Environmental Engineering Reference
In-Depth Information
mentioned in Chapter 2 indicates that the position and motion of a particle cannot
be known simultaneously.)
In spite of this, the Bohr model values for the electron energies
E
n
¼E
o
Z
2
/
n
2
,
spectral line wavelengths, and the characteristic atomic size,
a
o
¼
h
2
/
mk
c
e
2
¼
0.053
nm, are all exactly preserved in the fully correct treatment based on nanophysics,
described in the next section.
3.2
Charge Motion in Periodic Potential
Semiconductors and their electrical conduction are important in photovoltaic cells
and are strongly inuencedbythewavepropertiesofelectrons.Chargemotionis
acentralissueinefcient solar cells. We discussed the idea of the mean free path
L
L¼
1/(
n
s
s
scatt
)
¼
1/
n
s
pr
2
, for a scattering center of radius
r
2
present at density
n
s
.Thisformulais
useful also in connection with the electrical conductivity, for which the conven-
tional symbol is
for protons in the sun in Chapter 2, using formula (2.23),
s ¼
1
=r;
ð
3
:
7
Þ
m)
1
.
The resistivity of pure metals tends to zero at low temperature, and the ratio of the
resistivity at 300 K to that at 4.2 K is called the residual resistance ratio. It can be as
large as a million in a pure crystalline sample.
The large residual resistance ratio for extremely pure metals implies large mean
free path
where
r
is the resistivity, and with
s
in units Siemens or (
V
at low temperature. The formula for resistivity
r¼
1/
s
can be written in
several forms:
L
mÞ ¼ m
e
=ðNeetÞ¼m
e
n
F
=ðNe
2
r ¼
1
=
Nem ¼
1
=½Neðet=
LÞ:
ð
3
:
8
Þ
Here,
N
is the number of electrons freely moving,
t
is the time between scattering
events for a given electron,
n
F
is a characteristic electron speed, and the
mobility m
is
de
ned as
m ¼ et=
m
:
ð
3
:
9
Þ
The mobility, whose units are m
2
/Vs, provides the
drift velocity
V
D
¼ m E
ð
3
:
10
Þ
of the electron in an applied electric
field
E¼V
/
L
given in volts per meter. Looking at
the
final form in (3.8), we see that the only way
r
can increase by 10
4
-
10
6
going from
increases, because other factors such as
N
and
n
F
are
constant. The data for pure metals in Figure 3.2 show that the interaction of electrons
and metal ions is surprisingly weak: the resistivity goes to zero at low temperature.
Lets look to nd the most important modication of free particle motion that
L
300 to 4 K is if mean free path
