Page 80 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
or shows a discontinuity. These are known as Van Hove singularities after
their discoverer [2.14].
Phonon Since the density of states is constant in -space, the Walker calculation
k
Density of method [2.15] (also called the root sampling method) requires perform-
States
ing the following steps:
Computation
Method • Form a uniform grid in -space. Only consider a volume that is non-
k
redundant, i.e., exploit all symmetries in the Brillouin zone of the
crystal to reduce the size of the problem. Ensure that no grid points lie
directly on a symmetry plane or on a symmetry line, as this would
complicate the counting.
ω
• For each point on the grid, solve (2.71) for the eigenfrequencies .
• Compute ω , the highest frequency anywhere on the grid.
max
• Form a set of histogram bins from 0 to ω for each separate
max
branch of the computed dispersion curve, and calculate the histo-
grams of branch frequencies found on the grid. That is, each histo-
gram bin counts how many branch states occurred for a frequency
lying within the bin’s interval.

• Identify the singular points on the histograms. Smoothen the histo-
grams between the singular points and plot the resultant phonon den-
sity of states.

The method requires enough -space points spread evenly over the Bril-
k
louin zone and histograms with small-enough bin widths for a realistic
result. (The method can also be used to compute the density of electronic
states E ω() , from the electronic band structure E k() , as discussed in
Section 3.3). Applying the above algorithm to the linear chain of Section
2.4.1, we obtain the curves of Figure 2.24. Doing the same for the 2d lat-
tice results in the density-of-states diagram of Figure 2.25. For the silicon
dispersion diagram of Figure 2.23 we have numerically computed the
density of states by the root sampling method presented in Figure 2.26.

Semiconductors for Micro and Nanosystem Technology 77