A Hartree-Fock and Hartree-Fock-Bogoliubov study of a few body system of
spatially separated charge carriers was carried out. Using these variational
states, we compute an approximation to the correlation energy of a finite
system of electron-hole pairs. This energy is shown as a function of the
Coulomb coupling and the interplane distance. We discuss how the correlation
energy can be used to theoretically determine the formation of indirect
excitons in semiconductors which is relevant for collective phenomena such as
Bose-Einstein condensation (BEC).Comment: Conference EDISON16 (2009), 4 page

Boris A Rodriguez

Francisco E Lopez

I Mondragon-Shem

Ian Mondragon-Shem

Nielsen M A

Journal of Physics : Conference Series

English

arXiv

Mondragon-Shem I.

Rodriguez B.A.

Lopez F.E.

Repositorio Institucional ITM

Journal of Physics: Conference Series

Crossref

Coulomb correlations of a few body system of spatially separated charges

ABSTRACT: Hartree-Fock and Hartree-Fock-Bogoliubov study of a few body system of spatially separated charge carriers was carried out. Using these variational states, we compute an approximation to the correlation energy of a finite system of electron-hole pairs. This energy is shown as a function of the Coulomb coupling and the interplane distance. We discuss how the correlation energy can be used to theoretically determine the formation of indirect excitons in semiconductors which is relevant for collective phenomena such as Bose-Einstein condensation (BEC).COL000844

