Influence of Coulomb scattering of electrons and holes between ...

ARTICLE IN PRESSPhysica E 39 (2007) 137–149www.elsevier.com/locate/physeInfluence of Coulomb scattering of electrons and holes betweenLandau levels on energy spectrum and collectiveproperties of two-dimensional magnetoexcitonsS.A. Moskalenko a,1 , M.A. Liberman b , P.I. Khadzhi a , E.V. Dumanov a ,Ig.V. Podlesny a , V. Boţ an b,a Institute of Applied Physics of the Academy of Sciences of Moldova, Academiei 5, MD 2028 Chis-inău, Republic of Moldovab Department of Physics, Uppsala University, Box 530, SE 75121 Uppsala, SwedenReceived 3 July 2006; received in revised form 16 January 2007; accepted 25 February 2007Available online 6 March 2007AbstractThis study is concerned with a two-dimensional electron–hole system in a strong perpendicular magnetic field with special attentiondevoted to the influence of the virtual quantum transitions of interacting particles between the Landau levels. It is shown that virtualquantum transitions of two Coulomb interacting particles from the lowest Landau levels to excited Landau levels with arbitraryquantum numbers n and m and their transition back to the lowest Landau levels in the second order of the perturbation theory result inindirect attraction between the particles supplementary to their Coulomb interaction. The influence of this indirect interaction on thechemical potential of the Bose–Einstein condensed magnetoexcitons and on the ground state energy of the metallic-type electron–holeliquid (EHL) is investigated in the Hartree–Fock approximation. The supplementary electron–electron and hole–hole interactions beingaveraged with direct pairing of operators increases the binding energy of magnetoexciton and the energy per pair in the EHL phase. Theterms obtained in the exchange pairing of operators give rise to repulsion. Together with the Bogoliubov self-energy terms arising fromthe electron–hole supplementary interaction they both influence in the favor of BEC of magnetoexcitons with small momentum. Theinfluence of the excited exciton bands on the energy spectrum and on the wave function of the lowest magnetoexciton band is studied inthe second order of the perturbation theory. The BEC of magnetoexcitons in the superposition state is considered. The generalizedBogoliubov transformations, the BCS-type ground state wave function and the phase-space filling factors of the lowest and first excitedLandau levels are determined.r 2007 Elsevier B.V. All rights reserved.PACS: 71.35.Ji; 71.35.Lk; 71.35.EeKeywords: Magnetoexcitons; Bose–Einstein condensation1. IntroductionProperties of atoms and excitons are dramaticallychanged in strong magnetic fields, such that the distance Corresponding author. Fax: +46 18 4713524.E-mail address: vitalie.botan@fysik.uu.se (V. Boţ an).1 This research was supported by the common grant of the RussianFoundation for Fundamental Research and of the Academy of Sciences ofMoldova, as well as by the Swedish Royal Academy of Sciences. One ofthe authors (S.A.M.) gratefully acknowledges the hospitality of theDepartment of Physics of Uppsala University, where a part of this workhas been done.between Landau levels _o c , exceeds the corresponding pRydberg energies R y and the magnetic length l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi_c=eHis small compared to their Bohr radii [1,2]. Even moreinteresting phenomena are exhibited in the case of twodimensional(2D) electron systems due to the quenching ofthe kinetic energy at high magnetic fields, with therepresentative example being integer and fractional QuantumHall effects [3–5]. The discovery of the FQHE [6–8]changed fundamentally the established concepts aboutcharged elementary excitations in solids [5]. The notion ofthe incompressible quantum liquid (IQL) was introduced inRef. [7] as a homogeneous phase with the quantized1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physe.2007.02.004

ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149 141where V s;k is the 2D Fourier transform of the Coulombinteraction2pe 2V s;k ¼ p 0 Sffiffiffiffiffiffiffiffiffiffiffiffiffiffi . (9)s 2 þ k 2The application of these matrix elements for derivation ofthe lowest exciton bands will be shown in Section 4. Themain subject of the present paper concerns the simultaneousquantum transitions of two quasiparticles due totheir Coulomb scattering.Quantum transitions associated with Coulomb interactionsare allowed without changing the spins of interactingparticles. Since we are concerned only in the LLLs, whichcan accommodate all particles at zero temperature andhigh magnetic field, we consider virtual transitions, where aparticle is first promoted from the LLL to the ELL, andthen reverted to the LLL. We restrict ourselves with thesimultaneous virtual transitions of two particles, butallowing them to be excited in LLs with arbitrary n andm. Such virtual transitions correspond to matrix elementsF i2j ðp; 0; q; 0; p s; n; q þ s; mÞ and F i2j ðp; n; q; m; ps; 0; q þ s; 0Þ with i; j ¼ e; h. As a result, these transitionswill induce indirect interaction between particles in theLLLs and influence essentially on the BEC of magnetoexcitons.Such indirect interaction is attractive and appears inthe second order of the perturbation theory. Following thestatements of the paper [26], the main role is played by thesimultaneous quantum transitions with n 0 ¼ m 0 ¼ n. However,the quantum transitions with nam have to be alsotaken into account and we will show that the virtualquantum transitions with participation of the e–h pair giverise to the contributions of two types, which are bothdepended on the magnetoexciton wave vector k, butexhibitingvanishing or non-vanishing behavior at the pointk ¼ 0. The indirect e–e and h–h interaction lead tocontributions, which do not depend on k. It was shown[28] that for n ¼ m ¼ 1 this indirect interaction gives rise tothe shift of the magnetoexciton levels and influence onBEC. The aim of this section is to generalize the resultsobtained in Ref. [28] and to determine the influence of allELLs with arbitrary n and m.We start with rewriting the Hamiltonian Eq. (2) byseparating the term H LLLCoul , which contains only the LLLsand the term H ELLCoul , which describes the simultaneoustransitions ð0; 0Þ$ðn; mÞ discussed above, while all othersterms entering Eq. (2) will be neglected:H ¼ H 0 þ H LLLCoul þ HELL Coul . (10)From now on particle operators with n ¼ m ¼ 0willbedenoted as a y p ; a p; b y p and b p . The term H ELLCoul can beexcluded from the Hamiltonian Eq. (10) with the aid ofunitary transformation [32,33] ^U ¼ exp½i ^SŠ, where ^S ¼ ^S yis determined from the equationELLi½ ^H 0 ; ^SŠþHCoul ¼ 0. (11)Averaging the transformed Hamiltonian on the groundstate of electrons and holes in ELLs j0i ELL we obtain aneffective HamiltonianH eff ¼ ELL h0je iS ^He iS j0iXELL’ m e a y p a Xp m h b y p b p þ H LLLCoulpþ i 2 ELLh0j½HELL Coul ; ^SŠj0i ELL,which can be written asXH eff ¼ m e a y p a Xp m h b y p b p þ H LLLCoulpppð12Þ1 Xf2 e2e ðp; q; zÞa y p ay q a qþza p zp;q;z1 Xf2 h2h ðp; q; zÞb y p by q b qþzb p zp;q;zXf e2h ðp; q; zÞa y p by q b qþza p z . ð13Þp;q;zHere the indirect interaction matrix elements f i j ðp; q; zÞare given by the expressionsf i2j ðp; q; zÞ ¼ X f i2j ðp; q; z; n; mÞ,n_on;m ci þ m_o cjf i2j ðp; q; z; n; mÞ ¼ X F i2j ðp; 0; q; 0; p t; n; q þ t; mÞtF i2j ðp t; n; q þ t; m; p z; 0;q þ z; 0Þ.ð14ÞMaking use of the definition Eq. (5) and notations of Eq.(8) one can write Coulomb matrix elements in the followingform:F i2i ðp; n; q; m; p s; 0; q þ s; 0Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið 1Þm XpW2 nþm s;k f ðk; p q sÞð s þ ikÞ nþm l nþm ,n!m! kF i2i ðp; 0; q; 0; p s; n; q þ s; mÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið 1Þm XpW2 nþm s;k f ðk; p q sÞðs þ ikÞ nþm l nþm ,n!m! kF e2h ðp; 0; q; 0; p t; n; q þ t; mÞ1 X¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2 nþm t;k f ðk; p þ qÞ½t þ ikŠ n ½t ikŠ m l nþm ,n!m! kF e2h ðp t; n; q þ t; m; p z; 0; q þ z; 0Þ1 X¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2 nþm t z;s f ðs; p þ qÞ½ðt zÞþisŠ nn!m! s½ðt zÞ isŠ m l nþm . ð15ÞAfter straightforward calculation the indirect interactionmatrix elements take the following form:f i2i ðp; q; z; n; mÞ ¼l2ðnþmÞ X2 nþm W t;k W z t;s f ðk; p q tÞn!m!t;k;sf ðk; p q t zÞðt þ ikÞ nþmðt z þ isÞ nþm ,

ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149 143system, whereas the Coulomb exchange e–e and h–hinteractions are negative. They give rise to the attractionin the system and facilitate the creation of Coulombcorrelated e–h plasma and e–h liquid. The supplementaryattraction in the system increases the binding energy ofmagnetoexcitons and at the same time lowers the energyper one pair in the composition of EHL. These competingprocesses will be compared. On the contrary, the exchangepairing terms of the supplementary interaction are positive,giving rise to the repulsion in the e–h system. They act inthe favor of the BEC of magnetoexcitons tending tostabilize it and partially diminish the binding energy ofEHL. First, we will discuss the BEC of magnetoexcitons inHFBA.The ground state energy E g of the Bose–Einsteincondensed magnetoexcitons in Ref. [27] was calculatedbeyond the HFBA on the base of Pauli–Feynman theoremfollowing the proposal of Comte and Nozieres [35,36].Being applied to 2D magnetoexcitons the main formulareads asE g ¼N exXQW QXQZ 10Z_do e 2 dl2p 0 l Im 1.ðQ; o; lÞ(25)Here N ex is the average number of magnetoexcitons andðQ; o; lÞ is their dielectric constant in the case ofhypothetical e–h system with square electric charge equalto l. Substituting in Eq. (25) the dielectric constant in HFAcalled as HF ðQ; o; lÞ or in RPA denoted as RPA ðQ; o; lÞ inRef. [27] the results were obtained in HFBA or beyond itwith account for correlation energy.Two approximations for dielectric constant have differentdependencies on the polarizability 4pa HF0 ðQ; o; lÞ RPA ðQ; o; lÞ ¼1 þ 4pa HF0 ðQ; o; lÞ,1e HF ðQ; o; lÞ ¼ 14paHF 0 ðQ; o; lÞ. ð26ÞThe polarizability can be calculated in the approximationof a weak response, if the wave function of thesystem in zero order approximation is known. In the caseof BEC of magnetoexcitons as a ground state wavefunction was chosen the BCS-type wave function [27]jc g ðkÞi and as the excited wave functions the wavefunctions of the coherent excited states introduced inRef. [36] for e–h systems in a similar way as it wasdone by Anderson [37] in the theory of superconductors.The ground state wave function was introducedfollowing Keldysh–Kozlov method [38] by theactionpffiffiffiffiffiffiffiffiof the displacement unitary transformation^Dð N exÞon the vacuum state of the initially introducedelectron–hole operatorspffiffiffiffiffiffiffiffijc g ðkÞi ¼ ^Dð N exÞj0i;ap j0i ¼b p j0i ¼0. (27)The coherent excited states were generated as follows [27]:c e q Q x¼ a y2qþQ x =2 a q Q x =2jc g ðkÞi. (28)pffiffiffiffiffiffiffiffiThe unitary transformation ^Dð N exÞbreaks the gaugesymmetry of the initial Hamiltonian Eq. (13) transformingit to a new Hamiltonian ^DH eff ^D y , yielding the ground statepwaveffiffiffiffiffiffiffiffifunction Eq. (27) and macroscopic displacementof the exciton creation operatorN exX td y ðkÞ ¼p1 ffiffiffiffi e iQ ytl 2 a y kNx =2þt by k x =2 t . (29)Note that contrary to the Glauber coherent states [39] theexciton creation and annihilation operators are not pureBose operators but only quasi-boson operators [40].pffiffiffiffiffiffiffiffiThepffiffiffiffiffiffiffiffiunitary transformation ^Dð N exÞ¼exp½ N exðdy ðkÞ dðkÞÞŠ of the Hamiltonian implies theunitary transformations of the operators ^Da p ^D y k x a p ¼ ua p v p b þ k2 x p ,^Db p ^D y b p ¼ ub p þ v k xp a þ k2x p , ð30Þyielding inverse transformation k xa p ¼ ua p þ v p b y k2 x p ; b p ¼ ub p v k x2p a y k x p ,(31)with the coefficientsqffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffivðtÞ ¼ve ik ytl 2 ; v ¼ sinð 2pl 2 n ex Þ; u ¼ cosð 2pl 2 n ex Þ,n ex ¼ N ex =S.The restriction of the LLL implies the following equalities[27]:v 2 ¼ N ex =N; n ex ¼ v22pl 2 ,where v 2 is the filling factor of the LLL. The last lineimmediately brings us to the relations u ¼ cos v andv ¼ sin v, which can be satisfied only in the limit vo1.The theory developed in Ref. [27] and its application belowhas to be treated with the restriction vo1. To avoid thisconstraint it is necessary to generalize the structure of theexciton creation operator Eq. (29) including in