Understanding Electronic Properties and Building Crystals from Atoms

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) Lecture 4 – Electronic properties: Lecture 4 – Building a crystal from atoms

2. Module 2 – Properties and Characterization of Materials - Overview Diamond properties

3. Module 2 – Properties and Characterization of Materials - Overview

5. What explains the Periodic Table? Dmitri Mendeleev (1834 – 1907)

6. What explains the periodicity of the Periodic Table? • The Schrödinger equation • The Schrödinger equation + the Coulomb potential • The Schrödinger equation + the Coulomb potential + electron spin • The Schrödinger equation + the Coulomb potential + electron spin + the Pauli exclusion principle • What your viewers really want to hear about is how I’ve improved public transport in London Dmitri Mendeleev (1834 – 1907)

7. Classical physics fails to explain atoms - - - - - -

8. Electrons can behave like waves Electron gun Electron detector Louis de Broglie (1892 – 1987) crystal

9. Schrödinger’s equation is a wave equation: Boundary Condition Erwin Schrödinger (1887 – 1961) Elastic band video from Acoustics Group, University of Salford, Manchester

10. 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. 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. 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. Solve Schrödinger’s equation for electron in Coulomb potential and include spin Page 241, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

14. What explains the Periodic Table? Page 330, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

15. 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. Any questions so far?

17. 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. 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. 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

20. Potentials Atom Molecule Insulating crystal: tight binding model 1 Potential energy (V) 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. The tight-binding model Atom Molecule Insulating crystal tight binding model Schematics of the potential due to the ions in the crystal, Page 3, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

22. 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

23. The tight-binding model Group IA metal Group IIA metal e.g. sodium e.g. magnesium Atomic separation Atomic separation Energy Energy 3p 3s 2p 3p 3s 2p Schematic of the formation of tight binding bands as the spacing between atoms is increased. Page 36, Singleton

24. Which of these elements is not in Group IV of the periodic table? • C f) N • Si • Ge • Sn • Pb Dmitri Mendeleev (1834 – 1907)

25. Group IV Dmitri Mendeleev (1834 – 1907)

26. FCC with two atom basis Diamond crystal structure. Page 37, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001

27. 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. 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. The tight-binding model: Group IV Electron energy 4 6 2 4 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. Many strong directional bonds Coulomb forces with Pauli exclusion Low-Z atoms are smaller (their electrons are closer to their parent nucleus) Closer atoms are more strongly bound (less screening) Diamond has many strong bonds H-H bond is stronger than C-C, but you can’t make a crystal out of H-H bonds 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. Bond energy and cohesive energy 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). [1] CRC Handbook, Strengths of Chemical Bonds, 57th Edition, 1977 [2] C Kittel, Introduction to Solid State Physics, Wiley 1996, Chapter 3, Table 1

32. Many strong directional bonds Smalleratoms 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. 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. 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. 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)

36. PTFE (Teflon)  > 1018 -cm (room temperature) silicon  ~ 104 -cm (room temperature) Pure metal  ~ 10-10 -cm (1 K) Tin  ~ 10-5 -cm (room temperature) Superconductors  ~ 0 diamond  ~ 1016 -cm (room temperature) 10-10 1 1010 1020 Resistivity (ohm-cm)

37. Bandstructure Energy Eg Metal Insulator Semiconductor Schematic electron occupancy for allowed energy bands. See page 174, Kittel, Introduction to Solid State Physics, Wiley 1996

38. What explains the Periodic Table? Dmitri Mendeleev (1834 – 1907)

39. Metals • Which is the most advanced model of metals in the list below? • Drude model • Sommerfeld model • Nearly-free electron model • Tight-binding model • c) and d) are equally advanced • Lady Gaga Dolce & Gabbana Paco Rabanne

40. 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

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

42. 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)