Bose–Einstein condensation of excitons in ideal two-dimensional system in a strong magnetic field

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Physica E 19 (2003) 278–288

www.elsevier.com/locate/physe

Bose–Einstein condensation of excitons in idealtwo-dimensional system in a strong magnetic %eld�

S.A. Moskalenkoa ;∗, M.A. Libermanb, D.W. Snokec, V.V. Bot0anb, B. Johanssonb

aInstitute of Applied Physics of the Academy of Sciences of Moldova, Academic str. 5, Kishinev MD2028, Republic of MoldovabDepartment of Physics, Uppsala University, Box 530, SE-751 21 Uppsala, Sweden

cUniversity of Pittsburgh, 405 Allen Hall, 3941 O’Hara, St., PA 15260, USA

Received 13 December 2002; received in revised form 27 March 2003; accepted 1 April 2003

Abstract

We present theoretical study of ideal two-dimensional electron–hole (e–h) system in a strong magnetic %eld. The Bose–Einstein condensation of the correlated pairs takes place on a single particle state with an arbitrary wave vector k in asymmetric two-dimensional model. We show that the ground state energy per one exciton and the chemical potential at lowexciton damping rates are nonmonotonic functions versus the value of the %lling factor, and they form metastable statesof dielectric liquid phase with positive compressibility consisting of the Bose–Einstein condensate of magnetoexcitons andliquid drops. Since the dielectric liquid phase of the Bose condensed excitons with low damping rate corresponds to therelative minima of chemical potential, it is more stable than the e–h metallic liquid phase.? 2003 Elsevier B.V. All rights reserved.

PACS: 71.35.Ji; 71.35.Lk; 71.35.Ee

Keywords: Magnetoexcitons; Bose–Einstein condensation

1. Introduction

The observation of Bose–Einstein condensation(BEC) in atomic alkali and hydrogen gases usinga laser and magnetic trapping [1,2] has greatly ex-panded the related research in recent years. As is wellknown, under certain conditions excitons, i.e. boundstates of electron–hole (e–h) pairs in semiconductorshave bosonic properties [3]. Although theoreticallyrecognized many years ago [3], experiments on BEC

∗ Corresponding author.1 This research was supported by the Wenner-Gren Founda-

tion, by the Swedish Royal Academy of Sciences and by theCRDF-MRDA grant.

of excitons have made slow progress, because %nitelifetime e>ects, strong interactions between excitonsat high density, crystal imperfections and phonons inthe crystal all act to complicate the system. In re-cent years, the system of coupled quantum wells in astrong magnetic %eld has gained attention as a systemwith repulsive exciton–exciton interactions and longexciton lifetime, ideal for observation of BEC of exci-tons. Another advantage of the two-dimensional (2D)system is a possibility of much faster cooling of hotphotoexcited excitons compared with their bulk coun-terparts [4,5]. It has been shown [6] that the propertiesof atoms and excitons are dramatically changed in astrong magnetic %eld such that the distance betweenLandau levels e˝H=mec exceeds the Rydberg energy.

1386-9477/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.doi:10.1016/S1386-9477(03)00229-7

S.A. Moskalenko et al. / Physica E 19 (2003) 278–288 279

The diamagnetic excitons in bulk crystals were re-vealed in Ref. [7], and possibility of their BEC wasstudied in Ref. [8]. Even more attractive and worthinvestigating is the e–h system in two dimensions inthe presence of a strong perpendicular magnetic %eld.In the later case the energy spectrum of e–h systemis completely discrete and characterized by the num-ber of the Landau levels, which are N -fold degener-ated with N = S=2l2, where l is the magnetic length,l2 = ˝c=eH , and S is the 2D sample dimension.In the past two decades, a number of experimental

