B. Spivak, UWwith 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/rg )

Electrons (g=1) form Wigner crystals at T=0 and small nwhen rs >> 1 and Epot>>Ekin

3He and 4He (g are crystals at large n

2

2

2/

g

s

gpotkin

nr

nEnE

Electron interaction can be characterized by a parameter rs =Epot /Ekin

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

Ga1-xAlxAs

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)

SiO2

Si

2DEG

Metal “gate”

d

As the the parameter dn1/2 decreases the electron interaction changes from Coulomb V~1/r to dipole V~d2/r3 form.

MOSFET

Phase diagram of 2D electrons in MOSFET’s . ( T=0 )

n

correlated electrons.

WIGNER CRYSTAL

FERMI LIQUIDInverse 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 nW<n<nL near the critical nc where phase separation must occur

n

WL,

ncnW nL

dC

C

enCCinWLWL

1;

2

)(;

2)()()(

,,

LWcrystal liquid

phase separated region.

d

LLaL

ll

ldldadlE

SS

surf ln|'|

')(

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!

ld

> 0 is the microscopic surface energy

Coloumb case is qualitatively similar tothe dipolar case

nd

nn

ll

ldld

eldSSnnE

W

c

c

)d(

,densitiescriticalandaverageareand

phases,majority theandminority theofarea areS and S

,|'|

)(]][[

L

0

-

2

0

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

constNRd

RRdnneRNE LWsurf

2222 ;ln)(

2

12

WLLW dn

d

nne

deR

,1)(

4

,

isdropletstheofsizesticcharacteriThe

22

R

The minority phase.

Fermiliquid.

Wigner crystal

Stripes Bubblesof WC

Bubblesof FL

Mean field phase diagram of microemulsions

nnLnW

Transitions are continuous. They are similar to Lifshitz points.

A sequenceof more complicated patterns.

A sequenceof 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.

WLWLWLWL MHTSF ,||,,,

LWLW MMSS ;

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!

)2010( k

ps E

Er

Kravchenko et al

Factor of order 6.metal

insulator

T-dependence of the resistances of Si MOSFET at large rs andat different electron concentrations.

Kravchenko et al Gao at al, Cond.mat 0308003

T-dependence of the resistance of 2D electrons at large rs in the “metallic” regime (G>>e2/ h)

p-GaAs, p=1.3 10 cm-2 ; rs=30

Si MOSFET

Cond-mat/0501686

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 rs 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 !!

KEF 13

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

Tsui et al. cond-mat/0406566

G=70 e2/h

Gao et al

The slope of the resistance dR/dT is dramaticallysuppressed 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>> e2/h?

Why is the magneto-resistance positive at all?

Why does B|| so effectively quench the T dependenceof the resistance?

The electron mean free path lee ~n1/2 and hydrodynamics description of the electron system works !

Stokes formula in 2D case:

Connection between the resistance and the electronviscosity in the semi-quantum regime.

)/1ln()()(

222 aNne

NTT

i

i

In classical liquids (T) decreases exponentially with T. In classical gases increases as a power of T.

What about semi-quantum liquids?

u(r)

)/ln( nau

uF

a

If rs >> 1 the liquid is strongly correlated

is the plasma frequency

If EF << T << h<< 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 ofelectrons with rs >>1 and in liquid He

pots

potF E

r

EE

2/1

T <<U

U

Semi-quantum liquid: EF << T << h << U: (A.F. Andreev)

~ 1/T

Viscosoty of classical liquids (Tc , hD << T<< U) decreases exponentially with T (Ya. Frenkel) ~ exp(B/T)

h

Viscosity of gases (T>>U) increases as T increases

Comparison of two strongly correlated liquids: He3 and the electrons at EF <T < Epot

He4

Experimental data on the viscosity of He3 in the semi-quantum regime (T > 0.3 K) are unavailable!?

