Sveta Anissimova Sergey Kravchenko (presenting author) A. Punnoose A. M. Finkelstein Teun Klapwijk
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Transcript of Sveta Anissimova Sergey Kravchenko (presenting author) A. Punnoose A. M. Finkelstein Teun Klapwijk
2/21/2007 NNCI 2007
Sveta Anissimova
Sergey Kravchenko(presenting author)
A. Punnoose
A. M. Finkelstein
Teun Klapwijk
Interplay between disorder and interactions in two dimensions
2/21/2007 NNCI 2007
-4
-3
-2
-1
0
1
3D
2D 1D
MIT
ln(G)
d(lnG)/d(lnL) = (G)
One-parameter scaling theory for non-interacting electrons: the origin of the common wisdom “all states are localized in 2D”
Abrahams, Anderson, Licciardello, and Ramakrishnan, PRL 42, 673 (1979)
G ~ Ld-2 exp(-L/Lloc)
metal (dG/dL>0)insulator
insulator
insulator (dG/dL<0)
Ohm’s law in d dimensions
QM interference
2/21/2007 NNCI 2007
However, the existence of the quantum Hall effect is inconsistent with this prediction
Solution (Pruisken, Khmelnitskii…):two-parameter (xx, xy) scaling theory
2/21/2007 NNCI 2007
Do the electron-electron interactions modify the“all states are localized in 2D at B=0” paradigm?
(what happens to the Anderson transition in the presence of interactions?)
2/21/2007 NNCI 2007
Corrections to conductivity due to electron-electron interactions in the diffusive regime (T < 1)
always insulating behavior
However, later this result was shown to be incorrect
2/21/2007 NNCI 2007
Zeitschrift fur Physik B (Condensed Matter) -- 1984 -- vol.56, no.3, pp. 189-96
Weak localization and Coulomb interaction in disordered systems
Finkel'stein, A.M. L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR
0
02
2 1ln131ln
2 F
FT
e
Insulating behavior when interactions are weak Metallic behavior when interactions are strongEffective strength of interactions grows as the temperature decreases
Altshuler-Aronov-Lee’s result Finkelstein’s & Castellani-
DiCastro-Lee-Ma’s term
)45.01( 0 F)045.0( 0 F
2/21/2007 NNCI 2007
Same mechanism persists to ballistic regime (T> 1), but corrections become linear in temperature
where C(ns) < 0
This is reminiscent of earlier Stern-Das Sarma’s result
(However, Das Sarma’s calculations are not applicable to strongly interacting regime because at r s>1, the screening length becomes smaller than the separation between electrons.)
2/21/2007 NNCI 2007
What do experiments show?
2/21/2007 NNCI 2007
Strongly disordered Si MOSFET
(Pudalov et al.)
Consistent with the one-parameter scaling theory
2/21/2007 NNCI 2007
Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995
Clean Si MOSFET, much lower electron densities
2/21/2007 NNCI 2007
Klapwijk’s sample: Pudalov’s sample:
In very clean samples, the transition is practically universal:
103
104
105
106
0 0.5 1 1.5 2
0.86x1011 cm-2
0.880.900.930.950.991.10
resi
stiv
ity
r (O
hm)
temperature T (K)
(Note: samples from different sources, measured in different labs)
2/21/2007 NNCI 2007
103
104
105
106
0 0.5 1 1.5 2
0.86x1011 cm-2
0.880.900.930.950.991.10
resistiv
ity r
(Ohm
)
temperature T (K)
clean sample: disordered sample:
… in contrast to strongly disordered samples:
Clearly, one-parameter scaling theory does not work here
2/21/2007 NNCI 2007
Again, two-parameter scaling theory comes to the rescue
2/21/2007 NNCI 2007
finite rincreases
while reduces r the interplay of disorder and r and interaction changes the trend and gives non-monotonic R(T)
2 22
2
2
22 2
11 ( ) ln(1 ) 1
(1
)
)
1
2
(4v v
dnn
d
d
d
r r
r
cooperon singlet “triplet”
ln(1/ ) , 1T T
Two parameter scaling
to all orders in 2
(Finkelstein, 1983-1984;Castellani, Di Castro, Lee, and Ma, 1984;Punnoose and Finkelstein, 2002; 2005)
2/21/2007 NNCI 2007
Punnoose and Finkelstein, Science310, 289 (2005)
interactions
diso
rde
r
metallic phase stabilized by e-e interaction
disorder takes over
QCP
2/21/2007 NNCI 2007
Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions
1
2
Tk
Bg
B
B
22
22
2
1091.0
4,
T
B
k
g
h
eTB
B
B
Experimental test of the Punnoose-Finkelstein theoryFirst, one needs to ensure that the system is in the diffusive regime (T< 1).
