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Application of quantum Monte Carlo methods to excitonic and electronic systems

Robert Lee

Published Jul 12, 2011• 1 citation
Publisher

Abstract

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.

DOI
10.17863/cam.16574

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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.
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.
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 present calculations of the energy, pair-correlation function (PCF), static structure factor (SSF), and momentum density (MD) for the one-dimensional electron gas using the quantum Monte Carlo method. We are able to resolve 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 kF{k}_{\mathrm{F}}kF​, finding good agreement between the exponent of the ferromagnetic infinitely thin wire and the ferromagnetic harmonic wire.
Quantum Monte Carlo (QMC) methods are a class of stochastic techniques that can be used to compute the properties of electronic systems accurately from first principles. This thesis is mainly concerned with the development of trial wave functions for QMC. An extension of the backflow transformation to inhomogeneous electronic systems is presented and applied to atoms, molecules and extended systems. The backflow transformation I have developed typically retrieves an additional 50% of the remaining correlation energy at the variational Monte Carlo level, and 30% at the diffusion Monte Carlo level; the number of parameters required to achieve a given fraction of the correlation energy does not appear to increase with system size. The expense incurred by the use of backflow transformations is investigated, and it is found to scale favourably with system size. Additionally, I propose a single wave function form for studying the electron-hole system which includes pairing effects and is capable of describing all of the relevant phases of this system. The effectiveness of this general wave function is demonstrated by applying it to a particular transition between two phases of the symmetric electron-hole bilayer, and it is found that using a single wave function form gives a more accurate physical description of the system than using a different wave function to describe each phase. Both of these developments are new, and they provide a powerful set of tools for designing accurate wave functions. Backflow transformations are particularly important for systems with repulsive interactions, while pairing wave functions are important for attractive interactions. It is possible to combine backflow and pairing to further increase the accuracy of the wave function. The wave function technology that I have developed should therefore be useful across a very wide range of problems.
Fermi liquid theory is the basic paradigm within which we understand the normal behavior of interacting electron systems, but quantitative values for the parameters that occur in this theory are currently unknown in many important cases. One such case is the two-dimensional homogeneous electron gas (2D HEG), which is realized in a wide variety of semiconductor devices. We have used quantum Monte Carlo (QMC) methods to calculate the Landau interaction functions between pairs of quasiparticles. We use these to study the Fermi liquid parameters, finding that finite-size effects represent a serious obstacle to the direct determination of Fermi liquid parameters in QMC calculations. We have used QMC data in the literature for other properties of the 2D HEG to assemble a set of ``best available'' values for the Fermi liquid parameters.
Variational and diffusion Monte Carlo (VMC and DMC) are widely used techniques in the Monte Carlo study of many-body quantum systems. Through efficient use of ran- dom sampling to solve many-dimensional integrals, they can be used to reliably solve the Schrödinger equation. Within the wider ecosystems of electronic structure calculation, quantum chemistry, and quantum simulation, quantum Monte Carlo (QMC) methods are used in tandem with density functional theory (DFT) and other computational methods to deliver insight into a vast array of condensed matter systems. Of particular interest to this work are systems in the intermediate correlation regime, where methods such as simple band theory and the Hubbard model break down. This thesis proposes and tests an explicit three-body backflow term to improve the accuracy of VMC and DMC calculations by incorporating higher-order correlation in order to reduce fixed-node er- ror in DMC, finding statistically significant improvements in ground state energies and improved variance in a water molecule. Then, QMC is used to calculate accurate charge densities for weak and intermediately correlated materials, to test commonly used dens- ity functional approximations in a controlled environment, providing insight into where these approximations perform well, and where they fail. Finally, DMC is used to generate ground state energy data for a dipolar Fermi gas of dressed Rydberg atoms interacting in the intermediate correlation regime. This data is parametrised in terms of interaction strength, for use in generating exchange-correlation functionals, to enable the DFT study of inhomogeneous systems of dressed Rydberg atoms.