De Haas-van Alphen Periods Research Articles

Thickness-induced violation of de Haas-van Alphen effect through exact analytical solutions at a one-electron and a one-composite fermion level

A systematic study of the energetics of electrons in an interface in a magnetic field is reported with exact analytical calculations based on a Landau level (LL) picture, by serious consideration of the finite thickness of the quantum well (QW). The approach is physically transparent and subtly different in its line of reasoning from standard methods avoiding any semi-classical approximation. We find “internal” phase transitions (at partial LL filling) for magnetisation and susceptibility that are not captured by other approaches and that give rise to nontrivial violations of the standard de Haas-van Alphen periods, in a manner that reproduces the exact quantal astrophysical behaviours in the limit of full three-dimensional (3D) space. Upon inclusion of Zeeman splitting, additional features are also found, such as global energy minima originating from the interplay of QW, Zeeman and LL Physics, while a corresponding calculation in a composite fermion picture with Λ-levels, leads to new predictions on magnetic properties of an interacting electron liquid. By pursuing the same line of reasoning for a topologically nontrivial system with a relativistic spectrum, we find evidence that similar effects might be operative in the dimensionality crossover of 3D strong topological insulators to 2D topological insulator quantum wells.

Formation of Condon domains as a phase transition

It is shown, for the first time, that approaching the Condon domain phase in temperature from above is accompanied by critical temperature growth of the susceptibility thereby indicating formation of Condon domains, which occurs as a phase transition. The study is based on a recent measurement of the temperature dependence of the susceptibility in silver in magnetic field of 25 T under conditions of the nonlinear de Haas-van Alphen effect. Experimental results for the positive and negative slopes of the magnetization in one de Haas-van Alphen period are in good agreement with results of the calculations presented here. The Curie–Weiss law for the susceptibility is derived for a needle-type specimen. The Curie constant is calculated.

Evidence for Condon domains in white tin with two de Haas–van Alphen periods

Evidence for Condon domains in white tin with two de Haas–van Alphen periods

Theory of the de Haas-van Alphen effect in quasi-two-dimensional conductors

According to the Peierls theory, the de Haas-van Alphen oscillations should have a saw-tooth form in an ideal two-dimensional electron gas when the chemical potential is pinned to a Landau level during the entire de Haas-van Alphen period (constant number of carriers) or when a reservoir of carriers exists (fixed chemical potential). Recent experiments indicate that neither of these conditions are fulfilled, during a single period in AsF 5 intercalated graphite. We have studied both: analytically and numerically the smoothing out of the saw tooth form of the magnetization, at finite temperature, due to an external reservoir of carriers.

Quantum oscillations of magnetoresistance in point contacts

The theory of electrical conductivity of metallic point contacts in quantizing magnetic field is presented. The contact resistance is shown to oscillate in the ballistic case as a function of the inverse magnetic field with the de Haas-van Alphen period, in agreement with recent experiments [4].

Minority carriers in graphite and theH-point magnetoreflection spectra

Analysis of the $H$-point magnetoreflection spectra of graphite is shown to be sensitive not only to the magnitude, but also to the sign of the Slonczewski-Weiss-McClure band parameter $\ensuremath{\Delta}$. The demonstration that $\ensuremath{\Delta}$ is small and negative provides a definite assignment for the larger of the minority de Haas-van Alphen periods to the Brillouin-zone corner, the $H$-point. This assignment is in good agreement with other experimental data.

Low-field de Haas-van Alphen effect in thallium

De Haas-Van Alphen periods of the order of 10 −7 and 10 −6 gauss −1 in thallium have been observed by torque and modulation methods. These periods are not explained by the Fermi surface in either the single-OPW model or Soven's relativistic-OPW model.

The Tight-Binding and the Nearly-Free-Electron Approach to Lattice Electrons in External Magnetic Fields

The energy structure for lattice electrons in an external magnetic field is treated by extension of either the tight-binding or the nearly-free-electron method. The secular problems resulting from both methods are identical, if the coupling integrals and the real lattice characterizing the tight-binding method are interchanged with the Fourier integrals and the orbit lattice characterizing the nearly-free-electron method. A Bloch-type transformation reducing these essentially two-dimensional secular problems to one dimension is performed, following the procedure introduced in a preceding paper. For rational fluxes $\frac{2\ensuremath{\pi}N}{M}$ per lattice cell, an even finite secular determinant is left, yielding a splitting of each zero-field band into $M$ subbands. Total agreement between the level structures resulting from the two approaches is found. The relation of these structures to the de Haas-van Alphen periods in the case of magnetic breakdown is discussed.

