Distortion modes in halide perovskites: to twist or to stretch, a matter of tolerance and lone pairs
DDistortion modes in halide perovskites: to twist or to stretch, a matter of toleranceand lone pairs
Santosh Kumar Radha a , Churna Bhandari b , and Walter R. L. Lambrecht a a Department of Physics, Case Western Reserve University,10900 Euclid Avenue, Cleveland, OH-44106-7079 and b Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
Using first-principles calculations, we show that CsBX halides with B=Sn or Pb undergo oc-tahedral rotation distortions, while for B=Ge and Si, they undergo a ferro-electric rhombohedraldistortion accompanied by a rhombohedral stretching of the lattice. We show that these are mutuallyexclusive at their equilibrium volume although different distortions may occur as function of latticeexpansion. The choice between the two distortion modes is in part governed by the Goldschmidttolerance factor. However, another factor explaining the difference between Sn and Pb comparedwith Ge and Si is the stronger lone-pair character of Ge and Si when forced to be divalent as is thecase in these structures. The lone-pair chemistry is related to the off-centering. While the Si-basedcompounds have not yet been synthesized, the Ge compounds have been established experimentally.As a final test of the importance of the tolerance factor we consider RbGeX , which has smallertolerance factor than the corresponding CsGeX because Rb is smaller than Cs. We find that it canlower its energy by both rotations or rhombohedral off-centering distortions but the latter lower theenergy slightly more efficiently. I. INTRODUCTION
The cubic perovskite structure is well known from theoxide perovksites to exhibit various possible phase tran-sitions. These fall in two main categories: ferro-electricdistortions, in which the B atom in ABX is displacedwithin its surrounding octahedron, and antiferro-electricdistortions, in which the octahedra rotate, possibly aboutmultiple axes. Depending on the type of displacement,for example along a cubic axis such as [001], or along twocubic axis or a [110] direction, or three cubic axes, cor-responding to the [111] axis, the resulting symmetry be-comes tetragonal, orthorhombic or rhombohedral. Like-wise for the rotation type instabilities, rotation aboutone cubic axis leads to a tetragonal structure, about twoorthogonal axes leads to an orthorhombic phase.The halide perovskites with B=Pb, Sn, Ge have re-cently garnered a lot of attention, mostly driven by thehybrid organic/inorganic halides’ demonstrated poten-tial for solar cell applications.[1–8] In particular methylammonium lead iodide (CH NH PbI or (MA)PbI orMAPI) and closely related materials have reached largerthan 20 % efficiencies in solar cells in a record develop-ment time frame. The interplay between the dipole char-acter and the orientation of the organic component andthe inorganic framework leads to interesting effects onthe above mentioned phase transitions.[9, 10] However,similar phase transitions also occur in the purely inor-ganic CsBX family. While these distortions, which arerelated to soft-phonon mode instabilities,[11] lead to mi-nor changes in the band structure, related to bond angledistortions, other phases are known in the halides, whichare far more disruptive of the band structure. Theselatter phases include edge-sharing octahedra and exhibitband structures with much wider band gaps than theirperovskite counterparts.[12] As an example, the struc-tural phases in CsSnI were studied in detail by In Chung et al. [13]. They fall generally in a set of three “blackphases”, cubic, tetragonal and orthorhombic, which cor-respond to rotated octahedral structures, and another or-thorhombic “yellow phase”, which has 1D chains of edge-sharing octahedra forming Sn I − structural motifs. Itis notable that the transitions from cubic to tetragonalto orthorhombic perovksite each time increase the den-sity and the yellow phase has an even higher density.The orthorhombic γ -phase is stable with respect to soft-phonons, but has been calculated to have an energy ei-ther lower[14] than or very close[15] to that of the yellowphase.Because the driving force for these transitions appearsto be the increasing density, the occurrence of the edge-sharing octahedral structures, which appears to be detri-mental for many of the sought applications, may perhapsbe already inferred from the behavior of the material un-der octahedral rotations, which in turn is related to therelative sizes of the ions. For example, for the CsGeX compounds, the sequence of tetragonal, orthorhombic oc-tahedral rotations is not observed and, to the best of ourknowledges, no edge-sharing octahedral phase is knownto occur, although a different, monoclinic phase occursfor the Cl members of the family. Instead of octahedralrotation phases, a ferro-electric rhombohedral distortionis found to occur in these materials, consisting of the dis-placement of the Ge along the body diagonal of the cubicunit cell, accompanied by a rhombohedral stretch of theunit cell.In this paper we examine the behavior of a family ofhalide perovskites computationally under both octahe-dral rotation and rhombohedral ferro-electric distortions.Hence the phrase in the title: “to rotate or to stretch”.We find that the Sn and Pb members of the family of cu-bic perovskites are unstable toward rotation of the octa-hedra but stable with respect to ferro-electric distortions.In contrast, the Ge and Si based compounds show the op- a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b posite behavior: they are unstable towards ferro-electricdistortion but are stable with respect to rotations. Fur-thermore we relate this distinct behavior to the Gold-schmidt tolerance factor,[16] which provides a convenientway to summarize the relative ionic sizes. Notably, weinclude here the Si based halide perovskites, which have,as far as we known, not yet been synthesized.The remainder of the paper is organized as follows.The details of our computational approach are given inSec. II. The relationships between the different crystalstructures and distortions to be studied are given in Sec.III. The results Sec. IV is divided in several subsections.First, we give a qualitative discussion in Sec. IV A es-tablishing the different behavior of Sn and Pb vs. Si andGe. Next, we consider full relaxations of the rotation-ally distorted structures of Sn and Pb based compoundsin Sec. IV B, then the full relaxations of the rhomobohe-dral structures of the Ge and Si based compounds in Sec.IV C. In Sec. IV D we study the competion between bothtypes of distortion as function of lattice expansion for theSn and Pb based systems. Finally in Sec. IV E we con-sider the RbGeX compounds and end with a summaryof the results in Sec.V. II. COMPUTATIONAL METHODS
