Gavin W Morley
Department of Physics
University of Warwick
Diamond Science & Technology
Centre for Doctoral Training, MSc course
Module 2 – Properties and Characterization of Materials
Module 2 – (PX904)
Lecture 4 – Electronic properties:
Lecture 4 – Building a crystal from atoms
2
Module 2 – Properties and Characterization of Materials
- Lecture
4 – Building a crystal from atoms
Overview
Diamond properties
3
Module 2 – Properties and Characterization of Materials
- Lecture
4 – Building a crystal from atoms
Overview
Lectures
Lecturer
1-3
Philip Martineau
Crystallography
4-6
Gavin Morley
Electronic properties
7-8
Stephen Lynch
Optical
9
Gavin Morley
Electronic characterization
10
Richard Beanland
Electron microscopy
11-12
Claire Dancer
Mechanical
13-14
Martin Kuball
Thermal
15-16
Gavin Morley
Magnetic
4
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Lectures
4
Electronic structure:
- Atomic physics
- Building crystals from atoms
- Tight binding model
- Drude model of metals
5 and 6
- Sommerfeld model of metals
Bandstructure:
- Bloch’s theorem
- Nearly free electron model
- Semiconductors and insulators
- Relative permittivity
- Intrinsic and extrinsic conductivity
- Metal-insulator transition
- Mobility
5
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
What explains the Periodic Table?
Dmitri
Mendeleev
(1834 – 1907)
6
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
What explains the periodicity of the
Periodic Table?
a) The Schrödinger equation
b) The Schrödinger equation + the Coulomb potential
c) The Schrödinger equation + the Coulomb potential +
electron spin
d) The Schrödinger equation + the Coulomb potential +
electron spin + the Pauli exclusion principle
e) What your viewers really want to hear about is how I’ve
improved public transport in London
Dmitri
Mendeleev
(1834 – 1907)
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
7
Classical physics fails to explain atoms
-
-
-
8
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Electrons can behave like waves
Electron
gun
Electron
detector
crystal
Louis de
Broglie
(1892 – 1987)
9
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Schrödinger’s equation
is a wave equation:
Boundary
Condition
Erwin
Schrödinger
(1887 – 1961)
Elastic band video from Acoustics Group,
University of Salford, Manchester
10
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Solve Schrödinger’s equation
for an electron in a box:
→ Discrete energy levels
Erwin
Schrödinger
(1887 – 1961)
Page 240, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
11
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Pauli’s exclusion principle:
Two electrons cannot occupy
the same quantum state
simultaneously
Wolfgang
Pauli (1900 –
1958)
Page 308, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
12
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Solve Schrödinger’s equation
for an electron in a box:
→ Discrete energy levels
Erwin
Schrödinger
(1887 – 1961)
Page 240, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
13
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Solve Schrödinger’s equation for electron in
Coulomb potential and include spin
n
1
l
0
0
1
0
ml
0
0
-1,0,+1
0
+½,-½
+½,-½
+½,-½
+½,-½
ms
2
+½,-½ +½,-½
3
1
2
-1,0,+1 -2,-1,0,+1,+2
Number of
degenerate
eignenfunctions
for each l
2
2
6
2
6
10
Subshell name
1s
2s
2p
3s
3p
3d
Page 241, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
14
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
What explains the Periodic Table?
Page 330, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
15
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Schematic of
subshell energy
levels:
The ionization
energy of atoms:
Pages 333-336, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
16
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Any questions so far?
17
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
An atom
Schematic drawing of wavefunction for an
electron on a hydrogen atom. Page 245, Kittel,
Introduction to Solid State Physics, Wiley 1996
Page 240, Eisberg
and Resnick,
Quantum Physics of
Atoms, Molecules,
Solids, Nuclei, and
Particles, Wiley 1985
18
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Two atoms
Page 240, Eisberg
and Resnick,
Quantum Physics of
Atoms, Molecules,
Solids, Nuclei, and
Particles, Wiley 1985
(a) Schematic drawing of wavefunctions for
electrons on two hydrogen atoms at large
separation. Page 245, Kittel, Introduction to Solid
State Physics, Wiley 1996
19
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Building a molecule from atoms
…a bond
Page 240, Eisberg
and Resnick,
Quantum Physics of
Atoms, Molecules,
Solids, Nuclei, and
Particles, Wiley 1985
(b) Ground state wavefunction at closer separation.
