1. Sergey Kravchenko Approaching an (unknown) phase transition in two dimensions A. Mokashi (Northeastern) S. Li (City College of New York) A. A. Shashkin (ISSP Chernogolovka) V. T. Dolgopolov (ISSP Chernogolovka) T. M. Klapwijk (TU Delft) M. P. Sarachik (City College of New York in collaboration with:

2. Low disorder: weak (logarithmic) insulator Non-interacting electron gas in two dimensions (boring): High disorder: strong (exponential) insulator

3. ~35 ~1 r s Wigner crystal Strongly correlated liquidGas strength of interactions increases Coulomb energy Fermi energy r s = Terra incognita

4. Suggested phase diagrams for strongly interacting electrons in two dimensions 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 clean sample strongly disordered sample

5. University of Virginia In 2D, the kinetic (Fermi) energy is proportional to the electron density: E F = (  h 2 /m) N s while the potential (Coulomb) energy is proportional to N s 1/2 : E C = (e 2 /ε) N s 1/2 Therefore, the relative strength of interactions increases as the density decreases:

6. 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 be easily reached in clean samples. For comparison, in n-GaAs/AlGaAs heterostructures, this would require 100 times lower electron densities. Such samples are not yet available.

7. 10/10/09University of Virginia Al SiO 2 p-Si 2D electrons conductance band valence band chemical potential + _ energy distance into the sample (perpendicular to the surface)

8. Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995 Metal-insulator transition in 2D semiconductors

9. In very clean samples, the transition is practically universal: (Note: samples from different sources, measured in different labs) Sarachik and Kravchenko, PNAS 1999; Kravchenko and Klapwijk, PRL 2000

10. (Hanein, Shahar, Tsui et al., PRL 1998) Similar transition has later 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)

11. The effect of the parallel magnetic field:

12. (spins aligned) Magnetic field, by aligning spins, changes metallic R(T) to insulating: Such a dramatic reaction on parallel magnetic field suggests unusual spin properties!

13. Spin susceptibility near n c

14. T = 30 mK Spins become fully polarized (Okamoto et al., PRL 1999; Vitkalov et al., PRL 2000) Magnetoresistance in a parallel magnetic field Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 BcBc BcBc BcBc

15. Extrapolated polarization field, B c, vanishes at a finite electron density, n  Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 Spontaneous spin polarization at n  ? nn

16.  gm as a function of electron density calculated using g*m* =  ћ 2 n s / B c  B ( Shashkin et al., PRL 2001) nn

17. 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

18. M nsns dMdM nsns dnsdns BB Magnetization of non-interacting electrons spin-down spin-up gBBgBB

19. 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  

20. 1 fA!! Raw magnetization data: induced current vs. gate voltage d  /dB = - dM/dn B || = 5 tesla

21. Bar-Ilan University 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. Spin susceptibility exhibits critical behavior near the sample-independent critical density n  :  ~ n s /(n s – n  ) insulator T-dependent regime

23. g-factor or effective mass?

24. Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB (2002) Effective mass vs. g-factor (from the analysis of the transport data in spirit of Zala, Narozhny, and Aleiner, PRB 2001) :

25. Another way to measure m*: amplitude of the weak-field Shubnikov-de Haas oscillations vs. temperature (Rahimi, Anissimova, Sakr, Kravchenko, and Klapwijk, PRL 2003) high densitylow density

26. Comparison of the effective masses determined by two independent experimental methods: (Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003) 15 11 8 rsrs

27. Thermopower

28. Thermopower : S = -  V / (  T) S = S d + S g =  T +  T s  V : heat either end of the sample, measure the induced voltage difference in the shaded region  T : use two thermometers to determine the temperature gradient

29. Divergence of thermopower

30. 1/S tends to vanish at n t

31. Critical behavior of thermopower (-T/S) (n s - n t ) x x=1.0+/-0.1 n t =7.8+/-0.1*10 10 cm -2 and is independent of the level of the disorder ∝

32. In the low-temperature metallic regime, the diffusion thermopower of strongly interacting 2D electrons is given by the relation T/S ∝ n s /m Therefore, divergence of the thermopower indicates a divergence of the effective mass: m ∝ n s /(n s − n t ) Divergence of the effective electron mass Dolgopolov and Gold, JETP Lett. 2011 We observe the increase of the effective mass up to m  25m b  5m e !!

33. i.using Gutzwiller's theory (Dolgopolov, JETP Lett. 2002) ii.using an analogy with He 3 near the onset of Wigner crystallization (Spivak, PRB 2003; Spivak and Kivelson, PRB 2004) iii.extending the Fermi liquid concept to the strongly-interacting limit (Khodel et al., PRB 2008) iv.solving an extended Hubbard model using dynamical mean-field theory (Pankov and Dobrosavljevic, PRB 2008) v.from a renormalization group analysis for multi-valley 2D systems (Punnoose and Finkelstein, Science 2005) vi.by Monte-Carlo simulations (Marchi et al., PRB 2009; Fleury and Waintal, PRB 2010) A divergence of the effective mass has been predicted…

34. Transport properties of the insulating phase If the insulating state were due to a single-partical localization, a several orders of magnitude higher electric field would be needed to delocalize an electron: eE c l > W b ~ 0.1 – 1 meV But if this is a pinned Wigner solid, the electric field “pulls” many electrons while only one is pinned – hence, much weaker electric fields are required to depin it and break localization.

35. SUMMARY: 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 The dramatic increase of the spin susceptibility is due to the divergence of the effective mass rather than that of the g-factor and, therefore, is not related to the Stoner instability. It may be a precursor phase or a direct transition to the long sought-after Wigner solid. However, the existing data, although consistent with the formation of the Wigner solid, are not enough to reliably confirm its existence.