07/11/11SCCS 2008 Sergey Kravchenko in collaboration with: AMAZING PROPERTIES OF STRONGLY CORRELATED ELECTRONS IN TWO DIMENSIONS A. Punnoose M. P. Sarachik A. A. Shashkin CCNY CCNY ISSP S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk NEU ISSP Texas A&M TU Delft

Outline Scaling theory of localization: “all electrons are localized in 2D” Samples What do experiments show? Magnetic properties of strongly correlated electrons in 2D Conclusions

Band theory of metals: Conduction band Valence band Conduction band InsulatorMetal But it turns out that even if the Fermi level lies in the conduction band, the system may be insulating.

Localization by disorder (Anderson localization) Low disorder High disorder Electrons can penetrate infinitely far (with some scattering) Electrons are localized at a certain distance, called “localization length”, L loc L lo c

“… very few believed in [localization] at the time, and even fewer saw its importance... It has yet to receive adequate mathematical treatment, and one has to resort to the indignity of numerical simulations to settle even the simplest questions about it." P.W. Anderson, Nobel Lecture, 1977

In 1979, a powerful theory was created by the “Gang of Four” (Abrahams, Anderson, Licciardello, and Ramakrishnan), according to which, there is no conductivity in 2D at low temperatures. This became one of the most influential paradigms in modern condensed matter physics. However, this prediction is valid for non-interacting electrons only.

~1 ~35 r s Gas Strongly correlated liquid Wigner crystal Insulator Insulator strength of interactions increases Coulomb energy Fermi energy r s = Terra incognita But electrons do interact via Coulomb forces!

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:

Scaling theory of localization: “all electrons are localized in two dimensions Samples What do experiments show? Magnetic properties of strongly correlated electrons in 2D Conclusions

07/11/11SCCS 2008 silicon MOSFET Al SiO 2 p-Si 2D electrons conductance band valence band chemical potential + _ energy distance into the sample (perpendicular to the surface)

SCCS 2008 Why Si MOSFETs? large m*= 0.19 m 0 two valleys low average dielectric constant  =7.7 As a result, at low electron 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

Scaling theory of localization: “all electrons are localized in two dimensions Samples What do experiments show? Magnetic properties of strongly correlated electrons in 2D Conclusions

Strongly disordered Si MOSFET ( Pudalov et al.)  Consistent (more or less) with the one-parameter scaling theory

S.V. Kravchenko, G.V. Kravchenko, W. Mason, J. Furneaux, V.M. Pudalov, and M. D’Iorio, PRB 1995 Clean sample, much lower electron densities

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

T = 30 mK Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 The effect of the parallel magnetic field:

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

Scaling theory of localization: “all electrons are localized in 2D” Samples What do experiments show? Magnetic properties of strongly correlated electrons in 2D Conclusions

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

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

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 Ohm

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

Spin susceptibility exhibits critical behavior near the sample-independent critical density n  :  ~ n s /(n s – n  ) Are we approaching a phase transition?

g-factor or effective mass?

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, (2002) Effective mass vs. g-factor Not the Stoner scenario! Wigner crystal? Maybe, but evidence is insufficient

SUMMARY: (i) There exists a metallic state in 2D, contrary to the 30-years old paradigm! (ii) Strong interactions in clean two-dimensional systems lead to strong increase and possible divergence of the spin susceptibility: the behavior characteristic of a phase transition (iii) The dramatic increase of the spin susceptibility is caused by the effective mass rather than by the g-factor