Statistical properties of electrochemical capacitance in disordered mesoscopic capacitors

0.44 MB·

by  Fuming Xu

Statistical properties ofelectrochemical capacitance indisordered mesoscopiccapacitors
Fuming Xu and Jian Wang∗
DepartmentofPhysicsandtheCenterofTheoreticalandComputationalPhysics,TheUniversityofHongKong,HongKong,China.
We numerically investigate the statistical properties of electrochemical capacitance in disordered two-
dimensionalmesoscopiccapacitors.Basedonthetight-bindingHamiltonian,theGreen’sfunctionformalismis
adoptedtostudytheaverageelectrochemicalcapacitance, itsfluctuationaswellasthedistributionofcapaci-
tanceofthedisorderedmesoscopiccapacitorsforthreedifferentensembles:orthogonal(symmetryindexβ=1),
unitary(β=2),andsymplectic(β=4). Itisfoundthattheelectrochemicalcapacitanceinthedisorderedsystems
7
1 exhibitsuniversalbehavior.Inthecaseofsingleconductingchannel,theelectrochemicalcapacitancefollowsa
0 symmetricGaussiandistributionatweakdisordersasexpectedfromtherandommatrixtheory. Inthestrongly
2 disordered regime, the distribution isfound to be asharply one-sided formwitha nearly-constant tail inthe
largecapacitance region. This behavior is due to the existence of the necklace statesin disordered systems,
n
whichischaracterizedbythemulti-resonancethatgivesrisetoalargedensityofstates. Inaddition,itisfound
a
thatthenecklacestatealsoenhances thefluctuationofelectrochemicalcapacitance inthecaseofsinglecon-
J
ductingchannel. Whenthenumberofconductingchannelsincreases,theinfluenceofnecklacestatesbecomes
6
lessimportant. Forlargenumberofconductingchannels,theelectrochemicalcapacitancefluctuationdevelops

aplateauregioninthestronglydisorderedregime. Theplateauvalueisidentifiedasuniversalelectrochemical
]
l capacitance fluctuation, which is independent of system parameters such as disorder strength, Fermi energy,
al geometriccapacitance, and systemsize. Importantly, theuniversal electrochemical capacitance fluctuationis
h thesameforallthreeensembles,suggestingasuper-universalbehavior.
-
s
PACSnumbers:72.10.-d,73.21.La,73.23.-b,73.63.-b
e
m
. I. INTRODUCTION anexperimentalstudyonchargerelaxationresistancewascar-
t
a riedoutonamesoscopiccapacitormadeofatwodimensional
m
electron gas (2DEG) quantum dot with the second metallic
Asthebasicbuildingblockofnanoscaleelectroniccircuit,
- plate on top of it.14 This setup is equivalent to a RC circuit
d nanoscalecapacitorhasbeenthefocusofintensestudysince
consistingofclassicalcapacitance,quantumcapacitanceand
n its fundamentalsignificance. Due to the small sample sizes,
thechargerelaxationresistance. Bymeasuringthefrequency
o the finite density of states (DOS) in nanostrutures can give
c dependentconductanceofthisRCcircuitinGHzregime,the
rise to a non-negligiblequantum correction to their physical
[ chargerelaxationresistancewasobtainedandfoundtobehalf
properties.Forinstance,amesoscopiccapacitorshowsdiffer-
oftheresistancequantuminthesingleconductingmode.14At
1 entdynamicresponsetoanacexternalbiasfromitsclassical
higherfrequency,quantuminductance becomesrelevantand
v counterpart. Atlowfrequencies,thetransportpropertiesofa
8 the effectivecapacitance of the mesoscopic capacitor can be
mesoscopiccapacitorischaracterizedbytheelectrochemical
1 negative,2 which has been seen experimentally in gated car-
capacitanceC ,chargerelaxationresistanceR ,andquantum
5 µ q bonnanotubes.7
inductance.1,2Numerousresearcheffortshavebeendevotedin
1
0 thestudyofmesoscopiccapacitorinthepastdecade.3–17 In mesoscopic systems, the sample-to-sample fluctuation
.
1 Itisknownthatforabulkmetal,thehugedensityofstates of physical quantities such as conductance exhibits univer-
0 resultsinaverysmallscreeninglengthwhileforamesoscopic sal behavior in the diffusive regime. This universal conduc-
7 conductor, the finite density of states gives rise to a finite tancefluctuation(UCF)isoneofthehallmarksofmesoscopic
1
screening length. As a result, the electrochemical potential physics. It was known that value of UCF depends only on
:
v controlledexperimentallyisnotequaltotheelectrostaticpo- thesymmetryanddimensionalityofthe systems. Inthe lan-
i tential that defines the classical capacitance. This leads to a guage of random matrix theory, the symmetry is character-
X
quantumcorrectiontotheclassicalcapacitancethathasbeen izedbysymmetryindexβ whereβ=1,2,4correspondsto or-
r confirmedexperimentally.18Thiselectrochemicalcapacitance thogonal, unitary, and symplectic ensembles. Not only the
a
C can be treated in a dynamic way using scattering matrix conductance fluctuation has universal values in the diffusive
µ
theory1 or non-equilibrium Green’s function.19 At low fre- regime, the conductance distribution also exhibits universal
quencies, C is determined by the serial combinationof the features in the localized regime.20,21 Ac transport properties
µ
geometricalcapacitance and the quantumcapacitance which of mesoscopic systems also show sample to sample fluctua-
isinducedbythefiniteDOS.ItwasshownthatC oscillates tionsin disorderedmesoscopicsystems. Studyonthe acad-
µ
asafunctionoftheFermienergy.9Inaddition,thedissipative mittance of a chaotic quantumdot using randommatrix the-
partofamesoscopiccoherentcapacitorisrelatedtothecharge ory (RMT) revealed that the admittance is governed by two
relaxation resistance which was predicted to be a universal time scales.22 The emittance fluctuation of mesoscopic con-
quantityequaltohalftheresistancequantumR =h/2e2for ductorswereinvestigatedincluding1D,2D,andquantumdot
q
singleconductingchannel.1Theuniversalvalueofchargere- systems. It was found that due to the existence of the neck-
laxationresistanceisindependentofFermienergyeveninthe lace states, the average emittance is negative from ballistic
presenceofstrongelectronelectroninteraction.8,14 Recently, regimeallthewaytothelocalizedregimeshowinginductive-
2
likebehavior.23UsingRMT,24thedistributionfunctionsofthe II. THEORETICALFORMALISM
electrochemicalcapacitanceforachaoticquantumdotsystem
with single conducting channel were derived for three sym- The parallel plate capacitor considered in this work is
metryclassesβ=1,2,4.4Tobestofourknowledge,thereisno shownschematicallyinFig.1whichcontainsonemesoscopic
theoreticalstudyofelectrochemicalcapacitanceindisordered plate of size 1600nm × 1600nm referred as quantum dot
systems when the number of conducting channels is large. (QD).Asemi-infiniteleadofwidth400nmconnectsthisplate
Therefore,itisinterestingtoconductsuchaninvestigation.In totheelectronreservoir. Theotherplateofthecapacitorisa
particular,weareinterestedinthedistributionfunctionofCµ largemetallicplateonthetopofthemesoscopicplate. Ithas
sincetherearetheoreticalresults4 tocomparewith. Also,in- a largeDOS so thatthe quantumcorrectiondue to DOS can
spiredbytheuniversalconductancefluctuation25 indiffusive beneglected. TheFermienergyofthe mesoscopicplate can
regimeof disorderedsystems, we are interested in exploring be tuned by a gate voltage V . According to Ref.[1], the
gate
thefluctuationbehaviorofelectrochemicalcapacitanceindif- electrochemicalcapacitanceofthissystemisdefinedas
ferentsymmetryclasses.
