Potential Of Impurity Ions Research Articles

Scattering of electrons at impurity ions at low temperatures in a superlattice with doped quantum wells

The longitudinal and transverse mobilities of electrons in a superlattice with doped quantum wells, limited by scattering of the electrons at impurity ions, are calculated. The case of low temperatures and low densities of charge carriers is considered. The Conwell-Weisskopf hypothesis of the minimum scattering angle is used to solve the Boltzmann equation for a nondegenerate electron gas in the region of weak screening of the Coulomb potential of impurity ions. The numerical calculations are performed for the symmetric superlattice GaAs/Al0.36Ga0.64As with a period of 10 nm and an electron concentration of 1014 cm−3 at T=20 K.

Comment on "Solution of a nonlinear Poisson's equation for the potential of impurity ions in semiconductors with a spatially variable dielectric constant by an equivalent variational method"

Comment on "Solution of a nonlinear Poisson's equation for the potential of impurity ions in semiconductors with a spatially variable dielectric constant by an equivalent variational method"

Effects of the impurity potential on the band structure of doped trans-polyacetylene

In order to understand the drastic effect of impurity doping on the transport properties of polyacetylene, the authors investigate in detail how the impurity ion potential alone affects the band structure of electrons and the order parameter in doped polyacetylene by a self-consistent numerical calculation based upon the Su-Schrieffer-Heeger Hamiltonian plus the impurity potential. Their results show that the potential has only a small effect on gap closure, which is mainly determined by the larger overlap between the electronic wavefunctions in the nearest soliton states. They also find that the order parameter does not go to zero in the heavily doped region where there is no gap, and it is almost unaffected by the impurity potential.

Solution of a nonlinear Poisson's equation for the potential of impurity ions in semiconductors with a spatially variable dielectric constant by an equivalent variational method.

A correct knowledge of the screening potential in heavily doped semiconductors is highly desirable for analyzing various electronic engineering problems. In this paper a method for the solution of a nonlinear Poisson's equation in the framework of an equivalent variational approach is presented. Strengths and weaknesses of the earlier methods, including those of Csavinszky and co-workers, have been discussed in some detail. On the basis of a comparison of the present method with the earlier ones, it is felt that the present one is very promising, and perhaps more general and unified than most, if not all, other methods available in the literature.

Reply to "Comment on: Electron mobilities based on an exact numerical analysis of the dielectric-function-dependent linearized Poisson's equation for the potential of impurity ions in semiconductors"

The adequacy of our physically based boundary condition is discussed relative to the theoretical boundary condition used by Csavinszky and Morrow. The impurity-ion potentials resulting from the two calculations are compared and contrasted on the basis of their physical behavior and that of the ionized-impurity-limited mobilities to be derived from each.

Comment on: "Electron mobilities based on an exact numerical analysis of the dielectric-function-dependent linearized Poisson's equation for the potential of impurity ions in semiconductors"

We believe that an assumption made in this paper, namely, forcing the potential to exhibit a prescribed behavior at a certain distance from a point ion, is not tenable. We present below results of our numerical and variational calculations made without the above assumption.

Electron mobilities based on an exact numerical analysis of the dielectric-function-dependent linearized Poisson's equation for the potential of impurity ions in semiconductors

The complete linearized Poisson's equation for the impurity-ion potential incorporating the spatial variation of the semiconductor dielectric function is treated numerically as a boundary-value problem with the finite-difference method. The impurity-ion potential so derived is shown by direct substitution to be a superior solution to the differential equation than that obtained by an equivalent variational-principle approach, or by the straightforward replacement of the dielectric constant by the dielectric function of the Dingle potential. Theoretical electron mobilities for silicon, germanium, and gallium arsenide based on this potential are compared with the Dingle mobility. It is found that the result of including the spatial variation of the dielectric function in the theory of ionized-impurity scattering is to reduce the electron mobility from that calculated using the Dingle potential. This effect is small, except in the cases of heavy and very heavy doping, where it is found that the difference increases monotonically as the doping density is likewise increased.

Addendum to "Variational principles for solving nonlinear Poisson equations for the potential of impurity ions in semiconductors with spatially variable dielectric constants"

In the above-named paper, a simplified form of the nonlinear Poisson equation for the impurity-ion potential has been solved approximately by making use of an equivalent variational principle. The difference between the exact and the simplified nonlinear Poisson equations is a term involving the derivative of the potential and the derivative of the spatially variable dielectric constant. The purpose of the present paper is the presentation of a more general variational principle, by which one might obtain an approximate solution of the exact nonlinear Poisson equation.

Corrections to the impurity-ion potential characterized by a spatially variable dielectric function; corresponding electron-conductivity mobility in silicon and germanium

The complete linearized Poisson equation for the potential of impurity ions in a semiconductor, including the spatial variation of the dielectric function, is solved numerically by an equivalent variational principle, incorporating a functional recently proposed by Brownstein. The resulting potential is compared and contrasted with previous formulations which neglect a term involving the derivative of the dielectric function. In the region where this function becomes a constant, the chosen potential approximates that of Dingle, while at the impurity site it is greater than the latter by a multiplicative factor equal to the dielectric constant. Within the linear approximation, the present potential is believed to contain the correct embodiment of spatial variation in the dielectric function in a self-consistent manner. The electron-conductivity mobility, in the Born approximation, is also determined with this potential. Comparison with analogous calculations and with some typical experimental data as reference is made. It is found that the present theory of ionized-impurity-limited mobility yields a significant improvement over the Dingle theory and over prior treatments of this problem.

