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Abstract
Ultrathin III-V solar cells with proper light management have become more attractive than their optically thick counterparts as they are less expensive and lightweight, can maintain photon absorption, and have high radiation tolerance for space-related applications. Comprehensive optical modeling efforts have provided pathways to improve device efficiency in ultrathin GaAs solar cells with light trapping structures. Usually, the absorption mechanism known as free-carrier absorption (FCA) is ignored in these models due to the ultrathin layers and the direct bandgap of GaAs. This manuscript reports the significance of considering FCA as a parasitic loss caused by the optical enhancement in highly doped non-active layers between the ultrathin solar cell and backside light trapping structures. We model FCA based on Drude theory in a p-type AlGaAs layer behind ultrathin GaAs solar cells with a planar mirror and cylindrical gratings. Our results show that, depending on the AlGaAs thickness and doping concentration, free carriers will absorb transmitted photons and reduce the backside reflectance, degrading the current and voltage output from ideal conditions. One example shows that for a 300 nm-thick GaAs solar cell, the Ag mirror's peak reflectance decreases nearly 12% when the AlGaAs back layer is 800 nm-thick at a doping concentration of 4x1019 cm−3. Notably, the cylindrical grating designs with 38.5%, 46.5%, and 64.9% AlGaAs coverage resulted in an absolute efficiency reduction of 0.6%, 1.8%, and 2.9% at a doping concentration of 4x1019 cm−3, respectively. This novel study demonstrates that FCA in non-active layers must be properly addressed in the device design to progress the efficiency of ultrathin III-V solar cells with light trapping structures.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultrathin III-V photovoltaics (PV) with light trapping structures present the opportunity to reduce material cost while achieving high efficiency, suitable for terrestrial and space applications [1–12]. Light management is essential in these devices since the physical active region thickness is not optically thick enough to complete photon absorption. To combat the transmission loss in ultrathin solar cells, employing back surface reflectors (BSR) will increase the optical path length (OPL) of unabsorbed photons for another chance to generate electron-hole pairs inside the active region. The BSR can be designed in many ways, including a planar mirror to achieve a twofold photon path length [2,8], a textured surface to increase total internal reflection (TIR) [5,7,13] or nanostructures to excite optical modes within the cavity of the active region [9,11,12,14]. Many designs have experimentally demonstrated increased current and voltage output in ultrathin solar cells, yet achieving the maximum efficiency based on the detailed-balance limit awaits [15,16].
Optical modeling has become a well-established and integral part of demonstrating the efficacy of light trapping structures as ultrathin III-V solar cells continue to push towards their maximum achievable efficiency. In particular, researchers have rendered potential pathways to reach higher voltage and current output in ultrathin GaAs solar cells [2,11,13–18]. All parasitic optical losses must be accounted for when modeling ultrathin GaAs solar cells, including absorption loss in the back mirror and non-active layers behind the photoactive region of the device. Absorption in the metal can be suppressed by using a low-index spacer to improve TIR [8,13,14,19,20] and using wide bandgap materials as the non-active back layers removes band-to-band absorption loss [2,8,9]. This back layer is called "non-active" since it is outside the photoactive region and does not proactively generate photocurrent but requires an elevated doping concentration for carrier transport to the external electrodes. This elevated doping concentration in the back layer may lead to the absorption mechanism known as free-carrier absorption (FCA). FCA is a parasitic optical process that occurs in heavily doped semiconductors where free carriers reduce the intensity of light passing through an absorbing medium but do not generate electron-hole pairs [21,22]. In photovoltaics, FCA is mainly considered in silicon solar cells due to its indirect bandgap [21–25] or in heavily doped and thick III-V substrates [26,27]. Intuitively, FCA is ignored when modeling III-V solar cells since the non-active layers are ultrathin and usually have a direct bandgap. This assumption changes when the solar cell optical cavity is related to a laser cavity where FCA is regarded as a design constraint due to the optical enhancement within these systems [28–33]. Moreover, light trapping development requires the non-active back layers to be thick enough for texturing and must preserve electrical contact using a high doping concentration [2,8]. These factors reveal the situation where absorption by free carriers may be present in the non-active back layers, initially designed for light management and carrier transport. In particular, transmitted photons after the first pass through the ultrathin GaAs solar cell will slowly attenuate in the non-active back layer due to FCA, eliminating the light trapping structure’s potential current and voltage enhancement benefits. For these reasons, it is crucial to consider the parasitic loss by FCA in non-active layers when designing light trapping structures in ultrathin III-V solar cells.
