652
F Review of Solid State Concepts
and a minimum value
E = E 0 − 2(A x + A y + A z ) ,
precisely analogous to our earlier result for a 1D crystal.
F.3 Concept of Effective Mass
We know that the energy of free electrons is given by
=
¯
h 2 k 2
2m e
,
(F.3.1)
from which we see that the coefficient of k 2 determines the curvature of the vs. k
curves. As the energy for free electrons is simply quadratic, the curvature is constant
and has the value ¯
h 2 /m e . More generally, we may restate this by saying that the
reciprocal mass, 1/m e , determines the curvature at any given point in k-space, and
require that a statement of this type applies also to electrons in conduction bands of
a solid, even though they are no longer free electrons. We have seen that according
to the Feynman model, the energy of an electron in a band in a solid is, in the weakcoupling limit, given by
= 0 − 2A cos(ka) ,
(F.3.2)
with A the coupling constant. The relationship between the energy of a wave and its
wave vector is called the dispersion relation: the dispersion relation for free electrons
is thus simply = ¯
h 2 k 2 /(2m e ), while that for an electron in an energy band in a
solid (in the weak-coupling case) is as given above in Eq. (F.3.2). Let us see what
this form for the dispersion relation can tell us.
The curvature of the energy vs. wave number plot for electrons in the energy
band will be given from Eq. (F.3.2) as
d 2
dk 2 = 2a
2 A cos(ka) ,
(F.3.3)
for any value of k lying between ±
π
a . Let us focus for the moment upon what
happens near the bottom of the energy band, where we may approximate the energy
of a conduction electron by
0 − 2A
1 −
1
2 k
2 a
2
+ · · ·
= (( 0 − 2A) + Aa
2 k
2
+ · · ·
(F.3.4)