6.1 Introduction

Transport of photo‐generated carriers is a critical factor in the operation of photovoltaic devices [1–3]. For carrier transport in solar cells, the most important physical parameter is carrier diffusion length defined by

(6.1)

where D and τ are the diffusion coefficient and carrier lifetime, respectively [1, 2]. The D is a simple linear function of carrier mobility μ, which is expressed by the Einstein relation:

(6.2)

where k_B, T, and q are the Boltzmann constant, temperature, and electron charge, respectively [1, 2]. Note that k_BT/q = 25.9 mV at 300 K. Consequently, L_D is determined by the pair (μ, τ), and numerous efforts have been made to determine these characteristic values for various hybrid perovskites, including methylammonium lead iodide (CH₃NH₃PbI₃; MAPbI₃) [4–35] and formamidinium lead iodide (HC(NH₂)₂PbI₃; FAPbI₃) [36–39].

Quite fortunately, hybrid perovskites show favorable high (μ, τ) that lead to large L_D (Table 1.1). The L_D of the perovskites is sufficiently longer than an absorber layer thickness of ∼500 nm, which is the typical value used for spin‐coated hybrid perovskites. One of the remarkable features of hybrid perovskites is their quite high τ that can be confirmed easily based on photoluminescence (PL) characterization (Figure 1.6 and Chapter 8) [40]. It is now accepted that the long τ observed in hybrid perovskites originates from the absence of deep defects [41, 42]. Specifically, density functional theory (DFT) calculations indicate that the generation of deep traps is energetically unfavorable in MAPbI₃ (see Figure 5.17). In fact, trap densities reported for MAPbI₃ single crystals are surprisingly low at ∼10¹⁰ cm⁻³ [10, 23, 24]. Some authors argue that the high τ is caused by the indirect‐gap formation (Rashba splitting in the conduction band) [43–47]; however, this is highly unlikely because the direct‐gap formation in the perovskites is evident experimentally (Section 5.4.2).

For the absolute values of (L_D, μ, τ) for hybrid perovskites [4–39], however, there is widespread confusion. The intense controversy over hybrid‐perovskite transport properties arises essentially from differences in characterization techniques; namely (μ, τ) depend strongly on how these quantities are measured. Figure 6.1 illustrates the principles of carrier transport characterization by PL, microwave conductivity (MC), terahertz (THz) spectroscopy, and Hall measurements. In the PL, MC, and THz techniques, electron and hole carriers are generated first by short‐wavelength laser pulses, following which the time decay of the photoemission (PL) or the photoconductivity (MC and THz) is characterized (i.e. the optical pump‐probe method). The PL mobility (μ_PL) is determined analytically from the time decay of the PL intensity [5–7]. The origin of the MC mobility (μ_MC) and THz mobility (μ_THz) is essentially similar, and these are evaluated by using free‐carrier absorption phenomena in the GHz and THz regions. Specifically, free carriers (i.e. free electrons in metals, for example) absorb light particularly in the THz and GHz regions as expressed by the Drude model [48–50]. The variation of the electromagnetic‐wave transmission (or reflection) in the GHz and THz ranges can be related directly to the conductivity [22, 51, 52], from which μ is extracted with a known carrier concentration estimated from the optical‐pump laser pulse [13, 15, 19].

The PL, MC, and THz methods are noncontact methods and μ can be determined relatively easily. However, μ deduced from these methods includes contributions from both electrons and holes; thus, data interpretation is slightly difficult. In contrast, only one carrier type (i.e. electrons or holes) is characterized in the Hall measurements. Among all the techniques shown in Figure 6.1, the Hall measurement is the most reliable method because the Hall mobility (μ_Hall) can be determined accurately based on the Hall effect [53]. This is the exceptional technique that further allows the simultaneous determination of carrier type (i.e. n‐ or p‐type) and carrier concentration.

Even though the Hall measurement is the favored technique, μ_Hall is affected severely by grain boundaries [49, 54, 55] because the conductivity is measured in the direction perpendicular to the grain boundaries (see Figure 6.1). In contrast, μ_MC and μ_THz are negligibly influenced by grain boundaries, as free carriers do not exist in the region of grain boundaries (see Figure 9.1). Accordingly, when grain boundaries make a significant contribution, we find μ_THz > μ_Hall; therefore, comparing μ_THz (or μ_MC) with μ_Hall elucidates the effect of grain boundaries on carrier transport.

