Abstract
The conversion of light into different types of energy is based upon the interaction of electromagnetic radiation with matter. The matter might be represented by atoms, molecules, small clusters, liquids, or solids, such as metals, semiconductors, or dielectrics, and the radiation may be formulated in the wave approach with Maxwell’s equations. The light-matter interaction also may be expressed in the particle picture with Hamiltonians for each of the relevant species and by the appropriate vector potential for the radiation.
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Notes
- 1.
Spectral splitting means the subdivision of the solar spectrum into different parts and their subsequent individual conversion.
- 2.
The Greek prefix ‘endo’ means ‘internal’, so the idea here is that the internal process is reversible.
- 3.
Heats and work are regarded as flowing magnitudes infinitesimally slowly fed to or extracted from the system, in order to allow for the application of thermal-equilibrium relations.
- 4.
Note, that \(\dot{E}\), \(\dot{S}\), and \(\dot{S_{\mathrm{gen}}}\) are time derivatives of the internal magnitudes of the receiver system.
- 5.
After rearrangement the relation for the Mueser efficiency reads
$$\displaystyle{ \left ( \frac{T_{\mathrm{rec}}} {T_{\mathrm{Sun}}}\right )^{5} -\left (\frac{T_{\mathrm{Earth}}} {T_{\mathrm{Sun}}} \right )\left ( \frac{T_{\mathrm{rec}}} {T_{\mathrm{Sun}}}\right )^{4} + \left (\eta _{\mathrm{ Mues}} - 1\right )\left ( \frac{T_{\mathrm{rec}}} {T_{\mathrm{Sun}}}\right ) + \left (\frac{T_{\mathrm{Earth}}} {T_{\mathrm{Sun}}} \right ) = 0 }$$(4.32) - 6.
The true planetary temperatures can depart substantially from those estimated here with the assumption of ideal ‘black’ receivers/emitters, due to their individual compositions and spectrally dependent absorptivities and emissivities.
- 7.
Here the corresponding solid angles for the light input depend on the distance d SP and are replaced by \(\varOmega _{\mathrm{in}}(d_{\mathrm{SP}}) = (1/4)\left (R_{\mathrm{Sun}}/d_{\mathrm{SP}}\right )^{2}\).
- 8.
An intuitive way to understand the formation of energetic regimes allowed for electron occupation is to consider a one-dimensional periodic arrangement of potential wells at a certain distance from one another. In an isolated single well of limited depth exist a certain finite number m of discrete energy levels, whereas in an arrangement of n such wells each of the m levels of one of the wells interferes with the companion levels of type m of the other n − 1 wells to form an ensemble of levels that are energetically split. The total energy separation of this splitting converges towards a particular value, even for an infinite number of wells n → ∞. The behavior of such an infinite one-dimensional arrangement of potential wells is formulated by the Kronig–Penney model [7, 8] which can be solved analytically showing the formation of individual energy bands for the electrons.
- 9.
As the solid absorbers of solar light are commonly operated in the neighborhood of T Earth = 300 K boson behavior of electrons characterizing super conductivity is excluded.
- 10.
In the Fermi–Dirac distribution function the Fermi energy ε F designates for moderate temperatures the energetic position of the transition of the occupation probability from high (f F > 0. 5) to low (f F < 0. 5), and in the particular case of T → 0, the kink in the step-like distribution function represents ε F.
- 11.
Undoped means except the electrons in VB and CB no additional charges introduced by impurities in the lattice exist. In undoped semiconductors and insulators for the temperature T → 0 states in the valence band (D VB(ε)) are completely occupied by electrons whereas conduction band states are completely unoccupied.
- 12.
Basically in the semiclassical approach free electrons provide for transport of charges. Their contribution to the electric current density in terms of velocity \(\mathbf{v}\) and wave vector \(\mathbf{k}\) reads \(\mathbf{j} = \left ((-e)(1/(4\pi ))\right )\int _{\mathrm{occup.}}\mathbf{v}(\mathbf{k})d\mathbf{k}\) with integration over all occupied states in the respective band. The contribution of unoccupied states in this band can be formulated by the integral of a completely occupied band which is vanishing (\((1/(4\pi ))\int _{\mathrm{all\ states}}\mathbf{v}(\mathbf{k})d\mathbf{k} = 0\)) subtracting the supply of the unoccupied states \(\mathbf{j} = \left ((+e)(1/(4\pi ))\right )\int _{\mathrm{unoccup.}}\mathbf{v}(\mathbf{k})d\mathbf{k}\); accordingly the current density can be formulated by the unoccupied states with apparent particles of positive elementary charge (holes) moving in opposite direction compared to electrons (see Fig. 4.19 and [7]).
