Interplay of octahedral rotations and breathing distortions in charge ordering perovskite oxides
Abstract
We investigate the structure–property relationships in O perovskites exhibiting octahedral rotations and cooperative octahedral breathing distortions (CBD) using group theoretical methods. Rotations of octahedra are ubiquitous in the perovskite family, while the appearance of breathing distortions – oxygen displacement patterns that lead to approximately uniform dilation and contraction of the O octahedra – are rarer in compositions with a single, chemically unique site. The presence of a CBD relies on electronic instabilities of the site cations, either orbital degeneracies or valencestate fluctuations, and often appear concomitant with charge order metal–insulator transitions or site cation ordering. We enumerate the structural variants obtained from rotational and breathing lattice modes and formulate a general Landau functional describing their interaction. We use this information and combine it with statistical correlation techniques to evaluate the role of atomic scale distortions on the critical temperatures in representative charge ordering nickelate and bismuthate perovskites. Our results provide microscopic insights into the underlying structure–property interactions across electronic and magnetic phase boundaries, suggesting plausible routes to tailor the behavior of functional oxides by design.
pacs:
61.50.Ks, 31.15.xh, 71.30.+hI Introduction
Perovskite oxides with chemical formula O and site transition metal (TM) cations exhibit a range of functional electronic transitions that are intimately tied to the structure of the fundamental building blocks Benedek et al. (2012): () the number of unique –O bonds within an octahedron, and () the tilting of cornerconnected octahedra. Adjacent O units typically fill space in perovskites through nearly rigid rotations, which produce deviations of the –O– bond angles away from the ideal 180 found in the cubic aristotype ( symmetry); the rotations are described by two threedimensional irreducible representations (irreps), and , of the highsymmetry structure.Howard and Stokes (1998) Combinations of these lattice instabilities – cooperative bond length distortions and octahedral rotations – interact across structural phase transitions through elastic stresses and symmetry allowed coupling invariants as described within Landau theory.
Beside changes to crystal symmetry, the transition from high temperature (high symmetry) to low temperature (lowsymmetry) can also produce electronic metal–insulator (MI) transitions. Perovskite oxides with site cations in (), () and () electronic configurations are particularly susceptible, because the lowenergy electronic structure is dictated by the octahedral crystalfield split antibonding orbitals—the atomiclike states that are spatially directed at the coordinating oxygen ligands. MItransitions which occur simultaneously with lattice distortions are common in lowdimensional materials, e.g. Peierls systems.Canadell et al. (2012) In threedimensional perovskite oxides, however, the most familiar electronic transitions with concomitant changes in the –O bond lengths and octahedral rotations result from cooperative firstorder JahnTeller effects: Tetragonal elongations of the O octahedra occur to remedy the orbital degeneracies that localized electrons encounter for particular site cation configurations. And through these distortions, the crystal maintains a uniform TM valence among all sites across the transition.Millis et al. (1995) The JahnTeller distortions are described by irreps , and , and their interaction with octahedral rotations are wellestablished.Carpenter and Howard (2009a); ?
Unusually highvalence states or electrons in delocalized band states will also produce structural distortions, but in most cases will preserve the uniformity of the –O bonds and the octahedral crystal field of the O units in the process.Goodenough and Rivadulla (2005) The cations will readily adopt mixed valence configurations, e.g. doped perovskites manganitesSalamon and Jaime (2001) (containing nominally both Mn and Mn) and also stoichiometric nickelatesGoodenough (2004) (Ni) and ferratesMatsuno et al. (2002) (Fe). The TM cation will not maintain an integer valence state () uniformly on all sites, but rather charge disproportionate (CDP); the simplest case being into two sites as
where is a fraction of an electron transferred between sites.
Electronic correlation and onsite Coulomb repulsion effects will prefer to order the valence
deviations so that the
inequivalent lattice sites, 1 and 2, form a periodic arrangement, changing the translational symmetry.
This socalled charge ordering (CO) lowers the potential energy of the crystal and gaps the Fermi surface.
