Statistics of the Charging Spectrum of a Two-Dimensional Coulomb Glass Island

5 Pages·0.14 MB·English

by  A. A. Koulakov

Statistics of the Charging Spectrum of a Two-Dimensional Coulomb Glass Island.
A. A. Koulakov,F. G. Pikus and B. I. Shklovskii
Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455
(February 1, 2008)
The fluctuations of capacitance of a two-dimensional island are studied in the regime of low
electronconcentrationandstrongdisorder,whenelectronscanbeconsideredclassicalparticles. The
universal capacitance distribution is found, with the dispersion being of the order of the average.
Thisdistributionisshowntobeclosely relatedtotheshapeoftheCoulombgapintheone-electron
densityofstatesoftheisland. Behaviorofthethecapacitancefluctuationsnearthemetal-insulator
transition is discussed.
7
9 PACS numbers: 71.55.Jv
9
1
Capacitance C is conventionally understood as a well energy2. The fluctuations of the spacing between the

n
defined geometrical property of a metallic sample. For nearest neighbor levels are of the order of the average
a
J example, for a metallic sphere of radius R capacitance spacingEF/N,whereEF isthe Fermienergy. Hence, for
C =R,andwhenthe sphere islarge,inthe firstapprox- a dot of radius R
5
imationcapacitancedoes not depend onthe distribution
1 of impurities inside the sphere or on its charge. How- δ EF/N rs (3)
∼ e2/R ∼ R
2 ever, in very small metallic samples fluctuations of ca-
v pacitancebecomeobservable. Recently,suchfluctuations where r = a is the screening radius of the two-
s B
6 were measured in the semiconductor quantum dots as a dimensionalelectrongas,a isasemiconductorBohrra-
3 B
functionofthetotalchargeofthedotusingtheCoulomb dius,whichiscloseto10nminGaAs. Forthisvalueofa
22 blockade phenomenon1,2. In such an experiment a small andforR 200nm(seeRef.2)Eq.(3)givesfluctuationBs
1 quantum dot is weakly coupled to current leads while a that are s∼ubstantially smaller than observed in the ex-
6 gate is placed in the proximity of the dot and is used periment. This discrepancy initiated the computer mod-
9 to vary its electrostatic potential. At low temperatures elling and the analytical calculations of δ (see Ref 2,3).
t/ the charge of the dot is typically quantized and there is When discussing theoretical results one should keep in
a no significant current. However the gate voltage can be mind that in all the theoretical works δ was obtained
m
tuned in such a way that the ground states with N and by averaging over the different realizations of disorder,
- N+1electronsaredegenerate. At this gate voltagecur- instead of number of electrons N. Below we will also
d
n rentcanflowthroughthedot. Theresultingconductance assume that in strongly disordered systems there is no
o vs. gatevoltagecomprisesaseriesofsharppeaks(charg- difference between these two definitions of fluctuations.
c ing spectrum). The spacing between two peaks ∆Vg can Analytical diagrammatic calculations based on RPA
v: be expressedinterms ofthe groundstate energiesEN of confirmEq.(3)3. Onthe other hand, the results of com-
i the dot with N electrons: puter modelling2,3 agree with Eq. (3) for weak interac-
X
tions(larger anda )andleadtolargerandinteraction
r eα∆Vg =∆N =EN+1−2EN +EN−1 =e2/CN (1) independentsδ for stBrong interactions, corresponding to
a
low density electron gas. This fact was identified as a
Here α is the geometricalcoefficient, C is the capac-
N failure of RPA in the the low density electron gas3. It
itance of the dot with N electrons. This equation may
canbe interpretedeasily in terms of revisionof equation
be considered as a definition of the capacitance. For a
r = a at low densities n a−2. Indeed r cannot
macroscopic body with the positive background charge s B ≪ B s
eN0, EN has a simple form: EN =e2(N −N0)2/2C and bne−1s/m2aalnledr tahtatnhethsemaavllerdaegnesidtiisetsanocneebshetowueldensuelbescttirtountes
Eq. (1) gives C = C = const. For the quantum dot
the charging eneNrgy ∆ was found2 to have surprisingly rs = n−1/2 into Eq. (3). It is not clear yet whether
N
such a simple modification of Eq. (3) can quantitatively
large relative fluctuations:
explain numerical and experimental data2, which seem
(<∆2 > <∆ >2)1/2 to indicate that δ is almost R independent. Hence, it
δ N − N 0.15. (2) is challenging to understand what happens with δ(R) in
≡ <∆ > ∼
N
the limit of a very low electron density.
