# Tutorial¶

Input files prepared for this tutorial are located in the example/ directory of the ALAMODE package.

## Silicon¶

Silicon. 2x2x2 conventional supercell

In the following, (anharmonic) phonon properties of bulk silicon (Si) are calculated by a 2x2x2 conventional cell containing 64 atoms.

### 1. Get displacement patterns by alm¶

Change directory to example/Si and open file si_alm.in. This file is an input for the code alm which estimate interatomic force constants (IFC) by least square fitting. In the file, the crystal structure of a 2x2x2 conventional supercell of Si is specified in the &cell and the &position fields as the following:

&general
PREFIX = si222
MODE = suggest
NAT = 64; NKD = 1
KD = Si
/

&interaction
NORDER = 1  # 1: harmonic, 2: cubic, ..
/

&cell
20.406 # factor in Bohr unit
1.0 0.0 0.0 # a1
0.0 1.0 0.0 # a2
0.0 0.0 1.0 # a3
/

&cutoff
Si-Si None
/

&position
1 0.0000000000000000 0.0000000000000000 0.0000000000000000
1 0.0000000000000000 0.0000000000000000 0.5000000000000000
1 0.0000000000000000 0.2500000000000000 0.2500000000000000
1 0.0000000000000000 0.2500000000000000 0.7500000000000000
1 0.0000000000000000 0.5000000000000000 0.0000000000000000
1 0.0000000000000000 0.5000000000000000 0.5000000000000000
1 0.0000000000000000 0.7500000000000000 0.2500000000000000


Replace the lattice constant of the supercell (20.406 Bohr) by your own value.

Then, execute alm

$alm si_alm.in > si_alm.log1  which creates a file si222.pattern_HARMONIC in the working directory. In the pattern file, suggested displacement patterns are defined in Cartesian coordinates. As you can see in the file, there is only one displacement pattern for harmonic IFCs of bulk Si. ### 2. Calculate atomic forces for the displaced configurations¶ Next, calculate atomic forces for all the displaced configurations defined in si222.pattern_HARMONIC. To do so, you first need to decide the magnitude of displacements $$\Delta u$$, which should be small so that anharmonic contributions are negligible. In ordinary case, $$\Delta u \sim 0.01$$ Å is a reasonable choice. Then, prepare input files necessary to run an external DFT code for each configuration. Since this procedure is a little tiresome, we provide a subsidiary Python script for VASP, Quantum-ESPRESSO (QE), and xTAPP. Using the script displace.py in the tools/ directory, you can generate the necessary input files as follows: QE $ python displace.py --QE=si222.pw.in --mag=0.02 si222.pattern_HARMONIC


VASP

$python displace.py --VASP=POSCAR.orig --mag=0.02 si222.pattern_HARMONIC  xTAPP $ python displace.py --xTAPP=si222.cg --mag=0.02 si222.pattern_HARMONIC


The --mag option specifies the displacement length in units of Angstrom. You need to specify an input file with equilibrium atomic positions either by the --QE, --VASP, or --xTAPP.

Then, calculate atomic forces for all the configurations. This can be done with a simple shell script as follows:

!#/bin/bash

# Assume we have 20 displaced configurations for QE [disp01.pw.in,..., disp20.pw.in].

for ((i=1;i<=20;i++))
do
num=echo $i | awk '{printf("%02d",$1)}'
mkdir ${num} cd${num}
cp ../disp${num}.pw.in . pw.x < disp${num}.pw.in > disp${num}.pw.out cd ../ done  Important In QE, please set tprnfor=.true. to get atomic forces. The next step is to collect the displacement data and force data by the Python script extract.py (also in the tools/ directory). This script can extract atomic displacements, atomic forces, and total energies from multiple output files as follows: QE $ python extract.py --QE=si222.pw.in --get=disp *.pw.out > disp.dat
$python extract.py --QE=si222.pw.in --get=force *.pw.out > force.dat  VASP $ python extract.py --VASP=POSCAR.orig --get=disp vasprun*.xml > disp.dat
$python extract.py --VASP=POSCAR.orig --get=force vasprun*.xml > force.dat  xTAPP $ python extract.py --xTAPP=si222.cg --get=disp *.str > disp.dat
$python extract.py --xTAPP=si222.cg --get=force *.str > force.dat  In the above examples, atomic displacements of all the configurations are merged as disp.dat, and the corresponding atomic forces are saved in the file force.dat. These files will be used in the following fitting procedure as DFILE and FFILE. (See Format of DFILE and FFILE). Note For your convenience, we provide the disp.dat and force.dat files in the reference/ subdirectory. These files were generated by the Quantum-ESPRESSO package with --mag=0.02. You can proceed to the next step by copying these files to the working directory. ### 3. Estimate force constants by fitting¶ Edit the file si_alm.in to perform least-square fitting. Change the MODE = suggest to MODE = fitting as follows: &general PREFIX = si222 MODE = fitting # <-- here NAT = 64; NKD = 1 KD = Si /  Also, add the &fitting field as: &fitting NDATA = 1 DFILE = disp.dat FFILE = force.dat /  Then, execute alm again $ alm si_alm.in > si_alm.log2