The electronic charge ordering is
distinguishableQuan et al. (2012); ?; ?
by an associated structural change in the local O building blocks (Figure 1). It appears as a –O bond disproportionation or “breathing distortion,” which causes the octahedra to either dilate (site 1) or contract (site 2) according to the charge transferred between sites,
The magnitude of the octahedral rotations are modified by these changes in the O bond lengths. The extent to which cooperation between octahedral breathing and rotation instabilities is necessary to stabilizing charge order and MItransitions, however, is not as well understood Matsuno et al. (2002); SahaDasgupta et al. (2005) as the role of JahnTeller distortions on electronic transitions Mizokawa et al. (1999); Carpenter and Howard (2009a, b). To this end, group theoretical methods are particularly powerful to address the interactions among the multiple octahedronderived instabilities. They provide a rigorous means to evaluate the symmetry allowed interactions between coupled lattice degrees of freedom, and thus, glean insight into the microscopic atomic structural contributions in the electronic CO transition in perovskite oxides.
In this work, we enumerate the space group and order parameter relationships for the octahedral
breathing distortions, which are associated with irreps and of the aristotype
cubic phase and the 15 simple
octahedral rotation patterns available to bulk O perovskites.
We provide a list of symmetries that perovskite oxides with charge order tendencies
and octahedral rotations could adopt—not all linear combinations of the instabilities are
anticipated to be symmetry allowed.
Ii Cooperative Breathing Distortions
The octahedral breathing distortions, which create two unique crystallographic sites, must tile in threedimensions to maintain the cornerconnectivity of the O framework—the defining feature of the perovskite crystal structure. Here we consider –O distortions with wavevectors commensurate with the lattice periodicity, i.e., those modes which occur on the edges and corner of the simple cubic Brillouin zone. The two main cooperative breathing distortions (CBD) are illustrated in Figure 2. Note that the irreps, which we describe next, are defined using the setting of the O perovskite with the cation located at the origin.
The first type of CBD consists of two O building blocks which are tiled to form a columnar arrangement of dilated and contracted octahedra [Fig. 2(a)], that splits the –O bond lengths into a doublet and quartet.
The second CBD consists of O octahedra which are tiled in a threedimensional checkerboard arrangement [Fig. 2(b)]. The distortion is described by a onedimensional irrep and occurs as a zonecorner lattice instability. Consequently, in the absence of other distortions, the CBD lifts the site equivalence while maintaining the symmetry of the octahedra through uniform contraction and elongation of the –O bonds about the trigonal axis, splitting the site Wyckoff position and forming a larger unit cell (Table 1). We note that this type of site ordering is common in double perovskites with multiple cations that show large ionic size and/or considerable oxidation state differences, e.g., BaMgWO. King and Woodward (2010) But here, we consider only single (chemical species) cations with different nominal valences.
(no. 123)  

Atom  Wyck. Site  
0  
(1)  0  0  0  
(2)  0  
O(1)  0  
O(2)  0  0  
O(3)  
(no. 225)  
Atom  Wyck. Site  
(1)  0  0  0  
(1)  
O  0  0 
Iii Octahedral Rotation and CBD Space Groups
iii.1 Methodology
To enumerate the allowed combinations of CBD and octahedral rotations in perovskite oxides we use the group theoretical program isotropy.Campbell et al. (2006) We follow the approach of Stokes et al.Howard and Stokes (2005) and consider the changes in lattice symmetry due to the superposition of each CBD pattern (Figure 2) with the 15 octahedral rotation systems derived from the and threedimensional irreps describing rotations.Howard and Stokes (1998) The advantage of this approach is that group–subgroup relationships can be established between structural variants, enabling the understanding of the structural and electronic CO transitions within a first or secondorder theory. It should be noted that the analysis of charge order at a particular site in the crystal is effectively the same as analyzing the effects of cation order due to a order–disorder transition, because both valence and chemical species split a Wyckoff position identically. In the latter case, the irrep describes a “composition” mode and reflects the site occupancy.
