Static and Dynamic Structures of Perovskite Halides

Selected Paper Static and Dynamic Structures of Perovskite Halides ABX3 (B = Pb, Sn) and Their Characteristic Semiconducting Properties by a Hückel Analytical Calculation Koji Yamada,*1 Satomi Hino,1 Satoshi Hirose,1 Yohei Yamane,1 Ivan Turkevych,2 Toshiyuki Urano,2 Hiroshi Tomiyasu,2 Hideo Yamagishi,2 and Shinji Aramaki2 1

Department of Applied Molecular Chemistry, College of Industrial Technology, Nihon University, 1-2-1 Izumi-cho, Narashino, Chiba 275-8575, Japan 2

Chemical Materials Evaluation and Research Base (CEREBA), 1-1-1 Higashi, AIST Central 2, Tsukuba, Ibaraki 305-8565, Japan E-mail: [email protected] Received: March 6, 2018; Accepted: May 7, 2018; Web Released: May 15, 2018

Koji Yamada Koji Yamada received his Ph.D. from Hiroshima University, Japan, in 1977. After a 2 year stay in Germany as a Humboldt Research Fellow (1981–1983), he obtained a faculty position at Hiroshima University. In 2006 he was promoted to professor at Nihon University.

Abstract Solid solutions of methylammonium lead iodide (CH3NH3PbI3, abbreviated as MAPbI3) and formamidinium lead iodide (CH(NH2)2PbI3, as FAPbI3), which have been expected to be suitable materials as a visible light absorber of solar cells, were characterized by differential thermal analysis (DTA), XRD, 1H, 207Pb NMR and 127I nuclear quadrupole resonance (NQR). Continuous solid solutions of MAPbI3 and FAPbI3 were confirmed to have a cubic perovskite structure at 298 K except the tetragonal MAPbI3. 127I NQR spectra as well as DTA for CH3NH3PbI3 showed successive phase transitions at 162 K and 333 K associated with the space group trans formation from Pnma, I4/mcm to Pm3m. FAPbI3 (Black phase) showed similar successive phase transitions at around 120140 K and 283 K. The motional narrowing phenomenon observed on the 207Pb NMR proved useful to evaluate the halide ion migration and the activation energies were estimated to be ca. 48 kJ/mol for APbBr3 (A = Cs and CH3NH3), while the narrowing phenomena could not be observed clearly for MAPbI3 below 500 K, suggesting a lower concentration of vacancies than bromide analogs. Finally, Hückel calculations were performed for ABX3 (B = Sn, Pb) to demonstrate their excellent performance as a visible light absorber of solar cells, i.e., the direct band gap transition with a tunable property and the small effective masses of electron and hole.

1196 | Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

Keywords: Perovskite solar cell j Hückel calculation j Photovoltaics

1. Introduction Halides of the main group element such as Ge(II), Sn(II), Sb(III) have been far from application because of their undesirable nature such as hygroscopic, volatile, reactive and corrosive properties. Some of these crystals having perovskite structure ABX3 (B = Pb, Sn), however, are relatively stable in ambient conditions and also show intense colors due to their characteristic band structures. Furthermore, their light absorption properties are tunable by replacing not only anions but also cations and cover a broad area of visible light.15 These mixed halides MAPbX3¹xYx (MA = CH3NH3) have been characterized by optical properties,1 XRD,1,2 and theoretical study of band alignment.3 Hence, these halides as well as Sn analogs4,5 are suitable materials for visible light absorbers of solar cells. Kojima et al. first adopted these materials, MAPbBr3 and MAPbI3, as visible right absorbers for dye sensitized solar cells in 2009.6 The maximum efficiency of perovskite solar cells (PVSCs) has been improved from 3.8% by Kojima et al. to 22.1% after only 8 years,7 since they are expected to offer low cost solar energy devices that are simple to process. Thereafter, the ABX3 family with perovskite structures has attracted great attention not only as light absorbers for solar cells but also various optical devises. Recent review of photovoltaic devices has shown that these perovskites exhibit not only suitable electronic band

© 2018 The Chemical Society of Japan

2. Experimental Synthesis. All powder samples of MA1¹xFAxPbI3 (x = 0, 1/3, 2/3, 1) were prepared by solid state reactions between raw materials, methylammonium iodide (MAI), formamidinium iodide (FAI) and PbI2. The stoichiometric mixture was heated at 423 K for three days using a sealed glass tube to get a homogeneous solid-solution. Characterization. XRD measurements were performed using a Bruker D8 ADVANCE or D2 diffractometer, and the temperature dependences were observed by a Rigaku Rad-B system equipped with a homemade high temperature attachment. The powder patterns were analyzed by a Rietveld method23 in order to evaluate lattice parameters and/or to refine Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