The calculations are performed within density func-tional theory in the local density (LDA) and/or gen-eralized gradient (GGA) approximations. Specifially,we use the Perdew-Burke-Ernzerhof (PBE) form ofGGA.[17] The full-potential linearized muffin-tin orbital(FP-LMTO) band-structure method is utilized.[18, 19]Within this method, the basis set consists of Blochsums of atom centered spherical waves as envelopefunctions, described by smoothed Hankel functions,[20]which are then augmented with solutions of the radialSchr¨odinger equation inside muffin-tin spheres and theirenergy derivatives. For the present calculations, a largebasis set of spdf − spd with two sets of Hankel functiondecay constants κ and smoothing radii is used. Insidethe sphere, augmentation is done to an angular momen-tum cut-off of l max = 4. The Cs 5 p states are treated asvalence electrons. Likewise for Rb, the semicore 4 p aretreated as local orbitals. The Brillouin zone integrationsare done with a 6 × × III. CRYSTAL STRUCTURES
We start from the cubic perovskite structure. In thisstructure, with a simple cubic Bravais lattice, for the θ x,x+1/2 FIG. 1. Rotation of octahedra in perovskite structure, largecircles: A atom, small open circle: B atom, smallest filledcircle: X atom, the black small circle corresponds to the cubicperovskite position, the red one the rotated one. The bluedash-dotted trangle indicates the rotation angle θ . composition ABX , the B atom occurs in the centerof the cubic unit cell and is octahedrally surroundedby X atoms on the face centers. The A atoms oc-cupy the corners of the cubic cell. The stability is gov-erned among other by the Goldschmidt tolerance factor, t = ( R A + R X ) / √ R B + R X ), where R A , R B , R X are theionic radii. Hence for t = 1 the ionic spheres are touchingand hence Goldschmidt’s original idea was that t shouldnot deviate too far from 1 for the perovskite structure tobe stable. When t <
1, the A ion is somewhat too smallfor the interstitial space between the octahedra. This iswhat leads to the rotations, which tighten the space forthe A ion. In contrast, when t >
1, the octahedral spaceis too large for the B ion, which might then be expectedto shift in its surroundings to make stronger bonds witha subset of the 6 neighbors. On the other hand, it is notso clear a-priori whether this is related to the tolerancefactor or to the lone-pair character of the B-cation.In terms of octahedral rotations, we consider both thein-phase and out-of-phase rotations about a single cu-bic axis. These both lead to a tetragonal structure, thefirst one having the space group No. 127, P /mbm or D h , the second one space group No. 140 , I /mcm or D h . They correspond to the Glazer tilt systems[21, 22] a a c + and a a c − respectively. Although other Glazertilt systems are possible and in fact occur in the Sn-halide perovskites,[11] (a rotation about a second axis(Glazer a + b − b − leads to the orthorhombic P nma or D h γ -phase), we here are primarily concerned with the in-stability either with respect to rotation of octahedra orferro-electric distortions and thus consider the tetrahe-dral rotation as the trigger toward rotation behavior. So,we do not consider other tilt systems. In the tetrag-onal P /mbm structure, the Wyckoff positions for theB-atoms is 2a, for the A-atom is 2c, for the X atoms, 2band 4h. The x parameter of the 4h positions is relatedto the rotation angle of the octahedron by tan θ = 1 − x as can be seen in Fig. 1. In fact, the blue rectangular TABLE I. Shannon ionic radii ( R i ) and tolerance factors ( t )of cubic perovksites. The last column indicates whether thecubic structure is unstable toward octahedron rotation.Ion R i (˚A)Cs 1.88Rb 1.52Si 0.4Ge 0.53Sn 0.69Pb 0.775Cl 1.81Br 1.96I 2.2Compound t rotationsCsSiI triangle marked by one corner at position ( x, x + ) hassides ( x − ) √ √ / a and hence their ratio gives tan θ .As far as the ferro-electric distortions, we only considerthe rhombohedral structure corresponding to a displace-ment of the central B ion along the [111] direction. Inthe prototypical ferro-electric oxide BaTiO this phaseoccurs at the lowest temperatures, with an orthorhom-bic and tetragonal phase occuring at higher temperaturesand eventually a cubic phase. Cooling from high temper-ature, the displacement thus acquires successively morecomponents along the cubic axes which deviate fromthe central position. While we presently do not excludethese other potential phases, our choice is guided by theCsGeX compounds, which have been found to exhibitthis rhombohedral phase at low temperatures and a cu-bic phase at high temperatures but no other phases inbetween. The rhombohedral symmetry distortion of theion is accompanied by a rhombohedral shear of the lat-tice vectors. Thus we will study the energy as function ofdisplacement of the ion for varying rhombohedral strain.The occurrence of this distortion in Ge based halidesbut not in Sn or Pb based systems, which we will demon-strate later, is not only related to the Goldschmidt ratioof ionic sizes but is also related to the lone-pair charac- ter of the bonding. As one goes down the column ofgroup-IV atoms, the valence s states become increas-ingly deeper relative to the valence p states. That iswhy carbon has s and p orbitals of similar extent andis extremely flexible in choosing different hybridizationschemes: sp in graphite, sp in diamond and so on. Siand Ge clearly prefer sp hybridization and thus tendto be tetravalent, while Sn and Pb become increasinglydivalent. Nonetheless, in the halide perovskite crystalstructure, it is clear that even Ge behaves as a divalention. Whether Si can also be forced to be divalent in thesecompounds remains to be seen. However, the s -electronsthen behave as a stereochemically active lone-pair, whichpromotes off-centering of the Ge in its surrounding oc-tahedron with an asymmetric bonding configuration inwhich the lone pair electrons are located opposite to thedirection of the displacement of the ion.