The fermi surface and optical properties of copper

A Fourier expanded model Hamiltonian is applied to copper. The parameters in this phenomenological Hamiltonian are adjusted to fit the de Haas-van Alphen periods, the cyclotron effective masses, and the position of the lowest direct optical interband absorption. The agreement with experiment suggests that the electron-phonon enhancement of the effective mass is small, and that the optical properties are consistent with direct wave-vector conserving transitions between one-electron states.

Energy Band and Fermi Surface of Divalent Metals under Pressure

The energy band structure of face-centered cubic metals Ca, Sr, and Ba was investigated by a pseudopotential-like approximation, using the screened model potential of Abarenkov, Heine, and Animalu. All three of the metals exhibit a negative energy gap at atomospheric pressure, the magnitude of which tends to zero with increasing pressure. According to a symmetry demanded degeneracy along the line WL of the Brillouin zone, the transition to semiconductors with positive energy gap seems to be unlikely. The Fermi surface consists of holes in the first zone around the points $W$ and electrons in the second zone around the points $L$. The dimensions of these surfaces are also decreasing under pressure. The Fermi surface of Ca was studied in detail. Numerical results for the de Haas-van Alphen periods cyclotron masses, are given. The pressure dependence of electrical resistivity of Ca was calculated and found to be in qualitative agreement with the experiments.

Electronic Structure of Zirconium

Augmented-plane-wave (APW) calculations of the electronic structure of zirconium have been carried out. The resulting Fermi surface is almost entirely different from the model proposed earlier by Altmann and Bradley on the basis of cellular calculations. The APW Fermi surface (after slight modification) can account for the angular dependence of the de Haas-van Alphen periods observed by Thorsen and Joseph; the theoretical values of the periods differ from the experimental results by no more than about 40%. Although this is by no means an outstanding example of close agreement between experimental and theoretical results, it is believed that the new Fermi-surface model represents a substantial improvement over previous work and is a better starting point for any future attempts to deduce an improved model whenever additional experimental data become available.

De Haas-van Alphen Effect in Pyrolytic and Single-Crystal Graphite

The de Haas-van Alphen (DHVA) oscillations have been observed in the differential susceptibility of well-annealed pyrolytic graphite (PG) with the low-frequency field-modulation technique. The results for PG are compared with similar studies of natural single crystal (SCG) by Soule et al., and by the present authors. Majority-electron and majority-hole DHVA periods in PG are respectively equal to the periods in SCG for $0\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\le}\ensuremath{\theta}\ensuremath{\lesssim}50\ifmmode^\circ\else\textdegree\fi{}$, where $\ensuremath{\theta}$ is the angle between the applied field and the $c$ axis of the sample. Observation of a long-period, minority-carrier DHVA oscillation in PG and SCG confirms Soule's discovery of such an oscillation in SCG and provides a sensitive test for the Slonczewski-Weiss band model. The minority-carrier oscillation in PG is qualitatively but not quantitatively similar to the one in SCG. This oscillation is attributed to magnetic energy levels passing through the Fermi level near the Brillouin-zone corner at ${k}_{z}=\frac{\ensuremath{\pi}}{{c}_{0}}$, where for $\ensuremath{\theta}=0\ifmmode^\circ\else\textdegree\fi{}$ there is no extremal area of the Fermi surface at zero field, when spin-orbit interactions are neglected. This interpretation is supported by the calculation of an oscillatory free energy and by the excellent agreement for PG between the predicted values and experimental results for the period, effective mass, period anisotropy, and phase. The phase is not the value which would be predicted from a free-electron model for either a maximal or a minimal cross-sectional area. The existence of minority-carrier oscillations in PG indicates that the hole and electron Fermi surfaces are closed and that the electron surface intersects the hexagonal face of the Brillouin zone. The band parameters of the Slonczewski-Weiss model for PG were evaluated from the DHVA data and the preliminary magnetoreflection measurements of Dresselhaus and Mavroides. A similar analysis for SCG was carried out successfully for the majority-carrier pockets, but did not provide a consistent picture for the minority-carrier surface. The effects of spin-orbit interaction on the Fermi surface are discussed. Neutron and x-ray diffraction studies show that the $c$ axes of the individual crystallites in a PG sample are aligned to no better than \ensuremath{\sim}1\ifmmode^\circ\else\textdegree\fi{}. The effects of this misalignment as well as of the finite crystallite size on the DHVA oscillations are considered.

