1. B. Spivak UW S. Kivelson UCLA Electronic transport in 2D micro-emulsions.

2. Schematic picture of MOSFET’s

3. The electron band structure in MOSFET’s d 2D electron gas. Metal Si oxide + + +- - - +- n -1/2 As the the parameter dn 1/2 decreases the electron-electron interaction changes from Coulomb V~1/r to dipole V~1/r 3 form. eV

4. E kin ~ n ; E pot ~n g/2 ( e-e interaction energy is V(r) ~1/r g ) The systems forms crystals at r s >> 1, when E pot >>E kin Electrons (g=1) form Wigner crystals at small n 3 He and 4 He (g  are crystals at large n Electron-electron interaction can be characterized by a parameter r s =E pot /E kin

5. Phase diagram of 2D electrons in MOSFET’s or in double layer electron system. ( n P > n A ; T=0 ) n 1/d correlated electrons. WIGNER CRYSTAL FERMI LIQUID Inverse distance between layers. In green areas where quantum effects are important. MOSFET’s important for applications.

6. a.As a function of density 2D first order phase transitions in systems with dipolar or Coulomb interaction are forbidden. b. Transitions between the liquid and the crystal should be of first order. (L.D. Landau, S. Brazovskii) There are 2D electron phases intermediate between the Fermi liquid and the Wigner crystal

7. Phase separation in the electron liquid. There is an interval of electron densities n W <n<n L near the critical n c where phase separation must occur n ncnc nWnW nLnL crystalliquid phase separated region.

8. To find the shape of the minority phase one must minimize the surface energy at a given area of the minority phase S In the case of dipolar interaction At large L the surface energy is negative!  > 0 is the microscopic surface energy

9. Single connected shapes of the minority phase are unstable. Instead there are new micro-emulsion phases. characteristic size of an elementary block in the micro-emulsions is :

10. minority phase. d x metallic gate R Elementary explanation: Finite size corrections to the capacitance R is the droplet radius This contribution to the surface energy is negative and proportional to –R ln (R/d) + +++ - --- majority phase. E~1/x

11. NR 2 =const. N is the number of the droplets Shape of the minority phase R The minority phase.

12. Mean field phase diagram Large anisotropy of surface energy. Wigner crystal Fermi liquid. Stripes (crystal conducting in one direction) n Lifshitz points L

13. Fermi liquid. Wigner crystal Stripes Bubbles of WC Bubbles of FL Mean field phase diagram Small anisotropy of surface energy n nLnL nWnW Transitions are continuous. They are similar to Lifshitz points. A sequence of more complicated patterns. A sequence of more complicated patterns.

14. A phase diagram of 2D Coulomb systems is similar : In quasi-2D case, depending on values of parameters, a devil’s staircase of transitions is possible

15. Transitions between uniform, bubble and stripe phases in 2D have been previously discussed in : a. 2D ferromagnetic films (T. Garrel, S. Doniach) b. lipid membranes ( M. Seul, D. Andelman) In these theories the transitions between the uniform, the bubble and the stripe phases were Incorrectly identified as first order. In 3D the macroscopic phase separation is impossible. only at large enough |d  W  dn-d  L /dn . d.HTS (J. Zaanen, S. Kivelson, V. Emery, E. Fradkin) e. neutron stars (Bethe) SB B n T

16. Sequence of intermediate phases at finite temperature. a. Rotationally invariant case. “crystal” “nematic” liquid n n “crystal” “smectic” liquid b. A case of preferred axis. For example, in-plane magnetic field.

17. a.As T and H || increase, the crystal fraction grows. b.At large H || the spin entropy is frozen and the crystal fraction is T- independent. T and H || dependences of the crystal’s area. (Pomeranchuk effect). The entropy of the crystal is of spin origin and much larger than the entropy of the Fermi liquid. S and M are entropy and magnetization of the system.

18. Several experimental facts suggesting non-Fermi liquid nature 2D electron liquid at small densities and the significance of the Pomeranchuk effect:

19. There is a metal-insulator transition as a function of n! Kravchenko et al Factor of order 6. metal insulator T-dependence of the resistances of Si MOSFET at large r s and at different electron concentrations.

20. T-dependence of the resistance of 2D p-GaAs layers at large r s in the “metallic” regime.

21. Pudalov et al. A factor of order 6. There is a big positive magneto-resistance which saturates at large magnetic fields parallel to the plane. B || dependences of the resistance of Si MOSFET’s at different electron concentrations.