One can distinguish between diffusive and ballistic regimes by studying magnetoconductance:
2
,
T
BTB
TBTB
2
,
- diffusive: low temperatures, higher disorder (Tt < 1).
- ballistic: low disorder, higher temperatures (Tt > 1).
The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):
2/21/2007 NNCI 2007
Experimental results (low-disordered Si MOSFETs; “just metallic” regime; ns= 9.14x1010 cm-2):
2/21/2007 NNCI 2007
Temperature dependences of the resistance (a) and strength of interactions (b)
This is the first time effective strength of interactions has been seen to depend on T
2/21/2007 NNCI 2007
Experimental disorder-interaction flow diagram of the 2D electron liquid
2/21/2007 NNCI 2007
Experimental vs. theoretical flow diagram(qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
2/21/2007 NNCI 2007
Quantitative predictions of the two-parameter scaling theory for 2-valley systems
(Punnoose and Finkelstein, Phys. Rev. Lett. 2002)
Solutions of the RG-equations:a series of non-monotonic curves r(T). After rescaling, the solutions are described by a single universal curve:
max
max max
ρ(T) = ρ R(η)
η = ρ ln(T /T)
r(T
)
(T)
rmax ln(T/Tmax)
Tmax
rmax
2 = 0.45
For a 2-valley system (like Si MOSFET),
metallic r(T) sets in when 2 > 0.45
2/21/2007 NNCI 2007
Resistance and interactions vs. T
Note that the metallic behavior sets in when 2 ~ 0.45, exactly as predicted by the RG theory
2/21/2007 NNCI 2007
Comparison between theory (lines) and experiment (symbols)(no adjustable parameters used!)
2/21/2007 NNCI 2007
Si-MOSFET vs. GaAs/AlGaAs heterostructures
Si-MOSFET advantages
Moderately high mobility: There exists a diffusive window T < 1/ < EF; 1/ = 2-3 K
Short range scattering: Anderson transition in a disordered Fermi Liquid (universal)
Two-valley system: Effects of electron-electron interactions are enhanced (“critical” 2=0.45 vs. 2.04 in a single-valley system)
GaAs/AlGaAs:
Ultra high mobility: Diffusive regime is hard to reach; 1/ < 100-200 mK
Long range scattering: Percolation type of the transition?
Very low density: Non-degeneracy effects; possible Wigner crystallization,..
2/21/2007 NNCI 2007
It is demonstrated, for the first time, that as a result of the interplay between the electron-electron interactions and disorder, not only the resistance but also the interaction strength exhibits a fan-like spread as the metal-insulator transition is crossed.
Conclusions:
Resistance-interaction flow diagram of the MIT clearly reveals a quantum critical point, as predicted by renormalization-group theory of Punnoose and Finkelstein.
The metallic side of this diagram is accurately described by the renormalization-group theory without any fitting parameters. In particular, the metallic temperature dependence of the resistance sets in once 2 > 0.45, which is in remarkable agreement with RG theory.
The interactions between electrons stabilize the metallic state in 2D and lead to the existence of a critical fixed point