1 1 e2 dS
= +( Tr[S† ])−1 (1)
C C 2πi dE
In this paper, we perform an extensive numerical investi- µ e
gationon the statistical propertiesof a mesoscopiccapacitor
where C is the classical geometrical capacitance and N=
e
shown schematically in Fig.1. In the numerical calculation, 1 S†dS is the Wigner-Smith time-delay matrix27 with S
the tight-bindingmodelis used with Anderson-typedisorder 2πi dE
the scattering matrix. The second term of this equation is
torealizedifferentrandomconfigurations. We studyelectro-
also calledquantumcapacitanceC , whichis determinedby
q
chemicalcapacitancefluctuationanditsdistributionfunction the density of states (DOS) of the system at low frequency1
forthreeensembleswithβ=1,2,4. Forthecaseofsinglecon- C = e2 Tr[S†dS] ≡ e2D. The electrochemical capaci-
ducting channel, it is found that the capacitance distribution q 2πi dE
tanceC is simplythe serialcombinationofgeometricalca-
showssymmetricGaussianshapeintheweakdisorderregime µ
pacitanceC andquantumcapacitanceC ,whichreads
andchangestoaone-sidedformcenteredinlargecapacitance e q
region as the disorder is increased. At strong disorders, the
1 1 1
distributionspreadsoutinthewholerangeofcapacitanceand = + (2)
C C e2D
eventually peaks in small capacitance area with a long tail µ e
thatcorrespondsto the largedensity ofstates. Thislongtail
Tostudythestatisticalpropertiesofelectrochemicalcapac-
can notbe accountedfor within random-matrixtheory.4 It is
itanceinthepresenceofdisorders,wewillconsiderthreeen-
dueto the existenceof necklacestates in disorderedsystems
sembles:(1).orthogonalensemblewithβ =1inthelanguage
which are characterized by multi-resonance with large den- ofrandommatrixtheory24whereboththemagneticfieldand
sity of states.26 As a result, the fluctuation of electrochemi-
the spin orbit interaction are absent. (2). unitary ensemble
calcapacitanceisgreatlyenhancedatsingleconductingchan-
withβ =2wherethemagneticfieldisnonzerowhilethespin
nelcase,whichismuchlargerthanthetheoreticalvaluepre-
orbitinteractioniszero.(3).symplecticensemblewithβ =4
dicted by RMT.4 As the number of conducting channels N
where the magnetic field is zero and the spin orbit interac-
increases,thenecklacestatesbecomelessprominentandthe
tion is nonzero. We use the following general tight-binding
fluctuationofelectrochemicalcapacitancedevelopsaplateau Hamiltonianfordifferentsymmetryclasses21
structure at large disorder strength. The electrochemicalca-
pacitance fluctuation at large N is found to be an universal H = Pnmσ(εnm+vnm)c†nmσcnmσ
venaleureg,y,wgheiocmheistriicndceappeancidteanntceofanthdesydsisteomrdesrizset.reRnegmtha,rkFaebrmlyi, −tPnmσ[c†n+1,mσcnmσe−imφ+c†n,m−1,σcnmσ+h.c.]
thisuniversalcapacitancefluctuationisthesameforallthree −tSOPnmσσ′[c†n,m+1,σ(iσx)σσ′cnmσ
ensembles. This super-universal behavior is very different −c†n+1,mσ(iσy)σσ′cnmσ′e−imφ+h.c.]
