# Figure 526 curves of equal energy inserted into the

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Figure 5.26.Curves of equal energy inserted into the first Brillouin zone for a two-dimensional square lattice.Figure 5.27.A particular surface of equal energy (Fermi surface, see Section 6.1) and thefirst Brillouin zone for copper. Adapted from A.B. Pippard,Phil. Trans. Roy. Soc. London,A 250, 325 (1957).60I. Fundamentals of Electron Theory
Problems1. What is the energy difference between the pointsL20andL1(upper) in the band diagram forcopper?2. How large is the “gap energy” for silicon? (Hint: Consult the band diagram for silicon.)3. Calculate how much the kinetic energy of a free electron at the corner of the first Brillouinzone of a simple cubic lattice (three dimensions!) is larger than that of an electron at themidpoint of the face.4. Construct the first four Brillouin zones for a simple cubic lattice in two dimensions.5. Calculate the shape of the free electron bands for the cubic primitive crystal structure forn¼1 and n¼ ´2 (see Fig.5.6).6. Calculate the free energy bands for a bcc structure in thekx-direction having the followingvalues forh1/h2/h3: (a) 111; (b) 001; and (c) 010. Plot the bands ink-space. Compare withFig.5.18.7. Calculate the main lattice vectors in reciprocal space of an fcc crystal.8. Calculate the bands for the bcc structure in the 110 [G´N] direction for: (a) (000);(b)ð0±10Þ; and (c) 111.9. Ifb1·t1¼1 is given (see equation (5.14)), does this mean thatb1is parallel tot1?5. Energy Bands in Crystals61
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CHAPTER 6Electrons in a CrystalIn the preceding chapters we considered essentially onlyoneelectron, whichwas confined to the field of the atoms of a solid. This electron was in mostcases an outer, i.e., a valence, electron. However, in a solid of one cubiccentimeter at least 1022valence electrons can be found. In this section weshall describe how these electrons are distributed among the availableenergy levels. It is impossible to calculate the exact place and the kineticenergy of each individual electron. We will see, however, that probabilitystatements nevertheless give meaningful results.6.1. Fermi Energy and Fermi SurfaceThe Fermi energy,EF, is an important part of an electron band diagram.Many of the electronic properties of materials, such as optical, electrical, ormagnetic properties, are related to the location ofEFwithin a band.The Fermi energy is often defined as the “highest energy that the elec-trons assume atT¼0 K”. This can be compared to a vessel, like a cup, (theelectron band) into which a certain amount of water (electrons) is poured.The top surface of the water contained in this vessel can be compared to theFermi energy. The more electrons are “poured” into the vessel, the higherthe Fermi energy. The Fermi energies for aluminum and copper are shownin Figs.5.21and5.22. Numerical values for the Fermi energies for somematerials are given in Appendix 4. They range typically from 2 to 12 eV.

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Term
Fall
Professor
Ohuchi,F
Tags
Electron Theory