Experimental coupling process efficiency and benefits of back surface reflectors in photovoltaic multi‐junction photonic power converters

Current matching is crucial to maximize the efficiency of two‐terminal multi‐junction photovoltaic devices. However, even in perfectly designed devices, deviation from the target operating temperature and consequent changes in the subcell absorptances causes current mismatch between the subcell currents even at constant spectral conditions. Fortunately, luminescence coupling from current‐overproducing subcells to current limiting subcells mitigates this effect. In this work, the coupling process efficiency in three‐junction photonic power converters based on GaAs/AlGaAs rear hetero‐junction subcells is experimentally quantified. A coupling process efficiency of 32% ± 9% from top and middle subcells to the limiting bottom subcell is found. Under constant monochromatic illumination, the observed coupling reduces the current mismatch, induced by raising the temperature from current matched conditions at 25°C to 70°C, from 4.4% to 1.6%. Furthermore, in this work, three‐junction photonic power converters with back surface reflectors are implemented. Those reflectors improve the device response at elevated temperatures by increasing the optical path length in the limiting subcell. It is shown experimentally how a back reflector effectively redirects photons that are emitted by the bottom subcell towards the upper subcells to reinforce luminescence coupling.

chip level can be realized following the well-known multi-junction [8] or multi-segment approach [9], or a combination of both [10]. In multi-junction PV cells the subcells are vertically stacked and series connected by tunnel diodes. Note that opposed to solar cells for multi-junction PPCs, the subcells are all based on the same absorber material tuned to the incident laser light [6,7,[11][12][13][14]. In contrast, in multi-segment PV cells the subcells (also called segments) are laterally interconnected with a metal layer that connects the front side of one subcell with the rear side of the adjacent one. As the overall current in a series-connected device is limited by the subcell with the smallest current, in both configurations, ideally all series-connected subcells generate the same photo current (current match). In multi-segments cells this requires that all subcells need to be exposed to the same fraction of the light [15,16]. However, in multi-junction cells all subcells need to absorb the same fraction of the incident light (assuming perfect collection of the photo-generated carriers). This means that for a PPC with N subcells, the subcell thicknesses must be carefully designed in a way that all subcells absorb just a fraction of 1/N of the light impinging of the device. Now, in real-world applications the operating conditions typically vary and, for example, temperature variations can cause current mismatch between the subcells even in a perfectly designed device [7,13]. This reduces the output current and hence the conversion efficiency of the device.
In the case of multi-junction cells, current mismatch can be mitigated by luminescence coupling, between the subcells. Luminescence coupling originates from radiative recombination within the subcells.
Under mismatch conditions the subcells with excess current are forward biased and operate at a voltage between maximum power point and open circuit voltage. Hence, radiative recombination of overproduced carriers emits photons close to the materials bandgap (luminescence spectrum) that can be re-absorbed in the same subcell (photon recycling) or in other subcells (luminescence coupling). Consequently, the overproduced photocurrent is partly redistributed among the other subcells. In the ideal case of purely radiative recombination (i.e., 100% radiative efficiency [17]) and complete absorption of the re-emitted photons in the other subcells (i.e., 100% luminescence coupling), eventually the mismatched device would become current matched due to this redistribution. Generally speaking, even though deviations from design conditions in temperature or, for example, in the illumination spectrum affect the subcells absorptance and thus cause current mismatch, luminescence coupling makes multi-junction cells less sensitive to it. In multi-junction solar cells the effects of luminescence coupling have been studied in a number of experimental and theoretical works [18][19][20][21][22][23][24]. In those devices, usually each subcell features a different bandgap, and therefore, luminescence coupling proceeds in only one direction (Figure 1 left), namely, from high to F I G U R E 1 Sketches of a three-junction solar cell (with Eg1 > Eg2 > Eg3) (left) and a three-junction photonic power converter (right). Arrows represent luminescence photons (different colors means different photon energies). Photons involved in luminescence coupling are designated by dashed line arrows. The luminescence photons that escape through the front side or absorption into the substrate are designated by dotted line arrows. Photons that are trapped in each subcell as consequence of total internal reflection on interfaces with different refraction index are illustrated by the solid lines arrows [Colour figure can be viewed at wileyonlinelibrary.com] low bandgap materials (i.e., from top to bottom subcells). In multijunction PPCs, however, all subcells are usually made of the same material (Figure 1 right), so luminescence coupling can occur in both directions [12][13][14]. The fact that luminescence coupling is also possible towards upper subcells makes the presence of back surface reflectors (BSR) especially relevant to redirect those photons in multijunction PPCs. In order to shed light on the operation of these devices, Wilkins et al. included luminescence coupling in driftdiffusion based simulations and predicted coupling efficiencies between the subcells [25]. However, to the best of our knowledge, coupling efficiencies so far have not been studied and quantified experimentally in real multi-junction PPC devices.
In this work the coupling process efficiency in three-junction PPCs based on GaAs is studied and quantified experimentally. It is derived from the measured spectral response (SR) under current mismatch conditions due to operation away from target wavelength or induced by high operating temperature (Section 3.1). Furthermore, for the first time, three-junction PPCs with BSR have been implemented (Section 2.1). The effects of those reflectors are analyzed in Section 3.2, where the SR of devices with and without reflectors is compared at different temperatures.

