12.2.1 Physical Model

The solar‐cell structure assumed under SQ theory is simple (Figure 12.2a). In this model, a p–n homojunction cell with an infinite thickness is assumed while forcing the reflectance R of this hypothetical structure to zero (i.e. R = 0). Figure 12.2b schematically shows the absorption coefficient (α) spectra for the SQ model and a conventional material. Under SQ theory, the α spectrum of the absorber is modeled as a simple step function that shows infinite absorption at the photon energy of E ≥ E_g. Note that the experimental materials exhibit tail absorption below E_g with Urbach energies of E_U > 0 (see Figure 4.17); in contrast, E_U = 0 is assumed for SQ theory.

In SQ theory, the current density–voltage (J–V) characteristic is modeled by the following simple diode equation [12, 13]:

(12.1)

where J₀ is the reverse saturation current density (i.e. dark current) and q, k_B, and T are the electron charge, Boltzmann constant, and temperature, respectively. In Eq.  (12.1), there are only two critical variables (i.e. J_sc and J₀) and the solar‐cell parameters V_oc, FF, and η can be calculated from the J–V curve based on (J_sc, J₀). In the SQ limit, J_sc is simply estimated by assuming that all electron–hole pairs generated by a 1‐sun illumination (ϕ_1Sun) are collected in the energy region E ≥ E_g. Compared to this simple photocurrent calculation, the estimation of the dark current (J₀) is more complex.

Figure 12.3 shows a physical representation of the SQ model, which is considered for the J₀ calculation. In this approach, a p–n homojunction solar cell is placed in a cavity surrounded by a spherical blackbody held at 300 K. A blackbody is a hypothetical object that absorbs all incoming light, regardless of the wavelength and angle of incidence (i.e. R = 0). At elevated temperatures, the blackbody emits electromagnetic waves, which are known as the blackbody radiation. For example, even though graphite is almost completely black, it emits a wide spectrum of blackbody radiation when heated. In the SQ model, the p–n cell is treated as a perfect blackbody in the energy region above E_g, while it is completely transparent at E < E_g.

In Figure 12.3, the p–n cell and a spherical blackbody radiator are in thermal equilibrium. If the spherical blackbody radiator is held at 300 K, it emits photons with energies corresponding to those of infrared light (see Section 12.2.2). The generated blackbody photons are absorbed by the solar cell, creating the electron–hole pairs within the absorber. Note that the solar cell in the figure is under the dark condition because it is surrounded completely by the spherical blackbody. In a room covered completely by a thick graphite at 300 K, human eyes cannot detect any light, but blackbody radiation still occurs, generating the dark current in solar cells.

As illustrated in Figure 12.3, the photon absorption in the cell generates carriers with a current flow of J₀. If the direction of the J₀ current is reversed, radiative recombination of carriers occurs, leading to photon radiation (emission). The key assumption of the SQ model is that the rate of photon absorption is exactly equal to the rate of photon emission; therefore, within the cavity, a detailed balance is maintained by the photon exchange, resulting in a complete thermal equilibrium of the system. In other words, in this model, J₀ is ultimately considered as a “virtual current” at V = 0, where the current flow generated by photon absorption is counterbalanced by the reverse current flow due to radiative recombination with a net current being J = 0. Consequently, under the SQ approach, J₀ is estimated from the absorption of blackbody photons at 300 K. By applying the (J_sc, J₀) values obtained from the above‐described procedures, the J–V characteristics of a maximum‐efficiency cell can be evaluated using Eq.  (12.1). This is the fundamental physics of SQ theory.

12.2.2 Blackbody Radiation

Blackbody radiation is governed by a single physical parameter – temperature T. As the physical model of blackbody radiation forms the core of solar‐cell limit calculations, the derivation of the Planck's law (http://www.ioa.s.u-tokyo.ac.jp/kisohp/STAFF/nakada/Jugyo/G=Blackbody.ppt) is explained fully in this section.

For this purpose, we consider a blackbody cube with equal sides of length L (i.e. the volume is V = L³). The interior of the cube is empty, but the space inside the cube is filled with blackbody radiation, which is under thermal equilibrium with the blackbody wall maintained at T. The blackbody radiation for a wavelength of λ at T is denoted by γ_BB(λ, T). To describe γ_BB, a wavelength space for the photons within the unit volume is defined (Figure 12.4a). This wavelength space differs from the real space and is defined by two factors: λ and the moving direction of the photon (Ω). The dΩ indicates the solid angle for a traveling photon and the area of the basal plane is A = λ² dΩ. When the frustum formed by dλ is considered, its volume becomes λ² dλ dΩ. If a photon number density of ρ_n is applied, the number of photons in the frustum is given as p = ρ_nλ² dλ dΩ per unit volume. In the corresponding real space (Figure 12.4b), on the other hand, the number of photons passing through the small area dS during the time interval dt is expressed as v = c dt dS, where c is the speed of light. Accordingly, the photon number (dN) can be calculated as the product of the real‐space volume (v = c dt dS) and the number of photons per unit volume in the wavelength space (i.e. p calculated above):

