Igor Aleiner (Columbia)

1. Theory of Quantum Dots as Zero-dimensional Metallic Systems Igor Aleiner (Columbia) Collaborators: B.L.Altshuler (Princeton) P.W.Brouwer (Cornell) V.I.Falko (Lancaster, UK) L.I. Glazman (Minnesota) I.L. Kurland (Princeton) Physics of the Microworld Conference, Oct. 16 (2004)

2. Outline: • Quantum dot (QD) as zero dimensional metal • Random Matrix theory for transport in quantum dots • Non-interacting “standard models”. • Peculiar spin-orbit effects in QD based on 2D electron gas. • Interaction effects: • Universal interaction Hamiltonian; • Mesoscopic Stoner instability; • Coulomb blockade (strong, weak, mesoscopic); • Kondo effect.

3. Number of electrons: 1) “Artificial atom” Description requires exact diagonalization. (Kouwnehoven group (Delft)) 2) “Artificial nucleus” (Marcusgroup (Harvard)) Statistical description is allowed !!! “Quantum dot” used in two different contents: For the rest of the talk:

4. Random Matrix Theory for Transport in Quantum Dots 2DEG Energy scales 2DEG QD Level spacing L Thouless Energy Conductance Assume:

5. Statistics of transport is determined only by fundamental symmetries!!! Reviews: Beenakker, Rev. Mod. Phys. 69, 731 (1997) Alhassid, Rev. Mod. Phys. 72, 895 (2000) Aleiner, Brouwer, Glazman, Phys. Rep., 309 (2002) Original Hamiltonian: Confinement, disorder, etc RMT

6. No magnetic field, no SO Magnetic field, no SO No magnetic field, strong SO Magnetic field + SO

7. Conductance of chaotic dot Jalabert, Pichard, Beenakker (1994) Baranger, Mello (1994) Mesoscopic fluctuations classical Weak localization I V

8. Conductance of chaotic dot Mesoscopic fluctuations classical Weak localization I V Universal quantum corrections [Altshuler, Shklovskii (1986)]

9. Naively: Peculiar effect of the spin-orbit interaction SO But the spin-orbit interaction in 2D is not generic.

10. Spin-orbit interaction in GaAs/AlGaAs (001) 2DEG - spin-orbit lengths [001] Rashba term Dresselhaus term Dyakonov-Perel spin relaxation

11. Approximate symmetries of SO in QD Aleiner, Fal’ko (2001) T - invariance But Spin relaxation rate Spin dependent flux Meir, Gefen, Entin-Wohlman (1989) Mathur, Stone (1992) Khaetskii, Nazarov (2000) Lyanda-Geller, Mirlin (1994)

12. Energy scales: Brouwer, Cremers,Halperin (2002) May be violated for

13. Orthogonal, !!! But no spin degeneracy; spins mixed: New energy scale: Effect of Zeeman splitting

14. 6 possible symmetry classes:

15. 6 possible symmetry classes:

16. Orbital effect of the magnetic field

17. Orbital effect of the magnetic field Observed in Folk, Patel, Birnbaum, Marcus, Duruoz, Harris, Jr. (2001)

18. Random matrix ???? In nuclear physics: from shell model random Interaction Hamiltonian Energies smaller than Thouless energy:

19. are NOT random !!! Universal Interaction Hamiltonian Energies smaller than Thouless energy: Kurland, Aleiner, Altshuler (2000) Only invariants compatible with the circular symmetry

20. Universal Interaction Hamiltonian Energies smaller than Thouless energy: Random matrix Not random Valid if: 2) Fundamental symmetries are NOT broken at larger energies 1)

21. One-particle levels determined by Wigner – Dyson statistics Interaction with additional conservations Universal Interaction Hamiltonian Energies smaller than Thouless energy: Zero dimensional Fermi liquid Valid if: 2) Fundamental symmetries are NOT broken at larger energies 1)

22. Universal Interaction Hamiltonian Singlet electron-hole channel. Triplet electron-hole channel. Particle-particle (Cooper) channel. Analogy with soft modes in metals

23. Universal Interaction Hamiltonian Cooper Channel: Renormalization: Normal Superconducting (e.g. Al grains)

24. Universal Interaction Hamiltonian Triplet Channel: is NOT renormalized may lead to the spin of The ground state S > ½. But

25. Spin is finite even for Typical S: FM instability Stoner (1935) random with known from RMT correlation functions Mesoscopic Stoner Instability Kurland, Aleiner, Altshuler (2000) Energy of the ground state: NO randomness interactions NO vs. Does not scale with the size of the system Also Brouwer, Oreg, Halperin (2000)

26. gate voltage But Universal Interaction Hamiltonian Singlet Channel: is NOT renormalized Q: What is charge degeneracy of the ground state

27. (isolated dot) degeneracy gap - half-integer Otherwise

28. Coulomb blockade of electron transport For tunneling contacts: Charge degeneracy Charge gap Term introduced by Averin and Likharev (1986); Effect first discussed by C.J. Gorter (1951).

29. conductance (e2/h) gate voltage (mV) Small quantum dots (~ 500 nm) M. Kastner, Physics Today (1993) E.B. Foxman et al., PRB (1993) In metals first observed in Fulton, Dolan, PRL, 59, 109, (1987)

30. Random phase but not period. Coulomb blockade (CB) (II) Strong CB Weak CB Mesoscopic CB (reflectionless contacts)

31. Statistical description of strong CB: Theory: Peaks: Jalabert, Stone, Alhassid (1992); Valleys: Aleiner, Glazman (1996); Reasonable agreement, But problems with values of the correlation fields Courtesy of C.Marcus

32. Mesoscopic Coulomb Blockade Aleiner, Glazman (1998) Based on technique suggested by: Matveev (1995); Furusaki, Matveev (1995); Flensberg (1993).

33. Cronenwett et. al. (1998) Experiment: Suppression By a factor of 5.3 Th: Predicted 4.

34. Spin degeneracy in odd valleys: Effective Hamiltonian: local spin density of conduction electrons magnetic impurity Even-Odd effect due to Kondo effect Predicted: Glazman, Raikh (1988) Ng, Lee (1988)

35. D. Goldhaber-Gordon et al.(MIT-Weizmann) S.M. Cronenwett et al.(TU Delft) J. Schmid et al.(MPI @ Stuttgart) 1998 200 nm van der Wiel et al. (2000) 15mK 800mK Observation:

36. Conclusions • Random matrix is an adequate description for • the transport in quantum dots if underlying • additional symmetries are properly identified. • 2) Interaction effects are described by the • Universal Hamiltonian (“0D Fermi Liquid”)