as discussed below.
where <nˆnˆ > = <nˆ nˆ >. The special case
m (l6=k)6=m l k Inordertodetermineaninterpolativeschemebetween
MN =−20).(NNo=ticMe t−hPa1t)thhaissetxopbreesstiroenatfeodr aGs(maNt)(=ω)1is(Nfull=y ˜ǫthme),twwohelirmeiαtsmle=t uslfi6=rmstn˜nl(o1ti−cen˜tlh),awthΣe(mn2)Γ→m/Uω2α→m0/.(ωO−n
determined by the average charges nl and the two-body the other hand, am →αm in the small U limit and thus
correlations < nˆlnˆk >. From G(mat)(ω) one can define an Σ(mat) → U2αm/(ωP−ǫm) in this case. These properties
“atomic” self-energy, Σ(at) = ω − ǫHF − G(at)(ω) −1. suggest to define the interpolative self-energy, replacing
m m m
UsingEqs. (2)and(3)itcanbe shownthat(cid:2)Σ(mat) can(cid:3) be ω−ǫm byαm/Σ(m2)+∆ǫm,where∆ǫm =ǫ˜m−ǫm,inEq.
written as the ratio of two polynomials in ω of the form (4) for Σ(mat), which yields
Σ(2) am+(am∆ǫm/U +bm/αm) Σ(m2)/U
m
Σ = . (6)
m αm !1+(2∆ǫ /U +c ) Σ(2)/α U +(c ∆ǫ (cid:16)/U +d(cid:17)) Σ(2)/α U 2
m m m m m m m m m
(cid:16) (cid:17) (cid:16) (cid:17)
Thisexpressionprovidesthegeneralizationtothemul- ∞ ∂Σm
f(ω)Im G (ω) dω =0, (8)
m
tilevel case of the interpolative self-energy first intro- ∂ω
Z−∞ (cid:20) (cid:21)
ducedin9 forthesinglelevelAndersonmodel. Thesingle
levelcase(M =2)isreadilyobtainedfromEq. (6)when
N =1taking<nˆnˆ >m=0. Itisevidentbyconstruction which at zero temperature reduces to the Luttinger
that Σm → Σ(mat) when U/Γm → ∞ and Σm → Σ(m2) in theorem15 ensuring the fulfillment of the FSR14 nm =
the small U limit. −Im[lnG (E )]/π. In Ref.5 we showed that the con-
m F
Thefinalstepinthisapproachistodeterminethelevel dition of consistency between the effective and the final
charges nm, the correlation functions < nˆlnˆk > and the charges is nearly equivalent to imposing the fulfillment
effective levels ǫ˜m selfconsistently. The charges and the of the FSR for the simple Anderson model. Similar self-
correlations functions are determined through the rela- consistency conditions have been proposed in Refs.10,11.
tions n =− ∞ f(ω)ImG (ω)dω/π and
m −∞ m We have first applied this formalism to the case of a
R −1 ∞ doubly degenerate level (four-fold degeneracy when in-
<nˆ nˆ >= f(ω)Im[Σ (ω)G (ω)]dω. (7)
l m m m cluding spin) which is a simple generalization of the sin-
πU
lX6=m Z−∞ gle level Anderson model. The inset in Fig. 1 shows
Eq. (7)isanexactrelationthatfollowsfromtheequa- the charge per level nm as a function of the leads Fermi
tion of motion of the retarded Green functions. It is im- energy EF. For the case shown in Fig. 1, correspond-
portanttostressthattheself-consistentdeterminationof ing to ǫm = 0, Γm/U = 0.075 (same for all levels) and
the two-body correlations is essential to get the correct zero temperature, one can observe a modulation in the
values of the charge for large U. The effective level is charge resembling the typical Coulomb staircase of the
determined by imposing the condition: Γm/U →0 limit (shown as dotted line for comparison).
2
tance with temperature. At temperatures large enough
to be above the Kondo temperature, T , which can be
K
estimated by the condition G(T ) ≃ G(0)/21, the con-
K
ductance tends to exhibit the usual CB peaks at the
charge degeneracy points. We should remark that our
approachdoesnotprovideanaccurateestimateofT as
K
afunctionofthemodelparametersastheexponentialde-
creaseoftheKondopeakweightforverylargeU18 isnot
strictly recovered. This limitation does not affect, how-
ever, the qualitative behavior of the conductance except
for a rescaling of the temperature values.
