Transcript Document
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 r s =E pot /E kin
E kin
n E pot
n g
/ 2
r s
n g
2 2 ( e-e interaction energy is
V(r) ~1/r g
) Electrons ( g=1 ) form Wigner crystals at T=0 and small n when r s >> 1 and E pot >>E kin
3 He
and
4 He ( 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
2DEG
Hetero-junction
Ga 1-x Al x As GaAs
Electrons interact via Coulomb interaction V(r) ~ 1/r
w
Schematic picture of a band structure in MOSFET’s (metal-oxide-semiconductor field effect transistor)
2DEG
MOSFET
Metal “gate” SiO 2 Si d As the the parameter dn changes from Coulomb 1/2 decreases the electron interaction
V~1/r
to dipole
V~d 2 /r 3
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.
L
,
W
Phase separation in the electron liquid.
W
L
crystal liquid phase separated region.
n
L
,
W
(
in
)
L
,
W
n W
n c (
C
)
;
n L (
C
)
(
en
)
2
; 2
C C
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!
Coloumb case is qualitatively similar to the dipolar case
E
[
n
0
n c
][
S
S
]
d l
(
)
2
e
|
l
l
d l l
' | , S
and S
-
are area of the minority and the majority phases,
n
0
and
n c
are average
and critical d(
L
W
)
d n
densities ,
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
E surf
N
2
R
1 2
e
2 (
n W
n L
) 2
d
2
R
ln
R d
;
N
is the number of the droplets
NR
2
const
The characteri stic size of
R
de
, the droplets is
4
e
2
(
n W
n L
)
2
1 ,
d dn
L
W
Mean field phase diagram of microemulsions
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
.
T and H || dependences of the crystal’s area. (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.
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 r s at different electron concentrations.
and Kravchenko et al
r s
E p E k
( 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 r s in the “metallic” regime (G>>e 2 / h) Kravchenko et al
Gao at al, Cond.mat 0308003 Si MOSFET p-GaAs, p=1.3 10 cm -2 ; r s =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.
1/3
B || dependence of 2D p-GaAs at large r s and small wall thickness.
Gao et al
Comparison T dependences of the resistances of Si MOSFET’s at zero and large B ||
E F
13
K
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 e 2 /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>> e 2 /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
l ee ~n 1/2
and hydrodynamics description of the electron system works !
Stokes formula in 2D case:
F
ln( h h /
u nau
)
u(r)
a (
T
) h (
T
)
N i e
2
n
2 ln( 1 /
N i a
2 ) In classical liquids h
(T) decreases
In classical gases h(T)
increases
exponentially with as a power of
T
.
T.
What about semi-quantum liquids?
If
r s
>> 1 the liquid is strongly correlated
E F
E pot r s
1 / 2
E pot
is the plasma frequency If
E F << T << h
<< E pot
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 r s >>1 and in liquid He
Viscosity of gases (T>>U) increases as T increases Viscosoty of classical liquids (T c , h
D exponentially with T (Ya. Frenkel)
h
~ exp(B/T) << T<< U) decreases Semi-quantum liquid: E F << T << h
<< U: (A.F. Andreev)
h
~ 1/T h
U T <
Comparison of two strongly correlated liquids: He 3 and the electrons at E F
h He 4 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):
h 1
T
T - dependence of the conductivity
s
(T) in 2D p- GaAs at “high” T>E F 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.
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)
D
V P I A
T-dependence of the drag resistance in double layers of p-GaAs at different B ||
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 2 0.4
B * 4 6 8 10 12 14 B || (T) 0.6
0.8
1 Pillarisetty et al.
PRL. 90, 226801 (2003)
D
T
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 E pot
1/r 3
>> E kin 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
*
mn c
R
2
b. If it is quantum rough, much less mass need to be redistributed.
m
*
m
(
n L
n W
)
R
2 1/d
nd
2 1 ;
m
*
m
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.
H
i
i a i
*
a i
ij J ij a i
*
a j
; 1
G
1 .
J ij
|
r i
1
r j
| ; J ij j i
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 f WC , 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 n c1 n c2 n c1 is the critical density at H=0; while n c2 at H>H*.
is the critical density
10 /cm 2 resistivity vs. temperature 2000 1600 1200 800 400 0 80 70 60 50 40 30 20 10 0 0.0
conductivity vs. temperature 0.2
0.4
0.6
0.8
T [K] 1.0
1.2
1.4
B || [Tesla] 0 0.5
1 1.5
2 3 4 1.6
1.8
Xuan et al
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.”
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
i A. The case of complete spin polarization. All amplitudes of tunneling along s
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
( s
ij
)
ij
s
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 s
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.
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.
Hopping conductivity regime in MOSFET’s Magneto-resistance in the parallel and the perpendicular tmagnetic field
Kravchenko et al (unpublished)
H ||
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
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
Elementary explanation: Finite size corrections to the capacitance
C
R
2
d
R
ln 16
R d
R is the droplet radius
E C
Q
2
C
2 (
enR
2 ) 2 2
C
(
en
) 2
R
2
d
(
en
) 2
R
ln
R 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)
1/3
B || dependence of 2D p-GaAs at large r s and small wall thickness.
Gao et al
T-dependence of the resistance of 2D p-GaAs layers at large r s in the “metallic” regime .
Cond.mat
0308003 P=1.3 10 10 cm -2 ; r s =30
Mean field phase diagram Large anisotropy of surface energy.
Wigner crystal Stripes (crystal conducting in one direction)
L
Fermi liquid.
n
(
L
)
G=70 e 2 /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: E
pot
~A/r 1
c x
;
E surf
n R
2 s
R
A
(
n
) 2
R
4
x E surf
s
R
2
R x A
If x ≥ 1 the micro-emulsion phases exist independently of the value of the surface tension.
c x
;
E surf
n R
2 s
R
A
(
n
) 2
R
4
x E surf
s
R
2
R x A
If x ≥ 1 the micro-emulsion phases exist independently of the value of the surface tension.