Hidden symmetry and correlated states of electrons

and holes in quantum dots

Pawel Hawrylak*

Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Ont., Canada K1A 0R6

Received 14 May 2003; accepted 30 June 2003 by D. Das Sarma

Abstract

We describe hidden symmetry and its application to the construction of exact correlated states of electrons and holes in

quantum dots. The hidden symmetry is related to degenerate single particle energy shells and symmetric interactions. Both can

be engineered in a quantum dot. We focus on hidden symmetry involving spin singlet pairing of electrons and spin singlet

pairing of holes. Detailed calculations for a third shell are presented to illustrate the mechanism of pairing.

q 2003 Elsevier Ltd. All rights reserved.

PACS: 73.21 La; 71.35 Cc; 71.10 Li

Keywords: A. Quantum dot; D. Excitons; D. Electronic correlations

1. Introduction

The number of configurations NC in Hilbert space of

a many-body system of N fermions distributed over p

orbitals NC ¼�

Np

�is very large. In systems where

electronic correlations are important the ground and

excited states are a complex superposition of many, if

not all, configurations of the Hilbert space. It is

therefore desirable to deal with many-body systems

which offer the tunability of many parameters, such as

N; or electron density, and p; number of orbitals, via

single particle spectrum and confining potential. It is

perhaps even more desirable to be able to construct

exact eigenstates of a many-body system, even if only

in special cases. This construction is possible if there

exists a symmetry of the full interacting many-particle

Hamiltonian, the so-called hidden symmetry. It was

recognized early on by Quinn and co-workers [1] that

the artificially structured low dimensional systems, and

in particular the two-dimensional electron gas (2DEG)

with tunable carrier density are an ideal tool for the test

of many-body theories. Much of the work has been

devoted to the many-body effects in these systems, in

particular in magnetic field. When only electrons are

present in a partially filled, spin-polarized lowest

Landau level, very complicated ground states of the

fractional quantum Hall effect exist [2]. These states are

a very sensitive function of the filling factor. By

contrast, adding to spin polarized electrons an equal

number of spin polarized particles with opposite

charges, i.e. valence holes, changes the ground state

drastically. Electrons and holes form a condensate of

magneto-excitons due to ‘hidden symmetries’ [3]. This

many-body ground state is exactly known for any

fractional filling [3]. The experimental manifestation of

this condensate should be the emission spectrum which

does not depend on the filling fraction. Unfortunately,

this has not been observed yet in 2D systems.

MacDonald and Rezayi, [4] Apalkov and Rashba, [5]

and Chen and Quinn [6] extended the concept of hidden

symmetry in charge neutral electron–hole systems to an

electron system and a small number of valence holes.

They showed that the emission spectrum of 2DEG in

strong magnetic field is controlled by a single magneto-

exciton isolated from 2DEG. It was later shown by

Wojs and Hawrylak [7] that, contrary to hidden

0038-1098/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0038-1098(03)00574-X

Solid State Communications 127 (2003) 793–798

www.elsevier.com/locate/ssc

* Tel./fax: þ1-613-993-9389.

E-mail address: pawel.hawrylak@nrc-cnrc.gc.ca

(P. Hawrylak).

symmetry arguments, there exists a bound state with

finite angular momentum of charged exciton in the

lowest Landau level. Palacios et al. [8] and Dzyubenko

et al. [9] showed that the charged exciton is long-lived.

Wojs et al. [10] suggested FQHE of charged excitons

and studied its interaction with excess electrons [11].

Breaking hidden symmetry and its effect on the

emission from 2DEG in quantum Hall effect regime is

discussed theoretically by Asano and Ando in this

volume. The experimental work on charged exciton and

its effect on emission from 2DEG is covered by

McCombe and co-workers, Bar-Joseph and co-workers,

and Pinczuk and co-workers in this volume. The 2D

system in a strong magnetic field has been and

continues to be a useful tool to study many-body

effects in a controlled way. In this paper we focus on a

0D system, a quantum dot, [12] as a laboratory for

correlated electron systems. We discuss the ground state

properties of quantum dots filled with N electrons and

holes, exploiting the analogy between the quantum Hall

and quantum dot physics. Both the magnetic field and

the confining potential turn the continuous spectrum of a

quasi-2D electron into a discrete spectrum of two

harmonic oscillators. The most important aspect of

these spectra is the presence of degeneracies. For free

electrons, there are macroscopically degenerate Landau

levels. In a dot with parabolic confining potential in

zero magnetic field dynamical symmetries are respon-

sible for the formation of shells of degenerate states

with different angular momenta.

