Download PX432 Functional Properties of Solids Part III: Electrical properties

PX432 Functional Properties of Solids
Part III: Electrical properties
Easter term 2014
Lecturer: Dr James Lloyd-Hughes,
Room: MAS3.06,
Email: [email protected]
2
Contents
1 Course description
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Textbooks/resources . . . . . . . . . . . . . . . . . . . . .
1.5 Handout/notes . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Commitment . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Outline Syllabus . . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Bandstructure: theory and experiment (3 lectures)
1.8.2 Transport in intrinsic and extrinsic semiconductors
1.8.3 Semiconductor optics (1 lecture) . . . . . . . . . .
1.8.4 Semiconductor devices (1 lecture) . . . . . . . . . .
1.8.5 Terahertz optoelectronics (3 lectures) . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7
7
7
7
7
8
8
8
8
8
9
9
9
9
2 Electrons in crystals
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Free electron model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Sommerfeld-Drude theory of metals . . . . . . . . . . . . . . . . . . . . .
2.2.2 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Heat capacity of the Fermi gas . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Transport properties: dc electrical conductivity . . . . . . . . . . . . . . .
2.2.5 Wiedemann Franz ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.7 Response to electromagnetic waves: ac conductivity . . . . . . . . . . . .
2.2.8 Experimental verification / success and failure of the free electron model .
2.3 The nearly free electron model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Electrons in a periodic potential . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The 1D empty lattice approximation . . . . . . . . . . . . . . . . . . . . .
2.3.4 Nearly free electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Zone boundaries, origin of energy gap, physical interpretation . . . . . . .
2.3.6 2D: band overlaps, density of states and semimetals . . . . . . . . . . . .
2.3.7 Electrical classification of crystalline solids . . . . . . . . . . . . . . . . . .
2.4 Alternative approaches to bandstructure calculation . . . . . . . . . . . . . . . .
2.4.1 Pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Kronig-Penney model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 The tight-binding approach . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Carrier dynamics and collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
11
11
12
14
15
16
18
18
20
21
22
22
22
24
25
26
29
31
32
32
32
33
35
36
3
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
(2 lectures)
. . . . . . .
. . . . . . .
. . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
36
36
37
38
40
41
3 Intrinsic and Extrinsic Semiconductors
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Direct and indirect band gaps . . . . . . . . . . . .
3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . .
3.2 Intrinsic semiconductors . . . . . . . . . . . . . . . . . . .
3.2.1 Electron concentration . . . . . . . . . . . . . . . .
3.2.2 Hole concentration . . . . . . . . . . . . . . . . . .
3.2.3 Mass action law . . . . . . . . . . . . . . . . . . .
3.2.4 Intrinsic carrier concentrations . . . . . . . . . . .
3.2.5 Conductivity . . . . . . . . . . . . . . . . . . . . .
3.3 Extrinsic semiconductors . . . . . . . . . . . . . . . . . . .
3.3.1 Hydrogenic levels: donor ionisation energies . . . .
3.3.2 Extrinsic carrier concentrations . . . . . . . . . . .
3.3.3 Temperature and dopant dependence of n, p, µ & σ
3.3.4 Conductivity [revisited] . . . . . . . . . . . . . . .
3.3.5 The Hall effect [revisited] . . . . . . . . . . . . . .
3.3.6 Carrier dynamics . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
43
43
43
45
46
46
47
48
48
48
51
52
53
56
56
57
57
2.6
2.5.1 Conservation of energy and crystal momentum
2.5.2 Scattering: classical versus quantum pictures .
2.5.3 Scattering mechanisms and Mattheissen’s rule .
2.5.4 Effective mass model . . . . . . . . . . . . . . .
2.5.5 Holes . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
4 Semiconductor Optics
59
4.1 Interband absorption and emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Band gap engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Semiconductor devices
5.1 Introduction . . . . . . . . . . . . . . . .
5.2 p − n junction . . . . . . . . . . . . . . .
5.2.1 The p − n junction under applied
5.2.2 Solar cell / LED / Laser . . . . .
5.2.3 High electron mobility transistor
5.3 Metal-semiconductor junction . . . . . .
5.4 MOSFET . . . . . . . . . . . . . . . . .
. . .
. . .
bias
. . .
. . .
. . .
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6 Terahertz optoelectronics
6.1 Terahertz time-domain spectroscopy . . . . . . . . . . . . . . .
6.1.1 Pulsed terahertz generation from biased semiconductors
6.1.2 Ultrafast spectroscopy: pump-probe techniques . . . . .
6.1.3 Time-resolved THz conductivity of bulk materials . . .
6.2 Quasiparticles in condensed matter . . . . . . . . . . . . . . . .
6.2.1 Electrons in nanoscale semiconductors . . . . . . . . . .
6.2.2 Plasmons, excitons, polarons . . . . . . . . . . . . . . .
6.3 Intersubband devices . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Intersubband transitions . . . . . . . . . . . . . . . . . .
6.3.2 Quantum cascade lasers . . . . . . . . . . . . . . . . . .
4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
61
61
62
66
69
70
71
74
.
.
.
.
.
.
.
.
.
.
77
77
77
77
79
79
79
79
79
79
79
7 Further Reading
7.1 Further Reading . . . . . . . . .
7.1.1 Books . . . . . . . . . . .
7.1.2 Review General/ Articles
7.2 Acknowledgements . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
81
81
81
81
81
6
Chapter 1
Course description
1.1
Introduction
This is one third of the PX432 ‘Functional Properties of Solids’ module, it introduces the fundamental
ideas of how the electrical properties of materials are determined. The course will draw on concepts introduced in PX262 ‘Quantum mechanics and its applications’, PX394 ‘Electrons in solids’, and PX393 ‘Crystal
physics’. Functional devices based on semiconductors will be described. Advanced experimental techniques
that investigate the electrical properties of solids will be outlined, such as angle-resolved photoemission spectroscopy and terahertz time-domain spectroscopy. The final lectures will discuss optoelectronic devices and
spectroscopy using terahertz (far-infrared) radiation.
1.2
Aims
• to provide an overview of theoretical ideas that underpin electronic band structure and conduction in
solids.
• to introduce the fundamental properties of intrinsic and extrinsic semiconductorss.
• to provide a basis for the understanding of modern solid-state electronics.
• to highlight advanced experimental techniques, materials and devices at the forefront of contemporary
research into electronic materials.
1.3
Objectives
At the end of the module you should
• understand the underlying band theory of how materials are classified as metals, insulators, semiconductors and semi-metals.
• understand and appreciate experimental methods that investigate the electronic properties of solids.
• be able to apply basic ideas of quantum mechanics and statistical mechanics to situations in condensed
matter (metals & semiconductors).
• calculate carrier densities in doped and undoped semiconductors under a variety of conditions.
1.4
Textbooks/resources
Handouts are available on the course website at go.warwick.ac.uk/PX432.
Recommended text books
7
• Hook and Hall
- The recommended textbook you are used to from PX394 ‘Electrons in solids’, which I will try
to refer to where possible.
• J. Singleton, ‘Band Theory and Electronic Properties of Solids’
- A modern and excellent discussion of bandstructure and electron transport.
• M. Fox, ‘Optical Properties of Solids’
- Accessible descriptions of the optics of semiconductors and heterostructures
Additional resources:
• H.P. Myers ‘Introductory Solid State Physics’.
• S.M. Sze ‘Physics of semiconductor devices’
- Comprehensive, giving plenty of detail on semiconductor devices beyond that needed for this
course.
• L. Solymar and D. Walsh, ‘Electrical properties of materials’
- Written for engineers, covers the basics in the context of modern devices
• www.NanoHub.org is an excellent website containing a lot of tutorials and applets related to semiconductor physics.
1.5
Handout/notes
This handout is based on the handout provided by the course’s previous lecturer (Dr. N. Wilson). It
contains, for ease of reference, some material that you will be familiar with from previous courses (e.g.
PX394 ‘Electrons in solids’). The concepts listed in the outline syllabus and discussed in lectures form the
basis of the examinable material for this course.
1.6
Commitment
10 lectures, 2 examples sheets.
1.7
Assessment
2 hour examination, in which you will be required to answer 3 questions. There will be one compulsory
question in Section A covering the whole course. You will then have a choice of two from three questions in
Section B. 50 % of the marks will be for Section A and 50 % for Section B. Note that this is a change in the
exam structure to previous years.
1.8
1.8.1
Outline Syllabus
Bandstructure: theory and experiment (3 lectures)
• Electrons in a periodic potential
- Tight-binding approach (linear combination of atomic orbitals)
- Nearly-free electron model
• Indirect and direct measurements of bandstucture
• Effective mass m*
- arbitrary band dispersion
- results for parabolic and non-parabolic bands (including linear bands, e.g. graphene)
- measuring m*: cyclotron resonance
• Angle-resolved photoemission spectroscopy
8
1.8.2
Transport in intrinsic and extrinsic semiconductors (2 lectures)
• Intrinsic carriers; mass action law
• Fermi energy and chemical potential
• Extrinsic semiconductors:
- Donor and acceptor ionisation - effective Hamiltonian
- Extrinsic carrier statistics (donor ionisation regime, exhaustion regime, intrinsic regime)
• Temperature and dopant concentration dependence of chemical potential
• Transport in semiconductors:
- Drude model for conductivity; frequency dependence
- Current density for energy bands
- Conductivity and mobility
- Conductivity of extrinsic semiconductors
- Conductivity tensor in novel materials
1.8.3
Semiconductor optics (1 lecture)
• Inorganic semiconductors:
- Direct gap/indirect gap semiconductors
- Conservation of crystal momentum and energy
- Matrix elements
- Absorption coefficient
- Photoluminescence
• Excitons: Wannier and Frenkel
• Bandstructure engineering – III-V system
1.8.4
Semiconductor devices (1 lecture)
• Conductive devices:
- p-n junction
- Transistors
• Optoelectronic devices:
- LEDs, lasers
- Photovoltaics, photodiodes
1.8.5
Terahertz optoelectronics (3 lectures)
• THz time-domain spectroscopy
- Pulsed terahertz generation from biased semiconductors
- Ultrafast spectroscopy: pump-probe techniques
- Time-resolved THz conductivity of bulk materials
• Quasiparticles in condensed matter
- Electrons in nanoscale semiconductors
- Plasmons, excitons, polarons
• Intersubband devices
- Intersubband transitions
- Quantum cascade lasers
9
10
Chapter 2
Electrons in crystals
2.1
Introduction
In this course we will be looking at functional electrical properties of crystalline solids. We will start by
considering conduction in metals which to a ‘zeroth order’ approximation can be treated as a ‘free electron
gas’ in a box. The ‘first order’ approximation introduces the effect of the background lattice as a periodic
potential which acts to perturb the free electron gas. The insight this model brings will then be applied to
understand semiconductors - the bulk of this course. In the final lectures we will introduce terahertz (farinfrared) optoelectronics. The justification for this course is simple; the impact of semiconductor technology
on life in the developed world is unrivalled. You should understand the basic science behind the technology.
