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Abstract
In the photovoltaic community, short circuit current (Isc) of a current mismatched multijunction photovoltaic (MJPV) cell was usually thought to be limited by the lowest subcell photocurrent (Imin). However, under certain conditions for multijunction solar cells, Isc≠Imin was observed by researchers, while this effect has not been studied in multijunction laser power converters (MJLPCs). In this work, we provide an in-depth analysis of the formation mechanisms for the Isc of the MJPV cell by measuring I-V curves of the GaAs and InGaAs LPCs with different number of subcells and simulating the I-V curves with the reverse breakdown of each subcell considered. It is found that Isc of an N-junction PV cell can be theoretically equal to any current value within a range from a current lower than Imin to the maximum subcell photocurrent, which is up to the number of subcell current steps in the forward biased I-V curve. An MJPV cell with a constant Imin will demonstrate a higher Isc if it has more subcells, smaller subcell reverse breakdown voltage and smaller series resistance. As a result, Isc tends to be limited by the photocurrent of a subcell closer to the middle cell and is less sensitive to the optical wavelength than Imin. This should be another possible reason why the measured EQE of a multijunction LPC exhibits a wider spectrum width than the calculated Imin-based EQE, whereas this was usually attributed to the luminescent coupling effect merely.
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1. Introduction
Short circuit current (Isc), defined as the current at V = 0 V, is a key parameter as an approximation of the photocurrent generated in a single-junction photovoltaic (PV) cell. For the multijunction PV cells (MJPV), including multijunction solar cells (MJSCs) [1,2] for converting the sunlight and multijunction laser power converters (MJLPCs) [3–11] for converting the monochromatic light, subcells are serially connected and currents in subcells are the same even when subcell photocurrents are different from each other. The Isc was always assumed to be equal to the lowest photocurrent (Imin) of the subcell absorbing the fewest photons [12–14]. Isc = Imin was the cornerstone for calculating I-V curves [15] and external quantum efficiencies (EQEs) [12–14] of an MJPV cell. Researches [16–19] in recent years have raised some exceptional circumstances to this and Isc≠Imin when some subcells in the PV cell demonstrate serious shunt losses or low reverse breakdown voltages. If such circumstances are unknown to researchers, the measured Isc-based EQE may deviate from the calculated Imin-based EQE, causing them to seek alternative explanations for this divergence. In our previous work [20], we observed the subcell photocurrent steps in the wide-voltage-range I-V curves in an MJLPC and the subcell breakdown characteristic is the key to comprehend the formation mechanism of the steps. However, few literatures have considered the reverse breakdown of a subcell when calculating the I-V curves of an MJPV cell, and mechanisms on formation of the Isc have not been thoroughly explored.
In this study, we modify the I-V expression of a single-junction PV cell by including an additional exponential function term, allowing its reverse breakdown to be described mathematically. I-V curves are calculated for an N-junction (NJ) LPC with different N, current step widths and series resistances (Rs); and measured for a 10J GaAs LPC under different laser power densities (Pin). We demonstrate that Isc of the NJ LPC is not always limited by Imin and can be higher or lower than Imin, which depends on the number of steps in the forward biased I-V curves determined by N, Pin, step width and Rs. Furthermore, EQEs calculated by the Isc as well as the Imin are also investigated under various conditions.
