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Application of quantum Monte Carlo methods to homogeneous electron and electron-hole systems

G. G. Spink

Published Dec 22, 2017• 0 citations
Publisher

Abstract

The properties of the macroscopic world around us, and of which we are a part, are largely determined by the low energy, collective behaviour of many interacting particles, including the nuclei and, especially, the electrons present. Although the fundamental laws governing the behaviour of these many-body systems are believed to be known in principle, the practical solution of the equations of quantum mechanics remains a challenging area of research. This thesis is concerned with the application of quantum Monte Carlo methods to two model systems: the spin-polarised homogeneous electron gas, and a hole-doped electron gas. Electronic structure theory is briefly reviewed before discussing in more detail the quantum Monte Carlo methods used in this thesis. A study of the three-dimensional spin-polarised homogeneous electron gas (HEG) is then reported, where the relatively new technique of twist averaging is investigated in detail and accurate energies and pair correlation functions are obtained over densities $r_s = 0.5 – 20$ a.u. and the full range of spin-polarisation, allowing comparison with the Perdew-Zunger interpolation scheme used in local spin density approximation exchange-correlation functionals. Following this, an impurity is added to the electron gas in the form of a positively charged hole, and the interaction is studied. Relaxation energies, pair correlation functions and momentum densities are reported. Trion formation is observed over a range of carrier densities and electron-hole mass ratios in agreement with experiment. Isolated trions are also studied, where the diffusion Monte Carlo method is exact. Methodological innovations developed while carrying out this work are discussed, including a variance reduction technique for twist-averaged calculations and a new trial wave function for impurity-in-HEG calculations.

