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By determining up to 16 GHz the effective dielectric constant for microstrip lines with various strip widths, it has been found that anisotropy in alumina substrates considerably affects the wavelength and is responsible for most of the discrepancy between theory and experiments

A Fabry-Perot resonator formed by parabolic cylindrical reflecting mirrors and filled with anisotropic medium is analyzed theoretically. For the resonance of the extraordinary wave in such a resonator, optimum shapes of the reflecting mirrors exist, which are derived from the boundary conditions on each mirror.

The paper deals with the improvement in the performance of an induction machine with a solid iron rotor slitted axially. The improvement is observed mainly in the full-load slip, the power factor, and the starting torque. In this paper, a one- dimensional model of a slitted rotor is presented based on the original and novel concept that the tooth and slit profile is a flux line. Based on this, the field distribution inside the iron is obtained from 1) a linear theory and 2) a limiting nonlinear theory. The results of the analyses based on these theories are found in better agreement than those based on the composite permeability theory.

This paper gives the theory of spinwave instability under parallel pumping in hexagonal Zn/sub 2/Y ferrites with planar anisotropy in the case of a sphere. The configuration considered in this paper is the one in which the dc field makes an angle /spl beta/ with the easy plane and the resultant magnetization lies at an angle /spl alpha/ to it. To obtain parallel pumping with this arrangement it is necessfly to align the RF field along the magnetization. This leads to a new spinwave spectrometer which allows the dependence of the spinwave linewidth upon the angle /spl alpha/ which the unstable spinwave makes with the c axis of the crystal to be determined. The experimental results obtained on two Mn-Zn/sub 2/Y single-crystal ferrite spheres show that the spinwave linewidth increases with the angle /spl alpha/.

We describe the steady-state electron distribution of a high temperature, spatially inhomogeneous semiconductor subject to a large magnetic and an infinitesimal electric field, using classical mechanics (strictly valid if the range of the potential describing the inhomogeneities is considerably larger than the thermal electron wavelength), by a modified Boltzmann equation retaining both spatial and velocity variables. Such a classical transport equation can also describe the behavior of electrons in a plasma subject to given spatially dependent density fluctuations. In the case of the semiconductor the collision term refers to collisions with phonons which we treat in this paper in the relaxation time approximation with a constant relaxation time, this being our assumption (1). For mathematical convenience we restrict ourselves in this paper to a stratified medium in which the inhomogeneities are distributed in sheets perpendicular to say the x-axis, this being our assumption (2). We obtain the asymptotic solution of the transport equation in reciprocal powers of the strength of the magnetic field H . The velocity moments corresponding to the first few terms of the solution are identical with the solution of the velocity moment (4, 10, etc.) equations obtained by retaining only 2, 3, etc. generalized Hermite polynomials in the electron velocity, in the representation of the solution by these polynomials. These moment solutions provide us, in principle, with a method of solving approximately a general linear transport equation for which assumption (1) is dropped. The leading term of the asymptotic solution corresponds to a macroscopic theory of the high field magnetoresistance obtained by Herring. The transverse magnetoresistance does not saturate but increases indefinitely with H 2, while there in no longitudinal magnetoresistance in our stratified medium. The general first correction to the macroscopic theory results are explicity obtained. The mathematical methods can in certain instances be extended to more realistic inhomogeneity distributions, i.e., not subject to assumption (2). These extensions will be the subject of a subsequent paper. The physical significance of these mathematical results is noted.

All the functions essential for the operation of a memory (viz., storage, access, write, read, and detection) can be performed within a bubble-domain memory chip. This enables the design of a memory with a minimum number of peripheral circuits and interconnections, and a short access time. It is conceivable to achieve a 106-bit memory with fewer than 20 circuits (and interconnections), and with a few hundreds of microseconds access time. The present paper describes the devices required to implement such a memory, experimental data to demonstrate the operability of such devices, and the design criteria.

In a recent paper, Larrabee et al. proposed the magnetoresistance measurement for a quick determination of electron mobility in a functional Gunn diode. It is pointed out in this correspondence that the formula used in the paper is applicable only to extremely thin Gunn diodes where the length-to-width ratio is negligibly small. It is also shown how to obtain the correct value of the mobility by taking the geometry effect into consideration.

In magnetoresistance measurements on compensated metals, both maxima and minima have been observed as the magnetic field B becomes perpendicular to the open orbit direction. In this paper we show that this is a natural consequence of the transformation of well known magnetoresistance tensors to the appropriate set of axes. The condition for observing a maximum or minimum is derived. A similar analysis applied to uncompensated metals indicates that there will always be a maximum in the magnetoresistance for B in an open orbit direction, unless B approaches the longitudinal situation, when the presence of open orbits can not be observed in either compensated or uncompensated metals.

The magnetoresistance in the simple one-mobility model is exactly zero, although the curved nature of the carrier trajectories and the statistical nature of scattering both appear to be dispersive mechanisms which should produce a positive magnetoresistance. The first effect is considered separately by analyzing an artificial version of the model in which the particle is always scattered after the same time interval has elapsed. It leads to a negative magnetoresistance which is about as large as the well-known positive effects which have been calculated for isotropic semiconductors with energy-dependent scattering. Thus the zero result actually reflects a cancellation of two opposing effects. The possibility that this different point of view regarding the fundamental nature of magnetoresistance might contribute to a better understanding of the negative magnetoresistance found experimentally in several classes of materials is mentioned, but is not explored further in this paper.

In a former paper[1], we have shown that the magnetoresistance coefficient in p -type GaSb (1+ξ) remains close to 1 at 77°K and that the mobilities ratio remains equal to 6 in the temperature range 77–300°K. We show from these results that between 30 and 300°K, the predominant scattering is a mixed scattering by lattice vibrations and ionized impurities. Interband scattering is the predominant process for light holes, while heavy holes undergo intraband scattering. In this temperature range, this mechanism accounts for the mobility variation, a result which had not been found so far on p -type GaSb.

