B. Spivak, UW with S. Kivelson, Stanford

1. 2D electronic phases intermediate between the Fermi liquid and the Wigner crystal (electronic micro-emulsions) B. Spivak, UW with S. Kivelson, Stanford

2. Electron interaction can be characterized by a parameter rs =Epot /Ekin ( e-e interaction energy is V(r) ~1/rg ) Electrons (g=1) form Wigner crystals at T=0 and small n when rs >> 1 and Epot>>Ekin 3He and 4He(g>2 ) are crystals at large n

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

4. Experimental realizations of the 2DEG

5. Ga1-xAlxAs 2DEG w GaAs Hetero-junction Electrons interact via Coulomb interaction V(r) ~ 1/r

6. Schematic picture of a band structure in MOSFET’s (metal-oxide-semiconductor field effect transistor)

7. Metal “gate” d SiO2 2DEG Si MOSFET As the the parameter dn1/2 decreases the electron interaction changes from Coulomb V~1/r to dipole V~d2/r3 form.

8. Phase diagram of 2D electrons in MOSFET’s . ( T=0 ) Inverse distance to the gate 1/d MOSFET’s important for applications. FERMI LIQUID correlated electrons. WIGNER CRYSTAL n Microemulsion phases. In green areas where quantum effects are important.

9. Phase separation in the electron liquid. crystal liquid phase separated region. n nW nc nL There is an interval of electron densities nW<n<nL near the critical ncwhere phase separation must occur

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

11. Coloumb case is qualitatively similar to the dipolar case

12. At large area of a minority phase the surface energy is negative. Single connected shapes of the minority phase are unstable. Instead there are new electron micro-emulsion phases.

13. Shape of the minority phase The minority phase. R N is the number of the droplets

14. Mean field phase diagram of microemulsions A sequence of more complicated patterns. A sequence of more complicated patterns. Wigner crystal Fermi liquid. Bubbles of WC Bubbles of FL Stripes n nW nL Transitions are continuous. They are similar to Lifshitz points.

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

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

17. Experiments on the temperature and the parallel magnetic field dependences of the resistance of single electronic layers.

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

19. T-dependence of the resistance of 2D electrons at large rs in the “metallic” regime (G>>e2/ h) Gao at al, Cond.mat 0308003 Kravchenko et al p-GaAs, p=1.3 10 cm-2 ; rs=30 Si MOSFET

20. Cond-mat/0501686

21. B|| dependences of the resistance of Si MOSFET’s at different electron concentrations. 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.

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

23. Comparison T-dependences of the resistances of Si MOSFET’s at zero and large B|| 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 !!

24. G=70 e2/h Tsui et al. cond-mat/0406566 Gao et al The slope of the resistance dR/dT is dramatically suppressed by the parallel magnetic field. It changes the sign. Overall change of the modulus is more than factor 100 in Si MOSFET and a factor 10 in P-GaAs !

25. If it is all business as usual: Why is there an apparent metal-insulator transition? Why is there such strong T and B|| dependence at low T, even in “metallic” samples with G>> e2/h? Why is the magneto-resistance positive at all? Why does B|| so effectively quench the T dependence of the resistance?

26. Connection between the resistance and the electron viscosity h(T) in the semi-quantum regime. The electron mean free path lee ~n1/2 and hydrodynamics description of the electron system works ! Stokes formula in 2D case: u(r) a In classical liquids h(T)decreases exponentially with T. In classical gases h(T)increases as a power of T. What about semi-quantum liquids?

27. If rs >> 1 the liquid is strongly correlated • is the plasma frequency • If EF << T << hq << Epot the liquid is not degenerate • but it is still not a gas ! It is also not a classical liquid ! • Such temperature interval exists both in the case of • electrons with rs >>1 and in liquid He

28. Viscosity of gases (T>>U) increases as T increases Viscosoty of classical liquids (Tc , hQD << T<< U) decreases exponentially with T (Ya. Frenkel) h ~ exp(B/T) Semi-quantum liquid: EF << T << h q << U: (A.F. Andreev) h ~ 1/T U hq T <<U

29. Comparison of two strongly correlated liquids: He3 and the electrons at EF <T < Epot h He4 1/T Experimental data on the viscosity of He3 in the semi-quantum regime (T > 0.3 K) are unavailable!? A theory (A.F.Andreev):

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

31. Experiments on the drag resistance of the double p-GaAs layers.

32. B|| dependence of the resistance and drag resistance of 2D p-GaAs at different temperatures Pillarisetty et al. PRL. 90, 226801 (2003)

33. T-dependence of the drag resistance in double layers of p-GaAs at different B|| Pillarisetty et al. PRL. 90, 226801 (2003)

34. If it is all business as usual: Why the drag resistance is 2-3 orders of magnitude larger than those expected from the Fermi liquid theory? Why is there such a strong T and B|| dependence of the drag? Why is the drag magneto-resistance positive at all? Why does B|| so effectively quench the T dependence of drag resistance? Why B|| dependences of the resistances of the individual layers and the drag resistance are very similar An open question: Does the drag resistance vanish at T=0?

