# Fundamental Photovoltaic Limits

## Asad Kalantarian October 24, 2010

### Submitted as coursework for Physics 240, Stanford University, Fall 2010

As our current low-cost sources of energy near their end, renewable energy sources become more valuable and worthier to pursue. The immensity of the energy that reaches earth along with its synchrony with the peak usage time of the day has made solar energy one of the most promising sources of energy for the future. Although cost per watt is the current driver in the solar cell industry, the environmental impact of the materials used as well as the footprint of the solar plants will be important in the future.

The solar efficiency rating of a photovoltaic cell, called external quantum efficiency, is a measure of the percentage of solar energy shining on the device that is harnessed with the cell. Higher efficiency cells are useful if they can help reduce the cost and/or footprint of solar plants. Research on solar cell harvesting has lead to various thermal and photovoltaic approaches. The photovoltaic approach relies on the photovoltaic effect to convert solar radiation into electricity, a process in which electrons are transferred between different bands (from valence to conduction band) within the material, resulting in the buildup of a voltage or flow of current between the two electrodes.

A broad range of photovoltaic devices ranging from single crystal to amorphous semiconductors, thin-film technologies, organic and inorganic materials, and dye-sensitized cells have been and continue to be studied. Fig. 1 shows a measure of the highest measured solar cell efficiencies for different cell types.

A good understanding of the maximum obtainable cell efficiency for different types of cells is essential for analyzing the potential of different conversion techniques. This maximum limit is clearly dependant on the solar spectrum of the radiation available to us on the surface of the earth. Since we are looking for finding the maximum power output from out device, we wish to obtain the maximum of the product of the output voltage and output current P = I V. The maximum theoretical current would be obtained if every absorbed photon produced an electron that contributed to the current of the cell. Thus the maximum possible current (electrons per second times the charge on an electron) is directly related to the rate of photon absorption (photons per second). The average solar intensity reaching the earth's surface is 1 kW/m2.

 Fig. 1: NREL compilation of best research solar cell efficiencies. [1] Source: Wikimedia Commons.

The maximum current density increases as the band-gap gets smaller since all photons would have sufficient energy to create electricity. A semiconductor with zero band-gap would have the largest possible short-circuit current (Jsc), but it would not be able to deliver any power to the external circuit (i.e. P = I V, so P would be 0 if V is 0). Now we consider what happens when no current flows at all.

In a semiconductor solar cell, when no charge flows out of the cell, all of the electrons generated accumulate in the n-type region and all of the holes in the p-type region, until ultimately the internal field disappears and there is no longer a barrier to internal recombination. The produces a voltage across the device called the open circuit voltage (Voc). This is also not a useful operating point since zero current means zero power.

The electrical power delivered by a cell is the product of the current that it delivers and the voltage at which it operates. The current density cannot be greater than Jsc and the voltage cannot be greater than Voc. Therefore the output power cannot be greater than Jsc × Voc. However, Jsc decreases as the bandgap increases and Voc increases. Therefore there is an inherent tradeoff and the theoretical maximum of Jsc × Voc is obtained when the band-gap is 1.1 eV, and amounts to 320 W/m2 using the AM1.5 solar spectrum, which amounts to a 30.0% external quantum efficiency (EQE). [2]

This number is known as the Shockley-Quiesser limit and represents the theoretical maximum for a cell with a single p-n junction. The power-conversion efficiency is calculated as follows: EQE = Jsc × Voc × FF / Psolar, where FF is a number less than one related to the device ideality.

There are structures and devices that can indeed beat the Shockley-Queisser Limit by changing the assumptions used in the Shockley-Queisser theory. An example would be using heterojunction devices that use multiple semiconductor materials to use the full range of the solar spectrum. These solar cells will have their own efficiency limits.

© Asad Kalantarian. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] S. Kurtz, "Opportunities and Challenges for Development of a Mature Concentrating Photovoltaic Power Industry," U.S. National Renewable Energy laboratory, Technical Report, NREL/TP-520-43208, November 2009.

[2] W. Shockley and H. J. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961).