fromUCF,whosevaluedependsonthesymmetryindex.The (3)
distribution function of electrochemicalcapacitance also ex- wherec†nmσ(cnmσ)istheelectroncreation(annihilation)op-
hibitsuniversalfeatures. We find thatatlargeN the capaci- eratoronthelatticesite(n,m). Hereεnm representstheon-
tancedistributionisauniversalfunctionthatdependsonlyon site energy with magnitude 4t in the 2D system and t is the
theaveragecapacitanceanddoesnotdependonthesymmetry nearest-neighborhopping energy; vnm is the Anderson-type
ofthesystem. disorder potential which has uniform distribution in the in-
terval[-W/2,W/2]withW thedisorderstrength;σ arethe
x/y
Paulimatricesandt isthespin-orbitcouplingstrength.The
SO
The paper is organizedas follows. In Sec.II, the theoreti- magneticfluxφisduetothemagneticfield perpendicularto
calformalismanditsnumericalimplementationarepresented. themesoscopicplate.Differentsymmetryclassesarerealized
In Sec.III, we show the numericalresults on the distribution byturningonoroffthemagneticfieldaswellasthespin-orbit
andfluctuationofelectrochemicalcapacitanceforvariousdis- interactionintheHamiltonian. Fornumericalcalculation,we
order strengths, Fermi energies, geometric capacitances and discretize the mesoscopic plate using a 80×80 square grid
three different ensembles, accompanied with discussion and with lattice spacing a = 20nm, while the semi-infinite lead
analysis. Finally,theconclusionisdrawninSec.IV. has a width of 20 sites along y-direction(see Fig.1). In this
3
A. Thedistributionofelectrochemicalcapacitance
We first study the statistical properties of electrochemical
QD capacitance when only one conducting channel is available,
y since it has been experimentally realized14 and theoretically
L Lead Electron investigated from the random-matrix theory.4 From the ran-
reservoir dommatrixtheory,thedistributionofα = C /C foragen-
x µ e
eral symmetry class with symmetry index β in the case of
singleconductingchannelreads4
Vgate
(β/2)β/2 (1−α)β/2
P (α)= e−β(1−α)/2ηα (5)
FIG.1: Sketchofaparallelplatemesoscopiccapacitorconstructed β,η Γ(β/2) η(β+2)/2α(β+4)/2
on2DEG.SquareregionofwidthLisamesoscopicplate. Asingle
narrow lead whose width isL/4 connects this plate to an electron where η = Ce∆/e2 is a dimensionless parameter with ∆
reservoir. Dash-dotlinestandsforalargeclassicalplatewhichacts the mean level spacing of the disordered sample. From its
asanotherelectrodethatthemesoscopicplateiscapacitivelycoupled definition,η is inverselyproportionalto the leveldensity and
withthroughtheappliedgatevoltageVgate. hencegoestozeroformacroscopicsample.Inadditionηcon-
tainsgeometricinformationofthesystemthroughgeometric
capacitance Ce. In Sec.IIIB, we find that η plays a similar
system we set the hopping energy t as the energy unit with roleasdisorderstrengthinourcalculation.
t = ~2/2m∗a2 = 1.42meV,wheretheeffectivemassm∗ is We start with the orthogonal ensemble. For one incom-
typically that of GaAs, m∗ = 0.067m . The geometric ca- ing conducting channel N = 1, we fix the Fermi energy
e
pacitanceischosentobeC = 17.7fF,whichisreasonable at E = 0.065. The numerically calculated distributions of
e
foraparallelplatecapacitorwithsize1600×1600nm2. electrochemicalcapacitance α at differentdisorderstrengths
IntermsoftheGreen’sfunction,DOScanbeexpressedas are shown in Fig.2 where the histogram is obtained from
1,000,000configurationsateachdisorderstrength. Thetheo-
1 reticalpredictionfromEq.(5)atdifferentη areshownasred
D= Tr[GrΓGa] (4)
lineswherewehaveselected η so thatthe resultingdistribu-
2π
tionresemblesthenumericalresult. WeseefromFig.2athat,
whereGr =[E−H−Σr]−1istheretardedGreen’sfunction, at very weak disorder W = 0.05 P(α) is approximately a
Ga =(Gr)†,Σristheself-energyofthelead,andΓistheline Gaussian distribution with the peak located at α = 0.967,
widthfunctiondescribingcouplingoftheleadtothequantum indicating that the system is in the ballistic regime. As in-
dot with Γ = i[Σr − Σa]. In the following, we will study creasingthe disorderstrength, the distributionshiftstowards
thedimensionlesselectrochemicalcapacitanceαwhichisthe small capacitance. At W = 1 we see the occurrence of
ratiobetweentheelectrochemicalcapacitanceandtheelectro- smallcapacitancesduetothequantumcorrectionofDOSand
static(geometrical)capacitance,α = C /C . Obviously,we P(α)changesgraduallyfromGaussiantoone-sideddistribu-
µ e
have0<α<1sinceDispositivedefinite.Forsystemswith tion shown in panelb. A theoreticalresult from Eq.(5) with
verylargeDOSwe haveα ∼ 1 while forsystemswith very η = 0.01 shows a similar behavior. For a larger disorder
smallDOSwehaveα∼e2D/C . strengthW = 5 (panelc), α distributesalmostuniformlyin
e
thewholerangeexceptthatthereisacounter-intuitivenarrow
Inthepresenceofdisorder,theaverageelectrochemicalca-
peak for α close to one. We foundthat Eq.(5) is not able to
pacitanceαanditsfluctuationaredefinedas
givesucha peak. Thispeakis suppressedwhendisorderin-
creasesfurtherto W = 9, and α starts to peak in the region
<α>=<C /C >
µ e ofsmallα,asshowninpaneld. ItisfoundthatP(α)remains
rms(α)=p<(α−<α>)2 > approximatelyconstantvalueintherangeα=(0.2,0.9)with
amagnitudeofonefifthofthepeakvalue.Ontheotherhand,
where < ··· > denotesthe averagingover differentrandom the theoreticalcurvedecreasesrapidlyat large α. Uponfur-
configurationswiththesamedisorderstrengthW. therincreasingthedisorder(W =15inpanele),thesystemis
localized.28 AtthisdisorderstrengthP(α) issharplypeaked
atsmallvalueofαwithalongtailstretchingtoα=1. From
the inset of Fig.2e we see that this tail has approximately a
III. NUMERICALRESULTSANDDISCUSSION constantdistributioninrangeα=[0.5,1]whichclearlydevi-
atesfromEq.5. InFig.2fwedrawthedistributionofthelog-
Inthissection,numericalresultsonthestatisticalproperties arithmic of α at this disorder, where an abnormal peak near
of dimensionless capacitance α are presented. We will first ln(α)≃0appears29.
