This lesson aims at showing you how to get the following physical properties, for an insulator :
This lesson should take about 1 hour to be done.
Before beginning, you might consider to work in a different subdirectory as for lesson_1 or lesson_2 . Why not "Work3" ?
The file ~ABINIT/Tutorial/t3x.files lists the file names and root names. You can copy it in the Work3 directory and change it as you did for the t1x.files and t2x.files files. You can also copy the file ~ABINIT/Tutorial/t31.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 a few seconds).
Then, read the output file, and note the total energy.
etotal -8.8662238960E+00
There is of course a convergence study associated to 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 8However, the associated number of k-points in the irreducible Brillouin zone grows very fast. It is
nkpt1 2 nkpt2 10 nkpt3 28 nkpt4 60ABINIT computes this number of k point, from the definition of the grid and the symmetries. You might define an input nkpt value, in which case ABINIT will compare its computed value 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/Tutorial/t32.in is an example.
Do not forget to change t3x.files,
if you are using that file name . The message that you get at the end of the
log file is :
inkpts : ERROR - The input value of nkpt= 2, does not match the number of k points generated by kptopt, ngkpt, shiftk, ane the eventual symmetries, that is, nkpt= 10. Action : change nkpt in your input file, or one of the other input variables.
This is a typical ABINIT error message. It announce clearly that you should use nkpt 10.
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. ABINIT is able to examine automatically different k point grids, and to propose the best grids for integration. This is described in the abinis_help file, see the input variable prtkpt, and the associated characterisation of the integral accuracy, described in kptrlen. The generation of lists of k-point sets is done in different test cases, in the directory Test_v2. You can directly have a look at the output files in ~ABINIT/Test_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 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 !
In order to understand k-point grids, you should read the Monkhorst and Pack paper, Phys. Rev. B 13, 5188 (1976) ... Well, maybe not immediately ... In the meantime, you can try the above-mentioned convergence study.
The input file ~ABINIT/Tutorial/t33.in is an example, while ~ABINIT/Tutorial/Refs/t33.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 -8.8726017429E+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 -8.872 Ha .
The input variable "optcell"
governs the automatic optimisation of cell shape and volume.
For the automatic optimisation of cell volume, use :
opcell 1 ionmov 3 ntime 10 dilatmx 1.05 ecutsm 0.5
You should read the indications about dilatmx
and ecutsm.
Do not test all the k point grids, only those with nkpt 2 and 10.
The input file ~ABINIT/Tutorial/t34.in is an example,
while ~ABINIT/Tutorial/Refs/t34.out is a reference output
file.
This run might last a few minutes ...
You should obtain the following evolution of the lattice parameters :
acell1 1.0230001904E+01 1.0230001904E+01 1.0230001904E+01 Bohr
acell2 1.0216682464E+01 1.0216682464E+01 1.0216682464E+01 Bohr
with the following very small residual stresses :
strten1 -2.5365388633E-08 -2.5365388633E-08 -2.5365388633E-08
0.0000000000E+00 0.0000000000E+00 0.0000000000E+00
strten2 -5.3567080431E-08 -5.3567080431E-08 -5.3567080431E-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 t3x.files file) is 10.217
Bohr, that is 5.407 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).
We fix the parameters acell
to the theoretical value of 3*10.217, 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 :
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 :
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.
The input file ~ABINIT/Tutorial/t35.in is an example, while ~ABINIT/Tutorial/Refs/t35.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) -3.78987 -1.16032 4.69394 4.69394 7.38389 9.23579 9.23579 13.45363 kpt# 2, nband= 8, wtk= 1.00000, kpt= 0.4500 0.0000 0.0000 (reduced coord) -3.92925 -0.95943 4.71018 4.71018 7.40286 9.25271 9.25271 13.48581 kpt# 3, nband= 8, wtk= 1.00000, kpt= 0.4000 0.0000 0.0000 (reduced coord) -4.25584 -0.44579 4.76451 4.76451 7.46440 9.30899 9.30899 13.57381 kpt# 4, nband= 8, wtk= 1.00000, kpt= 0.3500 0.0000 0.0000 (reduced coord) -4.64158 0.24736 4.85456 4.85456 7.56450 9.38020 9.38020 13.64227 ....
One needs a graphical tool to represent all these data ... (For the MAPR 2451 lecture : try with MATLAB)
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) -3.78987 -1.16032 4.69394 4.69394 7.38389 9.23579 9.23579 13.45363 kpt# 11, nband= 8, wtk= 1.00000, kpt= 0.0000 0.0000 0.0000 (reduced coord) -6.17102 5.91515 5.91515 5.91515 8.44524 8.44524 8.44524 9.17125 kpt# 23, nband= 8, wtk= 1.00000, kpt= 0.0000 0.5000 0.5000 (reduced coord) -1.96598 -1.96598 3.00347 3.00347 6.50928 6.50928 15.94976 16.44101 kpt# 40, nband= 8, wtk= 1.00000, kpt= 1.0000 1.0000 1.0000 (reduced coord) -6.17102 5.91515 5.91515 5.91515 8.44524 8.44524 8.44524 9.17125The last gamma is exactly equivalent to the first gamma. It can be checked that the top of the valence band is obtained at Gamma (=15.56202 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 diplaced 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 sixth lesson of the tutorial.
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).
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).