2. Device structure and simulation
The structures of an MJLPC and an MJSC are depicted in Fig. 1(a) and (b), respectively. For the MJLPCs, subcells are composed of the same material while those of the MJSCs are different in bandgaps. Calculations in this work are based on the LPC structure in Fig. 1(a) using parameters of GaAs with a bandgap Eg of 1.424 eV, [21] recombination lifetimes of τn∼10−9 s and τp∼10−6 s, and diffusion coefficients of Dn = 200 cm2/s and Dp = 10 cm2/s for minority carriers [22]. The absorption coefficient α of the active layer varies over the wavelength λ [23]
where Aα is a constant related to the specific material, h is the Planck constant and c is the light velocity in vacuum. For an NJ LPC, subcell thicknesses should be designed for absorbing the same number of photons at a given α for the λ of interestThe total current I in an ideal single-junction cell is equal to the photocurrent Iph minus the diode current determined by the Shockley Equation [24]
where I0 is the PN junction reverse saturation current, q is the elementary charge, V is the cell voltage, n is the ideality factor, k is the Boltzmann constant and T is the temperature. Equation (3) is the basis for calculating the forward biased I-V characteristics of a single-junction PV cell. However, this equation lacks the description of the reverse breakdown for a PV cell. Hereafter, we construct the breakdown model using the exponential function to simulate the I-V characteristic of a single-junction cell from the forward biased state to reverse breakdown stateIn an NJ PV cell, current of each subcell varies monotonously with its voltage, hence the current can be calculated uniquely at a given voltage Vi. Conversely, Vi can also be determined uniquely at a given current I, which can be numerically calculated by the computer program. Since subcells are serially connected, currents of them are equal, and their voltages Vi's can be determined at a given current I by solving the following equations
where $f({{I_{phi}},\; {V_i},{I_{0i}},{A_i},{B_i}} )$ is the current of subcell i as a function of its photocurrent Iphi, voltage Vi, reverse saturation current I0i, constants Ai and Bi, determined by Eq. (5). For the MJSCs, constants of Ai, Bi and I0i are different for various subcells, whereas they are the same for all subcells of a multijunction LPC. The total voltage V of the device is the sum of Vi'sBy trying different currents, a series of total voltages can be obtained. Therefore, we can calculate the I-V characteristic of an NJ PV cell over a current range of interest. As the voltage V approaches 0 V tightly, the current I used is considered as an approximation of Isc of the PV cell. The voltage states of subcells at the short circuit point of the PV cell are determined by the relationships between subcell photocurrents Iphi's and Isc. For a subcell, when Iphi > Isc, it goes into the forward biased state, and if Iphi≫Isc, Vi is closer to its open circuit voltage Voci; when Iphi < Isc, this subcell goes into the reverse biased state, and if Iphi≪Isc, Vi is closer to its reverse breakdown voltage Vbri.
3. Results and discussion
3.1. Number of subcells
In the following, we calculate I-V curves of the 1 cm2 NJ LPCs for N from 1 to 20 under 1 W/cm2 835 nm laser at room temperature (RT) with Rs = 0 Ω, as shown in Fig. 2(a). Thicknesses for subcells 1-N in NJ LPC are listed in Table 1. In this calculation, each subcell reversely breaks down at Vbri = -2.1 V determined by the selected Ai = -12.2 and Bi = 0.39 according to the GaAs LPC in experiments. The saturation current density J0i of each subcell is set as 5.14 × 10−20 A/cm3. A low absorption coefficient of α=9690 cm-1 for GaAs at 835 nm is used and subcells near the top of the LPC absorb fewer photons than that of those near the bottom. Therefore, subcell photocurrent (Iphi) increases monotonically from top to bottom, i.e., Iphi-1< Iphi (i ≥ 2).
Fig. 2. (a) Calculated I-V curves of an NJ LPC with an area of 1 cm2 at 1 W/cm2 835 nm laser with N from 1-20. The inset shows their forward biased I-V curves. (b) Isc and Iph1-Iph7 versus N. (c) Voc, Nf, Nsc and Voc/(Voci-Vbri) as functions of N. (d) Measured I-V curves for 3J, 6J, 10J, 12J, 20J GaAs LPC under 804 and 835 nm lasers, and 1J, 6J, 8J InGaAs LPC under 1520 nm laser.
