Sasha Kuntsevich, Nimrod Teneh, Vladimir. Pudalov, Teun Klapwijk Aknowlegments: A. Finkelstein Spin Susceptibility of a 2D Electron Gas M. Reznikov

V. Pudalov at al., 2001 Metal-Insulator Transition in a Silicon Inversion Layer

Theory: interaction and disorder enhanced susceptibility A. Finkelstein (1983), Castellani at al.,(1984) Temperature-dependent correction to Pauli Susceptibility due to electron-electron interaction ( A. Finkelstein, A. Shekhter, 2006 ) Motivation A. Chubukov, D. Maslov, 2009

Shubnikov - de Haas measurements of the Spin Polarization F. Fang and P. Stiles, (1968), T. Okamoto at al., (1999) S. Vitkalov at al. (2000), V. Pudalov at.al., (2001) rsrs Requires low temperatures and finite magnetic fields (  B B>k B T) Problematic in the vicinity of the MIT AdvantagesDisadvantages Straightforward separation between orbital (diamagnetic) and spin contributions

Analysis of the in-plane magnetoresistance Advantages: Does not require high magnetic field Critical density is accessible Disadvantages: Heavily interpretation-dependent: a) Saturation of the in-plane magnetoresistance full polarization incorrect at low B (e.g E. Tutuc at al.) b) The saturation field may be recovered on the basis of the low field magnetoresistance A. Shashkin et al. PLR, 2001, S. Vitkalov et al. PRL 2001

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

Samples Russian samples, beginning of 80 th, Holland samples, mid 80 th,  ¼ 3.4 x10 4 cm 2 Si Field effect transistors Typical energy scales  p ¼ 3 ps ¼ ~ /(k B ¢ 2K)

The Principle of the Measurements Maxwell relation:  eVGeVG 00 ee W Al W 2D

Experimental setup _ + VGVG Out Modulated magnetic field B+  Current Amplifier Ohmic contact Gate SiO 2 Si 2D electron gas AdvantagesDisadvantages Measures thermodynamic magnetization Accessibility of the “Insulating phase” Does not require low temperatures Measures thermodynamic magnetization Measures, which is unknown at small n; Requires assumptions for the integration

dm/dn, expectations for degenerate case no interactions B  gg Polarization field

dm/dn, expectations for a single spin

Raw data, low fields

Raw data

d  /dn(n), T=1.7 ¥ 13K, Russian sample

d  /dn(n), Holland sample

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!

 (n), T=1.7 ¥ 13K

 (n), T=1.7 ¥ 13K, offset 0.7V offset 0.7V offset 0.5V

Old results (Prus et al, 2003)

Susceptibility vs. Temperature

Field dependence of the magnetic moment

Susceptibility in at B=2T

Maximal 1.7K

Raw data

-dM/dn at cm -2 B

Comparison with Transport Measurements

Conclusions Low-field susceptibility is many times the Pauli one Susceptibility is strongly temperature dependent even at high densities (most surprising) Low temperature susceptibility is strongly nonlinear The field scale for the nonlinearity is B c ¼ k B T/6  B

ВЫВОД О ФМ НЕУСТОЙЧИВОСТИ; (B>1.5 Т, Т<0.6К) НЕТ ЗАВИСИМОСТИ ВОСПРИИМЧИВОСТИ ОТ Т. ТД измерения: ранние результаты. A. Shashkin et al., PRL (2006). Порядок величины!  Гц  Образец закрывается при малых n