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# ex03 - Solid State Theory Exercise 3 Exercise 3.1 6-Orbital...

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Solid State Theory Exercise 3 FS 2011 Prof. M. Sigrist Exercise 3.1 6-Orbital tight-binding model ( j - 1) a ja ( j + 1) a Figure 1: One-dimensional chain of atom cores. In this exercise we want to calculate the band structure of (fictitious) one-dimensional sodium in the tight-binding approximation. The (single-particle) Hamiltonian of the system is given by H = p 2 2 m + X j - Ze 2 | r - r j | , (1) with Z = 11 for Na and r j = ( x, y, ja ) and the lattice constant a . a) As a starting point for the tight-binding approximation, a formulation in terms of Wannier functions is more practicable. We define the Wannier function w n ( x, y ; z - ja ) of atom j in band n by Ψ n,k ( x, y ; z ) = 1 N X j e ikja w n ( x, y ; z - ja ) . (2) Note that the potential is periodic in the z -direction only. The different bands originate from the atomic orbitals. Since Na has 11 electrons we need to consider the 6 orbitals n = (1 s, 2 s, 2 p x , 2 p y , 2 p z , 3 s ). Show that within the tight-binding approximation taking only nearest-neighbor hopping into account, the hamiltonian can be written as H = X n H n + X n 6 = n 0 H n,n 0 , (3)

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ex03 - Solid State Theory Exercise 3 Exercise 3.1 6-Orbital...

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