Numerical Analysis of the Detailed Balance of Multiple Exciton Generation Solar Cells with Nonradiative Recombination

In this study, we analyzed the nonradiative recombination impact of multiple exciton generation solar cells (MEGSCs) with a revised detailed balance (DB) limit. The nonideal quantum yield (QY) of a material depends on the surface defects or the status of the material. Thus, its QY shape deviates from the ideal QY because of carrier losses. We used the ideal reverse saturation current variation in the DB of MEGSCs to explain the impact of nonradiative recombination. We compared ideal and nonideal QYs with the nonradiative recombination into the DB of MEGSCs under one-sun and full-light concentration. Through this research, we seek to develop a strategy to maintain MEGSC performance.


Introduction
Multiple exciton generation solar cells (MEGSCs) are promising future-generation solar cells that are capable of creating several electron and hole pairs (EHPs) via impact ionizations. This can overcome the theoretical single-junction efficiency limit by decreasing the carrier thermalization loss in nanostructured materials [1][2][3].
A quantum efficiency of >100% has been observed in silicon solar cells, which has motivated the development of theoretical and empirical approaches for MEGSCs [4,5]. The theoretical efficiency of MEGSCs has been calculated [5][6][7][8][9] based on the ideal quantum yield (QY), which is the number of EHPs generated under blackbody radiation.
In the detailed balance (DB) limit, single-junction solar cells (SJSCs) consider only one radiative recombination (Figure 1a), and the MEGSC undergoes multiple-carrier recombination (MR) via Auger recombination (AR). The M − 1 electrons directly recombine into holes and transfer their energy to the remaining electron, where M is the maximum generated EHPs. By absorbing the energy, the electron excites to a high-energy state. Thereafter, it emits single-photon energy (which is equal to the product of the maximum quantum yield and the bandgap energy) without losses (Figure 1b) [7]. The amount of photon energy emitted is comparable to that of the topmost junction in a tandem solar cell under blackbody radiation. An MEGSC under blackbody radiation shows efficient carrier management, undergoing no other nonradiative recombination processes. However, the MR in the actual materials describes nonradiative AR to occur via phonon-assisted electron or hole relaxation [10] (Figure 1c). Experiments show that the excited electrons or holes release their kinetic energy to reach the valence band. Subsequently, the carrier experiences direct recombination or capture at the trap. Therefore, this relaxation process limits the performance of an MEGSC as the carrier cooling rate is slowed down, moving from one state to another [11]. In the context of carrier dynamics of the MEGSC measurements [19][20][21][22][23][24][25] have limited the QY up to 300% in QD systems. The experimental results of measurements in QD MEGSCs are much lower than the theoretical limit owing to nonradiative recombination and insufficient light absorption [26][27][28][29][30]. Therefore, these shortcomings have initiated developments in the DB limit of an MEGSC for quantitative analysis of the impact of nonradiative recombination.
The conventional AR depends on the material parameters (carrier lifetime, doping concentration, and Auger constant) in the DB [31]. To generalize the nonradiative recombination, the rate-based calculation of DB considers the nonradiative generation and recombination and its ideal reverse saturation current [Error! Bookmark not defined.]. Thus, the SJSC can be made to achieve a semiempirical limit by varying the ideal reverse saturation current, which helps explain the nonradiative recombination impacts [Error! Bookmark not defined.]. Therefore, this method can be applied to the DB of MEGSC.  The discrepancy between ideal and actual performance depends on MEGSC idealities. Two parameters are crucial to test the characteristics of an MEGSC: QY and threshold energy (= E th ). The ideal quantum yield (IQY) increases according to a staircase step function, and the threshold energy describes the onset point over 100% of QY. The E th for IQY is twice that of the bandgap energy (E th = 2·E g ) without carrier losses [5][6][7][8][9], but the actual material shows a difference in QY and E th because of nonidealities related to surface state or defects. Thus, the nonideal quantum yield (NQY) shows a delayed E th and a linearly increasing QY. During initial measurements of QY in PbSe QDs up to 300% and 700% [12,13], various groups have examined the NQY by pump-probe measurements with CdTe [14], CdSe [15], InAs [16,17], and Si [18]. The uncertainties related to surface states and long-lived QD photocharging in the pump-probe measurements [19,25] have limited the QY up to 300% in QD systems. The experimental results of measurements in QD MEGSCs are much lower than the theoretical limit owing to nonradiative recombination and insufficient light absorption [26][27][28][29][30]. Therefore, these shortcomings have initiated developments in the DB limit of an MEGSC for quantitative analysis of the impact of nonradiative recombination.
The conventional AR depends on the material parameters (carrier lifetime, doping concentration, and Auger constant) in the DB [31]. To generalize the nonradiative recombination, the rate-based calculation of DB considers the nonradiative generation and recombination and its ideal reverse saturation current [1]. Thus, the SJSC can be made to achieve a semiempirical limit by varying the ideal reverse saturation current, which helps explain the nonradiative recombination impacts [1]. Therefore, this method can be applied to the DB of MEGSC.
In this paper, we present a study on the DB limit of an MEGSC, which involves a quantitative analysis of the nonradiative recombination impact. For simulation purposes, we altered the DB of MEGSC by varying the ideal reverse saturation current (J 0 ) from the nonradiative generation and recombination in order to derive the ideal reverse saturation condition for MEGSC. By varying this parameter, we were able to obtain a semiempirical limit and discuss the impact of the nonradiative recombination of MEGSC.

