Magnetic Breakdown Effect Research Articles

A de Haas-van Alphen study of niobium: Fermi surface, cyclotron effective masses, and magnetic breakdown effects

The Fermi surface of niobium has been investigated using the de Haas-van Alphen effect. Data were taken at temperatures as low as 0.3 K and in fields as high as 130 kG. An on-line minicomputer was used to Fourier-transform the digitized signals. Many new extremal area data have been obtained, including oscillations associated with the previously unobserved γ-centered hole octahedron and γ and N-centered orbits on the so-called jungle gym. An additional set of signals has been observed near [100], which are thought to be a result of magnetic breakdown between the second zone octahedron and third zone jungle gym. A separate low-frequency signal was observed and is believed to be a result of magnetic-breakdown-induced quantum interference oscillations. Anisotropies of the cyclotron effective mass have been determined for many orbits on all three of the Fermi surface sheets. Finally, the area data have been used to parametrize the Fermi surface in terms of scattering phase shifts in a Korringa-Kohn-Rostoker band structure formalism.

Magnetic breakdown in aluminium

Magnetic breakdown effects in Al were investigated by transverse magnetoresistance measurements on specimens with (100), (110), (111) and (112) current directions. The rotation diagrams of the magnetoresistance exhibit small anisotropies which agree with earlier measurements using different experimental techniques and with calculations on the basis of a closed Fermi surface model for Al. Superimposed on this small anisotropy characteristic peaks occur which are attributed to magnetic breakdown. In the field dependence a strong linear increase of the high-field magnetoresistance is observed for all specimen orientations affected by magnetic breakdown, thus demonstrating that magnetic breakdown is responsible for, or at least amplifiers, the linear high-field magnetoresistance in Al. For the magnetic field B//(001) the magnetoresistance oscillates periodically in 1/B. The various effects of magnetic breakdown on the magnetoresistance are explained by consideration of the different geometries of breakdown orbits which can occur in Al.

Effect of Magnetic Breakdown on the Magnetoconductivity of Cadmium

The magnetoconductivity and magnetoresistivity tensors of cadmium have been calculated by the path integral method for magnetic fields in the [Formula: see text] and [Formula: see text] directions. A uniform relaxation time and a modified nearly free electron Fermi surface were used. Magnetic breakdown across the HAL spin–orbit energy gap vitiated Kohler's rule and necessitated separate calculations for different relaxation times. Even with open orbits, cadmium was found to be a 'compensated' metal in that the Hall terms never dominated the conductivity tensor determinant. A single breakdown field was found to be inadequate to describe magnetic breakdown on the sheaf of orbits which touch the HAL plane. A range of breakdown fields was calculated across the sheaf of open orbits. From an explanation of induced torque data, it was found that the ratio of the spin–orbit energy gap in the HAL plane to the Fermi velocity is 1.4 × 10−8 eV s cm−1. The resulting field dependences of the magnetoconductivity and magnetoresistivity tensor components are presented and discussed. The use of the path integral technique when there is magnetic breakdown is also presented.

Size-Dependent Oscillatory Magnetoresistance in Cadmium

Size-dependent oscillations in the magnetoresistance and Hall effect in cadmium have been studied using the field-modulation method. Nine sets of oscillations were observed for the first time and three sets discovered previously were observed with greater resolution over a larger range of field orientations. Most of the oscillations are assigned to orbits on the second-band and third-band sheets of the Fermi surface. The results indicate a slight departure of the lens from axial symmetry. Considerable second-harmonic content was observed on some of the oscillations. Disappearance of some monster oscillations but not the lens oscillations in magnetic fields greater than 8 kOe is attributed to the effect of magnetic breakdown of the energy gap between the first and second bands. Results are discussed in terms of the size-effect theory of Bloomfield and the Fermi-surface model of Stark and Falicov.

Ultrasonic quantum oscillations in white tin

A study has been made of acoustic quantum oscillations in pure single crystals of white tin, using 50 MHz longitudinal ultrasound and magnetic fields of up to 6.7 T. The resulting oscillations have been analysed and interpreted in terms of the known Fermi surface of white tin, including magnetic breakdown effects. The results complement de Haas-van Alphen effect studies and include an observation interpreted as an oscillation in magnetic breakdown probability.

Electrical properties of NbO in high magnetic fields

The resistivity and Hall coefficients of polycrystalline samples, single crystals, and doped specimens of NbO x (0.98 ⩽ x ⩽ 1.02) were measured in magnetic fields up to 160 kG at 4.2, 78, and 295 K. The material is found to be an excellent conductor. The results are interpreted in terms of a nearly free electron model in which holes and electrons contribute jointly to conduction processes. Orbital switching, magnetic breakdown effects, and perturbations arising from impurities are invoked to explain the observed magneto-resistance anomalies.

