Bose-einstein Condensation of Excitons in Ideal 2d System in Strong Magnetic Field

We study ideal two-dimensional ( 2D ) electron-hole system in a strong perpendicular magnetic field using the Keldysh-Kozlov-Kopaev method and generalized random phase approximation. The Bose-Einstein condensation of the correlated pairs takes place on a single particle state with an arbitrary wave vector k in a symmetric 2D model. We show that the ground state energy per one exciton and the chemical potential at low exciton damping rates are nonmonotonous functions versus the value of the filling factor. They reveal the relative minima, so that the metastable states of the dielectric liquid phase with positive compressibility consisting of the Bose-Einstein condensate of magnetoexcitons and liquid drops can be formed by excitons with sufficiently high wave vectors and motional dipole moments. It is shown that the dielectric liquid phase of the Bose condensed excitons with low damping rate is more stable than the e h − metallic liquid phase. The observation of Bose-Einstein condensation ( BEC ) in atomic alkali and hydrogen gases using a laser and magnetic trapping [1,2] has greatly expanded the related research in recent years. As is well known, under certain conditions excitons, i.e. bound states of electron-hole pairs in semiconductors have bosonic properties [3]. Although theoretically recognized many years ago [3], experiments on BEC of excitons have made slow progress, because finite lifetime effects, strong interactions between excitons at high density, crystal imperfections and phonons in the crystal all act to complicate the system. In recent years, the system of coupled quantum wells in strong electric field has gained attention as a system with repulsive exciton-exciton interactions and long exciton lifetime, ideal for BEC of excitons. Another advantage of the two-dimensional ( 2D ) system is a possibility of much faster cooling of hot photoexcited excitons compared with their bulk counterparts [4,5]. Another approach is to use strong magnetic field. It has been shown [6] that the properties of atoms and excitons are dramatically changed in a strong magnetic field such that the distance between Landau levels e e H m c / h exceeds the Rydberg energy. The diamagnetic excitons in bulk crystals were revealed in [7]. Their Bose-Einstein condensation was studied in [8]. Even more attractive and worth investigating is the electron-hole ( e h − ) system in two dimensions ( 2D ) in the presence of a strong perpendicular magnetic field. In the latter case the energy spectrum of e-h system is completely discrete, is characterized by the number of the Landau levels, which are N -fold degenerated with 2 2 N S l π = / , where l is the magnetic length, 2 l c eH = / h , and S is the 2D sample dimension. In the past two decades, a number of experimental [9,10,11,12] and theoretical [13,14,15] efforts have been dedicated to the study of 2D systems in a strong magnetic field. Lerner and Lozovik [13,14] studied the coherent pairing of electrons and holes resulting in the formation of the Bose-Einstein condensate of excitons in a single-particle state with wave vector 0 k = . In the Hartree-Fock approximation, when the coupling to the higher Landau levels and the correlation energy are neglected, the magnetoexcitons with 0 k = represent at 0 T = an ideal excitonic gas. A surprising Moldavian Journal of the Physical Sciences, Vol.1, N4, 2002 6 result was that the fermionic e h − droplets of the metallic electron-hole liquid ( EHL ) with the maximal local filling factor of the lowest Landau level ( LLL ), can be considered as an aggregate of excitons sticked together. The coupling to higher Landau levels, however, makes the system weakly nonideal [13,14], which allows the Berezinskii-Kosterlitz-Thouless topological phase transition [16,17,18] at finite temperature. The results obtained by Lerner and Lozovik [13,14] were reproduced by Paquet, Rice and Ueda in [15] on the basis of simpler and more transparent approach using the BCS -type wave functions of the BEC excitons and calculating the ground state energy in the Hartree-Fock-Bogoliubov approximation. They considered the case of nonzero wave vectors 0 k ≠ and introduced into the Hamiltonian the indirect interaction of the particles on the LLL due to their virtual excitation to excited Landau levels. The aim of our paper is to investigate the properties of the system beyond the Hartree-FockBogoliubov approximation, taking into account the possibility of the Anderson-type coherent excited states, the corresponding polarizability of the Bose-Einstein condensate, the screening effects and the correlation energy due to just this channel of polarizability. Contrary to [13,14] our correlation energy is not related to excited Landau levels. Simultaneously we will take into account the first excited Landau level ( FELL ) in the way proposed by Paquet, Rice and Ueda [15]. We find that a metastable dielectric liquid phase formed by Bose-Einstein condensed magnetoexcitons with 0 k ≠ can exist. It is more stable than the droplet of EHL . The Hamiltonian describing the electron-hole system on the surface of an ideal 2D layer in a strong perpendicular magnetic field was derived in [15] in a second quantization representation. Here we will use a simpler and more concrete form taking into account only two Landau levels: the lowest Landau level ( LLL ) with quantum number 0 n = and the first excited Landau level ( FELL ) with 1 n = . For the very beginning we introduced the chemical potentials for electrons and holes e μ and h μ , respectively. The operators ˆ e N and ˆ h N for the full numbers of electrons and holes are expressed through the creation and annihilation operators p p a a + , and p p c c + , for electrons on the LLL and FELL correspondingly and through the operators p p b b + , and p p d d + , for holes on the states with 0 n = and 1 n = correspondingly ˆ ˆ p p p p p p p p e h p p p p a a c c b b d d N N + + + + = + ; = + . ∑ ∑ ∑ ∑ (1) The full Hamiltonian H consists from three parts: 0 LLL FELL Coul Coul H H H H = + + . (2) The zero order Hamiltonian 0 H contains the cyclotron frequencies ce ω and ch ω , which are supposed to be greater than the exciton ionization potential l I , and has the form: 0 ˆ ˆ ce p p ch p p e h e h p p H c c d d N N ω ω μ μ + + = + − − ∑ ∑ h h (3) The Coulomb interaction of electrons and holes situated on the LLL ( 0 n = ) is denoted as LLL Coul H 1 ( 0 0 0 0) 2 1 ( 0 0 0 0) 2 ( 0 0 0 0) LLL Coul e e p q q s p s p q s h h p q q s p s p q s e h p q q s p s p q s H F p q p s q s a a a a F p q p s q s b b b b F p q p s q s a b b a + + − + − , , + + − + − , , + + − + − , , = , ; , ; − , ; + ,