SOLID STATE PHYSICS
PART I
Transport Properties of Solids
M. S. Dresselhaus
6.732
Fall, 2001
Oded Rabin – Head TA; Room 13-3025
Marcie Black – TA assistant; Room 13-3041
Yu-Ming Lin – TA assistant; Room 13-3037
Laura Doughty – Support; Room 13-3005
Lectures: MWF 9-10
Room 13-4101
Recitation: F 11-12
Room 38-136
10 problem sets
3 quizzes
• Part I
Transport
• Part II
Optical
• Part III
Magnetism
• Part IV
Superconductivity
1
Contents
1 Review of Energy Dispersion Relations in Solids 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 One Electron E(
~
k) in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Nearly Free Electron Approximation . . . . . . . . . . . . . . . . . . 2
1.2.2 Tight Binding Approximation . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Weak and Tight Binding Approximations . . . . . . . . . . . . . . . 15
1.2.4 Tight Binding Approximation with 2 Atoms/Unit Cell . . . . . . . . 15
2 Examples of Energy Bands in Solids 20
2.1 General Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Alkali Metals–e.g., Sodium . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Polyvalent Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 PbTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.4 III–V Compound Semiconductors . . . . . . . . . . . . . . . . . . . . 36
2.3.5 “Zero Gap” Semiconductors – Gray Tin . . . . . . . . . . . . . . . . 36
2.3.6 Molecular Semiconductors – Fullerenes . . . . . . . . . . . . . . . . . 40
2.4 Semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Effective Mass Theory 46
3.1 Wavepackets in Crystals and Group Velocity of Electrons in Solids . . . . . 46
3.2 The Effective Mass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Application of the Effective Mass Theorem to Donor Impurity Levels in a
Semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Quasi-Classical Electron Dynamics . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Quasi-Classical Theory of Electrical Conductivity – Ohm’s Law . . . . . . . 55
4 Transport Phenomena 58
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2
4.4 Electrical Conductivity of Metals . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Electrical Conductivity of Semiconductors . . . . . . . . . . . . . . . . . . . 63
4.5.1 Ellipsoidal Carrier Pockets . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Electrons and Holes in Intrinsic Semiconductors . . . . . . . . . . . . . . . . 69
4.7 Donor and Acceptor Doping of Semiconductors . . . . . . . . . . . . . . . . 72
4.8 Characterization of Semiconductors . . . . . . . . . . . . . . . . . . . . . . . 76
5 Thermal Transport 82
5.1 Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.2 Thermal Conductivity for Metals . . . . . . . . . . . . . . . . . . . . 85
5.2.3 Thermal Conductivity for Semiconductors . . . . . . . . . . . . . . . 87
5.2.4 Thermal Conductivity for Insulators . . . . . . . . . . . . . . . . . . 88
5.3 Thermoelectric Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.1 Thermoelectric Phenomena in Metals . . . . . . . . . . . . . . . . . 93
5.3.2 Thermopower for Semiconductors . . . . . . . . . . . . . . . . . . . . 94
5.3.3 Effect of Thermoelectricity on the Thermal Conductivity . . . . . . 96
5.4 Thermoelectric Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4.1 Seebeck Effect (Thermopower) . . . . . . . . . . . . . . . . . . . . . 97
5.4.2 Peltier Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4.3 Thomson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4.4 The Kelvin Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4.5 Thermoelectric Figure of Merit . . . . . . . . . . . . . . . . . . . . . 100
5.5 Phonon Drag Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Electron and Phonon Scattering 102
6.1 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2 Scattering Processes in Semiconductors . . . . . . . . . . . . . . . . . . . . 105
6.2.1 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.2 Ionized Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.3 Other Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . 111
6.2.4 Screening Effects in Semiconductors . . . . . . . . . . . . . . . . . . 111
6.3 Electron Scattering in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.1 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.2 Other Scattering Mechanisms in Metals . . . . . . . . . . . . . . . . 118
