Page 89 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
temperature Θ
is introduced that characterizes the lowest temperature
D
at which all modes of the crystal are being excited. After some algebraic
manipulations (see e.g. [2.1], p. 458) we obtain the expression
Θ D
-------
T 4 x
 T  3 x e
C = 9Nk ------- --------------------- xd (2.91)
v B  Θ  ∫ x 2
D
0 ( e – 1)
Upon evaluation the Debye relation leads to the following rather awful
but exact analytical expression (computed using Mathematica®)
1  T  Θ D  4

C = – 9Nk ------------------------------------------------------------ 15exp ------- – 4 π------- +
v B Θ D   Θ  T 
4
T
 

15 exp ------- – 1 ------- D

 Θ D  T 
T  Θ D  4 Θ D T
4exp ------- π------- + 60-------ln 1 – exp ------- –
Θ  T  T Θ
D D
T Θ D T
60exp ------- -------ln 1 – exp ------- +
Θ T Θ
D D
 Θ D  2 T T  Θ D  2 T
–
180 ------- Li exp( ------- ) 180exp ------- ------- Li exp( ------- ) –
 T  2 Θ D Θ D  T  2 Θ D
 Θ D  3 T T  Θ D  3 T
360 ------- Li exp( ------- ) 360exp ------- ------- Li exp( ------- ) +
–
 T  3 Θ Θ  T  3 Θ
D D D
Θ 4 Θ 4 
 D  T T  D  T
–
360 ------- Li exp( ------- ) 360exp ------- ------- Li exp( ------- ) 
 T  4 Θ D Θ D  T  4 Θ D 
(2.92)
where Li z() is the polylogarithm function. It provides an excellent
m
approximation of the specific heat over all temperature ranges, and is
plotted in Figure 2.28.
An- The harmonic potential is in fact a truncated series expansion model of
harmonicity the true crystal inter-atom potential, and as such is not capable of
explaining all lattice phenomena satisfactorily. We briefly discuss one
important examples here: the expansion of the lattice with temperature.

86 Semiconductors for Micro and Nanosystem Technology