B. Spivak, UW with S. Kivelson, Stanford
-
Upload
aspen-rowland -
Category
Documents
-
view
45 -
download
1
description
Transcript of B. Spivak, UW with S. Kivelson, Stanford
![Page 1: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/1.jpg)
B. Spivak, UWwith S. Kivelson, Stanford
2D electronic phases intermediate between the Fermi liquid and the Wigner crystal (electronic micro-emulsions)
![Page 2: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/2.jpg)
( e-e interaction energy is V(r) ~1/rg )
Electrons (g=1) form Wigner crystals at T=0 and small nwhen rs >> 1 and Epot>>Ekin
3He and 4He (g are crystals at large n
2
2
2/
g
s
gpotkin
nr
nEnE
Electron interaction can be characterized by a parameter rs =Epot /Ekin
![Page 3: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/3.jpg)
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)
![Page 4: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/4.jpg)
Experimental realizations of the 2DEG
![Page 5: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/5.jpg)
Ga1-xAlxAs
GaAs
2DEG w
Electrons interact via Coulomb interaction V(r) ~ 1/r
Hetero-junction
![Page 6: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/6.jpg)
Schematic picture of a band structure in MOSFET’s(metal-oxide-semiconductor field effect transistor)
![Page 7: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/7.jpg)
SiO2
Si
2DEG
Metal “gate”
d
As the the parameter dn1/2 decreases the electron interaction changes from Coulomb V~1/r to dipole V~d2/r3 form.
MOSFET
![Page 8: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/8.jpg)
Phase diagram of 2D electrons in MOSFET’s . ( T=0 )
n
correlated electrons.
WIGNER CRYSTAL
FERMI LIQUIDInverse distance to the gate 1/d
Microemulsion phases.In green areas where quantum effects are important.
MOSFET’s important for applications.
![Page 9: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/9.jpg)
Phase separation in the electron liquid.
There is an interval of electron densities nW<n<nL near the critical nc where phase separation must occur
n
WL,
ncnW nL
dC
C
enCCinWLWL
1;
2
)(;
2)()()(
,,
LWcrystal liquid
phase separated region.
![Page 10: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/10.jpg)
d
LLaL
ll
ldldadlE
SS
surf ln|'|
')(
To find the shape of the minority phase one must minimize the surface energy at a given area of the minority phase
S
In the case of dipolar interaction
At large L the surface energy is negative!
ld
> 0 is the microscopic surface energy
![Page 11: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/11.jpg)
Coloumb case is qualitatively similar tothe dipolar case
nd
nn
ll
ldld
eldSSnnE
W
c
c
)d(
,densitiescriticalandaverageareand
phases,majority theandminority theofarea areS and S
,|'|
)(]][[
L
0
-
2
0
![Page 12: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/12.jpg)
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.
![Page 13: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/13.jpg)
N is the number of the droplets
Shape of the minority phase
constNRd
RRdnneRNE LWsurf
2222 ;ln)(
2
12
WLLW dn
d
nne
deR
,1)(
4
,
isdropletstheofsizesticcharacteriThe
22
R
The minority phase.
![Page 14: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/14.jpg)
Fermiliquid.
Wigner crystal
Stripes Bubblesof WC
Bubblesof FL
Mean field phase diagram of microemulsions
nnLnW
Transitions are continuous. They are similar to Lifshitz points.
A sequenceof more complicated patterns.
A sequenceof more complicated patterns.
![Page 15: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/15.jpg)
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.
WLWLWLWL MHTSF ,||,,,
LWLW MMSS ;
T and H|| dependences of the crystal’s area. (Pomeranchuk effect).
The entropy of the crystal is of spin origin and much larger than the entropy of the Fermi liquid.
S and M are entropy and magnetization of the system.
![Page 16: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/16.jpg)
Several experimental facts suggesting
non-Fermi liquid nature 2D electron liquid
at small densities and the significance of
the Pomeranchuk effect:
![Page 17: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/17.jpg)
Experiments on the temperature and the parallel magnetic field dependences of the resistance of single electronic layers.
![Page 18: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/18.jpg)
There is a metal-insulator transition as a function of n!
)2010( k
ps E
Er
Kravchenko et al
Factor of order 6.metal
insulator
T-dependence of the resistances of Si MOSFET at large rs andat different electron concentrations.
![Page 19: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/19.jpg)
Kravchenko et al Gao at al, Cond.mat 0308003
T-dependence of the resistance of 2D electrons at large rs in the “metallic” regime (G>>e2/ h)
p-GaAs, p=1.3 10 cm-2 ; rs=30
Si MOSFET
![Page 20: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/20.jpg)
Cond-mat/0501686
![Page 21: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/21.jpg)
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|| dependences of the resistance of Si MOSFET’s at different electron concentrations.
