# A weekend between chapters

Forsythia in France

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 $\epsilon(\omega)$ 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 $\Sigma(\mathbf{r}_S,\mathbf{r}_S')$, 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)).

Calculating current through molecule (red circles) attached to metal contacts (green)

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 $\Im\Sigma_{l/r}$ 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!

# Back to Basque

Keeping clear of the cold water!

I’m back in San Sebastian for another month, visiting the Theoretical Physics Centre here to get on with the book. It’s certainly easier to write here than back home, as the atmosphere is very conducive to study, there are other scientists to talk to and seminars to listen to. The last week has been glorious, cloudless skies and pleasant spring temperatures, though today – I’m writing this on Saturday afternoon – is cooler and cloudier, but still pleasant. Spring is several weeks ahead of South Cumbria, and the wall-to-wall sunshine (till today) is great for morale. But the sea is still too cold for swimming, though perhaps not for a tentative toe in the water.

As I said in my last post, I am now on with the chapter on applications of the embedding method to solving Maxwell’s equations, the equations which determine the propagation of light. Progress has not exactly been rapid, but I’m getting on with it. As always, several difficulties have cropped up. There is the ever-present problem of consistency of notation – this time it’s just a matter of suffices! My convention has been to use the suffices m and n in quoting matrix elements, like $A_{mn}$, but then in this chapter m is also the azimuthal quantum number in the vector spherical harmonics, which appear in electromagnetic waves when you’re dealing with spherical symmetry. Should I change all the m‘s and n‘s to i‘s and j‘s in the earlier chapters? Well, I’ve decided to stick with m and n as suffices in matrix elements throughout, as well as using m as the azimuthal quantum number in a suffix on the spherical harmonics, and and hope that context will make things clear. And then there is the symbol $\epsilon$ which I’m really over-working: the latest in this chapter is to use $\hat{\boldsymbol{\epsilon}}$ for the direction of polarisation of an electromagnetic wave (it also stands for the dielectric constant, an important constituent of electromagnetic theory!).

As well as getting on with the writing, I’ve actually made progress with the formalism of the embedding method applied to electromagnetism. There are two uses for embedding in this context. The first is to replace a dielectric object (a metal sphere, for example, in some photonics structure) by an embedding potential. This is useful, because the electric or magnetic field is then expanded only between the dielectric objects, where it is smoothly varying and without discontinuities. This makes the calculation much more efficient. The second use, analogous to what we do in surface electronic structure calculations, is to replace the extended background in which the photonic structure typically sits – for example the air which might surround some structure, into which the electromagnetic waves can leak. Well, in my original formulation of the problem, I assumed that one could only replace dielectric objects with a spatially constant dielectric constant (no pun intended) and magnetic permeability. I now realise that this restriction doesn’t apply; it hasn’t limited me in applications of the method so far, but it was certainly unsatisfactory from an aesthetic point of view!

I also sorted out another problem which arose, something which hadn’t struck me before. Is my derivation of the embedding potential valid when the dielectric constant varies with frequency, as it certainly does in metals for example (otherwise we wouldn’t get important phenomena like plasma oscillations)? There is one point in the derivation where I differentiate the Helmholtz equation with respect to $\omega^2$, in which I assume that only the fields depend on frequency, and don’t differentiate the dielectric constant. It turns out that the final result is correct, because embedding ensures that the parallel components of both the electric and magnetic fields are continuous across the embedding boundary, something which doesn’t depend on whether or not $\epsilon$ varies with frequency. I hadn’t discussed this before in articles, but writing the book certainly concentrates the mind wonderfully.

San Sebastian

As the finished book will be an “e-book”, there will be opportunity for using different media, including inserting small computer programs. I’m not sure about the multi-media side of things – I don’t really want to record myself giving a mini-lecture – but interactive computer programs would be useful. How am I going to do this? Well, I don’t know yet, but as a start I’ve decided to learn Python, as this is a computer language which is widely taught in universities, is easy to learn, and flexible. I doubt if it would be suitable for the programs I actually use in research, as being an “interpreted” language (so I am told) it runs much slower than a compiled language like Fortran. So for research I shall stick with Fortran-95, which I enjoy using very much. But I am enjoying learning Python, and I think it will come in very useful for writing mini-programs for the book. I’m not there yet, however. Of course I am also trying to learn some Spanish at the same time, and having spent October here, then a week in the Canaries last month, then another four weeks here I should be getting somewhere.

