It’s the last day of my month in San Sebastían, and I am still writing new programs for the way that electrons propagate round kinks in waveguides, using embedding to confine the electrons to the kink. In my last post I described the treatment of the kink using the box pictured at the right, with the grey regions II replaced by an embedding potential. This confines the electrons to region I, giving the kink in the waveguide, shown by the red lines. In the work I described in my previous post I calculated the Green function for region I, with basis functions given by
satisfying zero amplitude at y = 0 and a, and zero derivative at x = 0 and a; in other words the Green function is forced to have a zero derivative boundary condition at the entrance and exit to the kink. From this we can calculate the transmission and reflection properties of the kink, using a method in which the wave-functions in the left and right-hand waveguides are written in terms of incident + reflected waves, and transmitted waves, and then using Green’s theorem to match the reflection and transmission coefficients (Dix and Inglesfield, J. Phys.: Condens. Matter 10 5923-5941 (1998)).
However I realised last week that we can do the problem using a different, simpler approach, based on the fact that embedding allows us to treat mixed boundary conditions, by which I mean that we can calculate the wave-functions or Green function for a system with a combination of zero amplitude and zero derivative boundary conditions. (This has been implicit in our previous publications.)We can then treat the kink by considering the rectangular box shown on the left, with confinement – zero amplitude – at the blue lines, and open boundary conditions – zero derivative – at the dashed lines, to which the straight sections of waveguide are joined. Region I is now a very simple shape, and we add the confinement embedding potential over the blue lines, and zero embedding potential at the dashed lines to give the required mixed boundary conditions (at the top and bottom, the choice of basis function ensures zero amplitude). The resulting embedded Green function with zero derivative at the waveguide entrance and exit can then be used, once again, to find the transmission and reflection of the kink.
Results for the transmission of electrons through the kink are shown in the figure on the right, as a function of electron energy (I should say that the incident electrons are in the first channel, but transmission is into all the open channels). These results are just the same as the transmission results I obtain using the geometry shown at the top of this page – both are describing the same kink. (Actually not quite the same if we haven’t got fully converged results, which we never have in the real world.) First point to notice – for several energies the electrons go unimpeded through the kink, with 100% transmission. There are also energies at which transmission is zero – at such energies, the current density in the kink is as shown in the previous post, with electrons circulating in the kink and getting nowhere. But what is very interesting is all the structure, for example at an energy E = 7.5 a.u. there is a curious sharp feature, and at E = 8.7 there is a sharp dip.
To understand these features we calculate the density of states of the kink, that is, how many electronic states there are at a particular energy. Again we can use the rectangular geometry to describe the kink, but this time as well as the confining embedding potentials along the blue lines we must add embedding potentials on the dashed lines to take account of the waveguides, the fact that electrons can move out of the kink into the waveguides. The results are shown in this figure, and we see numerous very sharp features, as well as a continuous background which starts at an energy of E = 0.55 a.u., the energy at which electrons can start to move through the waveguide (which has width 3 a.u.). Just below this energy there is a very sharp peak, which is in fact a bound state, in which an electron is trapped in the kink. The other sharp peaks come from states which are trapped for a long time, but can ultimately leak out – resonances – the width of the peak corresponding to 1/lifetime. If we compare the transmission with the density of states, we see that all the sharp features, peaks and dips, correspond to resonances in the density of states, in other words electrons getting stuck in the kink.
I think I’ve got all the programs working, which relate to the confinement of electrons – famous last words – and there are lots of results and figures to put in chapter 6, which isn’t quite finished after all. I’ve also made progress in relating embedding to scattering theory, and the work which I’m most pleased with is (yet another) calculation of the transmission through the kink using equation 33 from Inglesfield, Crampin and Ishida (Phys. Rev. B 71 155120 (2005)), ,
which gives the wave-function inside the kink, , in terms of the incident wave-function in the left-hand waveguide, , the embedding potential for the left-hand waveguide, and the full kink Green function . This gives the same results for the transmission as in the figure above, as it should, but in a more straightforward way. So that’s it, for the time being, in San Sebastían. Back to Cumbria, and quite a lot of work in the garden, no doubt, catching up with lawn-mowing, cutting plants back for the winter etc.