Download Document

2D microemulsion phases in Bose and Fermi
systems with dipolar interaction
B. Spivak, UW
S. Kivelson, Stanford,
S. Sondhi, D. Huse, C. Lamman, Princeton
interaction energy can be characterized by a parameter
rs =Epot /Ekin
Ekin  n
rs  n
E pot  n
g /2
g 2
2
( interaction energy is V(r) ~1/rg )
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 quantum phases intermediate between
the liquid and the crystal (micro-emulsion phases)
Phase separation
 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 densities nW<n<nL near
the critical nc where phase separation must occur
Elementary explanation:
R2
16R
C
 R ln
d
d
Finite size corrections to the capacitance
R is the droplet radius
Part of the bulk energy
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)
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!
Mean field phase diagram in the case of
isotropic surface tension
crystal Bubbles
of L
A sequence
of more
complicated
patterns.
Stripes
A sequence
of more
complicated Bubbles liquid.
patterns.
of C
Transitions are continuous.
They are similar to Lifshitz points.
n
Effective free energy in the case of the first order phase transition
which is close to the second order.

2
)
)
F ( , n)  c( r )    u 4   n  nc )  n  n Vˆ n  n ;
2
Q
r3

n  n  Vˆ 1 r )
V


)
)

F n )  c(| k |  k * ) 2  t |  k |2  n  nc ) k  u 4
k *   2 / Qc
t    c( k * ) 2

   0   i exp( ik i r )  c.c.
k1
k2
k3
Mean field phase diagram
cristal
hexagons
stripes
hexagons
liquid.
n
 ( )   0   1 cos 
1
1
a

t
3/ 2

when t  0
Mean field phase diagram of microemulsions
cristal
hexagons
stripes
hexagons
liquid.
n
hexagons
stripes
of h
Transitions are continuous.
They are similar to Lifshitz points.
liquid.
zero temperature hydrodynamics
of all mictoemulcion phases is equavalent
to supersolid hydrodynamics
finite temperature hydrodynamics of the stripe
phases is equivalent to super-smectic
hydrodynamics
Coloumb case is qualitatively similar to
the dipolar case (Jamei, Kivelson, Spivak)
 


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
The electron band structure in MOSFET’s
oxide
eV
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
Phase diagram of 2D electrons in MOSFET’s . ( T=0 )
Inverse distance
to the gate 1/d
FERMI LIQUID
d  d *  38aB  1000 A
MOSFET’s
important
for applications.
correlated
electrons.
nd 2  1
WIGNER CRYSTAL
n
Microemulsion phases.
In green areas where quantum effects are important.
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.
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 !
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 dependence
of 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
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 ?
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.
Conclusion:
There are pure 2D electron phases which
are intermediate between the Fermi liquid
and the Wigner crystal .