Second Harmonic Generation

by Dmitrii Nabok for exciting sodium

(Jupyter notebook by Tejus Rohatgi, Mara Voiculescu & Martin Kuban)

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Purpose: In this tutorial you will learn how to initialize and perform calculations of the second order susceptibility tensor with exciting. As an example, we compute the second-harmonic generation (SHG) spectrum of GaAs.

Table of Contents

0. Before Starting

1. Second-Harmonic Generation

• Excercise

Literature

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0. Before Starting

Read the following paragraphs before starting with the rest of this tutorial!

Before running any Jupyter tutorials, please refer to the 00_before_starting.md document on how to correctly set up the environment. This only needs to be done once. After which, the venv can be (re)activated from exciting's root directory:

source $EXCITINGROOT/tools/excitingjupyter/venv/excitingvenv/bin/activate

As a first step, you may create a running directory for the notebook.

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1. Second Harmonic Generation

Copy and paste the following exciting input data into input.xml.

<input>
 
   <title>GaAs SHG</title>
 
   <structure speciespath="$EXCITINGROOT/species">
      <crystal>
         <basevect>5.3435 5.3435 0.0000</basevect>
         <basevect>5.3435 0.0000 5.3435</basevect>
         <basevect>0.0000 5.3435 5.3435</basevect>
      </crystal>
      <species speciesfile="Ga.xml" rmt="2.0">
         <atom coord="0.00 0.00 0.00"/>
      </species>
      <species speciesfile="As.xml" rmt="2.0">
         <atom coord="0.25 0.25 0.25"/>
      </species>
   </structure>
 
   <groundstate
      do="fromscratch"
      rgkmax="7.0"
      ngridk="8 8 8"
      xctype="LDA_PW"
      nempty="10"
      >
   </groundstate>
 
   <properties>
      <momentummatrix/>
      <shg
         wmax="0.3"
         wgrid="400"
         swidth="0.004"
         etol="1.d-4"
         scissor="0.0423"
         tevout="true"
         >
         <chicomp>1 2 3</chicomp>
      </shg>
   </properties>
 
</input>

%%bash
mkdir -p shg

Make sure to set $EXCITINGROOT to the correct exciting root directory in the speciespath attribute using the command

%%bash
cd shg
python3 -m excitingscripts.setup.excitingroot
cd ../

Please get first familiar with the input parameters related to the shg element. Note that all required components of the second-order susceptibility tensor $\chi^{abc}_{(2)}(-2\omega, \omega, \omega)$ can be specified by adding corresponding chicomp combinations where $a$, $b$, and $c$ stand for the Cartesian indices ($1=x$, $2=y$, $3=z$).

Run exciting in the usual way.

%%bash
cd shg
time $EXCITINGROOT/install/bin/exciting_smp &

cd ../

As an example, we present results for $\chi^{xyz}_{(2)}(-2\omega, \omega, \omega)$ (which can be found in the output file CHI_123.OUT). Note that this file contains information about the frequency dependence of the real, imaginary, and absolute value of the corresponding tensor component given in the units of $10^{-7}\ esu$. In the literature, one would often find the second-order susceptibility in the units of $nm/V$. Please notice the conversion factor: $1\ nm/V$ = $23.86 \times 10^{-7}\ esu$. For tutorial purposes, the results could be visualized by executing

%%bash
cd shg

#extracting the thetha_k and gamma_k values
awk '{print $1, $2}' CHI_123.OUT > real
awk '{print $1, $3}' CHI_123.OUT > imaginary
awk '{print $1, $4}' CHI_123.OUT > modulus

# PLOT-SHG.py CHI_123.OUT
python3 -m excitingscripts.plot.files -f real imaginary modulus -cy 2 -ll 'Real' 'Imaginary' 'Modulus'  -lx 'Energy [eV]'  -ly '$\chi^{(2)}(-2\omega, \omega, \omega)$ $[10^{-7}\ esu]$' -g 
cd ../

The resulting plot is saved in the file PLOT.png, which should look like the following image:

Excercise

• Converge the SHG spectrum with respect to ngridk.
• Try different components of the SHG susceptibility tensor noting that many of them are actually zero due to crystal symmetry.

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Literature

• J. L. P. Hughes and J. E. Sipe, Phys. Rev. B 53, 10751 (2003)
• S. Bergfeld and W. Daum, Phys. Rev. Lett. 90, 036801 (2003)
• S. Sharma and C. Ambrosch-Draxl, Physica Scripta T109, 128 (2004)

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