(12.2)

Because the energy of a single photon is given by E = hc/λ (Einstein relation), the energy transferred by dN is dE = (hc/λ)dN. Thus,

(12.3)

where

(12.4)

To derive ρ_n, we consider the quantum state within the cube (V = L³). For a photon to exist stably within the cube, its wavelength λ in the x, y, and z directions must be exactly the same as L and thus the number of the photon states per volume V is given by

(12.5)

Here, the factor of 2 indicates that the electromagnetic wave can be in two states with different polarizations. By setting ΔV_λ = Δλ_xΔλ_yΔλ_z in Eq.  12.5, we can estimate the density of the quantum state as ρ_Q = ΔU/ΔV_λ = 2/λ⁶. On the other hand, the quantum‐mechanical consideration provides the occupation number of photons in the quantum state of λ as follows (http://www.ioa.s.u-tokyo.ac.jp/kisohp/STAFF/nakada/Jugyo/G=Blackbody.ppt):

(12.6)

As a result, ρ_n is derived as ρ_n = ρ_QΘ and thus γ_BB of Eq.  12.4 becomes

(12.7)

If hemispherical integration is considered for the blackbody radiation in Figure 12.3, γ_BB is multiplied by π (see Section 12.2.3). In this case, the density of blackbody photons (ϕ_BB) per second per meter square can be estimated by dividing γ_BB by E(=hc/λ):

(12.8)

Often, a value of ϕ_BB derived for the photon energy is employed (i.e. ϕ_BB(E, T)). From Eq.  (12.3), γ_BB(E, T) = γ_BB(λ, T)|dλ/dE|. Since |dλ/dE| = hc/E², we obtain

(12.9)

Figure 12.5a shows the variation of ϕ_BB(E, T) with T, calculated from Eq.  (12.9). The spectrum of blackbody photon density changes drastically with T and extends toward the ultraviolet region (E ∼ 3 eV) as T increases. At 300 K, the ϕ_BB spectrum is nonuniform and ϕ_BB increases in the infrared region (E ∼ 1 eV); thus, J₀ increases significantly when E_g becomes lower. The spectrum of sunlight can be approximated by setting T = 6000 K. In Figure 12.5b, the photon density produced by 1‐sun illumination (ϕ_1Sun) under the AM1.5G condition is compared with the blackbody radiation density at T = 6000 K and a close match is observed.

12.2.3 SQ Limit

Once the fundamental blackbody concept discussed in Section 12.2.2 is understood, the SQ theoretical‐limit calculation becomes straightforward. Figure 12.6a shows the general procedure established under SQ theory for calculating the maximum efficiency. In this method, the optical model is first defined and then used to calculate the external quantum efficiency (EQE). Once the EQE spectrum is known, J_sc and J₀ can be calculated from ϕ_1Sun (Figure 12.5b) and ϕ_BB (Figure 12.5a at T = 300 K), respectively, as described below. Based on (J_sc, J₀), the J–V curve can be calculated from Eq.  (12.1), from which all the other solar‐cell parameters (i.e. V_oc, FF, and η) are deduced.

Figure 12.6b shows an example of estimating J₀ at E_g = 1.4 eV under SQ theory. In the SQ model, (i) an infinite absorber thickness, (ii) a step‐function α, and (iii) R = 0 are assumed, which lead to the step‐function EQE spectrum of Q_SQ = 1 (E ≥ E_g) and Q_SQ = 0 (E < E_g). Under SQ theory, J₀ is essentially calculated as an overlap between EQE and ϕ_BB at 300 K (see Figure 12.6b). Accordingly, J₀ increases drastically as E_g decreases.

The actual calculation of J₀ is conducted by considering a hemispherical configuration (Figure 12.7a), in which the blackbody radiation onto a p–n junction solar cell is integrated over a series of solid angles θ and rotation angles φ. We consider the blackbody radiation emitted from a small ring‐like area, defined by dθ (gray area in Figure 12.7a). For a hemisphere of radius r, the total gray area on the hemisphere is dS = 2πrsin θL, where L is the latitudinal distance L = r dθ. We further consider the projection of the area dS onto a p–n junction surface with the area of dS′ (Figure 12.7b). When the radiation from dS is projected onto dS′ at an incident angle of θ, we find a simple relation dS′ = cos θ dS.

