Born Effective Charges of Zincblende-Structure Crystals

by Sebastian Tillack for exciting oxygen

Purpose: In this tutorial, you will learn how to compute Born effective charges from the macroscopic polarization in zincblende crystals.

0. Theoretical background

The Born effective charge (or dynamical charge) ${\bf Z}_\alpha^\star$ of an atom $\alpha$ describes the effective dipole ${\bf p}$ that is induced in the unit cell due to a displacement ${\bf d}$ of the atom from its equilibrium position:

\begin{align} {\bf p} = {\bf Z}_\alpha^\star \cdot {\bf d}\;. \end{align}

Note, that in general, ${\bf Z}_\alpha^\star$ is a 3$\times$3 tensor and the dipole moment is not necessarily parallel to the displacement. The Born effective charge tensor is defined as the negative second derivative of the total energy with respect to an external electric field ${\bf \mathcal{E}}$ and the atomic position ${\bf \tau}_\alpha$ evaluated at the equilibrium position:

\begin{align} Z^\star_{\alpha,ij} = - \left.\frac{\partial^2 E}{\partial \mathcal{E}_j\, \partial \tau_{\alpha,i}}\right|_0\;. \end{align}

In practice, we compute the Born effective charges from the macroscopic polarization ${\bf \mathcal{P}}$ that is given by

\begin{align} \mathcal{P}_i = - \frac{1}{\Omega} \left.\frac{\partial E}{\partial \mathcal{E}_i}\right|_0\;, \end{align}

where $\Omega$ is the unit cell volume. With that, the charge tensor reads

\begin{align} Z^\star_{\alpha,ij} = \left. \Omega \frac{\partial \mathcal{P}_j}{\partial \tau_{\alpha,i}}\right|_0\;. \end{align}

In order to evaluate the macroscopic polarization, we employ the so-called Berry phase approach.

1. Prerequisites

We assume that you are familiar with basic phonon calculations using exciting as described in the tutorial Phonon Properties of Diamond-Structure Crystals. In this example, we will compute the Born effective charge tensors in cubic boron nitride. Create the new directory cBN-Borncharges and move into it.

$ cd /home/tutorials/
$ mkdir cBN-Borncharges
$ cd cBN-Borncharges

2. Calculation of Born effective charges

The derivatives of the macroscopic polarization with respect to the atomic positions are evaluated using finite differences. Therefore, we need to displace each atom in the unit cell in each of the Cartesian directions and evaluate the macroscopic polarization. This procedure is very similar to a Γ-point phonon calculation. Therefore, the Born effective charges are implemented as a side product of a regular Γ-point phonon calculation. I.e., in order to trigger a Born effective charge calculation, we need to include the phonons element in the input file.

   <title>cBN Born effective charges</title>
   <structure speciespath="$EXCITINGROOT/species/">
      <crystal scale="6.8408">
         <basevect>0.0   0.5   0.5</basevect>
         <basevect>0.5   0.0   0.5</basevect>
         <basevect>0.5   0.5   0.0</basevect>
      <species speciesfile="B.xml">
         <atom coord="0.00 0.00 0.00"/>
      <species speciesfile="N.xml">
         <atom coord="0.25 0.25 0.25"/>
      ngridk="4 4 4" 
      ngridq="1 1 1"
         <qpoint> 0.0 0.0 0.0 </qpoint>

Note, that we set the attribute ngridq="1 1 1" which corresponds to a phonon calculation at Γ only.

Start the calculation with

$ time exciting_smp

Once the calulcation is finished, you will find the result for the Born effective charges in the file ZSTAR.OUT. Its content looks like

# Born effective charges
# Note: Acoustic sum rule was automatically imposed.
# Species  1  Atom  1 (B  1) :      0.000000     0.000000     0.000000
        2.001681        0.000009        0.000009
        0.000015        2.002129        0.000015
        0.000069        0.000069        1.998837
# Species  2  Atom  1 (N  1) :      0.250000     0.250000     0.250000
       -2.001681       -0.000009       -0.000009
       -0.000015       -2.002129       -0.000015
       -0.000069       -0.000069       -1.998837
# acoustic sum rule correction
       -0.000004        0.000002        0.000002
        0.000001       -0.000004        0.000001
        0.000001        0.000001       -0.000005

For each atom in the unit cell, it gives the full charge tensor in units of the elementary charge. In cubic zincblende crystals, the charge tensor is diagonal and isotropic and can effectively be described by a scalar value. In this example, we obtain an effective charge of about ±2.00 for the boron and nitrogen atom, respectively.

In theory, the Born effective charge tensors obey the acoustic sum rule, i.e., that the sum over all atoms is zero. We automatically impose the acoustic sum rule by subtracting the actual sum of all charge tensors divided by the total number of atoms from the charge tensor of each atom. The last block in the file gives the correction that was necessary to impose the acoustic sum rule. Ideally, it should be zero.

Note, that the actual values for the macroscopic polarization for each displacement can be found the files POLARIZATION_fileext.OUT in the directories Q0000_0000_0000_S##_A###_P#, where fileext is the filename extension described in Phonon properties of diamond-structure crystals. See also the input element polarization.

  • The calculation presented here for cubic boron nitride can be repeated for silicon carbide. Use the PBE exchange-correlation functional and an equilibrium lattice parameter of 8.2392 Bohr.

  • Which result do you expect for the Born effective charges in diamond? Examine the file ZSTAR.OUT in the directory diamond-phonons/c-test/ in your home directory and explain the result.
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