[9–12] and theoretical [13–16] e>orts have been dedi-cated to the study of 2D systems in a strong magnetic%eld. In [13–15] the coherent pairing of electronsand holes resulting in the formation of the Bose–Einstein condensate of excitons in a single-particlestate with wave vector k = 0 was studied. Evenbeyond the Hartree–Fock approximation, it was pos-sible to obtain an exact solution, if coupling to thehigher Landau levels and the corresponding correla-tion energy can be neglected. In this case, the mag-netoexcitons with k = 0 represent at T = 0 an idealexcitonic gas. A surprising result is that e–h dropletsof the metallic electron–hole liquid (EHL) with themaximal local %lling factor of the lowest Landau level(LLL) can be considered as an aggregate of excitonssticked together. The coupling to higher Landau lev-els makes the system weakly nonideal [13,14], whichallows the Berezinskii–Kosterlitz–Thouless topologi-cal phase transition [17–19] at %nite temperature.The results obtained in Refs. [13–15] were re-

produced in Ref. [16] on the basis of more simpleand transparent approach using the BCS-type wavefunctions of the BEC excitons and calculating theground state energy in the Hartree–Fock–Bogoliubovapproximation. In Ref. [16] the case of nonzero wavevectors k �= 0 was considered and the indirect inter-action of the particles on the LLL due to their virtualexcitation to excited Landau levels was taken intoconsideration.The aim of the present paper is to investigate

properties of the system beyond the Hartree–Fock–Bogoliubov approximation, taking into account thepossibility of the Anderson-type coherent excitedstates, the corresponding polarizability of the Bose–Einstein condensate, the screening e>ects and thecorrelation energy. In the present study we take intoaccount the %rst excited Landau level (FELL). We

found that a metastable dielectric liquid phase formedby Bose–Einstein condensed magnetoexcitons withk �= 0 may exist and it is more stable than anotherpossible state of droplet of EHL.

2. Hamiltonian of the ideal 2D system

The Hamiltonian describing the e–h system in anideal 2D layer in a strong perpendicular magnetic %eldwas derived in Refs. [15,16] in a second quantizationrepresentation. Here we will use a more simple andconcrete form taking into account only: the LLL withquantum number n = 0 and the %rst excited Landaulevel with n=1. We introduce the chemical potentialsfor electrons and holes �e and �h, respectively. Theoperators N e and N h for the whole numbers of elec-trons and holes are expressed through the creation andannihilation operators a+p ; ap and c+p ; cp for electronson the LLL and FELL correspondingly and throughthe operators b+p ; bp and d+

p ; dp for holes on the stateswith n= 0 and n= 1 correspondingly

N e =∑p

a+pap +∑p

c+pcp;

N h =∑p

b+pbp +∑p

d+p dp: (1)

The whole Hamiltonian H consists of three parts:

H= H0 + HLLLCoul + H FELL

Coul : (2)

The zero-order Hamiltonian H0 contains the cyclotronfrequencies !ce and !ch, which are supposed to begreater than the exciton ionization potential, and hasthe form

H0 =∑p

˝!cec+pcp +∑p

˝!chd+p dp

− �eN e − �hN h : (3)

The Coulomb interaction of electrons and holes situ-ated on the LLL is denoted as HLLL

Coul

HLLLCoul =

12

∑p;q; s

Fe–e(p; 0; q; 0;p− s; 0; q+ s; 0)

×a+pa+q aq+sap−s

280 S.A. Moskalenko et al. / Physica E 19 (2003) 278–288

+12

∑p;q; s

Fh–h(p; 0; q; 0;p− s; 0; q+ s; 0)

×b+pb+q bq+sbp−s

−∑p;q; s

Fe–h(p; 0; q; 0;p− s; 0; q+ s; 0)

×a+pb+q bq+sap−s: (4)

The Hamiltonian H FELLCoul describes the Coulomb scat-

tering processes when two particles from the LLL un-dergo the quantum transitions to FELL and vice versa

H FELLCoul =

12

∑p;q; s

Fe–e(p; 0; q; 0;p− s; 1; q+ s; 1)

×a+pa+q cq+scp−s

+12

∑p;q; s

Fh–h(p; 0; q; 0;p− s; 1; q+ s; 1)

×b+pb+q dq+sdp−s

−∑p;q; s

Fe–h(p; 0; q; 0;p− s; 1; q+ s; 1)

×a+pb+q dq+scp−s + h:c: (5)

Such transitions conserve the total numbers of elec-trons and holes and the operators (4) and (5) com-mute with the operators (1). We restrict ourselves onlywith these virtual transitions, because they lead to theindirect supplementary interaction between the parti-cles on the LLL.The Coulomb matrix elements in theHamiltonians (4) and (5) are calculated in the Landaugauge using the wave functions of 2D electrons andholes in a strong magnetic %eld