High-field magnetoresistance of metals by Kubo-Mott formula

A completely quantum-mechanical calculation of the galvanomagnetic tensor σ μν (including magnetic breakdown effects) for nearly free-electron (NFE) metals in strong dc magnetic fields ( ω c τ > 1) is presented. The method employed is the Kubo formula for σ μν in a particularly convenient form introduced recently by Mott: this involves the matrix elements, 〈 ψ k′ | j | ψ k 〉, of the current operator, j = ( e m )(−i h ̵ ∇ + e A c ) , where curl A defines the applied field H, and ψ k is the one-electron wave function which we determine to first order in a weak pseudopotential, V, by NFE perturbation of Landau (harmonic oscillator) eigenstates for a free electron in a magnetic field. It is shown that the contributions to such matrix elements of the current can be represented by Feynman graphs for a second-order scattering process which involves an electron-coulomb vertex (given by V) and the emission or absorption of a fictitious excitation (characterized by a vertex given by a current ( e h ̵ m )( A n π ) 1 2 , A n being the area of the quantized cyclotron orbit in k-space, and an energy transfer of amount h ̵ ω c in the plane normal to H). For a pseudopotential, V, having any number of Fourier components, the method enables us to include phase coherence exactly in the Pippard network of electron orbits coupled by magnetic breakdown, and to exhibit three types of oscillatory effects due to phase coherence, electron density of states (Shubnikov-de Haas effect) and the so-called “zone” oscillations. Analytical results are given for the linear chain and the hexagonal network and compared with experiment.

A pseudopotential calculation of the Fermi surface and of magnetic breakdown effects in thallium

We have parameterized the Fermi surface of thallium by using a local pseudopotential model Hamiltonian with spin effects included in the initial matrix elements. The pseudopotential coefficients are adjusted to fit the existing de Haas-van Alphen (dHvA) data and we find U10macron10 = -0.1058, U0002 = -0.0603, U10macron11 = -0.0429, U10macron12 = -0.0195, λ5p = 0.1130, all in Ryd. These values give a band structure which agrees very well with experiment, having a maximum error of 3 × 10-3 Ryd at a Fermi energy of 0.6110 Ryd. The interpretation by Capocci et al. of their recent dHvA data as due to a fifth zone dumbbell at H is confirmed, and the very small orbits which they observed are reliably assigned to fourth zone pockets along capital GammaM. Recent cyclotron resonance data by Shaw and Everett are shown to agree well with our model. We use our model to calculate magnetic breakdown fields for breakdown across the AHL plane and obtain values reasonably consistent with the existing dHvA data, but much larger than those deduced from the magnetoresistance. We find B0 = 90 kG for the open orbit along [11macron20], whereas Young finds B0 = 10.3 kG by fitting his magnetoresistance data. Our calculations confirm that the fourth zone posts do not contact the capital GammaKM plane, but we find that they are sufficiently close to it for breakdown across the capital GammaKM plane to occur at fields of several hundred kilogauss, thus creating open orbits along [0001].

New results on the fermi surface of beta-brass

New de Haas-van Alphen-effect results are reported for beta-brass (β′-CuZn), including a limited study under hydrostatic pressure in liquid helium. A new frequency of magnitude 0.133 [in units of (2π/a)2] is observed at 〈110〉. The experimental evidence appears to be against the association of theF1 oscillations with holes in the second band, but is unable to show that this frequency derives from holes in the first band. No evidence was observed in the de Haas-van Alphen effect of magnetic breakdown in beta-brass.

Open-Orbit Resonances and Magnetic Field Dependence of the Ultrasonic Attenuation of Shear Waves in Magnesium

The open-orbit resonances in the ultrasonic attenuation of shear waves in magnesium have been studied. The ultrasonic open-orbit resonances have been observed for magnetic fields applied in the basal plane for fields as high as 1100 G. The existence of the open-orbit resonance provides direct evidence for the presence of a spin-orbit-induced energy gap in the $\mathrm{AHL}$ Brillouin-zone plane. The period of the open orbit is in excellent agreement with the Brillouin-zone dimension in the [0001] direction. The effects of magnetic breakdown are observed to be of importance in fields of about 1 kG.

