<div dir="ltr"><div>Funny: I have always called "primitive" the 2-atom cell. "conventional" the 8-atom one. Anyway, the 2-atom diamond structure (fcc lattice) is simply<br></div><div> ibrav=2, a=lattice parameter (A)<div><div>or<br></div>
</div><div> ibrav=2, celldm(1)=lattice parameter (a.u.)</div></div><div>(lattice parameter=side of the cube) plus<br></div>
<div>ATOMIC_POSITIONS (alat)</div>
<div>Si 0.00 0.00 0.00 <br></div><div>Si 0.25 0.25 0.25 </div><div>or<br></div><div>ATOMIC_POSITIONS (crystal)</div>
<div>Si 0.00 0.00 0.00 <br></div><div>Si 0.25 0.25 0.25 <br></div><div>or any of the various possible ways of specifying atomic positions. If you do things properly you will find that the 2-atom and 8-atom cells give exactly the same results.</div><div><br></div><div>Paolo<br>
</div>
</div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Sep 20, 2021 at 1:28 PM Schön, Carl-Friedrich <<a href="mailto:schoen@physik.rwth-aachen.de">schoen@physik.rwth-aachen.de</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
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Dear QE users, <u></u><u></u></div>
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<u></u> <u></u></div>
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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.<u></u><u></u></div>
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I have used QE to calculate the bands/eigenvalues of Silicon. I have done this ones with the conventional unit cell (from materials project):<u></u><u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i> </i></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>Si2</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>1.0</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>3.8681383004362986 0.0 0.0</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>1.9340686210409386 3.349905532609194 0.0</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
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<i>1.9340686210409386 1.1166353539288019 3.1583211568194507</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>Si</i><u></u><u></u></div>
</div>
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<i>2</i><u></u><u></u></div>
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<i>Direct</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.250000000 0.250000000 0.250000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.000000000 0.000000000 0.000000000<u></u><u></u></i></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i> </i></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
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As well as the primitive unitcell:<u></u><u></u></div>
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<i>Si8</i><u></u><u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>1.0</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>5.4687280655 0.0000000000 0.0000000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.0000000000 5.4687280655 0.0000000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.0000000000 0.0000000000 5.4687280655</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>Si</i><u></u><u></u></div>
</div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>8</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>Direct</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.250000000 0.750000044 0.250000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>-0.000000000 -0.000000000 0.500000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.250000000 0.250000000 0.750000044</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>-0.000000000 0.500000000 0.000000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.750000044 0.750000044 0.750000044</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.500000000 0.000000000 0.000000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.750000044 0.250000000 0.250000000</i><u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i>0.500000000 0.500000000 0.500000000<u></u><u></u></i></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<i> </i></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
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:<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
-20.46496343 0.00000000 0.00000000<u></u><u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
0.00000000 -20.46496343 0.00000000<u></u><u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
0.00000000 0.00000000 -20.46496343<u></u><u></u></div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
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).<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
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If I look at the conventional, 2 atom unit cell, I get something like:<u></u><u></u></div>
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0.88345374 -0.00018375 -0.00018375<u></u><u></u></div>
</div>
<div style="color:rgb(0,0,0)">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
-0.00018375 0.11245293 -1.46684926<u></u><u></u></div>
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-0.00018375 -1.46684926 2.17115005<u></u><u></u></div>
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with non-degenerate eigenvalues of -0.65, 0.88, 2.93.<u></u><u></u></div>
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<u></u><u></u></div>
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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. <u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
I therefore looked at the eigenvalues (energies in the conduction band) of the nscf calculation of said dense grid:<u></u><u></u></div>
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<span>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:<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span>Position:<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span>5.8000000E-03 0.0000000E+00 0.0000000E+00<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span>Eigenenergies:<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 1 -5.811100<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 2 5.989300<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 3 5.997400<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 4 6.001900<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 5 8.550800 </span><span><span style="font-family:Wingdings">à</span></span><span> Conductino Band<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 6 8.552000<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 7 8.556000<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 8 9.102700<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 9 13.711000<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 10 13.716400<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 11 13.883000<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 12 17.181200<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span>Position:<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 0.0000000E+00 -5.8500000E-03 -0.0000000E+00<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span>Eigenenergies<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 1 -5.811100<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 2 5.988800<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 3 5.998800<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 4 6.000700<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 5 8.549800 </span><span><span style="font-family:Wingdings">à</span></span><span>Conduction Band<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 6 8.554400<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 7 8.554800<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 8 9.102800<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 9 13.710800<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 10 13.716400<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 11 13.883100<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<span> 12 17.179600</span><u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
</div>
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<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
Here the input I used for the scf calculation:<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
<u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
&control<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
calculation = 'scf'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
prefix='Si_mp-149_computed_Relax_6'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
tstress = .true.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
tprnfor = .true.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
pseudo_dir='/rwthfs/rz/cluster/home/NC'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
outdir='tmp'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
disk_io='low'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
wf_collect=.true.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
verbosity= 'high'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
/ <u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
&system<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
ibrav =0,<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
nat=2<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
ntyp=1<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
ecutwfc = 100<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
ecutrho = 400<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
occupations = 'fixed'<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
/ <u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
&electrons<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
mixing_beta = 0.2<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
conv_thr = 1.0d-10<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
/ <u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
ATOMIC_SPECIES<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
Si 28.086 Si.upf<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
ATOMIC_POSITIONS {crystal}<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
Si 0.25 0.25 0.25<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
Si 0.0 0.0 0.0<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
CELL_PARAMETERS<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
7.309722 0.0 0.0<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">
3.65486 6.330404 0.0<u></u><u></u></div>
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3.65486 2.110135 5.968362<u></u><u></u></div>
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K_POINTS {automatic}<u></u><u></u></div>
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24 24 24 0 0 0<u></u><u></u></div>
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<u></u> <u></u></div>
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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…<u></u><u></u></div>
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<u></u> <u></u></div>
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Thank you and all the best, <u></u><u></u></div>
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<u></u> <u></u></div>
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Carl-Friedrich Schön, PhD Student, RWTH Aachen University</div>
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