Transcript Slide 1

The Band Theory of Solids
What are the limitations of the model of the free electron gas
?
We don’t understand the origin of semiconductors and insulators
Where are the fingerprints of discrete energy levels of the atoms in solids ?
V
Jellium model
periodic potential

generalize
V(r )  V(r  r n )
x
0
L
r n  n1a1  n2 a2  n3 a3
Single electron theory: for any electron everything in the crystal
represented by an effective periodic potential V(r )  V(r  r n )
Bloch functions
1 dimensional consideration:
translational invariance of V
translational invariance of electron density
V( x)  V( x  ma)
( x)  ( x  ma)
Schroedinger equation invariant under translation
Since ( x )  ( x )
2
(x  a)  J (x)
(x  m a)  Jm (x)
and
with |Jm|=1 ( x)  ( x  ma) is fulfilled
+
Applying periodic boundary conditions (x  Na )  (x)
L
JN=1
J  e2 iM / N
+
+
+
+
M: integer
Note: for a rigorous proof of the Bloch Theorem based on group theoretical methods compare e.g.:
M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill 1964, p.38
+
+
+
with
N
L
a
J  e2iMa / L  eika
J  e2 iM / N
k
(x  a)  J (x)
(x  a) 
Any function satisfying (x  a) 
k (x)  uk (x) eikx
eika
e
ika
(x)
2
M
L
Transformational property
imposed by the translational
symmetry group
(x) can be written in the form
where
celebrated Bloch
uk (x  a)  uk (x)
theorem
How do Bloch functions look like
?
k (x)  uk ( x) eikx
Re(eikx )  cos kx
a
Bloch functions straightforward generalized to three dimensions
k (r)  uk (r) eik r
We know already: Fourier expansion of translational invariant function
Expansion involving the orthogonal set
where
e iGr
G  Ghkl  hg1  kg2  lg3
Apply this to
reciprocal lattice vectors
uk (r)   Ck G e
uk (r)
iGr
G
k (r)  uk (r) eik r
k (r)   Ck G e
i(k G)r
G
Consequences of this representation
Bloch waves which differ by Ghkl are identical
kG (r )  k (r )
Proof: k G (r ) 
 Ck G G e
G


G  G G
iG r i(k G)r
e
  Ck G ei (G
 G ) r
ei ( k  G ) r
G
 C
G
k G
e
iG r
eikr
uk (r )
 k (r )
ˆ   E(k )
H
k
k
ˆ
H
k G  E(k  G)k G
ˆ   E(k  G)
H
k
k
E(k )  E(k  G)
Periodic potential
V(r )   VG eiGr
G
Periodic potential with VG  0
Let’s gradually “switch on” the VG
2
E(k )  E(k  G) 
(k  G)2
2m
Reduced Zone Scheme
Consider analogy to phonon dispersion
-standing waves at zone boundary
due to Bragg reflection
-degeneracy lifted for 2 atoms/unit cell
with m1  m2
-standing electron waves at zone boundary
due to Bragg reflection
-degenerated lifted for VG  0
degeneracy
Superposition of eikx and e ikx
Standing electron wave:

a
@ k
1
2
Potential energy of
increased with respect to
free electron
 

 


i x
ea

i x
e a

i x
ea

i x
e a

2

 

 


2
2


2
 cos 2

x
a
 sin 2

x
a
reduced with
respect to
free electron
Degeneracy lifted
2 VG
E
Splitting proportional to leading
Fourier coefficient VG of the
periodic potential
k
Qualitative discussion
more formal approach in 1 dimension
Schroedinger equation:
Use periodicity of
  2 d2



V
(
x
)

( x )  E( x )
2
2
m
dx


V( x )   VG eiGx
G
and plane wave expansion ( x) 
Ckeikx
k
 2k 2
ikx
i(kG) x
ikx
C
e

C
V
e

E
C
e
 2m k
 k G
 K
k
k,G
k
k : k   G
k  k  G
each coefficient=0
2 2





k
ikx 
 e  2m  E Ck   VGCkG   0


k
G

  2k 2



 2m  E Ck   VGCk G  0
G


Translation by a reciprocal lattice vector G
  2 (k  G)2


 E Ck G   VGCk GG  0

2m
G


Simplification: Leading x-dependent term of periodic potential
2
V( x )  V0  2V1 cos
x
a
2
 V0  V1e
where
i
a
x
 V1e
i
2
x
a
V1  V1 and G1 
2
a
Consistent approximation for k ( x )
k ( x)  CkG e
iGx
ikx
e
approx.


