1
J. Humlíček FKL II
3. Realistic band structures
A successful approach for valence and conduction bands uses pseudopotential Vp
2
( ) ,
2
p
p
H V r
m
 

(3.1)
where
( ) ( ) ( ) .p t t tk
t
V r V r E E b b   
 
(3.2)
Here tb are states of inner shells belonging to eigenenergies Et. This potential leads to the same
eigenenergies Ek of the valence and conduction bands as the actual potential V:
2 2
( ) ( ) ( ) .
2 2
p t t tk k k k k
t
p p
V r V r E E b b E
m m
  
   
        
   
    
 
(3.3)
However, the corresponding eigenfunctions are smooth in the regions of inner atomic shells. The reason is
the subtraction of pronounced variations of the actual potential in those regions, since the eigenstates
tb and k
  are orthogonal.
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„Successful“ calculations for Si, Ge, -Sn, and several III-Vs and, II-VIs (Chellikowsky & Cohen) use:
( ) ( ) ,iKr
p
K
V r V K e 



(3.4)
where
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1
( ) ( ) ( ) , ( ) ,jiKR
j
V K S K V K S K e
N

  
 

  
    
(3.5)
31
( ) ( ) ,jiKR a
p
a
V K e V r d r




 
   
(3.6)
Choosing a (small) set of the “formfactors” (V(K)) might be based on “ab-initio” calculations, or also on
empirical adjustment.
The work of Chellikowsky and Cohen (PRB 1976) includes also a few “nonlocal” terms.
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The resulting energies in high-symmetry points of the Brillouin zone are in fairly good agreement with
experimental data (partly due to the empirical adjustment).
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The resulting bandstructures along several high-symmetry directions for elemental semiconductors:
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Formfactors and electron energies for several III-V compounds:
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Bandstructures along high-symmetry directions:
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Valence charge densities of In compounds (covalent and ionic contribution to bonding)
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II-VI compounds (ZnSe and CdTe)
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Density of states
Compare DOS with the calculated bandstructures for ZnSe and CdTe.