itscomposition the creation operators of electrons and holesat least in a few number of ELLs. This improvement will bediscussed in the next section, where the FELL is included.The Hamiltonian of Eq. (13) after the unitary transformation(31) will contain operators a y p ; a p; b y p ; b p in arbitraryordering. Their normal ordering will generate a constant Uplaying the role of the ground state energy of HFBA, aquadratic term H 2 similar with the quadratic expressionEq. (36) of Ref. [27], and a quartic term H 0 . Hence, theterms entering H 0 contain only normal ordered operatorsa y p ; a p; b y p ; b p instead of a y p ; a p; b y p ; b p. The average value of H 0

144ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149on the ground state wave function Eq. (27) equals zeroeven for the term proportional to v 4 . Its contribution isnon-zero only in higher orders of the perturbation theory,when the coherent excited states were used, as it wasdemonstrated in the Ref. [27], where the correlation energycontain a factor v 4 u 4 . The role of smallness parameter isplayed by the filling factor v 2 ¼ 2pl 2 n ex o1. At a givenmagnetic field v 2 can be altered arbitrary within intervalð0; 1Þ by changing the exciton concentration n ex , or the totalnumber of excitons in the system N ex . Contrary to thesmall parameter r ¼ I l =_o c discussed above and relatedwith the intensity of the magnetic field there exists anotherindependent parameter n ex of different origin, which in ourconsideration must be small as compared to 1=2pl 2 .The quadratic Hamiltonian H 2 is given below for thecase of electrons and holes with equal masses m e ¼ m h ,cyclotron frequencies o ce ¼ o ch ¼ o c and chemical potentialsm e ¼ m h ¼ m=2:H 2 ¼ X ½Eðk; v 2 ; mÞþðB 2AÞv 2 ð1 2v 2 Þpþ 2v 2 ð1 v 2 ÞDðkÞŠða y p a p þ b y p b pÞþ X uv k xp b2kx pa p þ uv ppk x2a y p by k xf cðk; v 2 ; mÞþ2v 2 ðB 2A þ DðkÞÞ DðkÞg. ð32ÞFollowing the notations of Eqs. (40) and (41) of Ref. [27]we haveEðk; v 2 ; mÞ ¼2v 2 u 2 I ex ðkÞþI l ðv 4 v 2 u 2 Þm2 ðu2 v 2 Þ,cðk; v 2 ; mÞ ¼2v 2 I l þ I ex ðkÞð1 2v 2 Þþm, ð33Þwhereas the coefficients DðkÞ, A and B are determined byEqs. (17), (21) and (24), respectively. Putting to zero thelast bracket in Eq. (32), i.e. compensating the dangerousdiagrams describing the spontaneous creation and annihilationof e–h pairs in the vacuum state Eq. (27), one canobtain the chemical potential m of the system in the HFBA:m HFB ¼ ~I ex ðkÞþ2v 2 ðB 2A þ ~I ex ðkÞ I l Þ¼ ~I ex ðkÞþ2v 2 ðB 2A þ DðkÞ EðkÞÞ. ð34ÞHere the renormalized ionization potential of magnetoexcitons~I ex ðkÞ containing the correction due to influence ofall ELLs was introduced:~I ex ðkÞ ¼I ex ðkÞþDðkÞ; I ex ðkÞ ¼I l EðkÞ,E ex ðkÞ ¼ I ex ðkÞ. ð35ÞIntroducing the value m HFB in the remainder part of thefirst line of Eq. (32), Hamiltonian H 2 will take the formH 2 ¼ X p~I ex ðkÞða þ p2a p þ b þ p b pÞ. (36)This Hamiltonian describes the single-particle elementaryexcitations from a single-exciton state with wave vector kof the condensed magnetoexcitons. To extract from thecondensate one pair of new quasiparticles the energy costp~I ex ðkÞ is equivalent to unbinding energy, i.e. the excitationenergy for one quasiparticle equals to ~I ex ðkÞ=2. Notice thatthe chemical potential m HFB in the point v 2 ¼ 0 coincides onthe energy scale with the position of the renormalizedmagnetoexciton energy band~E ex ðkÞ ¼ ~I ex ðkÞ,while in the point 2v 2 ¼ 1 it equals to the value I l þ B2A and does not depend on k. The concentrationcorrections to m HFB are determined by the term2v 2 ðB 2A þ DðkÞ EðkÞÞ. (37)The term EðkÞ appears in the frame of the LLLs and wasobtained in the Refs. [26,27]. It determines the instability ofthe ground state within the HFBA, when the correctionsdue to ELL are neglected. The term B 2A appears inboth phases, not only in the case of BEC of magnetoexciton,but also in the case of EHL. The term 2A is relatedwith the average Hartree terms of the supplementary e–e,h–h and e–h interactions, whereas the term B with theaverage exchange terms of the supplementary e–e and h–hinteractions. The term 2v 2 DðkÞ is according to e–hinteraction and Bogoliubov u-v transformation