The optical properties of electrons and holes in quantum

dots have been investigated theoretically by a number of

groups [13–21]. We have focused on electrons and holes in

degenerate shells of a quantum dot and analogies with the

QHE physics. For electrons, exact diagonalization tech-

niques established ‘generalized Hund’s rules’ for quantum

dots in analogy to atoms [22]. However, we were not aware

of the existence in nature of atoms composed of particles of

opposite charges living on degenerate shells, and hence of a

guiding principle, such as Hund’s rules, in their electronic

properties. Here we summarize past work and present

results of a detailed calculation illustrating the hidden

symmetry rules for a third shell. Some of the rules are

familiar from the QHE physics, but also new symmetries

appear related to spin. These symmetries involve not just

electron and a hole but pairs of electrons and pairs of holes

as one finds in a biexciton. The unusual behavior of pairs of

carriers can also be found in the Hubbard models of high Tc

materials, [23] and we find it a rather unexpected and

exciting development.

The starting point in our discussion is the single particle

spectrum and interacting Hamiltonian of a quantum dot. The

simplified model of single-particle spectra of electrons and

holes in lens-shaped self-assembled quantum dots is used. In

these structures, the electron energies Emn ¼ V0ðn þ m þ

1Þ; eigenstates lmnl and angular momenta Lmn ¼ m 2 n are

well approximated by those of two harmonic oscillators with

energy V0 [24,25]. Whenever m þ n ¼ s; electronic shells

of degenerate states are formed. The valence band hole is

treated in the effective mass approximation as a positively

charged particle with angular momentum Lmn ¼ n 2 m;

opposite to the electron.

With a composite index j ¼ ½m; n;s� the Hamiltonian of

the interacting charge neutral electron–hole system may be

written in a compact form as:

H ¼X

i

Eei cþi ci þ

Xi

Ehi hþ

i hi 2Xijkl

kijlVehlkllcþi hþj hkcl

þ 12

Xijkl

kijlVeelkllcþi cþj ckcl þ12

Xijkl

� kijlVhhlkllhþi hþj hkhl: ð1Þ

The operators cþi ðciÞ; hþi ðhiÞ create (annihilate) the

electron or valence band hole in the state lil with the

single-particle energy Ei: The two-body Coulomb matrix

elements are kijlV lkll for electron–electron (ee), hole–hole

(hh) and electron–hole (eh) scattering, respectively [7,27].

We shall discuss the case of symmetric e–e, h–h, and e–h

interactions. For symmetric interactions, kijlVhhlkll ¼kijlVeelkll and kijlVhhlkll ¼ 2kiklVeeljll: It is important to

note that the intra-shell electron hole scattering matrix

elements kiilVehljjl are equal to equivalent e–e (h–h)

exchange matrix elements kijlVhhlijl:In extensive numerical work [14,15,18], and in what

follows, the eigenstates lnl of the electron–hole system with

N excitons have been expanded in products of the electron

and hole configurations lnl ¼ ðQN

i¼1 cþji ÞðQN

i¼1 hþkiÞlvl: The

configurations are labeled by the total angular momentum

Ltot; the z component of the total spin Stotz ; and the total spin

S: Of particular interest are states in the optically active

subspace of Ltot ¼ 0 and Stotz ¼ 0: This subspace contains

exactly known, multiplicative states, of the interacting

Hamiltonian [14,15,18,26].

The existence of exact eigenstates is a consequence of

‘hidden’ symmetries in the Hamiltonian of a degenerate

shell. There are two types of symmetries which involve

either single particle of each kind ðPÞ; or pairs of particles

ðQÞ: We start with construction of relevant operators and

symmetries of the P type. The P-type symmetries are

associated with the interband polarization operators ðPþs ;

P2s ; PzÞ; which form an algebra of angular momentum.

Pþð2Þs creates (annihilates) electron–hole pairs: Pþ

s ¼P

i �

cþishþi;2s ðP2s ¼

Pi hi;2scisÞ by annihilating (creating) pho-

tons with definite circular polarization [14]. The operator

Pz ¼12ðNe

s þ Nhs 2 NtotÞ measures the population inversion

on a shell with orbital degeneracy Ntot: The polarization

operator involves only one particle of a given type, i.e. one

electron and one hole. Possible symmetries are associated

with commutation properties of polarization operator Pþ

with the Hamiltonian [14]. This commutator for spinless

electrons and holes on a degenerate shell involves a family

P. Hawrylak / Solid State Communications 127 (2003) 793–798794

of two- and four-particle operators:

½H;Pþ� ¼X

i

ðEei þ Eh

i Þcþi hþi 2

Xijk

kijlVehlkklcþi hþj

þ 12

Xijkl

ðkijlVeelkll2 kiklVehljllÞðcþi hþl cþj ck

2 cþi hþk cþj clÞ þ12

Xijkl

ðkijlVhhlkll2 kiklVehljllÞ

� ðcþl hþi hþj hk 2 cþk hþi hþj hlÞ: ð2Þ

The degeneracy of single-particle levels combined with

the symmetry of (ee), (hh), and (eh) interactions causes a

remarkable cancellation of the four particle contribution and

lead to a very simple result: ½H;Pþ� ¼ EXPþ; where EX is

the exciton energy for a given shell. The commutation

relation enables the construction of exact eigenstates lNl ¼ðPþÞN lvl of the Hamiltonian by a multiple application of Pþ

on vacuum. The energy of these states depends linearly on

the number of excitons. Hence the energy of addition/

subtraction of excitons from these states does not depend on

the number of excitons N: This is the essence of the ‘hidden

symmetry’, a quantum-dot analog of hidden symmetries in

the QHE [3].

As pointed out by MacDonald and Rezayi [4] and Chen

and Quinn, [6] this hidden symmetry can be understood by a

mapping of the electron–hole system into a spin up/spin

down electron system. The symmetric interactions translate

into the spin-independent electron–electron interactions.

The hidden symmetry is equivalent to the total spin being a

conserved quantity.

Due to Zeeman energy, in 2D systems in strong magnetic

field one can neglect electron spin. Since quantum dots

emulate high magnetic field effects in terms of orbital

degeneracies only, spin cannot be neglected. Spin starts

playing a role when we have more than one carrier of the

same type. This has been anticipating by constructing

another relevant operator Qþ creating singlet biexcitons:

[14,18]

Qþ ¼ 12

Xi;j

ðcþi# cþj" þ cþj# cþi" Þðhþi" hþj# þ hþj" hþi# Þ: ð3Þ

The singlet operator involves creation of pairs of

electrons and pairs of valence holes with total spin S ¼ 0;

a much more complex object. Rather surprisingly, this

biexciton operator was shown to satisfy a similar commu-

tation relation as the polarization operator ½H;Qþ� ¼

EXXQþ; with EXX ¼ 2EX : Hence, the application of Qþ to

the vacuum generates a coherent state of singlet biexcitons,

and the energy of this biexciton state is twice the energy of a

single exciton. The energy of the multiplicative state

ðPþÞ2lvl and the energy of the biexciton state Qþlvl are

degenerate. However, any small perturbation lowers the

energy of the singlet–singlet state with respect to the

triplet–triplet state. The biexciton operator involves pairing

of electrons and pairing of holes, and plays an important role

in determining ground state of the interacting electron–hole

system. The remainder of the paper is devoted to the

analysis of the biexciton operator.

We have given numerical evidence supporting the notion

of hidden symmetry associated with the biexciton operator

[14] and discussed it on a simple example of a p-shell [18].

We argue that the third shell, the d shell, is the most general

case, and provide here a detailed analysis for a d shell by

explicit construction of many-exciton states and diagonali-

zation of the Hamiltonian.

Let us first examine possible classes of biexciton

configurations on different shells. We start with the s shell

consisting of a single state ð0; 0Þ: This state can only be

doubly-occupied by electrons and holes, as shown in Fig. 1.

In a second, p shell, consisting of ð1; 0Þ; ð0; 1Þ states we have

doubly occupied as well as singly occupied states. Moving

to the d shell consisting of ð2; 0Þ; ð1; 1Þ; ð0; 2Þ states allows

a new type of configuration, where electrons (holes) are in a

doubly occupied configuration and holes (electrons) are in a

singly occupied configuration. While the first two classes of

configurations ensure charge neutrality, the third class of

configurations does not. By moving to higher shells we

encounter only these three classes of configurations: doubly

occupied, singly occupied, and doubly/singly occupied. It is

clear that the lowest shell where all possible configurations

are encountered is the d shell. For this reason we focus on

detailed calculations of electrons and holes on a d shell.