This study primarily focuses on the optical modeling of ultrathin GaAs solar cells with different BSR designs, as seen in Fig. 1, to investigate trends in FCA based on the doping concentration and thickness of non-active layers behind the photoactive GaAs region. The electrical performance of the diode is assumed to remain consistent across all simulations, while the change in doping concentration in the back layer impacts the series resistance. A thorough investigation to determine an optimal doping concentration in the back layer targeting low series resistance for sufficient carrier transport and minimal FCA is necessary for device-specific designs. Focusing on optical modeling, the light trapping geometries investigated in this work include a planar reflector and three cylindrical geometries. These designs are modeled using a rigorous coupled-wave analysis (RCWA) included in the open-source Python 3 extension, RayFlare [34]. The free-carrier absorption coefficient used in the back layer is determined according to classical Drude theory as discussed in Section 2.2. By combining Drude theory to describe the absorption by free carriers in the back layer with RCWA, we show that the FCA increases as the thickness and doping concentration of the back layer increases. From these trends, we extract the reduction in the short-circuit current density ($J_{SC}$) and the open-circuit voltage ($V_{OC}$) from ideal conditions where FCA is not considered. The results indicate that FCA in non-active layers should not be underestimated as the reduction in device performance may be substantial in these devices.
Fig. 1. Plot (a) illustrates the GaAs solar cell with a planar Ag mirror and a SiO$_{2}$ interlayer. The three main parameters including the absorptance in the GaAs solar cell ($A_{GaAs}$), absorptance in the back layer due the FCA ($A_{BL}$), and the backside reflectance (R) are displayed to correlate these processes to the regions in which they occur. Plot (b) shows a unit cell of the cylindrical gratings, further discussed in Section 3.3. The AlGaAs radius (r), SiO$_{2}$ pitch (a), and height ($t_{BL}$) change according to each grating design.
2. Methods
2.1 Solar cell design and light trapping geometries
The first BSR investigated is a planar geometry, capable of achieving a twofold increase in the OPL as transmitted photons reflect off the back mirror into the optically thin GaAs solar cell. Figure 1(a) represents the GaAs solar cell with a planar reflector and a bi-layer anti-reflection coating (ARC) of MgF$_{2}$ and ZnS. Below the ARC is the wide bandgap front surface window, AlInP. These three layers assist in reducing the front surface reflection of incident light. The GaAs absorber has a photoactive region with thickness indicated by $t_{GaAs}$. For the planar BSR, three values of $t_{GaAs}$ are investigated: 100 nm, 300 nm, and 500 nm. Behind the GaAs region is the non-active back layer where absorption by free carriers is studied, dependent on the thickness ($t_{BL}$) and doping concentration ($N_{BL}$) of the back layer. Behind the back layer is a 500 nm-thick low-index SiO$_{2}$ spacer and a highly reflective Ag mirror. In this design, the n-i-p polarity is considered to investigate the worst-case scenario of FCA in the p-type back layer. Further discussed in Section 2.2, holes naturally have a lower mobility and result in a higher FCA than electrons [28]. Due to its wide bandgap with increasing Al composition, Al$_{x}$Ga$_{1-x}$ As is chosen as the material for the non-active p-type back layer. Additionally, Al$_{x}$Ga$_{1-x}$ As (hereafter, referred to as AlGaAs) is a common material used in GaAs solar cells due to its bandgap tunability, texturing and doping capabilities, and the fact that it can be grown nearly lattice-matched to GaAs [5,8,19]. The AlGaAs back layer varies in $t_{BL}$ and $N_{BL}$, and combinations between these factors with $t_{GaAs}$ are modeled to find trends in FCA. Figure 1(a) also displays the specific regions of interest where absorptance and the backside reflectance (R) occur. Specifically shown are the absorptance in the photoactive GaAs solar cell ($A_{GaAs}$) which aids in proactive photogenerated current, and the parasitic absorptance in the back layer ($A_{BL}$) due to FCA. Lastly, R is the measured amount of reflected photons, which is dependent on $t_{GaAs}$, $t_{BL}$, and $N_{BL}$. Section 2.2 discusses the computation to find $A_{GaAs}$, $A_{BL}$, and R.
Apart from the planar geometry, three cylindrical gratings consisting of AlGaAs cylinders embedded in SiO$_{2}$ are investigated. Figure 1(b) displays a unit cell of the cylindrical gratings with AlGaAs radius (r), SiO$_{2}$ pitch (a), and height ($t_{BL}$). In this design, the unit cell repeats periodically in the xy plane and replaces the back layer. This substitution is the only difference compared to the planar BSR design in Fig. 1(a). For the cylindrical geometry, the 300 nm-thick GaAs solar cell is the main focus since, under ideal Lambertian circumstances, it can absorb 98% of the available photons in the radiative limit [14]. The optical constants used in the model were measured on in-house grown or deposited non-active layers (MgF$_{2}$, ZnS, AlInP, and SiO$_{2}$) using a spectroscopic ellipsometer (RC2, J.A. Woollam Co., Inc.) and appropriate fits were performed using the CompleteEASE software. Since GaAs is a widely understood material, the optical constants were taken from the J.A. Woollam CompleteEase database. The optical constants for the materials used in the model are displayed in Fig. S1 in Supplement 1. This study ignores band-to-band absorption in the AlGaAs layer since FCA is the main focus. Instead, the extinction coefficient in the AlGaAs layer is dependent on the modeled free-carrier absorption coefficient, as discussed in Section 2.2. This assumption in the model is reasonable considering that in experimental designs that use AlGaAs as a backside textured layer, the Al fraction is high enough to make it transparent in terms of band-to-band absorption [5,7,14].
2.2 Modeling free-carrier absorption in the back layer
The classical Drude model, which is extensively used to calculate the free-carrier absorption coefficient ($\alpha _{{FCA}}$) in laser cavities [28,35–37], is used to describe the FCA in the AlGaAs layer. The $\alpha _{{FCA}}$ is calculated by,
where q is the electron charge, $\lambda$ is the emission wavelength, N is the carrier concentration, $\varepsilon _{0}$ is the electric constant, n is the index of refraction, and $\mu$ and m are the mobility and effective mass of the charge carriers, respectively. In this expression, absorption by free carriers is dependent on the carrier concentration and mobility and will be greater in p-doped semiconductors since the mobility is nearly two orders of magnitude lower than in n-type semiconductors at the same doping concentration [37]. As the FCA is studied at different doping levels, the mobility to determine $\alpha _{{FCA}}$ must change, too. This is accomplished by using the low-field empirical mobility model to calculate the mobility of p-type carriers [38]. The mobility at 25 $^{\circ }$C is defined by, where $\mu _{min}$, $\mu _{min}$, $\textit {N}_{ref}$, and $\lambda$ are fitting parameters specific to the carrier type and semiconductor [38]. To validate the mobility model against experimental measurements, multiple Al$_{0.3}$Ga$_{0.7}$ As films were grown on 2-inch (100) GaAs wafers with a 2-degree offcut <110> via metalorganic vapor phase epitaxy (MOVPE) and measured using Hall to determine N and $\mu$. Figure 2(a) displays the experimental results plotted against the mobility model using the fitting parameters described by Sotoodeh et al. [38]. The agreement between the experimental Hall measurements validates the use of the mobility model to determine $\mu$ in Eq. (1). The remaining parameters in Eq. (1) are based on values in literature, such as the effective mass of holes in AlGaAs as determined by Adachi [39], which remains constant for a given Al composition. Values for n were determined by spectroscopic ellipsometry on Al$_{0.3}$Ga$_{0.7}$As and appropriate fits were performed. Figure 2(b) shows the $\alpha _{{FCA}}$ of AlGaAs at doping concentrations between $5{\times}10^{18}$ cm$^{-3}$ and $4{\times}10^{19}$ cm$^{-3}$, which is the range used in the modeled structures. In this plot, the doping concentration has a direct influence on FCA, especially at longer wavelengths. The modeled FCA results in this work are based on Eq. (1) and are performed at standard temperature at 25 $^{\circ }$C. Notably, if the solar cells are designed for extreme temperature conditions, then the low-field mobility model including temperature must be applied to calculate the change in mobility as a function of temperature [38].Fig. 2. Plot (a) shows experimental Hall results taken on p-type AlGaAs samples compared to the mobility model. Plot (b) shows the $\alpha _{{FCA}}$ calculated by the Drude model, which is integrated with the mobility model.