the structures. DTA were measured by means of a homemade heat-flow type differential scanning calorimetry (DSC) in the temperature range from 100 to 450 K, in which a sample sealed in a glass tube was used. Broadline 1H, 207Pb NMR were measured at 6.4 T using a homemade spectrometer system in the temperature range from 100500 K. The spectrum was obtained by a Fourier transformation of a pair of quadrature-detected free induction decays. Typical dead times of our spectrometer were ca. 2.5 ¯s at 270.2 MHz and 7 ¯s at 56.5 MHz for 1H, 207 Pb NMR, respectively. 127I NQR was also measured in the temperature range from 77500 K using a similar type spectrometer without the magnetic field. 3. Results and Discussion DTA and XRD Characterizations of FAPbI3, MAPbI3 and Their Solid Solutions. Figure 1 shows DTA curves for a series of solid-solution MA1¹xFAxPbI3 (x = 0, 1/3, 2/3, 1). Although the color of FAPbI3 was black just after synthesis at 423 K, it changed to yellow at room temperature within several days. Similar reconstructive phase transition was also reported by Li et al. for a single crystal grown from γ-butyrolactone.13 Therefore, the DTA measurement for FAPbI3 was started from the yellow phase which is stable at RT. Two endothermic peaks were observed at 171.5 K (peak tem.) and 378 K. Above 378 K the sample color changed from yellow to black, suggesting a reconstructive phase transition to a perovskite phase. No peaks could be detected in the cooling process near 378 K and its black color could be maintained several days. Figure 1(b) shows the DTA curve for this metastable black phase, in which a complex endothermic peak at around 120140 K and a weak λ-type phase transition at Ttr = 283 K (peak tem.) were observed. Complex phase transitions around 120140 K suggest the stepwise motional processes of the FA+ cation in a perovskite A-site, because of its low symmetry. As stated below, the crystal belongs to a cubic perovskite above 283 K. The polymorphic behavior of FAPbI3 accompanied by a drastic color change is analogous to that found in CsSnI324,25 or CsPbI3.26 The structure of the yellow phase, however, is quite different from NH4CdCl3-type found in CsSnI3 or CsPbI3 as will be described below. a

0.5

b c

ΔT / K

structures but also extremely high quantum efficiency of luminescence and long lifetime of photocarriers.8 That is, the long carrier-diffusion with small effective masses and the low density of defects are indispensable properties for materials as a light absorber of solar cells. Among many perovskite halide families, formamidinium lead iodide, FAPbI3, has been proven a promising material due to its excellent optoelectronic properties, such as narrow band gap and large absorption coefficient as well as its thermal stability.913 However, cubic FAPbI3 perovskite is a meta-stable phase at ambient conditions, and hence the structure gradually changes to a yellow phase within several weeks. Therefore, in order to keep perovskite phase, solid solutions between MAPbI3 and FAPbI3 are expected to avoid phase separation since cation diffusion constant at A-sites is expected to be low. Jodlowski et al. demonstrated improved material stability by introducing a large organic cation, guanidium, into the MAPbI3 lattice and stabilized performance for 1000 h with conversion efficiencies over 19%.14 However, there exist still a large number of fundamental challenges in this field, such as hysteresis in the current-voltage behavior, charge transport and recombination, ionic conductivity, dynamic disorder, chemical and thermal stability, ferroelectricity and so on.15 Since most of these fundamental problems are closely related to the defect structures in the perovskite lattice, i.e., vacant associated ionic migration must be an inevitable problem for further development of PVSCs. Although many structural studies on MAPbI31619 and FAPbI310,13 have been reported, there still exists confusion about the structural descriptions. In this report we have reevaluated structural phase transitions for these compounds by means of DTA, XRD and 127I NQR spectroscopy. Secondly, the possibility of cation and/or anion migrations at higher temperatures were evaluated for MA1¹xFAxPbI3 and APbBr3 (A = Cs, CH3NH3) by means of 1H NMR and 207Pb NMR. Especially 207 Pb NMR has proven to be a good technique to evaluate ionic migrations in the perovskite lattice. Finally, we will present a Hückel analytical treatment20,21 for the anionic sublattice to understand the excellent optical properties of ABX3 (B = Pb, Sn), such as tunable semiconducting properties, direct transition nature and small effective masses of electron and hole. The effects of PbI6 tilting and Peierls-type distortion were also discussed. This Hückel analytical treatment for the linear -X-Pb(Sn)-X- bond can be applied by introducing a concept “hypervalent bonds”.22

d

0.0

e Phase III

Phase II

Phase I

-0.5

100

150

200

250

300

350

400

T/K

Figure 1. DTA curves of the heating processes for FAPbI3, MAPbI3 and solid solutions. (a) Yellow phase of FAPbI3 (Below 378 K), (b) Black perovskite phase of FAPbI3, which was observed after heating the yellow phase above 400 K, (c) Ma0.33FA0.67PbI3, (d) Ma0.67FA0.33PbI3, (e) MAPbI3.

© 2018 The Chemical Society of Japan | 1197

Figure 2. (a) XRD patterns for a series of solid solutions MA1¹xFAxPbI3. (b) Cubic lattice constants against x = FA/(FA + MA). Lattice constants for the tetragonal MAPbI3 are plotted as a = aT/20.5 and a = cT/2.


88

Ttr=283K

FAPbI3

I NQR/MHz

87

127

In contrast, the DTA curve for MAPbI3 shows one strong endthermic peak at 162 K and a weak one at 333 K, which are consistent with those reported by Onoda-Yamamuro (Ttr = 161.4 K and 330.4 K, respectively).27 The resultant three phases are abbreviated as Phase III, II and I from the low temperature side. According to precise heat capacity measurements by Onoda-Yamamuro et al., two phase transitions were accompanied by large transition entropies indicating that the phase transitions are of the order-disorder type. The total entropy values observed were interpreted with structural models in which each MA+ is disordered with respect to the orientations around the C-N axis and the orientation of C-N axis itself.27 As Figure 1(c) and (d) show, DTA curves for solid solutions do not show strong endothermic peaks or color changes over the observed temperature ranges. They show only weak endothermic peaks around 250300 K. These findings suggest that the orientational disorders of the cations survive down to 100 K in their perovskite lattices as will be discussed again at the 1 H NMR section. Although, homogeneous black perovskites of these solid solutions could be maintained at least over several months under ambient conditions, a gradual disproportionation was observed from their colors especially for repeatedly heated samples. Figure 2(a) shows XRD patters at 298 K for a series of solid-solutions MA1¹xFAxPbI3. All samples belong to a cubic perovskite except the tetragonal MAPbI3, which are consistent with the DTA shown in Figure 1. Figure 2(b) plots lattice constant against mole fraction of FA+. Since MAPbI3 belongs to a tetragonal system with z = 4, the lattice constants are plotted as aab = aT/20.5 and ac = cT/2. The cubic lattice constants a increase linearly as a function of the mole fraction of FA+. These findings suggest that the spherical FA+ cation behaves as a slightly larger cation than MA+. 127 I NQR for FAPbI3 and MAPbI3. NQR spectroscopy has been proved to be a sensitive method to judge crystal symmetry if the sample contains a suitable probe nucleus (quadrupolar nucleus with I > 1/2)28 such as 127I or 81Br. Temperature dependent NQR spectra offer the number of crystallographically different sites, the symmetry and the bonding nature at the probe nucleus. In the case of 127I NQR (I = 5/2), a pair of transitions ν1(m = «1/2 § «3/2) and ν2 (m = «3/2 § «5/2) is expected for each crystallographically equivalent site. If we can observe a pair of ν1 and ν2 transitions at the same temperature, both quadrupole coupling constant (e2Qq/h) and

MAPbI3

86

85 Ttr=333K

84

83

82

Ttr=162K 100

200

300

400

500

Tem / K

Figure 3. Temperature dependencies of 127I NQR ν1 for FAPbI3 (red open circle) and MAPbI3 (black filled circle). Just above the Ttr = 162 K no NQR were observed for MAPbI3. NQR signals could not be detected for FAPbI3 below 155 K.

asymmetry parameter (η) of the e2Qq/h tensor can be determined from the pair. Figure 3 shows the temperature dependencies of ν1 transitions for MAPbI3 and FAPbI3 (Black perovskite phase). Above Ttr (333 K for MAPbI3 and 283 K for FAPbI3), both samples show only one pair of ν1 and ν2 transitions and are summarized in Table 1. Their η values were determined to be zero within the experimental error, since ν2 = 2¢ν1. These findings supported that the highest temperature phases of these crystals belong With decreasto the cubic systems with a space group Pm3m. ing the temperature, ν1 transitions for both MAPbI3 and FAPbI3 split into two below Ttr with an intensity ratio 1:2 from the high frequency side. These two ν1 transitions could be assigned to I(1) and I(2) sites, respectively. It should be noted that the η values for I(1) sites are zero for both compounds, i.e., I(1) must sit on the C4 symmetry axis of the tetragonal system. Two possible space groups I4/mcm (#140, centrosymmetric) and I4cm (#108, non-centrosymmetric) have been proposed for MAPbI3 (Phase II). Our Rietveld analysis of the XRD data © 2018 The Chemical Society of Japan

Table 1.