and isnamed as Bogoliubov self-energy term [41]. As it will beshown below it does not appear in the case of EHL.The renormalized ionization potential in a dimensionlessform is represented in Fig. 1, when the parameter r wastaken equal to 1 and 1 2. Note the value 1 is the maximalpossible value because the theory is valid only for ro1. Theresulting influence of the ELLs on the chemical potentialmðk; v 2 Þ calculated in the HFBA is determined by thecoefficient ðB 2A þ DðkÞÞ as was mentioned above. In thedimensionless form it is represented in Fig. 2 also for twodifferent ratios r ¼ 0:5: 1. This influence, as well as theinfluence of FELLs discussed in the paper [28], is essentialonly in the range of small values of klo0:5, decreasingrapidly with the increasing of kl. The inset on this figurerepresents the coefficient ½B 2A þ DðkÞ EðkÞŠ, whichreflects the influence of both the LLLs and of the ELLs.Thus, the main result obtained so far, namely thedependence of the chemical potential m HFB in HFBAversus the filling factor v 2 of the LLL at different values ofthe dimensionless wave vector kl ¼ 0; 0:5; 1:0; 3:6 and fortwo different values of ratio r ¼ 0:5: 1 is presented in Fig. 3.One can see that the BEC of 2D magnetoexcitons withwave vector klo0:5 is stable in HFBA. As was realized inRef. [27,28] at greater values kl40:5 the influence ofcoherent excited states [37] is important and leads to theappearance of the metastable dielectric liquid phase.Now we consider the EHL formation in HFA. We startwith an effective Hamiltonian (13), but without chemicalpotentials m e and m h , and calculate the ground state energyE EHL of EHL at T ¼ 0 when the average values ofelectrons and holes numbers on the LLLs are equal toha y p a pi¼hb y p b pi¼v 2 . (38)

ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149 145Here v 2 is the filling factor of LLLs. Applying the Wicktheorem, we obtained the ground state energyE EHL ¼ N e2h ½v 2 I l þ v 2 ð2A BÞŠ; N e2h ¼ Nv 2 , (39)and the energy per one e–h pair e EHL of EHL in units of I le EHL¼ v 2 1 þ 2I lðS TÞ . (40)I l p_o cTaking into account the estimated values S ¼ 0:481 andT ¼ 0:216 we havee EHL¼ ð1 þ 0:168rÞv 2 ; r ¼ I l =_o c . (41)I lThe minimal value is achieved at filling factor v 2 ¼ 1, and itdetermines the energy per pair inside EHD equal toe EHD ¼ I l ð1 þ 0:168rÞ. (42)e EHL and e EHD depend on the ratio ro1. In spite of therestriction ro1 equivalent to a strong magnetic fieldcondition, we will make also calculations in the case ofmaximal possible value r ¼ 1. One can see that thecorrections due to ELLs lower the energy per pair insideEHD by the value 0:168I l for r ¼ 1 and by the value0:084I l for r ¼ 0:5. The energy per e–h pair of EHD ispresented in Fig. 3 for different values of the ratio r. Theenergy e EHD is of the same order of magnitude as thechemical potential of condensed excitons with small wavevectors k and the coexistence of these two states is possible.4. Excitonic approach: BEC in the superposition excitonstateThe aim of this section is to generalize the resultsexpressed by formulas (27)–(31), as well as to compare the(r = 0.5)(r = 1)(r = 0.5)(r = 1)kl=0kl=0.5kl=1.0kl=3.6Fig. 1. Renormalized exciton ionization potential versus dimensionlesswavevector kl for different values of the parameter r ¼ I l =_o c . Solid line:r ¼ 1; dashed line: r ¼ 0:5. Dash-dotted line: ionization potential with theinclusion of only FELL for r ¼ 1; dotted line: the same, but for r ¼ 0:5(see Section 4).0.160.120.08kl=0kl=0.5kl=1.0kl=3.60.060.00-0.04-0.08-0.12-0.160.0 0.5 1.0kl1.5 2.0Fig. 2. Coefficient B 2A þ DðkÞ versus wavevector k for different valuesof the parameter r ¼ I l =_o c . Solid line: r ¼ 1; dashed line: r ¼ 0:5. Inset:coefficient B 2A þ DðkÞ EðkÞ versus wavevector k. Solid line: r ¼ 1;dashed line: r ¼ 0:5.Fig. 3. Chemical potential versus filling factor v 2 for different values of theparameter r ¼ I l =_o c . Solid line: energy per e–h pair in EHD phase;dashed line: chemical potential of condensed excitons with k ¼ 0; dottedline: the same, but for kl ¼ 0:5; dash-dotted line: the same, but for kl ¼ 1;dash-dot–dot line: the same, but for kl ¼ 3:6.

146ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149results illustrated in Fig. 1 with the ones obtained in thissection. Putting the aim into practice we need to calculatethe wave functions and the energy spectrum of the lowestexciton band. We will confine ourselves into the frame offour exciton levels model. By this restriction we willdetermine the influence of nearly situated exciton levelswith the same wave vector on the lowest exciton level inwhich the BEC of magnetoexcitons takes place. Thesecalculations are needed to obtain more exact expressionsfor the exciton wave function as well as of the excitoncreation and annihilation operators. They will be representedas coherent superpositions of their zero orderexpressions and will lead us to the necessity to investigatethe BEC in the exciton superposition state.For the beginning we will define the magnetoexcitoncreation operator [25–27] characterized by the number n ofthe electron LL and by the number m of the hole LL, aswell as by the 2D wave vector k with two components k xand k yX y n;m;k ¼ p1 ffiffiffiffiX exp½ ik y tl 2 Ša y n;kNx =2þt by m;k x =2 t . (43)tThe state with the pair of numbers ðn; mÞ equals to (0,0) willfor simplicity be denoted as the state 1, the pair of numbers(1,1) gives rise to the magnetoexciton state 2, the pairs (1,0)and (0,1) will be mentioned as the states 3 and 4,correspondingly. The exciton wave function c i ðkÞ isobtained as action of this operator on the vacuum statej0i determined as a n;p j0i ¼0; b m;p j0i ¼0c i;k ¼ X y n;m;kj0i, (44)where i labels the quantum numbers i !ðn; mÞ. Thus, onecan straightforwardly prove the orthogonality and normalizationproperties of eigenstates:hc i;k jc j;k 0i¼d i;j d k;k 0. (45)The matrix elements of the Hamiltonian Eq. (2), where thechemical potentials m e and m h are omitted as we areconcerned in single exciton properties, on the wavefunctions Eq. (44) are denoted asH ij ðkÞ ¼hc i;k j ^Hjc j;k i¼d i;j hc i;k j ^H 0 jc i;k iþV i;j ðkÞ, (46)where V i;j ðkÞ ¼hc i;k j ^H Coul jc j;k i. Analytical expression forV i;j ðkÞ one can derive from Eqs. (6) to (7) (for details seee.g. in Ref. [43]). The magnetoexciton energy bands will bedetermined at first in the zeroth order of the perturbationtheory, when only the diagonal matrix elements H ii andtwo off-diagonal matrix elements V 12 ðkÞ and V 21 ðkÞ aretaken into account. The reason for the choice of the zeroorder approximation will be given below. All other offdiagonalmatrix elements, which are smaller than thevalues V 12 and V 21 will be introduced in higher orders ofthe perturbation theory. We consider first the nondegeneratecase when o ce ao ch ,so that the four zero ordermagnetoexciton bands accounted from the LLLs areE 1 exðkÞ ¼Eð0;0Þ ex ðkÞ ¼H 1;1 ðkÞ ¼V 1;1 ðkÞ ¼ I ð0;0Þex ðkÞ, ð47aÞE 2 exðkÞ ¼Eð1;1Þ ex ðkÞ ¼H 2;2 ðkÞ ¼_o ce þ _o ch I ð1;1Þex ðkÞ, ð47bÞE 3 exðkÞ ¼Eð1;0Þ ex ðkÞ ¼H 3;3 ðkÞ ¼_o ce I ð1;0Þex ðkÞ, ð47cÞE 4 exðkÞ ¼Eð0;1Þ ex ðkÞ ¼H 4;4 ðkÞ ¼_o ch I ex ð0;1Þ ðkÞ. ð47dÞThe ionization potentials of four bands were determined asI ðn;mÞex ðkÞ ¼ X sF e2h ðp; n; k x p; m; p s; n; k x p þ s; mÞ exp½ik y sl 2 Š¼ e2e 0 lx 2 exp 12Z 1dx0 nþmx 22J 0 ðklxÞ; n; m ¼ 0; 1. ð48ÞAn explicit expression for I ðn;mÞex ðkÞ can be found e.g. in Ref.