6.4 Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4.1 Phonon-phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4.2 Phonon-Boundary Scattering . . . . . . . . . . . . . . . . . . . . . . 120
6.4.3 Defect-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4.4 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Temperature Dependence of the Electrical and Thermal Conductivity . . . 122
3
7 Magneto-transport Phenomena 124
7.1 Magneto-transport in the classical regime (ω
c
τ < 1) . . . . . . . . . . . . . 124
7.1.1 Classical Magneto-transport Equations . . . . . . . . . . . . . . . . . 125
7.1.2 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Derivation of the Magneto-transport Equations from the Boltzmann Equation129
7.4 Two Carrier Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.5 Cyclotron Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.6 Effective Masses for Ellipsoidal Fermi Surfaces . . . . . . . . . . . . . . . . 133
7.7 Dynamics of Electrons in a Magnetic Field . . . . . . . . . . . . . . . . . . . 134
9 Two Dimensional Electron Gas, Quantum Wells & Semiconductor Super-
lattices 139
9.1 Two-Dimensional Electronic Systems . . . . . . . . . . . . . . . . . . . . . . 139
9.2 MOSFETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.3 Two-Dimensional Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.3.1 Quantum Wells and Superlattices . . . . . . . . . . . . . . . . . . . . 145
9.4 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.5 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.6 WKB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.7 Kronig–Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.8 1–D Rectangular Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.9 Resonant Tunneling in Quantum Wells . . . . . . . . . . . . . . . . . . . . . 155
10 Transport in Low Dimensional Systems 162
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2 Observation of Quantum Effects in Reduced Dimensions . . . . . . . . . . . 162
10.3 Density of States in Low Dimensional Systems . . . . . . . . . . . . . . . . 164
10.3.1 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.4 The Einstein Relation and the Landauer Formula . . . . . . . . . . . . . . . 166
10.5 One Dimensional Transport and Quantization of the Ballistic Conductance 169
10.6 Ballistic Transport in 1D Electron Waveguides . . . . . . . . . . . . . . . . 172
10.7 Single Electron Charging Devices . . . . . . . . . . . . . . . . . . . . . . . . 176
11 Ion Implantation and RBS 180
11.1 Introduction to the Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11.2 Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.2.1 Basic Scattering Equations . . . . . . . . . . . . . . . . . . . . . . . 183
11.2.2 Radiation Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.2.3 Applications of Ion Implantation . . . . . . . . . . . . . . . . . . . . 187
11.3 Ion Backscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
11.4 Channeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
A Time–Independent Perturbation Theory 200
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A.1.1 Non-degenerate Perturbation Theory . . . . . . . . . . . . . . . . . . 201
A.1.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . . . . . 205
4
B Harmonic Oscillators, Phonons, and Electron-Phonon Interaction 208
B.1 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.3 Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
C Artificial Atoms 213
C.1 Charge quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
C.2 Energy quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
C.3 Artificial atoms in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . 222
C.4 Conductance line shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
C.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5
Chapter 1
Review of Energy Dispersion
Relations in Solids
References:
• Ashcroft and Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976, Chap-
ters 8, 9, 10, 11.
• Bassani and Parravicini, Electronic States and Optical Transitions in Solids, Perga-
mon, 1975, Chapter 3.
• Kittel, Introduction to Solid State Physics, Wiley, 1986, pp. 228-239.
• Mott & Jones – The Theory of the Properties of Metals and Alloys, Dover, 1958 pp.
56–85.
• Omar, Elementary Solid State Physics, Addison–Wesley, 1975, pp. 189–210.
• Ziman, Principles of the Theory of Solids, Cambridge, 1972, Chapter 3.
