Extracting LDA band structure parameters

 

To calculate transport parameters, including the optical conductivity as well as to calculate the photoemission spectra one needs make momentum summation of Greenís function (GF) or their convolutions.To do so one needs to make k-summation of DOS or the local Greenís function of a type:

††††††††††††††††††††††††††††††††††††††††

where is the chemical potential, is the overlap matrix, is the Hamiltonian and the is the self-energy matrix. In the LDA calculation the self-energy is represented by the Hartree-Fock constant while in the DMFT calculations it is frequency dependent function.Surely this integration/summation is the part of the LDA program we described in the previous section but very often it is quite convenient to have a separate routine/driver which does it outside one big program as it is much easier to change it in accord to a specific task.

 

But to make the summation we still use some information from LDA program i.e. the Hamiltonian, the overlap matrix, information about symmetry of a studied material. Program which prints all the necessary files for k-summation driver is the same as we described in the previous section with the only difference in printouts included. It means that one can make the self-consistent LDA calculation whether using this program or the previous one.And then one needs to make optical calculations to extract all the parameters. Run file should look something like this:

/home/user/LMTO/main <<!

latio3

ini+str+scs

scf+out

!

mv latio3.scf latio3.scs

mv latio3.out latio3.out_scf

/home/user/LMTO/main <<!

latio3

ini+str+scs

opt+out

!

mv latio3.out latio3.out_opt

 

(change /home/user/LMTO/ to the appropriate path to the programandlatio3 to the studied material)

Once the program run one should get the following files as an output:

1.     ham.dat

The Hamiltonian

2.     olp.dat

The overlap

3.     bz.dat

Brillouin zone data

4.     grp.dat

Space group data

5.     wig.dat

Space group data

6.     idx.dat

Auxiliary index pointer

7.     opar.dat

Factorization of the overlap

8.     Sk.dat

Structure constants

9.     vel.dat

Velocity matrix

 

1. Hamiltonian data (ham.dat).

This file contains the Hamiltonian matrix at each irreducible mesh point in the Brillouin zone. To facilitate and standardize the rotations of the Hamiltonian within the Brillouin zone the Hamiltonian is always assumed to be written in a spherical harmonics basis.

Quantity

Variable

Number of irreducible k-points

nkp

Dimension of Hamiltonian

ndim

Hamiltonian

hamf (1:ndim,1:ndim,1:nkp)

Table I:  Parameters available from ham.dat file.

Default energy unit on the Hamiltonian is Rydberg. Ndim=2 norb is the spin-orbital case is considered.

2. Overlap matrix data (olp.dat).

This file contains the overlap matrix at each irreducible mesh point in the Brillouin zone. The dimension of this matrix should be equal to the dimension of the Hamiltonian and the number of irreducible k-points should also match with the Hamiltonian data.

Quantity

Variable

Number of irreducible k-points

nkp

Dimension of overlap matrix

ndim

Overlap matrix

olap (1:ndim,1:ndim,1:nkp)

Table II: Parameters available from olp.dat file.

3. Brillouin zone data (bz.dat).

The file bz.dat contains the data on the Brillouin zone. It contains such information as the lattice constant, the mesh divisions, the irreducible k-points and tetrahedra as well as information about the group transformations that connect each k-point to its irreducible representative.††

Quantity

Variable

Lattice constant

latcon

Number of irreducible k-points

nkp

Number of irreducible tetrahedra

ntet

Total number of k-points

nktot

Number of k-points along each direction

ndiv (1:3)

Coordinate array for irreducible k-points

bk (1:3,1:nkp)

Integration weight array for irreducible k-points

wk (1:nkp)

Pointer from each irreducible tetrahedron to its multiplicity, irreducible k-point corners and type

itt (1:6,1:ntet)

kcut matrices for all tetrahedron types (see Figure)

kcut (1:3,1:4,1:6)

Pointer from irreducible k-point to the four nearest irreducible neighbors in each direction

nbr (1:3,1:4,1:nkp)

Pointer from irreducible k-point to the group operation that rotates the irreducible neighbor to the real neighbor

grop (1:3,1:4,1:nkp)

Bravais lattice vectors, multiplied by the division along the reciprocal direction in Brillouin zone

rvec (1:3,1:3)

Brillouin zone minicell vectors (i.e. )

mvec (1:3,1:3)

Pointer from each mesh points to its irreducible representative

kirrptr (0:ndiv(1)-1,0:ndiv(2)-1,

0:ndiv(3)-1)

Pointer from each mesh point to the group operation that takes the irreducible representative to that point

gropptr (0:ndiv(1)-1,0:ndiv(2)-1, 0:ndiv(3)-1)

Table III. Parameters available from bz.dat file.

