Pseudo potential density functional theory

1. 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
2
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
3
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

4. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
4
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
5
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
6
The Drude Model
A map of states in k-space, see also page
173, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001

7. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
7
The Drude Model
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
0
1
Potential
energy
(V)

8. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
8
The Drude Model
Dispersion relation for a free electron.
Page 177, Kittel, Introduction to Solid
State Physics, Wiley 1996

9. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
9
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
Number
of
electrons
Energy

10. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
10
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

11. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
11
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

12. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
12
Fermi-Dirac distribution function, Pages 8&9,
Singleton, Band Theory and Electronic Properties
of Solids, OUP 2001
the Sommerfeld
model

13. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
13
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
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
Arnold Sommerfeld
(1868 – 1951)

14. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
14
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
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
Potential
energy
(V)

15. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
15
Bloch’s theorem
Bloch’s theorem, Page 16, Singleton, Band Theory
and Electronic Properties of Solids, OUP 2001
Drude-Sommerfeld potential real ionic potential
0
1
Potential
energy
(V)
“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.”

16. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
16
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
17
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

18. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
18
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

19. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
19
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
20
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
21
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
22
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
Kittel page 238

23. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
23
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
24
Electronic Bandstructure of diamond
W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen, Physical
Review Letters 16, 354 (1966)
Indirect bandgap

25. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
25
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
26
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
27
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).)

28. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
28
Any questions?

29. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
29
Effect of an electric field
Relative permittivity. Page 271, Kittel, Introduction
to Solid State Physics, Wiley 1996

30. 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
31
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
32
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)

33. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
33
Dielectric permittivity
- frequency-dependent
Dielectric properties of insulators, page
533, Ashcroft and Mermin, Solid State
Physics, Harcourt 1976.
- - - - - -
+ + + + + +
+
-
+
-
+
-
→ Dielectric loss

34. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
34
Temperature dependence
Energy
Metal Insulator
Intrinsic
Semiconductor
at room
temperature
Eg

35. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
35
Cooling semiconductors down
Energy
Metal Insulator
Intrinsic
Semiconductor
at room
temperature
Eg
Intrinsic
Semiconductor
at low
temperature

36. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
36
Cooling semiconductors down
Energy
Intrinsic Extrinsic
for kBT > Eg for Eg > kBT > donor binding energy

37. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
37
Intrinsic charge carriers
Semiconductor at
room temperature
holes
Energy
Intrinsic

38. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
38
Intrinsic charge carriers
Eg
Page 56, Singleton, Band Theory and
Electronic Properties of Solids, OUP 2001
Semiconductor at
room temperature
Energy
Intrinsic

39. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
39
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 eV
Si: Eg = 1.17 eV
GaAs: Eg = 1.52 eV

40. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
40
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

41. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
41
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

42. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
42
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

43. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
43
Donor Qubits in Silicon
Picture by Manuel Voegtli

44. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
44
Electron Qubits in diamond
Picture by Alan Stonebraker

45. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
45
Why is diamond an insulator?
Electron energy
Interatomic spacing
2
4
4
6

46. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
46
Page 240, Eisberg and Resnick,
Quantum Physics of Atoms, Molecules,
Solids, Nuclei, and Particles, Wiley 1985
Solve Schrödinger’s equation
for an electron in a box:
Binding energies
for phosphorous
donors:
Silicon: 46 meV
Diamond: 500 meV
-

47. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
47
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
48
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
49
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
50
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

51. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
51
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
52
Carrier mobilities at room temperature in
cm2/Vs. Page 221, Kittel, Introduction to Solid
State Physics, Wiley 1996

53. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
53
Resistivity (ohm-cm)
10-10 1 1010 1020
Diamond  ~ 1016 -cm
(room temperature)
PTFE (Teflon)
 > 1018 -cm
(room temperature)
Silicon
 ~ 104 -cm
(room
temperature)
Superconductors

~
0
Pure metal
 ~ 10-10 -cm
(1 K)
Tin  ~ 10-5 -cm
(room temperature)

54. Module 2 – Properties and Characterization of Materials
- Lectures 5 and 6 – Bandstructure of crystals
54
Diamond properties