1. Sergey Kravchenko Divergent effective mass in strongly correlated two-dimensional electron system S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk NEU ISSP Texas A&M TU Delft in collaboration with: A. Punnoose M. P. Sarachik A. A. Shashkin CCNY CCNY ISSP

2. ~1 ~35 r s Gas Strongly correlated liquid Wigner crystal strength of interactions increases Coulomb energy Fermi energy r s = Terra incognita

3. 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

4. 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:

5. 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. For comparison, in n-GaAs/AlGaAs heterostructures, this would require 100 times lower electron densities. Such samples are still not available.

6. 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)

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

8. 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

9. (Hanein, Shahar, Tsui et al., PRL 1998) 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)

10. The effect of the parallel magnetic field:

11. (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!

12. Spin susceptibility near n c

13. 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

14. Scaling of the magnetoresistance yields B c (n s )

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. Is it possible to separate g* and m*? Which of the two is responsible for the renormalization of  ?  (T)/  = 1 – A*k B T g*m* =  ћ 2 n s / B c  B A* = -(1+  F 0 s ) g*m* /  ћ 2 n s = -(1+  F 0 s ) / B c  B (Zala, Narozhny, and Aleiner, PRB 2001).

25. Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB (2002) Indeed, conductivity in the metallic regime is a linear function of T (not too close to the transition):

26. 1/A* ~ µB c yields g(n s ) = const. This conclusion is independent of the value of A* and of how accurately B c is determined (only functional forms are important) 1/A* µBcµBc

27. Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB (2002) Effective mass vs. g-factor:

28. 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

29. Comparison of the effective masses determined by two independent experimental methods: (Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003) recalculated ratio E e-e /E F

30. Thermopower

31. 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

32. Divergence of thermopower

34. In the low-temperature metallic regime, the diffusion thermopower is given by the relation T/S ∝ n s /m Therefore divergence of the thermopower indicates divergence of the effective mass: m ∝ n s /(n s − n t ) Divergence of the effective electron mass

35. 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

36. 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 sample 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