B. SpivakUniversity 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

+

++ -

--

+ -

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

If the inter-particle interaction energy is V(r) ~1/rg, then Epot ~ng/2 while Ekin~ n. (n~1/r2)

g=2 is critical.

At small n the electron kinetic energy (gcan be neglected and the system forms a Wigner crystal.

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

2

2

g

kin

pot nE

E

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

Bad 38

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

n

WL,

ncnW nL

dnn

dC

C

enWL

CCinWLWL

1;

1;

2

)(;

2)()()(

,,

LWcrystal liquid

phase separated region.

d

LLaL

ll

ldldadlE

SS

surf ln|'|

'

To find the shape of the minority phase one mustminimize 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

d

R

RdC

16ln

11

minority phase.

d

x

metallic gate

R

Finite size corrections to C

.2

)(;2

2)(2

C

enRRE in

tot R is the droplet radius

This contribution to the surface energy is negative andproportional to –R ln (R/d) !

+ + + +

- - - -

majority phase.E~1/x

NR2 =const. N is the number of the droplets,R is the size of the droplets.

Shape of the minority phase

d

RRdnneRNE LWsurf ln)(

2

12 222

.1)(

4,

22

LW nnedeR

R

The minority phase.

Fermiliquid.

Wigner crystal

Stripes Bubblesof WC

Bubblesof FL

Mean field phase diagram of electron system at T=0. (Small anisotropyof surface energy).

nnLnW

Transitions are continuous.

They are similar to Lifshitz points.

A sequenceof more complicated patterns.

A sequenceof more complicated patterns.

Mean field phase diagram of electron system at T=0. (Big anisotropyof surface energy). An example: the magnetic field parallel to the film.

Wigner crystal

Fermiliquid.

Stripes

n

Lifshitz points

L

)( 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

2

2

lnd

LTd c

Transitions between uniform, bubble and stripephases 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 |dWdn-dL/dn.

d. neutron stars (Bethe)e. HTS (J. Zaanen, S. Kivelson, V. Emery, E. Fradkin)

S BB 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.

WLWLWLWL MHTSF ,||,,,

LWLW MMSS ;

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!

)2010( k

ps E

Er

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 !!

KEF 13

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 EF<<T<<Epot.

At the Fermi liquid-Wigner crystal transition the critical rs= Epot / EF ~ 38.and the liquid is strongly correlated.

If EF << T << Epot the liquid is not degenerate but it is still not a gas. (Semi-quantum regime).

Such temperature interval exists in the case of liquidHe as well.

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.

In this case lee ~n1/2, and hydrodynamics works ! (lee 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.

)/1ln(

)()(

)/ln(

222 aNne

TNT

nau

uF

i

i

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(He3 and the electrons in the semi-quantum regime ( EF <T < Epot ).

He4

Factor 1.5

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

T

1A theory (A.F.Andreev):

1/TT

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) 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)=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

At T=0 bubble superlattices melted by quantum fluctuations

The interaction between the droplets decays as 1/r3. 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 crystalembedded in Fermi liquid.

tvtRdtttR

dtC

trenrdtdS F )(;

1

)(

1)),((22

0

2

WCR(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).)

)( 22* dnmRmnm cc

2/122* )()( dnmRnnmm cWL

What is the effective mass of the droplets m* ?

n

1/d

WC FL

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

mmnd *2 ;1

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

kij

Hij is independent

of H ; while all higher moments decrease with H. mkij )(

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

At large rs there are phases of pure 2D electronsystem which are intermediate between the Fermi liquid and the Wigner crystal .

Conclusion:

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)

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.