Spectral converters and luminescent solar concentrators
aa r X i v : . [ qu a n t - ph ] J u l SPECTRAL CONVERTERS AND LUMINESCENT SOLARCONCENTRATORS
Petra F. Scudo*, Luigi Abbondanza, Roberto FuscoEni S.p.A,
Research Center for Non-Conventional Energies- Istituto ENIDonegani
Via G.Fauser 4 - 28100 Novara (Italy)*email for correspondence:[email protected]
Abstract
In this paper we present a comprehensive theoretical description of molec-ular spectral converters in the specific context of Luminescent Solar Con-centrators (LSCs). The theoretical model is an extension to a three-levelsystem interacting with a solar radiation bath of the standard quantum the-ory of atomic radiative processes. We derive the equilibrium equations ofthe conversion process and provide specific examples of application of thisprinciple to the development of solar concentration devices.1.
Introduction
Luminescent solar concentrators (LSCs) have been first introduced in thelate seventies as one of the simplest methods of concentrating sunlight inthin polymeric slabs [1] and have been lately reconsidered in the light ofthe recent advances in material science and nano-optics as a cheap alterna-tive to standard photovoltaic modules [2, 3, 4]. The concentration processcan be understood through the principles of classic geometric optics. Thepolymeric slabs are doped with active molecules, which absorb a portion ofthe incident solar radiation and re-emit part of it at lower energies, thusperforming a spectral down-shift. The radiation is isotropically scatteredby the luminescent centers and is subjected to refraction upon reaching theinterface between the slab and the air. Consequently, depending on thevalue of the material’s refractive index, part of the radiation is trapped bytotal internal reflection and part of it is lost through the upper and lowersurfaces. The trapped radiation is wave-guided and reaches the edges ofthe slab, where it can be converted into electricity by standard solar cells.These systems do not require tracking and limit the surface of the high-costmaterials used for the solar cells, thus offering a cheap alternative to stan-dard photovoltaic modules. However, to this day, a wide-scale application of these concentrators has been inhibited mainly by their low concentra-tion/conversion efficiency, due to several loss mechanisms of which the mainone is probably the low efficiency of the dyes’ wave-guiding properties due tolow absorption and self-absorption. A comprehensive review on the currentresearch status of LSCs is presented in [5]. Different Authors consideredthe problem of modeling these devices, following either a thermodynamicapproach [6] (based on Chandrasekhar’s radiative transfer theory, [8]) or acomputational ray-tracing approach [7]. Both ways represent a broad-scale,macroscopic theoretical model of LSCs. In our paper, we focus instead on amicroscopic model, aimed at characterizing the working principles of the de-vice at the molecular scale, considering the interactions between the dopingdyes and the solar photons. Indeed, the correct physical description under-lying the trapping mechanism of a LSC relies on the principles of quantummechanics which govern the molecular spectral conversion process. In doingthis, we start off from the microscopic thermodynamic model developed byRoss [10] and later elaborated by Yablonovitch in his seminal works [11, 12].With respect to the latter models, we propose a modification of the molecu-lar rate equations of the system to correctly account for interaction processesinvolving multiple molecular levels. We explicitly derive the equilibrium ra-diation density fields for the molecular system interacting with the solarphoton bath and describe a method for calculating the parameters involvedin the relaxation process coupled to the radiative interactions.2.
