1. Interplay between disorder and interactions
in two dimensions
Sveta Anissimova
Sergey Kravchenko
(presenting author)
A. Punnoose
A. M. Finkelstein
Teun Klapwijk
2/21/2007
NNCI 2007

2. One-parameter scaling theory for non-interacting electrons:
the origin of the common wisdom “all states are localized in 2D”
d(lnG)/d(lnL) = b(G)
G ~ Ld-2 exp(-L/Lloc)
QM interference
metal (dG/dL>0)
Ohm’s law in d dimensions
insulator
insulator
insulator (dG/dL<0)
Abrahams, Anderson, Licciardello, and Ramakrishnan, PRL 42, 673 (1979)
2/21/2007
NNCI 2007

3. However, the existence of the quantum Hall effect is inconsistent with this prediction
Solution (Pruisken, Khmelnitskii…):
two-parameter (sxx, sxy) scaling theory
2/21/2007
NNCI 2007

4. 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
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5. However, later this result was shown to be incorrect
Corrections to conductivity due to electron-electron interactions
in the diffusive regime (Tt < 1)
always insulating behavior
However, later this result was shown to be incorrect
2/21/2007
NNCI 2007

6. Effective strength of interactions grows as the temperature decreases
Zeitschrift fur Physik B (Condensed Matter) vol.56, no.3, pp
Weak localization and Coulomb interaction in disordered systems
Finkel'stein, A.M.
L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR
Insulating behavior when interactions are weak
Metallic behavior when interactions are strong
Effective strength of interactions grows as the temperature decreases
Altshuler-Aronov-Lee’s result
Finkelstein’s & Castellani-DiCastro-Lee-Ma’s term
2/21/2007
NNCI 2007

7. Same mechanism persists to ballistic regime (Tt > 1),
but corrections become linear in temperature
This is reminiscent of earlier Stern-Das Sarma’s result
where C(ns) < 0
(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.)
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8. What do experiments show?
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9. Strongly disordered Si MOSFET
(Pudalov et al.)
Consistent with the one-parameter scaling theory
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10. Clean Si MOSFET, much lower electron densities
Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995
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11. In very clean samples, the transition is practically universal:
Klapwijk’s sample:
Pudalov’s sample:
(Note: samples from different sources, measured in different labs)
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12. … in contrast to strongly disordered samples:
clean sample:
disordered sample:
Clearly, one-parameter scaling theory does not work here
2/21/2007
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13. Again, two-parameter scaling theory comes to the rescue
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14. Two parameter scaling while g2 reduces r finite r increases g2
(Finkelstein, ;
Castellani, Di Castro, Lee, and Ma, 1984;
Punnoose and Finkelstein, 2002; 2005)
finite r increases g2
while g2 reduces r
the interplay of disorder and r and interaction g2 changes the trend and gives non-monotonic R(T)
to all orders in
cooperon
singlet
“triplet”
2/21/2007
NNCI 2007

15. metallic phase stabilized
disorder takes over
disorder
QCP
interactions
Punnoose and Finkelstein, Science
310, 289 (2005)
metallic phase stabilized
by e-e interaction
2/21/2007
NNCI 2007

16. Experimental test of the Punnoose-Finkelstein theory
First, one needs to ensure that the system is in the diffusive regime (Tt < 1). One can distinguish between diffusive and ballistic regimes by studying magnetoconductance:
- diffusive: low temperatures, higher disorder (Tt < 1).
- ballistic: low disorder, higher temperatures (Tt > 1).
The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):
2 valleys
for
Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions
2/21/2007
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17. Experimental results (low-disordered Si MOSFETs;
“just metallic” regime; ns= 9.14x1010 cm-2):
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18. 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
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NNCI 2007

19. Experimental disorder-interaction flow diagram of the 2D electron liquid
2/21/2007
NNCI 2007

20. Experimental vs. theoretical flow diagram (qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
2/21/2007
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21. Solutions of the RG-equations:
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:
rmax
r(T)
Tmax
g2(T)
For a 2-valley system (like Si MOSFET),
metallic r(T) sets in when g2 > 0.45
g2 = 0.45
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rmax ln(T/Tmax)

22. Resistance and interactions vs. T
Note that the metallic behavior sets in when g2 ~ 0.45,
exactly as predicted by the RG theory
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23. Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
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24. Si-MOSFET vs. GaAs/AlGaAs heterostructures
Si-MOSFET advantages
Moderately high mobility: There exists a diffusive window T < 1/t < EF; 1/t = 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” g2=0.45 vs in a single-valley system)
GaAs/AlGaAs:
Ultra high mobility: Diffusive regime is hard to reach; 1/t < mK
Long range scattering: Percolation type of the transition?
Very low density: Non-degeneracy effects; possible Wigner crystallization,..
2/21/2007
NNCI 2007

25. Conclusions:
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.
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 g2 > 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
2/21/2007
NNCI 2007