On the interplay of photons and charge carriers in thin-film devices
aa r X i v : . [ phy s i c s . op ti c s ] F e b On the interplay of photons and charge carriers in thin-film solar cells
Pyry Kivisaari, Mikko Partanen, Toufik Sadi, and Jani Oksanen
Engineered Nanosystems Group, Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland
Thin films are gaining ground in photonics and optoelectronics, promising improvements in theirefficiency and functionality as well as decreased material usage as compared to bulk technologies.However, proliferation of thin films would benefit not only from continuous improvements in theirfabrication, but also from a unified and accurate theoretical framework of the interplay of photonsand charge carriers. In particular, such a framework would need to account quantitatively andself-consistently for photon recycling and interference effects. To this end, here we combine thedrift-diffusion formalism of charge carrier dynamics and the fluctuational electrodynamics of pho-ton transport self-consistently using the recently introduced interference-extended radiative transferequations. The resulting equation system can be solved numerically using standard simulation tools,and as an example, here we apply it to study well-known GaAs thin-film solar cells. In additionto obtaining the expected device characteristics, we analyze the underlying complex photon trans-port and recombination-generation processes that represent a full solution to the inhomogeneousMaxwell’s equations and that are generated directly as a result of solving the self-consistent model.The methodology proposed in this work is general and can be used to obtain accurate physical in-sight into a wide range of planar optoelectronic devices, of which the thin-film single-junction solarcells studied here are only one example.
Keywords: Thin-film solar cells, Photon recycling, Fluctuational electrodynamics
I. INTRODUCTION
Photonics and optoelectronics have irrevocably trans-formed several areas of society, including energy produc-tion through solar cells, general lighting through light-emitting diodes, and high-speed Internet through globaloptical fibre networks.
As these technologies matureand become established, new structures and device con-cepts are emerging both to increase the efficiency of exist-ing technologies and to create new application areas. Onesuch emerging technology is thin and ultra-thin solar cellsincorporating 0.1-10 µ m thick semiconductor films thatenable, e.g., reusing the growth substrate several timesand taking advantage of optical cavity effects. How-ever, taking full advantage of these possibilities still re-quires improved theoretical understanding, in particularin self-consistently accounting for the interplay betweenthe photons and the charge carriers as well as in combin-ing wave optical resonance effects with the quantum op-tical loss mechanisms. Without quantitative descriptionof these effects it is not possible to fully access e.g. the in-tricate photon recycling effects present in high efficiencydevices. Developing such models for a tractable simula-tion framework is therefore one of the major challengesfor the further development of the field. To that end,in this paper we propose an accurate, fully self-consistentand generally applicable electro-optical modeling tool forplanar thin-film optoelectronic devices, including photonrecycling and all interference effects.Due to its general importance, electro-optical simula-tion of resonant optoelectronic devices has been a topicof research already for some decades, and recent yearshave witnessed increased activity thanks to advances inthin-film device fabrication. The wide range the topicmakes it unfeasible to give a complete literature review
Gold n m G a A s b a s e G a A s e m i tt e r n m n m G a I n P A l G a A s n m A l I n P n m GaAs n m A R c o a t i n g n m G a A s G o l d FIG. 1: Thin-film solar cell structure studied in this paper us-ing the proposed interference radiative transfer–drift-diffusion(IRTDD) model. of it in a research article, but to give an idea of the stateof the art, here we refer to two recent works with verysimilar objectives and device structures as in this work.In Ref. 11, Wilkins et al. used transfer matrices to cal-culate coupling matrices to account for photon recyclingand luminescent coupling in drift-diffusion (DD) simula-tions, and in Ref. 12 Walker et al. calculated position-dependent dipole emission in a layer structure to scale thelocal radiative recombination coefficient