Table Of Content7
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ThermopowerofKondoEffectinSingleQuantumDotSystemswithOrbital
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atFiniteTemperatures
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l R.Sakanoa,1,T.Kitaa andN.Kawakamia,b
l
a
aDepartmentofAppliedPhysics,OsakaUniversity,Suita,Osaka565-0871, Japan
h
bDepartmentofPhysics,KyotoUniversity,Kyoto606-8502, Japan
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s
e
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. Abstract
t
a
m WeinvestigatethethermopowerduetotheorbitalKondoeffectinasinglequantumdotsystembymeansofthenoncrossing
- approximation. It is elucidated how the asymmetry of tunneling resonance due to the orbital Kondo effect affects the
d thermopowerundergate-voltageandmagnetic-fieldcontrol.
n
o
c Keywords: quantumdot,Kondoeffect,transport
[ PACS:73.23.-b,73.63.Kv,71.27.+a,75.30.Mb
1
v
3
1. Introduction tunneling resonance around the Fermi level. So far, a
3
few theoretical studies have been done on the ther-
5
1 TheKondoeffectduetomagneticimpurityscatter- mopowerinQDsystems[9,10,11,12,13,14,15,16].Are-
0 inginmetalsisawellknownandwidely studiedphe- cent observation of the thermopower due to the spin
7 nomenon[1].Theeffecthasrecentlyreceivedmuchre- KondoeffectinalateralQDsystem[17]naturallymo-
0
newedattentionsinceitwasfoundthattheKondoef- tivatesustotheoreticallyexplorethistransportquan-
/
at fect significantly influences the conductance in quan- tityinmoredetail.Here,wediscusshowtheasymme-
m tum dot (QD) systems [2]. A lot of tunable parame- try of tunneling resonance due to the orbital Kondo
ters in QD systems have made it possible to system- effectaffectsthethermopowerundergate-voltageand
-
d aticallyinvestigateelectroncorrelations.Inparticular, magnetic-field control. By employing the noncrossing
n high symmetry in shape of QDs gives rise to the or- approximation(NCA)fortheAndersonmodelwithfi-
o
bitalproperties, which hasstimulated extensivestud- nite Coulomb repulsion, we especially investigate the
c
: iesontheconductanceduetotheorbitalKondoeffect KondoeffectofQDforseveralelectron-chargeregions.
v
[3,4,5,6,7,8].Thethermopowerwestudyin thispaper
i
X is another important transport quantity, which gives
r complementary information on the density of states 2. ModelandCalculation
a
to the conductance measurement: the thermopower
can sensitively probe the asymmetric nature of the
Let us consider a single QD system with N-
degenerateorbitalsinequilibrium,asshowninFig.1.
1 E-mail:[email protected] TheenergylevelsoftheQDareassumedtobe
PreprintsubmittedtoPhysicaE 6February2008
πT ∂f(ε)
dot L11= Γ dερσl(ε) − , (7)
h Z „ ∂ε «
Xσl
πT ∂f(ε)
L12= Γ dεερσl(ε) − , (8)
lead D lead h Z „ ∂ε «
e orb Xσl
d D
orb where ρσl(ε) is the density of states for the electrons
with spin σ and orbital l in the QD and f(ε) is the
Fig. 1. Energy-level scheme of a single QD system with Fermidistributionfunction.Inordertoobtainthether-
threeorbitalscoupledtotwoleads.
mopoweritisnecessarytoevaluateρσl(ε).
We exploit the NCA method to treat the Hamil-
εσl=εd+l∆orb, (1) tonian (2) [18,19]. The NCA is a self-consistent per-
l=−(Norb−1)/2,−(Norb−3)/2,··· ,(Norb−1)/2 turbation theory, which summarizes a specific series
of expansions in the hybridization V. This method is
whereεd denotesthecenteroftheenergylevelsandσ known to give physically sensible results at tempera-
(l)representsspin(orbital)indexandNorb represents tures around or higher than the Kondo temperature.