In this chapter, we review the carrier transport properties of hybrid perovskites, in an effort to gain consensus on the absolute values of (L_D, μ, τ). Section 6.2 explains the carrier properties of hybrid perovskites, using MAPbI₃ as an example. We focus on the carrier mobility of MAPbI₃ in Section 6.3, which presents controversial physical models to explain what limits the maximum mobility in MAPbI₃. Based on the understanding of measurement techniques, we further discuss the appropriate L_D of hybrid perovskites (Section 6.4). Section 6.5 explains unique transport properties of various hybrid perovskites.

6.2 Carrier Properties of Hybrid Perovskites

The semiconducting properties of hybrid perovskites arise essentially from the inorganic scaffolds (i.e. PbI₃⁻ in APbI₃ perovskites), and the unique defect characteristics of hybrid perovskites allow the control of carrier conductivity by the selection of process conditions. This section treats self‐doping in hybrid perovskites by the formation of shallow donor and accepter defects (Section 6.2.1) and discusses its effect on carrier mobility (Section 6.2.2).

6.2.1 Self‐Doping in Hybrid Perovskites

Hybrid perovskites are rarely doped intentionally because the carrier concentration can be varied widely from p‐ to n‐type by so‐called “defect engineering,” in which the carrier properties are controlled by introducing proper defects [9, 56, 57]. Figure 6.2 shows (a) the carrier concentration and the corresponding μ_Hall and (b) the energy separation between the Fermi level (E_F) and valence‐band maximum (i.e. ΔE = E_F − E_VBM) in MAPbI₃ thin films as a function of the precursor ratio (PbI₂/MAI) [9]. Here, the precursor ratio represents the molar ratio of PbI₂/MAI in the precursor mixture solution, adopted for a one‐step spin‐coating process. The results shown in Figure 6.2a,b were obtained from six‐contact Hall measurements with an electrode spacing of ≤1 mm and ultraviolet photoelectron spectroscopy (UPS), respectively. In the figure, the carrier type switches from p to n at the precursor ratio of ∼0.5 and the electron carrier density increases in the PbI₂‐rich film. Upon increasing the precursor ratio, the E_F position systematically shifts upward due to the enhanced n‐type character [9, 57].

Importantly, p‐type MAPbI₃ films can be converted to n‐type films by thermal annealing at temperatures T ≤ 150 °C [9, 56, 58]. This carrier‐type conversion by annealing originates from MAI desorption from the MAPbI₃ surface. Accordingly, it is now widely accepted that MAPbI₃ has n‐type conductivity in MAI‐poor (PbI₂‐rich) films, whereas p‐type conductivity occurs in PbI₂‐poor (MAI‐rich) layers [9][56–59]. Based on DFT results obtained for the point‐defect formation in MAPbI₃ (see Figure 5.17), Wang et al. concluded that the n‐type character of MAI‐deficient layers originates from shallow donor formation by I vacancies (V_I), whereas p‐type hole generation in the PbI₂‐deficient layers occurs because of shallow accepter formation due to Pb vacancies (V_Pb) [9]. At the present stage, no direct quantitative correlations between defect and carrier densities have been established. Nevertheless, the systematic variation of carrier concentration with the processing condition strongly supports the generation of shallow donor and accepter levels by simple point (or elementary) defects.

The perovskite defects mentioned earlier (V_I and V_Pb) are charged defects (see Figure 5.17) and thus free electrons and holes in the perovskites can be scattered by these defects through the Coulomb interaction (i.e. ionized‐defect scattering). The strong μ_Hall reduction with carrier concentration, shown in Figure 6.2a, can therefore be understood in terms of enhanced carrier scattering by ionized defects.

In Figure 6.2, the carrier properties are shown based on the PbI₂/MAI precursor ratio. However, this precursor ratio is only a relative value and the actual Pb/MA ratio in the perovskite layers varies according to (i) growth conditions (deposition methods, solvent) [9, 57], (ii) thermal annealing process [9, 56, 58], and (iii) substrate (or underlying layers) [57]. Thus, the precursor ratio must be considered as a process parameter, rather than as a physical parameter.

For Pb‐based hybrid perovskites, thin layers with low carrier concentrations can be fabricated. Nevertheless, Sn‐based hybrid perovskites often exhibit quite high p‐type carrier densities in the range 10¹⁸–10¹⁹ cm⁻³ [60–62]. This significant self‐doping effect has been attributed to the oxidation of the Sn²⁺ B‐site cation to Sn⁴⁺ [63, 64]. In fact, it has been demonstrated that the incorporation of Sn⁴⁺ in the form of SnI₄ notably enhances p‐type conductivity [65]. Importantly, the stability of the 2+ oxidation state deteriorates as the group‐IV B‐site cation becomes lighter [63]. Thus, Sn²⁺ in ASnI₃ is unstable and is easily oxidized into the more stable Sn⁴⁺ state. In photovoltaic devices, high carrier concentrations are quite detrimental and, to reduce the p‐type carrier density in ASnI₃ compounds, the additives of SnF₂ [62] and Sn metal powders [66] were used.