This time alm extract harmonic IFCs from the given displacement-force data set (disp.dat and force.dat above).

You can find files si222.fcs and si222.xml in the working directory. The file si222.fcs contains all IFCs in Rydberg atomic units. You can find symmetrically irreducible sets of IFCs in the first part as:

 *********************** Force Constants (FCs) ***********************
*        Force constants are printed in Rydberg atomic units.       *
*        FC2: Ry/a0^2     FC3: Ry/a0^3     FC4: Ry/a0^4   etc.      *
*        FC?: Ry/a0^?     a0 = Bohr radius                          *
*                                                                   *
*        The value shown in the last column is the distance         *
*        between the most distant atomic pairs.                     *
*********************************************************************

----------------------------------------------------------------------
Index              FCs         P        Pairs     Distance [Bohr]
(Global, Local)              (Multiplicity)
----------------------------------------------------------------------

*FC2
1       1     2.7617453e-01   1     1x     1x       0.000
2       2    -1.2195057e-03   2     1x     2x      10.203
3       3    -6.2496693e-04   2     1x    33x      10.203
4       4     8.5935598e-03   1     1x     3x       7.215
5       5     1.9655898e-03   1     1x     3y       7.215
6       6    -4.2490872e-03   1     1x    17x       7.215
7       7    -3.9172885e-03   1     1x    17z       7.215
8       8     7.9671376e-03   4     1x     6x      14.429
9       9    -2.3601630e-04   4     1x    34x      14.429
10      10    -6.7104831e-02   1     1x     9x       4.418
11      11    -4.7380801e-02   1     1x     9y       4.418
12      12     1.3282532e-03   1     1x    10x       8.460
13      13    -8.2116209e-04   1     1x    10y       8.460
14      14    -6.9561306e-04   1     1x    10z       8.460
15      15     4.5271736e-04   1     1x    26x       8.460
16      16    -4.0571255e-03   1     1x    11x      11.118
17      17     5.0517278e-04   1     1x    11y      11.118
18      18    -2.0691115e-04   1     1x    25x      11.118
19      19    -4.7362015e-04   1     1x    25z      11.118
20      20    -1.0042599e-03   2     1x    20x      12.496
21      21     1.2863153e-03   2     1x    20y      12.496
22      22    -1.6874372e-04   2     1x    20z      12.496
23      23    -2.0122242e-04   2     1x    35x      12.496
24      24     6.0327036e-04   1     1x    28x      13.254
25      25    -5.1714558e-04   1     1x    28y      13.254


Harmonic IFCs $$\Phi_{\mu\nu}(i,j)$$ in the supercell are given in the third column, and the multiplicity $$P$$ is the number of times each interaction $$(i, j)$$ occurs within the given cutoff radius. For example, $$P = 2$$ for the pair $$(1x, 2x)$$ because the distance $$r_{1,2}$$ is exactly the same as the distance $$r_{1,2'}$$ where the atom 2’ is a neighboring image of atom 2 under the periodic boundary condition. If you compare the magnitude of IFCs, the values in the third column should be divided by $$P$$.

