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then equation (5.98b) becomes

where Immediately we see that is periodic in T, i.e. in only if

and so we have

which shows that a bounded solution (as requires Thus the solution,
to this order, is

the construction of higher-order terms is left as a fairly routine exercise.

E5.17 A p-n junction
A p-n junction is where two semiconducting materials meet; such junctions may
perform different functions. The one that we describe is a diode. We analyse the
device for where the junction sits at x = 0 (and, by symmetry, it extends into
and an ohmic contact is placed at x = 1. In suitable non-dimensional,
scaled variables we have

where e is the electrostatic field, p the hole density and n the electron density. The
term ‘+1’ in (5.99a) is a constant ‘doping’ density and we will assume that the current
density, I(x) (appearing in (5.99b,c)), is given; indeed, in this simple model, we take
I(x) = constant. The boundary conditions are

and is our small parameter (typically about 0.001). (See Shockley, 1949; Roosbroeck,
1950; Vasil’eva & Stelmakh, 1977; Schmeisser & Weiss, 1986.) It is evident that the set
(5.99) exhibits the characteristics of a boundary-layer problem (§§2.6, 2.7) because the
small parameter multiplies the derivatives in each equation. However, a neat manoeuvre
allows one equation to be independent of