1. Quantum conductance I.A. Shelykh St. Petersburg State Polytechnical University, St. Petersburg, Russia International Center for Condensed Matter Physics, Brasilia, Brazil ICCMP

2. Outline Overwiew of the classical results Quantum Point Contacts 1D Ballistic Conductance and Landauer Buttiker formula Quantum interference and Aharonov-Bohm effect Integer and Fractional Quantum Hall effect “0.7 anomaly” and fractional quantization of ballistic conductance

3. Classical results Ohm Law L W Parallel G1G1 G2G2 G -1 =G 1 -1 +G 2 -1 Consequent G1G1 G2G2 G=G 1 +G 2

4. Quantum Point Contacts Let us consider a very small object (QPC or QWire) L<<L free, W~k F -1 The condition L<<L free means that there is no inelastic scattering within the region of the QPC Is G=∞ then?

5. Contact resistance The condition W~k F -1 means that in the region of QPC the motion in x- direction is quantized The origin of the resistance: redistribution of the current among the current-carrrrying modes at the interfaces For parabolic confinement μ E0E0 Left lead Right lead QPC

6. Ballistic conductance R. Landauer. IBM J. Res. Dev., 1, 233 (1957) μ1μ1 μ2μ2 V ds =0V ds >0 I T=0

7. Ballistic conductance staircase B. J. van Wees, Phys. Rev. Lett. 60, 848-850 (1988) D. A. Wharam et al, J. Phys. C 21 L209-L214 (1988) If there are N open subbands

8. The role of backscattering GiGi GcGc

9. Several scatterers ?

10. Effects of quantum interference Quantum interference term Fabry-Perot oscillations of quantum conductance N.T. Bagraev et al, Semiconductors, 34, 817 (2000) L<<L φ

11. Parallel connection No interference: N=N 1 +N 2, G=G 1 +G 2 To account for the round trips: scattering matrix With interference S=S=

12. Aharonov-Bohm effect How one can easily change phaseshift between the electrons propagating in the quantum ring? Possible way: apply a magnetic flux through the ring Φ Weak backscattering: AB oscillations Strong backscattering: AAS half-period oscillations

13. Classical Hall effect UHUH y x

14. Experimental configuration I W LV1V1 V2V2 V3V3 ρ xy ρ xx B

15. Landau quantization DOS Group velocity

16. Edge states x y v g ≠0

17. Ballistic conductance and QHE I+I+ I-I- Δμ=eV H μ Backscattering is supressed

18. Quantum Hall Effect (QHE) K. v. Klitzing, G. Dorda, and M. Pepper Phys. Rev. Lett. 45, 494-497 (1980) Classical resultIn the experiment

19. Fractional QHE D.C. Tsui et al, PRL 48, 1559 (1982)H.L. Stormer et al, PRL 50, 1953 (1982)

20. Interpretation of FQHE Laughlin wavefunction Composite fermions

21. Fractional quantization of the ballistic conductance (« 0.7 anomaly » K.J. Thomas et al, PRL 77, 135 (1996) Related with spin!

22. Singlet and Triplet Scattering V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869 (2000) T. Rejec et al, J. Phys. Cond. Matt. 12, L233 (2000) Localised and propagating electrons interact in the region of the QPC Eigenstates: singlet and triplet configurations. The probabilities of realization:

23. Singlet and Triplet Scattering V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869 (2000) T. Rejec et al, J. Phys. Cond. Matt. 12, L233 (2000) Localised and propagating electrons interact in the region of the QPC Eigenstates: singlet and triplet configurations. The probabilities of realization:

24. Singlet and Triplet Scattering V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869 (2000) T. Rejec et al, J. Phys. Cond. Matt. 12, L233 (2000) Localised and propagating electrons interact in the region of the QPC Eigenstates: singlet and triplet configurations. The probabilities of realization:

25. 0.75 structure: calculation Consider the case

26. Is fractional ballistic conductance universal? D.J. Reilly et al, PRB 63, R121311 (2001) For short constriction For long wire ?

27. Supposing the contact containing a total spin J : QPC with Large Spin I.A. Shelykh et al, PRB 74, 085322 (2005)

28. Fractional quantization: calculation The HamiltonianUsing the following basis One represents H in a block-diagonal form Diagonalised Hamiltonian reads

29. With increase of the length of the wire J increases and conductance decreases- as in experiment!

30. Spontaneous polarization of 1D electron gas Chuan-Kui Wang, K.-F. Berggren PRB 57, 4552 (1998) N.T. Bagraev et al PRB 70, 155315 (2004) Why big J can appear in long quantum wires? Due to exchange interaction! Qualitatively in 1D Dominant for high density Dominant for low density Calculation gives: 2 for unpolarized 1 for polarized Critical density

31. What happens with holes? Light and Heavy Hole Bands in a QPC Bands splitted in energy depending on the width of the QPC: Si / GaAs / Ge

32. Spin Dependent Scattering for Holes Initial state: Conductance at T = 0 (44 transmission amplitudes) :

33. Model: Matrix form (16x16): where

34. Physical Origin of the Plateaus States presenting total spin S T = 3: 7 states; S T = 1: 3 states; S T = 2: 5 states; S T = 0: 1 states. Potential Barriers Ferromagnetic Interaction Steps at: Antiferromagnetic Interaction Steps at:

35. Ferromagnetic Si

36. Antiferromagnetic Si Antiferromagnetic

37. Applying an Axial Magnetic Field Si Ferromagnetic

38. Experiment for the holes L.P. Rokhinson et al, 2006N.T. Bagraev et al 2002 Klochan et al, 2006 ????

39. Thank you for your attention Obrigado por a sua atenção Спасибо за внимание