Spin-Orbit Interaction in Graphite

Using symmetry arguments, the effective-mass Hamiltonian including spin-orbit interaction is derived for energy bands with extrema near the vertical edge of the hexagonal prism which represents the Brillouin zone of graphite. The energy bands in the plane normal to the vertical edge are described by k\ifmmode\cdot\else\textperiodcentered\fi{}p perturbation theory, whereas along the edge a Fourier expansion is used for all the matrix elements. It is shown that spin-orbit interaction lifts all band degeneracies (other than the Kramers degeneracy), and affects the graphite Fermi-surface topology at the Brillouin-zone boundary ${k}_{z}=\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{{c}_{0}}$, where two de Haas-van Alphen periods are predicted. Magnetic energy levels for a static magnetic field H\ensuremath{\parallel}c are obtained by solution of the effective-mass Hamiltonian. Selection rules for infrared interband transitions are discussed. An evaluation of the spin-orbit band parameters is suggested by analysis of structure in the low-quantum-limit magneto-reflection data and of the low-frequency de Haas-van Alphen oscillations.

Linear-Network Model for Magnetic Breakdown in Two Dimensions

A heuristic model is set up for magnetic breakdown in a two-dimensional rectangular lattice at right angles to an applied magnetic field. The model is an adaptation of a method due to Pippard to a set of coupled ordinary differential equations derived from the Schr\odinger equation. A linear-chain network is set up. This network can also be derived from Pippard's two-dimensional network and it suggests a simple way to compute the energy bands not only when the number $F$ of flux quanta through a unit cell is the reciprocal of an even integer (the case treated by Pippard) but also when $F$ is any rational fraction. Numerical computations of the energy bands suggest that in the latter case the electron wave moves on large orbits which might be called hyperorbits. These hyperorbits may be open in a rectangular lattice and may give a resonant open-orbit ultrasonic attenuation. It is also found that when a free-electron Landau level is broadened by the lattice, it splits into two bands separated by a gap. This gap moves through the states from the Landau level as $F$ is changed and may give rise to new de Haas-van Alphen periods. The physical cause of this gap is discussed.

Fermi Surface, Pseudopotential Coefficients, and Spin-Orbit Coupling in Lead

The de Haas-van Alphen periods in lead have been studied in 200 kG impulsive fields, employing techniques with sensitivity and selectivity which are far improved over those used in an earlier study of this metal, and several new sets of oscillations have been discovered. The results confirm in detail the correctness of a nearly-free-electron Fermi surface based on four conduction electrons per atom, and the experimentally determined Fermi surface has been described in terms of an interpolation scheme using four orthogonalized plane waves for each wave vector k. As would be expected for a heavy metal such as lead, it is necessary to allow for the large spin-orbit interaction in order to achieve an accurate description of all portions of the Fermi surface. The four adjustable parameters required in the description have been determined by a least-squares fit to eight observed extremal areas of cross section, and are found to be (in Ry): Fermi energy ${E}_{f}=0.718\ifmmode\pm\else\textpm\fi{}0.001$; Fourier coefficients of the pseudopotential ${V}_{111}=\ensuremath{-}0.084\ifmmode\pm\else\textpm\fi{}0.002$ and ${V}_{200}=\ensuremath{-}0.039\ifmmode\pm\else\textpm\fi{}0.002$; and spin-orbit interaction $\ensuremath{\lambda}=0.096\ifmmode\pm\else\textpm\fi{}0.002$. These values of the parameters refer to a specific interpolation scheme and assume that the mass in the kinetic-energy matrix elements is the free-electron mass (i.e., no attempt has been made to consider explicity many-body effects or the electron-phonon interaction). When the above parameters are used, the total occupied volume is calculated to correspond to 4.02\ifmmode\pm\else\textpm\fi{}0.02 electrons per atom, and the model confirms the experimental finding that the fourth zone is empty. The calculated dispersion curves $E(\mathrm{k})$ reflect the `inert-pair' behavior which is so well known in the chemistry of lead salts, in that there is a large energy gap between a filled $6s$-like band and the lowest branch of the $6p$-like bands which is never less than 0.13 Ry. The predictions of the model as regards the detailed orientation dependence of the de Haas-van Alphen periods are found to be in excellent agreement with experiment; comparison is also made with the Fermi-surface dimensions, cyclotron masses, magneto-resistance, etc., as determined by other experiments.

