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Title

A statistical model of quantum dot arrays with
Coulomb coupling

Author(s)Suzuki, Toshi‑kazu; Nomoto, Kazumasa; Ugajin,
Ryuichi; Hase, Ichiro

Citation

Journal of Applied Physics, 78(4): 2547‑2549

Issue Date

1995‑08‑15

Type

Journal Article

Text version

publisher

URL

http://hdl.handle.net/10119/4503

RightsCopyright 1995 American Institute of Physics.
This article may be downloaded for personal use
only. Any other use requires prior permission of
the author and the American Institute of Physics.
The following article appeared in T.Suzuki,
K.Nomoto, R.Ugajin, and I.Hase, Journal ofApplied Physics, 78(4), 2547‑2549 (1995) and may
be found at
http://link.aip.org/link/?JAPIAU/78/2547/1

Description

Japan Advanced Institute of Science and Technology

A statisticalmodel of quantum dot arrays with Coulomb coupling

Toshi-kazu Suzuki,a) Kazumasa Nomoto, Ryuichi Ugajin, and lchiro HaseSony Corporation Research Centeer;I74 Fujitsuka-cho, Hodogaya-ku, Yokohama 240, Japan(Received 16 May 1994; acceptedfor publication 5 May 1995)
We presenta study of a statistical model of arrays of quantumdots, in which electronsare confinedby semiconductorheterojunctions in all three dimensions, with Coulomb coupling. Our model
describes repulsively interacting localized electrons whose number can vary with changes inchemical potential. By meansof a Monte Carlo simulation, it is shown that, at low temperatures,somestable electron densitieshardly changingin certain rangesof chemical potential appear,which
are associatedwith suppressedfluctuation and characteristic spatial electron distributions. Spinfluctuation, which gives magneticsusceptibility,exhibits essentiallydifferent behaviordependingon
temperatureand Coulomb energies. Q 1995 American Institute of Physics.

been determined.@In the quantum dot array, it is assumed
that both the size of dots and the spacing between adjacent
Various phenomena taking place in artificial lowdots
are 510 nm. When such a system is fabricated, deviadimensional quantum structuressuch as quantum wires andtioninthe size of the dots is inevitable. Assuming 10 nm
quantum dots, which are realized by virtue of new technolocubed
GaAs
dots embeddedin AlGaAs, the confined groundgies, have been the subject of extensive studies and the posstateenergylevelsmeasuredfrom the edgeof the conduction
sibility of applying them to devices has been suggested.’
band
for
bulk
GaAs
are approximately lo2 meV. If the size
Since coupled quantumdot arrays createdby a grid potential
of
the
dots
has
a
deviationof severalp/o(deviation of several
imposed on a two-dimensional electron gas, also called latA),
these
energieshaveadeviationof severalmeV. Since theeral surface superlattices,are in some aspectsdifferent fromtransfer
energybetweentwo
dots
isabout 1O-’ meV assumordinary superlattices,they have been of interest and theiring
thespacing
is
5
nm
and
about
10B5
meV assumingthe
novel properties have been demonst.rated.23
On the otherspacing
is
10
run,
the
transfer
energiesaremuch smaller thanhand, by means of a semiconductorheterojunction, we canthedeviationoftheenergylevelsestimated
above and theconstruct quantum dots in which electrons are confined bywave
functionsofelectrons
tend
to
be
localized
at each dot
heterojunctions in all three dimensions. Since the confineduetotheeffectsofrandomness.
On
the
other
hand,
neglectment of electronsin thesedots is quite strong and the size ofing
thescreening
effect,theintradot
Coulomb
energyand thethem can be reduced,it will be possibleto fabricate quantum
interdot
Coulomb
energyforadjacentdots
areapproximately
dot arraysin which the dots are very close togetherand Cou10 meV and several meV, respectively.Thus, we can expect
lomb coupling between the dots plays an important role. In
that
the Coulomb coupling will play an important role in thethis article, we discuss some aspectsof the quantum dot arsystem.
Moreover, modulating the screeningeffect by conrays with Coulomb coupling. In particular, considering thattrolling
theposition of the positive charge layer, the Couthe number of electrons in the quantum dot array can belomb
coupling
may be tuned.
modulated via an applied voltage by means of a tunnel caIn
order
toinvestigatethe above situation, we employ apacitor structure,4-6we investigate a statistical model of thefinitetemperature
Monte Carlo simulation of a model of requantumdot array in which the number of electronscan vary
pulsivelyinteracting
electrons representedby the following
by changing the chemical potential.
Hamiltonian:
I. INTRODUCTION

II. MODELING

We assumea tunnel capacitor to consist of an electron
supply layer, a tunnel barrier, a two-dimensional quantum
dot array layer, a thick barrier layer, and an electrode (seeFig. 1). This structure allows us to change the number ofelectrons in the quantum dot array layer; an applied voltage
betweenthe electron supply layer and the electrodechanges
the energy difference between the confined energy levels of
the quantumdots and the Fermi energyof the electron supply
layer, i.e., it changesthe chemical potential of the quantum
dot array layer, and electrons can enter and exit the arraylayer through the tunnel barrier. For a single quantum dot,
modulation of the electron number in the dot is observedby
capacitancespectroscopyand energy levels in the dot have
4Electronic mail: tsuzuki@src.sony.co.jp