T

1A theory (A.F.Andreev):

h

1/T

Points where T = EF 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>EF 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

A

PD I

V

Pillarisetty et al.PRL. 90, 226801 (2003)

0.1 11

10

100

1000

0 2 4 6 8 10 12 141.2

1.5

1.8

2.1

2.4

2.7

B*

Exp

onen

t

B|| (T)

0.80.60.40.30.2

D (/

)

T (K)

TD

T-dependence of the drag resistance in double layers of p-GaAs at different B||

Pillarisetty et al.PRL. 90, 226801 (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 dependenceof 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 Epot 1/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.

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 crystalembedded 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.

2* Rmnm c

2* )( Rnnmm WL

effective droplet’s mass m*

n

1/d

WC FL

At T=0 the liquid-solid surface is a quantum object.

mmnd *2 ;1

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

Bubblemicroemulsion

Is the WC bubble phase ferromagnetic at the Lifshitz point ??

At T=0 and G>>1 the bubbles are not localized.

.11

;||

1;**

G

rrJaaJaaH

jiij

ijjiijii

ii

G is a dimensionless conductance.

Jij

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 largesusceptibility in what would have been the broken symmetry state - see, e.g., quantum Hall nematic stateof Eisenstein et al.)

Pomeranchuk effect is “local” and so robust: Since is an increasing function of fWC, it is an increasing function of T and B||, with scale of B|| set by T.

Vitkalov at allnc1 nc2

nc1 is the critical density at H=0; while nc2 is the critical density at H>H*.

The ratio is big even deep in metallic regime!

0

400

800

1200

1600

2000

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80

10

20

30

40

50

60

70

80

[O

hm/s

quar

e]

B|| [Tesla]

0 0.5 1 1.5 2 3 4

GaAs 2D hole system, 10nm wide quantum well, density 2.06*1010/cm2

resistivity vs. temperature

[e

2/h

]T [K]

conductivity vs. temperature

Xuan et al

Additional evidence for the strongly correlated nature of the electron system.

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.”

B. Castaing, P. NozieresJ. De Physique, 40, 257, 1979.(Theory of liquid 3He .)

Vitkalov et al

ijeff lnln

kij

kij

kkij

kij

kijkl

lij

kijk

kijk

kijij

HS

iAAAAAAij

;)exp(||;|||| *22

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.

kijS

kijH

ij is independent of H ; while all higher moments decrease with H. mk

ij )(

ijHH

rLL

;

Here is the localization radius, LH is the magnetic length, and rij 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 areorthogonal and the orbital mechanism of the magneto-resistance is suppressed.

2|| m l

lijij

mA

Index “lm” labes tunneling paths which correspond to the same final spinConfiguration, the index “m” labels different groups of these paths.

i

i

i jj

j

The Pomeranchuk effect.

The temperature dependence ofthe heat capacity of He3. The semi-quantum regime.

The Fermi liquid regime.

He3 phase diagram:

The liquid He3 is also stronglycorrelated liquid: rs; 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 phasesat finite temperature.

a. Rotationally invariant case.

“crystal” “nematic” liquidn

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

+

++ -

--

+ -

dCC

C

enCCinWLWL

12

)(;

0

2)()()(

,,

n-1/2

As the the parameter dn1/2 decreasesthe electron-electron interaction changes from Coulomb V~1/r to dipole V~1/r3 form.

eV

d

RR

d

RC

16ln

2

Elementary explanation: Finite size corrections to the capacitance

d

RRendRen

C

enR

C

QEC ln)()(

2

)(

2222

222

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 rs and small wall thickness.

Gao et al

1/3

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

Cond.mat0308003

P=1.3 1010 cm-2 ; rs=30

Mean field phase diagram Large anisotropy of surface energy.

Wigner crystal

Fermiliquid.

Stripes (crystal conducting inone direction)

n

Lifshitz points

L

)( L

G=70 e2/h

The slope of the resistance as a function of T is dramaticallysuppressed 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: Epot ~A/r x ;

1<x<2; n=nc

xsurf

xsurf

RA

RE

RnARRnE

2

422 )(

If x ≥ 1 the micro-emulsion phases exist independently of the value of the surface tension.