Weak-field Hall Research Articles

New Method for Computing the Weak-Field Hall Coefficient. II. Some Extensions and Modifications

An earlier paper described a simple method for computing the weak-field Hall coefficient through the use of a Fermi surface composed entirely of planar faces. That paper developed a set of rules which linked the general behavior of the Hall coefficient to two fundamental properties of transport models, Fermi-surface shape and scattering anisotropy. The present paper reformulates those rules by adding a third ingredient to the model description, shape evolution. Exceptions to the earlier rules are thereby eliminated. The present paper also extends the simple method to noncubic models. The results for "undulating cylinders" (a Fermi-surface approximation for some hexagonal metals) and toroidal Fermi surfaces (a possible model for the wurtzite lattice) are analyzed. Finally, the effect of rounding the sharp edges at which the planar Fermi-surface faces intersect is investigated. The results resolve an apparent paradox pointed out by Stern, and provide some insight into the general magnetic field dependence of the Hall coefficient.

Weak-Field Galvanomagnetic Properties of a New Type of Metallic Model

The weak-field Hall coefficient and magnetoresistance are computed for a metallic model in which the Fermi surface has the form of a cube with rounded edges and corners. Exact and relatively simple results are obtained as a function of a parameter which allows the shape of the Fermi surface to evolve continuously from a sphere to a cube with sharp edges and corners. In going from the one extreme to the other, the Hall coefficient decreases monotonically from $\frac{1}{\mathrm{ne}}$ to $\frac{\frac{1}{4}\ensuremath{\pi}}{\mathrm{ne}}$, while the Seitz magnetoresistance coefficients $b$, $c$, and $d$ increase monotonically from zero to infinity (for $b$ and $d$) and to $1\ensuremath{-}(\frac{8}{3\ensuremath{\pi}})$ (for $c$). The results are interpreted and compared with the galvanomagnetic properties of other types of models.

New Method for Computing the Weak-Field Hall Coefficient

The factor $r$ in the expression ${R}_{0}=\frac{r}{\mathrm{ne}}$ for the weak-field Hall coefficient is computed for a wide variety of metallic band models. The technique employs a planar-faced approximation to the true Fermi surface, and this greatly simplifies the mathematics. Strongly distorted Fermi surfaces, multivalley models, anisotropic scattering, and band-filling effects are studied, and the results are used to develop a general theory of the behavior of the factor $r$. The effects of shape and scattering anisotropies, the question of their equivalence, and the reason why a Hall coefficient changes sign are discussed. Finally, the results are applied to the model of Cooper and Raimes, to lead-pipe models, and to experimental Hall data on the noble metals.

Hall coefficient behavior in TiFe, TiCo, TiNi, and their alloys

Abstract Hall coefficient and resistivity measurements on the 3d transition metal compounds TiFe, TiCo, and TiNi, and on several of their alloys, were made at 295°, 77°, and 4.2°K. These materials contain between 6 and 7 valence electrons per atom (Nve). The Hall coefficient behavior was characterized by values as large as +35 × 10−4 cm3/C (in TiFe), by a strong temperature dependence at all compositions studied, and, with increasing Nve, by changes in sign from positive to negative and back to positive again. Negative Hall coefficients have not been observed before in this range of Nve values in any other 3d materials. The discussion showed, first of all, that a phenomenological two-band model could account qualitatively for the dependence of the Hall data on temperature and on Nve. It was also demonstrated that a small group of positive carriers was present in TiFe, not exceeding 1 40 per atom in number, which had a room-temperature mobility greater than 122 cm2/V-sec, and it was shown that this conclusion did not depend on the number of bands actually present. Secondly, it was shown that Mattheiss's theoretical band model calculation for non-magnetic Cr, Mo, and W contained a specific feature—a band maximum at which the wave function had a surprisingly p-like (rather than d-like) character—which could account for the small number of high-mobility holes in TiFe. The model was then applied to the remainder of the Hall data, and it was not difficult to say which of the four bands was responsible for the several major features of the experimental data. Finally, a comparison of the present results with earlier data on Cr and its alloys (Nve between 5.9 and 6.6) revealed distinctive differences in the temperature- and composition dependence of the Hall coefficient. These differences were explained using Lomer's proposal that antiferromagnetic ordering in Cr changes Mattheiss's fourband model to a two-band model with a considerably lower carrier concentration. It was assumed that chemical ordering in TiFe has essentially the same effect, and it was then shown that the Hall effect behavior was related to the different ways that the magnetic and chemical ordering depend on temperature and composition in the two sets of materials. It was concluded that weak-field Hall coefficient studies on transition metal elements and compounds, and their alloys, can be quite useful, especially in cases where low carrier mobilities and lack of single-crystal samples prevent application of sophisticated, high magnetic field techniques. This is due to the fact that the weak-field Hall coefficient is more sensitive than had been realized to the numbers and mobilities of the carriers of all the bands which cross the Fermi level.