The total number of blackbody photons entering the p–n solar cell through the area dS′ is expressed as

(12.10)

where the prefactor of λ/hc(=1/E) corresponds to the conversion of radiation into photon density, as explained in Eq.  (12.8). If r = 1, we obtain dS′ = 2πsin θ cos θ dθ, which can be substituted into Eq.  12.10 to obtain

(12.11)

In the above equation, θ is integrated over the range 0°–90° and the relation of is used. As , the spherical integrations of dθ and dφ produce a value of π (i.e. 2π ⋅ 1/2) and ϕ_BB(λ) in Eq.  12.11 is reduced to that in Eq.  12.8. Finally, ϕ_BB(λ) can be integrated over λ to estimate J₀:

(12.12)

as illustrated in Figure 12.6b. Under SQ theory, a perfect blackbody solar cell is assumed and the EQE is always unity above E_g, independent of θ and φ. On the other hand, J_sc of the cell can be calculated by assuming 100% carrier collection at normal incidence:

(12.13)

From the obtained (J_0,SQ, J_sc,SQ), the solar‐cell parameters can be calculated based on Eq.  (12.1).

Figure 12.8 shows the variation of J_sc, V_oc, and FF with the absorber E_g, estimated from the above SQ approach (solid lines), together with the experimental parameters obtained from record‐efficiency solar cells (circles) reported in Refs. [3–7]. The corresponding efficiencies are shown in Figure 12.1. All SQ parameters in Figures 12.1 and 12.8 are summarized in Appendix B. The J_sc result is calculated from Eq.  12.13 using ϕ_1Sun shown in Figure 12.5b and J_sc decreases as E_g becomes wider. The V_oc of the SQ model is calculated directly from V_oc = k_B/q[ln(J_sc,SQ/J_0,SQ) + 1]. Unlike J_sc, V_oc increases with E_g, primarily because J₀ decreases logarithmically at higher E_g (see Figure 12.6b). The dotted line shows V_oc derived directly from E_g (i.e. V_oc = E_g/q). The V_oc obtained under SQ theory is smaller than E_g/q by 0.2–0.3 V, while FF remains essentially constant. As a result, the conversion efficiency calculated under SQ theory reaches a maximum at E_g = 1.34 eV.

In Figure 12.8, the SQ calculation result reproduces the overall experimental trends well. However, all the experimental solar cells show lower values compared with the SQ results. The reduction from the theoretical J_sc can be attributed to (i) the unfavorable light absorption occurring in transparent conductive oxide (TCO) and doped layers of the cells and (ii) the nonzero light reflection in the experimental solar cells, which is neglected completely in the SQ model. The lower V_oc in the experimental cells originates from (i) the tail absorption (E_U > 0), which reduces the effective E_g and V_oc, and (ii) the effect of nonradiative recombination (Section 11.4.2). The reduction of FF in the experimental solar cells can be understood by finite series and shunt resistances as well as the increase in the nonideal diode factor, which is neglected in Eq.  (12.1), as explained in Section 11.4.2.

12.3.1 Concept of Thin‐Film Limit

For thin‐film efficiency calculation, a rigorous approach that incorporates all physical and optical aspects of real‐world solar cell has been developed by extending SQ theory [1]. In this scheme, a theoretical efficiency limit for a perfectly realizable thin‐film structure in Figure 12.9a is estimated by adopting the true absorption characteristics of the light absorber and constituent layers. In this thin‐film concept, a general layer‐stacked structure of TCO (70 nm)/absorber (p–n layers: 1 μm)/metal (Ag) is considered. To maximize the light absorption of the thin‐film solar cell, the entire cell structure is assumed to be textured. A dual antireflection coating of MgF₂/Al₂O₃ is further incorporated to minimize the front light reflection. In addition, a front metal‐grid electrode with a shadow loss of 5% is assumed.

In a thin‐film solar cell with a limited absorber thickness, strong optical confinement is crucial for increasing J_sc. In the optical calculation of the structure shown in Figure 12.9a, all optical confinement effects due to texture, antireflection coating, and backside reflection are fully incorporated. Moreover, the light reflection of the solar cell is not assumed to be zero (R ≠ 0) and thus a non‐blackbody cell is considered. For solar cells, the presence of TCO is generally problematic because the free carriers (electrons) in the TCO exhibit strong light absorption, which reduces J_sc rather significantly [15–19]. In the thin‐film model, to suppress unfavorable parasitic absorption in the TCO, a high‐mobility TCO layer (H‐doped In₂O₃ [20, 21]) with a mobility of μ = 100 cm²/(V s) and a low carrier concentration of 2 × 10²⁰ cm⁻³ is assumed. The use of high‐mobility TCO is particularly important in obtaining a sufficiently low TCO series resistance at low carrier concentrations [20]. The carrier concentration in commercial SnO₂:F substrates (TEC substrates) is 5 × 10²⁰ cm⁻³ (μ = 30 cm²/(V s)) [20, 22] and the free carrier absorption is substantially stronger. The optical constants of hybrid perovskites can be found in Appendix A and those of the constituent layers can be obtained from Refs. [1, 15].

