ABSTRACT
 Top of page
 ABSTRACT
 INTRODUCTION
 IDEAL EFFICIENCY LIMITS
 QUANTITATIVE MULTIJUNCTION MODEL
 MULTIJUNCTION STRUCTURES ON SILICON
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
Single junction Si solar cells dominate photovoltaics but are close to their efficiency limits. This paper presents ideal limiting efficiencies for tandem and triple junction multijunction solar cells featuring a Si subcell also serving as substrate. Subject to this Si bandgap constraint, we design optimum cell structures that we show depart from the unconstrained ideal. In order to progress to manufacturable designs, the use of III–V materials is considered, using a novel growth method capable of yielding low defect density III–V layers on Si. In order to evaluate the real potential of these proposed multijunction designs, a quantitative model is presented, the strength of which is the joint modelling of external quantum efficiency and current–voltage characteristics using the same parameters. The method yields a singleparameter fit in terms of the Shockley–Read–Hall lifetime. This model is validated by fitting experimental data of external quantum efficiency, dark current and conversion efficiency of world record tandem and triple junction cells under terrestrial solar spectra without concentration. We apply this quantitative model to the design of tandem and triple junction solar cells, yielding cell designs capable of reaching efficiencies without concentration of 32% for the best tandem cell and 36% for the best triple junction cell. This demonstrates that efficiencies within a few per cent of world records are realistically achievable without the use of concentrating optics, with growth methods being developed for multijunction cells combining III–V and Si materials. Copyright © 2014 John Wiley & Sons, Ltd.
INTRODUCTION
 Top of page
 ABSTRACT
 INTRODUCTION
 IDEAL EFFICIENCY LIMITS
 QUANTITATIVE MULTIJUNCTION MODEL
 MULTIJUNCTION STRUCTURES ON SILICON
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
Multijunction cells remain the most successful high efficiency design concept and the only one successfully commercialised. Nevertheless, these structures are still restricted to niche applications in space and under concentration, although generic terrestrial applications are tantalisingly close. The materials most suitable for multijunction designs are the III–V materials families, given their range of optical and material properties where incompatibilities, especially due to lattice constants, may be managed or even eliminated. The classic example is the growth of GaInP and GaInAs top and middle gap junctions on Ge substrates as bottom cell. This triple junction has achieved efficiencies [1] under concentration greater than 40% with both latticematched and latticemismatched approaches.
The main obstacles to the widespread use of multijunctions in renewable energy supply are materials and fabrication issues. The materials cost is partly due to the scarcity of some essential III–V materials such as In and Ga, but more importantly, it is due to the use of expensive substrates, because even Ge remains relatively costly in the competitive photovoltaic market.
Concerning cheap substrates, the clear frontrunner is Si, which brings with it the advantage of lowcost industrystandard complementary metaloxidesemiconductor (CMOS) processing technologies and more specifically brings the advantage of the most advanced development for commercial solar cells. The additional need for III–V materials is barely a few microns, reducing the materials cost to acceptable levels.
However, Si suffers from a lack of semiconductors lattice matched to it and from an indirect bandgap and low absorption coefficient. This is an obstacle for most optoelectronic applications including multijunction solar cell design. Although allSi structures have been proposed, using amorphous silicon and siliconbased metamaterials to fabricate multijunction cells [2], the most attractive and most flexible option remains growth of high quality III–V materials on Si.
An early attempt to address this issue by Yang [3] simply grew a thick AlGaAs buffer layer on an active Si substrate, with a simple threeterminal design. This approach did not exceed efficiencies of 20% and has not to date been developed further. Some years later, Taguchi et al. [[4]] adopted a not dissimilar structure with a high quality GaAs cell grown on a GaAs substrate and transferred to a Si substrate by liftoff. They chose a fourterminal design but again failed to demonstrate more than 19% efficiency.
Geisz et al. have since reported results on twoterminal designs. They have investigated both latticematched attempts using nitrides [5] and latticemismatched approaches using graded buffer techniques for strain relaxation minimising bulk defect densities [6]. The work of Lueck et al. [[7]] in the same year demonstrated similar techniques and achieved an efficiency of 17% under AM1.5G spectrum. Work in this field continues with Putyato et al. [[8]] developing graded buffer approaches. While these methods have made progress, the efficiencies achieved remain subject to materials quality limitations due to the fundamental problem of high defect densities, despite the use of buffers to reduce them to acceptable levels.