Anyway, back to the book. Well, it’s the weekend, so perhaps I’ll go for a walk along the cliff tops instead.

# Embedding for photonics

Walking on La Gomera, Canary Islands

As well as the Schrödinger equation, the embedding method can be used to solve Maxwell’s equations, to calculate electromagnetic waves. There has been renewed interest in solving Maxwell’s equations over the last 20 or so years, and this has led to the development of photonics, plasmonics, and perhaps most important of all, the new science of metamaterials – all of these are about the manipulation of electromagnetic waves and photons, in various structures and situations. In the Schrödinger equation context, the main use of embedding has been to replace the semi-infinite substrate in surface calculations by the embedding potential, added on to the surface Hamiltonian. It can be used in an analogous way in Maxwell’s equations, to take care of the rest of space into which the electromagnetic waves can escape from some structure. But there’s another important use of electromagnetic embedding, and that is to to replace a dielectric object in a photonic structure, such as a metal sphere or cylinder. Why should we wish to do this? It’s because the electric field jumps across the surface of a dielectric, and this makes it more to difficult to calculate. If we replace these objects by embedding potentials, we only have to calculate the electric field outside, where it is likely to be smoothly varying and easier to calculate (by expanding it in plane waves, for example).

Having finished the chapter on the embedding potential/self-energy in tight-binding (actually, not quite), I’ve started the chapter on “Embedding Maxwell’s Equations”. The chapter on transport, which immediately follows the tight-binding chapter, I’ve put off until I have done some more background work and reading. The Maxwell chapter should be relatively straightforward to write, as this is the stuff which I’ve been writing papers on most recently. Beginning the chapter has reminded me of the difficulties I met with when I first started on Maxwell embedding, connected with the fact that one is dealing with the vector electric and magnetic fields, rather than the scalar wave-function of the Schrödinger equation. For a start, the embedding potential is replaced by an embedding tensor, a more complicated object. But more seriously, the solutions of Maxwell’s equations can become corrupted by approximate solutions of Laplace’s equation (the equation of electrostatics), which being approximate appear at finite frequency, mixed up with the solutions we are looking for! It still isn’t completely clear to me why these Laplace solutions crop up, but fortunately I found a way of dealing with them, pushing them down to zero frequency where they belong – if this was not possible, the embedding method wouldn’t have been much use for Maxwell’s equations. Needless to say, in the course of writing this chapter I have realised that there are more calculations I want to do, more figures I want to generate, and this all takes up time, good writing time!

I mentioned that I haven’t quite finished the chapter on embedding in tight-binding, and what remains to be done are sections on embedding in quantum chemical calculations. This includes work by Pisani and other quantum chemists from Turin, who had been working on embedding long before me. Their formalism is quite involved, and I still have to fully understand the rationale behind their method. I made some progress in understanding during a recent visit to my alma mater, Cambridge, where the excellent libraries provided the ideal working environment.

Garden on La Gomera

I’m writing this blog during a week’s holiday on the island of La Gomera, in the Canaries. What a beautiful place this is, and how lovely to have spring sunshine and temperatures – though it is very windy at the moment. I’ve been doing a lot of walking in the mountains of this tiny island, mountains formed by volcanic activity so they are very rough and rugged. Needless to say, in such terrain the walking is very hard. The flowers are one of the great delights, and the centre of the island is characterised by a sort of cloud forest, with tree heathers and laurels, – laurisilva. I’m staying in the main town on the island, San Sebastián; next week it’s back to another San Sebastián, the one in the Basque Country, to visit the physics institute there for a month of writing (and doing calculations).

El Teide on Tenerife, from La Gomera

Snowdrops in bud, 1 February

The unsettled weather continues, wet, windy, and very little sunshine. Here in the north-west it’s been much better than in the south, thank goodness, and the mild weather has meant no problems with ice and snow (so far). Awful weather for gardening – I’ve a lot to do, chopping dead plants back and that sort of thing – and so far this year I have had only one afternoon in the garden. There are many signs of spring, though the snowdrops are still mostly in bud. A great pleasure last week – a link with the past – was to go to Cardiff for lunch with former colleagues from the physics department. Excellent journey through the beautiful countryside of the Welsh Marches, and every train arrived on time! (So much for my grumbles about the railways last time.)

