

In this exercise you will calculate the macroscopic polarization
in a polar material using the so-called Berry phase approach.
We will obtain the behavior of the polarization 
as a function of the atomic displacements.
With this information you can calculate the so-called dynamical
or Born charges that characterize the interaction between a
macroscopic electric field and the elastic deformations of the 
material. Experimentally the dynamical charges can easily 
be proved by measuring the LO-TO splitting. This is a splitting 
between vibrational modes (difference of their frequencies)
that, in principle, could be degenerate
due to the symmetry of the crystal when q-->0 (q being the
wavevector of the vibration). However, when long range electrostatic
forces are introduced these modes behave in a different way 
when q-->0. LO modes produce accumulations of charge along
the q direction and, for 3-D crystals, this translates
into a macroscopic electric field when q-->0. In contrast TO modes
are not charged and not affect for the long range electrostatic
part of the dynamical matrix. The LO-TO splitting is proportional
to the Born or dynamical effective charges and can be extremely
large for ferroelectric materials. 

Important note: These calculations only make sense for insulators!!!!!

POLARIZATION AS A FUNTION OF THE BOND LENGHT IN c-BN

Go to directory 
'c-BN.pol_vs_bond_elongation' ,
you will find there two SIESTA input files
called 'c-BN.init.fdf' and 'c-BN.elongated.fdf'. 

Edit file 'c-BN.init.fdf'. It contains  
the information to produce a calculation for c-BN. At the
end of the file you will notice the presence of a peculiar
block:

%block PolarizationGrids
10  4   4
4   10  4
4   4   10
%endblock PolarizationGrids

This block contains the information about the k-space grids
that will be used in order to compute the macroscopic polarization 
using the geometric Berry phase approach 
(R.D. King-Smith and D. Vanderbilt, PRB 47, 1651 (1993)).
Only if this block is present, the calculation of the macroscopic
polarization will be performed.
In this approach, to compute the polarization along a given
direction it is necessary to perform: i) a 1-D integral 
in k-space along the chosen direction and ii) a 2-D integration
of the result of the previous integral within the plane 
perpendicular (also in k-space) to the chosen direcction.
the block 'PolarizationGrids' specifies the k-space grids used
perform these operation.
Each line in block 'PolarizationGrids' corresponds to a 
grid used to calculate the polarization along the 
a given lattice vectors: the first row specifies the grid that
will be used to calculate the polarization along the direction of the
first lattice vector, the second row will be used for the calculation
along the direction of the second lattice vector, and the third
row for the third lattice vector.  The numbers in the diagonal of the
matrix specify the number of points to be used in the one dimensional
line integrals along the different directions. The other numbers
specify the mesh used in the surface integrals.  
If the number of point in one of the grids is zero, 
the calculation will not be performed for this
particular direction.

You can play with the numbers in this block to check 
for the convergence of the polarization as a function
of the fineness of the grid.

Notice, however, that the only the changes in polarization are
physically relevant. The file 'c-BN.elongated.fdf' contains the 
input for a c-BN system with one elongated BN bond. As a consequence
the system has lost its perfect cubic symmetry and will develop
a polarization along the direction parallel to the elongated
bond. 

You should calculate the behavior of a polarization as a function
of this distorsion. An interesting aspect is that the calculated 
curve might not be smooth. This is a consequence ill-definition
of the electric polarization. The polarization per unit cell
can only be defined modulo a quantum of polarization: 2eR, 
being e the electron charge and R and arbitrary vector of the
Bravais lattice. Once this effect is taken into account
you should obtain a smooth curve. 

A question you should answer:
we said that "only" the changes of polarization make sense,
therefore,  what is the meaning of this curve? 
where should we set the zero of this curve?

CALCULATE DYNAMICAL CHARGES FOR c-BN

If you apply small enough displacements you can calculate 
the Born or dynamical charges.

The Born charges are tensors Z_xy defined as:

Z(a)_xy =   dP_x/dy_a

where dP_x is the change in the polarization along the x direction
when the atom a is displaced by dy along the y direction.
For a cubic crystal, like the c-BN for which you will perform
the calculations, Z(a)_xy becomes a scalar Z(a). 

You should check that this is indeed the case performing calculation
of the Z(N) and Z(B) along different direction.

Furthermore, the dynamical charge obey the so-called acoustic sum-rule:
the sum of the dynamical charges for all the atoms in the cell should be
zero. In our case this means that Z(N)=-Z(B).
You should check to what extent this is true in our calculations, 
and how the fulfillment of this rule
depends on the fineness of the utilized k-space grids.


BORN CHARGES FOR h-BN

Hexagonal BN is layered material. Therefore, 'in plane' and
'out of plane' dynamical charges might differ. Directories
'h-BN.Z_parallel' and  'h-BN.Z_perpendicular' contain input files
to perform these calculations. 

BORN CHARGES FOR h-BN VERSUS LAYER-LAYER DISTANCE.

The distance between different layers in h-BN can influence
the values of the dynamical charges, particularly for out 
of plane movements. Check this dependence performing a calculation
for a larger value of the c/a ratio.



 
  