Detailed Balance Equations of the MEG
The DB equations of the MEG are represented by Equations (1) to (5). QY is the most crucial parameter for determining all the characteristics of the MEG [5][6][7][8][9]. Both the IQY (QY = 14; Figure 1a) and NQY (E th, from 2E g to 4E g ; Figure 1b) are shown in Equation (1) and Figure 2. The IQY follows a step function with a full generation of multiple EHPs per photon. The actual QY (the NQY case) extracted from the pump-probe measurements shows a deviation from the IQY after reaching the E th [12][13][14][15][16][17][18][19][20][21][22][23][24][25]32]. In the pump-probe measurements, the NQY is induced from the scale of the transient absorption signals (peak intensities and their decay signals). While pump intensities between multiple EHPs (high peak) and a single EHP (bleach signal) are compared, certain phenomena can be analyzed at the point of reaching E th in MEG, the potential number of generated EHPs, and MEG efficiency (the increment of additional EHPs by applied intensities) [11,12,33]. The modeling and related equations are reported in [12,30,[34][35][36].
where m is the number of multiple EHPs generated, M is the maximum number of EHPs, Eg is the bandgap, and E is the photon energy. A (=1) is the slope of the linearized QY, and Eth is the threshold energy for an MEG event. In NQY, E th is a significant parameter that explains the carrier extraction of MEGSCs (E th : the onset point of QY over 100%). The delayed E th (>2E g ) requires a higher photon energy for MEG and describes the status of MEG materials. Typically, E th depends on the effective mass of electrons and holes in a material. The surface state has a large impact on the MEG process because of fast carrier decay, which creates other decay paths at trap states [12]. Therefore, creating a near-perfect MEGSC is the first priority for maintaining its idealities.
To ensure minimal or no mathematical errors, the open-circuit voltage (V OC ) must be less than the bandgap energy. QY in Equation (5) relates to the generated number of EHPs for MEG, the excited high-energy state, and its photon energy emission through radiative recombination without losses. For instance, the optimum bandgap is 0.05 eV with QY = 200 and C = 46,200 suns. When 199 electrons recombine into holes, the energy is transferred to the 200th electron. This excites the electron to an energy state of 10 eV and causes the emission of photon energy without losses under blackbody radiation [6]. However, conventional AR is the nonradiative recombination in actual materials and Appl. Sci. 2020, 10, 5558 4 of 10 uses an alternative DB in silicon solar cells [10,31]. Thus, the MEGSC theory only considers blackbody radiation [6][7][8].
where m is the number of multiple EHPs generated, M is the maximum number of EHPs, E g is the bandgap, and E is the photon energy. A (=1) is the slope of the linearized QY, and E th is the threshold energy for an MEG event.
where φ is the particle flux given by Planck's equation for a temperature T, with a chemical potential (CP) µ in the photon energy range E 1 -E 2 ; h is Planck's constant; c is the speed of light in vacuum; and µ is the CP of an SJSC (q·V), where V is the operating voltage. µ MEG is the CP of MEG (q·QY(E)·V), k is the Boltzmann constant, J is the current density of the solar cell, q is the element of the charge, C is the optical concentration, f S is the geometry factor (1/46,200), T S is the temperature of the sun (6000 K), and T C is the temperature of the solar cell (300 K).

Numerical Analysis of the Nonradiative Recombination of an MEGSC
The deviation in QY depends on the material properties, such as effective mass, surface states, or defects of QD materials [37]. Thus, NQY describes the loss of EHPs from m·E g to (m + 1)·E g . To account for this, we reconfigured the DB of the MEGSC to include nonradiative recombination [1].
In the rate-based calculations, the DB of MEG is shown in Equation (6) [1]: where F S,MEG and F C,MEG (V) are the generation and recombination for the radiative term, respectively, and R MEG (0) and R MEG (V) are the nonradiative generation and recombination, respectively [1]. Equation (6) is reordered to show the net rate of generation and recombination in Equation (7) [1].