and to create aphoton redistribution matrix to be used in the DD simu-lation. Summarizing from these, one typically starts withoptical solution methods, such as transfer or scatteringmatrices or finite-difference time-domain simulations, tocreate coupling matrices or other constructions relatingemission and absorption locations with each other. Theseare then used in the carrier dynamics simulation to ac-count for electro-optical effects such as photon recycling.However, previous works have not allowed straightfor-ward modeling of all the spectral, directional and reso-nant effects in the generation-recombination profiles ofthe structures. The aim of this paper is to achieve thisby creating a direct modeling tool where the optical andelectrical properties are solved simultaneously, either us-ing a direct or iterative solver.The full electro-optical device modeling tool pro-posed in this paper is implemented by combining thewell-known DD model with fluctuational electrodynam-ics using the recently introduced interference-extendedradiative transfer (IRT) model. Unlike the conven-tional radiative transfer equation, the IRT model in-cludes all wave-optical and photon recycling effects self-consistently as determined by both the external opticalfield boundary conditions and the applied bias. The mainbenefit of the model is that instead of having to use in-termediate coupling structures and possible simplifica-tions, here we have a set of differential equations whoseself-consistent solution directly satisfies both Maxwell’sequations and DD equations. To demonstrate the modelwith a well-known device structure, here we apply it tosimulate the thin-film solar cell structure studied in Ref.15. By doing this, we wish to illustrate the insight thatcan be gained with the proposed interference radiativetransfer–drift-diffusion (IRTDD) model, and how it canbe used to gain detailed physical understanding of essen-tially any planar optoelectronic device. II. THEORY
In this Section, we describe the IRTDD model usedto carry out fully self-consistent electro-optical simula-tions of thin-film optoelectronic devices. Present versionof the model combines the quantized fluctuational elec-trodynamics (QFED) framework directly with charge-carrier dynamics as governed by the DD equations, butwe expect that a similar IRTDD model could also bederived from the classical fluctuational electrodynamics.We start by writing the IRT model in a somewhat sim-plified form for a given polarization as ddz φ ( z, K, ω ) = − α ( z, K, ω )[ φ ( z, K, ω ) − η ( z, ω )]+ β ( z, K, ω )[ φ ( z, K, − ω ) − η ( z, ω )] S ( z, K, ω ) = ¯ hωv ( z, K, ω ) ρ ( z, K, ω ) × [ φ ( z, K, ω ) − φ ( z, K, − ω )] . (1)The first part of Eq. (1) is the interference-extended ra-diative transfer equation for the photon number φ of amode described by its in-plane wave number K and angu-lar frequency ω (with − ω here denoting the modes prop-agating towards the negative direction, i.e. downwardsin Fig. 1), z is position coordinate in the normal direc- tion, α and β are the damping and scattering coefficientsformulated with help of the dyadic Green’s functions asin, and η ( z, ω ) is the Bose-Einstein distribution func-tion η = 1 / (cid:0) e (¯ hω − e ∆ E F ) / ( k B T ) − (cid:1) that acts as a localradiative source term. Here e is the elementary chargeand ∆ E F is the local quasi-Fermi level separation.The second part of Eq. (1) relates the photon numberwith the spectral radiance S in the direction determinedby K and ω . Here ρ is the local density of states asdefined in, and v is the generalized speed of lightdefined in Appendix A. As the quantities v , ρ , α and β are derived from fluctuational electrodynamics, they cap-ture all wave-optical effects such as internal reflections,constructive and destructive interferences and emissionenhancement and suppression, and they are geometry de-pendent. The local generation-recombination rate can becalculated with help of the derivative of S ( z, K, ω ) as R r = X pol. Z ∞ dω Z ∞ dK dSdz hω πK, (2)where the sum is taken over the two orthogonal polariza-tions of light. The derivative of S is trivial to calculateusing the chain rule and the equations provided in. Fur-thermore, we have recently introduced a straightforwardway to calculate the dyadic Green’s function, the localand nonlocal optical densities of states, and the α and β coefficients for arbitrary planar structures with help ofoptical admittances, and it is used also in this paper.The IRT can be readily coupled with the DD model ofcarrier transport, given by ddz (cid:0) − ε ddz U (cid:1) = e ( p − n + N d − N a ) ddz J n = ddz (cid:0) µ n n ddz E F n (cid:1) = e ( R r + R nr ) ddz J p = ddz (cid:0) µ p p ddz E F p (cid:1) = − e ( R r + R nr ) , (3)where ε is the static permittivity, U is the electrostaticpotential, e is the elementary charge, n, p are the elec-tron and hole densities, N d , N a are the ionized donorand acceptor densities, J n , J p are the electron and holecurrent densities, µ n , µ p are the electron and hole mo-bilities E F n , E
