Page 87 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
1
n 〈〉 =
------------------------------------------------------ =
j
( (
⁄
⁄
–
exp ( ( E – 1 µ) kT) 1 ---------------------------------------------------------- , (2.85)
exp
–
µ) kT) 1
—ω –
j j
We will delay any further discussion of particle statistics to Chapter 5.
Because of the similarity of this expression with the case of photons
(quantized electromagnetic energy, see Section 5.2.4), the quantized
crystal vibrations are called phonons.
Phonons The connection to the normal modes of the previous section is that the
th
excitation of the m mode (from branch with wavevector ) corre-
s
k
s
sponds to saying that there exist m photons (from branch with wave-
k
vector ) in the crystal [2.1].
Heat The vibrating crystal lattice is an internal energy store. A measure for this
Capacity energy storage ability is the heat capacity at constant volume, defined as
 ∂S   ∂E 
C V = T ------- =  ------- (2.86)
∂T 
∂T 

V V
S
in terms of the entropy and the internal energy . Classical statistical
E
mechanics (SM), which treats the lattice as 3N classical linear harmonic
oscillators, assigns an average of energy kT to each vibrational degree-
of-freedom, from
– E
---------
k B T
–
∫ e E Γ ∂ ---------
E
d
k B T
U = ------------------------ = – ------------------ln ∫ e d Γ
– E
1

--------- ∂ ---------  (2.87)
k B T
∫ e d Γ  k T 
B
= 3Nk T = 〈〉 Classical
E
B
and hence we obtain a simple expression for the heat capacity as
 ∂E  – 1 – 1
C = ------- = 3Nk ≈ 24 Jmol K (2.88)
V  ∂T  B
V
84 Semiconductors for Micro and Nanosystem Technology