[23] The lone-pairrelated trends in the series Pb-Sn-Ge have been addressedby Waghmare et al. [24] in the context of IV-VI com-pounds. We will show that even in the Sn-case this hap-pens under lattice expansion, as was previously shownby Fabini et al. [25] According to the latest insights intolone-pair chemistry, the hybridization with the anion p -orbitals play a crucial role in this. The important role ofthe Sn- s halogen- p hybridization on the band structureof CsSnX halides was already pointed out in our earlierwork.[12] We point out here that competition betweenrotation instabilities and lone-pair off-centering was pre-viously studied in CsPbF by Smith et al. [26] Lone pairphysics related to Pb also occurs when Pb is the A-cationin oxide perovskites.[27, 28]Finally, we should mention that the tolerance factordepends on the choice of ionic radii. Usually we use theShannon[29] ionic radii for this purpose. However, theseare themselves based on an analysis of bonding in dif-ferent coordinations and for example do not give us anyinformation on the behavior under hydrostatic pressure.One might conceivably think of the relative ion sizes tochange with pressure or wish to include other aspectsthan pure ionic size to predict structural stability.[30, 31]With these precautions, we used the Shannon ionic radiicalculated tolerance factors as a guide to our study. Theyare summarized in Table I. We note that our goal withthe tolerance factor is not so much to predict struc-tural maps in the sense of separating perovskite ver-sus non-perovskite forming compounds but rather thetype of structural distortion occurring within the per-ovskite. Also, because Shannon only provides ionic radiifor Pb(II) in the divalent state, but not for Sn, Ge orSi, we used instead the tetravalent radii for octahedralenvironment. This may seem to contradict the fact thatin these structures the B ion is supposed to be divalent.On the other hand, we should recognize that the bondingis partially covalent anyways. We find that within eachgroup of a given anion, the tolerance factor decreasesalong the sequence Si-Ge-Sn-Pb. The dividing criticalvalue between octahedral rotations being favored or not,depends actually on which anion (a similar point was also -15 -10 -5 0 5 10 15 θ (°)-100102030 T o t a l e n e r gy ( m e V ) θ −θ + CsSnI -0.06 -0.04 -0.02 0 0.02 0.04 0.06Displ. of Sn (unit of a)050100150200250300 T o t a l e n e r gy ( m e V ) Cubic CsSnI FIG. 2. left: Total energy (per formula unit in meV) vs. oc-tahedral rotation angle θ ( ◦ ) in CsSnI left. Here θ + standsfor out-of-phase rotation and θ − for in-phase rotation respec-tively. Right: total energy vs. displacement of Sn from bodycenter in unit of the cubic lattice constant a . made by Travis et al. [31]), but is close to 1 for all the Cscompounds. For the Cl compounds, it would be between1.115 and 1.044. For Rb which has a smaller radius, thevalue 1.006 is thus definitely on the small side and hencepredicts rotations to occur.Our goal in this paper is to study the instability ofthe cubic structure to these two types of distortion asfunction of the B atom and to correlate them with thetolerance factors in the above Table I. IV. RESULTSA. Qualitative discussion
First, we consider the CsSnI compound. In Fig. 2awe show its total energy as function of rotation angle θ ofthe octahedra. This calculation is done at the cubic ex-perimental volume although we know that the observed β -structure, corresponding to the P /mbm space grouphas higher density. We consider both the in-phase andout-of-phase rotations. The figure shows that their en-ergy is almost indistinguishable. More importantly, itshows clearly that the system prefers a rotation angle ofabout 6.9 ◦ . Of course, the rotation can be either clock-wise or counterclockwise. The energy barrier betweenthe two is of the order of a few meV/formula unit. So,this agrees with the well-established fact that CsSnI un-dergoes octahedral rotations of this type although theequilibrium optimum angle appears to be somewhat un-derrestimated compared to the experimental angle whichis 9 ◦ , corresponding to the Wyckoff parameter x = 0 . under theferro-electric rhombohedral distortion. We do this at zerostrain, so keeping the cubic lattice vectors. Clearly there -15 -10 -5 0 5 10 15 θ (°)02468 T o t a l e n e r gy ( m e V ) θ + θ -CsGeI -0.06 -0.04 -0.02 0 0.02 0.04 0.06Disp.of Ge atom(unit of a)-50-250255075100 T o t a l e n e r gy ( m e V ) η=1η=1.01η=1.015η=1.02η=1.025η=1.03η=1.035η=1.04η=1.045η=1.05 CsGeI FIG. 3. Left: Total energy vs. octahedral rotation angle θ ( ◦ )in CsGeI left. Here θ + stands for out-of-phase rotation and θ − for in-phase rotation respectively. Right: total energy vs.displacement of Ge from body center in units of the latticeconstant a . is only one minimum at exactly the central position ofthe Sn in its octahedral cage. So, there is no evidencefor a ferro-electric instability. Nonetheless, the curvesare clearly not parabolic but show a rather flat energyminimum region for the position of the central atom.In contrast, if we consider CsGeI , (Fig. 3a) for itsrotational stability, we find no evidence at all of a ro-tational instability. The preferred angle is 0. This istrue for both in-phase and out-of-phase rotations. Onthe other hand, in Fig.3b we see that now there is a clearinstability against the ferro-electric displacement. Again,it is symmetric with respect to the central position. Thedisplacement is given in units of the lattice constant a of the cubic cell. The optimimum position lies between0.52 and 0.54 or 0.46 and 0.48. In this case, we studythe optimum position and the energy barrier as functionof rhombohedral strain but initially keeping the volumefixed at that of the cubic structure. This is quantified bythe parameter η which gives the stretch along the [111]direction (when η >
1) and is compensated by a com-pression in the orthogonal directions, which conserves thevolume. Thus, we applied here a pure shear or tracelessstrain at fixed volume. We can see that the optimum po-sition varies slightly with the strain. The lowest overallenergy occurs for a strain of η = 1 .