Let us denote the three single particle states as lal ¼ l2; 0l;lbl ¼ l0; 2l; and lcl ¼ l1; 1l: We occupy these configurations

with two electrons and two holes, ensure that the total angular

momentum L ¼ ðme1 2 ne1Þ þ ðme2 2 ne2Þ2 ðmh1 2

nh1Þ2 ðmh2 2 nh2Þ equals zero, and construct wave func-

tions as a product of the singlet electron and singlet hole wave

function. In this way we can construct a total of three doubly-

occupied, three singly-occupied, and two doubly/singly

occupied configurations, for a total of NC ¼ 8 configurations

Fig. 1. Different types of possible biexciton configurations with

increasing shell index.

P. Hawrylak / Solid State Communications 127 (2003) 793–798 795

shown in Fig. 2. Below we explicitly write down the examples

of the three types of configurations:

l1l ¼ laaaal ¼ ðcþ20"cþ20#Þðh

þ20#h

þ20"Þl0l; ð4Þ

l2l ¼ lababl ¼1ffiffi2

p ðcþ20"cþ02# þ cþ02"c

þ20#Þ

1ffiffi2

p ðhþ20#hþ02"

þ hþ02#hþ20"Þl0l;

l7l ¼ lccabl ¼ ðcþ11"cþ11#Þ

1ffiffi2

p ðhþ20#hþ02" þ hþ02#h

þ20"Þl0l:

We see that the three types of configurations are normal-

ized differently. These configurations are mixed by the

interaction among electrons and holes. For example, we can

evaluate the matrix element k1lHl2l: The coupling between

the two configurations moves one electron–hole pair from

orbital a to orbital b. There are four possible contributions from

the singlet–singlet biexciton states. Two are cancelled by the

normalization constant, and the final result is given by

k1lHl2l ¼ 2ð2Þk20; 20lVehl02; 02l: But the electron–hole

scattering k20; 20lVehl02; 02l equals k20; 02lVeel20; 02lwhich we denote by Vab and k1lHl2l ¼ 22Vab: Note that

the electron–hole scattering is responsible for the coupling

and the matrix element is negative. To calculate matrix

element between configurations l1l and l7l we note that it

requires us to move more then two particles, and hence

k1lHl7l ¼ 0: However, configurations l2l and l7l have

identical hole configurations and require scattering of

electrons. The matrix element is k2lHl7l ¼ffiffi2

pðþÞk20; 02lVeel11; 11l ¼

ffiffi2

pVccab where plus sign comes

from scattering of like particles andffiffi2

pfrom normalization of

configurations.

In this way we can construct all eight configurations and

build the 8 £ 8 biexciton Hamiltonian HXX :

The Hamiltonian is organized in such a way that the left

upper corner corresponds to the doubly and singly occupied

charge neutral configurations (1–6) while the bottom right

corner corresponds to doubly/singly occupied charged

configurations (7–8). The matrix is not very transparent

and can be simplified further by forcing all configurations to

be charge neutral with identical electron–hole and exchange

interactions. In this way we are only left with one parameter

V which describes both electron–hole scattering and

electron–electron exchange between any of the orbitals of

the degenerate shell. Such simplified Hamiltonian H0 with

energy measured from doubly occupied configuration, is

given below:

H0 ¼

0 22V 0 22V 0 0 0 0

22V 2V 22V 2V 0 2Vffiffi2

pV

ffiffi2

pV

0 22V 0 0 0 22V 0 0

22V 2V 0 2V 22V 2V 2ffiffi2

pV 2

ffiffi2

pV

0 0 0 22V 0 22Vffiffi2

pV

ffiffi2

pV

0 2V 22V 2V 22V 2V 2ffiffi2

pV 2

ffiffi2

pV

0ffiffi2

pV 0 2

ffiffi2

pV

ffiffi2

pV 2

ffiffi2

pV V 0

0ffiffi2

pV 0 2

ffiffi2

pV

ffiffi2

pV 2

ffiffi2

pV 0 V

266666666666666666664

377777777777777777775

:

ð6ÞFig. 2. All possible biexciton singlet–singlet configurations for the

d shell.

