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75 lines
2.9 KiB
ReStructuredText
75 lines
2.9 KiB
ReStructuredText
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The BravaisLattice and TightBinding classes: definitions and example
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====================================================================
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The following example is aimed at demonstrating the use of **TRIQS
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Lattice tools**.
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BravaisLattice
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--------------
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A ``BravaisLattice`` is constructed as
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``BravaisLattice(units, orbital_positions )`` where
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- ``units`` is the list the coordinates (given as triplets) of the
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basis vectors :math:`\lbrace \mathbf{e}_i \rbrace _{i=1\dots d}`
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(:math:`d` is the dimension)
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- ``orbital_positions`` is a dictionary of the atoms forming the basis
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of the Bravais Lattice: the key is the name of the atom/orbital, the
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value is the triplet of its coordinates.
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TightBinding
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------------
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A tight-binding lattice is defined by the relation:
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.. math:: \mathbf{t}_k = \sum_{\mathbf{R}\in \mathrm{BL}} e^{i \mathbf{k}\cdot \mathbf{R}} \mathbf{t}_\mathbf{R}
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where :math:`\mathbf{t}_i` is the matrix of the hoppings from a
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reference unit cell (:math:`\mathbf{R}=O`\ ) to a unit cell indexed by
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:math:`\mathbf{R}`\ . :math:`(\mathbf{t}_\mathbf{R})_{n,m}` is the
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tight-binding integral between atom :math:`n` of site :math:`O` and atom
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:math:`m` of site :math:`\mathbf{R}`\ , ie
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.. math:: (\mathbf{t}_\mathbf{R})_{n,m} \equiv \int d^3\mathbf{r} \phi_n(\mathbf{r})^{*} V(\mathbf{r}) \phi_m(\mathbf{r}-\mathbf{R})
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where :math:`\phi_n(\mathbf{r}-\mathbf{R})` is the Wannier orbital of
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atom :math:`n` centered at site :math:`\mathbf{R}`\ . The corresponding
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class in **Lattice Tools** is the ``TightBinding`` class. Its instances
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are constructed as follows:
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``TightBinding ( bravais_lattice, hopping_dictionary)`` where
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- ``bravais_lattice`` is an instance of ``BravaisLattice``
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- ``hopping_dictionary`` is a dictionary of the hoppings
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:math:`\mathbf{t}_\mathbf{R}`\ , where the keys correspond to the
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coordinates of :math:`\mathbf{R}` in the unitary basis
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:math:`\lbrace \mathbf{e}_i \rbrace _{i=1\dots d}`\ , and the values
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to the corresponding matrix: :math:`(\mathbf{t}_\mathbf{R})_{n,m}`
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energies_on_bz_path
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-------------------
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The function ``energies_on_bz_path (TB, start, end, n_pts)`` returns a
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:math:`n_{at} \times n_{pts}` matrix :math:`E` such that
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``E[n,k]``:math:`= \epsilon_n(\mathbf{k})`
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where ``k`` indexes the ``n_pts`` :math:`\mathbf{k}`\ -points of the
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line joining ``start`` and ``end``, and :math:`\epsilon_n(k)` is the
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:math:`n`\ th eigenvector of :math:`t_\mathbf{k}`\ .
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Example
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-------
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The following example illustrates the usage of the above tools for the
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case of a two-dimensional, square lattice with various unit cells. We
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successively construct three Bravais lattices ``BL_1``, ``BL_2`` and
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``BL_4`` with, respectively, 1, 2 and 4 atoms per unit cell, as well as
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three tight-binding models with hopping dictionaries ``hop_1``,
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``hop_2`` and ``hop_4``
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.. plot:: reference/python/lattice/ex2.py
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:include-source:
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:scale: 70
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