Table 1. Subcell thicknesses of the NJ LPCs in the calculations
Figure 2(a) shows N current steps with the equivalent width appear in each I-V curve of the NJ LPC. These steps always appear from the lowest to the highest with their positions starting from the open circuit voltage (Voc) to the reverse breakdown voltage (Vbr). In each case, subcell 1 always generates the lowest photocurrent (Iph1 = 8 mA). While as indicated by the forward biased I-V curves in the inset of Fig. 2(a), Isc is not invariably limited by Iph1. According to the N dependence presented in Fig. 2(b), Isc demonstrates a step-wise growth with increasing N. For LPC with fewer subcells, Isc is firstly limited by the lowest Iph1 (N = 1-3). When N = 4, Isc is liberated from the restriction of Iph1 and is limited then by Iph2. At larger N, Isc will be limited by a higher Iphi or lie between two subcell photocurrents. Finally, Isc is higher than Iph5 of 15.9 mA at N = 20, which is almost two times of Iph1. These calculations can be justified by the measured I-V curves of 3J, 6J, 10J, 12J, 20J GaAs LPCs and 1J, 6J, 8J InGaAs LPCs in Fig. 2(d), where if an LPC contains more subcells, the short circuit point will correspond to a step with a higher current.
The inset in Fig. 2(a) shows that the forward biased I-V curve exhibits more steps as N increases. The number Nf, defined as the number of complete steps appear in the forward biased I-V curve, is determined by the Voc and the subcell current step width of Voci-Vbri
At the short circuit point, four conditions may occur, i.e., V = 0 V corresponds to (a) the right end, (b) inside, (c) left end and (d) left breakdown segment of a step. This step corresponds to a subcell with the Nscth lowest photocurrent. For (a) and (b), Nsc = Nf + 1; for (c), Nsc = Nf. Isc is equal to the Nscth lowest photocurrent for these three conditions. However, for (d), Isc reaches a value between the Nfth and (Nf + 1)th lowest photocurrents, and we use Nsc = Nf + 0.5 to imply this condition. Nf and Nsc for different N are extracted in Fig. 2(c), and Nsc increases addictively with increasing Nf. Figure 2(c) also gives Voc and the calculated item $\frac{{{V_{oc}}}}{{{V_{oci}} - {V_{bri}}}}$ as functions of N. When N ≤ 8, Vtr is negligible because the forward biased curve contains fewer steps (≤2), and ${N_f} = \frac{{{V_{oc}}}}{{{V_{oci}} - {V_{bri}}}}$. However, when N ≥ 9, more steps emerges and Vtr becomes larger, resulting in ${N_f} < \frac{{{V_{oc}}}}{{{V_{oci}} - {V_{bri}}}}$. In general, the growing Voc is the reason of the increasing Nf and Nsc, causing a rising Isc as N increases.
3.2. Input power density (Pin)
Further on, we investigate the impact of the input power density Pin on the Isc by measuring the I-V curves of a 10J GaAs LPC with an area of 0.36 cm2 under 835 nm laser from 0.002 W/cm2 to 5.222 W/cm2 as shown in Fig. 3(a) and 3(b). At 0.002 W/cm2, Isc is equal to the fifth lowest photocurrent of step 5. As Pin increases, steps 4-10 keep moving to a higher reverse voltage, and the left end of step 4 reaches V = 0 V at 0.022 W/cm2, where Isc is equal to the fourth lowest photocurrent (point A). When Pin exceeds 2.111 W/cm2 (point B), the right end of step 4 departs from V = 0 V and Isc is a value between the currents of step 3 and 4 as shown in Fig. 3(b).
Fig. 3. Measured I-V curves of a 10J GaAs LPC with an area of 0.36 cm2 under 835 nm with power density from (a) 0.002 W/cm2 to 0.175 W/cm2 and (b) 0.175 W/cm2 to 5.222 W/cm2. (c) Calculated I-V curves of a 2J GaAs LPC with different photocurrent differences between two subcells.
