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Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA)

Metal-Insulator Transition in 2D Electron Systems. Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA) Mariam Rahimi (now at UC Berkeley) Sergey Kravchenko Alexander Shashkin Valeri Dolgopolov Teun Klapwijk.

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Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA)

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  1. Metal-Insulator Transition in 2D Electron Systems Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA) Mariam Rahimi (now at UC Berkeley) Sergey Kravchenko Alexander Shashkin Valeri Dolgopolov Teun Klapwijk

  2. Si MOSFET: Silicon Metal-Oxide-Semiconductor Field-Effect Transistor 2-D QUANTUM SYSTEMS: • Silicon MOSFETs • GaAs/AlGaAs heterostructures • SiGe heterostructures • Surface of a material (liquid helium, graphene sheets)

  3. Why Si MOSFET? • largem* =0.19 m0 • average = 7.7 • twovalleysnv = 2 At low densities, ns ~ 1011 cm-2, Coulomb energy exceeds Fermi energy: EC >> EF electron density decreases strength of interactions increases rs = EC / EF >10 – strongly interacting regimecan easily be reached

  4. Metal-Insulator Transition in 2D High-Mobility Si MOSFETs • Similar transition is also observed in other 2D structures: • p-Si:Ge (Coleridge’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) B = 0 Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995 Hanein, Shahar, Tsui et al., PRL 1998

  5. In very clean samples, the transition is practically universal: Klapwijk’s sample: Pudalov’s sample: (Note: samples from different sources, measured in different labs)

  6. … in contrast to strongly disordered samples: disordered sample: clean sample:

  7. The effect of magnetic field

  8. Magnetic field, by aligning spins, changes metallic R(T) to insulating: (spins aligned)

  9. Such a dramatic reaction on parallel magnetic field suggests unusual spin properties

  10. How to study magnetic properties of 2D electrons?

  11. Transport measurements: standard four-terminal technique - Diagonal resistance - Hall resistance • Rotator equipped Oxford dilution refrigerator • Base temperature ~ 30 mK • High mobility (100)-Si MOSFET μ=3 m2/Vs at T=0.1 K • Excitation current 0.1 – 0.2 nA • f = 0.4 Hz

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

  13. Extrapolated polarization field, Bc, vanishes at a finite electron density, nc Vanishing Bc at a finite n  ncindicates aferromagnetic transition in this electron system The fact that n is sample independent and n  nc indicates that the MIT in clean samplesis drivenby interactions Shashkin et al, 2001 Pudalov et al, 2002 Vitkalov, Sarachik et al, 2001 nc

  14. ~ gm as a function of electron density calculated using Shashkin et al., PRL 2001 nc

  15. Effective Mass Measurements: amplitude of the weak-field Shubnikov-de Haas oscillations vs. temperature high density: ns= 5x1011 cm-2 low density: ns = 1.2x1011 cm-2 Rahimi, Anissimova, Sakr, Kravchenko, and Klapwijk, PRL 2003

  16. dots – ν = 10 squares – ν = 14 solid line – fit by L-K formula ns = 1.2x1011 cm-2 Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003 The amplitude of the SdH oscillations follows the calculated curve down to the lowest achieved temperature: the electrons are in a good thermal contact with the bath.

  17. Comparison of the effective masses determined by two independent experimental methods: * Therefore, the sharp increase of the spin susceptibility near the critical density is due to the enhancement of the effective massrather then g-factor, unlike in the Stoner scenario Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003

  18. Measurements of thermodynamic magnetization suggested by B. Halperin (1998); first implemented by Prus et al. (2003) LVC6044 CMOS Quad Micropower Operational Amplifier with noise level: 0.2 fA/(Hz)1/2 f = 0.45 Hz Bmod = 0.01 – 0.03 tesla R=1010  - Lock-in amplifier + Vg Gate Current-to-Voltage converter SiO2 Modulated magnetic field B + Bmod Si 2D electron layer Ohmic contact Maxwell relation: C – capacitance  - chemical potential

