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)
Lectures 5 and 6 – Electronic properties:
Lectures 5 and 6 – Bandstructure of crystals
2
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
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
3
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
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
4
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
The Drude Model:
1) Gas of electrons
2) Electrons sometimes collide
with an atomic core
3) All other interactions ignored
Paul Drude
(1863 –1906)
5
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
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)
6
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
The Drude Model
A map of states in k-space, see also page
173, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
The Drude Model
1
Potential
energy (V)
7
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
0
Drude-Sommerfeld potential
Schematics of the potential due to the ions in
the crystal, Page 3, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
8
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
The Drude Model
Dispersion relation for a free electron.
Page 177, Kittel, Introduction to Solid
State Physics, Wiley 1996
The Drude Model:
The Drude Model
Distribution
functions for a
typical metal at
room temperature,
Page 10, Singleton,
Band Theory and
Electronic
Properties of
Solids, OUP 2001
vs
Energy
Number of electrons
9
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
the Sommerfeld
model
fFD
Energy
10
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
the Sommerfeld
model
Zero
temperature
T=0
Finite
temperature
T << EF/kB
Fermi-Dirac distribution function, Page 9,
Singleton, Band Theory and Electronic Properties
of Solids, OUP 2001
11
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
the Sommerfeld
model
At any given moment, roughly how quickly
does one of the fast electrons travel around in
a typical metal at low temperatures?
a)
b)
c)
d)
e)
0 mm s-1
1 mm s-1
7 million mph (1% of c)
200 million mph (30% of c)
Officer, I’m so sorry: I’m afraid I wasn’t
looking at the speedometer
12
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
the Sommerfeld
model
Fermi-Dirac distribution function, Pages 8&9,
Singleton, Band Theory and Electronic Properties
of Solids, OUP 2001
13
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
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)
Explains temperature dependence
and magnitude of:
a) Electronic specific heat
b) Thermal conductivity (approx.)
c) Electrical conductivity (approx.)
But does not explain:
a) Insulators & semiconductors
b) Thermopower
c) Magnetoresistence
d) Hall Effect
Beyond the Sommerfeld Model:
1) Gas of electrons
2) Electrons are in a periodic
potential due to the ions
3) Electron-electron
interactions ignored
4) Electrons obey the
Schrödinger equation and
the Pauli exclusion principle
1
Potential
energy (V)
14
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
0
Drude-Sommerfeld potential
Schematics of the potential due to the ions in
the crystal, Page 3, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
real ionic potential
Bloch’s theorem
“Consider a one-electron
Hamiltonian with a periodic
potential:
1
The eigenstates can be
chosen to be a plane wave
times a function with the
periodicity of the lattice.”
Potential
energy (V)
15
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
0
Drude-Sommerfeld potential
Bloch’s theorem, Page 16, Singleton, Band Theory
and Electronic Properties of Solids, OUP 2001
real ionic potential
16
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
The nearly-free electron model
Drude-Sommerfeld potential
weak ionic potential
17
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
The nearly-free electron model
Nearly free electron has bands
Dispersion relation for free and nearly-free
electrons. Page 177, Kittel, Introduction to
Solid State Physics, Wiley 1996
18
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
The nearly-free electron model
First Brillouin zone
Nearly free electron has bands
Dispersion relation for free and nearly-free
electrons. Page 177, Kittel, Introduction to
Solid State Physics, Wiley 1996
19
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Representing bands
Three energy bands
of a linear lattice.
Page 238, Kittel,
Introduction to Solid
State Physics, Wiley
1996
20
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Diamond model
From the following list,
which is the best model of
diamond?
a) Drude model
b) Sommerfeld model
c) Nearly-free electron
model
d) Tight binding model
21
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
22
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electronic Bandstructure of diamond
Kittel page 238
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
23
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electronic Bandstructure of diamond
Heavy-hole band
Light-hole band
Effective mass derivation, Page 42, Singleton,
Band Theory and Electronic Properties of
Solids, OUP 2001
24
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electronic Bandstructure of diamond
Indirect bandgap
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
25
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
26
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
27
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Bandstructure of Si & diamond
Based on M. Cardona and F. Pollack,
Physical Review 142, 530 (1966).)
Bandstructure of Si, page 50, Singleton,
Band Theory and Electronic Properties of
Solids, OUP 2001
28
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Any questions?
