sol01 - Solid State Theory Exercise 1 Kronig-Penney model...

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Solid State Theory Exercise 1 FS 11 Prof. M. Sigrist Kronig-Penney model We study a simple model for a one-dimensional crystal lattice, which was introduced by Kronig and Penney in 1931. The atomic potentials are taken to be rectangular, where the minima correspond to the atomic cores. The model is simpliﬁed even more by replacing the rectangular potentials by Dirac delta functions, V ( x ) = V 0 X n = -∞ δ ( x - an ) . (1) This is the so-called Kronig-Penney potential which is shown in Fig. 1A. 0 a 2 a A 0 a 2 a B Figure 1: A Kronig-Penney potential V ( x ). B Interface between a constant potential U ( x ) and a Kronig-Penney potential. Exercise 1.1 Energy bands Using Bloch’s Ansatz for the wave function in a periodic potential Ψ( x + a ) = Ψ( x ) e ika , (2) show that the energy in the Kronig-Penney potential for a given k obeys the equation cos λ = v 2 β sin β + cos β, (3) where λ = ka , β = a p 2 mE/ ~ 2 and v = 2 mV 0 a/ ~ 2 . In general, this equation can only be solved graphically or numerically. Show that the resulting band structure has band gaps

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This note was uploaded on 10/04/2011 for the course PHYS fs11 taught by Professor Sigrist during the Spring '11 term at Swiss Federal Institute of Technology Zurich.

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sol01 - Solid State Theory Exercise 1 Kronig-Penney model...

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