discuss the distribution function of α and make comparison FromFig.2,wenoticetwointerestingphenomenathatdis-
with existing theory and then study the average capacitance agrees with the theoretical prediction. One is the counter-
anditsfluctuation. intuitive narrow peak around α=1 at W = 5, which means
4

180 25
6
(a) (b) (c)
150
20
W=0.05 W=1.0 5 W=5.0
)120 =0.01 =0.4
( 15 4
P
90
3
10
60 2
5
30 1
0 0 0
0.955 0.960 0.965 0.970 0.975 0.980 0.75 0.80 0.85 0.90 0.95 1.00 0.0 0.2 0.4 0.6 0.8 1.0
18 0.5
5 (d) (e) W=15 (f)
W=9.0 15 =9.3 0.4 W=15
4 =2.3 0.4 ) =9.3
) 12 n
P( 3 0.3 0.3 P(l
9 0.2
0.2
2
6 0.1
1 3 0.00.5 0.6 0.7 0.8 0.9 1.0 0.1

0 0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 -7 -6 -5 -4 -3 -2 -1 0
ln
FIG.2: Distributionof thedimensionless capacitance α = Cµ/Ce for orthogonal ensemble withsingleconducting channel (N=1) inthe
lead. WechooseE=0.065inthecalculation. Thegrayhistogramsinallthepanelsarefromnumericalcalculation,correspondingtodifferent
disorder strengths, whiletheredlinesaretheoreticallypredictions4 atvarious valuesofparameter η. Thetheoretical resultsareshownfor
qualitativecomparison.
thatsomedisorderconfigurationshaveverylargeDOSmak- where the necklace states are not considered. The peak in
ing quantum capacitance very small. The other is the unex- Fig.2fnearlnα=0atW =15isalsooriginatedfromneck-
pected long tail in the strong localized regime with a nearly lacestates. Wewishtopointoutthatnecklacestatesmanifest
constant distribution probability at large α, where P(α) is onlyfortransportpropertiesthataresensitivetoDOS.Forin-
theoretically expected to decay. We attribute these counter- stance,emittanceandelectrochemicalcapacitancearesignifi-
intuitiveresultsto theexistenceofAzbelresonancestatesor cantlyaffectedbynecklacestates. Forconductance,however,
necklacestates,26 whichexistin stronglydisorderedsystems necklacestateshavenoeffect.33
asaresultofmulti-resonanceprocesses. Thesestatesarerare
eventsandformasmallportionofthewholeensemble.30Al- Since the necklace states are due to the multi-resonance
though rare, these states can have significant contribution to process,theirinfluenceonthedistributionofelectrochemical
theactransportpropertiesofdisorderedsystems. Duetothe capacitanceisnotableatN =1. WhenthereareN propagat-
natureofresonantprocess,thenecklacestates(orprecursorof ingchannelsinthesystem,thelongitudinalvelocityofdiffer-
necklacestateinthediffusiveregime)giverisetolargeDOS. entchannelsaredifferentforagivenenergy. Asaresult,the
We note that the existence of necklace states was confirmed effectofnecklacestatesissuppressedbecauseonlyonechan-
experimentallyinaquasi-1Dopticalsystem.31 InRef.[32],it nelcanbeonorclosetoresonanceandhaslargeDOSwhile
was numerically found that the necklace states widely exist theDOSofotherchannelsaresmall. Hence,comparingwith
in disordered 2D and QD systems for all symmetry classes thesinglechannelsituation,thenumberofnecklacestatesis
β=1,2,4andleadstotheuniversalpower-lawbehaviorofthe expectedto decrease by a factor of N. Thereforewe expect
distributionofWignerdelaytimeτ atlargeτ: P(τ) ∼ τ−2. thattheinfluenceofnecklacestatesonthedistributionofca-
Here in the distribution of electrochemicalcapacitance for a pacitance α becomes much weaker for large N. To support
disorderedmesoscopic capacitor, we find againthe evidence thisargument,weprovidenumericalevidenceontheeffectof
ofnecklacestates. BecauseofthehugeDOSfromtheneck- necklace states in Fig.3. In Fig.3a we calculate the distribu-
lace states the distribution of α = C /C in Fig.2c shows tionoflnαforalargerFermienergywithconductingchannel
µ e
anunexpectedlargesharppeakinlargeαregionandexhibits N =18atdisorderstrengthW =17. ComparingwithFig.2f
a long tail distributed in a wide range of α with nearly con- which correspondsto N = 1 and W = 15, it is found that
stant probability (for W ≥ 9). It is very different from the thepeakatlargeDOSnearlnα=0disappears. Thefactthat
theoreticalpredictionsobtained from random-matrixtheory4 P(lnα)doesn’tdecaytozeroatthispointshowsthatthereare
stillnecklacestatesatlargenumberofconductingchannels.
5
icalcapacitancedistributionfunctionatlargeN dependsonly
1.0 4
(a) (b) on the averageelectrochemicalcapacitanceanddoesnotde-
0.8 =1 3 =2 pendonthesymmetryindex.
) N=18 W=17 N=1 W=7
n0.6 ) We brieflysummarizethe resultsonthe distributionofdi-
P(l0.4 P(2 mensionlesscapacitanceα. In the case of a single transmis-
sion channel or a few conducting channels, the distribution
1
0.2
ofcapacitancegraduallychangesfromGaussiantoone-sided
0.0 0 normaldistributionpeakedatlargeαforweakdisorders. For
-3 -2 -1 0 0.0 0.2 0.4 0.6 0.8 1.0
0.3 ln 1.0 medium disorders, the distribution P(α) spreads out in the
(c) (d) wholeintervalα∈(0,1]withasharppeakaroundα=1. At
=4 0.8 N=18,W=17
)0.2 N=1 W=15 ) =1 strongdisorders,thedistributionfunctionbecomesone-sided
n 0n.6 =2 againbut sharplypeaked at smallα, with longtail of nearly
P(l P(l =4
0.4 constantmagnitudein large α. Both the peakand tail in the
0.1
capacitancedistributionsarisefromtheexistenceofnecklace
0.2
states in disordered systems, which give rise to large DOS.
