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High-throughput electronic band structure calculations: challenges and tools

Wahyu Setyawan, Stefano Curtarolo

arXiv:1004.2974v1cond-mat.mtrl-sci

TL;DR

Reliable band-structure workflows require consistent lattice and reciprocal-space conventions. This paper presents AFLOW’s standardized lattice-vector and symmetry-point framework and illustrates it with band-structure examples across Brillouin-zone shapes.

  • Problem

    Band-structure calculations require explicit, consistent conventions for lattice vectors and reciprocal-space symmetry-point coordinates.

  • Method

    AFLOW specifies lattice vectors and represents symmetry k-point coordinates as fractions of primitive reciprocal-lattice vectors.

  • Results

    The paper presents selected band-structure examples for each Brillouin-zone shape, with the Fermi energy shifted to the valence-band maximum.

  • Takeaways & Limitations

    The framework supplies standardized lattice and reciprocal-space descriptions for constructing and comparing band-structure calculations.

Abstract

from arXiv · show

The article is devoted to the discussion of the high-throughput approach to band structures calculations. We present scientific and computational challenges as well as solutions relying on the developed framework (Automatic Flow, AFLOW/ACONVASP). The key factors of the method are the standardization and the robustness of the procedures. Two scenarios are relevant: 1) independent users generating databases in their own computational systems (off-line approach) and 2) teamed users sharing computational information based on a common ground (on-line approach). Both cases are integrated in the framework: for off-line approaches, the standardization is automatic and fully integrated for the 14 Bravais lattices, the primitive and conventional unit cells, and the coordinates of the high symmetry k-path in the Brillouin zones. For on-line tasks, the framework offers an expandable web interface where the user can prepare and set up calculations following the proposed standard. Few examples of band structures are included. LSDA+U parameters (U, J) are also presented for Nd, Sm, and Eu.

1. Appendix A … 1.3. Body-centered Cubic (BCC, cI)

Appendix A specifies AFLOW’s lattice-vector and reciprocal-space conventions, then lists symmetry k-points and Brillouin-zone paths for cubic, FCC, and BCC lattices. The BCC path is Γ-H-N-Γ-P-H|P-N, with an example band structure referenced.

  • 1. Appendix A: AFLOW defines conventional lattice parameters a, b, c, α, β, γ and expresses symmetry k-point coordinates as fractions of primitive reciprocal vectors b1, b2, b3.When primitive and conventional lattices coincide, AFLOW calls the structure simply “lattice.”
  • 1.1. Cubic (CUB, cP): The cubic lattice uses orthogonal conventional vectors a1 = (a, 0, 0), a2 = (0, a, 0), and a3 = (0, 0, a).
  • 1.1. Cubic (CUB, cP): Table 2 lists the symmetry k-points for the CUB lattice.
  • 1.1. Cubic (CUB, cP): The CUB Brillouin-zone path is Γ-X-M-Γ-R-X|M-R, with an example band structure provided in Figure 26.
  • 1.2. Face-centered Cubic (FCC, cF): Table 3 lists the symmetry k-points for the FCC lattice.
  • 1.2. Face-centered Cubic (FCC, cF): The FCC Brillouin-zone path is Γ-X-W-K-Γ-L-U-W-L-K|U-X, with an example band structure provided in Figure 27.
  • 1.3. Body-centered Cubic (BCC, cI): Table 4 lists the symmetry k-points for the BCC lattice.
  • 1.3. Body-centered Cubic (BCC, cI): The BCC Brillouin-zone path is Γ-H-N-Γ-P-H|P-N, with an example band structure provided in Figure 28.

1.4. Tetragonal (TET, tP) · 1.5. Body-centered Tetragonal (BCT, tI)

The tetragonal sections specify lattice constructions and standardized Brillouin-zone symmetry paths for TET and the two body-centered tetragonal variants, BCT1 and BCT2. Each listed path is paired with an example band structure in the corresponding figure.

  • 1.4. Tetragonal (TET, tP): TET uses lattice vectors a1 = (a, 0, 0), a2 = (0, a, 0), and a3 = (0, 0, c).
  • 1.4. Tetragonal (TET, tP): The TET symmetry path is Γ-X-M-Γ-Z-R-A-Z|X-R|M-A.An example band structure using this path is given in Figure 29.
  • 1.5. Body-centered Tetragonal (BCT, tI): BCT is defined with a3 = (a/2, a/2, −c/2).
  • 1.5. Body-centered Tetragonal (BCT, tI): BCT has two variants: BCT1 for c < a and BCT2 for c > a.
  • 1.5. Body-centered Tetragonal (BCT, tI): The BCT1 symmetry path is Γ-X-M-Γ-Z-P-N-Z1-M|X-P.An example band structure using this path is given in Figure 30.
  • 1.5. Body-centered Tetragonal (BCT, tI): The BCT2 symmetry path is Γ-X-Y-Σ-Γ-Z-Σ1-N-P-Y1-Z|X-P.An example band structure using this path is given in Figure 31.

