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Gavin W Morley Department of Physics University of Warwick

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)

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Gavin W Morley Department of Physics University of Warwick

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  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. Metals • Most elements are metals, particularly those on the left of the periodic table • Good conductors of electricity & heat • Tend to form in crystal structures with at least 8 nearest neighbours (FCC, HCP, BCC) • Malleable Schematic model of a crystal of sodium metal. Page 142, Kittel, Introduction to Solid State Physics, Wiley 1996

  3. Metals • The Drude Model: • Gas of electrons • Electrons sometimes collide with an atomic core • All other interactions ignored Paul Drude (1863 –1906)

  4. Metals Sommerfeld • The Drude Model: • Gas of electrons • Electrons sometimes collide with an atomic core • All other interactions ignored • Electrons obey the Schrödinger equation and the Pauli exclusion principle Arnold Sommerfeld (1868 – 1951)

  5. Metals Sommerfeld The Drude Model A map of states in k-space, see also page 173, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

  6. Metals Sommerfeld The Drude Model 1 Potential energy (V) 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

  7. Metals Sommerfeld The Drude Model Dispersion relation for a free electron. Page 177, Kittel, Introduction to Solid State Physics, Wiley 1996

  8. Metals The Drude Model: The Drude Model vs Distribution functions for a typical metal at room temperature, Page 10, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001 Energy the Sommerfeld model Number of electrons fFD Energy

  9. Metals 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

  10. Metals 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? • 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

  11. Metals the Sommerfeld model Fermi-Dirac distribution function, Pages 8&9, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

  12. Metals Sommerfeld • The Drude Model: • Gas of electrons • Electrons sometimes collide with an atomic core • All other interactions ignored • Electrons obey the Schrödinger equation and the Pauli exclusion principle • Explains temperature dependence and magnitude of: • Electronic specific heat • Thermal conductivity (approx.) • Electrical conductivity (approx.) • But does not explain: • Insulators & semiconductors • Thermopower • Magnetoresistence • Hall Effect Arnold Sommerfeld (1868 – 1951)

  13. Metals, Semiconductors & Insulators • Beyond the Sommerfeld Model: • Gas of electrons • Electrons are in a periodic potential due to the ions • Electron-electron interactions ignored • Electrons obey the Schrödinger equation and the Pauli exclusion principle 1 Potential energy (V) 0 Drude-Sommerfeld potential real ionic potential Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

  14. Bloch’s theorem “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.” 1 Potential energy (V) 0 Drude-Sommerfeld potential real ionic potential Bloch’s theorem, Page 16, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

  15. The nearly-free electron model Drude-Sommerfeld potential weak ionic potential

  16. 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

  17. 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

  18. Representing bands Three energy bands of a linear lattice. Page 238, Kittel, Introduction to Solid State Physics, Wiley 1996

  19. Diamond model • From the following list, which is the best model of diamond? • Drude model • Sommerfeld model • Nearly-free electron model • Tight binding model

  20. Electronic Bandstructure of diamond W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

  21. Electronic Bandstructure of diamond Kittel page 238 W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

  22. 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

  23. Electronic Bandstructure of diamond Indirect bandgap W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

  24. Electronic Bandstructure of diamond W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

  25. Electronic Bandstructure of diamond W. Saslow, T. K. Bergstresser, and Marvin L. Cohen, Physical Review Letters 16, 354 (1966)

  26. 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

  27. Any questions?

  28. Effect of an electric field Relative permittivity. Page 271, Kittel, Introduction to Solid State Physics, Wiley 1996

  29. Effect of an electric field - capacitor - - - - - - + + + - - - + + + + + + Dielectric properties of insulators, page 533, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.

  30. Effect of an electric field - Coulomb field Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

  31. 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.

  32. Dielectric permittivity - frequency-dependent - - - - - - + + + - - - + + + + + + → Dielectric loss Dielectric properties of insulators, page 533, Ashcroft and Mermin, Solid State Physics, Harcourt 1976.

  33. Temperature dependence Energy Eg Metal Insulator Intrinsic Semiconductor at room temperature

  34. Cooling semiconductors down Energy Eg Metal Insulator Intrinsic Semiconductor at room temperature Intrinsic Semiconductor at low temperature

  35. Cooling semiconductors down Energy Intrinsic Extrinsic for kBT > Eg for Eg > kBT > donor binding energy

  36. Intrinsic charge carriers Energy Intrinsic holes Semiconductor at room temperature

  37. Intrinsic charge carriers Energy Intrinsic Eg Semiconductor at room temperature Page 56, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

  38. 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

  39. Extrinsic charge carriers Energy Intrinsic Extrinsic (n-type) Extrinsic (p-type) donor impurities acceptor impurities Semiconductor at room temperature Semiconductor at room temperature Semiconductor at room temperature

  40. 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

  41. 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

  42. Donor Qubits in Silicon Picture by Manuel Voegtli

  43. Electron Qubits in diamond Picture by Alan Stonebraker

  44. Why is diamond an insulator? Electron energy 4 6 2 4 Interatomic spacing

  45. 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

  46. Why is diamond an insulator rather than a semiconductor? a) Wide band-gap means no intrinsic conductivity, deep dopants mean no extrinsic conductivity

  47. But doped diamond and silicon can be metals too Extrinsic conductivity Semiconductor at room temperature Semiconductor at low temperature

  48. 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

  49. 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

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