Many-body Correlation Effect on Mesoscopic Charge Relaxation

0.7 MB·English

by  Minchul Lee

Many-body CorrelationEffect onMesoscopicChargeRelaxation
MinchulLee,1 Rosa Lo´pez,2,3 Mahn-SooChoi,4 ThibautJonckheere,5 and ThierryMartin5,6
1DepartmentofAppliedPhysics,CollegeofAppliedScience, KyungHeeUniversity,Yongin446-701, Korea
2Departament deF´ısica, Universitat de lesIllesBalears, E-07122 PalmadeMallorca, Spain
3InstitutdeF´ısicaInterdisciplinarideSistemesComplexosIFISC(CSIC-UIB),E-07122PalmadeMallorca,Spain
4Department of Physics, Korea University, Seoul 136-701, Korea
5CentredePhysiqueThe´orique, UMR6207, Case907, Luminy, 13288 MarseilleCedex9, France
6Universite´ de la Me´diterrane´e, 13288 Marseille Cedex 9, France
1
Weinvestigateinanonperturbative waythedynamics ofacorrelatedquantum capacitor. Wefindthatthe
1
20 0m)an=y-bho/d4ye2coerrnesluarteiodnbsydothneoKtdoirsrtiunrgba-tShehiubnaivruerlesawllhoewre-farseqthueenicnyterreplraextaattiioonnroefsitshteanqcueapneturmchcaanpnaecl,itRanqc(eωC=q
n hinugteermpesaokfst(hweitdhenvsailtuyesoflasrtgaetersthfaainlshw/4heen2)satrto~ngωc≈orre±laΓti∗o,nwshaererepΓre∗seinstt.heThreenAorCmraeliszisetdanlceeveRl bq(roωa)deshnoinwgs.
a Thesepeaksaremergedtoasingleoneatω =0whenafiniteZeemanfieldisappliedcomparabletoΓ∗. The
J observedfeaturesofRq,beingmostevidentintheKondoregime,areattributedtothegenerationofparticle-hole
3 excitationsinthecontactsaccomplishedbyspin-flipprocessesinthedot.

] PACSnumbers: 73.63.-b,74.50.+r,72.15.Qm,73.63.Kv
l
l
a
h
Introduction.— Optoelectronic devices such as light thechargerelaxationtimeupontheactionoftime-dependent
-
s sources emitting single photons on demand are of an enor- potential.
e
mousinterestinquantuminformation.Recently,itssolid-state SofarseveralaspectsofACconductanceforRCquantum
m
analogue,aquantumcapacitor(QC),wascreatedbyFe`veet circuitshavebeenaddressed[18–25]focusingmostly onthe
.
t al. [1],inwhichaquantumdot(QD)wascoupledtoasingle spin polarized case. Therefore the understanding on the ef-
a
m reservoir via a quantum point contact. Fast time-controlled fectofmany-bodycorrelationsinRCcircuitsisstillmissing.
variationsoffractionsofnanosecondsonthe dotgatepoten- This Letter attempts to fill this gap, offering a physical pic-
-
d tialproducedthesingle-electronsourceofemittingelectrons tureoftheinfluenceofmany-bodyinteractionsonthedynam-
n in a desirable quantum state. Previously, Gabelli et al. [2] ics of QCs. We consider a QC formed by an ultrasmall QD
o showed that the QC could act as a RC circuit with a quan- andcalculate its C and R by using their relationswith the
q q
c
tizedresistanceaspredictedinaseriesofseminalworks[3– charge susceptibility in the linear response regime. We per-
[
5]. The experimentalinterest in the AC propertiesof meso- form a thorough study by tuning the QD from the resonant

1 scopic conductors [6–11] has been revived recently due to tunnelingregimetotheKondoregime. In theKondoregime
v
theexperimentalconfirmationofthequantizationofboththe virtualtunnelingtransitionsbetweenthedotandthereservoir
8
6 ACcurrent[1]andthequantumresistance[2]. Inacoherent flip efficiently the dot spin resulting in a many-body singlet
4 conductorthe ACtransportishighlysensitiveto the internal spin state with binding energy given by the Kondo energy
0 distribution of charges and potentials that need to be calcu- k T . First, we find that the Korringa-Shiba (KS) relation
B K
1. latedinaself-consistentlymannertoensureagaugeinvariant [26],validonlyintheFermi-liquidregime,ensuresthequan-
