1. Sasha Kuntsevich Nimrod Teneh Vladimir Pudalov Spin-droplet state of an interacting 2D electron system M. Reznikov Magnetic order in clean low- density systems Methods of magnetization measurements Recharging Technique Experimental results Implications Technion

2. Electron gas with interactions Short range repulsive interaction 2nd order phase transition into ferromagnetic ordered state For a single-valley system Stoner (1947) Stoner instability

3. Ferromagnetic Bloch Instability Decreasing density Energy

4. Phase diagram Attaccalite et al. (2001) First order transition at r s ~20: Senatore et al. (2001) r s ~26

5. Clean system B. Tanatar and D.C. Ceperley (1989) ferromagnetic

6. Clean system Very small energy difference! antiferromagnetic ferromagnetic B. Tanatar and D.C. Ceperley (1989)

7. Methods: Shubnikov - de Haas beatings F. Fang and P. Stiles (1968), T. Okamoto at al., (1999), S. Vitkalov at al. (2000), V. Pudalov at.al., (2001) 2 4 6 7 rsrs

8. V. Pudalov at al, (2001) Metal-Insulator Transition in a Silicon Inversion Layer  gg

9. In-plane magnetoresistance S. Vitkalov et al. PRL 2001A. Shashkin et al. PLR, 2001

10. In-plane magnetoresistance A. Shashkin et al. PLR, 2001 Possible FM transition ??

11. Samples: Si Field effect transistors Russian samples, beginning of 80 th, Holland samples, mid 80 th Typical parameters  3.4 x10 4 cm 2 /Vs @1.7K

12. The Principle of the Recharging Technique Maxwell relation: Small correction

13. Diamagnetic contribution Capacitance contribution

14. Recharging Technique _ + VGVG Out Modulated magnetic field B+  Current Amplifier Ohmic contact Gate SiO 2 Si 2D electron gas

15. Expected behavior T=0, finite magnetic field  gg Interactions M n No interactions n  Interactions Prus et al,2003 B>T

16. B (T) g  B B~2E F kT/4

17. Raw data, low fields Compare with single spins ∂M/∂n=  B tanh(b), b=g  B B/2T

18. 1

20. The same characteristic magnetic field

21. Interactions n n No interactions Interactions d  /dn(n), expectations

22. d  /dn(n), T=1.7-13K

23. d  /dn(n), T=0.6-4K

24. vs. Temperature

27.  (n), T=1.7-13K

28. Magnetic moment at B=2T

29. Comparison with Transport Measurements

30. Main observations Possible scenario: few electron droplets

31. Droplet scenario vs theory Fermi-liquid expectations: Spontaneous large spin droplets in disordered metal Diffusion enhanced interactions in quantum dots Mean Field treatment: Andreev, Kamenev (1998) Numerics: Shepelyansky (2001) Narozhny, B. N. and Aleiner, I. L. and Larkin, A. I. (2000)

32. Conclusion: Problems :

33. Problem O. Prus, Y. Yaish, M. Reznikov, U. Sivan, and V. Pudalov, PRB 2003 : Assumption: at large density the susceptibility is the renormalized Pauli one This assumption happened to be wrong!

34. Old results (Prus et al, 2003)

35. Field dependence of the magnetic moment

36. In-plane magnetoresistance A. Shashkin et al. PLR, 2001Fleury, Weintal, 2010.

37. Raw data

38. Susceptibility in at B=2T

39. d  /dn(n), Holland sample

40. Stoner Ferromagnetic Instability Stoner (1947) Finkelstein (1983) For a short range repulsive interaction Diffusion enhanced interactions in quantum dots Mean Field treatment: Andreev, Kamenev (1998) Numerics: Shepelyansky (2001)

41. Clean system Very small energy difference! antiferromagnetic ferromagnetic A. Finkelstein (1983), Castellani at al.,(1984) Shekhter, A. and Finkel'stein, A. M (2005) B. Tanatar and D.C. Ceperley (1989)

42. Real system S=0 Bhatt and Lee (1982)

43. Real system S=0 Bhatt and Lee (1982)

44. Real system S=0 Bhatt and Lee (1982) Andreev, A. V. & Kamenev, A. (1998) Kurland, I. L. and Aleiner, I. L. and Altshuler, B. L. (2000)