Figure 2.1: Photo copyright Alcatel-Lucent.
2.2
Free electron model
The theoretical approach to understanding conduction in metals was first introduced by Paul Drude around
1900, only three years after the discovery of the electron by J.J Thomson. Drude applied the kinetic theory
of gases to try to understand electrical and thermal transport properties of metals. For our purposes the
key assumptions are:
1. Free and independent electron approximation: between collisions the electrons do not interact with
either the ions or the other electrons, but instead move according to Newton’s laws subject to any
external fields that may apply. The independent electron approximation (electrons do not interact
11
with one another) is bewilderingly good in many situations, however as we shall see the free electron
approximation (electrons do not interact with the ions) is not so useful.
2. Scattering: Collisions are the only form of interaction of the electron with the rest of the material.
They are instantaneous, and afterwards the electron moves at a velocity not related to its original one,
but instead dependent on the temperature of the material at that point. The probability of scattering is
τ −1 per unit time. Drude attributed the scattering events to be due to collisions with the ion cores, we
shall see that this is not the case. However the mechanism of scattering does not effect the predictions.
This is an entirely classical picture. The resultant velocity distributions of the particles are given by the
Maxwell Boltzmann distribution. Unfortunately the predictions from it are wrong, in particular the heat
capacity predicted by this model should have a contribution of 32 kB per electron which is not observed.
Correcting this required the application of quantum mechanics, a further 25 year wait.
2.2.1
Sommerfeld-Drude theory of metals
In this semi-classical approach first the Pauli exclusion principle is applied to determine the velocity distribution of electrons in the metal, and then this velocity distribution is applied classically in the manner
developed by Drude. We will derive some of the basic results here.
We can write the Hamiltonian for free electrons as
HΨ
Ĥ
= Eψ
~2 2
p2
=−
∇
=
2m
2m
(2.1)
The solution to this is plane wave eigenstates, i.e. eigenstates of the momentum operator.
Ψk
=
ceik̄.r̄
p~Ψk
=
⇒ Ek
=
−i~∇Ψk = ~kΨk
~2 k 2
2m
(2.2)
For a finite crystal we impose periodic boundary conditions so that
~
Ψ(~r + L)
~
i~
k.L
⇒e
⇒ ~k
=
Ψ(~r)
=
1
2π
(nx , ny , nz )
L
=
(2.3)
i.e. there is one state per (2π/L)3 of ~k-space. We also need to normalise the wavefunction
Z
Ψ∗k Ψk d3 r = 1
= c2 V
1
⇒c = √
V
1 i~k.~r
⇒ Ψk = √ e
V
(2.4)
Fermi statistics must be obeyed, i.e. the Pauli exclusion principle excludes two electrons from occupying
the same state. Electrons are spin 12 so the wavefunctions are anti-symmetrised and each ~k corresponds to
2 states. The states are filled in order of increasing energy so that at T = 0K all states are filled up to the
’Fermi Energy’
12
Figure 2.2: Sketches of dispersion relation with filled and empty states, and 2D Fermi surface (circle) - a
sphere in 3D.
What are EF and kF ? We can calculate the volume of k space that lies in |k| < kF :
N
=
X
2
1
|k|<kF
Z
2V
d3 k
(2π)3 |k|<kF
2V 4π 3
=
k
(2π)3 3 f
2 31
3π N
=
V
=
⇒ kF
(2.5)
Here we have imposed the periodic boundary conditions, which quantised the states in k space. These are
then filled up to a total number of N states. We can estimate the Fermi wavevector kF
kF−1
∼
V
N
13
∼a
Where a is the typical distance between electrons. Hence in a metal kF−1 ∼ few Å. Having derived an equation
for kF we can use this to find the Fermi energy, EF .
EF
~2 kF2
2me
∼ 10eV
=
(2.6)
∼ 105 K
The energy scale of the Fermi energy is far higher than room temperature or any real thermal excitations,
i.e. the Pauli exclusion principle is the dominant effect governing behaviour in metals. The low temperature
13
approximation of T = 0K will be quite valid in many cases. We can also calculate the Fermi velocity
vF
=
=
~kF
m
' 108 cms−1
(2.7)
Thus far we have assumed that all states up to the Fermi energy are filled, and any state above the Fermi
energy is empty. However at finite temperature some of the electrons will be excited into higher states.
Roughly we expect that states within kB T either side of EF will have a non-zero probability of being
occupied. Let’s use nk as the probability that the state k is occupied, this is equal to the Fermi distribution
function as defined by
1
nk = f (Ek ) = (E −µ)/k T
(2.8)
B
k
e
+1
where µ is the (electronic) chemical potential. The chemical potential at T = 0K is the Fermi energy,
µ = EF . For finite temperature µ is determined by
N =2
X
k
1
e(Ek −µ)/kB T + 1
(2.9)
The Fermi function is sketched at zero temperature and finite temperature in Fig. 2.3 (dashed lines).
Figure 2.3: Sketch of Fermi distribution function and EF .
2.2.2
Density of states
Critical to many physical properties is the density of states, N (E)dE: the number of states with energy
E < Ek < E + dE.
The density of states is
V
2
4πk 2 dk
(2π)3
We can rewrite
dk
dE =
dk =
dE
r
2m 1
√ dE
~2 2 E
Putting this together we have
V 2mE
N (E) = 2 2
π ~
14
r
2m 1
√
~2 2 E
(2.10)
This can be simplified to give the important result
N (E) =
V
2π 2
2m
~2
3/2
√
E
(2.11)
The density of states for a parabolic band is sketched in Fig. 2.3, where it is labelled gc (E)
2.2.3
Heat capacity of the Fermi gas
The internal energy can be calculated from the energy of the states and the probability that they are occupied
X
U =2
Ek nk
(2.12)
k
Switching to a continuous representation
Z
U=
N (E) E f (E) dE
(2.13)
However what we are really interested in here is the density of states at EF :
N (EF )
=
=
=
dN
dEF
dN
kF dk
F
F
kF dE
dkF
3N
2EF
Where we have used the following relations
N
∝ kF 3
EF
∝ kF 2
⇒N
∝ EF 2
3
From this density of states we can make a rough estimate of the heat capacity by considering only the
number of ’active’ electrons within kB T of the Fermi energy. Each electron that is thermally excited is
gaining ∼ kB T , and the number that are doing so is ∼ kB T N (EF ).
So the increase in internal energy due to temperature can be approximated by
U (T ) − U (T = 0) ' N (EF )(kB T )2
which gives the heat capacity
∼ kB N (EF )kB T
kB T
∼ N kB
EF
C
(2.14)
A more thorough and careful calculation gives
Cv =
π2
kB T
N kB
2
EF
(2.15)
Note that this is substantially less than the 32 kB that was predicted classically, and is now temperature
dependent. Recalling from the structure course that the heat capacity due to phonons ∝ T 3 we can write
the heat capacity of a metal as
C = γT + αT 3
(2.16)
where γ is a measure of the density of states.
15
Figure 2.4: Heat capacity of Au from Myers p.150
2.2.4
Transport properties: dc electrical conductivity
The current density is related to the electric field via the conductivity (tensor):
~
J~ = σ E
(2.17)
We will now try to understand the magnitude of the conductivity, starting with a classical model applied to
the free electron model.
Drude model
We start by applying Newton’s second law to the motion of an electron
F
dv
⇒
dt
= ma
eE
= −
m
We include damping due to collisions through adding a term with the momentum relaxation rate τ
dv
eE
1
=−
− v
dt
m
τ
In the simple case of zero electric field the relaxation of the electron speeds is then given by
v(t) = v(0)e
−t
τ
In steady state the speed distribution must be constant,
dv
dt
⇒v
=
0
= −
eEτ
m
The electron speed is related to the current density via the charge on the electron, and the electron density,
n:
J = −nev
(2.18)
Combining these results gives
J=
ne2 τ
E
m
16
(2.19)
By comparison with 2.17 we obtain the Drude formula for the conductivity
σ=
ne2 τ
m
(2.20)
Note that this predicts Ohm’s Law.
A quantum perspective: the moving Fermi sphere
When subjected to an accelerating electric field in the x-direction all electrons will feel a force in the −xdirection (force F = qEx = −eEx ), their x-momentum will change by an amount ~∆k, and the Fermi sphere
will thus shift [see Figure 2.5].
Figure 2.5: Schematic of shift of Fermi sphere from Myers p.155
We can rewrite Newton’s second law
dp
dt
dk
= ~
dt
F
=
mv
τ
~k
= −eE −
τ
= −eE −
In steady state we can equate the movement of the Fermi sphere
~∆k = −eEτ
i.e. the average momentum increase is given by the electric force applied multiplied by the time it is applied
for (the time τ before a relaxation process occurs).
Again we can write the current density as
J
=
=
−nev
~k
−ne
m
Combining these results gives once more
ne2 τ
E
(2.21)
m
The small shift in the Fermi surface illustrates that the current is effectively carried by only a very small
fraction of the electrons, and that these are the electrons at the Fermi energy.
