Brillouin zone sampling

The k-points are always given relative to the basis vectors of the reciprocal unit cell.

Monkhorst-Pack

ase.dft.kpoints.monkhorst_pack(size)[source]

Construct a uniform sampling of k-space of given size.

The k-points are given as [MonkhorstPack]:

\[\sum_{i=1,2,3} \frac{2n_i -N_i - 1}{2N_i} \mathbf{b}_i,\]

where \(n_i=1,2,...,N_i\), size = \((N_1, N_2, N_3)\) and the \(\mathbf{b}_i\)‘s are reciprocal lattice vectors.

ase.dft.kpoints.get_monkhorst_pack_size_and_offset(kpts)[source]

Find Monkhorst-Pack size and offset.

Returns (size, offset), where:

kpts = monkhorst_pack(size) + offset.

The set of k-points must not have been symmetry reduced.

Example:

>>> from ase.dft.kpoints import *
>>> monkhorst_pack((4, 1, 1))
array([[-0.375,  0.   ,  0.   ],
       [-0.125,  0.   ,  0.   ],
       [ 0.125,  0.   ,  0.   ],
       [ 0.375,  0.   ,  0.   ]])
>>> get_monkhorst_pack_size_and_offset([[0, 0, 0]])
(array([1, 1, 1]), array([ 0.,  0.,  0.]))
[MonkhorstPack]Hendrik J. Monkhorst and James D. Pack: Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188–5192 (1976)

Chadi-Cohen

Predefined sets of k-points:

ase.dft.kpoints.cc6_1x1
ase.dft.kpoints.cc12_2x3
ase.dft.kpoints.cc18_sq3xsq3
ase.dft.kpoints.cc18_1x1
ase.dft.kpoints.cc54_sq3xsq3
ase.dft.kpoints.cc54_1x1
ase.dft.kpoints.cc162_sq3xsq3
ase.dft.kpoints.cc162_1x1

Naming convention: cc18_sq3xsq3 is 18 k-points for a sq(3)xsq(3) cell.

Try this:

>>> import numpy as np
>>> import pylab as plt
>>> from ase.dft.kpoints import cc162_1x1
>>> B = [(1, 0, 0), (-0.5, 3**0.5 / 2, 0), (0, 0, 1)]
>>> k = np.dot(cc162_1x1, B)
>>> plt.plot(k[:, 0], k[:, 1], 'o')
[<matplotlib.lines.Line2D object at 0x9b61dcc>]
>>> p.show()
../../_images/cc.png

Special points in the Brillouin zone

You can find the special points in the Brillouin zone:

>>> from ase.lattice import bulk
>>> from ase.dft.kpoints import ibz_points, get_bandpath
>>> si = bulk('Si', 'diamond', a=5.459)
>>> points = ibz_points['fcc']
>>> G = points['Gamma']
>>> X = points['X']
>>> W = points['W']
>>> K = points['K']
>>> L = points['L']
>>> kpts, x, X = get_bandpath([W, L, G, X, W, K], si.cell)
>>> print len(kpts), len(x), len(X)
50 50 6
ase.dft.kpoints.get_bandpath(points, cell, npoints=50)[source]

Make a list of kpoints defining the path between the given points.

points: list
List of special IBZ point pairs, e.g. points = [W, L, Gamma, X, W, K]. These should be given in scaled coordinates.
cell: 3x3 ndarray
Unit cell of the atoms.
npoints: int
Length of the output kpts list.

Return list of k-points, list of x-coordinates and list of x-coordinates of special points.