Fi–i(p; 0; q; 0;p− s; n; q+ s; n)

=∑�

Vs;�[(� − is)2l2=2]n

×exp[− (s2 + �2)l2=2 + i�(p− q− s)l2];

Fe–h(p; 0; q; 0;p− s; n; q+ s; n)

=∑�

Vs;�[(�2 + s2)l2=2]n

×exp[− (s2 + �2)l2=2 + i�(p+ q)l2];

i; j = e; h; n= 0; 1; (6)

where Vs;� is the 2D Fourier transform of the Coulombpotential

Vs;� =2e2

�0S√�2 + s2

; (7)

where �0 is the background dielectric constant. The inplane uni-dimensional wave vectorsp; q; s are orientedperpendicular to the direction of Landau quantization.We introduce the unitary transformation

operator [20]

U = eiS ; S+ = S (8)

to exclude from the Hamiltonian (2) the term H FELLCoul ,

which is small at the %rst order of the perturbation the-ory. This term contributes in the second order of theperturbation theory and makes supplementary indirectinteraction between the particles on the LLL in ad-dition to the term HLLL

Coul . The corresponding e>ectiveHamiltonian [20] is

He> = FELL〈0|e−iSHeiS |0〉FELL≈−�e

∑p

a+pap − �h

∑p

b+pbp

+HLLLCoul +

i2FELL〈0|[H FELL

Coul ; S]|0〉FELL; (9)

where the operator S satis%es the condition

H FELLCoul + i[H 0; S] = 0; (10)

and the vacuum state of the FELL is determined bythe equalities

cp|0〉FELL = dp|0〉FELL = 0: (11)


The transformed Hamiltonian has the form

He> =−�e

∑p

a+pap − �h

∑p

b+pbp + HLLLCoul

− 12

∑p;q; s

�e–e(p; q;p− s; q+ s)a+pa+q aq+sap−s

− 12

∑p;q; s

�h–h(p; q;p− s; q+ s)b+pb+q bq+sbp−s

−∑p;q; s

�e–h(p; q;p− s; q+ s)a+pb+q bq+sap−s;

(12)

where matrix elements �i–j are

�i–j(p; q;p− s; q+ s)

=1

˝!ci + ˝!cj

×∑t

Fi–j(p; 0; q; 0;p− t; 1; q+ t; 1)

×Fi–j(p− t; 1; q+ t; 1;p− s; 0; q+ s; 0):(13)

The diagonal Coulomb matrix elements of theHamiltonians (4) and (5) satisfy the electroneutralitycondition

12

∑q

Fe–e(p; 0; q; 0;p; n; q; n)

+12

∑q

Fh–h(p; 0; q; 0;p; n; q; n)

−∑q

Fe–h(p; 0; q; 0;p; n; q; n)

=0; n= 0; 1: (14)

The exciton ionization potential is

Iex(k) =∑s

Fe–h(p; 0; kx − p; 0;p

− s; 0; kx − p+ s; 0)eikysl2

= Ile−k2l2=4I0

(k2l2

4

); Il =

e2

�0l

√2: (15)

Here I0(z) is modi%ed Bessel function. Note that theexciton energy depends on magnetic length and is in-dependent of the original 2D band structure.Contraryto the initial Coulomb matrix elements (6), the matrixelements �i–j do not satisfy the electroneutrality con-dition (14). The supplementary indirect interaction isattractive and the sums over these nondiagonal matrixelements are

∑s

�i–i(p;p− s;p− s; p) =(Il)2

4˝!ciA2;22 ;

∑s

�e–h(p; kx − p;p− s; kx − p+ s)eikysl2

=(Il)2

2(˝!ce + ˝!ch)(A(kl))2: (16)

They are expressed through full elliptic integrals ofthe %rst kind K(m2) and of the second kind E(m2),where m2 is the square modulus, and degeneratehypergeometric function 1F1(a; b; z)

A2;22 =

∫ ∞

0

∫ ∞

0x2y2exp

[−x2 + y2

2

]J2(xy) dx dy

=5K(1=2)− 6E(1=2)

2√2

;

A(kl) =

√2e−k2l2=2

1F1

(−12; 1;

k2l2

2

): (17)

Though the e>ective Hamiltonian (12) involves ex-plicitly only the LLL, it reJects the inJuence of theFELL. This inJuence is equivalent to the correlationenergy discussed in Refs. [13,14], which is related tothe electron and hole excited states, namely, to theexcited Landau levels. Below we will consider thecorrelation energy due to coherent excited states ofthe Bose–Einstein condensed magnetoexcitons.

