Skip to main content
SpringerLink
Log in

Advanced Concepts: Beyond the Shockley–Queisser Limit

  • Chapter
Photovoltaic Solar Energy Conversion

Part of the book series: Lecture Notes in Physics ((LNP,volume 901))

  • 3150 Accesses

Abstract

The main difference between the theoretical limit of solar energy conversion, like that of a Mueser engine at maximum concentration (see Sect. 4.1.4, where we found η Mues = 0. 86) and the Shockley–Queisser efficiency , representing the radiative limit of a single-gap absorber illuminated by sunlight without concentration (η SQ = 0. 29, [1]) results from

  • the excess energy of photons \(\hslash \omega >\epsilon _{\mathrm{g}}\) which is converted into heat,

  • the amount of photons \(\hslash \omega <\epsilon _{\mathrm{g}}\) not absorbed,

  • the low photon solid angle \(\varOmega _{\mathrm{in}} =\varOmega _{\mathrm{Sun}} = 5.3 \times 10^{-6}\) of non-concentrated sunlight1 compared to the solid angle for emission Ω out (e.g., for flat absorbers with highly reflecting rear contacts Ω out = 2π)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Since the efficiency of ideal photovoltaic converters rises logarithmically with the absorbed photon flux (\(\sim \ln [\varGamma _{\gamma }/\varGamma _{\gamma,0}] \sim \ln [(\mathit{np})/n_{0}p_{0}]\)), the maximum achievable Γ γ which corresponds to Ω in = Ω out should be applied to the absorber to get nominally the highest conversion efficiency.

  2. 2.

    In solar thermal collectors, e.g., inhomogeneous light fluxes generate, of course, equivalent inhomogeneous temperature distributions, which in turn lead to a mixture of intensive variables and to generation of entropy.

  3. 3.

    The comprehensive derivation of the total reflection includes the distinction of the direction of photon polarization with respect to the orientation of the reflecting surface.

  4. 4.

    In molecular systems electrons undergo optical transitions for absorption and emission from and to electronic levels, to which only different vibrational and rotational terms are added. After absorption of a photon the electron in the excited state usually undergoes a relaxation to a lower vibrational energy level from which it may return to the initial electronic level by emission of a photon with lower energy compared to the energy of the absorbed photon.

  5. 5.

    A photonic crystal is the analog for photons of an ‘electronic’ crystal where the periodic arrangement of the ions with their electrostatic potentials determines the electronic band structure, i.e., the energy-wave vector relation \(\epsilon (\mathbf{k})\), in matter in general leading to energy gaps between allowed energy bands (see Sect. 4.2). The allowed energy states present solutions to the Schrödinger equation for electrons.

    By analogy, for photons whose propagation also obeys differential equations of second order in space and time, the Maxwell equations for \(\mathbf{E}(\mathbf{x},t)\), \(\mathbf{H}(\mathbf{x},t)\), a periodic arrangement of refractive indices have the solution with frequency/energy regimes in which light propagation is not allowed (in fact, the wave vectors become imaginary). These are called stop gaps. In three dimensions, the spatial arrangement of periodic refractive indices controls the energy and angular direction of these stop gaps.

  6. 6.

    Here, we have neglected the kinetic energy of electrons and holes (each of them amounting to (3∕2kT).

  7. 7.

    AMi abbreviation of Air-Mass index designates the attenuation of the solar light flux when travelling through the Earth’s atmosphere.

  8. 8.

    Recall that the entire energy of free electrons and holes after relaxation is composed of the band separation plus average kinetic energy (3∕2)kT of both type of carriers

    $$\displaystyle{\epsilon =\epsilon _{\mathrm{g}} + 3\mathit{kT}\;,}$$

    where the temperature is that of the lattice T = T phon.

  9. 9.

