Tight-binding model for graphene ribbons

The energy spectra and wavefunctions of graphene (or, you know, any material) can be easily calculated under the tight-binding (TB) model using the formalism of second quantization. In this basis, describing the Hamiltonian operator becomes a simple graphical puzzle you can solve with pen and paper, and the interesting numerical results fall out with the help of computers. If you haven’t seen it before, you should take a look at the energy spectrum of an infinite sheet of graphene online; here we’ll immediately consider interesting edge effects.

Graphene consists of a hexagonal (“honeycomb”) lattice of carbon atoms. Some of graphene’s interesting surface effects are due to the fact that it has four orbitals, but only three connections to its nearest neighors; therefore the spare electron orbital hangs off of the edge, normal to the graphene surface.

The graphene lattice shown below (which should be considered to extend infinitely left and right; this will be our continuing convention, placing boundaries on the top and bottom of the lattice) has top and bottom edges classified as “armchair” edges.

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Compare the following, which has so called “zigzag” edges on top and bottom:

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Now we will calculate the Hamiltonian matrix for the zigzag model. First we choose a unit cell which forms the basic unit of tesselation for our 1D periodicity (in the x direction). Consider a carbon atom on the top of the zigzag lattice; call it site 1. Site 2 is then chosen to be the atom to its bottom left; 3 is below 2; 4 is to the bottom right of 3; 5 is below 4, and so on, so that our unit cell consists of a “wavy” collection of carbon atoms stacked somewhat vertically.

In the second quantization basis, we deal with creation and destruction operators (which are adjoint). Assign such a construction operator to each site in the unit cell, so that we have a collection of operators graphene ribbons_5.gif,graphene ribbons_6.gif, and so forth. The Hamiltonian in the TB model is

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where the sum is taken over nearest neighbors in the lattice (the model may be naturally extended to next-nearest neighbors and so on) and t is the “hopping parameter” which controls the likelihood with which an electron transfers between carbons. We wish to Fourier transform this Hamiltonian so that we may work in momentum space, but we can only do this in the x-direction; the finite extent in y destroys the LCA group structure. The spacial creation operator above may be expressed as

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where j indexes the site in the unit cell (essentially: the location in the y direction) and graphene ribbons_9.gif gives the translation distance between unit cells. Now we have

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Now whenever graphene ribbons_11.gif, that summand is just graphene ribbons_12.gif. Summing over all nodes i gives graphene ribbons_13.gif.

Therefore the Hamiltonian in the graphene ribbons_14.gif basis is (e.g.)

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The eigenvalues are of course the allowable energy bands:

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The zero-mode band in the center corresponds to the edge state on the zigzag boundary; it is topologically robust. The wavefunction intensity is found by sum of square moduli of the eigenstate components; below we graph the two edge states:

The following grahps show one of the edge states (there is another, symmetric one, correspondingly large near site 200). The first graph shows detail up close; note the difference in the actual squared probability amplitiude between them. Clearly this edge state appears correspondingly with the zero-mode energy state in the above spectrum.

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Now for the armchair model, which I don’t feel like writing up at the moment.

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