F.3 Concept of Effective Mass
653
for ka 1. The term in brackets just identifies the bottom of the conduction band,
so we can replace 0 − 2A by E c , so that we have
= E c + Aa
2 k
2
+ · · · ,
(F.3.5)
from which we can see that conduction electrons near the bottom of the band would
behave like free electrons, but with an effective mass m ∗ defined as
¯
h 2
2m ∗ ≡ Aa
2 .
(F.3.6)
We might wonder whether this effective mass m ∗ is different from the value m e for a
free electron. We can get some idea of how different it can be by making an order-ofmagnitude estimate using values for A and a that are reasonable for a typical metal
or semiconductor. If we take A to be of order 10 −20 J to 10 −21 J, and a a typical
lattice distance of order 3 × 10 −10 m, then we can estimate the effective mass m ∗ as
m
∗
=
¯
h 2
Aa 2
(1.055 × 10 −34 ) 2 J 2 s 2
2 × (5 × 10 −20 J) × (3 × 10 −10 m) 2
≈
1.12 × 10 −68 kg m 2
10 −19 × 9 × 10 −20 m 2 ≈ 1.2 × 10
−30 kg ,
which can be compared with the free electron mass m e 9.1 × 10 −31 kg. For this
particular calculation, we see that the effective mass is about 1.3m e : in practice, the
effective masses of electrons in energy bands in solids can be anywhere from 100
times smaller to 100 times greater than m e ! More generally, the effective mass of an
electron in a band is defined in terms of the curvature of the dispersion relation via
m
∗
= ¯
h
2 /
d 2
dk 2
.
(F.3.7)
Thus, by analogy with a free electron which has a well-defined mass, m e , and
which, when subjected to an applied electric field E is accelerated according to
Newtonian mechanics, an electron within a crystalline solid can also be considered
to be accelerated by an electric field in a Newtonian manner, but with a mass m ∗ ,
rather than the free electron mass m e .
We know that if we think of an electron as a particle with a well-defined location
(remember, that we say that such an electron is localized), then we must in quantum
mechanics necessarily think of the electron as being represented by a wave packet,
i.e., a superposition of a very large number of waves with rather similar de Broglie
wavelengths. The electron travels with the group velocity associated with the wave
packet, rather than with any of the phase velocities associated with the individual
waves that make up the wave packet. While the group velocity is traditionally
defined by the derivative of the angular frequency with respect to wave number
via
653
for ka 1. The term in brackets just identifies the bottom of the conduction band,
so we can replace 0 − 2A by E c , so that we have
= E c + Aa
2 k
2
+ · · · ,
(F.3.5)
from which we can see that conduction electrons near the bottom of the band would
behave like free electrons, but with an effective mass m ∗ defined as
¯
h 2
2m ∗ ≡ Aa
2 .
(F.3.6)
We might wonder whether this effective mass m ∗ is different from the value m e for a
free electron. We can get some idea of how different it can be by making an order-ofmagnitude estimate using values for A and a that are reasonable for a typical metal
or semiconductor. If we take A to be of order 10 −20 J to 10 −21 J, and a a typical
lattice distance of order 3 × 10 −10 m, then we can estimate the effective mass m ∗ as
m
∗
=
¯
h 2
Aa 2
(1.055 × 10 −34 ) 2 J 2 s 2
2 × (5 × 10 −20 J) × (3 × 10 −10 m) 2
≈
1.12 × 10 −68 kg m 2
10 −19 × 9 × 10 −20 m 2 ≈ 1.2 × 10
−30 kg ,
which can be compared with the free electron mass m e 9.1 × 10 −31 kg. For this
particular calculation, we see that the effective mass is about 1.3m e : in practice, the
effective masses of electrons in energy bands in solids can be anywhere from 100
times smaller to 100 times greater than m e ! More generally, the effective mass of an
electron in a band is defined in terms of the curvature of the dispersion relation via
m
∗
= ¯
h
2 /
d 2
dk 2
.
(F.3.7)
Thus, by analogy with a free electron which has a well-defined mass, m e , and
which, when subjected to an applied electric field E is accelerated according to
Newtonian mechanics, an electron within a crystalline solid can also be considered
to be accelerated by an electric field in a Newtonian manner, but with a mass m ∗ ,
rather than the free electron mass m e .
We know that if we think of an electron as a particle with a well-defined location
(remember, that we say that such an electron is localized), then we must in quantum
mechanics necessarily think of the electron as being represented by a wave packet,
i.e., a superposition of a very large number of waves with rather similar de Broglie
wavelengths. The electron travels with the group velocity associated with the wave
packet, rather than with any of the phase velocities associated with the individual
waves that make up the wave packet. While the group velocity is traditionally
defined by the derivative of the angular frequency with respect to wave number
via