6.2.2 Effect of Carrier Concentration on Mobility

Figure 6.3 shows μ_Hall as a function of carrier concentration in MAPbI₃ [8–10] and MASnI₃ [8, 60, 61]. In the figure, the results obtained from bulk crystals (single crystals and polycrystals) and thin films are shown separately. The MAPbI₃ bulk crystals show a very low carrier concentration of ∼10¹⁰ cm⁻³ [8, 10], which is significantly less than that in the thin films [9], most likely due to the suppression of the shallow‐defect formation at equilibrium conditions. It can be seen that μ_Hall of MAPbI₃ bulk crystals is ∼100 cm²/(V s) and decreases significantly with increasing carrier concentration, independent of carrier type. As discussed in Section 6.3.1, μ_Hall of MAPbI₃ thin films decreases in the presence of grain boundaries. However, the result of Figure 6.2 shows a clear trend that μ_Hall of MAPbI₃ is also limited by strong carrier scattering by ionized defects, particularly at high carrier concentrations. Thus, L_D of MAPbI₃ decreases notably at high carrier densities, resulting in reduced solar cell performance. Consequently, the carrier concentration in the perovskite device needs to be controlled properly (see also Section 11.2).

As mentioned earlier, MASnI₃ shows strong p‐type character. Rather surprisingly, μ_Hall of MASnI₃ bulk crystals is substantially larger than that of MAPbI₃ bulk crystals and exhibits >300 cm²/(V s) at the carrier concentration of 10¹⁵ cm⁻³ [8]. Accordingly, μ increases notably when Pb is replaced with Sn. We attribute the higher μ_Hall in MASnI₃ to a smaller effective mass in MASnI₃ than in MAPbI₃ (see Section 6.3.3).

6.3 Carrier Mobility of MAPbI₃

The mobility μ is a universal physical parameter that allows for the direct evaluation of carrier transport properties. Nevertheless, the μ reported for MAPbI₃ varies in a quite wide range of 0.5–100 cm²/(V s), depending on the sample and characterization method [4–21]. Section 6.3.1 discusses the origin of variation in μ and what is the appropriate μ for MAPbI₃. In Section 6.3.2, we see the relationship between the effective mass and μ by making further comparisons with other solar cell materials. The temperature dependence of μ (Section 6.3.3) provides important information on the carrier scattering mechanism in MAPbI₃. Based on various results for μ, we discuss what determines the maximum μ of MAPbI₃ (Section 6.3.4).

6.3.1 Variation of Mobility with Characterization Method

Figure 6.4 summarizes μ of MAPbI₃, as determined by various characterization methods, which are categorized into three groups: (i) PL [5–7], (ii) Hall [8–10], and (iii) MC [11–16] and THz [16–20] measurements. Here, μ_THz and μ_MC are treated as the same group because these methods characterize μ based on the free carrier absorption that occurs primarily within the grains. In other words, μ_THz and μ_MC represent the intragrain mobility without the significant influence of grain boundaries. In addition to the aforementioned methods, μ obtained from a time‐of‐flight (TOF) measurement [21] is also shown in Figure 6.4.

For MAPbI₃ bulk crystals (single crystals and polycrystals), consistent μ values of ∼100 cm²/(V s) (66–135 cm²/(V s)) have been confirmed by Hall [8, 10], MC [15], and TOF [21] measurements, as indicated by the red circles in Figure 6.4. The result of μ_Hall in the figure was evaluated from samples with low carrier concentrations of 10¹⁰ cm⁻³ (see Figure 6.3) and the influence of charged‐defect scattering is minimized. Thus, μ ∼ 100 cm²/(V s) in Figure 6.4 represents the maximum μ of MAPbI₃ at room temperature.

