Transcript Document
Phase separation in strongly correlated 2d electron liquid
B. Spivak University of Washington
Schematic picture of MOSFET’s
Electrons in semiconductors can have metallic conductivity. The electron band structure in MOSFET’s.
L
,
W
(
in
)
L
,
W
(
C
) ; (
C
) (
en
) 2 2
C C
C
0 1
d
eV
Metal
As the the parameter dn 1/2 decreases the electron-electron interaction changes from Coulomb V~1/r to dipole V~1/r 3 form. + + + +
oxide 2D electron gas.
d n -1/2
Si
The role of the electron-electron interaction.
If the inter-particle interaction energy is
V(r) ~1/r
g
,
then
E
pot
~n
g/2
while
E
kin
~ n
. (n~1/r
2
)
E pot
n g
2 2
E kin
g=2 is critical.
At small
n
the electron kinetic energy
(g
1)
can be neglected and the system forms a Wigner crystal. (
3
He
and
4
He (g
>2 )
are crystals at large
n.)
Phase diagram of 2D electron system at T=0.
The inverse distance to the gate.
1/d d
38
a B
FERMI LIQUID MOSFET’s important for applications.
correlated electrons.
WIGNER CRYSTAL
n
Experiments by V. Pudalov, T. Klapwijk, S. Kravchenko, M. Sarachik, S. Vitkalov et al.
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.
L
,
W
Phase separation in the electron liquid.
W
L
crystal liquid phase separated region.
n n W n c n L
L
,
W
(
in
)
L
,
W
(
C
) ; (
C
) (
en
) 2 ; 2
C C
1 ;
d n L
n W
1
d
There is an interval of electron densities
n W
the critical
n c
where phase separation must occur near
To find the shape of the minority phase one must minimize the surface energy at a given area of the minority phase
E surf
S
In the case of dipolar interaction
dl
a
S
|
d l l
d l l
' ' |
L
a L
ln
d L
> 0 is the microscopic surface energy
S
d l
At large L the surface energy is negative!
Finite size corrections to C majority phase.
- - - -
d
+ + + +
E~1/x x
minority phase.
R metallic gate
C
1
d
1
R
ln 16
R d E tot
R
2 2
R
; (
in
) (
en
) 2 .
2
C
R is the droplet radius
This contribution to the surface energy is negative proportional to –R ln (R/d) !
and
Shape of the minority phase The minority phase.
R
E surf
N
2
R
1 2
e
2 (
n W
n L
) 2
d
2
R
ln
R d NR 2 =const
.
N
is the number of the droplets
R is the size of the droplets.
, R
de
, 4
e
2 (
n W
n L
) 2 1 .
Mean field phase diagram of electron system at T=0. (Small anisotropy of surface energy).
Wigner Bubbles crystal of FL
n W
A sequence of more complicated patterns.
Stripes A sequence of more complicated patterns.
Bubbles of WC Fermi
n L
liquid.
n
Transitions are continuous. They are similar to Lifshitz points .
Mean field phase diagram of electron system at T=0. (Big anisotropy of surface energy). An example: the magnetic field parallel to the film.
Wigner crystal Stripes Fermi liquid.
L
n
(
L
)
Finite temperature effects.
d
T c
ln
L
2
d
2 a. The case of the magnetic field parallel to the film.
“crystal” “cmectic” liquid n b. The case of strong anisotropy of the surface tension.
“crystal” “nematic” liquid
Transitions between uniform, bubble and stripe phases 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.
T B S B n In 3D the macroscopic phase separation is impossible.
only at large enough |d
W
/
dn-d
L /dn
|
.
d. neutron stars (Bethe) e. HTS (J. Zaanen, S. Kivelson, V. Emery, E. Fradkin)
T and H || dependences of the area of the crystal. (Pomeranchuk effect).
F L
,
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.
S W
>>
S L
;
M W
>>
M L
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.
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! Kravchenko et al
r s
E p E k
( 10 20 )
insulator metal Factor of order 6.
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.
E F
13
K
Vitkalov et al.
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 !!
Comparison of the magneto-resistance in the hopping regime in the cases when the magnetic field is parallel or perpendicular to the film.
Kravchenko et al (unpublished)
H ||
A comparison of conductivity of metals and viscosity of liquid He in semi-quantum regime
E F <
.
