Sergey Kravchenko in collaboration with:

1. Interactions and disorder in two-dimensional semiconductors 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 SCCS 2008

2. 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

3. 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 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)

4. Coulomb energy Fermi energy rs = GasStrongly correlated liquidWigner crystal Insulator ??????? Insulator strength of interactions increases ~1 ~35 rs

5. Local moments, strong insulator Local moments, strong insulator disorder disorder electron density electron density Suggested phase diagrams for strongly interacting electrons in two dimensions 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

6. 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

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

8. WhySi MOSFETs? • large m*=0.19 m0 • twovalleys • low average dielectric constant e=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 SCCS 2008

9. 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

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

11. 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

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

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

14. 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)

15. 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

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

17. Spin susceptibility exhibits critical behavior near the sample-independent critical density nc : c ~ ns/(ns – nc) insulator T-dependent regime Are we approaching a phase transition?

18. 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

19. g-factor or effective mass?

20. Effective mass vs. g-factor Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002) Not Stoner scenario! Wigner crystal?

21. 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)

22. 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

23. Corrections to conductivity due to electron-electron interactions in the diffusive regime (Tt < 1) • always insulating behavior However, later this prediction was shown to be incorrect

24. 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

25. 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)

26. Experimental test 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 In standard Fermi-liquid notations,

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

28. 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

29. Experimental disorder-interaction flow diagram of the 2D electron liquid S. Anissimova et al., Nature Phys. 3, 707 (2007)

30. 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)

31. Quantitative predictions of the one-loop RG for 2-valley systems(Punnoose and Finkelstein, Phys. Rev. Lett. 2002) Solutions of the RG-equations for r << ph/e2: a series of non-monotonic curves r(T). After rescaling, the solutions are described by asingleuniversalcurve: 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 rmax ln(T/Tmax)

32. Resistance and interactions vs. T Note that the metallic behavior sets in when g2 ~ 0.45, exactly as predicted by the RG theory

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

34. g-factor grows as T decreases ns = 9.9 x 1010 cm-2 “ballistic” value

35. 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!