Two large parallel plates of area A are placed at x=0 and x=d \ll \sqrt{A} in a semiconductor medium. The plate at x=0 is grounded, and the plate at x=d is at a fixed potential -V_{0}, where V_{0}>0. Particles of positive charge q flow between the two plates. You may neglect any dielectric effects of the medium.

a. For large V_{0}, the velocity of the positive charges is determined by a strong drag force, so that

where E is the local electric field and \mu is the charge mobility.

i. In the steady state, there is a nonzero but time-independent density of charges between the two plates. Let the charge density at position x be \rho(x). Use charge conservation to find a relationship between \rho(x), v(x), and their derivatives.

ii. Let V(x) be the electric potential at x. Derive an expression relating \rho(x), V(x), and their derivatives. (Hint: start by using Gauss’s law to relate the charge density \rho(x) to the derivative of the electric field E(x).)

iii. Suppose that in the steady state, conditions have been established so that V(x) is proportional to x^{b}, where b is an exponent you must find, and the current is nonzero. Derive an expression for the current in terms of V_{0} and the other given parameters.

b. For small V_{0}, the positive charges move by diffusion. The current due to diffusion is given by Fick’s Law,

Here, D is the diffusion constant, which you can assume to be described by the Einstein relation

where T is the temperature of the system.

i. Assume that in the steady state, conditions have been established so that a nonzero, steady current flows, and the electric potential again satisfies V(x) \propto x^{b^{\prime}}, where b^{\prime} is another exponent you must find. Derive an expression for the current in terms of V_{0} and the other given parameters.

ii. At roughly what voltage V_{0} does the system transition from this regime to the high voltage regime of the previous part?