4.2.1 Visible/UV Region

Figure 4.4 shows the dielectric function (ε = ε₁ − iε₂), (n, k) spectra, and α spectrum of MAPbI₃ in the visible/UV region [1]. The determination of accurate optical data for hybrid perovskites has been challenging because of the (i) extensive roughness, which corresponds to λ of the probe light (∼500 nm) [10, 60] and (ii) quite fast degradation that occurs by humid air [61]. To avoid these serious problems, the optical spectra of Figure 4.4 were obtained from an ultra‐smooth MAPbI₃ layer with a roughness of 5 nm and its measurement was performed in a N₂‐globe bag without exposing the sample to air at all [1].

In Figure 4.4, it can be seen that the spectra shapes of ε₁ and ε₂ resemble those of n and k, respectively, and the α spectrum is calculated from k by Eq.  4.2. The onsets of ε₂, k, and α spectra essentially correspond to the E_g of MAPbI₃ (1.61 eV). In the high energy region of MAPbI₃, there exist clear absorption peaks at 2.53, 3.24, and 5.65 eV, which are labeled as the E₁, E₂, and E₃ optical transitions, respectively. Note that a small peak at ∼4.5 eV is caused by the presence of PbI₂ at the surface and interface. This PbI₂ peak becomes stronger with the perovskite degradation that occurs near the surface region [61]. The thin lines for (ε₁, ε₂) represent the results of the parameterization using the Tauc–Lorentz model (Appendix A). In this scheme, the ε₂ spectrum is expressed as the sum of all the Tauc–Lorentz peaks (see Appendix A for more details).

To understand the light absorption phenomena in hybrid perovskites, quite extensive density functional theory (DFT) analyses have been performed [1–3, 17, 38, 42]. Nevertheless, the DFT calculation results vary significantly depending on the calculation (or approximation) method (Section 5.4). It has been found, however, that DFT optical spectra calculated based on a rather simple approximation (generalized gradient approximation within the Perdew–Burke–Ernzerhof, i.e. PBE scheme) demonstrate remarkable agreement with the experimental spectra (Figure 5.8), enabling us to perform detailed optical transition analyses.

Figure 4.5 summarizes the result of the optical transition analysis performed for MAPbI₃ by DFT (PBE): (a) band structure and (b) charge density profiles near the conduction band minimum (CBM) and valence band maximum (VBM) [1]. Importantly, the light absorption of MAPbI₃ in the visible/UV region can be expressed by semiconductor‐type optical transitions (i.e. the electron transition from the valence band to the conduction band). Moreover, the E₀, E₁, and E₂ transition peaks observed experimentally in Figure 4.4 are consistent with the direct optical transitions at high symmetry points at R, M, and X points in the Brillouin zone, respectively (see arrows in Figure 4.5). For the E₂ peak, however, there are three transition components labeled as E_2,a–c. The above assignment of the optical transitions was determined by calculating the ε₂ spectral contributions for each pair of the valence and conduction bands [1, 5].

The band structure of Figure 4.5 clearly shows the direct‐transition‐type band structure; the VBM and CBM are positioned at the same symmetry point of R. When the spin–orbit coupling (SOC) effect is considered in the DFT calculations, however, the conduction band splits into two bands (Rashba splitting) and the band structure becomes slightly indirect (see Figure 5.8). Nevertheless, the DFT optical spectra calculated by incorporating the Rashba splitting effect disagree with the experimental spectra; in particular, (i) the light absorption strength is seriously underestimated and (ii) the absorption tail broadens excessively when the Rashba splitting is considered (Figures 5.8 and 5.9). Because the DFT results change significantly depending on the approximation method, the DFT results must be justified according to the experimental results, particularly for hybrid perovskites that include the dynamic motion of organic cations. Consequently, the direct‐gap formation in MAPbI₃ is quite appropriate to interpret the experimental results correctly (see Section 5.4.2 for further details).

