Nanoelectronics Part II Many Electron Phenomena Chapter 10 Nanowires, Ballistic Transport, and Spin Transport

2 Nickel nanowires Si nanowires

A review of nanowire technology Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

Courtesy from C. Lieber “Nanowires: A Platform for Nanoscience Nanotechnology”

10.1 Classical and Semi-classical Transport classical theory of conduction – free electron gas model A particle has 3 degrees of freedom Mean thermal velocity

classical theory of conduction – free electron gas model At T = 0 K, v T =0 At room tem.: However, this motion is random and does not result in net current. Now consider applying a voltage

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

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

classical theory of conduction – free electron gas model For example, for copper at room temperature, τ=2.47× s, E = 1V/m: v d = 4.35×10 -3 m/s This is much smaller than thermal velocity: ~ 10 5 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.)

Classical theory of conduction – free electron gas model Classical current density So Conductivity: For electron and hole:

classical theory of conduction – free electron gas model For copper: Velocity:

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

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: 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 Semiclassical theory of electrical conduction – Fermi gas model

Semiclassical theory of electrical conduction – Fermi gas model

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.

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 Semiclassical theory of electrical conduction – Fermi gas model

Classical Resistance and Conductance ab

Classical Resistance and Conductance Ohm’s law If S = w 1 X w 2 and length L, field E is uniform throughout the material with magnitude ϵ

Classical Resistance and Conductance For a two-dimensional flat wire having length L and width w: where σ s is the sheet conductivity in S (not S/m).

Conductivity of Metallic Nanowires- the 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.

Conductivity of Metallic Nanowires- the influence of Wire Radius As the Cu nanowire was being oxidized, the Cu became more and more like Cu 2 O and the wire acted like a p-type semiconductor.

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 << L m, mean free path, will the classical model of resistance works? When L << L m, one would expect that no collisions would take place – rendering the classical collision- based model useless

10.2 Ballistic Transport At very small length scales, electron transport occurs ballistically. It is very important in nanoscopic devices.

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 L m is mean free path L φ 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)

So, inelastic collisions are called dephasing events. L φ 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 Electron Collisions and Length Scales

Ballistic Transport Model The reservoir is large and it energy states form essentially a continuum: infinite source and sink for electrons. reservoir

Ballistic Transport Model The ballistic channel and subbands y and z dimensions are small

Ballistic Transport Model Let w 1 =w 2 =w The total number of subbands at or below the Fermi energy: We assume:

Quantum Resistance and Conductance Fermi energy Left reservoir: E F – eV Right reservoir: E F the electrons in the wire have wavefunction: With an associated probabilistic current density

Quantum Resistance and Conductance Using:

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: 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, E f - eV, T) and f(E, F F, T) – Transmission probability: T n (E) Quantum Resistance and Conductance

Current flowing from left to right Right to left Quantum Resistance and Conductance

Quantum Resistance and Conductance Total current flowing Because

So, the temperature-dependent conductance: At very low temperatures 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 Quantum Resistance and Conductance

Quantum Resistance and Conductance reservoir

Note: Landauer formula can also be applied to tunnel junctions T(E F ) is the transmission coefficient obtained from solving Schrodinger’s equation. As T increases, the observed quantization tends to vanish. (k B T becomes large) Quantum Resistance and Conductance

Origin of the Quantum Resistance The resistance quantum, R 0, 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 << L m, L φ 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.