Page 88 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
The expression shows no temperature dependence, which turns out to be
wrong for all but the very high temperatures. This classical result is due
to Dulong and Petit. From classical mechanics (CM), we obtained the
3N normal modes of the crystal via a quadratic expression for the
Hamiltonian (the harmonic approximation). In CM, any motion of the
crystal is simply a linear combination of the normal modes, each with its
characteristic frequency ω , without regard to the energy. A quantum-
mechanical (QM) treatment of the linear harmonic oscillator, however,
⁄
assigns each mode with allowed energy levels E = ( n + 12)—ω ,
,,
n = 12 … . The internal energy must depend on how energy is
absorbed into the quantized phonon levels. We calculate the energy of the
ensemble of oscillators as a sum over the QM states

E i
– ---------
i ∑ E e 1
k B T
i
U = -------------------------------- ≈ U equilibrium ∑ ---—ω k()
+
E i 2
– --------- i
k B T
∑ E ⋅ e (2.89)
i
i
—ω k()
+ ∑ ---------------------------------------
E
i
i exp – --------- – 1
k T
B
from which we can evaluate the specific heat as
∂
—ω k()
=
C V ∑ ---------------------------------------------- (2.90)
∂T E
i
i exp – --------- – 1
k T
B
This quantum-corrected model now depends on the temperature, as
required. Since we know how to determine ω k() , (2.90) can in principle
be evaluated.
A particularly simple QM model was worked out by Debye. It replaces
the lattice’s dispersion relation with a simplified linear form,
ω k() = v k , where v is the scalar isotropic speed of sound in the crys-
s s
tal (the equation for the slope(s) in Figure 2.20). In addition, the Debye

Semiconductors for Micro and Nanosystem Technology 85