In contrast, μ obtained from MAPbI₃ thin films (blue circles in Figure 6.4) differs significantly depending on the measurement method; in particular, there exists a clear trend of μ_MC ∼ μ_THz > μ_Hall > μ_PL. The results of μ_MC ∼ μ_THz show clearly that the intragrain mobility of MAPbI₃ thin films is ∼30 cm²/(V s). Note, however, that μ_MC varies with optical pump intensity (fluence) [11–13, 15, 16], probe microwave frequency [11], and interface structures [11, 14], and thus some ambiguity remains. When bare MAPbI₃ samples are probed at the microwave frequency of ∼10 GHz, μ_MC agrees quite well with μ_THz. Quite importantly, in a THz ellipsometry measurement, which characterizes the Drude‐type free‐carrier absorption directly without the pump light, the same μ_THz of ∼30 cm²/(V s) is observed [20]. Accordingly, μ ∼ 30 cm²/(V s) can be taken as the maximum intragrain μ in MAPbI₃ thin films. This upper limit observed in the thin films (∼30 cm²/(V s)) is much less than that confirmed in the bulk crystal (∼100 cm²/(V s)), indicating that enhanced scattering centers exist within MAPbI₃ grains due to imperfect growth of MAPbI₃ thin films.

The μ_Hall of polycrystalline thin films is significantly affected by grain boundaries (see Figure 6.1), and μ_Hall of ∼5 cm²/(V s) observed in MAPbI₃ is notably lower than μ_MC and μ_THz. In hybrid perovskite thin films, grain sizes exceed the film thickness and there are no grain boundaries along the film‐thickness direction (see Figure 6.1). To represent the vertical carrier transport in the thin‐film devices, therefore, an intragrain μ of 30 cm²/(V s) is appropriate.

The result μ_PL ∼ 0.5 cm²/(V s) in Figure 6.4 was extracted by assuming a one‐dimensional diffusion model:

(6.3)

where n(x, t), D, and k(t) are the carrier density distribution, diffusion coefficient, and PL decay rate, respectively [5, 6]. The problem of determining μ by PL is that the carrier lifetime tends to be underestimated seriously in this technique. Figure 6.5 shows (a) the time decays of PL and MC signal intensities for a MAPbI₃ single crystal [22] and (b) the PL time decays of MAPbI₃/quartz, PCBM/MAPbI₃/quartz, and spiro‐OMeTAD/MAPbI₃/quartz samples [6]. What is striking in Figure 6.5a is that the PL lifetime (τ_PL < 10 ns) is drastically shorter than the MC lifetime (τ_MC = 15 μs). The τ_PL is strongly influenced by interface conditions [5, 6, 67, 68] and, in the example of Figure 6.5b, τ_PL decreases significantly due to interface formation with the PCBM and spiro‐OMeTAD layers. Importantly, τ_PL of the bare MAPbI₃ sample in Figure 6.5b is governed primarily by surface recombination and τ_PL improves drastically upon passivating the MAPbI₃ surface (Figure 1.6) [40]. As a result, if the nominal short PL decay is adopted for Eq.  6.3, D is seriously underestimated, leading to the drastic reduction in μ. When the lateral diffusion of carriers within a grain is analyzed by PL, however, μ_PL increases to 33 cm²/(V s) (indicated by “2D” in Figure 6.4) [7], which is consistent with the intragrain μ determined by the MC and THz techniques.

To investigate μ of MAPbI₃, the field effect transistors (FETs) were fabricated [28–32]. Nevertheless, the FET mobilities are surprisingly low (10⁻⁵–0.5 cm²/(V s)) [28–31] and thus an accurate determination of μ is difficult from FET measurements, which use quite high applied voltages of <60 V.

6.3.2 Temperature Dependence

Figure 6.6 shows the variation of μ_THz in a MAPbI₃ thin film with a measurement temperature T, reported in Ref. [19]. The μ_THz increases strongly with decreasing T and reaches a high value of 100 cm²/(V s). As explained in Figure 3.13, the crystal structure of MAPbI₃ shows distinct phase transitions at 165 K (orthorhombic → tetragonal) and 327 K (tetragonal → cubic). However, no significant discontinuity appears for μ_THz as a function of T. This is in sharp contrast with the MA⁺ reorientation time (Figure 3.16) and the band gap (Figure 4.19), which show irregular discontinuities at the orthorhombic‐tetragonal transition temperature. In other words, the organic cation reorientation and the Pb–I–Pb bond angle change have little effects on μ. The solid line in Figure 6.6 shows the result of fitting to the experimental data and a clear T^−1.5 dependence can be seen. Similar variations of μ ∝ T^m with m = −1.8 ∼ −1.4 have been confirmed from PL [7], MC [11, 13], and THz [19] analyses. For μ_THz of FAPbI₃, different μ variations (m = −1.5 ∼ −0.5) have been reported [38, 39].

