1. Introduction

Multi-junction solar cells (MJSCs) can achieve the highest conversion efficiency among the current photovoltaic devices, especially those fabricated by epitaxially growing several III-V compound subcells that are playing important roles in space field and terrestrial concentration system [1–6]. Typically, the well-known InGaP/GaAs/InGaAs triple-junction (3J) solar cell, regarded as one of the commercially successful MJSCs, has obtained a conversion efficiency of more than 37.8% and has been made into a flexible module. Despite III-V MJSCs exhibiting very excellent performance, the costs of materials and complex manufacturing processes are extremely high when compared with the other types of photovoltaic devices, which greatly hinder their wider application. Therefore, the research on cost reduction without performance loss has been attracting a lot of interest from MJSC experts.

Aiming to reduce the fabrication cost and growing time of III-V MJSCs, an effective approach is to minimize the effective thicknesses of these subcells in the stack under the condition that does not sacrifice any yield—in other words, to fabricate an MJSC that is as thick as possible in optics but as thin as possible in physics. According to previous studies on single-junction (1J) solar cells, light trapping design is a good scheme to obtain ultra-thin cells without sacrificing cell performance, such as surface randomly texturing [7–9], two-dimensional (2D) diffraction grating [10], 3D nanophotonic structure [11,12]. The main principle is that such a sophisticated and rough surface can randomize the traveling direction of incident photons that tend to be trapped inside the cell by total reflection due to the quite narrow escape cone of the semiconductor-air interface. Herein, such random light paths increase the effective length of absorption path, which enables to compress subcell absorber layer without any drops in sunlight absorption.

Typically, surface random texturization, regarded as a relatively simple light-trapping scheme, can be obtained via wet chemical etching [13] or laser process [14,15] and combined with anti-reflection (AR) coating or Ag/Al reflector, which have been widely used in the field of Silicon solar cells [16,15] and also been expended in ultra-thin GaAs solar cells [8,9]. Generally, it includes textured front surface only (TF), textured rear surface only (TR), and double textured surfaces (DT), respectively. Some theoretical studies [17–20] have presented that 1J solar cells with TF or DT would exhibit the best cell performance and the thinnest absorber layer among all types of surface design. Currently, an ultra-thin GaAs solar cell with TR has been fabricated by wet chemical etching, which empirically points out that a diffusive rear mirror significantly enhances the cell’s photocurrent and external radiative efficiency (ERE) [21]. If surface random texturization was integrated into the field of MJSCs, all subcell thicknesses in the stack may be cut down as much as possible, which has a direct benefit of not only reducing the fabrication cost and time but also enhancing the irradiation tolerance in space environment.

However, unlike 1J solar cells, there are very complex absorption and luminescence couplings among all subcells in MJSC, which lead to the photocurrent of the entire device being greatly sensitive to all subcell thicknesses. Thus, it is an essential issue to optimize the subcell thickness combination very carefully to realize better current-matching among the stack. Based on that, a rigorous optical model needs to be established before the experimental study can begin. Recently, optical simulations on all subcell absorptance and photocurrents have been presented in perovskite/Si tandem-junction and triple-junction thin-film solar cells with textured interfaces or rear-surface [22,23]. Besides the absorption model, it is also urgent to predict subcell voltage and yield for those light-trapping MJSCs, which will provide a reasonable guide to obtaining the best design with higher cost performance.

In light of this, our previous study focused on the simplest light-trapping scheme, i.e. MJSC with an ideal textured rear surface only (TR), and established its absorption model including complex absorption and luminescence couplings, which was applied on the rear-textured InGaP/GaAs/InGaAs-3J solar cells [24]. The results systemically reveal if the top-subcell thickness is less than 600 nm, such Lambertian scattering rear reflector can enhance light absorption so significantly that over 90% of bottom-subcell thickness and even 50% of middle-subcell thickness would be cut without any sacrifice on conversion efficiency when compared with the traditional smooth-surfaced MJSC. This study can provide a theoretical design for the experimental studies on arbitrary MJSCs with TR.

