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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 28
Ballistic Quantum Transport in Semiconductor Nanostructures
In this lecture you will learn:
• Electron transport without scattering (ballistic transport)
• The quantum of conductance and the quantum of resistance
• Quantized conductance
Rolf Landauer (19271999)
Lester F. Eastman (1928)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Conductors and Dissipation
Traditional View of Conductors:
V
I
L
A
σ
L
V
E
=
E
J
=
R
V
GV
V
L
A
E
A
AJ
I
=
=
=
=
=
Power Dissipation in Conductors:
E&M (energy continuity equation) tells us that the power dissipation per
unit volume of a resistor is:
2
2
J
E
E
J
=
=
=
Power dissipation in the entire resistor is:
()
G
I
R
I
A
L
A
J
AL
J
AL
E
J
P
2
2
2
2
2
=
=
=
=
=
R
L
A
G
1
=
=
⇒
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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Mean Free Path
The
mean free path
ℓ
is defined as the distance an electron travels before it scatters.
It is given by:
where
v
c
is the characteristic electron velocity and
τ
is the scattering time.
c
v
=
l
Mean Free Path:
Characteristic Velocity:
The characteristic velocity is the average velocity of those electrons that contribute to
the conductivity:
()
[]
k
v
E
k
v
E
f
k
d
e
J
o
d
d
r
r
r
r
r
r
r
.
2
2
FBZ
2
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∫
×
=
π
() ()
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∫
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∫
=
E
f
k
d
k
v
k
v
E
f
k
d
v
o
d
d
o
d
d
c
FBZ
FBZ
2
2
.
2
r
r
r
r
r
r
Recall the formula for the current density obtained from the Boltzmann equation:
For metals and heavily doped semiconductors at low temperatures:
For low doped semiconductors at high temperatures:
F
c
v
v
=
e
c
m
KT
d
v
=
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Ballistic Transport:
When the length
L
of the conductor becomes much smaller than the mean free path
ℓ
the transport is termed “ballistic” meaning that the electrons do not scatter during
the time it takes to travel through the conductor
Ballistic transport condition:
l
<<
L
Questions:
• What happens when
L
<<
ℓ
?
• The formulas for conductivity that have the scattering time
in them are clearly
no longer valid since there is no scattering
• What about dissipation?
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 Spring '08
 RANA

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