Hasebe, Katsuhiko

Kawamura, Yukihiko

増山, 博行

The cubic-tetragonal phase transition of CH_3NH_3PbI_3 was investigated by single crystal X-ray diffractometry. The crystal structure was refined at five temperatures in the teragonal phase. The PbI_6 octahedron rorates around the c-axis alternatively to construct the SrTiO_3-type tetragonal structure. A methylammonium ion is partially order

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Typeset with jpsj2.cls <ver.1.2> Full PaperStructural Study on Cubic-Tetragonal Transition of CH3NH3PbI3Yukihiko KAWAMURA, Hiroyuki MASHIYAMA and Katsuhiko HASEBEDepartment of Physics, Faculty of Science, Yamaguchi University, Yamaguchi 753-8512, Japan(Received 2002 )The cubic-tetragonal phase transition of CH3NH3PbI3 was investigated by single crystalX-Ray diﬀractometry. The crystal structure was reﬁned at ﬁve temperatures in the teragonalphase. The PbI6 octahedron rorates around the c-axis alternatively to construct the SrTiO3-type tetragonal structure. A methylammonium ion is partially orderd; 24 disordered statesin the cubic phase are reduced to 8. With decreasing temperature, the rotation angle of theoctahedron increases monotonically, which indicates it is an order parameter of the cubic-tetragonal transition.KEYWORDS: X-ray diﬀraction, structure analysis, order parameter, tetragonal phase, methy-lammonium triiodideplumbate1. IntroductionMethylammounium triiodideplumbate, CH3NH3PbI3 takes a cubic perovskite struc-ture (space group Pm3m, Z=1) at high temperature.1) Since the methylammoniumion,CH3NH+3 (hereafter abbreviated as MA+) has a lower symmetry of C3v, the orientation ofthe MA+ ion should be disordered to satisfy the Oh symmetry. The same type cubic crys-tals can be realized with replacing I by Cl or Br; on decreasing temperature, all three com-pounds perform the cubic-tetragonal transition.1,2) In MAPbI3, the transition temperatureis 327.4K. The tetragonal crystal system(space group I4/mcm, Z=4) becomes orthorhombicbelow 162.2K.2,3)In the cubic phase, a MA+ ion is in a cage made by four PbI6 octahedra. Because theoctahedron is rather large compared with the general perovstite-type crystal such as BaTiO3,it is considered that MA+ ions can move freely inside the cage at high temperatures.2)The speciﬁc heat of MAPbX3(X=Cl, Br, and I) was measured with high accuracy, andthe transition entropy was estimated from the speciﬁc heat and the latent heat.3) The entropyof the cubic phase was estimated to be Rlog 24. In order to explain the entropy change withaccompanying the seccessive transitions, the following three models have been proposed.Model A: The C-N axis of a MA+ ion aligns along <100>, six directions. Since the <100>axis has a 4-fold symmetry, the MA+ ion rotates around the axis to occupy 4-sites equivalently.The total degree of freedom is 6 times 4, i.e. 24.Model B: The C-N axis aligns along <110>, twelve directions. Since the <110> axis hasa 2-fold symmetry, the MA+ ion rotates around the axis to occupy 2-sites equivalently. The1/??J. Phys. Soc. Jpn. Full Papertotal degree of freedom is 12 times 2, i.e. 24.Model C: The C-N axis aligns along <111>, eight directions. Since the <111> axis hasa 3-fold symmetry, the MA+ ion rotates around the axis to occupy 3-sites equivalently. Thetotal degree of freedom is 8 times 3, i.e. 24.It was demonstrated that each three models explained the observed type of successive transi-tions: cubic phase - tetragonal phase - orthorhombic phase.3)We have analyzed the cubic structure by the use of X-ray diﬀraction data.4) Although theR-factor (the residue of the least-squares calculations) for each model gave almost the samegoodness, the bond length of C-N was most plausible for the model B. The diﬀerential Fouriersynthesis suggested that the halogen atoms were disordered to occupy four equivalent sitesrandomly.On the other hand, more than 2 decades ago, the cubic structures of CsPbCl3 and CsPbBr3were analyzed. The halogen atoms were assumed to vibrate within an anharmonic potentialand the picture of disordered halogen atoms was abandoned.5)Another way to