Physics 301 22-Nov-2002 28-1Semiconductor BasicsIn solids, the electron energy levels are organized into bands. How does this comeabout? Imagine a regular crystal lattice composed of identical atoms. Imagine that thesize of the crystal can be varied. So we can consider the crystal to have an atomic spacinganywhere from its actual spacing to very large separations between the atoms. What arethe energy levels when the atoms are far apart? The energy levels are just the atomic levelsbut the degeneracy of each atomic level is multiplied by N, the total number of atoms inour crystal. Now imagine bringing the atoms closer together. Eventually, they will beclose enough that the electrons in neighboring atoms can begin to have weak interactions.Each atomic energy level then gives rise to N very closely spaced levels corresponding tothe original energy. As the atoms are brought still closer together, the interactions amongneighboring atoms increase and the spreading out of the original atomic levels into a rangeof levels increases. In effect, each atomic level becomes a band of extremely closely spacedlevels in the crystal. There can be gaps between bands and bands can overlap. The bands are most important for the energy levels of the outer electrons. The wavefunctions of the inner electrons barely overlap from one atom to the next, so there is verylittle spreading out of the energy levels of the inner electrons. The inner electrons can bethought of as belonging to a specific atom. They are localized at the site of the atom. Onthe other hand the outer electrons are more properly thought of as bound to the crystalrather than any particular atom. Of the most interest for what we’re about to discuss arethe bands of the outermost electrons.What’s the difference between an insulator, a semi-conductor, and a conductor? Con-sider a crystal at τ = 0 (and suppose it doesn’t become a superconductor!). The energylevels available to the outer electrons are in bands as we’ve discussed. The number oflevels within a band is just the number of atoms in the crystal. At absolute 0, electronsfill up the energy levels until they reach the Fermi energy. Where the Fermi energy occurswith respect to a band edge is critical for determining whether the material is an insula-tor or a conductor. If a band is just filled, so there is an appreciable energy gap to thenext available level, the material will be an insulator. To make any change to the electrondistribution, for example, to produce a distribution of electrons with a non-zero average ve-locity, requires that electrons be given enough energy to overcome the gap between bands.If the gap is substantial (several electron volts, say) ordinary electric fields will not do it.(Several electron volts is the energy of a photon of visible light!) On the other hand, if aband is only partially filled, or if the gap from a filled band to the next higher empty bandis very small, then it is easy to give electrons enough energy to change their distributionand in particular to put them in a distribution with a net average velocity in which casethey are carrying an electric current. An insulator has a filled band with a big gap to thenext band. A conductor has a partially filled band or else a filled band with a very smallgap to the next band.If the temperature is not zero, then the electron distribution does not end abruptlyCopyrightc 2002, Princeton University Physics Department, Edward J. GrothPhysics 301 22-Nov-2002 28-2at the Fermi energy. Instead, there will be electrons in states with energies above theFermi level and there will be holes (missing electrons) in states at energies below theFermi level. If at τ = 0 a band was filled with a gap to the next band, then if the gap isnot too large, there may be an appreciable number of electrons thermally excited to thenext band (leaving the same number of holes behind). In this case, there aren’t as manycharge carriers as in a good conductor, but there are more than in an insulator, so wehave a semiconductor. Semiconductors are technologically useful because their electricalproperties are relatively easy to control. We will investigate some of the properties ofsemiconductors starting with the electron distribution.Electron Distribution in SemiconductorsWe will consider a situation where the valence band is completelyfull at τ = 0 but the conduction band is completely empty. This isin the absence of doping by impurities. With doping, there may besome excess electrons that must reside in the conduction band, evenat τ = 0 or there may be some holes (a deficit of electrons) so thevalence band is not completely full at τ = 0. The energy of the topof the valence band is denoted by v, the bottom of the conductionband is c, and the energy gap is g= c− v.The concentrations of electrons in the conduction band and holes in the valence bandare denoted by neand nh. With a pure semiconductor, they are equal so that the semi-conductor is electrically neutral. But most of the fun with semiconductors comes from thedoping. Impurity atoms which provide an extra electron are called donors. Silicon is thebasis of much of the semiconductor industry. An atom from the periodic table column justto the right of silicon is a donor. Phosphorus is often used. An atom from the periodictable column just to the left of silicon is an acceptor. Boron is often used. Of course, notall donors and acceptors will be ionized, so the concentration of excess electrons will notbe exactly the same as the nd, the concentration of donor atoms. Instead, consider theconcentration of donor and acceptor ions, n+dand n−a. Then∆n = n+d− n−a,is the net ionized donor concentration, or the net concentration of positive charges from theions. If the sample is electrically neutral, this must be balanced by the net concentrationof negative charges in electrons and holes in the conduction and valence bands,ne− nh=∆n = n+d− n−a.The electron distribution (that is, the probability that a state of energy is occupied)is the Fermi-Dirac distribution (see lectures 13 and 16),fe( )=1e( − µ)/τ+1.Copyrightc 2002, Princeton University Physics Department, Edward J. GrothPhysics 301 22-Nov-2002 28-3The chemical potential of the electrons is µ. At this point, we have to mention a littlejargon. Recall that when we discussed Fermi gases earlier, we used the term “Fermienergy,” and the symbol F, to denote the chemical potential of