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# ex04 - Solid State Theory Exercise 4 Excitons Exercise 4.1...

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Solid State Theory Exercise 4 SS 11 Prof. M. Sigrist Excitons Exercise 4.1 One-Dimensional Model of a Semiconductor Let us consider electrons moving on a one-dimensional chain. We use the so-called tight- binding approximation. Thus, we assume that each atom has a localized electron state and that the electrons are able to hop between neighboring atoms. This hopping process describes the kinetic energy term. It is most convenient to use a second-quantized language. For simplicity, we assume spinless electrons. Let c i and c i be the creation and annihilation operators for an electron at site i , respectively. The overlap integral between neighboring electron states is denoted by - t . Then, the kinetic energy operator is written as H 0 = - t X i c i c i +1 + c i +1 c i . (1) We assume that the chain contains N atoms and in the following we set the lattice constant a = 1. Furthermore, we assume that two consecutive atoms are nonequivalent which is modeled by an alternating potential of the form V = v X i ( - 1) i c i c i . (2)

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ex04 - Solid State Theory Exercise 4 Excitons Exercise 4.1...

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