J=
17
2.2.5
Wiedemann Franz ratio
Kinetic theory predicts the thermal conductivity of a gas of electrons to be
1
l C v vF
3
(2.22)
π2
kB T
N kB
2
EF
(2.23)
K=
Recall Cv from 2.15
Cv =
The Drude formula for the electrical conductivity can be written
σ=
ne2 τ
ne2 l
=
m
m vF
(2.24)
So we can use EF = 12 mvF2 to find the ratio
K
σ
=
L =
2
π 2 kB
T = LT
3e2
2
π 2 kB
= 2.4 × 10−8 Js−1 ΩK −2
2
3e
(2.25)
This result explains the Wiedemann-Franz law which states that for metals at not too low temperature the
ratio of the thermal to electrical conductivities is directly proportional to temperature. The Lorenz number,
L, is the constant of proportionality and as calculated in the free electron approximation above only consists
of fundamental constants. This is a surprisingly good prediction for most metals at room temperature. It
relies on the scattering processes that determine the electrical and thermal conductivities being the same,
and on the electrons dominating the thermal conductivity rather than the phonons.
2.2.6
Hall effect
Let’s consider a current flowing in the presence of a magnetic field. The dynamics will be governed by the
Lorentz force
d~v
e ~
~ − 1 ~v
=−
(2.26)
E + ~v × B
dt
m
τ
The final term is the damping or momentum relaxation term. We define the field to be in the z-direction,
~ = B ẑ, and solve for the steady state solution.
i.e. B
0
0
0
e
(Ex + vy B) −
m
e
= − (Ey − vx B) −
m
e
1
= − E z − vz
m
τ
= −
1
vx
τ
1
vy
τ
(2.27)
Suppose that vy = 0, and Jx = −nevx
⇒ Ex
= −
=
m
m
vx = 2 Jx
eτ
ne τ
1
Jx
σ
18
Figure 2.6: Schematic of Hall geometry Kittel p.165
The electric field in the y-direction is related to the magnetic field in the z-direction and the current in the
x-direction
ωc τ
Ey = vx B = −
Jx
(2.28)
σ
where we have defined the cyclotron frequency
ωc =
eB
m
(2.29)
In this situation the product ωc τ is the fraction of the cyclotron period completed before scattering.
Hall angle: The angle between the current and the applied field
tan ΘH =
Ey
= −ωc τ
Ex
(2.30)
Hall coefficient: This is the usually quoted result, and is defined by
RH =
19
VH
BI
(2.31)
where the Hall voltage VH is the voltage induced by the magnetic field. Using the results above we see that
the Hall coefficient is given by
1
RH =
(2.32)
nq
where n is the number of charge carriers per unit volume and q is the charge on them.
The Hall effect is small in metals due to the large carrier concentration, however as we shall see the carrier
concentration in semiconductors is much lower making the effect much larger. The Hall effect can be used
to measure the carrier concentration in semiconductors, alternatively a semiconductor with a known carrier
concentration can be applied in a Hall probe to measure the magnetic field. The experimentally measured
quantity is the Hall voltage
B
(2.33)
VH = − I
ne
whilst the magnetic field B and current I are controlled.
Importantly the Hall effect measures not only the magnitude of the carrier concentration, but also the
sign of the charge carriers.
2.2.7
Response to electromagnetic waves: ac conductivity
Let us consider the effect of an ac electric field of angular frequency ω
h
i
−iωt
~
~
E(t)
= Re E(ω)e
(2.34)
~
where the amplitude of the electric field at ω is E(ω).
The resultant velocity of the electrons will be of the
form
~v (t) = Re ~v (ω)e−iωt
(2.35)
and will be governed by the equation
d~v
~v
e ~
= − − E(t)
dt
τ
m
(2.36)
1
e ~
−iω~v (ω) = − ~v (ω) − E(ω)
τ
m
(2.37)
Combining these we have
~
which gives us ~v (ω) in terms of E(ω).
We can rewrite the current density as
~
J(ω)
= −ne~v (ω)
(2.38)
for the moment let us just consider the magnitudes as the directions are the same. Now we can rewrite the
current density in terms of the electric field using 2.37
⇒ J(ω)
=
=
ne2 τ
m E(ω)
1 − iωτ
σ0 E(ω)
1 − iωτ
(2.39)
where we have written the dc (Drude) conductivity as σ0 . From this we can ascertain the (complex) ac
conductivity
J(ω)
= σ(ω)E(ω)
σ0
σ(ω) =
1 − iωτ
20
(2.40)
The zero frequency limit of this expression is clearly the dc conductivity. At high frequencies the oscillation
of the electric field becomes too fast for the electrons to follow, and there is zero conductivity.
The ac conductivity is a complex quantity, the complex conductivity σ
e(ω) = σ1 + iσ2 , and can be
measured directly using terahertz time-domain spectroscopy [Section 6.1].
It is worth noting that the real part of the conductivity gives the in phase response, and the imaginary
part the out of phase response. The imaginary part can be interpreted as showing that the current lags
behind the driving electric field, due to the finite time taken to accelerate electrons.
2.2.8
Experimental verification / success and failure of the free electron model
The free electron model works well for properties dependent on the density of states alone, for example
correctly predicting Ohm’s Law and the Lorenz number. However there are a number of places where it falls
down1 . For example:
The Hall Coefficient The free electron model cannot even account for the positive sign of the Hall Coefficient
for some materials, let alone its field and temperature dependence.
The Wiedemann-Franz Law Why does this break down at low temperatures?
ac conductivity Although the free electron model predicts some aspects of this accurately, it cannot predict
for example the colour of copper and gold.
There are more fundamental problems as well, such as:
What determines the number of conduction electrons?
Why are some elements nonmetals?
The role of the metallic ions Drude’s original idea that scattering was from the ion cores gives scattering
times that are far too short.
1 For
more information see Ashcroft and Mermin, Chapter 3, or Singleton, Section 1.4
21
2.3
2.3.1
The nearly free electron model
Electrons in a periodic potential
To extend the free electron model we must take some account of the background lattice of atomic nuclei. The
simplest approximation is to add a periodic potential which represents these nuclei and in an approximate
manner the core electron states. To ruin the surprise, the amazing result is that much of solid state physics
can be explained in terms of a nearly free electron gas. i.e. that the electrons are effectively independent of
one another except during collisions.
We can write the Schrodinger equation as before
HΨ
H
= Eψ
~2 2
∇ + V (~r)
= −
2m
(2.41)
Recalling from the structure course that we can write the translation vectors of a 3D crystal lattice as
~ = n1~a1 + n2~a2 + n3~a3
R
(2.42)
then we demand that the crystal potential V (~r) has the periodicity of the lattice, i.e.
~ = V (~r)
V (~r + R)
(2.43)
Figure 2.7: Sketch of lattice potentials, Kittel p.178
2.3.2
Bloch’s theorem
Bloch’s theorem states that if the potential is periodic as defined above then eigenstates of the Hamiltonian
may be chosen such that
~ = ei~k.R~ Ψ(~r)
Ψ(~r + R)
(2.44)
for some ~k. In words this says that if you translate by a lattice vector you change the wavefunction only
through a change in phase. An equivalent statement is that eigenstates may be written as
~
Ψ(~r) = eik.~r unk (~r)
~
unk (~r + R)
= unk (~r)
The quantity ~~k is the crystal momentum, and ~k the wavevector, of the eigenstate.
22
(2.45)
(2.46)
Proof of Bloch’s theorem
With the periodic potential the Hamiltonian has discrete translational symmetry. Defining the translational
operator
~
T̂R~ Ψ(~r) = Ψ(~r + R)
(2.47)
~ is a lattice vector then operating T̂ ~ on the Hamiltonian gives
if R
R
T̂R~ ĤΨ(~r)
=
−
~2 2
~ Ψ(~r + R)
~
∇ + V (~r + R)
2m
~2 2
~
∇ + V (~r)Ψ(~r + R)
2m
= Ĥ T̂R~ Ψ(~r)
= −
(2.48)
This implies that Ĥ and T̂R~ commute, which in turn implies that simultaneous eigenstates of the Hamiltonian
and translation operator exist, i.e.
ĤΨ = Eψ
T̂R~ Ψ = cR Ψ
(2.49)
The order in which translations are made does not matter, so translation operators commute, i.e.
[T̂R1 , T̂R2 ] = 0
(2.50)
~ has been dropped. But since the order of translations does not matter
where the vector notation for R
T̂R1 +R2 Ψ = cR1 +R2 Ψ = T̂R1 T̂R2 Ψ = cR1 cR2 Ψ
(2.51)
this gives a relation between eigenvalues of the translational operator
cR1 +R2 = cR1 cR2
(2.52)
From this we can deduce that the eigenvalues must be of the form
cR = eik·R
(2.53)
Putting this together we arrive at Bloch’s theorem
T̂R Ψ(~r) = eik·R Ψ(~r) = Ψ(~r + R)
(2.54)
Properties of Bloch states
Momentum: Bloch states for a non-constant potential are not eigenstates of the momentum operator. In the
~
free electron model the wavefunction Ψ(r) = ceik·~r is an eigenstate of the momentum operator (p̂ = −i~∇)
with eigenvalue ~~k. However this is not generally the case in the nearly free electron model. If we consider
the representation of the Bloch states given in 2.45 above then
~
~
(−i~∇)Ψnk = (−i~∇)eik.~r unk (~r) = ~~kΨnk − i~eik.~r ∇unk (~r)
(2.55)
i.e. Ψnk is not an eigenstate of the momentum operator. The quantity ~~k is instead referred to as the crystal
momentum.
23
Crystal momentum and the Brillouin zone: The significance of the crystal momentum will hopefully emerge
~ to ~k. Recalling from the structure
in later lectures, here let us consider adding a reciprocal lattice vector, G
course that the dot product of a reciprocal lattice vector and a lattice vector is a multiple of 2π we have
~
~
~
~
ei(k+G).~r = eik.~r+i2πn = eik.~r
(2.56)
The phase is unchanged by the addition of the reciprocal lattice vector, or in other words the wavevector
~ We can choose the k with the lowest modulus; this is equivalent to choosing
can be written as any ~k + nG.
~k to be within the first Brillouin zone. 2
If we take all solutions of the Schrodinger equation associated with the nearly free electron model we
find that the energies of the states form a continuous function of ~k which are referred to as energy bands.
We also find that at each ~k there is an infinite number of energy states (due to equivalence under addition
of reciprocal lattice vectors). We can label each state with wavevector ~k with a second index n, called the
band index. By convention the lowest energy state at a given ~k is given by n = 1 the next by n = 2 etc. To
specify a Bloch state we must specify n and ~k, hence we wrote Ψnk .
The reduced zone scheme: Label each state by a wavevector within the first Brillouin zone.