In this work, we enumerate those mode combinations with welldefined cooperative breathing distortions that are most likely to be observed experimentally. Although the rock salt cation order has been studied previously in O perovskites, the layered ordering of cations, i.e. irrep , was examined only for the compositionally complex quadruple perovskite O oxides.Howard and Stokes (2004) Here, we report the results of the irrep on simple O perovskites in the context of the CBD behavior, where celldoubling occurs. Specifically for the irrep, we retain only those structures with an OP such that the second and third vector components are zero, unless an octahedral rotation permits it to be nonzero by symmetry. Furthermore, we follow the convention introduced in Ref. Howard and Stokes, 1998; Stokes et al., 2002 and only keep structures with coherent rotations—those with a fixed “sense” about each axis. We remove structure variants from our analysis which would allow for a modulation in the amplitude and sense of the rotations about a single axis.
iii.2 Space Groups
We enumerate the space groups (Table 2) and associated octahedral tilt pattern, irreps, lattice vectors, and origin allowed by group theory. We follow Glazer’s established notation Glazer (1972) to denote the magnitude and phase of octahedral tilt patterns in perovskites: The description of octahedral rotations are encoded using the syntax , where letters , , and indicate rotations of O units of equal or unequal magnitude about Cartesian , , and axes. Note that in the case of equal magnitude rotations about different axes, the equivalent letter is duplicated, e.g., . The superscript can take three values: 0, +, or , for no rotations, inphase rotations (neighboring octahedra along a Cartesian axis rotate in the same direction), or outofphase rotations (adjacent octahedra rotate in the opposite direction), respectively.
Order parameter direction  
space group  tilt pattern  lattice vectors  origin  
221  (0,0,0)  (0,0,0)  
127  (a,0,0)  (0,0,0)  
139  (a,0,a)  (0,0,0)  
204  (a,a,a)  (0,0,0)  
71  (a,b,c)  (0,0,0)  
140  (0,0,0)  (a,0,0)  
74  (0,0,0)  (a,0,a)  
167  (0,0,0)  (a,a,a)  
12  (0,0,0)  (a,0,b)  
15  (0,0,0)  (a,b,a)  
2  (0,0,0)  (a,b,c)  
63  (0,0,a)  (b,0,0)  
62  (0,a,0)  (b,0,b)  
11  (0,a,0)  (b,0,c)  
137  (0,a,a)  (b,0,0)  
127 P4/mbm 

(a,0,0) 
(0,0,0)  (0,0,0)  
83  (a,0,0)  (b,0,0)  (0,0,0)  
74  (a,0,0)  (0,0,b)  (0,0,0)  
69  (a,0,0)  (0,b,b)  (0,0,0)  
12  (a,0,0)  (b,c,d)  (0,0,0)  
124  (a,0,0)  (0,0,0)  (b,0,0)  
63  (a,0,0)  (0,0,0)  (0,0,b)  
51  (a,0,0)  (0,0,0)  (0,b,b)  
13  (a,0,0)  (0,0,0)  (c,b,b)  
15  (a,0,0)  (0,0,0)  (b,c,0)  
11  (a,0,0)  (,0,0) 
(0,c,d)  
2  (a,0,0)  (,0,0)  (c,d,e)  
52  (a,0,0)  (0,0,b)  (c,0,0)  
62  (a,0,0)  (0,0,b)  (0,c,0)  
11  (a,0,0)  (b,0,0)  (0,c,d)  
68  (a,0,0)  (0,b,b)  (c,0,0)  
225 Fmm 

(a)  (0,0,0)  (0,0,0)  
128  (a)  (b,0,0)  (0,0,0)  
201  (a)  (b,b,b)  (0,0,0)  
134  (a)  (b,b,0)  (0,0,0)  
48  (a)  (b,c,d)  (0,0,0)  
148  (a)  (0,0,0)  (b,b,b)  
87  (a)  (0,0,0)  (b,0,0)  
12  (a)  (0,0,0)  (b,b,0)  
2  (a)  (0,0,0)  (b,c,d)  
15  (a)  (b,0,0)  (0,0,c)  
86  (a)  (b,b,0)  (0,0,c)  
14  (a)  (b,0,0)  (0,c,c) 
Table 2 provides the possible space group symmetries compatible with CBD and octahedral rotations. The structural data is divided into three main blocks: The first section contains the space group symmetries in the absence of CBD; the second section enumerates the symmetries that result from the planar CBD with octahedral rotations; and the third section includes those obtained from the threedimensional CBD combined with rotations. Structures appearing in bold in Table 2 correspond to CBD without any octahedral rotations and are given first at the top of block two and three. Note that for perovskites with rotations and the CBD, there may be more than one structure possible for a given rotation, because the relative orientation of the tilt pattern with respect to the columnar arrangement of dilated and contracted octahedra alters the crystal symmetry differently.
