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).


In the heat of the Sun: efficiency and temperature of solar cells

7 02 2009

With the renewable energy needs being widely recognized, a careful assessment of the viable photovoltaic technologies is needed. Solar energy is clean, regular and widely available, but it arrives in the Earth surface with a low concentration. Devices for producing electricity from sunlight on a  large scale must cover a broad area, and should be produced at a low cost. One critical number to feature the diffferent photovoltaic technologies is efficiency: which percent of the radiant energy in the light impacting on the device is converted to electrical energy. Doubling the efficiency immediately doubles the energy production for the same device size.

Monocrystalline silicon (MCS) solar cells are a mature technology that currently dominates the photovoltaic market. Solar cell modules of MCS with 15-17% efficiency are mass produced and installed. Younger technologies, such as thin film amorphous silicon, provide module efficiency of about 7%, which looks rather poor in comparison to MCS.

However, efficiencies are reported following the result of standard measurement protocols, including a fixed solar cell temperature. In operation on the field, the energy conversion process causes abundant heat, and the solar cell temperature raises. This is specially acute in sunny places like Spain, where the high ambient temperature slows down the heat evacuation from the device.

MCS efficiencies dramatically fall down when the temperature raises. This is due to the fact that MCS solar cells, based on a relatively thick, single piece of crystalline Si material, obey almost perfectly the ideal model of semiconductor physics, and this implies that the voltage must go down as temperature increases, exactly as

dV/dT = -0.00288 Volt/ºC

This leads to a decrease of about 0.5 points of efficiency per ºC increase (M. B. Prince, Journal of Applied Physics, 1955) — which is hugue. Our own monitoring of MCS pannels instaled in the roof of Universitat Jaume I shows that the top conversion efficiency remains around 9%.

MCS pannels installed in Castelló, Spain

MCS pannels installed in Castelló, Spain

Alternative technologies, with lower nominal efficiencies, may therefore become seriously competitive when considering real peformance versus cost. Dye-sensitized solar cells (DSC), where developed in the early 1990s by swiss scientist Michael Grätzel and coworkers. DSC have top efficiencies of 11%, but the efficiency-temperature curve is nearly flat, since by the principles of operation DSC employ a combination of nanoscale organic and inorganic materials instead of a single inorganic semiconductor. Sony has already produced top class DSC devices, and reported the different temperature behaviour which gives DSC a potential advantage over other technologies.

DSC also show advantages of performance by captation of diffuse light, and by maintaining the efficiency event at low light intensity. This gives a great deal of energy captation when the full daily cicle of sunlight is considered. A study led by Taro Toyoda in Japan (Journal of Photochemistry and Photobiology A: Chemistry 164, 203–207, 2004) also showed that DSC may outperform MCS pannels when the energy production is compared on the basis of the same power producing area.

While MCS provides the option prefered by many for photovoltaics, if would be desirable to have a variety of competing technologies that provide alternative pathways towards the central goal of establishing available benign solar energy for the future of humankind. Discussions of the promise of new photovoltaic technologies must look not only at efficiency, and cost, but also to performance in the real conditions in the heat of the sun.