0.0 0.0 When the number of conducting channels is large, the peak
-6 -5 -4 -3 -2 -1 0 -3 -2 -1 0
ln ln andtailduetonecklacestatesaresuppressed.Thedistribution
FIG.3: Panel(a)plotsthedistributionoflnαatorthogonalsymme- functionsfor threedifferentensemblescollapse intoa single
try with N = 18 and W = 17. Panel (b) shows the distribution curveatlargenumberofconductingchannelsandstrongdis-
functionofasystemwithmagneticfield(β = 2)atsinglechannel orderstrengths.
anddisorderstrengthW =7. Panel(c)depictsP(lnα)atsymplec-
tic case (β = 4) for N = 1 and W = 15. Panel (d) shows the
distributionfunctionatN =18andW =17forallthreesymmetry
B. Thefluctuationofelectrochemicalcapacitance
classesβ =1,2,4.Panel(b)isalmostthesameasFig.2candPanel
(c)isverysimilartoFig.2f.
Itiswellknownthattheconductancefluctuationinthedif-
fusive mesoscopic system assumes an universal value25 that
So far, we have discussed the distribution of α = C /C isindependentofFermienergy,disorderstrength,andsample
µ e
for the orthogonal symmetry with single and multiple con- size. Similaruniversalbehaviorforthespin-Hallconductance
ducting channels. Our numerical calculations on symmetry fluctuation34wasalsoreportedindisorderedmesoscopicsys-
classes β = 2,4 give very similar results, i.e., the distribu- tems. Itwouldbeinterestingtoseewhetherthefluctuationof
tionsof electrochemicalcapacitancein unitaryandsymplec- electrochemicalcapacitanceexhibitsuniversalfeaturesinthe
ticensemblesfollowsimilartrendasinorthogonalensemble presenceofdisorders.
showninFig.2. Thenecklacestatesareimportantforthecase InFig.4,weplottheaveragecapacitanceanditsfluctuation
ofsingleconductingchannel(N = 1)andarelessimportant vs disorderstrengthsfor β = 1 when the numberof incom-
when N becomes large. For instance, in Fig.3b, the distri- ing channel N = 1. For single conducting channel situa-
butionfunctionP(α) isgeneratedfromlargenumberofdis- tion, our numerical result can be compared with theoretical
order samples in the presence of magnetic field B = 0.01 prediction obtained from Eq.(5). We present theoretical re-
(unitaryensemble). With single channelin the lead and dis- sultsfromEq.(5)in Fig.4a, wheretheaveragecapacitanceα
orderstrengthW = 7,thesharppeakinP(α)isverysimilar andits fluctuationare shown as functionsof η (definedafter
tothatshowninFig.2cforsymmetryclassβ = 1,highlight- Eq.(5)).Hereηplaystherolesimilartodisorderstrength.We
ing the importance of necklace states for N = 1. Similar see that in the clean sample the DOS is large and hence the
behaviorisfoundforsymplecticensemblewithsingletrans- average capacitance is close to hαi ≃ 1. In the presence of
missionchannel. InFig.3c,weshowthedistributionofloga- disorders,theaveragecapacitancedecreaseswiththeincreas-
rithmalαforsymplecticsymmetryclasswitht =0.2. We ingofη. Thecapacitancefluctuationrms(α)risesrapidlyas
SO
see thatP(lnα)atN = 1 alsoexhibitsnecklacestates sim- the disorderstrength is turned on. As η is increased further,
ilar to Fig.2f. We have also calculated distribution function rms(α)reachesaplateauwherethemagnituderms(α)∼0.2.
P(lnα) for β = 2,4 when the number of conductingchan- Whenη keepsincreasing,thefluctuationdecaysslowlyfrom
nelislarge(N=18)whilefixingdisorderstrengthatW =17. theplateauvalue.OurnumericalresultforN =1isshownin
Remarkably P(lnα) for three ensembles are on top of each Fig.4b,withhαiandrms(α)changingwithdisorderstrength
othershowingsuper-universalbehavior(seeFig.3d).Wenote W. Fig.4a and Fig.4b show similar behaviorsespecially the
that for conductance fluctuation in the diffusive regime, the plateau region. From Fig.4a and Fig.4b, two differencesare
universalconductancefluctuation(UCF)whichisobtainedas noted: (1). the average capacitance in Fig.4b decays much
the second momentof its distribution functiondependsonly slower than that in Fig.4a. (2). the plateau value is about
onthesymmetryofthesystemaswellasthedimensionality.21 rms(α) = 0.32 at disorder strength W = 7 which is much
Our results on capacitance distribution function suggest that largerthanthatofFig.4a.Thesedifferencescanbeunderstood
thedistributionfunctionandhencethecapacitancefluctuation fromthedistributionofαshowninFig.2. Numericallycalcu-
atW =17isindependentofsymmetryindexβatlargeN.In latedP(α)haswiderspreadinginthewholeαrangethanthe
thenextsubsection,wewilldemonstratethattheelectrochem- theoreticalfunctionespeciallyatlargeα,featuredbythenar-
6

1.0 0.20 1.0 0.35 0.5
0.30 0.30
0.8 0.8
<>00..46 <rm>s( ) rms() 00..1105 00<>..46 000...122505rms() 0.4 rms()00..2205 19N 1290
00..020.0 0(.a2)0.4 0.6 0.8 1. 0 1.2 1.4 1.6 1.8 2.000..0005 00..020 (b) 4N = 1<rma>s(a8) 12 16000...001050 ms()0.3 00..311.059100 3.9105 3.9110 3.9115 3.911280 18 N
0.8 W r
1.0
0.7 < >=< >*Ce/e2 0.8 0.20 0.2
0.6 0.15 17
> 0>.6 rm
<0.5 < 0.10s( 0.1 rms( )
0.4 0.4 ) Number of Modes
0.3 (c) 0.2 (d) N = 1<8> 0.05 16
0.20 2 4 6 8 10 0.00 5 10 rm1s5( ) 20 25 300.00 0.03.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00
W W E
FIG.4: Panel(a)plotstheaveragecapacitanceanditsfluctuationvs FIG.5: ThecapacitancefluctuationatfixeddisorderstrengthW =
η, obtained fromthe theoretical prediction Eq.(5). Panel (b) isthe 17 vs Fermi energy. Right axis indicates the channel number
numericalresultcalculatedatFermienergyE =0.065withchannel in the system and the inset picture highlights the interval E ∈
numberN =1inthesystem.Panel(c)showsthecalculatedηfrom [3.910,3.912]whereN changesfrom18to19.