In this work, our group furtherly investigates an arbitrary N-junction solar cell with an ideal textured front surface and perfect rear reflector (TF) and systematically models the subcell absorptances for incident sunlight and for the luminescence’s from all subcells in the stack. Still taking InGaP/GaAs/InGaAs-3J solar cell, for instance, the absorptance, photocurrent, voltage, and yield for all subcells, as well as the current-matching thickness combination (the thinnest limit) are estimated via the proposed models. Subsequently, the benefit resulting from such front texturization have been compared with two control 3J cells with rear texturization (TR) and smooth surfaces. Finally, the impacts of non-ideal texturization on cell performance and the thinnest limit are also discussed. Since light-trapping design only can cut down the active region of these subcells in the stack, for convenience, in this paper we just focus on the reduction of the total subcell thicknesses, ridding of the thickness of the other necessary layers, such as metamorphic (MM) buffer or tunnel diodes in those MJSCs fabricated by monolithic metamorphic technology.

2. Optical modeling

Figure 1(a) exhibits the sketch of cross sections for arbitrary N-junction solar cell (a) with ideal non-reflective fully-randomized textured front surface and ideal 100% reflecting rear reflector (the core cell, device A) and two control cells: (b) with ideal non-reflective front surface and textured 100% reflecting rear reflector (device B) and (c) with ideal non-reflective front surface and 100% reflecting rear reflector (device C). For an arbitrary N-junction device, the integer i (N ≥ i ≥1) represents a certain subcell in the stack ordered from top to bottom.

 

Fig. 1. (a) Cross sections of an arbitrary N-junction solar cells with the three types of surface designs: device A, B, and C. (b) Schematic diagram of the reflection and transmission paths after incident sunlight passing through device A with non-reflective fully-randomized front textures.

Download Full Size | PDF

2.1 Transmission and reflection of ideal non-reflective fully-randomized front textures

To reduce the computing complexity, statistical ray tracing method is used to derive subcell absorptance and transmittance in this work. As shown in Fig. 1(b), the ideal non-reflective fully-randomized textured front surface is assumed as an aggregate of numerous minute surfaces covered by perfect AR coating, whose orientations are completely random over 2π and as many as possible. With that, the perpendicular incoming sunlight can fully enter the MJSC without any reflection and therefore the initial net transmittance is unity. Typically, the minute surfaces whose normal direction are parallel to the plane of the front surface would get the maximum inflow angle (θ_max=π/2–θ_c), equaling to the complementary angle of the critical angle of total reflection on semiconductor-air interface θ_c = arcsin(1/n_i), where n_i is the refractive index of subcell i. As a result, all initial inflow light is isotopically distributed but within such light cone:

Thus, the angle-dependent transmittance of the incident sunlight at a certain inflow scattering angle (θ) is given by

In the above Equation, within Ω_max, the intensity of inflow light at a certain dΩ is assumed as proportional to cosθ and the normalization coefficient A is determined by

If maximal scattering (Lambertian limit) [25] on the ideal front textures is assumed, the inflow scattering light is isotopically distributed over all internal angles (Ω_max = 2π). In this case, Eq. (2) can be simplified by

Figure 1(b) exhibits the light path after the sunlight enters device A. Taking an arbitrary internal scattering angle θ for instance, the initial inflow ray is absorbed twice by subcell i with the same θ during one absorption round (the downward and upward passing’s) and then hit on the front textures again, where 4π scattering would occur: the light within escape cone of each minute surface emit from the device and the rest are trapped by total reflection to start the next absorption round with a new angle φ. Based on the ideal assumption of fully-randomized front textures, the scattering angle φ of each photon in each round is completely random over 2π, however, the net reflectance or net emittance for all photons in each round are identical. Since the net emittance through the textured front surface (e_f^net) is a fraction of internal light from all light paths within the cell that can escape to air, it can be calculated by the fraction of all light paths within the escape cone Ω_esc, expressed as

Meanwhile, the net internal reflectance of the ideal front textures in each round equal to

Since the light reflected by front textures is regarded to be scattered with angular Lambertian distribution, its angle-dependent internal reflectance (R_f^Tex) is given by

In addition, the net internal reflectance of all rear surfaces (textured or smooth) are set as unity in the following calculations on the three devices (R_b = 1).

2.2 Subcell absorptance for incident sunlight

Taking a 3J solar cell shown in Fig. 2(a) for instance, the three subcells in the stack share the incoming sunlight according to different photon energy (E). In the general case, the absorptance of subcell i (a_i) for incident sunlight is separated by these subcell bandgap energies of (E_g1, E_g2, …, E_gi), expressed as

 

Fig. 2. (a) Schematic diagram of the absorptance (a_ij) in an arbitrary triple-junction solar cell. (b) Optical paths of subcell i in a N-junction solar cell for absorbing incident sunlight, (c) absorbing its self-luminescence (a_i^self) via front and rear surfaces (a_i^f and a_i^r), and (d) absorbing the luminescence from an arbitrary upper subcell k (a_k->i).

