Shockley-Queisser limit

The Shockley-Queisser limit , also Shockley-Queisser limit , specifies an upper limit in solid-state physics for the efficiency with which solar cells can convert sunlight into electrical energy . William B. Shockley and Hans-Joachim Queisser looked at the absorption and remission of photons in 1961 in order to derive the limit. The special thing here is the purely thermodynamic approach and the idealization of all bodies involved as black radiation bodies .

description

Shockley-Queisser limit for maximum efficiency of a solar cell as a function of the band gap

Usable electrical energy of a solar cell in the black area, which is below the Shockley-Queisser limit. In addition, the various losses in colors in a solar cell as a function of the band gap

In a solar cell , light is converted into electrical energy by exciting electrons from the valence band into the conduction band. There are two key loss mechanisms here. On the one hand, this is the fact that an electron lifted into the conduction band can emit at most the band gap energy as usable energy, regardless of how strongly it was excited by the incident light. In addition, it must be taken into account that the solar cell itself also has a certain finite temperature and thus emits energy in the form of radiant recombination through the black body radiation emitted by it .

The following considerations apply to the case of a cell with a single pn junction . With multi -junction solar cells in which several pn junctions with different band gaps are combined, higher efficiencies can also be achieved.

Spectral Losses

The size of the band gap of the semiconductor is decisive for the energy that can be gained per excited electron . Regardless of how far the electron is excited over the lower edge of the conduction band, the maximum energy of the band gap per electron is obtained as electrical energy. The rest is lost to the semiconductor as phonons in thermal relaxation . With the electrical power that is obtained from all excited electrons, one has to take into account that more electrons are generated with a small band gap. With a large band gap, each individual electron has more energy. A compromise must therefore be found between the following borderline cases: ${\displaystyle E_{\mathrm {g} }}$

• Large band gap: only high-energy light (blue and ultraviolet light) can generate electron-hole pairs , since longer wavelengths are not absorbed. Because of the large band gap, each electron has a high energy.
• Small band gap: Even long-wave light can excite electrons, so that a large number of electrons are excited into the conduction band. However, due to collision processes with the crystal lattice (phonon excitation), these lose part of their energy within a few hundred femtoseconds until they only have the energy of the band gap.

Ultimate limit and Shockley-Queisser limit. Here idealized black bodies were assumed for the sun (6000 K) and the solar cell (300 K).

The energy in electromagnetic (solar) radiation is made up of the energy of a single photon and the total number of photons of the frequency , i.e. H. the spectrum given. ${\displaystyle h\nu }$ ${\displaystyle f(\nu )}$ ${\displaystyle \nu }$

${\displaystyle E_{\mathrm {Strahlung} }=\int \limits _{0}^{\infty }h\nu \cdot f(\nu )\,\mathrm {d} \nu }$

Since only the photons whose frequency is higher than are absorbed and each generates an electron which, after its relaxation processes, has an energy of , the total electrical energy of the electrons results in ${\displaystyle E_{\mathrm {g} }/h}$ ${\displaystyle E_{\mathrm {g} }}$

${\displaystyle E_{\mathrm {Elektronen} }=E_{\mathrm {g} }\int \limits _{E_{\mathrm {g} }/h}^{\infty }f(\nu )\,\mathrm {d} \nu }$

The resulting efficiency from the ratio of to is called the “ultimate efficiency limit” and describes the maximum efficiency of a solar cell at 0 K, which therefore does not emit its own radiation. The value depends crucially on the band gap and the spectrum . The adjacent orange curve describes the course of the ultimate limit as a function of the band gap of the semiconductor . No solar spectrum was used for this, but the black body spectrum of a body at 6000 K, which corresponds to the surface temperature of the sun . The maximum of approx. 44% can be found at a band gap of approx. 1.1 eV. ${\displaystyle E_{\mathrm {Elektronen} }}$ ${\displaystyle E_{\mathrm {Strahlung} }}$ ${\displaystyle E_{\mathrm {g} }}$ ${\displaystyle f(\nu )}$

Recombination losses

Sketch of all radiations involved in the Shockley-Queisser limit. Incoming radiation is that of the sun (decrease in intensity compared to the radiation density on the solar surface) and the environment. Outgoing radiation is only that of the solar cell to the environment (increased spectrum, since there are exponentially more free charge carriers due to voltage)

All radiation spectra that are necessary to calculate the Shockley-Queisser limit. It should be noted here that these are only used (green) or emitted (red) up to the band gap energy of the semiconductor material used in the solar cell.

