Question

Show Maxwell's equations in a semiconductor for a wavelength

A.) Below
B.) At
and
C.) Above

the band gap of the a semiconductor.

I'm very stuck on this, so insight on any part would be greatly appreciated.

Answer 1

Maxwell's Equations The Maxwell's equations are and the Lorentz force is where -(cp,j) Faa 0 0 3- B10 This corresponds to: 1 dE OB Four Potential

Four Potential The four potential is defined by: this corresponds to: OA Ot The Maxwell's equations can then be written as (note that the two eq. without sources are automatically satisfied by the four potential): where we have employed the Lorentz gauge o4 -0, Semiconductor Device Phvsics In general, the task is to find the five quantities: n(x, t), p(x, t), Jn(x, t), JAx, t), E(x, t) where n (p) is the electron (hole) concentration, J, (Jr) is the electron (hole) current density, e is the electric field. And we have five equations that relate them. We start with the continuity equation where the current density J is composed of electron and hole current densities: and the charge density p is composed of mobile

and the charge density is composed of mobile (electrons and holes) and fixed charges (ionized donors and acceptors) where n and p is the electron and hole concetration, c is the net doping concetration (c-p -na where Po is the concentration of ionized donors, charged positive, and ma is the concentration of ionized acceptors, charged negative) and is the electron charge (positive). We get Assuming the fixed charges c are time invariant, we get where R is the net recombination rate for electrons and holes (a positive value means recombination, a negative value generation of carriers). We get the carrier continuity equations: Ot op Then we need material relations that express how the current j is generated using E and n and . A drift- diffusion model is to assume a drift current E) and a diffusion (¡D,Vn), which gives:

where μ», μ, D», Dp are the carrier mobilities and diffusivities Final equation is the Gauss's law Equations Combining (2) and (1) we get the following three equations for three unknowns n, P and E: And it is usually assumed that the magnetic field is time independent, so E-Vo and we get: These are three nonlinear (due to the terms ,mnVo and lippVo) equations for three unknown functions n, p and φ. Example1 We can substract the first two equations and we get and using ρ-9(p_ n +C) and σ-glapp + μ,,n), we get

where μ», μ, D», Dp are the carrier mobilities and diffusivities. Final equation is the Gauss's law: Equations Combining (2) and (1) we get the following three equations for three unknowns n, p and E: Op And it is usually assumed that the magnetic field is time independent, so E- and we get: On These are three nonlinear (due to the terms ,nVo and pequations for three unknown functions n, p and