Definitions and notations
For a monocromatic light source S , we define the mean photon-flux density at frequency ν , ϕ ( ν ), to be(2.1) ϕ ( ν ) = I ( ν ) h ν , where h is the Planck constant and I the light intensity. The mean pho-ton flux has units photons/cm × s and may be expressed in terms of the radiation energy density per unit volume and unit bandwidth ρ as(2.2) ϕ ( ν ) = c ρ ( ν ) h ν , where c has to be substituted by c/n for a material of refractive index n .In our discussion we shall be focusing on the interactions between light andmatter, in the form of molecular or atomic transitions caused by a thermalradiation field impinging on a set of molecules. The transitions betweendifferent molecular energy levels due to electron excitations are related tophoton absorption and emission. A fundamental physical quantity of ourmodel is represented by the transition rate , which expresses the probabilitydensity per unit time of a transition between atomic energy eigenstates andis defined as(2.3) W = ϕ ( ν ) σ ( ν ) , PECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS 3 where W is the transition rate and σ ( ν ) is the transition cross-section , i.e.the effective cross-sectional atomic area (in cm ). As we shall see later, σ is determined by the strength of the dipole oscillations which, coupled tothe incident photon momentum, give rise to the transition. In a LSC thelight beam inducing molecular transitions comes from the sun and can beapproximated by a thermal radiation distribution from a black-body sourceof temperature T ≃ K . The spectral energy density per unit volumeand unit bandwidth is given by Planck’s formula for black-body radiation(2.4) ρ ( ν ) = 8 π h ν c e h ν/K T − . The same quantity can be expressed in terms of the wavelength λ as(2.5) ρ ( λ ) = 8 π h cλ e h c/λK T − . Interaction dynamics
As mentioned earlier, whereas the coarse features of a LSC can be treatedwith the principles of geometric optics, the correct dynamics of the molecularinteractions is described by the principles of quantum mechanics. The modeldeveloped here is of general validity and can be applied to all cases in whichfluorescent molecules act as individual spectral converters. In the notationof second quantization, the electromagnetic field is an ensemble of photonswhose state is defined by a momentum and polarization vector for eachfrequency ν . The field is specified by giving the number of photons in agiven state; we set n k ,ν to be the number of photons with momentum k andfrequency ν .In a volume of space V containing atoms or molecules the energy operator(total energy), or Hamiltonian H , of the system consists of three terms: theradiation field energy H em , the molecular energy H mol and the interactionenergy H int . The latter describes the intensity of the transition processesand is used to compute transition rates between different energy levels. Weintroduce the following notation borrowed from Yariv [9]. Let E k ν be theelectric field generated by photons of mode k ν (polarization and frequency)and r the position vector relative to a particle interacting with the field.The interaction Hamiltonian can be written as H int = − e E k ν · r = ie r hν k V ǫ h a † k ν e − i k · r − a k ν e i k · r i e k ν · r , (3.1)where ǫ is the electric permittivity, a, a † are the photon annihilation andcreation operator of a single radiation mode. In our system, an incident ra-diation beam coming from the sun has a distribution of different frequenciesand momenta, each associated to a specific molecular transition. In general,when a photon of mode k ν is absorbed, the molecular electrons are excited SPECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS to a higher energy level: the transition energy, which equals the electron’slevels energy difference, is given by(3.2) E − E = hν . We refer hereafter to a general chemical compound that can be schema-tized in terms of a three-level system: a fundamental state and two excitedstates relative to absorption and emission. We consider the case in whichthese levels may be treated as well-defined electronic eigenstates. Further-more, we make the assumption- common in dealing with fluorescence- thatspontaneous emission dominates over stimulated emission.During absorption, the number of