This paper presents a quantitative study of siliconbased multijunction solar cells for terrestrial applications developed by the Multispectral Solar Cells on Silicon (MULTISOLSI) project [9]. The study assumes an AM1.5G spectrum that corresponds to photovoltaic systems with no solar concentration. This is suitable for strategies prioritising a low system cost over cell efficiency and peak power. However, we are also aiming at relatively low cell cost with our siliconbased design, while nevertheless reaching high efficiencies. While a detailed cost analysis on this point is beyond the scope of this paper, we note that the cell design methodology we describe is equally applicable to concentrating photovoltaics under direct solar spectra. The choice of a global spectrum does not, therefore, limit the scope of the design methods we will describe.
A detailed growth investigation presented in a companion paper [10] reports on progress in the development of III–V growth on Si using novel lowcost threedimensional growth techniques within this project. This technique involves epitaxial lateral overgrowth of latticemismatched polar semiconductors on Si via growth of nanoseeds in apertures opened in thin SiO_{2} layers. This technique has demonstrated defectfree growth and an absence of antiphase domain formation [11] due to the small initial growth area. This paper therefore does not consider the formidable materials issues that remain to be resolved within MULTISOLSI but concentrates on the design stage only by evaluating the potential of III–V structures on Si using a validated, quantitative model.
We first examine the suitability of silicon for multijunction cells from the commonly used ideal theoretical viewpoint of the radiative limit but amended to maximise the radiative efficiency of multijunction cells with nonideal bandgaps and layer thicknesses. We then describe a quantitative analytical model capable of accurately modelling record multijunction cells. On this basis, we present the design basis for dual and triple junction cells being currently developed in the MULTISOLSI project.
IDEAL EFFICIENCY LIMITS
 Top of page
 ABSTRACT
 INTRODUCTION
 IDEAL EFFICIENCY LIMITS
 QUANTITATIVE MULTIJUNCTION MODEL
 MULTIJUNCTION STRUCTURES ON SILICON
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
With the objective of this study being cheap terrestrial multijunction photovoltaic systems, it is instructive to first look at the suitability of Si in the ideal limit under the AM1.5G spectrum that we perform by looking at the maximum efficiencies achievable using silicon in multijunction cells in the radiative efficiency limit, using the form published by Henry [12]. The ideal system considered consists of a stack of N subcells of index i, each of unit external quantum efficiency (EQE), and converts every photon absorbed into a current carrier. Each subcell of bandgap E_{gi} absorbs light in the wavelength range corresponding to energies between E_{gi}, and E_{gi + 1} with obvious modifications for the first and Nth cells. For this system, under spectral irradiance F, and adopting the photovoltaic sign convention of positive photocurrent, the short circuit current density J_{SCi} and the radiative saturation current density J_{0i} of each subcell take the form
 (1)
 (2)
where n_{i} is the subcell refractive index (taken as 3.5 here as in Henry's work [12] which is representative of relevant semiconductors, and in particular of Si) and where the other symbols have their usual meaning. For series connected subcells, again with the positive photocurrent sign convention, the overall multijunction bias V as a function of multijunction series current density J is given by
 (3)
thereby defining the J(V) current–voltage characteristic. The maximum of the corresponding power–voltage characteristic yields the maximum possible efficiency achievable for any structure of N ideal junctions. This is usually presented as a plot of efficiency as a function of bandgap showing isoefficiency contours that allows the potential of materials for high efficiency to be estimated at a glance. In Figure 1(a), we see the maximum efficiency achievable with two gaps, which is 42.2% in the conditions specified earlier for materials with bandgaps 0.96 and 1.64 eV.
Replacing the lower 0.96 eV material with slightly higher bandgap silicon has a very slightly lower efficiency limit of 41.9% for bandgaps 1.12 (Si) and 1.74 eV. The good match is underlined by the fact that the two solutions occupy two local neighbouring maxima on the isoefficiency contour.
This bandgap restriction may be relaxed by considering subcells with less than unit EQE, specifically with optically thinned subcells in mind. While this reduces the limiting efficiency achievable because of increased thermalisation, it may be useful in order to mitigate material constraints. Figure 1(b) shows the efficiency obtainable in the radiative limit by thinning the top cell such that the ideal bottom cell gap approaches Si. Having achieved this, the top cell ideal bandgap is one with 68% overall absorptivity, consequently transmitting 32% of the spectrum to the bottom Si. The optimum top cell bandgap of 1.46 eV in this case being fortuitously close to GaAs at 1.42 eV is an accidental bonus as we shall see subsequently.
