Page 200 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Semi-Classical Boltzmann Transport Equation
6.1.3 The BTE for Phonons
In the case of phonons we know that in equilibrium they follow a Bose–
Einstein distribution
1
(
g = n ω ) = --------------------- (6.7)
0
q
e β—ω q – 1
which is the average number of phonons as given in (5.28) with the fre-
q
quency ω = ω q() , where is the wave vector. From the average num-
q
ber of phonons we define a distribution function g qx t,,( ) , i.e we allow
for spatial variations of the phonon number density. The only term that
gives a driving force for phonons will be a gradient with respect to the
spatial coordinates. The BTE for phonons thus is given by

∂ ( ,, x ˙∇ g qx t,,( C qx t)
,,
(
t ∂ g qx t) + x ) = (6.8)
We know that the group velocity v = x ˙ = ∇ ω q() is the gradient of
q
q
the frequency with respect to the wave-vector . To go further we have to
q
take into account spatial variations of the phonon frequency δω qx t,,( , )
which then results in an equation of the form

(
,,
C qx t)
 ∂  (6.9)
,,
(
,,
(
= + ( v + ∇ δω qx t))∇ + ∇ δω qx t)∇ g qx t,,( )
t ∂  q q x x q 
The reason for the appearance of δω qx t,,( ) is a perturbation of the sys-
tem by a momentum dependent potential, e.g., long wavelength elastic
distortions. Therefore, an anharmonic theory of phonons must be per-
formed [6.6] and a phonon-phonon interaction included (see Section
7.1).

6.1.4 Balance Equations for Distribution Function Moments
In (6.1) the interpretation of the function f k t() x t() t,( , ) as a density
became clear by the integration over velocities and positions. If we only
integrate over the wave vector space we discard any information about

Semiconductors for Micro and Nanosystem Technology 197