Since the solar cell is operated at a finite temperature, it itself emits black body radiation into the environment. It is common for a black body that it absorbs the same radiation as it emits, provided that the body itself and its surroundings have the same temperature, as is the case with the solar cell in a first approximation. However, due to the voltage applied in the solar cell, its radiation power is far above that of a conventional black body (and thus above that of the environment), since with increasing voltage there are exponentially more free charge carriers that can recombine and thus contribute to the characteristic black body spectrum. This intrinsic, radiating recombination of the electrons only occurs at energies below the band gap, since transitions below the band gap are not possible due to the non-existent electronic states.

Overall view

In order to find the theoretical maximum efficiency of a solar cell at finite temperature, all the effects described must be considered superimposed. In the sketch opposite, all radiation involved is shown with their sources and receivers.

1. Radiation of the sun (6000 K, no voltage) on the solar cell (dark green): The black body spectrum on the surface of the sun has its maximum in the visible range at approx. 500 nm with a radiation output of approx. 10 ⁵  W / (m² · nm) . However, this radiation density only applies directly to the surface of the sun. For distant objects (such as the solar cell on earth) the radiation density decreases quadratically with the radius, since the sun not only shines on the solar cell, but radiates in all spatial directions. In the sketch opposite, this is indicated by the arrows in several directions. This effect is a factor by which the incoming radiation power was reduced compared to the transmitted radiation. This ratio of the sun's radius and the distance between the earth and the sun (astronomical unit ) multiplied by the full angle of 360 ° is also known as the opening angle . ${\displaystyle {\frac {r_{\text{Sonne}}}{AU}}\approx 2\cdot 10^{-5}}$ ${\displaystyle AU}$
2. Radiation of the environment (300 K, no voltage) on the solar cell (light green): The (earthly) environment also irradiates the solar cell with a certain radiation output. However, due to the lower temperature according to Wien's law of displacement, this is at longer wavelengths than that of the sun and is also by far not as radiation-intensive. ( Planck's law of radiation )
3. Radiation of the solar cell (300 K, voltage present) to the environment (red): Since the solar cell itself also has a finite temperature (here assumed to be 300 K), this also radiates to the environment. Its maximum is at the same wavelength as that of the ambient radiation, since it has the same temperature. However, the intensity of the spectrum is increased by approx. 17 orders of magnitude compared to the conventional black body spectrum, since a voltage of the order of magnitude is applied to the solar cell . The number of free charge carriers and thus the number of recombinations per time and thus the radiation power increases according to the factor . ${\displaystyle 1\,V}$ ${\displaystyle e^{\frac {q\,U}{k_{\text{B}}T}}\;{\stackrel {U\approx 1V}{\approx }}\;10^{17}}$

In order to put all radiations in relation to each other and add them up, all three are drawn in a plot (see right). It should be noted that the outgoing radiation of the solar cell was multiplied by a factor of 2, since radiation can be emitted on the front and back. This factor does not apply to both incoming radiation, as the solar cell can only absorb radiation arriving from the front. (Exceptions are bifacial solar cells)

Now the best possible band gap is sought, since each of the three processes is limited by the corresponding wavelength . In the adjacent plot this wavelength is marked as a black vertical. Above this wavelength and thus below this energy, neither electronic excitation (absorption of radiation) nor electronic emission (emission of radiation) can take place. (see band model ) The band gap is now searched for in which the entries (hatched green) of the sun and the surroundings (negligibly small compared to the sun) are maximally large compared to the radiation output of the solar cell (hatched red). ${\displaystyle E_{\mathrm {g} }}$ ${\displaystyle \lambda _{\text{g}}={\frac {hc}{E_{\text{g}}}}}$

The dependence of the efficiency is shown in blue in the sketch above. The deviation from the first sketch comes from the fact that an idealized black body spectrum at 6000 K was assumed as the radiation function instead of the exact solar spectrum . Illumination under earthly, unfocused sunlight (solar spectrum AM 1.5; opening angle 0.5 °) results in a maximum efficiency of about 33.2% with a band gap of 1.34 eV. If the light is focused on the solar cell with a lens (corresponds to 46,200 suns), the maximum efficiency increases to 41% with a band gap of 1.1 eV. ${\displaystyle f(\nu )}$

Individual evidence

1. ^ ^{A b} William Shockley, Hans J. Queisser: Detailed Balance Limit of Efficiency of pn Junction Solar Cells . In: Journal of Applied Physics . tape 32 , no. 3 , 1961, pp. 510-519 , doi : 10.1063 / 1.1736034 .
2. ^ Sven Rühle: Tabulated values ​​of the Shockley-Queisser limit for single junction solar cells . In: Solar Energy . tape 130 , p. 139-147 , doi : 10.1016 / j.solener.2016.02.015 .
3. ↑ Giovanni Palmisano, Rosaria Ciriminna: Flexible Solar Cells . Wiley-VCH, 2008, ISBN 978-3-527-32375-3 , pp. 43 ( limited preview in Google Book search).