clarify the disordered structure is to determine the tetragonal structure,which is considered to display the ordered conﬁgration. In this report we collect intensitydata of X-ray diﬀraction in the tetragonal phase of MAPbI3. Usually even a small peace ofcrystal consists of many tetragonal domains. However, we suceeded to ﬁnd a single crystalsample after a number of trials. The single crystal analysis has been performed successfully.It is found the rotation angle of the PbI6 octahedron increases with decreasing temperaturein the tetragonal phase, which demonstrates the displacive character of the octahedron at thecubic-tetragonal transition.2. ExperimentalSingle crystals prepared as described previously6) were kindly supplied by Prof. Gesi.Spherical samples were cut and polished into 0.3mm in diameter, among which a single crys-tal was selected and was mounted on an oﬀ-centered 4-circle diﬀractometer (HUBER 424)controlled by MXC (Mac Science). Cold nitrogen gas ﬂow (Oxford Crysteram) was employedto keep the temperature of the sample within ±0.5K. Graphite-monochromated Mo Kα radi-ation was used from a rotating anode generator with 50kV-250mA.The space group of the tetragonal phase (165K<T<328K) was conﬁrmed to be I4/mcm(Z=4), which is the same space group with the tetragonal phase of SrTiO3. The intensitydata were collected at 323K, 310K, 280K, 250K, 220K and 180K.Lorentz and polarization corrections were made automatically by MXC. Absorption cor-rection for spherical sample was made by numerical integration. Starting from the SrTiO3-type parameters, structure parameters for Pb and I were reﬁned by least-squares calculations(BDLS of AXS98 program7)). Atomic scattering factors were taken from International Ta-ble.8) The R-factor converged at R=0.151 with the use of isotropic thermal parameters. Then2/??J. Phys. Soc. Jpn. Full Paperdiﬀerential Fourier synthesis was made to ﬁnd the positions of C and N of MA+ ion. Thehydrogen atoms are neglected because of their poor contribution in scattering. The diﬀerenceof C and N is little in comparison to Pb and I. Since NH3 of MA+ is considered to be posi-tively charged, the closer positions to I are assigned to N. At the ﬁnal stage of calculations,anisotropic thermal parameters were introduced for Pb and I, and the calculations convergedat R=0.060. Crystal data at 220K are summarized in Table I.3. Results3.1 Lattice constantsFirstly, the temperature dependence of lattice constants a and c is shown in Fig. 1; wherea˜ = a/√2 and c˜ = c/2 are plotted for the tetragonal phase in order to compare with thecubic phase. With decreasing temperature, the parameter a˜ decreases monotonically. On theother hand, the parameter c˜ increases when the crystal transforms into the tetragonal phase.Since the unit cell volume V˜ or the equivalent lattice parameter d˜ = V˜ 1/3 depends linearlyon temperature, the diﬀerences of a˜ and c˜ from d˜ are considered to be the anomalous parts,δa˜ and δc˜, of the lattice parameters, and are plotted in Fig. 2. Although the cubic-tetragonaltransition accompanies latent heat and the character is a ﬁrst order transition,3) the latticeparameters seem to change continuously with the following relations:δa˜ or δc˜ ∝ (Tc − T )β˜ , (1)where the ﬁtted transition temperature was Tc =331K. The exponent is estimated asβ˜=0.42(2) both for δa˜ and δc˜.3.2 Superlattice reflectionsWith accompanying the cubic-tetragonal transition, superlattice reﬂections appear in thelow temperature phase. The temperature dependences of two reﬂections 125 and 411 areplotted in Fig. 3. As if a second-order transition, the structure factor can be written in apower low relation:F (hkl) = F0(1− T/Tc)β′ . (2)In the whole temperature range of the tetragonal phase, the exponent β′ is ﬁtted as 0.22(3).3.3 Tetragonal structureThe unit cell structure at 220K is shown in Fig. 4. The atomic parameters are listed inTable II. The MA+ ions align along the [101]c, [1¯01]c, [01¯1¯]c and [011¯]c direction of the cubicstructure. Therefore the model B of the cubic structure is appropriate. However, the MA+ions align not within (001) plane as expected previously, but they point <111> direction ofthe tetragonal phase. In this conﬁguration, NH+3 is closer to I− and the electrostatic energy3/??J. Phys. Soc. Jpn. Full Paperwill be lower, here the charge is just nominal. Furthermore, the atomic spacing within x-yplane will shrink and the spacing along the c-direction will expand; the probability to alignMA+ within x-y plane disappears.The iodine atom deviates from the line connecting Pb atoms in the x-y plane. As in thecase of SrTiO3, this deviation is considered to be induced by a rotation of the PbI6 octahedronaround the c-axis. At 220K (110K below the transition temperature), this rotation angle φis 10.5◦, which is larger than the angle 1.4◦ in SrTiO3 at 77K9) (28K below the transitiontemperature; the reduced temperature t = (105 − 77)/77 = 0.36 is comparable to 220K ofMAPbI3).The increase of the superlattice reﬂection in the tetragonal phase shown in Fig. 3 can beexplained either by the gradual increasing of the rotation angle, or by the gradual orderingof the disordered iodine atoms and MA+ ions. The former model is a displacive model withrespect to iodine, and the latter one is the partially ordered model. These two models wereanalyzed at every temperature. It was found that the occupation probability of the splittediodione converged to almost 1.0 if the anisotropic thermal parameters were adopted. Thereforewe conclude that the displacive model is appropriate to the tetragonal structure.The rotation angle φ is plotted in Fig. 5. The temperature dependence is also ﬁtted to therelationφ ∝ (Tc − T )β . (3)Since the number of data is a few, the transition temperature was ﬁxed at 331K. The exponentwas estimated as β=0.25(6).4. DiscussionIn the foregoing section, we described three kinds of temperature dependence: the deviationof lattice parameters, the intensity of superlattice reﬂections and the rotation angle φ of PbI6octahedron. Firstly we note a relation between δa˜ and φ. If the octahedron rotates around theaxis parallel to the c axis by φ, then the spacing between the neighboring Pb’s decreases bya0(1−cosφ), where a0 is the lattice parameter of the cubic phase. Therefore δa˜ is proportionalto φ2 for small φ. This is the reason why the exponent β˜ for δa˜ (β˜=0.42) is twice of β for φ(β=0.25).The PbI6 octahedron has large thermal vibration in cubic phase.4) The root mean squareddisplacement√u2⊥ is about 0.4A˚ perpendicular to the Pb-I-Pb bond, but√u2‖  0.14A˚parallel to the bond. After the cubic-tetragonal phase transition√u2⊥ decreases to 2/3, whichmakes the bond to elongate along the tetradonal c-axis. This is schematically drawn in Fig. 6to explain the expansion of the c-axis.The observed exponent β′ for superlattice reﬂections F (125) and F (411) coincides with4/??J. Phys. Soc. Jpn. Full Papereach other. The structure factor F (k) is given by the summation of the atomic position rj asF (k) ∑jfjeik·r0j + i∑jfj∆rj · keik·r0j , (4)where ∆rj is a small deriation from the cubic phase position r0j . The ﬁrst term of the righthand side of eq. (4) is zero for a superlattice reﬂection. In our crystal the deviation of iodineis dominant in F. Since iodine I(1) rotates by φ, it is displacement is ∆r = a0sinφ ∝ φ.Therefore the exponent for structure factor F (hkl) of superlattice reﬂections coincides tothat of the rotation angle φ. The order parameter of the tetragonal phase is considered to bethe rotation angle of the PbI6 octahedron, just as same as in SrTiO3, though the magnitudeof the angle is quite large in MAPbI3.In the early studies on MAPbX3, the ordering of the orientation of the MA+ ion hasbeen attributed to the trigger of the transition.2–4) The previous structure analysis of thecubic phase also suggested the disordered PbX6 octahedron in MAPbCl3, MAPbBr3 andMAPbI3.4) However, the anharmonic thermal vibration has been introduced to analyze thecubic structure of CsPbCl3 and CsPbBr3,5) where the lattice parameters are almost the sameas MAPbCl3 and MAPbBr3. Since the residual electron density was at most 1.0 e/A˚3 as shownin the diﬀerential Fourier map (Fig. 1 of