Eet of the Lead Dimensionality
Over Transport Properties in Quantum Dots
L. Crao 1
and G.Cuniberti 2
1
InstitutodeFsia\GlebWataghin", Uniamp,13.081-970 Campinas,SP,Brazil
2
Max{Plank{Institut f urPhysik komplexerSysteme,NothnitzerStr. 38,D-01187Dresden,Germany
Reeivedon23April,2001
We theoretiallyinvestigate the eetof thelead dimensionalityon thenon-equilibrium eletron
transport through quantum dots. More preisely, we study nonlinear transport in a quantum
dotoupled to leads ofdiverse dimensionality. We show that the preseneof the latterstrongly
determinestheresultingtransportproperties. Dierentlyfromhigherdimensionalleads(wideand
smooth bandlimit), vanHovesingularities inthe density of statesof low{dimensional reservoirs
determinesharp resonanesinthe dierentialondutaneatniteappliedvoltages aswellas in
the dot spetral density. It is also shownthat, dueto the niteness ofthe terminal bandwidth,
the dierentialondutane hange itssign at higher biases. These results learly indiate that
the environment does play an important r^ole in determining transport properties in mesosopi
systems.
Quantumdotsrepresentperhapstheultimatelimit
inthedesignofminiaturizedeletroniiruits. T
rans-port measurements oered a wide variety of
informa-tion forthe theoretialunderstanding of thequantum
dotphysisespeiallyasonerningtotheroleofsingle
eletronharging[1℄,spingroundstateseletroni
or-relation[2℄, andKondo eet[3℄. Oneoftheunsolved
issuesoftheforemostnanofabriationistheinterfaing
betweenquantum dots and the externalenvironment.
The byprodutsof thiseort angiveoperational
ri-teria for thepotential appliabilityof these strutures
inlargesaleintegration. Lowdimensionalsystemsare
thoughtaspossibletoolsforoveromingthisdiÆulty.
Examples omefrom thehybridationofbiologialand
eletronimaterial[4℄,andfromnovel1D-likematerials
suhasarbonnanotubes[5℄. Atheoretial
omprehen-sionoftheouplingbetweendierentdimensional
ele-tronisystemsintransportexperimentsisonsequently
needed.
To the purpose of understanding theinterplay
be-tweensmalldotsandlowdimensionalreservoirsoupled
to them, we onsider here a singlelevel quantum dot
(QD)onnetedtoleadsofdierentdimensionality. We
modeltheexperimentallyplausibleaseinwhihaQD
isoupledtotwovanHoveleads(vHL),thatarenon
in-terating 1Dreservoirs,andtohighdimensional
reser-voirswith paraboli (PL) density ofstates (DOS) [6℄.
WealsoonsiderthemixedongurationwheretheQD
is simultaneouslyonnetedtotwodierentterminals:
vHL andPLleads.
Herewetreatthesimplestmodelthattakesinto
a-ounttheresonanttunnelingthroughasinglequantum
state, more preisely, the noninterating two-orbital
Andersonimpuritymodel. This model hasthe
advan-tageofbeingreduedinomplexitysothatitpreserves
exatlysolvablefeatures, andonsequentlyit allowsa
deepinvestigation ofthe possible physialbehaviorin
thefull parameter range. Moreover,it grasps the
ba-siphysisofmoresophistiatedsystemsuhasKondo
systemsinthestrongouplinglimit[7℄.
The noninterating two-orbital Anderson impurity
modelisdesribedbytheHamiltonian
H =
X
k ;; "
k
y
k
k +
X
"
0 d
y
d
+ X
k ;; t
k
y
k d
+H..
; (1)
where "
k
represents the single partile energy in the
reservoir=L;R withtheirhemialpotential
dier-enebeingtheapplied voltage,that is
L
R =eV.
Theparameters"
0 andt
k
denotethesinglepartile
en-ergy in the single level QD and the oupling between
QDandreservoirstates,respetively.
