2.57 Fall 2004 – Lecture 21
1
2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 21
Last time we talked about the current density as
11
12
'
e
d
dT
J
L
q
L
dx
dx
Φ
⎛
⎞
⎛
⎞
=
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
,
For electrons,
q
e
= −
and
e
e
µ
ϕ
Φ =
−
. Here
e
ϕ
is electrostatic potential, which is related
to the electrical field. Chemical potential
µ
is related to diffusion. Their combination
Φ
is electrochemical potential, indicating the total driving force of charges. The current
density can be rewritten as
11
12
e
d
d
J
L
L
dx
dx
Φ
Τ
⎛
⎞
⎛
⎞
=
−
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
.
Note: The second term
12
12
1
d
d
L
L T
dx
T dx
Τ
Τ
⎛
⎞
⎛
⎞
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
is similar to
dQ
S
T
∆
=
and may be
compared with entropy flux.
The heat transferred is
(
)
21
22
2
x
y
z
q
x
k
k
k
d
dT
J
v
E
f
L
L
V
dx
dx
µ
Φ
⎛
⎞
=
−
=
−
+
⎜
⎟
⎝
⎠
∑∑∑
For open circuits, Je=0. We obtain
12
11
/
/
h
c
L
d
dx
V
S
dT
dx
T
T
L
Φ
−
=
=
=
−
,
where S is called the seebeck coefficient.
Note: (1) S is dependent on the density of states. Therefore, it can be enhanced by using
nanostructures such as thin films or nanowires. This effect is also the principle of thermal
couples. (2) In the summation, we cannot use integral for quantized directions.
x
y
Π
2
Π
1
1
2
q=J(
Π
2

Π
1
)
q=J(
Π
1

Π
2
)
J
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2.57 Fall 2004 – Lecture 21
2
(2) In
/
/
d
dx
S
dT
dx
Φ
= −
, T should be the electron temperature T
e
. In equilibrium cases, T
e
is
close to phonon T
p
and we can use this effect to measure T
p
. However, for extreme cases
such as laser ablation, the two temperatures are not in equilibrium. Cautions should be
taken.
(3) For an onchip thermocouple, the measured temperature does not correspond to the
junction point, but closer to the average temperature from 1 to 3. This is different from
the normal thermocouples.
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 Spring '10
 WRIGHT,J
 Thermodynamics, Equations, Energy, Fundamental physics concepts, dx dx dx

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