1.1 Introduction
The transport properties of solids are closely related to the energy dispersion relations E(
~
k)
in these materials and in particular to the behavior of E(
~
k) near the Fermi level. Con-
versely, the analysis of transport measurements provides a great deal of information on
E(
~
k). Although transport measurements do not generally provide the most sensitive tool
for studying E(
~
k), such measurements are fundamental to solid state physics because they
can be carried out on nearly all materials and therefore provide a valuable tool for char-
acterizing materials. To provide the necessary background for the discussion of transport
properties, we give here a brief review of the energy dispersion relations E(
~
k) in solids. In
this connection, we consider in Chapter 1 the two limiting cases of weak and tight binding.
In Chapter 2 we will discuss E(
~
k) for real solids including prototype metals, semiconductors,
semimetals and insulators.
1
1.2 One Electron E(
~
k) in Solids
1.2.1 Weak Binding or Nearly Free Electron Approximation
In the weak binding approximation, we assume that the periodic potential V (~r) = V (~r+
~
R
n
)
is sufficiently weak so that the electrons behave almost as if they were free and the effect
of the periodic potential can be handled in perturbation theory (see Appendix A). In this
formulation V (~r) can be an arbitrary periodic potential. The weak binding approximation
has achieved some success in describing the valence electrons in metals. For the core elec-
trons, however, the potential energy is comparable with the kinetic energy so that core
electrons are tightly bound and the weak binding approximation is not applicable. In the
weak binding approximation we solve the Schr¨odinger equation in the limit of a very weak
periodic potential
Hψ = Eψ. (1.1)
Using time–independent perturbation theory (see Appendix A) we write
E(
~
k) = E
(0)
(
~
k) + E
(1)
(
~
k) + E
(2)
(
~
k) + ... (1.2)
and take the unperturbed solution to correspond to V (~r) = 0 so that E
(0)
(
~
k) is the plane
wave solution
E
(0)
(
~
k) =
¯h
2
k
2
2m
. (1.3)
The corresponding normalized eigenfunctions are the plane wave states
ψ
(0)
~
k
(~r) =
e
i
~
k·~r
Ω
1/2
(1.4)
in which Ω is the volume of the crystal.
The first order correction to the energy E
(1)
(
~
k) is the diagonal matrix element of the
perturbation potential taken between the unperturbed states:
E
(1)
(
~
k)=hψ
(0)
~
k
| V (~r) | ψ
(0)
~
k
i =
1
Ω
R
Ω
e
−i
~
k·~r
V (~r)e
i
~
k·~r
d
3
r
=
1
Ω
0
R
Ω
0
V (~r)d
3
r =
V (~r)
(1.5)
where V (~r) is independent of
~
k, and Ω
0
is the volume of the unit cell. Thus, in first order
perturbation theory, we merely add a constant energy V (~r) to the free particle energy, and
that constant term is exactly the mean potential energy seen by the electron, averaged over
the unit cell. The terms of interest arise in second order perturbation theory and are
E
(2)
(
~
k) =
0
X
~
k
0
|h
~
k
0
|V (~r)|
~
ki|
2
E
(0)
(
~
k) − E
(0)
(
~
k
0
)
(1.6)
where the prime on the summation indicates that
~
k
0
6=
~
k. We next compute the matrix
element h
~
k
0
|V (~r)|
~
ki as follows:
h
~
k
0
|V (~r)|
~
ki=
R
Ω
ψ
(0)∗
~
k
0
V (~r)ψ
(0)
~
k
d
3
r
=
1
Ω
R
Ω
e
−i(
~
k
0
−
~
k)·~r
V (~r)d
3
r
=
1
Ω
R
Ω
e
i~q·~r
V (~r)d
3
r
(1.7)
2
where ~q is the difference wave vector ~q =
~
k −
~
k
0
and the integration is over the whole crystal.
