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2D electronic phases intermediate between
the Fermi liquid and the Wigner crystal
(electronic micro-emulsions)
B. Spivak, UW
with S. Kivelson, Stanford
Electron interaction can be characterized by a parameter
rs =Epot /Ekin
Ekin  n
rs  n
E pot  n
g /2
g 2
2
( e-e interaction energy is V(r) ~1/rg )
Electrons (g=1) form Wigner crystals at T=0 and small n
when rs >> 1 and Epot>>Ekin
3He
and 4He (g>2 ) are crystals at large n
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
Hetero-junction
Ga1-xAlxAs
2DEG
w
GaAs
Electrons interact via Coulomb interaction
V(r) ~ 1/r
Schematic picture of a band structure in MOSFET’s
(metal-oxide-semiconductor field effect transistor)
MOSFET
Metal “gate”
SiO2
d
2DEG
Si
As the the parameter dn1/2 decreases the electron interaction
changes from Coulomb V~1/r to dipole V~d2/r3 form.
Phase diagram of 2D electrons in MOSFET’s . ( T=0 )
Inverse distance
to the gate 1/d
FERMI LIQUID
MOSFET’s
important
for applications.
correlated
electrons.
WIGNER CRYSTAL
n
Microemulsion phases.
In green areas where quantum effects are important.
Phase separation in the electron liquid.
 L,W
W
crystal
L
liquid
phase separated region.
nW
 L,W  
( in )
L ,W

nc
(C )
; 
n
nL
(C )
(en )2

;
2C
1
C
d
There is an interval of electron densities nW<n<nL near
the critical nc where phase separation must occur
To find the shape of the minority phase one must minimize
the surface energy at a given area of the minority phase
In the case of dipolar interaction
Esurf
 
dl dl '
L
   ( ) dl  a     L  a L ln
d
S
S | l  l '|
> 0 is the microscopic surface energy
S

dl
At large L the surface energy is negative!
Coloumb case is qualitatively similar to
the dipolar case
 


dl dl
  ,
E  [n0  nc ][ S  S ]   dl  ( ) 

e | l  l '|
2
S and S- are area of the minority and the majority phases,
n0 and nc are average and critical densities ,
d( L  W )

dn
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.
Shape of the minority phase
The minority phase.
R
Esurf
1 2
R

2 2
 N 2R  e ( nW  nL ) d R ln ;
2
d

N is the number of the droplets
NR2  const
The characteri stic size of the droplets is

R  de ,
4
d
 L  W 
 2
 1,  
2
e ( nW  nL )
dn
Mean field phase diagram of microemulsions
Wigner Bubbles
crystal of FL
A sequence
of more
complicated
patterns.
Stripes
A sequence
of more
complicated Bubbles Fermi
patterns.
liquid.
of WC
nW
nL
Transitions are continuous.
They are similar to Lifshitz points.
n
T and H|| dependences of the crystal’s area.
(Pomeranchuk effect).
FL ,W   L ,W  TS L ,W  H || M L ,W
S and M are entropy and magnetization of the system.
The entropy of the crystal is of spin origin and much
larger than the entropy of the Fermi liquid.
SW >> S L ;
M W >> M L
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.
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.
T-dependence of the resistances of Si MOSFET at large rs and
at different electron concentrations.
Kravchenko et al
rs 
Ep
Ek
 (10  20)
insulator
metal
Factor of order 6.
There is a metal-insulator transition as a function of n!
T-dependence of the resistance of 2D electrons at large
rs in the “metallic” regime (G>>e2/ h)
Kravchenko et al
Gao at al, Cond.mat 0308003
p-GaAs, p=1.3 10 cm-2 ;
Si MOSFET
rs=30
Cond-mat/0501686
B|| dependences of the resistance of Si MOSFET’s at different electron
concentrations.
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|| dependence of 2D p-GaAs at large rs and small wall thickness.
Gao et al
1/3
Comparison T-dependences of the resistances of Si MOSFET’s
at zero and large B||
EF  13K
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 !!
G=70 e2/h
Tsui et al. cond-mat/0406566
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>> e2/h?
Why is the magneto-resistance positive at all?
Why does B|| so effectively quench the T dependence
of the resistance?
Connection between the resistance and the electron
viscosity h(T) in the semi-quantum regime.
The electron mean free path lee ~n1/2 and hydrodynamics
description of the electron system works !
hu
Stokes formula in 2D case: F 
ln( h / nau )
u(r)
a
Ni
 (T )  h (T ) 2 2
e n ln( 1 / N i a 2 )
In classical liquids h(T) decreases exponentially with T.
In classical gases h(T) increases as a power of T.
What about semi-quantum liquids?
If rs >> 1 the liquid is strongly correlated
E F   
E pot
1/ 2
s
r
 E pot
 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 of
electrons with rs >>1 and in liquid He
Viscosity of gases (T>>U) increases as T increases
Viscosoty of classical liquids (Tc , hD << T<< U) decreases
exponentially with T (Ya. Frenkel)
h ~ exp(B/T)
Semi-quantum liquid: EF << T << h  << U: (A.F. Andreev)
h ~ 1/T
U
h
T <<U
Comparison of two strongly correlated liquids:
He3 and the electrons at EF <T < Epot
h
He4
1/T
Experimental data on the viscosity of He3 in the
semi-quantum regime (T > 0.3 K) are unavailable!?
1
A theory (A.F.Andreev): h 
T
T - dependence of the conductivity s(T) in 2D p- GaAs
at “high” T>EF and at different n.
H. Noh, D.Tsui, M.P. Lilly, J.A. Simmons, L.N. Pfeifer, K.W. West.
Points where T = EF are marked by red dots.
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, 226801
(2003)
VP
D 
IA
T-dependence of the drag resistance in double layers of p-GaAs
at different B||
1000
Pillarisetty et al.
PRL. 90, 226801
(2003)

