# Third (basic) lesson¶

## Crystalline silicon.¶

This lesson aims at showing you how to get the following physical properties, for an insulator:

• the total energy
• the lattice parameter
• the band structure (actually, the Kohn-Sham band structure)

You will learn about the use of k-points, as well as the smearing of the plane-wave kinetic energy cut-off.

This lesson should take about 1 hour.

## Computing the total energy of silicon at fixed number of k points¶

Before beginning, you might consider to work in a different subdirectory as for lesson_base1 or lesson_base2 . Why not “Work3”?

The file ~abinit/tests/tutorial/Input/tbase3_x.files lists the file names and root names. You can copy it in the Work3 directory and change it as you did for the tbase1_x.files and tbase2_x.files files. You can also copy the file ~abinit/tests/tutorial/Input/tbase3_1.in in Work3. This is your input file.

You should edit it, read it carefully, have a look at the following “new” input variables, and their explanation:

Note also the following: you will work at fixed ecut (8Ha). It is implicit that in real life, you should do a convergence test with respect to ecut. Here, a suitable ecut is given to you. It will allow to obtain 0.2% relative accuracy on lattice parameters.

When you have read the input file, you can run the code, as usual (it will run for a few seconds).
Then, read the output file, and note the total energy:

   etotal   -8.8662238960E+00


## Starting the convergence study with respect to k-points¶

There is of course a convergence study associated with the sampling of the Brillouin zone. You should examine different grids, of increasing resolution. You might try the following series of grids:

ngkpt1  2 2 2
ngkpt2  4 4 4
ngkpt3  6 6 6
ngkpt4  8 8 8


However, the associated number of k points in the irreducible Brillouin zone grows very fast. It is

nkpt1  2
nkpt2 10
nkpt3 28
nkpt4 60


ABINIT computes automatically this number of k points, from the definition of the grid and the symmetries.
You might nevertheless define an input nkpt value in the input file, in which case ABINIT will compare its computed value (from the grid) with this input value. We take this opportunity to examine the behaviour of ABINIT when a problem is detected. Let’s suppose that with ngkpt1 4 4 4, one mentions nkpt1 2. The input file ~abinit/tests/tutorial/Input/tbase3_2.in is an example.

Do not forget to change tbase3_x.files, if you are using that file name. The message that you get a few dozen of lines before the end of the log file is:

--- !BUG
message: |
The argument nkpt=     2, does not match
the number of k points generated by kptopt, kptrlatt, shiftk,
and the eventual symmetries, that is, nkpt=    10.
However, note that it might be due to the user,
if nkpt is explicitely defined in the input file.
src_file: getkgrid.F90
src_line: 415
...

Action : contact ABINIT group.


This is a typical ABINIT error message. It states what is the problem that causes the stop of ABINIT, then suggests that it might be due to an error in the input file, namely, an erroneous value of nkpt. The expected value, nkpt 10. is mentioned before the notice that the input file might be erroneous. Then, the file at which the problem occured is mentioned, as well as the number of the line in that file.

As the computation of nkpt for specific grids of k points is not an easy task, while the even more important selection of specific economical grids (the best ratio between the accuracy of the integration in the Brillouin zone and the number of k-points) is more difficult, some help to the user is provided by ABINIT.

The code is able to examine automatically different k point grids, and to propose the best grids for integration. This is described in the abinit help file, see the input variable prtkpt, and the associated characterisation of the integral accuracy, described in kptrlen.

Tip

The generation of lists of k-point sets is done in different test cases, in ~abinit/tests/v2. You can directly have a look at the output files in ~abinit/tests/v2/Refs, the output files for the tests 61 to 73.

When one begins the study of a new material, it is strongly advised to examine first the list of k points grids, and select (at least) three efficient ones, for the k point convergence study. Do not forget that the CPU time will be linearly proportional to the number of k points to be treated: using 10 k points will take five more time than using 2 k points. Even for a similar accuracy of the Brillouin zone integration (about the same value of kptrlen), it might be easy to generate a grid that will fold to 10 k points in the irreducible Brillouin zone, as well as one that will fold to 2 k points in the irreducible Brillouin zone. The latter is clearly to be preferred!

## 3 Actually performing the convergence study with respect to k points¶

In order to understand k-point grids, you should read [Monkhorst1976]. Well, maybe not immediately. In the meantime, you can try the above-mentioned convergence study.

The input file ~abinit/tests/tutorial/Input/tbase3_3.in is an example, while ~abinit/tests/tutorial/Refs/tbase3_3.out is a reference output file.

In this output file, you should have a look at the echo of input variables. As you know, these are preprocessed, and, in particular, ngkpt and shiftk are used to generate the list of k points (kpt) and their weights (wtk). You should read the information about kpt and wtk.

From the output file, here is the evolution of total energy per unit cell:

    etotal1  -8.8662238960E+00
etotal2  -8.8724909739E+00
etotal3  -8.8726017432E+00
etotal4  -8.8726056405E+00


The difference between dataset 3 and dataset 4 is rather small. Even the dataset 2 gives an accuracy of about 0.0001 Ha. So, our converged value for the total energy, at fixed acell, fixed ecut, is -8.8726 Ha .

## 4 Determination of the lattice parameters¶

The input variable optcell governs the automatic optimisation of cell shape and volume.
For the automatic optimisation of cell volume, use:

optcell 1
ionmov 2
ntime 10
dilatmx 1.05
ecutsm 0.5


Do not test all the k point grids, only those with nkpt 2 and 10.

The input file ~abinit/tests/tutorial/Input/tbase3_4.in is an example, while ~abinit/tests/tutorial/Refs/tbase3_4.out is a reference output file.

You should obtain the following evolution of the lattice parameters:

     acell1   1.0233363682E+01  1.0233363682E+01  1.0233363682E+01 Bohr
acell2   1.0216447241E+01  1.0216447241E+01  1.0216447241E+01 Bohr


with the following very small residual stresses:

    strten1   1.8591719160E-07  1.8591719160E-07  1.8591719160E-07
0.0000000000E+00  0.0000000000E+00  0.0000000000E+00
strten2  -2.8279720007E-08 -2.8279720007E-08 -2.8279720007E-08
0.0000000000E+00  0.0000000000E+00  0.0000000000E+00


The stress tensor is given in Hartree/Bohr^3, and the order of the components is

                        11  22  33
23  13  12


There is only a 0.13% relative difference between acell1 and acell2.
So, our converged LDA value for Silicon, with the 14si.pspnc pseudopotential (see the tbase3_x.files file) is 10.216 Bohr (actually 10.21644…), that is 5.406 Angstrom. The experimental value is 5.431 Angstrom at 25 degree Celsius, see R.W.G. Wyckoff, Crystal structures Ed. Wiley and sons, New-York (1963).

## Computing the band structure¶

We fix the parameters acell to the theoretical value of 3*10.216, and we fix also the grid of k points (the 4x4x4 FCC grid, equivalent to a 8x8x8 Monkhorst-pack grid)
We will ask for 8 bands (4 valence and 4 conduction).

A band structure can be computed by solving the Kohn-Sham equation for many different k points, along different lines of the Brillouin zone.
The potential that enters the Kohn-Sham must be derived from a previous self-consistent calculation, and will not vary during the scan of different k-point lines.
Suppose that you want to make a L-Gamma-X-(U-)Gamma circuit, with 10, 12 and 17 divisions for each line (each segment has a different length in reciprocal space, and these divisions give approximately the same distance between points along a line).
The circuit will be obtained easily by the following choice of segment end points:

L     (1/2 0 0)
Gamma (0 0 0)
X     (0 1/2 1/2)
Gamma (1 1 1)


Note:

1. the last Gamma point is in another cell of the reciprocal space than the first one, this choice allows to construct the X-U-Gamma line easily;

2. the k-points are specified using reduced coordinates - in agreement with the input setting of the primitive 2-atom unit cell - in standard textbooks, you will often find the L, Gamma or X point given in coordinates of the conventional 8-atom cell: the above-mentioned circuit is then (½ ½ ½)-(0 0 0)-(1 0 0)-(1 1 1), but such (conventional) coordinates cannot be used with the 2-atom (non-conventional) cell.

So, you should set up in your input file, for the first dataset, a usual SCF calculation in which you output the density (prtden 1), and, for the second dataset:

• fix iscf to -2, to make a non-self-consistent calculation;
• define getden -1, to take the output density of dataset 1;
• set nband to 8;
• set kptopt to -3, to define three segments in the brillouin Zone;
• set ndivk to 10 12 17 (this means a circuit defined by 4 points, with 10 divisions of the first segment, 12 divisions of the second, 17 divisions of the third)
• set kptbounds to

          0.5  0.0  0.0 # L point
0.0  0.0  0.0 # Gamma point
0.0  0.5  0.5 # X point
1.0  1.0  1.0 # Gamma point in another cell.

