B' Spivak, UW

1
2D electronic phases intermediate between the
Fermi liquid and the Wigner crystal
(electronic micro-emulsions)
B. Spivak, UW with S. Kivelson,
Stanford
2
Electron interaction can be characterized by a
parameter rs
Epot /Ekin
( e-e interaction energy is V(r) 1/rg
) Electrons (g1) form Wigner crystals at T0
and small n when rs gtgt 1 and EpotgtgtEkin 3He
and 4He (ggt2 ) are crystals at large n
3

• 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)

4
Experimental realizations of the 2DEG
5
Hetero-junction
Electrons interact via Coulomb interaction
V(r) 1/r
6
Schematic picture of a band structure in
MOSFETs (metal-oxide-semiconductor field effect
transistor)
7
MOSFET
As the the parameter dn1/2 decreases the electron
interaction changes from Coulomb V1/r to
dipole Vd2/r3 form.
8
Phase diagram of 2D electrons in MOSFETs . ( T0
)
Inverse distance to the gate 1/d
MOSFETs important for applications.
FERMI LIQUID
correlated electrons.
WIGNER CRYSTAL
n
Microemulsion phases. In
green areas where quantum effects are important.
9
Phase separation in the electron liquid.
crystal
liquid
phase separated region.
n
nc
nW
nL
There is an interval of electron densities
nWltnltnL near the critical nc where phase
separation must occur
10
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
ggt 0 is the microscopic surface energy
S
At large L the surface energy is negative!
11
Coloumb case is qualitatively similar to the
dipolar case
12
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.
13
Shape of the minority phase
The minority phase.
R
N is the number of the droplets
14
Mean field phase diagram of microemulsions
A sequence of more complicated patterns.
A sequence of more complicated patterns.
Fermi liquid.
Wigner crystal
Bubbles of WC
Bubbles of FL
Stripes
n
nL
nW
Transitions are continuous. They are similar to
Lifshitz points.
15
T and H dependences of the crystals area.
(Pomeranchuk effect).
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.

• As T and H increase, the crystal fraction
  grows.
• At large H the spin entropy is frozen and the
• crystal fraction is T- independent.

16
Several experimental facts suggesting non-Fermi
liquid nature 2D electron liquid at small
densities and the significance of the
Pomeranchuk effect
17
Experiments on the temperature and the parallel
magnetic field dependences of the resistance of
single electronic layers.
18
T-dependence of the resistances of Si MOSFET at
large rs and at different electron concentrations.
Kravchenko et al
insulator
metal
Factor of order 6.
There is a metal-insulator transition as a
function of n!
19
T-dependence of the resistance of 2D electrons at
large rs in the metallic regime (Ggtgte2/ h)
Gao at al, Cond.mat 0308003
Kravchenko et al
p-GaAs, p1.3 10 cm-2 rs30
Si MOSFET
20
Cond-mat/0501686
21
B dependences of the resistance of Si MOSFETs
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.
22
B dependence of 2D p-GaAs at large rs and small
wall thickness.
Gao et al
1/3
23
Comparison T-dependences of the resistances of Si
MOSFETs at zero and large B
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 !!
24
G70 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 !
25
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 Ggtgt
e2/h? Why is the magneto-resistance positive at
all? Why does B so effectively quench the T
dependence of the resistance?
26
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 ! Stokes formula in 2D case
u(r)
a
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?
27

• If rs gtgt 1 the liquid is strongly correlated
• is the plasma frequency
• If EF ltlt T ltlt hq ltlt 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 gtgt1 and in liquid He