(c) Excited state wavefunction. Page 245, Kittel,
Introduction to Solid State Physics, Wiley 1996
Potentials
Atom
Molecule
Insulating crystal:
tight binding model
1
Potential
energy (V)
20
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
0
Schematics of the potential due to the ions in
the crystal, Page 3, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
21
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The tight-binding model
Atom
Molecule
Schematics of the potential due to the ions in
the crystal, Page 3, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
Insulating crystal
tight binding model
22
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The tight-binding model
Schematic of the formation of tight binding
bands as the spacing between atoms is
reduced. Page 35, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
The tight-binding model
Group IIA metal
e.g. magnesium
3p
3s
Energy
Group IA metal
e.g. sodium
Energy
23
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
3p
3s
2p
2p
Atomic separation
Atomic separation
Schematic of the formation of tight binding bands
as the spacing between atoms is increased. Page
36, Singleton
24
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Which of these elements is not in
Group IV of the periodic table?
a)C
b)Si
c) Ge
d)Sn
e)Pb
f) N
Dmitri
Mendeleev
(1834 – 1907)
25
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Group IV
Dmitri
Mendeleev
(1834 – 1907)
26
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
FCC with two atom basis
Diamond crystal structure. Page 37, Singleton,
Band Theory and Electronic Properties of
Solids, OUP 2001
27
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The tight-binding model: diamond
Carbon:
1s2 2s2 2p2
Schematic of the formation of sp3 hybrid
bonding states in diamond. Page 37, Singleton,
Band Theory and Electronic Properties of
Solids, OUP 2001
28
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The tight-binding model: diamond
Carbon:
1s2 2s2 2p2
Schematic of the formation of sp3 hybrid
bonding states in diamond. Page 37, Singleton,
Band Theory and Electronic Properties of
Solids, OUP 2001
29
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The tight-binding model: Group IV
Electron
energy
4
6
2
4
Element Eg (eV)
a (nm)
C
5.5
0.356
Si
1.1
0.543
Ge
1.0
0.566
Sn
metallic
0.646
Interatomic
spacing
Schematic of tight-binding band formation in the
group IV elements, Page 38, Singleton, Band Theory
and Electronic Properties of Solids, OUP 2001
30
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Many strong directional bonds
Coulomb forces with Pauli exclusion
Element
Single
bond
length
(Å)
Carbon
1.54
Silicon
2.34
Diamond has many strong bonds
Germanium
2.44
H-H bond is stronger than C-C, but you
can’t make a crystal out of H-H bonds
Tin
2.80
Lead
2.88
Low-Z atoms are smaller
(their electrons are closer to their
parent nucleus)
Closer atoms are more strongly bound
(less screening)
Jeremy K. Burdett, Chemical Bonding in Solids. New York:
Oxford University Press, 1995: 152.
J. J. Gilman, Why silicon is hard, Science 261, 1436 (1993)
F. Gao et al., Hardness of Covalent Crystals, Physical Review
Letters 91, 015502 (2003).
31
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Bond energy and cohesive energy
Bond
Bond energy
for diatomic
molecule at
298 K (kJ/mol)
[1]
Cohesive
energy of
crystal at
298 K at 1
atm (kJ/mol)
[2]
Bond energy
of crystal at
298 K at 1 atm
= cohesive
energy × ½
(kJ/mol)
H-H
436.002 ±0.004
O=O
498.340 ±0.2
N≡N
945.33 ±0.59
C-C
607 ±21 for C=C
715
357
Si-Si
450
225
Ge-Ge
376
188
[1] CRC Handbook, Strengths of Chemical Bonds, 57th
Edition, 1977
[2] C Kittel, Introduction to Solid State Physics, Wiley
1996, Chapter 3, Table 1
For diamond, see:
- L. A. Schmid,
Physical Review 92,
1373 (1953).
- B. Holland, H. S.
Greenside and M.