22. Xuan et al B || dependence of 2D p-GaAs at large r s and small wall thickness. 1/3

23. B || dependence of 2D p-GaAs at large r s and small wall thickness. Xuan et al 1/3

24. M. Sarachik, S. Vitkalov B || The parallel magnetic field suppresses the temperature dependence of the resistance of the metallic phase. The slopes differ by a factor 100 !! Comparison T-dependences of the resistances of Si MOSFET’s at zero and large B ||

25. Hopping conductivity regime in MOSFET’s Magneto-resistance in the parallel and the perpendicular tmagnetic field H || Kravchenko et al (unpublished)

26. B || dependence of the resistance and drag resistance of 2D p-GaAs at different temperatures Pillarisetty et al.

27. T-dependence of the drag resistance in double layers of p-GaAs at different B || Pillarisetty et al

28. 1. The drag resistance is 2-3 orders of magnitude larger that the Fermi liquid theory prediction. 2.B || dependences of the resistances of the individual layers and the drag resistance are very similar: they increase with B || and saturates at large B || when the holes are spin polarized. 3.Its T- dependence is not Fermi liquid one: the exponent of the temperature dependence decreases with B || and saturates at large B || 4. Does the drag resistance vanish at T=0 ? Summary of experimental facts about the drag resistance at large r s

29. At T=0 and G>>1 the bubbles are not localized. G is a dimentionless cunductance. J ij i j

30. How universal this picture is? Is the drag resistance is finite at T=0 and r s << 1 ? FL WC

31. T-dependence of the resistance at “high” temperatures E F <<T<<E pot ; Semi-quantum regime If r s = E pot / E F >> 1 the liquid is strongly correlated. If E F << T << E pot the liquid is not degenerate but it is still not a gas. Such temperature interval exists both in the case of electrons with r s >>1 and in liquid He

32. In this case the electron mean free path l ee ~n 1/2 and hydrodynamics description of the electron system works ! Stokes formula in 2D case: Connection between the resistance and the electron viscosity  in the semi-quantum regime. In classical liquids  (T) decreases exponentially with T. In classical gases  increases as a power of T. What about semi-quantum liquids? u(r)

33. Comparison of two strongly correlated liquids: He 3 and the electrons at E F <T < E pot He 4 Factor 1.5 Experimental data on the viscosity of He 3 in the semi-quantum regime (T > 0.3 K) are unavailable!? A theory (A.F.Andreev):  1/T T 

34. Points where T = E F are marked by red dots. H. Noh, D.Tsui, M.P. Lilly, J.A. Simmons, L.N. Pfeifer, K.W. West. T - dependence of the conductivity  (T) in 2D p- GaAs at “high” T and at different n.

35. Quantum properties of micro-emulsion bubble phases

36. At T=0 bubble superlattices is melted by quantum fluctuations The interaction between the droplets decays as 1/r 3. Therefore at small N droplets form a quantum liquid. At small temperatures droplets are characterized by their momentum. They carry mass, charge and spin. Thus, they behave as quasiparticles. Question: What are the statistics and the effective mass of the “droplet quasi-particles”?

37. Quantum properties of droplets of Fermi liquid embedded in the Wigner crystal : a.In this case droplets are topological objects with a definite statistics b.The number of sites in the crystal is different from the number of electrons. Such crystals can bypass obstacles and cannot be pinned c. The corresponding set of equations is a combination of the elasticity theory and the hydrodynamics. This is similar to the scenario of super-solid He (A.F.Andreev and I.M.Lifshitz). The difference is that in that case the zero-point vacancies are of quantum mechanical origin.

38. a.The droplets are not topological objects. b. The action for macroscopic quantum tunneling between states with and without a Wigner crystal droplet is finite. The droplets contain non-integer spin and charge. Therefore the statistics of these quasiparticles is unknown. Quantum properties of droplets of Wigner crystal embedded in Fermi liquid. WC R(t)

39. a. If the surface is quantum smooth, a motion WC droplet corresponds to redistribution of mass of order b.If it is quantum rough, much less mass need to be redistributed. What is the effective droplet’s mass m* in MOSFET’s ? n 1/d WC FL At T=0 the liquid-solid surface is a quantum object. In Coulomb case m ~ m*

40. At large r s there are phases of pure 2D electron system which are intermediate between the Fermi liquid and the Wigner crystal. Conclusion :

41. Unsolved problems: What are quantum properties of WC-FL surface? How big are quantum fluctuations of its position? What is the spin structure of the surface? Is the number of quasi-particles in the Fermi liquid conserved? How Luttinger theorem works in the electron micro-emulsions? Quantum hydrodynamics of the micro-emulsion phases.