As shown in Figure 4.5b, the CBM of MAPbI₃ consists of Pb 6p orbital, whereas the VBM is composed of Pb 6s and I 5p states (Figure 5.11a). Since the CBM and VBM of MAPbI₃ are anti‐bonding states (Figure 5.12), the charge profiles of Pb 6s, Pb 6p, and I 5p are separated and localized near the Pb and I atoms. The light absorption in semiconductors can be understood simply from the overlap of the electronic charge in the valence and conduction states [62]. In MAPbI₃, the Pb 6s charge in the VBM overlaps with the Pb 6p charge in the CBM and thus the light absorption at the VBM → CBM occurs by Pb 6s → 6p transition [62]. In the energy region slightly above CBM, the dispersive I charge is present [1] and its contribution becomes important. Importantly, there are no electronic charges (or states) of MA⁺ near CBM and VBM (Figure 5.11a); thus, the direct light absorption by MA⁺ is negligible in the visible/UV region. In other words, the visible/UV light absorption in hybrid perovskites occurs only in the inorganic framework (i.e. PbI₃⁻ scaffold in APbI₃).

4.2.2 IR Region

In hybrid perovskites, weak light absorption occurs in the IR region due to the dipole moments created by the vibration of the organic cations. Figure 4.6 shows the ε₂ spectrum of MAPbI₃ obtained in the IR region [55]. In MAPbI₃, the main IR peaks originate from vibrational modes: bending (∼1550 cm⁻¹) and stretching (∼3150 cm⁻¹) modes. As discussed by Glaser et al. [55], the peak position of the N–H stretching mode is sensitive to hydrogen bonding and the shift of the N–H stretching peak toward lower energy indicates the formation of the strong hydrogen bonding in MAPbI₃ (see also Figure 3.15).

The ε₂ peaks in the IR region have a Lorentz shape, and the dielectric function can be expressed as a sum of a Lorentz model peak:

(4.6)

where A_j, ω_0,j, and γ_j represent the amplitude, peak frequency, and broadening parameters of the jth peak, respectively. The peak positions (i.e. ω_0,j) obtained from the above analysis are summarized in Table 4.1. All the IR peak positions are essentially independent of the X‐site halide anion, although the IR peak shifts slightly toward a higher wavenumber in the order of I → Br → Cl [55].

Peak position (cm−1)	Vibrational mode
911	CH3– rocking
960	C–N stretching
1250	CH3– rocking
1469	bending (symmetric)
1577	bending (asymmetric)
3132	stretch (symmetric)
3179	stretch (asymmetric)

The result of the peak assignment is also shown.

Source: Modified from Glaser et al. [55].

4.2.3 THz Region

One of the important features of hybrid perovskite optical properties is the quite strong optical phonon absorption by the inorganic framework. Figure 4.7 shows the ε₂ and Im(−1/ε) spectra of MAPbI₃ [56] and GaAs [59]. The ε₂ peaks show transverse optical (TO) phonon modes, where Im(−1/ε) is called the loss function, representing the longitudinal optical (LO) phonon modes. The TO and LO phonon peaks can be described by the Lorentz line shape shown in Eq.  4.6. Based on a DFT study, the strong TO peak at ω_TO = 63 cm⁻¹ has been assigned to a Pb–I stretching‐like vibration, whereas the peak of ω_TO = 32 cm⁻¹ is due to a Pb–I–Pb rocking‐like vibration [56]. The peak positions of the two LO phonon modes are 40 and 133 cm⁻¹. The slight change in the phonon frequencies has been observed in MAPbBr₃ and MAPbCl₃ by the effect of ionic masses [56].

In Figure 4.7, the TO and LO absorption peaks of MAPbI₃ are significantly broader, compared with GaAs. In fact, the peak half width of ω_LO = 133 cm⁻¹ (25 cm⁻¹) is 10 times greater than that of the GaAs LO peak (2.4 cm⁻¹). The extraordinary phonon peak broadening observed in MAPbI₃ has been attributed to the dynamic disorder of PbI₆ octrahedra [56]. In other words, the “soft and flexible” inorganic cage (see Table 6.1) allows unusually strong phonon absorption. The enhanced phonon absorption in MAPbI₃ can be evidenced from the considerably larger integrated peak area in MAPbI₃, compared with GaAs.