| Devices
To study luminescence coupling effects, we analyze PPCs based on rear hetero-junctions, which provide elevated output voltages due to reduced diode saturation current which indicates more radiative devices [26,27]. Three vertically stacked GaAs/Al 0.30 Ga 0.70 As rear hetero-junction subcells (J1, J2, and J3) are grown upright by metalorganic vapor phase epitaxy (MOVPE) on 4 00 p-type GaAs substrates.
A sketch of the device structure is shown in Figure 1 (right). To avoid majority barriers at the GaAs/Al 0.30 Ga 0.70 As interface, the aluminum content is increased gradually. More details on this procedure can be found elsewhere [28,29]. Each subcell is electrically passivated with a GaInP-based front surface field layer and an Al 0.60 Ga 0.40 As back surface field layer. Table 1 lists the subcell thicknesses, which provide current match at 25 C at two different wavelengths, 809 nm (Structure A) and 845 nm (Structure B), according to a transfer matrix model [30].
The 4 00 wafers with epitaxial Structures A and B are processed to PV cell devices with nominal designated areas of 1.001 cm 2 (later in this manuscript referred to as 1 cm 2 cells) and 0.054 cm 2 in the clean room laboratory at Fraunhofer ISE. Evaporated front metal contacts are used, resulting in a 0.5% grid coverage for 1 cm 2 cells used in measurements at low irradiance and 2% grid coverage for 0.054 cm 2 cells used in measurements at high irradiances. A two layer anti-reflection coating based on Ta 2 O 5 /MgF 2 (74/79 nm) is applied to minimize the reflectivity in the target wavelengths range (809 and 845 nm). As to the rear side, either a full area Pd/Zn/Pd/Au metallization for electrical contact is applied (PPCs processed on substrate) or a thin film process is conducted and a BSR is applied (PPCs processed on BSR) following the processing scheme described in detail in Ref. [29].
For the thin film process, first the front side is processed and then bonded to a temporal sapphire carrier. Subsequently, the GaAs substrate is wet chemically removed with NH 4 :H 2 O 2 :H 2 O. This etching stops at a GaInP-based etch stop layer, which is subsequently removed with a hydrochloride dip. Then, 99.1% of the now exposed back surface area is coated with 200 nm thermally evaporated gold, whereas the remaining 0.9% is used to provide low ohmic rear contact (with hundreds of point contacts of 10 μm diameter) by evaporation of a Pd/Zn/Pd/Au stack. Finally, a 30 μm copper layer is electroplated on the rear side to provide physical stabilization in addition to good electrical and thermal conductivity. As last step, the sapphire carrier is removed and flexible PV cell wafers with BSR on thin copper foil are obtained.

