Page 141 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
Box 3.1. The tight binding LCAO method [3.6]–[3.8].
Tight Binding. If we make the bold assumption G = V g G = V g
1 ss 1 2 sx 2 
that the wavefunctions of crystal atoms are only 
G = V g G = V g
sx 4 
3
sx 3
4
slightly perturbed from their free state, and we  (B 3.1.4)
G = V g G = V g 
severely limit the interaction between atoms to 5 xx 1 6 xy 2 
those of nearest neighbors only, and we only con- G = V g G = V g 
7
8
xy 4
xy 3
sider the most essential of the “free” atom’s orbit- The Hamiltonian matrix now becomes
als, then a particularly straightforward calculation
E G 0 0 0 G G G
of the crystalline band structure becomes possible s 1 2 3 4
that correctly predicts the bands of the tightly- G ∗ E s – G – G – G 4 0 0 0
1
2
3
bound valence electrons. 0 – G ∗ E p 0 0 G G G 6
2
5
8
Hamiltonian. The crystal electron’s Hamiltonian 0 – G ∗ 0 E p 0 G G G 7
5
3
8
2
is written as H = p ⁄ 2m + k ∑ V r() . We H = 0 – G ∗ 0 0 E G G G
k
assume that the basis states are the 3s and 3 p 4 p 6 7 5
G ∗ 0 G ∗ G ∗ G ∗ E 0 0
states. Since each of these four states ( , x , p y , 2 5 8 6 p
sp
p z ) can occur for each of the two Si sites in the G ∗ 0 G ∗ G ∗ G ∗ 0 E p 0
5
8
3
7
unit cell, the Hamiltonian will be an 8 × 8 matrix. G ∗ 0 G ∗ G ∗ G ∗ 0 0 E p
4
5
6
7
Basis states. The crystal basis states are (B 3.1.5)
Computed Band Structure. We can now evalu-
()
m
(
⋅
1
m
m
φ () = -------- ∑ e i kR j ) ψ r – R () ) (B 3.1.1) ate the Hamiltonian matrix using numerical values
(
l l j
for the parameters (see [3.6]). If the eigenspec-
N j
,
for each of the two types of atoms, i.e., m = 12 . trum is plotted along the lines of high symmetry of
Hamiltonian Matrix. For the diamond structure the Brillouin zone we easily obtain the familiar
only four parameters are unique when we evaluate band structure shown in Figure B3.1.1. This
|
the matrix elements by H = φ 〈 |H φ 〉 , i.e.
ml m m
Ek()
V V
ss sx 
 (B 3.1.2)
V V
xx xy 
We also obtain the phase parameters
1 id 1 k id 2 k id 3 k id 4 k
⋅
⋅
⋅
⋅
g = --- e ( + e + e + e ) k
1 4
⋅
⋅
⋅
⋅
1 id 1 k id 2 k id 3 k id 4 k
g = --- e ( + e – e – e )
2 4
(B 3.1.3)
⋅
⋅
⋅
⋅
1 id 1 k id 2 k id 3 k id 4 k Figure B3.1.1. The LCAO approximation for the
g = --- e ( – e + e – e )
3 4 band structure of silicon. Note that the
conduction bands are quite wrong, for the figure
⋅
⋅
⋅
⋅
1 id 1 k id 2 k id 3 k id 4 k predicts an almost direct band gap.
g = --- e ( – e – e + e )
4
4
and the factors
should be compared to the actual band structure of
silicon shown in Figure 3.13.
138 Semiconductors for Micro and Nanosystem Technology