The extended zone scheme: Label each state by a single wavevector which may be larger than the first
Brillouin zone.
The repeated or periodic zone scheme: Use all possible wavevectors to label a single state.
We can illustrate these ideas by considering the empty lattice approximation.
2.3.3
The 1D empty lattice approximation
We can apply the ideas above to the free electron model in 1D. For free electrons V (r) = 0. We can ascribe
an arbitrary lattice constant of a, so trivially V (r + a) = V (r). The solutions to the Schrodinger equation
will be as in the free electron model, i.e.
−
~2 d2
Ψ(x)
2m dx2
=
Ψ(x)
=
ε =
εΨ(x)
1
√ eiq.x
V
2 2
~ q
2m
(2.57)
The energy dispersion diagram is parabolic as discussed before. We can now imagine that the system is
periodic with lattice parameter a. Let q = k + G, where the reciprocal lattice vectore G = n 2π
a and
− πa < k < πa . That is to say we define an arbitrary Brillouin zone and demand that the wavevector is within
it. Now we can write the wavefunction and energy as
Ψnk (x)
=
εnk
=
1
1
√ eik.x eiGn .x = √ eik.x un (x)
V
V
~2 (k + Gn )2
2m
(2.58)
2 Recall from the structure course that the first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It can be
constructed by taking all points closer to the origin of reciprocal lattice space than the nearest reciprocal lattice vectors, its
boundaries are thus constructed from the perpendicular bisectors between the origin and the nearest reciprocal lattice points.
24
2.3.4
Nearly free electron gas
In this approximation the atomic lattice and core electrons are included via a weak perturbative, periodic
potential acting on an otherwise free electron gas. We will use perturbation theory to analyse the effect of
this potential, so first we’ll briefly review perturbation theory.
Perturbation theory
We consider adding a small perturbation Ĥ I to a Hamiltonian Ĥ 0 for which the solutions are known. The
total Hamiltonian is
Ĥ = Ĥ 0 + Ĥ I
(2.59)
and we know the solutions
Ĥ 0 Ψ0n = En0 Ψ0n
(2.60)
Perturbation theory then tells us that the eigenstates of the full Hamiltonian are given by (to first order)
|Ψn i = |Ψ0n i +
X hΨ0 |Ĥ I |Ψ0 i
m
n
|Ψ0m i
0
En0 − Em
(2.61)
m6=n
and the corresponding eigenvalues or energies are given by
En = En0 + hΨ0n |Ĥ I |Ψ0n i +
X |hΨ0 |Ĥ I |Ψ0 i|2
n
m
0
En0 − Em
(2.62)
m6=n
Perturbing the free electron gas
Here we start from the free electron Hamiltonian in one dimension, which has the now familiar eigenstates
and energies
Ψ0q (x)
=
Eq0
=
1
√ eiq.x
V
~2 q 2
2m
(2.63)
We consider first the simplest possible periodic perturbation
X
Ĥ I = V (x) =
VG eiG.x
(2.64)
G
where VG is small. Let’s consider the first matrix element
Z
X
0
1
hΨq |V (x)|Ψq0 i = dx ei(q −q).x
VG eiG.x
V
G
(
⇒ hΨq |V (x)|Ψq0 i =
6 G for any G
if q − q 0 =
0
if q − q = G
0
VG
(2.65)
This gives the eigenstates and energies
Ψq
=
Ψ0q +
X
Eq0 +
X
G
Eq
=
VG
Ψ0q+G
0
Eq0 − Eq+G
Eq0
G
25
|VG |2
0
− Eq+G
(2.66)
Letting q = k + Gm we can rewrite this as
Ψmk
Ψ0mk +
X
0
= Emk
+
X
=
Gn
Emk
Gn
2.3.5
VGn
Ψ0(m+n)k
0 − E0
Emk
(m+n)k
|VGn |2
0 − E0
Emk
(m+n)k
(2.67)
Zone boundaries, origin of energy gap, physical interpretation
From this we can see that the situation becomes interesting where the states are degenerate, i.e. at the zone
0
boundaries where Bragg reflection occurs. At these positions Eq0 = Eq+G
, and the second term diverges.
Near these points we must apply instead degenerate perturbation theory; we start by writing the eigenstate
as a linear combination of the two degenerate states
Ψ = cq Ψ0q + cq+G Ψ0q+G
The Hamiltonian can be expressed in this two state basis as
0
VG
Eq
Ĥ =
0
V−G Eq+G
and we must solve
Ĥ
cq
cq+G
=E
cq
(2.68)
(2.69)
cq+G
(2.70)
For simplicity we consider first the situation at the degeneracy point where ε0q = ε0q+G = ε0 then the solutions
of the Hamiltonian are given by
0
ε −E
VG
=0
(2.71)
VG
ε0 − E This can trivially be solved
⇒ E = ε0 ± VG
(2.72)
v
!2
u
0
u Eq0 − E 0
Eq0 + Eq+G
q+G
t
±
+ |VG |2
E=
2
2
(2.73)
In general
The effect of the periodic potential is to open up a gap in the density states at the zone boundaries, breaking
the degeneracy of the states.
26
Figure 2.8: Sketch of nearly free electron model in periodic, extended and reduced zone scheme, Ashcroft
and Mermin p. 160, c.f. Myers p.183
Why does the band gap appear?
We can also solve for the wavefunction of the two states at the zone boundaries, i.e. where k = ± πa . At the
zone boundaries the energies are E = ±|VG | + ε0 , so to find the eigenvectors we must solve
c+ πa
∓|VG |
VG
=0
c− πa
V−G ∓|VG |
(2.74)
This can be trivially solved to give the eigenvectors
c+ πa
c− πa
1
=
1
27
or
1
−1
(2.75)
Explicitly we can write these two solutions as
Ψ−
=
Ψ+
=
π
x
a
π
sin x
a
cos
(2.76)
In the absence of a potential these states have identical energy. In the presence of the periodic potential Ψ−
has a higher charge density in the low points of the potential, and hence a lower potential energy. The low
points of the potential correspond to where the atoms in the lattice would be.
Figure 2.9: Sketch of wavefunctions and background potential, Kittel p.178
How many states are in a band?
x
In a finite crystal only discrete wavevectors are allowed corresponding to kx = ± 2πn
Lx . There is only one
3
(2π)
allowed ~k state per volume V in ~k-space. The total number of electron states in a band is
no. of states = 2
V (2π)3
= 2N
(2π)3 Vcell
(2.77)
where N is the number of unit cells in the crystal. The factor of 2 comes from spin degeneracy. At the
simplest level we can then see that if there are an odd number of electrons per unit cell then a half filled
band will result, whilst an even number of electrons will result in a filled band. This gives a hint to why
some materials are metallic, and some materials are insulators.
A material with a filled band can still be metallic if there is band overlap, as we now discuss.
28
2.3.6
2D: band overlaps, density of states and semimetals
Let’s consider the next most complicated case after the 1D lattice, the 2D square lattice. The direction of
the k vector in the energy dispersion diagram must be considered. This can result in band overlap. First let
us consider the bandstructure looking in different directions
Figure 2.10: Sketch of bandstructure in 2D, Needs 4.12, Myers p.186
Alternatively we can draw a map of equal energy contours in reciprocal space
Figure 2.11: Sketch of equal energy contours in reciprocal space, Needs 4.12, Myers p.187
Or we can plot out the density of states
Figure 2.12: Sketch of the density of states, Needs 4.12, Myers p.189
29
Example: band structure of Al
We can compare the bandstructure of Al (face centred cubic) derived from detailed calculations with that
predicted by an empty lattice model. 3
Figure 2.13: A comparison of the empty lattice model and detailed calculations for the bandstructure of Al,
Myers p.197
There is a clear similarity between the free electron model and the detailed calculation. Note that the
effective size of the Fourier components of the potential can be extracted from a comparison of the two. As
discussed before these will be significantly weaker than those one would estimate from the atomic potentials,
and instead reflect the magnitude of the appropriate pseudopotential [see Sec. 2.4.1].
3 The Fermi surface can be probed experimentally by e.g. ARPES (see Lecture 3) and the de Haas Van Alphen effect – see
Myers Chapter 9.
30
2.3.7
Electrical classification of crystalline solids
Metals: finite density of states at the Fermi level, partially filled bands or overlapping bands
Insulators: filled band with a large energy gap Eg to the next empty band, Eg & 2 − 3eV
Semi-conductors: filled band with a small energy gap Eg to the next empty band, Eg . 2 − 3eV
Semi-metals: filled band but with touching or small overlap of bands at the Fermi level
Figure 2.14: Sketch of band structures of metals, semi-metals, semi-conductors, insulators, c.f. Kittel p.194
31
2.4
2.4.1
Alternative approaches to bandstructure calculation
Pseudopotential
In reality the potential experienced by electrons in a solid is hugely complicated with contributions from
the lattice and the other electrons, both core and valence. We have reduced all this to a simple periodic
potential, so it is worth considering how valid an exercise this is. For some metals this simple picture is
surprisingly accurate. The overlapping atomic potentials result in a strong and fast varying potential, far too
strong for the nearly-free electron model to apply. However when considering the valence electrons we must
also consider the effect of the core electrons which will be more tightly localised to the atomic lattice. The
valence electron states must be orthogonal to these core states. We can construct an effective potential for
the valence states which includes the ‘Pauli repulsion effect’ forcing this orthogonality. The real potential is
strongly attractive near each atom, but the Pauli exclusion repels the valence electrons from the atom cores
so that they see a less attractive potential. This ‘pseudopotential’ is weaker and the nearly free electron
model becomes applicable. The wavefunction contains fewer nodes (see figure), meaning far fewer Fourier
components are needed to describe the wavefunction – plane wave basis sets then become more appropriate.
Figure 2.15: Sketch of real potential and pseudopotential, (Wikimedia commons)
The nearly free electron model seems such an outrageous simplification that even using the pseudopotential justification it is still not compelling evidence on its own that crystalline solids will form bandstructures with allowed energy bands and gaps between them. However this is generically true. We can look
at two further relatively similar models which demonstrate this.
2.4.2
Kronig-Penney model
A useful model for investigating how/why energy bands are formed in solids was introduced in the 1930s by
Kronig and Penney. It is treated in reasonable detail in Kittel (pp. 180-182), here the results are reviewed
without a full derivation.