Schlüter, physica
status solidi (b) 126,
511 (1984).
- X. Jiang et al., Sci.
Rep. 3, 1877 (2013).
- H. Shin et al., The
Journal of Chemical
Physics 140, 114702
(2014).
32
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Many strong directional bonds
Smaller atoms get closer together
- can we make a crystal with stronger bonds?
- BN, BC2N and B are almost as hard as diamond
Jeremy K. Burdett, Chemical Bonding in Solids. New York:
Oxford University Press, 1995: 152.
J. J. Gilman, Why silicon is hard, Science 261, 1436 (1993)
F. Gao et al., Hardness of Covalent Crystals, Physical Review
Letters 91, 015502 (2003).
33
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Hardness (and brittleness)
Three things make a covalent crystal hard:
- High bond density (electronic density)
- Short bond length
- High degree of covalent bonding
See:
F. Gao et al., Hardness of Covalent Crystals,
Physical Review Letters 91, 015502 (2003)
See Claire
Dancer’s
lectures
(11 and 12
in this
module)
The covalent bonds in diamond are very directional, so the atoms do
not move out of the way if indented, unlike in a metal. Eventually, the
crystal must break (with broken bonds) rather than bend, i.e. it is brittle.
34
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Many strong directional bonds
Many strong directional bonds
hard
brittle
chemically inert
incompressible (i.e. high bulk modulus)
High speed of sound
See Claire
Dancer’s
lectures
(11 and 12
in this
module)
35
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Electronic Bandstructure of diamond
Mini-Summary:
- Atomic physics  bandstructure
…by assuming the electrons in crystals
are generally stuck in their atomic
potentials
- Metals next: we will assume that the
electrons are not stuck, and still get
bandstructure
W. Saslow, T. K. Bergstresser, and
Marvin L. Cohen, Physical Review
Letters 16, 354 (1966)
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
36
PTFE (Teflon)
 > 1018 -cm
(room temperature)
Superconductors  ~ 0
silicon
 ~ 104 -cm
(room
temperature)
Pure metal
 ~ 10-10 -cm
(1 K)
Tin  ~ 10-5 -cm
(room temperature)
diamond  ~ 1016 -cm
(room temperature)
10-10
1
1010
1020
Resistivity (ohm-cm)
37
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Bandstructure
Energy
Eg
Metal
Semiconductor
Insulator
Schematic electron occupancy for allowed
energy bands. See page 174, Kittel, Introduction
to Solid State Physics, Wiley 1996
38
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
What explains the Periodic Table?
Dmitri
Mendeleev
(1834 – 1907)
39
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Which is the most advanced model of metals
in the list below?
a)
b)
c)
d)
e)
f)
Drude model
Sommerfeld model
Nearly-free electron model
Tight-binding model
c) and d) are equally advanced
Lady Gaga
Paco Rabanne
Dolce & Gabbana
40
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
1) Most elements are metals,
particularly those on the left
of the periodic table
2) Good conductors of
electricity & heat
3) Tend to form in crystal
structures with at least 8
nearest neighbours (FCC,
HCP, BCC)
4) Malleable
Schematic model of a crystal of sodium
metal. Page 142, Kittel, Introduction to
Solid State Physics, Wiley 1996
41
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The Drude Model:
1) Gas of electrons
2) Electrons sometimes collide
with an atomic core
3) All other interactions ignored
Paul Drude
(1863 –1906)
42
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
The Drude Model:
1) Gas of electrons
2) Electrons sometimes collide
with an atomic core
3) All other interactions ignored
4) Electrons obey the
Schrödinger equation and
the Pauli exclusion principle
Arnold Sommerfeld
(1868 – 1951)
43
Module 2 – Properties and Characterization of Materials
- Lecture 4 – Building a crystal from atoms
Lectures
4
Electronic structure:
- Atomic physics
- Building crystals from atoms
- Tight binding model
- Drude model of metals
5 and 6
- Sommerfeld model of metals
Bandstructure:
- Bloch’s theorem
- Nearly free electron model
- Semiconductors and insulators
- Relative permittivity
- Intrinsic and extrinsic conductivity
- Metal-insulator transition
- Mobility