thedegreeoftheorbitaldegeneracy.Theenergy-level TheNCAbasicequationscanbeobtainedintermsof
splitting between the orbitals ∆orb is induced in the coupledequationsfortheself-energiesΣm(z)ofthere-
presence of magnetic field B; ∆orb ∝ B. In addition, solventsRm(z)=1/[z−εm−Σm(z)],
the Zeeman splitting is assumed to be much smaller
Γ σl 2 σl 2
than the orbital splitting, so that we can ignore the Σm(z)= Mm′m + Mmm′
Zeeman effect. In practice, this type of orbital split- π Xm′ Xσl »“ ” “ ” –
tinghasbeenexperimentallyrealized asFock-Darwin
× dεRm′(z+ε)f(ε), (9)
statesinverticalQDsystemsorclockwiseandcounter- Z
clockwisestatesincarbonnanotubeQDsystems.Our where the index m specifies the eigenstates of Hd
QDsystem is described bythemultiorbital Anderson and the mixing width is Γ = πρcV2. The coefficients
impuritymodel, Mmσlm′ are determined by the expansion coefficients
of the Fermion operator d†σl = mm′Mmσlm′|mihm′|.
H=Hl+Hd+Ht (2) We compute the density of statPes by this method to
Hl= εkσlc†kσlckσl, (3) investigatethethermopower.
Xkσl
Hd= εσld†σldσl+U nσlnσ′l′
Xkσl σlX=6 σ′l′ 3. Results
−J Sdl·Sdl′, (4)
Xl=6 l′ 3.1. Gatevoltagecontrol
Ht=V c†kσldσl+H.c. , (5)
Xkσ “ ” ThethermopowerfortwoorbitalsisshowninFig.2
asa function of theenergylevel εd (gate-voltage con-
where U is the Coulomb repulsion and J(> 0) repre- trol).TherearefourCoulombpeaksaround−εd/U ∼
sentstheHundcouplingintheQD. 0,1,2,3 at high temperatures (see the inset of Fig.
Thenon-equilibriumGreen’sfunctiontechniqueal- 2(a)).Asthetemperaturedecreases,thethermopower
lows us to study general transport properties, which in the region of −1 < εd/U < 0(−3 < εd/U < −2)
gives the expression for the T-linear thermopower as with nd ∼ 1(3) is dominated by the SU(4) Kondo ef-
[14], fect.Thethermopowerhasnegativevaluesintheregion
−1 < εd/U < 0, implying that the effective tunnel-
S =−(1/eT)(L12/L11), (6) ingresonance,suchastheKondoresonance,islocated
abovetheFermilevel.Atlowenoughtemperatures,the
withthelinearresponsecoefficients, SU(4) Kondo effect is enhanced with decrease of en-
2
(a) 1.2 0.6
e) 00..48 kkkkkBBBBBTTTTT=====00000.....0001246800GGGGG e) 00..24 kkkkkBBBBBTTTTT=====00000.....1000008642GGGGG
1/ 0 1/ 0
S/( h) 2 S/(
-0.4 2e/ -0.2
2
-0.8 G/( 0 -0.4
-3 -2 -1 0
-1.2 -0.6
-3 -2 -1 0 -1 -0.5 0
(b) 1.2 ed/U ed/U
J=0G
0.8 JJ==00..2400GG Fig. 3. The thermopower due to the ordinary spin Kondo
J=0.60G effectasafunctionofthedotlevel.WesetU =6Γ.
0.4 J=0.80G
e)
1/ 0
S/( 0
-0.4
-0.8 e) -0.2
1/
-1.2-3 -2 ed/U -1 0 S/( -0.4 kkkkkBBBBBTTTTT=====00000.....2100000864GGGGG
Fig. 2. The thermopower for the two orbital QD system -0.6
0 0.2 0.4 0.6 0.8 1
withfiniteCoulombrepulsionU =8Γasafunctionofthe
energy level of the QD. (a) The temperature dependence D orb/G
forJ =0.Theinsetshowstheconductanceasafunctionof
Fig.4. ThethermopowerforthetwoorbitalQDsystem,in
t(hbe)TdohtelHevuenlda-tcokuBpTlin=g0d.e2p0eΓnd(CenocuelofmorbkrBesTon=an0c.0e4pΓe.aks). caseofεd=−U/2,asafunctionoforbitalsplitting∆orb.
