Homework problem set 2

Here follow some homework problems that are meant to be illustrative but simple. You are supposed to solve the problems with the help of books that you can easily find. Collaboration with other participants is encouraged.

Each set has five problems, which when solved fully can give 3 points. The 0-15 points that you achieve are stored in the student server under the heading H1. The result in this column and the other 9, 7 or 3 performance columns (for 5p-, 4p-, and 2p-students, respectively) are summed and divided by 10, 8, and 4, respectively. If you are a Chalmers student, who should be given grades; a result of 6 points then gives a good basis for grade 3, 9 ponts for grade 4, and 12 points for grade 5. However, there should be the concluding dialogue, which might change the examination result up or down a little.

We expect to get your solutions to this homework set, when the course is coming to an end.

  1. For a qualitative discussion of sp-bonded systems and hybrid states one often first looks at a diatomic molecule, like N2, where the atoms contain electrons in p- as well as s-states. In tetrahedral semiconductors the atomic hybrid orbital |h1> consists of ½ times an s-state plus \sqrt(3)/2 times a p-state, which is \sigma-oriented with respect to the neighbor in the [111] direction. |h1'> on the [1,1,1] neighbor is of the form |h1>, but with the opposite sign for the amplitude of the p-states. The net coupling between thes two hybrids, <h1|H|h1'>, can be written in terms of the \sigma-coupling matrix element from the simplest LCAO expression for the total energy by collecting the contributions for the coupled pairs of s- and p-\sigma states. This so-called covalent energy V2 is negative and has a magnitude equal to

        V2 = -(1/4)(Vss\sigma - 2\sqrt(3) Vsp\sigma - 3 Vpp\sigma) /4 = 3.22 (\hbar)2/(md2),               (1)

    where the last equality indicates the dependence on the distance d between the atoms.

    a) Show the first equality of Eq. (1)!

    b) The second equality of Eq. (1) is for the geometry of tetrahedral semiconductors. Show that Vss\sigma = -[(\pi)2/8] (\hbar)2/(md2) for the simpler case of a one-dimensional linear chain, e.g.,  a row of N lithium atoms, spaced at d, with s-states coupled by Vss\sigma, with periodic boundary conditions, by comporing results for the band structure in the tight-binding and free-electron approximations.

    c) Typical values of V2 are 10.35, 4.44, 4.12, and 3.13 for diamond, Si, Ge, and Sn, respectively (d = 1.54, 3.35, 2.44, and 2.80 Å, respectively).  How would you explain this trend qualitatively?




Textansvarig: Bengt Lundqvist