Environmental Engineering Reference
In-Depth Information
Figure 3.3 Sketch of weak scattering of an electron wave by atoms at spacing a. (Courtesy of M.
Medikonda).
2
a ¼ nl ¼ nl
2
p=
k
;
ð
3
:
11
Þ
so that the wavenumber condition for backscattering is
k¼np
/
a
, where
n
is any
integer.
Sketched in Figure 3.4 is the curve
E¼
2
k
2
/
m
) versus
k
, which can be regarded
as themomentum
p
/
h
. The curve ismodi
ed from that for a free particle, near
np
/
a
because the scattering produces linear combinations,
y¼
e
ikx
1
/
2
(
h
þ
e
ikx
¼
(1/2) cos
kx
or e
ikx
e
ikx
¼
(
i
/2) sin
kx
. These combinations are standing waves, and the energy
is reduced for the cos
kx
versus the sin
kx
choice. For Si, where
a¼
0.543 nm,
k ¼p
/
a¼p
/0.543 nm
¼
5.78
10
9
m
1
2
k
2
/2
m¼
1.27 eV, a value similar to mea-
sured values. Near
k¼p
/
a
the backscattered wave is as strong as the forward wave:
the result is a standing wave (Figure 3.5). It is generally true, for Schrodingers
equation or any linear differential equation, that if one has two solutions, such as
exp(
ikx
), then linear combinations of these, such as sin
kx
and cos
kx
are equally
valid, with speci
c choices to be based on physical reasoning. (It is also generally true
that the wavefunction can be a complex quantity, such as
y¼
exp(
ikx
), because the
measured probability of finding the particle is the product of
y
with its complex
conjugate,
P¼y
y
, which is a positive number.) In this case, for
k
just below
p
/
a
, the
cos
kx
form is more stable, because the electron spends more time near the ions
and
h
)
2
/2m at k¼p/a, using the
lattice constant for crystalline silicon
a¼0.543 nm, with m¼m
e
. This figure shows
the origin of electron energy bands and gaps in a
periodic lattice. (Courtesy of M. Medikonda).
Figure 3.4 Sketch of E(k) for electrons
influenced by weak scattering from atoms at
spacing a. The background curve corresponds
to the free particle condition E¼ (k
h
)
2
/2m. The
discontinuities at k¼p/a arise by coherent
backscattering, see text. The energy 1.27 eV
corresponds to (k
h