A linear relation has been found between the ESR g-value anisotropy, the electrical resistivity ρ300K, the resistivity ratio ρ77K/ρ300K, the thermoelectric power and the magnetoresistance of a large number of carbon fibres based on polyacrylonitrile precursors and processed in a variety of ways (including hot stretching techniques) above 1750 °C. These correlations also appear to extend to cellulose-based fibres. Accordingly a term `relative graphitic order' is introduced in this paper, and is defined empirically as the resistivity ratio in order to represent the structural order of the graphite more effectively than the heat treatment temperature which has been used by previous workers. The relative graphitic order also shows a linear variation with x-ray crystallite size. The data suggest that those fibres with the highest relative graphitic order resemble the stage attained by soft (graphitizing) carbons after heat treatment close to 2200 °C; that is, the stage immediately preceding the appearance of 3-dimensional order and the onset of rapid graphitization It has also been found that the magnetoresistance is related to the alignment of these fibres, and can be used to deduce the preferred orientation.

The magnetoresistance of avalanching semiconductor diodes is analyzed in terms of the Lorentz force acting on a Maxwellian distribution of carriers, assuming a scattering mechanism characterized by a constant mean free path. At low current levels, the magneto-resistance only arises in the generation region close to the junction interface, and is a result of the reduction in the hot carrier temperature caused by the deflection of the carriers from their original trajectories. The fractional change in the maximum field strength in the junction to maintain the carrier temperature at its original value is shown to be ΔEm/Em = 0.388 (μ(Em) H)2 where μ(Em) is the conductivity mobility corresponding to the average scattering rate of carriers in equilibrium with the maximum field in the junction. This magnetoresistance may be determined through measurement of the change in reverse bias required to maintain a constant avalanche current through the device. At high current levels a correction must be made for the magnetoresistance of the bulk semiconductor in series with the exhaustion region. From measurements on a series of diodes with different breakdown field strengths, it is possible to measure the field dependence of the scattering rate of the majority carriers on the lower doped side of the junction at field strengths of interest in device operation. The predictions of the paper are in good agreement with experimental observations. Es wird der Magnetowiderstand von Halbleiter-Avalanche-Dioden mit der Lorentzkraft, die auf eine Maxwellverteilung von Ladungstragern wirkt, analysiert, wobei ein Streumechanismus angenommen wird, der durch eine konstante mittlere freie Weglange charakterisiert ist. Bei niedrigen Stromen entsteht der Magnetowiderstandseffekt in dem Generationsbereich in der Nahe der Ubergangsgrenzflache und ist ein Ergebnis der Verringerung der Temperatur der heisen Ladungstrager, die durch die Ablenkung der Ladungstrager von ihren ursprunglichen Bahnen verursacht wird. Es wird gezeigt, das die Teilanderung der maximalen Feldstarke am Ubergang, die die Ladungstragertemperatur auf ihrem ursprunglichen Wert halt, ΔEm/Em = 0,388 (μ(Em) H)2 ist, wobei μ(Em) die Leitfahigkeitsbeweglichkeit ist, die der mittleren Streurate der Ladungstrager im Gleichgewicht mit dem Maximalfeld im Ubergang entspricht. Dieser Magnetowiderstandseffekt kann durch Messung der Anderung der Vorspannung bestimmt werden, die erforderlich ist, um einen konstanten Lawinenstrom in der Diode aufrechtzuerhalten. Bei hohen Stromen mus eine Korrektur fur den Magnetowiderstandseffekt des Halbleitervolumens in Reihe mit der Vorarmungs region angebracht werden. Aus Messungen einer Reihe von Dioden mit unterschiedlicher Durchbruchsfeldstarke ist es moglich, die Feldabhangigkeit der Streurate der Majoritatstrager an der niederdotierten Seite des Ubergangs bei Feldstarken, die fur die Arbeitsbereiche des Elements interessant sind, zu messen. Die Vorhersagen der Arbeit befinden sich in guter Ubereinstimmung mit experimentellen Beobachtungen.

A practical extremely low frequency (ELF) horizontal antenna requires a site of low effective conductivity. This paper describes the application of certain dc resistivity sounding methods in rapidly evaluating potentially useful areas within geologically interesting regions, and in determining both basement anisotropy and the direction of the principal (maximum) resistivity axis. Apparent resistivity data from dc array expansion is interpreted with a horizontal layered earth model using a "generalized inverse" method of fit. With layered models over principal axes the effective conductivity to be experienced with a horizontal wire antenna of arbitrary orientation is readily calculated. Salient results for three geographical regions are presented.

A technique for increasing the linear range of a magnetoresistor displacement transducer by means of shaped pole pieces is developed. The method of determining the required profiles is discussed and observations on the machining of the pole pieces are made. Practical results given in the paper show a sevenfold increase in the linear range. The technique described may have applications in other fields.