    The reduction of the dimensions in condensed matter systems leads to discrete energy levels for electrons and holes, separated on the average, by a typical ‘level spacing’ Δ ε. As a rule of thumb \(\varDelta \epsilon \approx \chi /N\) where N is the number of respective electrons (e.g., in the valence band) and χ the energetic width of the according band. For sufficient energy separation of these levels compared with longitudinal optical phonon energies \(\varDelta \epsilon > \hslash \omega _{\mathrm{phon,\,LO}}\) the cooling of ‘hot electrons’ requires the generation of more than one phonon (multi-phonon emission), which is less likely compared to one-phonon emission. Thus the transfer of electron energies to the lattice is reduced and might increase the relaxation time by more than one order of magnitude. This effect is called the phonon bottleneck .

  10. 10.

    In order to collect hot electrons in each of the directions in the \(\mathbf{k}\)-space the respective exits would have to be attached to each CB minimum of the Brillouin zone [in Si exist six CB minima (ellipsoidal pockets with long axes directed along the < 100 > direction) or eight CB minima in Ge (half ellipsoids with long axes along < 111 > )].

  11. 11.

    An electron-hole-pair coupled by Coulomb interaction has to be regarded as single quasi-particle (exciton); two such pairs (a bi-exciton) with sufficient spatial overlap of their wave functions behave, like a ‘molecule’ composed of two quasi-particles.

  12. 12.

    A quantum dot (QD) results from the reduction of the geometrical size of matter in three dimension with the effect of sufficient separation of energy levels (see footnote 5 in Sect. 6.5.1).

  13. 13.

    Since AM1.5 spectra contain scattering and absorption of photons in the terrestrial atmosphere, e.g., by water vapor, ripples occur in the solar light flux as well as in the spectral performance of converters.

  14. 14.

    We remember that polarizable sites may be introduced generally by bound and by free electrons al well as by ions; the polarization effect to be described also in terms of the appropriate susceptibility tensor as \(\mathbf{P} =\varepsilon _{\mathrm{0}}\overline{\boldsymbol{\chi }}\).

References

  1. W. Shockley, H.-J. Queisser, J. Appl. Phys. 32, 510 (1961)

    Article  ADS  Google Scholar 

  2. W.T. Welford, R. Winston, The Optics of Non-imaging Concentrators (Academic, New York, 1978)

    Google Scholar 

  3. W.H. Weber, J. Lambe, Appl. Opt. 15, 2299 (1976)

    Article  ADS  Google Scholar 

  4. A. Goetzberger, W. Greubel, Appl. Phys. 14, 123 (1977)

    Article  ADS  Google Scholar 

  5. E. Yablonovich, J. Opt. Soc. Am. 70, 1362 (1980)

    Article  ADS  Google Scholar 

  6. G. Smestad et al., Sol. Energy Mater. 21, 99 (1990)

    Article  Google Scholar 

  7. T. Markvart, J. Opt. A Pure Appl. Opt. 10, 015008 (2008)

    Article  Google Scholar 

  8. T. Markvart, J. Appl. Phys. 99, 026101 (2006)

    Article  ADS  Google Scholar 

  9. U. Rau et al., Appl. Phys. Lett. 87, 171101 (2005)

    Article  ADS  Google Scholar 

  10. E. Yablonovitch, J. Opt. Soc. Am. 72, 899 (1982)

    Article  ADS  Google Scholar 

  11. C. Ulbrich et al., Phys. Status Solidi (a) 205, 2831 (2008)

    Google Scholar 

  12. A. Bielawny et al., Phys. Status Solidi (a) 205, 2796 (2008)

    Google Scholar 

  13. S. Knabe et al., Phys. Status Solidi (RRL) 4, 118 (2010)

    Google Scholar 

  14. L. Shaffer, Sol. Energy 2, 21 (1958)

    Article  ADS  Google Scholar 

  15. C. Liebert, R. Hibbard, Sol. Energy 6, 84 (1962)

    Article  ADS  Google Scholar 

  16. A. De Vos, J. Phys. D: Appl. Phys. 13, 839 (1980)

    Article  ADS  Google Scholar 

  17. P. Baruch, J. Appl. Phys. 57, 1347 (1985)

    Article  ADS  Google Scholar 

  18. G.H. Bauer et al., in Proceedings of the 2nd World Conference on Photovoltaic Solar Energy Conversion, European Commission ∕ Directorate General Joint Research Centre Environment Institute Renewable Energies Unit Ispra (I), (ISBN 92-828-5179-6) 1998, p. 132

