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07/11/11

PROFOUND EFFECTS OF ELECTRON-ELECTRON CORRELATIONS IN TWO DIMENSIONS. Sergey Kravchenko in collaboration with:. S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk NEU ISSP Texas A&M TU Delft.

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07/11/11

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  1. PROFOUND EFFECTS OF ELECTRON-ELECTRON CORRELATIONS IN TWO DIMENSIONS Sergey Kravchenko in collaboration with: S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk NEU ISSP Texas A&M TU Delft A. Punnoose M. P. Sarachik A. A. Shashkin CCNY CCNY ISSP 07/11/11 SCCS 2008

  2. Why 2D? • Quantum Hall effect (Nobel Prize 1985) • High-Tc superconductors (Nobel Prize 1988) • FQHE (Nobel Prize 1998) • Graphene (Nobel Prize 2010) 07/11/11

  3. Outline • Scaling theory of localization: “all electrons are localized in 2D” Samples What do experiments show? Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder) Interplay between disorder and interactions in 2D; flow diagram Conclusions 07/11/11

  4. One-parameter scaling theory for non-interacting electrons: the origin of the common wisdom “all states are localized in 2D” d(lnG)/d(lnL) = (G) G ~ Ld-2 exp(-L/Lloc) QM interference Ohm’s law in d dimensions metal (dG/dL>0) insulator G = 1/R insulator L insulator (dG/dL<0) Abrahams, Anderson, Licciardello, and Ramakrishnan, PRL 42, 673 (1979) 07/11/11

  5. Coulomb energy Fermi energy rs = GasStrongly correlated liquidWigner crystal Insulator ??????? Insulator strength of interactions increases ~1 ~35 rs 07/11/11

  6. Suggested phase diagrams for strongly interacting electrons in two dimensions Local moments, strong insulator Local moments, strong insulator disorder disorder electron density electron density Attaccalite et al. Phys. Rev. Lett. 88, 256601 (2002) Tanatar and Ceperley, Phys. Rev. B 39, 5005 (1989) strongly disordered sample Wigner crystal Ferromagnetic Fermi liquid Paramagnetic Fermi liquid, weak insulator Wigner crystal Paramagnetic Fermi liquid, weak insulator clean sample strength of interactions increases strength of interactions increases 07/11/11

  7. Scaling theory of localization: “all electrons are localized in two dimensions • Samples What do experiments show? Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder) Interplay between disorder and interactions in 2D; flow diagram Conclusions 07/11/11

  8. silicon MOSFET Al SiO2 p-Si conductance band 2D electrons chemical potential energy valence band _ + distance into the sample (perpendicular to the surface) 07/11/11 SCCS 2008

  9. WhySi MOSFETs? • large m*=0.19 m0 • twovalleys • low average dielectric constant =7.7 As a result, at low electron densities, Coulomb energy strongly exceeds Fermi energy: EC >> EF rs = EC / EF >10 can easily be reached in clean samples 07/11/11 SCCS 2008

  10. Scaling theory of localization: “all electrons are localized in two dimensions Samples • What do experiments show? Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder) Interplay between disorder and interactions in 2D; flow diagram Conclusions 07/11/11

  11. Strongly disordered Si MOSFET (Pudalov et al.) • Consistent (more or less) with the one-parameter scaling theory 07/11/11

  12. Clean sample, much lower electron densities S.V. Kravchenko, G.V. Kravchenko, W. Mason, J. Furneaux, V.M. Pudalov, and M. D’Iorio, PRB 1995 07/11/11

  13. In very clean samples, the transition is practically universal: Klapwijk’s sample: Pudalov’s sample: (Note: samples from different sources, measured in different labs) 07/11/11

  14. The effect of the parallel magnetic field: T = 30 mK Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 07/11/11

  15. Magnetic field, by aligning spins, changes metallic R(T) to insulating: Such a dramatic reaction on parallel magnetic field suggests unusual spin properties! (spins aligned) 07/11/11 SCCS 2008

  16. Scaling theory of localization: “all electrons are localized in 2D” Samples What do experiments show? • Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder) Interplay between disorder and interactions in 2D; flow diagram Conclusions 07/11/11

  17. How to study magnetic properties of 2D electrons? 07/11/11 Pisa 2006

  18. Method 1: magnetoresistancein a parallel magnetic field T = 30 mK Bc Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 Bc Bc Spins become fully polarized (Okamoto et al., PRL 1999; Vitkalov et al., PRL 2000) 07/11/11

  19. Method 2: weak-field Shubnikov-de Haas oscillations high density low density (Pudalov et al., PRL 2002; Shashkin et al, PRL 2003) 07/11/11 SCCS 2008