Mondragon Shem, Ian

Rodríguez Rey, Boris Ánghelo

López Giraldo, Francisco Eugenio

Universidad de Antioquia Institutional Repository

This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 190.159.203.49This content was downloaded on 19/04/2017 at 03:22Please note that terms and conditions apply.Coulomb correlations of a few body system of spatially separated chargesView the table of contents for this issue, or go to the journal homepage for more2009 J. Phys.: Conf. Ser. 193 012133(http://iopscience.iop.org/1742-6596/193/1/012133)Home Search Collections Journals About Contact us My IOPscienceYou may also be interested in:Diagrammatic perturbation theory: the contribution of triply excited states to correlation energiesS WilsonTotal and correlation energy of the uniform polarized electron gas at finite temperature: Directpath integral simulationsV S Filinov, V E Fortov, M Bonitz et al.Multiparticle–multihole Gogny energy density functional for low-lying spectroscopy applicationsN Pillet, V Zelevinsky, M Dupuis et al.Relativistic Hartree-Fock-Bogoliubov predictions of superheavy magic nucleiJ J Li, W H Long, J Margueron et al.Deformation properties with a finite range simpleeffective interactionB Behera, X Viñas, T R Routray et al.Self-consistent studies of the dipole response in neutron rich nuclei using realistic potentialsF Knapp, N Lo Iudice, P Veselý et al.Di-neutron correlation in monopole pair transfer modes near the neutron-separation energy in Sn,Ni, Ca, Zr isotopesH Shimoyama and M MatsuoDiscovering correlated fermions using quantum Monte CarloLucas K Wagner and David M CeperleyCoulomb correlations of a few body system ofspatially separated chargesIan Mondragon-Shem and Boris A. Rodriguez!Instituto de F́ısica, Universidad de Antioquia, AA 1226, Medelĺın, ColombiaE-mail: !banghelo@fisica.udea.edu.coFrancisco E. LopezCentro de Investigaciones, Instituto Tecnológico Metropolitano, Calle 73 No 76A -354,Medelĺın, ColombiaAbstract. A Hartree-Fock and Hartree-Fock-Bogoliubov study of a few body system ofspatially separated charge carriers was carried out. Using these variational states, we computean approximation to the correlation energy of a finite system of electron-hole pairs. This energyis shown as a function of the Coulomb coupling and the interplane distance. We discuss how thecorrelation energy can be used to theoretically determine the formation of indirect excitons insemiconductors which is relevant for collective phenomena such as Bose-Einstein condensation(BEC).1. IntroductionThe physical properties displayed by charge carriers in spatially confining settings could havea diverse range of potential applications relevant to technological developments in the nearfuture. Systems with confined charge carriers such as semiconductor quantum wells, quantumwires and quantum dots, will likely have an important impact in the development of quantumcomputation and quantum information [1] and in producing new technological devices inelectronics, spintronics and optoelectronics [2]. In order to take advantage of this importantpractical potential, it is necessary to have a thorough theoretical understanding of the physicsof such systems.In particular, it is well known that the physics of electron-hole pairs in quantum systems,such as quantum wells and quantum dots, is fundamentally a!ected by Coulomb correlations(see, for example, [3] and [4]). As a consequence, a complete theoretical study of the physicalproperties of charge carriers in confining settings must take into account the very importantlong-ranged Coulomb interaction.With this in sight, we focus our study on a few-body system of electron-hole pairs confined tovertically stacked parabolic quantum dots. This type of system has been studied recently in thetheory of quantum dot molecules (e.g. [5], [6]). Our calculations will involve twelve particles:six electrons and six holes, for a total of six electron-hole pairs. It is clear that such a systemis a long way from the O(102) pairs of real systems. However, the correlations that arise hereshould give us a first reasonable insight into the real physical system.EDISON 16 IOP PublishingJournal of Physics: Conference Series 193 (2009) 012133 doi:10.1088/1742-6596/193/1/012133c© 2009 IOP Publishing Ltd 1dr1r2q1q2!Fig 1: Schematic of our system with two chargecarriers q1 and q2. They are in two di!erentplanes, spatially separated by a vertical distanced. The angle ! is the angle between the positionvectors r1 and r2.2. Theoretical frameworkWe compute the ground state energy of the system of electron-hole pairs using two variationalapproximations used in finite system studies: the Hartree-Fock (HF) state and a Hartree-Fock-Bogoliubov-type (HFB) state. These states di!er in that the HF state fails to provide theso-called correlation energy of the few-body system, whereas the HFB state is able to yield afirst approximation this energy. With this information, we will be able to assess under whichconditions the correlation energy due to electron-hole pair formation is the greatest i.e. to assesswhen the formation of spatially indirect excitons is favoured.To achive our goal, it is necessary to have an e"cient way of computing the representationof the Coulomb interaction in the 2D harmonic oscillator basis. The Coulomb matrix elementsread: !ij|V̂ (d)|kl" = !ij| 1#r212+d2|kl", where d denotes the separation distance (see Fig.(1)) andthe indices in the bras and kets label the 2D harmonic oscillator eigenstates . We have beenable to compute the matrix elements [7] by expanding the integrals in terms of a single infiniteseries followed by a series acceleration algorithm to achieve rapid convergence.The hamiltonian of the electron-hole pairs in this system is given byH =!i{"(e)i e†iei + "(h)i h†ihi} + #ee!ijkl!ij|V (0)|kl"e†ie†jelek ++#hh!ijkl!ij|V (0)|kl"h†ih†jhlhk $ #eh!ijkl!ij|V (d)|kl"e†ih†jhlek. (1)The #ee, #hh and #eh parameters are dimensionless quantities that measure the relative strengthof the Coulomb interaction and the harmonic potential: #xx ="me4!2xxh̄2 /(h̄$o)#1/2. In this work,we take #ee = #hh = #eh = #, because we only seek to understand the immediate physicale!ect of having spatial separation between the charge carriers. A more realistic calculation willneed to take into account the fact that the # factor dependends on the dielectric constant ofthe materials involved in the semiconductor and, thus, their values are going to be di!erent ingeneral. Through out this work, we take the energy scale to be the harmonic potential energyh̄$ and the length scale to be the harmonic oscillator length l =$h̄m" .2.1. The Hartree-Fock ground stateIn the first step of our calculation, we minimized the average energy using a slater determinantin order to compute the ground state energy of the few body system. The HF state is givenby: |%HF " =%µ=1 e†#µ%$=1 h†%!