As known well [1, 2], when μ is limited by lattice scattering, μ decreases with increasing T because stronger lattice vibrations occur at higher T. In contrast, when μ is determined by ionized‐impurity (or defect) scattering, μ increases with T [1, 2], because higher thermal speeds of the carriers at high T reduce the Coulombic interaction between charged scattering center and carrier. The negative exponent observed for μ_THz of MAPbI₃ (μ ∝ T^−1.5) clearly shows that carrier scattering is dominated by thermal‐induced lattice vibrations (see Section 6.3.4 for more discussion).

6.3.3 Effect of Effective Mass

Physically, μ is determined by two factors: (i) an effective mass m* and (ii) a mean time interval between carrier scattering (〈τ〉), as follows [2, 48, 49]:

(6.4)

If there are a large number of defects or scattering centers, the time interval between scattering shortens, thus reducing μ. However, even when the relaxation time 〈τ〉 is same, we observe higher μ for smaller m*. Thus, semiconductors with low m* are quite advantageous in terms of carrier transport. Note that m* is an inherent material value derived from the band structure (Section 5.4.4) and, in semiconductor materials, m* becomes smaller than the electron mass in vacuum (m₀) due to the band formation (i.e. m*/m₀ < 1).

Table 6.1 summarizes the effective mass of electrons () and holes (), the maximum electron mobility (μ_e) and hole mobility (μ_h), and the bulk modulus of representative solar cell materials. The results of  = 0.32m₀ and  = 0.36m₀ for MAPbI₃ correspond to the results of DFT in Table 5.3 [70], whereas μ_e and μ_h of MAPbI₃ indicate μ_Hall in Figure 6.3 [8, 10]. All other values in Table 6.1 are taken from Ref. [69]. The bulk modulus shows how a material compresses under external pressure and represents the hardness (or softness) of materials. It can be seen that MAPbI₃ is a very soft material (Table 1.1) [69][71–73].

Material			μe (cm2/(V s))	μh (cm2/(V s))	Bulk modulus (GPa)
Si	0.32	0.54	1500	500	98
GaAs	0.06	0.53	8000	400	75
CdTe	0.11	0.72	1100	100	42
CuInSe2	0.16	1	150	20	75
MAPbI3	0.32	0.36	66	105	<20

All the numerical data were taken from Ref. [69], except for the effective mass [70] and mobility [8, 10] of MAPbI₃ (see Table 5.3 for m* of MAPbI₃).

Figure 6.7 shows the variation of μ with and obtained from Table 6.1. This figure contains a plot for MASnI₃ that is based on the results of Refs. [8, 74, 75]. All the tetrahedral‐bonding semiconductors show the trend of  <  and thus μ_e > μ_h. In particular, for these materials, due to the large difference between and , μ_h is significantly less than μ_e. In contrast, MAPbI₃ has the unique property of  ∼  and thus we find μ_e ∼ μ_h. In solar cells, since both electron and hole carriers need to be collected, well‐balanced μ_e ∼ μ_h in MAPbI₃ provides a great advantage for efficient carrier collection. Nevertheless, the maximum μ_e and μ_h of MAPbI₃ (∼100 cm²/(V s)) are rather low. Comparing μ_e values for Si and MAPbI₃ shows that μ_e of MAPbI₃ is one order of magnitude smaller, even though both materials have similar  ∼ 0.3m₀. The reason for low μ in MAPbI₃ is discussed in Section 6.3.4.

We have seen in Figure 6.3 that μ_h of MASnI₃ is substantially larger than that of MAPbI₃. This result can be interpreted from a lower of MASnI₃ (see Figure 6.7), which in turn increases μ according to Eq.  6.4. Thus, the effect of m* is significant in hybrid perovskite materials.

6.3.4 What Determines Maximum Mobility of MAPbI₃?

The experimental μ of MAPbI₃ described earlier reveals the important facts that (i) the maximum μ of nondoped MAPbI₃ single crystals at room temperature is ∼100 cm²/(V s), (ii) the intragrain μ of MAPbI₃ thin films at room temperature is ∼30 cm²/(V s), and (iii) the T dependence of μ is μ ∝ T^−1.5. The μ in semiconductor materials is influenced by a variety of factors and, by incorporating carrier scattering models proposed for MAPbI₃ [76–88], the experimental μ of hybrid perovskites can be expressed as

(6.5)

The above equation is based on Matthiessen's rule [2], which indicates that the observable μ is given by the inverse relation and is limited by the lowest μ factor. The major factors determining μ in semiconductors include ionized‐impurity scattering (μ_ion) [1, 89], longitudinal optical (LO) phonon scattering (μ_LO) [76–80], acoustic phonon scattering (μ_AP) [1][81–83], and grain‐boundary scattering (μ_grain) [49, 54, 55]. In Eq.  6.5, μ_TF is incorporated to express the reduced μ in the perovskite thin films. The two terms denoted as “Model” represent the proposed scattering mechanisms of polaron formation (μ_polaron) [84–86] and anharmonic structural disorder (μ_dis) [69, 87], as explained below.

