Edge-type Dislocations Research Articles

Electrical Properties ofN-Type Germanium

Measurements of conductivity and Hall effect from 11\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K on a set of $n$-type germanium samples covering the range from intrinsic to degenerate are reported. The purity and uniformity of the samples and extensiveness of the data permit a more thorough-going comparison with theory than has been possible in previous work.The theory of mobility is reviewed briefly. The treatment of impurity scattering by Brooks and Herring is presented, and their formula for the impurity mobility is used throughout. An analytical formula for obtaining the mobility from lattice and impurity mobilities is included. The effect of electron-electron collisions on the mobility is considered in a qualitative manner.The principal conclusions concerning the mobility are as follows: (1) Over the range 20.4\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K the lattice mobility varies as ${T}^{\ensuremath{-}1.64}$ rather than the theoretically predicted ${T}^{\ensuremath{-}1.5}$. (2) The impurity mobility increases with temperature less rapidly than the theoretical formula predicts, the exponent of $T$ in the numerator being apparently between 1.0 and 1.5. (3) The Erginsoy formula for neutral impurity scattering appears to fit the experimental data reasonably well for a value of effective mass about one-third the free electron mass. (4) Dislocation scattering is negligible, leading to the conclusion that the density of edge-type dislocations is less than ${10}^{6}$/${\mathrm{cm}}^{2}$.In fitting the concentration data, the parameters involved are activation energy, acceptor concentration, and effective mass. An effective mass in the neighborhood of one-quarter the free electron mass gives the best fit for all samples. The values of acceptor concentration obtained in this way for this effective mass agree well with those calculated from the low-temperature mobility values. The activation energy obtained for the purer samples is 0.0125 ev, in agreement with the value calculated from the hydrogen-like model for one-quarter the free electron mass.The variation of activation energy with concentration does not agree with that observed by Pearson and Bardeen for $p$-type silicon. The effects which have been proposed to explain the variation are: residual potential energy of attraction between free electrons and ionized donors, screening of trapping centers by the free electrons, polarization of neutral centers by free electrons. It is concluded that a combination of the three effects will probably be required to explain the results.The ratio of Hall to drift mobility is shown to agree with the theoretically predicted value within about 10 percent in the range 78\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K.

The Small Angle Scattering of X-Rays from Cold-Worked Solids

The small angle scattering of x-rays or thermal neutrons from cold-worked crystals is calculated on the basis of two models, according to which the scattering arises predominantly from the density variation associated either with small cavities or with edge-type dislocations. The elastic distortions surrounding the cavities are unimportant, so that the scattering from cavities in a uniform medium of density $n$ is the same, to a good approximation, as that from partices (in vacuum) of density $n$ of the same size as the cavities. Thus the usual formulas of small angle scattering obtain, the scattered intensity has the familiar Gaussian-like dependence on scattering angle, and earlier results on multiple scattering may be applied. Around edge-type dislocations, on the other hand, the density variation is proportional to $\frac{sin\ensuremath{\xi}}{r}$, where $\ensuremath{\xi}$ is the angle measured from the slip direction in the plane perpendicular to the dislocation line and $r$ is the distance from the dislocation line. This angular variation results in a complete modification of the usual formulas, and, in fact, all of the terms ordinarily present disappear for this case, and conversely. The scattering shows a parabolic increase from zero at small angles, a maximum, and finally a monotonic decrease with increasing scattering angle. There is a large degree of anisotropy in the scattering, depending on the direction of the incident beam relative to the slip and dislocation axes. Multiple scattering from an array of dislocations in even a thick specimen is shown to be negligible. The theory is compared with Blin and Guinier's preliminary experimental results, and it is concluded that dislocations are incapable of explaining their data, although it is expected that under suitable conditions the measurement of the scattering from dislocations should be experimentally feasible.