Figure I: One of four different ways of splitting the unit cube into six tetrahedral. The common diagonal is chosen from to . The coordinates of a given corner of each tetrahedron are given by the pointer kcut(m,,) where .

4. Group operations (grp.dat).

The file grp.dat contains information about the space group operations of the crystal. Here we read in both the number of atoms and the number of group operations that exist. In principle the maximum number of symmetry operations is 48 (cubic) but in low symmetry crystals they could be less and if one is using a reducible (i.e. not minimal) unit cell they could be more. To each of the symmetry operation corresponds a group operation element but since the maximum number of group elements is 48, two or more symmetry operations could correspond to a given group element. In addition to knowing the correspondence between the symmetry operations and the group elements one must know how the atoms move under the action of the symmetry operation if there are more than one atom of a given type in the unit cell. There is also the possibility that the symmetry operation takes the atom out of the unit cell and then it must be translated back by a lattice vector to the original unit cell. Since the Hamiltonian is in the Brillouin zone these translations must be known because they introduce phase factors into the equation for the transformation of the Hamiltonian.

 

Quantity

Variable

Number of symmetry operations

ngroup

Number of atoms in unit cell

natom

Symmetry operation to group element pointer

ikov (1:ngroup)

Atom to atom mapping for each symmetry operation

igatom (1:natom,1:ngroup)

Back to cell translation vectors for each atom and symmetry operation

rdc (1:3,1:natom,1:ngroup)

Table IV. Parameters available from grp.dat file.

5. Group operations (wig.dat).

The file wig.dat contains Wigner matrices for spherical harmonics angular momenta up to. This is needed to both rotate the Hamiltonian and the overlap matrix and to create the vector rotation matrices for each group operation. It contains \the number of group operations and the maximum angular momentum (should be always 3). The Wigner matrix is stored as a vector in the computer and the mapping from the angular momentum labels,, to the vector component, , is given by: ††

Quantity

Variable

Number of group operations

ng

Maximum angular momentum

lmax 3

Wigner matrices

wigmat (1:imax,1:48)

Table V. Parameters available from wig.dat file.

 

Figure II: Left figure: An illustration of the information encoded in the arrays nbr and grop along the 3rd () direction. Right figure: An illustration of the back-to-cell vectors rdc that take an atom that is rotated by the transformation igr out of the unit cell back to its equivalent position in the unit cell.

In addition to the parameters in Table V available from the file bz.dat there are two more parameters that are initialized using information in bz.dat. One of them is the total number of tetrahedra which is computed from the variable itt but should of course be equal to. The other one is a pointer from each corner of every tetrahedron to its irreducible representative.

Quantity

Variable

Total number of tetrahedra

mtet

Pointer from each tetrahedron corner to its irreducible representative

coridx (1:4,1:6,1:ndiv(1),1:ndiv(2),1:ndiv(3))

Table VI: Additional Parameters available initialized from data in bz.dat.

 

6. Auxiliary index data (idx.dat).

The file idx.dat contains pointers from the computer representation of the physical quantities (Hamiltonian etc.) as array to the corresponding atoms and angular momenta. Here we also get information about how many sorts of atoms there are, a pointer from each atom to its sort and what is the maximum angular momentum for a given atomic sort. We also have a logical array that indicates if some angular momenta are omitted from the Hamiltonian. This is useful if we want to consider simplified models where certain angular momenta have been ignored, usually because they are of much to low energy to be of interest.

 

Quantity

Variable

Number of atoms in unit cell

natom

Number of atomic sorts

nsort

Pointer from each atom to its sort

is (1:natom)

Pointer from each sort to its maximum angular momentum

ilsb (1:nsort)

Logical array indicating whether a given angular momentum is present for a given sort of atoms

ltpan (1:lmax,1:natom)

A pointer from each orbital state of every to its position in the Hamiltonian matrix

lmtind (1:(lmax+1),1:natom)

Table VII: Parameters available from idx.dat file.

7. Factorization of the overlap (opar.dat):

Files opar.dat and Sk.dat are used to orthonormalize (almost) the overlap matrix.

Quantity

Variable

Maximum angular momentum used

l_max

Number of atomic sorts

nsort

Number of spins

nspin (=2 for spin-orbit)

Transformation matrixes

moo(4,0:l_max,nsort,nspin)

Table VIII. Parameters available from opar.dat file.

8. Structure constants (Sk.dat):

The file Sk.dat contains the structure constants data.

Quantity

Variable

Maximum angular momentum used +1 squared

(l_max+1)

Number of atoms in unit cell

natom

Lattice constant

latcon

Structure constants

Stk(1:ndim,1:ndim,1:nkp)

Table VIII. Parameters available from Sk.dat file.

 

9. Velocity matrix (vel.dat).:

The vel.dat is the velocity matrix. It is not used in calculation of the k-integral. This information is used in the optical calculations.