We now exploit the periodicity of V (~r). Let ~r = ~r
0
+
~
R
n
where ~r
0
is an arbitrary vector in
a unit cell and
~
R
n
is a lattice vector. Then since V (~r) = V (~r
0
)
h
~
k
0
|V (~r)|
~
ki =
1
Ω
X
n
Z
Ω
0
e
~
iq·(~r
0
+
~
R
n
)
V (~r
0
)d
3
r
0
(1.8)
where the sum is over unit cells and the integration is over the volume of one unit cell.
Then
h
~
k
0
|V (~r)|
~
ki =
1
Ω
X
n
e
~
iq·
~
R
n
Z
Ω
0
e
~
iq·~r0
V (~r
0
)d
3
r
0
. (1.9)
Writing the following expressions for the lattice vectors
~
R
n
and for the wave vector ~q
~
R
n
=
P
3
j=1
n
j
~a
j
~q=
P
3
j=1
α
j
~
b
j
(1.10)
where n
j
is an integer, then the lattice sum
P
n
e
~
iq·
~
R
n
can be carried out exactly to yield
X
n
e
~
iq·
~
R
n
=
"
3
Y
j=1
1 −e
2πiN
j
α
j
1 −e
2πiα
j
#
(1.11)
where N = N
1
N
2
N
3
is the total number of unit cells in the crystal and α
j
is a real number.
This sum fluctuates wildly as ~q varies and is appreciable only if
~q =
3
X
j=1
m
j
~
b
j
(1.12)
where m
j
is an integer and
~
b
j
is a primitive vector in reciprocal space, so that ~q must be a
reciprocal lattice vector. Hence we have
X
n
e
i~q·
~
R
n
= N δ
~q,
~
G
(1.13)
since
~
b
j
·
~
R
n
= 2πl
jn
where l
jn
is an integer.
This discussion shows that the matrix element h
~
k
0
|V (~r)|
~
ki is only important when ~q =
~
G
is a reciprocal lattice vector =
~
k −
~
k
0
from which we conclude that the periodic potential
V (~r) only connects wave vectors
~
k and
~
k
0
separated by a reciprocal lattice vector. We note
that this is the same relation that determines the Brillouin zone boundary. The matrix
element is then
h
~
k
0
|V (~r)|
~
ki =
N
Ω
Z
Ω
0
e
i
~
G·~r
0
V (~r
0
)d
3
r
0
δ
~
k
0
−
~
k,
~
G
(1.14)
where
N
Ω
=
1
Ω
0
(1.15)
and the integration is over the unit cell. We introduce V
~
G
= Fourier coefficient of V (~r)
where
V
~
G
=
1
Ω
0
Z
Ω
0
e
i
~
G·~r
0
V (~r
0
)d
3
r
0
(1.16)
3
so that
h
~
k
0
|V (~r)|
~
ki = δ
~
k−
~
k
0
,
~
G
V
~
G
. (1.17)
We can now use this matrix element to calculate the 2
nd
order change in the energy
E
(2)
(
~
k) =
X
~
G
|V
~
G
|
2
k
2
− (k
0
)
2
Ã
2m
¯h
2
!
=
2m
¯h
2
X
~
G
|V
~
G
|
2
k
2
− (
~
G +
~
k)
2
. (1.18)
We observe that when k
2
= (
~
G +
~
k)
2
the denominator vanishes and E
(2)
(
~
k) can become
very large. This condition is identical with the Laue diffraction condition. Thus, at a
Brillouin zone boundary, the weak perturbing potential has a very large effect and therefore
non–degenerate perturbation theory will not work in this case.