Download Full Size | PDF

Based on the absorption path shown in Fig. 2(b), the general formula of a_ij is expressed as:

The first term in Eq. (9) represents the absorptance of subcell i during the first absorption round before hitting on the textured front surface, while the last term is the exact sum of the absorptance through all paths after the first internal diffuse reflection by front textures, where the symbol φ represents the randomly reflected angle in each round. Typically, T_i is the transmittance passing through subcell i with the corresponding inflow angle θ_i (related to subcell radiative index n_i), which is also determined by the energy-dependent absorption coefficient α_i(E) and the thickness L_i of subcell i. Moreover, V_ij stands for the absorptance of subcell i for the sunlight of E_gj-1 > E ≥ E_gj that have been absorbed by these subcells with the order from j to i-1, and Λ_ij represents that with the order of j, j+1, …, N, N, …, i+1. O_j is the total transmittance for the photons of E_gj-1 > E ≥ E_gj during one light-passing round (absorbed by subcell j, j+1, …, N, N, …, j+1, j). Thus, for a given subcell combination (subcell materials and thicknesses in the stack), all values of a_ij in device A can be calculated by Eq. (9). To simplify this calculation, we assume all subcell inflow angles θ_i are almost the same, because the refractive indexes of these materials are relatively close. Meanwhile, a_ij in device B and C also can be calculated by the corresponding models established in our previous study [24]. With them, each subcell photocurrent (J_i^Sun) in the N-junction device can be evaluated by

2.3 Subcell absorptance for self-luminescence

Aiming to predict the voltage and conversion efficiency of all subcells in N-junction device A, it is necessary to model the emittance of self-luminescence from all subcells in the stack. With that, the carrier losses caused by external radiative emission are known, which is very important for establishing each subcell carrier balance equation under light condition. According to detailed balance theory, the front- or rear-surface emittances of a certain subcell can be obtained from its absorptance for background blackbody radiation incident from all angles at thermal equilibrium. Figure 2(c) exhibits the absorption paths for blackbody radiation respectively coming from the front and rear surface of subcell i. Here, θ and β stand for the angle of each photon before and after the first diffuse reflection by front textures, respectively. With that, the general formulas of front- and rear-surface absorptance (a_i^f and a_i^r) for the inflow angle θ are given by

2.4 Subcell absorptance for upper subcells’ self-luminescence

Due to luminescence coupling always presenting in MJSCs, the absorptance of a certain subcell for the photons emitting from these upper subcells in the stack are also important when the photocurrent is evaluated. Figure 2(d) shows the absorption paths of subcell i for the photons emitting from an arbitrary upper subcell k (1 ≤ k < i) in N-junction device A. The general formula of such absorptance (a_k→i) is given as

The two terms in Eq. (16) are initial absorption at inflow angle θ before diffuse reflection and the net absorptions for all paths after it.

2.4 Performance prediction with subcell absorption and luminescence couplings

Until now, we can establish the carrier balance equation for each subcell in N-junction device A to predict their current density versus voltage (J-V) characteristics under light condition, given by

3. Results

In this section, not only the performance parameters but also the ultra-thin design of the InGaP/GaAs/InGaAs-3J device A are predicted via the proposed model and are compared with those of device B and C, where the same subcell bandgap energy and the function of absorption coefficients α_i(E) presented in our previous paper [24] were used. But note here, in this paper, the front surface of devices B and C are also assumed to be covered by the same ideal AR coatings (R_f=0). Therefore, the following results of devices B and C (such as subcell absorptance, J_sc, V_oc, conversion efficiency, C-M thickness combination, etc.) slightly differ from the presented results in the previous paper [24], in which an experimental data of front reflectance (R_f $\ne$ 0) measured from an actual IMM-3J cell were used for these calculations.