F p are the conduction and valence bandquasi-Fermi levels such that ∆ E F = E F n − E F p , and R nr is the nonradiative recombination rate calculated asa sum of Shockley-Read-Hall and Auger recombinationas in. Radiative recombination-generation R r is calcu-lated from the photon numbers using Eq. (2), and there-fore the resulting IRTDD model constructed by Eqs. (1)and (3) solves photon and charge carrier dynamics self-consistently.The boundary conditions and integration limits re-quired for the IRTDD model can be chosen as follows.For each photon energy, the IRT simulation is run for anequally-spaced set of K between zero and suitably cho-sen K > N k , where N is the largest refractive indexpresent in the semiconductor layers. According to theresults of this paper, spectral radiances and generation-recombination rates all go to zero at K > N k as ex-pected, indicating that the simulations span all relevantoptical modes. For the DD model, the boundary condi-tions for the electrostatic potential and quasi-Fermi levelsare determined by the built-in potential and applied biasin the conventional way. In particular, the quasi-Fermilevels of electrons and holes are equal with each other atthe contacts, corresponding to infinitely fast contact re-combination of minority carriers that reach the oppositecontact. For incoming solar light, we use the light inten-sity according to the typical ASTM G173 global refer-ence spectrum to enable direct comparison with devicecharacteristics reported elsewhere. In this work, the in-coming solar intensity is fully projected into the normallyincident modes for simplicity. III. RESULTS & DISCUSSION
To demonstrate the IRTDD model with a widelyknown and well-studied example device structure, wesimulate a thin-film GaAs solar cell structure similar tothe one studied in Ref. 15 and shown schematically inFig. 1. The device operation is simulated both underdark current conditions and under direct sunlight hittingthe sample at normal incidence. The material parame-ters and the dielectric functions used in the simulationsare specified in Appendix B. We start by presenting theoverall device characteristics resulting from the IRTDDsimulation and proceed to studying the internal photontransport and generation-recombination properties un-der illumination and in dark current conditions. For thepresent purposes, we limit the geometry to be fully planarand homogeneous in the lateral direction. Most impor-tantly, here this means that similar to Ref. 12, the sparsetop contact grid is not included in the optics calculation.It is, however, expected that the results and the modelcan be generalized for higher dimensional geometries orlight scattering surfaces by separately accounting for lat-eral effects and extending the boundary conditions.
A. Overall device characteristics
Beginning with the overall device operation, Fig. 2shows the current-voltage characteristics from the fullIRTDD simulation both under illumination and underdark current conditions. The photocurrent curve showsthe expected high-efficiency GaAs solar cell characteris-tics with a short-circuit current of 30.47 mA/cm , open-circuit voltage of 1.09 V, fill factor of 0.88 and maxi-mum power efficiency of 0.29. All these values are closeto the experimental and theoretical values reported in, with the main differences likely to result from slightlydifferent simulation or material parameters, and possi-bly also from optical interference and carrier transporteffects that were not included in the calculations of. By a closer inspection, the current-voltage characteris-tics follow roughly the superposition principle J ( V a ) = J sc + J dark ( V a ), where J is the total current, V a is the Bias voltage (V) -30-20-100 C u rr en t ( m A / c m ) PhotocurrentDark current
Max. e (cid:1) ciency: 0.29Fill Factor: 0.88
FIG. 2: Current-voltage characteristics of the structure sim-ulated with the IRTDD model in dark and under normallyincident sunlight corresponding to the ASTM G173 globalreference spectrum.
