Page 78 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
Box 2.4. Computing the carbon structure’s dynamical matrix.
⁄
⁄
,
,
⁄
,
⁄
⁄
⁄
,
Atom Contributions. The dynamical matrix D { 14 14 – 14} and –{ 14 – 14 14} , at,
⁄
describes the effect that perturbing the position of r = 34 , and so on. The sets of equally distant
0
atom has on the force acting on atom , for all neighbors are positioned in “onion shells” about
i
0
0
the interacting neighbors of atom . Clearly this atom . For each of these sets we can write down
is a direct function of the crystal structure, so that a force-constant tensor that is equal to the tensor
we can write the dynamical matrix in terms of the product of its normalized position vector w.r.t.
bonding strengths between the adjacent sites. We atom . For example for the first atom, the tensor
0
now do this for the diamond lattice, depicted in equals
0
Figure B2.4.1. From the viewpoint of atom , we
⁄
⁄
⁄
13 13 – 13
Γ = γ 1 13⁄ 13 – 13 (B 2.4.1)
⁄
⁄
1
⁄
⁄
⁄
–
– 13 13 13
i
and can be interpreted as follows: If atom expe-
j
riences a unit displacement in direction , then
0
atom experiences a force in direction of size
k
()
i0
Γ jk . Each unique separation r l is associated
Atom with a force constant γ l . A complete formulation
0
includes all force constant tensors and hence all
neighboring atoms involved in an interaction with
0
atom .
The Dynamical Matrix. The net result of all
atom interactions is computed from
Figure B2.4.1: The structure of the diamond
⋅
(
1
lattice w.r.t. the centered atom 0, marked by an D = – ---- ∑ Γ () i kr i ) (B 2.4.2)
i0
e
arrow. Bookkeeping becomes critical. jk m jk
i
The force constants are evaluated by comparing
have 4 equally distant neighbors at the positions the long-wavelength dynamical matrix with the
⁄
,
⁄
⁄
,
⁄
⁄
{ 14 14 – 14} , 14⁄ ,{ – 14 14} , values obtained from elasticity theory, (2.80).
,
constants of silicon. The qualitative agreement with the measured curve
is fairly good, as all major features are represented.
Phonon A more useful form of the phonon density of states as a function of the
Density of phonon frequency D ω() and not as a function of the wave vector coor-
States
k
dinates. The density of states in -space is reasonably straightforward to
L
formulate. For the case of a 1D monatomic lattice of length with lat-
⁄
⁄
tice constant , it is the constant value w k() = L π for k ≤ π a and
a
zero otherwise. This amounts to saying that for each of the N atoms in
Semiconductors for Micro and Nanosystem Technology 75