Quantum Description and Classical Transport in Metals

Introduction

• Second lecture on transport in metals.
• Brief recap of classical transport in normal metals discussed previously.

Classical Transport in Metals

• Diffusion Equation: Applicable to any material, involves the diffusion constant.
• Density of States (DoS): Relates to the number of states at the Fermi level.
• Spectrum Surface: Involves the density and velocity at the Fermi surface.
• Material Specifics: Diffusion constant and DoS are material-specific, influencing transport properties.

Quantum Description of Metals

• Green Function: Central in quantum description, inversely related to energy and potential.
• Disorder Averaging: Introduction of self-energy (Σ) in Green functions.
• Diagrammatic Expansion: Illustrates self-energy through diagrams, efficiency in summing specific diagrams due to singularities.

Perturbation Theory

• Self Consistent Approximation: A method to efficiently sum diagrams by avoiding crossing diagrams which introduce complexity.
• Sigma (Σ): Self-energy approximated through consistent averaging of Green functions.

Transport Properties

• Diffusion Propagator: Calculated using Green functions, relates to classical diffusion equation.
• Kubo Formula: Relates transport properties like conductivity to quantum mechanics.

Conductivity and Resistance

• Conductance vs. Resistance: Conductance is inverse of resistance; important for understanding metallic properties.
• Scaling Hypothesis: Proposes that derivative of conductance with respect to system size is a function of conductance itself.

Metal vs. Insulator

• Conductance Behavior:
  • Metals: Conductance scales with cross-sectional area and is finite as size increases.
  • Insulators: Conductance decreases exponentially with sample size.
• Scaling Law: Universal behavior depending on dimensions; metals and insulators show distinct transitions.

Conclusion

• Scaling Hypothesis: Suggests a universal function for conductance behavior across different dimensions and materials.
• Phase Transition Point: Marks the transition from metallic to insulating behavior in materials.