Here < ... > denotes the averaging over N. Much ef- Inthispaperwetheoreticallystudythe fluctuationsof
fort has been done to explain such large fluctuations. capacitance of the island in the extreme classical limit
First, the experimental data were compared with the when the quantum kinetic energy of electrons is much
so-called constant interaction model in which ∆ = smallerthanboththedisorderstrengthandCoulombin-
N
e2/C+ηN+1 ηN, where ηN is the N-th single-electron teractions. Weconsiderthecaseofalargedisorderwhen
−
1
thegroundstateoftheislandisaCoulombglass4. Below all the possible states of N electrons on M M lattice
×
we show that for a piece of Coulomb glass or, in other sites and find one with the lowest energy. In the second
words,aCoulombglassisland,thefluctuationsarelarge, method we employ the simulated annealing technique,
δ is of the order of unity, does not depend on R, i.e. is running the finite-temperature classical Monte-Carlo for
universal for a given shape of the island. For the square some time and taking the lowest energy state. The con-
samplewefindδ =0.32. (Previouslyasimilarstatement vergence of the solution to the ground state has been
aboutgiant,ofthe orderofunity,relativefluctuationsof checkedbydoublingthetimeofthesimulationandmak-
the polarizability of the Coulomb glass island was made ing sure that the solution is not affected. The reliable
in Ref. 5). results have been obtained by this method for M 8.
≤
Westudyprobabilitydensityof∆ andshowthatitis ∆ has been calculated typically for 1000-2000different
N N
a universal function of the ratio x= ∆ / < ∆ >. We realizationsofdisorder. Wehaveobtainedδ =0.32. Nor-
N N
also discuss how the transition from the Coulomb glass malizeddistributionsF(x)oftheratiox=∆ /<∆ >
N N
to metal is reflected in the capacitance fluctuations. for M = 4,5,7,8 are shown in Fig. 1. We have found
When an electron is added to a large metallic sam- < ∆ >=2.3/M, in a good agreement with the inverse
N
ple its charge is distributed in the unique way according capacitanceofmetallicsquareofthe samesize. Remark-
to the electrostatic theory. For this reason with addi- ably, F(x) does not depend on M. We emphasize also
tion of every electron the electric potential grows by the that contrary to the predictions of the constant interac-
same amount e2/C, or in other words C = const. In tion model the inverse capacitance of the Coulomb glass
the Coulomb glass the electronic states are localized, so island can be both larger and smaller than that of the
that every new electron is put into some localized site. metallic island of the same shape.
Then electrons rearrange themselves in the vicinity of
this site. However it was shown in Ref. 5 that such rear- 1.6
rangementhappensonlywithprobabilitycloseto1/2on
everyscale. Asaresultthe addedchargeinthe majority ... .
of cases is localized in the region smaller than the size 1.2 . . 4x4
of the island. When next electron enters the island its .. 5x5
.
charge is centered near another site in the island. The . 7x7
distance between this site and the position of the previ- (x)0.8 . . 8x8
ous electron fluctuates between 0 and 2R. As a result F .
the difference between energies required to bring two se- .
.
.
quential electrons, ∆ and capacitance C experience
N N 0.4 .
roughly speaking hundred percent fluctuations.
. .
Toverifythis reasoningwestudy the capacitancefluc- ..
tuations numerically. We use the lattice model of the . .........
Coulomb glass suggested by Efros6 and widely used to 0.0
0.0 0.5 1.0 1.5 2.0 2.5
study the Coulomb gap in the density of states (DOS). x
The Coulomb glass island is modelled by the square
FIG. 1. The inverse capacitance distribution is pre-
M M lattice, with every site being either empty ( oc-
× sentedfordifferentsamplesizes. ThelineisthefitbyEq.
cupation number n = 0 ) or occupied by one electron
i (5).