The very large ε_s of MAPbI₃, shown in Figures 4.2 and 4.3, can also be understood from the phonon absorption spectra. In general, ε_s is estimated by a simple theory known as the Lyddane–Sachs–Teller relation [59]:

(4.7)

In non‐polar materials (Si, for example), ω_LO = ω_TO; thus, we obtain ε_s = ε_∞ from Eq.  4.7. In MAPbI₃, the peak separation between ω_TO and ω_LO is considerably greater than that in GaAs, leading to a remarkably large ε_s, which has been estimated to be ε_s = 33.5 from Eq.  4.7 [56].

Important points in MAPbI₃ phonon absorption include the (i) low energy peak position of ω_LO and (ii) strong LO‐phonon absorption that leads to ε_s ≫ ε_∞. Carriers in semiconductors interact with LO phonons (phonon coupling), and the small ω_LO and ε_s ≫ ε_∞ observed in MAPbI₃ are indicative of the carriers being scattered by LO‐phonon vibration (see Eq. (6.6)). In fact, it has been proposed that the low mobility of ∼100 cm²/(V s) observed in MAPbI₃ (see Figure 6.4) originates from the strong LO phonon scattering, although different interpretations exist (Section 6.3.4).

4.3.1 Band Gap Analysis of MAPbI₃

Table 4.2 summarizes various E_g values of MAPbI₃ reported using different measurement and analysis techniques [1, 10, 12][18–27]. The reported E_g depends strongly on the measurement technique, independent of the sample forms (i.e. thin film or single crystal). In high‐precision ellipsometry characterization, consistent values of 1.60–1.61 eV have been obtained [1, 8, 10, 12, 18]. The transmission (T) measurements result in slightly reduced E_g values (1.57–1.61 eV) [20–23]. Note that E_g can be underestimated seriously, particularly when the baseline of the α spectrum is elevated by roughness scattering (Section 4.4.2). In the reflectance (R) measurements, α spectrum is deduced using a relation known as the Kubelka–Munk function: α ∝ F(R) = (1 − R)²/(2R). The E_g values obtained from this scheme are, however, low (1.50–1.52 eV) [25–27].

The E_g of MAPbI₃ is often represented by 1.55 eV based on the work of Lee et al. [24]. This E_g is deduced as the onset of the external quantum efficiency (EQE) spectrum at λ ∼ 800 nm. However, the E_g obtained from this indirect approach reflects the absorption onset, which shifts toward a reduced energy by tail absorption. The EQE spectrum varies by external effects including film thickness, light scattering, and back‐side reflection, and accurate E_g determination is rather difficult from EQE measurements.

In general, the most reliable E_g analysis is performed by critical point analysis, where E_g is extracted from the theoretical fitting of the second derivative (ε₁, ε₂) spectra [54, 63]:

(4.8)

where A, φ, E_p, and Γ are the amplitude, phase, position, and width of the peak, respectively. Depending on the band structure, the critical point is classified into one dimension (j = −1/2), two dimensions (j = 0), three dimensions (j = 1/2), or excitonic (j = −1).

Figure 4.8 shows the results of the MAPbI₃ critical point analysis performed for the experimental dielectric function of Figure 4.4 [1]. In the second derivative spectra, the spectral feature in only the transition peak (critical point) region is extracted, enabling the reliable determination of the transition energy. From the fitting analysis based on Eq.  4.8, E_g has been determined to be 1.61 ± 0.01 eV. The exact same E_g was reported in another critical point analysis [18]. The transition absorption (TA) spectroscopy also shows the negative TA peak at 1.61 eV due to photobleaching observed for a direct‐gap semiconductor [19]. Thus, the appropriate E_g of MAPbI₃ is 1.61 eV.

4.3.2 Band Gap of Basic Perovskites

The E_g values of various hybrid perovskites reported in Refs. [1, 2, 4, 18, 20][32–36] are summarized in Table 4.3. Due to the variation of the measurement and analysis methods, inconsistent E_g values have been reported, particularly for Sn‐based materials including FASnI₃ [32, 34, 37], MASnI₃ [20, 34][38–42], and CsSnI₃ [33, 34][43–46]. Moreover, strong excitonic peaks are present in the absorption spectra of APbBr₃ and APbCl₃. In these cases, E_g can no longer be determined from the onset of the absorption; rather it must be deduced by separating the contribution of the excitonic peak from the band‐to‐band (interband) absorption. Such analyses have been performed for MAPbBr₃ [35] and MAPbCl₃ [36] in Table 4.3. Note that, if conventional E_g analyses are performed for excitonic absorption spectra, E_g is seriously underestimated.