our previous report4)), the cubic structure may beanalyzed well by the use of anharmonic thermal vibration model.In summary, the tetragonal structure of MAPbI3 was determined successfully with thedisplacive model for iodine atoms by means of single crystal diﬀractometry. The MA+ ionsstill retain disordered in their orientations. Protons are considered to be motionally disorderedaround the C-N axis. The rotation of the PbI6 octahedron is considered to be the orderparameter of the cubic-tetragonal transition.Finally, the reason why the value of the exponent β is not 1/2 but about 0.25 can beexplained in three ways.(1) The critical behavior appears as similar to SrTiO3, where the exponent is about 1/3.9)(2) The transition takes place accidentally near the tricritical point. Then the classical valueof the exponent is 1/4.(3) Since the transition is a ﬁrst order one, a small discontinuity eﬀectively suppresses theexponent if it is estimated by the continuous function of eq. (3).Two of these reasons may cooperate to decrease the exponent from the clasisical value of 1/2.AcknowledgementsThe authers are grateful to Prof. Gesi of Iwaki Meisei University for providing the singlecrystals. They also were indebted to Center for Instrumental Analysis, Yamaguchi University,for the use of X-ray equipment.5/??J. Phys. Soc. Jpn. Full PaperReferences1) D. Weber: Z. Naturforsch. 33b (1978) 1443.2) A. Poglitsch and D. Weber: J. Chem. Phys. 87 (1987) 6373.3) N. Onoda-Yamamuro, T. Matsuo and H. Suga: J. Chem. Solids. 51 (1990) 1383.4) H. Mashiyama, Y. Kurihara and T. Azetsu: J. Korean Phys. Soc. 32 (1998) S156.5) M. Sakata, J. Harada, M. J. Cooper and K. D. Rouse: Acta Cryst. A36 (1980) 7.6) K. Gesi: Ferroelectrics 203 (1997) 249.7) H. Mashiyama: J. Phys. Soc. Jpn. 60 (1990) 220.8) Int. Union of Crystallogr: International Tables for Crystallography Vol. C (Kluwer Acad. Pub.,Dordrecht, 1999).9) K. A. Mu¨ller and W. Berlinger: Phys. Rev. Letter. 26 (1971) 13.6/??J. Phys. Soc. Jpn. Full PaperTable I. Crystal data of CH3NH3PbI3 at 220K.Phase IISpace group I4/mcmZ 4a(A˚) 8.800(9)c(A˚) 12.685(7)Index range 0 ≤ h ≤ 120 ≤ k ≤ 120 ≤ l ≤ 16Number of independent reﬂections 360(Fo > 5σFo)Radius of the sample (mm) 0.125Absorption coeﬃcient (mm−1) 26.1R−factor 0.060∆/σ 0.0387∆ρ[e/A˚3] −3.31 ≤ ρ ≤ 1.72Table II. Atomic parameters at 220K in the tetragonal phase of CH3NH3PbI3. The e.s.d.’s are givenin parentheses for independent parameters.x y z Ueq (A˚2)Pb 0.0 0.0 0.0 0.0206(5)I(1) 0.2039(8) 0.2961 0.0 0.05551(10)I(2) 0.0 0.0 0.25 0.0521(18)C 0.555(17) -0.055 0.264(11) 0.0404(33)N 0.459(20) 0.041 0.202(9) 0.0582(41)U11 U33 U12Pb 0.0200(9) 0.0218(7) 0.0I(1) 0.0327(19) 0.0700(15) 0.0329(20)I(2) 0.0708(36) 0.0149(13) 0.07/??J. Phys. Soc. Jpn. Full Paper??? ??? ??? ??? ?????????????????????????????????????????????????????????Fig. 1. The temperature dependence of lattice parameters in CH3NH3PbI3.8/??J. Phys. Soc. Jpn. Full Paper??????????????????????? ??? ??? ??? ???????????????????????????????Fig. 2. The anomalous part of lattice constants in the tetragonal phase of CH3NH3PbI3.9/??J. Phys. Soc. Jpn. Full Paper???????????? ??? ??? ??? ???? ???????????????????????????????????Fig. 3. T5che temperature dependence of structure factor for superlattice reﬂections in the tetragonalphase of CH3NH3PbI3.10/??J. Phys. Soc. Jpn. Full PaperFig. 4. The half of the tetragonal unit cell (0<z<1/2) of CH3NH3PbI3 projected on the x-y plane(top). Directions of the methylammonium ion around (1/2, 0, 1/4) are indicated schematically inbottom.11/??J. Phys. Soc. Jpn. Full Paper024681012150 200 250 300 350 400Angle(degree)Temperature(K)?????????????????Fig. 5. The temperature dependence of the rotation angle of the PbI6 octahedron in the tetragonalphase of CH3NH3PbI3. The arrow indicates the rotation angle in the cubic phase analyzed by thedisordered model (after ref. 4).12/??J. Phys. Soc. Jpn. Full PaperFig. 6. Schematic picture of the Pb-I-Pb bond along the tetragonal c-axis.13/??

Structural study on cubic-tetragonal transition of CH_3NH_3PbI_3

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