Figure 3 compares the three-subcell absorptance in 3J-device A, B, and C with the same thickness combinations of (L₁, L₂, L₃). For a very thin combination of (300 nm, 200 nm, 15 nm), the top subcell absorptance (a₁ = a₁₁) for front textures (device A, solid) is higher than that with the smooth front surface (device B and C, dashed and dotted). The middle- and bottom-subcell absorptance (a₂ and a₃) are divided into two and three energy regions, respectively. Firstly, a₂₂ (the main absorption in the middle subcell) of device A is close to 80%, slightly higher than that of B, and almost twice as high as the average value of C. Notably, although L₃ is as thin as 15 nm, both front and rear textures can greatly enhance the average value of a₃₃ (the main absorption in the bottom subcell) up to about 50% absolute, by contrast, that in C is below 5% absolute. For this thin combination, the parasitic absorptions of middle and bottom-subcells for these high-energy photons that have passed through the upper subcells (a₂₁, a₃₁, a₃₂) are also clearly presented in Fig. 3(a), which will increase the photocurrents of the middle and bottom subcells and save more materials. By contrast, the two thicker combinations of (500 nm, 350 nm, 25 nm) and (500 nm, 1000 nm, 100 nm) exhibit the higher main absorptions (a₁₁, a₂₂, a₃₃) and the lower parasitic absorptions (a₂₁, a₃₁, a₃₂), whose subcell photocurrents thus primarily come from their main absorption (a_ii). Typically, Fig. 3(c) reveals that device A significantly improves a₁₁, a₂₂, and a₃₃ over 80% absolute, while B only enhances that of the bottom cell greatly, when compared with C.

 

Fig. 3. Three subcell absorptances of InGaP/GaAs/InGaAs-3J device A, B, and C, when their combinations of (L₁, L₂, L₃) are fixed as (300 nm, 200 nm, 15 nm), (500 nm, 350 nm, 25 nm) and (500 nm, 1000 nm, 100 nm), respectively.

Download Full Size | PDF

The results shown in Fig. 3 are consistent with common sense that the front textures (device A) can effectively enhance the sunlight absorption of all subcells in the stack, while the rear textures (device B) only improve those of the thinner bottom subcells strongly, especially the bottom-most one. This can be explained by the following reasons. In the smooth-surfaced device C, the incoming sunlight is always perpendicular to the front surface, so that each subcell thickness must be thicker than the average absorption length along the normal direction. In device B, after the initial perpendicular absorption, the sunlight hit on the rear textures and then redistribute uniformly over 2π. In the following paths, these diffuse photons with large angles travel a longer path in each absorption round, which would be absorbed more strongly than those with small angles. Notably, such randomly reflected light will become more and more concentrated toward the normal direction as they are going deeper through the entire device and will be randomized again by the rear textures in the following rounds. Therefore, the absorption enhancement effect caused by rear textures is particularly noticeable in a few bottom subcells for the thicker combinations. By contrast, the front textures in device A can randomize the initial incident sunlight directly, resulting in the absorption enhancement effect starting from the first absorption round, and is therefore more effective design for thickness reduction for all subcells, especially for those thinner combinations. No doubt, for those thicker subcell combinations or those with much more junctions, the absorption enhancement effect produced by the front textures would become weaker in a few bottom subcells.

When taking the thickness combination of (L₁, L₂, L₃) as a variable, the three subcell photocurrents (J_1,2,3^sun) were estimated via integrating their absorptances with solar spectrum. Figure 4 exhibits the entire-cell J_sc (the minimum of J_1,2,3^sun) of the three devices as a function of L₂ and L₃, where all L₁ was fixed as 500 nm. In Fig. 4, J_sc increases as both L₂ and L₃ increase, and the values located at the up-right side are limited by L₁, while that at the below-right side are limited by L₃ and at the up-left are limited by L₂. The dotted curves present the combinations of (L₂, L₃) making J₂^sun = J₃^sun, and the blue crosses show the special thickness combinations that can get current matching through the whole stack (J₁^sun = J₂^sun = J₃^sun, called C-M combination). Specifically, such C-M combinations in device A reveal that L₂ = 825 nm combined with L₃ = 50 nm is thick enough to match L₁ = 500 nm, whose J_sc at this time (C-M current, remarked by J_CM) can reach up to 15.1 mA/cm². By contrast, the C-M combinations in B and C are (500 nm, 392 nm, 28 nm) and (500 nm, 1089 nm, 347 nm) respectively, and their J_CM are only 14.2 mA/cm². Therefore, we can conclude that the J_CM in A for L₁ = 500 nm increases by over 6% but requires 2.1 times of L₂ and 1.8 times of L₃ in B or 76% of L₂ and 14% of L₃ in C. That is because J₁^sun in A primarily coming from the initial diffuse absorption is much higher than those in B and C primarily coming from the initial perpendicular absorption. To match it, device A requires the thicker L₂ and L₃ to generate more J₂^sun and J₃^sun. It is worthy to mention that such C-M combination represents the thinnest subcell combination for getting a certain J_sc, however, since the contours of J_sc change drastically near the blue crosses, it is risky to just choose the C-M combination for ultra-thin design, while a thicker combination of L₂ and L₃ (close to the blue cross but at the right-up side) is strongly recommended for experimental study. But the C-M combination is still regarded as a very important reference for the ultra-thin design of MJSCs in theoretical study.

