This post offers an intuitive, non-mathematical explanation of the Shockley Queisser (SQ) limit. All research in solar cells today that is focused on energy efficiency , is based on understanding and tackling this limit. Thus a grasp of this limit can help in the analysis, critique and appreciation of research in multi-junction cells, III-V group semiconductors, quantum dot and concentrator solar cells.

The SQ limit is much like the carnot efficiency of heat engines. It puts an upper limit on the maximum achievable efficiency of a solar cell based on fundamental physics, material properties and on the nature of the incident solar radiation. For example, under the assumptions of the limit, commercially popular crystalline solar cells can never exceed an efficiency of 33% in converting sunlight to electricity on earth’s surface. Other considerations further lower this limit, but several techniques can also work around the assumptions and can thus create cells with greater efficiencies. We will go over all these aspects in steps. Some familiarity with the basic working of a solar cell is assumed on the reader’s part.

**Assumptions/Conditions:**

1. Each incoming photon excites only one electron-hole pair

2. Photovoltaic material consists of a single band gap energy

3. Photons with energy lower than band gap energy are not absorbed. All photons with energy greater than band gap are absorbed (infinite absorption coefficient). Any excess energy is transferred to the the generated electron-hole pair and is subsequently lost by thermal relaxation/thermalisation (collisions with molecules which generate heat)

4. Ideal, infinite mobility of charge carrier. They are able to travel to the probes regardless of where they are generated

5. Loss of electron-hole pairs is through radiative recombination only

6. Illumination of the solar cell is by unconcentrated sunlight under standard test conditions (AM 1.5, 1000 W/m2 and 300K cell temperature)

7. The sun is a black body at 6000K and the solar cell is a black body at 300K

**Limiting phenomena:**

**Spectral mismatch:** This aspect can be best illustrated by the diagram below. The sunlight reaching earth has a characteristic spectrum with intensity peaks in the visible and infra-red range. It does not have uniform irradiance across wavelengths and hence, certain wavelengths have poor irradiance even though a solar cell may be capable of absorbing them.

Image courtesy: commons.wikimedia.org

The wavelengths with energy lower than the band-gap energy pass unabsorbed and unutilized. The higher energy wavelengths are completely absorbed but only the ‘band-gap’ amount of energy is used to generate electricity. The excess energy is lost by electron-hole pairs to thermalisation. Thus, in both wavelength ranges we have energy which is lost and not converted to electric current. Additionally, energy lost by thermalisation heats up the solar cell, which in-turn further lowers its efficiency.

This spectral mismatch means that a band-gap of 1.2ev-1.34ev gives the highest efficiency (33.7%) for AM 1.5 solar spectrum. Any deviation from this band-gap gives a lower efficiency. Spectral mismatch accounts for about 48% of the SQ limit.

**Radiative recombination:** Just as an electron-hole pair can be excited by a photon, they can also recombine to emit a photon. Since this process cannot be stopped, a certain base-line recombination loss is always present in the solar cell. It is quantitatively studied in the detailed balance. It is the simplest form of recombination loss.

**Black-body radiation:** The solar cell is assumed to be a black-body at 300K, emitting a characteristic radiation that is an internal function of its temperature. It is in thermal equilibrium with its surroundings. This means that the molecules of a solar cell emit the same amount of energy as they thermally absorb. This represent a 7% loss of the incoming radiation which cannot be converted to electricity. This phenomena follows from Prevost’s law and Kirchoff’s theory of black-body radiation.

**Voltage losses (Voc):** As discussed in the previous section on spectral mismatch, only the photon energy equivalent to the band-gap energy is used to generate current via the collection of the generated electron from the conduction band and the hole from the valence band. However, the maximum voltage a solar cell can generate is rarely ever the band-gap energy voltage, but a lower value due to recombination currents. In this calculation of the SQ limit, only radiative recombination is considered. Nevertheless, this amounts to a loss in the power efficiency of a cell. This open-circuit voltage (Voc) can be up to 40% lower than band-gap voltage (eg: 1V for BG of 1.7V).

**Fill factor:** Unfortunately, even a cell voltage of Voc is difficult to achievable. A solar cell is rarely operated in an open cirucuit condition. This is usually operated at the point of peak power in the fill factor curve. This gives maximum power, but a voltage that is lower than Voc.

**The SQ limit is determined by adding up all these loss mechanisms for a given solar cell.**

**Other practical losses not accounted for in this limit:**

1. If other mechanisms such as shockley-read-hall (SRH) and auger recombination are considered, then efficiencies lower than the SQ limit are predicted. Both of these are non-radiative mechanisms. In indirect band gap materials such as Silicon, auger recombination is more dominant than radiative recombination, thus further reducing the limit to 29.4%. In this regard, the SQ limit is more suited for GaAs single junction cells

2. Optical losses: Reflection, shading, transmission (finite absorption coefficient)

3. Non-ideal fill factor due to series resistance in the bulk of the junction and leakage currents from shunt resistance or local defects.

By looking at the assumptions made in the derivation of this limit, it is apparent that some conditions on a solar cell can be changed in order to exceed the shockley queisser limit. In my next post, I will explain these and hence, touch upon the latest research in PV cells.

**Link to the original paper:** William Shockley and Hans J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells”, *Journal of Applied Physics*, Volume 32 (March 1961), pp. 510-519; doi:10.1063/1.1736034