In the log file si_alm2.log, the fitting error is printed. Try

$grep "Fitting error" si_alm2.log Fitting error (%) : 1.47766  The other file si222.xml contains crystal structure, symmetry, IFCs, and all other information necessary for subsequent phonon calculations. ### 4. Calculate phonon dispersion and phonon DOS¶ Open the file si_phband.in and edit it for your system. &general PREFIX = si222 MODE = phonons FCSXML =si222.xml NKD = 1; KD = Si MASS = 28.0855 / &cell 10.203 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.0 / &kpoint 1 # KPMODE = 1: line mode G 0.0 0.0 0.0 X 0.5 0.5 0.0 51 X 0.5 0.5 1.0 G 0.0 0.0 0.0 51 G 0.0 0.0 0.0 L 0.5 0.5 0.5 51 /  Please specify the XML file you obtained in step 3 by the FCSXML-tag as above. In the &cell-field, you need to define the lattice vector of a primitive cell. In phonon dispersion calculations, the first entry in the &kpoint-field should be 1 (KPMODE = 1). Then, execute anphon $ anphon si_phband.in > si_phband.log


which creates a file si222.bands in the working directory. In this file, phonon frequencies along the given reciprocal path are printed in units of cm-1 as:

#  G X G L
# 0.000000 0.615817 1.486715 2.020028
# k-axis, Eigenvalues [cm^-1]
0.000000   1.097373e-10   1.097373e-10   1.097373e-10   5.053714e+02   5.053714e+02   5.053714e+02
0.012316   6.132848e+00   6.132848e+00   1.033887e+01   5.052979e+02   5.052979e+02   5.053619e+02
0.024633   1.225469e+01   1.225469e+01   2.066997e+01   5.050779e+02   5.050779e+02   5.053331e+02
0.036949   1.835448e+01   1.835448e+01   3.098558e+01   5.047130e+02   5.047130e+02   5.052842e+02
0.049265   2.442116e+01   2.442116e+01   4.127802e+01   5.042056e+02   5.042056e+02   5.052137e+02
0.061582   3.044357e+01   3.044357e+01   5.153973e+01   5.035592e+02   5.035592e+02   5.051196e+02
0.073898   3.641049e+01   3.641049e+01   6.176326e+01   5.027783e+02   5.027783e+02   5.049995e+02


You can plot the phonon dispersion relation with gnuplot or any other plot software.

For visualizing phonon dispersion relations, we provide a Python script plotband.py in the tools/ directory (Matplotlib is required.). Try

$python plotband.py si222.bands  Then, the phonon dispersion is shown by a pop-up window as follows: You can save the figure as png, eps, or other formats from this window. You can also change the energy unit of phonon frequency from cm-1 to THz or meV by the --unit option. For more detail of the usage of plotband.py, type $ python plotband.py -h


Next, let us calculate the phonon DOS. Copy si_phband.in to si_phdos.in and edit the &kpoint field as follows:

&kpoint
2  # KPMODE = 2: uniform mesh mode
20 20 20
/


Then, execute anphon

$anphon si_phdos.in > si_phdos.log  This time anphon creates files si222.dos and si222.thermo in the working directory, which contain phonon DOS and thermodynamic functions, respectively. For visualizing phonon DOS and projected DOSs, we provide a Python script plotdos.py in the tools/ directory (Matplotlib is required.). The command $ python plotdos.py --emax 550 --nokey si222.dos


will show the phonon DOS of Si by a pop-up window:

To obtain more sharp DOS, try again with a denser $$k$$ grid and a smaller DELTA_E value.

### 5. Estimate cubic IFCs for thermal conductivity¶

Copy file si_alm.in to si_alm2.in. Edit the &general, &interaction, and &cutoff fields of si_alm2.in as the following:

&general
PREFIX = si222_cubic
MODE = suggest
NAT = 64; NKD = 1
KD = Si
/


Change the PREFIX from si222 to si222_cubic and set MODE to suggest.

&interaction
NORDER = 2
/


Change the NORDER tag from NORDER = 1 to NORDER = 2 to include cubic IFCs. Here, we consider cubic interaction pairs up to second nearest neighbors by specifying the cutoff radii as:

&cutoff
Si-Si None 7.3
/


7.3 Bohr is slightly larger than the distance of second nearest neighbors (7.21461 Bohr). Change the cutoff value appropriately for your own case. (Atomic distance can be found in the file si_alm.log.)

Then, execute alm

$alm si_alm2.in > si_alm2.log  which creates files si222_cubic.pattern_HARMONIC and si222_cubic.pattern_ANHARM3. Then, calculate atomic forces of displaced configurations given in the file si222_cubic.pattern_ANHARM3, and collect the displacement (force) data to a file disp3.dat (force3.dat) as you did for harmonic IFCs in Steps 3 and 4. Note Since making disp3.dat and force3.dat requires moderate computational resources, you can skip this procedure by copying files reference/disp3.dat and reference/force3.dat to the working directory. The files we provide were generated by the Quantum-ESPRESSO package with --mag=0.04. In si_alm2.in, change MODE = suggest to MODE = fitting and add the following: &fitting NDATA = 20 DFILE = disp3.dat FFILE = force3.dat FC2XML = si222.xml # Fix harmonic IFCs /  By the FC2XML tag, harmonic IFCs are fixed to the values in si222.xml. Then, execute alm again $ alm si_alm2.in > si_alm2.log2


which creates files si222_cubic.fcs and si222_cubic.xml. This time cubic IFCs are also included in these files.

