6.9 Electron Dispersion
159
v =
1
∇ k E(k) ,
(6.35)
where ∇ k is the gradient with respect to k. Through the dispersion relation the influence of the crystal
and its periodic potential on the motion enters the equation.
An electric field E acts on an electron during the time δt the work δ E = −eEv g δt. This change in
energy is related to a change in k via δ E = dE/dk δk = v g δk. Thus, we arrive at dk/dt = −eE.
For an external force we thus have
dk
dt
= −e E = F .
(6.36)
Thus, the crystal momentum p = k takes the role of the momentum. A more rigorous derivation can
be found in [451].
In the presence of a magnetic field B the equation of motion is
dk
dt
= −e v × B = −
e
(∇ k E) × B .
(6.37)
The motion in a magnetic field is thus perpendicular to the gradient of the energy, i.e. the energy of the
electron does not change. It oscillates therefore on a surface of constant energy perpendicular to B.
6.9.2 Effective Mass of Electrons
From the free-electron dispersion E =
2 k
2
/(2m) the mass of the particle is inversely proportional to
the curvature of the dispersion relation, i.e. m =
2
/(d
2 E/dk
2
). This relation will now be generalized
for arbitrary dispersion relations. The (inverse) tensor of the effective mass is defined as
(m
∗−1
) i j =
1
2
∂
2 E
∂k i ∂k j
.
(6.38)
The equation F = m
∗
˙
v must be understood as a tensor equation, i.e. for the components of the force
F i = m
∗
i j a j . Force and acceleration must no longer be collinear. In order to find the acceleration from
the force, the inverse of the effective-mass tensor must be used, a = (m
∗
)
−1 F.
In Fig. 6.33 the energy dispersion of the (lowest) conduction band in a typical semiconductor, the
related electron velocity and the effective mass are shown schematically.
In (6.22) the ratio of the effective mass and the free-electron mass is of the order of m
∗
/m ≈ U/λ,
the ratio of the free particle energy and the band gap. For typical semiconductors, the width of the
(valence) band is of the order of 20 eV, and the gap is about 0.2–2 eV. Thus, the effective mass is
expected to be 10–100 times smaller than the free-electron mass. Additionally, the relation m
∗
∝ E g
is roughly fulfilled (Fig. 6.34).
From so-called k · p theory [510] (see Appendix H) the effective electron mass is predicted to be
related to the momentum matrix element p cv
p cv = =c|p|v =
0
u
∗
c,k (r) p u c, k (r) d
3 r ,
(6.39)
with 0 being the unit cell volume and the Bloch functions |c and |v of the conduction and valence
band, respectively, given as
159
v =
1
∇ k E(k) ,
(6.35)
where ∇ k is the gradient with respect to k. Through the dispersion relation the influence of the crystal
and its periodic potential on the motion enters the equation.
An electric field E acts on an electron during the time δt the work δ E = −eEv g δt. This change in
energy is related to a change in k via δ E = dE/dk δk = v g δk. Thus, we arrive at dk/dt = −eE.
For an external force we thus have
dk
dt
= −e E = F .
(6.36)
Thus, the crystal momentum p = k takes the role of the momentum. A more rigorous derivation can
be found in [451].
In the presence of a magnetic field B the equation of motion is
dk
dt
= −e v × B = −
e
(∇ k E) × B .
(6.37)
The motion in a magnetic field is thus perpendicular to the gradient of the energy, i.e. the energy of the
electron does not change. It oscillates therefore on a surface of constant energy perpendicular to B.
6.9.2 Effective Mass of Electrons
From the free-electron dispersion E =
2 k
2
/(2m) the mass of the particle is inversely proportional to
the curvature of the dispersion relation, i.e. m =
2
/(d
2 E/dk
2
). This relation will now be generalized
for arbitrary dispersion relations. The (inverse) tensor of the effective mass is defined as
(m
∗−1
) i j =
1
2
∂
2 E
∂k i ∂k j
.
(6.38)
The equation F = m
∗
˙
v must be understood as a tensor equation, i.e. for the components of the force
F i = m
∗
i j a j . Force and acceleration must no longer be collinear. In order to find the acceleration from
the force, the inverse of the effective-mass tensor must be used, a = (m
∗
)
−1 F.
In Fig. 6.33 the energy dispersion of the (lowest) conduction band in a typical semiconductor, the
related electron velocity and the effective mass are shown schematically.
In (6.22) the ratio of the effective mass and the free-electron mass is of the order of m
∗
/m ≈ U/λ,
the ratio of the free particle energy and the band gap. For typical semiconductors, the width of the
(valence) band is of the order of 20 eV, and the gap is about 0.2–2 eV. Thus, the effective mass is
expected to be 10–100 times smaller than the free-electron mass. Additionally, the relation m
∗
∝ E g
is roughly fulfilled (Fig. 6.34).
From so-called k · p theory [510] (see Appendix H) the effective electron mass is predicted to be
related to the momentum matrix element p cv
p cv = =c|p|v =
0
u
∗
c,k (r) p u c, k (r) d
3 r ,
(6.39)
with 0 being the unit cell volume and the Bloch functions |c and |v of the conduction and valence
band, respectively, given as