photons in the radiation field decreasesby one unit (we consider only single-photon processes) while the molecularelectronic state is shifted to an upper level. If E , E are the lower andhigher electronic energy eigenvalues respectively, the rate of absorption isgiven by Fermi’s golden rule (3.3) W abs = h |h E , n k ν |H int | E , n k ν − i| δ ( E − E − hν ) , where δ denotes Dirac’s delta function. The number of photons in a givenmode is related to the energy density by(3.4) n k ν = ρ ( k , ν ) Vhν , and the average absorption rate is(3.5) ¯ W abs = 2 π e ν h ǫ | µ , | ρ ( ν ) , where µ , is the dipole moment of the transition, and ν the associatedfrequency. The same formula expresses also the rate of induced emission,where the electronic states return to a lower energy level by emitting onephoton to the external field. The coefficients of W can be related to Ein-stein’s coefficients of laser theory as follows. According to Einstein [13], therate of absorption in the presence of thermal radiation is(3.6) W = Bρ ( ν ) , with ρ ( ν ) given by (2.5). The constant B is expressed in terms of the abovecoefficients by(3.7) B = 2 π e ǫh | µ | , and is termed absorption coefficient . For an exhaustive discussion of Ein-stein’s derivation and its implications on the quantum theory of light-matterinteractions, see for example Dirac [14]. PECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS 5 Radiative processes
The theoretical model developed hereafter applies in general to spectralconversion procedures in which absorption and emission involve differentenergies and different atomic configurations. In most fluorescent species ab-sorption and emission take place in two different parts of the molecule andinvolve different energy levels. The first rigorous thermodynamic descriptionof such spectral converters was formulated by Ross [10] and later extendedby Yablonovitch [11, 12]. However, their model was based on the assump-tion that both absorption and emission are fully reversible processes and canshare chemical equilibrium with the radiation field. As we shall see later, amore correct description of a general molecular spectral converter, involvesat least three, rather than only two, energy levels and therefore is charac-terized by a double-equilibrium condition between radiation-absorption andradiation-emission. These two processes, both reversible and thus entropy-preserving, are related to each other by an internal energy (or charge) trans-fer which requires the introduction of two different chemical potentials. As-sociated to these processes there is an increase of entropy of the electronicsystem coupled with the photon bath: the entropy acquired upon absorbinga photon of energy hν is only partially rendered upon emission, as the emit-ted photon has an energy hν , with ν < ν . This entropy difference canbe more rigorously described referring to the change of molecular electronicstates between absorption and emission by considering the transition from apure electronic state to a statistical mixture of states due to a superpositionof different nuclear wavefunctions. We can then apply the formula for the von Neumann entropy S of a quantum system [17] described by a densityoperator Σ(4.1) S (Σ) = − T r
Σ log(Σ) , where Σ is the electronic density matrix corresponding to the emission ex-cited states and the log function refers to its diagonal form. Note that thisanalysis assumes that the state of the system before emission is describedby a superposition of nuclear and electronic wavefunctions of the form(4.2) Ψ = X l,m c l,m Ξ l ⊗ Φ l,m , where Ξ , Φ denote the electronic, nuclear wavefunctions respectively. Withthe above notation, we have(4.3) Σ l,l ′ = X m h Φ l,m , ΨΨ † Φ l ′ ,m i , where ΨΨ † is the projector on the state (4.2). The dynamics of the threeprocesses (absorption, emission, thermalization) can be described by a setof differential equations for the electronic occupations of the absorption and SPECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS emission excited states respectively, here labelled as N , N ddt N = B ρ ( ν ) N − ( B ρ ( ν ) + A ) N − qN (4.4) ddt N = B ρ ( ν ) N − ( B ρ ( ν ) + A ) N + qN , where B is given by (3.7), BA = λ πh , and q is the energy transfer intensity,specified below. Figure 1.
