B. Spivak, UW S. Kivelson, Stanford, S. Sondhi, D. Huse, C. Lamman, Princeton

2D microemulsion phases in Bose and Fermi systems with dipolar interaction

( interaction energy is V(r) ~1/rg )

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

22

2/

g

s

gpotkin

nr

nEnE

interaction energy 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 quantum phases intermediate between the liquid and the crystal (micro-emulsion phases)

Phase separation

There is an interval of densities nW<n<nL near the critical nc where phase separation must occur

n

WL,

ncnW nL

dC

CenCCin

WLWL1;

2)(;

2)()()(

,,

LWcrystal liquid

phase separated region.

dRR

dRC 16ln

2

Elementary explanation: Finite size corrections to the capacitance

dRRendRen

CenR

CQEC 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)

Part of the bulk energy

dLLaL

llldldadlE

SSsurf 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

liquid. crystal Stripes Bubblesof C

Bubblesof L

Mean field phase diagram in the case of isotropic surface tension

n

Transitions are continuous. They are similar to Lifshitz points.

A sequenceof more complicated patterns.

A sequenceof more complicated patterns.

..)exp(

)(/

||)|(|

ˆ

;ˆ)(),(

0

2*2*

422*

1

3

422

ccrki

kctQck

unntkkcnF

rVnnrQV

nnVnnnnucnF

ii

kck

cr

Effective free energy in the case of the first order phase transitionwhich is close to the second order.

1k 2k

3k

liquid. cristal stripes

Mean field phase diagram

n

hexagonshexagons

01cos)(

2/31

10

twhenta

liquid. cristal stripes

Mean field phase diagram of microemulsions

n

Transitions are continuous. They are similar to Lifshitz points.

stripesof h

hexagonshexagons

liquid.hexagons

zero temperature hydrodynamicsof all mictoemulcion phases is equavalent to supersolid hydrodynamics

finite temperature hydrodynamics of the stripephases is equivalent to super-smectic hydrodynamics

Coloumb case is qualitatively similar tothe dipolar case (Jamei, Kivelson, Spivak)

nd

nn

llldld

eldSSnnE

W

c

c

)d(

,densitiescriticalandaverageareandphases,majority theandminority theofarea areS and S

,|'|

)(]][[

L

0

-

2

0

The electron band structure in MOSFET’s

d

2D electron gas.Metal Si

oxide

+++ -

--

+ -

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

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

Aadd B100038*

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.

12 nd

Experiments by V. Pudalov, S. Kravchenko, M. Sarachik, et al.

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

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 !

Do materials exist where the resistance has dielectric values R>>h/e2 and yet still increases as the temperature increases ?

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?

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*

Expo

nent

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 ?

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.

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

Conclusion: