Bias-dependence of luminescent coupling efficiency in multijunction solar cells

Jieyang Jia; Yu Miao; Yangsen Kang; Yijie Huo; Mojgan Mazouchi; Yusi Chen; Li Zhao; Huiyang Deng; Pakapol Supaniratisai; Sara H. AlQahtani; James S. Harris

doi:10.1364/OE.23.00A219

1. Introduction

Luminescent coupling effects have attracted substantial research interest recently due to their significant impact on multijunction solar cell performance, which has been experimentally demonstrated [1–3]. Luminescent coupling is commonly described as the process in which the photons emitted by the radiative recombination of the higher band gap subcells generate photocurrent in the lower band gap subcells [1]. Previous research has shown that luminescent coupling can have significant effects on the characteristics of the cell, and thus must be considered carefully in modeling and simulation in order to generate accurate results [1,2]. Therefore, it is crucial to understand the physics of luminescent coupling and develop methods to accurately model luminescent coupling effects in multijunction solar cell simulations.

Previous research work, such as [2] and [4], have established different methods to model luminescent coupling effects. In both of these reports, a standard two-diode equivalent-circuit model is used to simulate the cell characteristics and the luminescent coupling efficiency is defined as the ratio between the luminescent coupling current of the bottom junction and J_1,top (the n = 1 recombination current density of the top junction) [1,2,5]. In this two-diode model, the coupling efficiency is assumed to be a constant, determined by the materials and the structure of the solar cell. However, the two-diode model may not be sufficiently accurate in simulating the behavior of cells with luminescent coupling effects. In addition, to the best of our knowledge, very little research effort has been put into either verifying this assumption of constant luminescent coupling efficiency, or investigating an appropriate variation of luminescent coupling under different operating conditions.

In this work, we discuss the limitations of the two-diode constant-coupling-efficiency model and suggest an alternative analytical method to simulate multijunction cell behavior in the presence of bias dependent luminescent coupling. We investigate the radiative and non-radiative recombination rates at different physical locations of the cell under different operation conditions, and thereby introduce a bias-dependent luminescent coupling efficiency. We use an analytical method with bias-dependent coupling efficiency to simulate the variation of luminescent coupling efficiency as well as the cell performance under different operating conditions, and compare them to the results from the standard two-diode model.

2. Multijunction cell J-V characteristics in the presence of luminescent coupling

A. The standard two-diode model and its limitations

In the standard two-diode equivalent-circuit model used in previous research work [2,5], each junction of the solar cell is modeled as a current source connected in parallel with a shunt resistance and two diodes with ideality factors n = 1 and n = 2 respectively. For a cell with good quality, the shunt resistance is close to infinite, and thus will not be considered in this work. In this model, the independent J-V characteristics of the i-th junction can be expressed as [1]:

J_{i} (V_{i}) = J_{01, i} [\exp (\frac{e V_{i}}{k_{B} T}) - 1] + J_{02, i} [\exp (\frac{e V_{i}}{2 k_{B} T}) - 1] - J_{L, i}^{e x t}

where

V_{i}

is the bias voltage across the junction,

J_{01, i}

and

J_{02, i}

are the saturation current density of the two diodes and

J_{L, i}^{e x t}

is the photocurrent density from external illumination [1]. Friedman et al. in [1] and Steiner et al. in [5] showed that the luminescent coupling current density of the i-th junction can be modeled as proportional to the

J_{01, i - 1}

:

J_{L C} = η_{i - 1, i} J_{01, i - 1} [\exp (\frac{e V_{i - 1}}{k_{B} T}) - 1]

where

η_{i - 1, i}

is the coupling efficiency between the two junctions. Figure 1 shows a schematic of the two-diode model of a two-junction solar cell with the luminescent coupling current modeled as a current-control-current-source.

Fig. 1 Two-diode equivalent circuit model of a two-junction solar cell. The components are as follows: photocurrent from external illumination $(J_{e x t}$ ), luminescent coupling current ( $J_{L C}$ ), recombination current with ideality factor n = 1 and n = 2 ( $J_{1}$ and $J_{2}$ ), parallel resistance ( $R_{p}$ ), and series resistance due to the tunnel junction ( $R_{s}$ ).

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While this model has been shown good agreement with experiments under most conditions [1,2,5], it has certain limitations. First, this model assumes that all the radiative recombination current comes from electrically injected carriers, but in a cell under illumination, a significant amount of radiative recombination current will be generated by the photogenerated carriers, especially at low bias voltage. As we will show later, ignoring the recombination currents from the photogenerated carriers leads to inaccurate coupling efficiency calculations. In addition, the two-diode model also ignores other non-ideal effects of the solar cell, such as the bias-voltage dependence of series resistance and the non-idealities of the P-N junction [6]. Finally, the two-diode model requires accurate fitting of $J_{01}$ and $J_{02}$ , which are not always easy in practice. Any fitting error my lead to significant error in the simulation result as $J_{L C}$ is strongly dependent on $J_{01}$ .

B. Calculating multijunction J-V from subcell J-V’s and bias-dependent coupling efficiency

To overcome the limitations of the two-diode model, we take an alternative approach, in which we derive the J-V characteristics of the multijunction cell directly from the subcell J-V’s, without $J_{01}$ and $J_{02}$ fitting. Here the subcell is referred to a single junction cell that is fabricated to mimic the characteristics of a junction in a multijunction cell.

To simplify the discussion, in this work we assume the cell we are modeling is a two-junction cell (2J). The discussion can be easily extended to cells with more than two junctions. In the typical cell design process, each single-junction subcell is fabricated and characterized. The J-V characteristics of the first junction subcell and the second junction subcell can be expressed as $V_{1} (J)$ and $V_{2} (J)$ respectively. If the two cells are serially connected without luminescent coupling, the J-V characteristics of the two-junction cell can be expressed as:

V_{2 J, n o L C} (J) = V_{1} (J) + V_{2} (J)

With the presence of luminescent coupling, extra current $J_{L C}$ is generated in the second junction, causing the J-V curve to move up by $J_{L C}$ :

J_{2} (V_{2}) = J_{2, n o L C} (V_{2}) + J_{L C}

Rewrite Eq. (4) as:

V_{2, L C} (J) = V_{2} (J - J_{L C})

Figure 2 illustrates the J-V characteristics of a two-junction cell with and without luminescent coupling effects, which helps to visually understand the relationships in Eqs. (3)–(5).

Fig. 2 J-V characteristics of an example two-junction cell. The cell is shown to be limited second junction current. The red solid line and blue dashed line are the J-V curves of the first and second junction subcells respectively. The black dot-dashed line and black dotted are the J-V curve for the two-junction cell with and without luminescent coupling effects respectively. The two-junction tandem cell has a higher current output than the second junction, indicating the presence of luminescent coupling current.

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Since $J_{L C}$ physically originates from the $J_{r e c, 1}$ , the recombination current of the first junction, we express $J_{L C}$ as

J_{L C} (V_{1}) = η (V_{1}) J_{r e c, 1} = η (V_{1}) (J_{s c, 1} - J), J < J_{s c, 1}

where

J_{s c, 1}

is the short-circuit current of the first junction, or the current that satisfies

V_{1} (J_{s c}) = 0

;

J_{r e c, 1}

is the recombination current of the first junction; and

η (V_{1})

is the coupling efficiency. Since the luminescent coupling current can be present only if the top junction is not current limiting, Eq. (6) requires

J < J_{s c, 1}

. Note that this definition is similar to the

η

defined in [1], however, here

η (V_{1})

is not a constant but a function of

V_{1}

. The derivation of the

V_{1}

-dependent coupling efficiency

η

will be discussed in detail in the next section. With

V_{1} (J)

,

V_{2} (J)

and

η (V_{1})

known, we can solve the V-J characteristics of the two-junction cell:

V_{2 J} (J) = V_{1} (J) + V_{2} (J - η (V_{1}) (J_{s c, 1} - J))

Using this method, we can solve the characteristics of the two-junction cell without fitting

J_{01}

and

J_{02}

.

3. Deriving bias-dependent luminescent coupling efficiency

As shown in the previous section, determining the $V_{1}$ -dependent coupling efficiency $η (V_{1})$ is crucial for solving the two-junction cell characteristics. In this section we will show the simulation of typical bias-dependent luminescent coupling efficiency and the physical reason behind it.

A. Relating luminescent coupling efficiency to recombination currents

In a two-junction cell, the luminescent coupling current of the second junction is directly proportional to the radiative recombination current of the first junction:

J_{L C} (V_{1}) = α_{2} γ_{1, 2} J_{r a d, 1} (V_{1})

where

α_{2}

is the average quantum efficiency of the second junction to the photons emitted by the first junction,

γ_{1, 2}

is the fraction of luminescent photons transmitted into the second junction, and

J_{r a d, 1}

is the radiative recombination current of the first junction. Relating Eq. (6) and Eq. (8), we can express

η (V_{1})

as

η (V_{1}) = α_{2} γ_{1, 2} \frac{J_{r a d} (V_{1})}{J_{r e c, J 1} (V_{1})}

Both

α

and

γ

are determined by the optical properties of the cell materials. For typical solar cell materials, it is reasonable to assume that

α

and

γ

are not significantly dependent on either the bias voltage or the photocurrent density. Therefore,

η

can be considered directly proportional to

J_{r a d} / J_{r e c, 1}

. In the standard two-diode model,

J_{r a d} (V_{1})

is usually assumed to have an ideality factor

n = 1

. Under such an assumption, if the

n = 1

recombination current in

J_{r e c, 1}

is considered dominant,

η

will be constant, as shown in [1]. However, we will show that in accurate modeling of typical cells,

J_{r a d} / J_{r e c, 1}

will not be constant, thus

η

will also vary with

V_{1}

.

To show the relationship of $J_{r a d} / J_{r e c, 1}$ to bias voltage, we decompose $J_{r a d}$ and $J_{r e c, 1}$ into different current components:

J_{r a d} = J_{r a d, l i g h t} + J_{r a d, b i a s}

J_{r e c, 1} = J_{r a d, l i g h t} + J_{r a d, b i a s} + J_{S R H, l i g h t} + J_{S R H, b i a s}

where

J_{r a d, l i g h t}

and

J_{r a d, b i a s}

are the radiative recombination current densities from photogenerated carriers and electrically injected carriers, respectively, and

J_{S R H, l i g h t}

and

J_{S R H, b i a s}

are the Shockley-Read-Hall (SRH) recombination current densities from photogenerated carriers and electrically injected carriers, respectively. Here we are assuming SRH recombination is the dominant non-radiative recombination mechanism. Other recombination mechanisms, such as Auger recombination are neglected. We can then derive the dependence of

J_{r a d} / J_{r e c, 1}

on bias voltage by investigating the variation of each current component with bias voltage.

To further demonstrate this, we simulated a typical GaAs single-junction solar cell with Sentaurus, which represents the middle junction of typical three-junction solar cells. The illumination spectrum is AM 1.5G with a low-pass filter such that the cell produces a $J_{s c}$ of 13.8 mA/cm² under 1 sun. The simulated cell is assumed to have a $1.5 μ m$ thick emitter with n-doping concentration of $N_{D} = 1 \times 10^{17} c m^{- 3}$ , and $2.5 μ m$ thick base with p-doping concentration of $N_{A} = 1 \times 10^{16} c m^{- 3}$ . The cell is assumed to have an ideal window layer and contact so that surface recombination does not affect the subsequent discussion. The cell has a minority carrier SRH lifetime $τ_{p} = 3 \times 10^{- 6} s$ and $τ_{n} = 1 \times 10^{- 7} s$ , both are within a reasonable range [7–9].

Figure 3 shows the simulated change of $J_{r a d, l i g h t}$ , $J_{r a d, b i a s}$ , $J_{S R H, l i g h t}$ , and $J_{S R H, b i a s}$ with bias voltage. Figure 4 shows the ratio $J_{r a d} / J_{r e c, 1}$ versus bias voltage.

Fig. 3 Simulation results of different recombination currents in a GaAs single-junction cell. The red dashed line and solid-circle line are the SRH recombination current densities from photogenerated carriers and electrically injected carriers, respectively. The blue dot-dashed line and dotted line are the radiative recombination current densities from photogenerated carriers and electrically injected carriers, respectively. The total recombination current density is shown as the black solid line. The J_sc is $13.8 m A / c m^{2}$ , and V_oc is $0.97 V$ .

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Fig. 4 Simulation results of $J_{r a d} / J_{r e c, t o t a l}$ at different bias voltages under different illumination intensities. The solid, dashed, dot-dashed and dotted lines represent the results when the illumination is 1, 10, 100 and 1000 suns, respectively. The circles mark the voltages at the maximum power points and the squares mark the open-circuit voltages.

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As shown in Fig. 3 the general trend of variation of $J_{r a d} / J_{r e c, 1}$ can be described as below: when the bias voltage is low and close to the short-circuit condition, $J_{r a d, l i g h t}$ and $J_{S R H, l i g h t}$ will be the dominant terms of $J_{r a d}$ and $J_{r e c, 1}$ ; when the bias voltage is high and exceeds the open-circuit voltage, $J_{r a d, b i a s}$ and $J_{S R H, b i a s}$ become dominant. Therefore,

{\frac{J_{r a d}}{J_{r e c, 1}} |}_{s c} \approx \frac{J_{r a d, l i g h t}}{J_{r a d, l i g h t} + J_{S R H, l i g h t}} = \frac{1}{1 + \frac{J_{S R H, l i g h t}}{J_{r a d, l i g h t}}}

{\frac{J_{r a d}}{J_{r e c, 1}} |}_{s c} \approx \frac{J_{r a d, b i a s}}{J_{r a d, b i a s} + J_{S R H, b i a s}} = \frac{1}{1 + \frac{J_{S R H, b i a s}}{J_{r a d, b i a s}}}

Note that $J_{r a d, l i g h t}$ and $J_{S R H, l i g h t}$ are not significantly dependent on $V_{1}$ , and thus cannot be modeled with either diode in the two-diode model. Nevertheless, $J_{r a d, l i g h t}$ and $J_{S R H, l i g h t}$ are very important in deciding the luminescent coupling efficiency. The two-diode model simulation of luminescent coupling effects does not take these currents into account and thus results in a physical inaccuracy of the model.

From Fig. 4 we can see that in the low bias region (0 – 0.7V), $J_{r a d} / J_{r e c, 1}$ is nearly a constant. When the bias voltage increases and approaches the maximum-power point, $J_{r a d} / J_{r e c, 1}$ significantly decreases, and reaches a minimum value near the maximum-power point. Then $J_{r a d} / J_{r e c, 1}$ will increase as the bias voltage further increases. From Eq. (9) we know that the luminescent coupling efficiency, $η$ , is directly proportional to $J_{r a d} / J_{r e c, 1}$ , thus $η$ should follow the same variation trend described above. This result is not based on any assumption about the second junction. Different characteristics of the second junction will affect the constants $α$ and $γ$ in Eq. (9), as well as the first junction bias voltage range under operating conditions (as shown later). However, the characteristics of the second junction will not affect the variation of $η$ with bias voltage of the first junction.

The increase in coupling efficiency at bias higher than the $V_{o c}$ is mainly due to saturation of SRH recombination as the traps are filled. Since solar cells never operate in this region, the variation of coupling efficiency at bias higher than $V_{o c}$ will not be discussed in this work.

B. Distribution of recombination rate and decreased coupling efficiency near $V_{M P P} ~ V_{o c}$

The change in coupling efficiency when the bias voltage approaches $V_{o c}$ is very important, especially because this happens on the typical operating voltage range of the first junction in a two-junction cell. Therefore it is very important to understand the physical reason for the change in coupling efficiency.

To first order, the radiative recombination and net SRH recombination rate in a typical P-N junction can be expressed as:

R^{r a d} = B (n p - n_{i}^{2})

R_{n e t}^{S R H} = \frac{n p - n_{i}^{2}}{τ_{p} (n + n_{i} \exp (\frac{E_{t r a p}}{k T})) + τ_{n} (p + n_{i} \exp (\frac{- E_{t r a p}}{k T}))}

where

τ_{p}

and

τ_{n}

are the SRH minority carrier lifetimes of holes and electrons, respectively,

B

is a constant that can be calculated from the absorption of the semiconductor materials [10]. Equations (14) and (15) show that

R_{n e t}^{S R H}

has a strong dependence on doping type and density in the material, while

R^{r a d}

does not.

For a typical III-V materials used in multijunction solar cells, $τ_{n}$ is significantly larger than $τ_{p}$ [7–9]. In addition, for typical solar cells, the N-doped emitter is more heavily doped than the P-doped base. Consequently, with the same level of minority carrier injection, $R_{n e t}^{S R H}$ tends to be significantly higher in the base than in the emitter, while $R^{r a d}$ is roughly the same. As a result, it can be shown that

{\frac{R_{S R H}}{R_{r a d}} |}_{e m i t t e r} < {\frac{R_{S R H}}{R_{r a d}} |}_{b a s e}

Figure 5 plots the simulated results of the radiative and SRH recombination rates at different depths through the cell. It can be seen that $\frac{R_{S R H}}{R_{r a d}}$ is significantly lower in the emitter than in the base. Since the light is incident on the top surface of the cell, and the material has a typical absorption depth of hundreds of nanometers, more photogeneration takes place in the emitter than in the base. Electrical injection of minority carriers, on the other hand, takes place at the junction, but is more favorable to electrons from the emitter into the base due to the doping ratio.

Fig. 5 Simulation results of the rates of radiative recombination (black solid: short-circuit condition; green dot-dashed: open-circuit condition) and SRH recombination (blue dashed: short-circuit condition; red dotted: open-circuit condition). Note the 0-0.5 μm region is the emitter and the 0.5-3 μm region is the base.

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Therefore, under low-bias or short-circuit condition, as $J_{r a d}$ and $J_{r e c, 1}$ are dominated by the recombination current from photogenerated carriers, the $J_{r a d} / J_{r e c, 1}$ ratio is primarily determined by ${\frac{R_{S R H}}{R_{r a d}} |}_{e m i t t e r}$ . Under high-bias conditions, since $J_{r a d}$ and $J_{r e c, 1}$ are dominated by the recombination current from electrically injected carriers, the $J_{r a d} / J_{r e c, 1}$ ratio is determined by the spatial average of ${\frac{R_{S R H}}{R_{r a d}} |}_{e m i t t e r}$ and ${\frac{R_{S R H}}{R_{r a d}} |}_{b a s e}$ . From the inequality in Eq. (16), we can derive that

{\frac{J_{S R H, l i g h t}}{J_{r a d, l i g h t}} |}_{s c} < {\frac{J_{S R H, b i a s}}{J_{r a d, b i a s}} |}_{o c}

Which when combined with Eq. (12) and (13), we have

{\frac{J_{r a d}}{J_{r e c, J 1}} |}_{s c} < {\frac{J_{r a d}}{J_{r e c, J 1}} |}_{o c}

Hence the coupling efficiency

η \propto \frac{J_{r a d}}{J_{r e c, J 1}}

decreases when the bias voltage increases and approaches

V_{o c}

, as shown in Fig. 4.

In some special cases, such as in a cell with a lightly-doped emitter, the inequality sign of Eqs. (16)–(18) may be inverted, resulting in an increase in coupling efficiency at $V_{o c}$ . The derivation of bias-dependent coupling efficiency in this situation is identical to the above. In this work we assume the simulated cell has a typical configuration and thus the coupling efficiency decreases near $V_{o c}$ .

C. Effect of illumination intensity on coupling efficiency

Since the coupling efficiency exhibits a strong dependence on the recombination current ratio, it also varies with illumination intensity in addition to bias voltage. Figure 6 shows the coupling efficiency at different bias voltage points versus the illumination intensity.

Fig. 6 Simulation results of the $J_{r a d} / J_{r e c, t o t a l}$ ratio at different illumination intensities and different voltages. The black circle, blue square and red triangle symbols each represent the ratio under short-circuit, maximum-power and open-circuit conditions.

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Changing the illumination intensity will not significantly affect the recombination currents from electrically injected carriers, thus $J_{S R H, b i a s}$ and $J_{r a d, b i a s}$ will not significantly change. It will also not affect $\frac{J_{S R H, l i g h t}}{J_{r a d, l i g h t}}$ , thus the coupling efficiency at low bias will also not change. However, higher illumination intensity requires a higher electrical injection level to make the recombination in the base region important. Therefore, as the illumination intensity increases, the coupling efficiency starts to decrease at a higher bias voltage, and the minimal coupling efficiency will also increase, as shown in Fig. 4.

In previous work on luminescent coupling [2,4], using a two-diode model to analyze the cell behavior, the coupling efficiency is extracted from the measurement data by calculating the ratios of $J_{s c}$ of each of the subcells and tandem cells under different spectral conditions. This method may introduce error into the modeling results because the coupling efficiency at the MPP can be significantly different from that at the short-circuit point, and the coupling efficiency is also very sensitive to the illumination spectrum condition. A detailed discussion of experimental characterization of luminescent coupling efficiency will be included in a future publication.

4. Effects on luminescent coupling efficiency variation on solar cell performance

In the previous section we have shown that the luminescent coupling efficiency can change significantly with bias voltage of the first junction. To analyze the effects of such variation on the performance modeling of the two-junction cell, we performed a series of simulations of a two-junction cell using the modeling technique introduced in the previous section, and compare the results with that of the two-diode equivalent circuit model.

Simulations using both modeling techniques were performed with the assumptions below. The first junction, GaAs, has the same material and structure described in the previous section and the second junction is assumed to have typical performance of the bottom junction of a lattice-match (LM) multijunction cell [3], with band-gap ~1.0eV. The first and second junctions are connected in tandem via an ideal tunnel junction, which is assumed to be both optically and electrically lossless. Such ideality assumption of the tunnel junction should not affect the generality of subsequent discussions. Similar to the previous section, the illumination spectrum is filtered AM 1.5G, under which the first and second junction both produces a $J_{s c}$ of 13.8 mA/cm² under 1 sun.

For all simulations, the first junction J-V characteristic is generated by Sentaurus simulations using the same parameters as in the previous section. The second junction J-V characteristic is generated from the simulation result of a typical LM bottom cell. The characteristic of the second junction should not be critical for comparison of the two simulation methods. For two-junction simulations using the standard two-diode model, we first found $J_{01}$ and $J_{02}$ for each junction by logarithmic fitting, and then use HSPICE to solve the equivalent circuit. The values of the luminescent coupling efficiency of the two methods are equal up to 0.88 at the short-circuit bias point.

A. Comparing the simulation results of models under current-matching conditions

Figure 7 shows the simulation results of the first junction subcell, the second junction subcell and the two-junction cell with the two-diode equivalent circuit model and the analytical method. The major performance metrics are listed in Table 1. The two-junction cell simulation results of the two methods have negligible difference if the luminescent coupling effects are omitted ( $η = 0$ is enforced).

Fig. 7 Simulation results of a two-junction cell. The J-V curves of the first junction subcell (blue dashed), the second junction subcell (red dotted), two-junction cell with constant coupling efficiency (green dot-dashed) and two-junction cell with bias-dependent coupling efficiency (brown solid) are shown. The luminescent coupling efficiencies of the two cases are equal to 0.88 in the short-circuit condition.

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Table 1. Key simulation performance of the first, second junction subcell and two-junction cell

View Table

From the above results we can see that there is a noticeable difference between the two-junction cell simulation results using the two different methods near the maximum-power point (MPP). Since the analytical method uses a variable coupling efficiency which has a rapid decrease near the MPP, the analytical method predicts a lower degree of luminescent coupling at the MPP. As a result, the analytical method using variable coupling efficiency predicts a smaller output current, voltage and power at the MPP. For this simulated cell, the $P_{m p p}$ calculated by the two-diode model is 0.9% higher than by the analytical method.

This result indicates that the two-diode model with constant coupling efficiency can lead to an overestimation of the cell efficiency. Such overestimation may increase when the modeled cell has three or more junctions or operates under non-uniform illumination [2].

B. Effects of current mismatch on luminescent coupling efficiency

In a two-diode model with constant coupling efficiency, the coupling efficiency is not affected by the photocurrent mismatch of the two junctions. However, when simulating a cell using the analytical method with bias-dependent coupling efficiency, different current mismatch conditions will change the operating voltage point of the first junction, and thereby change the luminescent coupling efficiency. The change of luminescent coupling efficiency with the two-junction cell applied voltage under different current-mismatch conditions is shown in Fig. 8.

Fig. 8 Change of the coupling efficiency with applied voltage for the two-junction cell under different photocurrent mismatch conditions. The blue solid, red dashed and green dotted lines represent photocurrent mismatch of 0, 0.2 and 1 mA/cm², respectively,

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In this set of simulations, the photocurrent mismatch, $Δ J_{p h} = J_{p h, 1} - J_{p h, 2}$ , is introduced by filtering the spectrum in such a way that $J_{p h, 2}$ decreases while $J_{p h, 1}$ remains constant. It can be seen that the coupling efficiency changes significantly over the entire range of applied voltage with different current mismatch. Under real operating conditions, such current mismatch can result from a change in the incidental light spectrum or non-uniform illumination. Therefore, the cell will have a different degree of luminescent coupling under different spectral conditions. The overall luminescent coupling efficiency will decrease as the current mismatch becomes larger.

Figure 9 shows the relative difference of $P_{m p p}$ simulated by the two-diode model and the analytical method. It can be seen that the different of $P_{m p p}$ significantly increases as the photocurrent mismatch becomes larger, indicating a larger efficiency overestimation by the two-diode model. This result suggests that the two-diode model with constant coupling efficiency can result in a significant simulation error with illumination spectral variations.

Fig. 9 Relative difference in the simulated maximum output power by the two methods versus the photocurrent mismatch.

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From above results, we can conclude that the two-diode model and the analytical method have noticeable difference in simulating two-junction cell characteristics. We suggest from a theoretical perspective that modeling with constant coupling efficiency leads to reduced accuracy in predicting cell performance. To further demonstrate that the analytical method with bias-dependent coupling efficiency generates more accurate simulation results, experimental results on real devices are needed. Comparison of the modeling and experimental results will be included in a future work.

5. Conclusion

In summary, to accurately simulate multijunction solar cell performance, the bias-voltage dependence of the luminescent coupling efficiency must be introduced in order to account for the recombination current from photogenerated carriers. We discussed the limitations of the two-diode model and assumption of a constant coupling efficiency, and introduced an analytical method with a bias-dependent coupling efficiency. We theoretically derived and numerically simulated cells with a bias-dependent coupling efficiency and showed that the coupling efficiency may change significantly near the maximum power point due to the change of distribution of recombination rates in the top junction. We compared the two-diode modeling method with the analytical method in simulating a two-junction cell and showed a significant difference in the simulated performance. This work may lead to further discussion on the accuracy of different methods in modeling multijunction solar cells with luminescent coupling effects and provide guidance for higher efficiency multijunction solar cells.

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10. J. Nelson, The Physics of Solar Cells (Imperial College Press, 2003), Chap. 4.

	J_sc (mA/cm²)	V_oc(V)	J_mpp (mA/cm²)	V_mpp (V)	P_mpp (mW/cm²)	FF (%)
the first junction	13.8	0.973	13.4	0.880	11.8	87.7
the second junction	13.8	0.617	13.0	0.534	7.0	81.6
two-junction cell (constant coupling efficiency)	13.8	1.608	13.3	1.429	19.0	85.6
two-junction cell (bias-dependent coupling efficiency)	13.8	1.597	13.3	1.415	18.8	85.3

Bias-dependence of luminescent coupling efficiency in multijunction solar cells

Abstract

1. Introduction

2. Multijunction cell J-V characteristics in the presence of luminescent coupling

A. The standard two-diode model and its limitations

B. Calculating multijunction J-V from subcell J-V’s and bias-dependent coupling efficiency

3. Deriving bias-dependent luminescent coupling efficiency

A. Relating luminescent coupling efficiency to recombination currents

B. Distribution of recombination rate and decreased coupling efficiency near $V_{M P P} ~ V_{o c}$

C. Effect of illumination intensity on coupling efficiency

4. Effects on luminescent coupling efficiency variation on solar cell performance

A. Comparing the simulation results of models under current-matching conditions

B. Effects of current mismatch on luminescent coupling efficiency

5. Conclusion

References and links

Cited By

Figures (9)

Tables (1)

Equations (18)

Optics Express

Abstract

1. Introduction

2. Multijunction cell J-V characteristics in the presence of luminescent coupling

A. The standard two-diode model and its limitations

B. Calculating multijunction J-V from subcell J-V’s and bias-dependent coupling efficiency

3. Deriving bias-dependent luminescent coupling efficiency

A. Relating luminescent coupling efficiency to recombination currents

B. Distribution of recombination rate and decreased coupling efficiency near VMPP~Voc

C. Effect of illumination intensity on coupling efficiency

4. Effects on luminescent coupling efficiency variation on solar cell performance

A. Comparing the simulation results of models under current-matching conditions

B. Effects of current mismatch on luminescent coupling efficiency

5. Conclusion

References and links

Cited By

Figures (9)

Tables (1)

Equations (18)

Optics Express

B. Distribution of recombination rate and decreased coupling efficiency near $V_{M P P} ~ V_{o c}$