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

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

3. 07/11/11 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

4. 07/11/11 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 ~ L d-2 exp(-L/L loc ) metal (dG/dL>0) insulator insulator (dG/dL<0) Ohm’s law in d dimensions QM interference L G = 1/R

5. 07/11/11 ~1 ~35 r s Gas Strongly correlated liquid Wigner crystal Insulator ??????? Insulator strength of interactions increases Coulomb energy Fermi energy r s =

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

7. 07/11/11 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

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

9. 07/11/11SCCS 2008 Why Si MOSFETs? large m*= 0.19 m 0 two valleys low average dielectric constant  =7.7 As a result, at low electron densities, Coulomb energy strongly exceeds Fermi energy: E C >> E F r s = E C / E F >10 can easily be reached in clean samples

10. 07/11/11 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

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

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

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

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

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

16. 07/11/11 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

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

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

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

20. 07/11/11 2D electron gas Ohmic contact SiO 2 Si Gate Modulated magnetic field B +  B Current amplifier VgVg + - Method 3: measurements of thermodynamic magnetization suggested by B. I. Halperin (1998); first implemented by O. Prus, M. Reznikov, U. Sivan et al. (2002) i ~ d  /dB = - dM/dn s 10 10 Ohm

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

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

23. 07/11/11 Magnetic field of full spin polarization vs. electron density: electron density (10 11 cm -2 ) data become T-dependent

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

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

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

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

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

29. 07/11/11 Effective mass as a function of r s -2 in Si(111) and Si(100) Si (111) Si (100) Shashkin, Kapustin, Deviatov, Dolgopolov, and Kvon, PRB (2007) Si(111): peak mobility 2.5x10 3 cm 2 /Vs Si(100): peak mobility 3x10 4 cm 2 /Vs

30. 07/11/11 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

31. 07/11/11SCCS 2008 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

32. 07/11/11 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

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

34. 07/11/11SCCS 2008 Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions 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): In standard Fermi-liquid notations,

35. 07/11/11 Experimental results (low-disordered Si MOSFETs; “just metallic” regime; n s = 9.14x10 10 cm -2 ): S. Anissimova et al., Nature Phys. 3, 707 (2007)

36. 07/11/11SCCS 2008 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

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

38. 07/11/11SCCS 2008 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)

39. 07/11/11 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/e 2 : a series of non-monotonic curves  (T). After rescaling, the solutions are described by a single universal curve:  (T)   (T)  max ln(T/T max ) T max  max  2 = 0.45 For a 2-valley system (like Si MOSFET), metallic  (T) sets in when  2 > 0.45

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

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

42. 07/11/11 g-factor grows as T decreases n s = 9.9 x 10 10 cm -2 “ballistic” value

43. 07/11/11 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!