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Electronic Conduction of Mesoscopic Systems and Resonant States

Explore electronic conductance and resonant states in mesoscopic systems, collaboration by researchers from University of Tokyo and University of Texas. Understand Landauer formula, interference phenomena, and implications in quantum mechanics.

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Electronic Conduction of Mesoscopic Systems and Resonant States

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  1. Electronic Conduction of Mesoscopic Systems and Resonant States Naomichi Hatano Institute of Industrial Science, Unviersity of Tokyo Collaborators: Akinori Nishino (IIS, U. Tokyo) Takashi Imamura (IIS, U. Tokyo) Keita Sasada (Dept. Phys., U. Tokyo) Hiroaki Nakamura (NIFS) Tomio Petrosky (U. Texas at Austin) Sterling Garmon (U. Texas at Austin)

  2. Contents Conductance and the Landauer Formula Definition of Resonant States Interference of Resonant States and the Fano Peak

  3. What are mesoscipic systems? T. Machida (IIS, U. Tokyo) T. Machida (IIS, U. Tokyo) S. Katsumoto (ISSP, U. Tokyo)

  4. Theoretical modeling lead lead Scatterer (Quantum Dot, …) Cross section of a lead

  5. Perfect Conductor L   k Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems”  m2

  6. Conductance of a Perfect Conductor  m2 k Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” Voltage difference Spin Density n =1/L

  7. Be aware! does not hold! So, what was the conductance? Conductance is the inverse of the resistance.

  8. Perfect Conductor   Contact resistance Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems”

  9. L   Conductance in general Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” Gate voltage Scatterer Probability T Linear response Calculates at the Fermi energy

  10. L   Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” Scatterer Probability T Contact resistance “Raw” resistance of a scatterer

  11. V V Conductance (Inverse Resistance) Note does not hold. Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” Scatterer Transmission probability: T(E)

  12. Example: 3-state quantum dot Keita Sasada: Ph. D. Thesis (2008) Trans. Prob. T Resonance Peak (Asymmetric Fano Peak) Conductance Fermi Energy

  13. Contents Conductance and the Landauer Formula Definition of Resonant States Interference of Resonant States and the Fano Peak

  14. Pole → or , where Definition of resonance: 1 Resonance: Pole of Trans. Prob. (S-Matrix) where

  15. Definition of resonance: 2 Siegert condition (1939) Resonance: Eigenstate with outgoing waves only. V(x) x

  16. Definition of resonance: 2 Even solutions: B C, F G Odd solutions: B C, F G V(x) x

  17. Definition of resonance: 2 Bound state Eigen-wave-number Eigenenergy

  18. where Non-Hermiticity of open system N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187 

  19. Non-Hermiticity of open system N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187  “Anti-resonant state as an eigenstate “Resonant state” as an eigenstate

  20. Definition of resonance: 2 Bound state Eigen-wave-number Eigenenergy Anti-resonant state Resonant state

  21. Eigenfunction of resonant state N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187

  22. Particle-number conservation N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187 x

  23. (Far left) K E Bound, resonant, anti-resonant states (Far right) Anti-resonant state Bound state Bound state Continuum Continuum Branch point Anti-resonant state Resonant state Branch cut Resonant state

  24. is an eigenstate for Dispersion relation: Tight-binding model energy band  k p p Continuum limit Impurity bound state

  25. (Far left) K E Bound, resonant, anti-resonant states (Far right) Anti-resonant state Bound state Bound state t t Continuum Continuum -p p Branch point Resonant state Branch cut Anti-resonant state Resonant state

  26. Fisher-Lee relation S. Datta “Electronic Transport in Mesoscopic Systems” Complex effective potential: H. Feshbach, Ann. Phys. 5 (1958) 357 Complex potential eikx

  27. Conductance and resonance Green’s function: Inverse of a finite matrix ↓ Conductance for real energy Resonance from poles in complex energy plane

  28. Contents Conductance and the Landauer Formula Definition of Resonant States Interference of Resonant States and the Fano Peak

  29. N-state Friedrichs model Keita Sasada: Ph. D. Thesis (2008) • All leads are connected to the site d0 • Time reversal symmetry is not broken (no magnetic field)

  30. (where) N-state Friedrichs model Keita Sasada: Ph. D. Thesis (2008) Conductance formula Maximum conductnace from leadto lead Sign depends on the inner structure of the dot and E Bound st., Res. st., Anti-res. st. Local DOS of discrete eigenstates: Local DOS of leads:

  31. Interference of discrete states Keita Sasada: Ph. D. Thesis (2008) Discrete eigenstates Bound states Resonance pair (Res. and Anti-res.) : Interference between B and R : Interference between R and R Asymmetry of a conductance peak q: Fano parameter

  32. T-shape quantum dot (N=2) Bound state: 2 Resonant state: 1 Anti-resonant state: 1 Anti-resonant state Interference between each bound state and the resonace pair determines the asymmetry of the conductance peak. Bound state 1 Bound state 2 Resonant state

  33. 3-state quantum dot (N = 3) Keita Sasada: Ph. D. Thesis (2008) Bound state: 2 Resonant state: 2 Anti-resonant state: 2 Anti-resonant state 1 Anti-resonant state 2 Interference between the resonance pairs 1 and 2 determines the asymmetry of the conductance peal. Bound state 1 Bound state 2 Resonant state 2 Resonant state 1

  34. Fano parameter : Interference between B and R : Interference between R and R Large when close

  35. Summary • - Electronic conduction and resonance scattering • Definition and physics of resonant states • Particle-number conservation • - Interference between resonant states

  36. Discetization of Schrödinger equation Tight-binding model

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