Figure 2(a) shows similar evaluations of Si for triple junction cells, where the ideal limit has a middle junction ideal bandgap of 1.21 eV for an efficiency of 47.2%. Figure 2(b) shows the efficiency profile for the maximum achievable with a silicon middle bandgap cell. We see that a slightly lower efficiency limit of 45.4% is achievable with a silicon middle gap cell. Although the loss is greater than for the tandem case, this retains the recordbreaking potential.
These introductory remarks set out the suitability of silicon for multijunction cells in the ideal limit. In the following sections, we go on to describe a realistic, quantitative model of triple junction cells in order to design multiple junction cells on silicon substrates.
QUANTITATIVE MULTIJUNCTION MODEL
 Top of page
 ABSTRACT
 INTRODUCTION
 IDEAL EFFICIENCY LIMITS
 QUANTITATIVE MULTIJUNCTION MODEL
 MULTIJUNCTION STRUCTURES ON SILICON
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
In order to design test structures for the novel growth method, we use an analytical model SOL that has been reported in more detail elsewhere [13], but of which we sketch the main aspects here firstly for a single junction. With material parameters for the majority of semiconductors in the III–V family, drawn from the literature, including Si(Ge), the model evaluates the EQE and photocurrent J_{PH} as a function of bias by standard analytical solutions [12] of transport and continuity equations in the depletion approximation for onedimensional structures with abrupt interfaces. The point of interest in this model is the detailed accounting of loss mechanisms by explicit solution of the nonradiative and radiative recombination losses in the different regions of the cell, and the inclusion of series and shunt resistances for each subcell. As such, the radiative and nonradiative recombination currents in the chargeneutral layers are described by the Shockley diffusion or injection current [14] as follows:
 (4)
where n_{ip} is the intrinsic carrier concentration in the player doped at a level N_{A}, of surface recombination velocity S_{n}, and corresponding parameters n_{in} and N_{D} in the ndoped layer with its recombination velocity S_{p}.
The nonradiative recombination currents in the spacecharge region are described by the Shockley–Read–Hall (SRH) recombination current [[15]] with the standard expression
 (5)
where x_{1} and x_{2} are the edges of the spacecharge region, n(x) and p(x) are the electron and hole concentration as a function of position, n_{t} and p_{t} are the electron and hole trap occupation densities for midgap trap levels and τ_{n} and τ_{p} are the electron and hole nonradiative lifetimes, respectively.
The radiative recombination currents in spacecharge layers are evaluated from the absorption coefficients and quasiFermi level separation in each region of the cell as
 (6)
where n is the refractive index of the material, ∆φ is the quasiFermi level separation, and the other symbols have their usual meanings. The absorptivity α(E,θ,s) is the line integral over position through the different layers of the cell along the optical path of radiation at angle θ with the normal exiting or entering surface S, the total emitting surface in three dimensions [16].
The sum of these three recombination currents therefore describes the radiative and nonradiative recombination dark currents in the cell. Following this analytical method, a little more analysis can evaluate the radiative efficiency of the structures [13].
A noteworthy feature of this approach combining the modelling of the radiative and nonradiative currents together with the EQE is that all the parameters determining these currents except for the SRH lifetimes are determined by the EQE, thereby minimising the number of free parameters. The combination of dark current and EQE fitting therefore leaves, as only free parameters, the SRH lifetimes. In the absence of better knowledge, and consistently with current continuity, the electron and hole nonradiative SRH lifetimes are assumed equal in practice, resulting in a single free parameter: the SRH lifetime.
The efficiency and fill factor (FF) of the cell is then evaluated from the light current J_{L} assuming superposition of dark current and photocurrent J_{PH} as
 (7)
by finding the maximum power point on the light current curve.
The corresponding multijunction structure requires series connection of subcells via tunnel junctions. At this design stage of the MULTISOLSI project, we rely on published values as a guideline and on numerical calculations for Si tunnel junctions we have performed with SILVACO (St. Yves, Cambridge, PE27 5JL UK) software. These show that in GaAs tunnel junctions [17], for example, peak tunnelling currents above 10^{4} A/m^{2} are readily obtainable. The total current density obtainable with the global AM1.5G terrestrial spectrum is 726 A/m^{2}. The multijunction current flow is therefore at least an order of magnitude less than that of a suitable GaAs tunnel junction, which therefore serves to cause a small voltage loss, as it is designed to. As such, in this work, we treat the tunnel junctions as absorbing layers from an optical point of view and as series resistance elements from the electrical point of view. In addition, we note that the losses incurred because of the tunnel junctions are of the order of 0.1% absolute due to a combination of optical and electrical penalties.