Table 2 directly reveals the effect of the superposition of octahedral rotation patterns and the CBD patterns on the crystal symmetry lowering. Consider the middle block (). The seemingly similar tilt patterns and , depending on the crystallographic axes they act upon, yield two different space groups, and , respectively.
Furthermore, there are a number of space groups appearing in each of the blocks of Table 2: appears both in the rotation only block (sans any CBD) and in the third block with the CBD. Formally the crystallographic symmetries are identical in each case; however, physically the rotation patterns adopted by the crystals are different. For instance, when only rotations associated with irrep and order parameter are present in the crystal structure, space group allows the outofphase rotations about the  and axes to be of different magnitude (tilting pattern is ). Now, consider the direct sum for a vector space corresponding to the order parameter that includes CBD. Even though the resulting space group is , the corresponding octahedral tilt pattern changes to , where the outofphase rotations about the  and axes are of the same magnitude. On the other hand, the direct sum for a vector space corresponding to the order parameter is found to preserve the tilting pattern, but the overall symmetry reduces to . Also note that, in the case of rotations without CBD, the tilt pattern corresponds to the higher symmetry space group. Such restrictions imposed on the rotation axes equivalence are described next, and for the reason previously described explains why “none” appears as an entry in Table 3.
iii.3 Compatible CBD and Rotation Symmetries
Tilt system  
(Glazer notation)  No Breathing  
none  
none  
none  
none  
no change  no change  
none  
none  
In Table 3, we aggregate the results of the change in space group symmetry due to the superposition of tilt patterns of and irreps with each of the 15 simple octahedral tilt systems. This information is schematically shown in Figure 3. Only one symmetry exists, , where the breathing distortion is geometrically compatible with the symmetry of the perovskite structure with octahedral rotations alone (), meaning “no change” in unit cell or translational symmetry is required to accommodate the multiple distortions. Table 3 also reveals that the symmetry is compatible with two different octahedral rotation patterns and the CBD: and . While is compatible with both and irreps, only exists for the irrep.
For entries containing “none,” the direct sums, and , do not yield an isotropy subgroup. This is made clear by examining the rotation pattern, which corresponds to an equal amplitude of outofphase rotations about each axis, or equivalently a single outofphase rotation about the threefold axis. The trigonal symmetry is incompatible with a distortion that would require a loss of the threefold axis and therefore need at a minimum a symmetry reduction to a lattice with tetragonal geometry. Thus, ‘none’ appears in the column corresponding to the row with the rotation pattern. Such incompatibility with the CBD is alleviated if the rotation pattern about any two crystallographic axes are of different magnitudes, e.g., , yielding space group (Table 3).
Iv Structural Transitions
Although the microscopic origin for charge disproportionation results from an electronic instability related to the electronic configuration of a particular metal center, the cooperative ordering of the CDP leads to a macroscopic bonddisproportionation. In the displacive limit, the two unique and nearly uniform octahedra result, dilating and contracting in proportion to the magnitude of charge transfer. Across the electronic phase transitions, the breathing distortions can couple directly to octahedral rotations. (We do not consider here indirect coupling through a common strain component.)
Here we describe the effect of superposition of the common orthorhombic rotation pattern () obtained from irreps and , which gives the sixdimensional order parameter with two unique directions as
The superposition of the threedimensional CBD and the rotation pattern gives a sevendimensional order parameter space, i.e., (), which we contract to an effective fourdimensional space to obtain the space group (cf. Table 2). Without assuming which transition occurs first—whether the rotations precede or follow the CBD—we construct a Landau free energy expansion about the cubic phase () as
where and describe the homogeneous quadratic and quartic terms for each order parameter, and are the coefficients coupling the CBD to the in or outofphase rotations, and describes the biquadratic coupling of the different O rotation “senses” in the tilt pattern. The group–subgroup relationships are depicted in Figure 4.
