Chapter 6
Band Structure
Silicon is a metal.
A.H. Wilson, 1931 [74]
Abstract A treatment of electron states in one-dimensional potentials introduces into the concepts
of band gap and effective mass. The band structures of various semiconductors are reviewed. The
systematics of band gaps, symmetry considerations, band gaps in alloys, amorphous semiconductors
and the effect of strain and temperature are discussed. Electron and hole dispersions are treated and
the density of states in various dimensions is derived.
6.1 Introduction
Valence electrons that move in the crystals feel a periodic potential
U (r) = U (r + R)
(6.1)
for all vectors R of the direct lattice. The potential
1 is due to the effect of the ion cores and all
other electrons. Thus a serious many-body problem is present. In principle, the band structure can be
calculated from the periodic arrangements of the atoms and their atomic order number. We note that
for some problems, e.g. the design of optimal solar cells, a certain band structure is known to be ideal
and a periodic atomic arrangement, i.e. a material, needs to be found that generates the optimal band
structure. This problem is called the inverse band structure problem.
6.2 Electrons in a Periodic Potential
6.2.1 Bloch’s Theorem
First, we will deduce some general conclusions about the structure of the solution as a consequence of
the periodicity of the potential. We first investigate the solution of a Schrödinger equation of the type
1 In this book the form of the potential will never be explicitly given.
© Springer Nature Switzerland AG 2021
M. Grundmann, The Physics of Semiconductors, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-51569-0_6
135
Band Structure
Silicon is a metal.
A.H. Wilson, 1931 [74]
Abstract A treatment of electron states in one-dimensional potentials introduces into the concepts
of band gap and effective mass. The band structures of various semiconductors are reviewed. The
systematics of band gaps, symmetry considerations, band gaps in alloys, amorphous semiconductors
and the effect of strain and temperature are discussed. Electron and hole dispersions are treated and
the density of states in various dimensions is derived.
6.1 Introduction
Valence electrons that move in the crystals feel a periodic potential
U (r) = U (r + R)
(6.1)
for all vectors R of the direct lattice. The potential
1 is due to the effect of the ion cores and all
other electrons. Thus a serious many-body problem is present. In principle, the band structure can be
calculated from the periodic arrangements of the atoms and their atomic order number. We note that
for some problems, e.g. the design of optimal solar cells, a certain band structure is known to be ideal
and a periodic atomic arrangement, i.e. a material, needs to be found that generates the optimal band
structure. This problem is called the inverse band structure problem.
6.2 Electrons in a Periodic Potential
6.2.1 Bloch’s Theorem
First, we will deduce some general conclusions about the structure of the solution as a consequence of
the periodicity of the potential. We first investigate the solution of a Schrödinger equation of the type
1 In this book the form of the potential will never be explicitly given.
© Springer Nature Switzerland AG 2021
M. Grundmann, The Physics of Semiconductors, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-51569-0_6
135