Bad weather for gardening, good weather for writing, as I’m forced to stay inside by the rain. Since my last post I’ve been working hard on the book, but I am still on with chapter 7 on the self-energy in tight-binding calculations of electronic structure. The chapter is rather dense, to put it mildly, with 105 equations in about 25 pages, far more than in any other chapter. I can see no way round this, because there are several important ways of calculating the self-energy, and each of them involves quite a lot of formalism and working stuff out. I also want to set the scene for chapter 8 on transport, the way that electrons move through a molecule between metal electrodes, for which the self-energy is a vital tool.

Molecule (blue) between electrodes A (brown) and B (green). This can be replaced by the “extended molecule” (lower figure), with embedding potentials added at each end to simulate (exactly) bulk A and B.

It would be better if I had a few more examples in chapter 7, nice graphs of the density of states of atoms adsorbed on surfaces (treated à la Grimley-Newns-Anderson), or of the charge density on a molecule sandwiched between electrodes. At the moment there are just my schematic diagrams of tight-binding systems, as in the figure on the left. Chapter 7 is broken up into numerous sections and subsections, and I hope that that will make it clear and fairly digestible.

In the last post but one, “So much to write ….”, I discussed how the expressions for the embedding potential and the self-energy could be shown to be equivalent, by discretizing space and in this way converting the usual Schrödinger equation into tight-binding form. In this post I’m concerned with the methods which are actually used to calculate the self-energy in electrodes, and seeing the links between these and ways of calculating the embedding potential. Let’s suppose that we want to calculate $\Sigma_{\mathrm{A}}$ to replace the material to the right of the surface $S_{\mathrm{A}}$ in the figure above. The self-energy can be found from the Green function $\mathcal{G}_{\mathrm{A}}$ for “semi-infinite” A (this is all of A to the right of the surface), using the formula $\Sigma_{\mathrm{A}}=-h\mathcal{G}_{\mathrm{A}}h$, where h measures how an electron hops from one layer to the next. So we must find $\mathcal{G}_{\mathrm{A}}.$

Treat two layers together and repeat the process

We start by dividing the material into atomic layers, and then one way of proceeding is illustrated in the sketch above. Starting off with the separate layers (stage 1) we calculate the Green function for two layers brought together into a single entity (stage 2); we can repeat this, bringing two of these new layers together (stage 3), and so on, till after n stages we have the Green function for $2^{n-1}$ layers brought together. It doesn’t need many repeats of this process to get a pretty accurate representation of the Green function for the semi-infinite system (M. P. López Sancho, J. M. López Sancho, and J. Rubio, J. Phys. F: Metal Phys. 15 851 (1985)). This method is the tight-binding equivalent of the layer-doubling method to find the way that electrons are reflected by a metal surface in low-energy electron diffraction, which was the basis of the first method for calculating the embedding potential for a metal surface. A nice link!

The Green function for the semi-infinite solid can also be constructed from the Green function for the bulk, which is relatively straightforward to calculate.

Remove layer two in the bulk to give two semi-infinite systems.

This figure shows how to do this – all the layers together make up bulk A, but if we remove layer 2, the red layer, we are left with two semi-infinite pieces of A, with Green functions $\mathcal{G}_{\mathrm{A}}$ on each side. The formula corresponding to this operation is $\mathcal{G}_{\mathrm{A}}=G_{33}-G_{32}(G_{22})^{-1}G_{23}$, where $G_{22}$ is the bulk Green function in layer 2, $G_{33}$ is the bulk Green function for layers 3, and $G_{23},\; G_{32}$ are Green functions which couple layers 2 and 3 in the bulk. So starting from the Green function for the bulk we can find the Green function for the semi-infinite solid, which is what we need to build up the self-energy. This construction was devised by A. R. Williams, P. J. Feibelman, and N. D. Lang (Phys. Rev. 26 5433 (1982)), and is sometimes called the ideal construction, because the surface of the semi-infinite solid is an “ideal” surface – ideal in the sense that it doesn’t interact with anything on the left-hand side (at least I think that’s why). For me this construction is very reminiscent of the “Matching Green Function method” I worked on in the 1970′s, following the work of García Moliner and Rubio, in which one Green function can be built up from another – another link.

