Fall 2010
ECE 453 Lecture Notes #4
For Purdue ECE 453 students ONLY
NOT FOR CIRCULATION
ECE 453 Lecture Notes#4
Excerpted from
(LNE) Lessons from nanoelectronics:
A new perspective on transport
1. The bottomup approach
4
Introductory concepts
2. Why electrons flow
6
3. The elastic resistor
10
4. The new Ohm's law
14
Notes #1
5. Where is the resistance?
19
*****************************************************************
6. Transverse modes
27
7. Drude formula
33
8. Kubo formula
38
Notes # 2
9. How realistic is an elastic resistor?
41
*****************************************************************
Semiclassical and quantum transport
10. Beyond low bias: The nanotransistor
44
Notes#3
11. Semiclassical Transport and the Scf method
50
*****************************************************************
12. Resistance and uncertainty
56
13. Quantum Transport: Schrodinger to NEGF
63
Notes#4
14. Resonant tunneling and Anderson localization
68
*****************************************************************
15. Coulomb blockade and Mott transition
72
16. Hall effect / QHE
xx
Beyond voltages and currents
17. Thermoelectricity
xx
18. Heat flow
xx
19. Spin flow
xx
20. Spin transistor
xx
21. Electronic Maxwell's demon
xx
22. Physics in a grain of sand
xx
Supriyo Datta, [email protected]
Purdue University
World Scientific (2011), to be published
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Copyright S. Datta
[email protected]
All Rights Reserved
56
12. Resistance and uncertainty
In Chapter 4 we saw that the conductance of a short ballistic conductor is given by
(q
2
/h)*M, where M is the number of independent conduction channels. What about a
really short "molecular" conductor? There cannot be more than a few conduction
channels at best, and so we would expect a maximum conductance ~ a few (q
2
/h). Indeed
experimentalists have found the conductance of a hydrogen molecule to be ~ 2 (q
2
/h).
Would our theoretical treatment predict this? The answer is no. Let me explain why.
For something like a hydrogen molecule the energy level diagram should look just like
the onelevel resistor that we discussed in Section 3 where we obtained the following
expression for the conductance
G
one level
q
2
t
F
T
(
0
)
(same as Eq.(3.2))
This
suggests
a
maximum
conductance of
(
q
2
/
t
)*(1/4
kT
)
if
the energy level
sits right on the
electrochemical potential μ
0
, dropping
down to zero as the level moves away
from
μ
0
.
Clearly
this
is
not
the
quantized conductance
q
2
/
h
that we
are looking for, since there is no
reason to expect that 4kT*t < h. After
all, kT and t are two independent physical quantities: kT is related to the temperature,
while t is the transfer time related to the quality of the contacts if the transit time through
the channel is negligible. For any given t, we can always find a temperature below which
the conductance would exceed our "maximum."
What Eq.(3.2) misses is the broadening of the
energy
level:
Quantum
mechanically
the
process of coupling to a level, effectively
spreads it out giving a picture more like the
one shown in Fig.6.2. The onelevel resistor
then effectively becomes a resistor with a
density
of
states,
D
~1/
,
being
the
broadening
of
the
energy
level.
The
conductance even
for
a
onelevel
resistor
should then be obtained from the multilevel result
G
q
2
D
/
t
(see Eq.(3.3c)) which
gives
G
one level
q
2
t
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 Spring '10
 SupriyoDatta
 Copyright S. Datta, Supriyo Datta

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