[QE-users] Effective Mass Tensor unit-cell dependency

[Schön, Carl-Friedrich]

Dear QE users,

I have a question regarding the effective mass tensor, that I cannot seem to find the solution for. One could also rephrase this as a general question regarding the energy eigenvalues of k-points within different representations of unit-cells.
I have used QE to calculate the bands/eigenvalues of Silicon. I have done this ones with the conventional unit cell (from materials project):

Si2
1.0
3.8681383004362986 0.0 0.0
1.9340686210409386 3.349905532609194 0.0
1.9340686210409386 1.1166353539288019 3.1583211568194507
Si
2
Direct
0.250000000 0.250000000 0.250000000
0.000000000 0.000000000 0.000000000


As well as the primitive unitcell:
Si8
1.0
5.4687280655 0.0000000000 0.0000000000
0.0000000000 5.4687280655 0.0000000000
0.0000000000 0.0000000000 5.4687280655
Si
8
Direct
0.250000000 0.750000044 0.250000000
-0.000000000 -0.000000000 0.500000000
0.250000000 0.250000000 0.750000044
-0.000000000 0.500000000 0.000000000
0.750000044 0.750000044 0.750000044
0.500000000 0.000000000 0.000000000
0.750000044 0.250000000 0.250000000
0.500000000 0.500000000 0.500000000


Let's look at the gamma point only at this point: I then constructed (using the emc.py script) a cartesian k-point grid around the Gamma point in order to get the effective mass tensor for the gamma point for each definition of the unit cell (Conduction Band). For the primitive, 8 atom unit cell, I get something like:

-20.46496343 0.00000000 0.00000000
0.00000000 -20.46496343 0.00000000
0.00000000 0.00000000 -20.46496343

with all Eigenvalues being -20.465 (the tensors are given in units of 1/m*). As far as I am aware, this is what it should look like in terms of symmetry/degeneracy (the absolute values are not of interest at this point).

If I look at the conventional, 2 atom unit cell, I get something like:
0.88345374 -0.00018375 -0.00018375
-0.00018375 0.11245293 -1.46684926
-0.00018375 -1.46684926 2.17115005
with non-degenerate eigenvalues of -0.65, 0.88, 2.93.
I do not understand how this can be. I would understand that the tensors might look different due to different (absolute) orientation in k-space, but the eigenvalues should remain identical, should they not? Otherwise the physics would have changed. Especially in the example of the Gamma point above, I would have assumed to get the exact same tensor, as the effective masses are equal in all directions. Again, the dense grid around the Gamma point is constructed as cartesian cube in both cases.

I therefore looked at the eigenvalues (energies in the conduction band) of the nscf calculation of said dense grid:
Points that should be equivalent (and are indeed identical in the primitive 8 atom case) are not in the 2 atom case, e.g. E(0.01,0,0)!=E(0,0.01,0). I varied the spacing of the grid, but to no effect. The differences are also too big to be numeric errors:

Position:
5.8000000E-03   0.0000000E+00   0.0000000E+00
Eigenenergies:
   1       -5.811100
   2        5.989300
   3        5.997400
   4        6.001900
   5        8.550800 --> Conductino Band
   6        8.552000
   7        8.556000
   8        9.102700
   9       13.711000
  10       13.716400
  11       13.883000
  12       17.181200

Position:
  0.0000000E+00  -5.8500000E-03  -0.0000000E+00
Eigenenergies
   1       -5.811100
   2        5.988800
   3        5.998800
   4        6.000700
   5        8.549800 -->Conduction Band
   6        8.554400
   7        8.554800
   8        9.102800
   9       13.710800
  10       13.716400
  11       13.883100
  12       17.179600

Here the input I used for the scf calculation:

&control
calculation = 'scf'
prefix='Si_mp-149_computed_Relax_6'
tstress = .true.
tprnfor = .true.
pseudo_dir='/rwthfs/rz/cluster/home/NC'
outdir='tmp'
disk_io='low'
wf_collect=.true.
verbosity= 'high'
/
&system
ibrav =0,
nat=2
ntyp=1
ecutwfc = 100
ecutrho = 400
occupations = 'fixed'
/
&electrons
mixing_beta = 0.2
conv_thr = 1.0d-10
/
ATOMIC_SPECIES
Si 28.086 Si.upf
ATOMIC_POSITIONS {crystal}
Si  0.25  0.25  0.25
Si  0.0  0.0  0.0
CELL_PARAMETERS
7.309722  0.0  0.0
3.65486  6.330404  0.0
3.65486  2.110135  5.968362
K_POINTS {automatic}
24 24 24 0 0 0

I am not sure what I am missing… In my understanding the results should be (generelly) independent of the definitino of the used unit-cell…

Thank you and all the best,

Carl-Friedrich Schön, PhD Student, RWTH Aachen University
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