For
~
k values near a Brillouin zone boundary, we must then use degenerate perturbation
theory (see Appendix A). Since the matrix elements coupling the plane wave states
~
k and
~
k +
~
G do not vanish, first order degenerate perturbation theory is sufficient and leads to
the determinantal equation
¯
¯
¯
¯
¯
¯
E
(0)
(
~
k) + E
(1)
(
~
k) − E h
~
k +
~
G|V (~r)|
~
ki
h
~
k|V (~r)|
~
k +
~
Gi E
(0)
(
~
k +
~
G) + E
(1)
(
~
k +
~
G) −E
¯
¯
¯
¯
¯
¯
= 0 (1.19)
in which
E
(0)
(
~
k)=
¯h
2
k
2
2m
E
(0)
(
~
k +
~
G)=
¯h
2
(
~
k+
~
G)
2
2m
(1.20)
and
E
(1)
(
~
k)=h
~
k|V (~r)|
~
ki = V (~r) = V
0
E
(1)
(
~
k +
~
G)=h
~
k +
~
G|V (~r)|
~
k +
~
Gi = V
0
.
(1.21)
Solution of this determinantal equation (Eq. 1.19) yields:
[E − V
0
− E
(0)
(
~
k)][E − V
0
− E
(0)
(
~
k +
~
G)] −|V
~
G
|
2
= 0, (1.22)
or equivalently
E
2
−E[2V
0
+ E
(0)
(
~
k) + E
(0)
(
~
k +
~
G)] + [V
0
+ E
(0)
(
~
k)][V
0
+ E
(0)
(
~
k +
~
G)] −|V
~
G
|
2
= 0. (1.23)
Solution of the quadratic equation (Eq. 1.23) yields
E
±
= V
0
+
1
2
[E
(0)
(
~
k) + E
(0)
(
~
k +
~
G)] ±
r
1
4
[E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)]
2
+ |V
~
G
|
2
(1.24)
and we come out with two solutions for the two strongly coupled states. It is of interest to
look at these two solutions in two limiting cases:
4
case (i) |V
~
G
| ¿
1
2
|[E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)]|
In this case we can expand the square root expression in Eq. 1.24 for small |V
~
G
| to
obtain:
E(
~
k) = V
0
+
1
2
[E
(0)
(
~
k) + E
(0)
(
~
k +
~
G)]
±
1
2
[E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)] ·[1 +
2|V
~
G
|
2
[E
(0)
(
~
k)−E
(0)
(
~
k+
~
G)]
2
+ . . .]
(1.25)
which simplifies to the two solutions:
E
−
(
~
k) = V
0
+ E
(0)
(
~
k) +
|V
~
G
|
2
E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)
(1.26)
E
+
(
~
k) = V
0
+ E
(0)
(
~
k +
~
G) +
|V
~
G
|
2
E
(0)
(
~
k +
~
G) − E
(0)
(
~
k)
(1.27)
and we recover the result Eq. 1.18 obtained before using non–degenerate perturbation
theory. This result in Eq. 1.18 is valid far from the Brillouin zone boundary, but near
the zone boundary the more complete expression of Eq. 1.24 must be used.
case (ii)
|V
~
G
| À
1
2
|[E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)]|
Sufficiently close to the Brillouin zone boundary
|E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)| ¿ |V
~
G
| (1.28)
so that we can expand E(
~
k) as given by Eq. 1.24 to obtain
E
±
(
~
k) =
1
2
[E
(0)
(
~
k)+E
(0)
(
~
k+
~
G)]+V
0
±
·
|V
~
G
|+
1
8
[E
(0)
(
~
k) − E
(0)
(
~
k +
~
G)]
2
|V
~
G
|
+...
¸
(1.29)
∼
=
1
2
[E
(0)
(
~
k) + E
(0)
(
~
k +
~
G)] + V
0
± |V
~
G
|, (1.30)
so that at the Brillouin zone boundary E
+
(
~
k) is elevated by |V
~
G
|, while E
−
(
~
k) is
depressed by |V
~
G
| and the band gap that is formed is 2|V
~
G
|, where
~
G is the reciprocal
lattice vector for which E(
~
k
B.Z.
) = E(
~
k
B.Z.