Integrated in Rayflare, the Pol method established in the Stanford Stratified Structure Solver (S$^{4}$) was used in the RCWA computations [34,40] to study the impacts of FCA in the AlGaAs layer (simply referred to as the back layer, hereafter) behind the ultrathin GaAs solar cells. The four main factors contributing to the change in transmitted photons during each pass through the solar cell cavity include $t_{GaAs}$, $\alpha _{GaAs}$, $t_{BL}$, and $\alpha _{{FCA}}$. With the RCWA methods and absorption coefficients determined in the GaAs absorber and back layer, various combinations depending on the thicknesses of each layer are simulated to determine the variation in FCA. The total reflectance (R), absorptance (A), and transmittance (T) between 350 nm - 900 nm at normal incidence are computed as follows:
where A$_{GaAs}$ and A$_{BL}$ present the absorptance in the GaAs and back layer based on their absorption coefficients, respectively. The fractional R, T, A$_{GaAs}$ and A$_{BL}$ are determined separately by normalizing each parameter to the total RAT computation. Additionally, the summation of each fractional parameter always equals unity. In the ideal case where FCA is not present, A$_{BL}$ is ignored so that all absorption occurs within the GaAs solar cell. Initially, the ideal conditions are computed without including FCA in the back layer. By introducing the $\alpha _{{FCA}}$ to find A$_{BL}$ in the back layer, the total RAT changes where some level of FCA occurs in the back layer. This absorption takes away from potential photogenerated current in the GaAs absorbing region. The fractional free carrier absorption (FFCA) in the back layer is extracted by normalizing A$_{BL}$ to the RAT computation and is used to describe the loss in current and voltage output compared to ideal conditions, as discussed in Section 3.Due to the three-dimensional nature of the cylindrical gratings, convergence tests were conducted to find accuracy in the RCWA computation. The number of Fourier orders increased in the xy plane of the grating layer, and the two main responses including the $J_{SC}$ and FFCA in the back layer converge at higher Fourier orders. The maximum deviation for the $J_{SC}$ and FFCA between 169 orders and 225 orders is less than 0.01% and 0.05%, respectively. To accommodate for computation time, 169 Fourier orders was used, resulting in nearly a 3x reduction in computation time. The convergence results are shown in Fig. S2 and Fig. S3 in Supplement 1.
3. Results and discussion
3.1 Current loss due to free-carrier absorption with a planar BSR
An example of the reduction in absorptance in a 300 nm-thick GaAs solar cell with a planar BSR, as depicted in Fig. 1(a), when FCA is considered in a 800 nm-thick AlGaAs back layer is shown in Fig. 3. Specifically, Fig. 3(a) shows the reduction in A$_{GaAs}$ and increase in A$_{BL}$ as $N_{BL}$ increases in the non-active back layer. Focusing on Fig. 3(a), the shortest wavelengths are easily absorbed towards the front of the optically thin GaAs solar cell (blue curves) during the first pass. Around 500 nm in wavelength, AGaAs decreases as photons transmit through the GaAs region and the thin-film interference pattern forms as these photons reflect off the back mirror. As $N_{BL}$ increases in the back layer, the photon absorption by free carriers in the back layer increases, too (green curves). This absorption in the back layer is especially noticeable towards the longer wavelengths as $N_{BL}$ increases. This increased FCA is especially limiting to the absorption of near band edge photons not easily absorbed during the first pass in the GaAs solar cell. In Fig. 3(b), the reduction in R from the planar BSR is observed as $N_{BL}$ increases in the back layer and is more pronounced at longer wavelengths. In the ideal case where FCA is not present, the backside reflectance is near 98% at 900 nm in wavelength (dark brown curve) and the reflected photons experience a twofold path length enhancement through the GaAs absorber. The photons that reflect off the mirror will experience a path length enhancement in the back layer before reaching the GaAs region again. Therefore, when $N_{BL}$ and the associated FCA is considered, the path length enhancement in the back layer will reduce the measured R as photons parasitically absorb in the back layer. For example, at $N_{BL}$ equal to $4{\times}10^{19}$ cm$^{-3}$, the backside R is measured to be 86% at 900 nm in wavelength (light pink curve), indicating a significant loss of photons due to the absorption by free carriers. This observation is especially detrimental not only to the $J_{SC}$, but also the $V_{OC}$, as discussed in Section 3.2.