mean level spacing as a function of W. Panel (d) corresponds to
averageαanditsfluctuationversusW inthecaseofN = 18. All
resultsareobtainedfromtheorthogonalensemble.
with a nearly constantmagnituderms(α) ≈ 0.20 indicating
that rms(α) is independentof W in this interval. The max-
row peak at W = 5 and long tails in W = 9 and 15. This imum plateau value is usually identified as the value of uni-
directly leadsto largeraverage capacitancehαi and stronger versalconductancefluctuation(UCF)inthenumericalinves-
fluctuationrms(α). Ourpreviousanalysissuggestedthatthe tigationofUCF in disorderedmesoscopicsystems.33 For in-
necklace states are responsiblefor the peak and long tails in stance,theuniversalspin-Hallconductancefluctuationinthe
thedistributionfunctionatlargeαarea.Hereweseethatthese presenceofspinorbitinteractionwaspredictedusingthemax-
necklacestatesgreatlyenhancethefluctuationofcapacitance imumplateauvalue.35 Exactvaluewaslaterconfirmedtheo-
forthecaseofsingletransmissionchannel(N =1). retically by random matrix theory (RMT).36 Fig.4d suggests
that there may exist a universal behavior of the capacitance
To make some connectionbetween theoreticalresult from
fluctuation in the strong disorder regime. Since there is no
Eq.(5) and our numericalcalculation, we calculate the mean
theoreticalpredictionofcapacitancedistributionanditsfluc-
level spacing ∆ of our system when the lead is removed,
which in turn gives the parameter η since η = C ∆/e2. In tuationforN 6=1,wewillcarryoutextensivenumericalcal-
e
culationforthecaseoflargeN.
Fig.4c we find that η versus disorder strength W showing a
nearlylineardependence.FromFig.4b,wefindthemaximum Fig.5 plots the maximum value of capacitance fluctuation
valueoftheplateaulocatingatW =7. Thedisorderstrength versusFermienergyforafixeddisorderstrengthW =17lo-
W = 7 corresponds to η = 0.57 from Fig.4c. Indeed, the catinginthemiddleoftheplateauinFig.4d,whererms(α)is
maximumplateauvalueinFig.4a isatη = 0.57. Henceour independentof W. We have also shown the numberof con-
result agrees with the prediction of RMT (Eq.(5)) shown in ductingchannelsforthecleansamplevsFermienergyinthe
Fig.4a, i.e., the capacitancefluctuationreachesits maximum samefigure.Clearlytherms(α)remainsaconstantvalue0.20
atW = 7 butwith a largermaximumvalue dueto the exis- inthislargeenergyrangeE ∈[3.6,4].Thisstronglyconfirms
tanceofnecklacestates. thatthefluctuationplateauvaluesaturatesatlargenumberof
Since the necklace states enhance the capacitance fluctu- channels. We highlightin the inset of Fig.5 the behavior of
ation and the probability of occurrence of necklace states is rms(α)inasmallenergyinterval,correspondingtothenum-
suppressed by increasing the number of conducting channel ber of modes changing suddenly from 18 to 19. Based on
N, we expectthat the fluctuation of α at large N should be theseevidencesandFig.4d,wecansafelystatethatthecapac-
smallerthanthecaseofN = 1. Wehavecalculatedrms(α) itance fluctuation rms(α) is a constant in the disorder win-
forN =18whichisshowninFig.4d. Weseethatthefluctua- dow 15 ≤ W ≤ 18, and independent of Fermi energy in a
tionrms(α)vsdisorderstrengthisverysimilartothatshown widerangeE ∈[3.6,4]whereN islarge. Tosubstantiatethe
in Fig.4b and the plateau value of rms(α) is indeed smaller physicalpictureofuniversalcapacitancefluctuation,wenow
than that of N = 1. Interestingly, this plateau value is ap- examine the dependence of capacitance fluctuation on other
proaching0.20,whichistheoreticallypredicted4 forN = 1. parameters.
OurnumericalresultsshowthatatevenlargerN,theplateau Wefirstchecktheeffectofdifferentgeometriccapacitances
valueofrms(α)doesnotchange. C on rms(α). In all previous calculations we have cho-
e
We also noticed that in Fig.4d the capacitance fluctuation sen C =17.7 fF for a mesoscopic capacitor with plate size
e
exhibitsaplateaustructureindisorderintervalW = [15,18], 1600×1600nm2. Nowweinvestigatethecapacitancefluctu-
7

1.0 0.25 0.25 0.25
N=18 1.0
0.8 0.20 0.20
0.20
(a) ) (b) )
<>000...246 CCee==81 7f.F7 fF 000...011505 rms( N =CCC1eee8===812 76fF. 7fF fF 000...011505 rms( <>00..68 (a) N= 1==812 00..1105 rms()
0.0 Ce=26 fF 0.00 0.00 =4
0 10 W2 0 30 400.25 0.0 0.2 0.4< >0.6 0.8 1.00.25 0.4 0.05
1.0
0.20 0.20 0.2 0.00
<>00..68 (c) NNN===123898 fffooorrr LLL===81102600 00..1105 rms() (d) NN==1289 ffoorr LL==81020 00..1105 rms() 0.200 10 W 20 30
N=38 for L=160
0.4 0.05 0.05 0.15 (b)
)
0.20 5 10 15 20 250.00 0.2 0.4 0.6 0.8 1.00.00 ms( N=18
W < > r0.10 =1
FIG.6: Theinfluenceofgeometriccapacitanceonelectrochemical =2
0.05 =4
capacitancefluctuation. Panel(a)showstherms(α)and<α>vs
disorderstrengthWfordifferentgeometriccapacitancesbyvarying
0.00
separationbetweentwoplatesofthecapacitor. Herewehaveused 0.0 0.2 0.4 0.6 0.8 1.0
80gridpointsalongeachdimensionofthemesoscopiccapacitorand < >
fixedFermienergytobeE =3.85.Panel(b)isobtainedfrompanel FIG.7: Panel(a)showsrms(α)and<α>vsdisorderstrengthW
(a) by eliminating W from rms(α) and < α >. Panels (c) and fordifferentensemblesβ=1,2,and4atN =18.Panel(b)givesthe
(d) aresimilar to(a) and (b) except that thegeometric capacitance rms(α)versus< α>curvesatchannelnumberN = 18showing
ischangedbyvaryingthecross-sectionareaofthecapacitor. Three universalbehaviorforallsymmetryclasses.