The resistivity $\ensuremath{\rho}({H}_{\mathrm{ext}},T)$ and Hall effect were studied in ten $n$-type EuSe single-crystal samples with room-temperature carrier concentrations from 4.2 \ifmmode\times\else\texttimes\fi{} ${10}^{18}$ to 3.5 \ifmmode\times\else\texttimes\fi{} ${10}^{19}$ electrons/${\mathrm{cm}}^{3}$. The electrical measurements were performed at temperatures $1.6\ensuremath{\le}T\ensuremath{\le}300$ K and external magnetic fields $0\ensuremath{\le}{H}_{\mathrm{ext}}\ensuremath{\le}150$ kOe, and were supplemented by magnetization and differential susceptibility measurements to aid in the interpretation. The following phenomena were observed: (a) A very large peak in the zero-field resistivity occurred at a temperature ${T}_{max}$ which varied from \ensuremath{\sim} 7 to \ensuremath{\sim} 13 K, depending on the sample. (b) A negative magnetoresistance was observed for most temperatures and was very large near ${T}_{max}$. (c) However, in a limited temperature interval well above ${T}_{max}$, a positive magnetoresistance was observed at low fields (followed by a negative magnetoresistance at higher fields). (d) Below \ensuremath{\sim} 3 K, the zero-field resistivity increased rapidly with decreasing $T$, except for the sample with the highest carrier concentration. (e) The anomalous Hall effect was negligible at temperatures $T\ensuremath{\le}4.2$ K. Much of the data are interpreted in terms of band conduction. The mobility is then limited by spin-disorder scattering and it increases with increasing ${H}_{\mathrm{ext}}$, resulting in a negative magnetoresistance. This description fails when both $T\ensuremath{\lesssim}{T}_{max}$ and ${H}_{\mathrm{ext}}l1 \mathrm{to} 10$ kOe. Under these conditions, the very high resistivity is attributed to the localization (trapping) of electrons by the $s\ensuremath{-}f$ (or $d\ensuremath{-}f$) interaction. At temperatures for which the electrons are localized at ${H}_{\mathrm{ext}}=0$, a field of 1 to 10 kOe delocalizes the electrons and, therefore, leads to band conduction and a much lower resistivity. At higher fields there is a negative magnetoresistance (smaller than at low fields) due to the reduction in spin-disorder scattering of the band electrons. A model for the positive magnetoresistance, which focuses on the spin splitting of the conduction band by a magnetic field, is presented in the following paper.

The purpose of this discussion is the extension of some results derived by Lax on the small coupling limit of transport theory. Our results extend the perturbation treatment for small external electric fields to magnetic fields in classical and quantum cases. The quantum results for Ohm's current and the Hall current are derived and compared to the classical results and the conditions for classical limit to exist are examined. A quantum mechanical expression for the Hall coefficient is derived along with the usual expression for the conductivity tensor. This method could be extended to give similar results for magnetoresistance. The single particle approximation is used throughout both sections of the paper.

Synopsis The purpose of this paper is to review and extend the explanation of the resistivity anomalies observed in certain magnetic metals and alloys in terms of spin-dependent scattering of the conduction electrons, For this purpose, the usual theory of electrical conductivity has been extended to include inelastic collisions with and without spin flip. Assuming the existence of an exchange interaction between the conduction electrons and unpaired electrons localized on particular atoms, the resistivity has been calculated for scattering by (i) isolated magnetic ions, (ii) pairs of exchange-coupled magnetic ions and (iii) 'spin-disorder in a ferromagnetic lattice. For models (i) and (ii) the magnetoresistance has also been investigated.

We have obtained exactly the excess dissipation due to a small spherical inclusion in a semiconductor, using the same model of the semiconductor which was investigated in previous papers of this series, and the ``four moment approximation'' to the electron‐distribution function. We have also calculated the high‐field limit of the transverse magnetoresistance for a dilute sprinkling of spherical inclusions, which is proportional to the high‐field limit of the excess dissipation, and varies as |H| as H→∞, where H is the magnetic field. Previously, Herring obtained approximately the excess dissipation of a cylindrical inclusion with axis parallel to the magnetic field and made plausible a high‐field transverse magnetoresistance which varies as |H|/ln|H| as H→∞. In the case of spherical inclusion, we can investigate in detail perturbations produced by the external specimen boundaries, and we suggest that analogous results hold for a cylindrical inclusion.

The conventional treatment of magnetoresistance is based upon the Boltzmann equation for a distribution function f(k), and an equation which gives the average current in terms of f(k). With the usual definition of f(k) neither equation is exact: the derivation of the Boltzmann equation requires ωτ < 1 where ω is the cyclotron frequency and τ is the relaxation time. In this paper a representation is proposed which exploits the properties of homogeneous systems in a uniform magnetic field. We redefine f(k) as the diagonal element of the density matrix in the new representation. Then the usual equation for obtaining the current becomes exact. Proceeding from the equation of motion of the density matrix we then give a weak-coupling derivation of the Boltzmann equation in the manner used by Kohn and Luttinger in 1957 for field-free systems. The derivation is restricted to the case of non-interacting carriers elastically scattered by impurities or lattice vibrations, and moving in a lattice which in the absence of the magnetic field gives rise to non-overlapping bands. The conductivity yielded by this method is found to be accurate as long as hω, h/τ are small compared with the Fermi energy, and hω is less than the thermal energy. The condition ωτ < 1 does not arise.

The Hall resistivities of polycrystalline samples of the series Ag(Cd → Sb) have been measured in the temperature range 4.2–80 °K, and in the field range 2–15.17 kG. The solute concentration ranges between 273 and 6715 atomic p.p.m. Special consideration was given to the establishment of the low-field condition, for which the negative Hall coefficients of the Ag(Cd → Sb) series at 4.2 °K are found to be 9.49, 9.59, 9.67, and 9.77 × 10−11 m3 A−1 s−1 respectively, with an accuracy of ± 0.04 in these units. Using the model of Dugdale and Firth, the corresponding values of the ratio τN/τB are 0.64, 0.57, 0.52, and 0.47 respectively. It is shown that there is qualitative agreement between the dependence of τN/τB upon solute valence and that expected from the known electronic structure of the solutes. The paper also includes results for the field dependence of the transverse magnetoresistance shown by the samples at 4.2 °K, which have been used to establish the low-field condition.