J. Appl. Phys. 78 (4), 15 August 1995.%=C

Eij?2~-knl~)+~i+izj

UiPt~nl~i

Vij(n’+nzXnlf+nJ~),

where the dynamical variable n+(nJ is assumedto be 0 or
1, which denotesthe number of up (down) spin electronsin
the ith dot. The parametersei and Ui20 denote the quantized energy level and the intradot Coulomb energyof the ith
dot, respectively.The interdot Coulomb energy between theith dot and $h dot is representedby Vij= Vji~O. In. thismodel, electronsare assumedto be localized at each dot andwe have taken into accountonly one bound statelocalized ateach dot, which has twofold degeneracywith respect to thefreedom of spin. This is justified when the dot size is sufficiently small, because of a large separation between the

002%8979/95/78(4)/2547/3/$6.00

Q 1995 American institute of Physics

2547

(p), (Sp) vs. p (B&20.0,

Quantumt

Dot Array Layer

10.0, 5.0, Wu=O.l)i.

FIG. 1. Conduction band profile of the tunnel capacitor structure. EF is the
Fermi level in the electron supply layer.

ground-stateenergy and an excited-stateenergy or the absence of excited states. The total number of electrons
N=Z2,i(n++nz~) cm vary with change in the chemical
potential and the statistical properties of the modelare determined by the grand partition function ZG=z,:=o,lexp[-P(~-~N)],
where p and JL are theinverse temperatureand the chemical potential, respectively.
In this model, we are interested in the behavior of thestatistical averageof the electron density, i.e., the electron
number per dot, p=(N)INd,,=(8 i(n” + nZr))lN,,,, where(e) denotesthe statistical averageand Ndotis the total numberof dots. FJluctuationof the electron density is given by&NMs,t- @W%J2) = d$%%?J d%i,in which the factor l/Kt
arises from the law of large
numbers. We define the electron density fluctuation
Sp= dm= &(N- (N)) 2)/Nd,,t,which is related tothe capacitanceof the system and from which the factor fromthe law of large number is divided. Since our model has no
spin interaction, the statistical averageof the magnetizationof the system M=Zi(n+ - nf) always vanishes.However,the fluctuation of the magnetization,which gives the magnetic susceptibility, exhibits a characteristicbehavior. Thus
we define the spin fluctuation Sm = ~((M-(M))2)/Ndob, from which the factor from the law of large numbersis also
divided.
Ill. SIMULATION

Hereafter, we restrict our attention to the case thatelectron-electron interaction is short range; only the intradot
Coulomb energy with the same value U and the interdot
Coulomb energy for nearest neighbor dots with the samevalue V are taken into account. Similar situations were considered in order to investigate some materials with a fixed
numberof electrons.7However, we are interestedin a system
in which the number of electrons can be changed by thechemical potential; this situation can be realized in the quantum dot arrays. Figure 2 shows the behavior of the electron
density and its fluctuation as the chemical potential changes
for square-latticearrays of quantumdots. The energy levelsof the dots are assumed to randomly distribute obeying
Gaussiandistribution whose averageis zero (without loss ofgenerality) and standarddeviation is denotedby SE We use
ten random samplesof 36X36 squarelattices with the peri2548

J. Appl. Phys., Vol. 78, No. 4, 15 August 1995

-3

-2

-1

0

1

2

3

4

5

6

w
FIG. 2. The behavior of the ekctron density and its Huctuation as functionsof the chemical potential. At low temperatures, there exist some plateaus in
the electron density with suppressed fluctuation. For V/U==O.2 and V/U=0.3, a very similar behavior is seen.

odic boundarycondition and thermal averagesis again averaged for the samples. The parameter is taken to be PlY=20.0,
10.0, 5.0, and VllJ=O.2, 0.3, and &IU=O.l. If theintradot Coulomb energy is assumed to be 10 meV, PU=20.0, 10.0, and 5.0 correspondto temperaturesof about 6,

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0X0X0X

0e000000e0000.0*0*
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em00*0000000000000000@00000000Q00000000000xxxxxxxxxxxx(4(0

FIG. 3. Various characteristic spatial electron distributions. Open and closed
circles denote quantum dots occupied by up- and down-spin electrons, respectively. A quantum dot occupied by two electrons (both up and downspin) is represented by an open circle with a closed circle inside. An unoccupied quantum dot is represented by a cross. (a) Every dot is unoccupied
by electrons. Both the electron density and the spin fluctuation vanish. (b) A
dot occupied by one electron and an unoccupied dot alternate. The electron
density equals a half and the spin Huctuation equals l/fi. (c) Every dot is
occupied by one electron. The electron density equals unity and the spin
fluctuation takes the maximum value of unity. (d) A dot occupied by twoelectrons and an unoccupied dot alternate. The electron density equals unity
and the spin fluctuation vanishes. (e) A dot occupied by one electron and a
dot occupied by two electrons alternate. The electron density equals threehalves and the spin fluctuation equals 116. (f) Every dot is occupied by twoelectrons. The electron density equals two and the spin fluctuation vanishes.