Note

In the above example, we obtained cubic IFCs by least square fitting with harmonic IFCs being fixed to the value of the previous harmonic calculation. You can also estimate both harmonic and cubic IFCs simultaneously instead. To do this, merge disp.dat and disp3.dat ( and force files) as

$cat disp.dat disp3.dat > disp_merged.dat$ cat force.dat force3.dat > force_merged.dat


and change the &fitting field as the following:

&fitting
NDATA = 21
DFILE = disp_merged.dat
FFILE = force_merged.dat
/


### 6. Calculate thermal conductivity¶

Copy file si_phdos.in to si_RTA.in and edit the MODE and FCSXML tags as follows:

&general
PREFIX = si222
MODE = RTA
FCSXML = si222_cubic.xml

NKD = 1; KD = Si
MASS = 28.0855
/


In addition, change the $$k$$ grid density as:

&kpoint
2
10 10 10
/


Then, execute anphon as a background job

$anphon si_RTA.in > si_RTA.log &  Please be patient. This can take a while. When the job finishes, you can find a file si222.kl in which the lattice thermal conductivity is saved. You can plot this file using gnuplot (or any other plotting softwares) as follows: $ gnuplot
gnuplot> set logscale xy
gnuplot> set xlabel "Temperature (K)"
gnuplot> set ylabel "Lattice thermal conductivity (W/mK)"
gnuplot> plot "si222.kl" usi 1:2 w lp


Calculated lattice thermal conductivity of Si (click to enlarge)

As you can see, the thermal conductivity diverges in $$T\rightarrow 0$$ limit. This occurs because we only considered intrinsic phonon-phonon scatterings in the present calculation and neglected phonon-boundary scatterings which are dominant in the low temperature range. The effect of the boundary scattering can be included using the python script analyze_phonons.py in the tools directory:

$gnuplot gnuplot> set logscale x gnuplot> set xlabel "L (nm)" gnuplot> set ylabel "Cumulative kappa (W/mK)" gnuplot> plot "cumulative_300K.dat" using 1:2 w lp  Cumulative thermal conductivity of Si at 300 K (click to enlarge) To draw a smooth line, you will have to use a denser $$q$$ grid as shown in the figure by the orange line, which are obtained with $$20\times 20\times 20\ q$$ points. #### Thermal conductivity spectrum¶ To calculate the spectrum of thermal conductivity, modify the si_RTA.in as follows: &general PREFIX = si222 MODE = RTA FCSXML = si222_cubic.xml NKD = 1; KD = Si MASS = 28.0855 EMIN = 0; EMAX = 550; DELTA_E = 1.0 # <-- frequency range / &cell 10.203 0.0 0.5 0.5 0.5 0.0 0.5 0.5 0.5 0.0 / &kpoint 2 10 10 10 / &analysis KAPPA_SPEC = 1 # compute spectrum of kappa /  The frequency range is specified with the EMIN, EMAX, and DELTA_E tags, and the KAPPA_SPEC = 1 is set in the &analysis field. Then, execute anphon again $ anphon si_RTA.in > si_RTA2.log


After the calculation finishes, you can find the file si222.kl_spec which contains the spectra of thermal conductivity at various temperatures. You can plot the data at room temperature as follows:

$awk '{if ($1 == 300.0) print $0}' si222.kl_spec > si222_300K.kl_spec$ gnuplot
gnuplot> set xlabel "Frequency (cm^{-1})"
gnuplot> set ylabel "Spectrum of kappa (W/mK/cm^{-1})"
gnuplot> plot "si222_300K.kl_spec" using 2:3 w l lt 2 lw 2


Spectrum of thermal conductivity of Si at 300 K (click to enlarge)

In the above figure, a computational result with $$20\times 20\times 20\ q$$ points is also shown by dashed line. From the figure, we can see that low energy phonons below 200 cm$$^{-1}$$ account for more than 80% of the total thermal conductivity at 300 K.