![Page 22: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/22.jpg)
Gao et al
B|| dependence of 2D p-GaAs at large rs and small wall thickness.
1/3
![Page 23: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/23.jpg)
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 !!
KEF 13
Comparison T-dependences of the resistances of Si MOSFET’s at zero and large B||
![Page 24: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/24.jpg)
Tsui et al. cond-mat/0406566
G=70 e2/h
Gao et al
The slope of the resistance dR/dT is dramaticallysuppressed 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 !
![Page 25: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/25.jpg)
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 dependenceof the resistance?
![Page 26: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/26.jpg)
The electron mean free path lee ~n1/2 and hydrodynamics description of the electron system works !
Stokes formula in 2D case:
Connection between the resistance and the electronviscosity in the semi-quantum regime.
)/1ln()()(
222 aNne
NTT
i
i
In classical liquids (T) decreases exponentially with T. In classical gases increases as a power of T.
What about semi-quantum liquids?
u(r)
)/ln( nau
uF
a
![Page 27: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/27.jpg)
If rs >> 1 the liquid is strongly correlated
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 ofelectrons with rs >>1 and in liquid He
pots
potF E
r
EE
2/1
![Page 28: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/28.jpg)
T <<U
U
Semi-quantum liquid: EF << T << h << U: (A.F. Andreev)
~ 1/T
Viscosoty of classical liquids (Tc , hD << T<< U) decreases exponentially with T (Ya. Frenkel) ~ exp(B/T)
h
Viscosity of gases (T>>U) increases as T increases
![Page 29: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/29.jpg)
Comparison of two strongly correlated liquids: He3 and the electrons at EF <T < Epot
He4
Experimental data on the viscosity of He3 in the semi-quantum regime (T > 0.3 K) are unavailable!?
T
1A theory (A.F.Andreev):
h
1/T
![Page 30: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/30.jpg)
Points where T = EF are marked by red dots.
H. Noh, D.Tsui, M.P. Lilly, J.A. Simmons, L.N. Pfeifer, K.W. West.
T - dependence of the conductivity (T) in 2D p- GaAs at “high” T>EF and at different n.
![Page 31: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/31.jpg)
Experiments on the drag resistance of the double p-GaAs layers.
![Page 32: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/32.jpg)
B|| dependence of the resistance and drag resistance of 2D p-GaAs at different temperatures
A
PD I
V
Pillarisetty et al.PRL. 90, 226801 (2003)
![Page 33: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/33.jpg)
0.1 11
10
100
1000
0 2 4 6 8 10 12 141.2
1.5
1.8
2.1
2.4
2.7
B*
Exp
onen
t
B|| (T)
0.80.60.40.30.2
D (/
)
T (K)
TD
T-dependence of the drag resistance in double layers of p-GaAs at different B||
Pillarisetty et al.PRL. 90, 226801 (2003)
![Page 34: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/34.jpg)
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 dependenceof 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?
![Page 35: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/35.jpg)
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.
Quantum aspects of the theory of micro-emulsion electronic phases
![Page 36: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/36.jpg)
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?
![Page 37: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/37.jpg)
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.
![Page 38: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/38.jpg)
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.
Quantum properties of droplets of Wigner crystalembedded in Fermi liquid.
![Page 39: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/39.jpg)
a. If the surface is quantum smooth, a motion WC droplet corresponds to redistribution of mass of order
b. If it is quantum rough, much less mass need to be redistributed.
2* Rmnm c
2* )( Rnnmm WL
effective droplet’s mass m*
n
1/d
WC FL
At T=0 the liquid-solid surface is a quantum object.
mmnd *2 ;1
In Coulomb case m ~ m*
![Page 40: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/40.jpg)
There are pure 2D electron phases which are intermediate between the Fermi liquid and the Wigner crystal .
Conclusion:
![Page 41: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/41.jpg)
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 ?
![Page 42: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/42.jpg)
Conclusion # 3:
Are bubble microemulson phases related to recently Observed ferromagnetism in quasi-1D GaAs electronic channels ( cond-mat ……….. ) ?
Wigner crystal
Bubblemicroemulsion
Is the WC bubble phase ferromagnetic at the Lifshitz point ??
![Page 43: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/43.jpg)
At T=0 and G>>1 the bubbles are not localized.
.11
;||
1;**
G
rrJaaJaaH
jiij
ijjiijii
ii
G is a dimensionless conductance.
Jij
i
j
![Page 44: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/44.jpg)
The drag resistance is finite at T=0
FL WC
![Page 45: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/45.jpg)
What about quenched disorder?