Fig. 3. The plots above show the change in absorptance in a 300 nm-thick GaAs solar cell ($A_{GaAs}$) and in a 800 nm-thick back layer ($A_{BL}$) along with the change in backside reflectance (R) of a planar BSR as a function of $N_{BL}$. Specifically, plot (a) shows the reduced AGaAs in the GaAs photoactive region (blue curves) and the increased $A_{BL}$ in the back layer (green curves) as $N_{BL}$ increases. Plot (b) shows reduction in R as $N_{BL}$ increases in the back layer.
The example explained above provides a glimpse of the negative impacts on device performance due to FCA. However, there are various combinations of the $t_{GaAs}$, $t_{BL}$, and $N_{BL}$ available to explore. First, the ideal absorption for the three GaAs solar cells with $t_{GaAs}$ of 100 nm, 300 nm, and 500 nm are determined without including FCA in the back layer. The solar cells are modeled in the radiative limit and A$_{GaAs}$ is used to determine the external quantum efficiency (EQE) where A$_{GaAs}$ ($\lambda$) = EQE($\lambda$). From this absorption, the ideal $J_{SC}$ ($J_{SC}^{ideal}$) is found by integrating the GaAs absorption against AM1.5G reference spectrum, E($\lambda$),
This computation is performed for various combinations of $t_{BL}$ and $N_{BL}$, and contour maps of the $J_{SC}^{loss}$ for the three ultrathin GaAs solar cells are shown in Fig. 4(a-c). The N$_{BL}$ between $5{\times}10^{18}$ cm$^{-3}$ and $4{\times}10^{19}$ cm$^{-3}$ and the t$_{BL}$ between 0.2 $\mu m$ to 3 $\mu m$ are investigated. These ranges are common for sufficient carrier transport and texturing capabilities as reported in literature [5,7,8,13].
Fig. 4. Contour maps displaying the $J_{SC}^{loss}$ in GaAs solar cells with a planar BSR at various combinations between the back layer doping concentration (N$_{BL}$) and back layer thickness (t$_{BL}$). Specifically, plot (a) represents t$_{GaAs}$ = 100 nm, (b) t$_{GaAs}$ = 300 nm, and (c) t$_{GaAs}$ = 500 nm. The contour lines display the FFCA corresponding to specific back layer conditions. Plot (d) displays the reduction in $J_{SC}$ in the GaAs solar cells from ideal conditions vs. t$_{BL}$ at three values of N$_{BL}$.
The scale bar to the right displays the $J_{SC}^{loss}$ and the contour lines display the FFCA at the specified back layer conditions. In all cases, the most detrimental current loss occurs as the back layer thickness and doping concentration increase. Intuitively, this is expected to occur as the attenuation of photons due to the FCA in the back layer increases at elevated doping levels. Comparing the FFCA contour lines in plots a, b, and c, the rise in the $J_{{SC}}^{{loss}}$ is more prominent in the 100 nm-thick GaAs absorber since the transmitted photons after the first pass through GaAs have a higher intensity compared to the thicker absorbers. At the maximum values investigated for $t_{BL}$ and $N_{BL}$, the FFCA over 25% results in nearly 5 mA/cm$^{2}$ current loss for the 100 nm-thick GaAs device. Any transmitted photons that survive the first pass through the back layer and reflect off the planar mirror experience a twofold increase in OPL through the back layer. Figure 4(d) displays the reduction in $J_{SC}$ with increasing back layer thickness at three specific doping concentrations for the three GaAs solar cells under investigation. For each doping concentration, the slope is greater in the 100 nm-thick GaAs solar cell as t$_{BL}$ increases from 0 $\mu m$ to 2.5 $\mu m$. This trend illustrates the significance of GaAs absorber thickness with the loss of photons due to FCA after the first pass.
