How can I get spontaneous polarization and piezoelectric constants for III-V wurtzite nitride?

[hongkiha]

I tried to reproduce spontaneous polarization and piezoelectric constants for III-V nitride in Phys. Rev. B. Vol. 56, R10024 {Vanderbilt's paper}.

My reusults on piezoelectric constants are somewhat different from reference.
I don't know how to obtain spontaneous polarization value from OUTCAR.

1. Spontaneous polarization,
I followed the procedure of Berry phase calculation part in manual. Then, it is estimated that Born effective charges are -2.72 for GaN and -2.62 for AlN.
Although I got to know Born effective charge value, I don't know how to obtain spontaneous polarization values with Born effective charge. Because the volume difference between AlN and GaN, simple dividing by volume does not produce the reference data (-0.081 for AlN and -0.029 for GaN)

2. Piezoelectric constants
In case of GaN, I composed the input files which are represented in the bottom.

In OUTCAR

PIEZOELECTRIC TENSOR for field in x, y, z (e Angst)
XX YY ZZ XY YZ ZX
--------------------------------------------------------------------------------
x -0.00005 -0.00033 -0.00014 0.00000 0.00000 1.51383
y -0.00004 -0.00013 -0.00044 0.00000 1.51480 0.00000
z 1.39418 1.39418 -2.84098 0.00000 0.00000 0.00000

PIEZOELECTRIC TENSOR for field in x, y, z (C/m^2)
XX YY ZZ XY YZ ZX
--------------------------------------------------------------------------------
x 0.00000 0.00000 0.00000 0.00000 0.00000 0.51746
y 0.00000 0.00000 0.00000 0.00000 0.51749 0.00000
z 0.47642 0.47642 -0.97082 0.00000 0.00000 0.00000

PIEZOELECTRIC TENSOR IONIC CONTR for field in x, y, z (C/m^2)
XX YY ZZ XY YZ ZX
--------------------------------------------------------------------------------
x -0.00003 0.00003 0.00000 0.00000 0.00000 -0.66156
y -0.00004 0.00004 0.00000 0.00000 -0.64968 0.00000
z -0.73543 -0.73543 1.44461 0.00000 0.00000 0.00000

When we add Ionic contribution to piezoelectric tensor,
e31 = -0.27 C/m^2
e33 = 0.47 C/m^2

These values are quite different from the reference (e31=-0.49 and e33=0.73)
Could you check my calculation procedure and I am very happy to know me how to obtain Spontaneous and piezoelectric polarization of III-V nitride.


###################
INCAR
###################
ITE = 1
LWAVE = F
LCHARG = F

PREC = accurate
ENCUT = 600.000
ENAUG = 800.000

LEPSILON = T
LRPA = F
LOPTICS = F
IBRION = 8

ISTART = 0
ISPIN = 1
LREAL = A
ISIF = 2
EDIFF = 1.E-6
EDIFFG = -.1e-2
NSW = 1
ISMEAR = 0
SIGMA = 0.05

###################
KPOINTS
###################
Automatic mesh
0
Monkhorst-Pack
6 6 4
0.5 0.5 0.5

###################
POSCAR
###################
GaN
1.00000000000000
3.2152414869952199 0.0000000000012499 0.0000000000000000
-1.6076207439954446 2.7844808083999704 0.0000000000000000
0.0000000000000000 0.0000000000000000 5.2369782223497880
N Ga
2 2
Selective dynamics
Direct
0.6666672861897567 0.3333350505174906 0.4384005495276355 T T T
0.3333332564105405 0.6666665126141780 0.9384007388822226 T T T
0.6666672861897567 0.3333350505174906 0.0615996398269587 T T T
0.3333332564105405 0.6666665126141780 0.5615979178493191 T T T

###############
Pontentials....
###############
PAW_PBE N 08Apr2002
PAW_PBE Ga_d 06Sep2000

[gaojin06]

IN the calculation of spontaneous polarization,you should bulit a reference with out polarizaiton,for example, the ZB structure