28
Viscosity of gases (TgtgtU) increases as T
increases
Viscosoty of classical liquids (Tc , hQD ltlt Tltlt
U) decreases exponentially with T (Ya.
Frenkel) h
exp(B/T)
Semi-quantum liquid EF ltlt T ltlt h q ltlt U (A.F.
Andreev)
h 1/T
U
hq
T ltltU
29
Comparison of two strongly correlated liquids
He3 and the electrons at EF ltT lt Epot
h
He4
1/T
Experimental data on the viscosity of He3 in the
semi-quantum regime (T gt 0.3 K) are unavailable!?
A theory (A.F.Andreev)
30
T - dependence of the conductivity s(T) in 2D
p- GaAs at high TgtEF 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.
31
Experiments on the drag resistance of the double
p-GaAs layers.
32
B dependence of the resistance and drag
resistance of 2D p-GaAs at different
temperatures
Pillarisetty et al. PRL. 90, 226801 (2003)
33
T-dependence of the drag resistance in double
layers of p-GaAs at different B
Pillarisetty et al. PRL. 90, 226801 (2003)
34
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 T0?
35
Quantum aspects of the theory of micro-emulsion
electronic phases
At large distances the inter-bubble interaction
decays as Epot 1/r3 gtgt 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.
36
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?
37
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.

38
Quantum properties of droplets of Wigner
crystal embedded in Fermi liquid.

• 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.
39
effective droplets mass m
At T0 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

• If it is quantum rough, much less mass need to be
  redistributed.

1/d
FL
WC
n
In Coulomb case m m
40
Conclusion
There are pure 2D electron phases which are
intermediate between the Fermi liquid and the
Wigner crystal .
41
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 T0 ? 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 ?
42
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 ??
43
At T0 and Ggtgt1 the bubbles are not localized.
G is a dimensionless conductance.
Jij
j
i
44
The drag resistance is finite at T0
WC
FL
45
What about quenched disorder?
Pomeranchuk effect is local and so robust
Since r is an increasing function of fWC,
it is an increasing function of T and B, with
scale of B set by T. (1T 1K)
46
The ratio is big even deep in metallic regime!
Vitkalov at all
nc1
nc2
nc1 is the critical density at H0 while nc2 is
the critical density at HgtH.
47
Fig.1
Xuan et al
48
Additional evidence for the strongly correlated
nature of the electron system.
Vitkalov et al
B. Castaing, P. Nozieres J. De Physique, 40,
257, 1979. (Theory of liquid 3He .)
E(M)E0aM2bM4.. 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.
49
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.

is independent of H
while all higher moments decrease
with H.
The phases are random quantities.
j
i
The magneto-resistance is big, negative,and
corresponds to magnetic field corrections to the
localization radius.
Here is the localization radius, LH is
the magnetic length, and rij is the typical
hopping length.
50

B. The case when directions of spins of localized
electrons are random.
j
i
i
j
j
i
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.
51
He3 phase diagram
The Pomeranchuk effect.
The liquid He3 is also strongly correlated
liquid rs m/m gtgt1.
The temperature dependence of the heat capacity
of He3. The semi-quantum regime. The Fermi
liquid regime.
52
Hopping conductivity regime in MOSFETs
Magneto-resistance in the parallel and the
perpendicular tmagnetic field
Kravchenko et al (unpublished)
H
53
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
54
The electron band structure in MOSFETs
oxide
eV
2D electron gas.
Metal
Si
d

-
As the the parameter dn1/2 decreases the
electron-electron interaction changes from
Coulomb V1/r to dipole V1/r3 form.

-

-

-
n-1/2
55
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)
56
B dependence of 2D p-GaAs at large rs and small
wall thickness.
Gao et al
1/3
57
T-dependence of the resistance of 2D p-GaAs
layers at large rs in the metallic regime .
Cond.mat 0308003
P1.3 1010 cm-2 rs30
58
Mean field phase diagram Large anisotropy of
surface energy.
Wigner crystal
Stripes (crystal conducting in one direction)
Fermi liquid.
L
n
Lifshitz points
59
G70 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 !
60
M. Sarachik, S. Vitkalov
61
More general case Epot A/r x 1ltxlt2 nnc
If x 1 the micro-emulsion phases exist
independently of the value of the surface
tension.