3.2 Voltage loss due to free-carrier absorption with a planar BSR
Along with improved current output, light trapping structures in ultrathin GaAs solar cells allow an enhancement in the $V_{OC}$ to occur and numerous studies have shown increased voltage output by introducing highly reflective mirrors [8,15,41]. The voltage enhancement is especially noticeable in high-quality ultrathin GaAs solar cells when radiative recombination dominates and the internal luminescent efficiency ($\eta _{int}$) is close to or at unity [42]. The $\eta _{int}$ is represented by,
where $U_{rad}$ and $U_{nr}$ are the radiative and nonradiative recombination rates, respectively. For GaAs solar cells operating in the radiative limit, an ideal reflector allows internally emitted photons to cycle within the active region and increase the external luminescent efficiency ($\eta _{ext}$), which in return, increases $V_{OC}$. Embedded in $\eta _{ext}$ are optical properties, which can be changed according to the efficacy of the light trapping geometry. $V_{OC}$ can be expressed in terms of the ideal open-circuit voltage, ($V_{OC}^{ideal}$), as calculated using detailed-balance [15,41–43], and $\eta _{ext}$ asIn this equation, $\eta _{ext}$ is determined by the $\eta _{int}$ and the photons probability of escape ($\overline {P}_{esc}$) and absorption ($\overline {P}_{abs}$):
These probabilities depend on the front ($R_{f}$) and backside reflectance ($R_{b}$) and their expressions are further explained by Steiner et al. [41]. In these probabilities, $R_{f}$ is calculated from the solar cell escape cone and $R_{b}$ is calculated from the backside reflectance. In an ideal design, both $\eta _{int}$ and $R_{b}$ are equal to 1, which allows $\eta _{ext}$ to reach unity and sets the measured $V_{OC}$ to equal the $V_{OC}^{ideal}$ in Eq. (7).
When FCA is considered in the back layer, the ideal conditions diminish as $R_{b}$ reduces. Building on the work of Steiner et al., an adjusted back reflectance ($R_{adj}$) that now considers the FFCA in the back layer is introduced into the model. The $R_{adj}$ is calculated by,
where $R_{b}$ is set to 1 to represent an ideal reflector and FFCA is determined using the same methods described in Section 2.2. For all combinations of N$_{BL}$ and t$_{BL}$, the $R_{adj}$ determines the loss in $V_{OC}$ using Eq. (7–8). In all calculations, the $\eta _{int}$ is set to unity to model the GaAs cell operating in the radiative limit. The $R_{f}$ is set to 96% which considers the escape cone in GaAs [44]. Under the detailed-balance limit for a 3 $\mu m$-thick GaAs cell, 1.12 V is used for $V_{{OC}}^{ideal}$ in Eq. (7) [43].The contour maps in Fig. 5(a-c) display the loss in voltage from $V_{{OC}}^{ideal}$ for GaAs solar cells with $t_{GaAs}$ equal to (a) 100 nm, (b) 300 nm, and (c) 500 nm. The contour lines display the $R_{adj}$ at the specified back layer parameters to represent the reduced reflectance from the ideal $R_{b}$. The scale bar represents the voltage loss from $V_{{OC}}^{ideal}$, where a thicker and highly doped back layer results in a larger voltage drop from ideal conditions. The voltage loss is less extreme in the 500 nm-thick GaAs solar cell since there are less transmitted photons after the first pass through the absorbing region. For the 100 nm-thick GaAs solar cell, the higher amount of transmitted photons result in a larger voltage loss as more photons are parasitically absorbed by free carriers in the back layer. At the maximum conditions explored for t$_{BL}$ and N$_{BL}$, the voltage loss approaches 60 mV in the 100 nm-thick design. For each solar cell, the reduction in $V_{OC}$ as the back layer thickness increases at three values of N$_{BL}$ are seen Fig. 5(d). The decay in voltage is greater in the 100 nm-thick GaAs solar cell as t$_{BL}$ increases from 0 $\mu m$ to 3 $\mu m$. Across all solar cells, the drop in $V_{OC}$ is more prominent at higher doping concentrations. The three ultrathin absorbers require near 100% reflected photons at the band edge for maximum voltage benefits to occur, and the parasitic FFCA in the back layer removes any chance of achieving $V_{{OC}}^{ideal}$. In realistic planar BSR designs, achieving a backside reflectance equal to 100% is complex, and experimental results found in literature have shown mirrors performing with a 98-99% peak reflectance [30,32,33]. The realistic peak mirror reflectance reduces $R_{b}$ in Eq. (9), ultimately dropping $R_{adj}$ even further. Therefore, the expected $R_{b}$ for specific BSR designs must be accounted for when measuring the adjusted reflectance.
Fig. 5. Contour maps displaying $V_{OC}^{loss}$ in GaAs solar cells with a planar BSR at various combinations between the back layer doping concentration (N$_{BL}$) and back layer thickness (t$_{BL}$). Specifically, plot (a) represents t$_{GaAs}$ = 100 nm, (b) t$_{GaAs}$ = 300 nm, and (c) t$_{GaAs}$ = 500 nm. The contour lines display the $R_{adj}$ corresponding to specific back layer conditions. Plot (d) displays the reduction in $V_{OC}$ from ideal conditions in the GaAs solar cells vs. t$_{BL}$ at three values of N$_{BL}$.