Figure 4.9 shows the variation of E_g with the (a) A‐site cation and (b) X‐site anion in hybrid perovskites. In APbI₃ materials, E_g increases slightly in the order of FA → MA → Cs. A systematic DFT study reveals that this E_g variation with the A‐site cation is caused by the change in the Pb–I–Pb bond angle; the larger A‐site cation (Table 3.3) leads to a Pb–I–Pb angle nearer to 180°, which in turn reduces the E_g (Figure 5.14). This phenomenon has been interpreted by the anti‐bonding character of the Pb and I orbitals (see Section 5.5.1).

In contrast, there is no clear trend for E_g when the A‐site cation is changed in ASnI₃. However, unlike MASnI₃, which exhibits the pseudo‐cubic structure, FASnI₃ and CsSnI₃ have orthorhombic structures (see Table 3.2) and their E_g values cannot be compared directly. The DFT study of CsSnI₃, for example, shows that the crystal structure changes E_g significantly [64]. Moreover, due to the small ionic radius of Sn²⁺, the Sn–I–Sn bond angle in MASnI₃ is almost 180° [33]. In ASnI₃ perovskites, therefore, the E_g variation by the lattice expansion is considered more important [65].

It can be observed from Figure 4.9b that the influence of the X‐site anion is considerably greater than those of the A‐site and B‐site cations; the variation of the halide anion from I⁻ to Br⁻ increases E_g drastically by ∼0.8 eV. The increase in E_g by the order of I → Br → Cl follows the general trend that a lighter atom leads to E_g widening.

4.3.3 Band Gap Variation in Perovskite Alloys

To control E_g in a wide range, the alloying by I–Br and Pb–Sn mixing has been performed for hybrid perovskites. Figure 4.10a shows the E_g variations with the Br content x in Cs_0.17FA_0.83Pb(I_1−xBr_x)₃ and Cs_0.05FA_0.8MA_0.15Pb(I_1−xBr_x)₃. The E_g values have been extracted from the optical data in Appendix A using the Tauc analysis. Up to x = 0.4, the E_g of the perovskites increases linearly with x:

(4.9)

Figure 4.10b shows the E_g variation of MAPb(I_1−xBr_x)₃ in a wider x range (x = 0–1), which indicates a similar E_g increase, yet with a slight E_g bowing [22]:

(4.10)

For the E_g determination, a simple analysis has been employed (see E_g = 1.57 eV of Ref. [22] in Table 4.2). For FAPb(I_1−xBr_x)₃ and CsPb(I_1−xBr_x)₃, similar E_g variations have been confirmed (see Figure 18.7). In the case of MAPb(Br_1−xCl_x)₃, a slight E_g bowing expressed by E_g = 2.415 + 0.53x + 0.30x² has also been reported [36].

Interestingly, for the Sn–Pb mixed alloys (i.e. A(Sn,Pb)I₃), a significant E_g bowing has been confirmed. Figure 4.10c summarizes the E_g change for the Sn content x in MA(Pb_1−xSn_x)I₃ for which E_g is expressed by the following equation [42]:

(4.11)

where and show the E_g values of MAPbI₃ (1.6 eV) and MASnI₃ (1.3 eV), respectively. Notably, E_g bowing has also been confirmed when different A‐site cations are adopted in A(Pb,Sn)I₃ (see Figures 10.8 and 18.12).

The origin of the monotonous E_g increases in APb(I,Br)₃, and the strong E_g bowing in A(Sn,Pb)I₃ can be explained by the energy alignment of the VBM and CBM [42] (Figure 4.10d). When the X‐site anion is changed in MAPbX₃, for example, the energy position of the CBM moves upward, whereas that of the VBM shifts downward by I → Br exchange. When Br is incorporated into APbI₃, therefore, the E_g widening occurs by the energy shift of VBM and CBM. In contrast, the band alignment between MAPbI₃ and MASnI₃ is staggered (type‐II alignment) and the Sn alloying with Pb leads to a smaller gap material, whose optical transition occurs from the Sn‐derived VBM state to the Pb‐based CBM state. This E_g reduction is helpful for an all‐perovskite tandem application, as the E_g reduction in the Sn‐based alloy increases the photosensitivity in the long λ region (see Chapter 18).

