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Solid State From Chapter 12 of Modern Physics Contents for Chapter 12 1. 2. 3. 4. 5. 6. 7. Bonding in Solids Classical Free Electron Quantum Theory of Metals Band Theory of Solids Semiconductor Devices Superconductors* (advanced reading) Lasers (extra reading) In the beginning … • Solid state or condensed matter physics is the largest branch of physics; both experimental and theoretically. • The solid state has entranced humans ever since we have found diamonds and gems thousands of years ago. • What is really interesting is that we really do need quantum mechanics to correctly model the behaviour of these solids. • In particular the wave-like nature of the electrons in these solids turns out to be very important! 12.1 Bonding in Solids • Ionic Solids:- They form stable hard solids and are poor conductors; they have a high melting point and are transparent to visible radiation (in the infra-red they absorb radiation due to lattice vibrations) • Covalent Bonds:- What properties do we have as compared to ionic solids? • Metallic Solids: A free electron sea? • Molecular Crystals:- Via the Van de Waals force due to dipole molecules • Amorphous Solids (Glass):- Ionic Solids: Ionic attraction: Coulomb law is balanced by repulsions from inner electron orbitals leading to following potential The minimum leads to Covalent Solids These form from atoms sharing outer orbital electrons (remember the wave nature of an electron) For example in Diamond these bonds can be stronger than ionic bonds *Table 12.2 should be multiplied by a factor of 2 to compare with Table 12.1 Metallic Solids Generally weaker than ionic or covalent bonds. There are a large number of mobile electrons in a metal. A metal can be viewed of as a lattice of positive ions surrounded by an electron gas. States of matter Fig. 12-5, p.410 12.2 Classical Free Electron Model • We have briefly discussed this in Electricity & Magnetism (E&M) • Thomson, Drude and Lorentz developed this model at the turn of the 20th Century • Ohm’s Law: 𝐽 = 𝜎𝐸 is observed experimentally in many metals and semiconductors and can be derived as in (E&M) 𝑛𝑒 2 𝜏 𝑚𝑒 𝑛𝑒 2 𝐿 𝑚𝑒 𝑣𝑟𝑚𝑠 • Recall that 𝜎 = = where 𝜏 = 𝐿/𝑣𝑟𝑚𝑠 implies that 𝑛𝑒 2 𝜏 𝜎= 1/2 , but this leads to a too (3𝑘𝐵 𝑇𝑚𝑒 ) small for the conductivity (see Ex 12.1) Wiedeman-Franz Law • Thermal Conductivity K: defined in much same way as conductivity 𝐽 = −𝜎Δ𝑉/Δ𝑥, Δ𝑉 where E = − Δ𝑥 • The W-F law shows that and this ratio as a constant is predicted by classical theory although wrong magnitude • So we need to move onto a QM theory 12.3 Quantum Theory of Metals • We need to use Fermi-Dirac (FD) distribution not Maxwell-Boltzmann • In a semi-classical approximation we just replace 𝑣𝑟𝑚𝑠 → 𝑣𝐹 : use the 2 𝑛𝑒 𝐿 1 Fermi velocity for 𝜎 = and 𝐾 = 𝐶𝑣 𝑣𝐹 𝐿 (see next slide) 𝑚𝑒 𝑣𝐹 3 • Using now the fact the FD only has a small fraction of electrons near 𝐸𝐹 (see next-next slide) leads a modified Wiedemann-Franz Law (see pg 422 & cf. Table 12.7): • Note the quantum mean free path is ~100 times larger than the classical one: Electrons have infinite path length in a perfect ion lattice Replacement of 𝑣𝑟𝑚𝑠 → 𝑣𝐹 Fig. 12-14, p.421 Electron fraction near Fermi surface Fig. 10-12, p.358 12.4 Band Theory of Solids • Much like the 𝐻2± bonding/antibonding orbitals in a molecule the same thing can happen with solids Fig. 12-16, p.426 Fig. 12-17, p.426 Fig. 12-18, p.426 Fig. 12-19, p.427 Fig. 12-20, p.428 Table 12-8, p.428 Fig. 12-21, p.428 Fig. 12-22, p.429 Energy Bands from Electron Wave Reflections • This is almost real quantum mechanics and very elegant Fig. 12-23, p.430 Fig. 12-24, p.430 Fig. 12-25, p.432 Fig. 12-26, p.432 12.5 Semiconductor Devices • In progress Fig. 12-27, p.433 Fig. 12-28, p.434 Fig. 12-28a, p.434 Fig. 12-29, p.435 Fig. 12-30, p.436 Fig. 12-31, p.437 p.437 Fig. 12-32, p.438 Fig. 12-33, p.438 p.439 Fig. 12-34, p.440 Fig. 12-35, p.441 Fig. 12-36, p.442 Fig. 12-37, p.442 Fig. 12-38, p.443 12.6 Superconductivity • A little out of the scope of this book, but let’s note that we have • A critical temp 𝑇𝑐 (𝐾) where material super conducts (see Table 12.9) • The Meissner effect: magnetic fields are expelled from a superconductor • Industrial applications range from transport to computing! • Origin of basic superconductivity comes from the work of Bardeen, Cooper and Scheiffer (BCS) theory:• Two electrons can actually interact with each other attractively … Table 12-9, p.443 Fig. 12-39, p.444 Fig. 12-40, p.445 12.7 Lasers: Interaction of light with atoms • Absorption, Spontaneous Emission & Stimulated Emission Fig. 12-41, p.448 Fig. 12-42, p.450 Fig. 12-43, p.451 Table 12-10, p.451 Semiconductor Lasers • Extra reading …. Fig. 12-44, p.452 Fig. 12-45, p.453 Fig. 12-46, p.453