Quantum Monte Carlo Simulations of Exciton Condensates J. Shumway a and D. M. Ceperley b a Dept. of Physics and Astronomy, Arizona State University, Tempe, AZ 8583 b Dept. of Physics, University of Illinois, 1110 W. Green St., Urbana, IL 61801 Abstract We have studied scattering states and thermodynamic properties of electron-hole systems. Starting from the constituent electrons and holes and Coulomb interactions, we have used quantum Monte Carlo simulation techniques to sample properties of wavefunctions and thermal density matrices. We have studied three types of systems: (1) the scattering of two excitons, with full-quantum treatment of the four constituent particles, () the thermodynamic equilibrium of 14 electron-hole pairs having two spin states for each particle, which form excitons and biexcitons at low temperatures, and (3) the thermodynamic equilibrium of 7 spin-polarized electron-hole pairs, which form a dilute exciton gas that undergoes Bose condensation at low temperatures. We compare our results with predictions of the Saha equation for exciton and biexciton formation, and Bogoliubov theory for the energy of the dilute Bose gas of excitons. We also discuss the outlook for future quantum Monte Carlo simulations on these systems. Key words: A. Semiconductors, D. Excitons, D. Bose-Einstein Condensation, E: Computer Simulations PACS: 71.35.Cc, 36.10.Dr 1 Introduction Electron-hole systems in semiconductors have been a source of interesting new physics for forty years. The bound state of an electron and a hole, an exciton, is a neutral bosonic excitation of a semiconductor. Formation of a Bose-Einstein condensate (BEC) of excitons has been a target of many experiments [1,], though none have produced clear proof of Bose condensation. Some issues that arise concern the exciton-exciton scattering length, the rate of Auger decay [3], the formation of biexcitons or electron-hole liquids, and the nature of the BEC transition at high densities [4]. Preprint submitted to Elsevier Science June 004
Theoretically, an electron-hole system is an interacting quantum system. Interactions typically make exact solutions impossible, leading theorists to consider perturbative solutions or numerical simulations. Quantum Monte Carlo (QMC) techniques use random sampling to estimate properties of interacting quantum systems. The accuracy of QMC is often limited only by computer time and by small errors from fermion fixed-node approximations. Good scaling of computer time with the number of particles (usually low-order polynomial or even linear scaling) and the ability to treat detailed models has made QMC appealing for nanometer scale systems, such as semiconductor heterostructures. In this work we summarize our QMC calculations on bulk electron-hole systems. Numerical results have appeared previously in Refs [5 7], but some of the analysis presented here is new. We have taken as a starting model for interacting electrons and holes the effective-mass Hamiltonian H = N i=1 p i + q i q j, (1) m i i<j r ij with m e = m h = 1, q e = q h = e = 1, and = 1. In these units the exciton has an energy E X = 0.5 and radius a X =. We describe the density by a radius r s that satisfies 4 3 πr3 S = V/N eh, where V is the simulation volume and N eh is the number of electron-hole pairs. This simple Hamiltonian omits many effects, such as radiative recombination, multiple-band coupling, anisotropic effective masses, band non-parabolicity, and frequency and momentum dependence of the dielectric function. Even with these approximations, this interacting Hamiltonian still dictates a rich variety of states, including excitons, biexcitons, ionized plasmas, and excitonic Bose condensation. It is sensible to first study this simplified model; we anticipate that future QMC will utilize more realistic Hamiltonians. To study the properties of the electron-hole system, Eq. (1), we have constructed QMC simulations that use random walks to sample quantum and thermal expectation values of wavefunctions or density matrices. We have conducted three series of simulations: Exciton-exciton scattering. As described in detail in Ref. [7], we have calculated phase shifts and scattering lengths for elastic collisions of two excitons. The formalism starts from four-particle variational wavefunctions describing low-energy scattering states and any bound states. We fix a node (Ψ = 0) in the relative wavefunciton at a series of relative separation distance in order to discretize the scattering spectrum and then probe a series of relative momenta. The essentially exact energies of these scattering states were sampled using an excited state diffusion Monte Carlo algorithm [8,9]. The phase shifts and scattering lengths were inferred from the calculated relationship between energy and the node position.