Solving the Schrödinger equation in a square well, by matching solutions inside and outside the well.

The self-energy and embedding potential can both be built up out of all the solutions of the Schrödinger equation which are allowed in the semi-infinite solid, at the energy at which we are working, and these involve determining the band structure. This was discussed by Sanvito et al. (Phys. Rev. B 59 11 936 (1999)) in the tight-binding self-energy context, and by Ishida (Surface Sci. 388 71 (1997)) for the embedding potential. But building up the wave-function out of the solutions at fixed energy is just what we do when we solve the square-well problem for example, as in this figure. What we do is to match $\phi_1$ inside the well with $\phi_2$ outside in amplitude and derivative. So there is a direct link between embedding and self-energy, and the earliest problems we solve in our quantum mechanics courses.

Hamamelis mollis, 1 February

I shall finish this post at this point, because it’s time for the last two hours of the superb Danish-Swedish detective thriller, “The Bridge”. With these programmes on BBC4 on Saturday evenings, and endless “Midsomer Murders” and Poirot on ITV3 it’s a wonder I have any time at all to write a blog, let alone a book.

# So much to learn …..

It’s only when you start writing that you realise how much you don’t know. Of course it’s the same when you are preparing a lecture course (it must be the same in schools as in universities) – all sorts of questions and difficulties arise. Now that I have more-or-less finished the chapter on LCAO embedding, in other words self-energies in tight-binding calculations of impurities, adsorbates etc., the next logical step is a chapter on the transport of electrons through molecules, the molecule being connected at each end to a metal contact described by a self-energy. Unfortunately I still have a lot of background reading to do on this subject, not just an up-to-date literature survey, but also some many-electron theory. In particular, I want to know the limitations of adding the self-energy of metallic contacts, calculated in a one-electron approach (in principle density-functional theory), to a molecule in which electron-electron interactions are important. Is this a well-defined approximation?

Electrodes A and B, attached to the molecule, can be replaced by self-energies.

To help me with this and other problems I went last week to the Netherlands, to chat with a friend who is an expert in the theory of molecular transport, and the answers were given in the thesis of one of his former students. In the Netherlands, theses are actually published as little books, and these are an invaluable reference. Before I start to write this chapter, and before I do a proper literature survey on the subject, I am going through several textbooks, including “Quantum Transport”, by S. Datta. This is a model of what a textbook should be like, starting at a simple, undergraduate level, but full of the physics of Green functions and self-energies. It has given me many ideas of what I should include in my interpretation of the embedding potential, and not only in the chapter on transport.

While I was in the Netherlands I visited the village south of Nijmegen where I used to live, a real sentimental journey. In a blog partly about transport (admittedly, electron transport!) perhaps I can remark that my train journeys there, and back to Schiphol Airport, were superb, arriving and departing to the minute, and no difficulty with connections. How different from my journey to Bristol from the Lake District earlier this week. Every train was late, every connection missed, so that I arrived back home over two hours late. But I mustn’t let this blog become a rant (it is supposed to be about writing a book), so that’s enough grumbling for the moment.

While I decide what to put in the transport chapter, which will eventually be chapter 8, I’ve started the chapter on embedding in photonics and plasmonics. The embedding potential can be used in two ways in solving Maxwell’s electromagnetic equations: firstly, to replace the infinite regions of space around the photonic structures, and into which the electromagnetic waves can escape, but also to replace the dielectric structures themselves. The first application is analogous to using the embedding potential to replace the semi-infinite substrate in the calculation of surface electronic structure, which was one of the first applications of embedding. But why should we want to replace the dielectric structure? Well, if we have a structure made of metal spheres, for example, the electric field will have discontinuities (or jumps, in ordinary language) at the surface of the spheres, and it’s always more difficult to calculate this sort of behaviour than to deal with smoothly varying properties. However, if we replace the sphere by an embedding potential over its surface, we only have to find the field outside the sphere, and the embedding potential takes care of the rest. It turns out that this is a very accurate and economical way of solving Maxwell’s equations in such structures. My most recent paper has been on photonics in metallic structures, so as all this is fresh in my mind I should be able to write this chapter without too many problems (famous last words). Of course there will be a lot of literature to check up on.