3. The coherent pairing of electrons and holes.The Hartree–Fock–Bogoliubov approximation

The coherent pairing of electrons and holes in asingle-exciton state with %nite wave vector k is stud-ied using the Keldysh–Kozlov–Kopaev method [21].The coherent macroscopic state is introduced into


Hamiltonian by the unitary transformation operator

D(√

Nex) = exp[√

Nex(d+k − dk)]; (18)

where d+k ; dk are the exciton creation and annihilation

operators [15,16]:

d+k =

1√N

∑t

e−ikytl2a+kx=2+tb+kx=2−t : (19)

Here√Nex is the amplitude of the coherent macro-

scopic state, and k is a quantum number describingexciton translational motion, with components kx andky. In the Landau gauge ky also characterizes excitoninternal bound state of relative motion, which forms acontinuous energy spectrum [22]. The correspondingunitary transformation of the operators ap; bp is

DapD+ = )p = uap − v

(p− kx

2

)b+kx−p;

DbpD+ = ,p = ubp + v

(kx2

− p)

a+kx−p; (20)

where the coeKcients are

v(t) = ve−ikytl2 ; v= sin(√2l2nex);

u= cos(√2l2nex);

u2 + v2 = 1; nex =Nex

S=

v2

2l2;

Nex = N · v2: (21)

The applicability of the theory due to the restrictionof the LLL can be expressed as

v2 ≈ sin2(v): (22)

The transformed Hamiltonian He> =DHe>D+ must

be expressed in terms of new operators )+p ; )p; ,+p ; ,p

using the inverse Bogoliubov u; v transformation

ap = u)p + v(p− kx

2

),+kx−p;

bp = u,p − v(kx2

− p)

)+kx−p: (23)

The BCS-type wave function |-g(k)〉 of the new co-herent macroscopic state is

|-g(k)〉=D(√

Nex)|0〉; (24)

where |0〉 is the vacuum state for the operators ap andbp, whereas |-g(k)〉 is the vacuum state for the newFermi operators )p and ,p

ap|0〉= bp|0〉= 0;

)p|-g(k)〉= ,p|-g(k)〉= 0: (25)

Expanding the Hamiltonian He> in terms of the newoperators after their normal ordering we obtain

He> = U + H2 + H ′: (26)

The %rst term U does not contain the operators andplay the role of the ground state energy. The secondterm is quadratic in the operators )+p ; )p and ,+

p ; ,p.They appear after the transposition inside the initialquaternion forms. The Hamiltonian H2 contains thediagonal and nondiagonal quadratic terms which aregiven below, for the sake of simplicity, in the case ofelectrons and holes with the same masses me=mh, cy-clotron frequencies !ce =!ch and chemical potentials�e = �h = �=2:

H2 =∑p

E(k; v2; �)()+p)p + ,+p,p)

−∑p

[uv

(kx2

− p)

(k; v2; �),kx−p)p

+ uv(p− kx

2

) (k; v2; �))+p,

+kx−p

]: (27)

The term H ′ contains the remaining normal-orderedterms with four operators, which is treated as perturba-tion. The quadratic form H2 contains the nondiagonalterms, which describe the spontaneous creation of thepair of new Fermi quasiparticles from vacuum stateand their annihilation. Such terms in the new Hamil-tonian He> are named dangerous and they lead to theinstability of the new ground state (24). To avoid thisinstability we use the condition of the compensationof dangerous diagrams of H2, which gives us chemi-cal potential in the Hartree–Fock–Bogoliubov approx-imation and energy spectrum E(k; v2; �) of the singleparticle elementary excitations. Taking into account


Eqs. (15)–(17) we obtain

�HFB =−I ex(k)− 2v2(Il − I ex(k))

+v2I 2l

2˝!c(A2;2

2 − 1)

=−I ex(k)− 2v2(Il − Iex(k))

