Schrödinger's Cat A cat is placed in an airtight box with an oxygen supply and with a glass vial containing cyanide gas to be released if a radiation detector is struck by a particle from a radioactive source. The radioactive sample is a quantum system for which we can accurately predict the probability a particle will we released in a given time interval. We propose to place the cat in the box just long enough for the probability to be one-half, then we open the box and see what happened. According to quantum mechanics, what is in the box just before we open it is neither a dead cat, nor a live cat, but a wave function describing equal probabilities of finding a live cat or a dead cat. When we open the box, the wave function collapses and we observe one state or the other.

3.1 Bonding Forces and Energy Bands in Solids 3.1.1 Bonding Forces in Solids Ionic bonding Alkali halides such as NaCl, LiF, KBr, KCl are ionic solids formed by ionic bonding. stable and hard crystals; high vaporization temperatures; good insulators. complete transfer of valence electrons Covalent bonding Semiconductors such Si, Ge, ZnS and insulators such as diamond are formed by covalent bonding, where each atom shares its valence electrons with its neighboring atoms. hard; high melting points; insulators. sharing of valence electrons among neighboring atoms

Figure 3—1 Different types of chemical bonding in solids (a) an example of ionic bonding in NaCl; (b) covalent bonding in the Si crystal, viewed along a <100> direction (see also Figs. 1–8 and 1–9).

Metallic bonding Valence electrons are contributed to the crystal as a whole. The bonding force is the attractive force between the positive ions and the electron gas. Not as strong as ionic and covalent bonding; good conductors. Sharing of valence electrons within the whole crystal Molecular bonding e.g. organic solids, ice, inert gas crystals. The bonding force is van de Waals forces. Weak bonds, low melting and boiling points. NOTE: Only valence electrons participate in bond formation! When solids are formed from isolated atoms or molecules, the total energy of the system is reduced!

Energy Bands and Energy Gaps 3.1.2 Energy Bands in Semiconductors The interaction between the valence electrons and the ions (nucleus + core electrons) leads to on particularly important effect: Energy Bands and Energy Gaps - there are two complementary ways of understanding the quantum behaviour of electrons electrons in solids: 1. tight-binding, or covalent-bonding picture 2. nearly-free electron model - together these two models give us a reasonably clear picture of the quantum behaviour of valence electrons in semiconductors.

Energy Bands —What will happen when two isolated atoms (e.g., H) are brought together? Wave functions Energy levels The formation of new bonding and anti-bonding orbitals. Energy degeneracy is brokenthe splitting of energy level 1s and 2s The lowering of energy of the bonding state gives rise to the cohesion of the system. These results can be obtained by solving the Schrödinger equation with the LCAO approximation. LCAO liner combination of atomic orbitals.

Figure 3—2 Linear combinations of atomic orbitals (LCAO): The LCAO when 2 atoms are brought together leads to 2 distinct “normal” modes—a higher energy anti-bonding orbital, and a lower energy bonding orbital. Note that the electron probability density is high in the region between the ion cores (covalent “bond”), leading to lowering of the bonding energy level and the cohesion of the crystal. If instead of 2 atoms, one brings together N atoms, there will be N distinct LCAO, and N closely-spaced energy levels in a band.

—What will happen when many (N) Si atoms are brought together to form a solid? Energy bands are formed Conduction band Valence band Forbidden band (band gap Eg) Electronic configuration of Si 1s22s22p63s23p2

Figure 3—3 Energy levels in Si as a function of inter-atomic spacing. The core levels (n = 1,2) in Si are completely filled with electrons. At the actual atomic spacing of the crystal, the 2 N electrons in the 3 s sub-shell and the 2 N electrons in the 3 p sub-shell undergo sp 3 hybridization, and all end up in the lower 4 N states (valence band), while the higher lying 4 N states (conduction band) are empty, separated by a band gap.

3.1.3 Metals, Semiconductors, and Insulators Metals have free electrons and partially filled valence bands, therefore they are highly conductive (a). Semimetals have their highest band filled. This filled band, however, overlaps with the next higher band, therefore they are conductive but with slightly higher resistivity than normal metals (b). Examples: arsenic, bismuth, and antimony. Insulators have filled valence bands and empty conduction bands, separated by a large band gap Eg(typically >4eV), they have high resistivity (c ). Semiconductors have similar band structure as insulators but with a much smaller band gap. Some electrons can jump to the empty conduction band by thermal or optical excitation (d). Eg=1.1 eV for Si, 0.67 eV for Ge and 1.43 eV for GaAs

Typical band structures at 0 K. Figure 3—4 Typical band structures at 0 K.