Consider a 1-D square-well periodic potential such as that shown in the figure below.
The time-independent Schrodinger equation must be obeyed in all space
−
~2 d2 Ψ
+ U (x)Ψ = EΨ
2m dx2
(2.78)
This can be solved by considering the form of the function within and outside a given barrier, and then
constructing Bloch states from it. Let us consider limits of the situation.
U0 small: this is equivalent to the nearly free electron model we described before, so energy bands will be
created with small gaps between them.
32
Figure 2.16: Square-well periodic potential as introduced by Kronig and Penney, see e.g. Kittel p. 182
U0 large: each well is isolated from its neighbours, and so we have an array of one-dimensional particle in a
box problems. The resultant wavefunctions are
Ψn = sin(
nπ
)
a
with energy levels
~2 n 2 π 2
2m a2
We can create Bloch states from these by writing them as
X
Ψkn =
eikaj Ψnj
En =
(2.79)
j
This is equivalent to flat energy ’bands’ where the allowed states are the atomic energy levels.
moderate U0 : in between these situation we get large energy gaps and relatively flat bands, with the band
width dependent on the depth of the potential. The large U0 atomic-like case suggests that the bands can
be linked to the atomic orbitals, a link which is not apparent in the nearly free electron model.
Exercise 2.1 Log-in to NanoHub and use the Periodic Potential Lab to experiment with the Kronig-Penney
model. Try changing the barrier heights and comparing the resultant dispersions to the free electron model.
2.4.3
The tight-binding approach
Although the nearly-free electron model makes some conceptual sense for metals such as aluminium, where
a traditional viewpoint would be of a sea of electrons around a lattice of positive nuclei, it does not fit at all
with covalently bound solids such as diamond or graphite. Here the electrons are bound tightly to the atoms,
and a more reasonable approach would seem to be to start with atomic orbitals and perturb them, rather
than starting with a free electron and perturbing that. This is the origin of the ‘tight-binding’ approach.
The tight-binding model is an approximate method for calculating the electronic band structure of a
solid. It works on an approximation to the full Hamiltonian of a structure (e.g. a crystalline structure)
analogous to the linear combination of atomic orbitals used for molecular structure calculations. The atoms
are treated first as independent to find the atomic orbitals. In the crystal there will be some overlap between
atomic orbitals so that they are not true eigenfunctions of the full crystal Hamiltonian. However, the tightbinding approximation assumes that the overlap is small and treats it as a perturbation to the atomic
wavefunction - often only nearest neighbour interactions are considered. Starting from the atomic orbitals,
Bloch wavefunctions are then constructed and the resultant dispersion relation is calculated dependent on
the overlap of the atomic states, and on the crystal structure. The tight-binding model is discussed in Hook
and Hall, and at a more advanced level in Ashcroft and Mermin.
33
The tight-binding model will be discussed in lectures for a real system [graphene], here for illustration
purposes let’s look at a tight-binding calculation for a cubic lattice with one atom per unit cell, with each
atom having only one valence orbital, φ(~r), (e.g. an s-state). We can make a Bloch state by writing the
wavefunction
1 X i~k·R~ m
~ m)
e
φ(~r − R
(2.80)
Ψk (~r) = √
N m
~ m is the position of the mth atom in the lattice.
where R
Exercise 2.2 Confirm that this is a Bloch function by applying a translation operator (T̂ ).
The expectation energy of the Hamiltonian is given by
1 X i~k·(R~ n −R~ m )
h~k|Ĥ|~ki =
e
hφm |Ĥ|φn i
N m,n
(2.81)
~ m ). The term hφm |Ĥ|φn i is the overlap between states on atom n and
where |~ki = Ψk (~r) and φm = φ(~r − R
atom m. We are assuming that the states are tightly bound, i.e. localised on the atoms, and so the overlap
will be large if n = m but will decrease rapidly as the separation increases. Here we consider only nearest
neighbour interactions and so write
hφn |Ĥ|φn i = −α,
hφm |Ĥ|φn i = −γ
hφm |Ĥ|φn i =
0
(2.82)
if n and m are nearest neighbours,
otherwise.
(2.83)
(2.84)
With this assumption we have
E~k = h~k|Ĥ|~ki = −α − γ
X
~ ~
eik.Rneighbour
(2.85)
n
The first term
−α
= hφn |Ĥ|φn i
Z
~ n )Ĥφ(~r − R
~ n ) d~r
=
φ∗ (~r − R
Z
=
φ∗ (~r)Ĥφ(~r) d~r
(2.86)
(2.87)
(2.88)
reflects the binding energy of the atom (note that there may be some corrections to the Hamiltonian as
compared to an isolated atom in free space). The second term depends on
−γ
= hφm |Ĥ|φn i
Z
~ m )Ĥφ(~r − R
~ n ) d~r
=
φ∗ (~r − R
Z
~ neighbour )Ĥφ(~r) d~r
=
φ∗ (~r − R
(2.89)
(2.90)
(2.91)
i.e. the overlap between orbitals on neighbouring atoms. This is often called the overlap integral, and its
value will depend on the type of orbitals, orientation of the bonds etc. The tight-binding model can be
implemented at different levels, from the semi-empirical where α and γ are found from fits to experiments,
to the ab-initio where they calculated explicitly from calculated wavefunctions and derived Hamiltonians.
~ neighbour are the
The other part of the second term involves the structure of the system more explicitly (R
vectors joining each atom to its nearest neighbours). For the example we are considering here of a simple
cubic lattice with one s-state
~ neighbour } = {(a, 0, 0), (−a, 0, 0), (0, a, 0), (0, −a, 0), (0, 0, a), (0, 0, −a)}
{R
34
(2.92)
and the overlap integral is the same in each direction, giving
E~k = −α − 2γ(cos(kx a) + cos(ky a) + cos(kz a))
(2.93)
This is periodic, but as before only wavevectors within the first Brillouin Zone are unique so we can
sketch it in the reduced, or periodic zone schemes. The ’band width’ is the energy difference between the
top and bottom of the band, which here is 12γ, i.e. proportional to the degree of overlap. For |k| |π/a|
the dispersion is approximately E ∝ k 2 , i.e. free electron like. Other bands of states can be constructed
from different atomic orbitals. The result is a series of bands with energy gaps similar to the result derived
for the nearly free electron case.
When bands arise from p and d states the additional degeneracy associated with the allowed angular
momentum of the bands must be included, and the relative orientations and shapes of the bands must be
considered. The accuracy of the tight-binding model can be increased by including next-nearest neighbour
interactions and further, and by adding overlaps between different orbitals.
Exercise 2.3 Log-on to NanoHub and look at the bandstructure lab - this calculates the bandstructure of
various materials using the tight-binding approach.
2.4.4
Density functional theory
As the number of atoms increases the number of wavefunctions and interactions scales exponentially, so that
for large systems the tight-binding approach becomes computationally excessive. Instead approaches based
instead on the electron density rather than the individual wavefunctions are used: these are the basis for
density functional theory calculations - see e.g. PX441.
35
2.5
2.5.1
Carrier dynamics and collisions
Conservation of energy and crystal momentum
Collisions between particles underlie many solid state phenomena, for example collisions between the electrons
and nuclei which form the crystal, between photons and electrons, or neutrons and nuclei. The normal rules
of conservation of energy and momentum apply to the system as a whole, however as we pointed out earlier
the individual electron states in the presence of a periodic potential do not have a defined momentum.
Instead we introduced the concept of the crystal momentum, ~~k to describe the electron states. The lattice
vibrations, or phonons, also have a defined crystal momentum. For spectroscopic purposes incident beams
of photons, electrons and neutrons with well defined momentum can be created. It is worthwhile to consider
scattering processes involving particles with defined crystal momentum. The law of conservation of crystal
momentum asserts that 4
X
X
~
~~ki =
~~kf + ~G
(2.94)
i
f
~ is any reciprocal lattice vector.
where i labels the initial crystal momenta of the particles, f the final, and G
This makes sense as the wave vector is only defined to within a reciprocal lattice vector anyway. We should
be clear however that crystal momentum is only conserved for periodic systems. Let’s now consider the
implications of this for two important scattering processes, the interaction of an electron and a photon, and
between an electron and a phonon.
Electronic transition with absorption of a photon
What is the momentum of a 1 eV photon? For the photon ~ω = ~cq so q =
ω
c
' 0.5 × 107 m−1 .
What is a typical crystal momentum of an electron? Typical lattice constant is a ∼ 4 Å, so the Brillouin
10 −1
.
zone width is 2π
a ' 1.6 × 10 m
The photon wavevector is thus less than a thousandth the width of the Brillouin zone, whilst its energy is
typical for a semiconductor band gap. So the photon induced transition is essentially vertical when plotted
on the bandstructure.
Electronic transition with absorption of a phonon
What is the momentum of a typical phonon? The crystal momenta of the phonon similarly to the photon is
in the range q < πa .
What is the energy of a typical phonon? The largest frequency will be ωmax ' vs πa , where vs is the velocity
of sound (typically vs = 103 ms−1 ). The energy is then given by ~ωmax ' 0.5 × 10−2 eV .
The phonon crystal momentum is thus in the range of the Brillouin zone, but the energy scales are small
compared with most transitions of interest. The phonon induced transition is essentially horizontal when
plotted on the bandstructure.
You should check the numbers estimated above.
2.5.2
Scattering: classical versus quantum pictures
In Drude’s original model (Sec. 2.2.4) the positive ions were the scattering points, and were regarded as randomising the direction of the charge carrier’s momentum every time τ . In this classical picture, scattering
4 For
a derivation of this see Ashcroft and Mermin Appendix M
36
is thus detrimental to electrical conduction. However, in the quantum picture (of electrons as quasiparticles/wavepackets, with defined energy bands) scattering is actually required for a material to be conductive!
This surprising (and somewhat counterintuitive) conclusion can be reached from the following argument:5 .
1. Consider an electron at the bottom of a single (e.g. tight-binding) band, k = 0. If a constant electric
field is applied in the x−direction, the electron accelerates under this force (in the −x−direction), and
gains momentum.
2. As the electron gains momentum the effective mass begins to increase, and therefore the acceleration
decreases.