[44].A more exact expression of the magnetoexciton wavefunction is the linear combinationc n;k ¼ X4i¼1a in c i;k . (49)The new functions c n;k are denoted by Greek symbolsn ¼ a; b; g; d, whereas the initial zero order functions c ik byLatin letter i ¼ 1; 2; 3; 4. They obey to the Schro¨dingerequation^Hc n;k ¼ E n ðkÞc n;k (50)what leads to four linear equations which determine theenergy spectrum E n ðkÞ and the coefficients a in . Theequations contain the matrix elements Eq. (46) and havethe formX 4i¼1a in H ij ðkÞ ¼E n ðkÞa jn ; j ¼ 1; 2; 3; 4. (51)The energy spectrum E n ðkÞ can be found solving secularequationH 11 ðkÞ E n ðkÞ V 12 ðkÞ V 13 ðkÞ V 14 ðkÞV 21 ðkÞ H 22 ðkÞ E n ðkÞ V 23 ðkÞ V 24 ðkÞ¼ 0.V 31 ðkÞ V 32 ðkÞ H 33 ðkÞ E n ðkÞ V 34 ðkÞ V 41 ðkÞ V 42 ðkÞ V 43 ðkÞ H 44 ðkÞ E n ðkÞ (52)Neglecting 10 off-diagonal matrix elements denoted as afirst order infinitesimals ejV 13 jjV 14 jjV 23 jjV 24 jjV 34 je (53)we will obtain the zero order solutions of Eqs. (51) and (52)E 0 a;b ðkÞ ¼ H 11ðkÞþH 22 ðkÞ2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH 11 ðkÞ H 22 ðkÞÞ 2 þ 4V 2 122ðkÞ ,E 0 g ðkÞ ¼H 33ðkÞ; E 0 d ðkÞ ¼H 44ðkÞ. ð54Þ

ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149 147In the limit V 12 ðkÞo_o ce þ _o ch two magnetoexcitonbands look as follows:E 0 a ðkÞ ¼E1 ex ðkÞ V 2 12 ðkÞ ,_o ce þ _o chE 0 b ðkÞ ¼E2 ex ðkÞþ V 2 12 ðkÞ .ð55Þ_o ce þ _o chThe coefficients a 0 in can be also determined in the zerothorderja 0 1a j2 ¼ja 0 ðH 22 ðkÞ H 11 ðkÞÞ 22b j2 ¼ðH 22 ðkÞ H 11 ðkÞÞ 2 þ V 2 12 ðkÞV 2 12 1ðkÞð_o ce þ _o ch Þ 2 ,ja 0 2a j2 ¼ja 0 V 21b j2 12¼ðkÞðH 22 ðkÞ H 11 ðkÞÞ 2 þ V 2 12 ðkÞV 2 12ðkÞð_o ce þ _o ch Þ 2 ,a 0 2a V 12 ðkÞ; a 0 1b_o ce þ _o V 12ðkÞ,ch _o ce þ _o cha 0 3n ¼ a0 4n ¼ 0 for n ¼ a; b. ð56ÞThe different signs of the coefficients a 0 1b and a0 2aare takento obey Eqs. (51) and will be important in the discussionsbelow. We are interested in a more exact expressions foronly the lowest exciton band taking into account theinfluence of three exciton bands situated upper on theenergy scale. For these three bands involved in this schemethe starting wave function (49) is not sufficient becausethere are another exciton bands which do not affect toomuch the lowest exciton bands, but can stronger influenceon the upper exciton bands. It can be easily shown that firstorder corrections give zero contribution to the energy andthe following contribution to the coefficients a in :a 0 1n ¼ a0 2n ¼ 0,a 0 3n ¼ðV 31a 0 1n þ V 32a 0 2n ÞðH 33 E 0 n Þ ; n ¼ a; b,a 0 4n ¼ ðV 41a 0 1n þ V 42a 0 2n ÞðH 44 E 0 n Þ . ð57ÞThe perturbation theory used here gives rise to the secondorder correction to the lowest exciton band E 00aE 00a ðkÞ ¼ jV 13 ðkÞj 2E 0 a ðkÞ H 33ðkÞ þ jV 14ðkÞj 2 E 0 a ðkÞ H ja 0 1a44ðkÞj2jV 23 ðkÞj 2þE 0 a ðkÞ H 33ðkÞ þ jV 24 ðkÞj 2 E 0 a ðkÞ H ja 0 2a44ðkÞj2þ a 01a a0 2aþ a 02a a0 1aV 13 ðkÞV 32 ðkÞE 0 a ðkÞ H 33ðkÞ þ V 14ðkÞV 42 ðkÞE 0 a ðkÞ H 44ðkÞV 23 ðkÞV 31 ðkÞE 0 a ðkÞ H 33ðkÞ þ V 24ðkÞV 41 ðkÞE 0 a ðkÞ H . ð58Þ44ðkÞSubstituting the coefficients a 0 1a and a0 2a, given by (56) andthe matrix elements V ij ðkÞ from (46) into (58) we obtain thedependence of E 00aðkÞ on the dimensionless wave vector kl.The dependence of the ionization potential of the lowestexciton band on the wave vector k with the account of thesecond order corrections is represented in Fig. 1 by thedotted and dash-dotted lines. This corrections are 40%smaller as compared with the influence of all ELL.In the frame of four level models the magnetoexcitoncreation operator has the formX y ak ¼ a0 1a X y 0;0;k þ a0 2a X y 1;1;k þ a0 3a X y 1;0;k þ a0 4a X y 0;1;k . (59)On its base it is possible to construct a new displacementoperator ^DðN ex Þ and to discuss the phenomenon of BECtaking into account explicitly the LLLs and FELLs.Consider the BEC in the lowest exciton superpositionstate in a simplified variant, when only two main terms inexpression (59) are taken into account and the coefficientsof superposition are a 0 1a a 1 and a 0 2a a 2, with thecondition a 2 1 þ a2 2 ¼ 1. The expression Eq. (31) now canbe rewritten as:X y k ¼ a 1X y 0;0;k þ a 2X y 1;1;k . (60)pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiThe unitary transformation ^Dð N exÞ¼exp½N exðXy ðkÞXðkÞÞŠ leads do generalized Bogoliubov’s u-v transformationa p ¼ ^Da p ^D y¼ cosðga 1 Þa p sinðga 