  19. Magnetic field of the full spin polarization Bc vs. ns spontaneous spin polarization at nc: non-interacting system mBns B/Bc forB < Bc Bc = ph2ns/mB g*m* Bc = ph2ns/2mBmb M = mBx ns = mBns forB > Bc dM Bc dns B > Bc 0 B ns nc B < Bc 0 ns

  20. Raw magnetization data: induced current vs. gate voltage dm/dB = - dM/dn B|| = 5 tesla 1 fA!! the onset of complete spin polarization d/dB = 0 Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

  21. Raw magnetization data: induced current vs. gate voltage Integral of the previous slide gives M (ns): complete spin polarization at ns=1.5x1011 cm-2 B|| = 5 tesla

  22. dm/dB vs. ns in different parallel magnetic fields: Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

  23. Magnetic field of full spin polarization vs.electron density from magnetization measurements Spontaneous spin polarization at nc?

  24. Measurements of thermodynamic density of states LVC6044 CMOS Quad Micropower Operational Amplifier with noise level: 0.2 fA/(Hz)1/2 f = 0.3 Hz Vg = 0.09V C0 = 624 pF R=1010  - Lock-in amplifier + Vg Gate Current-to-Voltage converter SiO2 Si 2D electron layer Ohmic contact Modulated gate voltage Vg + dVg C0– geometric capacitance A – sample area

  25. Polarization field from capacitance measurements: Jump in the density of states signals the onset of full spin polarization D-1 fully spin-polarized electrons ns spin-unpolarized electrons Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

  26. Magnetic field of full spin polarization vs. electron density: data become T-dependent, possibly due to localized band-tail electron density (1011 cm-2) Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

  27. Spin susceptibility exhibits critical behavior near the metal-insulator transition: c ~ ns/(ns – nc) insulator cannot measure Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, cond-mat/0409100

  28. g-factor or effective mass?

  29. dm/dB vs. ns in perpendicular magnetic field

  30. g-factor measurementsin perpendicular fields: Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, cond-mat/0503123

  31. g-factor and effective mass: g-factor: Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, cond-mat/0503123

  32. Summary of the results obtained by four (or five) independent methods

  33. spin susceptibility critically grows near the metal-insulator transition • the enhancement of the g-factor is weak and practically density independent • the effective mass becomes strongly enhanced as the density is decreased Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, Phys. Rev. Lett. 96, 036403 (2006); Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, Phys. Rev. Lett. 96, 046409 (2006) Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, Phys. Rev. Lett. 91, 046403 (2003)

  34. Temperature-dependent corrections to conductivity due to electron-electron interactions Low temperatures, Diffusive regime (Tt<<1)

  35. Zeitschrift fur Physik B (Condensed Matter) -- 1984 -- vol.56, no.3, pp. 189-96 • Weak localization and Coulomb interaction in disordered systems • Finkel'stein, A.M. • L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR (always “insulating” behavior) • Insulating behavior when interactions are weak • Metallic behavior when interactions are strong • Magnetic field destroys metal

  36. Same corrections persist to ballistic regime (Higher temperatures; Tt>>1) • Insulating behavior when interactions are weak • Metallic behavior when interactions are strong • Magnetic field destroys metal

  37. Theory of the metal-insulator transition (diffusive regime)

  38. …the point of the metal to insulator transition correlates with the appearance of the divergence in the spinsusceptibility… note that at the fixed point the g-factor remains finite Punnoose and Finkelstein, Science, Vol. 310. no. 5746, pp. 289 - 291 These conclusions are in agreement with experiments

  39. Punnoose and Finkelstein, Science Vol. 310. no. 5746, pp. 289 - 291

  40. SUMMARY: • Pauli spin susceptibility critically grows with a tendency to diverge near the critical electron density • We find no sign of increasing g-factor, but the effective mass is strongly (×3) enhanced near the metal-insulator transition and… Punnoose-Finkelstein theory gives a quantitatively correct description of the metal-insulator transition in 2D

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