Tomakeontatwithexperimentswealulatethe
urrentusingtheLandauer[8℄formula
I = 2e
~ X
Z
d! ~
(!)[ f
L (!) f
R (!)℄
(!); (2)
where ~
(!)=
L (!)
R (!)=(
L (!)+
R (!)),
(!)=
P
jt
j 2
Æ(! "
k
levelandthelead. InEq.(2)thetransmission
prob-ability span the produt of ~
(!), the Fermi funtion
of leads f
(!) = 1=(e (! )
+1) and the spetral
density of states (DOS) of the eletron in the QD:
(!)=
1
ImG
(!).
Theretarded one-partile Green's funtion for the
noninteratingAndersonimpuritymodelisgivenby
G
(!)=
1
! "
0
(!)
; (3)
where (!)= P
(!)is theself-energydue tothe
tunnelinginto theleads,whihisgivenby
(!)=
X
k 2 jt
k j
2
! "
k
: (4)
In what follows we will assume an k-independent
oupling between the leads and the single level QD:
t
k t
=0:25W
, W
isthe half with of ondution
bandin thereservoir. Indoingso,thek-dependene
inEq. (4)isrestritedtothetight-bindingenergies"
k
and one annaturally replaethe sum overk by and
integralin energyintheform
(!)=t
2
Z
d
0 ()
!+
; (5)
where
0
()istheunorrelatedDOSforthereservoirs.
IntheaseofvHL,weonsider a1Dsystemforwhih
0 ()=
(W jj)
p
W 2
2
[9℄. Todesribeasystem wherethe
eet of the van Hovesingularities is absent, we
on-sider an hyperubi lattie (PL) whose orbital is
de-sribedby
0 ()=
3
4W (1
2
W 2
)forjjW
andzero
otherwise[10℄.
Letus nowturnourattentionto thenumerial
re-sults. Withoutlossofgenerality,wewillonsider here
thesymmetri "
0
=0 (zerogatevoltage)ase atzero
temperature. The van Hove singularities observed in
the density of states (DOS) of the QD at Figs. (1a)
and (1b) are related with the presene of the same
singularities in the leads. It is well known that the
singularities in a free eletron 1D system appears in
the border of the band. Sinein theequilibrium ase
eV = 0W (W
W)the bands of the left and right
leadsoinide,wethereforeobserveonlytwovanHove
singularities in DOS of the single level quantum dot,
see Fig. (1a). One the applied voltage is onsidered
eV 6= O, as in Fig. (1b), the Fermi level of the two
leadsareshiftedfromtheequilibriumvalueandthe
to-talDOSoftheexternalbath(!)forthepure1D
envi-ronmentpresentsfourvanHovesingularities.Sinethe
dotfeelsthepreseneofbothreservoirssimultaneously,
weonsequentlyobservefoursingularitiesfornite
ap-pliedvoltage. AsoneanseeinFigs. (1a)and(1b)the
spetralDOSof thedotis symmetriaroundthezero
in both equilibrium and non-equilibrium ases. This
symmetri property breaks down in the mixed
ong-urationforeV 6=0. Inthis regime theentral peakis
shiftedtopositiveenergiesandduetothebiasvoltage
the two vH singularities of the 1D reservoir are also
shiftedtopositiveenergies,asoneanseein Fig.(1b).
−2
−1
0
1
2
eV
0
1
2
3
ρ(ω)
−2
−1
0
1
2
eV
0
1
2
3
ρ(ω)
(a)
(b)
Figure1. SpetralDOSfor (a)eV =0and(b)eV =0:75
inthethreedierentenvironmentongurations: paraboli
leads (dottedline), 1D-leads(dashed line)and the hybrid
ase(solidline).
Ourresultsforthedierentialondutaneandthe
urrentareshownin Figs.(2a)and (2b),respetively.