03 and δu = 0 . u and η but also thevolume and the results of such a full relaxation are givenin Table III in Sec. IV C.We thus see a mutually exclusive behavior of the twotype of distortion modes. Either the material is unstableunder rotations, or it is unstable under the ferro-electricdistortion but not both. We found that these structuralinstabilities already occur at the cubic structure equi-librium volume but once the distortion takes place andfull relaxation is allowed, a new equilibrium is found.We should remember though that the mutual exclusiv-ity correspond to the experimental volume. This mightchange as function of pressure. For example, in SrTiO ,Zhong and Vanderbilt[32] predict an interplay betweenthe two types of distortions, leading eventually to a com-plex phase diagram as function of pressure and tempera-ture. We will discuss the distortion behavior for CsSnI as function of lattice constant later.Having established the basic two types of behavior, wenow consider the variation with anion. In the CsSnBr and CsSnCl cases, we again find the structure to be sta-ble against ferro-electric distortion, but unstable towardrotations. The energies as function of rotation angle aregiven in Suppelementary Material.[33] For the CsGeBr ,CsGeCl cases, we find the structures to be stable underrotation as expected but we do find a ferro-electric dis-tortion in both cases.[33] Next we show that Pb behavessimilar to Sn and Si behaves similar to Ge.[33] For theSi case, where no experimental results are known, we ini-tially used the LDA optimized lattice constants for thecubic CsSiX case but in the next section for our fullyrelaxed structures, we use GGA-PBE for improved accu-racy. B. Full structural relaxation for Sn and Pb basedrotations.
In this section we study the fully relaxed tetragonal P /mbm structure corresponding to the rotational dis-tortions. The optimum rotation angles are summarizedin Table II. Because we found LDA to underestimate thelattice constants more than GGA overstimates them, weperformed the full structural relaxations in GGA-PBE.In Table II we show both the results for the rotationangle when fixing the lattice constants to be “rotated cu-bic” and fully relaxing the tetragonal structure, i.e. alsorelaxing c/a . By “rotated cubic” we mean we consider a √ × √ c/a ratio exactlyat a factor √ , there are two sets of experimental results,by Yamada et al. [34] and by In Chung et al. [13]. Ya-mada et al. give a = 8 .
772 ˚A, c = 6 .
261 ˚A, V = 240 . for the β -structure and a = 6 .
219 ˚A, V = 240 .
526 ˚A for the cubic structure, in other words, almost the samevolume. In contrast, In Chun et al. [13] give a = 8 . c = 6 . V = 235 .
27 ˚A for the tetragonal andand a = 6 . V = 238 .
99 ˚A for the cubic struc-ture. These results correspond to 500 and 380 K respec-tively and clearly show a smaller volume for the tetrago-nal structure. Our calculated results agrees qualitativelybetter with those of In Chun et al. [13] in finding a volumereduction induced by the α → β transition. Our GGAcalculations overestimate the experimental volumes byabout 5.6% and 3.6 % for the cubic and tetragonal struc-tures compared to In Chung et al. [13] We find systemati-cally the same trend in volumes for the other compounds.We may note that the optimum rotation angle dependsstrongly on volume. It is typically larger in the relaxed TABLE II. Structural relaxation results for rotation for theCsSnX and CsPbX compounds: α (cid:48) means “rotated cubic”and β means fully relaxed tetragonal. All results obtainedwithin GGA-PBE. Volume is per formula unit. ∆ E is theenergy barrier between the optimum angle structure and thecubic structure at rotation angle θ = 0.Compound CsSnI CsSnBr CsSnCl Structure α (cid:48) β α (cid:48) β α (cid:48) βa (˚A) 8.935 8.800 8.372 8.282 8.033 7.942 c (˚A) 6.318 6.300 5.920 5.944 5.78 5.710 V (˚A ) 252.19 243.92 207.47 203.84 183.25 180.10∆ V /V (%) -3.28 -1.75 -1.72 θ ( ◦ ) 6.93 10.1 3.61 8.85 2.49 8.32∆ E (meV) 9.9 11.2 4.6 6.6 0.7 8.9Compound CsPbI CsPbBr CsPbCl Structure α (cid:48) β α (cid:48) β α (cid:48) βa (˚A) 9.065 8.610 8.514 8.367 8.160 8.034 c (˚A) 6.410 6.245 6.020 6.085 5.77 5.82 V (˚A ) 263.37 231.520 218.17 213.039 192.10 187.98∆ V /V (%) -12.09 -2.35 -2.15 θ ( ◦ ) 10.75 12.36 9.13 12.36 8.58 11.77∆ E (meV) 33 258 22 50 17 39 Cl Br I180.096200.915221.735242.555263.375 V o l u m e ( ˚A ) Cl Br I2.4924.9597.4279.89412.361 A n g l e ( θ ◦ ) CsPbX rigid RotationCsSnX rigid Rotation CsPbX RelaxedCsSnX Relaxed
FIG. 4. Optimized volume and rotation angle as function ofhalogen in CsSnX and CsPbX . tetragonal β -structure than if we keep the volume fixedat the cubic volume. We may also note that it decreaseswith decreasing volume along the series CsSnI , CsSnBr ,CsSnCl and similarly in the Pb based series. The rota-tion angles are larger in the Pb-based compounds thanin the Sn-based compounds. This means the smaller thetolerance factor, the larger the rotation. Our optimumangle of octahedral rotation for CsSnI agrees well withthe experimental value of 9.09 ◦ .[34]Finally we may note that for the larger cubic volumes,the rotation angle for the Sn-based compounds becomes TABLE III. Optimized cubic and rhomobohedral structuresfor CsBX with B=Ge, Si. The ∆ E are the barriers betweenthe cubic structure with δu = 0 and η = 1 and the optimizedrhombohedral structure each at their own equilibrim volume.Compound CsGeCl CsGeBr CsGeI cubic V GGA (˚A ) 155.72 177.50 216.00cubic V Expt.