+v2I 2l

2˝!c(A2(kl) + A2;2

2 − 1); (28)

and

E(k; v2; �) =I ex(k)2

;

I ex(k) = Iex(k) +I 2l

4˝!cA2(kl): (29)

When the BEC of magnetoexcitons takes place on thestate with k=0, then the concentration corrections tothe chemical potential are positive. This e>ect is dueto FELL and it corresponds to the positive compress-ibility and to stability of the system. When the wavevector k increases function A(kl) tends to zero andthe concentration corrections to the chemical potentialexpressed by the last term in Eq. (28) becomes neg-ative as well as the second term. The system remainsunstable in the HFBA, as earlier, when the FELL wasnot taken into account. The energy of a single particleelementary excitation (29) is equal to one half of theexciton e>ective ionization potential I ex(k), becausean e–h pair can appear due to the unbinding of themagnetoexciton. The energy spectrum of the elemen-tary excitations does not depend on the wave vectorp, because the kinetic energy is absent on the LLL.The mean value of the energy per one exciton in

the state of BEC di>ers from the chemical potential(28), which is nothing but the necessary energy foradding one more exciton. Below we will determinealso the energy per one e–h pair in the EHL and in theelectron-hole drop (EHD). In both cases we start withthe e>ective Hamiltonian of the Coulomb interaction(12), but without chemical potentials �e and �h

He> =12

∑p;q; s

Fe–e(p; q;p− s; q+ s)a+pa+q aq+sap−s

+12

∑p;q; s

Fh–h(p; q;p− s; q+ s)b+pb+q bq+sbp−s

−∑p;q; s

Fe–h(p; q;p−s; q+s)a+pb+q bq+sap−s;

(30)

where the matrix elements

F i–i(p; q;p− s; q+ s)

= Fi–i(p; 0; q; 0;p− s; 0; q+ s; 0)

−�i–i(p; q;p− s; q+ s); i = e; h;

Fe–h(p; q;p− s; q+ s)

=Fe–h(p; 0; q; 0;p− s; 0; q+ s; 0)

+�e–h(p; q;p− s; q+ s); (31)

are determined by the expressions (6) and (13). In thecase of BEC the average of the Hamiltonian (30) istaken on the BCS-type wave function (24), whereasin the case of EHL it is taken on the Fermi degener-ated states of electrons and holes on the LLL, but inboth cases we use HF approximation. For the BEC ofmagnetoexcitons the averages are

〈-g(k)|a+pap|-g(k)〉= 〈-g(k)|b+pbp|-g(k)〉= v2

〈-g(k)|a+pb+q |-g(k)〉

=/kr(q; kx − p)uv∗(p− kx

2

): (32)

Using Wick’s theorem we obtain

〈-g(k)|a+pb+q bq+sap−s|-g(k)〉

=/kr(s; 0)v4 + /kr(q; kx − p)u2v2eikysl2; (33)

and the ground state energy is

Eg(k) = 〈-g(k)|He> |-g(k)〉=−Nv2 I ex(k)− Nv4(Il − Iex(k))

+Nv4I 2l

4˝!c(A2;2

2 − 1 + A2(kl)): (34)

The derivative dEg(k)=dNex determines the chemicalpotential �HFB in full accordance with (28), whereas


the rate Eg(k)=Nex is the mean energy per one excitonin the BEC state

Eg(k)Nex

=−I(k)− v2(Il − Iex(k))

+ v2I 2l

4˝!c(A2;2

2 − 1 + A2(kl)): (35)

In the case of EHL the electrons and holes at T = 0are Fermi degenerated on the LLL and their averagevalues are

〈a+pap〉= 〈b+pbp〉= v2; (36)

where v2 = Ne–h=N is the %lling factor of the LLL.Applying the Wick’s theorem we obtain

E=EEHL

Ne–h=−v2Il + v2

I 2l4˝!c

(A2;22 − 1): (37)

The minimal value of this energy is achieved at the%lling factor v2 = 1, which determines the density ofthe EHD. One can see that the corrections due to FELLin Eqs. (35) and (37) lower the energy per one pair ofEHL more than the energy per one exciton of BEC.We show below that the taking into account the cor-relation energy makes the ground state of the BECof magnetoexcitons more stable than that one of EHLand EHD.

