Page 71 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
Since the potential energy is located in the bonds and not the lattice site,
and therefore depends on the positions of the neighboring lattice sites,
2
the term d udX⁄ 2 is replaced by its lattice equivalent for the lattice site
⁄
i ( 2u – u – u ) a 2 . This finite difference formula expresses the
i i 1 i + 1
–
u
i
fact that the curvature of at the lattice site depends on the next-neigh-
bor lattice positions. This step is necessary for a treatment of waves with
a wavelength of the order of the interatomic spacing. If we use the sim-
pler site relation, we only obtain the long wavelength limit of the disper-
sion relation, indicated by the slope lines in Figure 2.20. We now look for
solutions, periodic in space and time, of the form
(
,
(
uX t) = exp [ jkX – ωt)] (2.53)
X = ia X = ia
which we insert into equation (2.52) and cancel the common exponential
2 E
(
– ρω = – ----- 2 – exp [ kia] – exp – [ kia]) (2.54)
2
a
Reorganizing equation (2.54), we obtain

(
2E 1 – cos [ ka]) E ka
ω = ----------------------------------------- = 2 -------- sin ------ (2.55)
2
2
ρa ρa 2
Equation (2.55) is plotted in Figure 2.20 on the left, and is the dispersion
relation for a monatomic chain. The curve is typical for an acoustic wave
in a crystalline solid, and is interpreted as follows. In the vicinity where
⁄
⁄
ω is small, the dispersion relation is linear (since sin [ ka 2] ≈ ka 2 )
and the wave propagates with a speed of E ρ⁄ as a linear acoustic
wave. As the frequency increases, the dispersion relation flattens off,
causing the speed of the wave ∂ω ∂k⁄ to approach zero (a standing wave
resonance).

1D Diatomic Crystals with a basis, i.e., crystals with a unit cell that contains different
Dispersion atoms, introduce an important additional feature in the dispersion curve.
Relation
We again consider a 1D chain of atoms, but now consider a unit cell con-
taining two different atoms of masses m and M .

68 Semiconductors for Micro and Nanosystem Technology