650
F Review of Solid State Concepts
say when we are able to ignore the possibility of a direct electron transfer from a
given atom to any atom twice removed or more from it. For this case, we can write
i ¯
h
dw k
dt
= E 0 w k − Aw k−1 − Aw k+1 .
(F.2.1)
As before, let us assume a solution of the form
w k = C k e
−iEt/¯ h ,
(F.2.2)
so that we obtain, upon substitution into Eq. (F.2.1) the algebraic equation
EC k = E 0 C k − A(C k−1 + C k+1 ) .
(F.2.3)
Now, as atom k in the line is located at x k while atoms k − 1 and k + 1 are located at
x k−1 = x k − a and x k+1 = x k + a, we may treat the coefficients C k as parameters
that depend upon the positions of the atoms, rather than as true constants: thus, we
may write
EC(x k ) = E 0 C(x k ) − A[C(x k + a) + C(x k − a)] ,
(F.2.4)
which is a difference equation that is solvable by the same methods that one uses
to solve differential equations. Suppose, then, that we assume a trial solution of the
form
C(x k ) = e
ikx k
(F.2.5)
which, upon substitution into Eq. (F.2.4) gives us the result
Ee
ikx k = E 0 e
ikx k − A
e
ik(x k +a)
+ e
ik(x k −a)
= E 0 e
ikx k − Ae
ikx k
e
ika
+ e
−ika
or
E = E 0 − 2A cos ka .
(F.2.6)
This result is shown in Fig. F.3, and we see that indeed the allowed energies
fall into an energy band centred on energy E 0 and 4A wide: energies below E 0 −
2A and above E 0 + 2A are forbidden to the electrons. Notice that this argument
could equally well apply to holes, and to other types of particles. Moreover, it is
relatively readily generalizable to three-dimensional solids: rectangular structure,
with distances a, b, c in the directions of the coordinate axes x, y, z, respectively.
If we designate translations along the three axes by distance a, b, and c by a, b,
c, we may also take advantage of the conciseness of employing vector notation.
F Review of Solid State Concepts
say when we are able to ignore the possibility of a direct electron transfer from a
given atom to any atom twice removed or more from it. For this case, we can write
i ¯
h
dw k
dt
= E 0 w k − Aw k−1 − Aw k+1 .
(F.2.1)
As before, let us assume a solution of the form
w k = C k e
−iEt/¯ h ,
(F.2.2)
so that we obtain, upon substitution into Eq. (F.2.1) the algebraic equation
EC k = E 0 C k − A(C k−1 + C k+1 ) .
(F.2.3)
Now, as atom k in the line is located at x k while atoms k − 1 and k + 1 are located at
x k−1 = x k − a and x k+1 = x k + a, we may treat the coefficients C k as parameters
that depend upon the positions of the atoms, rather than as true constants: thus, we
may write
EC(x k ) = E 0 C(x k ) − A[C(x k + a) + C(x k − a)] ,
(F.2.4)
which is a difference equation that is solvable by the same methods that one uses
to solve differential equations. Suppose, then, that we assume a trial solution of the
form
C(x k ) = e
ikx k
(F.2.5)
which, upon substitution into Eq. (F.2.4) gives us the result
Ee
ikx k = E 0 e
ikx k − A
e
ik(x k +a)
+ e
ik(x k −a)
= E 0 e
ikx k − Ae
ikx k
e
ika
+ e
−ika
or
E = E 0 − 2A cos ka .
(F.2.6)
This result is shown in Fig. F.3, and we see that indeed the allowed energies
fall into an energy band centred on energy E 0 and 4A wide: energies below E 0 −
2A and above E 0 + 2A are forbidden to the electrons. Notice that this argument
could equally well apply to holes, and to other types of particles. Moreover, it is
relatively readily generalizable to three-dimensional solids: rectangular structure,
with distances a, b, c in the directions of the coordinate axes x, y, z, respectively.
If we designate translations along the three axes by distance a, b, and c by a, b,
c, we may also take advantage of the conciseness of employing vector notation.