(n =1). ElectronsinteractwitheachotherbyCoulomb
i
interaction. The interaction energy between the near- The fit in Fig. 1 is given by the following equation:
est lattice sites is chosen to be the unit of energy and
0, x<x0
the lattice constant is the unit of length. Disorder is F (x)= 4(x x0)exp 2(x x0)2 , x x0, (5)
introduced by the randomsite energies φ which are dis- (cid:26) − − − ≥
i
tributeduniformlybetween 1and1. Thecorresponding where x0 = 0.37. Two int(cid:0)eresting featu(cid:1)res of F (x) are
−
Hamiltonian has the form clearly seen. Firstly, this function has a termination
1 point at x = x0. One can easily check that ∆N cor-
Hclass = (φi+ui)ni+ ninj/rij (4) responding to this point is equal to the smallest possible
Xi 2Xi,j Coulombinteraction: 1/rmax,wherermax =(M −1)√2
is the maximum distance between sites in the square
Here ui is the potential due to the uniform background M M. Secondly, F(x+x0) is identical to the Wigner
×
chargemakingthe systemelectricallyneutralforN elec- surmise for the nearest neighbor distance distribution of
trons. WefindthegroundstateenergiesofN 1,N,N+1 the levels ofarandommatrix. We shallshowbelow that
electrons where N is the integer part of M2−/2 and then this is only an interesting coincidence.
calculate ∆ using Eq. (1). To find the ground state Now we will interpret both features establishing the
N
we use two different methods: the exhaustive enumera- relation between F(x) and the one-electron DOS g(ǫ) of
tion and the simulated annealing. The first one is used theCoulombglassisland. Theenergyoftheone-electron
for relatively small samples with M 5. We enumerate excitation localized at i-th site can be written as
≤
2
n
ǫ =φ +u + j. (6) andcapacitance. Theexcitationsrelevanttocapacitance
i i i
rij are electronic polarons introduced by Efros6. Their en-
j
X
ergies ǫ˜ can be used to calculate ∆ . Indeed, ǫ˜ is de-
i N i
For an empty site, for example, it is the energy required fined as the energy required to bring an electron to the
tobringanelectronfrominfinitytothissite. Whenaver- site i and rearrangethe other electrons in order to reach
agingthis DOS overdifferentrealizationsof the disorder minimum of the total energy. From this definition it is
potential or, in other words, over different samples, we obvious that the minimum polaron energy of the empty
match the chemical potential in them (µ - averaging)4. states is EN+1 EN, and the maximum polaron energy
−
The chemical potential of the island is situated halfway ofthe occupiedstatesis EN EN−1. Itis clearthat∆N
−
between the largest energy of the occupied states and is just the energy gap between these two polaron states.
the lowest energy of the empty ones. The corresponding Itisknown,however,thatintwodimensionsthepolaron
DOS is shown in Figure 2. energies are very close to the one-electron ones4. This
means that ∆ can be well approximated by the energy
N
0.3 0.5 difference between the lowest empty and the highest oc-
cupied one - electron states. This immediately explains
the existence of the termination point of F(x): ∆ can-
N
0.0 notbesmallerthanthesmallestinteractionbetweentwo
0.2 -2 0 2 electrons within the island. Moreover the function F(x)
can be related to the one-electron DOS g(ǫ) in the fol-
) lowing way.
(
g As the inverse capacitance of an island is equal to
the difference between the lowest empty and the high-
0.1
est occupied states’ energies, our problem is to find the
distribution function of this difference. The probability
P(∆ ǫ) to have it bigger or equal than certain value ǫ
≥
isequaltotheprobabilitynottofindenergylevelsinthe
0.0
region ǫ < ǫ′ < ǫ. Assuming the Poissonian statistics
-0.4 -0.2 0.0 0.2 0.4 −2 2
of the level distribution we arrive at:
FIG.2. DOS for the8×8 sample averaged over disor- ǫ
2
der. The main picture shows the region near the Fermi P(∆ ǫ)=exp M2 g(ǫ′)dǫ′ . (9)
level. TheCoulomb gapinthedensityofstatesforanin- ≥ − Z−2ǫ !
finitelylarge sample[Eq.(7)] ispresentedbythestraight
lines. The inset shows thegeneral view of the DOS. Here g(ǫ) is assumed to be normalized to unity:
∞ g(ǫ′)dǫ′ = 1. The expression in the exponential is
TheimportantfeatureofthisDOSisalinearCoulomb −∞
the average number of electrons in the band of energies
ggaapp,rwelhaitcehdattotthheesfimniatlelseinzeeregffieesctcsro4s.seLsinoevaerrdtoeptehnedheanrcde R−2ǫ,2ǫ . Theprobabilitydensityof∆is,hence,equalto
of the DOS for ǫ > 0.2 agrees with the analytical ex- (cid:0) (cid:1)
| | dP(∆ ǫ)
pression for an infinitely large sample F(ǫ) = ≥
− dǫ
ǫ (10)
2 ǫ 2
g(ǫ)= ǫ (7) =2Ng exp 2M2 g(ǫ′)dǫ′
π | | 2 − 0 !