Formation of excitons and biexcitons. To study the equilibrium phases of the e-h system, we developed QMC simulations based on Feynman path integrals [10,11]. To handle the fermion sign problem we used a free-particle fixed node approximation [1]. From these path integral simulations, we computed spin-dependent pair correlation functions for electrons and holes. These correlation functions indicate exciton formation as short-range attraction of electrons and holes, and biexciton formation as short-range attraction between two electrons (or holes) with opposite spin. A more accurate identification of excitons and biexcitons could be obtained by sampling the two-body density matrix and identifying natural orbitals. We conducted a simulation of 14 unpolarized electron-hole pairs and studied their pair correlation functions at different temperatures. Bose-condensation of excitons. Path integrals are a natural language for discussing BEC of interacting particles [10,11]. The phase transition appears as permuting paths that extend many times the interparticle spacing, and the phase transition itself resembles classical percolation [10,11]. Superfluidity can be calculated from path winding, condensate fraction can be calculated from the off-diagonal electron-hole density matrix, and other thermodynamic quantities such as energy and specific heat can indirectly indicate the BEC transition. We studied the energetics and superfluid order parameter for 19 spin-polarized electron hole pairs at different densities. Results of Computer Simulations Our computer simulation data were previously published in [5 7]. Here we report on new analysis of the data. Exciton-exciton scattering. Since publication of Ref. [7] we have learned of a better analysis technique based on an effective range expansion [13,14], k cot δ 0 (k) = 1 a + 1 r ek. () We have refit our data from Ref. [7] using effective range theory and present the values in Table. 1. Formation of excitons and biexcitons. Path integral QMC calculations of 14 unpolarized electron-hole (eh) pairs at a density r s /a X = 5 showed evidence of exciton and biexciton formation in pair correlation functions [5]. To extend this analysis, we now estimate the fraction of e-h pairs in excitons and biexcitons by integrating the pair correlation function out to a radius of.5 a X and 3.0 a X. We take an enhancement of electron-electron pairing as a clear evidence of a biexciton, and any additional enhancement of electron-hole pairing (beyond 3
Table 1 Scattering length for various elastic scattering processes, in units of excitonic radius. Here para and ortho refer to the exciton singlet and triplet state, repectively. From Ref. [7], with new analysis using effective range theory [14]. process a s (excitonic radii) para-para.18(7) ortho-orth (S=0) 3.759() ortho-orth (S=).18(7) ortho-para 0.706(14) ortho-ortho para-para -1.411(17) that expected in the biexciton) as evidence of isolated excitons. Figure 1 shows our estimated exciton and biexciton fractions. We compare these results to the Saha equation. Assuming three phases: an ionized e-h plasma (P), an exciton gas (X), and a biexciton gas (XX), classical gas equations give the following chemical potentials for eh pairs, [ ] N µ p eh = k n p BT log V λ3 e µ X eh = k B T log N V = k BT log N V µ XX eh ( λe ) 3 n X E X 4 ( ) 3 λe n XX E XX where n p, n X, and n XX are the percentage of e-h pairs in each phase, and E X and E XX are exciton and biexciton binding energies (E XX =.064E X ). Equating chemical potentials gives n X = 3 n p ( N V λ3 e n XX = n4 p ) exp E X k B T ( ) N 3 V λ3 e exp E XX k B T, which, along with the normalization condition n p + n X + n XX = 1, can be solved for n p, n X, and n XX as a function of temperature and density. We show the Saha equation predictions for excitons and biexcitons in Fig. 1, where we see good agreement with path integral QMC results. Bose condensation of excitons. In Ref. [6] we reported superfluid densities and energetics of the BEC transition. To see BEC in our QMC simulations we found it was necessary to add pairing the the fixed-node restriction. In Fig. we show one of our most convincing results, the energy of an eh system as the (3) (4) 4
Fig. 1. Fraction of electron-hole pairs bound in excitons (solid line and circles) and biexcitons (dashed line and s) as a function of temperature at a density r s /a x = 5. Data points are estimated by integrating PIMC pair correlation functions previously published in Ref. [6]. Lines are predictions of the Saha equation. Fig.. Energy as a function of density at constant temperature, k B T = 1 8 E x. Inset shows the free particle transition temperature T C (solid line) and the k B T = 1 8 E x isotherm, which crosses T C at r s = 4.04 a X. Adapted from Ref. [6]. density is varied across the BEC transition. The theoretical curves show the energies of a dilute classical gas and a Bose gas with Bogoliubov theory [15], using the a s =.18a X value from our scattering calculations. The energetics are strongly suggestive of a BEC transition at r s 4a X. 5
3 Conclusions and Outlook These simulations illustrate the ability of QMC to identify and quantify features in the phase diagram of an electron-hole system. The most crucial technical issue to be addressed is a study of the fixed-node approximation in the context of the BEC transition. A natural extension of these calculations would be to consider more detailed models, especially coupled quantum well structures. Anisotropic mass and three-dimensional confining potentials are straightforward to add to QMC and would allow direct contact with current experiments. Acknowledgments: We thank Jim Mitroy for his comments to us about scattering calculations and the use of effective range theory. Work supported by NSF grants DMR 98-0373, DGE 93-54978, DMR-039819 and computer resources at NCSA. References [1] D. W. Snoke, J. P. Wolfe, A. Mysyrowicz, Phys. Rev. B 41 (1990), 11171. [] E. Fortin, S. Fafard, A. Mysyrowicz, Phys. Rev. Lett. 70 (1993), 3951. [3] K. E. O Hara, J. R. Gullingsrud, J. P. Wolfe, Phys. Rev. B 60 (1999), 1087. [4] A. J. Leggett, in: Modern Trends in Condensed Matter Physics, edited by A. Pekalski, J. Przystawa, pp. 14 7 (Springer, Berlin, 1980). [5] J. Shumway, Quantum Monte Carlo Simulations of Electrons and Holes, Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (1999). [6] J. Shumway, D. M. Ceperley, J. Phys. IV France 10 (000), Pr5. [7] J. Shumway, D. M. Ceperley, Phys. Rev. B 63 (001), 16509. [8] D. M. Ceperley, B. Bernu, J. Chem. Phys. 89 (1988), 6316. [9] B. Bernu, D. M. Ceperley, W. A. Lester, Jr., J. Chem. Phys. 93 (1990), 55. [10] R. P. Feynman, Statistical Mechanics (Addison-Wesley, Reading, MA, 197). [11] D. M. Ceperley, Rev. Mod. Phys. 67 (1995), 79. [1] D. M. Ceperley, Phys. Rev. Lett. 69 (199), 331. [13] G. F. Chew, M. L. Goldberger, Phys. Rev 75 (1949), 1637. [14] I. A. Ivanov, Phys. Rev. A 67 (003), 03704. [15] A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963). 6