The rotations and the CBD couple biquadratically, which indicates that the interactions across the transitions could be either cooperative or antagonistic. It is possible that the particular rotations could suppress CO by eliminating the structural octahedral breathing distortion altogether through the interaction terms containing the coefficients. However, in most cases, the rotation amplitudes are weakly modified across the electronic CO transition, suggesting that the strength of the coupling is in general small. We explore this in the V through a statistical approach.
We also note that while the order of the coupling is important, the difference in temperature scales at which the rotation and charge ordering occurs is also important in determining how the structural order parameters influence each other. The strongest interaction occurs when the temperatures are similar, while if they are far apart, the two structural transitions will weakly couple.
V Structure–Functionality Relationships
In this section, we apply our group theory results to quantitatively explore the relationship between structure and physical properties of experimentally known rareearth nickelate and bismuthate perovskites. There is significant interest in developing strategies – both experimentally and theoretically – to rationally control octahedral distortions through the interplay of chemical pressure, epitaxial strain engineering, and ultrathin superlattice heterostructure formationTorrance et al. (1992); May et al. (2010); Rondinelli et al. (2012); Chakhalian et al. (2011b); BlancaRomero and Pentcheva (2011); Boris et al. (2011) for property control. Although several octahedral distortion metrics, e.g., the crystallographic tolerance factor or bending of the –O– bond angle,Zhou et al. (2005); Catalan (2008) have played an important role in the understanding of the electronic and magnetic properties of perovskite oxides,Goodenough (2001) they have had limited success in materials design of nonthermodynamic phases in thin film geometries. Knowledge of quantitative structure–property octahedral distortion relationships are required to accelerate materials discoveries.
Distortionmode decomposition analysis is an alternative approachCampbell et al. (2006); Carpenter and Howard (2009a, b); PerezMato et al. (2010) (widely practiced and followed in the crystallography literature) to study displaciveDove (1997) phase transitions in perovskites. It involves describing a distorted (lowsymmetry) structure as arising from a (highsymmetry) parent structure with one or more static symmetrybreaking structural distortions.PerezMato et al. (2010) In the undistorted parent structure, each symmetry breaking distortionmode has zero amplitude. The lowsymmetry phase, however, will have finite amplitudes for each irrep compatible with the symmetry breaking. Said another way, the lowsymmetry phase is rigorously described through a series expansion of static symmetry breaking structural modes that “freeze” into the parent structure. Critically, the weights or amplitudes assigned to each irrep are obtained according to the contribution that each irrep is present and the requirement that linearity is maintained.
What is of particular utility in formulating quantitative relationships connecting octahedral distortions, which are now described mathematically, to macroscopic properties for materials design is that each irrep carries a physical representation of the displacive distortions—the unique atomic coordinates describing various symmetryadapted structural modes. The relative importance of these modes on properties may then be mapped by means of ab initio computational methods. Diéguez et al. (2011) Accessibility to computational methods make the distortionmode analysis powerful, because it is possible to independently study various distortions and directly assess their role in structural and electronic phase transition mechanisms. Furthermore, the distortionmode analysis relies solely on crystal structural data, which enables both bulk and thin film stabilized structures with identical compositions to be evaluated on equalfooting. Such direct comparison is not possible through aggregate parameters such as the tolerance factor, i.e. when the composition is fixed, or other metrics widely followed in the literature.
In the remainder of this section, we use the distortionmodes to form the basis for the quantitative description of octahedral distortions and CBD on material properties. Using bulk NiO, where is a rareearth element, and BaKBiO perovskite compounds as prototypical chargeordering materials, we decompose available lowsymmetry structural data into symmetryadapted structural distortion modes. We then evaluate and correlate the amplitudes of the distortionmodes to macroscopic materials behavior to uncover trends linking octahedral distortions to the structural and physical properties. We use the grouptheory program isodistort Campbell et al. (2006) for distortionmode decomposition analysis and R R Core Team (2012); Sarkar (2008) for the statistical analysis. For readers interested in reproducing our work, we have deposited the raw data and the Rscript in the supplementary materials section available at http://link.aps.org/supplemental/XYZ.