Cyclotron Resonance in Aluminum

A systematic study of Azbel'-Kaner cyclotron resonance in aluminum at 35 Gc/sec has clarified and extended previous work. Results on the orientation dependence and relative values of the cyclotron mass for different electron orbits correspond closely with expectations based on Harrison's free-electron model. However, the observed mass values are consistently 55-60% higher than those predicted by the free-electron model. Since the nearly-free-electron model agrees with ab initio band-structure calculations which exist for aluminum and which are thought to be reliable within the independent-particle model, the bulk of this discrepancy is presumed to be attributable to the many-body effects of correlation and electron-phonon interactions. The anomalous cyclotron resonance spectrum attributed in previous work to a cyclotron mass of $3.10m$ ($m$ is the free electron mass) is elucidated in terms of current sheets which reproduce the skin currents deep within the metal. These sheets are produced by the spatially periodic magnetic focusing of limiting-point electrons on the Fermi surface. This mechanism leads to a cyclotron mass for these electrons of $1.55m$, just one half the value given on the Azbel'-Kaner model. Quantum oscillations in the surface impedance analogous to the de Haas-van Alphen oscillations in the susceptibility are observed at \ensuremath{\sim}1.5\ifmmode^\circ\else\textdegree\fi{}K. Observed periods agree with the de Haas-van Alphen periods measured by Gunnersen.

The Fermi surface and conductivity of calcium

A comparison is proposed between a Fermi surface for calcium calculated by the cellular method and de Haas-van Alphen periods recently measured by Condon and Marcus. The calculations predict calcium metal to be a semiconductor at high pressure and this agrees semiquantitatively with the recent results of Stager and Drickamer.

Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas

An electron gas with short-range interactions is considered in the presence of a uniform magnetic field. It is shown that (1) the cyclotron resonance frequency is independent of the interaction; (2) for a two-dimensional gas, the de Haas-van Alphen period is independent of the interaction. The low-lying excited states are briefly discussed.

Effect of Interactions on Determination of Fermi Surfaces

The effect of both electron-electron and electron-phonon interactions on a degenerate electron gas in a uniform positive background is considered. It is shown that when electron-electron interactions alone are considered the free-electron mass is still measured by cyclotron resonance, the Faraday effect, and optical constants. However, the period of the de Haas-van Alphen oscillations is changed from what one calculates neglecting interactions and is changed in the same way that the specific heat is. When electron-phonon interactions are added everything changes. In particular, it is shown that the cyclotron mass is no longer the free value, and the de Haas-van Alphen period and the specific heat are changed in different ways. Comparison with measurements on aluminum which approximates the model used shows that both electron-phonon and electron-electron effects are important and of the same magnitude.

The fermi surface in white tin

Abstract The orientation dependence of the de Haas-van Alphen periods in tin single crystals has been studied in high pulsed magnetic fields. As many as seven periods have been found at some orientations, indicating that the Fermi surface is much more complicated than the 3-ellipsoid model suggested by Fawcett. The general degree of complexity of the results is much the same as that predicted by the free-electron model, but serious discrepancies are found when a detailed comparison is made. Effective mass values have been determined at the principal orientations from the temperature dependence of the amplitudes; these are compared with the results of cyclotron resonance experiments and with the free-electron orbit masses.