Revolutions of steps 4-10 are the result of the increasing Vtr caused by the rising photocurrent difference (ΔIphi,j) between every two subcells as Pin increases
where Iphi and Iphj are the photocurrent of subcell i and j, respectively, S is the area of the device, Rphi and Rphj are the photocurrent responsivities for subcell i and j, respectively. Schematically, Fig. 3(c) shows the calculated I-V curves of a 2J LPC with different ΔIphi,j. The voltage of the transient area between two steps (Vtri,j) can be expressed asIn contrast to steps 4-10, steps 1 and 2 keep moving to the right of I-V curves instead with increasing Pin and step 3 almost stays at a fixed voltage for the Pin used. Revolutions of steps 1 and 2 can be attributed to the boosted Voc at a higher Pin, because the steps appear in the I-V curve always begin from the right end, i.e., Voc of the I-V curve. Furthermore, the increased Voci makes a step wider, which will also shift the steps to the left of I-V curves. Finally, the stationary step 3 is the result of the competitions between the three factors of the increasing ΔIphi,j, Voc and Voci.
3.3. Laser wavelength ($\lambda$)
A noteworthy consequence of Isc≠Imin is the inconsistency between the EQEs calculated by them. We subsequently calculate the wavelength (λ) dependent I-V curves for 1 cm2 2J, 3J, 6J, 10J and 20J LPCs in Fig. 4(a) under a low Pin of 0.01 W/cm2. The five LPC structures are designed to be current matched at 805 nm with total absorber thickness fixed at 3121 nm. Subcell photocurrents and Isc's for these LPCs are extracted in Fig. 4(b). It seems that Nsc's for 2J, 3J, 6J, 10J and 20J LPCs almost remain unchanged at 1, 1.5, 2, 4 and 7.5, respectively, for each λ used. Although full electrical simulation for the accurate determination of EQE for an MJPV cell has been demonstrated in literature [25], for the sake of simplicity in this work, the EQE of an LPC at a certain λ is calculated by
where I(λ) is the current of the device at λ, R(λ) is the current responsivity at λ. The calculated EQEs of the five LPCs based on Isc and Imin are shown in Fig. 4(c). Isc-based and Imin-based EQE both peak at 805 nm because Isc and Imin peak at this point simultaneously [Fig. 4(b)]. For 2J LPC, Nsc = 1, i.e., Isc = Imin, and the Isc- and Imin-based EQE are almost overlapped. For LPCs from 3J to 20J, Nsc ≥ 1.5, i.e., the Isc is gradually higher than Imin, which makes the Isc- and Imin-based EQE depart more significantly when the wavelength deviates from 805 nm, especially to a shorter one. As a result, the Isc-based EQE is less sensitive to the wavelength than the Imin-based. Furthermore, a greater spectrum sensitivity of the Imin-based EQE is observed as N increases.Fig. 4. Calculated (a) I-V curves, (b) Isc, subcell photocurrents, (c) EQEs based on Isc and Imin for 1 cm2 2J, 3J, 6J, 10J and 20J GaAs LPCs, respectively, under 0.01 W/cm2 laser of different wavelengths. (d) Measured EQEs of 10J and 20J GaAs LPC as well as the calculated EQEs using each subcell photocurrent (Iphi) and the lowest photocurrent (Imin) for 10J and 20J GaAs LPC structures, respectively.
Results above have been further confirmed by comparing the measured EQE of the 10J and 20J GaAs LPCs to the calculated Imin and Iphi-based EQEs in Fig. 4(d), corresponding to the structures of the two LPCs. The wavelength at which each measured EQE peaks is assumed as where the LPC current matched and the α value adopted in the design is, thus, corresponding to this wavelength. Therefore, Aα's in Eq. (1) for the two LPCs are obtained and the simple α-λ relationships are determined then for the calculations. It is found that the measured EQEs illustrates a great difference from the Imin-based EQE both for the 10J and 20J LPCs, this phenomenon has also been widely reported in MJLPCs [12–14]. In these previous studies, this disagreement was attributed to the luminescent coupling (LC) effect solely as it improves the current mismatching condition of the devices. However, we found in Fig. 4(d) that the measured EQE is consistent with one of the calculated Iphi-based EQEs, i.e., Iph4 for the 10J LPC and Iph7 or Iph8 for the 20J LPC. This also agrees with the calculated results in Fig. 4(b) that Nsc for 10J and 20J are 4 and 7.5, respectively, at 0.01 W/cm2. Therefore, depending on the exact measurement routine, the measured EQE deviating from the Imin-based EQE could also be caused by Isc > Imin, and further investigations are needed to verify the presence of the LC effect.