When carrier density is low, the value of μ_ion becomes quite large [1, 89] and the term 1/μ_ion vanishes (i.e. 1/μ_ion → 0). In single crystals of MAPbI₃ with low carrier densities, the terms 1/μ_ion, 1/μ_grain, and 1/μ_TF disappear. Moreover, DFT calculations show that μ_AP of MAPbI₃ is quite high (>1500 cm²/(V s)) [81–83] and 1/μ_AP is also eliminated. Consequently, several studies have proposed that the maximum μ is determined by μ_LO (i.e. μ ∼ μ_LO in Eq.  6.5) [76–80]. Note that LO phonons displace the atomic positions in a way that can create local electric fields, which further scatter free carriers through strong Coulomb interactions. This coupling between LO phonons and free carriers is referred to as the Fröhlich interaction. The strength of the electron–phonon coupling can be expressed by a dimensionless Fröhlich parameter α [77, 84]:

(6.6)

where ε_∞ and ε_s show high frequency and static dielectric constants (see Figure 4.2). The quantities R_y, ℏ, c, and ω_LO are the Rydberg constant (R_y = 13.6 eV), Dirac's constant, speed of light, and LO–phonon frequency. The prefactor (1/ε_∞ − 1/ε_s) reflects the effect of the ionic screening.

As explained in Chapter 4, MAPbI₃ shows unique phonon absorption characteristics, including (i) quite strong phonon absorption, which leads to ε_s ≫ ε_∞ (Figure 4.2) and (ii) very low peak frequency of ω_LO (Figure 4.7). These specific features lead to a quite high coupling constant α, which is estimated to be 1.72–2.68 in MAPbI₃ [77, 84, 85]. This value is significantly larger than those of tetrahedral‐bonding semiconductors, such as GaAs (α = 0.07) and CdTe (α = 0.31) [90], and is comparable to crystals with high ionicity, including AgCl (α = 1.8) and KI (α = 2.5) [90]. Accordingly, the LO–phonon coupling in the hybrid perovskite is indeed strong and μ_LO likely governs the room temperature μ of MAPbI₃. Theoretical attempts show that the maximum μ_LO ranges from 30 to 200 cm²/(V s) with a T dependence of T^−1.37 to T^−1.5 [78–80].

To explain the maximum μ of ∼100 cm²/(V s) in MAPbI₃, two different models have been proposed further: (i) the formation of polarons [84–86] and (ii) carrier scattering by anharmonic lattice vibration [69, 87]. The physical pictures of these models are shown in Figure 6.8. When a hole conducts inside a polar (ionic) semiconductor, for example, the positive charge of the hole attracts anions, while expelling the cations in the opposite direction. Thus, a conduction hole (or electron) displaces the constituent ions from their equilibrium positions, creating a quasiparticle called a polaron (Figure 6.8a). The formation of a polaron lowers μ through a widespread lattice deformation, which can be interpreted as an increase in m*. The polaron effective mass (m_p) is derived from the Fröhlich parameter α and m_p increases with α [84, 85]. For MAPbI₃, m_p is 28–64% heavier than m* [84–86] and the polaron‐limited mobility (μ_polaron in Eq.  6.5) is estimated to be 133–200 cm²/(V s) [84, 85]. The corresponding polaron radii range from 26 to 51 Å [84, 85]. However, although μ_polaron quantitatively agrees with the maximum μ of ∼100 cm²/(V s) at room temperature, the theoretical calculation shows a T dependence of μ_polaron ∝ T^−0.46 [85]. Thus, μ variation for T is underestimated with the polaron concept.

If the PbI₃⁻ cage deforms anharmonically, a local potential fluctuation is created, scattering carriers (see Figure 6.8b). A molecular‐dynamics‐based calculation shows that the maximum μ_e of MAPbI₃, limited by such an anharmonic lattice disorder (μ_dis in Eq.  6.5), is ∼135 cm²/(V s) with the T dependence of ∼T⁻² [87]; thus, this model provides the correct transport parameters for MAPbI₃. It has been argued that carrier scattering by dynamic lattice disorder becomes particularly important in compounds with “soft lattices” [69, 87] (see bulk modulus in Table 6.1).