3. Eventually the effective mass flips in sign, because the curvature of the band is negative at higher
energies. In this case the electron is now accelerated in the opposite () direction! It loses momentum,
returning towards k = 0.
4. As it approaches k = 0 the curvature returns to positive, and the electron is accelerated back in the
original (−x) direction.
5. This process will repeat - the electron therefore only oscillates in real space (and reciprocal space), and
there is no current flow, i.e. zero conductivity.
Scattering creates conduction as follows: the action of an electric field gives the electron a small velocity
for a short time. The electron then scatters, randomising it’s momentum, and ‘reseting’ the electron to an
average value of k = 0. Each short period of restricted velocity gain (before scattering) gives electrons a
finite drift velocity, and a current flows.
2.5.3
Scattering mechanisms and Mattheissen’s rule
In Drude’s model, and our previous discussion, no real mention of the possible mechanisms that contribute to
electron scattering was made. In the quantum picture of electrons moving in a periodic potential, anything
that interrupts the crystal’s perfection can act as a scattering centre. Thus, phonons (displacements of
atoms from their equilibrium positions) will contribute at non-zero temperatures, as will vacancies (missing
atoms) and impurities (such as donors or acceptors). Quantum mechanical scattering rates can be derived
for these different electron scattering mechanisms,6 but here we restrict our discussion to their temperature
dependence, which can often identify the dominant scattering mechanism. Controlling τ , and thus the
mobility and conductivity, is obviously of key interest in optimising semiconductor materials for efficient
devices.
1.) Impurities: Scattering from impurities is roughly independent of temperature, and will depend on the
number of impurities present
1
∝ concentration of impurities
τimp
(2.95)
2.) electron-phonon scattering: Electron-phonon scattering scattering will be temperature dependent as the
rate of scattering will depend on the number of phonons which at high temperature is proportional to
temperature (c.f. structure course)
1
⇒
∝T
(2.96)
τph
At high temperature
3.) electron-electron scattering: This process cannot contribute to the momentum relaxation as momentum
must be conserved in the collisions.
5 See
6 See
Singleton, Sec. 9.1 for a fuller discussion
e.g. Yu and Cardona’s ‘Fundamentals of Semiconductors’ for a full treatment.
37
Mattheissen’s Rule
The scattering rates add, i.e.
1
1
1
=
+
τ
τph
τimp
(2.97)
m 1
1
= 2
σ
ne τ
(2.98)
Rewriting the conductivity as a resistivity
ρ=
For instance, for a metal such as Cu and Al (see Fig. 2.17) the resistivity increases linearly with temperature at room temperature, suggesting that electron-phonon scattering dominates.
Figure 2.17: Sketch of temperature dependence of conductivity for a good metal, Myers p.154
For a typical metal at room temperature the mean free path is l ∼ 10 − 100 Å corresponding to a momentum
relaxation rate of τ ∼ 10−14 − 10−15 s. For carefully prepared samples at low temperature the mean free
path can be increased up to about the µm length scales.
2.5.4
Effective mass model
Let us consider the conductivity of a metal in the free electron approximation. Although we might expect
the lattice to strongly scatter the electrons and result in a significantly higher resistivity, this is already
included in the Bloch functions. The electrons propagate through the periodic potential without attenuation
due to constructive interference of the waves scattered from the individual ions. A perfect metallic crystal
with static ions would have no electrical resistance. In a real metal scattering is due to phonons and defects
as discussed before in section 2.2.4.
~
The mean velocity of a free electron is ~v = ~mk . The mean velocity of a Bloch electron in the state Ψnk with
7
energy εnk is
1 ∂εnk
vnk =
(2.99)
~ ∂~k
You should check that for the free electron case this gives the expected result. We can think of the electrical
current being carried by a wavepacket of Bloch states, the actual form of which is not important.
X
Φn (~r, ~k, t) =
g(~k, ~k 0 )Ψnk0 (~r)e−iεnk0 t/~
(2.100)
~
k0
7 see
Appendix E of Ashcroft and Mermin for a proof of this result
38
For the moment let’s consider a Gaussian of width ∆~k. The wavepacket is also peaked in real space, and
will move with the group velocity 2.99. We expect the spatial extent of the wavepacket to be intermediate
between the unit cell of the crystal and the wavelength of the applied field (∆~k will be small compared to
the width of the Brillouin zone so it is much bigger than the unit cell).
Figure 2.18: Sketch of wavepacket and unit cell, p.217 Ashcroft and Mermin
Since the external fields vary only slowly over the wavepacket the dynamics of the wavepacket can be
described classically, whilst the effect of the ions on the electrons is treated quantum mechanically. This is
called the semiclassical model.
Effective mass
Classically, Newton’s Laws state that the rate of change of momentum is equal to the force, F = dp/dt. Here
the rate of change of crystal momentum is equal to the external force8 .
F =~
d~k
d~v
= m∗
dt
dt
(2.101)
Note that the right-hand equality is only true if dm∗ /dt = 0, which is only the case close to band extrema.
The more general definition of m∗ = p/dE/dp was given in lectures. Recalling 2.99
!
∂~k
∂~vnk
∂~v ∂~k
1 ∂ 2 εnk
~
=
= 2
(2.102)
∂t
~ ∂~k 2
∂t
∂~k ∂t
From this we can define an effective mass, m∗
⇒
1
1 ∂ 2 εnk
= 2
∗
m
~ ∂~k 2
(2.103)
When acted upon by an external field the electron will react as though it were a free particle with mass m∗ .
The effect of the force due to the periodic lattice is incorporated into the definition of the effective mass,
under certain conditions the effective mass can even be negative such that the electron accelerates in the
“wrong” direction. Newton’s Laws are still being obeyed, but the whole system must be considered. The
effective mass gives a more convenient approach.
In general semiconductors are anisotropic, and the effective mass is a tensor.
1 ∂ 2 εnk
1
= 2
∗
mij
~ ∂~ki~kj
8 This
result was derived in lectures, see also Kittel chapter 8
39
(2.104)
Motion in a uniform electric field
In an external electric field the semiclassical equation of motion is
~
d~k
dt
~k(t)
~
= −eE
~
eEt
= ~k(0) −
~
(2.105)
i.e. the wavevector changes with a rate proportional to the applied field, but independent of ~k. In a full
band the electrons move out of the first Brillouin zone in one direction and in from the other side. There
is no net current flow and as a result the material is an insulator. In real space the electrons are oscillating
about fixed points, known as Bloch oscillations. In reality this is very difficult to observe experimentally due
to scattering, which we have temporarily ignored.
2.5.5
Holes
The current density from electrons in a band (isotropic, and in 3D) is given by
~j = (−e)2
Z
~vk
d3~k
(2π)3
(2.106)
occupied
where the factor of two arises from spin degeneracy (two spins per k-point). If all the states in the band are
full the net current will be ~j = 0. So let’s consider an almost full band
Z
~vk
~j = (−e) 2
d3~k
(2π)3
occupied
Z
Z
~vk
~vk
3~
d
d3~k
= (−e) 2
k
−
(−e)
2
3
(2π)
(2π)3
unocc.
f ull
Z
~vk
d3~k
(2.107)
= 0 + (+e) 2
(2π)3
unocc.
i.e. we can consider the current to be carried not by the electrons, but by the unoccupied states known as
holes. This leads to two sources of current
R
~
vk
3~
~j =
(−e) 2
electrons in the conduction band
(2π)3 d k
unocc.
~j =
0 + (+e) 2
R
unocc.
~
vk
(2π)3
d3~k
holes in the valence band
(2.108)
Properties of holes
Let’s look at the properties of a single hole in the valence band, compared to those of the corresponding
electron.9
Charge ofPa hole, qh : Consider
P taking one electron out of a full band. The total charge is given by
Q = full (−e) − (−e) = full (−e) + qh . So
qh = e
9 See
Kittel chapter 8, p.206-208
40
(2.109)
Energy of the hole state, εh : A similar argument gives
εh = −εe
(2.110)
k~h = −~ke
(2.111)
Wavevector of the hole state, ~kh
Velocity of the hole state, ~vh Consider the current due to the full band less one electron:
X
X
~j =
(−e)~v − (−e)~ve =
(−e)~v + (+e)~vh
full
full
As a result we see that
~vh = ~ve
(2.112)
Effective mass, m∗h since ~vh = ~ve but k~h = −~ke then we see that10
m∗h = −m∗e
(2.113)
One way to look at this is to consider a hole band constructed by inverting the hole band
Figure 2.19: Inverted hole band, Kittel p.208
In short the hole behaves like a positively charged particle, and thus provides an explanation for the
anomalous positive values of the Hall coefficient.
2.6
Conclusions
First we looked at the free electron model where electrons are independent of one another and do not interact
with the background lattice. We saw that this couldn’t account for experimental results such as positive
values of the Hall coefficient for some metals. The next step accounted for the atomic lattice through the
introduction of a periodic potential, resulting in Bloch states, the reduced zone scheme and band gaps. This
accounted for crystals behaving as metals, semi-metals, semi-conductors and insulators. The implications of
10 Note
that at the top of the valence band me is negative so the hole effective mass is positive
41
the Bloch states led to the introduction of important concepts such as the crystal momentum, group velocity,
and the effective mass. The effective mass tensor is dependent on the state, for physical quantities average
values of the effective mass can be used. For example the dc conductivity is due to electrons near the Fermi
energy, defining an average effective mass the conductivity can be written
σ=
ne2 τ
m∗
(2.114)
Finally we introduced the concept of a hole; an empty electron state which follows the equations of motion
of a positively charged particle.
42
Chapter 3
Intrinsic and Extrinsic
Semiconductors
3.1
Introduction
We have shown that a periodic lattice causes band gaps and briefly discussed how this can result in metals,
semiconductors and insulators. In this section we will look in more detail at the physical properties of
semiconductors whose properties are both interesting and technologically very important.
Crystal
Copper
Silicon
Diamond
Resistivity at room temperature (Ωm)
10−8
103
1014
Clearly the conductivity of silicon whose band gap is (Eg = 1.1eV ) is intermediate between that of metallic
copper and insulating diamond (Eg = 5eV ). As we shall see, conduction in semiconductors is dominated by
carrier concentrations.
