Friday, February 22, 2013

Plasmons, polarization, and intuition

This is one of those once-every-six-months self-promotion posts about a new paper from our group.  The result is sufficiently surprising, while illustrating a generalizable idea, that I think it's worth sharing.

I've written before about plasmons, the collective "normal modes" of the electronic fluid in a metal.  Like many phenomena in condensed matter physics, there are times when it is useful to think of plasmon modes in some metal structure as generic oscillators, each analogous to a mass on a spring (only the natural frequency of the plasmon has to do with the complex dielectric function of the metal, while the natural frequency of the mass on a spring is set by the mass and the spring constant). In particular, if you take two identical oscillators, nominally of the same natural frequency \( \omega_{0} \), and you couple them together, it often makes sense to describe the coupled system in terms of two "new" normal modes "built" from linear combinations of the uncoupled modes, the symmetric ( \( \omega_{\mathrm{s}}  < \omega_{0} \) ) (the individual oscillators move in phase with each other)  and antisymmetric modes ( \( \omega_{\mathrm{as}}  > \omega_{0} \) ) (the two oscillators move \( \pi \) out of phase with each other).  In quantum mechanics we see the same idea; for example, when two 1s orbitals are coupled together, it can make more sense to think instead about "hybridized" bonding ( \(\sigma\) ) and antibonding ( \(\sigma* \) ) molecular orbitals.   As my colleagues showed almost ten years ago in this highly cited paper, plasmons can hybridize, too, and hybridization can provide real insights into the plasmonic modes of complicated structures.

In my lab, we have spent quite a bit of time over the last several years playing with and looking at the local plasmon modes that live at nanoscale gaps between lithographically fabricated Au electrodes.  In many ways, these structures look a bit like two scanning tunneling microscope tips pointing at each other.  Many other groups have made similar structures, and it has been known for a long time that placing metal tips in close proximity to each other or a tip pointing down at a very nearby metal plane leads to "tip plasmon" modes that can be useful for various spectroscopies.  In the plasmon hybridization language, the local tip modes result from the hybridization of (delocalized) surface plasmon modes of the two electrodes, thanks to their very local coupling.  For those interested in these nanogap plasmon effects, by the way, I want to point out our recent review article about this, which will appear in an issue of Phys Chem Chem Phys focusing on plasmonics.

We had lingering mysteries, however, in our own particular geometry.  For example, why did we get such good reproducibility in the resonant wavelength of the modes (always near our laser line of 785 nm), and more dramatically, why did we observe our particular polarization dependence?  It's tricky to explain what I mean without a diagram, but I'll try.  "Common sense" and intuition suggest that light polarized with the electric field across the gap between the electrodes should be best at exciting modes that are localized to the gap.  That's proven to be true in many experiments (cited in the paper).  However, in our devices we find that we get the best optical response when the light is polarized with the electric field pointing along the gap (!), and that the emitted light also is polarized along the gap.

After a series of very careful experiments and calculations (collaboration with Mark Knight of the Halas group), we know the answer.  In our system, the metal wire in which the nanogap sits has a transverse plasmon mode (because of our particular choice of material and transverse dimensions) that is well matched to our laser, and optically "bright" in the sense of having a big electric dipole coupling.  Because a given nanogap is not perfectly symmetric, the higher order, multipolar modes localized to the gap (ordinarily optically "dark" because they lack a dipole coupling) get hybridized with that bright mode.  This explains our counterintuitive polarization dependence (the dipole-active transverse piece is what couples to both the incoming and outgoing far field light), and the reproducibility of the plasmon energy (it's set largely by the wire width, not the details of the gap).  Cute stuff, and it is a good example of how even well-known physics (after all, deep down this is a matter of solving Maxwell's equations) can give interesting surprises.

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