| Methods
External quantum efficiency (EQE) of the multi-junction device is measured with a LOANA solar cell analysis system at 25 C and 70 C on the 1 cm 2 PV cells with a minimal front metal grid coverage of 0.5% (to avoid measurement uncertainty due to shading effects). The Furthermore, the spectral absorptances of the top, middle, and bottom subcells are modeled using the transfer matrix formalism [30], that is, purely optical consideration without accounting for luminescence coupling effects. For this purpose, the spectral optical data It is noted that the obtained GaAs optical data mostly agree well with literature data [31] but differs close to the bandgap above 840 nm, presumably because the optical bandgap is affected by the doping [32]. The data are used to determine intensity-dependent spectral response (SR) at the laser wavelength. The laser is temperature controlled and calibrated to maintain its emission wavelengths for all studied irradiance levels. The laser beam is homogenized by two micro lens arrays to a square spot of 3.5 × 3.5 mm 2 (larger than the area of the measured cell), with a spatial uniformity of 98.8%. More details about the spatial light intensity uniformity in the test plane can be found in Ref. [33]. The temperature of the multi-junction PPCs is defined at 25 C or 70 C using a temperature controlled in the raw signal are reduced to deviations ≤ ±0.16% during one laser pulse [33]. Note that this procedure to compensate small irradiance variations is only valid when the current generated by the device under study is linear with irradiance. Rigorously, this is not the case in multi-junction devices, where luminescence coupling occurs. However, as the laser intensity fluctuations are small compared with the signal itself, the influence of luminescence coupling on the correction procedure is neglected. In order to determine the absolute irradiance in the test plane (G in ) at different laser intensities, a calibrated GaAs-based single-junction reference cell is used.

| Coupling process efficiency
In this work we introduce the coupling process efficiency (η C ) as the effective efficiency of redistributing the photocurrent under mismatched conditions from current-overproducing subcells to the current limiting subcell. It includes all losses in the coupling process within the multi-junction semiconductor structure, and it can be determined experimentally without knowledge about the actual number of photons emitted from each individual subcell. Note that this definition differs from the "total coupling efficiency" defined in Ref. [25], which describes per subcell the probability for photons emitted by radiative recombination from that subcell to be re-absorbed in any of the other junctions. The coupling process efficiency η C is where η lum is the probability that excess carriers recombine radiatively and emit photons which escape the subcell where they are generated, reaching another subcell where they can be absorbed; η abs is the probability to absorb photons generated by luminescence in another subcell and subsequent hole-electron pair generation; η col is the probability of collecting the corresponding minority carriers. The impact of different coupling process efficiencies on the current density-voltage (J-V) characteristic of a three-junction PPC is illustrated in Figure 2. The excess carriers in two overproducing subcells recombine as consequence of the series connection imposed. This is illustrated in Figure 2 by arrows labeled as R 1 and R 2 , which represent the reduction in current from the photogenerated current to the current of the operating point. Depending on the material quality and its radiative efficiency, part of this recombination results in photons that can be re-absorbed in the limiting subcell and in turn raise its current.
The gain in current in the limiting subcell (illustrated in Figure 2 by the arrow labeled as G) effectively raises the device current in dependence of the luminescence coupling process efficiency η C .
Referring to the nomenclature of Figure 2, the coupling process efficiency can be expressed as follows: where the values for G, R 1 , and R 2 will depend on η lum , η abs , and η col corresponding to the specific case under study (direction of luminescence propagation and features of the subcells involved). Equation 2 can be reformulated if the reduction in the current produced by the limiting subcell (D 3 ) equals the excess of generation in the overproducing subcells (D 1 and D 2 ), which means that D 3 = D 1 + D 2 . Then It is remarked that in fact coupling takes place in all directions; that is,      [12,33]. These effects feature different bias light dependences: whereas the parallel resistances artifact diminishes with increasing bias light intensity and thus current density, the luminescence coupling effect intensifies with increasing intensity because the device becomes more radiative. Hence, we conclude that parallel resistance affects the EQE for bias light intensities below 0.04 suns.
At higher intensities the effect of luminescence coupling becomes apparent. As the EQE measurement with a 0.04 suns bias approaches the modeled bottom subcell absorptance best, we consider this measurement to be least affected by luminescence coupling and parallel resistance artifacts. Note that a similar study was done in Ref. [35], where the optimal bias light intensity to minimize the EQE measurement artifact results also from the trade-off between the shunt and the luminescence coupling effects.  Within the measurement uncertainty it is independent of irradiance, as expected for this case without luminescence coupling. For the same device but at elevated temperature of 70 C (red circles) the measured SR is reduced and a minor increase with irradiance is observed (1.2% increase in SR per irradiance decade). A similar but much more pronounced trend is observed in Structure B measured at 25 C (5.3% increase in SR per irradiance decade), which is strongly current mismatched under 809 nm laser light. The increase in SR with irradiance is caused by luminescence coupling, as the subcells become more and more radiative as irradiance increases [13].
In the current mismatch cases evaluated in Figure 4  (without BSR), η col is expected to be close to 100%, whereas η abs is expected to be around 80% for the emitted photon energies close to the GaAs bandgap [29]. In the current mismatch cases analyzed in this section, η lum is the probability that excess carriers in the top and middle subcells recombine radiatively and emit photons which escape these subcells to be absorbed in the bottom subcell. Due to the thin thickness of the top and middle subcells, η lum can be analyzed like an external luminescence efficiency emitted from a thin semiconductor heterostructure, like the ones studied in Ref. [38], but considering the photons that escape through the back side, not through the front side.
This parameter is very sensitive to small non-radiative and parasitic optical losses, such as free carrier absorption in highly doped semiconductor layers, so even with highly radiative subcells we can obtain relatively small η lum , compared with the internal radiative efficiency.
To identify if η lum is dominated by non-radiative losses or parasitic optical losses, a similar study to the one reported in Ref. [38] should be done in the future.