3.1.1
Direct and indirect band gaps
Typical band gaps of semiconductors are in the range 0.2 − 3eV . For comparison light with energy 1eV has
a wavelength of 1200nm (i.e. in the infrared), 2eV is 600nm (orange), and blue light (475nm) is 2.6eV . In
other words optical absorption is a good way of probing the band gap of semiconductors. Experimentally
two types of result are observed:
Direct band gaps
The top of the valence band is directly below the bottom of the conduction band, the threshold for optical
absorption is ~ω = Eg .1
Indirect band gaps
The bottom of the conduction band is separated from the top of the conduction band in k space. The optical
absorption threshold is still at ~ω = Eg , but it requires the simultaneous absorption of a phonon which is
correspondingly much less probable. A second threshold is reached at the direct gap.
1 recall
that the relative crystal momentum of a photon is negligible so the transition is vertical
43
Figure 3.1: Schematics of direct and indirect band gaps. (Kittel Ch. 8 p.202)
44
Figure 3.2: Optical absorption of InSb (Kittel Ch. 8 p.203)
3.1.2
Examples
Crystal
Si
Ge
GaAs
InSb
group
IV
IV
III V
III V
type
i
i
d
d
Band gap at 300K (eV )
1.1
0.7
1.4
0.2
Figure 3.3: (Left) structure of silicon, (Right) structure of gallium arsenide, NanoHub Crystal viewer tool.
A second important distinction to make is between intrinsic and extrinsic semiconductors. The properties
of a perfect crystal of a pure element or perfectly stoichiometric compound are called intrinsic properties,
whereas the influence of added impurities or defects give rise to extrinsic properties.2
2 Definition
from Myers Ch. 10 p.275
45
3.2
Intrinsic semiconductors
For intrinsic semiconductors the carrier concentrations are determined by thermal excitation of carriers
across the band gap. At T = 0K the valence band will be full, and the conduction band empty. The
chemical potential is midway through the band gap, and there would be no conductance. As the temperature
increases electrons are thermally excited across the band gap, as a result the concentration of electrons in
the conduction band (n = p) is equal to the concentration of holes in the valence band. These electrons and
holes are free to move through the crystal and can carry an electric current and so are known as carriers.
Figure 3.4: Sketch of density of states and Fermi function, Needs 7.3
3.2.1
Electron concentration
The electron density is given by
Z∞
1
no.electrons
=
n=
V
V
dE Nc (E) f (E)
(3.1)
Ec
Recall from 2.11 that the density of states for electrons in the conduction band is
V
Nc (E) = 2
π
2m∗c
~2
3/2 p
E − Ec
(3.2)
where we have used the effective mass for electrons in the conduction band, m∗c . We introduced the Fermi
function in 2.8. Here we will make the assumption that the semiconductor is non-degenerate. We will explain
in more detail what this means when we discuss extrinsic semiconductors, but for the moment we assume
that kB T Eg , and that µ is away from the band gap edges.
f (E) =
1
' e−(E−µ)/kB T
e(E−µ)/kB T + 1
(3.3)
Putting these together we find
n =
Z∞
1
V
dE Nc (E) f (E)
Ec
=
=
1
π2
2m∗c
~2
n0 (T )e
3/2 Z∞
dE
p
E − Ec e−(E−µ)/kB T
Ec
−(Ec −µ)/kB T
46
(3.4)
where we have defined
1
n0 (T ) =
2π 2
2m∗c
~2
3/2 Z∞
dE 0
√
E 0 e−E
0
/kB T
(3.5)
0
We can simplify this using the relation
Z∞
√
√
−x
xe
π
2
dx =
0
giving
n = n0 (T )e−(Ec −µ)/kB T
n0 (T ) = 2
m∗c kB T
2π~2
3/2
(3.6)
This defines the electron concentration as a function of the chemical potential for non-degenerate semiconductors. Note that the exponential will dominate the temperature dependence.
3.2.2
Hole concentration
We can construct a similar argument to calculate the hole density in the valence band.
ZEv
no.”missing” − e
1
p=
=
V
V
dE Nv (E) (1 − f (E))
(3.7)
−∞
The density of states in the valence band is
Nv (E) =
V
π2
2m∗v
~2
3/2 p
Ev − E
(3.8)
Again we make the assumption that the semiconductor is non-degenerate i.e. that kB T Eg , and that µ is
away from the band gap edges.
1 − f (E) = 1 −
1
e(E−µ)/kB T
+1
=
e(E−µ)/kB T
' e(E−µ)/kB T
+1
e(E−µ)/kB T
(3.9)
Putting these together we find
p
=
1
V
ZEv
dE Nv (E) f (E)
∞
=
1
2π 2
2m∗v
~2
3/2 ZEv
dE
p
Ev − E e(E−µ)/kB T
−∞
(Ev −µ)/kB T
= p0 (T )e
(3.10)
simplifying as above we find
p = p0 (T )e(Ev −µ)/kB T
47
p0 (T ) = 2
m∗v kB T
2π~2
3/2
(3.11)
This defines the hole concentration as a function of the chemical potential for non-degenerate semiconductors.
3.2.3
Mass action law
If we look at the product of the electron and hole concentrations
np
= n0 (T )e−(Ec −µ)/kB T p0 (T )e(Ev −µ)/kB T
= n0 (T ) p0 (T ) e(Ev −Ec )/kB T
(3.12)
This gives the ”Mass action law”
np = n0 (T ) p0 (T ) e−Eg /kB T
(3.13)
which is true in general for non-degenerate semiconductors. For the case of intrinsic semiconductors we have
the added constraint that ni = pi so that
p
√
ni = pi = np =
n0 (T ) p0 (T )e−Eg /2kB T
3/2
kB T
3/4
(m∗c m∗v ) e−Eg /2kB T
(3.14)
= 2
2π~2
This defines the intrinsic carrier concentration as a function of temperature for a nondegenerate semiconductor.
3.2.4
Intrinsic carrier concentrations
For intrinsic semiconductors the mass action law applies, and n = p. At T=300K, and for m∗ = m then
n0 = p0 = 2.5 × 1025 m−3 . So we can estimate the carrier concentration for a semiconductor of band gap
1eV at room temperature. n ' 2.5 × 1025 ∗ e−20 ' 1017 m−3 . Using the fact that n = p it is easy to calculate
the chemical potential as a function of temperature
∗
Eg
mh
3
+ kB T ln
(3.15)
µ=
2
4
m∗e
i.e. close to the centre of the band gap. We can also calculate the carrier concentrations for a given intrinsic
material and compare them to metals and semi-metals (see over page).
Let’s quickly illustrate why these carrier concentrations are so important.
3.2.5
Conductivity
The conductivity of a semiconductor can be written as
σ=
ne2 τe
pe2 τh
+
m∗e
m∗h
(3.16)
which looks very similar to the equation for a metal.
However here the temperature dependence is dominated by the carrier concentrations as opposed to
the scattering mechanisms (as we found for metals). We can rewrite the above as
σ = (n(T ) e)µe + (p(T ) e)µh
48
(3.17)
Figure 3.5: Carrier concentrations at room temperature, Kittel p.198.
where the mobilities, µ, are
(
µe =
µh =
eτe
m∗
e
eτh
m∗
h
electron mobility
hole mobility
(3.18)
The conductivity is then determined by the mobility times the carrier concentration, the mobility will be
higher in materials with low rates of scattering. Typical mobilities at room temperature are
Crystal
Si
Ge
GaAs
InSb
electron mobility (cm2 (V s)−1 )
1350
3600
8000
800
49
hole mobility (cm2 (V s)−1 )
480
1800
300
450
50
3.3
Extrinsic semiconductors
The tremendous importance of semiconductors is reliant on the ability to control the carrier concentrations
through the incorporation of impurity atoms. By doping pure semiconductors with small concentrations of
impurities we can increase the electron concentration or the hole concentration independently. Let’s consider
silicon, a group IV element with 4 valence electrons (3s2 3p2 ). If some of the silicon atoms are replaced by a
group V element like phosphorus they will contribute an extra electron and so act as a donor. The potential
of the group V element is otherwise not too different from the silicon, so as long as the concentration of
phosphorus is low the conduction and valence bands will be basically unchanged, and the extra electron will
occupy a state near the bottom of the conduction band. If instead some of the silicon was replaced by a
group III atom such as boron they will contribute one fewer electron (i.e. a hole) and so act as an acceptor
and create an energy level for a hole near the top of the valence band.
III
B
Al
Ga
In
IV
C
Si
Ge
Sn
V
N
P
As
Sb
A donor atom will have an extra positive charge compared to the lattice, this will attract an electron.
Ignoring the lattice this looks very similar to a hydrogen atom (see figure over page).
Figure 3.6: Schematic of arsenic atom in germanium lattice (A&M p.577)
51
3.3.1
Hydrogenic levels: donor ionisation energies
Recall the Hamiltonian for a hydrogen atom
H=
p2
e2
−
2m 4π0 r
(3.19)
The ground state is symmetric with a characteristic size of the Bohr radius (a0 ) and energy of the Rydberg
Energy (E0 )
a0
=
E0
=
4π0 ~2
= 0.5Å
me2
e2
= 13.6eV
8π0 a0
(3.20)
(3.21)
In the semiconductor the Coulomb attraction is reduced (screened) by the electronic polarisation of the
medium, and of course the mass should be replaced by the effective mass. This results in a Hamiltonian
Hef f =
e2
p2
−
2m∗
4π0 r
(3.22)
This is valid for a donor with charge Z = +e. By analogy with the hydrogen atom we can see that the
ground state will have characteristic size ad and energy Ed
ad
=
Ed
=
4π0 ~2
m
= ∗ a0
∗
2
m e
m
e2
m∗ 1
E0
=
8π0 ad
m 2
(3.23)
(3.24)
Note that the ionisation energies depend only on the medium (i.e. the semiconductor) and not on the donor
atom (for donors with the same charge). Obviously a similar argument can be made for acceptor levels to
estimate the acceptor ionisation energy Ea .
Exercise 3.1 Find the donor energy for a Group V element in silicon. The effective mass for silicon is
small, m∗e ' 0.2m, and the dielectric constant is large: = 12. [You should find that Ed = 0.02 eV and
ad = 3 nm].