 

Fig. 4. Photocurrent (J_sc) as functions of L₂ and L₃, in 3J-device A, B, and C with L₁ = 500 nm.

Download Full Size | PDF

Figure 5(a-b) exhibits the C-M L₂ and L₃ for the three devices with L₁ varying from 200 nm to 700 nm. Firstly, all curves increase superlinearly as L₁ increasing, particularly noticeable in the C-M L₂ of device A and B. Taking the middle subcell for instance, the superlinear trend can be explained by the fact when all subcells in the stack are relatively thin (L₁<500 nm), the parasitic absorption of these higher-energy photons by the middle subcell are significant, which would no doubt generate more J₂^sun so that a thinner necessary L₂ are presented. As a result, the C-M thickness in the thin cases (L₁<500 nm) are compressed stronger than those in the thick cases (L₁ > 500 nm). Moreover, the excellent superlinear trend indicts the absorption enhancement effect works very well in the two textured cells, especially in device A.

 

Fig. 5. (a, b) Current-matching L₂ and L₃ and (c) C-M current (J_CM) of 3J-device A, B, and C with various L₁.

Download Full Size | PDF

Figure 5(c) compares the C-M currents (J_CM) for the three devices. They increase gradually and then become saturated as the subcell thickness increases. For the thin combinations (L₁<500 nm), J_CM of B is much higher than that of C, while becomes the same when the subcells are thick enough (L₁>500 nm). As L₁ increases from 200 nm to 700 nm, J_CM of A is always higher than that of B and C by 4-6%. If get the same values of J_CM=15.1 mA/cm², the total C-M thickness (L_sum= L₁+ L₂ + L₃) in device A, B, and C are 1407 nm, 1987nm, and 3275 nm, respectively. In other words, the lower limit of L_sum in B and C are respectively 1.4 and 2.3 times higher than that in A, which clarifies that front texturization has more potential for reducing the subcell thicknesses than rear texturization only.

By using the C-M combinations exhibited in Fig. 5(a, b), the conversion efficiency (η_3J) and open circuit voltage (V_oc) of the three devices with various subcell internal radiative efficiency (η_int) were estimated and plotted in Fig. 6(a-b). From an overall perspective, η_3J of device A is 1%-2% absolute higher than that of B and 2%-6% absolute higher than that of C. For a certain η_int, η_3J of A increases sharply as L₁ increasing, and then coverage to the maximum at L₁ ≥ 640 nm. As η_int decreasing from 1 to 10⁻³, the maximum (L₁ ≥ 640 nm) decrease from 45% to 34% for these thicker C-M combinations. In B and C, for a certain η_int, η_3J keep adding up as L₁ increase from 200 nm to 700 nm. For a thick C-M combination of L₁ = 500 nm, η_3J of A decreases from 44% to 35% as η_int decrease from 1 to 10⁻³, while that of B and C decrease from 41% to 32%-33%, quantitatively reflecting the effects of the non-radiative recombination in all subcells on the entire yield that can be used to predict the realistic yields of the three devices. On the other hand, their V_oc show similar trends influenced by L₁ and η_int. They decrease almost linearly with η_int in semi-logarithmic coordinates when η_int is less than 0.1. Specifically, for the typical value of L₁ = 500 nm, V_oc of A decreases from 3.30 V to 2.64 V as η_int decreases from 1 to 10⁻³, while that of B and C decrease from 3.30 V to 2.67 V and from 3.32 V to 2.58 V, respectively.

