Observation and implications of the Franz‐Keldysh effect in ultrathin GaAs solar cells

Voltage‐dependencies were observed in the external quantum efficiency (EQE) spectra of ultrathin GaAs solar cells. The subbandgap tail was shown to increase going from forward to reverse bias, while at energies above the bandgap, voltage‐dependent oscillations in the EQE were measured. Using optical simulations, it is irrefutably shown that the voltage‐dependencies are caused by the Franz‐Keldysh effect, that is, an electric field‐dependent absorption coefficient near the bandgap. The dependency on voltage of the subbandgap tail is demonstrated to be strongest in thin‐film cells with a textured rear mirror, since the absorptivity below the bandgap is enhanced by light trapping. The voltage‐dependent subbandgap tail has important implications for the use of the reciprocity relation between photovoltaic quantum efficiency and electroluminescence. It is shown that the radiative limit for the open‐circuit voltage of thin‐film cells integrated with light management schemes can be underestimated by more than 25 mV. Consequently, these cells may be assumed to be closer to the radiative limit than they really are.


INTRODUCTION
In 1958, Franz and Keldysh described in two separate publications the effect of a uniform electric field on the absorbing properties of semiconductors and dielectrics. 1,2 The electric field distorts the band structure of the material, thereby changing the joint density of states in the valence and conduction band. This leads to oscillations in the absorption coefficient at energies above the bandgap while below the bandgap an exponentially decaying tail is introduced as a result of photon-induced tunneling. When the electron and hole wave functions are confined in a quantum-well structure, the effect is known as the quantum-confined Stark effect. 3 Tharmalingam 4 showed that the absorption coefficient near the bandgap in the presence of an electric field can be described by Airy functions, which capture both the oscillating nature of the above-bandgap absorption coefficient as well as the exponentially decaying subbandgap tail. In Figure 1, we plot the calculated absorption coefficient of GaAs for a range of electric fields using these Airy functions. It shows that both the period and amplitude of the oscillations as well as the extent of the subbandgap tail strongly depend on the magnitude of the electric field.
The Franz-Keldysh effect can therefore be one of the contributors using an Urbach energy, which characterizes the slope of the subbandgap absorption coefficient. Typical values for the Urbach energy range from 5 meV for high-quality GaAs, 9 15 meV for most chalcogenide and other polycrystalline materials, to >42 meV for amorphous silicon. 10 In general, a narrow Urbach tail-and hence low Urbach energy-is advantageous for solar cell performance because it minimizes radiative emission rates, allowing higher open-circuit voltages (V OC ) to be achieved in the radiative limit.
The field-induced oscillations in the absorption coefficient, known as Franz-Keldysh oscillations, contain a wealth of information about the electronic properties of a material, including internal fields and band structure. This is exploited in modulation spectroscopy, where the internal field is periodically modulated either electrically (e.g., in electroabsorption measurements) or optically by a chopped pump beam. 11,12 The resulting changes in the absorptivity or reflectivity can then be detected by a lock-in amplifier. The technique has found wide application, amongst others in the analysis of solar cells based on direct bandgap semiconductors. Examples include the analysis of internal fields and doping levels in GaAs 13-15 and amorphous Si 16 solar cell structures, analysis of absorber bandgaps in GaInP/GaInAs/Ge triple-junction cells and several chalcogenide solar cells, [17][18][19] and to determine material parameters such as exciton binding energy of perovskites. 20 However, a significant impact of the Franz-Keldysh effect on solar cell performance metrics has not been observed thus far to the best of our knowledge.
In this work, we report observations of significant voltagedependencies in the external quantum efficiency (EQE) spectra of several ultra-thin GaAs solar cell structures. First, using on-substrate ultrathin heterojunction cells with different emitter doping levels, we show irrefutably that the voltage-dependencies are caused by the Franz-Keldysh effect. Then, we analyze a thin-film structure with a textured rear mirror and show that in this case, the dependence on voltage of the subbandgap tail is significantly more pronounced due to efficient trapping of sub-bandgap photons. We discuss the implications of this observation for the use of the reciprocity relation between photovoltaic quantum efficiency measured at short circuit and electroluminescence.