To estimate the J₀ of a solar cell with the structure of Figure 12.9a, the solar cell is placed in a spherical cavity surrounded by a blackbody radiator (Figure 12.9b). For a non‐blackbody thin‐film cell (R ≠ 0), the optical response is incident‐angle‐dependent and the optical calculation must be carried out by considering the p‐ and s‐polarized waves (see Figure 4.11a for definitions of these polarizations). Specifically, the incident‐angle‐dependent EQE spectrum (i.e. Q(λ, θ)) is estimated as a simple average of the EQE spectra calculated for the p‐polarization [Q_p(λ, θ)] and s‐polarization [Q_s(λ, θ)]:

(12.14)

In the configuration of Figure 12.9b, ϕ_BB(λ) in Eq.  (12.10) is expressed using the θ‐dependent function ϕ_BB(λ, θ). Over a small area dθ dφ, we obtain ϕ_BB(λ) = ∫∫ϕ_BB(λ, θ)dθ dφ. By inserting this equation and Eq.  (12.14) into Eq.  (12.12), the blackbody radiation from all the internal surface of the cavity can be integrated using the following exact formula:

(12.15)

where S is the shadow loss of the front metal‐grid electrode (S = 0.05). On the other hand, the J_sc of the solar cell is calculated using the standard equation of

(12.16)

where Q_θ=0(λ) is the EQE spectrum at θ = 0°. From J_sc and J₀ above, the thin‐film solar‐cell limit can be obtained using Eq.  (12.1).

12.3.2 EQE Calculation Method

The EQE spectra Q_p(λ, θ) and Q_s(λ, θ) in Eq.  (12.15) can be calculated by applying the antireflection condition (ARC) approach [15–17]. This scheme allows for the incorporation of all possible optical effects generated in textured solar cells. Figure 12.10a shows an example of calculating EQE using the ARC method for a MAPbI₃ absorber at θ = 0°. In the ARC method, the R spectrum is first calculated by considering a flat multilayer structure using a conventional optical admittance method [15, 16, 23]. The result obtained assuming this flat structure is indicated as R_flat in Figure 12.10a. Next, the minimum R points in the R_flat spectrum (open circles) are selected and connected linearly. In the ARC method, the resulting spectrum represents the R spectrum of a textured structure (R_tex) that incorporates the widely observed antireflection effect in random textured structures. By adopting this R_tex spectrum, the EQE spectrum of a random texture can be approximated based on the optical calculation using a flat optical model [15–17]. This simple approach has been applied successfully in reproducing the optical response of numerous submicron‐textured solar cells, including hybrid perovskite [16], CIGSe [17], CZTSSe [16], and CdTe [16]. A free Windows‐based software has been released to calculate EQE based on the ARC method (see Figure 11.13) [24].

Figure 12.10b shows an ARC‐calculated EQE spectrum plotted on a logarithmic scale. In the visible region, the EQE is almost 100% but is reduced in the ultraviolet region (λ ∼ 300 nm) due to parasitic absorption by the TCO layer. The figure also shows ϕ_BB at 300 K, indicating the drastic increase in this factor in the longer λ region. In Figure 12.10c, the λ‐dependent J₀ contribution (J_0,λ) calculated from J_0,λ = (1 − S)qQ_θ=0(λ)ϕ_BB(λ) is shown. Note that J₀ is determined by integrating the J_0,λ spectrum. The result in the figure illustrates the importance of the near‐band‐edge EQE in determining J₀. In an absorber with extensive tail absorption, the EQE spectrum is extended toward the longer λ region, which in turn increases J₀ and reduces V_oc significantly [1].

In the above example, θ = 0° is assumed and J₀ is calculated as

(12.17)

This simple approach has been employed widely in the estimation of approximate J₀ values [25–27]. Importantly, it has been confirmed that the J₀ values obtained using the detailed integration of Eq.  (12.15) and the simplified calculation of Eq.  (12.17) fall into similar ranges (Figure 12.11) [1]. Accordingly, although an exact calculation using Eq.  (12.15) is preferable, Eq.  (12.17) can still be adopted to estimate an approximate value.

12.3.3 Maximum Efficiencies of Single Solar Cells

Figure 12.12a shows thin‐film efficiency limits obtained from the strict J₀ integration performed by Eq.  12.15 assuming 1‐μm absorber thickness [1]. Table 12.1 lists the numerical solar‐cell parameters derived from these calculations [1]. The calculation results show that the maximum thin‐film efficiencies are ≤31%, which are lower than the SQ limits by ∼3%. In particular, the realistic maximum efficiencies of MAPbI₃ and FAPbI₃ solar cells are 28%, while the corresponding SQ limits are 30–31%. As shown in Table 12.1, the estimated maximum V_oc values of MAPbI₃ and FAPbI₃ are 1.32 and 1.28 V, respectively. The V_oc limit can also be determined empirically if the experimental EQE spectrum is applied using Eq.  12.17. The maximum V_oc of MAPbI₃ obtained using this semi‐empirical procedure is 1.32–1.33 V [25–27], which is consistent with the result of Table 12.1. In the above thin‐film limit, the series and shunt resistances are not considered and the obtained maximum FF is much higher (∼0.9) than the experimental values ( Figure 11.18).

