My Community

Dear VASP community:

I try to understand optical matrix elements <phi(n,k)|d/dk_i|phi(m,k)> printed in WAVEDERF, and specifically their units. There were posts in the past but no clarity yet.
http://cms.mpi.univie.ac.at/vasp-forum/ ... =4&t=14986
https://cms.mpi.univie.ac.at/vasp-forum ... =3&t=17302

I selected GaAs as it is a well studied compound. Here is the part of WAVEDERF related to the transition at Gamma-point (band 8 top VB and band 9 bottom CB) calculated with HSE06 (vasp.5.4.4):
8 2.1314902 1.0000000 9 3.1088767 0.0000000 -0.6042883396 2.4010763168 -1.9962309599 0.4970015585 2.4682097435 -2.9514818192

Let's make sure I understand the output correctly:
[8 2.1314902 1.0000000] stands for n, E(n,k), occupancy of the VB edge
[9 3.1088767 0.0000000] stands for m, E(m,k), occupancy of the CB edge.
The band gap is ~1 eV vs experimental 1.5 eV.
The matrix elements are:
[-0.6042883396 2.4010763168] are Re and Im parts of the <phi(n,k)|d/dk_1|phi(m,k)>
[-1.9962309599 0.4970015585] are Re and Im parts of the <phi(n,k)|d/dk_2|phi(m,k)>
[2.4682097435 -2.9514818192] are Re and Im parts of the <phi(n,k)|d/dk_3|phi(m,k)>

|P_i|^2=|<phi(n,k)|d/dk_i|phi(m,k)>|^2 = 6.1, 4.2, and 14.8 for i = 1, 2, and 3
Here are my first questions: Should I expect any symmetry, e.g., equal values of |P_i|^2? Do i = 1, 2, and 3 refer to reciprocal lattice vectors or cartesian coordinates?
Here are the reciprocal lattice vectors:
-0.174690685 0.174690685 0.174690685
0.174690685 -0.174690685 0.174690685
0.174690685 0.174690685 -0.174690685

Now about the magnitude of |P_i|^2. According to the textbook (Yu & Cardona, Fundamentals of Semiconductors, chap. 2.6) 2P^2/m0 ~ 20 eV, where m0 is the electron mass (not the effective mass). Basically I would like to verify this result with VASP.

Let's assume that VASP uses atomic units m0=1, then energies should be in Ha. Using the matrix elements above, we get 2*6.1/1 = 12.2 Ha = 330 eV, which is too much. Any help with rationalizing the values is appreciated.

Thanks in advance
Oleg

Dear Oleg,

Did you find the answers for your questions? I am also interested in matrix elements units and whether i = 1, 2, and 3 refer to reciprocal lattice vectors or cartesian coordinates.

Best wishes,
Tomasz