PDOS¶

sisl.physics.electron.PDOS(E, eig, state, S=None, distribution='gaussian', spin=None)[source]¶

Calculate the projected density of states (PDOS) for a set of energies, E, with a distribution function

The \(\mathrm{PDOS}(E)\) is calculated as:

\[\mathrm{PDOS}_\nu(E) = \sum_i \psi^*_{i,\nu} [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i)\]

where \(D(\Delta E)\) is the distribution function used. Note that the distribution function used may be a user-defined function. Alternatively a distribution function may be aquired from distribution.

In case of an orthogonal basis set \(\mathbf S\) is equal to the identity matrix. Note that DOS is the sum of the orbital projected DOS:

\[\mathrm{DOS}(E) = \sum_\nu\mathrm{PDOS}_\nu(E)\]

For non-collinear calculations (this includes spin-orbit calculations) the PDOS is additionally separated into 4 components (in this order):

Total projected DOS
Projected spin magnetic moment along \(x\) direction
Projected spin magnetic moment along \(y\) direction
Projected spin magnetic moment along \(z\) direction

These are calculated using the Pauli matrices \(\boldsymbol\sigma_x\), \(\boldsymbol\sigma_y\) and \(\boldsymbol\sigma_z\):

\[\begin{split}\mathrm{PDOS}_\nu^\Sigma(E) &= \sum_i \psi^*_{i,\nu} \boldsymbol\sigma_z \boldsymbol\sigma_z [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i) \\ \mathrm{PDOS}_\nu^x(E) &= \sum_i \psi^*_{i,\nu} \boldsymbol\sigma_x [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i) \\ \mathrm{PDOS}_\nu^y(E) &= \sum_i \psi^*_{i,\nu} \boldsymbol\sigma_y [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i) \\ \mathrm{PDOS}_\nu^z(E) &= \sum_i \psi^*_{i,\nu} \boldsymbol\sigma_z [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i)\end{split}\]

Note that the total PDOS may be calculated using \(\boldsymbol\sigma_i\boldsymbol\sigma_i\) where \(i\) may be either of \(x\), \(y\) or \(z\).

Parameters

Earray_like: energies to calculate the projected-DOS from
eigarray_like: eigenvalues
statearray_like: eigenvectors
Sarray_like, optional: overlap matrix used in the \(\langle\psi|\mathbf S|\psi\rangle\) calculation. If None the identity matrix is assumed. For non-collinear calculations this matrix may be halve the size of len(state[0, :]) to trigger the non-collinear calculation of PDOS.
distributionfunc or str, optional: a function that accepts \(E-\epsilon\) as argument and calculates the distribution function.
spinstr or Spin, optional: the spin configuration. This is generally only needed when the eigenvectors correspond to a non-collinear calculation.

Returns

numpy.ndarray: projected DOS calculated at energies, has dimension (state.shape[1], len(E)). For non-collinear calculations it will be (4, state.shape[1] // 2, len(E)), ordered as indicated in the above list.