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2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 3
8. Micro & Nanoscale Phenomena
8.1 Classical size effects
In section 7, the characteristic length of the box is much longer than the mean free path
Λ
. Therefore, the collisions between molecules and the wall are neglected in our
derivation and thermal conductivity is regarded as the bulk property of the gas. However,
there are many applications in which
Λ
becomes comparable or larger than the size of the
system. The classical size effects occur in such situations.
Example 1:
Λ>
d
for a disk drive
Example 2:
Λ
~
d
or
d
for
thin films
d=10~20 nm
Λ
=100 nm
Thin film
k
Disk drive
In example 2, we can further reduce the film thickness to enhance the size effects. With
measured data for
k
and specific heat
c
, the mean free path in silicon can be estimated by
cv
Λ
k
=
3
,
where
v
is sound velocity. The approximated mean free path
Λ
is around 40 nm, while
the actual value is around 300 nm. The size effects occur for silicon films with thickness
less than
Λ
.
Note: in some thermal insulation applications, we also use porous materials whose pore
sizes are comparable to or less than
Λ
. The thermal conductivity of the air trapped in the
pores will be significantly reduced.
8.2 Quantum size effects
According to quantum mechanics, electrons and phonons are also material waves; the
finite size of the system can influence the energy transport by altering the wave
characteristics, such as forming standing waves and creating new modes that do not exist
in bulk materials.
For example, electrons in a thin film can be approximated as standing waves sitting
inside a potential well of infinite height. The condition for the formation of such standing
waves is that the wavelength,
λ
, satisfies the following relation
D
=
n
/2
(
n
=1,2,…),
where
D
is the width of the potential well. According to the de Broglie relation, the
wavelength is
/
=
hp
,
2.57 Fall 2004 – Lecture 3
1
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View Full Documentwhere
h
is Planck constant (
h
=6.6x10
34
Js),
p
is momentum. The energy of the electron
n
is thus
E
=
p
2
/2
m
or
E
=
p
2
=
h
2
()
2
.
2
m
8
m
D
n=1
n=2
D
1) For a free electron,
D
=1 mm, we have
2
−
3
4
2
E
=
n
(
6.6
×
10
−
3
4
)
2
~
1
0
n
<
<
k
T
=
4
.
1
×
1
−
2
1
J
at room temperature.
×
10
−
3
B
8
×
9.1 10
−
31
2) For
D
=1e8 m, we calculate
Ek
. Further reducing
D
results in more observable E.
>
B
8.3 Fast transport
For many materials, we have
τ
=
10
−
12
−
10
−
11
s
. Laser pulse can be as short as a few
femtosecond, we cannot use diffusion theory when the time scale is shorter than the
relaxation time.
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 Fall '04
 GangChen

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