(˚A ) 163.67 184.22 221.44rhombohedral V (˚A ) 168.19 189.11 228.46rhombohedral a (˚A) GGA 5.52 5.74 6.11rhombohedral a (˚A) Expt 5.434 5.635 5.983 δu δ δ η α GGA 89.17 88.64 88.61 α Expt. 89.72 88.74 88.61∆ E (meV) 75 65 56 E g (eV) GGA 2.01 1.31 1.05Compound CsSiCl CsSiBr CsSiI cubic V V (˚A ) 170.53 187.06 222.91rhombohedral a (˚A) 5.54 5.71 6.06 δu δ δ -0.057 0.033 0.018 η α E (meV) 357 235 142 E g (eV) GGA 2.02 1.31 0.605 rather small. Below, in Sec. IV D, we show that underlattice constant expansion it actually goes to zero, and,at some critical volume, the ferro-electric distortion be-comes preferable instead.The energy barriers ∆ E between the tetragonal energyminimum and the cubic unrotated structure are seen tobe significantly larger for the Pb compounds than the Sncompounds and within each family decrease from I to Brto Cl, except for the fully relaxed CsSnBr and CsSnCl . C. Full structural relaxation for Ge and Si basedrhombohedral distortions.
In this section we further study the fully relaxed rhom-bohedrally distorted structures. In Table III we first givethe optimum GGA volume of the cubic structure. It iscompared with the experimental values at elevated tem-perature where that phase is stable, from Thiele et al. [35] at 170, 270 and 300 ◦ C respectively for the Cl, Br,I cases. Clearly these values are larger than our GGAbecause of the lattice expansion at elevated temperature.Next, we applied a rhombohedral strain along the cubicstructure, allowed the central Ge atom to go off-center by
TABLE IV. B-X bond length ( in ˚A) compared betweenthe perfect cubic structure and the relaxed structure, whereB=Ge,Si and X=Cl,Br,ICompound cubic relaxed % changeCsGeCl3 2.69 2.49 -7.88%CsGeBr3 2.81 2.65 -5.93%CsGeI3 3 2.86 -4.77%CsSiCl3 2.63 2.31 -13.91%CsSiBr3 2.75 2.50 -9.80%CsSiI3 2.94 2.68 -9.62% a displacement δu and allowed the volume to relax. Thestrain η = 1+2 (cid:15) is applied along the [111] cubic directionwhile perpendicular to it, the distances are multiplied by1 / √ η ≈ − (cid:15) , thus maintaining the volume. The straintensor can be written to linear order (cid:15) = (cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15) The cubic lattice vectors a [1 , ,
0] are thus distorted intovectors of a (1 , (cid:15), (cid:15) ) with length a √ (cid:15) which to firstorder in (cid:15) means they stay unchanged. The results for theGe and Si based compounds are given in Table III. Wecan see that for the Br and I cases, our relaxed latticeconstant for the rhombohedral phase in GGA slightlyoverestimates the experimental value, even though thelatter is measured at 20 ◦ C while our calculated volumeis in principle at 0 K. For the Cl case the calculated latticeconstant is slightly underestimated.The displacement from the 0.5 value is almost the samein all cases. The rhombohedral angle extracted from theshear η using cos α = η min − . − δ , − δ , . δ ), in otherwords, it moves inward toward the displaced Ge as shownin Fig.7. The motion of the other anions is similarlydetermined by symmetry. The corresponding parame-ters are given in Table III. Table IV shows that the B-Xbond lengths are shortened upon relaxation in spite ofthe overall volume being expanded in the rhombohedraldistortion.The energy differences ∆ E between the cubic undis-torted structure and the rhombohedral optimized struc-ture each at their own equilibrium volume are also shownin Table III. They indicate an increase from Cl to Br toI and much larger values for the Si then the Ge basedcompounds.The band gaps, which must be underestimates becauseof the GGA, are also included in Table III and show theexpected trend of decreasing from Cl to Br to I, in other Cl Br I145.532164.509183.486202.463221.44 V o l u m e ˚A Cubic Cl Br I160.451177.454194.458211.461228.464 V o l u m e ˚A RhombohedralCl Br I0.0020.0140.0260.0390.051 D i s t a n ce f r o m [. ,. ,. ] ( a ) Rhombohedral Cl Br I88.00488.43388.86289.29189.72 R h o m b o h e d r a l a n g l e ( α ◦ ) RhombohedralCl Br I56.783131.992207.2282.408357.616 E n e r g y b a rr i e r ( m e V ) Rhombohedral Cl Br I0.6050.9611.3171.6732.029 GG A B a nd G a p ( e V ) RhombohedralCsSiX CsGeX Exp value for CsGeX FIG. 5. Trends in structural relaxation parameters for the CsGeX and CsSiX halides corresponding to the data in Table IIIFIG. 6. Unit cell of the relaxed structure of CsSiCl with thecolored atoms at the relaxed positions and the gray atoms atthe unrelaxed cubic positions. Green pink and violet spheresrepresent Cs, Cl and Si respectively. words decreasing with decreasing ionicity. They are alsosmaller in the Si than the Ge compounds. The gaps inthe GW approximation at the experimental rhombohe-dral structures for the CsGeX compounds were given inRef. 14 and are 4.304, 2.654 and 1.619 eV for the Cl, Brand I cases respectively. For the Si- based compounds,they remain to be determined but assuming a similar gap correction, we can already see that both CsSiI andCsSiBr may have gaps suitable for photovoltaics. Thetrends of the data in Table III are visualized in Fig. 5.Although the energy barriers increase from Cl to Brto I, they do not show a clear correlation with the tran-sition temperatures, which are 277-283 ◦ C, 238-242 ◦ C,and 155 ◦ C respectively for CsGeI , CsGeBr , CsGeCl .The problem here is that our calculations consider a ho-mogeneous transformaton, which is forced to be the samein each unit cell. In the actual phase transition, there isa competition between the interaction energies of atomsin neighboring cells and the double-well anharmonic po-tential well in each unit cell. The phase transition couldbe either displacive or order-disorder type.