5.62 Spring 2008 Lecture summary 26 Page 1 Band Theory of Solids In the free electron theory we ignored all effects of the nuclei in the lattice, utilizing a particle-in-the-box approach sans a potential. In the band theory of solids considered here, we include a very simple potential representing the nuclei that leads to “bands” of potentially occupied states that are separated by gaps. The forces on the electrons are the regularly spaced, positively charged, essentially stationary nuclei and they are represented by delta functions. Dirac Comb Potential: The simple periodic structure depicted below reproduces many interesting aspects of the band theory of metals. It is referred to as a Dirac comb, whereas a more sophisticated model, the Kronig-Penny model, employs a comb of rectangular shapes. The actual shape is not so important for our purposes. The potential periodic repeats itself after some distance that we can write the potential as a (the lattice spacing) so V (x) = V (x + a) This is the same idea that we used previously (the approach of Born and von Kármán) in our treatment of Debye solids and in the free electron theory of metals. Bloch’s Theorem - Felix Bloch [Z. Physik 52, 555 (1928)] suggested an ingenious approach to treating this problem that is today know as Bloch’s theorem. For the potential like the periodic comb above, the solution to the time independent Schrödinger equation (TISE) 5.62 Spring 2008 Lecture summary 26 Page " !2 d 2 % $ ! 2m 2 + V (x)' ( = E( dx # & can be taken to satisfy the condition ! (x + a) = eiKa! (x) for some constant K. K could depend on E, but is generally independent of x. We assume we have an operator D̂ that moves us along the chain for “a” units such that D̂f (x) = f (x + a) For a periodic potential, D̂ commutes with Ĥ -- !# D̂, Ĥ $& = 0 . Therefore, we can " % choose eigenfunctions of Ĥ that are simultaneously eigenfunctions D̂ . D! = "! or ! (x + a) = "! (x) Therefore We will see below that K is real. ! = eiKa For a macroscopic crystal (containing Na = Avogodro’s number of sites) we can neglect the edge effect, and use the Born-von Kármán method to impose the boundary condition ! (x) = ! (x + Na) Therefore it follows that ! (x + Na) = eiNKa! (x) Since eiNKa = 1 = cos(NKa) + isin(NKa) and therefore NKa = 2! n and 2! n % # Na '& K = "$ () n = 0, ± 1, ± 2,.....*+ 2 5.62 Spring 2008 Lecture summary 26 Page 3 Thus, K is real and we need to solve the TISE within a single cell (for example, for the interval 0 ! x ! a . Potential V(x) – We assume that V(x) consists of a long string of delta function spikes (the Dirac comb) depicted above. This is represented algebraically as N #1 V (x) = ! $ " (x # ja) j =0 Thus, the x-axis of the comb has been “wrapped around” so the Nth spike occurs at x= -a. 0 < x < a : In this region the potential is zero so ! 2 d 2" ! = E" 2m dx 2 d 2! 2 2 = "k ! dx 1 where 2mE 2 k = !# 2 $& " ! % per usual. And the general solution is ! (x) = Asin(kx) + Bcos(kx) 01. These regions, which arise from the sinc term, correspond to “gaps” and are forbidden energies. These are separated by “bands” which are allowed energies. Within a band any energy is allowed since 2! n % Ka = "$ # N '& recall that N is a very large number and n = an integer. The figure below illustrates the bands and gaps. The oscillating function descends from on high ( ! =10 in this example) and decays to a constant cos (ka). The shaded regions between ±1 correspond to the bands and the unshaded to the gaps. 5.62 Spring 2008 Lecture summary 26 Page 6 For a filled band (with q=2) it takes a large energy to excite an electron across the gap to the conduction band – this is an insulator ! However, if the band is partially filled it generally takes less energy to excite an electron and this material is typically a conductor. If we dope an insulator with a few atoms with either larger or smaller q, then we place extra electrons into the next high band or create holes in the previously filled band. This permits weak currents to flow and the materials are referred to as semiconductors. Summary: In the free electron theory all solids are conductors because there are no gaps in the energy level scheme. It takes the periodic potential of band theory to account for the factors of ~1030 that are observed experimentally in resistivity/conductivity.

Band Theory of Solids

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