0 andcurrentconservingdescription.Foramacroscopiccapaci- tization of R even in the Kondo regime as long as the fre-
q
1 torthelow-frequencydynamicalconductancedependsontwo quencyis lowenough,~ω ≪ k T . However,it ispointed
B K
1
elements: the geometrical capacitance C, and the resistance outthattherelationof C withthe localizedDOSshouldbe
: q
v R. Classically,theelectricfieldonthesurfaceofthemetallic revisedwhenthechargedynamicsisfrozenduetoKondocor-
i platesiscompletelyscreenedandCischaracterizedsolelyvia
X
Coulombforces. However,Bu¨ttikeretal. pointedoutthatfor
r (a) (b)
a acoherentnanoscalesystemthatrulestatementisnotlonger !$#
valid, since electric fields penetrate at distances of the order
of the Thomas-Fermi screening length [3, 4, 12, 13]. Con-
sequently, the capacitance for a coherent conductor, termed
aselectrochemicalcapacitance,dependsonthegeometry,on
itsphysicalpropertiesandparticularlyonitsdensityofstates
!"#
(DOS)throughthequantumcapacitanceC . Evenmoresur-
q
prisingly,the resistancebecomesquantizedindependentlyof
the value of the transmission through the mesoscopic con- FIG. 1: (color online) (LEFT) Illustration of a quantum capacitor
ductor [14–17]. To distinguish it from the DC resistance, and(RIGHT)second-ordertunnelingprocessesthatgenerateasingle
R is called charge relaxation resistance, and together with particle-holepairintheconductionbandwithout[(a)]andwith[(b)]
q
C = (C−1+C−1)−1 definestheRCtimeforaQC,thatis, aspinflipinthedot.Filledandemptyarrowsinthecontactindicates
µ q particlesandholes,respectively.Hereweassume∆Z >0.
2
relations.Moreimportantly,theACresistanceRq(ω)displays 1(cid:144)2HaL 30 0.2HbL
bifpnoreoratatkhhdseeeaKnCtioon~nugωdlooom∼frtbeh±geiniΓmmte∗ear,.ajoncTrotithotuinprn.rdneH,seealeinrndetgeΓipnd∗aoaritstumlterhveeeaefrnlre-aonfinmoedrlmdtΓha∗delei=usznceridkviBpleetrTivsoKeanll 2HL@(cid:144)DR0heq1(cid:144)40 12000qC0eDHL@(cid:144)D2 HLΧΩc--0000...211. Εd=-0.3 IRme@@ΧΧccDD
-1 0 -1.5-1.-0.5 0. 0.5 1. 1.5
value for Rq(ω = 0) is achieved when an applied Zeeman Εd Ω
-0.02 0 0.02
field∆Z iscomparabletoΓ∗. TheenhancementofRq atzero G* HcL 1.HeL
tahonodMltehfieo(npdsi-etthrleo.)—nfergxeqcCiTuoteahuntelicooeimnessssbeainiscntcaitoaettmlrraifpcbetauaintotueyndri,enstsgoeoeatfhFsaepigicmn.r-1eefl(asibotpi)so.cinnopothifcepRdaorCtticdcluiere-- @HLDImΧΩ-c00..404 Εd=-0.1 2HL@(cid:144)DRΩheq00000....2468.Εd=-0.1 G*
cAptiulmneidtdeei-ntdrosetohpaneesnmpidnroeegdnsleeet-nlvccwohelahtoanefgnreeceolaVrner(leeitlnc)attte=rioroannVcstraieccnsagcenorsvsbinoeωgiwtrl.ese-TulllebhcvjeeaecpHlttQautmroDeidalitsiowncneotiahaukne- -@HLDImΧΩ11c00--033 Εd=-0.3 TK HdL 2HL@(cid:144)DRΩheq1125050HΕfdL=-0.3 TK
of this system is H = HL + HD + HT. The lead part, -2´10-4 Ω0 2´10-4 -0100-10-2-10-4 Ω 10-410-2 100
HL = kµ[ǫk + eV(t)]c†kµckµ, describes the noninteract-
ing condPuction electrons with energy ǫk (measured with re- FIG.2:(coloronline)(a)Zero-frequencylimitsofRqandCqversus
specttotheFermienergyǫF =0)andspinµinthereservoir, ǫd.(b)Typicalspectralstructureoftherealandimaginarypartsofthe
and the tunneling of electrons between the reservoir and the chargesusceptibilityχc.(c,d)EnlargedviewsofIm[χc]nearω=0
dot is modeled by HT = kµ tkd†µckµ+(h.c.) in terms ftaonrgtewnotvlianleusesatofinǫfidn.iTtehseimdaalslhyesdm(ablllueω)aanndddaottpteodin(tgsre(iennd)iclianteesdabrye
of energy-independent tunPneling(cid:2) matrix element,(cid:3)tk = t. arrow)wheretheslopeismaximal,respectively. (e,f)Rq(ω)inthe
The hybridizationbetween the dot and the lead is character-
logarithmicscale.HereΓ=0.04,andEC=0.5.
