[Pw_forum] problem of Bi2Se3 with spin orbit coupling parity analysis at TRIM points

Andrea Dal Corso dalcorso at sissa.it
Thu Jun 1 19:47:59 CEST 2017


Note that when there is inversion symmetry the spin-orbit bands always
comes in pairs of degenerate bands, so if you simply count how many
minus you have, you will always find an even number. It seems to me that
among the valence states (removing the d states of Bi), in the NOSOC
case you have 6g and 8u, in the SOC case you have 7+ (where 7 is the
number of couples of states) and 7-. 

HTH,

Andrea

On Thu, 2017-06-01 at 23:55 +0800, balabi wrote: 
> Thank you very much for explanation.
> 
> 
> I found another problem. According to the paper "Topological
> insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on
> the surface"  DOI: 10.1038/NPHYS1270  
> There is a band inversion at gamma point for Bi2Se3. So the parity
> product at gamma point is different for no SOC case and SOC case.
> 
> 
> However, I count the valence parity as below
> 
> 
> no SOC
> 
> 
> uugggugguuguguguguugguu       product=+
> 
> 
> SOC
> 
> 
> ++--++----++++++----++--++--++--++--++--++----    product=+
> 
> 
> They are the same. But I calculated the surface band, dirac cone do
> exist in SOC case.
> 
> 
> What is wrong with my counting?
> 
> 
> below is the output of bands.x for no SOC and SOC at gamma point
> --------------------------
> 
> 
> no SOC gamma symmetry, highest occupied level 7.8539
> 
> 
> Band symmetry, D_3d (-3m)  point group:
> 
> 
>      e(  1 -  2) =    -15.33380  eV     2   --> E_u  L_3'      
>      e(  3 -  4) =    -15.33248  eV     2   --> E_g  L_3       
>      e(  5 -  5) =    -15.32190  eV     1   --> A_1g L_1       
>      e(  6 -  6) =    -15.31898  eV     1   --> A_2u L_2'      
>      e(  7 -  8) =    -15.30949  eV     2   --> E_g  L_3       
>      e(  9 - 10) =    -15.30663  eV     2   --> E_u  L_3'      
>      e( 11 - 11) =     -6.57512  eV     1   --> A_1g L_1       
>      e( 12 - 12) =     -5.32387  eV     1   --> A_2u L_2'      
>      e( 13 - 13) =     -5.22664  eV     1   --> A_1g L_1       
>      e( 14 - 14) =     -2.00339  eV     1   --> A_2u L_2'      
>      e( 15 - 15) =     -1.26798  eV     1   --> A_1g L_1       
>      e( 16 - 16) =      5.51014  eV     1   --> A_2u L_2'      
>      e( 17 - 17) =      5.51634  eV     1   --> A_1g L_1       
>      e( 18 - 19) =      5.81972  eV     2   --> E_u  L_3'      
>      e( 20 - 21) =      6.26517  eV     2   --> E_g  L_3       
>      e( 22 - 23) =      6.96103  eV     2   --> E_u  L_3'      
>      e( 24 - 24) =      7.91582  eV     1   --> A_2u L_2'      
>      e( 25 - 25) =      8.11337  eV     1   --> A_1g L_1       
>      e( 26 - 27) =      9.02988  eV     2   --> E_g  L_3       
>      e( 28 - 28) =     10.34421  eV     1   --> A_2u L_2'      
>      e( 29 - 30) =     10.59126  eV     2   --> E_u  L_3'      
>      e( 31 - 31) =     13.58144  eV     1   --> A_1g L_1       
>      e( 32 - 32) =     14.17587  eV     1   --> A_2u L_2' 
> 
> 
> ------------------------
> 
> 
> SOC gamma symmetry, highest occupied level 7.7444
