Environmental Engineering Reference
In-Depth Information
where b¼V o wma / h
2 and q ¼ (2 mE ) 1/2 / h . (This simpli ed form of Equation 3.14 is
actually obtained in a limiting process where the potential barriers are simulta-
neously made higher and narrower, preserving the value b . This can be described as
Nd functions of strength b .) The parameter b is a dimensionless measure of the
strength of the periodic variation.
We can test Equation 3.14 by examining what happens in the simple cases of
vanishing and extremely strong potential barriers V .
One can see that Equation 3.14 in the zero-barrier limit, 0, the condition
cos ka¼ cos qa leads to (2 mE ) 1/2 / h , which recovers the free electron result,
h
2 k 2 /2 m .
Next, if b becomes arbitrarily large, the only way the term b (sin qa / qa ) can remain
finite, as the equation requires, is for sin qa to become zero. This requires qa¼np ,or
a (2 mE ) 1/2 / h ¼np , which leads to E¼n 2 h 2 /8 ma 2 (see Equation 2.15). These are the
levels for a 1D square well of width a (recall that in the limiting process, the barrier
width w goes to zero, so that each atomwill occupy a potential well of width a , which is
the atomic spacing).
The new interesting effects of band formation occur for nite values of b . Figure 3.7
gives a sketch of the right-hand side, R ( E ), of Equation 3.14, versus Ka ( qa in the text).
Solutions of this equation are possible only when R ( E ) is between 1 and þ 1, the
range of the cos ka term on the left. Solutions for E , limited to these regions,
correspond to allowed energy bands. Note that allowed solutions are possible for
1
<p . More generally, boundaries of
the allowed bands are at k ¼ ( ) np / a , 1, 2, 3 ... . It is conventional to collect the
allowed bands as shown in Figure 3.8 into the range between ( ) p / a .
The result of this analysis, then, is that in the allowed bands no scattering occurs ,as
long as the potential is periodic! This can explain why the resistivity approaches zero
at low temperature for a pure crystalline metal like Au. It also explains the band
<
cos ka
<
1, which corresponds to p<
ka
Figure 3.7 Kronig
Penney model, plot of
ordinate R(E) versus abscissa Ka, for P¼4.712.
Shaded areas in this plot denote ranges of Ka
that do not allow travelingwave states. Traveling
wave solutions occur only when ordinate R(E)
has magnitude unity or less. The bottom of the
conduction band E o corresponds to R(E) ¼1,
the first crossing, and the band edge ka¼p
corresponds to R(E)¼1. In this plot dark
areas correspond to bandgaps. In the text the
symbol b is used for P.
-
 
Search WWH ::




Custom Search