Observation and simulation of the Franz-Keldysh effect in ultrathin heterojunction cells
The layer structure of the investigated ultrathin heterojunction solar cell is schematically depicted in Figure 2. The structure consists of a 300-nm n-type GaAs emitter interfaced with a p-type InGaP base/back surface field (BSF) and a wide-bandgap n-type AlInP window. The heterojunction architecture allows us to observe band tailing effects originating solely from the n-type emitter, while in an ultrathin homojunction architecture different contribution from the emitter and base may complicate the interpretation of results. Furthermore, due to the low absorber thickness, the depletion region makes up a significant fraction of the absorber layer. Since the Franz-Keldysh effect (electric field-dependence of the absorption coefficient) only affects the absorption of photons in the depletion region, we expect it to be more significant in an ultrathin cell structure. We processed two batches of solar cells according to the structure shown in Figure 2 with emitter doping levels (N d ) of 6x10 16 Table 1. Although the difference in performance between the two architectures is not the focus of this work, the lower J SC of the highly doped cell can be explained by a slightly lower broadband spectral response, which might be related to a higher interface recombination at the window/emitter interface.
The EQE spectra of the two heterojunction cells are shown in Figure 3, for externally applied bias voltages ranging from a reverse bias of -2.4 V to a forward bias of 0.8 V. Beyond 0.8 V, the voltage-induced dark current starts to become of comparable magnitude to the measured signal which therefore diminishes (see Section 4). Figure 3A shows clear oscillations above the bandgap energy in the EQE of the lightly doped cell, which are largely absent in the highly doped cell. Fur- Abbreviations: FF, fill factor; PCE, power conversion efficiency thermore, in Figure 3A,B, a significant dependency of the subbandgap tail on bias voltage can be observed, again more strongly in the lightly doped cell than in the highly doped cell. In addition, Figure 3B reveals that at long wavelengths, the EQE of the highly doped cell becomes states is expected to be higher in the highly doped cell, and therefore the voltage-dependent effects should be more prominent. This is, however, in stark contrast to the observations.
In order to irrefutably show that the observed voltage-dependent effects are due to the Franz-Keldysh effect, we simulated the absorptance of the GaAs absorber layer for a range of bias voltages using the transfer-matrix method. The detailed mathematical approach is outlined in the Supplementary Information; however, the general procedure is as follows. First, the inhomogeneous electric field profile in the GaAs absorber layer was calculated. 24,25 The changing bias voltage affects both the field strength as well as the width of the depletion region. Therefore, the 300-nm-thick GaAs absorber layer was divided into 60 slices of 5-nm thickness and the local absorption  In the simulations, this is due to the convolution of the Franz-Keldysh absorption coefficient with a regular Urbach tail, for which a higher Urbach energy was taken in case of the highly doped cell.
The oscillations in the EQE spectra as a function of bias voltage imply that at some wavelengths, the absorptance increases with voltage while at other wavelengths the absorptance decreases with voltage.
The slope of a monochromatic IV-curve near short-circuit condition can therefore be both positive as well as negative, depending on the illumination wavelength, as was theoretically modeled by Aeberhard. 29 Here, we present measurements of this phenomenon in Figure 5.
The most striking behavior is measured in the lightly doped cell explaining the steeper slope of the curves. Figure 5B shows the same monochromatic measurements on the highly doped cell. It is clear that the voltage-dependence of the absorptance is less pronounced, in line with the EQE measurements shown in Figure 3.