4.4.1 Principles of Optical Measurements

Figure 4.11 shows the principles of (a) ellipsometry and (b) T/R optical measurements. Ellipsometry is a high‐precision optical characterization technique, in which the optical constants are determined by the change in polarized light upon reflection [54]. This technique employs p‐ and s‐polarized light waves, whose oscillation directions of the electric field are parallel and perpendicular to the incident plane, respectively (see Figure 4.11a). Depending on the (n, k) values, the polarized light undergoes different changes upon light reflection and ellipsometry measures two angles (ψ, Δ), which express the relative amplitude and phase difference between the p‐ and s‐polarized waves, respectively. The ψ is essentially related to the n component, whereas the phase variation Δ is generated by the light absorption (i.e. k). In ellipsometry, therefore, two defining values of optical constants can be determined from a single measurement.

Although ellipsometry allows the determination of highly accurate values, its data analysis tends to be complicated. In particular, the exact optical modeling of an entire sample structure as well as all the (n, k) values of the constituent layers is necessary [54]. Moreover, ellipsometry spectra (particularly Δ spectrum) are extremely sensitive to surface/interface structures; improper modeling of these structures generates strong artifacts and, often, the adjustment of roughness thickness on a few‐Å scale is required [5].

In T/R measurements, on the other hand, the absolute light intensities (i.e. square of the electric field; I ∝ |E|²) of a thin‐film/substrate structure are generally characterized. In this case, the phase information obtained in ellipsometry is lost and, therefore, α is obtained analytically. The simplest approach often adopted to determine α is the application of Beer's law: I(d) = I₀ exp(−αd), where I₀ and I(d) show the light intensity on the surface (d = 0) and at the film thickness of d, respectively. In this case, α is obtained from the measured absorbance (A_b) based on a simple relation:

(4.12)

Thus, from a known d, α can be determined using A_b. In another approach, the following relation has been adopted [14]:

(4.13)

where A is the absorptance that can be estimated from A = 1 − R − T (i.e. T + R + A = 1). Eq.  4.13 is particularly useful to eliminate the effect of interference.

In the above simple T/R approaches, however, the estimation of the accurate α of hybrid perovskites is difficult because the (i) light scattering that occurs on a film surface and (ii) optical interference caused by the light reflection at a thin‐film/substrate interface are completely neglected. In particular, hybrid perovskite layers formed by one‐step spin coating generally exhibit very large roughness with a size comparable to the wavelength of incident light (λ ∼ 500 nm) [10, 60]. Accordingly, the strong light scattering occurs in these layers, which in turn enhances the nominal α due to the increased light pass length. Note that, in Eqs.  (4.12) and (4.13), the αd component is considered and, when the analytical film thickness d is fixed, the enhanced absorption by light scattering is misinterpreted as high α in the film. The optical interference effect, caused by the finite light reflection on the surface and at the interface, also enhances the effective optical pass length within the layer and, similarly, nominal α increases if a simple layer thickness d is adopted in the analysis.

As a result, the α values derived particularly from simple T/R analyses are generally overestimated. The T/R measurements themselves are straightforward and are useful for obtaining the overall shape of the α spectra and rough α values; however, the extra care is necessary to interpret the absolute α value. On the other hand, ellipsometry measurements are far less sensitive to light scattering as only the specular reflection component is measured [6, 7]. In ellipsometry analysis, the optical interference effect is also modeled explicitly. Nevertheless, rigorous optical modeling of the layer structures is necessary for accurate optical‐constant determination by ellipsometry.

4.4.2 Interpretation of α Variation

Figure 4.12a shows the α spectra of MAPbI₃ determined by spectroscopic ellipsometry [1][8–12] and T/R‐based techniques [13–16]. The reported MAPbI₃ α spectra are quite different, although the spectral shape is rather similar. The highly controversial α results do not originate from measurement techniques; all the ellipsometry and T/R results are essentially inconsistent. In particular, some results show very high α values even below the E_g of MAPbI₃.

