1. 3/13/2016Bilkent University Sergey Kravchenko in collaboration with: S. Anissimova, V.T. Dolgopolov, A. M. Finkelstein, T.M. Klapwijk, A. Punnoose, M.P. Sarachik, A.A. Shashkin INTERPLAY OF DISORDER AND INTERACTIONS IN TWO-DIMENSIONAL SEMICONDUCTORS

2. 3/13/2016Bilkent University Outline Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D” Samples What do experiments show? What do theorists have to say? Interplay between disorder and interactions: experimental test “Clean” regime: diverging spin susceptibility Summary

3. 3/13/2016Bilkent University 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

4. 3/13/2016Bilkent University ~1 ~35 r s Gas Strongly correlated liquid Wigner crystal Insulator ??????? Insulator strength of interactions increases Coulomb energy Fermi energy r s =

5. 3/13/2016Bilkent University 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

6. 3/13/2016Bilkent University Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D” Samples What do experiments show? What do theorists have to say? Interplay between disorder and interactions: experimental test “Clean” regime: diverging spin susceptibility Summary

7. 3/13/2016University of Virginia silicon MOSFET Al SiO 2 p-Si 2D electrons conductance band valence band chemical potential + _ energy distance into the sample (perpendicular to the surface)

8. 3/13/2016University of Virginia Why Si MOSFETs? It turns out to be a very convenient 2D system to study strongly-interacting regime because of: large effective mass m*= 0.19 m 0 two valleys in the electronic spectrum low average dielectric constant  =7.7 As a result, at low 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

9. 3/13/2016Bilkent University Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D” Samples What do experiments show? What do theorists have to say? Interplay between disorder and interactions: experimental test “Clean” regime: diverging spin susceptibility Summary

10. 3/13/2016Bilkent University Strongly disordered Si MOSFET (data of Pudalov et al.)  Consistent (more or less) with the one-parameter scaling theory strong insulator disorder electron density weak insulator

11. 3/13/2016Bilkent University Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995 Clean sample, much lower electron densities

12. 3/13/2016Bilkent University Klapwijk’s sample:Pudalov’s sample: In very clean samples, the transition is practically universal: (Note: samples from different sources, measured in different labs)

13. 3/13/2016Bilkent University clean sample:disordered sample: … in contrast to strongly disordered samples:  Clearly, one-parameter scaling theory does not work here

14. 3/13/2016Bilkent University T = 30 mK Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 The effect of the parallel magnetic field:

15. 3/13/2016Bilkent University (spins aligned) Magnetic field, by aligning spins, changes metallic R(T) to insulating:

16. 3/13/2016Bilkent University Referee A: “The paper should not be published in PRL. Everyone knows there is no zero-temperature conductivity in 2-d.” Referee B: “The reported results are most intriguing, but they must be wrong. If there indeed were a metal-insulator transition in these systems, it would have been discovered years ago.” Referee C: “I cannot explain the reported behavior offhand. Therefore, it must be an experimental error.” Reaction of referees:... I remember being challenged over that well-known fact that all states were localized in two dimensions, something that made no sense at all in light of the experiments I had just shown. R. B. Laughlin, Nobel Lecture

17. 3/13/2016Bilkent University (Hanein, Shahar, Tsui et al., PRL 1998) However, later similar transition has also been observed in other 2D structures: p-Si:Ge (Coleridge’s group; Ensslin’s group) p-GaAs/AlGaAs (Tsui’s group, Boebinger’s group) n-GaAs/AlGaAs (Tsui’s group, Stormer’s group, Eisenstein’s group) n-Si:Ge (Okamoto’s group, Tsui’s group) p-AlAs (Shayegan’s group)

18. 3/13/2016Bilkent University (Physics Today “Search and Discovery” July 1997) (Nature 389, “News and views” 30 October 1997)

19. 3/13/2016Bilkent University (Nature 400, “News and views” 19 August 1999) (The Economist “Science and Technology” July 1997)

20. 3/13/2016Bilkent University Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D” Samples What do experiments show? What do theorists have to say? Interplay between disorder and interactions: experimental test “Clean” regime: diverging spin susceptibility Summary

