1. 08/09/2005Leiden 2005 Sveta Anissimova Ananth Venkatesan Mohammed Sakr (now at UCLA) Sergey Kravchenko (presenting author) Alexander Shashkin Valeri Dolgopolov Teun Klapwijk Spin susceptibility near the 2D metal-insulator transition

2. 08/09/2005Leiden 2005 Klapwijk’s sample:Pudalov’s sample: In very clean 2D samples, there exists a practically universal metal-insulator transition: (Note: samples from different sources, measured in different labs)

3. 08/09/2005Leiden 2005 (spins aligned) Magnetic field, by aligning spins, changes metallic R(T) to insulating:

4. 08/09/2005Leiden 2005 Simonian et al., PRL 1997 Magnetoresistance in a parallel magnetic field Such a dramatic response to parallel magnetic field suggests unusual spin properties

5. 08/09/2005Leiden 2005 How to study magnetic properties of 2D electrons?

6. 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

7. 08/09/2005Leiden 2005 Method 2: weak-field Shubnikov-de Haas oscillations (Pudalov et al., PRL 2002; Shashkin et al, PRL 2003) high densitylow density

8. 08/09/2005Leiden 2005 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

9. 08/09/2005Leiden 2005 Magnetic field of the full spin polarization 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  

10. 08/09/2005Leiden 2005 1 fA!! Raw magnetization data: induced current vs. gate voltage d  /dB = - dM/dn B || = 5 tesla insulator

11. 08/09/2005Leiden 2005 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

12. 08/09/2005Leiden 2005 Spin susceptibility  = M /B

13. 08/09/2005Leiden 2005 d  /dB vs. n s in different parallel magnetic fields:

14. 08/09/2005Leiden 2005 Magnetic field of full spin polarization vs. electron density: Spontaneous spin polarization at n  ?

15. 08/09/2005Leiden 2005 2D electron gas Ohmic contact Si Gate Modulated gate voltage Vg +  Vg Current amplifier VgVg + - Method 4: Measurements of the thermodynamic density of states i ~ d  /dn s + const. 10 10 Ohm SiO 2

16. 08/09/2005Leiden 2005 Jump in the density of states signals the onset of full spin polarization D -1 nsns fully spin-polarized electrons spin-unpolarized electrons

17. 08/09/2005Leiden 2005 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

18. 08/09/2005Leiden 2005 disorder electron density Anderson insulator paramagnetic Fermi-liquid Wigner glass? Disorder increases at low density due to reduced screening Density-independent disorder

19. 08/09/2005Leiden 2005 Summary of the results obtained by four (or five) independent methods

20. 08/09/2005Leiden 2005 Spin susceptibility exhibits critical behavior characteristic of the existence of a phase transition:  ~ n s /(n s – n  ) insulator cannot measure

21. 08/09/2005Leiden 2005 g-factor or effective mass?

22. 08/09/2005Leiden 2005 d  /dB vs. n s in perpendicular magnetic field

23. 08/09/2005Leiden 2005 d  /dB vs. n s in perpendicular magnetic field

24. 08/09/2005Leiden 2005 g-factor:g-factor and effective mass:

25. 08/09/2005Leiden 2005 g*, m*, and  vs. n s

26. 08/09/2005Leiden 2005 Very recent theoretical development:

27. 08/09/2005Leiden 2005 …our solution gives a physical picture that is in qualitative agreement with the experimental situation. In particular, it is shown that 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 Punnoose and Finkelstein, preprint (2005)

28. 08/09/2005Leiden 2005 CONCLUSIONS: In strongly interacting 2D electron system in silicon, Pauli spin susceptibility critically grows near a sample-independent density n  : behavior that is characteristic of the existence of a phase transition. In clean samples, n  practically coincides with the metal-insulator transition point We find no sign of increasing g-factor, but the effective mass is strongly (×3) enhanced near n 

29. 08/09/2005Leiden 2005 Appendix: Difference between our results and those of Prus, Reznikov, Sivan et al. (PRB 2003): Raw data shifted up to obtain correspondence with transport results

30. 08/09/2005Leiden 2005 Data of Prus, Reznikov, Sivan et al. PRB 2003

31. 08/09/2005Leiden 2005 Data of Prus, Reznikov, Sivan et al. PRB 2003 Only three data points are relevant to the metallic regime

32. 08/09/2005Leiden 2005 Data of Prus, Reznikov, Sivan et al. PRB 2003 Indeed, the (inverse) spin susceptibility has a Curie form characteristic of local moments