Quantum Physics

28.13 Energy Bands in Solids 925Energy2s 2s 2s1sEnergy1sEnergyEquilibriumseparation1sFigure 28.23 (a) Splitting of the1s and 2s states when two atoms arebrought together. (b) Splitting of the1s and 2s states when five atoms arebrought close together. (c) Formationof energy bands when a large numberof sodium atoms are assembled toform a solid.rrr(a)(b)r 0(c)separated by forbidden gaps. The separation and electron population of thehighest bands determines whether a given solid is a conductor, an insulator, or asemiconductor.Consider two identical atoms, initially widely separated, that are brought closerand closer together. If two identical atoms are very far apart, they do not interact,and their electronic energy levels can be considered to be those of isolated atoms.Hence, the energy levels are exactly the same. As the atoms come close together,they essentially become one quantum system, and the Pauli exclusion principle demandsthat the electrons be in different quantum states for this single system. Theexclusion principle manifests itself as a changing or splitting of electron energylevels that were identical in the widely separated atoms, as shown in Figure 28.23a.Figure 28.23b shows that with 5 atoms, each energy level in the isolated atom splitsinto five different, more closely spaced levels.If we extend this argument to the large number of atoms found in solids (onthe order of 10 23 atoms/cm 3 ), we obtain a large number of levels so closely spacedthat they may be regarded as a continuous band of energy levels, as in Figure28.23c. An electron can have any energy within an allowed energy band, but cannothave an energy in the band gap, or the region between allowed bands. Notethat the band gap energy E g is indicated in Figure 28.23c. In practice we are onlyinterested in the band structure of a solid at some equilibrium separation of itsatoms r 0 , and so we remove the distance scale on the x-axis and simply plot the allowedenergy bands of a solid as a series of horizontal bands, as shown in Figure28.24 for sodium.3pConductors and InsulatorsFigure 28.24 shows that the band structure of a particular solid is quite complicatedwith individual atomic levels broadening by varying amounts and some levels(3s and 3p) broadening so much that they overlap. Nevertheless, it is possible togain a qualitative understanding of whether a solid is a conductor, an insulator, ora semiconductor by considering only the structure of the upper or upper two energybands and whether they are occupied by electrons.Deciding whether an energy band is empty (unoccupied by electrons), partiallyfilled, or full is carried out in basically the same way as for the energy-level populationof atoms: we distribute the total number of electrons from the lowest energylevels up in a way consistent with the exclusion principle. While we omit the detailsof this process here, one important case is that shown in Figure 28.25a (page 926),where the highest-energy occupied band is only partially full. The other importantcase, where the highest occupied band is completely full, is shown in Figure 28.25b.Notice that this figure also shows that the highest filled band is called the valenceband and the next higher empty band is called the conduction band. The energyband gap, which varies with the solid, is also indicated as the energy difference E gbetween the top of the valence band and the bottom of the conduction band.3s2p2s1sFigure 28.24 Energy bands ofsodium. Note the energy gaps (whiteregions) between the allowed bands;electrons can’t occupy states that liein these forbidden gaps. Bluerepresents energy bands occupied bythe sodium electrons when the atomis in its ground state. Gold representsenergy bands that are empty. Notethat the 3s and 3p levels broaden somuch that they overlap.