Suzuki et al.

.

interdot interaction dominates.Our simulation demonstrates
that, at high temperatures@ U = 5.0), this difference disappears. Notice that the distributions shown in Figs. 3(c) and3(d) have very different entropies;the one shown in Fig. 3(c)has entropy kB log 2 per dot due to the freedom of spin, andthe other shown in Fig. 3(d) has no entropy. Therefore, thelatter is rather weak in thermal fluctuation and, at high temperatures,even if V/U> l/z, the probability of the appearanceof the distribution shown in Fig. 3(c) can be rather large
due to its entropy.

1

8

y

IV. CONCLUSION
0

-3

-2

-1

0

1

2

3

4

5

6

cl/u
PIG. 4. The behavior of the spin fluctuation as a function of the chemical
potent& For V/U=02 and V/U=O.3, a remarkably different behavior isseen at low temperatures. At high temperatures this difference vanishes.

12, and 24 K, respectively,and &/U=O.l correspondsto a
deviation of the quantizedenergiesof 1 meV. In the regionsof very low and very high chemical potential, the electron
density vanishescorrespondingto Fig. 3(a), and equals twocorrespondingto Fig. 3(f), respectively.In the intermediate
region, electron density plateaus exist at low temperatures@U=20.0, lO.O), namely one half, unity, a@ three halves,
which is especially clear in the case of lower temperatures@U=20.0). The behavior of the electron density in the caseof V/U=O.2 and 0.3 is very similar. There is, however, adistinction between them, which is elucidated from the behavior of the spin fluctuation, which is shown in Fig. 4. In
the plateaus of the electron density of one half and threehalves, the spin fluctuation is approximately l/&. In fact, inthe one-half and three-halvesplateau, the probability of theappearanceof the distributions shown in Figs. 3(b) and 3(e)is very large, respectively.The spin fluctuation of both these
distributions is l/a. In contrast, in the plateau of the electron density of unity, the spin fluctuation is unity for the caseof VlU=O.2, and suppressedfor VlU=O.3. For the case of
V/U-0.2,
the intradot Coulomb interaction dominates.As aresult, in the unity plateau, the probability of the appearanceof the distribution shown in Fig. 3(c), in which the spin
fluctuation takes the maximum value unity, is very large. On
the other hand, for VlU=O.3, the interdot interaction isdominant. Thus, in the unity plateau, the probability of theappearanceof the distribution shown in Fig. 3(d), in which
the spin fluctuation vanishes,is very large. In the result of
our simulation, although the spin fluctuation does not completely vanish, suppressionof the spin fluctuation at low
temperaturesis seen.Thiseffect can appearfor bipartite lattices with the nearestneighbor Coulomb interaction.At very
low temperatures,whether the intra- or interdot interaction isdominant is determinedby the relation between v/U and thenumber of nearestneighborsz (z =4 for the squarelattice); if
V/U< l/z, intradot interaction dominates,and if VIlJ> l/z,

J. Appl. Phys., Vol. 78, No. 4, 15 August 1995

As discussedabove, we can expect, for a quantum dot
array with Coulomb coupling at low temperatures,somestable electron densities associatedwith suppressedfluctuation and a large probability of the appearanceof characteristic spatial electron distributions, even in the presenceof randomness.This can be viewed as many-valuedstability taking
place in the system.In order to suppressthe fluctuation, conventional electron devicesmake use of the law of large numbers. However, the suppressedfluctuation in the system isdue to a kind of charging effect. lf we make use of thismany-valuedstability, we can suppressthe fluctuation without the law of large numbers,i.e., even if Ndotis not so large.
This many-valued stability will be useful for new electron
devices. Similar effects of suppressedfluctuation due to acharging effect were experimentallyinvestigatedin the earlydays4and also more recently.s*6V8
However, there is a distinction between them and our case. In our cases not only the
intradot Coulomb interaction but also the interdot one, which
induces characteristicsspatial distributions, is important.
The effect of spin fluctuation discussedabove is similar
to the Kubo effect,’ which takes place in independentfine
metallic particles due to the charging effect. Arranging Coulomb coupled semiconductorquantumdots using heterojunctions, the new effects related to chemical potential, Coulomb
energies,and temperatureshould be observed.
ACKNOWLEDGMENTS

We would like to thank K. Taira, M . Ukita, and J. Westwater for encouragementand useful conversations. This
work was performedunder the managementof FED as a partof the M IT1 R&D program (Quantum Functional Device
Project) supportedby NEDO.
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