PHY 752 Solid State Physics
11-11:50 AM MWF Olin 103
Plan for Lecture 20:
Review: Chapters 1-6 in GGGPP
1.
2.
3.
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Bloch theorem
Kronig-Penny model
Group theory
PHY 752 Fall 2015 -- Lecture 20
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Bloch theorem for solution  k (r ) to the Schroedinger equation for
an electron in a periodic solid in terms of translation vector T:
 k (r  T) =eik T  k (r )
Consider an electron moving in a one-dimensional model
potential (Kronig and Penney, Proc. Roy. Soc. (London)
130, 499 (1931)
a
V0
Schroedinger equation for electron:
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 2 2

 V ( x)   ( x)  E  ( x)

 2m PHY 752 Fall 2015 -- Lecture 20
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a
V0
b
Matching conditions reduce to:
cos( ka )  F ( E )cos  b   ( E ) 
1/ 2


V
2
F ( E )  1 
sinh   ( a  b  
 4E  E  V 

0


V0  2 E
tan  ( E ) 
tanh   (a  b) 
2 E V0  E 
2
0
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a
V0
b
Details for V0 
Let
E
f 
V0
2
 2 
2
2m 16b
49
2

16
; b a
17

1/ 2
 sinh 0.687 1  f 
 cos 11.7 f   ( f )
cos( ka )  1 


4 f 1  f 


1 2 f
tan  ( f ) 
tanh 0.687 1  f
2 f 1  f 
2

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PHY 752 Fall 2015 -- Lecture 20



5


1/ 2
 sinh 0.687 1  f 
 cos 11.7 f   ( f )
cos( ka )  1 


4 f 1  f 


1 2 f
tan  ( f ) 
tanh 0.687 1  f
2 f 1  f 
2




v
Forbidden
states
f
v
Forbidden
states


1/ 2
 sinh 0.687 1  f 
 cos 11.7 f   ( f )
X ( f )  1 


4 f 1  f 


2
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
PHY 752 Fall 2015 -- Lecture 20

6
f
Band gap
Band gap
ka/
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Classification of lattice types –
14 Bravais lattices
230 space groups
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Short digression on abstract group theory
What is group theory ?
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Group theory – some comments
• The elements of the group may be abstract; in
general, we will use them to describe
symmetry properties of our system
Representations of a group
A representation of a group is a set of matrices (one
for each group element) -- ( A), ( B)... that satisfies
the multiplication table of the group. The dimension
of the matrices is called the dimension of the
representation.
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The great orthogonality theorem
Notation: h  order of the group
R  element of the group
i ( R )  ith representation of R

denote matrix indices
li  dimension of the representation
   ( R)    ( R)
i
*
j
R
Analysis shows:
l
2
i
h
  ij  
li
h
i
Note: only irreducible representations are used.
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Simplified analysis in terms of the “characters” of the
representations
lj
 j ( R )   j ( R) 
 1
Character orthogonality theorem
   ( R)  
i
*
j
( R)  h ij
R
Note that all members of a class have the same
character for any given representation i.
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Reciprocal lattice
real lattice
reciprocal lattice
Unit vectors of the reciprocal lattice
General reciprocal lattice vector
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Reciprocal lattice
real lattice
reciprocal lattice
Unit vectors of the reciprocal lattice
General reciprocal lattice vector
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Quantum Theory of materials
Electronic
Exact Schrodinger equation:
coordinates
a
H ({ri },{R a })  ({ri },{R a })  E  ({ri },{RAtomic
})
coordinates
where
H ({ri },{R a })  H Nuclei ({R a }) + H Electrons ({ri },{R a })
Born-Oppenheimer approximation
Born & Huang, Dynamical Theory of Crystal Lattices, Oxford (1954)
Approximate factorization:
Nuclei
 ({ri },{R a })  X 
({R a }) Electrons ({ri },{R a })
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Quantum Theory of materials -- continued
Electronic Schrodinger equation:
H Electrons ({ri },{R a }) Electrons ({ri },{R a })  U ({R a }) Electrons ({ri },{R a })
2
a 2
2
Z
e
e
2
H Electrons ({ri },{R a })  



i 
a
2m i
a ,i | ri  R |
i  j | ri  r j |
Nuclear Hamiltonian:


(Often treated classically)
Nuclei
a
H Nuclei {R a } X 
({R a })  W X Nuclei
({
R
})

H
Nuclei

a

{R }
Pa2

a
a 2M
 U ({R a })
Effective nuclear interaction
provided by electrons
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First consider electronic Hamiltonian
Electronic Schrodinger equation:
H Electrons ({ri },{R a }) Electrons ({ri },{R a })  U ({R a }) Electrons ({ri },{R a })
2
a 2
2
Z
e
e
2
H Electrons ({ri },{R a })  



i 
a
2m i
a ,i | ri  R |
i  j | ri  r j |
Replace by “jellium”
Independent electron
contributions
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Jellium model of metals
• Nuclear potential represented by a uniform
background of positive charge with charge density
n0=Z/V (V representing volume per atom)
• Electrons represented as independent 2free
electrons
2
occupying free-electron states E (k )  k for 0  k  k
2m
Figure from GGGPP
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F
Calculation of the Fermi level for jellium
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Density of states analysis of free electron gas
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Practical modeling schemes – based on density functional theory
Kohn-Sham formulation of density functional theory
Results of self-consistent calculations
Variationally determined -Ground state energy
Ev [n]
Electron density
n(r )
Some remaining issues
• Theory for Eexc[n] still underdevelopment
• This formalism does not access excited states
• Strongly correlated electron systems are not well
approximated
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Practical solution of Kohn-Sham equations in solids
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