Conductance through strongly interacting rings in a magnetic field

0.12 MB·English

by  Julián Rincón

9
0 Conductancethroughstronglyinteractingringsinamagneticfield
0
2 ∗
Juli´anRinc´on,A.A.AligiaandK.Hallberg

n
InstitutoBalseiro,CentroAto´micoBariloche,CNEAandCONICET,8400Bariloche,Argentina
a
J

3

] Abstract
l
e
WestudytheconductancethroughfiniteAharonov-Bohmringsofinteractingelectronsweaklycoupledtonon-
-
r interacting leads at two arbitrary sites. This model can describe an array of quantum dots with a large charging
t
s energycomparedtotheinterdotoverlap.Asaconsequenceofthespin-chargeseparation,whichoccursinthesehighly
t. correlatedsystems,thetransmittanceisshowntopresentpronounceddipsforparticularvaluesofthemagneticflux
a piercing the ring. We analyze this effect by numerical and analytical means and show that the zero-temperature
m
equilibriumconductanceinfactpresentsthesestrikingfeatureswhichcouldbeobservedexperimentally.
-
d
n Keywords: charge-spinseparation,conductancethroughnanoscopicsystems
o PACS:75.40.Gb,75.10.Jm,76.60.Es
c
[

1 1. Introduction and a spin part for any size of the system. [10] There
v
have been several experiments reporting indirect in-
8
dications of charge-spin separation [11,12,13], and it
3 Oneofthechallenges of nanotechnology isthepos-
could also be potentially observed in systems such as
3 sibility of fabricating new artificial structures with
cuprate chains, ladder compounds, [14] and carbon
0 tailored properties. For example, the Kondo effect
. was achieved in a system consisting of one quantum nanotubes.[15]
1
Several theoretical approaches tackled this phe-
0 dot connected to leads[1,2,3]; systems of a few QD’s
nomenonin theringgeometry. Forexample,thereal-
9 havebeenproposedtheoreticallyasrealizationsofthe
time evolution of electronic wave packets in Hubbard
0 two-channel Kondo model [4,5], the ionic Hubbard
rings has shown a splitting in the dispersion of the
: model, [6] and the double exchange mechanism. [7]
v spin and charge densities as a consequence of the dif-
Also,thecorrelation-drivenmetal-insulatortransition
i
X hasbeenstudiedinachainofquantumdots.[8] ferent charge and spin velocities. [16,17] Pseudospin-
charge separation has also been studied in quasi-one-
r Another interesting phenomenon in strongly corre-
a lated systemsiswhat isknownascharge-spin separa- dimensionalquantumgasesoffermionicatoms.[18,19]
Other theoretical approaches concern the transmit-
tion. It is well known that strong correlations in one
tance through Aharonov-Bohm rings modeled by
dimensioninvalidatetheFermiliquidconventionalde-
Tomonaga-LuttingerliquidorothercorrelatedHamil-
scription of electrons. In particular, correlations can
toniansliketheHubbardort−J models.[20,21,22,23]
leadtothefractionalizationoftheelectronintocharge
In these systems, noticeable dips are observed in the
and spin degrees of freedom.[9] This separation is an
conductanceforfractionalvaluesofthefluxwhichcan
asymptotic low-energy property in an infinite chain.
be attributed to destructive interference between de-
However, exact Bethe ansatz results for the Hubbard
generatestatesasweexplainbelow.Wehaverecently
model in the limit of infinite Coulomb repulsion U
discussed the extension of these results to ladders of
show that the wave function factorizes into a charge
twolegsasafirststeptohigherdimensions.[24]
In this work, we analyze the origin of the dips in
∗ Tel:[phone]; e-mail:[email protected] thetransmittanceasafunctionofappliedfluxinfinite
PreprintsubmittedtoPhysicaB 3January2009
(a) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) 3. Conductance
t’ t’
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Φ (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
When the ground state of the isolated ring is non-
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)t,J(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) degenerate,andthecouplingt′ betweentheleadsand
