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Physical review. B, Condensed matter

Self-consistentGWfor a quasi-one-dimensional semiconductor

H. J. de Groot, P. A. Bobbert, W. van Haeringen

Published Oct 15, 1995• 35 citations
Publisher

Abstract

We present self-consistent GW calculations for a quasi-one-dimensional model semiconductor, using multipole representations for W, the screened interaction, and G, the electron Green function. In the case of G, we distinguish between the quasiparticle contribution and the so-called incoherent contribution. We consider various strengths of the external potential. For two different starting points of the self-consistency cycle, local-density approximation and Hartree-Fock, the band gaps converge to the same values. The self-consistent GW band gaps differ very little from the Hartree-Fock band gaps. There is considerable disagreement with quantum Monte Carlo calculations for the same model, indicating the importance of vertex corrections.

DOI
10.1103/physrevb.52.11000

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We present self-consistent GW calculations for a quasi-one-dimensional model semiconductor, using multipole representations for W, the screened interaction, and G, the electron Green function. In the case of G, we distinguish between the quasiparticle contribution and the so-called incoherent contribution. We consider various strengths of the external potential. For two different starting points of the self-consistency cycle, local-density approximation and Hartree-Fock, the band gaps converge to the same values. The self-consistent GW band gaps differ very little from the Hartree-Fock band gaps. There is considerable disagreement with quantum Monte Carlo calculations for the same model, indicating the importance of vertex corrections.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00
The concepts of quasiparticle and collective excitation are considered in the context of continued-fraction expansions. This leads to a practical scheme allowing a highly accurate description of the screened interaction function W and the electron self-energy \ensuremath{\Sigma}, which can be achieved at a reasonable computational cost. The usefulness of this scheme is demonstrated on a simple two-band model of a semiconductor, where it leads to an unambiguous calculation of the electron self-energy in the GW approximation. Conceptually, the theory clarifies some of the limitations of the standard quasiparticle concept in many-body theory. Furthermore, it allows a proper assessment of the quality of the standard approximations employed in GW calculations.
We extend the quasiparticle self-consistent approach beyond the GWGWGW approximation by using a range-separated vertex function. The developed approach yields band gaps, dielectric constants, and band positions with an accuracy similar to highest-level electronic-structure calculations without exceeding the cost of regular quasiparticle self-consistent GWGWGW. We introduce an exchange-correlation kernel that accounts for the vertex over the full spatial range. In the long range it complies with the Ward identity, while it is approximated through the adiabatic local density functional in the short range. In this approach, the renormalization factor is balanced and the higher-order diagrams are effectively taken into account.
Recently several schemes have been proposed to perform self-consistent GW calculations within a one-particle (quasiparticle) approximation. These so-called quasiparticle self-consistent GW schemes have been found to be successful in reproducing band gaps of semiconductors but there is some arbitrariness in the choice of the ``effective'' one-particle Hamiltonian in these schemes and their validity has not been assessed. To avoid ambiguity in choosing the one-particle Hamiltonian, we propose a scheme which is based on L\"owdin's method of symmetric orthogonalization. In our approach we first calculate the true quasiparticle wave functions and energies with the real part of the frequency-dependent self-energy, and then orthonormalize these states using L\"owdin's procedure to construct the effective quasiparticle Hamiltonian. L\"owdin's procedure ensures that the obtained orthonormal orbitals are the closest to the original nonorthogonal quasiparticle wave functions in the least-square sense and uniquely defines the one-particle Hamiltonian. Unlike previous approaches, this approach takes into account the full frequency dependence and the off-diagonal elements of the self-energy without ambiguity. As test cases, we perform quasiparticle self-consistent GW calculations on NiO and Gd. We find that our results compare well with previous results obtained using a different effective Hamiltonian.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.