In Figure 12.12a, the thin‐film limits for CZTS, CZTSe, and CZTSSe are significantly lower than the SQ limits because of the extensive tail absorption in these compounds [28, 29], which lowers the V_oc drastically [1]. In fact, the E_U values of these materials are quite high (∼80 meV), whereas hybrid perovskites show quite sharp absorption edges with E_U ∼ 15 meV (Figure 4.18). Accordingly, the maximum V_oc values attainable for hybrid perovskites are sufficiently high.

One limitation of the above thin‐film limit estimation is that it assumes a substrate‐type solar cell as a basic structure, while superstrate‐type (i.e. glass substrate type) solar cells are typically adopted for hybrid perovskite single cells ( Figure 11.2). In conventional hybrid perovskite single cells, the shadow loss arising from a front grid electrode can be neglected and J_sc could be potentially higher than those listed in Table 12.1. In superstrate perovskite cells, however, the front TCO layer is thicker and the free carrier absorption in the TCO reduces J_sc. To estimate the maximum J_sc, therefore, more exact optical modeling is necessary. Note that, for hybrid perovskite cells, an almost 100% carrier collection efficiency has been confirmed experimentally under the short‐circuit condition ( Figure 11.11).

Solar cell	(mA/cm2)	(V)	FFmax	Thin‐film limit (%)	SQ limit (%)
FAPbI3	24.5	1.286	0.903	28.4	31.4
MAPbI3	23.6	1.320	0.905	28.2	30.4
GaAs	30.7	1.121	0.892	30.7	33.1
Sia)	39.4	0.861	0.868	29.4	33.3
CIGSe	39.9	0.876	0.870	30.4	33.5
CdTe	28.6	1.189	0.897	30.5	32.2
CZTSSe	45.5	0.723	0.850	27.9	33.5

The maximum J_sc (), V_oc (), and FF (FF^max) in this table correspond to those of the thin‐film limits indicated by the closed circles in Figure 12.12a. The optical model of Figure 12.9a is adopted for the thin‐film limit calculations. For the J_sc estimation, a metal‐grid electrode with a shadow loss of 5% is assumed.

^a)Si absorber thickness is assumed to be 150 μm.

It should be emphasized that, for the SQ efficiency, there is always an ambiguity for the correct choice of material E_g, as experimental E_g changes with the characterization method and definition (see Table 4.1). In contrast, in the derivation of the thin‐film limit, E_g is not required and the theoretical efficiency is obtained directly from the thin‐film absorption characteristics (i.e. the EQE spectrum). Accordingly, for evaluating the absolute performance limit of a solar cell, the thin‐film limit is more appropriate.

Figure 12.12b shows the variation of the MAPbI₃ and GaAs thin‐film limits with absorber thickness [1]. In both cases, the maximum efficiencies saturate at >0.5 μm. Thus, when the optical confinement is strong, an absorber thickness of 0.5 μm is sufficient. This optimum absorber thickness is consistent with experimental absorber thicknesses adopted typically for the perovskite single cells [4, 5].

12.3.4 Performance‐Limiting Factors of Hybrid Perovskite Devices

The potential efficiency calculations for a thin‐film configuration further enable the evaluation of the performance‐limiting factors in record‐efficiency photovoltaic devices. Figure 12.13 compares the calculated maximum efficiencies under the thin‐film limit (η_TF) with the experimental efficiencies of world record cells (η_ex) reported in Refs. [3, 4, 6, 7, 30]. To perform this calculation, the solar‐cell structure of Figure 12.9a is adopted but its absorber thickness is adjusted to match those of the experimental devices [1]. In Figure 12.13, the difference between the theoretical and experimental efficiencies is further categorized as the efficiency reduction Δη caused by the V_oc deficit (Δη^Voc), J_sc deficit (Δη^Jsc), and FF deficit (Δη^FF) relative to the calculated maximum V_oc, J_sc, and FF values, respectively. The efficiencies of the hybrid perovskite cells employed in the analysis are 23.7% and 21.2% for FAMAPb(I,Br)₃ [30] and MAPbI₃ cells [4], respectively. The calculation of the FAMAPb(I,Br)₃ cell is performed assuming a pure FAPbI₃ phase, as the MA and Br contents are very small [31]. In FAPbI₃‐based solar cells, however, higher record efficiencies of ∼25% have been reported [3, 5] and, for these cells, the efficiency deficits are more suppressed.