Position ( m) -0.500.511.5 E ne r g y ( e V ) n-GaAs emitterp-GaAsbasei-GaInPp-AlGaAsp-GaAs n-AlInP n-GaAscontact EE EE
CFn FpV
FIG. 3: Band diagram of the structure simulated with theIRTDD model under illumination and an applied bias of 0.9V. applied bias, and J sc and J dark are the short-circuit anddark currents. B. Characteristics under illumination
To study the device operation under illumination, Fig.3 shows the band diagram of the structure under illumi-nation at an applied bias of 0.9 V. The voltage 0.9 V ischosen for all the following figures to represent an exam-ple case where the device still operates fully as a solarcell, and the different layers are marked in the figure foreasier interpretation. The band diagram starts at the p-doped GaAs bottom layer and ends at the n-doped GaAstop layer. The left side of the band diagram is clearly p-type with large potential barriers for electrons to preventthem from leaking to the p-contact. The GaAs pn junc-tion formed by the GaAs base and emitter layers is clearlyvisible in the middle of Fig. 3, and the AlInP layer onthe right creates a potential barrier to the valence bandto prevent holes from leaking to the n-contact.For a detailed look at the absorption of sunlight inthe structure, Fig. 4 shows a colormap of the generationrate due to solar photons as a function of photon en-ergy and position. At low photon energies below roughly P osition ( (cid:0) m)0 20.5 1 1.531.522.5 E n e r g y ( e V ) Aup-GaAs p-AlGaAsi-
G(cid:2)(cid:3)(cid:4)(cid:5) p-GaAs n-GaAs n-
A(cid:6)(cid:7)(cid:8)(cid:9)
ZnS MgF Air
FIG. 4: Generation rate (a.u.) due to incoming solar photonsas a function of photon energy and position (with positivevalues denoting generation). Here, the ZnS and MgF layersconstitute the antireflective (AR) coating marked in Fig. 1. P osition ( μ m) 20.5 1 1.5(a) ((cid:10)(cid:11) -100.5-0.5 E n e r g y ( e V ) E n e r g y ( e V ) FIG. 5: Total recombination-generation rate (a.u.) due tointernal emission as a function of photon energy and positionfor (a) TE polarization and (b) TM polarization at an ap-plied bias of 0.9 V under illumination (with positive valuesdenoting recombination). For this device configuration andbias, there is a very clear spatial separation between regionsof net generation and net recombination. Stronger absorp-tion of sunlight towards the top surface creates an imbalancein carrier densities, which is balanced here by recombinationat around 2 µ m and a corresponding generation at around1 µ m. P osition ( μ m)(a) (cid:12)(cid:13)(cid:14) K / k K / k FIG. 6: Recombination-generation rate (a.u.) due to internalemission as a function of K and position at photon energy1.43 eV at an applied bias of 0.9 V under illumination for (a)TE and (b) TM polarization (with positive values denotingrecombination). The rates include the integration factor 2 πK . Turning our attention more towards photon recy-cling, all the figures from now on describe the netrecombination-generation (RG) after subtracting thegeneration created by the incident light. This will alsoenable a more direct comparison with the correspond-ing figures under dark current, which are studied laterin Subsection III C. To look at the resulting internal RGcharacteristics, Fig. 5 shows the RG rate as a function ofposition and photon energy at an applied bias of 0.9 V un-der solar illumination. More specifically, Fig. 5(a) showsthe RG rate normalized with its maximum value for TEpolarization, and Fig. 5(b) shows the same rate for theTM polarization. Comparing TE and TM in Figs. 5(a)and (b), they share the same qualitative characteristics,in which the internal net RG rate is positive (correspond-ing to net recombination) close to the the right side ofthe figure and negative (corresponding to net generation)towards the left side of the figure. This photon recyclingprocess is caused by the spatially uneven absorption ofsolar photons in Fig. 4, which causes ∆ E F to be slightlylarger next to the top contact than elsewhere in the de-vice. Additionally, the internal recombination-generationin Fig. 5 mostly takes place at photon energies notablybelow 1.5 eV, as the Fermi-Dirac distributions decay fastas a function of the recombination energy. There are alsovisible quantitative differences between the TE and TMRG rates in Fig. 5 due to wave-optical effects. For ex-ample, the maxima and minima of the RG rate occurat somewhat different locations in (a) and (b) due to thedifferent mode structure of the TE and TM polarizations.To take a more detailed look at one relevant photon en-ergy, Fig. 6 shows the RG rate as a function of K at 1.43eV (a value close to the GaAs bandgap energy) for (a) TEand (b) TM polarization. To compare with the previousfigure, the row at 1.43 eV in Fig. 5 represents the inte- P osition ( μ m)(a) (cid:15)(cid:16)(cid:17) K / k K / k -(cid:18) (cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) FIG. 7: Spectral radiance S × πK (a.u.) as a function of K and position at photon energy 1.43 eV at an applied bias of0.9 V under illumination for (a) TE and (b) TM polarization. gral of Fig. 6 over all K . Figure 6 naturally exhibits thesame trend in the spatial distribution as Fig. 5, wherethe RG rate is positive towards the top contact on theright side and negative towards the bottom contact onthe left side. Moreover, here the interference effects areclearly visible as spatial oscillations in the RG rate dueto constructive and destructive interferences and possi-bly also due to the Purcell effect. The RG rates approachzero roughly at K/k = 3 .
