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Nanoelectronics
Part II Many Electron Phenomena
Chapter 10 Nanowires, Ballistic
Transport, and Spin Transport
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Nickel nanowires
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Si nanowires
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A review of nanowire technology
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10.1 Classical and Semi-classical
Transport
10.1.1 classical theory of conduction – free
electron gas model
• A particle has 3 degrees of freedom
Mean thermal velocity
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10.1.1 classical theory of conduction –
free electron gas model
At T = 0 K, vT=0
• At room tem.:
• However, this motion is random and does not
result in net current.
• Now consider applying a voltage
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10.1.1 classical theory of conduction –
free electron gas model (Drude model)
will velocity increase infinitely as time increases?
• Collision between electrons and material lattice
• Mean time between collision is τ
It is called relaxation time
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10.1.1 classical theory of conduction –
free electron gas model
• Therefore, velocity is accelerated from 0 to τ:
• More precisely, mean time between collision
and momentum relaxation time are different.
• µ is electrical mobility
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10.1.1 classical theory of conduction –
free electron gas model
• For example, for copper at room temperature,
τ=2.47×10-14 s, E = 1V/m: vd = 4.35×10-3 m/s
• This is much smaller than thermal velocity: ~
105 m/s.
• Despite the small value of drift velocity,
electrical signals propagate as electromagnetic
waves at the speed of light.
(EM signal propagates in the medium exterior to
the wire, such air or other dielectric insulation.)
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10.1.1 Classical theory of conduction –
free electron gas model
• Classical current density
• So
• Conductivity:
• For electron and hole:
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10.1.1 classical theory of conduction –
free electron gas model
• For copper:
• Velocity:
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
• Classical model is simply a classical free
electron gas model with the addition of
collisions.
• It is better to use quantum physics principles.
• We should consider Fermi velocity. All
momentum states within Fermi sphere are
occupied, and outside the Fermi sphere are
empty
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
• For copper:
• Fermi velocity is many orders of magnitude
larger than drift velocity, in general. It is also
randomly directed in the absence of an
applied field.
• From the viewpoint of Fermi surface:
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
• The increase in momentum in a time
increment dt is
• Fermi gas model: use Fermi velocity rather
than thermal velocity to describe conduction
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
• Mean-free path: the mean distance an
electron travels before a collision with the
lattice.
• Copper:
• Using Fermi velocity is better than thermal velocity, in fact,
the electron can pass through a perfect periodic lattice
without scattering, where the effect of lattice merely leads to
the use of effective mass.
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10.1.2 Semiclassical theory of electrical
conduction – Fermi gas model
• Mobility is a strong function of temperature.
• As T decreases, mobility increases due to
diminished phonon scattering.
• At sufficiently low temperature, scattering is
mostly due to impurities.
• For relatively pure copper at 4K, it is possible
to obtain long τ = 1 ns.
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10.1.3 Classical Resistance and
Conductance
a
b
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10.1.3 Classical Resistance and
Conductance
• Ohm’s law
• If S = w1 X w2 and length L, field E is uniform
throughout the material with magnitude ϵ
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10.1.3 Classical Resistance and
Conductance
• For a two-dimensional flat wire having length
L and width w:
• where σs is the sheet conductivity in S (not
S/m).
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10.1.4 Conductivity of Metallic Nanowiresthe influence of Wire Radius
• Wire resistance increases with decreasing radius
• For radius between 1-20nm, the wire resistance
increases significantly, due to scattering from wire
surface, from grain boundaries, defects/impurities.
• For 5-10 nm, a quantum wire model that accounts
for transverse quantization is needed.
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10.1.4 Conductivity of Metallic Nanowiresthe influence of Wire Radius
As the Cu nanowire was being oxidized, the Cu became more and more
like Cu2O and the wire acted like a p-type semiconductor.
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10.2 Ballistic Transport
• When L is large, conductivity is derived assuming a
large number of electrons and a large number of
collision between electrons and phonons, impurities,
imperfections.
• As L becomes very small L << Lm , mean free path,
will the classical model of resistance works?
• When L << Lm , one would expect that no collisions
would take place – rendering the classical collisionbased model useless
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10.2 Ballistic Transport
• At very small length scales, electron transport occurs
ballistically. It is very important in nanoscopic
devices.
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10.2.1 Electron Collisions and Length Scales
• An electron can collide with an object such that
there is no change in energy – elastic collision.
another type, the energy of electron changes –
inelastic collision.
• L is system length
• Lm is mean free path
• Lφ is the length over which an electron can travel
before having an inelastic collision. It is also
called phase-coherence length, since it is the
length over which an electron wavefunction
retains its coherence (i.e., its phase memory)
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10.2.1 Electron Collisions and Length Scales
• So, inelastic collisions are called dephasing
events. Lφ is about tens – hundreds nm
• During ballistic transport, no momentum or
phase relaxation. Thus, in a ballistic material,
the electron wavefunction can be obtain from
Schrodinger’s equation.
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10.2.2 Ballistic Transport Model
• The reservoir is large and it energy states form
essentially a continuum: infinite source and
sink for electrons.
reservoir
reservoir
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10.2.2 Ballistic Transport Model
• The ballistic channel and subbands
y and z dimensions are small
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10.2.2 Ballistic Transport Model
• Let w1=w2=w
• The total number of subbands at or below the
Fermi energy:
We assume:
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10.2.3 Quantum Resistance and Conductance
• Fermi energy
Left reservoir: EF – eV
Right reservoir: EF
the electrons in the wire have wavefunction:
With an associated probabilistic current density
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10.2.3 Quantum Resistance and Conductance
Using:
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10.2.3 Quantum Resistance and Conductance
• Assume the wavefunction can be represented
by a traveling state, indicating left-to-right
(positive k) movement of electron
Such that:
Because spin up and down:
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10.2.3 Quantum Resistance and Conductance
• We don’t know if a certain state will be filled
or not. The probability that the electron
makes it into the channel from the left
reservoir, out of the channel into the right
reservoir must be considered.
– Fermi-Dirac probability: f(E, Ef - eV, T) and f(E, FF, T)
– Transmission probability: Tn(E)
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10.2.3 Quantum Resistance and Conductance
• Current flowing from left to right
• Right to left
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10.2.3 Quantum Resistance and Conductance
Total current flowing
Because
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10.2.3 Quantum Resistance and Conductance
• So, the temperature-dependent conductance:
• At very low temperatures
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10.2.3 Quantum Resistance and Conductance
• If there are N electronic channels, and the
transmission probability is one for each
channel.
This is Landauer formula
Since N is number of conduction channels
Resistance of each channel is
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10.2.3 Quantum Resistance and Conductance
reservoir
reservoir
reservoir
reservoir
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10.2.3 Quantum Resistance and Conductance
• Note: Landauer formula can also be applied to
tunnel junctions
T(EF) is the transmission coefficient obtained
from solving Schrodinger’s equation.
• As T increases, the observed quantization
tends to vanish. (kBT becomes large)
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10.2.4 Origin of the Quantum Resistance
• The resistance quantum, R0, arises from perfect
(infinitely wide) reservoirs in contact with a single
electronic channel (i.e., a very narrow physical
channel).
• Indeed, the resistance of a ballistic channel is
length independent, as long as L << Lm, Lφ
• Ballistic metal nanowires have been shown to be
capable of carrying current densities much higher
than bulk metals, due to absence of heating in the
ballistic channel itself.
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