All the above discussions, however, are based on a quite simple assumption that μ is determined totally by a single scattering mechanism. In the LO–phonon scattering model, for example, μ(T) = μ_LO(T) is assumed in Eq.  6.5. However, this is highly doubtful, because the T dependence of μ(T) ∝ T^−1.5 is reported for MAPbI₃ thin films [7, 11, 13, 18, 19], in which the intragrain μ is lowered by μ_TF and μ_ion. In particular, it is likely that the intragrain μ of MAPbI₃ thin films (∼30 cm²/(V s)) is described by 1/μ = 1/μ_ion + 1/μ_TF + 1/μ_LO, or other similar models. Accordingly, more quantitative experimental data, including μ(T) of single crystals, are necessary to conclude the absolute limiting factor of μ in MAPbI₃ at room temperature. We mention, however, that the strong LO–phonon carrier scattering is quite likely because the LO–phonon absorption is extraordinarily strong in hybrid perovskites, leading to a large Fröhlich parameter.

6.4 Diffusion Length

One of the greatest advantages of hybrid perovskite solar cells is long L_D of carriers [10][13–15, 19][22–27]. Nevertheless, absolute L_D values differ significantly depending on the characterization technique. Table 6.2 summarizes L_D in MAPbI₃ single crystals and thin films, together with their corresponding μ and τ. Figure 6.9 further summarizes the relation of μ and τ for the values of Table 6.2. For the single crystals, L_D varies in a range of 8–175 μm, while L_D of the thin films is 0.1–23 μm.

In Figure 6.9, we can confirm μ ∼ 100 cm²/(V s) for MAPbI₃ single crystals, even though the PL result shows much lower μ, as discussed earlier. In other words, the large variation in L_D observed for the single crystals originates from the variation in τ. The MC measurements consistently show τ ∼ 10 μs [15, 22], whereas the transient photovoltaic (TPV) measurements [10] and transient absorption (TA) spectroscopy [23] give different τ values. On the other hand, L_D can be determined directly from spatially resolved techniques, including scanning photocurrent microscopy (SPCM) [15] and the electron beam‐induced current (EBIC) technique [27]. The SPCM characterization of MAPbI₃ single crystals results in L_D of 10–28 μm [15], which is consistent with ∼50 μm determined by MC [15, 22]. Accordingly, L_D ∼ 50 μm with μ ∼ 100 cm²/(V s) and τ ∼ 10 μs can be considered as a representative L_D of MAPbI₃ single crystals.

For MAPbI₃ thin films, we observe μ = 10–30 cm²/(V s), with μ_PL ∼ 1 cm²/(V s), which are consistent with the results of Figure 6.4. However, τ reported for the thin films is quite different (τ = 5 × 10⁻³–30 μs in Table 6.2), most likely due to the strong influence of interfaces in the thin‐film geometry. The τ_PL is particularly sensitive to the interface (see Figure 6.5b) and the PL technique seriously underestimates L_D. The direct characterization of L_D in MAPbI₃ thin‐film solar cells has been performed by EBIC, resulting in L_D ∼ 1 μm [27]. The MC results of the thin films provide L_D in a similar range (L_D ∼ 5 μm) [14]. Accordingly, L_D of 1–5 μm appears to be appropriate for MAPbI₃ thin films. Nevertheless, an extremely long τ_PL of 8 μs has been reported for a MAPbI₃ thin film with optimized surface [40] (see Figure 1.6). If τ_PL = 8 μs and μ = 30 cm²/(V s) are applied, we obtain L_D ∼ 25 μm, which is comparable to that of the single crystals. Thus, the L_D of 4–23 μm is shown in Table 1.1.

It should be emphasized that the above L_D sufficiently exceeds the absorber thicknesses of MAPbI₃ solar cells (300–500 nm). In this condition, the effect of carrier recombination within the bulk component becomes less significant and the interface recombination becomes more important. In fact, the open‐circuit voltage (V_oc) of hybrid perovskite solar cells depends critically on the interface structure (see Section 11.5.1). Accordingly, it can be concluded that the carrier transport characteristics of MAPbI₃ thin films are quite satisfactory, which is one of the main reasons why solar cells with over 20% efficiencies can be fabricated using MAPbI₃ as a light absorber (Chapters 1 and 11).