In semiconductors with extremely low majority carrier densities, the conductivity of the surface space charge region can be very different from the conductivity of the bulk. This paper demonstrates that the conductivity of this surface space charge region can be materially altered by the application of a magnetic field which is perpendicular to the direction of current flow. A tentative prototype model of this effect is presented which not only illustrates the mechanisms of importance, but also is sufficiently accurate to enable one to estimate the order of magnitude of the effect for certain experimental conditions. This model suggests that this effect may find application in magnetoresistance and/or negative resistance devices and might prove useful in the measurement of surface parameters.

PH YSICAL REVIEW' VOLUM E 182, NUMBER 10 JUN E 1969 H —Impurity States and Nuclear Magnetic Relaxation* GREGoRY BENPQRD Laurence Radiation Laboratory, University of California, Lieermore, California 94550 AND NQRMAN RosTQKERt' Un& ersity of California, San Diego, La Jolla, California 9Z037 (Received 31 January 1969) It is shown that a hydrogenic impurity will bind a second electron in high magnetic fields, because of contraction of the electron wave functions about the lattice sites. The binding energy of the second electron exceeds kT for semiconductors at helium temperatures when p=kco, /2 Ry»1. This H ion may then provide a new density of states g+(E) for electrons with spin up (o. = 1). Nuclei which relax by mutual spin-flip processes with conduction electrons have a relaxation rate 1/T1 proportional to g+(E). In the last Landau state n = 0, 0-= — 1, the conduction electron g+(E) vanishes so the additional g+(E) from the H state can produce a large change in T1. The magnitude of this effect is estimated for InSb and comparison is made with the experiments of Bridges and Clark, with some success. Alternative explanations involving plasma modes are investigated and found wanting. The presence of H impurities in concentrations on the order of one-tenth of the conduction electron concentration may also affect the negative magnetoresistance and the theory of scattering from localized spins. I. INTRODUCTION 'HE contact interaction with conduction electrons mecha- is a well-known nuclear-spin-relaxation nism in metals, semimetals, and semiconductors. At low temperatures in InSb it is dominant until the tempera- ture exceeds =10'K, at which point phonon relaxation coupling through the nuclear quadrupolar moment be- comes more favored. In order for nuclear relaxation to proceed via the contact interaction, there must exist a density of states for the final electron eigenstate. During the relaxation process, the electron and nucleus flip their spin (con- serving the total spin of the system), so the final density of states for electrons must be of opposite spin from the initial state. This paper deals with the possibility that additional densities of states may arise in semicon- ductors, due to the formation of hydrogenic states which bind tzvo electrons in a high magnetic field. We shall limit ourselves to consideration of a strongly degenerate conduction electron gas in a solid having a simple nondegenerate energy band with an isotropic, quadratic dispersion law, within the effective-mass ap- proximation. The conduction electron Quid is in equi- librium. The applied magnetic field is taken to be so strong that quantization of the electron motion (Landau quantization) is important. Generally, the phenomena we treat will only be important when the electrons all occupy the lowest Landau level and the lowest spin state within that level. In Sec. II, we give a short review of electron dynamics in high fields. This work was originally prompted by the experi- * Work performed under the auspices of the U. S. Atomic Energy Commission. A portion of this work is based upon material submitted by Gregory Benford in partial fulfillment of the require- ments for the Ph. D. degree, University of California, San Diego. t Present address: Applied Physics Department, Clark Hall, Cornell University, Ithaca, N. Y. ments of Bridges and Clark' (thereafter referred to as BC), which determined the nuclear relaxatjon times of InSb nuclei in high magnetic fields. Before describing our work, we therefore brieQy summarize the BC results. This is done in Sec. III. Feher' has suggested that the BC experiments in very high fields may be explicable in terms of plasma effects in the conduction electron gas. In Sec. IV, we discuss various plasma modes which have been proposed, and the schemes necessary for them to relax nuclear spins in a semiconductor. We conclude that all the mechanisms thus far advanced fail to agree qualitatively with experiment. In Secs. V and VI, the possibility of additional densities of states in semiconductors is examined, and the binding energy of the high 6eld H ion is calculated. The formation of this ion makes a new density of states available for conduction electrons with spin index g = i. It is then possible for this new state to participate in relaxation of neighboring nuclear spins by making energetically possible a spin-Qip process between con- duction electrons and nuclei. The relaxation rate due to this impurity is estimated in Sec. VII. A He-like ion could also contribute a 0 = 1 density of states. We consider the binding energy of this impurity in Sec. VIII. We compare our model with the BC ex- penments in Sec. IX. Although the BC work is the only data currently available for comparison, it should be noted that our theory applies to any semiconductor in which the H— binding energy may exceed kT. (This will usually mean the dielectric constant X&)1. ) The important point of this work is that formation of the H state for high fields (y) 1) does occur, and it may have important consequences in many areas of current interest, such as the theory of scattering from 'F. Bridges and W. G. Clark, Phys. ' G. Feher (private communication). Rev. 182, 463 (1969).

In this paper the author has studied the variation of Lorentz number, Hall coefficient, magnetoresistance, and Hall mobility with impurity parameter. The effect of ionized impurity scattering has also been considered.

This paper reports an advanced undergraduate laboratory experiment concerning the helicon solid-state plasma wave. Student results are reported. The experiment is performed as an inexpensive dip in a liquid helium storage vessel. The experiment is made possible by the design of a small, superconducting magnet. The magnet is capable of attaining 18 kG and costs about $25. The whole experiment can be built by students in a week's time. When the experiment is completed, the student not only understands helicon propagation but also understands the use of the helicon to measure magnetoresistance and Hall coefficients in pure metals.