Disorder is a relevant perturbation in d < 4! No macroscopic symmetry breaking
(However, in clean samples, there survives a largesusceptibility in what would have been the broken symmetry state - see, e.g., quantum Hall nematic stateof 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.
![Page 46: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/46.jpg)
Vitkalov at allnc1 nc2
nc1 is the critical density at H=0; while nc2 is the critical density at H>H*.
The ratio is big even deep in metallic regime!
![Page 47: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/47.jpg)
0
400
800
1200
1600
2000
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80
10
20
30
40
50
60
70
80
[O
hm/s
quar
e]
B|| [Tesla]
0 0.5 1 1.5 2 3 4
GaAs 2D hole system, 10nm wide quantum well, density 2.06*1010/cm2
resistivity vs. temperature
[e
2/h
]T [K]
conductivity vs. temperature
Xuan et al
![Page 48: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/48.jpg)
Additional evidence for the strongly correlated nature of the electron system.
E(M)=E0+aM2+bM4+……..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.”
B. Castaing, P. NozieresJ. De Physique, 40, 257, 1979.(Theory of liquid 3He .)
Vitkalov et al
![Page 49: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/49.jpg)
ijeff lnln
kij
kij
kkij
kij
kijkl
lij
kijk
kijk
kijij
HS
iAAAAAAij
;)exp(||;|||| *22
The magneto-resistance is big, negative,and corresponds to magnetic field corrections to the localization radius.
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.
The phases are random quantities.
kijS
kijH
ij is independent of H ; while all higher moments decrease with H. mk
ij )(
ijHH
rLL
;
Here is the localization radius, LH is the magnetic length, and rij is the typical hopping length.
j
i
![Page 50: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/50.jpg)
B. The case when directions of spins of localized electrons are random.
In the case of large tunneling length “r” the majority of the tunneling amplitudes areorthogonal and the orbital mechanism of the magneto-resistance is suppressed.
2|| m l
lijij
mA
Index “lm” labes tunneling paths which correspond to the same final spinConfiguration, the index “m” labels different groups of these paths.
i
i
i jj
j
![Page 51: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/51.jpg)
The Pomeranchuk effect.
The temperature dependence ofthe heat capacity of He3. The semi-quantum regime.
The Fermi liquid regime.
He3 phase diagram:
The liquid He3 is also stronglycorrelated liquid: rs; m*/m >>1.
![Page 52: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/52.jpg)
Hopping conductivity regime in MOSFET’s Magneto-resistance in the parallel and the perpendicular tmagnetic field
H||
Kravchenko et al (unpublished)
![Page 53: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/53.jpg)
Sequence of intermediate phasesat finite temperature.
a. Rotationally invariant case.
“crystal” “nematic” liquidn
n
“crystal” “smectic” liquid
b. A case of preferred axis. For example, in-plane magnetic field.
![Page 54: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/54.jpg)
The electron band structure in MOSFET’s
d
2D electron gas.
Metal Si
oxide
+
++ -
--
+ -
dCC
C
enCCinWLWL
12
)(;
0
2)()()(
,,
n-1/2
As the the parameter dn1/2 decreasesthe electron-electron interaction changes from Coulomb V~1/r to dipole V~1/r3 form.
eV
![Page 55: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/55.jpg)
d
RR
d
RC
16ln
2
Elementary explanation: Finite size corrections to the capacitance
d
RRendRen
C
enR
C
QEC ln)()(
2
)(
2222
222
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)
![Page 56: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/56.jpg)
B|| dependence of 2D p-GaAs at large rs and small wall thickness.
Gao et al
1/3
![Page 57: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/57.jpg)
T-dependence of the resistance of 2D p-GaAs layers at large rs in the “metallic” regime .
Cond.mat0308003
P=1.3 1010 cm-2 ; rs=30
![Page 58: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/58.jpg)
Mean field phase diagram Large anisotropy of surface energy.
Wigner crystal
Fermiliquid.
Stripes (crystal conducting inone direction)
n
Lifshitz points
L
)( L
![Page 59: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/59.jpg)
G=70 e2/h
The slope of the resistance as a function of T is dramaticallysuppressed by the parallel magnetic field. It changes the sign. Overall change of the modulus is more than factor 10 !
![Page 60: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/60.jpg)
M. Sarachik,S. Vitkalov
![Page 61: B. Spivak, UW with S. Kivelson, Stanford](https://reader036.fdocuments.net/reader036/viewer/2022081519/56812d37550346895d9237a2/html5/thumbnails/61.jpg)
More general case: Epot ~A/r x ;
1<x<2; n=nc
xsurf
xsurf
RA
RE
RnARRnE
2
422 )(
If x ≥ 1 the micro-emulsion phases exist independently of the value of the surface tension.