1 Þ exp½ ik y ðp k x =2ÞŠb y k x p ,b p ¼ ^Da p ^D y¼ cosðga 1 Þb p þ sinðga 1 Þ exp½ ik y ðk x =2 pÞŠa y k x p ,g p ¼ ^Dc p ^D y¼ cosðga 2 Þa p sinðga 2 Þ exp½ ik y ðp k x =2ÞŠd y k x p ,d p ¼ ^Dd p ^D y¼ cosðga 2 Þd p sinðga 2 Þ exp½ ik y ðk =2 pÞŠc y k x p . ð61ÞpHere g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pl 2 n ex ; n ex ¼ N ex =S; Fermi operators c p ; d pand c y p ; dy p describe the annihilation and creation of anelectron and a hole in the FELL, respectively. Then thenew ground state wave function acquires the followingform:pffiffiffiffiffiffiffiffijC g ðkÞi ¼ ^Dð N exÞj0i¼ Y ðcosðgÞþsinðgÞ expð ik y tl 2 Þ½a 1 a y k x =2þt by k x =2 ttþ a 2 c y k x =2þt dy k x =2 tŠÞ, ð62Þwhich plays the role of vacuum state for the operatorsa p ; b p ; g p ; d p , i.e.a p jC g ðkÞi ¼ b p jC g ðkÞi ¼ g p jC g ðkÞi ¼ d p jC g ðkÞi ¼ 0.With the above definitions one can calculate the averagenumber of excitons (which is equal to the average number

148ARTICLE IN PRESSS.A. Moskalenko et al. / Physica E 39 (2007) 137–149of electrons or holes) as follows:hC g ðkÞj X ða y t a t þ c y t c t ÞjC g ðkÞi ¼ N½sin 2 ðga 1 Þþsin 2 ðga 2 ÞŠ.t(63)Owing to the definition of g one arrives at the concentrationrelationg 2 ¼ sin 2 ðga 1 Þþsin 2 ðga 2 Þ, (64)which in case of a 2 ¼ 0 leads to the previous expression Eq.(36) of Ref. [27]. Another special case is the following:pa 1 ¼ a 2 ¼ 1=ffiffip2 ; g 2 =2 ¼ sin 2 ðg= ffiffi2 Þ, (65)what implies g 2 o2. This relation generalize the descriptionof the BEC of magnetoexciton gas on its superpositionstate, with electrons and holes residing both in the LLLsand the FELLs.5. ConclusionsThe virtual excitations due to Coulomb scattering of twocharged particles from their LLLs to ELLs with arbitraryindices n and m and their return back to LLLs lead in thesecond order of the perturbation theory to supplementaryindirect interaction between the particles side by side withtheir Coulomb interaction. General expressions for thecorresponding matrix elements were obtained. On thesebase the influence of indirect interaction on the chemicalpotential of the condensed magnetoexcitons and on theenergy per pair in the components of EHL and EHD wererevealed in HFA. The indirect supplementary e–e and h–hinteraction being averaged in HFA gives rise to directpairing terms and exchanges pairing terms. The first termsbeing negative increase the binding energy of magnetoexcitonsand energy per pair in the EHL phase, whereas thesecond terms are repulsive. They diminish the influence ofthe direct pairing terms, but do not surpass them, so thatthe resulting influence of both terms remains attractive.The supplementary e–h attraction after the u-v transformationin the case of BEC of magnetoexcitons in the statewith wave vector k gives rise to repulsive-type Bogoliubovself-energy terms [41]. They stabilized the BEC in the smallregion of wave vectors klo0:5. Such terms do not appearin the case of EHL.The energy per one e–h pair inside the EHD found to besituated on the energy scale very close to the value of thechemical potential of the Bose–Einstein condensed magnetoexcitonswith wave vector k ¼ 0 calculated in the HFBA.These two phases can coexist. Coexistence of the degenerateBose gas with k ¼ 0 and of the droplets of dielectricliquid phase formed by magnetoexcitons with non-zerowave vector k was revealed in Ref. [42], so that one canexpect the coexistence of three phases simultaneously.The wave functions of the lowest exciton levels in thesecond order of the perturbation theory represent thesuperpositions of the zero order exciton wave functionsrelated with definite Landau levels. The BEC of magnetoexcitonsin such superposition state involving differentLandau levels was considered. 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