Page 69 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
Normal
Modes Normal modes are the natural eigen-shapes of the mechanical system.
We already know these from musical instruments: for example from the
shapes of a vibrating string (one-dimensional), the shapes seen on the
stretched surface of a vibrating drum (two-dimensional), or a vibrating
bowl of jelly (three-dimensional). Normal modes are important, because
they are orthogonal and span the space of the atomic movements and
thereby describe all possible motions of the crystal. To obtain the normal
modes of the crystal, we will assume time-dependent solutions that have
the same geometric periodicity of the crystal.

We will now derive an expression for the equations of motion of the crys-
h
tal from the mechanical Lagrangian L = T ∗ – U (see Box 2.3.) From
M
h T
⁄
d
the previous section we have that U = 12 V ∫ ( ε :E:ε) V . The kinetic
co-energy is added up from the contributions of the individual atoms
n
1 2
T ∗ = --- ∑ mr ˙ i (2.45)
2
i = 1
As for the potential energy, this sum can also be turned into a volume
integral, thereby making the transition to a continuum theory. We con-
sider a primitive cell of volume V

n n
m
1  2 1 2 1 2
--- ρr ˙ Vd
T ∗ = --- ∑ ---- r ˙ V = --- ∑ ρr ˙ V ≅ 2∫ i (2.46)
i
i
2  2
V
i = 1 i = 1 V
Thus we have for the mechanical Lagrangian that
2
2∫
T
--- (
d
L M = ∫ L d V = 1 ρr ˙ – ε :E:ε) V , with
i
M
V V
1 2 T
L = --- ρr ˙ –( ε :E:ε) (2.47)
M i
2
The continuum Lagrange equations read [2.7]
d ∂L ∂L d ∂L
------------- = -------- – -------------------------------------- (2.48)
dt∂x˙ j ∂x j dX ∂∂x ∂X⁄( j k )
k
66 Semiconductors for Micro and Nanosystem Technology