Typical shape of the quantum oscillations in the magnetostriction of tin single crystals as a function of magnetic field. The marker to the left of the upper curve indicates the magnetostriction scale. Temperature , : (a) for and (b) for tilted at 5° in the (010) plane. The lower curve reveals two groups of oscillations with a frequency ratio owing to two extremal cross-sections of the FS. Note the sawtooth shape of the oscillations. Reproduced from Ref. 6.
Schematic diagram of the Landau levels in the presence of Zeeman splitting . The left and right ladders of Landau levels are for spin-up and spin-down subbands .
The crossed magnetic field set-up. The main superconducting solenoid produces an in-plane magnetic field up to . The superconducting split coils, positioned inside the main solenoid, produce the normal field , which can be as high as for , and decreases gradually down to for . The sample (Si-MOSFET) is attached to the cold finger of the mixing chamber, with its plane perpendicular to the axis of the coils. Reproduced from Ref. 25.
Shubnikov–de Haas oscillations for (Si-MOSFET sample) at .27
Examples of curve fitting with Eq. (7): (a) , , , (the data correspond to Fig. 1b); (b) , , ; (c) , , , . The data are shown as solid curves, the fits (with parameters shown) as dashed curves. All are normalized to .
Shubnikov–de Haas oscillations versus for (i.e. ) and ; 0.5; 0.6; 0.7; : (a) and (b). The insets show the temperature dependences of the fitting parameters .
Parameters , , and for different samples as a function of (dots). The solid line in Fig. 7a shows the data from Ref. 21. The solid and open dots in Figs. 7b and 7c correspond to two different methods of finding (see the text). The solid and dashed curves in Fig. 7b are polynomial fits for the two dependences . The values of shown in Fig. 7c were obtained by dividing the data by the smooth approximations to the experimental curves shown in Fig. 7b.
Renormalized spin susceptibility measured by SdH effect in tilted or crossed fields on n-Si-MOS by Okamoto et al.,21 Pudalov et al.,27 and on by Zhu et al. 30 The horizontal bars indicate the upper and lower limits on the values, determined from the sign of the SdH oscillations measured at in Ref. 31. The dashed and dotted curves show two examples of extrapolation of the data.27
(a) Overall view of the SdH oscillations in low fields at different densities. The hollow circles show the oscillations for a high-mobility Si-MOSFET sample in high fields, corresponding to the reentrant QHE–insulator transitions.32 (b) Expanded view of one of the curves ( (right axis) and its oscillatory component normalized to (left axis)).27 The dashed segment indicates the region of the SdH measurements in Refs. 27 and 31.
(a) The dashed curve corresponds to extracted from the SdH data27 using the LK theory, the dash-dotted curve, to the empirical approach used in Ref. 27. The symbols depict obtained by fitting the transport data with the theory.43 The shaded regions in panels (a) and (b) show (with the experimental uncertainty) obtained by fitting our SdH data27 with the theory.41 (b) Comparison of recalculated from the available data: dots—Ref. 42, triangles and diamonds—Refs. 45 and 46, respectively.
SdH oscillations normalized to : (a) sample Si6-14, , ; (b) sample Si1-46, , . Dots show the data, solid curves—the theoretical dependences Eq. (7) modified for a finite and for samples Si6-14 and Si1-46, respectively. Panels (c) and (d) show the T-dependences of the amplitude of the SdH oscillations for Si6-14 and Si1-46 , the solid curves show the noninteracting LK-model Eq. (7), and the dashed curves, a fit based on the interaction theory (17).
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