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.. module:: ase.dft.kpoints
:synopsis: Brillouin zone sampling
=======================
Brillouin zone sampling
=======================
The **k**-points are always given relative to the basis vectors of the
reciprocal unit cell.
Monkhorst-Pack
--------------
.. autofunction:: monkhorst_pack
The k-points are given as [MonkhorstPack]_:
.. math::
\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.
.. autofunction:: get_monkhorst_pack_size_and_offset
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)
:doi:`10.1103/PhysRevB.13.5188`
Special points in the Brillouin zone
------------------------------------
.. data:: special_points
The below table lists the special points from [Setyawan-Curtarolo]_.
.. toctree:: bztable
.. include:: bztable.rst
.. [Setyawan-Curtarolo]
High-throughput electronic band structure calculations:
Challenges and tools
Wahyu Setyawan, Stefano Curtarolo
Computational Materials Science,
Volume 49, Issue 2, August 2010, Pages 299–312
:doi:`10.1016/j.commatsci.2010.05.010`
You can find the special points in the Brillouin zone:
>>> from ase.build import bulk
>>> si = bulk('Si', 'diamond', a=5.459)
>>> lat = si.cell.get_bravais_lattice()
>>> print(list(lat.get_special_points()))
['G', 'K', 'L', 'U', 'W', 'X']
>>> path = si.cell.bandpath('GXW', npoints=100)
>>> print(path.kpts.shape)
(100, 3)
.. autofunction:: get_special_points
.. autofunction:: bandpath
.. autofunction:: parse_path_string
.. autofunction:: labels_from_kpts
Band path
---------
The :class:`~ase.dft.kpoints.BandPath` class stores all the relevant
band path information in a single object.
It is typically created by helper functions such as
:meth:`ase.cell.Cell.bandpath` or :meth:`ase.lattice.BravaisLattice.bandpath`.
.. autoclass:: BandPath
:members:
Band structure
--------------
.. autoclass:: ase.spectrum.band_structure.BandStructure
:members:
Free electron example:
.. literalinclude:: bs.py
.. image:: cu.png
Interpolation
-------------
.. autofunction:: monkhorst_pack_interpolate
High symmetry paths
-------------------
.. data:: special_paths
The :mod:`ase.lattice` framework provides suggestions for high symmetry
paths in the BZ from the [Setyawan-Curtarolo]_ paper.
>>> from ase.lattice import BCC
>>> lat = BCC(3.5)
>>> lat.get_special_points()
{'G': array([0, 0, 0]), 'H': array([ 0.5, -0.5, 0.5]), 'P': array([0.25, 0.25, 0.25]), 'N': array([0. , 0. , 0.5])}
>>> lat.special_path
'GHNGPH,PN'
In case you want this information *without* providing the lattice parameter
(which is possible for those lattices where the points do not depend on the
lattice parameters), the data is also available as dictionaries:
>>> from ase.dft.kpoints(import special_paths, sc_special_points,
... parse_path_string)
>>> paths = sc_special_paths['bcc']
>>> paths
[['G', 'H', 'N', 'G', 'P', 'H'], ['P', 'N']]
>>> points = sc_special_points['bcc']
>>> points
{'H': [0.5, -0.5, 0.5], 'N': [0, 0, 0.5], 'P': [0.25, 0.25, 0.25],
'G': [0, 0, 0]}
>>> kpts = [points[k] for k in paths[0]] # G-H-N-G-P-H
>>> kpts
[[0, 0, 0], [0.5, -0.5, 0.5], [0, 0, 0.5], [0, 0, 0], [0.25, 0.25, 0.25], [0.5, -0.5, 0.5]]
Chadi-Cohen
-----------
Predefined sets of **k**-points:
.. data:: cc6_1x1
.. data:: cc12_2x3
.. data:: cc18_sq3xsq3
.. data:: cc18_1x1
.. data:: cc54_sq3xsq3
.. data:: cc54_1x1
.. data:: cc162_sq3xsq3
.. data:: cc162_1x1
Naming convention: ``cc18_sq3xsq3`` is 18 **k**-points for a
sq(3)xsq(3) cell.
Try this:
>>> import numpy as np
>>> import matplotlib.pyplot 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') # doctest: +SKIP
[<matplotlib.lines.Line2D object at 0x9b61dcc>]
>>> plt.show()
.. image:: cc.png
|