v.1 RNiO Nickelates
Rareearth perovskite nickelates, NiO, where
Y, Ho, Er, Dy, Lu, Pr, or Nd, exhibit nontrivial changes in structure and
physical properties, including sharp firstorder temperaturedriven MItransitions,
unusual antiferromagnetic order in the ground state, and site or bondcentered
charge disproportionation.Medarde (1997); Alonso et al. (1999); Zhou et al. (2005)
At the MIT temperature (), the crystal symmetry lowers from orthorhombic to monoclinic symmetry
Moreover, the insulating groundstate displays a complex antiferromagnetic order (Type) below a Néel temperature (). Several previous studies Medarde et al. (1998); Anisimov et al. (1999); Zhou et al. (2005); Mazin et al. (2007); Catalan (2008); Lau and Millis (2013) have suggested the likely existence of a complex interplay between octahedral rotations, transport, and magnetic properties. To extract deeper insight into these interrelationships, we () identify all active distortion modes in each composition, () determine the individual amplitudes for each modes, and () explore the statistical correlation between individual distortionmodes and the physical properties, specifically and .
Structure Decomposition
We follow the procedure outlined by Campbell et al. Campbell et al. (2006) to decompose the monoclinic crystal structure data, obtained from previously published diffraction studies, Alonso et al. (2000); Medarde et al. (2008); GarcíaMuñoz et al. (2009); Munoz et al. (2009) into the orthonormal symmetrymodes. Diffraction data for YNiO, ErNiO, LuNiO, and HoNiO were measured at 295 K; DyNiO at 200 K; NdNiO at 50 K; and finally, PrNiO at 10 K. and were obtained from the review article by Catalan Catalan (2008).
(a) Undistorted octahedra  (b) type JahnTeller 



(c) inphase rotation  (d) outofphase tilting 


(e) type JahnTeller  (f) outofphase rotation 


(g) bending/buckling mode  (h) inphase tilting 


The structure decomposes into eight algebraically independent symmetrymodes corresponding to the following irreps: , , , , , , , and . The relative amplitudes for each of the distortions are given in Table 4, with the physical representation of each irrep and its consequence on the octahedral framework schematically illustrated in Figure 5.
also present in  

NiO  
HoNiO  1.499  0.195  0.826  0.048  1.153  0.172  0.024  0.025  2.081 
ErNiO  1.535  0.203  0.842  0.048  1.179  0.145  0.022  0.020  2.127 
LuNiO  1.648  0.248  0.907  0.063  1.229  0.153  0.004  0.016  2.268 
YNiO  1.502  0.196  0.835  0.055  1.161  0.128  0.011  0.072  2.089 
NdNiO  1.179  0.117  0.445  0.030  0.751  0.126  0.050  0.283  1.505 
PrNiO  1.093  0.063  0.362  0.004  0.690  0.091  0.012  0.000  1.347 
DyNiO  1.442  0.207  0.780  0.073  1.144  0.241  0.025  0.168  2.041 
Std. Err.  0.029  0.009  0.031  0.003  0.032  0.007  0.002  0.015  – 
The sequence of structures involved in the phase transition could be written as follows: . Five distortions participate in the transition, namely , , , , and . Within Landau theory, the structure results from the condensation of two zoneboundary phonons of different wavevectors: one at the point of the cubic Brillouin zone that transforms as the irrep , and the other at the point, which transforms as the irrep . Not surprising, the two most common octahedral distortionmodes that describe the rotation pattern, and , emerge as the primary modes. A secondary mode with symmetry also appears with relatively high amplitude, and it contains cation displacements. Such antiparallel displacements of the cations are established to be correlated with the amplitudes of the octahedral rotations in perovskites.Mulder et al. ()
In the symmetry lowering structural phase transition, irrep with order parameter is the primary order parameter capturing the CDP behavior. The and are secondary distortions in the phase transition that accompany . This is seen in Table 4, where the largest amplitude for modes not already present from the transition to is almost always given by . NdNiO is found to be the exception to this rule and the origin of phase transition in NdNiO could be different compared to other bulk RNiO compounds.
