Quantum Physics

926 Chapter 28 Atomic PhysicsFigure 28.25 (a) Half-filled bandof a metal, an electrical conductor.(b) An electrical insulator at T 0Khas a filled valence band and anempty conduction band. (c) Bandstructure of a semiconductor atordinary temperatures (T 300 K).The energy gap is much smaller thanin an insulator, and many electronsoccupy states in the conduction band.Conduction bandEnergy gapE gE gConduction bandValence bandValence bandMetal(a)InsulatorE g 10 eV(b)SemiconductorE g 1 eV(c)With these ideas and definitions we are now in a position to understandwhat determines, quantum mechanically, whether a solid will be a conductor or aninsulator. When a modest voltage is applied to a good conductor, the electrons accelerateand gain energy. In quantum terms, electron energies increase if there arehigher unoccupied energy levels for electrons to jump to. For example, electrons near thetop of the partially filled band in sodium need to gain very little energy from theapplied voltage to reach one of the nearby, closely spaced, empty states. Thus, it iseasy for a small voltage to kick electrons into higher energy states, and chargeflows easily in sodium, an excellent conductor.Now consider the case of a material in which the highest occupied band is completelyfull of electrons and there is a band gap separating this filled valence bandfrom the vacant conduction band, as in Figure 28.25b. A typical case might be diamond(carbon), in which the band gap is about 10 eV. When a voltage is applied,electrons can’t easily gain energy, because there are no vacant energy states nearbyto which electrons can make transitions. Because the only empty band is the conductionband, an electron must gain an amount of energy at least equal to theband gap in order for it to move through the solid. This large amount of energycan’t be supplied by a modest applied voltage, so no charge flows and diamond isa good insulator. In summary then, a conductor has a highest-energy occupiedband which is partially filled, and in an insulator, has a highest-energy occupiedband which is completely filled with a large energy gap between the valence and conductionbands.SemiconductorsTo this point, we have completely ignored the influence of temperature on theelectronic populations of energy bands. Recalling that the average thermal energyof a particle at temperature T is 3k B T/2, we find that an electron at room temperaturehas an average energy of about 0.04 eV. Because this energy is about 100 timessmaller than the band gap in a typical insulator, very few electrons would haveenough random thermal energy to jump the energy gap in an insulator and contributeto conduction. However things are different for a semiconductor. As we seein Figure 28.25c, a semiconductor is a material with a small band gap of about1 eV whose conductivity results from appreciable thermal excitation of electronsacross the gap into the conduction band at room temperature. The most commonlyused semiconductors are silicon and gallium arsenide, with band gaps of1.14 eV and 1.43 eV, respectively, at 300 K. As you might expect, the resistivity ofsemiconductors usually decreases with increasing temperature, because k B T becomesa larger fraction of the band gap energy.It is interesting that the electrons in the conduction band of a semiconductordon’t carry the entire current when a voltage is applied, as Figure 28.26 shows.(It might be said that conduction electrons do not constitute the “whole” story.)The missing electrons in the valence band, shown as a narrow white band in the