(cid:16) (cid:17) Z
derived in Ref. 6,7. The total width of the hard gap is
This expression establishes the general relationship be-
equalto1/rmax,wherermax isthemaximumdistancebe-
tween the one-electron DOS and the inverse capacitance
tween two points in the island. Indeed, the energy that
distribution function for the Coulomb glass island. Hav-
isrequiredtotransferanelectronfromsiteitoanempty
ingbeenappliedtothenumericallyobtainedDOSitgives
site j is equal to:
averygoodagreementwiththeactualF(ǫ). Substituting
1 Eq.(7)intoEq. (10)onearrivesattheGaussianasymp-
∆i→j =ǫj −ǫi− r r ≥0 (8) totic behavior of F(x): lnF(x) x2, 1 x M.
i j ∝ − ≪ ≪
| − | This asymptotic, however, does not persist in three di-
The minimum difference between the energies of empty mensions, where the DOS has a different form.
andoccupiedstatescannotexceedtheminimuminterac- Letus nowexaminewhathappens withδ whenahop-
tion energy within the island and, hence, is greater than pingtermisaddedtotheclassicalHamiltonian,givenby
or equal to 1/rmax. Eq. (4):
Let us now explain how the one-electron DOS can be
used to find F(x). Strictly speaking the one-electronen- H =Hclass−J a†iaj. (11)
ergiesarenotdirectlyrelatedtothegroundstateenergies
hXi,ji
3
Here J is the hopping matrix element and a† is the cre- Eq. (12) is valid only if ξ R, or J < J ∆J, where
i ≪ c −
ationoperatoroftheelectronatsitei. Thesummationis ∆J = (r /R)1/ν . At ξ = R the relative capacitance
s
carried over the neighboring sites. The ground states of fluctuations saturate at
thisHamiltonianwerefoundnumericallyusingtheLanc-
zos algorithm. The system considered was of size 4 4 δ (r /R)2 (13)
s
× ∼
lattice sites, with up to 7, 8 and 9 electrons. The results
are depicted in Fig. 3. Eq.(13)isthethree-dimensionalanalogofEq.(3). Itcan
alsobeobtainedfromtheassumptionthatfluctuationsof
∆ are equal to the fluctuations of the spacing between
N
the one-electron quantum levels. Predicted behavior of
0.3 function δ(J) is schematically depicted in Fig. 4.
1
0.2
0.1
0.0 0.2 0.4 0.6
J (r/R)2
s
FIG. 3. The relative inverse capacitance fluctuations
of4×4islandasafunctionofthehoppingmatrixelement
J. 0 J - J J J
c c
Asitisseenthatδdecreasesbyafactorof2fromJ =0 FIG. 4. The relative inverse capacitance fluctuations
to J =0.4. Such a decay is consistentwith the tendency of the three-dimensional island as given by Eq. (12) for
of metallic samples to have smaller capacitance fluctua- J <Jc−∆J and by Eq (13) for J >Jc−∆J.
tions. AtthesametimetheshapeofF(x)becomesmore
Weconjecturethatinthetwo-dimensionalcase,where
symmetricinagreementwithRef.2,3. Unfortunately we
as it is commonly believed ξ grows monotonically with
were not able to do such calculations for M >4. Hence,
J, the crossover from δ 1 to Eq. (3) happens in the
the existing numerical data leave open the challenging ∼
similar way: δ r /ξ at r ξ R and δ r /R at
question of how the crossover happens between δ 1 ∼ s s ≪ ≪ ∼ s
∼ ξ R. Schematically this behavior is similar to the one
in the classical case and Eq. (3) in the quantum one for ≫
showninFig.4. Ournumericalresults donotcontradict
largeenoughsamples. Belowwe try to answerthis ques-
to this prediction.