I NQR and crystallographic parameters for FAPbI3 and MAPbI3

127

292.7

ν1/MHza 87.294 85.205 86.61

ν2/MHz 174.59 170.34 173.22

300.0

57.315

87.935

85.973 84.895 83.430 82.073 82.13

160.192 159.161 166.846 164.114 164.28

Sample

T/K

FAPbI3 (Black)

189.5

FAPbI3 (Yellow)

77.4d MAPbI3

297.3 350.8

Ratio 1 2

e2Qqh¹1/MHz 581.96 567.84 577.40

ηb 0.00 0.018 0.00

306.50

0.508

540.07 536.05 556.16 547.06 547.59

0.241 0.229 0.00 0.01 0.00

1 2 1 2

Crystallographic parameters Tetragonal, I4/mcmc a = 6.354(1) ¡ at 298 K Cubic, Pm3m, Trigonal, P31c a = 8.675(1) ¡, c = 7.926(1) ¡ at 298 K Orthorhombic, Pnma Tetragonal, I4/mcm a = 8.870(1) ¡, b = 12.661(1) ¡ at 298 K a = 6.329(1) ¡ at 400 K Cubic, Pm3m,

(SI-2) and the neutron powder diffraction study by Weller et al.19 supported a tetragonal system with a centrosymmetric space group I4/mcm. Below 155 K we could not detect 127I NQR for FAPbI3, probably due to the disordered structure in which some orientation disorders of the FA+ are frozen. On the other hand, two 127 I NQR lines with an intensity ratio 1:2 were observed for MAPbI3 at Phase III (T < Ttr = 162 K). The η values were determined to be 0.229 and 0.241 for these sites at 77 K.29 These observations suggest that the Phase III belongs to an orthogonal system with a space group Pnma (#62)1719 rather than Pna21 (#33),16 since there are only two iodine sites in the crystal. The relatively larger asymmetry parameters than that expected from perovskite lattice suggest not only the tilting of the PbI6 octahedra but also the hydrogen bonds such as N-H£I at Phase III. We have observed 127I NQR up to 500 K to evaluate the effect of the anionic diffusion. However, there was no abrupt change in their intensities up to 500 K for both compounds. Table 1 summarizes NQR parameters for the respective phases of FAPbI3 and MAPbI3 together with the possible space groups suggested by the 127I NQR spectroscopy. Table 1 also shows NQR parameters for the yellow FAPbI3, for which only one pair of ν1 and ν2 transitions with large η was observed suggesting a non-linear Pb-I-Pb bond. The NQR observations for the yellow phase are consistent with the structure as will be stated below. Crystal Structures of FAPbI3 (Yellow and Black Phase) and MAPbI3 (Phase II and I). According to single crystal Xray diffraction studies,9 FAPbI3 (Phase I at 293 K) and MAPbI3 (Phase I at 400 K) belong to a trigonal system with P3m1 and a tetragonal system with P4mm, respectively. However, as was already discussed in our previous section, both FAPbI3 (at 298 K) and MAPbI3 (at 400 K) belong to the cubic perovsk where FA+ and MA+ cations ites with a space group Pm3m, behave as spherical cations as will be discussed in the next NMR section. Since many structural data have been reported for FAPbI3 (Black phase) and MAPbI3, our Rietveld refinement plots and refined structural parameters are summarized in SI-1 and SI-2. Only their space groups and lattice parameters are summarized in Table 1 together with 127I NQR parameters. Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

Intensity / cps

a) ν1, ν2 are m = «1/2 § «3/2 and «3/2 § «5/2 transitions, respectively. FWHM line widths for ν1 are 2030 kHz. b) η = «(e2Qqyy ¹ e2Qqxx)/e2Qqzz«. c) Expected from 127I NQR. d) Ref. 29.

30000

20000

10000

0

10

20

30

40

50

60

70

Figure 4. Rietveld refinement plot for FAPbI3 (Yellow phase) at 298 K and the structure (Trigonal system).

We will show briefly the result of the Rietveld analysis for FAPbI3 (Yellow phase) at room temperature. Figure 4 shows the refinement plot and the structure at the trigonal system. The crystal structure of this phase was already reported by Stoumos et al. to be a hexagonal system with a space group P63mc (#186) in which an orientational disorder was introduced at the FA+ site along the c-axis.9 In our Rietveld analysis, however, a simple trigonal system was adopted, in which a dummy atom having a large anisotropic atomic displacement parameter was introduced at the cation site for simplicity instead of the disordered model. Both anionic structures are identical to each other. The anionic structure in the yellow phase consists of the face sharing PbI6 octahedra forming one-dimensional chains along the c-axis. The crystallographic and structural parameters for FAPbI3 are summarized SI-1. Dynamic Structures of Ions in the Perovskite Studied by Broadline 1H, 207Pb NMR. The dynamic structures of ions in a perovskite lattice such as the reorientations of organic cations and the migrations of component ions, have become a major issue concerning the degradation of perovskite solar cells. Broadline 1H and 207Pb NMR spectroscopies are sensitive methods to detect ion dynamics. Especially 207Pb NMR is expected to reflect the migration of iodide ions, since the dipole-dipole interactions between 207Pb and 127I disappear as a result of the migration.


