3.3 Landauer formalism and conductance quantization

The Landauer formalism revolutionizes our understanding of electron transport in tiny systems. It links conductance to transmission probability, revealing that conductance comes in neat little packages. This idea is key to grasping how electricity flows in the nano-world.

Conductance quantization is like a staircase of electric flow in super-narrow channels. As we widen these channels, the conductance jumps up in fixed steps. This weird behavior shows up in quantum point contacts and other nano-sized structures, giving us a peek into the quantum nature of electricity.

Landauer Formalism

Fundamental Concepts of Landauer Formula

• Landauer formula relates conductance to transmission probability in mesoscopic systems
• Expresses conductance as $G = \frac{2e^2}{h} T$, where T represents transmission coefficient
• Applies to systems where electron transport occurs without scattering (ballistic transport)
• Reveals quantization of conductance in integer multiples of $\frac{2e^2}{h}$
• Assumes perfect contacts between leads and conductor

Transmission and Ballistic Transport

• Transmission coefficient quantifies probability of electron passing through conductor
• Ranges from 0 (no transmission) to 1 (perfect transmission)
• Ballistic conductors exhibit minimal scattering of electrons
• Mean free path of electrons exceeds device dimensions in ballistic regime
• Enables study of quantum effects in electron transport

Chemical Potential and Energy Considerations

• Chemical potential represents energy required to add or remove an electron from system
• Plays crucial role in determining electron flow between reservoirs
• Difference in chemical potential between reservoirs drives current flow
• Fermi level aligns with chemical potential at equilibrium
• Landauer approach considers transport as transmission between reservoirs with different chemical potentials

Conductance Quantization

Principles of Conductance Quantization

• Conductance quantization occurs in narrow constrictions or channels
• Manifests as stepwise increases in conductance as channel width increases
• Each step corresponds to an increase of $\frac{2e^2}{h}$ in conductance
• Results from discrete nature of electronic states in confined systems
• Observed experimentally in various nanostructures (quantum point contacts, carbon nanotubes)

Quantum Point Contacts and Device Structure

• Quantum point contacts (QPCs) serve as primary devices for observing conductance quantization
• Created by applying gate voltages to deplete regions of a two-dimensional electron gas
• Form narrow constriction between two electron reservoirs
• Width of constriction controlled by gate voltage
• Allow precise manipulation of number of conducting channels

Subbands and Energy Level Discretization

• Subbands represent discrete energy levels in confined systems
• Arise from quantization of electron motion perpendicular to transport direction
• Number of occupied subbands determines number of conducting channels
• Energy spacing between subbands increases as confinement increases
• Subband structure directly influences conductance quantization steps

Fermi Energy and Transport Mechanisms

• Fermi energy marks highest occupied energy level at absolute zero temperature
• Determines which subbands contribute to electron transport
• Conductance quantization observed when Fermi energy crosses subband edges
• Increasing Fermi energy (through gate voltage) leads to occupation of more subbands
• Interplay between Fermi energy and subband structure governs conductance quantization behavior