4. Coherent excited states and correlation energy

We will study the BEC beyond the Hartree–Fock–Bogoliubov approximation taking into account theAnderson type coherent excited states. Here wedetermine the correlation energy due to the coherentexcited states on the base of Hamiltonian (12) neg-electing the corrections �i–j to the Coulomb potentialdue to FELL. The wave functions of the coherentexcited states are written following the method pro-posed by Anderson in the theory of superconductivity[23] and using density Juctuation operator 1Q

1Q =∑t

eiQytl2

×(a+t−Qx=2at+Qx=2 − b+−t−Qx=2b−t+Qx=2): (38)

It describes the creation of one excited e–h pair withresultant wave vector kx + Qx from the BEC. The

coherent excited state wave function is

|n〉=∣∣∣∣-e

(p± Px

2

)⟩

=1uv

a+p+Px=2ap−Px=2|-g(k)〉

=1vv(p− Px

2− kx

2

)

×)+p+Px=2,+kx−p+Px=2|-g(k)〉: (39)

They form a normalized and orthogonalized full setof wave functions⟨-e

(p± Px

2

) ∣∣∣∣-e(q± Qx

2

)⟩

=/kr(Px; Qx)/kr(p; q): (40)

The excitation energy in the HFB approximationcan be obtained using expression (27) for theHamiltonian H2

E(p± Px

2

)=

⟨-e

(p± Px

2

)∣∣∣∣H2

∣∣∣∣-e(p± Px

2

)⟩

= Iex(k): (41)

The excitation energy is the same for all coherent ex-cited states and does not depend on the wave vectorsp and Px, i.e. has no dispersion. For convenience weuse the following notation:

|0〉= |-g(k)〉; ˝!n0 = E(p± Px

2

): (42)

The matrix elements of the density Juctuation operator1Q (38) are

(1+Q)n;0 = (1−Q)n;0

= uv/kr(Px; Qx)e−iQypl2 [1− e−i(kyQx−kxQy)l2 ];

|(1+Q)n;0|2 = |(1−Q)n;0|2 = 4u2v2/kr(Px; Qx)

×sin2[(kyQx − kxQy)l2=2]: (43)

The last expression does not depend on the wave vec-tor p and contains the coherence factor sin2[(kyQx −kxQy)l2=2]. The polarizability 4)HF0 (Q; !) of theBose–Einstein condensed system can be found in ap-proximation of weak response to external longitudinalperturbation, characterized by a wave vector Q and


frequency !. In the HFBA the general expression forthe polarizability is

4)HF0 (Q; !)

=− WQ

˝∑n

[|(1+

Q)n;0|2!− !n;0 + i/

− |(1Q)n;0|2!+ !n;0 + i/

]; (44)

where WQ=VQ exp(−Q2l2=2). After straightforwardcalculations we obtain

4)HF0 (Q; !)

=/→+0

− 4u2v2WQN sin2(kyQx − kxQy

2l2)

×[

1˝!− Iex(k) + i/

− 1˝!+ Iex(k) + i/

]: (45)

The polarizability has a resonance frequency equal tothe ionization potential Iex(k) of the magnetoexcitonwith wave vector k, because the only excitation mech-anism involved into the polarizability is the unbindingof one e–h pair from the condensate. It does not con-tain the small parameter Il=˝!c related with FELL.The polarizability vanishes when the wave vectork tends to zero and the magnetoexcitons behave asan ideal noninteracting gas in the HFBA, when thecorrections due to the FELL are neglected. Thepolarizability is an anisotropic function of the wavevector Q, periodically depends on Q at its smallvalues and decreases exponentially when Q goes toin%nity. To avoid singularity in expression (45) weintroduced phenomenologically the damping rate 4of the exciton level by changing in%nitesimal value/ → +0 by 4¿ 0 in Eq. (45). Then the real andimaginary parts of the polarizability can be written as

4)HF0 (Q; !) = 4)HF0;1(Q; !) + i4)HF0;2(Q; !)