WesetU =8Γ.
level,whichcausesthesignchangeofthethermopower.
ergy leveldown toεd/U =−1/2, which resultsin the
enhancementofthethermopower.However,ifthetem- Aroundεd/U =−3/2,evensmallperturbationscould
easilychangethesignofthethermopoweratlowtem-
peratureofthesystemislargerthantheSU(4)Kondo
peratures.Notethatthesepropertiesarequitesimilar
temperature, the Kondo effect is suppressed and the
to those for the ordinary spin Kondo effect shown in
thermopower has a minimum in the regime −1/2 <
Fig. 3, because the filling is near half in both cases.
εd/U < 0. As the energy level further decreases, the
ForlargeHundcouplingsJ,thetripletKondoeffectis
SU(4)Kondoeffectandtheresultingthermopowerare
realized and the resulting Kondo temperature is very
bothsuppressed.NotethattheHundcouplinghardly
small, so that the thermopower shown in Fig. 2(b) is
affects the thermopower because of nd ∼ 1 in this
dramaticallysuppressed.
regime,asshowninFig.2(b).Sincetheregionof−3<
εd/U < −2 can be related to −1 < εd/U < 0 via an
electron-holetransformation,wecandirectlyapplythe 3.2. Magneticfieldcontrol
abovediscussionsontheSU(4)Kondoeffecttothefor-
merregionbychangingthesignofthethermopower. Let us now analyze the effects of orbital-splitting
Let usnow turn totheregion of −2< εd/U < −1, causedbymagneticfields.Thecomputedthermopower
where nd ∼ 2. At J = 0, the Kondo effect due to for εd/U = −1/2 is shown in Fig. 4 as a function
six-folddegeneratestatesoccurs.Althoughtheresult- of the orbital splitting ∆orb. It is seen that magnetic
ingKondoeffectisstronglyenhancedaroundεd/U = fields dramatically suppress the thermopower, which
−3/2inthiscase,thethermopowerisalmostzerobe- iscausedbythefollowingmechanism.Inthepresence
causetheKondoresonanceislocatedjustattheFermi of magnetic fields, the Kondo effect changes from the
level.Therefore,whenthedotlevelischanged,theposi- SU(4)orbitaltypetotheSU(2)spintypebecausethe
tionoftheKondoresonanceisshiftedacrosstheFermi orbitaldegeneracyislifted.Asaconsequence,theres-
3
onancepeakapproachestheFermilevelandtheeffec- thepresenceofmagneticfields.
tive Kondo temperature is reduced, so that the ther- For εd/U ∼ −3/2, where nd ∼ 2, the Kondo effect
mopoweratfinitetemperaturesisreducedinthepres- duetosix-folddegeneratestatesoccursforJ =0.How-
enceofmagneticfields. ever,thethermopowerisstronglyreducedbecausethe
Notethat,inourmodel,magneticfieldschangethe resonance peak is located nearthe Fermilevel. When
lowestenergylevelεσ−21 from−U/2to−(U+∆orb)/2. the Hund coupling is large, the triplet Kondo effect
Accordingly,thepeakpositionoftherenormalizedres- is dominant. The resulting small Kondo temperature
onance shifts downward across the Fermi level (down suppresses the thermopower around εd/U ∼ −3/2 at
toalittlebelowtheFermilevel).Thus,thelargeneg- finitetemperatures. In this region, magnetic fields do
ative thermopower changes to a small positive one as notaffecttheasymmetryoftheresonancepeakandthe
the magnetic field increases at low temperatures. In resulting thermopower remains almost zero because
strong fields,theeffectiveKondoresonance islocated thefillingisfixed.
aroundtheFermilevelwithsymmetricshape,sothat
evensmallperturbationscouldgiverisetoalargevalue
ofthermopowerwitheithernegativeorpositivesign.
Finallyabriefcommentisinorderforotherchoices
oftheparameters.Thethermopowerforεd/U =−5/2 AcknowledgementWethankS.Tarucha,A.C.Hew-
showssimilarmagnetic-fielddependencetotheεd/U = son, A. Oguri and S. Amaha for valuable discussions.
−1/2caseexceptthatitssignischanged.Forεd/U = RS was supported by the Japan Society for the Pro-
−3/2, the thermopower is almost zero and indepen-
motionofScience.
dentof magnetic fields,because theKondoresonance
is pinned at theFermi level and gradually disappears
withincreaseofmagneticfields.
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