In Fig. 3(a), the measured Nsc for 10J GaAs LPC is 5 at 0.002 W/cm2, which is higher than that of 4 observed in Fig. 4(d). This is because the power of the light shining at the surface of the LPC is quite low (∼10 µW) for each wavelength after splitting and chopping for light of the Xenon lamp in the EQE measurement. Such low power may further lower the Voc of the LPC, shorten the step width and reduce the current differences between steps, which are the three competitive factors as discussed in Fig. 3. In general, Nsc = 4 is reached with the combined effect of these three factors. Besides, the low power poses a great requirement for the precision of the I-V model used in the calculations, hence the calculated Isc-based EQEs under ∼10 \W are not given in Fig. 4(d). Instead, we give the Iphi-based EQE since it is independent of the power as indicated by Eq. (12).
3.4. Subcell reverse breakdown voltage (Vbri)
As shown in Eq. (9), step width plays a vital role in determining the Nf. Subcell reverse breakdown voltage Vbri is an important parameter in tuning the step width and usually affected by the material bandgaps and doping concentrations [26]. Figure 5(a) shows I-V curves of the 1 cm2 10J LPCs with different subcell Vbri's under 0.01 W/cm2 835 nm laser. As Vbri increases, each current step becomes wider whereas the Voc remains the same. Nf and Nsc extracted in Fig. 5(b) basically follow the same trend as the calculated Voc/(Voci-Vbri) with increasing Vbri. When Vbri = 0 V, though not possible for a real PV cell, the step width is only Voci, and almost all the steps, besides the highest one, are found in the forward biased I-V curve, i.e., Nsc = 9.5. This value is lower than 10 because Vtr between steps cannot be neglected. As the steps become wider at a higher Vbri, those steps with higher currents are driven to a higher reverse voltage, hence a reduced Nsc is observed. Furthermore, Nsc is more likely to be limited to a smaller value over a larger Vbri range with an increase in Vbri, as Nf depends inversely on Vbri. Therefore, as shown in Fig. 5(c), Isc falls rapidly in the beginning, while it is gradually limited by Iph4, Iph3 and then Iph2 within a lager voltage range. Once the step width exceeds Voc, i.e., Voci+|Vbri|≥Voc, Isc remains to be limited by Imin, which can be seen as the Vbri reaches around -9 V in Fig. 5(c).
Fig. 5. Calculated (a) I-V curves (the inset gives the forward biased I-V curves), (b) Nf, Nsc, Voc/(Voci-Vbri), (c) Isc, Im, Iph1-Iph9, Eff, FF, (d) Isc-based and Imin-based EQE of 10-junction LPCs with subcell Vbri from 0 V to -9 V at 0.01 W/cm2 835 nm laser.
Figure 5(c) also gives the Vbri dependences of the current at the maximum power point (Im), power conversion efficiency (Eff) as well as the fill factor (FF). It shows that Im is limited by Imin and independent of the Vbri, resulting in an almost constant Eff. However, FF is deeply correlated to Vbri and increases with an increase in Vbri due to the decreased number of steps in the forward biased I-V curve.
The Isc and Imin based EQE of the 10J LPCs with different Vbri's are calculated in Fig. 5(d). Imin based EQE remains unchanged at any Vbri's, whereas the Isc-based EQE is highly relevant to the subcell Vbri. For the case of Vbri = -1.2 V, the Isc-based EQE shows the least spectrum sensitivity because its Nsc is 5, corresponding to the very middle subcell in the LPC structure. As Vbri > -1.2 V or < -1.2 V, Nsc departs from the middle value of 5, and Isc-based EQE becomes more sensitive to the wavelength. This can be explained that the photocurrent for a subcell near the top or bottom is more sensitive to the wavelength than the middle cell, as demonstrated previously in Fig. 4(b). A reversed EQE shape is observed for a smaller Vbri, because Isc is always equal to a higher Iphi. Notably, Isc-based EQE remains unchanged for Vbri = -1.8 V and -2.1 V, and for Vbri = -2.7 V and -3 V, as Nsc's of the two pairs of Vbri's are limited to the same values of 4 and 3, respectively. The Isc and Imin based EQEs are observed to overlapped at Vbri = -9 V where Nsc = 1 due to Isc = Imin.
