Gavin W Morley Department of Physics University of Warwick
Diamond Science & TechnologyCentre for Doctoral Training, MSc course Module 2 – Properties and Characterization of MaterialsModule 2 – (PX904)Lectures 5 and 6 – Electronic properties: Lectures 5 and 6 – Bandstructure of crystals
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals2
Lectures
4 Electronic structure: - Atomic physics - Building crystals from atoms - Tight binding model - Drude model of metals
5 and 6 - Sommerfeld model of metalsBandstructure: - Bloch’s theorem - Nearly free electron model - Semiconductors and insulators - Relative permittivity - Intrinsic and extrinsic conductivity - Metal-insulator transition - Mobility
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals3
Schematic model of a crystal of sodium metal. Page 142, Kittel, Introduction to Solid State Physics, Wiley 1996
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
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals4
The Drude Model:1)Gas of electrons2)Electrons sometimes collide with an atomic core3)All other interactions ignored
Paul Drude (1863 –1906)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals5
The Drude Model:1)Gas of electrons2)Electrons sometimes collide with an atomic core3)All other interactions ignored4)Electrons obey the Schrödinger equation and the Pauli exclusion principle
Sommerfeld
Arnold Sommerfeld (1868 – 1951)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals6
The Drude ModelSommerfeld
A map of states in k-space, see also page 173, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals7
The Drude ModelSommerfeld
Drude-Sommerfeld potentialSchematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
0
1
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals8
The Drude ModelSommerfeld
Dispersion relation for a free electron. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals9
The Drude Model
vs
fFD
Energy
Distribution functions for a typical metal at room temperature, Page 10, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
The Drude Model:
the Sommerfeld model
Energy
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals10
Fermi-Dirac distribution function, Page 9, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
the Sommerfeld model
Zero temperature
T = 0
Finite temperature
T << EF/kB
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals11
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)0 mm s-1
b)1 mm s-1
c)7 million mph (1% of c)d)200 million mph (30% of c)e)Officer, I’m so sorry: I’m afraid I wasn’t looking at the speedometer
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals12
Fermi-Dirac distribution function, Pages 8&9, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
the Sommerfeld model
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals13
The Drude Model:1)Gas of electrons2)Electrons sometimes collide with an atomic core3)All other interactions ignored4)Electrons obey the Schrödinger equation and the Pauli exclusion principle
SommerfeldExplains temperature dependence and magnitude of: a)Electronic specific heat b)Thermal conductivity (approx.) c)Electrical conductivity (approx.)
But does not explain: a)Insulators & semiconductorsb)Thermopowerc)Magnetoresistenced)Hall EffectArnold Sommerfeld
(1868 – 1951)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals14
Beyond the Sommerfeld Model:1)Gas of electrons2)Electrons are in a periodic potential due to the ions3)Electron-electron interactions ignored4)Electrons obey the Schrödinger equation and the Pauli exclusion principle
Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Drude-Sommerfeld potential real ionic potential
0
1
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals15
Bloch’s theorem, Page 16, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Drude-Sommerfeld potential real ionic potential
0
1
“Consider a one-electron Hamiltonian with a periodic potential:The eigenstates can be chosen to be a plane wave times a function with the periodicity of the lattice.”
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals16
The nearly-free electron model
Drude-Sommerfeld potential weak ionic potential
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals17
The nearly-free electron model
Dispersion relation for free and nearly-free electrons. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996
Nearly free electron has bands
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals18
The nearly-free electron model
Dispersion relation for free and nearly-free electrons. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996
Nearly free electron has bands
First Brillouin zone
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals19
Representing bands
Three energy bands of a linear lattice. Page 238, Kittel, Introduction to Solid State Physics, Wiley 1996
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals20
Diamond model
From the following list, which is the best model of diamond?a)Drude modelb)Sommerfeld modelc)Nearly-free electron modeld)Tight binding model
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals21
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals22
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)
Kittel page 238
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals23
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
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals24
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)
Indirect bandgap
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals25
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals26
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals27
Bandstructure of Si & diamond
Bandstructure of Si, page 50, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Based on M. Cardona and F. Pollack, Physical Review 142, 530 (1966).)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals28
Any questions?
