L4 ECE-ENGR 4243/ FJain 1 Derivation of current-voltage relation in 1-D wires/nanotubes (pp A) Ballistic, quasi-ballistic transport—elastic and inelastic length and phase coherence (pp ) Universal Conductivity Fluctuation (UCF) in in 1-D wires/nanotubes in the presence of quasi-ballistic transport (pp. 102B-104). Overview

Derivation of current due to 1D subband or channel i. page Show conductance quantized as e 2 /h, g is 2 due to spin

3 (1) n = half of the number of carriers per unit length (carrier density) e = electron charge = increase in velocity due to constriction. N(E) =  1D(E) = Here, and f(E f )=1 at 0°K. (1b) Fermi energy is expressed in terms of carrier velocity V f Substituting Eq. 1c in Eq. 1b, (1c) 2(a) Now we need an expression for the increase in velocity due to constriction or applied voltage. e*external voltage V = eV= (2b)

4 Derivation of current-voltage relation in 1-D wires/nanotubes continued e * external voltage V = eV = (3) If << V f,, eV =as we neglect the first term in Eq. 3. Or = (4) Substituting equations (2a) and (4) into equation 1: (5a) (non magnetic field case where g s = 2) (5b) Multiple sub-bands or multiple channels The total resistance R is expressed as: G= (6)

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Ballistic, quasi-ballistic, and diffusive Transport 6 L and W are the length and the width of the wire or constriction; l e is the elastic mean free path between impurity scattering processes, and l  is the inelastic or phase-breaking mean free path between phonon scattering events.

7 Ballistic: l  >> l e >> L, W. Electrons can only experience the boundaries of the wire, and quantum states extend from one end to the other. Any occupied states carry current from one end to the other. When there is no applied voltage, the left and the right currents cancel each other. If a small voltage V is applied the only states present on the left and the right ends are those who have chemical potentials  L and  R on the left and on the right respectively. This imbalance gives a net current that is proportional to the chemical potential difference,  L   R = eV. For the electrons in the quantum wire, their paths are determined by scattering on the potential walls of the wire in a perfectly ballistic way. If W is small compared with the Fermi wavelength, then only one of a few channels can be occupied. Quasi-ballistic and Universal Conductance Fluctuation (UCF) regimes: Refer to figure 1(b). In this regime, there are a few impurities in the wire, and transport is via channel but scattering introduced by the impurities mixes the modes, and increase the reflection probability of electrons entering the wire. It is also possible for electrons via multiple scattering on the walls and on a few impurities to be trapped in states that are localized on the scale of l e. These states have no contact to her reservoirs, and they do not contribute to the transport. The conductance in this regime depends on the precise positions of the impurities and the potentials that define the wire, and conductance changes in the order of e 2 /h when the potentials or the sample is changed. Quasi-ballistic regime: l  >> L >> l e ; UCF: l  >> L >> l e >> W.

8 Weakly Localized Regime: l  >> L >> W >> l e In this situation, multiple scattering on the impurities dominates, so wire modes no longer have meaning. Electrons are localized both longitudinally and transversely on the length scale l e. The electrons no longer see the one-dimensionality of the wire. NO states exist that extend from one end of the wire to the other end. This 2D weakly localized regime has no conductivity at low temperatures. Diffusive regime: L, W >> l  Electrons diffuse through the wire, and the transport of electrons through the wire is by scattering between the localized states, and this requires inelastic scattering. Mobility is determined by the average density of impurities, but at high temperatures when inelastic length is smaller than elastic length, phonon scattering determine mobility. Refer to Figure 1(a).

L5 Universal conductance fluctuation (UCF) 9 pp. 120B Fig. 1. Biased quantum wire between two reservoirs (top). The location of Fermi levels (bottom). Fig. 2. Density of states in wire and point contact (represented by quantum well like reservoirs). When energy states extend from left hand reservoir to the right hand reservoir all across the wire, conductance is determined by quantum mechanical transmission probability of state between µ R +µ L wire represent a barrier between two reservoir

10 Scattering of carriers: Impurity/defect/grain boundary scattering, Ionized impurity –scattering is elastic (no energy exchange and maintain phase relationship) Phonon scattering (acoustical phonons and optical phonons—quantum of sound waves)- electrons scattering with phonon is inelastic. Energy can increase or decrease. Electron wave no longer the same. It is known as phase breaking scattering. No longer ballistic.

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13 Ballistic or quasi-ballistic transport is wave like. Like microwaves in a waveguide. Non-locality or global nature of transport. (page 103)

14 (16) The carrier mobility μ n =, it depends on the scattering processes. Chapter 2 ECE 4211

15 (12) Without f(E) we get density of states Quantization due to carrier confinement along the x and z-axes. Looking at the integration Carrier density n or p 13 This simplifies 19 Go back to Eq. (12), the density of states in a nano wire is where E=E-E nx -E nz Density of state (quantum wires) review L3 p85 Density of state (quantum wells) review L3 p85 Density of states

Density of states in 0-D (Quantum dots) The k values are discrete in all three directions 16

Energy levels in nanowire (discrete due to nx and nz) Y-axis gives energy width Discrete value of nx=1 and then add nz= 1, 2, 3 2. discrete value of nx=2 and then add nz= 1, 2, 3 3. Add to each discrete value of nx, nz an energy width as shown in density of state plot (for one level).