1.6. Orthorhombic (ORC, oP) · 1.7. Face-centered Orthorhombic (ORCF, oF)

The orthorhombic sections specify standardized lattice conventions and Brillouin-zone symmetry paths for ORC and the ORCF variants. ORCF is subdivided into cases by reciprocal-lattice inequalities, with corresponding k-point tables and example band-structure paths.

  • 1.6. Orthorhombic (ORC, oP): ORC uses the conventional-lattice ordering a < b < c.Its lattice vectors are a1 = (a, 0, 0), a2 = (0, b, 0), and a3 = (0, 0, c).
  • 1.6. Orthorhombic (ORC, oP): The ORC symmetry path is Γ-X-S-Y-Γ-Z-U-R-T-Z|Y-T|U-X|S-R.The path is illustrated in the ORC Brillouin zone, with an example band structure provided in Figure 32.
  • 1.7. Face-centered Orthorhombic (ORCF, oF): ORCF uses the conventional-lattice ordering a < b < c.The section also defines a3 = (a/2, b/2, 0).
  • 1.7. Face-centered Orthorhombic (ORCF, oF): ORCF1 is defined by 1/a2 > 1/b2 + 1/c2.This reciprocal-lattice inequality distinguishes the ORCF1 variation.
  • 1.7. Face-centered Orthorhombic (ORCF, oF): ORCF2 is defined by 1/a2 < 1/b2 + 1/c2.This reciprocal-lattice inequality distinguishes the ORCF2 variation.
  • 1.7. Face-centered Orthorhombic (ORCF, oF): Table 9 lists symmetry k-points for ORCF1 and ORCF3.The associated Brillouin-zone paths are presented separately for these variants.
  • 1.7. Face-centered Orthorhombic (ORCF, oF): The ORCF1 path is Γ-Y-T-Z-Γ-X-A1-Y|T-X1|X-A-Z|L-Γ, with an example band structure in Figure 33.This path corresponds to the ORCF1 Brillouin zone.
  • 1.7. Face-centered Orthorhombic (ORCF, oF): The ORCF3 path is Γ-Y-T-Z-Γ-X-A1-Y|X-A-Z|L-Γ, with an example in Figure 35; ORCF2 uses Γ-Y-C-D-X-Γ-Z-D1-H-C|C1-Z|X-H1|H-Y|L-Γ, exemplified in Figure 34.Tables 9 and 10 provide the corresponding symmetry k-points for ORCF1/ORCF3 and ORCF2.

1.8. Body-centered Orthorhombic (ORCI, oI)

This section specifies the conventional-cell ordering for the body-centered orthorhombic ORCI lattice and gives its Brillouin-zone symmetry path for band-structure calculations.

  • The conventional lattice is ordered as a < b < c.
  • Table 11 lists the symmetry k-points of the ORCI lattice.
  • The ORCI Brillouin-zone path is Γ-X-L-T-W-R-X1-Z-Γ-Y-S-W|L1-Y|Y1-Z, with an example band structure provided in Figure 36.

1.9. C-centered Orthorhombic (ORCC, oS)

For the C-centered orthorhombic lattice, the conventional lattice is ordered with a < b, and its Brillouin-zone symmetry path is specified for band-structure calculations.

  • The conventional lattice ordering is a < b.
  • Table 12 lists the symmetry k-points of the ORCC lattice.
  • The Brillouin-zone path is Γ-X-S-R-A-Z-Γ-Y-X1-A1-T-Y|Z-T.An example band structure using this path is given in Figure 37.

1.10. Hexagonal (HEX, hP) √

For the hexagonal lattice, the framework specifies symmetry k-points and the Brillouin-zone path Γ-M-K-Γ-A-L-H-A|L-M|K-H. Figure 38 provides an example band structure calculated using this path.

  • Hexagonal (HEX, hP): The hexagonal lattice has a defined set of symmetry k-points.These symmetry k-points are listed in Table 13.
  • Hexagonal (HEX, hP): The Brillouin-zone path is Γ-M-K-Γ-A-L-H-A|L-M|K-H.The path is shown for the hexagonal lattice in Figure 13.
  • Hexagonal (HEX, hP): An example band structure using this path is given in Figure 38.The example applies the specified hexagonal-lattice path.

1.11. Rhombohedral (RHL, hR)

The rhombohedral lattice section specifies its lattice-vector construction and provides symmetry k-points and Brillouin-zone paths for the RHL1 and RHL2 variants.