The multijunction current–voltage characteristic is therefore evaluated by separately calculating the current–voltage characteristics of the subcells optically and electrically connected in series, including parallel and series resistance losses and numerically evaluating the resulting multijunction solar cell current. This approach has shown itself capable of quantitatively reproducing world record EQE, dark current and light current characteristics of tandem and triple junction cells published by Japan Energy Corp. [[18]] and Spectrolab [19] (Tables 1 and 2). While complete details including fits of EQE, dark and light current–voltage characteristics are available elsewhere [13], here we show the good fit to the EQE of the triple junction cell (Figure 3(a)). As described in detail in Ref. [[13]], the modelling in both cases relies on minority carrier transport as a function of doping given in the literature.
Table 1. Experimental data and SOL modelling of a record tandem solar cell [18] under an AM1.5G spectrum.  J_{SC} (A/m^{2})  V_{OC} (V)  FF (%)  Efficiency (%) 

Japan Energy Corp.  142.5  2.49  85.6  30.3 
Model SOL  139.5  2.32  87.0  29.4 
Table 2. Experimental data and SOL modelling of a Spectrolab triple junction record cell [19] under an AM1.5G spectrum.  J_{SC} (A/m^{2})  V_{OC} (V)  V_{mp} (V)  FF (%)  Efficiency (%) 

Spectrolab  143.7  2.62  2.30  85  32.0 
Model SOL  143.2  2.62  2.31  86.0  32.4 
It is important to note that the experimental EQE of this Spectrolab triple junction record cell is close to 100%. This in part is due to the very low reflectivity achievable with multilayer antireflection (AR) coats, such as the doublelayer AR coat used by the modelling here. The high EQE is also due to the relaxation of minority carrier transport constraints in multijunction cells, which is a consequence of each subcell in the stack needing to absorb a shorter range of incident wavelengths close to the bandedge. In particular, the impact of front surface recombination is reduced.
The corresponding singleparameter dark current fitting is shown in Figure 3(b). It is worth underlining that this dark current fit uses the transport parameters validated by the EQE fitting as we have discussed, with the single SRH lifetime in the spacecharge region the only fitting parameter for the dark current fit. The strength of this approach is illustrated by the close dark current fit, shown in Figure 3(b), to available data for the triple junction subcells, in terms of the radiative and nonradiative recombination mechanisms.
This combined validation of transport by fitting EQE and dark current gives confidence in the transport parameters used and confirms that the very high EQE predicted is to some degree overoptimistic only in the analytical calculation of the doublelayer AR coat reflectivity. We cannot avoid this slight overestimation, however, without arbitrarily reducing cell transport efficiency below published values, or by including external parasitic losses that are present in real AR coats, such as absorption losses due to impurities or backscattering losses due to surface roughness for example. This point is further confirmed by comparison with the EQEs achieved in practice by Spectrolab that also nearly reach 100%, as shown in Figure 3(a).
The principal transport parameters of the EQE and dark current fits are summarised in Table 3 for the Japan Energy Corp. tandem and Spectrolab triple cell fits. The quoted diffusion lengths in both cases are obtained from the literature as stated earlier for each emitter and base regions in each of the subcells in both cases. The diffusion lengths in all cases are of the order of microns, with the exception of Ge that benefits from significantly longer diffusion lengths as experimental studies in the literature have shown.
Table 3. Transport parameters: pdoped layer electron diffusion length diffusion lengths L_{n} and ndoped layer hole diffusion length L_{p}, electron and hole recombination velocities S_{n} and S_{p} determine the EQE.Tandem and triple cell transport parameters  L_{n} (µm)  S_{n} (m/s)  L_{p} (µm)  S_{p} (m/s)  τ_{SRH} (ns) 


JEC tandem—GaInP  0.4  750  2  50  40 
JEC tandem—GaAs  1.124  20  7.6  1  20 
Spectrolab triple—GaInP  5.11  100  1.56  35  50 
Spectrolab triple—GaInAs  0.28  3000  1  20  40 
Spectrolab triple—Ge  62.3  140  499  5  1000 
Also indicated in Table 3 are surface recombination velocities. The impact of these on the EQE cannot be easily decoupled from the diffusion lengths without a numerical fitting parameter exploiting the different wavelength dependence of recombination velocities and diffusion lengths. Here, we use a manual method and, using a fixed diffusion length determined by the material composition and doping, we vary the front and back recombination lengths to fit the short wavelength and long wavelength response, respectively. Following this procedure, these transport parameters are fixed by the EQE fit and determine the Shockley injection and surface recombination dark current mechanisms.