 

Fig. 6. (a) Conversion efficiency (η_3J) and (b) V_oc of the 3J-device A, B, and C with C-M thickness combination as functions of subcell internal radiative efficiency (η_int) from 10⁻⁴ to 1 and L₁ from 200 nm to 700 nm. (c) (left axis) η_3J predicted with η_int =1,0.1, 0.01 as a function of L₁; (right axis) normalized C-M thickness of (L₁, L₂, L₃, L_sum) as a function of L₁ by taking the values at L₁ = 700 nm as standards.

Download Full Size | PDF

To find the most promising ultra-thin design (high cost performance), the tradeoff between η_3J and the thinnest thickness (the total C-M thickness, L_sum) was also analyzed. The dashed curves (left axis) in Fig. 6(c) plot η_3J of the three devices with η_int =1, 0.1, 0.01 as a function of L₁. Note that the up-to-data efficiency record of the smooth-surfaced GaInP/GaAs/InGaAs-3J with L₁ of 550-600 nm is 37.9%, which approaches the predicted η_3J for η_int = 0.1 shown as the purple star in Fig. 6(c). According to this experimental benchmark (37.9% at η_int = 0.1), the recommended design for device A is the C-M combination at L₁ ∼ 350 nm, i.e. (350 nm, 315 nm, 28 nm). By contrast, device B and C require at least 500 nm and 600 nm of L₁, i.e. their thinnest combinations are (500 nm, 411 nm, 27 nm) and (600 nm, 1559 nm, 411 nm), respectively. This result quantitatively points out that for the same benchmark, device A is the thinnest among the three devices, and saves the most top- and middle-subcell thickness but costs almost the same bottom-subcell materials when compared with device B. Moreover, the circle-solid curves (right axis) in Fig. 6(c) plot the normalized C-M thicknesses of (L₁, L₂, L₃, L_sum) as a function of L₁, by taking the values at L₁ = 700 nm as standards. The super-linear features of the normalized L_sum (black curves) in device A and B also can be used to judge the most promising ultra-thin designs for the low-cost or anti-irradiation MJSCs with light-trapping, while that of the normalized L_sum in device C is almost linear. Specifically, as L₁ decreases from 350 nm to 200 nm, the η_3J of device A for η_int =0.1 gradually decreases from 38.2% to 35.3%, meanwhile the normalized L_sum decreases sharply from 100% to 14% with the decrease in L₁ from 700 nm to 350 nm and then gradually decreases to 6% as L₁ decreasing from 350 nm to 200 nm. Therefore, it is recommended to choose the thinner combinations of 200nm ≤ L₁≤350 nm, which sacrifice a little yield (η_3J are still greater than 35%) but only cost 6%-20% III-V materials when taking the L_sum of L₁ = 700 nm as standards. For device B, the C-M combinations of 300nm ≤ L₁≤500 nm are recommended greatly, which cost 21%-47% III-V materials but still maintained η_3J of over 35%. Such ultra-thin device A or B is very favorable for enhancing the irradiation tolerance when working in space, because the GaAs-subcell in the smooth-surfaced 3J cell that usually has the thickest absorption layer and the highest lattice quality in the stack is very sensitive to irradiation damage, leading to terrible drops in J_sc, FF, and V_oc of the entire cell. Also, the top- and bottom-subcell thicknesses need to be compressed as much as possible to furtherly reduce the performance degradations caused by collisions of their lattice atoms with high-energy particles.

Table 1 compares the performance parameters in detailed balance limit and the cut-down ratio of subcell thickness in device A to those in B or C when they obtain the same J_CM. For each benchmark of J_CM varying from 13.5 to 15.1 mA/cm², L_sum in A is obviously lower than that of B and C, especially for top- and middle-subcell thicknesses, but almost the same V_oc and η_3J can be achieved. In more detail, for the same benchmarks of J_CM or η_3J, ideal front textures can at least reduce InGaP-subcell thickness (L₁) by 28%, GaAs-subcell thickness (L₂) by 56%, and InGaAs-subcell thickness (L₃) by 90%, compared with the smooth-surfaced device C, and save more than 28% of L₁, 22% of L₂, and 2% of L₃, compared with the rear-textured device B. To get the same J_CM of 15.1 mA/cm and η_3J of over 44%, the minimum of L_sum in A is only 1453 nm, which is even 71% of that in B. In addition, only A can reach the benchmark of more than 15.5 mA/cm² of J_CM and 45.6% of yield, while B and C cannot. In summary, the cut-down ratios on subcell thicknesses in device A quantitatively clarifies front texturization have excellent potential for ultra-thin MJ designs.