HXX ¼

Eaaaa 22Vab 0 22Vca 0 0 0 0

22Vab Eabab 22Vab 2Vca 0 2Vca

ffiffi2

pVccab

ffiffi2

pVccab

0 22Vab Ebbbb 0 0 22Vca 0 0

22Vca 2Vca 0 Eacac 22Vca 2Vca 2ffiffi2

pVccab 2

ffiffi2

pVccab

0 0 0 22Vca Ecccc 22Vca

ffiffi2

pVccab

ffiffi2

pVccab

0 2Vca 22Vca 2Vca 22Vca Ebcbc 2ffiffi2

pVccab 2

ffiffi2

pVccab

0ffiffi2

pVccab 0 2

ffiffi2

pVccab

ffiffi2

pVccab 2

ffiffi2

pVccab Eccab 0

0ffiffi2

pVccab 0 2

ffiffi2

pVccab

ffiffi2

pVccab 2

ffiffi2

pVccab 0 Eabcc

266666666666666666664

377777777777777777775

: ð5Þ

P. Hawrylak / Solid State Communications 127 (2003) 793–798796

The purpose of writing this Hamiltonian explicitly is to

observe directly that for all six rows (1–6) the Hamiltonian

satisfies the following relation:P

j¼1;2;…;6 Hi;j ¼ 24V : This

relation immediately implies that there is an eigenvector of

Hamiltonian 6 in the space of doubly and singly occupied

configurations of the form lGSl ¼ 1=ffiffi6

pð1; 1; 1; 1; 1; 1; 0; 0Þ

with eigenvalue EGS ¼ 24V : By direct inspection, this

state, apart from the normalization constant, is equal to the

multiplicative state lGSl ¼ Qþl0l: In Fig. 3a we show

the energies of configurations of H0 (left column) and the

eigenvalues of H0 (right column). In Fig. 3b we show the

amplitude of all NC ¼ 8 configurations in the ground state.

The energy spectrum of H0 consists of three bands with

energy and degeneracy 0 £ V (3), 1 £ V (2) and 2 £ V (3).

By contrast, the spectrum of interacting H0 has a well

isolated ground state which is exactly the one calculated

analytically using the Qþ operator. Let us now turn to the

spectrum of the biexciton on the d shell. We calculate [27]

the following interaction strengths and energies measured

from Eaaaa ¼ Ebbbb : Vab ¼ 0:1025; Vca ¼ Vccab ¼ 0:1113;

Ecccc ¼ 20:0527; Ecaca ¼ 0:1963; Eccab ¼ 0:2813: These

values, except for the energies of doubly/singly occupied

configurations, are very close to the ones that enter H0:

In Fig. 3c we show the calculated spectrum of HXX ; and in

Fig. 3d we show amplitudes of each configurations in the

ground state.

The energy spectrum, measured from configuration (1) is

shown in Fig. 3c before and after mixing of configurations is

allowed. We see that without mixing there are three low

energy states, corresponding to singlets occupying each of

the three orbitals, i.e. configurations (1), (3), and (5). The

lowest energy configuration is (5) as it corresponds to

singlet on zero angular momentum state ð1; 1Þ which

maximizes electron–hole attraction. The higher energy

band at 0.2, corresponding to singly occupied singlet

configurations (2), (4) and (5), is shifted up by twice the

repulsive exchange energy (singlet configurations). Finally,

the highest energy band corresponds to doubly/singly

occupied configurations (7) and (8). The blueshift of energy

of these configurations is due to both repulsive exchange

terms as well as due to lack of local charge neutrality. The

correlated spectrum, i.e. the one obtained after mixing of

configurations has been allowed, is much broader in energy,

with one well-isolated ground state. The amplitudes Ai of

the ground state wave function lGSl <P

i¼1;6 Ailil are

shown in Fig. 3. We see that these coefficients are almost

exactly equal to 1=ffiffi6

p¼ 0:408 for the first six configur-

ations, and almost zero for the last two configurations. There

is a small enhancement of the amplitude corresponding to

configuration (5) as the l1; 1l orbital leads to slightly

different Coulomb matrix elements. The overlap of the

ground state and the state created by operator Qþ equals

0.996. When pairs of holes shadow pairs of electrons, they

cannot be scattered in or out of this coherent state, and the

state remains a ground state. By the same analysis we can

construct a second and third multiplicative state ðQþÞ2l0land ðQþÞ3l0l as ground states of the four and six excitons on

the d shell.

To summarize, we discussed here the hidden symmetry

and its application to the construction of exact correlated

states of electrons and holes in quantum dots. We focused on

partially filled electronic shells and showed the conden-

sation of electrons and holes into coherent multiplicative

states due to relevant symmetries. We identified the relevant

‘hidden symmetries’ and showed that they replace Hund’s

rules in two-component systems. The third shell discussed

here in detail illustrates nicely a rather unusual pairing of

electrons and holes.

The signatures of hidden symmetries in a doubly-

degenerate shell were used as a fingerprint of excitonic

artificial atoms observed in ‘single dot spectroscopy’

experiments by Bayer et al. [26]. A challenge would be to

extend these measurements to higher shells to probe directly

with electron–hole pairing.

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