The large α variation observed for the ellipsometry results can be explained by the extensive roughness in MAPbI₃ layers. Specifically, the α spectrum of Shirayama et al. which is shown in Figure 4.4, was obtained from an ultra‐smooth MAPbI₃ layer where the effect of the roughness is minimized. Notice that the α values of Shirayama et al. show the smallest values among the results. Figure 4.12b shows the α simulation result obtained when a hypothetical roughness thickness (d_r) is assumed for the Shirayama's result [1]. In the figure, the α spectrum of d_r = 0 nm corresponds to the spectrum reported by Shirayama et al. and the nominal α (pseudo‐absorption coefficient, 〈α〉) increases largely with increasing d_r. As explained previously in Figure 4.11a, Δ in ellipsometry can be related to the light absorption. However, the roughness also induces the large phase lag and, in simplified ellipsometry analyses, the Δ change generated by the roughness is misinterpreted as the α increases [5]. As confirmed from Figure 4.12, the ellipsometry results of Refs. [8, 9] are well reproduced by the roughness simulation. In fact, when the roughness correction is performed for the α spectra of Refs. [8, 9], the recalculated α spectra show excellent agreement with Shirayama's result [5]. In some ellipsometry studies, on the other hand, (T, R) spectra are analyzed simultaneously, in addition to (ψ, Δ) ellipsometry spectra [10, 66, 67]. However, when extensive sample roughness modifies the T/R spectra, the validity of such (ψ, Δ, R, T) analyses must be examined carefully.

It is highly likely that the large α confirmed for the T/R results is caused by strong light scattering due to large roughness. In fact, some α spectra obtained from the T/R analyses show large non‐zero α values in the region below E_g, which can be understood by the light scattering effect of the roughness. Note that any transmission loss generated by roughness light scattering can be misinterpreted as light absorption in the simple analysis described above. In a well‐cited α spectrum reported by De Wolf et al. [14], the α was determined from Eq.  4.11; therefore, the influence of the light scattering and optical interference was not included in this analysis, and the large analysis error is expected. It is often argued that MAPbI₃ shows the highest α among solar cell materials based on the result of De Wolf et al. However, when accurate α spectrum is compared, the α of MAPbI₃ is comparable to those of GaAs and CdTe (see Figure 1.5). In the past, CuInSe₂ (CISe) had been considered to have the highest α (∼10⁵ cm⁻¹) among all the solar cell materials [68]. Nevertheless, this was also artifact caused by the roughness scattering in the T/R measurements [5], and actual α of CISe is comparable to other solar cell materials [69].

To justify the α result of Shirayama et al., complementary analyses were performed. Figure 4.13 shows the experimental α spectrum of Shirayama et al., and additional analysis results were obtained from spectroscopic ellipsometry (SE), EQE, and DFT analyses. In SE, a multi‐sample analysis was performed [1] and the calculated (ψ, Δ) spectra (solid lines) show almost perfect agreement with the experimental spectra (open circles).

In the EQE analysis, a semi‐transparent MAPbI₃ solar cell reported in Ref. [71] was analyzed using the α spectrum of Shirayama et al. In this solar cell, a rear metal electrode is not present and the light absorption properties can be studied with greater accuracy due to the lack of metal back‐side reflection. As shown in the figure, the calculated EQE spectrum (solid line) shows excellent agreement with the measured spectrum (open circles) and the short‐circuit current density (J_sc) obtained from the analysis agrees quite well with the experimental value [1]. It should be emphasized that, when an α spectrum with greater absolute values is applied for the EQE simulation, J_sc is seriously overestimated [1].

In Figure 4.13, the experimental α spectrum of Shirayama et al. is further compared with the DFT α spectrum [70]. For the DFT calculation, the PBE functional was employed (see Section 5.2). The magnitude of α determined theoretically by DFT shows excellent overall agreement with the experimental spectrum; therefore, the experimental light absorption is consistent with the dipole transition expected from rigorous theoretical calculation.