20 15 10

270.20

100

200

300

400

T/K

56.52

56.54

56.56

Frequency / MHz

56.58

20 FWHM / kHz

502 483 464 446 426 408 390 369 349 331 319

x=0.33 (316K)

MAPbI3 x=0.33 x=0.67 FAPbI3

4

300

Figure 5. (a) Temperature dependence of the broadline 1 H NMR spectra for MAPbI3 from 100 K up to 400 K with an interval of 20 K, (b) FWHM linewidth as a function of temperature for MA1¹xFAxPbI3. (a)

x=0.33 (497K)

0

0

270.25

Frequency / MHz

(b)

6

2 5

270.15

(a)

8

0$3E, [ [ )$3E,

FWHM / kHz

(b) FWHM / kHz

(a)

350

400 T/K

MAPbI3 (500K)

450

500

270.18

270.20

Frequency / MHz

Figure 6. (a) FWHM as a function of temperature for MA1¹xFAxPbI3 at high temperature regions for quenched samples from 500 K. (b) 1H NMR spectra at selected temperature.

(b)

15 10 5 0 300

MAPbBr3 (Virgin) MAPbBr3 (Qenched) CsPbBr3 (Single crystal)

350

400

450

500

T/K

Figure 7. (a) Motional narrowing of 207Pb NMR for CsPbBr3, (b) FWHM as a function of temperature for bromide analogs, (c) Repeatedly measured FWHM linewidth for MAPbI3 as a function of temperature. (d) Simulations of FWHM as a function of temperature according to eqs (1) and (2), where τ0(n/N)¹1 = 10¹8 s, 10¹7 s, 10¹6 s, from the left, respectively.

At first, we will discuss dynamic structures of the perovskite from 1H NMR. Figure 5(a) shows a typical example of 1H NMR spectra for MAPbI3 from 100 K to 400 K with an interval of ca 20 K. Figure 5(b) summarizes FWHM linewidth for all samples including solid-solutions. At the phase transition temperatures of MAPbI3 (162 K) or FAPbI3 (120140 K, 283 K), the linewidth decreased discontinuously with increasing temperature, suggesting the stepwise changes of the dynamic structures of the cation. Similar stepwise changes were not observed clearly for solid-solutions, that is, the motional activities of cations are maintained down to the lower temperatures. In general, the linewidth observed by a broadline 1H NMR depends on the local magnetic field i.e., dipole-dipole interactions between neighboring like and unlike nuclear spins. Therefore, the dominant contributions to the 1H NMR linewidth in these perovskites arise from the interactions between intra cationic 1H-1H and inter ionic 1H-127I ones. The intra cationic 1 1 H- H interactions, which are the largest contribution to the 1 H NMR linewidth, are averaged out to zero in a stepwise manner depending on the motional type and its rate, so called “motional narrowing”. Xu et al. discussed motional state of MA+ in MAPbI3 by means of 1H NMR T1 and concluded that the C3-axis reorientation of CH3NH3+ are excited already in the lowest temperature Phase III. Furthermore, they suggested that an isotropic reorientation takes place in Phase II and I with an activation energy of 11 kJ/mol.29 Therefore, the linewidths of ca. 1617 kHz observed below Ttr = 162 K for MAPbI3 suggest a fully excited C3-axis reorientation of CH3NH3+. On the other hand, the constant linewidth of about 7 kHz above Ttr, suggests an isotropic reorientation of the cation. Recent inelastic neutron scattering experiments on MAPbI3 also supported these dynamical structures of MA+.30 1200 | Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

Since all samples show similar linewidths of 7 kHz above 290 K, the isotropic reorientations of MA+ and FA+ are excited. This resultant linewidth of ca. 7 kHz could be assigned mainly to the contribution from 1H-127I interaction. Since no further narrowing was observed up to 480 K for all samples, other motions such as the cation and/or anion diffusions were nor excited faster than the NMR timescale (1/T2 , i.e., line width). According to the early electrochemical and conductivity measurements by Mizusaki et al., the ionic conductivity of CsPbX3 (X = Cl, Br) was caused by the migration of halideion vacancies.31 Senocrate et al. also suggested by applying spectroscopic and many electrochemical techniques that iodide ions were shown to be the mobile species in MAPbI3.32 On the other hand, recent first-principles calculations suggested a vacancy associated iodide ion migration than that of MA+ or Pb2+. However, the migration activation energies were different from each other from 0.1 to 0.6 eV.3135 Hence, we have measured 1H NMR in the temperature range from 300 K up to 500 K several times using the same sample, since the motional narrowing is expected on the 1H NMR spectrum as a result of the vacancy migration. As Figure 6 shows, the linewidths for FAPbI3 and MAPbI3 did not show remarkable narrowing under the repeated measurements up to 500 K. However, the quenched solid-solutions from 500 K showed the narrowing at relatively low temperatures. For example, as Figure 6(b) shows, the profile for x = 0.33 sample at 316 K did not show a pure Gaussian shape, suggesting the migration of the iodide ion and/ or cation as a result of the increased vacancy concentration. In order to evaluate the migration of halide ions more quantitatively, we have observed 207Pb NMR as a function of temperature. Figure 7 summarizes the narrowing phenomena of the 207Pb NMR for CsPbBr3, MAPbBr3 and MAPbI3. A © 2018 The Chemical Society of Japan