21. 3/13/2016Bilkent University non-Fermi liquid:

22. 3/13/2016Bilkent University continued.... (superconductivity)

23. 3/13/2016Bilkent University continued.... (superconductivity)

24. 3/13/2016Bilkent University continued.... (new quantum phase prior to Wigner crystallization)

25. 3/13/2016Bilkent University continued.... (spin-orbit interaction)

26. 3/13/2016Bilkent University continued.... (percolation)

27. 3/13/2016Bilkent University Charging/discharging of interface traps: Temperature-dependent screening: continued.... Spin-orbit scattering:

28. 3/13/2016Bilkent University …and, finally, an unspecified mechanism: continued....

29. 3/13/2016Bilkent University Now, a more reasonable approach

30. 3/13/2016Bilkent University 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 (0<F<1, N.B.: F is not the Landau Fermi-liquid constant)

31. 3/13/2016Bilkent University 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

32. 3/13/2016Bilkent University Recent theory of the MIT in 2D

33. 3/13/2016Bilkent University Punnoose and Finkelstein, Science 310, 289 (2005) interactions disorder metallic phase stabilized by e-e interaction disorder takes over QCP

34. 3/13/2016Bilkent University Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D” Samples What do experiments show? What do theorists have to say? Interplay between disorder and interactions: experimental test “Clean” regime: diverging spin susceptibility Summary

35. 3/13/2016 Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions 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,

36. 3/13/2016Bilkent University Experimental results (low-disordered Si MOSFETs; “just metallic” regime; n s = 9.14x10 10 cm -2 ):

37. 3/13/2016Bilkent University 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

38. 3/13/2016Bilkent University Experimental disorder-interaction flow diagram of the 2D electron liquid

39. 3/13/2016Bilkent University Experimental vs. theoretical flow diagram (qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)

40. 3/13/2016Bilkent University 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

41. 3/13/2016Bilkent University Resistance and interactions vs. T Note that the metallic behavior sets in when  2 ~ 0.45, exactly as predicted by the RG theory

42. 3/13/2016Bilkent University Comparison between theory (lines) and experiment (symbols) (no adjustable parameters used!)

43. 3/13/2016Bilkent University g-factor grows as T decreases n s = 9.9 x 10 10 cm -2 “non-interacting” value

44. 3/13/2016Bilkent University Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D” Samples What do experiments show? What do theorists have to say? Interplay between disorder and interactions: experimental test “Clean” regime: diverging spin susceptibility Summary

45. 3/13/2016Bilkent University 2D electron gas Ohmic contact SiO 2 Si Gate Modulated magnetic field B +  B Current amplifier VgVg + - Magnetic measurements without magnetometer 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

46. 3/13/2016Bilkent University 1 fA!! Raw magnetization data: induced current vs. gate voltage d  /dB = - dM/dn B || = 5 tesla

47. 3/13/2016University of Virginia Raw magnetization data: induced current vs. gate voltageIntegral of the previous slide gives M (n s ): complete spin polarization B || = 5 tesla at n s =1.5x10 11 cm -2

48. 3/13/2016Bilkent University Spin susceptibility exhibits critical behavior near the sample-independent critical density n  :  ~ n s /(n s – n  ) insulator T-dependent regime

49. 3/13/2016Bilkent University disorder electron density Anderson insulator paramagnetic Fermi-liquid Wigner crystal? Liquid ferromagnet? Disorder increases at low density and we enter “Finkelstein regime” Density-independent disorder

50. 3/13/2016Bilkent University SUMMARY: Competition between electron-electron interactions and disorder leads to the existence of the metal-insulator transition in two dimensions. The metallic state is stabilized by the electron-electron interactions. In the insulating state, the disorder takes over Modern renormalization-group theory (Punnoose and Finkelstein, Phys. Rev. Lett. 2002; Science 2005) gives quantitatively correct description of the metallic state without any fitting parameters In the clean regime, spin susceptibility critically grows upon approaching to some sample-independent critical point, n , pointing to the existence of a phase transition. Unfortunately, residual disorder does not allow to see this transition in currently available samples