|0". As is well known, the Hartree-Fock energy provides thebest energy for uncorrelated single particle wave functions. Deviations from this result will yieldinformation of the correlations beyond a single particle picture of the system. This transitionto a correlated state will involve, in a first approach, two-particle correlations between electronsand holes, thus generating a picture of bound electron-hole pairs.EDISON 16 IOP PublishingJournal of Physics: Conference Series 193 (2009) 012133 doi:10.1088/1742-6596/193/1/0121332Figure 1. Contour plots of the HF and the BCS ground states in the (#, d) parameter space.2.2. The BCS ground stateIn order to include correlation e!ects, in the second step of our calculations we made use of theBCS (Bardeen-Cooper-Schrie!er) state, given by: |%BCS" =%i(ui + vie†ih†ī)|0" which is one ofa various types of Hartree-Fock-Bogoliubov correlated variational ground state approximations.Such a state introduces pair correlations between electrons and holes from the outset. Thecoe"cients vi are the variational parameters in the system and are a measure of the likelihoodfor an ith correlated pair to appear in the system.3. Results and discussionIn Fig.(1), we show the HF and BCS energies in the parameter space (#, d). Both of theseplots tell us how, by a suitable combination of the Coulomb coupling # and the interplanedistance, we can go from energies below the energy of the noninteracting particles to energieswell above this value. The appearance of the blue region shows that the attractive interactionbetween electrons and holes is dominant over that of the repulsive electron-electron and hole-holeones, when # increases or when d decreases. However, as we separate the planes, the electron-hole energy gradually loses its predominance until we achieve a state in which the contributionfrom repulsive and attractive interactions cancels out, and we are left with the energy value ofthe noninteracting system.Note that the contour level corresponding to the particular situation of equal contributionsfrom attractive and repulsive energies is given approximately by a small region of constantdistances, independent of the Coulomb coupling. This region is shifted to the right in the BCSplot, with respect to that in the Hartree-Fock plot.The correlation energy, which is approximately given by EC(d,# ) % EBCS(d,# )$EHF (d,# ),is shown in Fig.(2); both plots in this figure provide di!erent views of the correlation energy.We arrive at two important conclusions from the behaviour of EC(d,# ).First, the system electron-hole pairs show the strongest excitonic behaviour at a non-trivialvalue of interplane distance. This can be understood as follows: when d = 0, the interactionsbetween the particles are e!ectively screened since they all exist in the same space i.e. inEDISON 16 IOP PublishingJournal of Physics: Conference Series 193 (2009) 012133 doi:10.1088/1742-6596/193/1/0121333Figure 2. (Left) Correlation energy EC for various values of the distance d. (Right) Contourplot of the correlation energy in the (d,# ) plane.the two-dimensional plane. However, when there is a finite distance between the planes, thescreening between electrons and holes is reduced and, thus, the electron-hole attractive Coulombinteraction is reinforced. This enhanced attraction e!ect competes with the electron-electronand hole-hole repulsive interactions and with the suppression of the Coulomb interaction dueto the finite distance between the electrons and holes. The attractive interactions eventuallylose this competition when the repulsive interaction dominates the ground state of the system,which happens at a critical distance, as can be noted in the plot of the correlation energy.This last physical turning point brings us to the second important feature of the correlationenergy: at the aforementioned critical distance, the energy of the BCS state looses its superiorityover the Hartree-Fock energy, since the BCS state energy becomes larger than that of theHartree-Fock state. This means that the system is no longer suitably described throughcorrelated pairs i.e. the system has su!ered a structural change of quantum state, which isreminiscent of the quantum phase transitions that occur in large many-body systems. Thistransition point, together with the minimum of the correlation energy for finite d, are importantpieces of information for understanding the BEC state of indirect excitons in semiconductors.AcknowledgmentsThe authors acknowledge partial financial support from the CODI- Universidad de Antioquiaand the Centro de Investigaciones- Instituto Tecnológico Metropolitano de Medelĺın. I.M.S. isgrateful for useful discussions with J. L. Sanz on the Hartree-Fock method.References[1] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information, Cambridge UniversityPress 2000.[2] I. Zutic, J. Fabian and S. Das Sarma, 2004 Rev. Mod. Phys. 76 323.[3] M. Alatalo, M.A. Salmi, P. Pietilainen and Tapash Chakraborty, 1995 Phys. Rev. B 52 7845.[4] J. P. Eisenstein, G.S. Boebinger, L.N. Pfei!er, K.W. West and Song He, 1992 Phys. Rev. Lett. 68 1383.[5] S. K. Saikin, C. Emary, D. G. Steel and L. J. Sham, 2008 Phys. Rev. B 78 235314.[6] M. F. Doty, M. Scheibner, A. S. Bracker, I. V. Ponomarev, T. L. Reinecke, and D. Gammon, 2008 Phys. Rev.B 78 115316.[7] I. Mondragon-Shem, F. E. Lopez and B.A. Rodriguez. Submitted.EDISON 16 IOP PublishingJournal of Physics: Conference Series 193 (2009) 012133 doi:10.1088/1742-6596/193/1/0121334

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Coulomb correlations of a few body system of spatially separated charges

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