5, when no more propagatingmodes exist in the structure. It is noteworthy that therather complex spatial RG rates shown here are prac-tically impossible to calculate without detailed opticalmodels of emission and photon recycling. As such, theyhave not previously been reported in this level of detaildue to the difficulty to simultaneously account for all theunderlying physical processes. In the model proposedhere, the RG rates are obtained as a direct result of solv-ing the coupled equation system. We expect this to bebeneficial for understanding and optimizing the internalelectrical and optical properties of various thin-film de-vices, such as the emerging ultra-thin GaAs photovoltaicdevices, and to complement existing advanced mod-eling and characterization frameworks of multi-junctionsolar cells.
To study how the optical power propagates within thesolar cell structure studied here, Fig. 7 shows the the K -resolved spectral radiance of internally emitted photonsas a function of position and K at the same photon en-ergy 1.43 eV for (a) TE and (b) TM polarization, againfor 0.9 V. It can be seen that in the escape cone de-termined by K/k ≤
1, there is a clear positive opticalpower flow out of the device in the bottom right corner ofthe figures. However, based on a closer inspection of theresults it is only around 0.1 % of the incoming solar opti-cal power, indicating that the structure is still operatingfully as a solar cell at this bias. At larger K values, thepower escaping the structure becomes zero, but the spec- tral radiance is negative in the middle of the GaAs layers,corresponding to the RG profiles in the previous figureswhich show that light is emitted close to the top contacton the right side and absorbed in the lower GaAs layerson the left side due to the imbalance in ∆ E F createdby sunlight. This illustrates once more how the internalrecombination-generation evens up part of the imbalancein the carrier densities created by solar absorption. C. Characteristics under dark current
In this Section, we repeat the internal RG results forthe dark current case to show that even if the superpo-sition principle holds roughly for the total current, theinternal properties of the solar cell under illuminationand under dark current conditions can be quite different.The information in Fig. 5 is repeated for dark currentconditions in Fig. 8, which again shows the RG rate den-sity as a function of photon energy and position for (a)TE and (b) TM at the bias voltage of 0.9 V. Interestingly,the result is very different from the one shown in Fig. 5calculated at a similar voltage under solar illumination.In the GaAs region of Fig. 8, we only see positive netRG rates, whereas in Fig. 5 under illumination, therewas also a negative RG rate towards the bottom of theGaAs layer on the left, corresponding to net generationthere. In the dark current case shown in Fig. 8, there isno clear imbalance in the electron-hole densities createdby solar absorption. Therefore net emission is createdprimarily through the process of radiation out from thestructure, and partly through photons that are absorbedby the bottom contact. This difference in the internal op-tical properties between illuminated and dark operatingconditions has been discussed also in Ref. 25.The RG rate in dark current conditions can be stud-ied in more detail with help of Fig. 9, which repeatsthe RG rates of Fig. 6 in dark as a function of K andposition for (a) TE and (b) TM for a single energy 1.43eV, again at 0.9 V. It can be seen that there is signif-icant nonzero net recombination only in the modes upto K/k = 1, corresponding primarily to photons thatradiate out from the structure and partly to photons ab-sorbed by the bottom contact. Above K/k = 1, thereis only modest recombination resulting in photons thatare absorbed by the bottom contact on the left. Thisobservation is further backed up in Fig. 10, which re-peats Fig. 7 in dark for (a) TE and (b) TM modes as afunction of K and position. Here the spectral radiancehas its clearly largest absolute values at K/k <