=−4u2v2WQN sin2(kyQx − kxQy

2l2)

·(6(!; k; 4) + i7(!; k; 4)); (46)

where the dispersion and absorption parts are

6(!; k; 4) =[

˝!− Iex(k)(˝!− Iex(k))2 + 42

− ˝!+ Iex(k)(˝!+ Iex(k))2 + 42

]

7(!; k; 4) = 4[

1(˝!+ Iex(k))2 + 42

− 1(˝!− Iex(k))2 + 42

]: (47)

The damping rate 4 is supposed to be independent onk, but in reality it depends essentially on the excitonscattering mechanism.We can use obtained expression (46) to calculate

the dielectric constant �(Q; !) in the random phaseapproximation (RPA) and in the HF approximation:

�RPA(Q; !) = 1 + 4)HF0 (Q; !);

1�HF(Q; !)

= 1− 4)HF0 (Q; !): (48)

This allows us to calculate the correlation energy of theBose–Einstein condensed excitons using the method[24]. This method considers simultaneously the bind-ing processes and screening e>ects. This formalism,known as generalized RPA, is based on the Pauli–Feynman theorem [24,25] for the ground state energy,which in our case has the form

Eg =∫ e2

0Eint(8)

d88

; (49)

where Eg is the ground state energy of the Hamiltonian(4) and Eint(8) is a mean value of the Coulomb inter-action HLLL

Coul(8) with 8 being the hypothetical squaredelectric charge, which changes from zero to real valuee2. Here the kinetic energy is absent and the ELL isneglected. The expression Eint(8) takes the form

Eint(8) =−Nex

∑Q

WQ(8)

+12

∑Q

WQ(8)∑n(8)

|(1+Q)n;0|2; (50)

which can be expressed through the imaginary partof the polarizability 4)HF0 (Q; !) obtained in the HF


approximation∫ ∞

0

˝ d!2

Im 4)HF0 (Q; !)

=−∫ ∞

0

˝ d!2

Im1

�HF(Q; !; 8)

=12WQ

∑n(8)

|(1+Q)n;0|2: (51)

It was proposed in Ref. [24] to calculate the groundstate energy using expression

Eg =−Nex

∑Q

WQ

−∑Q

∫ ∞

0

˝ d!2

∫ e2

0

d88Im

1�(Q; !; 8)

: (52)

The choice of the approximation for �(Q; !; 8) deter-mines the accuracy of the energy Eg given by Eq. (52).When �(Q; !; 8) is taken in the form �HF(Q; !; 8) weobtain the results equivalent to (28) without the cor-rection due to FELL. Taking �RPA(Q; !; 8) instead of�HF(Q; !; 8) we obtain

�(Q; !; 8) = �1(Q; !; 8) + i�2(Q; !; 8);

�1(Q; !; 8) = 1 + 4)HF0;1(Q; !; 8);

�2(Q; !; 8) = 4)HF0;2(Q; !; 8): (53)

Supposing �2 about 8, �1 about 1 and �2(Q; !; 8)¡�1(Q; !; 8), we obtain

∫ e2

0

d88Im

1�(Q; !; 8)

≈ − �2(Q; !)�21(Q; !)

: (54)

Then we substitute Eq. (53) into Eq. (54) and after ex-panding in a power series we obtain (4)HF0;2(Q; !)−2·4)HF0;2(Q; !)4)HF0;1(Q; !)), where the %rst term cor-responds to HFB approximation, and the second termdetermines the correlation energy

Ecorr =−2∑Q

∫ ∞

0

˝ d!2

×4)HF0;2(Q; !)4)HF0;1(Q; !): (55)

Substituting Eq. (46) into (55) we obtain the correla-tion energy in the form

Ecorr =−N (u2v2)2√

I 2lIex(k)

F(kl)T (k; 4); (56)

where

T (k; 4) =− 2

4Iex(k)I 2ex(k) + 42

+2arctan

(Iex(k)

4

);

F(kl) = 3 + e−k2l2=2I0

(k2l2

2

)

− 4e−k2l2=8I0

(k2l2

8

);

Iex(k) = IlG(kl); G(kl) = e−k2l2=4I0

(k2l2

4

):

(57)

The correction to the chemical potential is

�corr =dEcorr

dNex

=− 2√

I 2l F(kl)Iex(k)

T (k; 4)v2

×(1− v2)(1− 2v2): (58)