ized by a tunneling amplitude Γ = πρ0|t|2 (ρ0 is the con-
tact DOS at the Fermi energy). The dot Hamiltonian reads
HD = µ[ǫµ+eV(t)]nµ+2ECn↑n↓,wherenµ =d†µdµis While the imaginarypartof the susceptibility is obtainedby
thedotPoccupationoperatorandEC =e2/2CistheCoulomb theNRGprocedureitsrealpartiscalculatedviatheKramers-
chargingenergy. Theorbitallevelǫ = ǫ −µ∆ /2isspin- Kronig relation. Note that the NRG results for the finite-
µ d Z
dependentduetoaZeemanenergy∆ . Thetime-dependent frequencylinearresponseintheKondoregimeareknownto
Z
voltageV(t)inducesthepolarizationchargesN (t)between be reliable as long as the perturbation is weak enough [30].
U
thedotandthegate,whichinturnleadstothetime-dependent We focus on the zero-temperature case and use the contact
potentialU(t) = |e|N (t)/C inside the dot. Consequently, bandwidthDastheenergyunit. Wesetk =1hereafter.
U B
theappliedvoltagenotonlygeneratesacurrentI(t)between No Zeeman splitting, ∆ = 0.— Figure 2(a) shows our
Z
the lead and the dot, but also induces a dot-gate displace- main results for R and C for the spin-degenerate case.
q q
ment current I (t) = e(dN /dt) = −C(dU/dt). Charge First, the zero-frequency limit of the relaxation resistance,
d U
conservation requires I(t)+ Id(t) = 0. Assuming that the Rq(ω → 0)isalwaysclosetotheuniversalvalueh/4e2, re-
gate-invariantperturbationV(t)−U(t)issufficientlysmall, gardlessofvaluesofǫdandEC. Thisvaluecanbeinterpreted
the linear response theory leads to the relation, I(ω) = as the compositeresistance of two parallelresistors of resis-
g(ω)(V(ω)−U(ω)), where g(t) = (ie/~)h[I(t),N]iΘ(t) tanceh/2e2,thewell-knownuniversalresistanceperchannel
is the equilibrium correlation function between the occupa- [3–5, 20]. The NRG results showthe quantizationofcharge
tion operator N = n and the current operator I = relaxationevenintheKondoregimewheremany-bodycorre-
µ µ
e(dN/dt). NotethatPthecurrent-densitycorrelationfunction lationsareeffective. Theobservedsmalldeviationsfromthe
g(ω) is directly related to the charge susceptibility χc(t) = exactvalueh/4e2,persistingeveninthenoninteractingcase,
−ih[N(t),N]iΘ(t),whichispreferablefornumericalcom- are attributed to the finiteness of the contact bandwidth D,
putation,viatherelationg(ω)= iω(e2/~)χ (ω). Then,with whichintroducesafrequency-dependentrealpartintothedot
c
the help of I(ω) = −Id(ω) = −iωCU(ω), the dot-lead selfenergy,Re[Σ(ω)]=−(Γ/2π)ln|(D−~ω)/(D+~ω)|.