narrowing phenomena was clearly observed for a single crystal of CsPbBr3 which was grown from the melt (Tm = 567 °C36). MAPbBr3 also showed the narrowing especially for the quenched sample from 500 K. However, as Figure 7(c) shows, the narrowing could not be observed clearly for MAPbI3 in spite of the similar thermal treatment. In general, motional correlation time ¸c of the mobile species could be empirically evaluated from the linewidth at the transition temperature region using a following equation,37 ³ðH 2 A2 tan 1 2ðB2 A2 Þ ; ð1Þ c ¼ ¼ kc 2³¡¢H where ¦H is a FWHM linewidth at the transition temperature region, A and B are linewidths below and above the narrowing region and α is a constant near 1. In the case of a vacant-diffusion mechanism, the motional correlation time estimated from eq (1) is not that of the vacancy itself but supposed to be an averaged value over the space surrounding the probe nucleus. Therefore, the motional correlation time ¸c or the correlation rate kc could be expressed using a modified Arrhenius equation as follows, n 1 1 Ea ¢ exp ¸c ¼ ¼ ¸0 ¢ ; ð2Þ kc N RT where (n/N) means a vacancy concentration, τ0 is a preexponential parameter of the order of 10¹1410¹10 s. Figure 7(d) shows a simulation of FWHM as a function of temperature, in which (n/N) changed 2 orders under the constant activation energy (49 kJ/mol). These simulations suggested that the motional narrowing phenomena reflect not only the activation energy but the vacancy concentration (n/N). Following kinetic parameters were obtained for bromide analogs, single crystal CsPbBr3 and virgin sample of MAPbBr3, respectively. 0 1 kJ 48:8¢ @ mol A ¸ c ðCsPbBr3 Þ ¼ 3:2 108 s¢ exp ð3Þ RT 0 1 kJ @ 48:5¢ mol A 7 ð4Þ ¸ c ðMAPbBr3 Þ ¼ 3:3 10 s¢ exp RT These activation energies for bromide analogs are roughly consistent with those for MAPbI3 from the first-principles calculations.33,34 It is interesting to note that single crystal CsPbBr3 crystallized from the melt shows the narrowing phenomena at relatively low temperature, suggesting the high concentration of the vacancies. In contrast, 207Pb NMR on MAPbI3 used in this experiment demonstrated the lower concentration of the vacancies than those of bromide analogs. Electronic Band Structures of Perovskite Halides ABX3 (B = Sn(II), Pb(II)) Based on a Hückel Analytical Calculation. In our previous report, we presented an electronic band structure for ABX3 (B = Pb, Sn) perovskites based on a Hückel method.5 Using this simple model we showed the tunable bandgap having a direct transition property. In this report we will discuss further the reduced effective masses of carriers as a function of the resonant integral (β) and the effect of bond alternation. We have introduced the following two approximations, which are derived from the characteristic bonding nature Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

of main group elements such as Ge(ll), Sn(ll) or Pb(ll) having an s-electron lone-pair. (1) Central metal ions (Ge2+, Sn2+ and Pb2+) form octahedral coordination with six halide ions, in which only three orthogonal p-orbitals of the metal ion can be available for the bonding, since the contribution from an inert s-electron lonepair and outer d-orbitals are ignored. (2) Hence, the linear -X-M-X-M- bond in an anionic sublattice consists of one p-orbital from each ion. The importance of ns2 configuration of the B2+ ion on the ABX3 electronic structure has also been suggested recently by DFT calculations.38,39 The chemical bond stated above is a orbital deficient bond and is called a “hypervalent bond”,22 which was first introduced by Pimentel as a “three-center fourelectron (3c-4e) bond” for the linear bond in I3¹ anion.40 Since -X-B-X-B- consists of one p-orbital from each ion, it is easy to simulate the electronic state using a Hückel method. In general, a ¨ © ¨ secular equation must be solved for an infinite chain to get the solution. If we introduce two Bloch equations for the central B and the X ions, however, the ¨ © ¨ secular equation reduces to a 2 © 2 as follows;20 1 ºB ðkÞ ¼ pffiffiffiffi ½ »B ðr þ aÞ expðikaÞ þ »B ðrÞ N ð5Þ þ »B ðr aÞ expðikaÞ ; 1 h a a ºX ðkÞ ¼ pffiffiffiffi »X r þ exp ik 2 2 N a a i þ»X r exp ik ; ð6Þ 2 2 HBB EðkÞ HBX ¼ 0: ð7Þ H H EðkÞ BX