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To show the radiative and nonradiative limits, the following change is made to the part of Equation (7) inside brackets [1]: where f NR denotes the ratio between radiative recombination and nonradiative recombination, which indicates the contribution of radiative recombination in the MEGSC [1]. If the nonradiative recombination fits the ideal rectifying equation, f NR can show the ideal reverse saturation current (J 0 ) (Equation (9)) [1]: where 0 < f NR ≤ 1 and J 0 = (F C0,MEG + R MEG (0)).
In an ideal rectifying diode, J 0 is a voltage-dependent parameter. Therefore, the term for the recombination current, exp(q·QY·V/k·T C ), is included as shown in Equation (10): where F S,MEG = C·f S ·φ MEG (E g ,∞,T S ,0) + C·(1 − f S )·φ MEG (E g ,∞,T C ,0), and F C0,MEG = φ MEG (E g ,∞,T C ,0). In the DB equations above, V OC depends on f NR , which decreases as J 0 increases. Its equation is shown in Equation (11). This leads to reduced theoretical efficiencies by increased nonradiative recombination for producing a smaller fraction of radiative recombination [1].
The correlated theoretical efficiency is shown in Equation (12): where mpp is the maximum power point, P in is the input power, and σ is the Stefan-Boltzmann constant (= 2π 5 k 4 /(15c 2 h 3 ) = 5.670373 × 10 −8 Wm −2 K −4 ). A similar approach has been discussed for nonradiative recombination in QD solar cells. This involves investigating the external radiative efficiency (ERE) and V OC [38]. The ERE parameter depends on the radiative and nonradiative recombination rates, which affect the voltage losses of quantum dot solar cells (QDSCs). Smaller ratios of ERE reduce the maximum available operating voltage. The voltage losses on QD GaAs or perovskite solar cells have also been discussed [38][39][40]. Further, the current results of QDSCs have achieved 16.6% efficiency of Cs 1−x FA x PbI 3 QDSCs by minimal nonradiative recombination [41].

Results and Discussion
We tested seven cases of f NR (10 −10 , 10 −5 , 10 −4 , 10 −3 , 10 −2 , 10 −1 , and 1) to see the impact of nonradiative recombination. The results are summarized in Figures 3-5, Tables S1 and S2, and Figures  S1 and S2 (in the supplementary materials). J 0 is inversely proportional to f NR (Equation (9)). A small f NR represents a high nonradiative recombination rate. For all cases, f NR = 1 is the case for the IQY of MEGSC. We used both IQY (see  Table S1, and Figures S1a and S2a) and NQY under one-sun (C = 1; Figures 3a and 4, Table S1, and Figure S1b Table S2, and Figure S2b-d). For NQY, we chose E th = 2E g , 3E g , and 4E g to determine the impact of the nonradiative recombination. Equation (10) can predict the status of the QD MEGSC with NQY because of the nonidealities in the DB of an MEGSC. The MEGSC effect disappears after f NR = 10 −3 (1000 times J 0 ) (Figure 3a). We determined the critical point to maintain MEGSC at f NR = 10 −2 and E th = 2E g under one-sun illumination (Figure 3a and Figure S1b). Finally, increasing the light concentration shifts the critical point (from MEG to SJSC) after f NR = 10 −5 (10 5 of J 0 ) for both IQY and NQY (Figures 3b and 5d and Figure S2). Cs1−xFAxPbI3 QDSCs by minimal nonradiative recombination [41].

