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

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 ,