Page 90 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
20 C V J mole-K⁄
Figure 2.28. The specific heat C
V
of a harmonic crystal versus
T Θ computed using the Debye 10
⁄
D
interpolation formula. The high
T
temperature limit of DuLong and -------
Θ
Petit is approximately 24 D
0
⁄
J mole-K . 0 1 2 3
The other important example, the lattice thermal conductivity, will be
discussed in Section 7.1.2.

Thermal The thermal expansion coefficient is defined by
Expansion
1 ∂V  1 ∂P 


α = ---- ------- () = ------- ------- () (2.93)

3B ∂T 
V ∂T 

T
V
P
(
⁄
B
where is the bulk modulus defined as B = – V ∂P ∂V) T () , and equal
T
to the inverse of the compressibility, is some length, is the tempera-
l
ture, is the pressure and is the volume. From thermodynamics, we
V
P
have that
 ∂S   ∂U   ∂F 
T ------- = ------- , P = – ------- and F = U – TS (2.94)
 ∂T  ()  ∂T  ()  ∂V  T ()
V
V
S
for the entropy , the internal energy U and the Helmholtz free energy
F . Combining these, we can write the pressure only in terms of the inter-
nal energy, volume and temperature as
∂  T ∂ d T′ 

P = – ------- U – T 0 ∫ --------UT′ V,( )--------  (2.95)
∂V  ∂T′ T′ 
All that remains is to insert the expression (2.89) for the internal energy
into (2.95), and then to insert this result into (2.93). Evaluating this
Semiconductors for Micro and Nanosystem Technology 87