Winter sunset, Aude: Pyrenees in the background.

I’ll have no excuse not to get on with the book after this weekend, as I am having a few days in the south of France, enjoying unseasonably mild and sunny weather. Very different from the dark, wet days of winter in Cumbria! This afternoon was spent working in my little jardin – not that there is much to do at this time of year, apart from pulling out a few weeds and giving the lavender a trim. In fact there are already some very early flowers on my Forsythia here in France. But there is a lot to gladden one’s spirits in my garden in Cumbria: the snowdrops are nearly out (a few are fully out in some gardens, perhaps early varieties), and one of the treats of the garden in winter is the superb winter-flowering honeysuckle (Lonicera x purpusii).

Winter-flowering honeysuckle, January 2014.

# So much to write …..

December in the Aude – Pyrenean foothills

I’m getting behind in my blog – six weeks behind! – reflecting the state of the book. This is because I’m now dealing with embedding in a tight-binding representation, or the self-energy as it is usually called, and there is a huge amount of literature to absorb and to describe. Moreover, there are different notions of embedding in tight-binding, and somehow I’ve got to make sense of it all, tying it in with what I mean by embedding, and writing a coherent chapter. It’s not so much writer’s block, as doing a good literature search, and trying to understand other people’s work. It isn’t all gloom and doom, as I enjoyed the nice weather in autumn, and had a week back in France in early December. I cannot even blame my slow progress on spending too much time outside, as the weather in the last few weeks, from before Christmas till into the New Year has been terrible. But not as bad here in South Cumbria as in much of the country – no floods, and at least in my village, no power black-outs.

Chapter 6, on confining electrons with an embedding potential, is now finished. I’ve included many new figures and examples of electron transmission through a kink, and in the chapter on transport (possibly chapter 8) there will be some new results to include based on this work. But I’ve got to get the tight-binding chapter finished first, as this is the basis for most of the transport work. The title for chapter 7 is “Embedding in tight-binding”: it’s important to consider how embedding works in a local orbital/tight-binding representation, as local orbitals are used in many computational schemes, quantum chemistry codes, some condensed matter codes, and also in simplified or model Hamiltonians. Moreover, the embedding potential appears in calculations of electron transport through molecules, usually in a local orbital representation, as a way of including the metallic leads connected at each end of the molecule; in these calculations the embedding potential $\Sigma$ is invariably called the self-energy. The self-energy is exactly the same as the embedding potential: the concept of self-energy has been used for a long time in many-body theory, and something I want to do in this book is to see to what extent the properties of the embedding potential are the same as self-energies in general.

Among the first applications of self-energy were the papers by Grimley and Newns on the chemisorption of atoms on a metal surface (T.B. Grimley, Proc. Phys. Soc. 90 751 (1967), and D.M. Newns, Phys. Rev. 178 1123 (1969)), based on the treatment of magnetic impurities in non-magnetic metals by Anderson (P.W. Anderson, Phys. Rev. 124 41 (1961)). In all these papers, the local density of states on the impurity or adsorbate atom was found from the atomic Green function embedded into the substrate – the atomic density of states as a function of energy E looks like
$n(E)=\frac{1}{\pi}\frac{\Delta(E)}{(\epsilon_a+\Lambda(E)-E)^2+\Delta(E)^2}$,
where $\epsilon_a$ is the energy level on the isolated atom, and $\Lambda,\Delta$ are the real and imaginary parts of the self-energy,
$\Lambda(E)=\Re\Sigma(E),\quad\Delta(E)=\Im\Sigma(E),\quad\text{with}\;\;\Sigma(E)=VG_{\mathrm{sub}}(E)V$
- here $V$ is the hopping integral between the substrate and the adsorbate atom, and $G_{\mathrm{sub}}(E)$ is the substrate Green function in the region adjoining the adsorbate. $\Lambda$, the “shift function” shifts the centre of gravity of the adsorbate atomic energy level, and $\Delta$, the “chemisorption function” either broadens the atomic energy level into a Lorentzian in the case of weak adsorbate-substrate coupling, or gives rise to discrete energy levels on either side of the substrate continuum in the case of strong coupling. The expression for $\Sigma$ in terms of the adsorbate-substrate hopping and the substrate Green function crops up time and time again in applications of the self-energy, most notably these days in transport theory.

