Electronic velocities of degenerate states

[Vahid]

Dear EPW Users,

With a view to extract RTA thermoelectric properties, I used the diamond test case with fine grids of 10x10x10 to extracted the electronic velocities using the following expression:

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vkk(:,ibnd) = 2.0 * REAL (dmef(:,ibndmin-1+ibnd,ibndmin-1+ibnd, ikk))
 

Looking at k-point 2 (0.0,0.0,0.1), the energies and velocities for the 4 valence bands are:

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ENERGIES
           2           1 -0.566406269532078     
           2           2  0.896037640310406     
           2           3  0.980593932883515     
           2           4  0.980593934077102  
VELOCITIES
      2            1          -0.1750304088   0.1750304046  -0.1750304033
      2            2           0.7796141206  -0.7796141151   0.7796140748
      2            3           0.0961879298  -0.1261303222   0.2925564189
      2            4           0.2470620776  -0.2171194495   0.0506931518

The velocities of the degenerate bands 3 and 4 are different. The same trend is observed in all k points that have degenerate bands. In addition, symmetry-equivalent points disagree in the velocities of bands 3 and 4. For example for k-point 10 (0.0,0.0,0.9) I get:

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ENERGIES
          10           1 -0.566406268602839     
          10           2  0.896037639699794     
          10           3  0.980593932623874     
          10           4  0.980593934017855  
VELOCITIES
     10            1           0.1750304539  -0.1750304667   0.1750305038
     10            2          -0.7796141798   0.7796142709  -0.7796141136
     10            3          -0.1994467913   0.2396866194  -0.0757409008
     10            4          -0.1438025082   0.1035625485  -0.2675091844

The velocities for the first two bands for k-points 2 and 10 are consistent and differ by +/- sign as expected. However, bands 3 and 4 are inconsistent.

I tested other equivalent k-points with no degeneracies and they are fine. For example for equivalent k-points 37 (0.0,0.3,0.6) and 48 (0.0,0.4,0.7), we have:

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ENERGIES (same for both k-points)
          37           1  4.187508120565024E-003
          37           2  0.143849038349193     
          37           3  0.360023104630337     
          37           4  0.640827982129482     
VELOCITIES 
     37            1           0.3805651532   0.6128871404   0.3805651419
     37            2           0.1168264368   0.3803884837   0.1168264280
     37            3          -0.3425894618  -1.3808010707  -0.3425895230
     37            4          -0.3285271749  -0.1390662617  -0.3285265241

     48            1           0.3805650766  -0.6128868454   0.3805650628
     48            2           0.1168262835  -0.3803885162   0.1168262721
     48            3          -0.3425895170   1.3808018967  -0.3425894063
     48            4          -0.3285268578   0.1390662045  -0.3285267317

I would appreciate any thoughts on how these inconsistencies in electronic velocities for degenerate states may be resolved.

Best,

Vahid

Vahid Askarpour
Department of Physics and Atmospheric Science
Dalhousie University
Halifax, NS Canada

[sponce]

Hello,

EPW is working as expected. Degenerate subspace have a gauge freedom. Any linear combination is therefore valid.
Only the average can be compare with an observable.

I notice that for your kpt 2 you have (averaging band 3 and 4):
0.1716250037 -0.17162488585 0.17162478535

For kpt 10 you have (averaging band 3 and 4):
-0.17162464975 0.17162458395 -0.1716250426

I do not know why you have a sign opposition between the two though?
But the values are the same.

Cheers,

Samuel

[Vahid]

Dear Samuel,

Thank you for your quick response and clarification. The sign difference is probably imposed because of symmetry requirements. This sign difference also exists for bands 1 and 2 and are consistent with the sign difference of the average values you quoted for bands 3+4.

Best,

Vahid