1, corre-sponding to photons exiting through the AR coating onthe right side of the figure and at angles close to escapecone boundaries in particular. However, as seen in thecurrent-voltage characteristics, these internal differencesbetween the dark current case and the case under illumi-nation do not cause the superposition principle to faltervisibly, but they certainly introduce effects that mightbe relevant for some geometries with, e.g., a smaller to- P osition ( μ m) 20.5 1 1.5(a) !" -100.5-0.5 E n e r g y ( e V ) E n e r g y ( e V ) FIG. 8: Information in Fig. 5 repeated for dark current con-ditions: recombination-generation rate (a.u.) due to internalemission as a function of photon energy and position at anapplied bias of 0.9 V under dark current conditions, includingall propagation directions: (a) TE and (b) TM polarization. (a) $%& P o’)*+,. / μ m0 K / k K / k FIG. 9: Information in Fig. 6 repeated for dark current con-ditions: recombination-generation rate (a.u.) due to internalemission as a function of K and position at photon energy1.43 eV at an applied bias of 0.9 V under dark current condi-tions for (a) TE and (b) TM polarization. The rates includethe integration factor 2 πK . tal device thickness. In the case under illumination, there-emission and recycling of absorbed solar photons is notprimarily a loss process in the absence of substantial non-radiative recombination, as it merely redistributes partof the solar generation more evenly throughout the wholeGaAs active region. P osition ( μ m)0 20.5 1 1.5 K / k K / k =>?@ BCDEFHIJK FIG. 10: Information in Fig. 7 repeated for dark currentconditions: spectral radiance S × πK (a.u.) as a function of K and position at photon energy 1.43 eV at an applied biasof 0.9 V under dark current conditions for (a) TE and (b) TMpolarization. IV. CONCLUSIONS
To improve the theoretical insight into various thin-film optoelectronic devices, we introduced an accurateand fully self-consistent model of photon and charge car-rier transport applicable to planar devices. The modelwas based on connecting the drift-diffusion formalismof carrier transport and fluctuational electrodynamics ofphoton transport by making use of the newly developedinterference-extended radiative transfer equations. Todemonstrate the thus obtained IRTDD modeling tool, weapplied it to study the device characteristics and internaloptical properties of a thin-film GaAs solar cell. The re-sults indicate that the IRTDD model not only reproducesthe expected device characteristics, but also directly pro-vides detailed physical insight into complex nonlocal ef-fects such as photon recycling, which is a relevant pro-cess particularly in high-efficiency thin-film solar cells.The IRTDD framework presented here is general, and itcould be implemented in a wide range of planar resonantdevices for detailed studies of photon recycling, lumines-cent coupling and other complex forms of electro-opticalinteraction.
Acknowledgments
We acknowledge financial support from the Academyof Finland (grant number 315403) and the European Re-search Council (Horizon 2020 research and innovationprogramme, grant agreement No 638173).
Appendix A: Modified photon numbers and relatedquantities
To use the IRT model in solar cells and other devices,it is convenient if the incoming photon numbers can beset to zero at the outer boundaries of the computational domain as a boundary condition. To this end, we slightlymodify the definition of the interference density of states(IFDOS) as compared to the one used in. Using thenotation of, we begin by writing the spectral radianceas h ˆ S σ ( z, K, ω ) i = ¯ hωv σ ( z, K, ω ) Z ∞−∞ ρ IF,σ ( z, K, ω, z ′ ) h ˆ η ( z ′ , ω ) i dz ′ , (A1)where the generalized velocities of light are given by v T E = 2 ck | µ |ℜ ( k z /µ ) | k | + | k z | + K , v T M = 2 ck | ε |ℜ ( k z /ε ) | k | + | k z | + K . (A2)These changes in the velocities are balanced by scaling the IFDOSs with a