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Numerical methods for the computation of densities of states of periodic operators

Ewen Lallinec, Antoine Levitt

Abstract

We present a comparative study of numerical methods for computingelectronic densities of states (DOS) in periodic systems. We provide a detailed analysis of the domain of validity of the Brillouincomplex deformation (BCD), a recently-proposed method promising exponential convergence without need for smearing. We compare on a range of systems the BCD with several methods, including the standard smearing and linear tetrahedron methods, as well as an adaptive integration method. Our results establish clear performance regimes for each method, offering practical guidance for DOS computations across a range of systems and accuracy requirements.

Numerical methods for the computation of densities of states of periodic operators

Abstract

We present a comparative study of numerical methods for computingelectronic densities of states (DOS) in periodic systems. We provide a detailed analysis of the domain of validity of the Brillouincomplex deformation (BCD), a recently-proposed method promising exponential convergence without need for smearing. We compare on a range of systems the BCD with several methods, including the standard smearing and linear tetrahedron methods, as well as an adaptive integration method. Our results establish clear performance regimes for each method, offering practical guidance for DOS computations across a range of systems and accuracy requirements.

Paper Structure

This paper contains 36 sections, 1 theorem, 55 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Theoretical settings
  3. Tight-binding Hamiltonian and band structure
  4. Density of states
  5. Smearing methods
  6. Periodic Trapezoidal Rule
  7. Iterated Adaptive Integration
  8. Smearing-free methods
  9. Linear Tetrahedron
  10. Brillouin Complex Deformation
  11. The Green function
  12. Complex deformation
  13. Choosing the deformation
  14. Choosing the parameters
  15. Validity of the BCD
  16. ...and 21 more sections

Key Result

Lemma 4.1

Let $I$ be a $\mathcal{R}^*$-periodic function, analytic in an open set $U=\mathbb{R}^d+i[-\eta,\eta]^d$. Then, for all $\mathcal{R}^*$-periodic and continuously differentiable functions $\mathbf{h}(\mathbf{k}):\mathbb{R}^d\rightarrow[-\eta,\eta]^d$, we have

Figures (16)

  • Figure 1: Examples of deformation for the monatomic chain in 1D (on the left) and graphene in 2D (on the right). We represent as vectors the complex deformation in 2D.
  • Figure 2: DOS computed with the BCD, displaying a systematic underestimation near $0$. With a finite sampling, oscillations occur near the transition.
  • Figure 3: Crossings between the third and fourth band, protected by the mirror symmetry. The color indicates the energy of the crossing along the nodal line.
  • Figure 4: Bands and DOS for the 8-bands graphene with a zoom between -7.5 and 9 eV. Zoom: The blue area indicates the energy range on which the crossings from \ref{['fig:nodal_line']} occur, the dark-red range corresponds to the BCD energy smearing range for $\Delta E=0.4$ eV.
  • Figure 5: Left panel: Three first bands ($G_1=0$, $G_2=-1$, $G_3=1$ ) with corresponding exact DOS $E^{-1/2}$ and DOS computed with the BCD for $\alpha=0.1$ and $\Delta E= 0.3$ eV. The blue line indicates $E=0.2$ eV. Right panel: Deformation field for the 1D free electron gas at $E=0.2$ eV with parameters $\Delta E=0.3$ and $\Delta E=0.2$ eV. The reference deformation field is computed by taking into account only the first band with $\Delta E=0.2$ eV.
  • ...and 11 more figures

Theorems & Definitions (1)

  • Lemma 4.1