| Impact of the BSR
The implementation of BSRs in single-junction and multi-junction PV cells has been studied for solar and thermophotovoltaic applications [29,[39][40][41]. However, to the best of our knowledge, these reflectors  [29,42]. In addition, in the multi-junction case the BSR also impacts luminescence coupling between the subcells and thus the output current of the device in current mismatched conditions. This effect will be analyzed in more detail in the following.  Figure 1). So those photons can reach upper subcells and be absorbed there. This is why at temperatures below 25 C ( Figure 5 bottom), when the top subcell is limiting the SR, the cell with a BSR increases its SR in respect to the cell without BSR at irradiances above $10 W/cm 2 . At lower irradiances, the increase in SR with G in is not affected by the presence of a BSR, which indicates that the luminescence generating this increase does not originate from the bottom subcell but from the middle one. Here, the coupling process efficiency cannot be quantified, because the considerations used to obtain Equation 3 are not fulfilled in this case: The current in the limiting top subcell is reduced with respect to the current match value due to insufficient absorption and that cannot be compensated by absorption in the other subcells, so D 3 ≠D 1 + D 2 . However, the results shown in Figure 5 show the influence of a BSR on luminescence coupling from the bottom to the top subcells in a three-junction PPC.

| CONCLUSIONS
In this work a methodology has been presented to assess the coupling process efficiency in multi-junction PV cells based on one absorber material only, as used in photonic power converters for one monochromatic wavelength. The procedure is based, on one hand, on absorptance modeling and measurements at low irradiance conditions (bias-dependent EQE) to estimate the spectral response in absence of luminescence coupling at a target wavelength. On the other hand, the procedure also requires measurements at high irradiance conditions, like I-V measurements under high-power laser illumination to quantify the coupling process efficiency under current mismatch conditions.
For photonic power converters based on three stacked GaAs/AlGaAs rear hetero-junctions, the coupling process efficiency has been experi-