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Appendix A Number of electrons in square cross-section For a wire with quadratic cross section, like in Paper I, we have standing waves in the cross section with eigenenergies 2π2 En = 2md2 (n21 + n2) = E0(n21 + n2), (A.1) 0 wheren1 andn2 isthequantumnumbers= 1,2,3,...,andd0 isthesideinthesquare cross section. m is the electron mass. The degeneration is twofold (not counting spin) unless n1 = n2. Note that E0 here is not the same as the one used in Paper I. To obtain the number of electrons under the Fermi level EF from this equation we use the Matlab/Octav code shown below. We will in the code let both n1 and n2 go from 1 to Nmax > EF/E0. The double loop then gets all possible energy levels the correct number of times so that we obtain the correct number of electrons below Fermi level directly and no special care must be taken in the program to account for different degeneration of different energy levels. We can see this by making a table with n1 = 1, 2, 3, . . . horizontal and n2 = 1, 2, 3, . . . vertical. Every position in the matrix can then ad one electron to the sum, if En < EF . The of-diagonal elements like E(1, 2) and E(2, 1) then each give one electron to the sum so we have here twofold degeneracy of this energy value E(1, 2) = E(2, 1). This in contrast to an on-diagonal element like E(2, 2) which only yields one single electron. If we instead would have made the outer loop for n1 go from 1 to Nmax but letting the inner loop for n2 go from 1 to n1, care would have to be taken to get the degeneration right. 43PDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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