Simple Derivation of the Hall Anisotropy Factors for Cubic and Octahedral Constant-Energy Surfaces

An elementary method is described for the derivation of the weak-field Hall anisotropy factor ($r$ in ${R}_{0}=\frac{r}{\mathrm{ne}}$) for the cases of cubic and octahedral energy surfaces. The magnetoconductivity approach is used (i.e., an electric field which is fixed in direction and magnitude, and a current which rotates when the magnetic field is turned on). The method simply computes the longitudinal current due to the electric field and the transverse current due to the magnetic force. The latter results entirely from carriers which drift across an edge of the energy surface, thereby changing the direction of their velocity. Then the Hall angle, Hall field, and finally the Hall coefficient are easily determined, and the results agree with the results of the much more complicated approach (the kinetic method of Shockley, McClure, and Chambers) previously used to derive the Hall anisotropy factor for these two cases.

Microwave free carrier Faraday effect in semiconductors--perturbation theory for guided waves

At frequencies approaching the electron-lattice collision frequency, the weak field Faraday effect can be used to study details of scattering mechanisms that are manifested in the complex Hall terms of the permittivity tensor. Plane wave theories that may apply to experiments using infrared or visible light do not apply rigorously to the guided wave experiments that have been performed in this frequency range. This paper describes a guided wave theory relating the real and imaginary parts of the weak field Hall terms to microwave measurements performed on a degenerate TE mode waveguide containing a longitudinally magnetized cubic semiconductor. The theory assumes only that the magnetic field is small and takes losses and reflections into consideration exactly.

Valence Bands in Lead Telluride

The magnetic field dependence of the Hall coefficient at 296° and 77°K, and the temperature dependence of the weak-field Hall coefficient and the resistivity between 296° and 77°K were studied in single-crystal samples of p-type PbTe having carrier concentrations ranging from 4.9×1017 to 1.7×1019 per cm3. The Hall data at 77°K are quantitatively consistent with magnetoresistance data which have previously established the presence of 〈111〉 ellipsoids in the valence band. They are not consistent with a low-temperature two-band model, proposed by Stiles from de Haas-van Alphen data at 4.2°K, unless the band edges lie at approximately the same energy (as Stiles found) and unless the carrier mobilities in the two bands are nearly alike. On the other hand, both the Hall and resistivity data above about 150°K do exhibit two-carrier effects suggesting the presence of a lower mobility band at an energy about 0.1 ev below those bands which are occupied at low temperatures.

Theoretical Transport Coefficients for Polar Semiconductors

The mobility, thermoelectric power and weak field Hall coefficient have been calculated when the charge carriers of a polar semiconductor are scattered by an optical mode of vibration of the lattice. The collision operator derived by Frohlich, and Howarth and Sondheimer was used, and the Boltzmann transport equation was integrated directly on an electronic computer. Results are given for 2θ ≥ T ≥ θ/2 and 5 ≥ η ≥ -2. (η is the reduced Fermi level, θ = ν1h/k, and ν1 is the frequency of the longitudinal optical mode.) These results are compared where possible with those obtained from a different method of calculation by Howarth, Lewis and Sondheimer. Rough values are also given for the Lorenz number and weak field Ettingshausen coefficient.