One can see that the chemical potential being a di>er-ential value determines properly the energy needed tochange the number of excitons by one [26]. It di>ersessentially from the arithmetic mean value, which ismore rough characteristic. By this reason we study thechemical potential instead of the mean energy per oneexciton. We can rewrite the total chemical potentialwith corrections due to FELL (28) and with correla-tion corrections (58) in terms of the exciton ionizationpotential Il�Il=−G(kl)− 2v2(1− G(kl))

− 2√

F(kl)G(kl)

T (k; 4)v2(1− v2)(1− 2v2)

− Il4˝!c

A2(kl)

+Il

2˝!c[A2(kl) + A2;2

2 − 1]v2: (59)

The dependence of the correlation energy and of thecorresponding part of the chemical potential �corr on


0.0 0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

γ = 0.12Il

γ = 0.08Il

γ = 0.05Il

Dim

ensi

onle

ss c

hem

ical

pot

entia

l

Filling factor ν2

Fig. 1. Chemical potential of the Bose–Einstein condensed mag-netoexcitons at kl = 4:6 versus %lling factor v2. Solid line: thetotal value with corrections due to correlation energy, FELL anddamping rate 4= 0:05Il; dashed line: the same, but damping rateis 4 = 0:08Il; dotted line: at 4 = 0:12Il.

the wave vector k is governed by three factors. The%rst one is the coherence factor, which turns to bezero when k equals to zero and increases with the in-creasing of kl. The second factor is the denominatorIex(k), which tends to zero when kl increases. Thethird factor is T (k; 4), which is particularly importantin the range of large kl, where it decreases much fasterthan the denominator due to the dependence T (k; 4) ≈(Iex(k)=4)3, when Iex(k) is less than 4. The chemicalpotential as function of the %lling factor v2 is shownin Fig. 1 for the case of suKciently large wave vec-tor k and small damping rates. For large kl, it be-comes a nonmonotonous function of %lling factor v2

with a well-pronounced local minimum. The %rst lo-cal minimum appears for kl=2:8 and 4=0:05Il. Thisminimum becomes deeper and more pronounced withthe increase of the dipole moment kl2, due to the in-crease of the coherence factor and the decrease of theionization potential Iex(k), up to the moment whenthe ionization potential is approximately equal to thedamping rate. Further increase of the absolute valueof the chemical potential is suppressed by the factorT (k; 4), which reJects the fact that damping prevailsin the system. At larger values of the damping fac-tor, 4¿ 0:1Il, the minimum does not appear for anykl. The relative minimum of the chemical potential

implies the formation of a metastable dielectric liquidphase with positive compressibility in this range ofthe %lling factor v2. This state is more stable than thee–h metallic liquid state. For suKciently low dampingrates 4¡ 0:04Il this state becomes more stable thanthe Bose–Einstein condensed gas of magnetoexcitonswith k=0. The minimum of the chemical potential ofmagnetoexcitons with kl= 4:6 is lower than that onewith k =0 at the value of %lling factor v2 = 0:28. Themean distance between magnetoexcitons for the di-electric liquid drop d=

√2l=vm is of the same order

as the absolute value of the dipole moment 10 = kl2 atkl=4:6 and v2m = 0:28, which are presented in Fig. 1.At the same time the drop formation being a slowlyinhomogeneous in space distribution of excitons willgive rise to a small uncertainity of the condensate wavevector k depending on the drop parameters.

5. Conclusions

In conclusion, we have shown that a newmetastabledielectric liquid phase of Bose-condensed magnetoex-citons exists in a symmetric 2D quantum well system,with large dipole moment and low damping rate. Al-though textbook case of BEC does not yield a spatialcondensation when the system is translationally invari-ant, the unique properties of magnetic excitons meanthat the BEC state in this case will also have someproperties of a liquid, including droplets. This phaseis more stable than the electron–hole metallic liquidphase with maximal local concentration v2 = 1. Thecomparison with the state formed by Bose–Einsteincondensed excitons with the wave vector k=0 is alsoneeded, because they have a lower energy level onthe energy scale with maximal ionization potential. Inthis case the repulsion between excitons with k = 0prevails due the inJuence of the FELL and the gasof condensed magnetoexcitons coexists with dielectricliquid phase at considered exciton damping rates.

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Bose–Einstein condensation of excitons in ideal two-dimensional system in a strong magnetic field