At the Fermi liquid-Wigner crystal transition the critical
r s = E pot / E F
~ 38 .
and the liquid is strongly correlated.
If
E F << T << E pot
the liquid is not degenerate but it is still not a gas. (Semi-quantum regime).
Such temperature interval exists in the case of liquid He as well.
He 3 phase diagram: The Pomeranchuk effect.
The liquid He 3 is also strongly correlated liquid: r s ; m*/m >>1.
The temperature dependence of the heat capacity of He 3 .
The semi-quantum regime.
The Fermi liquid regime.
A connection between the resistance and the viscosity of electrons in the semi-quantum regime. In this case l ee ~n 1/2 , and hydrodynamics works !
( l ee is the electron-electron mean free path.) Stokes formula in 2D case:
F
ln( /
u nau
) (
T
)
N i
(
T
)
e
2
n
2 ln( 1 /
N i a
2 )
u(r)
Viscosity
(T)
of classical liquids
decreases
exponentially with
T
. Viscosity of classical gases
increases
What about semi-quantum liquids?
as a power of T.
Comparison of viscosities of two strongly correlated liquids (He 3 and the electrons in the semi-quantum regime ( E F
He 4 Factor 1.5
1/T
Experimental data on the viscosity of He 3 in the semi-quantum regime (T > 0.3 K) are unavailable!?
A theory (A.F.Andreev):
1
T
T
T - dependence of the conductivity
(T) of 2D holes in GaAs at “high” T and at different n. H. Noh, D.Tsui, M.P. Lilly, J.A. Simmons, L.N. Pfeifer, K.W. West.
Points where T = E F are marked by red dots.
Additional evidence for the strongly correlated nature of the electron system.
Vitkalov et al
E(M)=E 0 +aM 2 +bM 4 +……..
M is the spin magnetization.
B. Castaing, P. Nozieres J. De Physique, 40, 257, 1979.
(Theory of liquid
3 He
.)
“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.”
At T=0 bubble superlattices melted by quantum fluctuations The interaction between the droplets decays as 1/r
3
. 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.
Quantum properties of droplets of Wigner crystal embedded in Fermi liquid. WC
R(t)
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.
S
dtd r
(
en
(
r
,
t
)) 2
C
0
dt
1
R
2 (
t
) 1
t
2
dt
;
R
(
t
)
v F t
The droplets contain
non integer
spin and charge. Therefore the statistics of these quasiparticles is
unknown
.
What is the effective mass of the droplets m* ?
At T=0 the liquid-solid surface is a quantum object.
a.
If the surface is quantum smooth, motion of a Wigner crystal droplet is associated with a redistribution of the liquid mass of order
m
*
mn c
R
2
m
(
n c d
2 )
b. If the surface is quantum rough much less mass need be redistributed.
m
*
m
(
n L
n W
)
R
2
m
(
n c d
2 ) 1 / 2
(A.F.Andreev, A.I. Shalnikov (unpublished).)
1/d
nd
2 1 ;
m
*
m
WC FL n
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
eff
ln
ij
i A. The case of complete spin polarization. All amplitudes of tunneling along
ij H
different tunneling paths are coherent.
|
k A k ij
| 2
k
|
A ij k
| 2
kl A k ij A ij
*
l
;
S ij k
j
A k ij
|
A ij k
| exp(
i
ij k
ij k
) ;
k
HS k ij
The phases are random quantities.
ij
(
ij
)
ij
ij
is independent The magneto-resistance is big, negative,and corresponds to magnetic field corrections to the localization radius.
L H
;
L H
H
r ij
is the magnetic length, and r ij is the typical hopping length.
B. The case when directions of spins of localized electrons are random.
i j i
ij
j i
l A ij l m
| 2 j Index “l m ” 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.
Conclusion
: At large r
s
there are phases of pure 2D electron system which are intermediate between the Fermi liquid and the Wigner crystal .
T-dependence of the drag resistance of 2D GaAs holes.
Tsui at al.
1000 100 10 1 0.1
0.2
0.3
T (K) 2.7
2.4
2.1
1.8
1.5
1.2
0 B * 0.4
2 4 6 8 10 12 14 B || (T) 0.6
0.8
1
H-depemdence of the resistance and drag resistance of 2D GaAs.
Tsui at al.