(b) 1(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Φ(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)2 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)M(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) =(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) 3(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) orpifenrtgahtiesurwreeetcaaakrnd,etbhdeeeGexqrpeureeinlsi’bsserdifuutmnocscteioconondnufdocrotratdnhecereiiasnottlza′etinreodtetrreimmngs-
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) 0(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) tpbhaeetrwttirecaelennwssmiittihetsteaninecarengdyreωaj:d=sG−Ri,j2(tωc)o.s[k6,a2n0d] Fmoormaenntinucmid±enkt,
Fig.1.Schematicrepresentationofthesystemsstudiednumer-
ically.(a)L=6,(b)L=7.InbothcasesM =3. 4t2sin2k|t˜(ω)|2
T(ω,V ,φ)= , (5)
g [ω−ǫ(ω)+teik]2−|t˜2(ω)|2
rings described by the t−J model. We discuss the
˛ ˛
conditions for which the intensity of the lowest-lying whereǫ(ω)=t˛′2GR(ω)andt˜(ω)=t′2GR˛ (ω)repre-
00 0M
peakinthezero-temperatureequilibriumconductance sentacorrectiontotheon-siteenergyattheextremes
asafunctionofthegatevoltagepresentscharacteristic oftheleadsandaneffectivehoppingbetweenthemre-
dipsforcertainfluxvalues.Thesewillbeshowntobe spectively.
aconsequenceofspin-chargeseparation. This equation is in fact exact for a non-interacting
system,however,itlosesvalidityforanoddnumberof
electrons,wherethegroundstateisKramersdegener-
ate.Inthiscasethemethodmissescompletelythein-
2. ModelHamiltonian
teresting physicsarising from theKondo effect. [6,25]
However, the Kondo temperature is small compared
ThebasicmodelisdepictedinFig.1.Wehavecon-
to the other energy scales of the system and we can
sidered a ring of L sites, weakly connected to non- assume safely that theKondoeffect is destroyed bya
interactingleadsatsites0andM.
smalltemperature.Thisapproachisjustifiedforsmall
TheHamiltonianreads: enought′sincethecharacteristicenergyofthisKondo
effectdecreasesexponentially.
H =H +H +H . (1)
ring leads links The conductance is G = (ne2/h)T(µ,V ,φ), where
g
n=1or2dependingifthespindegeneracy isbroken
Thefirsttermdescribestheisolatedring,withon-site
ornot,[6]andµistheFermilevel,whichwesetaszero
energygivenbyagatevoltageV ,andhoppingsmod-
g
(half-filled leads). When the gate voltage V is varied
ified bythephaseexp(iφ/L) duetothecirculation of g
a peak in the conductance is obtained when there is
thevectorpotential.Formostoftheresultsofthispa-
a degeneracy in the ground state of the ring for two
perweusethet−J modeltodescribethering,
consecutivenumberofparticles:E (N+1)=E (N),
g g
H =−eV c† c −t c† c eiφ/L+H.c. whereEg(N)isthegroundstateenergyofHring with
ring g iσ iσ i+1σ iσ Nelectrons.Withoutlossofgenerality,weassumethat
Xiσ “ ”
westart with N +1electrons in theringand applya
1
+J S ·S − . (2) negativegatevoltageinsuchawaythatapeakinthe
Xi „ i i+1 4« conductanceisobtainedatacriticalvalueVgcwhenthe
numberofelectronsintheringchangesfromN+1to
where φ =2πΦ/Φ , /Φ = h/2e is thefluxquantum,
0 0 N electrons.
S isthespinoperatoratsiteianddoubleoccupancy
i
isnotallowedatanysiteofthering.Thesecondterm
corresponds to two tight-binding semi-infinite chains
fortheleftandrightleads 4. Analyticalandnumericalresults
−1 ∞
H =−t a† a −t a† a +H.c. (3) For J = 0 the model is equivalent to the Hubbard
leads i−1,σ i,σ i,σ i+1,σ
i=X−∞,σ i=X1,σ model with infinite on-site repulsion U, for which the
wavefunctioncanbefactorizedintoaspinandacharge
ThethirdterminEq.(1)describesthecouplingofthe part,evidencingcharge-spinseparation.[10,22,26]For
left(right)leadwithsite0(M)ofthering eachspinstate,thesystemcanbemappedintoaspin-
lessmodelwithaneffectivefluxwhichdependsonthe
Hlinks=−t′ (a†−1,σc0σ+a†1,σcMσ+H.c.). (4) totalspin.ForasystemofNparticlesonecanconstruct
Xσ spin-wave functions with wave vectors ks = 2πns/N,
2
wheretheintegerns characterizesthespinwavefunc- 0.003
tion. The total energy and momentum (in an appro-
0.002
priate gauge) of any state of the ring have simple ex-
pressions: M = 1
0.001
M = 2 N = 4
M = 3
E=−2t N cos(k), k = 2πnl+φeff, (6) Ifp 0.0003
l l L
Xl=1 0.002
K= k =[2π(n +n )+Nφ]/L, (7)
l c s
X 0.001
2π N = 5
φ =φ+k =φ+ n , (8)
eff s N s 0
0 0.2 0.4 0.6 0.8 1
φ/π