Synopsis This paper reviews the status of experiments bearing on the band structure of semiconductors, with particular reference to germanium and silicon. The situation in n-Ge is especially clear, as both cyclotron resonance and magnetoresistance experiments show that the energy surfaces at the bottom of the conduction band are prolate spheroids oriented along the [111] directions in k-space; the spheroids are highly anisotropic, with a mass ratio of about 15:1. In p-Ge the two isotropic lines in cyclotron resonance suggest that the band edge is at the center of the Brillouin zone; the multiplicity is explained by spin-orbit splitting. It appears that this model may account, qualitatively for the remarkable infra-red absorption of p-Ge, but it seems to be difficult to explain transport data. From magnetoresistance and from cyclotron resonance it is found that the energy surfaces in n-Si are prolate spheroids directed along [100] axes, while the structure of p-Si is like p-Ge.

The paper describes an instrument employing the Hall effect in indium arsenide crystals for measuring magnetic fields from 20 gauss to 25 kilogauss. Various aspects of choosing indium arsenide crystals have been considered. Advantages of this instrument over the conventional fluxmeter have been enumerated. Errors on account of magnetoresistance, temperature variation have also been described.

This paper is concerned with an investigation of the electrical, the galvanomagnetic, and the thermoelectric properties of pyrolytic graphites whose morphological features are conditioned by the deposition temperature, the heat treatment, and the doping level.(1) Basal-plane magnetoresistance and c-direction specific resistance of deposits prepared at temperatures ranging from 1900° to 2500°C point to a remarkable improvement of the crystallites' alignment with rising deposition temperature. In both crystallographic directions the Seebeck coefficient closely follows semi-empirical predictions based on the two-dimensional model of the π-electron bands. The Fermi level of a standard deposit (2100°C) is at 0.025 eV below the valence-band edge and thus indicates that crystal defects trap about 7.5×1018 electrons/cm3 at room temperature; this figure is in accord with a Hall coefficient of 0.33 cm3/C. The average in-plane mobility (930 cm2/V-sec) corresponds to a mean free path of the order of the crystallite diameter (250 A).(2) Post-deposition treatment at temperatures above 2500°C results in (a) a rapid drop of the room-temperature basal-plane resistivity down to 50 μΩ-cm or less, (b) a Hall effect conversion from p to n type in the early stages of graphitization, and (c) a trend toward negative Seebeck coefficients in the layer planes. In conjunction with low-field magnetoresistance measurements these results can be described in terms of semimetallic concepts, the simultaneous presence of holes and electrons with equal concentrations (6×1018 cm−3 at room temperature) stemming from a slight band overlap. Average mobilities imply that the carrier behavior approaches single-crystal characteristics ( ≈ 104 cm2/V-sec at room temperature) after heat treatment above 3000°C. Normal to the layers, the specific resistance always exceeds 0.1 Ω-cm, which points to a molecular conduction process in this direction.(3) An incorporation of boron into the carbon-hexagon networks lowers the electrical resistance of graphite particularly in the c direction (twenty-fold decrease at a composition of 0.6 at.%B); concurrently the two temperature coefficients become approximately equal to zero. In the rigid-lattice approximation band-population figures derived from the resistivity temperature dependence reflect the Hall coefficient behavior, the peak occurring at an equivalent boron content of 0.04%. The ionization efficiency is of the order of 50% with a Fermi level depressed by more than 0.1 eV. Thermoelectric power measurements in the c direction accord with the view that charge transport across the layer planes involves most of the excess holes, and reveal that boron enhances the Seebeck anisotropy of graphite.

This paper is designed to supplement the existing extensive literature on the conductivity of a randomly inhomogeneous medium, by treating the effects of inhomogeneities on piezoelectric, galvanomagnetic, and thermoelectric measurements. The scale of the inhomogeneities is supposed small compared with the dimensions of the specimen being measured, but large compared with mean free path, Debye length, etc. Formulas for all the effects are derived which are asymptotically exact in the limit of small fractional fluctuations in the local conductivity, etc. Comparison with other approximations and application to various exactly soluble cases show that these formulas are often roughly valid for quite sizable fluctuations. For material which, if uniform, would show a high field saturation of transverse magnetoresistance, the presence of appreciable inhomogeneities in the Hall constant will cause the magnetoresistance to increase indefinitely with field. This effect is due to the current distortions arising from ...

The paper describes the application of magnetoresistance of indium antimonide to frequency mixing in microwave range. The experimental results were found in good agreement with the theory, of a deviation less than 15 per cent.

This paper describes measurements of the magnetoresistance and of the planar Hall effect of p -type PbTe crystals; it also describes an interpretation of the electronic band structure of the crystal. The measurements were made at 90°K as a function of the magnetic field strength and of the angle between the magnetic field and the current through the specimen. The coefficients relating to the galvanomagnetic effects were determined from the experimental data. Using these coefficients the electronic band structure of the crystal is shown to be {111} energy spheroids, presumably prolate with a mass ratio of 0.3.