impedanceZ(ω) = V(ω)/I(ω),whichisexperimentallyac- Its presence slightly violates the Fermi-liquid assumptions
cessible, is givenby Z(ω) = 1/(−iωC)+1/g(ω). The re- and, consequently, the KS relation is not exactly fulfilled so
laxationresistance andthe quantumcapacitanceare thenex- thattheuniversalvalueisnotrecovered: SinceRe[Σ(ω)] in-
pressedintermsofthechargesusceptibilityas creaseswith ω in magnitude,the deviationsare largeras the
R (ω) 1 e2/h 1 resonantlevelbecomesfarfromǫF = 0. Theuniversalvalue
q
=Re , =Im . (1) canberestoredbysettingalltherelevantenergyscalestobe
h/e2 (cid:20)2πiωχ (ω)(cid:21) C (ω) (cid:20)2πiχ (ω)(cid:21)
c q c muchsmallerthanD[31]. Second,thequantumcapacitance,
Thenumericalnormalizationgroup(NRG)[27,28]treatsthe C exhibitstwo remarkableconsiderations: (i) at the degen-
q
Coulombinteractioninanonperturbativewaybeingthemost erate points, ǫd ∼ ǫF and ǫd + 2EC ∼ ǫF, Cq shows two
adequatemethodforcomputingthechargesusceptibility[29]. pronounced peaks [see Fig. 2(a)], which is consistent with
3
the known understanding that Cq is proportional to the dot 25HaL 22..05 80HbL
tDhOeSKoρnddoto(ǫrFeg)im[3e, 4a,lth1o2]u,gahntdhe(iiD)OCSq raetmthaeinKsoqnuditoersemsoanllainnt 2@(cid:144)Dhe112050 0101....0055 10-4 10--02.050100 2@(cid:144)DeD4600 -00.05
level pinned at the Fermi level achievesits maximum value. Rq -0.15 -0.1 Cq
5 20
It implies that the Kondo resonant level, even though it can -0.15-0.1
open a tunneling channel, is not a real level which can hold 100-6 10-5 10-4 1D0Z-3 10-2 10-1 100-6 10-5 10-4 1D0Z-3 10-2 10-1
realchargesandcannotcontributetothecapacitance. Hence, 0 RqHΩL@h(cid:144)e2D 17
inthepresenceofmany-bodycorrelationsCq isnot,always, 10-1HcL 0.7
directlyrelatedtotheDOS. 10-2 0.6HdL 0.5
inTFihge.f2rewqhueernectyhdeetpheenbdeehnacveioorfoRfqt(hωe)reaanldaCndq(iωm)aigsinanaraylypzaerdt DZ111000---453 2@(cid:144)DRheq0000....2345 00..00015 00..12
ǫodft=hec−h0a.r3g.esTuhseceipmtiabgiilnitayryχcp(aωr)t oisfsχhco(wωn),rseefleeFcitgs.t2h(ebc)ofuo-r 10-6 -10-2-10-4-10-Ω610-6 10-4 10-2 0.1010Ε-d3=0 10-2DZ 10-1
plingbetweenthegroundstateandp-hexcitationsduetothe
dot-leadhybridization. Sincethespectraldensityofmultiple FIG.3: (coloronline)(a,b)Rq(ω = 0)andCq(ω = 0)versus∆Z
p-h excitations increases with energy, |Im[χc]| would grow for annotated values of ǫd with EC = 0.2. The enlarged view of
monotonically with |ω|. However, a finite D puts an upper Rq(ω = 0)forlarge∆Z isshownintheinset. (c)Contourplotof
limittotheenergyforp-hexcitations[|ω|&O(D)]resulting Rq asafunctionofωand∆Z inthelogarithmicscaleintheKondo
in the observed nonmonotonic behavior for Im[χ ]. More- regime(ǫd =−0.1). (d)Rq(ω =0)versus∆Z fordifferentvalues
c ofECasannotated. HereΓ=0.02isused.