XX

In this Hückel approximation, the coulomb integrals HBB and HXX of the atomic ºB(k) and ºX(k) orbitals are defined as, HBB ¼ hºB jHjºB i ¼ ¡B ;

ð8Þ

HXX ¼ hºX jHjºX i ¼ ¡X :

ð9Þ

While the non-diagonal element HBX, which contains the nearest neighbor interactions between B and X, is reduced to, ka HBX ¼ hºB jHjºX i ¼ 2¢¢ cos ; ð10Þ 2 where a is a cubic lattice constant (X-B-X), k is a wave number, β is a resonant integral between adjacent p-orbitals between B and X. Then, the final secular equation becomes, ¡B EðkÞ 2¢ cos ka 2 ð11Þ ¼ 0: 2¢ cos ka EðkÞ ¡ X 2 Solution of this determinant leads to, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ka ð¡B þ ¡X Þ ð¡B þ ¡X Þ2 4ð¡B ¡X 4¢2 cos2 Þ 2 E¼ : 2 ð12Þ Resultant dispersion behavior of the energies as a function of k is shown in Figure 8 together with the corresponding DOS in a schematic fashion. Since the chain contains 2N electrons from N halide ions, the lower conduction band is fully occupied © 2018 The Chemical Society of Japan | 1201

1.2

2

(a) m */m

Bandwidth / eV

Figure 8. Dispersion behavior of the p-orbitals and corresponding density of state.

0.8

(b)

0 -1

0.4 0.0 0.0

1

-2 0.2

0.4 0.6 β / eV

0.8

1.0

0.2

0.4

0.6

0.8

1.0

β /eV

Figure 9. (a) Band width as a function of «β«. (b) Effective masses of the electron and hole as a function of «β« at the band edge, k = π/a.

with 2N electrons and the upper valence band is empty. There is a nondegenerate energy level between αB and αX, which corresponds to the band gap. Figure 8 shows clearly that the perovskite halide is a semiconductor with a direct and a tunable band gap energy (Eg = «¡B ¹ αX«) at k = π/a. Effective Masses m of the Electron and Hole as a Function of β and Bond Alternation. According to this model stated above, the band gap does not directly depend upon the β. On the other hand, the width of the valence or conduction band depends upon β as shown in Figure 9(a). Hence, the effective masses m for the electron and hole could be calculated as a function of β using the following definition. 2 h 2 m ¼ ð13Þ d2 E dk2 Figure 9(b) shows calculated results for MAPbI3 assuming Eg = «¡B ¹ αX« = 1.61 eV,8 where effective masses m at the band edges (k = π/a) are plotted as a function of β. The positive and negative masses correspond to that of the electron and hole, respectively. Since the resonant integral «β« in perovskite halide is supposed to be ca. 1 eV, m /m is expected to decrease of the order of 0.10.2. These values are consistent with those from the optical and magnetoabsorption spectra,41 the transient absorption spectra,42 and also consistent with the Density Function Analysis.43 If the cation radius is smaller than that expected for a cubic perovskite, a tetragonal or an orthorhombic perovskite appears as the result of the sublattice shrinking such as a tetragonal MAPbI3 at RT. Although the numerical simulations for the sublattice shrinking, i.e., the tilting of PbI6 octahedra, are impossible using this simple treatment, the resonant integral «β« is simply expected to decrease. Therefore, increases of the effective masses are expected with the PbI6 tilting angles. 1202 | Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

(Figure 9). On the other hand, it is also an interesting problem to compare Eg between tetragonal MAPbI3 and cubic FAPbI3. The Eg for MAPbI3 and FAPbI3 were determined to be 1.54 eV and 1.46 eV, respectively, from our diffused reflectance spectra (SI-3). These observations suggested that the PbI6 tilting results in an increase of the Eg. This is a reasonable result since Eg = «¡B ¹ αX« is supposed to be proportional to the ionic character of the bond. The higher 127I NQR frequency for FAPbI3 than that of the MAPbI3 supported this idea, since 127 I NQR frequency reflects the covalent character of the bond. Hence, the PbI6 tilting results in the increases not only of the effective masses but also the band gap energy, Eg. However, it is interesting to note that the Eg did not show noticeable changes for MAPbI3 at the phase transition to the cubic phase (Ttr = 333 K) as shown in SI-3. One possible reason is an extremely large anisotropic atomic displacement parameter of the iodine which is perpendicular to the Pb-I-Pb bond. That is, the cubic MAPbI3 is supposed to appears as the result of the orientational disorder of the PbI6 octahedra as shown in SI-2. Next we will discuss the effect of bond alternation, i.e., a bond deformation from -X-B-X-B- to -X£B-X£B-. This type of deformation is a characteristic feature for the hypervalent compounds such as Ge(II) and Sn(II) crystals. For example, a cubic perovskite CH3NH3SnBr3 at 298 K exhibits a strong distortion at 215 K associated with the drastic color change.44,45 A pyramidal GeX3¹ anion could be also recognized in the distorted perovskite structures.46,47 In order to simulate the bond alternation effect on the electronic band structure, the non-diagonal element HBX in the secular equation, which contains nearest neighbor interactions between B and X, must be modifies as,20 HBX ¼ hºB jHjºX i ¼ ¢1 ¢eikxa þ ¢2 ¢eikð1xÞa ;