6.5 Carrier Transport in Various Hybrid Perovskites

A variety of perovskite compounds and alloys have been used to fabricate hybrid perovskite solar cells. Here, we see the carrier transport properties of representative perovskite materials. Figure 6.10 summarizes μ_THz values reported for various Pb‐based hybrid perovskite thin films [17–19, 37][91–94]. In the figure, the maximum and minimum ranges of μ_THz reported for each material are indicated by bars. This result shows an important result; the upper limit of μ_THz is ∼30 cm²/(V s) for all the perovskite thin films. Based on the discussion of Section 6.3.4, the observed limit on μ_THz could be expressed as

(6.7)

Moreover, μ_THz values for MAPbI₃ [19], FAPbI₃ [37], and Cs_0.17FA_0.83PbI₃ [92] are quite similar, confirming that the effect of the A‐site cation is essentially negligible.

Nevertheless, the A‐site cation affects μ indirectly through the perovskite crystal stability. Figure 6.11 shows (a) μ_THz and the width of the (100) X‐ray diffraction (XRD) peak and (b) tolerance factor (t), as a function of the Cs content y in Cs_yFA_1−yPb(I_0.6Br_0.4)₃ thin films. The result of Figure 6.11a is taken from Ref. [92], whereas t in Figure 6.11b is calculated by using Eq. (3.1) with the values of Table 3.3. It can be seen from Figure 6.11a that the crystallinity of the CsFAPb(I,Br)₃ alloys improves notably upon the slight incorporation of Cs (y ∼ 0.2) into FAPb(I,Br)₃, as evidenced by the sharper XRD peak. More importantly, the crystallinity confirmed by XRD shows an almost perfect correlation with the transport characteristics and μ_THz reaches a maximum when the peak width is minimized.

The high μ_THz and small peak width at y ∼ 0.2 can be interpreted by t, which is a global indicator that describes the stability of perovskite crystals (Section 3.3). For Pb‐based hybrid perovskites, t = 0.945 is an ideal value (Figure 3.10), allowing the formation of the stable perovskite phase. Note that secondary‐phase materials, such as PbI₂, δ‐CsPbI₃, and δ‐FAPbI₃ (see Figure 3.5), form when t deviates largely from the optimum t range of 0.91–0.98 (Figure 3.10), as indicated in Figure 6.11. For CsFAPb(I_0.6Br_0.4)₃, t ∼ 0.95 is obtained when y = 0.24 (Figures 6.11b and 3.11c). Thus, the Cs content of y ∼ 0.2 not only provides higher crystallinity but also prevents the inclusion of the secondary phase. Moreover, τ_PL improves at this Cs composition [92]. As a result, the proper control of the A‐site cation species is critical for the improvement of μ_THz and L_D. Note that the variation of μ_THz in CsFAPb(I,Br)₃ thin films can best be represented by μ_TF in Eq.  6.7, provided that the crystallinity of the single crystals is much better than that of the thin films.

For the variation of the Br content x in FAPb(I_1−xBr_x)₃ and Cs_0.17FA_0.83Pb(I_1−xBr_x)₃ thin films, a large change in μ_THz is also reported (Figure 6.12) [37, 92]. In the case of FAPb(I_1−xBr_x)₃ thin films, μ_THz has a minimum value of ∼1 cm²/(V s) at x = 0.3–0.4, which is attributed to the formation of an amorphous phase in this region. The μ_THz of Cs_0.17FA_0.83Pb(I_1−xBr_x)₃ also varies strongly with x. For this alloy, μ_THz decreases monotonically from 40 to 11 cm²/(V s) with increasing x. This reduction is suggested to originate from the strong LO–phonon scattering in Br‐based perovskite compounds [4]. However, μ of a MAPbBr₃ single crystal evaluated from the TOF measurement (115 cm²/(V s) in Ref. [24]) is comparable to μ_TOF of a MAPbI₃ single crystal (135 cm²/(V s) in Figure 6.4). In addition, the Fröhlich parameter of MAPbBr₃ (α = 1.69) is almost the same as that of MAPbI₃ (α = 1.72) [84]. Accordingly, μ reduction by the Br incorporation is not evident in MAPbX₃. The Br content has negligible effect on t in Cs_0.17FA_0.83Pb(I_1−xBr_x)₃ (see Figure 3.11). Since the absolute μ_THz at x = 1.0 is rather small (11 cm²/(V s)), it is possible that μ_TF or μ_ion plays a dominant role in the carrier scattering of the Cs_0.17FA_0.83PbBr₃ thin film. For the increase of x in these thin films, the large reduction in L_D from 4.4 μm (x = 0) to 0.8 μm (x = 1) is also confirmed [92].

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