Erratum: Variational principles for solving nonlinear Poisson equations for the potential of impurity ions in semiconductors with spatially variable dielectric constants

Erratum: Variational principles for solving nonlinear Poisson equations for the potential of impurity ions in semiconductors with spatially variable dielectric constants

Variational principles for solving nonlinear Poisson equations for the potential of impurity ions in semiconductors with spatially variable dielectric constants

The theory of the potential of a (point) impurity ion in a semiconductor involves an expansion of the screening-charge density in terms of the impurity-ion potential. In a recent paper, this problem has been reexamined by taking into consideration the spatial variation of the dielectric constant of the host medium. In this paper, the linearized Poisson equation, with the neglect of a small term, has been solved approximately by making use of an equivalent variational principle. In another recent paper, the spatial variation of the dielectric constant has been ignored and variational principles have been formulated for obtaining approximate solutions to nonlinear Poisson equations of any given order in the impurity-ion potential. The present paper aims at the unification of the above two approaches and presents variational principles for obtaining approximate solutions of nonlinear Poisson equations for the potential of an impurity ion which is located in a medium characterized by a spatially variable dielectric constant.

Asymptotic corrections to the potential of impurity ions in semiconductors with spatially variable dielectric constants

In a recent paper, the problem of the screening of a (point) donor ion in Si and Ge has been reexamined by introducing into the theory the spatial dielectric functions of Azuma and Shindo and Okuro and Azuma, respectively. Poisson's equation, with the neglect of a small term, has been solved approximately by making use of a variational principle. The result was an impurity-ion potential that consists of a linear combination of two exponentially screened Coulomb potentials (with two different screening lengths) which is scaled by the static dielectric constant of the medium. The present paper shows that, at distances larger than Dingle's screening length ${R}_{0}$ the term neglected in Poisson's equation would add to this impurity-ion potential two types of correction. One type consists of terms which are proportional to Dingle's potential, but with proportionality constants so small that the correction terms are rendered utterly negligible. The other type consists of terms each one of which is proportional to functions of the form $(\frac{1}{{r}^{m}})\mathrm{exp}(\ensuremath{-}{q}_{i}r)$, where $m=2,3,4,\dots{}$ and the parameters ${q}_{i}$ are related to the constants appearing in the spatial dielectric functions of Si and Ge. The finding that the correction terms to the impurity-ion potential fall off faster than $\frac{1}{r}$ does, in the author's opinion, establish the asymptotic validity of the previous variational theory.

Variational principles for solving nonlinear Poisson equations for the potential of impurity ions in semiconductors

The present paper concerns itself with corrections to the potential of an impurity ion obtained by Dingle for charged impurities in semiconductors with spherical energy surfaces. In Dingle's theory, the Fermi-Dirac integral ${\mathcal{F}}_{\frac{1}{2}}$, appearing in the density of the screening charge, is expanded in powers of the potential. Dingle, retaining only terms up to and including the linear term in the potential, has obtained a linearized Poisson equation. The solution of this differential equation results in a screened Coulomb potential for the impurity ion which is scaled by the static dielectric constant of the medium. In the present paper, a variational principle is suggested that is equivalent to the Poisson equation to any given order in the impurity-ion potential. The analytical groundwork is carried through to include Poisson equations resulting from the retention of quadratic and cubic terms in the expansion of ${\mathcal{F}}_{\frac{1}{2}}$. Use of the variational principle permits the representation of the approximate solutions to the nonlinear Poisson equations as linear combinations of exponentially screened Coulomb potentials which are scaled by the static dielectric constant of the semiconductor. This feature has the advantage of permitting a straightforward modification of theories of ionized impurity scattering which are based on the use of a single screened Coulomb potential.

Theory of the screening of impurity ions in semiconductors with spatially-variable dielectric constants

In this paper, Dingle's theory of the screening of impurity ions in semiconductors is generalized. This is accomplished by inserting in the theory of screening the spatial dielectric function of the medium. Poisson's equation for the potential of an impurity ion follows from one of Maxwell's equations. With the analytical form for the spatial dielectric function for Si and Ge, obtained for these materials by Azuma and Shindo and by Okuro and Azuma, Poisson's equation assumes a specific form. The asymptotic form of this differential equation, the region where the theory is expected to be valid, is solved approximately by an equivalent variational principle. The end result of the present theory is an impurity-ion potential which is represented by a linear combination of two exponentially screened Coulomb potentials scaled by the static dielectric constant of the medium. Each of the screening lengths appearing in the screened Coulomb potentials is related to Dingle's screening length. Numerical data, as functions of charge carrier concentration, are given both for the screening lengths, and for the coefficients involved in the linear combination of the two screened Coloumb potentials. It is concluded that the form obtained in the present paper for the potential of an impurity ion leads to modifications in theories of ionized-impurity scattering in semiconductors.

Magnetic-Field-Induced Mott Transition in Semiconductors

The influence of an applied magnetic field on semiconductor impurity states is calculated, including the effects of screening of the impurity-ion potential by free carriers. When the screening is sufficiently great so that no bound states occur at zero field, they are introduced at some critical field. If the semiconductor is extrinsic, then the appearance of bound states at the critical field results in a Mott transition as the field is increased. Free carriers are entrapped at impurity sites and the electrical conductivity is markedly reduced as the magnetic field is increased beyond the critical value.