Landauer-Büttiker Formalism Examples in Physics
Explore the Landauer-Büttiker formalism examples in physics including differences in potentials and temperatures, scattering matrices, and various scatterer configurations. Understand the principles and applications behind conductance in nanostructures.
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Presentation Transcript
1. Landauer-Bttiker formalism Examples Farkas D niel Gergely Msc physics student 1
Contents Difference of potencials Difference of temperatures Scattering matrix 1x1, 2x2, 3x3 Scatterer with two leads Scatterer with potencial contact Scatterer embedded in ring 2
Difference of potencials Starting point: If pot. difference << temperature: Than the current is: T = 0, easy case. V > 0: S12(E) E-dependence is important. where: 3
Difference of temperatures Similar case, now the temperature differs: Fermi functions: Current, and thermoelectric conductance: 2 4
Scattering matrix 1x1 1 reservoir, 1 scattering channel, => 1x1 scattering matrix: Total reflection, S11: reflection coefficient. 5
Scattering matrix 2x2 2 reservoir, 2x2 matrix, 8 parameter + unitarity. 6
Scattering matrix 3x3 3 reservoirs, 9 matrix elements Example: Reflection probabilities can be different! T depends on transmission with the third lead. 7
Scatterer with two leads 2 reservoirs, but NL and NR sub-bands, T = 0, sum up to the sub-bands. 8
H dependence of conductance Our goal: Some quantities: Identity + unitarity => Symmetry => Summary: 9
Scatterer with potencial contact 3 reservoir special case, Motivation: measurement in the sample => + contact, + problem G12 depends on scattering between 1,2 and 3! 10
Scatterer embedded in ring 2 reservoir special case, H occurs, Thin sample, wave function: + boundary conditions: Magnetic flux equivalent with assymetry! 11
Scatterer embedded in ring Spectrum of free electrons: + some nasty equations Dispersion relation: Current: 12