- 13.
For the sake of simplicity, in order to be able to evaluate the integral analytically, the upper limit of the integral is commonly taken to be ε → ∞ rather than the upper edge of CB, the vacuum level ε vac, and for typical temperatures, the Fermi distribution function reduces the contribution of ε > ε vac to n CB to negligible values. An analogous simplification is applied for the lower boundary value of the integral for holes ε → −∞.
- 14.
At the minima and maxima of \(\epsilon =\epsilon (\mathbf{k})\) that correspond to the top of the valence band ε = ε V and the bottom of the conduction band ε = ε C, the energy of electrons or holes depends to a good approximation quadratically on the wave vector ε ∼ k 2. Using this dependence for a three-dimensional solid, the density of states, again in the vicinity of ε V and ε C, yields \(D \sim \sqrt{\epsilon }\).
- 15.
The effective mass represents the effect of the periodic potential of the crystal on the motion of electrons (and holes) in externally applied electric fields, and it is derived from the dispersion relation \(\epsilon =\epsilon (\mathbf{k})\) for the electrons [9] by
$$\displaystyle{ \frac{1} {m^{{\ast}}} = \frac{1} {\hslash ^{2}}\left (\begin{array}{ccc} \frac{\partial ^{2}\epsilon } {\partial k_{x}^{2}} & \frac{\partial ^{2}\epsilon } {\partial k_{x}\partial k_{y}} & \frac{\partial ^{2}\epsilon } {\partial k_{x}\partial k_{z}} \\ \frac{\partial ^{2}\epsilon } {\partial k_{y}\partial k_{x}} & \frac{\partial ^{2}\epsilon } {\partial k_{y}^{2}} & \frac{\partial ^{2}\epsilon } {\partial k_{y}\partial k_{z}} \\ \frac{\partial ^{2}\epsilon } {\partial k_{z}\partial k_{x}} & \frac{\partial ^{2}\epsilon } {\partial k_{z}\partial k_{y}} & \frac{\partial ^{2}\epsilon } {\partial k_{z}^{2}} \end{array} \right )\;.}$$ - 16.
For the Fermi energy ε F sufficiently separated from VB- and CB-edge, explicitly \((\epsilon _{\mathrm{C}} -\epsilon _{\mathrm{F}}) > 3\mathit{kT}\) and \((\epsilon _{\mathrm{F}} -\epsilon _{\mathrm{V}}) > 3\mathit{kT}\), the Boltzmann-energy distribution function is a reasonable approximation of the Fermi distribution function.
- 17.
The effective masses in the upper relation may be regarded as an abbreviation of the respective density of states.
- 18.
Due to the \(\epsilon \left (\mathbf{k}\right )\) relation in crystals, wave-vector and energy relaxation times are in general not equal, τ k ≠ τ ε .
- 19.
For recombination lifetimes as small as the intraband relaxation times, we do not need to consider such semiconductors for photovoltaic applications since the photoexcited state is not conserved sufficiently long and thus the splitting of the quasi-Fermi levels becomes negligible.
- 20.
Whereas the densities of electrons in CB and of holes in VB are formulated with quasi-Fermi distribution functions, their counterparts, the electrons in VB and the holes in CB are derived via conservation of states \(n_{\mathrm{VB}}(\epsilon ) = D_{\mathrm{VB}}(\epsilon )\left [1 - f_{\mathrm{p}}(\epsilon,\epsilon _{\mathrm{Fp}})\right ]\), and \(p_{\mathrm{CB}}(\epsilon ) = D_{\mathrm{CB}}(\epsilon )\left [1 - f_{\mathrm{n}}(\epsilon,\epsilon _{\mathrm{Fn}})\right ]\).
- 21.
This mode of operation is called “open circuit”.