Sketch of the process dynamics: absorption (be-tween levels E and E ), spontaneous emission (between lev-els E and E ) and internal energy transfer (between levels E and E ). Processes occurring with low probability arerepresented by dashed lines and the energy levels refer to theelectronic states exclusively. A is Einstein’s coefficient of spontaneous emission, introduced in orderto account for thermal equilibrium and is simply given by the inverse ofthe spontaneous life-time of the atomic level. The system (4.5) can besolved by considering that, as thermodynamic equilibrium is reached, thepopulation fractions referred to the ground state are the relative Boltzmanndistributions N N = exp[ − ( hν − µ ) /κT m ](4.5) N N = exp[ − ( hν − µ ) /κT m ] , where T m is referred to the molecular temperature after the interaction and µ , µ are the equilibrium chemical potentials relative to ν , ν respectively.The set of equations can be solved for the energy densities yielding ρ ( ν ) = 8 πhν /c + q/B e ( hν − µ ) /κT m − ρ ( ν ) = 8 πhν /c − q/B e ( hν − µ − hν + µ ) /κT m e ( hν − µ ) /κT m − . PECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS 7
In the above formulas, we have introduced a frequency-dependent chemi-cal potential of the molecular system in equilibrium with the radiation field[15]. With regard to the latter quantity (borrowed from the theory of lasers),it is important to observe that from a statistical point of view the molecularsystem is described by a Grand Canonical ensemble in thermal and chemicalequilibrium with the photon bath of the radiation field. In this case, thestandard concept of particle exchange is substituted by the more generalconcept of energy quanta exchange. Indeed, the chemical potential of a sys-tem constitutes a measure of its ability to exchange particles with a bath atassigned values of the other thermodynamic parameters.From (4.6) one can note that the presence of the dissipative term q mod-ifies the standard energy distributions by increasing the density of the in-cident radiation field, while decreasing the emitted one. Thus, the term q denotes the rate of energy transfer from radiative to thermal modes due tointeraction between light and particles.This term can be related to the quantum-mechanical description of molec-ular dynamics as follows. Our molecular system can be decomposed in threeconstituents: the electronic states involved with absorption, those involvedwith emission, and the remaining nuclear degrees of freedom related to equi-librium rotational and vibrational motions. The above electronic states aredifferently localized in the compound; emission states are generally relatedto a metallic ion.The total molecular wave-function can be computed within the frame ofthe so called Born-Oppenheimer approximation . The underlying idea of thisapproximation method consists in computing separately the electronic andthe nuclear wavefunctions, the separation being justified by their differentkinetic energies. In the first step, the electronic wavefunctions are found bysolving Schr¨odinger’s equation for fixed values of the nuclear coordinates(4.7) H e ( r, R )Ξ l ( r, R ) = E el Ξ l ( r, R ) , where E el = E el ( R ) are the electronic eigenvalues, dependent on the nuclearcoordinates R . By varying adiabatically the value of R one can determinethe potential energy surface E el ( R ). In the second step, the nuclear energiesand wavefunctions are derived in correspondence of a given electronic eigen-value by introducing in the Schr¨odinger equation the kinetic term which wasinitially neglected(4.8) [ T n + E el ( R )]Φ l,m ( R ) = E nm Φ l,m ( R ) , where T n contains partial derivatives with respect to R , which is now avariable and no longer a parameter.Thus, the state of the electrons depends parametrically on the nuclei posi-tions, whereas the one of the nuclei is labeled by the electronic eigenvalues.On the basis of these considerations, we can write the total approximatestate of our molecule as a factorized wavefunction(4.9) Ξ l ( r, R ) ⊗ Φ l,m ( R ) , SPECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS where r, R denote the electronic and nuclear coordinates respectively, Ξ theelectronic component and Φ the nuclear one.Upon absorbing a photon of frequency ν , the electronic energy levels ofthe molecule undergo a transition from a state Ξ l to a state Ξ l , with hν = E el − E el . In parallel, interaction with the incident photons causes a changein the nuclear momentum with a relative increase of the kinetic energy, froma value E m to a value E m . This transition corresponds to a vibrationalmotion of the nuclei away from their original equilibrium positions and arearrangement of the global molecular coordinates. In turn, nuclear motioncauses a perturbation of the electronic wavefunctions which determines achange in charge distribution. The latter may be described as an electrontransfer which involves a (possibly) non-radiative transition from the originallevel Ξ l , acquired after absorption, to an intermediate state Ξ l + δ Ξ l ,before finally reaching the emission state Ξ l . The transition rate of thisinternal energy transfer can be estimated from the principles of perturbationtheory applied to the action of the nuclear kinetic energy operator on theunperturbed wavefunctions (4.9)(4.10) q = | Ψ i , T n Ψ f | δ ( E − E ) , where Ψ i = Ξ l ( r, R ) ⊗ Φ l ,m ( R ) is the total molecular state after absorption,Ψ f = Ξ l ( r, R ) ⊗ Φ l ,m ( R ) the one before emission, and the delta function δ ( E − E ) refers to the total energy balance of the transfer including boththe electronic and the nuclear shifts, with E , = E m , + E el , . The order ofmagnitude of q can be computed using quantum chemical methods, whichallow to determine the overlap between the two molecular wavefunctionsinvolved in the process.After reaching the state Ξ l , the electrons return to their original energylevel by emitting a photon of lower frequency ν ′ , such that hν ′ = E el − E el .5. Incident and emitted radiations
The incident and emitted energy densities are represented by (4.6), withchemical potentials(5.1) µ i = E i (cid:18) − T T (cid:19) , where T , T are the ambient temperature and the incident field black-bodytemperature respectively. In a real setting, the incident radiation causesthermal energy exchanges between the particles and the photon field. Afterabsorption, the system transfers energy to a lower electronic level and thusloses kinetic energy in terms of heat by raising its temperature from T to ahigher value T m . (Note that this temperature has been already introducedabove as the correct equilibrium temperature of the molecules). This processcorresponds to thermalization of the molecular system due to non radiativeinternal energy transfer (the rate of which is given by the term q above). PECTRAL CONVERTERS AND LUMINESCENT SOLAR CONCENTRATORS 9 T m can be estimated thanks to the equipartition theorem of statistical me-chanics, which states that in a system at equilibrium, the temperature isequally distributed between the different forms of internal energy. It followsthat the molecular temperature T m can be computed by the internal energyaverage using the relation(5.2) ¯ E rv ∝ κT m , where κ is Boltzmann’s constant and E rv is the molecular roto-vibrationalenergy. The latter quantity can also be calculated by quantum chemicalsimulations, as well as experimentally measured. The relative numericalvalues and figures are included in a separate file.6. Concentration and efficiency
From (4.6), we see that the energy density of the emitted field at theemission frequency is higher than the one of the incident field at the absorp-tion frequency. On the basis of the above treatment, we could estimate anideal spectral concentration factor directly related to the conversion process,defined by the relation(6.1) C M = ρ ρ , which, neglecting stimulated emission, yields(6.2) C M = 8 πhν /c − q/B e ( hν − µ − hν + µ ) /κT m πhν /c + q/B e µ + µ − hν − hν /κT m . Ideal efficiencies of LSCs are however limited by the combination of fourloss mechanisms: absorption efficiency ( η a ), quantum efficiency of fluores-cence ( η f ), self-absorption coefficient ( θ q ) and total internal reflection ( η t ).To give a concrete example, let us consider a slab of refractive index n = 1 . η a = 0 . η f = 0 . θ q =0. Defining the geometric concentration factor G as the ratio between theupper, absorbing surface and the edge surface, we obtain the total flux gainas(6.3) G Φ = Gη a η f η t (1 − θ q ) . Above we report a plot of the incident solar spectrum as a function ofwavelength and of the emitted fluorescence spectrum modulated by themolecular blackbody radiation which propagates toward the edges of theslab. 7.
Conclusions
In this work we presented a microscopic model for the interaction pro-cesses taking place in a LSC. These processes involve multiple molecularlevels, as they are usually coupled to non-radiative, besides than radiative,energy transfer mechanisms. We derive the rate equations for the molecularsystem in equilibrium with the solar photon bath and highlight a way of
Figure 2.
Energy density vs wavelength of incoming (left)and wave-guided (right) radiations.determining the parameters involved in the non-radiative processes, whichare a subject of a further investigation currently in progress. Furthermore,we provide examples of the type of spectral distributions involved. In orderto fully understand the efficiency of LSCs it appears fundamental to han-dle not only the macroscopic, ray-tracing aspect of the device, but also themolecular-scale efficiencies involved. In these respects, it would be of greatimportance to be able to predict on the basis of molecular properties theentity of thermal relaxation and fluorescence efficiency.
Acknowledgements.
P.F.S. would like to thank L.C. Andreani for helpfuldiscussions and for useful comments on the manuscript.
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