^{1,a)}

### Abstract

The quantum oscillation effect was discovered in Leiden in 1930, by W. J. de Haas and P. M. van Alphen when measuring magnetization, and by L. W. Shubnikov and de Haas when measuring magnetoresistance. Studying single crystals of bismuth, they observed oscillatory variations in the magnetization and magnetoresistance with magnetic field. Shoenberg, whose first research in Cambridge had been on bismuth, found that much stronger oscillations are observed when a bismuth sample is cooled to liquid helium temperature rather than liquid hydrogen, which had been used by de Haas. In 1938 Shoenberg went from Cambridge to Moscow to study these oscillations at Kapitza’s Institute where liquid helium was available at that time. In 1947, J. Marcus observed similar oscillations in zinc and that persuaded Schoenberg to return to this research. After that, the dHvA effect became one of his main research topics. In particular, he developed techniques for quantitative measurement of this effect in many metals. A theoretical explanation of quantum oscillations was given by L. Onsager in 1952, and an analytical quantitative theory by I. M. Lifshitz and A. M. Kosevich in 1955. These theoretical advances seemed to provide a comprehensive description of the effect. Since then, quantum oscillations have been widely used as a tool for measuring Fermi surface extremal cross-sections and all-angle electron scattering times. In his pioneering experiments of the 1960’s, Shoenberg revealed the richness and deep essence of the quantum oscillation effect and showed how the beauty of the effect is disclosed under nonlinear conditions imposed by interactions in the system under study. It was quite surprising that “magnetic interaction” conditions could cause the apparently weak quantum oscillation effect to have such strong consequences as breaking the sample into magnetic (now called “Shoenberg”) domains and forming an inhomogeneous magnetic state. With his contributions to the field of quantum oscillations and superconductivity, Shoenberg is undoubtedly one of the 20th century’s foremost scientists. We describe experiments to determine the quantitative parameters of electron–electron interactions in line with Shoenberg’s idea that quasiparticle interaction parameters can be found by analyzing quantum oscillations as modified by interactions.

The author has benefited from fruitful collaboration with M. Gershenson, H. Kojima, E. M. Dizhur, G. Bauer, G. Brunthaler, O. E. Omel"yanovskii, N. N. Klimov, D. A. Knyazev, and A. Yu. Kuntsevich in performing measurements. The work was partially supported by grants from RFBR, the Russian Academy of Sciences, and the Russian Ministry of Education and Science (under contracts Nos. 02.552.11.7093, 14.740.11.0061, P2306, P798, P1234).

I. INTRODUCTION: CORRESPONDENCE WITH DAVID SHOENBERG

II. INTERACTING TWO-DIMENSIONAL ELECTRON SYSTEM

A. Renormalization of the quasiparticle parameters

B. Quantum oscillations in 2D electron gases as a tool for extracting interaction constants

C. The idea of measuring, , and

D. Crossed field technique

E. Data analysis

F. Comparison with other data

1. High density/weak interaction regime

2. Low density/strong interaction regime

III. MAGNETOOSCILLATIONS IN STRONGLY INTERACTING 2D ELECTRON SYSTEM

A. Refinement of the derived values

B. Other quasiparticle parameters derived from SdH data

1. Valley splitting

2. Drude scattering time

IV. CONCLUSION

### Key Topics

- Magnetic fields
- 26.0
- Fermi liquid theory
- 15.0
- Magnetic field measurements
- 14.0
- Effective mass
- 10.0
- Collective excitations
- 9.0

## Figures

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.

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 .

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.

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}

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 .

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 .

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.

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}

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) 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 data^{27} 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 data^{27} with the theory.^{41} (b) Comparison of recalculated from the available data: dots—Ref. 42, triangles and diamonds—Refs. 45 and 46, respectively.

(a) The dashed curve corresponds to extracted from the SdH data^{27} 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 data^{27} 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).

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).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content