  • Rhombohedral lattice: Rhombohedral lattice vectors are defined using the lattice parameter a and angle α.The vectors are expressed through trigonometric functions of α and α/2.
  • Symmetry k-points: Symmetry k-points are tabulated separately for RHL1 and RHL2.The corresponding lists are provided in Tables 14 and 15.
  • RHL1: RHL1 uses the path Γ-L-B1|B-Z-Γ-X|Q-F-P1-Z|L-P for band-structure calculations.The path follows the Brillouin zone of the RHL1 lattice; an example band structure is given in Figure 39.
  • RHL2: RHL2 uses the path Γ-P-Z-Q-Γ-F-P1-Q1-L-Z for band-structure calculations.The path follows the Brillouin zone of the RHL2 lattice; an example band structure is given in Figure 40.

1.12. Monoclinic (MCL, mP)

The monoclinic (MCL, mP) lattice is defined by an ordered set of lattice parameters and a specific primitive-vector representation. Its Brillouin-zone treatment uses standardized symmetry k-points and the path Γ-Y-H-C-E-M1-A-X-H1|M-D-Z|Y-D for band-structure calculations.

  • 1.12. Monoclinic (MCL, mP): The lattice ordering is a, b ≤ c, with α < 90° and β = γ = 90°.The primitive vectors are a1 = (a, 0, 0), a2 = (0, b, 0), and a3 = (0, c cosα, c sin α).
  • 1.12. Monoclinic (MCL, mP): The MCL Brillouin zone is associated with a defined set of symmetry k-points.These points are listed in Table 16.
  • 1.12. Monoclinic (MCL, mP): The standardized MCL band-structure path is Γ-Y-H-C-E-M1-A-X-H1|M-D-Z|Y-D.An example band structure using this path is given in Figure 41.

1.13. C-centered Monoclinic (MCLC, mS)

The C-centered monoclinic lattice is organized into five cases, MCLC1–MCLC5, using lattice conventions and reciprocal-space conditions. Each case has specified symmetry k-points and a Brillouin-zone path for band-structure calculations.

  • Lattice convention: MCLC is defined with a, b ≤ c, α < 90°, and β = γ = 90° in the conventional lattice.The conventional lattice vectors include a2 = (−a/2, b/2, 0) and a3 = (0, c cosα, c sin α).
  • MCLC3 and MCLC4: MCLC3 and MCLC4 are distinguished by the reciprocal-space inequality involving b cosα/c + b2 sin2 α/a2, with equality defining MCLC4.Both satisfy kγ < 90°, and I is equivalent to F in MCLC4.
  • MCLC5: MCLC5 satisfies the greater-than condition and has its own symmetry k-points and path, Γ-Y-F-L-I|I1-Z-H-F1|H1-Y1-X-Γ-N|M-Γ.An example band structure using this path is given in Figure 46.
  • MCLC1 and MCLC2: MCLC1 and MCLC2 have distinct symmetry k-points and Brillouin-zone paths, with Y equivalent to X in MCLC2.The paths are Γ-Y-F-L-I|I1-Z-F1|Y-X1|X-Γ-N|M-Γ for MCLC1 and Γ-Y-F-L-I|I1-Z-F1|N-Γ-M for MCLC2.

1.14. Triclinic (TRI, aP)

The triclinic section defines the lattice and distinguishes TRI1a/2a and TRI1b/2b variants by reciprocal-angle conditions. It specifies symmetry-point tables and a common Brillouin-zone path, with example band structures referenced for TRI1a and TRI1b.

  • Lattice definition: The triclinic lattice is parameterized by vectors a1, a2, and a3 using lattice lengths and interaxial angles.The supplied lattice expression gives a1 = (a, 0, 0) and a2 = (b cosγ, b sin γ, 0), while the a3 expression is truncated.
  • Triclinic variations: The section distinguishes TRI1a by reciprocal-angle conditions kα > 90°, kβ > 90°, and kγ > 90°.The passage also introduces variations, but only the TRI1a conditions are supplied in full.
  • Symmetry k-points: Symmetry k-points are tabulated separately for TRI1a and TRI2a.These variants are covered in Table 20.
  • Brillouin-zone paths: The Brillouin-zone path for the triclinic variants is X-Γ-Y|L-Γ-Z|N-Γ-M|R-Γ.Figures 22–25 show this path for TRI1a, TRI2a, TRI1b, and TRI2b; example band structures using it are referenced for TRI1a and TRI1b.
  • Symmetry k-points: Symmetry k-points are also tabulated separately for TRI1b and TRI2b.These variants are covered in Table 21.

2. Appendix B

Appendix B presents selected band-structure examples spanning the Brillouin-zone shapes associated with the 14 Bravais lattices. The figures shift the Fermi energy to the valence-band maximum and plot orbital-projected total density of states logarithmically.

  • Appendix B: The figures shift the Fermi energy to the valence-band maximum at zero.This zero establishes the energy reference used for the displayed band structures.
  • Appendix B: Each figure plots the orbital-projected total density of states N(E) in the right panel on a logarithmic scale.The density-of-states display accompanies the band-structure panels.
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