Table 1. Comparison of C-M thicknesses and performance parameters in detailed-balance limit among the three devices when get the same benchmarks of photocurrent (J_CM) varying from 13.5 to 15.5 mA/cm².

View Table

Additionally, it would be necessary to consider the effect of the metamorphic (MM) layer including physical thickness and parasitic optical absorption when we perform the ultra-thin design on those MJSCs fabricated by IMM or UMM technology. Taking the actual InGaP/ GaAs/InGaAs-3J cell for instance, the MM layer is usually fabricated by AsGaInP- or InGaP-graded-layers (E_g ≥1.7 eV), whose total thickness is more than 3 µm and cannot be reduced by light-trapping design. Therefore, in these relatively thicker 3J cells (the cases of L₁≥500 nm in device A, for example) where the light absorption of the top- and middle-cells mainly originate from the first absorption round, if their thicknesses were designed properly, the parasitic absorption of the MM layer can be neglected, and therefore the impact of the MM buffer on the bottom-subcell thickness also can be neglected (that means the presented C-M thicknesses design do not change for these thick combination). In contrast, in these ultra-thin 3J cells (L₁≤300 nm in device A, for example), the effect of the MM layer on the thickness design of all subcells cannot be neglected. In this case, with the actual absorption coefficient and the thickness of all sub-layers in the MM buffer, the three subcell absorptances, and the C-M thickness design also can be evaluated by using these general formulas proposed in this work.

4. Discussion

Impacts of non-ideal front textures

In the above simulation, the fully-randomized front textures, regarded as ideal Lambert surface, can uniformly scatter the internal photons within all angles, on which the angle-dependent reflection obeys Cosine’s law. However, the textured surface, usually prepared by anisotropic chemical etching technique or laser processing, would probably not achieve ultimate randomness, which may lead to more scattered photons tending to the normal direction (imperfect angular scattering) or discrete several directions rather than uniformly over all angles. Thus, it is also necessary to model the effect of non-ideal front textures on these key performance parameters of MJSC. Aiming at this issue, we introduce a parameter, defined as angular concentration factor (ξ), to represent various scattering degrees on initial transmitted rays or internally reflected rays after passing through or hitting on such non-ideal front textures with various roughness, schematically shown in Fig. 7, where ξ is set as 1, 1.5, 2, 3, 4 and 5 in this work. This factor has been presented in our previous study of rear-textured MJSC [24]. Notably, for non-ideal front textures, not only the angular scattering of initial transmission or internal reflection decrease with increasing ξ, but also their intensities decrease accordingly.

 

Fig. 7. Sketch of scattering degrees on initial transmitted rays or internally reflected rays against concentration factor (ξ), resulting from various surface roughness’s of the front textures.

Download Full Size | PDF

Following this assumption, the intensity of initial transmitted ray at inflow angle θ is modeled as proportional to cos^ξθ (within θ_max). Thus, the angle-dependent initial transmittance of imperfect front textures is modified as

Meanwhile, the internal reflectance of imperfect front textures is rewritten by

When calculating the performance parameters and the C-M thickness combination for device A with non-ideal front textures, we will replace Eq. (2, 4, 6–7) with Eq. (20–23).

Figure 8 plots the effects of imperfect angular scattering on the C-M L₂ and L₃ and the corresponding photocurrent (J_CM) for device A and B, whose L₁ are fixed as 500 nm (thick cases). In these subplots, the corresponding values for device C (L₂ = 1160 nm, L₃ = 364 nm and J_CM = 14.2 mA/cm²), being independent from ξ, are regarded as reference values without light trapping. As ξ increases, the increasing C-M L₂ and L₃ in device A and B quantitatively reflect the absorption enhance effects in them reduce with decreasing scattering degrees on textured front or rear surfaces. Specifically, the C-M L₂ of A increases gradually from 869 nm to 1019 nm, while that of B increases from 446 nm to 840 nm with ξ increasing from 1 to 5. Meanwhile, the C-M L₃ of A and B increase from 52 nm to 134 nm and from 30 nm to 110 nm, respectively. There is no doubt that they are still obviously thinner than those in device C. At the same time, the J_CM of B for L₁ = 500 nm almost equals that of C, only with a tiny deviation at the ideal textured case (ξ = 1). By contrast, the J_CM of A drops from 15.1 to 14.7 mA/cm² as ξ increasing from 1 to 5, but still much higher than that of B or C. The above results indicate even that the surface scattering degrees of front textures is not ideal enough, taking ξ ≤5 for instance, it is still promising to obtain a photocurrent add-on in device A, but needs more middle- and bottom-cell materials when compared with B with the same L₁. In summary, non-ideal scattering textures would weaken the light-trapping effects in MJSCs (both A and B), which in turn requires thicker subcells to compensate for the photocurrent loss accordingly in the stack.

