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SJSU PHYS 175A - Chapter 7

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Chapter 7Energy BandsPhys 175ADr. Ray KwokSJSUNearly Free Electron Model+=axUUxUπ2cos)(10010<< UUUU = 0atomic potentialFEGNFE modelsmall perturbation, U1λ = aUxU1U0Electronic Wave in NFE modelConsider the following cases:Electrons wavelengths much larger than a, so wavefunctionsand energy bands are nearly the same as above01=U)( tkxiAeωψ−=mkE222h=akUπ<<≠01(FEG) wavefunctions are plane waves and energy bands are parabolic:akUπ=≠01Electrons waves are strongly back-scattered (Bragg scattering) so standing waves are formed:[][]tiikxikxtkxitkxieeeAeeCωωωψ−−−−−±±=±=21)()(akUπ≤≠01Electrons wavelengths approach a, so waves begin to be strongly back-scattered :)()( tkxitkxiBeAeωωψ−−−±±=AB<Standing wave at zone boundaryThere are two such standing waves possible:[]ti21tiikxikx21e)kxcos(A2eeeAω−ω−−+=+=ψ[]ti21tiikxikx21e)kxsin(iA2eeeAω−ω−−−=−=ψ)(cos222*axAπψψ=++)(sin222*axAπψψ=−−These two approximate solutions to the Schrodinger Equation at k=π/a have very different potential energies. |ψ+|2has its peaks at x = a, 2a, 3a, …at the positions of the atoms, where U is at its minimum (low energy wavefunction). The other solution, |ψ−|2has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between atoms, where U is at its maximum (high energy wavefunction).Ux|ψ+|2|ψ-|2Electronic EnergyWe can do an approximate calculation of the energy difference between these two states as follows. Letting U0= 0 for simplicity, and remembering U1< 0:( ) ( )∫∫==++−−+−−−=−≅−axaxaxaxaxdxUAdxxUEE0222120**)(sin)(cos)cos(2)(πππψψψψ()axπ2cosaA1=∫=+−−≅−axaxadxUEE02212)(cosπwith()xx 2cos1cos212+=( ) ( )gaaxaaUaxaxaUEUxdxEE =−=+−=+−≅−∫=+− 104404)sin()cos(111πππnormalizationenergy gap at zone boundary!!Origin of the Energy GapIn between the two energies there are no allowed energies; i.e., an energy gap exists. We can sketch these 1-D results schematically:aπaπ−The periodic potential U(x) splits the free-electron E(k) into “energy bands” separated by gaps at each BZ boundary.EkxE−E+EgFEGNFEEnergy Bands1GviˆG)k(E)Gk(Ean2π==+vvvv(i) extended zone scheme: plot E(k) for all k (bold curve)(ii) periodic zone scheme: redraw E(k) in each zone and superimpose(iii) reduced zone scheme: all states with |k| > π/a are translated back into 1stBZNumber of electrons per bandIt is easy to show that the number of k values in each BZ is just N, the number of primitive unit cells in the sample. Thus, each band can be occupied by 2N electrons due to their spin degeneracy.aπaπ−A monovalent element with one atom per primitive cell has only 1 valence electron per primitive cell and thus N electrons in the lowest energy band. This band will only be half-filled.EFThe Fermi energy is the energy dividing the occupied and unoccupied states, as shown for a monovalent element.EkxE−E+EgFEGNFEMetal, Insulator & Semiconductor• Metals are solids with incompletely filled energy bands• Semiconductors and insulators have a completely filled or empty bands and an energy gap separating the highest filled and lowest unfilled band. Semiconductors have a small energy gap (Eg< 2.0 eV).In 1-D, yes, but not necessarily in 2-D or 3-D! Bands along different directions in k-space can overlap, so that electrons can partially occupy both of the overlapping bands and thus form a metal.But it is true that only crystals with an even number of valence electrons in a primitive cell can be insulators.Does this mean a divalent element will always be an insulator?Bloch’s TheoremThis theorem is one of the most important formal results in all of solid state physics because it tells us the mathematical form of an electron wavefunction in the presence of a periodic potential energy. What exactly did Felix Bloch prove in 1928? In the independent-electron approximation, the time-independent Schrodinger equation for an electron in a periodic potential is:ψψErUm=+∇− )(222vh)()( rUTrUvvv=+where the potential energy is invariant under a lattice translation vector Bloch showed that the solutions to the Schrodinger Equation are the product of a plane wave and a function with the periodicity of the lattice (~atomic function):rkikkerurvvvvvv⋅= )()(ψ“Bloch wave functions”)()( ruTrukkvvvvv=+wherewhere T is the translational vector describes the lattice symmetryProof of Bloch’s Theorem in 1-DAnother way to write Bloch’s TheoremTkirkikkeeTruTrvvvvvvvvvv⋅⋅+=+ )()(ψTkirkikeeruvvvvvv⋅⋅= )(Tkikervvvv⋅= )(ψOr just:TkikkerTrvvvvvvv⋅=+ )()(ψψNow consider N identical lattice points around a circular ring (periodic boundary), each separated by a distance a. To prove the Bloch’s Theorem is to prove:ikaexax )()(ψψ=+12 N3)()( xNaxψψ=+Built into the ring model is the periodic boundary condition:The symmetry of the ring implies that we can find a solution to the wave equation:)()( xCaxψψ=+Proof (cont.)If we apply this translation N times we will return to the initial atom position:Now that we know C we can rewrite)()()( xxCNaxNψψψ==+This requires1=NCAnd has the most general solution:,...2,1,02±±== neCniNπikaNnieeC ==/2πOr:where we define the Bloch wavevector:Nankπ2=...)()()( DEQxexCaxikaψψψ==+It is not hard to generalize this to 3-D:TkikkerTrvvvvvvv⋅=+ )()(ψψrkikkerurvvvvvv⋅= )()(ψor equivalentlyThis result gives evidence to support the nearly-free electron approximation, in which the periodic potential is assumed to have a very small effect on the plane-wave character of a free electron wavefunction. It also explains why the free-electron gas model is so successful for the simple metals!BlochWavefunctionsConsequence of Bloch’s TheoremFor phonons, the propagation speed of the vibrational wave isSimilarly, it can be shown using Bloch’s theorem that the propagation speed of an electron wavepacket in a periodic crystal can be calculated from a knowledge of the energy band along that direction in reciprocal space:dkdvgω=group velocity: (1-D)dkdEdkdvgh1==ωelectron velocity: (1-D))()( kkvkgvvvvvω∇=(3-D))(1)( kEkvkgvvhvvv∇=(3-D)This means that an electron (with a specified wavevector) moves through a perfect periodic lattice with a constant velocity; i.e., it moves without being scattered or in any way having its velocity affected! (E = ћ2k2/2m)But wait…does Bloch’s theorem prove too much? If this result is true,


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SJSU PHYS 175A - Chapter 7

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