  20. Method 3: measurements of thermodynamic magnetization suggested by B. I. Halperin (1998); first implemented by O. Prus, M. Reznikov, U. Sivan et al. (2002) 1010 Ohm - + Gate Current amplifier Vg SiO2 Modulated magnetic field B + B Si 2D electron gas Ohmic contact i ~ d/dB = - dM/dns 07/11/11

  21. Raw magnetization data: induced current vs. gate voltage d/dB = - dM/dn 1 fA!! B|| = 5 tesla 07/11/11 SCCS 2008

  22. d/dB vs. ns in different parallel magnetic fields: 07/11/11

  23. Magnetic field of full spin polarization vs. electron density: data become T-dependent electron density (1011 cm-2) 07/11/11

  24. Summary of the results obtained by four independent methods (including transport) 07/11/11

  25. Spin susceptibility exhibits critical behavior near the sample-independent critical density n :  ~ ns/(ns – n) insulator T-dependent regime Are we approaching a phase transition? 07/11/11 SCCS 2008

  26. Anderson insulator Disorder increases at low density and we enter “Punnoose-Finkelstein regime” disorder paramagnetic Fermi-liquid Wigner crystal? Liquid ferromagnet? Density-independent disorder electron density 07/11/11

  27. g-factor or effective mass? 07/11/11 SCCS 2008

  28. Effective mass vs. g-factor Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002) Not the Stoner scenario! Wigner crystal? Maybe, but evidence is insufficient 07/11/11 SCCS 2008

  29. Effective mass as a function of rs-2 in Si(111) and Si(100) Si (111) Si(111): peak mobility 2.5x103 cm2/Vs Si(100): peak mobility 3x104 cm2/Vs Si (100) Shashkin, Kapustin, Deviatov, Dolgopolov, and Kvon, PRB (2007) 07/11/11

  30. Scaling theory of localization: “all electrons are localized in 2D” Samples What do experiments show? Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder) • Interplay between disorder and interactions in 2D; flow diagram Conclusions 07/11/11

  31. Corrections to conductivity due to electron-electron interactions in the diffusive regime (T < 1) • always insulating behavior However, later this prediction was shown to be incorrect 07/11/11 SCCS 2008

  32. 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 07/11/11

  33. Recent development: two-loop RG theory disorder takes over disorder QCP interactions metallic phase stabilized by e-e interaction Punnoose and Finkelstein, Science 310, 289 (2005) 07/11/11

  34. Experimental test First, 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: - 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 In standard Fermi-liquid notations, 07/11/11 SCCS 2008

  35. Experimental results (low-disordered Si MOSFETs; “just metallic” regime; ns= 9.14x1010 cm-2): S. Anissimova et al., Nature Phys. 3, 707 (2007) 07/11/11

  36. 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 07/11/11 SCCS 2008

  37. Experimental disorder-interaction flow diagram of the 2D electron liquid S. Anissimova et al., Nature Phys. 3, 707 (2007) 07/11/11 SCCS 2008

  38. Experimental vs. theoretical flow diagram(qualitative comparison b/c the 2-loop theory was developed for multi-valley systems) S. Anissimova et al., Nature Phys. 3, 707 (2007) 07/11/11 SCCS 2008

  39. Quantitative predictions of the one-loop RG for 2-valley systems(Punnoose and Finkelstein, Phys. Rev. Lett. 2002) Solutions of the RG-equations for  << h/e2: a series of non-monotonic curves (T). After rescaling, the solutions are described by asingleuniversalcurve: max (T) Tmax (T) For a 2-valley system (like Si MOSFET), metallic (T) sets in when 2 > 0.45 2 = 0.45 max ln(T/Tmax) 07/11/11

  40. Resistance and interactions vs. T Note that the metallic behavior sets in when 2 ~ 0.45, exactly as predicted by the RG theory 07/11/11 SCCS 2008

  41. Comparison between theory (lines) and experiment (symbols) (no adjustable parameters used!) S. Anissimova et al., Nature Phys. 3, 707 (2007) 07/11/11

  42. g-factor grows as T decreases ns = 9.9 x 1010 cm-2 “ballistic” value 07/11/11

  43. SUMMARY: • Strong interactions in clean two-dimensional systems lead to strong increase and possible divergence of the spin susceptibility: the behavior characteristic of a phase transition • Disorder-interactions flow diagram of the metal-insulator transition clearly reveals a quantum critical point: i.e., there exists a metallic state and a metal-insulator transition in 2D, contrary to the 20-years old paradigm! 07/11/11

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