The Franz-Keldysh effect in cells with a textured rear mirror
The Franz-Keldysh effect was shown to have a significant impact on the EQE spectra of the on-substrate cell architecture shown in Figure 2. Because it is an ultrathin cell structure without a rear mirror, the long-wavelength absorptivity and therefore the maximum efficiency is low, and as a result, the cell structure may not be of much practical significance. However, in general, ultrathin GaAs cells possess advantages over thicker cells in terms of radiation hardness and production cost. 30 When produced in a thin-film architecture, the long-wavelength absorptivity of ultrathin GaAs solar cells can be increased by incorporating a rear mirror, especially when the mirror is textured. In this case, the impact of the Franz-Keldysh effect can be significantly higher due to the strongly increased path length of scattered subbandgap photons, as we will show in the following. We recently described an effective light trapping approach for ultrathin GaAs solar cells based on simple wet chemistry. 31 The cell structure that was employed is schematically depicted in Figure 6. It consists of an ultrathin homojunction GaAs absorber sandwiched between passivating, charge-selective layers and highly doped contact layers.
Local Ohmic contact points were fabricated at the rear side to conduct the current while at the same time, the Al 0.3 Ga 0.7 As contact layer in between these contact points was manipulated in order to engineer the cell optics. On one half of the wafer, the Al 0.3 Ga 0.7 As contact layer was textured using a simple 1-min wet-etching step in-between the contact points. On the other half of the wafer, reference cells with a planar rear mirror were produced by completely removing the Al 0.3 Ga 0.7 As contact layer in-between the contact points using a regular polishing etchant. Since the planar and textured cells were processed on the same wafer and therefore originate from the same growth run, the initial material quality was identical.

The performance of these cells is characterized in detail in Van
Eerden et al, 31 however, in short, the J SC and PCE of the best planar reference cell were 21.7 mA cm −3 and 18.5%, respectively, compared with 24.8 mA cm −3 and 21.4% for the best rear-side textured cell.
In the present study, we merely show the voltage-dependent EQE spectra of the two architectures in Figure 7. The figure shows that the oscillations above the bandgap energy are again present as shown in the insets in Figure 7B, albeit less prominent than in the EQE spectra   Figure 3A. This is inline with our earlier observation that when the variation of the depletion region is small in comparison to the absorber layer thickness, the impact of the Franz-Keldysh on the absorptance is less significant. This is the case for this homojunction architecture, due to the higher doping levels employed. The dependency of the subbandgap tail, however, is markedly stronger in these cells, particularly in the cell with a textured rear mirror (note that the measurements were only made down to V = -1.2 V, instead of -2.4 V in Figure 3). This is related to the longer path length of subbandgap photons in these cells. Especially in the textured cell, scattered subbandgap photons can be trapped inside the cell, strongly increasing the absorptance. Small changes in the subbandgap absorption coefficient due to the Franz-Keldysh effect therefore result in larger absolute changes in absorptance and show up more prominently in the EQE spectra. It is noteworthy that the morphology of the textured rear mirror is random, 31 and therefore its optical properties do not exhibit sharp spectral features. As a result, the EQE near the bandgap is rather smooth and its spectral variation mainly depends on the absorption coefficient. This renders cells employing this rear-side textured mirror particularly suited to study band tailing effects.