> 
> 
> Band symmetry, D_3d (-3m)  double point group:
> 
> 
>      e(  1 -  4) =    -16.63750  eV     4   --> G_5+  L_4+     
>      e(  1 -  4) =    -16.63750  eV     4   --> G_6+  L_5+     
>      e(  1 -  4) =    -16.63750  eV     4   --> G_5-  L_4-     
>      e(  1 -  4) =    -16.63750  eV     4   --> G_6-  L_5-     
>      e(  5 -  6) =    -16.62527  eV     2   --> G_4+  L_6+     
>      e(  7 -  8) =    -16.62265  eV     2   --> G_4-  L_6-     
>      e(  9 - 10) =    -13.66107  eV     2   --> G_4-  L_6-     
>      e( 11 - 12) =    -13.65905  eV     2   --> G_4+  L_6+     
>      e( 13 - 14) =    -13.64175  eV     2   --> G_4+  L_6+     
>      e( 15 - 16) =    -13.63940  eV     2   --> G_5+  L_4+     
>      e( 15 - 16) =    -13.63940  eV     2   --> G_6+  L_5+     
>      e( 17 - 18) =    -13.63741  eV     2   --> G_4-  L_6-     
>      e( 19 - 20) =    -13.63672  eV     2   --> G_5-  L_4-     
>      e( 19 - 20) =    -13.63672  eV     2   --> G_6-  L_5-     
>      e( 21 - 22) =     -6.40199  eV     2   --> G_4+  L_6+     
>      e( 23 - 24) =     -5.17683  eV     2   --> G_4-  L_6-     
>      e( 25 - 26) =     -5.15601  eV     2   --> G_4+  L_6+     
>      e( 27 - 28) =     -1.79777  eV     2   --> G_4-  L_6-     
>      e( 29 - 30) =     -1.02045  eV     2   --> G_4+  L_6+     
>      e( 31 - 32) =      5.18513  eV     2   --> G_4-  L_6-     
>      e( 33 - 34) =      5.40747  eV     2   --> G_4+  L_6+     
>      e( 35 - 36) =      6.03590  eV     2   --> G_4-  L_6-     
>      e( 37 - 38) =      6.14972  eV     2   --> G_4+  L_6+     
>      e( 39 - 40) =      6.19631  eV     2   --> G_5-  L_4-     
>      e( 39 - 40) =      6.19631  eV     2   --> G_6-  L_5-     
>      e( 41 - 42) =      6.51310  eV     2   --> G_5+  L_4+     
>      e( 41 - 42) =      6.51310  eV     2   --> G_6+  L_5+     
>      e( 43 - 44) =      6.80580  eV     2   --> G_4-  L_6-     
>      e( 45 - 46) =      7.20470  eV     2   --> G_5-  L_4-     
>      e( 45 - 46) =      7.20470  eV     2   --> G_6-  L_5-     
>      e( 47 - 48) =      7.50637  eV     2   --> G_4+  L_6+     
>      e( 49 - 50) =      8.05825  eV     2   --> G_4-  L_6-     
>      e( 51 - 52) =      9.13333  eV     2   --> G_4+  L_6+     
>      e( 53 - 54) =      9.27043  eV     2   --> G_4-  L_6-     
>      e( 55 - 56) =      9.86223  eV     2   --> G_5+  L_4+     
>      e( 55 - 56) =      9.86223  eV     2   --> G_6+  L_5+     
>      e( 57 - 58) =     11.09081  eV     2   --> G_4-  L_6-     
>      e( 59 - 60) =     11.46791  eV     2   --> G_5-  L_4-     
>      e( 59 - 60) =     11.46791  eV     2   --> G_6-  L_5-     
>      e( 61 - 62) =     13.77176  eV     2   --> G_4+  L_6+     
>      e( 63 - 64) =     14.35855  eV     2   --> G_4-  L_6-  
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-- 
Andrea Dal Corso                    Tel. 0039-040-3787428
SISSA, Via Bonomea 265              Fax. 0039-040-3787249
I-34136 Trieste (Italy)             e-mail: dalcorso at sissa.it






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