D  T
2.7
2.4
10
Exponent
D (W/o)
100
2.1
1.5
1.2
1
0.1
*
B
1.8
0 2 4 6 8 10 12 14
B|| (T)
0.2
0.3
0.4
T (K)
0.6
0.8 1

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?
Quantum aspects of the theory of
micro-emulsion electronic phases
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.
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.
Quantum properties of droplets of Wigner crystal
embedded in Fermi liquid.
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.
effective droplet’s mass m*
At T=0 the liquid-solid surface is a quantum object.
a. If the surface is quantum smooth, a motion WC droplet corresponds to
redistribution of mass of order
m*  mncR 2
b. If it is quantum rough, much less mass need to be redistributed.
m  m(nL  nW )R
*
nd 2  1;
2
m*  m
1/d
WC
FL
n
In Coulomb case m ~ m*
Conclusion:
There are pure 2D electron phases which
are intermediate between the Fermi liquid
and the Wigner crystal .
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.
1
H    i a a   J ij a a j ; J ij 
;

| ri  rj |
i
ij
*
i i
*
i
1
  1.
G
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 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 fWC,
it is an increasing function of T and B||, with
scale of B|| set by T. (1T ~ 1K)
The ratio is big even deep in metallic regime!
Vitkalov at all
nc1
nc2
nc1 is the critical density at H=0; while nc2 is the critical density
at H>H*.
10
GaAs 2D hole system, 10nm wide quantum well, density 2.06*10 /cm
2
resistivity vs. temperature
2000
 [Ohm/square]
1600
1200
800
B|| [Tesla]
400
0
0.5
1
1.5
2
3
4
0
80
70
conductivity vs. temperature
Xuan et al
60
2
s [e /h]
50
40
30
20
10
0
0.0
0.2
0.4
0.6
0.8
T [K]
1.0
1.2
1.4
1.6
1.8
Additional evidence for the strongly correlated nature
of the electron system.
Vitkalov et al
E(M)=E0+aM2+bM4+……..
M is the spin magnetization.
B. Castaing, P. Nozieres
J. De Physique, 40, 257, 1979.
(Theory of liquid 3He .)
“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.”
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 : ln s eff  ln s ij
A. The case of complete spin polarization. All amplitudes of tunneling along
different tunneling paths are coherent.
s ij | k Aijk |2  k | Aijk |2 kl Aijk Aij*l ; Aijk | Aijk | exp( iijk   k );
ij
ijk  HS ijk
The phases  ijk are random quantities. s ij is independent
of H ; while all higher moments (s k ) m decrease with H.
H
i
ij
S
k
ij
j
The magneto-resistance is big, negative,and corresponds to magnetic field
corrections to the localization radius.
 

;   LH  rij

LH
Here  is the localization radius, LH is the magnetic length, and rij is the typical
hopping length.
B. The case when directions of spins of localized electrons are random.
j
i
i
j
j
i
s ij  m | l Aijl |2
m
Index “lm” labes tunneling paths which correspond to the same final spin
Configuration, the index “m” labels different groups of these paths.
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.
He3 phase diagram:
The Pomeranchuk effect.
The liquid He3 is also strongly
correlated liquid: rs; m*/m >>1.
The temperature dependence of
the heat capacity of He3.
The semi-quantum regime.
The Fermi liquid regime.
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
b. A case of preferred axis. For example, in-plane magnetic field.
“crystal”
“smectic”
liquid
n
The electron band structure in MOSFET’s
oxide
 L ,W   L(in,W)   (C ) ;  (C )
C  C0 
(en) 2

2C
eV
1
d
2D electron gas.
Metal
Si
d
As the the parameter dn1/2 decreases
the electron-electron interaction changes
from Coulomb V~1/r to dipole V~1/r3
form.
+
+
+
+
-
n-1/2
Elementary explanation:
Finite size corrections to the capacitance
R2
16R
C
 R ln
d
d
R is the droplet radius
Q 2 (enR 2 )2
R
2 2
2
EC 

 (en ) R d  (en ) R ln
2C
2C
d
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.mat
0308003
P=1.3 1010 cm-2 ;
rs=30
Mean field phase diagram
Large anisotropy of surface energy.
Wigner
crystal
Stripes (crystal conducting in
one direction)
L
Fermi
liquid.
n
Lifshitz points
( L)
G=70 e2/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: Epot ~A/r x ;
1<x<2; n=nc
Esurf   n R 2  sR  A(n )2 R 4 x
Esurf  sR 
2
A
Rx
If x ≥ 1 the micro-emulsion phases exist independently
of the value of the surface tension.