The simple approach just outline is correct to the order of magnitude, as a comparison between the result
of the exercise and the table below reveals. Importantly, Ed Eg for dopants [and similarly for acceptors],
so we have energy states as shown in Fig. 3.7 that are close to either the conduction band edge (for donors)
or valence band edge (for holes).
Note that the degeneracy of a band can influence the donor/acceptor energies – degeneracy here in the
sense of multiple levels at certain points in k-space, such as the heavy-hole and light-hole bands in GaAs.
Degenerate bands have several effective masses.3 The discrepancy between the predicted donor ionisation
energy for silicon and the observed can be largely accounted for by the degeneracy of the conduction band
of silicon so that the effective mass should be much higher.
Ed (meV)
Si
Ge
P
45
12
As
49
13
Sb
39
10
Ea (meV)
Si
Ge
B
45
10.4
Al
57
10.2
Ga
65
10.8
Tables of (left) donor and (right) acceptor ionisation energies in meV, values from Kittel p.224-5
3 See
for example Kittel, Singleton or Ashcroft and Mermin for more details.
52
Figure 3.7: Myers p.286
Occupation of donor and acceptor levels
Note that the ionisation energies are similar to thermal energies at room temperature; thermal ionization
of the donors is an important contribution to the conductivity at room temperature. By comparison Eg kB T300K so thermal excitation across the band gap at room temperature is much less likely.
Ignoring electron-electron interactions, the donor level could be empty, singly occupied by an electron of
either spin, or doubly occupied. However Coulomb repulsion between the electrons energetically prohibits the
doubly occupied state. The average occupation can then be calculated using the grand canonical ensemble
P
−(Ei −Ni µ)/kB T
i Ni e
fd = P
(3.25)
−(Ei −Ni µ)/kB T
ie
with the corresponding states
i
1
2
3
Ni
0
1 (spin up)
1 (spin down)
Ei
0
εd
εd
which gives
fd =
1
1 (εd −µ)/kB T
2e
+1
(3.26)
The unusual factor of 1/2 comes from the prohibition of the doubly occupied state. The occupation of the
acceptor state can be similarly calculated. For both the situation is further complicated by the degeneracy
of the conduction and valence bands which changes the 1/2 to 1/g where g is the degeneracy.
3.3.2
Extrinsic carrier concentrations
Many physical properties are dominated by the electron concentration in the conduction band (n) and the
hole concentration in the valence band (p). These come from ionization of the donors and acceptors, and
53
excitations from the valence to the conduction band. As long as the semiconductor is nondegenerate the
mass action law 3.13 will hold independent of whether impurities are present. The effect of the impurities
is to alter the chemical potential, they remove the constraint n = p. If the number of ionised donors (Nd+ )
is much greater than the number of ionised acceptors (Na− ), i.e. Nd Na (if they have comparable binding
energy), then n p and the material is called n-type: the majority carriers are electrons. If p n then the
material is called p-type: the majority carriers are holes. The mass action law controls the product np; by
controlling the impurities we can control the total number of carriers n + p, with the minimum being the
undoped (intrinsic) case where n = p.
In reality it is the values of n and p that we want to know. In a practical situation we may know or be
able to measure Nd , Na , εd , εa , m∗e , m∗h , gd , ga , & T . The unknowns are Nd+ , Na− , n, p, & µ, and these are
determined by the 4 equations we have already derived
3/2
∗
mc kB T
e−(Ec −µ)/kB T
n = 2
2π~2
∗
3/2
m v kB T
p = 2
e(Ev −µ)/kB T
2π~2
Nd − Nd+
Nd
=
Na − Na−
Na
=
1
1 (εd −µ)/kB T
gd e
+1
1
1 −(εa −µ)/kB T
ga e
+1
(3.27)
along with the final condition of overall charge neutrality
Nd+ − Na− + n − p = 0
(3.28)
These can be solved numerically, but it is also worth looking at some limiting cases at specific temperatures.
We will consider systems with Nd Na (and we will neglect Na ). In Fig. 3.8 the temperature dependence
of n is shown for lightly-doped silicon, and reference to this figure should be made with respect to the three
following regimes/ranges.
Extrinsic regime/freeze-out range (low temperature)
At T = 0 donors are all singly occupied, the chemical potential is between the donor levels and the conduction
band, and n = p = 0. As the temperature increases the donors become ionised, it can be shown in this
regime (0 < kB T Ed ) that the electron concentration is given by4
p
n = n0 Nd e−Ed /2kB T
(3.29)
This equation predicts that ln n ∝ −1/kB T , as seen in Fig. 3.8.
The excitation of electrons into the conduction band only requires thermal excitation over the relatively
low barrier of Ed . In this regime the chemical potential is close to the majority carrier level.
Exhaustion regime /saturation range (intermediate temperatures)
As the temperature becomes greater than the donor ionisation energy the system enters the exhaustion
regime (saturation range in Fig. 3.8). The donors are ionised but the probability of thermal excitation over
the full band gap is still small. In this case
n ' Nd
(3.30)
and the majority (electron) carrier concentration is roughly independent of temperature. Recall that the
concentration of electrons is given by
n = n0 (T )e−(Ec −µ)/kB T
4 See
Myers for a derivation
54
In the exhaustion regime we have
µ = Ec − kB T ln
N0
Nd
(3.31)
i.e. the chemical potential is in the upper half of the band gap, it will move downwards towards the centre
of the gap as the minority carrier concentration increases in line with the mass action law.
Intrinsic regime/range (high temperature)
As the temperature increases further the intrinsic carrier concentration (as defined by the mass action law
3.13) exceeds the donor concentration (ni Nd ) and the electron concentration becomes
p
(3.32)
n = n0 (T ) p0 (T )e−Eg /2kB T
in this regime n ' p ' ni Nd and the chemical
found before. i.e.
Eg
µ=
+
2
potential will be close to the middle of the band gap as
∗
3
mh
(3.33)
kB T ln
4
m∗e
Figure 3.8: ln(n) vs 1000/T , From Singleton p61.
Degenerate and nondegenerate semiconductors
All our analysis so far has been based on the assumption that the semiconductor is non-degenerate. We
stated that this required kB T Eg , and that µ is away from the band gap edges.
f (E) =
1
' e−(E−µ)/kB T
e(E−µ)/kB T + 1
55
(3.34)
A degenerate semiconductor has high carrier concentrations, although still lower than in conventional metals.
Usually this will be due to high dopant concentrations. If the dopant concentrations are too high they form
an impurity band and the material no longer behaves as a semiconductor (e.g. the conductivity no longer
increases with temperature). We estimated the characteristic size of the donor states earlier, from this the
degenerate dopant concentrations can be predicted.
Exercise 3.2 Predict the dopant concentration at which an impurity band is formed in Si.
3.3.3
Temperature and dopant dependence of n, p, µ & σ
You should login to NanoHub and experiment with the simulation ‘Carrier statistics lab’ to see the effects of
doping and temperature for yourself. However here we shall sketch out some of the dependencies. First let’s
consider the effect of dopant concentration on the chemical potential at fixed temperature. We will consider
again an n-type semiconductor with donor concentrations Nd and donor energy levels Ed .
Figure 3.9: Sketch of µ vs Nd − Na
Figure 3.10: Sketch of µ vs T , from Myers p.289.
3.3.4
Conductivity [revisited]
Recall our previous discussion of the conductivity of materials with electrons and holes where we found the
conductivity to be (3.17)
σ = (n(T ) e)µe + (p(T ) e)µh
(3.35)
The mobilities are only weakly dependent on temperature in comparison to the carrier concentrations. Recall
that the mobilities are
(
e
µe = eτ
electron mobility
m∗
e
(3.36)
eτh
µh = m∗
hole mobility
h
and their temperature dependence is dominated by the scattering times τe and τh . The main scattering
mechanisms are scattering by phonons which increase with increasing temperature (τph ∝ T −3/2 ), and
scattering by ionized impurities which decreases with temperature so that (τi ∝ T 3/2 ). To find the total
scattering time we add the inverse scattering rates.
56
For intrinsic materials, or materials demonstrating intrinsic behaviour, the conductivity will increase
exponentially with temperature in line with the carrier concentration and hence gives a measure of the band
gap. Doped materials will show a temperature dependence dominated by the carrier concentration, but with
some contribution from the temperature dependence of the mobility.
3.3.5
The Hall effect [revisited]
Let’s consider a heavily doped sample in the exhaustion regime, where p n and can be ignored. A Hall
effect measurement will give us
1
RH =
(3.37)
nq
i.e. the type of the carriers (RH negative implying electrons, as q = −e), and the carrier concentration.
Measuring also the conductivity gives us
σ = n eµe
(3.38)
so that combining with the Hall measurement we can extract the mobility. Under the assumption that
the transport properties are dominated by the impurities we then have experimentally found the dominant
carrier type (electrons), the donor concentration (n), and the mobility (µe ).
Now consider the more complicated case where both electrons and holes are contributing to the transport.
The Hall coefficient at moderate magnetic fields then becomes (you should derive this yourselves)5
RH =
p µ2h − n µ2e
e(p µh + n µe )2
(3.39)
where e is the magnitude of the electron charge. In the intrinsic regime, where n = p >> Nd or Na then the
Hall coefficient is dominated by the electrons, as µe >> µh . A question in the Problem Sheets explores the
Hall coefficient for doped InSb, where both holes and electrons contribute.
The conductivity is now
σ = (n e)µe + (p e)µh
(3.40)
implying that care must be taken in extracting carrier concentrations, mobilities etc. from Hall effect and
conductivity data.
3.3.6
Carrier dynamics
In considering the conductivity we looked at the drift of charge carriers induced by an electric field, i.e.
situations in which the carrier concentrations are effectively uniform in space. When looking at device
structures we will see that this is not always the case. When the concentration is not constant diffusion
of charge carriers must be included as well as drift. Other important effects include carrier generation and
recombination which may create or destroy charge carriers. For example photoexcitation of charge carriers
across the band gap will generate carriers and create a system out of thermal equilibrium. We will consider
these effects in more detail when we look at semiconductor devices in Chapter 5. Non-equilibrium carrier
dynamics after photoexcitation will be explored in detail in Chapter 6. Beforehand, however, let’s look at
the light-matter interaction in semiconductors in more detail.
5 If
you get stuck then it is in Hook and Hall section 5.5.2, or Singleton Section 10.2.2.
57
58