FIG. 1. Density of states for a fully degenerate two-level
dotwithΓm/U =0.075andǫm =0fordifferentvaluesofthe
leads Fermi energy EF. The inset shows the charge per level
and spin as a function of EF (full line) as compared to the
Γm/U →0case(dottedline). Energiesaremeasuredinunits
of U.
Fig. 1 also shows the density of states (DOS) associ-
ated with the dot levels for different values of E /U =
F
0,0.5,1.0,1.5 corresponding to the steps and the center
of the plateaus in the charging curve. For E /U = 0,
F
n ≃ 0.125 (the total charge on the dot being ∼ 0.5e)
m
the systemis in the so-calledmixed-valence regime. The FIG. 2. Total conductance for the same case of Fig. 1 as
spectrum is in this case similar to the one found us- a function of EF for different temperature values T/U: a)
ing large N-expansions in the U → ∞ limit16 show- 0.0005, b) 0.0025, c) 0.005, d) 0.01 and e) 0.03.
ing a resonance at a renormalized level just above E .
F
For E /U = 0.5 and 1.5 there are approximately one So far we have analyzed the case of a completely de-
F
and two electrons respectively inside the dot. For these generatelevel. InanactualQDgeometricalasymmetries
cases the system is in the Kondo regime, the DOS ex- couldresultinaneffectivesplitting∆E ofthedotlevels.
hibitingasharppeakaroundE (Kondoresonance)half When∆E ≫Γthe physicalsituationcouldbedescribed
F
way between two broader resonances (of width ∼Γ) ap- by a single level Anderson model. It is interesting to
proximately separated by U, corresponding to the ele- analyze the evolution from this situation to the quasi-
mentary charge excitation energy. It is worth noticing degenerate case ∆E → 0 previously discussed. This
that the weight of these resonances is the same for the evolution is depicted in Fig. 3 and 4 where a splitting
E /U = 1.5 case due to the electron-hole symmetry in ∆E =0.1U and∆E =0.5U isintroduced. Onlythefirst
F
the half-filled dot, while there is a pronounced asymme- half of the charging curve is shown as the total conduc-
try for E /U = 0.5. For a half-integer occupation of tanceissymmetricalwithrespecttoE =(3U+∆E)/2.
F F
the dot like in the E /U = 1.0 case the DOS exhibits Ascanbeobservedinthechargingcurvesshownasinsets
F
a more complex structure with three broad resonances in these figures,the splitting tends to block the charging
aroundE =0,U and 2U,and a narrowKondo peak still of the upper levels. This blocking effect is nearly com-
present at E . The overall shape is reminiscent of an plete for ∆E =0.5U and is already apparent in the case
F
averagebetween the cases with one and two electrons in of Fig. 3 where ∆E ∼ Γ. The total conductance at low
the dot. temperatures(casea)isobservedtodecreaseforincreas-
The evolution of the Kondo peak as a function of E ingsplittingapproachingthebehavior2(e2/h)sin2[πn ],
F m
and temperature should be reflected in the dot linear where m denotes the lower level. The overall effect of
conductance, G, which can be readily obtained from the temperature is again to reduce the total conductance in
Greenfunctions(seeRef.17). Theconductanceasafunc- therangeforE wheretheKondoeffectispresent. Inthe
F
tion of E and different temperatures is shown in Fig. case of Fig. 4 the conductance increases with tempera-
F
2. When approachingzero temperature the conductance tureintheregionE ∼1.5U wherethehigherlevelstarts
F
behaves like G = 4(e2/h)sin2[πn ] as expected from to be populated. One can also notice for this case that
m
the FSR. The conductance decreases very rapidly with theconductancepeaksassociatedwiththelowerlevelare
temperature in the region 0.5 < E < 2.5 where the not completely symmetrical due to the influence of the
F
DOS at E is controlled by the Kondo peak. Outside second level. Both effects, the increase of conductance
F
this region one can notice a slight increase of conduc- with temperature between each pair of peaks as well as
3
the asymmetry, are present in the experimental results the Coulomb interaction U.
of Ref.1 and can be considered as a manifestation of the A.L.Y. and A.M.R. thank J.J. Palacios and C. Teje-
multilevel structure of an actual QD. dor for fruitful discussions. This work has been funded
by the Spanish CICyT under contracts PB97-0028 and
PB97-0044.
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where orbital and spin quantum numbers are conserved
FIG.4. Same as Fig. 3 for ∆E/U =0.5.
in tunneling. Generalization of the present approach for
including indirect coupling between dot states can be ob-
The most remarkable consequences of this multilevel
tained along the same lines introducing the off-diagonal
structure would be observed in a situation correspond-
Green function components (to bepublished).
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