Application of translationally invariant wavefunctions in a magnetic field to frequency and wavelength dependent transport phenomena

Density matrix techniques are used to analyze magneto transport effects in solids. The representation used consists of wave functions that are simultaneous eigenfunctions of the magnetic Bloch Hamiltonian and the magnetic translation operators. This procedure yields results for both ordinary magnetic field dependent transport phenomena in general band structures and those due to magnetic breakdown. The conductivity tensor governing ultrasonic Doppler-shifted cyclotron resonance is analyzed, i.e. the wave vector q is parallel to the magnetic field B, the latter being parallel to the z-axis. Using a simplified periodic potential model in which V( x, y, z) = V( x, y) + V( z), the contributions to σ( q, ω, B) are separated into a part expressing magnetic breakdown effects and a part that is largely semiclassical in composition. Of significance here is the explicit dependence of some of the terms on magnetic quasi-particle parameters such as the magnetic field dependent effective mass M xx . The magnetic quasi-particle contribution to σ( q, ω, B) is analyzed in a regime where the electron mean free path Λ satisfies the condition qΛ ⪢ 1. It is found that increments appear in the conductivity that are periodic in 1/ B. These increments lead to spikes in the attenuation of ultrasound and helicons outside the absorption edge.

Effect of Magnetic Breakdown and Nonlinear Magnetization on the High-Field Magnetoresistance of Be

The high-field magnetoresistance of Be exhibits large-amplitude quantum oscillations as a function of magnetic field for the field parallel to the hexagonal axis. A low frequency, which appears only at temperatures below 4.2\ifmmode^\circ\else\textdegree\fi{}K, cannot be related to a Fermi-surface area. It is demonstrated how this frequency is generated through the interaction of magnetic breakdown and the magnetic domains reported by Condon.

Measurements of the thermal magnetoresistance of aluminum

The transverse thermal magnetoresistance of single-crystal plates of pure aluminum has been measured at liquid helium temperatures. It is shown that the high field Lorentz ratio is practically independent of field in agreement with theoretical predictions, and the value of the Lorentz ratio in high fields is discussed in terms of thermal and electrical relaxation times. It is also shown that the previously observed linear increase of the electrical resistivity in high fields is reflected in the thermal resistivity, and the origin of this effect is discussed. It is believed that the linear behavior is not caused by a magnetic breakdown effect in this case, but is due to other intrinsic or nonintrinsic effects.

Azbel'-Kaner Cyclotron Resonance in Thallium

The angular dependence of Azbel'-Kaner cyclotron resonance in single crystals of high-purity thallium has been investigated at 24 GHz and 1.4\ifmmode^\circ\else\textdegree\fi{}K for sample surfaces perpendicular to the three principal crystal-lographic directions. These measurements have revealed sections of Fermi surface not previously reported. Detailed comparisons are made with the nearly free-electron Fermi surface and with Soven's relativistic orthogonalized-plane-wave calculations. While several orbits were observed to be compatible with these models, other orbits were observed which we are unable to assign using either model. Cyclotron resonance for magnetic field directions making oblique angles with a ($10\overline{1}0$) surface has also been investigated. These studies revealed highly anisotropic, large-amplitude resonances for magnetic field directions making angles of up to 80\ifmmode^\circ\else\textdegree\fi{} with the sample surface. These resonances have been attributed to extended orbits that are possible because of magnetic breakdown effects. The extended-orbit mechanism proposed by Khaikin and Cheremisin was found to account qualitatively for the observed behavior. The experimentally determined masses were quantitatively compared with the nearly free-electron predictions, resulting in estimates of the mass enhancement factors ranging from 1.4 to 1.8. These values were found to disagree with the most recent determinations of the specification-heat mass enhancement factor.

Band Structure and Fermi Surface of Ferromagnetic Nickel

The band structure and Fermi surface of ferromagnetic Ni were studied using Mueller's combined interpolation scheme, extended to include spin-orbit and exchange interactions. Uniform exchange splittings were included in the molecular-field approximation. Semiempirical band structure were obtained which gave detailed agreement with experimental data where they were available, and over-all agreement with the results of ab initio calculations. In particular, good fits were obtained to the $X$ pocket and $L$-neck de Haasvan Alphen data, magnetic breakdown effects, optical edges, and valence (i.e., 10.00 electrons/atom). It was possible to fit the $X$ pocket Fermi surface using a "partial" two-center approximation to the tight-binding $d$ parameters. The ${X}_{5}$ quasiellipsoidal pocket was found to have an unusual shape, resulting from anisotropic interaction with the neighboring ${X}_{2}$ band. Moreover, the $d$ bands were sufficiently high in energy that the $X$ pocket had some $s\ensuremath{-}p$ character along its major axis. Many-body mass enhancement factors were found to be nearly uniform (\ensuremath{\approx} 2) after the light-mass $s\ensuremath{-}p$ character was added to the $X$ pocket. For the $d$-band exchange splittings, values of 0.4-0.6 eV were found, depending on assignments of optical edges. The density of states showed a sharp multipeaked structure, which, when smeared out, gave qualitative agreement with the density of states obtained from recent ultraviolet photoemission studies.