One of the remarkable features of hybrid perovskite solar cells is the negligible J_sc loss due to the very small optical losses in the devices (Section 11.4.2) [16, 18]. In other words, the record‐efficiency perovskite cells are limited primarily by V_oc and FF losses. Note that the rapid increase in the perovskite cell efficiency observed over the past decade is caused mostly by the improvements of the V_oc and FF (see Figure 1.3). The V_oc of the current record‐efficiency perovskite cells ranges 1.12–1.18 V [3–5, 30], which are lower than the theoretical limits by ΔV_oc = 100–200 mV. The relatively large ΔV_oc can be interpreted mainly by the presence of nonradiative recombination at the cell interfaces ( Figure 11.14). However, a quite high V_oc of 1.26 V, with ΔV_oc of only 60 mV, has been obtained in a V_oc‐optimized MAPbI₃ p‐i‐n cell [32], although the J_sc of this cell is limited by the parasitic light absorption in the front PTAA layer (see also Figure 11.3).

In contrast, the FF values of the record‐efficiency cells (80–85%) [3–5, 30] are notably lower than their theoretical counterparts (90%). The reduced FF in the experimental cells can be explained by the presence of cell resistance and deterioration of the diode factor ( Figure 11.12). Interestingly, FF shows a clear correction with the grain size of the perovskite layers, while V_oc is independent of the grain size (see Figure 11.18). Consequently, the FF loss in the record‐efficiency cells can be primarily interpreted by the small grain size (∼1 μm) of the polycrystalline absorber.

As shown in Figure 12.13, among all experimental solar cells, the GaAs cell exhibits the highest conversion efficiency [3]. The overall efficiency loss in the GaAs cell is quite small and its experimental efficiency (29.1%) is very close to the thin‐film limit (30.9%). Remarkably, for the GaAs cell, ΔV_oc ∼ 0 and ΔFF is only 2%; in particular, this cell is simply limited by the parasitic light absorption of the front layers [1]. In Si solar cells, in contrast, the V_oc deficit is quite large due to the significant Auger recombination in the indirect‐transition Si, while the defect‐induced nonradiative recombination plays only a minor role [33, 34].

The results of Figure 12.13 show clearly that the performance of all thin‐film solar cells is essentially limited by V_oc and FF losses [1]. In general, V_oc can be related to FF [35] and a V_oc loss tends to increase with a FF loss. The Δη^Voc and Δη^FF increase notably in polycrystalline absorbers (i.e. hybrid perovskite, CIGSe, CdTe, and CZT(S)Se) and the formation of larger polycrystalline grains is important in MAPbI₃ (see Figure 11.18), CIGSe [36], CdTe [37], and CZTSSe [38] solar cells. Note that all of the chalcogenide‐based solar cells (CIGSe and CZTSSe) have rather large values of Δη^Jsc due to the strong parasitic absorption in the Mo back electrode [15–17]. In hybrid perovskite cells, the incorporation of high‐reflectivity Ag and Au rear electrodes has been a great advantage in improving J_sc.

12.4.1 Optical Model and Assumptions

Figure 12.14a shows an optical model assumed for a two‐terminal monolithic CsFAPb(I,Br)₃/Si tandem device with a fully textured structure. To derive realistic efficiency limits for the tandem cells, the multilayer structure has been constructed based on experimental perovskite tandem devices [8–10]. In the structure, for a CsFAPb(I,Br)₃ top cell, the electron transport layer (ETL) of C₆₀ and the hole transport layer (HTL) of spiro‐OMeTAD are adopted. The LiF layer acts as a shunt‐blocking layer [9], with the LiF also exhibiting a passivation effect ( Table 11.2) [40]. On the C₆₀ ETL, a thin SnO₂ layer is incorporated to reduce the sputtering damage that occurs during TCO (InZnO: IZO) deposition. For the top cell, a single antireflection layer (MgF₂) is adopted.

The perovskite top cell is formed on an n‐i‐p type Si heterojunction solar cell with a Si wafer thickness of 260 μm [8]. In the bottom cell, the Si wafer is sandwiched between hydrogenated amorphous silicon (a‐Si:H) n‐i front and i‐p rear layers [41–43]. To construct the tandem cell, the top and bottom cells are combined using recombination junction layers of nano‐crystalline Si (nc‐Si:H) [8, 10]. The optical constants of all layers in the model can be found in Refs. [15, 39, 44, 45].

The J_sc of two‐terminal tandem cells is maximized when the J_sc values of the top and bottom cells are equal (i.e. the current matching condition). In a perovskite/Si tandem cell, the E_g of the Si bottom cell is fixed (i.e. E_g,bottom = 1.11 eV [20]), leaving the E_g and thickness of the CsFAPb(I,Br)₃ layer as the critical parameters of the tandem‐cell design. In this section, we calculate the maximum potential efficiencies using both the true perovskite absorption spectrum and a simplified step‐function spectrum and discuss how the results differ.

Figure 12.14b shows the absorption coefficient (α) spectrum of experimental CsFAPb(I,Br)₃ with E_g = 1.65 eV [39], together with a step‐function α spectrum assumed for simplified calculation. Experimentally, the band‐edge α of CsFAPb(I,Br)₃ is 2 × 10⁴ cm⁻¹ (see also Figure 4.14) and the calculation of the light penetration depth (d_p = 1/α = 500 nm) immediately gives a rough optimum thickness of the CsFAPb(I,Br)₃ top cell. For the step‐function α, a constant α of 10⁵ cm⁻¹ is assumed at E ≥ E_g.

