1
J. Humlíček FKLii
5. Spin-orbit interaction, top of valence bands
Electron states in atoms are influenced by the inner magnetic field, resulting from the orbital motion. The
field tries to orient the spin magnetic moment. This is a relativistic effect, vanishing in the limit of c → ∞.
The spin operator,
0 1 0 1 0
, , ,
1 0 0 0 1
  
     
       
     
x y z
i
i (5.1)
enters the spin-orbit contribution to the Hamiltonian operator:
2 2
1
( ) .
2 2

   


SOH V p
m c (5.2)
An alternate notation use the magnetic moment related to the spin of electron,
= ,
2
B
e e S S
S
mc mc S S
 
 
 
 
 
  (5.3)
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the energy in the magnetic field,
,  

SOH H (5.4)
where the intensity H due to the orbital motion is
.  
 v
H E
c (5.5)
Magnetic moment of free electron, B (Bohr magneton) is 9.2740154E-21 erg/gauss, its value in SI is 9.2740154E-24 J/T (1 erg = 1E-7 J, 1 gauss
= 1E-4 T); using the appropriate units of energy for the microstructure of matter, it amounts to 58 eV/T. Magnetic field at the surface of Earth is
in the range from 25 to 65 T; strong permanent neodymium (neodymium-iron-boron) magnets achieve up to 1.3 T.
The spin-orbit part of Hamiltonian in an isolated atom is
( )( ) ( ) ,     
  
SOH r r p S r L S (5.6)
where L is orbital angular momentum. Let us consider the atomic p-state (that of l=1). The total angular
momentum is represented by the operator
. 
 
J L S (5.7)
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The scalar product (the magnitude squared) involves three terms:
( ) ( ) ( ) ,            
            
J J L S L S L L S S L S S L (5.8)
where L and S commute (they operate in different vector spaces). However, their projections (ml and ms) are
not good quantum numbers, since they are coupled by the SO interaction; on the other hand, l and s remain
good quantum numbers. The appropriate state vectors, allowing us to estimate the product of L and S are
therefore
, , , .jj l s m (5.9)
From Eq. (5.8), the diagonal matrix element of the square of total angular momentum is
2
2
( 1) ( 1) ( 1) ,      


j j l l s s L S (5.10)
i.e., the expectation of the product LS is
 
2
( 1) ( 1) ( 1) .
2
      
 
L S j j l l s s (5.11)
For atomic p-states (l=1, s=1/2), possible values of j are 3/2 or ½; consequently
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2
2
pro 3/ 2 , pro 1/ 2 .
2
      
  
L S j L S j (5.12)
Spin-orbit interaction leads to the energy splitting of the states with the total angular momentum of 3/2 a 1/2.
Atomic s-state is not influenced by the SO interaction, the spin degeneracy is retained. Further, atomic dstate
splits into 6-fold degenerate state D5/2 and 4-fold degenerate D3/2.
Atomic numbers and SO splitting of the top of valence bands.
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Bandstructure of Ge without the SO interaction (Dresselhaus). Note
the erroneous dispersion of the lowest valence band.
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Bandstructure of Ge with the SO interaction included (Dresselhaus).
Note the erroneous dispersion of the lowest valence band.
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Double groups – include symmetry operations of the spin variable (rotations by o 2 change the sign of the
state vector).
Characters of irreducible representation of the double group of the
zincblene structure at the center of the Brillouin zone, .
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Dispersion of the topmost valence states close to  („warping“):
2 2 4 2 2 2 2 2 2 2
( ) ( ) .x y z y x zE k Ak B k C k k k k k k      (5.13)
Parameters of the topmost valence states close to 
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Contours of constant energy around the Brillouin zone center .
Heavy- and light-hole bands, effective masses averaged over directions:
2
* 2 2
21 1
2 2 1 ,
15hh
C
A B
m B
  
      
    
 (5.14)
2
* 2 2
21 1
2 2 1 .
15lh
C
A B
m B
  
      
    
 (5.15)