ð14Þ

where β1 and β2 are resonant integrals at X£B and B-X, and xa (x < 1/2) and (1 ¹ x)a are bond lengths at the trans position, respectively. The sum of the resonant integrals at the trans positions are kept constant as 2β for small distortions. The solution of the modified secular equation is expressed as,

1 ð¡B þ ¡X Þ E¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð¡B þ ¡X Þ2 4ð¡B ¡X ¢1 ¢2 2¢1 ¢2 cosðkaÞÞ : ð15Þ Figure 10 shows the dispersion behavior as a function of k, where the calculations were done changing β1/(β1 + β2) from 0 to 0.5 in the interval of 0.05. As is simply expected from the model, the band width of an equally spaced bond (β1/ (β1 + β2) = 1/2) shows a maximum, and hence the effective masses show a minimum. These calculations suggested that the cubic A(Pb,Sn)X3 perovskite having a large β value is a promising material for the photovoltaic cell. Conclusion MAPbI3 and FAPbI3 form continuous solid solutions with the cubic perovskite structures except the tetragonal MAPbI3. The successive phase transitions for MAPbI3 were confirmed at 162 K and 333 K by means of DTA and 127I NQR spectroscopy.


a β1/(β1+β2) = 0.5 β1/(β1+β2) < 0.5

xa (1-x)a x ≤ 1/2 2

1.2 0.5

*

E /eV

1

(b)

0

CB

m /m

(a)

0 -1

0.5

VB

0.0

0.4

0

-2 0.4

0.8

a*k/π

0.8

0.0 1.2

0.0

0.1

0.2

0.3

0.4

0.5

β1/(β1 +β2)

Figure 10. (a) Bond alternation effect on the dispersion behavior of the p-orbitals. (b) Effective masses of the electron and hole as a function of β1/(β1 + β2). β1/(β1 + β2) = 0.5 means equally spaced bond. These calculations were done assuming β1 + β2 = 2 eV.

The space groups of the perovskite are Pnma, I4/mcm and from the low temperature phase. The phase transitions Pm-3m for FAPbI3 were also confirmed at 120140 K and 283 K and it belongs to a cubic perovskite with a space group Pm-3m at 298 K. The dynamic structures of component ions were observed as a function of temperature by 1H, 207Pb NMR. The motional narrowing phenomena on 207Pb NMR suggested the vacancies associated ionic migrations. However, the defect concentration for MAPbI3 was suggested to be lower than the bromide analogs, since the onset temperature of the narrowing was higher. In the ABX3 (B = Pb, Sn) perovskite halides, the anionic sublattice consists of an infinite -X-B-X- chain formed by hypervalent bonds. This orbital deficient and infinite bond formed in the perovskite is a key point for the excellent semiconducting material as a visible right absorber. An infinite linear bond having a similar bonding character is found in PbS48,49 or SnS50 with a rock salt structure. These materials have been also expected to be good candidates for visible light absorbers in photovoltaics. This work was partially supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan. Supporting Information Describe concisely what is in the material. This material is available on http://dx.doi.org/10.1246/bcsj.20180068. References 1 N. Kitazawa, Y. Watanabe, Y. Nakamura, J. Mater. Sci. 2002, 37, 3585. 2 T. Zhang, M. Yang, E. E. Benson, Z. Li, J. van de Lagemaat, J. M. Luther, Y. Yan, K. Zhu, Y. Zhao, Chem. Commun. 2015, 51, 7820. 3 K. T. Butler, J. M. Frost, A. Walsh, Mater. Horiz. 2015, 2, 228. 4 K. Yamada, H. Kawaguchi, T. Matsui, T. Okuda, S. Ichiba, Bull. Chem. Soc. Jpn. 1990, 63, 2521.

Bull. Chem. Soc. Jpn. 2018, 91, 1196–1204 | doi:10.1246/bcsj.20180068

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