[36] In the for-mer case, corresponding to a large interaction energy be-tween neighboring cells, the positions of the atoms vi-brate about an average near the barrier maximum (cor-responding to the cubic structure) at high temperatureand settle into one or the other minimum below the tran-sition temperature. A nucleation process occurs wheregroups of neighboring atoms settle into one of the two lo-cal minima. In contrast in the order-disorder model, cor-responding to a strong double well potential but weakerintercellular interactions, the atoms are always in one ofthe two minima but at high temperature, they are equallylikely to be in the left or right well. From our present cal-culations, we do not have access to the inter-cell energiesin such a model, and thus we cannot draw conclusions θ FIG. 7. Relative energy as function of rotation angle at var-ious lattice expansions for CsSnI as % expansions in lat-tice constant The energies are considered relative to the un-rotated energy at each lattice parameter about the nature of the phase transition. Experimentally,it was established by Thiele et al. [35] that for CsGeBr and CsGeI the phase transition is first-order, while forthe Cl-case it is second order. This would indicate a dis-placive transition for the latter case but an order-disordertype for the former. D. Rotation and rhombohedral distortion undervolume expansion
As we already mentioned, the tendency toward octahe-dral rotation in the Sn and Pb halides decreases, that isto say the rotation angle decreases, with increasing vol-ume for a given material. We therefore further studiedthe behavior under lattice expansion and compression,which one might think of as occurring by thermal expan-sion and under high-pressure respectively. First, we showthe energy curves for CsSnI as function of rotation an-gle for various lattice expansions in Fig. 7. Even withoutvolume expansion, we see that the curves shows two localminima, one at zero angle and one at about 7 ◦ . As weincrease the volume, the local minimum corresponding tothe finite rotation moves up in relative energy and even-tually,beyond 3 % expansion of the lattice constants, itdisappears, at which point the curve becomes very flat.Although they still show a very shallow finite angle min-imum, we may essentially consider this as a sign thatthe rotation is no longer preferred. On the other hand,under compression, the local minimum appears to shifttoward smaller angle and becomes deeper relative to theunrotated structure.The optimum angle of rotation is shown as the redcurve as function of lattice expansion in Fig. 8. The in- FIG. 8. Percentage lattice-constant expansion vs. angle of ro-tation and distance of Sn from [ . , . , .
5] for CsSnI . The sizeand darkness of the rotation markers (circular ones) representthe size of energy barrier w.r.t the perfect cubic perovskitestructure creasing values for lattice expansion actually correspondto a very low energy barrier, as is indicated by the smallsign of the symbols marking each point and may to firstapproximation be ignored. Under compression, the rota-tion angle clearly is reduced and the barrier increases,meaning the energy of the rotated minimum becomesdeeper.Next, we examine the possibility of off-centering of theSn atom as function of lattice expansion. As we cansee in Fig. 8 in the blue dashed curve, the off-centeringdisplacement stays zero until 1 % expansion at whichpoint it starts increasing linearly. Eventually it collapsesagain beyond 6 % expansion. Similar results are also ob-tained for the other halogens and for the Pb compoundsas shown in Fig. 9. In summary, we find that beyond agiven lattice expansion the CsSnX and CsPbX materi-als undergo a rhombohedral distortion with off-centeringof the Sn (or Pb) rather than the octahedral rotation.This type of behavior was reported earlier for CsSnBr by Fabini et al. [25] and related to the active lone-pairbehavior of the s electrons which was studied in detail.We thus see that there is indeed a competition betweenthe two types of distortion behavior, rotation or rhom-bohedral off-centering. The lone pair character promotesthe off-centering and is strongest for Ge and Si if thelatter are required, as in this structure, to behave diva-lent but it also occurs in Sn and to a smaller degree inPb. However, in Sn and Pb this mechanism of distor-tion is in competition with rotations, while in Ge andSi it is not. Finally, it should be pointed out that theoff-centering in CsSnBr was experimentally observed byFabini et al. [25] but occurs dynamically. It was observedonly through analysis of the pair distribution functions.In other words, it does not occur coherently throughoutthe sample, which means that a rhombohedral crystallo-graphic phase is not found for this compound. Insteadit is hidden in the cubic phase but is apparent from the FIG. 9. Displacement of Sn/Pb from the center [ . , . , .
