9.2. Example: PDOS of graphene

We now turn to partial density of states calculations and use graphene to illustrate how to set them up. PDOS calculations are fundamentally similar to DOS calculations, the difference being that the energy levels constituting the DOS are weighted by atomic orbital coefficients. Suppose that the self-consistent ground state density of graphene may be found in ./results/gr_lcao_scf.h5 and ./results/gr_lcao_scf.mat, one may compute the PDOS of graphene using the following input script

info.calculationType = 'dos'
info.savepath = 'results/gr_lcao_pdos'
kpoint.sampling = 'tetrahedron'
kpoint.gridn = [45,45,1]
rho.in = 'results/gr_lcao_scf'
dos.range = [-0.75,0.35]
dos.projL = [0,1,2]
dos.projM = -2:2

The relevant keywords in PDOS calculations are the following

dos.range gives the energy range over which the DOS and PDOS are calculated. This is relative to the Fermi energy. The default units are Hartree.
dos.projAtom is an array containing the atom indices on which to project the DOS.
dos.projL is an array containing the orbital angular momenta on which to project the DOS.
dos.projM is an array containing the projected orbital angular momenta on which to project the DOS.
dos.projSpecies is an array containing the species (element) indices on which to project the DOS.
dos.projZ is an array containing the \(\zeta\)-numbers on which to project the DOS.

It is not necessary to project the DOS onto all valid values of a given quantum number. For instance, we could have set dos.projL equal to [0,1]. Note however that the sum of the PDOS would not equal the DOS in this case since part of the DOS has d-orbital character. Whereas the values of dos.projAtom, dos.projL, dos.projSpecies and dos.projZ are self-explanatory, dos.projM depends on the convention for the real spherical harmonics used by RESCU, which may be found in Table 2.8.1.

In the present example, we decompose the DOS using both dos.projL and dos.projM. In this case, RESCU will decompose the DOS into all valid – meaning that, say, dos.projL = 3 would be ignored since there are no \(f\)-orbital in our atomic orbital basis – combinations of the specified constraints. There are thus nine PDOS calculated here. After loading the results, the dos data structure should look like

       desc: [1x1 struct]
     dosvec: [1100x1 double]
     nrgvec: [1100x1 double]
    pdosdef: [9x5 double]
    pdosvec: [1100x1x9 double]
   projAtom: []
      projL: [0 1 2]
      projM: [-2 -1 0 1 2]
projSpecies: []
      projZ: []

In particular, the field dos.pdosdef contains the constraints specific to each PDOS. The field dos.pdosdef has nine rows, corresponding to the nine PDOS and five column corresponding to the value of the atomic orbital indices. A negative value out of bound means that the index is summed over. For instance, there is no minus-oneth atom, so projAtom = -1 means that the contribution of all atoms is accumulated. Similarly, projM = -4 means that the contribution of \(Y_{l,m}\)-orbitals is summed over \(m\). In the latter case, projM = -1 would simply refer to \(Y_{l,-1}\)-orbitals however.

dos.pdosdef
#projAtom projL projM projSpecies projZ
    -1     0     0    -1    -1
    -1     1    -1    -1    -1
    -1     1     0    -1    -1
    -1     1     1    -1    -1
    -1     2    -2    -1    -1
    -1     2    -1    -1    -1
    -1     2     0    -1    -1
    -1     2     1    -1    -1
    -1     2     2    -1    -1

The total DOS along with the \(s\), \(p_y\), \(p_z\) and \(p_x\) PDOS (first four rows of dos.pdosdef respectively), are plotted in Fig. 9.2.1. The low lying state have \(s\) character while the Dirac cone is strictly composed of \(p_z\) orbitals. The \(p_x\) and \(p_y\) PDOS are virtually indistinguishable as expected from symmetry considerations.