3.5. Series resistance (Rs)
Last, the Rs dependence of Isc for a multijunction LPC is investigated by calculating the I-V curves of a 1 cm2 10J GaAs LPC under 1 W/cm2 835 nm laser with Rs from 0 Ω to 260 Ω, as shown in Fig. 6(a). Rs plays a role in expanding Vtr by
Fig. 6. Calculated (a) I-V curves (the inset gives the forward biased I-V curves), (b) Isc and Iph1-Iph4 of a 10-juncton LPC under 1 W/cm2 835 nm laser with different Rs's.
4. Formation mechanism of the Isc for an NJ LPC
The formation mechanism of the Isc of an NJ LPC is depicted as a diagram in Fig. 7. Isc is determined by two aspects, i.e., the subcell photocurrents and the number of current steps appear in the forward biased I-V curve. On one hand, each subcell photocurrent (Iphi) is generated based on the input laser power density Pin, area S of the cell and its current responsivities Ri(λ) depending on the number of junctions N, subcell thickness di, absorption coefficient α(λ) and other material parameters. On the other hand, the Iphi-related Voci and the reverse breakdown voltage Vbri simultaneously determines the width of a current step. The Voc of the LPC depends on Voci and N. Besides, the transient voltage range (Vtr) between steps in the forward biased I-V curve is correlated to the series resistance Rs, current differences between Iphi's and the Imin. Furthermore, the number Nf of current steps completely appear in the forward biased I-V curve can be estimated by step width, Voc and Vtr. According to Nf and the corresponding step position at V = 0 V, the Nscth lowest current step can be confirmed. Ultimately, Nsc and Iphi give birth to the Isc together.
For an ordinary MJSC containing a limited number of subcells, the widths of steps may be different from each other due to the various bandgaps of subcells and their doping concentrations. If there exists at least one current step whose width is shorter than Voc of the total device, and when this step corresponds to the subcell generating the lowest photocurrent, this step will completely appear in the rightmost of the I-V curve, i.e., near the Voc point. Hence, the rest voltage range of the forward biased I-V cure will be taken by another step with a higher photocurrent, deviating Isc from the Imin. The subcell of the lowest bandgap in the bottom of the MJSC tends to demonstrate a shorter step width due to its lower Voci and Vbri. Isc > Imin is more likely to happen in EQE measurements where at a shorter wavelength the bottom subcell may absorb less photons than the top subcell and generate the lowest photocurrent, which needs to be further investigated.
5. Conclusions
Isc's of the MJLPCs are studied by calculating the wide-voltage-range I-V curves considering the reverse breakdown of each subcell. Isc of an NJ LPC is not necessarily limited by Imin, and the specific subcell photocurrent to limit the Isc is dependent on N, Pin, Voci, Vbri and Rs. Based on the number of steps that the forward biased I-V curve able to contain, Isc can be equal to any value between the Imin and the highest subcell photocurrent (reached when Vbri = 0 and Vtr = 0). For a higher Rs, Isc may be below Imin. Once Isc is higher than Imin and closer to Iphi of the middle subcell, the Isc-based EQE becomes less sensitive to λ than Imin-based EQE. Therefore, the wider spectrum width for the measured EQE compared to the calculated Imin-based EQE for a multijunction LPC should not be solely attributed to the luminescent coupling effect between subcells.
Funding
National Natural Science Foundation of China (62275262).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data that support the findings of this study are available from the corresponding author upon reasonable request.
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