As expeted thedierentialondutaneis dereasing
when inreasing thebias voltage in the vHL, PLand
mixed senarios. Nevertheless when the bias voltage
approahes the ritial value W
W=e the lead
di-mensionalitymayberevealedbynonequilibrium
alu-lations. Infat,thepreseneofatleastonevHlead
in-duesaresonaneinthedierentialondutane,whih
doesnotappearin thePLase,seeFig.(2a). By
fur-ther inreasingthe bias voltagea negative dierential
ondutaneisobtainedforallsystems. Intheaseof
pure vHL reservoirs, we also observe alarge negative
resonaneat theborderof theband, whilefor thePL
isharaterizedbyasuppressionofdI=dV atthesame
energy.
InFig.(2b)weshowourresultsfortheurrent. As
one an see, the urrent rst inreasesup to energies
near W
and after start to derease due to the
nite-nessoftheleadbandwidth. Theresonaneobservedin
thedierentialondutaneattheritialbiasvoltage
belowW
. Itisimportanttomentionherethatsimilar
type of eet has been reently observed in
urrent-voltage measurements of gold point ontats oupled
to (BCS)superondutingaluminumleads[11℄.
0
0.2
0.4
0.6
0.8
1
eV
−1
−0.5
0
0.5
1
dI/dV(2e
2
/h)
0
0.2
0.4
0.6
0.8
1
eV
0
0.1
0.2
0.3
I
(a)
(b)
Figure2. Dierentialondutane(a)andtheurrent(b),
respetively, for eV = 0 in the three dierent
environ-mentongurations: parabolileads(dottedline),1D-leads
(dashedline)andthehybridase(solidline).
Summarizing,wehaveinvestigated the
nonequilib-rium eletron transport through a QD onneted to
leadsofdierentdimensionality. Thepeuliarnatureof
the leadsis reetedinaresonantbehaviorofthe
dif-ferentialondutaneinduedbyadisontinuityinthe
urrent. Contrary to paraboli DOS leads (emerging
in systems with large oordination number), the
ur-rent and the dierentialondutane manifest abrupt
hangesforvaluesoftheexternalappliedvoltagewhih
math theone-dimensionalhalfbandwidth.
Aknowledgments
LCwasalsosupportedbytheFunda~aodeAmparo
researh of GC at MPI is sponsored by the
Shloe-mann Foundation. We aknowledge the kind help of
theeditorialoÆe of theBrazilianPhysial Soietyin
theeditionoftheProeedingsoftheBWSP-10.
Referenes
[1℄ Single ChargeTunneling, Vol. 294 of NATO Advaned
StudyInstituteseriesB,editedbyH.GrabertandM.H.
Devoret(PlenumPress, NewYork,1992).
[2℄ T. H. Oosterkamp et al., Phys. Rev. Lett. 80, 4951
(1998).
[3℄ D. Goldhaber{Gordon et al., Nature 391, 156 (1998);
S.M.Cronenwett,T.H.Oosterkamp,andL.P.
Kouwen-hoven,Siene281,540(1998).
[4℄ D. Porath, A. Bezryadin, S.de Vries, and C.Dekker,
Nature403, 635 (2000); B. Crone et al.,Nature 403,
521(2000);S.S.Wongetal.,Nature394,52(1998).
[5℄ R.Saito, G.Dresselhaus, andM.S.Dresselhaus,
Phys-ial Properties of Carbon Nanotubes (World Sienti
PublishingCo.Pte.Ltd.,London,1998).
[6℄ L.CraoandK.Kang,Phys.Rev.B59,12244(1998).
[7℄ Byusingtheslavebosonrepresentationwithmeaneld
approximationthesolutionthestronglyorrelated
An-dersonimpuritymodelismappedontothe
noninterat-ingonewiththehybridisationandtheboundstate
self-onsistently renormalized, for more details see for
ex-ample, G. M. Zhang et al., Phys. Rev.Lett. 86, 704
(2001).
[8℄ Y.MeirandN.S.Wingreen,Phys.Rev.Lett.68,2512
(1992).
[9℄ E.N.Eonomou,Green'sFuntionsinQuantumPhysis
(Springer,Berlin,1990).
[10℄ L.G.Brunet,R.M.RibeiroTeixeira,andJ.R.Iglesias,
SolidStateComm.68,477(1988).
