velocity_matrix

sisl.physics.electron.velocity_matrix(state, dHk, energy=None, dSk=None, degenerate=None, degenerate_dir=(1, 1, 1))[source]

Calculate the velocity matrix of a set of states

These are calculated using the analytic expression (\(\alpha\) corresponding to the Cartesian directions):

\[\mathbf{v}^\alpha_{ij} = \frac1\hbar \langle \psi_j | \frac{\partial}{\partial\mathbf k_\alpha} \mathbf H(\mathbf k) | \psi_i \rangle\]

In case of non-orthogonal basis the equations substitutes \(\mathbf H(\mathbf k)\) by \(\mathbf H(\mathbf k) - \epsilon_i\mathbf S(\mathbf k)\).

Although the matrix \(\mathbf v\) should be Hermitian it is not checked, and we explicitly calculate all elements.

Parameters

state (array_like) – vectors describing the electronic states, 2nd dimension contains the states. In case of degenerate states the vectors may be rotated upon return.
dHk (list of array_like) – Hamiltonian derivative with respect to \(\mathbf k\). This needs to be a tuple or list of the Hamiltonian derivative along the 3 Cartesian directions.
energy (array_like, optional) – energies of the states. Required for non-orthogonal basis together with dSk. In case of degenerate states the eigenvalues of the states will be averaged in the degenerate sub-space.
dSk (list of array_like, optional) – \(\delta \mathbf S_k\) matrix required for non-orthogonal basis. This and energy must both be provided in a non-orthogonal basis (otherwise the results will be wrong). Same derivative as dHk
degenerate (list of array_like, optional) – a list containing the indices of degenerate states. In that case a prior diagonalization is required to decouple them. See degenerate_dir for details.
degenerate_dir ((3,), optional) – a direction used for degenerate decoupling. The decoupling based on the velocity along this direction