B. Spivak University of Washington Phase separation in strongly correlated 2d electron liquid

Schematic picture of MOSFET’s

Electrons in semiconductors can have metallic conductivity. 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

If the inter-particle interaction energy is V(r) ~1/r g, then E pot ~n g/2 while E kin ~ n. (n~1/r 2 ) g=2 is critical. At small n the electron kinetic energy (g  can be neglected and the system forms a Wigner crystal. ( 3 He and 4 He (g  are crystals at large n.) The role of the electron-electron interaction.

Phase diagram of 2D electron system at T=0. n 1/d correlated electrons. MOSFET’s important for applications. WIGNER CRYSTAL FERMI LIQUID Experiments by V. Pudalov, T. Klapwijk, S. Kravchenko, M. Sarachik, S. Vitkalov et al. The inverse distance to the gate.

a.Transitions between the liquid and the crystal should be of first order. (L.D. Landau, S. Brazovskii) b.First order phase transitions in 2D systems with dipolar interaction are forbidden.

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.

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

minority phase. d x metallic gate R Finite size corrections to C 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

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

Fermi liquid. Wigner crystal Stripes Bubbles of WC Bubbles of FL Mean field phase diagram of electron system at T=0. (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.

Mean field phase diagram of electron system at T=0. (Big anisotropy of surface energy). An example: the magnetic field parallel to the film. Wigner crystal Fermi liquid. Stripes n Lifshitz points L

Finite temperature effects. a. The case of the magnetic field parallel to the film. n “crystal” “cmectic” liquid b. The case of strong anisotropy of the surface tension. “crystal” “nematic” liquid

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 of first order. In 3D the macroscopic phase separation is impossible. only at large enough |d  W  dn-d  L /dn . d. neutron stars (Bethe) e. HTS (J. Zaanen, S. Kivelson, V. Emery, E. Fradkin) SB B n T

a.As T and H || increase, the crystal fraction grows and the resistance of the system increases. b.At large H || the spin entropy is frozen and the resistance should be independent of T. T and H || dependences of the area of the crystal. (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.

Several experimental facts suggesting a non-Fermi liquid nature of the electron liquid in MOSFET’s at small densities and the significance of the Pomeranchuk effect:

There is a metal-insulator transition as a function of n! Kravchenko et al Factor of order 6. metal insulator

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.

Vitkalov et al. (unpublished) H || The magnetic field parallel to the plane suppresses the temperature dependence of the resistance of the metallic phase. The slopes differ by a factor 100 !!

Comparison of the magneto-resistance in the hopping regime in the cases when the magnetic field is parallel or perpendicular to the film. H || Kravchenko et al (unpublished)

A comparison of conductivity of metals and viscosity of liquid He in semi-quantum regime E F <<T<<E pot. At the Fermi liquid-Wigner crystal transition the critical r s = E pot / E F ~ 38. and the liquid is strongly correlated. If E F << T << E pot the liquid is not degenerate but it is still not a gas. (Semi-quantum regime). Such temperature interval exists in the case of liquid He as well.

The Pomeranchuk effect. The temperature dependence of the heat capacity of He 3. The semi-quantum regime. The Fermi liquid regime. He 3 phase diagram: The liquid He 3 is also strongly correlated liquid: r s ; m*/m >>1.

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

Comparison of viscosities of two strongly correlated liquids (He 3 and the electrons in the semi-quantum regime ( 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 

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) of 2D holes in GaAs at “high” T and at different n.

Additional evidence for the strongly correlated nature of the electron system. E(M)=E 0 +aM 2 +bM 4 +…….. 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.” B. Castaing, P. Nozieres J. De Physique, 40, 257, (Theory of liquid 3 He.) Vitkalov et al

At T=0 bubble superlattices 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 quasiparticles”?

Quantum properties of droplets of Fermi liquid embedded in the Wigner crystal : a. In this case droplets are topological objects. The droplets have DEFINITE statistics. It is determined by the number of electrons which should be changed in the system to create droplets. 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.

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)

a.If the surface is quantum smooth, motion of a Wigner crystal droplet is associated with a redistribution of the liquid mass of order b. If the surface is quantum rough much less mass need be redistributed. (A.F.Andreev, A.I. Shalnikov (unpublished).) What is the effective mass of the droplets m* ? n 1/d WCFL At T=0 the liquid-solid surface is a quantum object.

The magneto-resistance is big, negative,and corresponds to magnetic field corrections to the localization radius. 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. The phases are random quantities. is independent of H ; while all higher moments decrease with H. Here is the localization radius, L H is the magnetic length, and r ij is the typical hopping length. j i

B. The case when directions of spins of localized electrons are random. 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. Index “l m ” labes tunneling paths which correspond to the same final spin Configuration, the index “m” labels different groups of these paths. i i i jj j

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

T-dependence of the drag resistance of 2D GaAs holes. Tsui at al.

H-depemdence of the resistance and drag resistance of 2D GaAs. Tsui at al.