29
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Effect of an electric field
Relative permittivity. Page 271, Kittel, Introduction
to Solid State Physics, Wiley 1996
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
30
Effect of an electric field
- capacitor
-
+
-
+
-
-
-
-
+
+
+
-
-
-
+
+
+
+
Dielectric properties of insulators, page
533, Ashcroft and Mermin, Solid State
Physics, Harcourt 1976.
31
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Effect of an electric field
- Coulomb field
Page 240, Eisberg and Resnick, Quantum
Physics of Atoms, Molecules, Solids,
Nuclei, and Particles, Wiley 1985
32
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Dielectric permittivity
- static
See J. C. Phillips, Physical Review Letters 20, 550 (1968)
Dielectric constants, page 553, Ashcroft
and Mermin, Solid State Physics, Harcourt
1976.
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
33
Dielectric permittivity
- frequency-dependent
-
+
-
+
-
-
-
-
+
+
+
-
-
-
+
+
+
+
→ Dielectric loss
Dielectric properties of insulators, page
533, Ashcroft and Mermin, Solid State
Physics, Harcourt 1976.
34
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Temperature dependence
Energy
Eg
Metal
Intrinsic
Semiconductor
at room
temperature
Insulator
35
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Cooling semiconductors down
Energy
Eg
Metal
Intrinsic
Semiconductor
at room
temperature
Intrinsic
Semiconductor
at low
temperature
Insulator
36
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Cooling semiconductors down
Energy
Intrinsic
for kBT > Eg
Extrinsic
for Eg > kBT > donor binding energy
37
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Intrinsic charge carriers
Energy
Intrinsic
holes
Semiconductor at
room temperature
38
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Intrinsic charge carriers
Energy
Intrinsic
Eg
Semiconductor at
room temperature
Page 56, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
39
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Intrinsic charge carriers
Ge:
Eg = 0.74 eV
Si:
Eg = 1.17 eV
GaAs: Eg = 1.52 eV
Calculated intrinsic carrier densities versus
temperature. Page 59, Singleton, Band Theory
and Electronic Properties of Solids, OUP 2001
40
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Extrinsic charge carriers
Energy
Intrinsic
Extrinsic (n-type)
donor impurities
Semiconductor
at room
temperature
Semiconductor
at room
temperature
Extrinsic (p-type)
acceptor impurities
Semiconductor
at room
temperature
41
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Extrinsic charge carriers
Si:P
binding energy = 46 meV
Page 240, Eisberg
and Resnick,
Quantum Physics of
Atoms, Molecules,
Solids, Nuclei, and
Particles, Wiley 1985
42
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Extrinsic charge carriers
20 ppb
Dopants in diamond have larger
binding energies so are not
ionised at room temperature
Temperature dependence of the electron density in
silicon with a net donor density ND-NA=1015 cm-3.
Page 61, Singleton
43
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Donor Qubits in Silicon
Picture by Manuel Voegtli
44
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electron Qubits in diamond
Picture by Alan Stonebraker
45
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Why is diamond an insulator?
Electron energy
4
6
2
4
Interatomic spacing
46
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Solve Schrödinger’s equation
for an electron in a box:
Binding energies
for phosphorous
donors:
Silicon: 46 meV
Diamond: 500 meV
-
Page 240, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
47
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Why is diamond an insulator rather
than a semiconductor?
a) Wide band-gap means no intrinsic conductivity,
deep dopants mean no extrinsic conductivity
48
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
But doped diamond and silicon can
be metals too
Extrinsic
conductivity
Semiconductor
at room
temperature
Semiconductor
at low
temperature
49
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Doped silicon can be a metal
Observed “zero
temperature” conductivity
versus donor
concentration n for Si:P,
after T F Rosenbaum et
al. Page 285, Kittel,
Introduction to Solid State
Physics, Wiley 1996
50
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Doped diamond can be a metal
Charge transport in heavily Bdoped polycrystalline diamond
films, M. Werner et al Applied
Physics Letters 64, 595 (1994)
Sample A has 8 x 1021 cm-3 boron
51
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Electrical conductivity of semiconductors. Page
127, Singleton, Band Theory and Electronic
Properties of Solids, OUP 2001
52
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Carrier mobilities at room temperature in
cm2/Vs. Page 221, Kittel, Introduction to Solid
State Physics, Wiley 1996
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
53
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)
54
Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
Diamond properties