As discussed above, there are inherent difficulties for obtaining reliable optical spectra of hybrid perovskites and the extracted experimental optical spectrum must be justified based on other complementary methods including optical device simulations and DFT calculations. The key points for accurate optical‐constant characterization are the (i) preparation of smoother perovskite layers to reduce the roughness effect, (ii) optical measurements without exposing the samples to air to avoid rapid surface degradation, and (iii) application of a high‐precision ellipsometry technique with exact optical modeling.

4.5.1 Variation with Center Cation

Figure 4.14 summarizes the α spectra obtained with the variation of the A‐site cations. All the α spectra were determined by high‐precision ellipsometry characterization. These results show that the α in the band‐edge region changes with the choice of organic cation; FAPbI₃ shows a low α, whereas the α in MAPbI₃ increases. The incorporation of a small amount of Cs (17 at.%) into FAPbI₃ increases the band‐edge α. High α is also maintained in a CsFAMAPb(I,Br)₃ triple‐cation perovskite even though the Cs and MA contents in the material are relatively low.

The unexpected large influence of the A‐site cation on α has been discussed based on DFT calculations [2]. As mentioned above, in the visible/UV region, there is no direct light absorption by the organic cations and the optical transition is characterized by Pb(6s) + I(5p) → Pb(6p). Nevertheless, the distribution of the I‐5p charge densities varies largely depending on the cation species due to the hydrogen bonding interaction (I⋯HN). Interestingly, when the hydrogen bonding is formed, the I‐5p charge disappears due to the reduction of the N–I distance. This electrostatic interaction (the anti‐coupling effect [2]) can be adopted to explain the α variation in Figure 4.14. In particular, the α in the perovskite increases in the order of FA → MA → Cs due to the weakening hydrogen bonding interaction (see Figure 10.11). Nevertheless, what is less apparent is the high α observed for the FA‐based mixed‐cation perovskites. The molecular dynamics simulations of CsFAPbI₃ show that rather strong distortion occurs for the FAPbI₃ cubic structure by incorporation of Cs⁺ [72]. Thus, a large‐scale DFT calculation is necessary to further verify the effect of the mixed A‐site cations on α.

4.5.2 Variation with Halide Anion

Figure 4.15 summarizes the variations of hybrid perovskite α spectra with the X‐site anion, determined by spectroscopic ellipsometry characterization. All the optical constant data of the figure are summarized in Appendix A. In MAPbX₃, the entire spectrum shifts toward higher energy as the X‐site halide anion becomes lighter [3]. In the case of MAPbBr₃ and MAPbCl₃, the presence of the excitonic absorption increases the α in the E_g region. For the APb(I_1−xBr_x)₃ alloys in Figure 4.15, the entire α spectrum slides toward higher energy without an amplitude change. A similar trend can be confirmed for the A(Sn,Pb)I₃ material shown in Figure 4.1. This simple horizontal α‐shift can be explained by applying the sum rule [73]:

(4.14)

The sum rule requires that, when the ε₂ spectrum is shifted toward a higher energy by ΔE, the ε₂ amplitude is reduced by a factor of f = E/(E + ΔE) [2, 74].

Figure 4.16a shows an example of the ε₂ shift calculation based on the sum rule. In this figure, the ε₂ spectrum of MAPbI₃ is shifted toward higher energy while reducing the ε₂ amplitude using f. From the shifted ε₂ spectra, the corresponding ε₁ spectra are calculated by applying the Kramers–Kronig integration [54, 73]:

(4.15)

where P shows the principal value of the integral. From a set of (ε₁, ε₂) obtained from the above calculation, α(E) can further be deduced. Figure 4.16b shows the α spectra calculated from the above approach. It can be seen that the resulting α spectrum shifts horizontally in the energy direction without the amplitude change. The above scheme can be applied to calculate the optical constants of an alloy with an arbitrary composition (or ΔE) and is particularly helpful in designing a perovskite top cell in a tandem architecture.

When the dielectric function is modeled by the Tauc–Lorentz model (see Figure 4.4), the alloy dielectric function can be modeled by simply increasing the corresponding peak energy positions toward higher energy. From the Tauc–Lorentz model parameters described in Appendix A, the optical constants of various perovskite alloys can be calculated. Such an approach has previously been adopted in the EQE simulation program (e‐ARC software) [75], explained in Figure 11.13.