tion concentrating on the three-dimensional case where
UptonowwehavedealtwiththeunscreenedCoulomb
the insulator-metal transition happens at some critical
interaction between electrons. In case if a metallic gate
value J = J . We assume that this transition is accom-
c is situated at a small distance d from the plane of the
paniedbythe divergencyofthe wavefunctioncorrelation
island the long-range part of the Coulomb interaction
length ξ and the dielectric constant κ: ξ = J J −ν
and κ = J J −ζ. It was argued that κ | c(−ξ/r|)2, is screened. Let us now discuss how this screening af-
| c − | ∼ s fects our results. Consider first the case of an extremely
where r is the screening radius ofthe three-dimensional
s close gate when one can completely ignore the interac-
degenerate Fermi-gas, or ζ = 2ν ( see Ref. 8 ). When
tions between electrons. In this case the constant (zero)
J approaches J from the insulator side the growth of κ
c interaction model is valid, N electrons occupy N lowest
plays a very important role in the distribution of charge
of the added electron, even when ξ is still much smaller one-electronlevelsηk,and∆N =ηN+1−ηN isjustanear-
est neighbor level spacing. At J = 0 it has the Poisson
than a sample size R. It is well knownthat if a localized
distribution,thereforeδ =1. Inthemetallicphase(large
charge is put inside a dielectric sample with κ 1 the
≫ J) the random matrix theory is applicable and δ = 0.52
sample becomes polarized in such a way that almost all
(seeRef.2). Westudiedthiscrossoverforupto104 real-
of the added charge appears on its surface in the form
izationsofdisorderinthesquare4 4,withtheCoulomb
of induced charge. Only a small fraction of charge e/κ interactionreplaced by 1/r 1/(r2×+(2d)2)1/2, d=1/2.
remains localized inside. If the second electron is added −
We obtained δ = 0.67 at J = 0 and δ = 0.39 at J = 0.4
to the sample at the distance r from the first the in-
in the qualitative agreement with our expectations. At
teraction energy between these two consecutively added
J = 0 we also observed a drastic change in the form of
electrons fluctuates by e2/κR e2/R, provided that r
≪ F(x): thegapatsmallpositivexdisappearsandF(x)ap-
fluctuates in the range 0<r<2R for a sphere of radius
proaches the Poisson distribution. The most remarkable
R. Therefore
feature of the observeddistribution is that there exists a
very small tail of F(x) at x < 0, so that the differential
δ 1/κ (r /ξ)2 (J J)2ν (12) capacitance of a strongly screened disordered island can
s c
∼ ∼ ∼ −
4
be negative. The possibility of the negative capacitance 1P. L. McEuen, E. B. Foxman, U. Meirav, M. A. Kast-
in the presence of a gate was recently demonstrated in ner,Y. Meir, N. S. Wingreen, and S. J. Wind, Phys. Rev.
Ref.9. Theobservedprobabilitytohavenegativecapaci- Lett.67, 2862 (1991).
tanceisconsistentwiththecalculationsoftheseauthors. 2U.Sivan,R.Berkovits,Y.Aloni,O.Prus,A.Auerbuch,and
For our set of parameters it is equal to 3 10−4. G. Ben-Yoseph,Phys.Rev. Lett. 77, 1123 (1996).
In conclusion, we have found the large·universal rela- 3R. Berkovits and B. L. Altshuler, preprint, cond-
tive fluctuations of capacitance of the Coulomb glass is- mat/9610214.
4B.I.Shklovskii,A.L.Efros, Electronic Properties of Doped
land and described a scenario of their decay when the
Semiconductors, Springer, Heidelberg (1984).
system undergoes the insulator-metal transition. We
5S. D. Baranovskii, B. I. Shklovskii, A. L. Efros, Sov. Phys.
are grateful to M. M. Fogler for valuable discussions.
JETP, 60, 1031 (1984).
This work was supported by the NSF Grant No. DMR-
6A.L. Efros, J. Phys.C, 9, 2021 (1976).
9321417and by University of Minnesota Supercomputer 7A.L. Efros, B. I.Shklovskii, J. Phys. C, 8, L49 (1975).
Institute. 8Y. Imry, Y. Gefen, and D. J. Bergman, Phys. Rev. B , 26,
3436 (1982).
9M.E.Raikh,L.I.Glazman,L.E.Zhukov,Phys.Rev.Lett.
, 77, 1354 (1996).
5