35. Quantum aspects of the theory of micro-emulsion electronic phases At large distances the inter-bubble interaction decays as Epot1/r3 >> Ekin Therefore at small N (near the Lifshitz points) and the superlattice of droplets melts and they form a quantum liquid. The droplets are characterized by their momentum. They carry mass, charge and spin. Thus, they behave as quasiparticles.

36. Questions: What is the effective mass of the bubbles? What are their statistics? Is the surface between the crystal and the liquid a quantum object? Are bubbles localized by disorder?

37. Properties of “quantum melted” droplets of Fermi liquid embedded in the Wigner crystal : • Droplets are topological objects with a definite statistics • The number of sites in such a crystal and the number of electronsare different . • Such crystals can bypass obstacles and cannot be pinned • 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. Quantum properties of droplets of Wigner crystal embedded in Fermi liquid. • 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 and the properties of the ground state are unknown.

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

40. Conclusion: There are pure 2D electron phases which are intermediate between the Fermi liquid and the Wigner crystal .

41. Conclusion #2 (Unsolved problems): 1. Quantum hydrodynamics of the micro-emulsion phases. 2. Quantum properties of WC-FL surface. Is it quantum smooth or quantum rough? Can it move at T=0 ? 3. What are properties of the microemulsion phases in the presence of disorder? 4. What is the role of electron interference effects in 2D microemulsions? 5. Is there a metal-insulator transition in this systems? Does the quantum criticality competes with the single particle interference effects ?

42. Conclusion # 3: Are bubble microemulson phases related to recently Observed ferromagnetism in quasi-1D GaAs electronic channels ( cond-mat ……….. ) ? Wigner crystal Bubble microemulsion Is the WC bubble phase ferromagnetic at the Lifshitz point ??

43. At T=0 and G>>1 the bubbles are not localized. G is a dimensionless conductance. Jij j i

44. The drag resistance is finite at T=0 WC FL

45. Disorder is a relevant perturbation in d < 4! No macroscopic symmetry breaking (However, in clean samples, there survives a large susceptibility in what would have been the broken symmetry state - see, e.g., quantum Hall nematic state of Eisenstein et al.) What about quenched disorder? Pomeranchuk effect is “local” and so robust: Since r is an increasing function of fWC, it is an increasing function of T and B||, with scale of B|| set by T. (1T ~ 1K)

46. The ratio is big even deep in metallic regime! Vitkalov at all nc1 nc2 nc1 is the critical density at H=0; while nc2 is the critical density at H>H*.

47. Fig.1 Xuan et al

48. Additional evidence for the strongly correlated nature of the electron system. Vitkalov et al B. Castaing, P. Nozieres J. De Physique, 40, 257, 1979. (Theory of liquid 3He .) E(M)=E0+aM2+bM4+…….. M is the spin magnetization. “If the liquid is nearly ferromagnetic, than the coefficient “a” is accidentally very small, but higher terms “b…” may be large. If the liquid is nearly solid, then all coefficients “a,b…” as well as the critical magnetic field should be small.”

49. Orbital magneto-resistance in the hopping regime. (V.L Nguen, B.Spivak, B.Shklovski.) To get the effective conductivity of the system one has to average the log of the elementary conductance of the Miller-Abrahams network : A. The case of complete spin polarization. All amplitudes of tunneling along different tunneling paths are coherent. is independent of H ; while all higher moments decrease with H. The phases are random quantities. j i The magneto-resistance is big, negative,and corresponds to magnetic field corrections to the localization radius. Here is the localization radius, LH is the magnetic length, and rij is the typical hopping length.

50. B. The case when directions of spins of localized electrons are random. j i i j j i Index “lm” labes tunneling paths which correspond to the same final spin Configuration, the index “m” labels different groups of these paths. In the case of large tunneling length “r” the majority of the tunneling amplitudes are orthogonal and the orbital mechanism of the magneto-resistance is suppressed.