Page 241 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
y
origin (a scattering centre) on the sphere, say , makes at x
with respect
0
to the negative x-axis, then that phonon has a velocity component along
the x-axis of vcos θ . Like the temperature, the internal energy is also
assumed to vary only along the x-axis. This means that the phonons
arrive at x with an internal energy density of ux –( lcos θ) . The aver-
0 0
age energy current in the x-direction is now the solid angle integral of the
(
the product of velocity with energy density v cos θ ux – lcos θ) . To
p 0
θ
obtain the differential solid angle as a function of we take the ratio of
the differential surface area to the total spherical area, i.e.,
( 2πr) rsin θ)dθ 1
(
---------------------------------------- = ---sin θdθ (7.2)
2
4πr 2
Thus we have to evaluate
π 1
(
q = v cos θ ux – lcos θ)---sin θdθ
x 0 ∫ p 0 2
(7.3)
v p 1
2 ∫
d
= ----- – 1 µux –( 0 lµ) µ
Note the change of variables µ = cos θ and the new integration limits.
We next expand the energy density about the position x . We drop the
u
0
⁄
quadratic and higher terms to obtain u = u + ∂u ∂x µl , and insert
x 0 x 0
this into (7.3) and evaluate
v 1  ∂u 2  v l ∂u
p
---
q = 2 – ∫ 1  u x 0 µ + ------ µ l d µ = 0 + ------------- (7.4)

x
∂x
3 ∂x
x 0 x 0
(
⁄
⁄
⁄
Remembering that ∂u ∂x⁄ = ( ∂u ∂T) ∂T ∂x) = c ∂T ∂x , we can
v
relate the derived to the phenomenological equation
∂T v l ∂T
p
q = – κ------- = -------c ------- and hence (7.5a)
v
∂x 3 ∂x
1 1 2
κ = ---v lc = ---v τ c (7.5b)
p p v
v
p
3 3
238 Semiconductors for Micro and Nanosystem Technology