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

( e-e interaction energy is V(r) ~1/r g ) Electrons (g=1) form Wigner crystals at T=0 and small n when r s >> 1 and E pot >>E kin 3 He and 4 He (g  are crystals at large n Electron interaction can be characterized by a parameter r s =E pot /E kin

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)

Experimental realizations of the 2DEG

Ga 1-x Al x As GaAs 2DEG w Electrons interact via Coulomb interaction V(r) ~ 1/r Hetero-junction

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

SiO 2 Si 2DEG Metal “gate” d As the the parameter dn 1/2 decreases the electron interaction changes from Coulomb V~1/r to dipole V~d 2 /r 3 form. MOSFET

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

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

Coloumb case is qualitatively similar to the dipolar case

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.

N is the number 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 microemulsions n nLnL nWnW Transitions are continuous. They are similar to Lifshitz points. A sequence of more complicated patterns. A sequence of more complicated patterns.

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.

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

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

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.

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

Cond-mat/

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.

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

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 ||

Tsui et al. cond-mat/ G=70 e 2 /h 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 !

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>> e 2 /h? Why is the magneto-resistance positive at all? Why does B || so effectively quench the T dependence of the resistance?

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) a

If r s >> 1 the liquid is strongly correlated  is the plasma frequency If E F << T << h  << E pot 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 r s >>1 and in liquid He

T <<U U Semi-quantum liquid: E F << T << h  << U: (A.F. Andreev)  ~ 1/T Viscosoty of classical liquids (T c, h  D << T<< U) decreases exponentially with T (Ya. Frenkel)  ~ exp(B/T) hh Viscosity of gases (T>>U) increases as T increases

Comparison of two strongly correlated liquids: He 3 and the electrons at E F <T < E pot He 4 Experimental data on the viscosity of He 3 in the semi-quantum regime (T > 0.3 K) are unavailable!? A theory (A.F.Andreev): h 1/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) in 2D p- GaAs at “high” T>E F and at different n.

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

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

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

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?

At large distances the inter-bubble interaction decays as E pot 1/r 3 >> E kin 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. Quantum aspects of the theory of micro-emulsion electronic phases

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?

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 electrons are 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.

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 and the properties of the ground state are unknown. Quantum properties of droplets of Wigner crystal embedded in Fermi liquid.

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. effective droplet’s mass m* n 1/d WC FL At T=0 the liquid-solid surface is a quantum object. In Coulomb case m ~ m*

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

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 ?

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 ??

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

The drag resistance is finite at T=0 FL WC

What about quenched disorder? 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.) Pomeranchuk effect is “local” and so robust: Since  is an increasing function of f WC, it is an increasing function of T and B ||, with scale of B || set by T. 

Vitkalov at all n c1 n c2 n c1 is the critical density at H=0; while n c2 is the critical density at H>H*. The ratio is big even deep in metallic regime!

Fig.1 Xuan et al

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

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

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.

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

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.

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

Elementary explanation: Finite size corrections to the capacitance R is the droplet radius This contribution to the surface energy is due to a finite size correction to the capacitance of the capacitor. It is negative and is proportional to –R ln (R/d)

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

T-dependence of the resistance of 2D p-GaAs layers at large r s in the “metallic” regime. Cond.mat P= cm -2 ; r s =30

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

G=70 e 2 /h The slope of the resistance as a function of T is dramatically suppressed by the parallel magnetic field. It changes the sign. Overall change of the modulus is more than factor 10 !

M. Sarachik, S. Vitkalov

More general case: E pot ~A/r x ; 1<x<2; n=n c If x ≥ 1 the micro-emulsion phases exist independently of the value of the surface tension.