- 22.
Here it is assumed that each photon with energy beyond the threshold for absorption, viz., \(\hslash \omega \geq \epsilon _{\mathrm{g}} =\epsilon _{\mathrm{C}} -\epsilon _{\mathrm{V}}\), generates one electron–hole pair, which recombines after a certain time and creates one photon. Any nonlinear generation and/or recombination process, such as impact ionization and Auger recombination, is neglected.
- 23.
The above approximation is based upon neglecting the contribution of photons from the Universe. Despite the large solid angle from which the absorbers might receive such photons, (4π −Ω in), the low background temperature of the Universe T Univ = 3 K makes this part negligible. The marginal contribution of photons from stars and the Moon are also neglected. The modified balance after replacing the Bose–Einstein distribution function by the Boltzmann energy distribution and assuming the solid angle for emission Ω out = 4π becomes
$$\displaystyle\begin{array}{rcl} & & \varOmega _{\mathrm{in}}\int _{\epsilon _{\mathrm{g}}}^{\infty }\left (\hslash \omega \right )^{2}\exp \left (- \frac{\hslash \omega } {\mathit{kT}_{\mathrm{Sun}}}\right )\mathrm{d}(\hslash \omega ) {}\\ & & \quad = 4\pi \int _{\epsilon _{\mathrm{g}}}^{\infty }\left (\hslash \omega \right )^{2}\exp \left (-\frac{\hslash \omega -\mu _{\mathrm{np,oc}}} {\mathit{kT}_{\mathrm{abs}}} \right )\mathrm{d}(\hslash \omega )\;. {}\\ \end{array}$$Introducing the analytical solution of
$$\displaystyle{\int x^{2}\exp (\beta x)\mathrm{d}x = \left (\frac{x^{2}} {\beta } + \frac{2x} {\beta ^{2}} + \frac{2} {\beta ^{3}}\right )\exp (\beta x)\;,}$$one arrives at
$$\displaystyle{\exp \left ( \frac{\mu _{\mathrm{np,oc}}} {\mathit{kT}_{\mathrm{abs}}}\right ) = \left (\frac{\varOmega _{\mathrm{in}}} {4\pi } \right )\left (\frac{\epsilon _{\mathrm{g}}^{2}\mathit{kT}_{\mathrm{Sun}} + 2\epsilon _{\mathrm{g}}\left (\mathit{kT}_{\mathrm{Sun}}\right )^{2} + 2\left (\mathit{kT}_{\mathrm{Sun}}\right )^{3}} {\epsilon _{\mathrm{g}}^{2}\mathit{kT}_{\mathrm{abs}} + 2\epsilon _{\mathrm{g}}\left (\mathit{kT}_{\mathrm{abs}}\right )^{2} + 2\left (\mathit{kT}_{\mathrm{abs}}\right )^{3}} \right )\left (\exp \left (- \frac{\epsilon _{\mathrm{g}}} {\mathit{kT}_{\mathrm{Sun}}} + \frac{\epsilon _{\mathrm{g}}} {\mathit{kT}_{\mathrm{abs}}}\right )\right )\;,}$$and finally one obtains
$$\displaystyle{\mu _{\mathrm{np,oc}} = \mathit{kT}_{\mathrm{abs}}\ln \left [\frac{\varOmega _{\mathrm{in}}} {4\pi } \frac{\mathit{kT}_{\mathrm{Sun}}} {\mathit{kT}_{\mathrm{abs}}} \exp \left (- \frac{\epsilon _{\mathrm{g}}} {\mathit{kT}_{\mathrm{Sun}}} + \frac{\epsilon _{\mathrm{g}}} {\mathit{kT}_{\mathrm{abs}}}\right )\right ]\;.}$$ - 24.
Ideal here means that all solar photons arriving at the absorber are absorbed (no reflection of solar light); absorption of one photon generates one electron–hole pair and vice versa for recombination and light emission.
- 25.
Here we have again neglected the photon contribution from the environment/Universe.
- 26.
We approximate \(-(A - B) \approx -A\) since A ≫ B.
- 27.
Here we neglect the fact that, for thermodynamic reasons, any crystal at temperature T > 0 will contain a certain amount of defects, such as point defects, dislocations, interstitial site occupation, etc., not to mention unavoidable chemical impurities.