k ( x)  Ck  Ck G1 eiG1x eikx
G
Ck G  0
Study
  (k  G)


Ck G   VGCk GG  0 for G=0 and G  G1  2

E


a
2m
G


2
only for G=0
2
and
G  G1 
2
a
2 coupled equations for Ck and Ck-G1

a
We are interested in the splitting at E(k  )
for G=0
and V0=0 for simplicity
1
for G  G1 
2
a
2
more accurate approach requires qm perturbation theory
k

a
  2   2


   E / a C / a  VG1C / a  0
 2m  a 



  2   2


    E  / a C / a  VG1C / a  0
 2m  a 



What do we learn from the 2 linear equations
C / a  C / a
?
2




i x  i x
i x
i x
a
a


e  C / ae a  C / a e a
substitution in k  / a ( x )   C / a  C / a e



Standing waves :   

i x
ea
From secular equation of
we obtain:
E1  E / a 
VG1
VG1
0
E1  E / a 

i x
e a
  2   2




E
 
 / a C / a  VG1C  / a  0
 2m  a 



1
  2   2





E


 / a C   / a  VG1C  / a  0
 2m  a 



2
where
2   
E1 
 
2m  a 
2
Free electron energy @ k 

a
Solution: E1  E / a 2  VG
1
2
0
2
VG1  VG1 VG1
E(k 

)  E1  VG1
a
E
2   
E1 
 
2m  a 
2
VG1
VG1

a
2|VG1|
k
Bandstructure can be derived from two extreme limiting cases
1
2
Free electrons in periodic potential “switching on” VG
From insulated atoms “switching on” interaction v(Ri  R j ) for Ri  R j  a
Tight-Binding Approximation
LCAO-method: Linear Combination of Atomic Orbitals
Energy splitting:
- coupled pendulums
- covalent bond in H2-molecule
Motion of Electrons in Bands and the Effective mass
v
Free electron in vacuum: p   k  m0 v
2
2 k
E
2m0
v
1
v  k E(k )

group velocity of wave packet
k
m0
1
k E(k )

Remains useful description for
electrons in a crystal
band structure
E

in a crystal: k  k  Ghkl
Crystal momentum:

k
m0
 2k 2
E  EB 
2m
P  k  m v
effective mass

a
k
Bands in general not isotropic
effective mass becomes a tensor
Consider the free electron
 2k 2
to remember this formula: E 
1  E
 1 
   2
 m ij  k ik j
2
 2E  2

2m0
k 2 m0
1
1  2E

m0  2 k 2
E


a


a


a
0

a
k
0

a
k
0

a
k
v
m*
Currents in Bands and the Concept of Holes
Remember:
Current density
for homogeneous velocity
jx  q n v x

k

V
 2
3
d k
3
q
jx 
V
generalized
jx 
q
23
 v x (k)
Current density
for k-dependent velocity
k
3
v
(
k
)
d
 x k
e
3
j

E
(
k
)
d
k
3  k
8 
v
1
k E(k )

1st BZ
for a fully occupied band
Only one spin
species considered
In this formula
Full band: for each velocity v(k ) 
because:
1
k E(k ) there is also v( k )  1 k E( k )


E(k )  E(k )
for crystals with
inversion symmetry
in general: E(k )  E(k )
Current density carried by a full band is zero:
j(partially filled band)  0
j(full band)  0
Partially filled band:
0
e
e
3
3
e
3 

E
(
k
)
d
k


E
(
k
)
d
k
k
k
j

E
(
k
)
d
k


3
3
k
8  st
8 
83 k occupied
1 BZ
k empty
e
j   3  kE(k )d3 k
8 
Current density of holes
k empty
behave like positively charged particles
Metals, Insulators and Semiconductors
Only partly filled electronic band can contribute to electric current
E
conduction band
EC
Eg
EF
EV
valence band
core electrons
Metal
Semiconductor
Insulator
Why is Na or Li a metal
?
e.g.
Li+
e-
metal
Half filled band
SEA OF MOBILE VALENCE ELECTRONS
dk x 
1 valence electron/ atom
band is filled with 2N electrons


a
0
2
L

a
L=Na
k
2 / a L
 N
2 / L a
Spin degeneracy
# of primitive unit cells