systemswithwidthL=80,120,and160areexaminedwithfixedlat-
ticespacing a. Filledsymbols represent < α >and unfilledones
correspondtorms(α).
where differentcurvesof conductancefluctuationversus av-
erageconductanceatdifferentenergiescollapseintoasingle
curve.21InRef.21itwasfoundthattheuniversalbehaviorof
ationasafunctionofdisorderstrengthforaseriesofgeomet- conductance fluctuation versus average conductance reflects
rical capacitances. Different Ce can be realized by varying theuniversalityofconductancedistribution: theconductance
theseparationbetweenthetwoplateswhilekeepingtheplate distributiondependsonlyontheaverageconductance.Hence
area unchanged.37 The Fermi energy is fixed at E = 3.85 ourresultssuggestthattheelectrochemicalcapacitancedistri-
(N = 18 in the lead). The numerical results are depicted butionhasthefollowingform: P(α,< α >),i.e.,thecapac-
in Fig.6a. We see fromFig.6a thatfor differentCe the elec- itance distribution depends only on the average capacitance.
trochemicalcapacitanceandits fluctuationare verydifferent Thisresult is consistentwith Eq.(5), which says thatcapaci-
as functions of W. However, when we plot rms(α) versus tancedistributionforN = 1dependsonlyonβ andηandso
< α > in Fig.6b by eliminatingthe parameterW in Fig.6a, is<α>=R dαP(α). EliminatingηfromEq.(5)and<α>,
we find all three curvesalmost collapse into one with maxi- wefindthedistributionfunctionasP(α,β,<α>).
mumrms(α) ∼ 0.20. Thisfactsuggeststhatthecapacitance So far, our numericalresults show that for the orthogonal
distributiondoesnotdependonthegeometriccapacitance.In ensemble there exists a universal capacitance fluctuation for
Fig.6c we show the calculated results of electrochemicalca- theelectrochemicalcapacitanceindisorderedsystems,which
pacitance by changing the area of the mesoscopic capacitor is independentof disorderstrength, Fermienergy,geometric
while fixingthe separationbetweentheplatesandthelattice capacitance and system size. This conclusion can be gener-
spacinga. Whentheareaischanged,wekeeptheratioofdi- alized to othersymmetryclasses. In the previoussubsection
mensionofleadandquantumdottobe1/4sothatforagiven Sec.IIIA we already see that the distribution of α in all the
Fermienergy,largerareacontainsmoresubbandsinthelead. three ensembles are exactly the same for large N and W.
Three system size with width L=80,120, and 160 are exam- Since the fluctuation rms(α) is directly calculated from the
ined. InFig.6dweplotrms(α)versus<α>. Hereagainwe distributionfunction,oneexpectsthatthefluctuationofelec-
seethreecurvesfallingintoasingleoneandthemaximumca- trochemicalcapacitancefororthogonal,unitaryandsymplec-
pacitancefluctuationcloseto0.20. Thisphenomenonfurther ticensemblesshouldbethesame. InFig.7weprovidethenu-
confirmstheconclusiondrownfromFig.6b,thegeometricca- mericalevidencetosupportthispoint. InFig.7a,theaverage
pacitance has no influence on the distribution of α. Fig.6c electrochemicalcapacitance and its fluctuation as a function
andFig.6dalso showthatthe distributionofelectrochemical of disorder strength W for β = 1, 2, and 4 are numerically
capacitanceanditsfluctuationplateauarenotaffectedbythe calculated with channel number N = 18 and other system
systemsize. parameters are kept the same as in Sec.IIIB. From this fig-
The behaviors in Fig.6b and Fig.6d are very similar to ure, we see thatthree curvesare almostidenticalat largeW
the distribution of conductance in the presence of disorders, whichisconsistentwiththeobservationinSec.IIIAandthey
8
alldevelopaplateaubehaviorwithrms(α)≈0.20.Forsmall berofconductingchannelsN isincreased,theeffectofneck-
W betweenW = (5,8), we dosee somedifferenceinthree lace statesgraduallydiminishes. At largeN, itis foundthat
curves. When eliminating W from rms(α) and < α >, we instronglydisorderedregimetheelectrochemicalcapacitance
obtainaperfectsinglecurveinFig.7bforallthreeensembles. shows universalfluctuation with the magnitudearound0.20,
Based on the above analysis, we can conclude that the elec- which is independentof the disorderstrength, Fermienergy,
trochemicalcapacitancefluctuationexhibitsauniversalvalue geometricalcapacitance and system size. More importantly,
0.20atlargeN fororthogonal,unitaryandsymplecticclasses. theuniversalcapacitancefluctuationis thesame forallthree
symmetryclasses. Finally,thedistributionofelectrochemical
capacitanceatlargeN isfoundtobea super-universalfunc-
IV. CONCLUSION tion that depends only on the average capacitance and does
notdependonthesymmetryindexβ.