DOI
10.17863/cam.22988

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The properties of the macroscopic world around us, and of which we are a part, are largely determined by the low energy, collective behaviour of many interacting particles, including the nuclei and, especially, the electrons present. Although the fundamental laws governing the behaviour of these many-body systems are believed to be known in principle, the practical solution of the equations of quantum mechanics remains a challenging area of research. This thesis is concerned with the application of quantum Monte Carlo methods to two model systems: the spin-polarised homogeneous electron gas, and a hole-doped electron gas. Electronic structure theory is briefly reviewed before discussing in more detail the quantum Monte Carlo methods used in this thesis. A study of the three-dimensional spin-polarised homogeneous electron gas (HEG) is then reported, where the relatively new technique of twist averaging is investigated in detail and accurate energies and pair correlation functions are obtained over densities rs=0.5–20r_s = 0.5 – 20rs​=0.5–20 a.u. and the full range of spin-polarisation, allowing comparison with the Perdew-Zunger interpolation scheme used in local spin density approximation exchange-correlation functionals. Following this, an impurity is added to the electron gas in the form of a positively charged hole, and the interaction is studied. Relaxation energies, pair correlation functions and momentum densities are reported. Trion formation is observed over a range of carrier densities and electron-hole mass ratios in agreement with experiment. Isolated trions are also studied, where the diffusion Monte Carlo method is exact. Methodological innovations developed while carrying out this work are discussed, including a variance reduction technique for twist-averaged calculations and a new trial wave function for impurity-in-HEG calculations.
We have studied the spin-polarized three-dimensional homogeneous electron gas using the diffusion quantum Monte Carlo method, with trial wave functions including backflow and three-body correlations in the Jastrow factor, and we have used twist averaging to reduce finite-size effects. Calculations of the pair-correlation function, including the on-top pair density, as well as the structure factor and the total energy, are reported for systems of 118 electrons in the density range rs=0.5{r}_{\mathrm{s}}=0.5rs​=0.5--20 a.u., and for spin polarizations of 0, 0.34, 0.66, and 1. We consider the spin resolution of the pair-correlation function and structure factor, and the energy of spin polarization. We show that a control variate method can reduce the variance when twist averaging, and we have achieved higher accuracy and lower noise than earlier quantum Monte Carlo studies.
This thesis comprises a trio of separate yet thematically-linked investigations within the field of electronic structure that address challenging problems to which quantum Monte Carlo, the most accurate electronic structure method capable of treating large extended systems, has been applied only rarely or indeed never before. The first is the study of polarization and localization in many body systems. Density functional and variational quantum Monte Carlo methods are used to examine the behaviour of the many-electron localization length near band insulator to metal transitions in various oneand two-dimensional model systems. We examine when the localization length, known to be infinite in metals and finite in insulators and assumed to diverge at a metal-insulator transition, serves as a useful diagnostic of the transition. We also present a novel derivation of the formulae for polarization and localization in many body systems that expresses the connection of these two quantities to the concept of Many Body Wannier Functions. The second is a study of the surface energy of the electron gas. The electron gas, or ‘jellium’, is an important benchmark system for electronic structure methods and serves as a simple model for metals. We apply diffusion Monte Carlo (DMC), and a new method for determining surface energies without an unreliable comparison of bulk and slab results, to the calculation of the surface energy. Our method improves on previous calculations in a number of ways and removes many sources of error that plagued previous calculations. The final study applies DMC to the properties, specifically the formation energy, of point defects in Alumina. Alumina is an important technological material, and there are many important unanswered questions relating to its properties that stem from the behaviour and concentration of its point defects. We address many of the problems associated with the calculation of formation energies with electronic structure calculations, and present the first calculations of formation energies of charged point defects in DMC.
The work in this thesis is concerned with the application and development of quantum Monte Carlo (QMC) methods. We begin by proposing a technique to maximise the efficiency of the extrapolation of DMC results to zero time step, finding that a relative time step ratio of 1:4 is optimal. We discuss the post-processing of QMC data and the calculation of accurate error bars by reblocking, setting out criteria for the choice of block length. We then quantify the effects of uncertainty in the correlation length on estimated error bars, finding that the frequency of outliers is significantly increased for short runs. We then report QMC calculations of biexciton binding energies in bilayer systems. We have also calculated exciton-exciton interaction potentials, and radial distribution functions for electrons and holes in bound biexcitons. We find a larger region of biexciton stability than other recent work [C. Schindler and R. Zimmermann, Phys. Rev. B 78, 045313 (2008)]. We also find that individual excitons retain their identity in bound biexcitons for large layer separations. Finally, we give details of a QMC study of the one-dimensional homogeneous electron gas (1D HEG). We present calculations of the energy, pair correlation function, static structure factor (SSF), and momentum density (MD) for the 1D HEG. We observe peaks in the SSF at even-integer-multiples of the Fermi wave vector, which grow as the coupling is increased. Our MD results show an increase in the effective Fermi wave vector as the interaction strength is raised in the paramagnetic harmonic wire; this appears to be a result of the vanishing difference between the wave functions of the paramagnetic and ferromagnetic systems. We have extracted the Luttinger liquid exponent from our MDs by fitting to data around the Fermi wave vector, finding good agreement between the exponents of the ferromagnetic infinitely-thin and harmonic wires.
The two-dimensional (2D) homogeneous electron gas (HEG) is a fundamental model in quantum many-body physics. It is important to theoretical and computational studies, where exchange-correlation energies computed in it serve as the foundation for density-functional calculations. It is also of direct relevance to a variety of experimental settings, especially with the rapid recent growth in 2D materials and moir\'e systems. In these experiments, metallic gates are often present, which screen the Coulomb interaction between electrons. The effect of the screening can qualitatively change the behavior of the 2D HEG, and requires accurate many-body computations to capture. In this paper, we perform state-of-the-art diffusion Monte Carlo (DMC) calculations in the 2D HEG subjected to symmetric dual-gate screening. We systematically compute the correlation energy across a range of densities and gate separations for both spin unpolarized and fully polarized systems. A global fit is obtained for the correlation energy, using these data and imposing various limiting behaviors obtained from perturbation analysis. The functional will allow density-functional calculations to be performed for a variety of realistic experimental setups, which can accurately account for the presence of gates. We also investigate how the gate screening affects the bulk modulus, pair correlation function, and the structure factor of the 2D HEG, which can potentially be probed in experiments.
We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.
Electron and hole excitations in semiconductors may be approximated as particles with effective masses which interact via Coulomb potentials. We study systems of electrons and holes with quantum Monte Carlo (QMC) techniques, covering three related areas: (1) elastic scattering of excitons, (2) thermodynamics of electron-hole plasmas, and (3) electrons confined in a quantum dot. Excitons are bound states of an electron and a hole, and obey Bose statistics. A low density exciton gas is an experimentally realizable dilute Bose gas. The scattering length a8 of a dilute Bose gas determines its properties, but is difficult to calculate. We present an essentially exact QMC treatment of exciton-exciton scattering, and find scattering lengths for different spin orientations of the excitons. At some mass ratios mh/me the scattering lengths diverge in conjunction with the appearance of biexciton vibrational states, an effect not found by earlier perturbative treatments. Path integral Monte Carlo (PIMC) is used to model the thermodynamics of the electron-hole plasma. Our primary interest is the Bose condensation of an excitonic gas. At low density and low temperature the spin-unpolarized system forms biexcitons. Since we are interested in Bose condensation, we study a spin-polarized system, which has no biexcitons. Restricted paths are used to handle the Fermion sign problem. With an appropriate choice of paired nodes for the restricted path approximation we find an excitonic Bose condensate. The energy of the low temperature, low density exciton gas determined from PIMC agrees well with the theory of dilute Bose gases, in which our previously calculated scattering length is used to model the exciton-exciton interactions. At higher densities the excitons are less well defined and the transition changes character. Finally, we study electrons in the inhomogeneous environment of a self-assembled InAs-GaAs quantum dot. We combine our ground state QMC treatment with another method, density functional theory (DFT) within the local spin density approximation (LSDA). Our comparison shows that LSDA is acceptable for treating interactions in the case considered, but recommend further tests for application of LSDA to larger dots or coupled dot systems.