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

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

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

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

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

Effective strength of interactions grows as the temperature decreases 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 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

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.) 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

Clean Si MOSFET, much lower electron densities Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995 2/21/2007 NNCI 2007

In very clean samples, the transition is practically universal: Klapwijk’s sample: Pudalov’s sample: (Note: samples from different sources, measured in different labs) 2/21/2007 NNCI 2007

… in contrast to strongly disordered samples: clean sample: disordered sample: 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

Two parameter scaling while g2 reduces r finite r increases g2 (Finkelstein, 1983-1984; 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

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

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 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

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 2/21/2007 NNCI 2007 rmax ln(T/Tmax)

Resistance and interactions vs. T Note that the metallic behavior sets in when g2 ~ 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/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. 2.04 in a single-valley system) GaAs/AlGaAs: Ultra high mobility: Diffusive regime is hard to reach; 1/t < 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

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