Inconclusion,weperformlargescalenumericalcalculation
to investigatethe statistical propertiesof electrochemicalca-
pacitancein a disorderedmesoscopiccapacitorforthreedif-
V. ACKNOWLEDGMENTS
ferent symmetry ensembles. It is found that in the strongly
disordered regime, the existence of necklace states greatly
influence the statistical behavior of electrochemical capaci- ThisworkwasfinanciallysupportedbytheResearchGrant
tance for the case of single conducting channel. Due to the Council (Grant No. HKU 705212P), the University Grant
necklacestates,thecapacitancedistributionexhibitsasharply Council (Contract No. AoE/P-04/08) of the Government of
one-sided shape with a nearly-constant tail in large capaci- HKSAR, and LuXin Energy Group. We thank the informa-
tance regionat strongdisorder, whichdeviatesfromtheoret- tiontechnologyservicesofHKUforprovidingcomputational
icalpredictionbytherandom-matrixtheory. Whenthenum- resourceonthehpcpower2system.
∗ Electronicaddress:[email protected] 23 W. Ren, F.M. Xu, and J. Wang, Nanotechnology 19, 435402
1 M. Bu¨ttiker, J. Phys.: Condens. Matter 5, 9361 (1993); M. (2008).
Bu¨ttiker,H.Thomas,andA.Pretre,Phys.Lett.A180,364(1993). 24 C.W.J.Beenakker,Rev.Mod.Phys69,731(1997).
2 J. Wang, B. G. Wang, and H. Guo, Phys. Rev. B 75, 155336 25 B.L.Altshuler,Pis’maZh.Eksp.Teor.Fiz.41,530(1985)[JETP
(2007). Lett.41648(1985)];P.A.LeeandA.D.Stone,Phys.Rev.Lett.
3 M. Macucci, Karl Hess, and G. J. Iafrate, J. Appl. Phys. 7, 77 55, 1622 (1985); S. Washburn and R. A. Webb, Adv. Phys. 35,
(1995). 375(1986).
4 V. A. Gopar, P. A. Mello, and M. Bu¨ttiker, Phys. Rev. Lett. 77, 26 M.Y.Azbel, Phys.Rev. B27, 3852(1983); M.Y.Azbel, Phys.
3005(1996). Rev.B28,4106(1983); C.Basu,A.Mookerjee, A.K.Sen,and
5 M.Bu¨ttiker,J.KoreanPhys.Soc.34,121(1999). A.K.Thakur,J.Phys.:Condens.Matter3,9055(1991).
6 J.Wanget.al.,Phys.Rev.Lett80,4277(1998). 27 E.P.Wigner,Phys.Rev.98,145(1955); F.T.Smith,Phys.Rev.
7 L.Latessa,A.Pecchia,A.DiCarlo,andP.Lugli,Phys.Rev.B72, 118,349(1960).
035455(2005). 28 It is not straigthforward to calculate the localization length of a
8 S.E.Nigg,R.Lopez,andM.Bu¨ttiker,Phys.Rev.Lett.97,206804 quantum dot system. Based on the previous calculation on a 2-
(2006). dimentional80×80grid,thesystementersthelocalizedregime
9 M.Bu¨ttikerandS.E.Nigg,Nanotechnology18,044029(2007). whendisorderstrengthW islargerthan4forN=1case.Thenu-
10 M. Moskalets, P.Samuelsson, and M. Bu¨ttiker, Phys.Rev. Lett. mericalevidence can befound inRef.[32].Henceitisanatural
100,086601(2008). deductionthatatlargerdisorderW=15,thequantumdotsystem
11 S.E.NiggandM.Bu¨ttiker,Phys.Rev.B.77,085312(2008). isalreadystronglylocalized.
12 C.MoraandK.LeHur,NaturePhysics6,697(2010). 29 More numerical data can be found inthe M.Phil thesisby F.M.
13 A.DiCarloet.al.,Phys.Rev.B63,235305(2001). Xu,theUniversityofHongKong,2008.
14 J.Gabelliet.al.,Science313,499(2006). 30 J.B.Pendry,J.Phys.C:SolidStatePhys.20,733(1987).
15 S.Ilani,L.A.K.Donev,M.Kindermann,andP.L.Mceuen,Na- 31 J.Bertolotti,S.Gottardo,andD.S.Wiersma,Phys.Rev.Lett94,
turePhysics2,687(2006). 113903(2005).
16 S.KobayashiandYCho,Phys.Rev.B82,245427(2010). 32 F.XuandJ.Wang,Phys.Rev.B.84,024205(2011).
17 T.Freyet.al.,Phys.Rev.B86,115303(2012). 33 W.Ren,J.Wang,Z.S.Ma,Phys.Rev.B72,195407(2005).
18 T.P.Smith,B.B.Goldberg,P.J.StilesandM.Heiblum,Phys.Rev. 34 Z. H. Qiao, W. Ren, and J. Wang, Mod. Phys. Lett. B 25, 359
B32,2696(1985);T.P.SmithIIIandW.I.Wang,Phys.Rev.B34, (2011).
2995(1986). 35 W.Ren, Z.H.Qiao, J.Wang, Q.F.Sun, andH. Guo, Phys. Rev.
19 Y.D.WeiandJ.Wang,Phys.Rev.B79,195315(2009). Lett.97,066603(2006).
20 A. Garcia-Martin and J. J. Saenz, Phys. Rev. Lett. 87, 116603 36 J.H.Bardarson,I.Adagideli,andP.Jacquod,Phys.Rev.Lett.98,
(2001). 196601(2007).
21 Z.H. Qiao, Y.X. Xing, and J. Wang, Phys. Rev. B. 81, 085114 37 Thegeometriccapacitanceinoursystemcanbeapproximatedas
(2010). thatofaparallelplatecapacitor,whichisCe ≈ε0A/d,whereε0
22 P.W.BrouwerandM.Buttiker,Europhys.Lett.37,441(1997). isthevacuumpermittivity.Aanddaretheareaofthemesoscopic
9
plateandtheseparationbetweenitandthemacroscopicmetalic for Ce = 26fF. Larger capacitance can be realized by sand-
plate,respectively.WithfixedareaA = 1600×1600nm2,dif- wichingdielectricmaterialwithhighpermittivitybetweenthetwo
ferentCeisacheivedbychangingthedistanced.Onesimplygets plates.
thatd=2.84nmcorrespondstoCe =8.0fF andd=0.87nm