Already I’m half-way through my stay in San Sebastián, and conveniently I have just finished the chapter on embedding in photonics. I shall now start the chapter on electron transport, which I had put on one side. So this is a really good time to spend a long weekend at my holiday home in the Languedoc, where I am writing this. It was very spring-like when I arrived here on Thursday, and yesterday was also warm and sunny, but today is dull, cool, and rain is on the way. A good moment to get on with the next blog-post. Yesterday I spent getting my little garden sorted out, and in anticipation of rain decided to patch up what passes for a lawn (mostly dandelions) with some grass-seed. Needless to say the rain hasn’t arrived yet, just when I want it to water in the seed. The forsythia in the garden is magnificent – I don’t think I’ve ever seen it so good – and the pear tree is also covered with blossom. Everything is just showing the first signs of coming into leaf, and there is cherry blossom in the woods and hedgerows. Despite the change in the weather, I think I have arrived in France at just the right time.
In writing the photonics chapter, really solving Maxwell’s equations with embedding, I have learnt quite a lot, and found some gaps in the method which had to be put right. I described these in the previous post. I still have to do one or two more calculations to illustrate various points, but the programs I need are on my Mac back home, so they will have to wait another couple of weeks. In particular, I want to re-calculate the normal modes in a one-dimensional lattice of metal cylinders, including the frequency-dependence of the dielectric function in the frequency-dependence of the embedding potential. It sounds an esoteric point, but it is quite important. The point is that the frequency-dependence of changes the normalisation of the fields. I wish I knew whether it has any physical significance: it’s analogous to the way that an energy-dependent pseudopotential changes the normalisation of the electron wave-function, but what does it mean in the case of the electromagnetic field?
The chapter on electron transport should be interesting: to start off with it will describe the use of self-energies in transport calculations based on linear-combination of atomic orbitals (LCAO) methods. This is the way most calculations on electron transport through molecules are carried out, but the connection with embedding is that the self-energy is the same as the embedding potential (I must have said this before in the blog), only expressed in terms of atomic orbitals rather than in terms of , something which depends on spatial coordinates over the embedding surface. But in fact the expressions for the current through a molecule or whatever can all be written in terms of the spatial embedding potential, in expressions which have exactly the same form as the self-energy equations. This was shown by Daniel Wortmann, Hiroshi Ishida, and Stefan Blügel in Phys. Rev. B 66 075113 (2002), and using a somewhat different method by Simon Crampin, Hiroshi Ishida, and myself (Phys. Rev. B 71 155120 (2005)).The result is shown in the figure: the total transmission probability T across a molecule attached to metal contacts can be written in terms of the imaginary part of the embedding potential which describes the left/right contact and the Green function G which describes motion through the molecule. This result, with the self-energy for the embedding potential, has been used for a long time by people doing transport calculations in an LCAO framework.
I’ve just come back from a cold, blustery walk – sunshine and showers – for my last afternoon during this visit to the Aude. What an interesting walk, as there are already many spring flowers in bloom – but most interesting was a fire salamander, very dead in the middle of the road, unfortunately, but with startling yellow and black stripes. I have never seen one before, and they are not native to Britain. I only wish that I had seen it alive. The train journey here from San Sebastián was also very interesting, as near Bayonne there was a group of three or four storks in a field, and nearby were some deer (fallow, I think). A wonderful combination of wild-life.
Back in San Sebastián – picking up the blog again – and the weather has been terrible since my return, cold, windy, and very wet. Fortunately it shows signs of picking up again for my last weekend here. Writing on transport has gone well, and things are clearer in my own mind. Whether they are clearer on paper is a different matter! What I’m going to do next is to demonstrate the result for T using the waveguide kink as as example. (I described the kink in my post “Electrons getting stuck”, which I wrote during my last stay in the Basque Country.) I hope it will work out!