 

Fig. 8. (a, b) C-M L₂ and L₃, and (c) photocurrent J_CM as a function of ξ in 3J-device A, B, and C with L₁ = 500 nm.

Download Full Size | PDF

Figure 9(a-d) exhibit each subcell V_oc and their sum in the three devices with the C-M thickness combinations illustrated in Fig. 8(a-b). For A and B, the top- and middle-subcell V_oc are insensitive to ξ, while V_oc of their bottom-subcell basically in the same way increase by about 10 mV as ξ increasing from 1 to 5. At the same time, the entire V_oc in A and B increases from 3.299 V to 3.311 V and to 3.307 V, respectively. Taking the values of C as standards (V_oc1 = 1.527 V, V_oc2 = 1.080 V, V_oc3 = 0.712 V, V_oc = 3.319 V), top- and middle-subcell V_oc in A are lightly higher by less than 2 mV, but bottom-subcell V_oc is lower by 11-22 mV, resulting in 8-20 mV lower than C for the entire V_oc. Meanwhile, the entire V_oc of B is less than C by 13-20 mV when ξ is varied from 1 to 5. As a conclusion, less poor scattering degrees (ξ ≤5) by textured surface has a weak effect on the C-M V_oc, but a strong effect on the C-M photocurrent. Figure 9(e) also shows their corresponding η_sc in detailed-balance limit. As ξ increases from 1 to 5, η_sc of front textures decrease from 44.2% to 43.1%, while that of rear textures basically remain 41.7%-41.8%, primarily determined by the trend of their C-M photocurrents. The above results reveal that less poor front textures (ξ ≤5) still can boost the performance of MJSCs compared with device B and C with the same L₁. For the same thickness combinations (especially for these ultra-thin designs), the front textures will achieve the highest conversion efficiency and photocurrent, thus having great practicality when compared with the rear textures or smooth surfaces. This simple non-ideal model is universal and useful to pre-evaluate the quantitative relationship between different subcell designs and device performance for those MJSCs with various light-trapping strategies before starting the expensive experimental studies, and thus is a good tool for further development of low-cost and anti-irradiation III-V MJSCs.

 

Fig. 9. (a-c) Three subcell V_oc, (d) entire V_oc, and (e) η_3J as a function of ξ in 3J-device A, B, and C with L₁ = 500 nm.

Download Full Size | PDF

Conclusion

In this work, the general formulations on all subcell absorptance and I-V characteristics for arbitrary MJSCs with ideal non-reflective fully randomized front textures and perfect rear-reflector were established systematically, including absorption and luminescence coupling among the subcells in the stack. Such model was also applied to InGaP/GaAs/InGaAs-3J solar cell to evaluate the performance parameters and the optimal ultra-thin combination under current-matching condition. Our results clarified that to get the same benchmark of photocurrent or conversion efficiency, ideal front textures can significantly cut down all subcell thicknesses in the stack, while ideal rear textures only obviously reduce the middle- and bottom-subcell thicknesses, compared with the traditional smooth-surfaced 3J cell. Typically, (350 nm, 315 nm, 28 nm) is recommended as the lower limit of the subcell thicknesses for the front-textured 3J with an experimental efficiency of over 38%. Moreover, textured front surface, even with imperfect angular scattering (ξ≤5), is still promising and feasible for achieving ultra-thin and high-efficient MJSCs compared with rear textures or smooth-surfaced cells. This work provides theoretical guidance for the development of low-cost and anti-irradiation III-V MJSCs and can be used for the light-trapping design on other types of MJSCs, such as perovskite, organic or hybrid MJSCs.

Author Contributions

L. Z. supervised the project and established the physics models. L. Z. and Y. W. performed the calculations and wrote the paper. X. P. and H. A. joined the discussion and commented on the manuscript.