Note that, when experimental optical properties of hybrid perovskites are modeled by the Tauc–Lorentz model (see Figure 4.4 and Appendix A), the optical functions of their alloys can be approximated by simply shifting the corresponding peak energy positions (Section 4.5.2). Figure 12.14c shows the α spectra of the CsFAPb(I,Br)₃ alloys obtained from these energy shift calculations, which show that the α spectrum slide horizontally along the energy scale as E_g changes, a result that has been confirmed experimentally (see Figure 4.15).

12.4.2 Calculation of Tandem‐Cell EQE Spectra

For fully pyramid‐textured solar cells, the optical confinement effect needs to be evaluated accurately. Although a ray‐tracing method is frequently applied in the optical calculation of Si textures [46], the computational cost of this method is relatively high and it is relatively difficult to accurately estimate the light absorption within the thin top cell [47]. To overcome these problems, a double‐reflection model (DRM), shown in Figure 12.15, has been developed [48]. The essential concept underlying the DRM is to represent the optical absorption in a multilayer structure formed on a Si texture by assuming two flat optical models with different incident angles of θ = 50° (Model1) and 30° (Model2), as shown in Figure 12.15a. These incident angles are derived simply from the top angle of the texture (80°) [49, 50] and the overall optical response is expressed by combining the calculation results of Model1 and Model2 (Figure 12.15b). Under the DRM, the light scattering effect is also considered by introducing an adjustable parameter (optical confinement factor). The EQE spectra of Si single cells calculated using the DRM show excellent agreement with the experimental EQE spectra [48].

Figure 12.16 shows the DRM‐calculated EQE spectra of the CsFAPb(I,Br)₃ top and Si bottom cells of the tandem structure in Figure 12.14a (solid lines), together with experimental EQE spectra reported for a 25.2% efficiency cell (open circles) [8]. The calculation results further show the variation of the EQE spectrum with the CsFAPb(I,Br)₃ top‐cell thickness. To perform the EQE calculation, the perovskite α spectrum with E_g = 1.59 eV (i.e. −0.06 eV shift in Figure 12.14c) is employed. With increasing the perovskite absorber thickness, the EQE response of the top cell becomes larger, while that of the bottom cell decreases. The calculated EQE spectrum is in excellent agreement when the perovskite thickness is d = 600 nm.

12.4.3 Maximum Efficiencies of Tandem Devices

Once the EQE spectra of the top and bottom cells are calculated, their corresponding (J₀, J_sc) values can be estimated separately using Eqs.  (12.16) and (12.17). By applying a simple diode equation (i.e. Eq.  (12.1)), we can further calculate the J–V curve for each cell. From these J–V curves, the J–V curve of the tandem cell is determined [48]:

(12.18)

where V_top(J) and V_bottom(J) are the inverse functions of J_top(V) and J_bottom(V) calculated for the top and bottom cells, respectively. In estimating V(J) using Eq.  (12.18), the current matching condition is also considered; namely, the J_sc of the tandem cell is limited by either the top cell or the bottom cell that shows lower J_sc. From the J(V) obtained from Eq.  (12.18), the theoretical solar‐cell parameters and conversion efficiency of the tandem cell are obtained.

Figure 12.17 shows the mapped results of the CsFAPb(I,Br)₃/Si tandem‐cell efficiency calculated by varying the top‐cell E_g and layer thickness in the cell structure of Figure 12.14a. For these calculations, the shadow loss of the front metal‐grid electrode is assumed to be zero (i.e. S = 0). In Figure 12.17a, the simulation results obtained using the true CsFAPb(I,Br)₃ optical properties are shown, whereas the calculation results derived assuming the step‐function α are indicated in Figure 12.17b. The optical properties of the top cell absorber modify the maximum cell efficiency rather significantly; specifically, when an exact perovskite α is used, the optimum E_g varies with the top‐cell absorber thickness (d) due to limited light absorption. In particular, when the top cell is thin, the J_sc of the tandem cell is limited by the top cell and, with decreasing d from 1000 to 400 nm, the optimum top‐cell E_g is reduced from E_g = 1.63 to 1.53 eV in an effort to increase the top‐cell light absorption. In contrast, when a step‐function α is assumed, the light absorption in the top cell becomes unrealistically high, pushing the optimum E_g up to 1.7 eV, essentially independent of the top‐cell absorber thickness.

The open circles in Figure 12.17a,b show the highest efficiencies obtained in each simulation with d ≤ 1000 nm. Although the maximum efficiencies of these calculations are similar (η ∼ 40%), the optimum E_g differs (i.e. E_g = 1.63 eV for the true perovskite α and E_g = 1.69 eV for the step‐function α). The EQE spectra obtained for these optimum conditions are shown in Figure 12.17c, which confirms that the EQE of the top cell becomes much higher when the step‐function α is considered due to high α particularly in the band‐edge region (see Figure 12.14b).