Wave packet approach to lattice electrons in external galvanomagnetic fields

The probability for Bragg reflections of wave packets in a lattice and an external electric fieldF is no longer invariant against inversion: (a) Reflections anti-parallel toF are favored compared to reflections parallel toF owing to the energy gain involved. (b) Reflection amplitudes caused in successive cyclotron periods cancel by interference, except when the respective energy gain equals an integer number of magnetic quanta. The effective current parallel toF is calculated up to terms of second order in the lattice potential. It is found proportional to 1/B 2 for scattering times shorter (low field limit) and proportional to 1/B for scattering times longer (high field limit) than one cyclotron period. The effect of magnetic breakdown interpolates between the results of a band model for closed and open electron orbits. From the reflection condition (b) one can deduct oscillations of the effective current inF/B 2 in the 1/B-region.

Magnetic Breakdown in Ferromagnetic Nickel

Using Mueller's interpolation scheme we had previously obtained good agreement with the dHvA measurements on the X pocket Fermi surface of Ni. We have extended these calculations to include the magnetic breakdown effects observed in the same experiment. Our calculations support the results of Ruvalds and Falicov who used k̄·p̄ perturbation theory and group theory to determine band degeneracies in symmetry planes normal to the applied magnetic field. Our model Hamiltonian yields the required accidental degeneracies between bands of opposite spin in these symmetry planes. Our band structure now also gives good agreement with the Fermi surface neck dHvA data.

Magnetic Breakdown Effects in Helicon Propagation in White Tin

The propagation of helicon waves through white tin has been studied at low temperatures and related to topological properties of the Fermi surface. The large-amplitude quantum oscillations observed at 1.2\ifmmode^\circ\else\textdegree\fi{}K as the field was varied arise from magnetic breakdown. They provide a sensitive means of investigating some features of the Fermi surface. Comparisons are made with the results of a recent band-structure calculation by Weisz.

Oscillatory Field Dependence of the Knight Shift in a Monocrystal of Tin

A detailed experimental investigation of the magnetic-field dependence of the Knight shift in a monocrystal of white tin has been performed. The measurements were made at 1.2\ifmmode^\circ\else\textdegree\fi{}K and in magnetic fields ranging from 10.5 to 16.2 kG. The Knight shift is found to oscillate at the de Haas-van Alphen (dHvA) frequency, and in the range of fields investigated, exhibits two periods of 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$ and 5.8\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$ ${\mathrm{G}}^{\ensuremath{-}1}$. The values and angular variations of these periods are in satisfactory agreement with the data on the tin Fermi surface obtained from other experiments and from a pseudopotential calculation of the band structure. The magnetic-field dependence of the amplitude of the oscillations has been investigated for both periods. The amplitude of the shorter-period component increases, and that of the longer-period component decreases, with increasing magnetic field. The decrease in amplitude of the longer-period oscillation is attributed to the effects of magnetic breakdown. The observed amplitudes are in satisfactory agreement with the assumption that the Knight-shift oscillations are due to oscillations in the ground-state wave functions of the contributing electrons rather than oscillations in the susceptibility as has been previously suggested. Measurements of the dHvA susceptibility oscillations in the same sample used for the Knight-shift measurements give further support to this viewpoint, since the field variation of the amplitude of the susceptibility oscillations is drastically different from the corresponding field dependence of the oscillatory Knight shift.

Magnetoresistance and Coupled Orbits in Tin

We have observed large quantum oscillations in the magnetoresistance of pure white tin at liquid-helium temperatures and in fields up to 150 kG. The oscillations are best developed when H is approximately 5\ifmmode^\circ\else\textdegree\fi{} from the $c$ axis and perpendicular to [110], and are interpreted as a magnetic breakdown effect which gives rise to a linear chain of coupled orbits. A calculation of the resistance, based on Pippard's suggestion that phase coherence is important only around small orbits, is in good agreement with the experimental results when fitted with a relaxation time for phase coherence of 2.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ sec.