over, Im[χc] has two kinks at |ω| = min(|ǫd|,|ǫd + 2EC|)
sincebeyondthisfrequencyp-hexcitationsaccompaniedwith
achargeexcitationcontributestoIm[χc]aswell. Aninterest- Rq(ω =0)|max ∼100×h/4e2forǫd =−0.15.Furthermore,
ing structure appears in Im[χc] near ω = 0, see Figs. 2(c) the peak is exactly located at ∆Z = Γ∗(TK) for the fluc-
and (d) for two dot level positions, ǫd = −0.1 (correspond- tuating valence (Kondo) regime. The peak height increases
ingtothefluctuatingvalenceregime)andǫd = −0.3(Kondo as the the effective hybridizationdecreases so it is the high-
regime). Close to ω = 0, Im[χc] depends linearly with ω, est in the Kondo regime. In the meanwhile, Cq(ω → 0) re-
mainly due to single p-h excitations [see tangent (dashed) mainsratherconstant,exceptattheresonanttunnelingregime
lines in Figs. 2(c) and (d)]. However, Im[χc] departs from (ǫd ≈ 0) where it displays a small peak, see Fig. 3(b). The
linearity when ω becomes of the order of the effective hy- evolutionofthespectraldistributionofR (ω)with∆ isdis-
q Z
bridizationΓ∗ (= TK in the Kondo regime). The renormal- played in Fig. 3(c) for ǫd = −0.3 (Kondo regime). As ∆Z
izedhybridizationΓ∗(TK) isextractedfromthewidthofthe increases,thelow-frequencypartofRq(ω)for|~ω|<kBTK
resonancecloseto(at)ǫF inρdot. Besides,wefoundthatthe keeps going up until ∆Z reaches kBTK; the side peaks are
slope of Im[χc(ω)]/ω is the largest at ~ω = Γ∗, while the merged into the central peak. With increasing ∆Z further,
change in Re[χc(ω)] is marginal. As a consequence, Rq(ω) thecentralpeakdiminishesgraduallyand,eventually,together
(seeEq.(1))exhibitstwosidepeaksat~ω = ±Γ∗ asshown withthesidepeakslocatedat~ω =±k T ,disappearcom-
B K
inFigs.2(e)and(f). NoticethatintheKondoregimeRq(ω) pletely. We haveobservedasimilartransitionofRq(ω)with
becomesmuch largerin orderof magnitudethan the univer- ∆ in the resonant tunneling regime (ǫ = −0.05,0) ex-
Z d
sal value, see Fig. 2(f). Remarkably, such peak structure in cept that the variation of the central part is smaller. Finally,
Rq(ω)isabsentinthenoninteractingcase. Foranoninteract- Fig. 3(d) compares the zero-frequencyvalues of Rq for dif-
ing system, the analytical expression of Rq(ω) in the wide- ferentvaluesoftheCoulombinteractionintheresonanttun-
band limit is given by [32] Rnon(ω) = (h/e2){[G(ω) + nelingregime. Inthenoninteractingcase,thereisnopeakat
q
(~ω/Γ)F(ω)]/[G(ω)2 + F(ω)2]} with G(ω) = ln{[(ǫd + all,withRq(ω →0)equaltoh/4e2. However,assoonasthe
~ω)2 + Γ2][(ǫd − ~ω)2 + Γ2]/[ǫ2d + Γ2]} and F(ω) = chargingenergy2ECbecomescomparabletoΓ∗ ∼Γ,apeak
2{tan−1[Γ/(ǫd − ~ω)] − tan−1[Γ/(ǫd + ~ω)]}. Rqnon(ω) startstoriseupandmanifestsitselffor2EC ≫ Γ. Itimplies
increases monotonically with increasing |ω|, and the only thatthe existence of the peak structureobservedin Fig. 3(d)
characteristic energy scale is ǫd. Hence, the peaks seen in definitelyhasitsoriginintheCoulombinteraction.
Figs. 2(e) and (f) are a genuine many-body effect. In sum- Discussion.— Now we have two questions to be an-
mary many-body correlations, apparently having no impact swered: (1) How can Coulomb interaction increase the re-
onthezero-frequencyvaluedoaffectRq(ω)atfinitefrequen- laxationresistancefarbeyondtheuniversalvalue,h/4e2 and
cies, by forming a pronouncedpeak at ~ω = Γ∗(TK). The (2) Why does it take place noticeably at ~ω = ±Γ∗(TK)
explanationfortheseobservationswillbegivenlater. in the fluctuating valence (Kondo)regime for ∆ = 0 or at
Z
Finite Zeeman splitting, ∆ 6= 0.— The spin-split case ~ω =0for∆ = Γ∗(T )? Thechargerelaxationresistance
Z Z K
(∆ > 0) in the presence of external magnetic fields is il- isattributedtop-hpairgenerationintheconductionbandas
Z
lustratedin Fig. 3. Interestingly,R (ω → 0)versus∆ ex- shown in Fig. 1. Such processes are put in action when the
q Z
hibits a peak structure reaching values much larger than the dot-leadtunnelingis switched on. The tunnelingin turn hy-
quantizedresistanceinthespin-degeneratecase:forexample, bridizesdotandconductionbandelectrons,resultinginlow-
4
ering of the ground state by the effective binding energy Γ∗ levelcannotcontributetothechargingofrealcharges.