• set enunit to 1, in order to have eigenenergies in eV

• the only tolerance criterion admitted for non-self-consistent calculations is tolwfr. You should set it to 1.0d-10 (or so), and suppress toldfe.

The input file ~abinit/tests/tutorial/Input/tbase3_5.in is an example, while ~abinit/tests/tutorial/Refs/tbase3_5.out is a reference output file.

You should find the band structure starting at (second dataset):

 Eigenvalues (   eV  ) for nkpt=  40  k points:
kpt#   1, nband=  8, wtk=  1.00000, kpt=  0.5000  0.0000  0.0000 (reduced coord)
-3.78815  -1.15872   4.69668   4.69668   7.38795   9.23867   9.23867  13.45707
kpt#   2, nband=  8, wtk=  1.00000, kpt=  0.4500  0.0000  0.0000 (reduced coord)
-3.92759  -0.95774   4.71292   4.71292   7.40692   9.25561   9.25561  13.48927
kpt#   3, nband=  8, wtk=  1.00000, kpt=  0.4000  0.0000  0.0000 (reduced coord)
-4.25432  -0.44393   4.76726   4.76726   7.46846   9.31193   9.31193  13.57737
kpt#   4, nband=  8, wtk=  1.00000, kpt=  0.3500  0.0000  0.0000 (reduced coord)
-4.64019   0.24941   4.85732   4.85732   7.56855   9.38323   9.38323  13.64601
....


One needs a graphical tool to represent all these data … (For the MAPR 2451 lecture: try with MATLAB). In a separate file (_EIG), you will find the list of k-points and eigenenergies (the input variable prteig is set by default to 1).

Even without a graphical tool we will have a quick look at the values at L, Gamma, X and Gamma again:

 kpt#   1, nband=  8, wtk=  1.00000, kpt=  0.5000  0.0000  0.0000 (reduced coord)
-3.78815  -1.15872   4.69668   4.69668   7.38795   9.23867   9.23867  13.45707

kpt#  11, nband=  8, wtk=  1.00000, kpt=  0.0000  0.0000  0.0000 (reduced coord)
-6.17005   5.91814   5.91814   5.91814   8.44836   8.44836   8.44836   9.17755

kpt#  23, nband=  8, wtk=  1.00000, kpt=  0.0000  0.5000  0.5000 (reduced coord)
-1.96393  -1.96393   3.00569   3.00569   6.51173   6.51173  15.95524  15.95524

kpt#  40, nband=  8, wtk=  1.00000, kpt=  1.0000  1.0000  1.0000 (reduced coord)
-6.17005   5.91814   5.91814   5.91814   8.44836   8.44836   8.44836   9.17755


The last gamma is exactly equivalent to the first gamma. It can be checked that the top of the valence band is obtained at Gamma (=5.91814 eV). The width of the valence band is 12.09 eV, the lowest unoccupied state at X is 0.594 eV higher than the top of the valence band, at Gamma.

The Si is described as an indirect band gap material (this is correct), with a band-gap of about 0.594 eV (this is quantitatively quite wrong: the experimental value 1.17 eV is at 25 degree Celsius). The minimum of the conduction band is even slightly displaced with respect to X, see kpt # 21. This underestimation of the band gap is well-known (the famous DFT band-gap problem). In order to obtain correct band gaps, you need to go beyond the Kohn-Sham Density Functional Theory: use the GW approximation. This is described in the first lesson on the GW approximation.

For experimental data and band structure representation, see
M.L. Cohen and J.R. Chelikowski
Electronic structure and optical properties of semiconductors
Springer-Verlag New-York (1988).

Important

There is a subtlety that is worth to comment about. In non-self-consistent calculations, like those performed in the present band structure calculation, with iscf=-2, not all bands are converged within the tolerance tolwfr. Indeed, the two upper bands (by default) have not been taken into account to apply this convergence criterion: they constitute a “buffer”. The number of such “buffer” bands is governed by the input variable nbdbuf.

It can happen that the highest (or two highest) band(s), if not separated by a gap from non-treated bands, can exhibit a very slow convergence rate. This buffer allows to achieve convergence of “important”, non-buffer bands. In the present case, 6 bands have been converged with a residual better than tolwfr, while the two upper bands are less converged (still sufficiently for graphical representation of the band structure). In order to achieve the same convergence for all 8 bands, it is advised to use nband=10 (that is, 8 + 2).