Despite the several experimental and theoretical studies that have been devoted to the investigation of the Fermi surface of beryllium, certain problems remain elusive in connection with the behavior of the electrons in beryllium single crystals. Only recently, for example, has magnetic breakdown in beryllium been investigated in detail, although its onset in fields of 50 kOe was discovered as far back as 1963.1'2 Beryllium is well known as one of the lightest of the elements; it has a low atomic number, and its atoms have only four electrons. Elements of this nature exhibit good agreement between the theoretical calculations of the Fermi surface and the experimental data. Metallic beryllium has relatively few carriers in the conduction band and, accordingly, a low density of carrier states at the Fermi level. Beryllium also has a small electron heat capacity; the coefficient associated with the linear term of its heat capacity has a value ~ = 0.4 x 10 -4 cal/mole • deg2. 3 Beryllium is a metal characterized by a high Debye temperature, which, according to Ref. 3, is equal to (1480 _+ 16)°C. The end result of the low density of states of the electrons and high Debye temperature is that a number of quantum effects are far more pronounced in beryllium than in most other metals. This feature renders beryllium a useful and fascinating medium for the investigation of such effects as magnetic breakdown. It has proven feasible in beryllium, for example, to study magnetic breakdown over a wide range of temperatures and to observe the socalled "thermal" breakdown effect) '4 A fair number of papers have been written to date on the subject of magnetic breakdown (see, e.g., the survey in Ref. 5), but a great many striking characteristics of this intriguing effect have not been unequivocally explained and are in need of further study. As the magnetic fields in which measurements are possible are increased, new possibilities are opened up for the investigation of magnetic breakdown, which is rapidly becoming one of the most interesting problems of the modern physics of l~etals. Another absorbing facet of magnetic breakdown is that it is one of many phenomena in which coherent quantum effects are strongly manifested. The study of various relevant factors and how they influence magnetic breakdown is there

It is well known that in a magnetic field bismuth shows a greater change of resistance than any other substance, and it is also known that in the case of a crystal this phenomenon varies very much with the orientation of the crystal. A great deal of literature exists on this subject. The general view of the phenomenon is that the increase of resistance is largest when the cleavage plane of the crystal is parallel to the magnetic field, and when the current is flowing perpendicular to it. It is also known that the resistance in a magnetic field increases very rapidly with decreasing temperature. A complication in all these phenomena arises through certain time lags. When a current is passed through bismuth placed in a magnetic field, the resistance at the first moment is large, and then gradually decreases to its final value. This time lag accounts for the fact, first discovered by Lenard, that bismuth has a larger resistance for alternating currents than for direct currents. This phenomenon also depends on the crystal state of the bismuth.

In previous papers we have studied the high magnetic field magnetoresistance of a classical model semiconductor (or a plasma) produced by the presence in the sample of more or less random spatial inhomogeneities on a scale small compared to the size of the sample. We extend our computations of the leading terms of the components of the resistivity tensor to the two‐dimensional case in which the external magnetic field H is inclined (but not perpendicular) to the plane of variation of the spatial inhomogeneity distribution. The field H is applied along the z axis, and the inhomogeneity distribution is characterized by a sufficiently smooth potential φ(x, z + λy). While the magnetoresistance ratios saturate in the y and z directions, one finds that this ratio is proportional to H2 (as H → ∞) in the x direction.

When plasmas are immersed in static magnetic fields a number of plasma instabilities are likely to develop. In turns out that in many cases the type of instability which occurs is in some way related to the method used to generate the plasma. One of the easiest and most common ways of generating a plasma is to pass a radio-frequency current through it. On the other hand, very little is known about the behavior of such RF-generated plasmas when immersed it static magnetic fields. In this paper experiments dealing with the behavior of RF magnetoplasmas are reviewed. It is observed in many different RF experiments that once the static magnetic field exceeds a critical value, the rate at which plasma is transported across the magnetic field is enhanced. This phenomenon, which is known as anomalous diffusion, is accompanied by an onset of fluctuations it plasma density and potential. The characteristics of the experimental onset data are reviewed and compared with the onset conditions of a number of particularly applicable theories. On the basis of this comparison, it is concluded that the ion-acoustic instability comes closest to explaining the experimental behavior of high-frequency discharges in static magnetic fields.

Hot carrier galvanomagnetic characteristics of n-type germanium of intermediate carrier concentration are studied theoretically. The range of carrier concentration considered ensures that the intercarrier collisions enforce a Maxwellian distribution on the carriers, and the momentum loss of the carriers occurs mainly through scattering by the lattice vibrations. Numerical values of Hall mobility obtained from the present analysis are found to be in agreement with the earlier published results for low and high carrier concentrations. Values of magnetoresistance, however, while agreeing with the low concentration case, are found to be different from those for the case of high carrier concentration. The physical origin of this difference is discussed and a comparison between the available experimental data and the values calculated in this paper is also made

This paper describes, in theory and experiment, three different types of application of the magnetoresistance effect at microwave frequency: the first is a microwave wattmeter, employing the multiplying action between the electric field and magnetic field components of the wave; the second a microwave mixer, employing the multiplying action between the electric field and magnetic field of the different frequencies; the third a microwave amplifier, employing the multiplying action between a d.c. current and a microwave magnetic field.

This paper presents a study of the magnetoresistance of several single crystals of high-purity thallium in magnetic fields up to 100 kG at liquid-helium temperatures. The experimental results are interpreted in terms of Soven's relativistic orthogonalized-plane-wave model for the Fermi surface of thallium. At low fields the magnetoresistance is in better agreement with the model than was previous work, while no evidence is found for an open orbit parallel to [0001]. Detailed results up to 54 kG are consistent with the model if incomplete magnetic breakdown is assumed and high-field results show that breakdown is not complete below 100 kG.

The construction and properties of certain superconductive amplifiers in which the resistance of a gate element is controlled by the electromagnetic field of a signal coil are described in this paper. Vacuum-deposited films of tin, which have narrow resistive transition widths, have been used as the gate elements, and the field has been applied perpendicularly to the film surface. The parameters of small-signal analysis are determined, and the influence of temperature, bias magnetic field, and zero-signal gate current upon power gains are shown. The limitations and merits of the device in its present form are discussed.