One of the interesting features of the self-energy in tight-binding is that it’s the substrate Green function which appears in the expression, whereas in my formulation of the embedding potential, $\Sigma(r_S,r'_S)$ is given by the surface inverse of the Green function. This apparent paradox was explained by Fisher (A.J. Fisher, J. Phys.: Condensed Matter 2 6079 (1990)), who showed that the embedding potential as a function of real-space coordinates can also be found without taking the inverse, namely,
$\Sigma(r_S,r'_S)=-\frac{1}{4}\frac{\partial^2 G_0}{\partial n_S\partial n'_S}$, where $G_0$ is the substrate Green function with zero amplitude on the embedding surface. There is in fact a direct connection between this formula and the tight-binding embedding formula, because if we discretize space – in other words calculate the wave-function or Green function or whatever at discrete points – the Schrödinger equation becomes tight-binding in form, and the resulting tight-binding embedding potential looks exactly the same as Fisher’s result.

One of the things I’ve included in this chapter is a derivation of tight-binding embedding in a way analogous to my original embedding method, as a variational principle. This is quite easy to do if the overlap matrix between the local orbitals is diagonal, but is a little more involved in the general case. I have also included the “re-normalization” of the embedded wave-functions in the local orbital representation. What this means is that if we normalize a wave-function in region I, with the embedding potential taking care of region II, the wave-function in region I must be re-normalized to take account of region II. This looks like:
$\psi^2_{\mathrm{ren}}=\psi^2_{\mathrm{orig}}/(1-\langle\partial \Sigma/\partial E\rangle)$,
which is a completely general result for self-energies (and also pseudopotentials, by the way – hence energy-independent pseudopotentials give the correctly normalized wave-functions). Before trying to prove this, I thought I ought to test it on the computer for a simple tight-binding system. And of course it didn’t work – conclusion, perhaps not such a general formula after all. I then realised my mistake in the computer program, the Lapack eigenvalue routine overwrites the overlap matrix, something which I had forgotten. Mistake corrected, and re-normalization works after all.

Another topic still to be included in this chapter is the Pisani method of embedding, which is used in important quantum chemical programs. There’s a lot of original literature on this which I still have to read – in fact get hold of first. This is one of the problems of writing a book when retired, especially living in a village a fair distance from the nearest university library, getting hold of some of the articles. I do have remote access to a lot of e-literature, but this is by no means complete. Actually, using journals on-line convinces me that there is nothing as good as browsing the literature with real paper journals, a big pile of Phys. Rev.’s or J. Phys.: CM’s on the library table. That’s what I call a proper literature search. Anyway, now that we are into the New Year – the year I must finish the book – it is back to writing with a vengeance.

End of the year on Whitbarrow (South Lakeland)

Poor weather for walking, over the holiday period. But a walk on the limestone of Whitbarrow was wonderful, with fine views of the estuaries and the Lake District fells.

Inglesfield on Ingleborough (Yorkshire Dales), 29 December 2013

A later walk on Ingleborough (one of the Three Peaks of the Yorkshire Dales) was disappointing for views, with mist on the top, but it was very enjoyable – a sentiment not really conveyed by this photograph.

# Electrons getting stuck

Kink in waveguide (red lines): the Green function is calculated for region I, the square with grey regions II removed by embedding.

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
$\chi_{mn}=\cos(m\pi x/a)\sin(n\pi y/a)$
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.)

Kink treated as a rectangular box. Grey area is region I, with confinement at the blue lines, and open boundary conditions at the dotted lines.

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.

Transmission through the kink vs. electron energy

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.

Density of state of kink joined on to waveguides.

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)), $\chi(r)=2i\int_S dr_s\int_s dr'_s G(r,r_s)\Im\mathrm{m}\Sigma(r_s,r'_s)\psi(r'_s)$,
which gives the wave-function inside the kink, $\chi$, in terms of the incident wave-function in the left-hand waveguide, $\psi$, the embedding potential $\Sigma$ for the left-hand waveguide, and the full kink Green function $G$. 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.