Page 74 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
1
4
1
1
1
 −
2
------
---
ω = E  ---- + ----- +  ---- + -----  2 – ---------sin  ka  2 (2.62)
a  m M   m M  mM  2 
The two solutions are plotted in Figure 2.20 on the right.
Discussion The case when the two masses of the unit-cell atoms differ only by a
µ
small amount, i.e., M = m + µ with small, is instructive. The optical
and acoustic branches approach then each other at the edge of the recip-
µ
⁄
+
rocal cell, i.e., at k = − π a . As the mass difference goes to zero, the
lattice becomes a monatomic lattice with lattice constant a 2⁄ , so that the
branches touch each other at the reciprocal cell edge.
The interpretation of the two branches is as follows. For the lower
branch, all the atoms move in unison just as for an acoustic wave, hence
the name acoustic branch. In fact, it appears as a center-of-mass oscilla-
tion. For the upper or optical branch, the center of mass is stationary, and
the atoms of a cell only move relative to each other. Its name refers to the
fact that for ionic crystals, this mode is often excited by optical interac-
tions.

2D Square The next construction shows the richness in structure that appears in the
Lattice dispersion relation when an additional spatial dimension and one level of
Dispersion
interaction is added to the 1D monatomic lattice. It serves as an illustra-
Relation
tion that the anisotropy of the interatomic binding energy enables more
involved crystal vibrational modes and hence additional branches in the
dispersion curves. At the same time it shows that most of the essential
features of the expected structure is already clear from the simple 1D
models.

The model considers a 2D monatomic square lattice, and includes the
interaction of the four nearest and four next-nearest neighboring lattice
atoms, i.e., 8 interactions in all. Furthermore, the forces are assumed to
be linear (the harmonic assumption) and the next-nearest neighbor spring

Semiconductors for Micro and Nanosystem Technology 71