Implications of the occurrence of the Franz-Keldysh effect for PV research
In recent years, for virtually all PV technologies, significant research efforts have been directed to increase cell efficiencies towards the Shockley-Queisser limit. Therefore, it is becoming increasingly relevant to accurately determine the performance limits and to be able to gauge how closely a cell operates to them. The radiative (or ''detailed-balance'') limit for the open-circuit voltage (V db ) is one of such performance limits, and it can be calculated using the photovoltaic reciprocity relation between external quantum efficiency and electroluminescence 32 as follows: where k is the Boltzmann constant, T is the temperature of the cell, q is the elementary charge, J SC is the short-circuit current, J rad 0 is the radiative saturation current density of a solar cell in the radiative limit, sun (E) is the solar spectrum, and BB (E) is the blackbody spectrum at the temperature of the cell. EQE is the weighted average over all angles of incidence, which is usually similar to the regular EQE for normal incidence. 33,34 When the radiative limit for the open-circuit voltage of a solar cell is known, one can assess how closely a given solar cell is operating to this limit by comparing it with a measured V OC .
Any losses of the measured V OC with respect to the radiative limit can be related to an external luminescent efficiency ext lower than unity according to 35 where ext represents the fraction of all recombination events that leads to emission through the front side of the cell. In recent years, In the following evaluation, we demonstrate the impact of the Franz-Keldysh effect on the determination of V db and ext using measured EQE spectra of the planar and rear-side textured cells shown in Figure 7. In our recent publication 31  It can be noted that in reality, the differences will be even larger since we used EQE spectra measured at 0.8 V to calculate V db . At actual open-circuit condition (V >1 V for our cells); however, the subbandgap tail will likely be even narrower. Therefore, the actual V db and ext will be even higher and thus lower, respectively. Nevertheless, by measuring the EQE as close to open circuit as possible and using these spectra to calculate V db and ext , more accurate values can be obtained than when the standard practise of using short-circuit EQE spectra is employed.

CONCLUSIONS
In this study, the Franz-Keldysh effect was shown to have a significant It is demonstrated that the voltage-dependencies have to be taken into account when employing the standard reciprocity relation between photovoltaic quantum efficiency measured at short circuit and electroluminescence. This relation no longer holds when the absorption coefficient is voltage dependent. With device measurements, it is shown that the radiative limit for the V OC of thin-film cells that incorporate light management schemes can be underestimated by more than 25 mV when it is calculated using EQE spectra measured at short circuit. In turn, this leads to an overestimation of the external luminescent efficiency by roughly a factor of 3. Consequently, one may assess these solar cells to be closer to the radiative limit than they truly are. By using EQE measurements made at forward bias to calculate the radiative V OC limit and external luminescent efficiency, these metrics can be determined more accurately. This may yield a better understanding of the true optoelectronic quality of a solar cell.

EXPERIMENTAL
Solar Cell Fabrication. The solar cells used in this study were grown using low-pressure metal-organic chemical vapor deposition on 2 inch diameter (100) GaAs wafers, 2 • off to (110) orientation. Zn was used as p-type dopant and Si as n-type dopant except for the highly doped n-GaAs contact layer, which was doped using Te. More details on the growth can be found elsewhere. 49,50 For the on-substrate heterojunction cells, cleaning steps in acetone, water, diluted HCl, water, and isopropanol were followed by standard lithographic techniques to define a resist mask for the front grid. A 250 nm Au front grid and subsequently a 100 nm Au rear contact were then evaporated using electron beam evaporation. The n-GaAs contact layer was then etched in between the front grid using NH 4 OH:H 2 O 2 :H 2 O (2:1:10 by volume).
After another lithography step, etching of the MESA was performed in 25% H 2 SO 4 :H 2 O 2 (4:1). The solar cells were finished by evaporating a ZnS/MgF 2 antireflection coating using thermal evaporation.
More details on the growth and processing of the thin-film cells with a planar and textured rear mirror can be found in Van Eerden et al. 31 EQE measurements. EQE measurements were made using a ReRa SpeQuest system operated with ReRa photor 3.1 software. Light from a Xenon or halogen light source was directed through a monochromator (LOT MSH-300) and chopper and subsequently focussed on the cell with a spot size of about 3 mm. The cell response was measured using a lock-in amplifier. Spectra were measured in 5-nm intervals. In the case of the planar and textured thin-film cells, at a bias of 0.8 V, the dark current started to noticeably diminish the measured photocurrent, which is why the EQE was not measured at higher bias voltages.
The EQE spectra measured at 0.8 V were normalized to the spectra Transfer-matrix simulations. Calculations of the bias-dependent absorptance spectra were performed using a freely available Matlab code from Stanford University. 51