pythtb.Lattice.k_path#

Lattice.k_path(k_nodes, nk, report=False)[source]#

Interpolates a path in reciprocal space.

Interpolates a path in reciprocal space between specified k-points. In 2D or 3D the k-path can consist of several straight segments connecting high-symmetry points (“nodes”). The interpolated path is constructed so that the k-points are (nearly) equidistant in Cartesian k-space.

Parameters:
k_nodesarray-like, str

Array of k-vectors in reduced units, between which the interpolated path will be constructed. In 1D k-space, value may be a string:

  • “full”: Implies [0.0, 0.5, 1.0] (full BZ)

  • “fullc”: Implies [-0.5, 0.0, 0.5] (full BZ, centered)

  • “half”: Implies [ 0.0, 0.5] (half BZ)

nkint

Total number of k-points along the path.

reportbool, optional

Optional parameter specifying whether printout is desired (default is False).

Returns:
k_vecnp.ndarray

Array of (nearly) equidistant interpolated k-points. Shape is (nk, dim_k).

k_distnp.ndarray

Array giving accumulated k-distance to each k-point in the path. This array is useful when plotting bands along the path. The distances between the points can be used to ensure that the plot accurately reflects the k-space geometry. Shape is (nk,).

Changed in version 2.0.0: The returned k_dist now includes the factors of \(2\pi\). See notes for further details.

k_nodenp.ndarray

Array giving accumulated k-distance to each node on the path in Cartesian coordinates. This array can be used to reference the nodes on the 1D k_dist array, e.g., when plotting high-symmetry points. Shape is (n_nodes,).

Notes

  • The distance between the points is calculated in the Cartesian frame, however coordinates themselves are given in dimensionless reduced coordinates! This is done so that this array can be directly passed to function pythtb.TBModel.solve_ham().

  • Unlike array k_vec, k_dist has dimensions! Units are defined so that for a one-dimensional crystal with lattice constant equal to for example \(10\) the length of the Brillouin zone would equal \(2\pi/10\).

Examples

Construct a path connecting four nodal points in k-space Path will contain 401 k-points, roughly equally spaced

>>> path = [[0.0, 0.0], [0.0, 0.5], [0.5, 0.5], [0.0, 0.0]]
>>> (k_vec, k_dist, k_node) = my_model.k_path(path,401)

Solve for eigenvalues on that path and plot the band structure

>>> import matplotlib.pyplot as plt
>>> evals = tb.solve_ham(k_vec)
>>> for n in range(evals.shape[1]):
...     plt.plot(k_dist, evals[:, n], 'b-')
>>> for node_dist in k_node:
...     plt.axvline(x=node_dist, color='k', linestyle='--')
>>> plt.xlabel('k')
>>> plt.ylabel('Energy (eV)')
>>> plt.show()
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