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals29
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 crystals30
Effect of an electric field- capacitor
- - - - - -
+ + + + + +
+
-
+
-
+
-
Dielectric properties of insulators, page 533, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals31
Effect of an electric field- Coulomb field
Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals32
Dielectric permittivity- static
Dielectric constants, page 553, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.
See J. C. Phillips, Physical Review Letters 20, 550 (1968)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals33
Dielectric permittivity- frequency-dependent
Dielectric properties of insulators, page 533, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.
- - - - - -
+ + + + + +
+
-
+
-
+
-
→ Dielectric loss
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals34
Temperature dependence
Energy
Metal InsulatorIntrinsic Semiconductor
at room temperature
Eg
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals35
Cooling semiconductors down
Energy
Metal InsulatorIntrinsic Semiconductor
at room temperature
Eg
Intrinsic Semiconductor
at low temperature
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals36
Cooling semiconductors down
EnergyIntrinsic Extrinsicfor kBT > Eg for Eg > kBT > donor binding energy
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals37
Intrinsic charge carriers
Semiconductor at room temperature
holes
Energy
Intrinsic
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals38
Intrinsic charge carriers
Eg
Page 56, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Semiconductor at room temperature
Energy
Intrinsic
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals39
Intrinsic charge carriers
Calculated intrinsic carrier densities versus temperature. Page 59, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Ge: Eg = 0.74 eVSi: Eg = 1.17 eVGaAs: Eg = 1.52 eV
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals40
Extrinsic charge carriers
Energy
Semiconductor at room
temperature
Intrinsic Extrinsic (n-type) Extrinsic (p-type) donor impurities acceptor impurities
Semiconductor at room
temperature
Semiconductor at room
temperature
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals41
Extrinsic charge carriers
Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985
Si:P binding energy = 46 meV
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals42
Extrinsic charge carriers
Temperature dependence of the electron density in silicon with a net donor density ND-NA=1015 cm-
3. Page 61, Singleton
20 ppb
Dopants in diamond have larger binding energies so are not ionised at room temperature
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals43
Donor Qubits in Silicon
Picture by Manuel Voegtli
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals44
Electron Qubits in diamond
Picture by Alan Stonebraker
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals45
Why is diamond an insulator?Electron energy
Interatomic spacing
2
4
4
6
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals46
Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985
Solve Schrödinger’s equationfor an electron in a box:
Binding energiesfor phosphorous donors:Silicon: 46 meVDiamond: 500 meV
-
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals47
Why is diamond an insulator rather than a semiconductor?
a) Wide band-gap means no intrinsic conductivity, deep dopants mean no extrinsic conductivity
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals48
But doped diamond and silicon can be metals too
Extrinsic conductivity
Semiconductor at room
temperature
Semiconductor at low
temperature
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals49
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
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals50
Doped diamond can be a metal
Charge transport in heavily B-doped polycrystalline diamond films, M. Werner et al Applied Physics Letters 64, 595 (1994)
Sample A has 8 x 1021 cm-3 boron
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals51
Electrical conductivity of semiconductors. Page 127, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals52
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 crystals53
Resistivity (ohm-cm)
10-10 1 1010 1020
Diamond ~ 1016 -cm(room temperature)
PTFE (Teflon) > 1018 -cm(room temperature)
Silicon ~ 104 -cm(room temperature)
Sup
erco
nduc
tors
~
0
Pure metal ~ 10-10 -cm (1 K) Tin ~ 10-5 -cm
(room temperature)
Module 2 – Properties and Characterization of Materials- Lectures 5 and 6 – Bandstructure of crystals54
Diamond properties