factor ζ v,σ = c/ ( v σ n r ). The value of thespectral radiance is unchanged, but the IFDOSs and the related propagating photon numbers are slightly modifiedas specified below. Making use of the photon numbers propagating in the positive or negative direction instead, thespectral radiance is written as in Eq. (1). With this choice of v σ , we obtain that outside the simulation domain, α + = const. in the negative direction, α − = const. in the positive direction, β + = 0 in the negative direction, and β − = 0 in the positive direction. In addition, the mentioned constant values of α + and α − are zero in lossless media.This allows one to set the input photon numbers to zero in all lossy and lossless cases when there are no sourcesoutside.Another difference with is the definition of the IFDOSs, which are here given by ρ IF,T E ( z, K, ω, z ′ ) = − ω π c v T E ℑ (cid:2) ε ′ i g ee g ∗ me + µ ′ i g mm g ∗ em + µ ′ i g mm g ∗ em (cid:3) , (A3) ρ IF,T M ( z, K, ω, z ′ ) = ω π c v T M ℑ (cid:2) µ ′ i g mm g ∗ em + ε ′ i g ee g ∗ me + ε ′ i g ee g ∗ me (cid:3) . (A4)To derive the IRT coefficients, the derivatives of the IFDOSs are given by ∂ρ IF,T E ∂z = − ω π c v T E (cid:20) ℑ (cid:20) εk z k (cid:21) (cid:0) ε ′ i | g ee | + µ ′ i | g em | + µ ′ i | g em | (cid:1) + µ i (cid:0) ε ′ i | g me | + µ ′ i | g mm | + µ ′ i | g mm | (cid:1)(cid:21) + ω n r π c ℑ (cid:20) ε i g ee + µ i g mm + µ i µ | µ | g mm (cid:21) δ ( z − z ′ ) , (A5) ∂ρ IF,T M ∂z = − ω π c v T M (cid:20) ℑ (cid:20) µk z k (cid:21) (cid:0) µ ′ i | g mm | + ε ′ i | g me | + ε ′ i | g me | (cid:1) + ε i (cid:0) µ ′ i | g em | + ε ′ i | g ee | + ε ′ i | g ee | (cid:1)(cid:21) + ω n r π c ℑ (cid:20) µ i g mm + ε i g ee + ε i ε | ε | g ee (cid:21) δ ( z − z ′ ) . (A6)Substituting these to the general equations in allows simplifying the damping and scattering coefficients to theirfinal forms given by α ± ,T E = k r k ρ T E /ρ ℑ (cid:20)(cid:18) ζ e,T E ζ v,T E + ζ v,T E ε i (cid:19) g ee + (cid:18) ζ m,T E ζ v,T E + ζ v,T E µ i (cid:19) g mm ± ζ ex,T E ( g me − g em ) + ζ v,T E µ i µ | µ | g mm (cid:21) , (A7) β ± ,T E = k r k ρ T E /ρ ℑ (cid:20)(cid:18) ζ e,T E ζ v,T E − ζ v,T E ε i (cid:19) g ee + (cid:18) ζ m,T E ζ v,T E − ζ v,T E µ i (cid:19) g mm ± ζ ∗ ex,T E ( g me − g em ) − ζ v,T E µ i µ | µ | g mm (cid:21) , (A8) α ± ,T M = k r k ρ T M /ρ ℑ (cid:20)(cid:18) ζ m,T M ζ v,T M + ζ v,T M µ i (cid:19) g mm + (cid:18) ζ e,T M ζ v,T M + ζ v,T M ε i (cid:19) g ee ± ζ ex,T M ( g em − g me ) + ζ v,T M ε i ε | ε | g ee (cid:21) , (A9) β ± ,T M = k r k ρ T M /ρ ℑ (cid:20)(cid:18) ζ m,T M ζ v,T M − ζ v,T M µ i (cid:19) g mm + (cid:18) ζ e,T M ζ v,T M − ζ v,T M ε i (cid:19) g ee ± ζ ∗ ex,T M ( g em − g me ) − ζ v,T M ε i ε | ε | g ee (cid:21) (A10)with the parameters and their equations given in the Supplemental Material of. Appendix B: Simulation parameters and other simulation details
Parameters used in the DD simulations of electron-hole transport are listed in Table I. The radiative recombinationcoefficient B is not used in the final IRTDD simulations, but an initial value of 7 . × − m /s was used to getinitial conditions. The A and C recombination coefficients are set to selected reasonable values for GaAs, and withthese values the open-circuit voltage is still determined by radiative recombination. The A and C coefficients are thesame for all the materials for simplicity, as recombination takes place predominantly in GaAs due to the relativelylow injection levels.For the optical simulations, we use photon energy-dependent complex dielectric functions reported in the literaturefor Au, GaAs, AlGaAs, GaInP, AlInP, ZnS, and MgF , with help from. The relative permeability isassumed to be 1 in all materials at all photon energies.
TABLE I: Parameters used in the DD simulations of electron-hole transport.
The band offsets dE c and dE v are given withrespect to the band edges in GaAs. Property GaAs AlGaAs InGaP AlInP Unit ε ε E g dE v dE c µ n /(Vs) µ p
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