3.3 Free-carrier absorption in back cylindrical gratings
Besides the planar BSR design, cylindrical gratings consisting of AlGaAs embedded in SiO$_{2}$, as shown in Fig. 1(b), are also considered in this study. This geometry is quite attractive for light trapping in ultrathin GaAs solar cells given the contrast in index of refraction between the two materials, making it suitable to enhance resonance modes and photogenerated current. Recently, Buencuerpo et al. demonstrated strategies to obtain an optically thick, but physically thin 300 nm-thick GaAs solar cell using these materials in a backside nanostructured layer [14]. In experimental designs, some method of backside carrier transport and collection is required and one approach includes using a top-bottom contact design with highly doped AlGaAs cylinders to serve as ohmic pathways, or vias, to the metal mirror. Based on the promising results in Buencuerpo’s optimized nanostructured design, three similar designs are investigated for a 300 nm-thick GaAs solar cell operating in the radiative limit while considering different doping concentrations in the AlGaAs cylinders. In this design, the SiO$_{2}$ spacer above the Ag mirror reduces parasitic absorption in the mirror, but blocks carrier transport. To explore similar designs to Buencuerpo et al. with maximum reflectance from the backside metal, the SiO$_{2}$ spacer remains in these simulations. In realistic design, a top-top contact approach with a lateral conduction layer (LCL) allows the SiO$_{2}$ space to remain intact, but will increase FCA as the LCL requires a high doping concentration and its thickness must scale with increasing solar cell active area to reduce sheet resistance [7]. For researchers interested in using the top-top contact design with light trapping structures, optimization must be made between the thickness and doping concentration in the LCL to minimize FCA while providing sufficient carrier transport.
In the top-bottom contact design, highly doped AlGaAs cylinders with larger radii may provide adequate carrier transport, but also increase FCA as the doped semiconductor coverage scales up. Grating B refers to the optimized light trapping structure while Grating A and C look at a low and higher AlGaAs coverage, respectively. For each grating, the AlGaAs percent coverage varied across the unit cell by adjusting the radius of the cylinder and changing the pitch, as shown in Fig. 1(b). Each grating achieves $J_{{SC}}^{ideal}$ that is 3-4 mA above the planar BSR design (refer to Section 3.1). Table 1 shows the unit cell grating specifications labeled A, B, and C, with AlGaAs and SiO$_{2}$ percent coverage and $J_{{SC}}^{ideal}$ where FCA is not considered.
Table 1. Cylindrical grating unit cell specifications and $J_{{SC}}^{ideal}$ without FCA
An example of the absorptance in the 300 nm-thick GaAs solar cell with Grating B is shown in Fig. 6(a) where $A_{BL}$ now represents the absorptance in the AlGaAs cylinders within Grating B. Similar trends to the planar BSR design show that higher values of $N_{BL}$ reduces the $A_{GaAs}$ in the GaAs solar cell. At wavelengths shorter than 500 nm, the $A_{GaAs}$ at different $N_{BL}$ remains the same. After this point, transmitted photons interact with the nanostructured layer and as the doping concentration increases, the $A_{GaAs}$ decreases. Due to the nature of propagating light in nanostructured designs, the FCA can be higher at specific wavelengths and a dramatic spike in $A_{BL}$ is evident near 900 nm in wavelength, near the band edge of GaAs. Figure 6(b) displays the reduction in backside reflectance as the $N_{BL}$ increases in Grating B. At $N_{BL}$ equal to $4{\times}10^{19}$ cm$^{-3}$, the backside reflectance dramatically drops below 80% after the GaAs band edge (light pink curve), which is near 100% in the ideal case (dark brown curve).
Fig. 6. The plots above show the change in absorptance in the 300 nm-thick GaAs solar cell ($A_{GaAs}$) and in the AlGaAs regions of Grating B ($A_{BL}$) along with the change in backside reflectance (R) as a function of $N_{BL}$. Specifically, plot (a) shows the reduced $A_{GaAs}$ (blue curves) and the increased $A_{BL}$ (green curves) as $N_{BL}$ increases. Plot (b) shows reduction in R as $N_{BL}$ increases in the AlGaAs regions in Grating B.
