Formation time of a screening charge at boundaries of illuminated spots of different configuration
E. M. Uyukin
A. V. Shubnikov Institute of Crystallography, Academy of Sciences of the USSR, Moscow
(Submitted December 28, 1978)
Fiz. Tverd. Tela (Leningrad) 21, 1578-1580 (May 1979)
PACS numbers: 72.40. + w
The photoconductivity of high-resistivity insulating crystals is estimated in Ref. 1 from measurements of the Maxwellian relaxation time. The photoconductivity of high-resistivity crystals can then be found by a contactless method, which is very desirable because the conventional methods are difficult. A laser beam is formed into an elliptic spot1 with its major axis perpendicular to an external electric field applied along the x axis, normal to the optic axis of the crystal z. The beam direction is parallel to the z axis. The problem of screening of the external field is solved in Ref. 1 for the situation when an elliptic spot reduces to a rectangular strip, which can be regarded as the limiting case of an ellipse.
We shall solve the problem of the screening of an external electric field in the case of a homogeneously illuminated elliptical spot oriented in various ways relative to the external field. We shall show that the Maxwellian relaxation time differs very considerably for the cases when the major axis of the ellipse is perpendicular and parallel to the electric field.
We shall assume that the formation of a space charge is due to charging of deep centers whose concentration is fairly high. Under these conditions, which correspond to real situations, the screening charge which appears at the boundary of a uniformly illuminated region can be regarded as distributed on the surface. It is quite clear that the field of such a charge has the same coordinate dependence as the field of a homogeneously polarized ellipsoid. This can be demonstrated by direct substitution of the proposed solutions into the complete system of equations for this problem. The potential of this field outside the illuminated spot2 can be represented conveniently in terms of elliptic cylindrical coordinates3
The total potential outside the illuminated region is , where sh u sin v is the potential of the external field. Inside the illuminated region (where the space-charge field is homogeneous), the potential issh u sin v. The explicit form of the function is best determined not by finding the limit of the three-dimensional problem,2 but directly by solving the two-dimensional Laplace equation. The function is then described by
FIG. 1. Major axis of an elliptic illuminated spot perpendicular to an external electric field E0.
FIG. 2. Major axis of an elliptic illuminated spot parallel to an external electric field E0.
Using Eq. (1) and applying condition that the potential vanishes at infinity, we find that . The conditions of continuity of the potential at the boundary of the illuminated spot lead to the following expression for :
where Ei is so far an undetermined field inside the spot. This field can be found from the law of conservation of charge
and the boundary condition
which is written down for the points denoted by o in Figs. 1 and 2.
In the case corresponding to Fig. 1, we find from Eqs. (4) and (2) that
Equations (5), (2), and (3) yield
The solution of this equation subject to the initial condition is
It is clear from Eq. (8) that in the limit the Maxwellian relaxation time is governed by the conductivity in the illuminated region, in agreement with the results of Ref. 1. When the major axis of the ellipse has the same direction as the external field (Fig. 2), the corresponding solutions are
The case of a circular spot is obtained by assuming b = c in Eq. (11), and it is also described by Eq. (8).
It follows that the time to establish a screening charge depends strongly on the ratio of the semiaxes of the ellipse. When the major axis of an elliptic spot is perpendicular to the electric field and also if , this time is equal to the Maxwellian relaxation time of a homogeneous medium whose conductivity is . However, if the major axis of the ellipse is directed along the electric field and if , the time is equal to the Maxwellian relaxation time for a homogeneous medium whose conductivity is .
The author is grateful to A. P. Levanyuk, V. A. Pashkov, and N. M. Solov'eva for a valuable discussion of the topics considered here, and also to Kh. S. Bagdasarov for his encouragement.
1V. A. Pashkov, N. M. Solov'eva, and E. M. Uyukin, Fiz. Tverd. Tela (Leningrad) 21. 1879 (1979) [Sov. Phys. Solid State 21,1079 (1979)].
2L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford (1960).
3 G. Arfken, Mathematical Methods for Physicists, Academic Press, New York (1966).
Article credit: E.M. UYUKIN