Photons at nonzero chemical potential

14 02 2009

“A system far removed from its condition of equilibrium is the one chosen if we wish to harness its processes for the doing of useful work” [G. N. Lewis and M. Randall, Thermodynamics, 1923, p. 111].

The basic way of doing work is a Carnot cycle, that converts a quantity of heat Q partially into a quantity of work W. The work is energy free of entropy, so the initial entropy Q/T must be sent to a lower temperature sink, then Q’/T’=Q/T, and the efficiency W/Q is (1-T’/T). This process not only can happen, it does happen (with less efficiency) because it increases the total entropy (Q’/T’>Q/T). We switch on the machine removing some barrier that impedes the process from happening, this is our control over the machine. But a “machine” is a way of speaking (these concepts were discussed some 150 years ago, when thermal machines were still a scientific goal, a work of imagination). This kind of processes also happen naturally out of our control, for example in plants, creating chemical fuels (stored energy able to produce work by combustion in another machine) from sunlight.

We use solar cells to do something similar as plants do. The source of energy for photovoltaic devices is the solar spectrum, that is distributed in frequency closely to blackbody radiation as shown in the figure.

fig-3The Planck spectrum is a combination of the density of states of the radiation and the photon occupation numer of the radiation modes. The occupation formula for photons (bosons) is shown in the figure. This is the Bose-Einstein distribution function at zero chemical potential .

One can have many kinds of photon distributions, from the thermal distribution described by Planck’s formula, to a thin lineshape in laser emission (which has very small entropy). It is interesting to note that photons can be in a thermal distribution with nonzero chemical potential. One example is the setup of the next figure, that shows a diode in contact with the heat sink, and receiving light from a front emitter [P. Berdahl, J. Appl. Phys. 58, 1369 (1985)].


Assuming that the semiconductor only absorbs in a narrow frequency strip near the bandgap energy E, the photons generated by radiative recombination and received from the emitter must be conserved. These photons arrive to equilibrium with the electron-hole pairs, thus the chemical potential of the photons is the same as that of the carriers. The modified formula for the number of photons in mode E (the gap energy as assumed) is shown below. This is the occupation function of the radiation mode, when the chemical potential is mu


Everybody working with electronic devices is familiar with the (Fermi-Dirac)  distribution of electrons at (electro)chemical potential V. This function is 1 below V, and above V it vanishes progressively as the energy increases, with exponential dependence that approaches very closely the Boltzmann distribution. Note how different is the distribution for photons: No photons with energy less than V, and the photons tend to accumulate at the lowest energy available, which is V. Well above V, it does’nt matter if the particles are photons or electrons.

When electrons and holes in a semiconductor like Cu2O are bound by electrostatic attraction they form an exciton, and these particles also follow the Bose-Einstein distribution. So one can have bosons in a semiconductor, and their number fixes the chemical potential. Excitons display the distribution in energy shown above, and this is measurable [Hulin et al., Phys. Rev. Lett. 45, 1970 (1980)]. By increasing the concentration of the ground state beyond a certain threshold, a special state of matter called Bose-Einstein condensation can be formed in the semiconductor [ J. Kasprzak et al., Nature 443, 409 (2006)].

By the way, radiation is heat? The answer depends on the properties of the radiation. One can attribute a temperature to radiation flux, considering the energy and entropy that it carries: T=(dQ/dt)/(dS/dt). One can then proove that for the Planck distribution the radiation is in thermal equilibrium  with the furnace emitting that radiation, so blackbody radiation qualifies as “pure” heat [P. T. Landsberg and D. A. Evans, Phys. Rev. 166, 242 (1968); C. E. Mugan, Am. J. Phys. 73, 315 (2005)].

When the radiation tends to monochromatic, the entropy of the radiation decreases, and the temperature increases, and according to Carnot’s criterion it is a much better source of energy to extract work from it, using a machine like the diode drawn above, a solar cell.

So 150 years ago people was dreaming of optimizing combustion machines that provide mechanical work buring fuels that originated from sunlight. Now such machines are in cars and everywhere but we view them also as carbon dioxide producing machines. The dream now is to optimize the Carnot cycle to produce work from the available original source, the light source. This is a high quality heat focus (T = 5000 K), however the energy is very disperse when it arrives in the earth. That is why it is critical to produce very efficient and cheap solar cells.


J. Bisquert
Excitons diffusion and singlet–triplet occupation at high Bose–Einstein chemical potential
Chemical Physics Letters, 462, 229-233 (2008).



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