Sunday, June 27, 2010


A reader emailed me and asked if I had done a posting about excitons.  Looking back, I see that I haven't, so here is an attempt to rectify the situation.  As I've written previously, condensed matter physicists are fond of giving specific names to excitations of solid state systems when those excitations have well-defined quantum numbers (and are in that sense "particle-like").  An exciton comprises an electron and a "hole" bound together by the attractive Coulomb interaction (since an electron has charge -e and a hole has charge +e).  It is the (negative!) binding energy of the exciton that makes it different than a generic "electron-hole" excitation in which an electron is kicked out of an occupied state (leaving behind a hole) and into a previously empty state.

Excitons can exhibit very rich physics.  In a 3d crystalline system, excitons can be very analogous to hydrogen-like atoms, or more accurately, positronium, the bound state of an electron and a positron.  One can think of the electron and hole as having center-of-mass momentum, and having an exciton wavefunction that describes the relative displacement of the electron and hole, which would look like a hydrogenic orbital (s-like, p-like, etc.).  Like positronium, the electron and hole can annihilate each other and emit a photon.  Several important features crop up, however, due to the fact that the exciton exists within a solid host.  For example, one cannot ignore the screening of the electron-hole Coulomb interaction by the surrounding host.  One approximation commonly shown in textbooks is to treat this screening by using the bulk (relative) dielectric constant of the host material when solving for the exciton wavefunctions.  As a result, the exciton is much larger, spatially, than positronium - say 5 nm in extent rather than 0.5 nm.  (Note that this had better be true!  Otherwise the assumption that the bulk material can screen the interaction would not be internally consistent....)  Large excitons like this are called Wannier excitons.  In contrast, if the screening is relatively weak, the exciton can be small compared to a unit cell of the crystal.  Such a small exciton is called a Frenkel exciton.

Furthermore, the electron and hole parts of the exciton wavefunction are really "built" out of the Bloch wave electronic states of the solid.  In a semiconductor, the hole states "live" in the valence band, while the electron states live in the conduction band.  Hole states often exhibit stronger spin-orbit effects, and as a result, confinement can affect the exciton energy levels quite strongly.

Excitons may be produced by the absorption of light of appropriate energy, and therefore are of intense interest in photovoltaic research.  The comparatively strong screening in traditional semiconductors that gives large exciton spatial sizes also leads to modified binding energies.  Wannier exciton binding energies in materials like silicon can be on the order of 10 meV (as opposed to electron volts for positronium!).  In materials with weaker screening (with Frenkel-like excitons), the exciton binding energy can be higher, more like hundreds of meV.  These binding energies are of critical importance.  In a silicon pn junction, for example, the built-in electric field due to the junction is large enough to rip apart any light-produced excitons - that's how charge separation happens in a silicon solar cell.  In organic semiconductors, in contrast, the binding energies are stronger, and built-in fields are too weak to take apart excitons.  Thus, there are no homojunction organic solar cells, and this is one of a number of reasons why organic photovoltaics is challenging. 


Larry said...

I enjoyed the post and your excellent writing style. I would like to say that in Si, the thermal energy at room temperature is more than enough to rip apart any excitons. The role of the internal electric field, whether provided by a Schottky or p-n junction, is to provide the asymmetric carrier flow that is necessary to create the photovoltaic effect. Without a junction, diffusion is the only mechanism for current flow and a very small voltage will result from the slight difference in diffusivity for electrons and holes.

CAN said...

Thanks! I need share this with my BIOE course

CarlBrannen said...

To the high energy theorist, there's nothing odd about the screening of electric charge at long distances. The same thing is observed with the electron.

The energy of positronium is 6.8eV in the ground state. To see significant changes in the electric coupling constant alpha, you need vacuum polarization which sort of Feynman diagrams will show up around 1 MeV.

And hey, I just got my first quantum field theory paper published, 1006.3114.

Raja Vinay Chandra said...

Thanks for the intro on exciton. Im doing a small report and wanted to know how you'd approach an exciton in 3D quantum well. How do you write and solve the Hamiltonian.

Will be great if you can help me out here..

Thanks again!

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