1. Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA) Mariam Rahimi (now at UC Berkeley) Sergey Kravchenko Alexander Shashkin Valeri Dolgopolov Teun Klapwijk

2.  Silicon MOSFETs  GaAs/AlGaAs heterostructures  SiGe heterostructures  Surface of a material (liquid helium, graphene sheets)

3. At low densities, n s ~ 10 11 cm -2, Coulomb energy exceeds Fermi energy: E C >> E F electron density decreases strength of interactions increases r s = E C / E F >10 – strongly interacting regime can easily be reached  large m* = 0.19 m 0  average  = 7.7  two valleys n v = 2

4. Hanein, Shahar, Tsui et al., PRL 1998 Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995 Similar transition is also observed in other 2D structures: p-Si:Ge (Coleridge’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)

9. Such a dramatic reaction on parallel magnetic field suggests unusual spin properties

11. - Diagonal resistance - Hall resistance Rotator equipped Oxford dilution refrigerator Base temperature ~ 30 mK High mobility (100)-Si MOSFET μ=3 m 2 /Vs at T=0.1 K Excitation current 0.1 – 0.2 nA f = 0.4 Hz

12. (Okamoto et al., PRL 1999; Vitkalov et al., PRL 2000) Shashkin, Kravchenko, Dolgopolov, Klapwijk, PRL 2001 BcBc BcBc BcBc

13. Shashkin et al, 2001 Vitkalov, Sarachik et al, 2001 Pudalov et al, 2002 nn Vanishing B c at a finite n   n c indicates a ferromagnetic transition in this electron system The fact that n  is sample independent and n   n c indicates that the MIT in clean samples is driven by interactions Extrapolated polarization field, B c, vanishes at a finite electron density, n 

14.  gm as a function of electron density calculated using Shashkin et al., PRL 2001 nn

15. Effective Mass Measurements: amplitude of the weak-field Shubnikov-de Haas oscillations vs. temperature Rahimi, Anissimova, Sakr, Kravchenko, and Klapwijk, PRL 2003 high density:low density: n s = 5x10 11 cm -2 n s = 1.2x10 11 cm -2

16. dots – ν = 10 squares – ν = 14 solid line – fit by L-K formula The amplitude of the SdH oscillations follows the calculated curve down to the lowest achieved temperature: the electrons are in a good thermal contact with the bath. Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003 n s = 1.2x10 11 cm -2

17. Comparison of the effective masses determined by two independent experimental methods: Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003 Therefore, the sharp increase of the spin susceptibility near the critical density is due to the enhancement of the effective mass rather then g-factor, unlike in the Stoner scenario *

18. 2D electron layer Ohmic contact SiO 2 Si Gate Modulated magnetic field B + B mod Current-to-Voltage converter VgVg + - Measurements of thermodynamic magnetization suggested by B. Halperin (1998); first implemented by Prus et al. (2003) C – capacitance  - chemical potential Maxwell relation: R=10 10  Lock-in amplifier LVC6044 CMOS Quad Micropower Operational Amplifier with noise level: 0.2 fA/(Hz) 1/2 f = 0.45 Hz B mod = 0.01 – 0.03 tesla

19. Magnetic field of the full spin polarization B c vs. n s BcBc nsns 0 B c =  h 2 n s /2  B m b nsns 0 dMdM dnsdns M =  B  n s =  B n s B/B c for B < B c  B n s for B > B c B > B c B < B c B B c =  h 2 n s /  B g*m* nn non-interacting system spontaneous spin polarization at n  

20. 1 fA!! Raw magnetization data: induced current vs. gate voltage d  /dB = - dM/dn B || = 5 tesla the onset of complete spin polarization d  /dB = 0 Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

21. 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

22. d  /dB vs. n s in different parallel magnetic fields: Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

23. Spontaneous spin polarization at n  ? Magnetic field of full spin polarization vs. electron density from magnetization measurements

24. 2D electron layer Ohmic contact SiO 2 Si Gate Modulated gate voltage Vg +  Vg Current-to-Voltage converter VgVg + - Measurements of thermodynamic density of states C 0 – geometric capacitance A – sample area R=10 10  Lock-in amplifier LVC6044 CMOS Quad Micropower Operational Amplifier with noise level: 0.2 fA/(Hz) 1/2 f = 0.3 Hz  V g = 0.09V C 0 = 624 pF

25. Jump in the density of states signals the onset of full spin polarization D -1 nsns fully spin-polarized electrons spin-unpolarized electrons Polarization field from capacitance measurements: Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

26. Magnetic field of full spin polarization vs. electron density: electron density (10 11 cm -2 ) data become T-dependent, possibly due to localized band-tail Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

27. Spin susceptibility exhibits critical behavior near the metal-insulator transition:  ~ n s /(n s – n  ) insulator cannot measure Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

29. d  /dB vs. n s in perpendicular magnetic field

30. g-factor measurements in perpendicular fields: Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, cond-mat/0503123

31. g-factor:g-factor and effective mass: Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, cond-mat/0503123

32. Summary of the results obtained by four (or five) independent methods

33.  spin susceptibility critically grows near the metal-insulator transition  the enhancement of the g-factor is weak and practically density independent  the effective mass becomes strongly enhanced as the density is decreased Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, Phys. Rev. Lett. 96, 036403 (2006); Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, Phys. Rev. Lett. 96, 046409 (2006) Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, Phys. Rev. Lett. 91, 046403 (2003)

35. 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  Magnetic field destroys metal

36.  Insulating behavior when interactions are weak  Metallic behavior when interactions are strong  Magnetic field destroys metal

38. …the point of the metal to insulator transition correlates with the appearance of the divergence in the spin susceptibility… note that at the fixed point the g-factor remains finite These conclusions are in agreement with experiments Punnoose and Finkelstein, Science, Vol. 310. no. 5746, pp. 289 - 291

39. Punnoose and Finkelstein, Science Vol. 310. no. 5746, pp. 289 - 291

40. Pauli spin susceptibility critically grows with a tendency to diverge near the critical electron density We find no sign of increasing g-factor, but the effective mass is strongly (×3) enhanced near the metal-insulator transition and… Punnoose-Finkelstein theory gives a quantitatively correct description of the metal-insulator transition in 2D