A diffusion problem in semiconductor technology

Abstract

This paper deals with a mathematical model of a SEM-EBIC experiment devised to measure the diffusion length of semiconductor materials. In the model the semiconductor material occupies a half-space, of which the plane bounding surface is partly covered by a semi-infinite current-collecting junction, the Schottky diode. A scanning electron microscope (SEM) is used to inject minority carriers into the material. It is assumed that injection occurs at a single point only. The injected minority carriers diffuse through the material and recombine in the bulk at a rate proportional to their local concentration. Recombination also occurs at the free surface of the material, not covered by the junction, where its rate is determined by the surface recombination velocity v. The mathematical model gives rise to a mixed-boundary-value problem for the diffusion equation, which is solved by means of the Wiener-Hopf technique. An analytical expression is derived for the measurable electron-beam-induced current (EBIC) caused by the minority carriers reaching the junction. The solution obtained is valid for all values of v, and special attention is given to the limiting cases v = o0 and v = 0. Asymptotic expansions of the induced current, which are usable when the injection point is more than a few diffusion lengths away from the edge of the junction, are derived as well.

Cathodoluminescence Studies of Electron Injection Effects in Wide-Band-Gap Semiconductors

“…As the beam is moved away from the barrier/junction in a line-scan mode, the current decays as fewer and fewer minority carriers are able to diffuse to the space-charge region. The mathematical model for EBICurrent [20], shows that the decay of current, I EBIC , can be described by the following expression:…”

“…It should be noted that in Ref. [20], the authors analyzed only the two asymptotic cases, namely v s = 0 and v s = ∞, and found that  = -1/2 for the former and -3/2 for the latter. Later, Chan et.…”

A brief historical perspective of the Wiener–Hopf technique

“…Depending on the given problem, such boundary conditions may be defined in space or time, and may range over discrete or continuous independent variables. The Wiener-Hopf technique has found application in an very wide variety of research areas, including the diffraction of acoustic, elastic and electromagnetic waves, crystal growth [60], fracture mechanics, flow problems [61], diffusion models [62], geophysical applications [63] and mathematical finance. A simple search of the science literature will reveal the many thousands of articles that have employed Wiener and Hopf's elegant method.…”

Charge Collection From Within a Collecting Junction Well

“…The infinite summation in (16) was approximated by using 400 terms. For the L-shaped geometry, the investigated parameters were the junction depth h, the diffusion length L, and the depth of the generation source z.…”

“…Donolato, on the other hand, derived the expression for the case of zero surface recombination velocity [15]. The expression for any value of surface recombination velocity was derived by Boersma et al [16]. This expression was used in the analysis and development of measurement techniques for the planarcollector configuration [17], [18].…”