- 28.
In this mode of operation the absorber only emits thermal equilibrium radiation, as it is kept at ambient temperature by an appropriate heat reservoir.
- 29.
AM0 relates to non-concentrated sunlight at the Earth‘s position without the influence of the atmosphere on the spectral distribution, such as photon energy dependent scattering, absorption, and reflection.
- 30.
This mode of operation is called ‘maximum power point’ (mpp).
- 31.
For T → 0 and corresponding \((\epsilon _{\mathrm{Fn}} -\epsilon _{\mathrm{Fp}}) \rightarrow \epsilon _{\mathrm{g}}\) we have neglected inversion and the respective transition to lasing; in addition we did not consider that doping by incorporation of impurities for T → 0 will not work.
- 32.
In an ideal two-band system, the depletion of the initial state (electrons in the VB) corresponds directly to the depletion of the final state (holes in CB).
- 33.
In metals there are often several bands overlapping each other and crossing ε F.
- 34.
For a typical lattice constant of 0. 3 nm, the first Brillouin zone spans from
$$\displaystyle{- \frac{\pi } {0.3\,\mathrm{nm}} \leq k \leq \frac{\pi } {0.3\,\mathrm{nm}},}$$whereas photon wave vectors, e.g., for ℏ ω = 2 eV, correspond to λ ≈ 600 nm and relate to wave vectors
$$\displaystyle{k_{\mathrm{phot,\,2\,eV}} \approx \frac{2\pi } {600\,\mathrm{nm}} \approx 0.01(1/\mathrm{nm}) \ll k\;.}$$ - 35.
Remember that the effective mass m ∗ may be interpreted as an abbreviation for the density of states in the parabolic band approximation. This abbreviation is usually applied for the density of states in the valence D VB and in the conduction band D CB as well as for their combined density of states D c.
- 36.
In amorphous and microcrystalline semiconductors, such as a-Ge:H, a-Si:H, μc-Si etc., the spectral absorption for low absorption coefficients is strongly governed by states in band tails and at midgap. Due to the contribution of these states it is difficult to determine the pseudo-optical band gap from experimental absorption coefficients α(ℏ ω); a more appropriate evaluation of the pseudo-band gap according to the proposal of Tauc [18] is derived when plotting α 2 versus ℏ ω for direct, or equivalently \(\sqrt{\alpha }\) versus ℏ ω for indirect semiconductors; in these plots the linear extrapolation of α 2, or \(\sqrt{\alpha }\) respectively at sufficiently high photon energies resulting from band-to band transitions yields consistent values for the pseudo-band gaps by the intercept with the photon-energy axis.
- 37.
The rate of carriers extracted from the absorber is composed of electrons and holes moving in opposite directions and thus contribution by particular fractions to the entire electrical output current.
- 38.
Note, that the free energy of those thermal equilibrium photons with respect to a converter at identical temperature disappears.
- 39.
This heat \(\varDelta \dot{q}\) (generated by fast relaxation of ‘hot’ photoexcited electrons and holes which subsequently recombine with emission of photons with lower energy) may easily be translated into an entropy density term by \((1/T_{\mathrm{abs}})\dot{q} =\dot{ s}\).
- 40.
The directional dispersion of light is commonly treated by the difference in the etendue of the propagating light before and after the interaction with matter. The etendue \(\varepsilon\) describes the “radiance” of the light beam, which is conserved in non-scattering, non-absorbing media, where
$$\displaystyle{\updelta \varepsilon = n^{2}\cos \theta \updelta \varOmega \updelta A,}$$with angle θ to the normal to the area element \(\updelta A\), solid angle for emission \(\updelta \varOmega\), and refractive index n of the matter [16]. By scattering of photons and/or by absorption and subsequent emission with Stokes shift into a larger solid angle than that of the incoming photon flux, due to the lower photon flux density of the ‘bundle’ of rays in the ‘beam’, the ability of the photon gas to perform valuable work is reduced. In the language of statistical physics, it is the increase in etendue that reduces the free energy of the photons.
- 41.
For the sake of simplicity we use a flat profile and assume moreover negligible surface recombination at the rear and front as well as density-independent carrier lifetimes τ.
