Getting out more than you put in: an overly efficient LED

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Semiconductor light emitting diodes (LEDs) have been around for decades and they're used in a wide variety of high-tech applications. When an electrical potential is applied across an LED, work is done to each electron in the system, in an amount equal to the product of the electron's charge and the potential difference. This work excites electrons and creates holes, and some of these electron-hole pairs—an amount called the external quantum efficiency—recombine, producing a photon.

LED's have a second efficiency, called the wall-plug efficiency, that's a measure of the ratio of energy that is emitted as photons to the electrical energy that gets put in. If one wishes to write an equation for this, it would be the energy of the emitted photon(s) times the external quantum efficiency (the fraction of hole-electron pairs that combine into photons), divided by the product of the electron charge and the applied voltage.

Recently, researchers made the news because they managed to create an LED with a wall-plug efficiency that's greater than one—it emitted more energy as photons than the researchers put into it as electricity. Unfortunately, many of the reports were short on details. Have no fear: the gods of thermodynamics have their say, this isn't violating any laws of the Universe. We've taken a look at the Physical Review Letter that those reports were based on.

We've known for a while that an LED can emit photons at an energy much higher than the electrical energy supplied to each electron. When operating in this mode, the electrons are pumped not only by the electrical work, but also by the Peltier heat in the semi-conductor lattice. Typically, the external quantum efficiency in these situations is quite small, so overall, the LED is not putting out more energy than is being pumped into it.

Work done in the 1960s had concluded that in order for the wall-plug efficiency to be greater than unity, its external quantum efficiency would need to be very high as well—that is a large fraction of the electrons and holes would need to recombine and kick out a photon. More recent work concluded that, when the electrical energy delivered to a single electron was much smaller then the band gap (the energy difference between conducting and valence electrons) of the material, the external quantum efficiency should approach zero.

The authors of the PRL paper showed that this is not exactly true. At very low applied voltages, where the ratio of energy supplied by the electric source is several hundred times less than the thermal energy in the system, the external quantum efficiency becomes voltage independent. By simply lowering the voltage enough, this term becomes small and constant, about 3x10^-4 in the device described in the paper,

This allowed the authors to develop a device where the total energy of the light output was over twice that supplied by the input electrical energy. More generally, if the external quantum efficiency is constant, then the theoretical wall-plug efficiency becomes unbounded on the upper end.

All of the energy involved, however, is so low that you won't be seeing it in any consumer level products. The device, based on In_0.15Ga_0.85As_0.13Sb_0.87, is an infrared LED. When it's pumped with 29.9±0.1pW of power, the authors were able to detect the emission of 69±11pW of optical power from the emitted photons.

The equations that describe how and why this works don't imply that this is due to anything special about this particular material, though. The numbers rely on three key energy scales: the thermal energy in the system, the electrical energy, and the width of the band gap in the semiconductor. This particular semiconducting material has a narrow band gap relative to the thermal energy of the system (at 135 ^oC), with a ratio of about 15. Other recent studies used GaAs and InP, which have a much larger ratio of band gap relative to thermal energy (roughly 50 at 25 ^oC). This difference would theoretically reduce the optical output by as much as 15 orders of magnitude.

There's a way to tie this back to thermodynamics. If this system were made in a way that allowed it to be truly reversible—removing all lattice defect sites and non-radiative electron-hole recombinations—it would represent a Carnot-efficient heat engine where the lattice and photon fields would act as the hot and cold heat reservoirs. In other words, even if we could make an absolutely perfect semi-conductor, it would still never be more efficient than a Carnot-cycle engine, and hence is not violating thermodynamics by producing more energy than it takes in.

While some will find it impressive that we can make an LED that outputs more power than the electricity that's put in, others will ask "what can you do with it?" The authors suggest that the device would be immediately useful, and efficient, as a source of 4130 cm^-1 light in a 135 ^oC environment. One case where this might make sense: infrared spectroscopy of the exhaust gases from combustion processes. They also point out that doping the material in order to further reduce the ratio of the band gap to the thermal energy will widen the applicable temperature ranges and increase the power output.

Physical Review Letters, 2012. DOI: 10.1103/PhysRevLett.108.097403

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