Each grating is tested in the ideal case where no FCA is present in the AlGaAs regions, and again for AlGaAs at the same doping concentrations modeled with the planar BSR. Figure 7(a) shows the reduction in $J_{SC}$ from the ideal conditions as $N_{BL}$ increases. When FCA is not considered, Grating C outperforms Grating A in terms of $J_{SC}$. However, once the FCA is included in the model, Grating C significantly reduces the amount of photogenerated current in the GaAs solar cell at $N_{BL}$ greater than $1{\times}10^{19}$ cm$^{-3}$ due to the increased AlGaAs coverage. The normalized $J_{SC}$ plot in Fig. 7(b) highlights the significant reduction in the photogenerated current as AlGaAs coverage increases across the grating designs. As a result, Grating A becomes more effective as a light trapping structure than Grating C to improve photgenerated current at doping concentrations greater than $1{\times}10^{19}$ cm$^{-3}$.
Fig. 7. Plot (a) displays the reduction in $J_{SC}$ and increase in FFCA and plot (b) displays the normalized $J_{SC}$ as $N_{BL}$ increases in the AlGaAs regions of the three gratings behind a 300 nm-thick GaAs solar cell. Plot (c) displays the voltage loss from $V_{OC}^{ideal}$ with associated FFCA and plot (d) displays the decrease in efficiency as $N_{BL}$ increases.
The same method to determine the loss in $V_{OC}$ as described in Section 3.2 is done for the grating geometries by changing the R$_{adj}$ based on the FFCA. As shown in Fig. 7(c), similar trends to the loss in $J_{SC}$ are observed, and Grating C results in the largest voltage loss. The solar cell efficiency is calculated using AM1.5G conditions with a total irradiance of 100 mW/cm$^{2}$, and a realistic fill factor of 84% for GaAs solar cells [18]. Figure 7(d) shows the decrease in efficiency as $N_{BL}$ increases in each grating geometry. Noticeably, Grating A degrades the least as $N_{BL}$ increases since it has the lowest doped semiconductor coverage. At low doping concentrations, Grating B remains as the optimized light trapping design. At doping concentrations greater than $2.7{\times}10^{19}$ cm$^{-3}$, Grating A outperforms Gratings B and C due to its steady decline in efficiency, allowing it to become the favored light trapping design. At $4{\times}10^{19}$ cm$^{-3}$, Grating B and C lose an absolute efficiency of 1.8% (6.9% relative) and 2.9% (11.5% relative), respectively, while Grating A losses an absolute efficiency less than 0.6% (2.2% relative), making it the better light trapping structure at higher doping concentrations. This study suggests that when designing nanostructured geometries for light trapping, the doped semiconductor regions should lean towards smaller unit cell dimensions to reduce the absorption by free carriers. While increased doped semiconductor coverage leads to improved carrier transport in a top-bottom contact design, the optical benefits of the light trapping structure will reduce at high doping concentrations. Therefore, an optimal point between sufficient carrier transport and effective light trapping must be found in device-specific designs as research aims to improve the efficiency in ultrathin III-V solar cells.
4. Conclusions
This research focuses on the optical modeling of FCA in non-active layers behind ultrathin GaAs solar cells with planar and nanostructured grating designs. The results show that FCA increases as the thickness and doping concentration of the back layer increase. The FCA is more noticeable as the GaAs absorber thickness reduces since more transmitted photons after the first pass interact with free carriers. The FCA optical loss degrades the light trapping benefits as it reduces the $J_{SC}$ and $V_{OC}$ from ideal conditions, and in return, reduces the device efficiency. When designing nanostructured gratings, the doped semiconductor region must be optimized to balance minimal FCA and sufficient carrier transport to the external electrodes. Recognizing the potential optical loss by FCA in non-active layers will help set guidelines for careful material selection when designing ultrathin III-V solar cells with light trapping structures and will support these devices in reaching their maximum efficiency based on the detailed-balance limit.
Funding
Air Force Research Laboratory (FA9453-19-C-0592); BlueHalo (FA9453-14-D-0312 (SVAT)/TO 04).
Acknowledgments
The authors would like to thank both Microlink Devices and the Air Force Research Laboratory for their support and funding under the SBIR FA9453-19-C-0592, as well as a subcontract from BlueHalo under FA9453-14-D-0312 (SVAT)/TO 04. Approved for public release; distribution is unlimited. Public Affairs release approval AFRL-2022-0655. The authors also thank Phoebe Pearce from the University of Cambridge for the insightful discussions working with Rayflare.
Disclosures
The authors declare no conflicts if interest.
Data availability
Data underlying the presented results are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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