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 140 
University of California, Berkeley
EE230  Solid State Electronics
Prof. J. Bokor
Quantum transport
(Read Kittel, 8th ed., pp. 533554)
When we have structure in which many collisions take place as carriers transport across it,
the quantum mechanical phase of the electron wavefunctions is essentially randomized, and
the semiclassical approach that we have been following is appropriate. In nanoscale elec
tronic devices, especially at low temperatures:
• device dimensions begin to approach the meanfree path between collisions
• scattering processes play an decreasingly important role
This is the regime of
quantum transport
.
Electron transport becomes governed by princi
ples of quantummechanical wave propagation and interference, rather than the classical dif
fusive transport.
Conductance quantization
Consider the limit of a onedimensional conducting wire with no collisions.
Does it have
resistance?
Connect the wire between to contacts.
We model the contacts as large, 3D reservoirs with a
voltage difference V between them.
The rightgoing states of the wire are populated up to the electrochemical potential (quasi
Fermi energy)
μ
1
and the leftgoing states will be populated up to the electrochemical poten
tial
μ
2
where the potential difference is
μ
1

μ
2
=
qV.
The net current flowing through the
wire is due to the excess rightmoving carrier density
Δ
n is limited by the density of states in
the wire.
For the lowest transverse subband (or mode) of the wire:
μ
2
μ
1
μ
2
qV
+
=
left contact
right contact
wire
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View Full Document  141 
University of California, Berkeley
EE230  Solid State Electronics
Prof. J. Bokor
We obtain the
differential conductance,
,
as the small change in current for a small
change in applied voltage. Then we can write the density of excess rightmoving carriers as
.
We divide by 2 to get the density of states only for right moving carriers, and we divide by L
to get the carrier density per unit length. We can rewrite
D(E)
in terms of the carrier veloc
ity,
v:
.
The net current (for electron flow) is then:
and we obtain the famous result of quantized conductance:
.
The resistance quantum,
R
Q
=
1/G
Q
=
12.906 k
Ω.
This is the conductance for carriers in the
lowest subband of the wire.
As additional subbands are populated, each adds one addi
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This note was uploaded on 08/01/2008 for the course EE 230 taught by Professor Bokor during the Spring '08 term at University of California, Berkeley.
 Spring '08
 BOKOR

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