- 42.
Starting from the fundamental relation of thermodynamics
$$\displaystyle{\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V +\mu _{i}\mathrm{d}N_{i}\;,}$$where U, T, S, p, V, μ, N are the internal energy, temperature, entropy, pressure, volume, chemical potential of a particular species, and number N i = N i (z), for spatial rearrangement of particles with no change in total internal energy dU = 0, constant temperature T = constant, no volume change dV = 0, and solely an exchange of the chemical potential of particles/species at particular locations, we get in essence
$$\displaystyle{\mathrm{d}S = \frac{1} {T}\left (\mathrm{d}U + p\mathrm{d}V -\mu _{i}\mathrm{d}N_{i}\right ) = -\frac{\mu _{i}} {T}\mathrm{d}N_{i}\;.}$$ - 43.
Since the electron system of the absorber has no memory, how the excitation to the excess carrier concentration has been performed, the excitation state also could have been established by the very same photon flux for \(\hslash \omega \geq \epsilon _{\mathrm{g}}\) from the Sun.
- 44.
We approximate the radiation out
$$\displaystyle{ \frac{\varOmega _{\mathrm{out}}} {c_{0}^{2}4\pi ^{3}\hslash ^{3}}\int _{\epsilon _{\mathrm{g}}}^{\infty } \frac{\left (\hslash \omega \right )^{3}} {\exp \left ( \frac{\hslash \omega -\mu } {\mathit{kT}_{\mathrm{abs}}}\right ) - 1}\mathrm{d}(\hslash \omega )}$$by
$$\displaystyle{ \frac{\varOmega _{\mathrm{out}}} {c_{0}^{2}4\pi ^{3}\hslash ^{3}}\int _{\epsilon _{\mathrm{g}}}^{\infty }\left (\hslash \omega \right )^{3}\exp \left ( \frac{\hslash \omega -\mu } {\mathit{kT}_{\mathrm{abs}}}\right )\mathrm{d}(\hslash \omega )}$$and get the fraction
$$\displaystyle{(\varGamma _{\epsilon,\mathrm{out}}(oc))/(\varGamma _{\epsilon,\mathrm{out}}(mpp)) = \frac{\int _{\epsilon _{\mathrm{g}}}^{\infty }\left (\hslash \omega \right )^{3}\exp \left (-\frac{\hslash \omega -\mu _{\mathrm{oc}}} {\mathit{kT}_{\mathrm{abs}}} \right )\mathrm{d}(\hslash \omega )} {\int _{\epsilon _{\mathrm{g}}}^{\infty }\left (\hslash \omega \right )^{3}\exp \left (-\frac{\hslash \omega - (\mu _{\mathrm{oc}} - 3\mathit{kT}_{\mathrm{abs}})} {\mathit{kT}_{\mathrm{abs}}} \right )\mathrm{d}(\hslash \omega )},}$$which simplifies to
$$\displaystyle{\exp \left ( \frac{\mu _{\mathrm{oc}}} {\mathit{kT}_{\mathrm{abs}}}\right )/\exp \left (\frac{(\mu _{\mathrm{oc}} - 3\mathit{kT}_{\mathrm{abs}})} {\mathit{kT}_{\mathrm{abs}}} \right ) =\exp \left (\frac{3\mathit{kT}_{\mathrm{abs}}} {\mathit{kT}_{\mathrm{abs}}} \right ) =\exp \left (3\right ).}$$ - 45.
Only a weak dependence of \(\epsilon _{\mathrm{C}}(\mathbf{x})\), \(\epsilon _{\mathrm{Fn}}(\mathbf{x})\), and \(T(\mathbf{x})\) is assumed, justifying the idea that only the gradients are spatial functions, while the temperature dependence of the effective density N C of electrons in the conduction band is also neglected (N C = constant).
- 46.
The electron mobility is represented here by the asterixed \(\mu _{\mathrm{n}}^{{\ast}}\) and might not be confused with the electron chemical potential μ n.
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Bauer, G.H. (2015). Theoretical Limits for Solar Light Conversion. In: Photovoltaic Solar Energy Conversion. Lecture Notes in Physics, vol 901. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46684-1_4
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