The EQE calculation therefore fixes the transport parameters. As a result, the only fittting parameter for the dark current calculation is the SRH lifetime in the spacecharge region, given in Table 3. The values in all cases, including the much longer SRH lifetime for the Ge subcell, are determined from the available high bias data available from the Spectrolab publication [19]. The trends observed as a function of material, and in particular the long SRH lifetime we see for Ge, are confirmed by a brief review of net SRH lifetimes in the literature. Somewhat fortuitously, the data available, while not supplying separate IV data for the Ge subcell, provide data for the middle and top bandgap subcells, in a bias range that allows us to fit the total triple junction dark current, and therefore the Ge subcell dark current, with a high degree of confidence that results from the exact simultaneous fit obtained for middle GaInAs subcell, top GaInP subcell and the combined triple dark currents seen in Figure 3(b).
Summarising this modelling, Tables 1 and 2 show the main cell performance figures of merit under AM1.5G in the cases of both the Japan Energy Corp. tandem record [18] and the Spectrolab triple junction record [19], showing good agreement with, in the worst case a ≈ 3%, relative underestimation of efficiency by SOL in the case of the tandem cell.
In the following sections, we will use this modelling approach to look at multijunction designs combining III–V materials on silicon. We will assume that material quality equal to that shown in these record cells is achievable using the low defect density III–V on silicon growth methods we have mentioned [9, 10].
CONCLUSIONS
 Top of page
 ABSTRACT
 INTRODUCTION
 IDEAL EFFICIENCY LIMITS
 QUANTITATIVE MULTIJUNCTION MODEL
 MULTIJUNCTION STRUCTURES ON SILICON
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
Investigating the use of Si for multijunction cells, we have looked at optimal yet nonideal designs in the ideal radiative limit that are constrained by the inclusion of a Si subcell. These designs show that high efficiencies are achievable by balancing competing compromises of material restrictions and nonideal optical properties.
In order to design real cells, we have developed a quantitative analytical model. We have validated this model by quantitatively reproducing stateoftheart light and dark current characteristics of record tandem and triple junction solar cells. The strength of the model lies in the minimisation of free parameters via consistent modelling of EQE and dark current characteristics.
This leads to dark current fitting in terms of a single free parameter, which is the SRH nonradiative lifetime in the spacecharge region. With the exception of the reflectivity calculations that assume lossfree dual layer MgF/ ZnS AR coats, the approach to modelling has been to use published experimental values for transport parameters, in order to maximise agreement with experimental data. The quantitative modelling of the record tandem and triple junction solar cells, together with the compatibility of transport parameters used with values in the literature, gives a high degree of confidence in this analytical modelling methodology. The minority carrier properties and SRH lifetimes used in the modelling are therefore reliable benchmarks for the material to be grown on Si by the growth methods developed within the MULTISOLSI project.
Applying this quantitative model to realistic solar cell designs combining Si and III–V materials, we first find that an appropriate tandem design of thinned GaAs on Si can deliver an efficiency of 29%, close to the 30% world record achieved with a GaAs substrate under a terrestrial spectrum with no concentration.
Furthermore, we find that a more challenging growth of ternary materials delivers tandem efficiencies greater than 32% and triple junction efficiencies greater than 36%.
These high efficiencies are remarkable in that they are achieved without solar concentration. They nevertheless reach efficiencies just a few per cent below the maximum efficiencies achieved to date, at concentrations of 500 suns and above. This lack of concentrating optics invites the clear advantage of simple photovoltaic systems. More importantly, from the cell design point of view, it significantly relaxes the design criteria for tunnel junctions between subcells, due to the significantly reduced and less demanding current flow achievable with a terrestrial global spectrum.
This work has been carried out within the MULTISOLSI project, which has demonstrated the growth of GaAs on Si without the formation of antiphase domains, and without the generation of bulk defects. Work continues to develop a finer theoretical understanding of the threedimensional nature of the structures involved, of the corresponding tunnel junctions, and to optimise the growth and processing techniques required to fabricate these potentially groundbreaking designs.