2.57 Fall 2004 – Lecture 18
1
2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 18
In last lecture, we deal with the Boltzmann transport equation
s
f
dd
f
ff
td
t
d
t
t
∂∂
⎛⎞
+•
∇
∇=
⎜⎟
⎝⎠
rp
where the subscripts (
r
and
p
) in the gradient operators represent the variables of the
gradient. The scattering term
o
s
f
f
f
t
τ
−
∂
=−
∂
was discussed based on two particle
interactions. Here
()
,
τω
k
is the relaxation time. For equilibrium distribution
f
0
, we have
2
0
3/2

2
1
BoseEinstein distribution (phonon)
1
1
f
FermiDirac distribution (electron)
1
Displaced Maxwell velocity distribution (molecules)
2
B
B
B
kT
E
m
B
e
e
m
ne
ω
µ
π
−
−
⎧
⎪
⎪
⎪
−
⎪
⎪
=
⎨
⎪
+
⎪
⎪
⎪
⎪
⎩
vu
=
Note: The relaxation time is due to combined factors and can be evaluated numerically
by adding all possible influence together. This idea can be used in calculating the band
structure by the Boltzmann transport equation.
For twoparticle interactions (above figure (a)), we have
12
3
ωω
+=
==
=
(energy conservation),
31
+
2
G+k =k
k
(momentum conservation),
where zero
G
corresponds to normal process, otherwise it is umklapp scattering.
Generally speaking, electrons will collide with electrons, phonons, and impurities in the
crystals. The electronphonon scattering causes the electrical resistance.
k
1
,
ν
1
k
2
,
ν
2
ν
3
=
ν
1
+
ν
2
k
3
ν
1
,
k
1
ν
2
,
k
2
ν
3
,
k
3
(a)
(b)
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View Full Document2.57 Fall 2004 – Lecture 18
2
Note: At low temperatures, phonon has low energy and the electronphonon scattering is
negligible. Therefore, the impurityelectron scattering is the main cause of electrical
resistance.
For gas molecules, in Lecture 2 we have derived the collision obeys
2
1
2
nD
π
Λ=
,
v
τ
.
Note: (1) To simplify, most time we view
( )
,
τω
k
as
( )
. (2) The time
()
,
k
is only
applicable to elastic scattering (see chapter 8), such as electronelectron scattering
(energy conserved). It is not accurate for electronphonon scattering. In this situation, we
have
o
ep
s
ff
f
gT T
t
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 Spring '10
 WRIGHT,J
 Statistical Mechanics, Fundamental physics concepts, relaxation time, Boltzmann transport equation

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