For pt. I see ibid., vol. 4, 458 of 1971. Large-amplitude (5-50% of the total resistance) quantum oscillations in the magnetoresistance of tin are described, and discussed in terms of the coupled orbit networks produced by magnetic breakdown on the pseudopotential model of the Fermi surface, and the specimen calculations of the preceding paper (to be referred to as I). Slow oscillations (f=1.7*106G) from orbit delta were restricted to localities: (a) H near (001) and perpendicular to (110) or (100) and (b) H at 22 degrees from (001). Both line shapes, resonant for (a) and sinusoidal for (b), and field dependences agreed with predictions of I, although very careful field setting was required to obtain the best T* value, 0.4 K, or the largest amplitudes. A little nearer the c axis than (b), the slow oscillations were principally second harmonic of delta with a comparable amplitude of fast oscillations, reflecting complex networks in this region.

In a magnetic field H and a thermal gradient $\ensuremath{\nabla}T$, a conductor develops a Nernst field $\ensuremath{-}B\mathrm{H}\ifmmode\times\else\texttimes\fi{}\ensuremath{\nabla}T$ and its thermoelectric power $Q$ depends on H. These two effects have been measured for a number of single-crystal six-armed samples of high-purity $n$-type germanium, with various orientations, at fields up to 18 000 gauss and at a number of temperatures from the liquid-hydrogen range to the onset of intrinsic conduction; the most detailed measurements were made near liquid air temperature. Both the Nernst coefficient $B$ and the quantity $\ensuremath{\Delta}Q=Q(\mathrm{H})\ensuremath{-}Q(0)$ are compounded additively out of terms arising from diffusion of the electrons and terms arising from their anisotropic scattering by the phonons moving from hot to cold. The latter "phonon-drag" terms are strongly predominant at low temperatures. They give rise to a positive contribution ${B}_{p}$ to the low-field $B$ which outweighs the negative electron-diffusion contribution for $Tl175\ifmmode^\circ\else\textdegree\fi{}$K, and which can be expressed in terms of Hall mobility ${\ensuremath{\mu}}_{H}$ and phonon-drag thermoelectric power ${Q}_{p}$ by ${B}_{p}=\frac{{\ensuremath{\zeta}}_{p}|{Q}_{p}|{\ensuremath{\mu}}_{H}}{c}$ with ${\ensuremath{\zeta}}_{p}\ensuremath{\approx}0.25$ over most of the range. Theoretically and experimentally, $\mathrm{BH}\ensuremath{\rightarrow}0$ as $H\ensuremath{\rightarrow}\ensuremath{\infty}$. As expected, $\ensuremath{\Delta}Q(\mathrm{H})$ resembles the magnetoresistance both in its anisotropy and in its variation with H. All the results are understandable theoretically in terms of a model which assigns to each ellipsoidal energy shell in crystal-momentum space an anisotropic phonon-drag Peltier tensor with principal components ${\ensuremath{\Pi}}_{p\mathrm{II}}$ in the high-mass direction, ${\ensuremath{\Pi}}_{p\ensuremath{\perp}}$ in the low-mass direction. The data show that $\frac{{\ensuremath{\Pi}}_{p\mathrm{II}}}{{\ensuremath{\Pi}}_{p\ensuremath{\perp}}}\ensuremath{\gg}1$ but the mass-relaxation-time anisotropy $\frac{{{m}_{\mathrm{II}}}^{*}{\ensuremath{\tau}}_{\ensuremath{\perp}}}{{{m}_{\ensuremath{\perp}}}^{*}{\ensuremath{\tau}}_{\mathrm{II}}}\ensuremath{\approx}17$. The effective average of ${\ensuremath{\Pi}}_{p\mathrm{II}}$ and ${\ensuremath{\Pi}}_{p\ensuremath{\perp}}$ increases slightly with decreasing energy of the shell, as it should if the relaxation times of the phonons vary with wave number $q$ at a rate between ${q}^{\ensuremath{-}1}$ and ${q}^{\ensuremath{-}2}$. A more quantitative analysis is possible and will be given in a forthcoming paper.

A twistor is an anisotropic ferromagnetic cylindrical wire with nonmagnetic core. The intrinsic magnetization flux curls in helical sense in the wire and has an air return path. Many field problems must be solved for their successful use as information storage elements. For instance, the demagnetizing field in the wire causes instability of storage and therefore must be reduced by suitable geometry of the twistor. The flux lines emanating from a bit link neighboring windings and also impose a magnetic field intensity in neighboring bits. The interactions, although undesirable in packing bits in a memory array, can be used to advantage as operating forces in logical devices.1 This paper analyzes the demagnetizing field in a twistor bit, based on the geometrical model of a confocal hollow prolate spheroid and the magnetic characterization of the material by B? = ?0(H+M) where M? is the intrinsic magnetization, constant in magnitude, but oriented by the external field. Demagnetizing factors for confocal hollow prolate spheroids are plotted against length-to-diameter ratio and wall thickness. Expressions for field intensities outside a twistor bit are given. Analogies between twistors and thin films are examined.

There are many practical applications of galvanomagnetic effects of metallic and intermetallic semiconductors in electronic devices, most of them utilizing the Hall effect. On account of its characteristics as a two dimensional multiplier in nature and its simple circuit assembly, the Hall effect element has attracted more attentions than the magnetoresistance element which inevitably needs an extra biasing magnetic field of a considerable magnitude to improve the non-linearity of its characteristic. In order to overcome this disadvantage a type of push-pull arrangement with two magnetoresistance elements in circuit is recommended, and the result is a three dimensional multiplier. A few examples of the applications of this kind of multiplier will be described in the paper, such as: (1) Analog computing element which may carry out a large number of mathematical operations; (2) Polyphase power meter which is capable of showing the total power (real, reactive or apparent), in an electrical network with a single indication; (3) Amplifiers and oscillators; (4) Mixer, modulator and demodulator, and (5) d.c. current-transformers, etc. In comparison with similar devices based on Hall effect, magnetoresistance-electronic devices have the advantages of higher sensitivity, better efficiency and greater ease of adjustment. All the devices will be operating at ordinary room temperature. The construction of the magnetoresistance element will be discussed and some experimental results are given.