(T in the Kondoregime). It means that the p-h generation
K Conclusion.— In closing, we have investigated the dy-
starts when the energy supplied by the source is larger than
namics of a many-body quantum capacitor. Using the rela-
Γ∗. This argument explains the observed peak in R (ω) at
q tionofchargerelaxationresistanceandquantumcapacitance
~ω = ±Γ∗ in the absence of the Zeeman splitting. In the
withthechargesusceptibility,wefindthatinthedeepKondo
presenceoffinitebutsmallZeemansplitting,theenergycost regimetheKSruleensuresaquantizedR (ω = 0)= h/4e2.
q
can be compensated by the Zeeman splitting. The p-h pair
Besides we show that the interpretation of C in terms of
q
excitationstatesshowninFig. 1(b)arenowloweredby∆ ,
Z the DOS fails when many-body effects are present. Here,
andwhen∆ ≈ Γ∗ theybecomealmostdegeneratewiththe
Z Cq(ω =0)becomesverysmallevenwhenρdot(ǫF)becomes
groundstate,allowingp-hpairgenerationwithnegligibleen-
largeduetoacompletelyfrozenchargedynamicsinthepres-
ergy cost. Hence, R (ω) exhibits a single peak at ~ω = 0
q ence of Kondo correlations. Finally, we find that R (ω) is
when∆ =Γ∗. Thisargumentworkssolelywhen∆ .Γ∗ q
Z Z builtbydot-leadtunnelingeventsconnectingp-hexcitations
inwhichthegroundstateisnotyetcompletelypolarizedand
inthereservoirswithspin-flipprocessesinthedot. Thisin-
thereexistsafinitecouplingamongspin-downdotstatesand
terpretationexplainsourresultsforR (∆ = 0,ω)showing
q Z
spin-updotstatesaccompanyingwithap-hpairgenerationin peaks at ~ω = ±Γ∗(T ) and for R(∆ = Γ∗(T ),ω) ex-
K Z K
thereservoirs,seeFig.1.Theimportanceofthespinflipinthe hibitingapeakat~ω =0.
boostingoftherelaxationresistancealsoexplainswhyR (ω)
q
can reach higher values in the Kondo regime. The Kondo M.L.wassupportedbytheNRFGrants(2009-0069554and
ground state is built from spin fluctuations due to spin-flip 2010-0015416). R. L. was supported by MEC-Spain (Grant
scattering among the localized dot electron and the delocal- No. FIS2008-00781). T.M. and T.J. acknowledge support
ized electronsin the reservoirs, thusspin-flipprocesseshave from ANR 2010 BLANC 0412 02. M.-S.C. was supported
largeamplitudesinitswavefunction. Hencetheprocessesas bytheNRFGrants(2009-0080453and2010-0025880).
shown in Fig. 1(b) can happenmore frequently, leading to a
large R . Similarly, the answer for the first question is now
q
ready. Thespectralweightforthechargecorrelationfunction
isproportionalto|hα|N|gsi|2,whereαrepresenttheexcited
[1] G.Feve`etal.,Science316,1169(2007).
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2
2 µ 1 1 [5] S.E.Niggetal.,Phys.Rev.Lett.97,206804(2006).
2 4
|hα|N|gsi| =t (cid:12) − + (cid:12) (2) [6] J.B.PieperandJ.C.Price,Phys.Rev.Lett.72,3586(1994).
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[14] R.Landauer,Phil.Mag.21,863(1970).
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↑ ↓ [16] R.Landauer,Z.Phys.B68,217(1987).
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q Z
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Z
[23] J.Splettstoesseretal.,Phys.Rev.B81,165318(2010).
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with a dot level at the Fermi energy with the effective hy- [28] H.R.Krishnamurthyetal.,Phys.Rev.B21,1003(1980);ibid.,
bridizationT . However, this pictureis only suitable in the 21,1044(1980).
K
[29] For better accuracy, we adopt Hofstetter’s algorithm [33] ac-
Fermi-liquid regime in which p-h excitations accomplished
companiedwiththeimprovedz-averagingmethod[34].
by spin-flip events in the dot are not allowed. Besides, this
[30] M.Sindeletal.,Phys.Rev.Lett.94,196602(2005).
effectivemodelpredictsanenhancedmesoscopiccapacitance
[31] A.C.Hewson,J.Phys.C18,1815(2006).
Cq(ω → 0) due to the presence of the resonant level at the [32] A.-P.Jauhoetal.,Phys.Rev.B50,5528(1994).
Fermi level. As noted before, however, the Kondo resonant [33] W.Hofstetter,Phys.Rev.Lett.85,1508(2000).
5
[34] V. L. Campo and L. N. Oliveira, Phys. Rev. B 72, 104432 (2009).
(2005); R. Zitko and T. Pruschke, Phys. Rev. B 79, 085106