+
~
G) and
V
~
G
=
1
Ω
0
Z
Ω
0
e
i
~
G·~r
V (~r)d
3
r. (1.31)
From this discussion it is clear that every Fourier component of the periodic potential
gives rise to a specific band gap. We see further that the band gap represents a range of
energy values for which there is no solution to the eigenvalue problem of Eq. 1.19 for real k
(see Fig. 1.1). In the band gap we assign an imaginary value to the wave vector which can
be interpreted as a highly damped and non–propagating wave.
5
Figure 1.1: One dimensional electron energy bands for the nearly free electron model shown
in the extended Brillouin zone scheme. The dashed curve corresponds to the case of free
electrons and solid curves to the case where a weak periodic potential is present.
We note that the larger the value of
~
G, the smaller the value of V
~
G
, so that higher
Fourier components give rise to smaller band gaps. Near these energy discontinuities, the
wave functions become linear combinations of the unperturbed states
ψ
~
k
=α
1
ψ
(0)
~
k
+ β
1
ψ
(0)
~
k+
~
G
ψ
~
k+
~
G
=α
2
ψ
(0)
~
k
+ β
2
ψ
(0)
~
k+
~
G
(1.32)
and at the zone boundary itself, instead of traveling waves e
i
~
k·~r
, the wave functions become
standing waves cos
~
k · ~r and sin
~
k · ~r. We note that the cos(
~
k · ~r) solution corresponds to a
maximum in the charge density at the lattice sites and therefore corresponds to an energy
minimum (the lower level). Likewise, the sin(
~
k · ~r) solution corresponds to a minimum in
the charge density and therefore corresponds to a maximum in the energy, thus forming the
upper level.
In constructing E(
~
k) for the reduced zone scheme we make use of the periodicity of E(
~
k)
in reciprocal space
E(
~
k +
~
G) = E(
~
k). (1.33)
The reduced zone scheme more clearly illustrates the formation of energy bands (labeled
(1) and (2) in Fig. 1.2), band gaps E
g
and band widths (defined in Fig. 1.2 as the range of
energy between E
min
and E
max
for a given energy band).
We now discuss the connection between the E(
~
k) relations shown above and the trans-
port properties of solids, which can be illustrated by considering the case of a semiconductor.
An intrinsic semiconductor at temperature T = 0 has no carriers so that the Fermi level
runs right through the band gap. On the diagram of Fig. 1.2, this would mean that the
Fermi level might run between bands (1) and (2), so that band (1) is completely occupied
6
Figure 1.2: (a) One dimensional electron energy bands for the nearly free electron model
shown in the extended Brillouin zone scheme for the three bands of lowest energy. (b) The
same E(
~
k) as in (a) but now shown on the reduced zone scheme. The shaded areas denote
the band gaps and the white areas the band states.
and band (2) is completely empty. One further property of the semiconductor is that the
band gap E
g
be small enough so that at some temperature (e.g., room temperature) there
will be reasonable numbers of thermally excited carriers, perhaps 10
15
/cm
3
. The doping
with donor (electron donating) impurities will raise the Fermi level and doping with accep-
tor (electron extracting) impurities will lower the Fermi level. Neglecting for the moment
the effect of impurities on the E(
~
k) relations for the perfectly periodic crystal, let us con-
sider what happens when we raise the Fermi level into the bands. If we know the shape
of the E(
~
k) curve, we are in a position to estimate the velocity of the electrons and also
the so–called effective mass of the electrons. From the diagram in Fig. 1.2 we see that the
conduction bands tend to fill up electron states starting at their energy extrema.
Since the energy bands have zero slope about their extrema, we can write E(
~
k) as a
quadratic form in
~
k. It is convenient to write the proportionality in terms of the quantity
called the effective mass m
∗
E(
~
k) = E(0) +
¯h
2
k
2
2m
∗
(1.34)
so that m
∗
is defined by
1
m
∗
≡
∂
2
E(
~
k)
¯h
2
∂k
2
(1.35)
and we can say in some approximate w