5] forCsSnI , CsSnCl , CsSnBr and CsPbI as function of latticeconstant expansion. large atomic displacements which are coherent only ona local scale. This is an important difference from Gewhere the rhombohedral phase is the actually observedequilibrium crystal structure. E. Rb instead of Cs
TABLE V. Energy barrier and angle of rigidly rotatedRbGeX with X=Cl, Br, I.The Energy barrier is the bar-riers between the cubic structure with 0 ◦ and local/globalminimum at the given angleCompound RbGeCl3 RbGeBr3 RbGeI3Angle ( θ ) 4.71 6.93 10.21Energy barier (meV) -70.0 13.5 36.4 In this section we consider the RbGeX compoundscompared with the CsGeX compounds. From Table Iwe expect that because of the smaller size of the Rb ion,these compounds would be unstable toward octahedralrotation. The results for rotation in Table V show indeedthat octahedral rotation lowers the energy for a finiterotation angle for I and Br but not for Cl. In the lattercase, there is still a local minimum at a finite angle but itsenergy is actually higher than at the zero angle rotation.The energy lowering is comparable and even larger thanfor the corresponding CsSnX compounds and the angleof rotation is larger for I than for Br.On the other hand, from the previous sections, it is alsoclear that from the point of view of lone-pair physics, Geis prone to off-centering. Therefore we also study thepossibility of lowering the energy by the rhombohedraldisotortion. The results are shown in Table VI. This TABLE VI. Optimized cubic and rhomobohedral structuresfor RbGeX with X=Cl, Br, I. The ∆ E are the barriers be-tween the cubic structure with δu = 0 and η = 1 and theoptimized rhombohedral structure each at their own equilib-rim volume.Compound RbGeCl3 RbGeBr3 RbGeI3Cubic a (˚A) GGA 5.345 5.57 5.97Cubic V (˚A ) 152.70 172.80 212.77Cubic bond length (˚A) 2.67 2.78 2.98Rhombohedral a (˚A ) 5.44 5.65 5.99Rhombohedral V (˚A ) 161.27 180.44 214.98Rhombohedral bond length (˚A) 2.31 2.46 2.69∆ V /V (%) 5.31% 4.23% 1.02%Change in bond length (%) -13.37% -11.55% -9.84% δu δ δ η α E (meV) GGA 455.8 367.2 304.7Band Gap (eV) GGA 1.72 1.05 0.43 shows that the off-centering and related rhombohedraldistortion lowers the energy significantly more efficientlythan the octahedral rotation. For RbGeCl the rotationactually does not lower the energy, while the distortiondoes. For RbGeBr and RbGeI , the energy lowering bythe off-centering is significatnly larger than by rotationof the octahedra. Thus comparing to the Cs case, this in-dicates that the off-centering of Ge is not so much deter-mined by the tolerance factor but rather by the lone-pairphysics. The relaxation parameters, barriers and energygaps in GGA for these compounds are given in Table VIin the same way as for the other compounds.Finally, we illustrate the lone-pair character in this caseby plotting the charge density for this in Figs. 10, 11.The firts one shows a 3D view of isosurfaces, the sec-ond one shows the valence charge density along the bodydiagonal. V. CONCLUSIONS
In this paper we have explored the stability of inorganichalide perovskites ABX with X a halogen (Cl, Br, I), A a large alkali ion(Cs or Rb) and B a group IV element, (Si,Ge, Sn, Pb), under two types of distortion: an antiferro-electric distortion corresponding to octahedral rotationand a ferro-electric off-centering of the central IV ion in-side its halogen octahedron. At first, we find that thereis a clear trend that the Pb and Sn cases prefer rotationwhile Ge and Si prefer ferro-electric distortion. We alsofind that the rotation, when fully optimizing the struc-tures, is accompanied by a reduction of the volume. Theoff-centering is accompanied by rhombohedral distortionand volume increase. The tendency toward rotation is0 FIG. 10. Total valence charge density for the relaxed distortedRbGeCl shown as a superposition of 8 isosurfaces with valuesranging frm 0.058 to 0.071 e/a . Each isosorface is shown as amesh of different color. One can clearly distinguish the Ge- s like lobe in the direction opposite to the displacemnt. Thepink sphere is Rb, the blue one Ge and the green ones Cl. Electron density (ea0-3)
D i s t a n c e a l o n g [ 1 , 1 , 1 ] ( a )0 . 5