For the experimental CsFAPb(I,Br)₃/Si tandem cell, the optimum top‐cell E_g is 1.60 eV [8], with the effective thickness of 600 nm analyzed in Figure 12.16. This experimentally determined optimum E_g is indeed the best value at d = 600 nm in Figure 12.17a and thus the calculation result is consistent with the experimental result. In previous simple optical simulations [51–53], a much wider optimum E_g of ∼1.73 eV has been reported. It should be emphasized that V_oc loss becomes larger in wide‐gap perovskite cells with E_g > 1.65 eV (see Figure 1.10) and a low‐E_g perovskite top cell is advantageous in realizing higher efficiencies. Accordingly, the detailed calculation is vital to accurately establish the best tandem‐cell design.

In the above analysis, the Si wafer thickness is fixed at 260 μm. When this thickness is varied, however, the optimum top‐cell E_g also varies. Figure 12.18 shows the optimum E_g of the perovskite top cell as a function of Si wafer thickness. In Figure 12.18, several results calculated for different top‐layer thicknesses in a range of 400–1000 nm are shown. If the Si wafer becomes thin, the J_sc of the bottom cell decreases and thus the E_g of the top cell widens to establish the current matching for the lower J_sc. In contrast, when the top cell is thick, there is sufficient light absorption and the E_g of the top cell increases so that the V_oc gain becomes larger. In fact, for a very thick perovskite top cell (∼1 μm), a wide‐gap perovskite with E_g = 1.68 eV has been adopted [54]. As shown above, the tandem‐cell design varies with the numerous parameters and device simulation is beneficial for efficient optimization of the solar‐cell architecture.

12.4.4 Realistic Maximum Efficiency of Tandem Cell

The efficiency limit obtained for the CsFAPb(I,Br)₃/Si tandem cell is ∼40% (Figure 12.17). In this calculation, however, the critical influences of cell series resistance, shadow loss of the front metal‐grid electrode, and Auger recombination in the Si absorber [33, 34] are neglected. In this section, we calculate more realistic potential efficiency by accounting for these effects.

Table 12.2 summarizes the V_oc, J_sc, FF, and conversion efficiency of a CsFAPb(I,Br)₃/Si tandem cell obtained under different models (assumptions). The calculation result of Model1 shows the maximum efficiency indicted by the open circle in Figure 12.17a (i.e. η = 39.55%). For this tandem cell, the V_oc of the top cell is 1.38 V, while that of the bottom cell is 0.82 V. However, this bottom‐cell V_oc is unrealistically high because it is calculated without considering the strong Auger recombination in the indirect Si absorber. By applying the J–V analysis method shown in Figure 11.12, the increase in J₀ in the Si bottom cell due to the Auger recombination was determined based on experimental J–V characteristics reported in Ref. [41]. In particular, the J₀ of the Si bottom cell can be expressed as J₀ = J_0,rad + J_0,Aug, where J_0,rad indicates the radiative recombination calculated from Eq.  12.17 and J_0,Aug is the Auger recombination contribution determined experimentally. The actual values are J_0,rad = 3.1 × 10⁻¹³ mA/cm² and J_0,Aug = 1.3 × 10⁻¹¹ mA/cm², indicating that the J₀ of the Si bottom cell is governed by J_0,Aug. The calculation result obtained by incorporating Auger recombination in the Si cell is shown as Model2 in Table 12.2. In this case, the V_oc of the bottom cell decreases to a realistic value of 0.725 V. As a result of the Si Auger recombination, the tandem efficiency is reduced by ∼2%.

In Model3, the series resistance of the Si bottom cell (R_s,bottom) is further considered by applying Eq. (11.3). For this calculation, R_s,bottom = 0.44 Ωcm² extracted from an experimental Si heterojunction cell [41] is applied. It can be confirmed that R_s,bottom does not affect the overall result notably. In Model4, the effect of R_s in the perovskite top cell (R_s,top) has been incorporated by assuming R_s,top = 1.8 Ωcm² obtained from the analysis of Figure 11.12. In this case, the FF decreases slightly, reducing the efficiency to 36.77%. In the final model (Model5), the shadow loss caused by the front metal‐grid electrode is considered using S = 0.023, as adopted in the experimental perovskite/Si tandem cell [8]. This model results in η = 35.93% with J_sc of 19.6 mA/cm². In Figure 12.19, the J–V characteristics calculated from Model5 are shown. As indicated by this result, a high efficiency of ∼36% can be achieved realistically in a hybrid perovskite/Si tandem cell with the practical structure of Figure 12.14a. To realize such a high efficiency, it is necessary to suppress the V_oc and FF losses in the perovskite top cell, as indicated in Figure 12.13.