hsugawa8651/brillouin.jl

Brillouin zones and paths for dispersion calculations in Julia.

Brillouin

Brillouin.jl is a Julia package that provides tools to generate and visualize k-space paths and Brillouin zones for eigenvalue problems in crystals. The k-path functionalities are inspired by the Python SeeK-path package (and return equivalent paths in 3D).

Examples

Wigner-Seitz cells & Brillouin zones

To generate the Brillouin zone of a crystal in space group 147 (Hermann-Mauguin symbol, P-3; Bravais type, hP), we first define its reciprocal basis Gs (e.g., using Bravais.jl) and then call Brillouin's wignerseitz:

julia> using Brillouin, 
julia> using Bravais: reciprocalbasis
julia> Rs = ([1.0, 0.0, 0.0], [-0.5, sqrt(3)/2, 0.0],   [0, 0, 1.25]) # direct basis for space group 147
julia> Gs = reciprocalbasis(Rs) # using Bravais to create the reciprocal basis
julia> cell = wignerseitz(Gs)   # construct associated Brillouin zone
Cell{3} (8 faces, 12 vertices):
  verts: [0.666667, -0.333333, -0.5]
         [0.333333, -0.666667, -0.5]
         [0.666667, -0.333333, 0.5]
         [0.333333, 0.333333, 0.5]
         [0.333333, 0.333333, -0.5]
         [0.333333, -0.666667, 0.5]
         [-0.333333, 0.666667, -0.5]
         [-0.666667, 0.333333, -0.5]
         [-0.333333, -0.333333, -0.5]
         [-0.333333, -0.333333, 0.5]
         [-0.666667, 0.333333, 0.5]
         [-0.333333, 0.666667, 0.5]
  faces: [5, 4, 3, 1]
         [8, 9, 10, 11]
         [2, 1, 3, 6]
         [2, 6, 10, 9]
         [7, 5, 1, 2, 9, 8]
         [4, 12, 11, 10, 6, 3]
         [4, 5, 7, 12]
         [11, 12, 7, 8]
  basis: [6.283185, 3.627599, -0.0]
         [0.0, 7.255197, 0.0]
         [0.0, -0.0, 5.026548]

The returned vertices are in the coordinates of the reciprocal basis (to convert, see cartesianize(!)); this is the default behavior in Brillouin. The basis is accessible with basis(cell).

The Brillouin zone can be plotted using e.g. PlotlyJS.jl (or 3D-capable backends of Makie.jl such as GLMakie.jl):

julia> using PlotlyJS
julia> plot(cell)

Examples of interactive visualizations are included in the documentation.

Minimal k-paths in the irreducible Brillouin zone

Given a symmetry setting and a lattice, specified by a space group number sgnum and a conventional direct basis Rs (respecting the conventions of the International Tables of Crystallography, Volume A), irrfbz_path will return a "minimal" k-path in the irreducible Brillouin zone. E.g.,

julia> sgnum = 147
julia> kp = irrfbz_path(sgnum, Rs)
KPath{3} (7 points, 3 paths, 13 points in paths):
 points: :M => [0.5, 0.0, 0.0]
         :A => [0.0, 0.0, 0.5]
         :H => [0.333333, 0.333333, 0.5]
         :K => [0.333333, 0.333333, 0.0]
         :Γ => [0.0, 0.0, 0.0]
         :L => [0.5, 0.0, 0.5]
         :H₂ => [0.333333, 0.333333, -0.5]
  paths: [:Γ, :M, :K, :Γ, :A, :L, :H, :A]
         [:L, :M]
         [:H, :K, :H₂]
  basis: [6.283185, 3.627599, -0.0]
         [0.0, 7.255197, 0.0]
         [0.0, -0.0, 5.026548]

Returned k-vector coordinates are referred to the basis of the primitive reciprocal cell (in the CDML setting). The associated transformation matrix between conventional and primitive bases can be obtained via Bravais.jl's primitivebasismatrix.

The resulting object kp can be interpolated, using either interpolate(kp, N) or splice(kp, N) which return a KPathInterpolant iterable whose values interpolate the connected paths (and enable convenient plotting of band structure diagrams). See also visualization examples in documentation.