Recent two publications by Y osida and Y oshimori have clarified the detailed electronic structure of the singlet collective bound state developed so far by them and by Okiji. In this paper, by means of their method, the magnetic-field dependence of the local electronic struc­ ture of the ground state is investigated. In particular, it is shown that the z-component of the spin correlation which sticks around the impurity remains unchanged even in the limit of increasing field. From this calculation, an expression for the magnetoresistance is inferred with the aid of the Friedel sum rule : 1t .d R(.d)=Rocos22- (Eo2+.J2)t/2 ' where E0 denotes the binding energy of the singlet state and .J the Zeeman energy of the free spin. It is to be noticed that this has no logarithmic field dependence. It is further shown that the relation

This paper describes the application of the magnetoresistance effect to oscillators using the high mobility semiconducting compound InSb as the active element. The first stable oscillator consisted of an InSb Corbino disk in a ferrite pot core reduced to a temperature of -40°C. The frequency was 210 cps and the open-circuited ac voltage was 540 mv, peak to peak. In order to construct a stable oscillator at room temperature, the shape of the semiconductor specimen was modified to a rectangular form composed of narrow strips of InSb joined together with metallic indium. This specimen has a comparatively higher internal resistance and a more pronounced magnetoresistance effect at low magnetic flux densities. A second oscillator using this specimen as an active element in an E-form ferrite core was tested and found to operate stably at a temperature of 25°C, in a frequency range from a few cps up to 300 cps. The maximum output power available was 2.5 mw, at 70 cps. There is no heating-up time required by the oscillator and the oscillation can start immediately under certain initial conditions. The elimination of the equipment to provide the low temperature, i.e., below 0° C for the oscillator has shown that the magnetoresistance effect can be made good use of in electronic devices without requiring a complicated circuit arrangement.

General equations with unknown parameters are obtained for the effective electric conductivity tensor of polycrystalline mixtures. Different methods of obtaining the unknown parameters are discussed. One of these methods, called in the paper the Bruggeman-Odelevsky method, has been applied to polycrystalline metals in a magnetic field. The influence of three different factors 1) fluctuations of the electron density 2) open orbits, 3) extended orbits, on the galvanomagnetic properties of a polycrystal has been investigated. In the first case agreement is found with Herrng's results, although it is shown that the range of applicability of his method is very restrained. The second case has not been considered by other authors. In the third case it is shown that the averaging method used by earlier authors needs very strong assumptions and this makes their explanation of the linear growth of the magnetoresistance doubtful.

A numerical analysis and an experimental study of Cerenkov radiation from an anisotropic ferrite are reported in this paper. Extensive curves of the interaction resistance R/spl lambda/, measured per unit wavelength of interaction distance, are presented as functions of the several ferrite and geometric parameters. Values in excess of 10/sup 3/ ohms per wavelength of interaction distance are noted. The conditions for Cerenkov radiation in the ferrite are derived from consideration of plane-wave propagation through the ferrite. X-band output powers of approximately one watt were observed using a 0.88 Mev bunched electron-beam with a peak current of 18 ma.

The paper describes the measurement of magnetoresistance in the semiconducting compound InSb at a frequency of 10 Gc/sec. The microwave value of |dR/μdH|H0 is the same order of magnitude as the dc value.

The simple model of a semiconductor, based on a single effective mass for the charge carriers and a spherical shape for the surfaces of constant energy, is now known to be inadequate for most of the semiconductors which have been extensively studied experimentally. However, some of these do correspond to what may be called the “many-valley” model, a model for which the band edge occurs at a number of equivalent points K(i) in wave number space, and for which the surfaces of constant energy are multiple ellipsoids, one centered on each of these points. This paper develops, for models of this type, the theory for: mobility (Section 2) and its temperature dependence (Section 3); thermoelectric power (Section 4); piezoresistance (Section 5); Hall effect (Sections 6 and 9); high-frequency dielectric constant (Section 7); and magnetoresistance (Sections 8 and 9). These phenomena are treated, for cases to which Maxwellian statistics apply, on the assumption that the scattering of the charge carriers is describable by a relaxation time which depends on energy only, but is otherwise unrestricted. This assumption can be shown to be justified in a large class of cases, although for some cases it fails, notably when ionized impurity scattering predominates and at the same time the effective mass is very anisotropic. Special attention is given to the role of inter-valley lattice scattering, i.e., to processes whereby a charge carrier is scattered from the neighborhood of one of the band edge points

Synopsis The measurements of the resistance and magneto-resistance at low temperatures of dilute alloys of gold with cobalt (up to 0.44 at.%) demonstrate that striking anomalous effects are absent in the relations between resistance, magneto-resistance and temperature or magnetic field. The influence of the transition metal, however, is observable as small deviations from a normal resistance temperature curve. The Kohler relation for the magneto-resistance fails for the larger cobalt concentrations. A survey of the available experimental data on alloys of this kind, including magnetic properties, concludes the paper.

This paper derives and analyzes the dispersion relation for wave propagation along a drifted electron‐hole plasma in a semiconductor in the presence of a transverse magnetic field. In short samples of high‐mobility semiconductors (InSb, in particular), this transverse magnetic field and the boundary condition at the sample contacts (vanishing of the tangential electric field) introduce a strong magnetoresistive effect. This effect is shown to result in a plasma instability when the applied fields exceed certain threshold values. The instability is found to have many features in common with the experimental results on microwave emission from InSb and it is proposed here as a possible explanation of this phenomenon.