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. 1/34 Contents 1.Conductance and the Landauer Formula 2.Definition of Resonant States 3.Interference of Resonant States and the Fano Peak

3. 2/34 What are mesoscipic systems? T. Machida (IIS, U. Tokyo) S. Katsumoto (ISSP, U. Tokyo) T. Machida (IIS, U. Tokyo)

4. 3/34 Theoretical modeling lead Scatterer (Quantum Dot, …) Cross section of a lead lead

5. 4/34 k Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems”    Perfect Conductor  L

6. 5/34 Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” Conductance of a Perfect Conductor Spin Density n =1/L Voltage differenc e

7. 6/34 So, what was the conductance? Conductance is the inverse of the resistance. Be aware! does not hold!

8. 7/34  Perfect Conductor  Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” Contact resistance

9. 8/34 Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems”   L Scatterer Probability T Conductance in general Linear response Calculates at the Fermi energy Gate voltage

10. 9/34 Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems”   L Contact resistance “Raw” resistance of a scatterer Scatterer Probability T

11. 10/34 Landauer formula S. Datta “Electronic Transport in Mesoscopic Systems” VV VV Transmission probability: T(E) Scatterer Conductance (Inverse Resistance) Note does not hold.

12. 11/34 Example: 3-state quantum dot Keita Sasada: Ph. D. Thesis (2008) Resonance Peak (Asymmetric Fano Peak) Trans. Prob. T Fermi Energy Conductance

13. 12/34 Contents 1.Conductance and the Landauer Formula 2.Definition of Resonant States 3.Interference of Resonant States and the Fano Peak

14. 13/34 Definition of resonance: 1 Pole → or, where where Resonance: Pole of Trans. Prob. (S-Matrix)

15. 14/34 Definition of resonance: 2 Siegert condition (1939) Resonance: Eigenstate with outgoing waves only. x V(x)V(x)

16. 15/34 Definition of resonance: 2 x V(x)V(x) Even solutions: B  C, F  G Odd solutions: B  C, F  G

17. 16/34 Definition of resonance: 2 Eigen-wave-numberEigenenergy Bound state

18. 17/34 Non-Hermiticity of open system N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187  where

19. 18/34 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. 19/34 Definition of resonance: 2 Eigen-wave-numberEigenenergy Bound state Resonant stateAnti-resonant state

21. 20/34 Eigenfunction of resonant state N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187

22. 21/34 Particle-number conservation N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187 x

23. 22/34 Bound, resonant, anti-resonant states K Bound state Resonant state Anti-resonant state Continuum E Bound state Resonant state Anti-resonant state Continuum Branch point Branch cut (Far left) (Far right)

24. 23/34 Tight-binding model is an eigenstate for Dispersion relation: k   Continuum limitImpurity bound state energy band

25. 24/34  t tt Bound, resonant, anti-resonant states K Bound state Resonant state Anti-resonant state Continuum E Bound state Resonant state Anti-resonant state Continuum Branch point Branch cut (Far left) (Far right) 

26. 25/34 Fisher-Lee relation Complex effective potential: H. Feshbach, Ann. Phys. 5 (1958) 357 S. Datta “Electronic Transport in Mesoscopic Systems” Complex potential e ik  x

27. 26/34 Conductance and resonance Green’s function: Inverse of a finite matrix ↓ Conductance for real energy Resonance from poles in complex energy plane

28. 27/34 Contents 1.Conductance and the Landauer Formula 2.Definition of Resonant States 3.Interference of Resonant States and the Fano Peak

29. 28/34 N-state Friedrichs model Keita Sasada: Ph. D. Thesis (2008) All leads are connected to the site d 0 Time reversal symmetry is not broken (no magnetic field)

30. 29/34 Conductance formula N-state Friedrichs model Local DOS of discrete eigenstates: Local DOS of leads: Maximum conductnace from lead  to lead  Sign   depends on the inner structure of the dot and E (where ) Bound st., Res. st., Anti-res. st. Keita Sasada: Ph. D. Thesis (2008)

31. 30/34 Interference of discrete states Bound statesResonance pair (Res. and Anti-res.) : Interference between B and R Keita Sasada: Ph. D. Thesis (2008) Discrete eigenstates Asymmetry of a conductance peak q: Fano parameter : Interference between R and R

32. 31/34 T-shape quantum dot (N=2) Bound state: 2 Resonant state: 1 Anti-resonant state: 1 Interference between each bound state and the resonace pair determines the asymmetry of the conductance peak. Bound state 1Bound state 2 Anti-resonant state Resonant state

33. 32/34 3-state quantum dot (N = 3) Keita Sasada: Ph. D. Thesis (2008) Bound state 1Bound state 2 Anti-resonant state 2 Anti-resonant state 1 Resonant state 1 Resonant state 2 Interference between the resonance pairs 1 and 2 determines the asymmetry of the conductance peal. Bound state: 2 Resonant state: 2 Anti-resonant state: 2

34. 33/34 Fano parameter Large when close : Interference between B and R : Interference between R and R

35. 34/34 Summary - Electronic conduction and resonance scattering - Definition and physics of resonant states - Particle-number conservation - Interference between resonant states

36. 35/34 Discetization of Schrödinger equation Tight-binding model