Environmental Engineering Reference
In-Depth Information
Figure 3.8 Schematic of bands E versus k in a
periodic potential, based on Kronig
basis of Bragg reflections at k¼(p/a). Each
band accommodates exactly 2N electrons, so
that one electron per atom gives a half-filled
band and a metal, while two electrons per atom
give a filled band, an insulator.
Penney
model. The bands are restricted in k to values
less than p/a. Energy gaps occurring at k ¼
(p/a) are also physically understood on the
-
structure of semiconductors that are important in their application in photovoltaic
devices. This effect has an analogue in electrical circuitry, where a periodic
transmission line of lumped inductor and capacitor elements can have a pass
band and a stop band. So the Kronig
-
Penney model explains the observation, as in
Figure 3.2, of very long mean free path for electrons in pure metals, and also in
semiconductors.
Each band (
p
/
a
<p
/
a
) contains
N
atoms and can accommodate 2
N
electrons.
In gold, one
finds a half-
lled band (metal) because Au releases only one electron. On
the other hand, an
even
number of electrons per atom
lls
one or more bands, leading
to an
insulator
because electrons cannot respond by changing
k
in an
E
field. Si has
valence 4, will have
two
filled bands
, and is thus a band insulator.
<
k
3.3.1
Properties of a Metal: Electrons in an Empty Box (I)
Some essential properties of a metal such as sodium or gold can be obtained from a
simple 3Dmodel. This idealized picture is of electrons con
ned in a box of side
L
,with
in
nite potential outside. More realistically, the potential barrier is the work function
j
, with typical values between 3 and 5 eV. This is a classic case of the situation
mentioned at the end of Chapter 2 where the effects of other particles can be
summarized in a useful way by a potential or just by a con
ning boundary condition.
The Schrodinger equation can be solved inside the box (see Equation 2.16) because
of the boundary condition
y¼
0 at the walls, and the solutions inside an empty cube