Effects of Dislocations on Mobilities in Semiconductors

The scattering of electrons or holes in semiconductors by the dilation of the lattice around rigid, randomly-arranged edge-type dislocations is treated by the method of the deformation potential. The contribution of this scattering to the electrical resistance is determined from the Boltzmann transport equation by the method of Mackenzie and Sondheimer, except for the use of Maxwell-Boltzmann, rather than Fermi-Dirac statistics. The temperature dependence of the reciprocal of the mobility $\ensuremath{\mu}$ is given by $\frac{1}{\ensuremath{\mu}}={\ensuremath{\alpha}}_{l}{T}^{\frac{3}{2}}+{\ensuremath{\alpha}}_{i}{T}^{\ensuremath{-}\frac{3}{2}}+{\ensuremath{\alpha}}_{d}{T}^{\ensuremath{-}1}$ where the $\ensuremath{\alpha}$'s are temperature-independent quantities referring to scattering from the lattice, impurity atoms, and dislocations, respectively. A discussion is given of the relative magnitudes of these three terms, and of the possibility of obtaining experimental information concerning dislocations in semiconductors by electrical measurements.

Scattering of Electrons in Metals by Dislocations

The problem of the change in electrical conductivity in metals upon cold-working is treated by the method of Koehler in which the scattering of electrons by pairs of parallel edge-type dislocations is assumed to be the major effect. A comparison is made between the electronic shielding assumed in the scattering potential used by Koehler and the shielding assumed by Landauer in a somewhat different treatment. The large effect of the discontinuity in the ionic displacement across the plane connecting the dislocation axes is shown to cancel that of a previously neglected term in the scattering potential. The scattering matrix element is evaluated, and the change in resistivity is computed by the perturbation method of Mackenzie and Sondheimer. Finally, the anisotropy in the resistivity is discussed, and it is shown that the slip direction is the direction of low resistivity.

THE MACROMOSAIC STRUCTURE OF TIN SINGLE CRYSTALS

Single crystals of high purity tin grown from the melt by a modified Bridgman method are shown to be partitioned into bands, or striations. A difference of orientation exists between the striations such that the lattice of one striation may be brought into coincidence with the lattice of its neighbor by a pure rotation about an axis parallel to the specimen axis. The properties of the striations have been shown to be dependent on both the rate of growth and the crystallographic orientation relative to the direction of heat flow. A tentative explanation of the origin of this effect is advanced in terms of the formation of edge type dislocations from the condensation of vacant lattice sites.

Conductivity of Cold-Worked Metals

In the vicinity of an edge type dislocation, the density of electrons, and therefore the width of the filled portion of the conduction band is not uniform. To keep the electrons in equilibrium, an electrostatic potential is required. The scattering caused by this potential is used to calculate the increased resistance of isotropically cold-worked copper.

The Theory of the Change in the Conductivity of Metals Produced by Cold Work

An attempt is made to calculate the increase in the residual electrical resistance of cold-worked metals by attributing the effect to an elastic distortion similar to that produced by the presence of edge type dislocations. It is shown that, in single crystals, the direction of the dislocation axes, the slip direction and the direction normal to the slip plane represent a set of principal axes of the conductivity tensor. The dislocation resistance parallel to the dislocation axes is zero, while the resistance in the slip direction exceeds the resistance normal to the slip plane by a factor which depends upon Poisson's ratio and which lies between 1 and 3. The mean density of dislocations in a highly cold-worked polycrystalline specimen of copper is estimated from the observed resistance change and is found to be in reasonable agreement with the value derived from the energy stored during work hardening. The formulation of the present theory follows the lines of a recent paper by J. S. Koehler; the results differ considerably from Koehler's, however, and it is shown that his treatment cannot be regarded as satisfactory.

LXXXII. Edge dislocations in anisotropic materials

Abstract Burgers has discussed dislocations in materials with cubic symmetry for the general case in which the dislocation axis is an arbitrary curve. The results of the present paper are limited to dislocations of edge type with an infinite straight line as axis, but apply to a material with the symmetry of any of the crystal classes. The axis of the dislocation may be arbitrarily inclined to the symmetry axes of the material. Expressions are given for the elastic displacements and the energy of the dislocation. Nabarro's calculation of the width of a dislocation is extended to the anisotropic case.