    Google Scholar 

  19. A. Marti, G.L. Araujo, Sol. Energy Mater. Sol. Cells 43, 203 (1996)

    Article  Google Scholar 

  20. W.H. Bloss et al., in Proceedings of 3rd European Photovoltaic Solar Energy Conference (Reidel Publishing Company, Dordrecht, 1981), p. 401

    Book  Google Scholar 

  21. J. Fischer et al., J. Appl. Phys. 108, 044912 (2010)

    Article  ADS  Google Scholar 

  22. E. Klampaftis et al., Sol. Energy Mater. Sol. Cells 93, 1182 (2009)

    Article  Google Scholar 

  23. A. Luque et al., J. Appl. Phys. 96, 903 (2004)

    Article  ADS  Google Scholar 

  24. A. Luque, A. Marti, Phys. Rev. Lett. 78, 5014 (1997)

    Article  ADS  Google Scholar 

  25. R.T. Ross, A.J. Nozik, J. Appl. Phys. 53, 3813 (1982); A.J. Nozik, Physica E 14, 115 (2002)

    Google Scholar 

  26. V.S. Vavilov, J. Phys. Chem. Solids 8, 223 (1959)

    Article  ADS  Google Scholar 

  27. O. Christensen, J. Appl. Phys. 47, 689 (1976)

    Article  ADS  Google Scholar 

  28. F.J. Wilkinson et al., J. Appl. Phys. 54, 1172 (1983)

    Article  ADS  Google Scholar 

  29. S. Kolodinsky et al., Appl. Phys. Lett. 63, 2405 (1993)

    Article  ADS  Google Scholar 

  30. P. Würfel et al., Progr. Photovolt. Res. Appl. 13, 277 (2005)

    Article  Google Scholar 

  31. R.J. Ellington et al., Nano Lett. 5, 865 (2005)

    Article  ADS  Google Scholar 

  32. G. Allan, C. Delerue, Phys. Rev. B 73, 205423 (2006)

    Article  ADS  Google Scholar 

  33. M.C. Beard et al., Nano Lett. 10, 3019 (2010)

    Article  ADS  Google Scholar 

  34. J.-W. Luo et al., Nano Lett. 8, 3174 (2008)

    Article  ADS  Google Scholar 

  35. F. Hallermann et al., Phys. Status Solidi (a) 205, 2844 (2008)

    Google Scholar 

  36. S. Link et al., J. Phys. Chem. B 103, 3529 (1999)

    Article  Google Scholar 

  37. S. Link et al., J. Phys. Chem. B 103, 4212 (1999)

    Article  Google Scholar 

  38. S. Pillai et al., J. Appl. Phys. 101, 093105 (2007)

    Article  ADS  Google Scholar 

  39. T.J. Coutts, Sol. Energy Mater. Sol. Cells 66, 443 (2000)

    Article  Google Scholar 

  40. N.W. Ashcroft, N.D. Mermin, Solid State Physics/International Edition (W.B. Saunders Company, Philadelphia, 1976)

    Google Scholar 

  41. J.C.C. Fan, G.W. Turner, R.GP. Gale, C.O. Bozler, in Conference Record of 14th IEEE Photovoltaic Specialists Conference (IEEE, New York, 1980), p. 1102

    Google Scholar 

  42. Khz. Seeger, Semiconductor Physics. Springer Series in Solid State Sciences, vol. 40 (Springer, Berlin, 1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Physik Carl von Ossietzky Universität AG Halbleiterphysik, Oldenburg, Germany

    Gottfried H. Bauer

Authors
  1. Gottfried H. Bauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bauer, G.H. (2015). Advanced Concepts: Beyond the Shockley–Queisser Limit. In: Photovoltaic Solar Energy Conversion. Lecture Notes in Physics, vol 901. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46684-1_6

Download citation

Publish with us

Policies and ethics

Search

Navigation