where the integers n characterize the charge part of
l
thewavefunctionandn = n.
c l Fig.2.IntensityofthefirstpeakinthetransmittanceIfp,as
Thevalues of thefluxφd fPor which dipsor reduced a function of applied magnetic flux, for a ring of 6 sites and
conductancesareexpected,correspondtosomepartic- (a)N+1=5and(b)N+1=6electronsinthegroundstate,
ularcrossings oftheenergylevelsofN electrons.One three different configurations M = 1, 2 and 3, t′ = 0.3t and
J=0.001t.
canseethisfromthegeneralformoftheGreen’sfunc-
tionsGR(ω)enteringthetransmittance[Eq.(5)]when
0j
a particle is destroyed. Using the Lehman’s represen- withninteger.Thesearealsothepositionswherecross-
tation,therelevantpartoftheGreen’sfunctionis: ingsinthe(experimentallyaccessible)groundstatefor
N particlestakeplace(n′ −n =±1).
s s
hg|c† |eihe|c |gi To check these predictions we have performed nu-
GR0j(ω)= ωj+σ E −E0σ . (9) mericalcalculationsforthetransmittance,obtainedby
Xe e g diagonalizing the ring using Davidson’s method [27].
Once the Green’s functions were obtained, they were
Noting that the ring has translational symmetry and
replaced in Eq. (5), to obtain the transmittance. The
namingK thewavevectorofthestate|νi,oneobtains:
ν systems studied are represented in Fig. 1. In contrast
to previous work, [20,21,22,24] we concentrate on the
GR(ω)= e−ij(Ke−Kg)|he|c0σ|gi|2. (10) first peak in the transmittance as the gate voltage is
0j Xe ω+Ee−Eg decreasedsincethisisthefeaturewhichisexperimen-
tallyaccessibleatequilibriumandlowtemperatures.
At certain flux values, two states of N electrons, In Fig. 2 we show numerical results for a ring with
|ei and |e′i, become degenerate. Assuming that the L = 6 sites and N = 4 and 5 particles in the inter-
corresponding matrix elements entering Eq. (10) are mediate state for the three non-equivalent configura-
nonzero,onlythesetwostates,contributesignificantly tionscorrespondingtoM =1,2and3.Toquantifythe
to the Green’s function GR(ω) at the Fermi energy relative intensity of the conductance we integrate the
0j
(ω = µ = 0) when Vg (which displaces all Ee rigidly transmittancegiven byEq. (5) overawindow of gate
with respect toEg) is tunedin such a way that Ee = voltageVg ofwidth0.002tcenteredaroundthedegen-
Ee′ ∼ Eg. Denoting by β = exp[iM(Ke′ −Ke)] the eracy point between the ground state for N +1 and
relativephasebetweenthetwointerveningdegenerate N electrons. This corresponds to the intensity of the
statesweseethatifβ6=1,thetransmittance,whichis first observablepeak in thetransmittance as thegate
proportional to |GR (ω)|2, (Eq. (5)), is reduced near voltage is lowered. As the curve is symmetric under
0M
the crossing (note that this results is independent of changeofsign ofφweshowonlytheinterval0≤φ≤
anyspecificmodel). π.For J → 0and N = 4thedipsappear asexpected
Wecanpredictthepositionsofthedipsinthetrans- (Eq. (11)) at φ /π = 0.25 and 0.75. For N = 5 the
d
mittance without having to resort to the calculation dipsshould occur at φ /π = 0.2, 0.6 and 1. However,
d
of the matrix elements entering the Greens functions near 0.6 the total spin in the ground state for 5 elec-
Eq.(10).Thecrossingsofenergylevelsatlowenergies trons changes from S = 1/2 to S = 3/2, which is not
take place at φ = −π(ns+n′s)/N.[23] When ns+n′s accessible by destroying an electron in the 6-electron
is odd (even) the relative phase β = exp[iL(Ke′ − singletgroundstate.Therefore,thetransmittancevan-
Ke)/2]=exp[i(n′s−ns)] =−1(1)andthereis(there ishes for 0.6 < φ ≤ π.Asa consequence, in theinter-
is not) a dip in the integrated transmittance. There- valshown there is only one dip present. In this figure
fore,thepositionsofthedipsarelocatedat onecanseetheeffectofthedifferentsource-draincon-
figurations:thelowestconductanceisachievedforthe
φ =π(2n+1)/N, (11) symmetriccase(M =3forL=6)whereβ =−1and
d
3
0.003
0.002
N = 1 N = 2 Acknowledgments
0.001
This investigation was sponsored by PIP 5254 of
0 CONICETandPICT2006/483 oftheANPCyT.
I 0.002
fp
0.001 N = 3 N = 4
0
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0.002
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4