Macroscopic polarization by Berry's phase

The macroscopic electric polarization of a bulk system can be calculated based on the Berry phase formalism [12]. As an example, let us illustrate a calculation of a Born effective charge of Na in a NaCl bulk via the macroscopic polarization.

(1) SCF calculation
First, perform a conventional SCF calculation using an input file NaCl.dat in the directory 'work'. Then, the following keyword 'HS.fileout' should be switched on

    HS.fileout               on     # on|off, default=off

When the calculation is completed normally, then you can find an output file 'nacl.scfout' in the directory 'work'.

(2) Calculation of macroscopic polarization
The macroscopic polarization is calculated by a post-processing code 'polB' of which input data is 'nacl.scfout'. In the directory 'source', compile as follows:

    % make polB 
  

When the compile is completed normally, then you can find an executable file 'polB' in the directory 'work'. Then, move to the directory 'work', and perform as follows:

    % polB nacl.scfout
   or
    % polB nacl.scfout < in > out
  

In the later case, the file 'in' contains the following ingredients:

    9 9 9
    1 1 1

In the former case, you will be interactively asked from the program as follows:

******************************************************************
******************************************************************
 polB:
 code for calculating the electric polarization of bulk systems
 Copyright (C), 2006-2007, Fumiyuki Ishii and Taisuke Ozaki 
 This is free software, and you are welcome to         
 redistribute it under the constitution of the GNU-GPL.
******************************************************************
******************************************************************

Read the scfout file (nacl.scfout)
 Previous eigenvalue solver = Band
 atomnum                    = 2
 ChemP                      = -0.156250000000 (Hartree)
 E_Temp                     = 300.000000000000 (K)
 Total_SpinS                =  0.000000000000 (K)
 Spin treatment             = collinear spin-unpolarized

 r-space primitive vector (Bohr)
  tv1=  0.000000   5.319579   5.319579
  tv2=  5.319579   0.000000   5.319579
  tv3=  5.319579   5.319579   0.000000
 k-space primitive vector (Bohr^-1)
  rtv1= -0.590572   0.590572   0.590572
  rtv2=  0.590572  -0.590572   0.590572
  rtv3=  0.590572   0.590572  -0.590572

  Cell_Volume=301.065992 (Bohr^3)

 Specify the number of grids to discretize reciprocal a-, b-, and c-vectors
 (e.g 2 4 3)
   k1    0.00000    0.11111    0.22222    0.33333    0.44444  ...
   k2    0.00000    0.11111    0.22222    0.33333    0.44444  ...
   k3    0.00000    0.11111    0.22222    0.33333    0.44444  ...

 Specify the direction of polarization as reciprocal a-, b-, and c-vectors
 (e.g 0 0 1 ) 1 1 1

Then, the calculation will start like this:

calculating the polarization along the a-axis ....
The number of strings for Berry phase : AB mesh=81

  calculating the polarization along the a-axis ....   1/  82
  calculating the polarization along the a-axis ....   2/  82 
  .....
  ...

*******************************************************
              Electric dipole  (Debye) : Berry phase
*******************************************************

 Absolute dipole moment      163.93373639

               Background        Core             Electron          Total

 Dx           -0.00000000       94.64718996       -0.00000338       94.64718658
 Dy           -0.00000000       94.64718996       -0.00000283       94.64718713
 Dz           -0.00000000       94.64718996       -0.00000317       94.64718679


***************************************************************
              Electric polarization (muC/cm^2) : Berry phase
***************************************************************

               Background        Core             Electron          Total

 Px           -0.00000000      707.66166752       -0.00002529      707.66164223
 Py           -0.00000000      707.66166752       -0.00002118      707.66164633
 Pz           -0.00000000      707.66166752       -0.00002371      707.66164381


Elapsed time = 77.772559 (s) for myid=  0

Since the Born effective charge $Z_{\alpha\beta}^{*}$ is defined by a tensor:

$$Z_{\alpha\beta}^{*} = \frac{V_{c}}{\vert e \vert}\frac{\Delta P_{\alpha}}{\Delta u_{\beta}}$$

where $V_{c}$ is the volume of the unit cell, $e$ the elementary charge, $\Delta u_{\beta}$ displacement along $\beta$-coordinate, $\Delta P_{\alpha}$ the change of macroscopic polarization along $\alpha$-coordinate, therefore we will perform the above procedures (1) and (2) at least two or three times by varying the $x$, $y$, or $z$-coordinate of Na atom. Then, for example $x$-coordinates, we have

    Px = 94.39497736 (Debye/unit cell) at x= -0.05 (Ang)
    Px = 94.64718658 (Debye/unit cell) at x=  0.0  (Ang)
    Px = 94.89939513 (Debye/unit cell) at x=  0.05 (Ang)

Thus,

$$Z^{*}_{xx} = \frac{(94.89939513-94.39497736)/(2.54174776)}{0.1/0.529177}$$

$$= 1.050$$

Note that in the NaCl bulk the off-diagonal terms in the tensor of Born charge are zero, and $Z^{*}_{xx}=Z^{*}_{yy}=Z^{*}_{zz}$. In Table 5 we see that the calculated value is in good agreement with the other calculation [67] and an experimental result [68]. The calculation of macroscopic polarization is supported for both the collinear and non-collinear DFT. It is also noted that the code 'polB' has been parallelized for large-scale systems where the number of processors can exceed the number of atoms in the system.

Table: Calculated Born effective charge of Na in a NaCl bulk. The input file is NaCl.dat in the directory 'work'. Another theoretical value (FD: Ref. [67]) and experimental value (Ref. [68]) are also shown for comparison.

	OpenMX	FD	Expt.
$Z^*$	1.05	1.09	1.12