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 e2iMa / 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
kG (r ) k (r )
Proof: k G (r )
Ck G G e
G
G G G
iG r i(k G)r
e
Ck G ei (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 eiGx
G
and plane wave expansion ( x)
Ckeikx
k
2k 2
ikx
i(kG) 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 VGCkG 0
k
G
2k 2
2m E Ck VGCk G 0
G
Translation by a reciprocal lattice vector G
2 (k G)2
E Ck G VGCk GG 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
V1e
i
2
x
a
V1 V1 and G1
2
a
Consistent approximation for k ( x )
k ( x) CkG e
iGx
ikx
e
approx.
k ( x) Ck Ck G1 eiG1x eikx
G
Ck G 0
Study
(k G)
Ck G VGCk GG 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 VG1C / 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
VG1
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 VG1C / 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 VG1
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 ik 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
23
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
83 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