FIG. 11. Plot of the valence charge density along the [111]body diagonal. One can see the asymmetry of the chargedensity near the Ge position, at a position larger than 0.5,again reflecting the lone-pair character. clearly related to the Goldschmidt tolerance factor. Onthe other hand, we find that upon volume expansion,the rotation angle decreases and beyond a certain expan-sion off-centering becomes favorable even for Sn and Pbbased compounds. The origin of the off-centering is thusmore related to the lone-pair physics. The Ge and Sibased compounds, in which Ge and or Si are forced tobehave as a divalent ion, strongly favor lone-pair inducedoff-centering or ferro-electric distortion. In the Rb case,both distortion modes tend to lower the energy (exceptfor the Cl case) but the ferro-electric distortion nonethe-less lowers the energy signiricantly more efficiently. Thusthe lone-pair physics dominates the RbGeX based com-pounds rather than the tolerance factor related rotation.The two distortion mechanisms can thus be in competi-tion with each other and the off-centering for the Si andGe cases occurs even if the tolerance factor would allowfor rotations as a mechanism to lower the energy. ACKNOWLEDGMENTS
This work was supported by the U.S. Department ofEnergy (Basic Energy Sciences) DOE-BES under grantNo. DE-SC0008933. The calculations made use of theHigh Performance Computing Resource in the Core Fa-cility for Advanced Research Computing at Case WesternReserve University. [1] A. Kojima, K. Teshima, Y. Shirai, andT. Miyasaka, Journal of the American Chem-ical Society , 6050 (2009), pMID: 19366264,http://dx.doi.org/10.1021/ja809598r. [2] M. M. Lee, J. Teuscher, T. Miyasaka, T. N. Murakami,and H. J. Snaith, Science , 643 (2012).[3] J. M. Ball, M. M. Lee, A. Hey, and H. J. Snaith, EnergyEnviron. Sci. , 1739 (2013). [4] G. E. Eperon, V. M. Burlakov, P. Docampo, A. Goriely,and H. J. Snaith, Advanced Functional Materials , 151(2014).[5] M. Liu, M. B. Johnston, and H. J. Snaith, Nature ,395 (2013).[6] J. Burschka, N. Pellet, S.-J. Moon, R. Humphry-Baker,P. Gao, M. K. Nazeeruddin, and M. Gr¨atzel, Nature , 316 (2013).[7] H.-S. Kim, S. H. Im, and N.-G. Park, The Journal ofPhysical Chemistry C , 5615 (2014).[8] N.-G. Park, Materials Today , 65 (2015).[9] C. Quarti, E. Mosconi, and F. D. Angelis, Chem. Mater. , 6557 (2014).[10] J. Li and P. Rinke, Phys. Rev. B , 045201 (2016).[11] L.-y. Huang and W. R. L. Lambrecht, Phys. Rev. B ,195201 (2014).[12] L.-y. Huang and W. R. L. Lambrecht, Phys. Rev. B ,165203 (2013).[13] I. Chung, J.-H. Song, J. Im, J. Androulakis,C. D. Malliakas, H. Li, A. J. Freeman, J. T.Kenney, and M. G. Kanatzidis, Journal of theAmerican Chemical Society , 8579 (2012),http://pubs.acs.org/doi/pdf/10.1021/ja301539s.[14] L.-y. Huang and W. R. L. Lambrecht, Phys. Rev. B ,195211 (2016).[15] E. L. da Silva, J. M. Skelton, S. C. Parker, and A. Walsh,Phys. Rev. B , 144107 (2015).[16] V. M. Goldschmidt, Naturwissenschaften , 477 (1926).[17] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[18] M. Methfessel, M. van Schilfgaarde, and R. A. Casali,in Electronic Structure and Physical Properties of Solids.The Use of the LMTO Method , Lecture Notes in Physics,Vol. 535, edited by H. Dreyss´e (Berlin Springer Verlag,2000) p. 114.[19] T. Kotani and M. van Schilfgaarde, Phys. Rev. B ,125117 (2010).[20] E. Bott, M. Methfessel, W. Krabs, and P. C. Schmidt,Journal of Mathematical Physics , 3393 (1998), http://dx.doi.org/10.1063/1.532437.[21] A. M. Glazer, Acta Crystallographica Section B , 3384(1972).[22] P. M. Woodward, Acta Cryst. B53 , 44 (1997).[23] A. Walsh, D. J. Payne, R. G. Egdell, and G. W. Watson,Chem. Soc. Rev. , 4455 (2011).[24] U. V. Waghmare, N. A. Spaldin, H. C. Kandpal, andR. Seshadri, Phys. Rev. B , 125111 (2003).[25] D. H. Fabini, G. Laurita, J. S. Bechtel, C. C.Stoumpos, H. A. Evans, A. G. Kontos, Y. S. Rap-tis, P. Falaras, A. Van der Ven, M. G. Kanatzidis,and R. Seshadri, Journal of the American Chem-ical Society , 11820 (2016), pMID: 27583813,http://dx.doi.org/10.1021/jacs.6b06287.[26] E. H. Smith, N. A. Benedek, and C. J. Fennie, In-organic Chemistry , 8536 (2015), pMID: 26295352,https://doi.org/10.1021/acs.inorgchem.5b01213.[27] U. V. Waghmare and K. M. Rabe, Phys. Rev. B , 6161(1997).[28] P. Ghosez, E. Cockayne, U. V. Waghmare, and K. M.Rabe, Phys. Rev. B , 836 (1999).[29] R. D. Shannon, Acta Cryst. A , 751 (1976).[30] J. A. Brehm, J. W. Bennett, M. R. Schoen-berg, I. Grinberg, and A. M. Rappe, The Jour-nal of Chemical Physics , 224703 (2014),http://dx.doi.org/10.1063/1.4879659.[31] W. Travis, E. N. K. Glover, H. Bronstein, D. O. Scanlon,and R. G. Palgrave, Chem. Sci. , 4548 (2016).[32] W. Zhong and D. Vanderbilt, Phys. Rev. Lett. , 2587(1995).[33] A more complete set of results as function of rotationand/or ferro-electric distortion is provided in the Sup-plementary Materials accompanying this paper.[34] K. Yamada, S. Funabiki, H. Horimoto, T. Matsui,T. Okuda, and S. Ichiba, Chemistry Letters , 801(1991).[35] G. Thiele, H. W. Rotter, and K. D. Schmidt, Z. anorg.allg. Chem. , 148 (1987).[36] M. T. Dove, American Mineralogist82