Results and Discussion
We tested seven cases of fNR (10 −10 , 10 −5 , 10 −4 , 10 −3 , 10 −2 , 10 −1 , and 1) to see the impact of nonradiative recombination. The results are summarized in Figures 3-5, Tables S1 and S2, and Figures S1 and S2 (in the supplementary materials). J0 is inversely proportional to fNR (Equation (9)). A small fNR represents a high nonradiative recombination rate. For all cases, fNR = 1 is the case for the IQY of MEGSC. We used both IQY (see Figures 3-5, Table S1, and Figures S1a and S2a) and NQY under onesun (C = 1; Figures 3a and 4, Table S1, and Figure S1b-d) and full-light concentration (C = 46,200;  Figures 3b and 5, Table S2, and Figure S2b-d). For NQY, we chose Eth = 2Eg, 3Eg, and 4Eg to determine the impact of the nonradiative recombination. Equation (10) can predict the status of the QD MEGSC with NQY because of the nonidealities in the DB of an MEGSC. The MEGSC effect disappears after fNR = 10 −3 (1000 times J0) (Figure 3a). We determined the critical point to maintain MEGSC at fNR = 10 −2 and Eth = 2Eg under one-sun illumination (Figure 3a and Figure S1b). Finally, increasing the light concentration shifts the critical point (from MEG to SJSC) after fNR = 10 −5 (10 5 of J0) for both IQY and NQY (Figures 3b and 5d and Figure S2).  For this simulation, we used ideal quantum yield (IQY) and with three different threshold energies (E th = 2E g , 3E g , and 4E g ) for NQY.   For this simulation, we used IQY and NQY (E th = 2·E g , 3·E g , 4·E g ). f NR is 1, 10 −1 , 10 −3 , and 10 −5 ). Figure 4. Eg vs. theoretical efficiencies of the MEGSC with fixed fNR under one-sun illumination (C = 1). For this simulation, we used IQY and NQY (Eth = 2•Eg, 3•Eg, 4•Eg). fNR is 1, 10 −1 , 10 −3 , and 10 −5 ). NQY under light-concentration conditions shows dual peaks after E th = 3·E g ( Figure 5 and Figure S2b-d). This represents the transition from the MEGSC to the SJSC, where the first peak is for the MEGSC and the second peak is for the SJSC. Thus, the delayed E th results in less efficient MEGSCs because of the shifting of the optimum point from the MEGSC to an SJSC under light-concentration conditions. Therefore, E th must be maintained at 2·E g with a low nonradiative recombination rate (e.g., 10 times J 0 ), typically under one-sun illumination. The material condition for MEGSC is crucial in that the nonradiative recombination impact must be small to maximize the radiative limit. Overall, the results explain the low efficiency of the QD MEGSCs. For instance, the theoretical efficiency is 30.6% for E g = 0.98 eV at E th = 3E g when f NR is 1. In the proposed MEGSC detailed balance approach, the theoretical efficiency is 4.6% at f NR = 10 −10 , which is similar to 4.5% for PbSe QDs [27]. We compared other materials such as PbSe and CdTe and summarized the results in Table 1 [27][28][29]. This shows that even if a QDSC can present excellent conditions for creating multiple EHPs, its high nonradiative recombination decreases the expected results [27][28][29].  Table 1 presents a comparison of the experimental results (PbSe [27,28] and CdTe [29]) and MEGSC DB approaches considering the effect of nonradiative recombination. For this simulation, we used the ratio of f NR up to 10 −12 to find an appropriate range of f NR to compare with the low experimental results of an MEGSC. These results indicate low efficiencies of the colloidal QDs of an MEGSC due to the high nonradiative recombination.
As shown in Figure 3 and Table S1, and in agreement with a previous study that used ERE approaches in GaAs QD solar cells [38], we detected similarities between the MEGSC DB and the experimental results. If the GaAs QD bandgap is 1.4-1.5 eV, the efficiency range at f NR = 10 −4 and 10 −5 is 23%-25%. These values show similar efficiency ranges at 10 −4 -10 −5 of ERE, even if the GaAs QD solar cell does not consider the MEG effects [38]. A small ratio of ERE has a significant impact on V oc owing to the increased nonradiative recombination rate. For instance, if the ERE is in the order of 10 −8 , the overall voltage drop in V OC is 0.5-0.6 V. Thus, its corresponding efficiency also decreases [38]. Other studies have also explained the voltage drop of QDSCs due to nonradiative recombination [38][39][40].

Conclusions
We analyzed the DB of the MEGSC with nonradiative recombination. In the ideal MEGSC, the excited electron after AR emits high photon energy under the stringent blackbody radiation condition (=QY max ·E g , where QY max is the maximum QY). However, an excited electron in the actual materials loses its kinetic energy through a phonon-assisted cooling process. The NQY from the pump-probe measurements depends on the material status (defects and effective mass), so its related parameters, E th and QY, deviate from IQY.
In the DB of the MEGSC, we introduced the ratio f NR between radiative and nonradiative recombination to see the impact of the ideal reverse saturation current. Typically, E th = 2E g with f NR = 10 -1 is the critical point to regard the QD status for the effect of MEG under one-sun illumination. Increasing the light concentration improves MEG. The minimum point for the MEGSC under NQY (in actual material systems) is f NR = 10 −4 at E th = 2E g , 3E g , and 4E g . J 0 must be lower than 10 4 times J 0 under full-light concentration to maintain the performance of the MEGSC. We compared the proposed approach with the experimental results to explain the effect of nonradiative recombination on MEGSCs. Minimizing nonradiative recombination can significantly improve V OC by reducing surface defects.
Supplementary Materials: The following are available online at http://www.mdpi.com/2076-3417/10/16/5558/s1, Table S1: The theoretical maximum efficiency with the variation of f NR under one-sun illumination (C = 1), where η is the efficiency. Table S2: The theoretical maximum efficiency for different values of f NR under full-light concentration (C = 46,200), where η is the efficiency. Figure S1: Variations of f NR with theoretical efficiencies of the MEGSC. For this simulation, we use IQY and NQY (E th = 2·E g , 3·E g , and 4·E g ) under one-sun illumination. Figure S2: Variations of f NR with theoretical efficiencies of the MEGSC. For this simulation, we use IQY and NQY (E th = 2·E g , 3·E g , and 4·E g ) under full-light concentration.