In the nickelate literature, generally NdNiO is discussed in conjunction with PrNiO, because together they represent a unique case where and coincide. In NdNiO, we find that the distortion amplitude for is twice that of , indicating that irrep acts as the primary distortion mode instead of . Indeed our group theoretical analysis shows that for the direct sum, , whose tilt system correspond to the order parameter containing only three free parameters:, the required space group is monoclinic structure, an isotropy subgroup of , indicating that could be the primary distortion mode in the phase transition if it also provides the greatest energetic stability to the structure. The latter constraint requires evaluation of the lattice dynamics using a technique beyond group theory. Interestingly for the experimental bulk PrNiO structure, irrep is found to have zero amplitude.
To summarize this discussion, we have introduced an alternative symmetrymode description of the structural distortions in nickelates that identifies bulk NdNiO compound as anomalous in the series. Combining temperaturedependent diffraction studies with mode decomposition analysis could provide key insights necessary to fully understand the evolution of the primary and secondary distortions with the thermodynamic origin of phase transition.
We now shift our attention to the weak secondary distortion modes, particularly to the observation of two types of JahnTeller distortions to the NiO units, and (Figure 5). Although and are practically negligible and are the secondary distortion modes in the transition, their presence could still have significant implications on the physical properties. The motivation for critically analyzing secondary distortion modes in nickelates comes from the study of improper phase transitions,Dvořák and Petzelt (1971); Levanyuk and Sannikov (1974); Tolédano and Tolédano (1987) e.g., where spontaneous electric polarization arises as a secondary effect accompanying complex nonpolar distortions through anharmonic interactionsPerezMato et al. (2008); Etxebarria et al. (2010); Benedek and Fennie (2011); Stroppa et al. (2013) in improper ferroelectrics. Similarly, we are interested in exploring the role of secondary order parameters on the electronic properties of nickelates.
Experimental observations based on giant oxygen OO isotope effect Medarde et al. (1998) indicate the presence of dynamic JahnTeller polarons; however, the interpretation of orbitalordering has lacked conclusive support from diffraction experiments Catalan (2008). More recently, ultrathin films of nickelates are being increasingly investigated for rational control of orbital polarization Chakhalian et al. (2011a); Freeland et al. (2011); Chen et al. (2013). The identification of a JahnTeller bond elongations and contractions from our symmetry analysis in the bulk compounds suggests the nickelates should have tendencies to orbital (ordering) polarizations. Misfit strain, superlattice formation, or symmetry mismatch at the thin film–substrate interface may be used to selectively enhance the contributions of JahnTeller modes, described by irreps and , as recently proposed in Ref. Tung et al., 2013. Alternatively, orbitalordering could emerge in nickelates, e.g., LaNiO where the JahnTeller modes are prohibited by symmetry in the bulk groundstate structure, through quantum confinement effects in the limits of ultrathin films Chakhalian et al. (2011a); Freeland et al. (2011); although in this case, straininduced JahnTeller distortions would likely result in larger energetic penalties.
Statistical Analysis
We now evaluate the statistical correlation between the amplitude of the symmetryadapted modes and the macroscopic , and transition temperatures. The physical motivation behind the statistical analysis is to uncover hidden associations between the cooperative atomic displacements and electronic/magnetic phase transitions. The goal is to quantitatively identify structure–functionality relationships that could be rigorously evaluated at the ab initio level.
We begin by constructing a dataset containing seven NiO nickelates, where each rareearth nickelate in our set is described using eight distortionmodes, , and , resulting in a matrix. We scaled the data by subtracting the mean of each column from its corresponding columns (this process is also known as centering), and then divided the centered columns by their standard deviations. To evaluate the degree of linear relationship between the structural distortions, , and , we calculate the sample covariance of the centered and scaled dataset as
where is the total number of NiO compounds in our dataset; and are the centered and scaled column vectors of our data matrix , respectively. Results from the covariance analysis are summarized as a correlation heat map in Figure 6.
A strong positive correlation is found between irreps that describe distortions to the NiO octahedra: (a planar JahnTeller mode), , , , and modes. Even though the distortionmodes are orthonormal by construction, when they are collectively evaluated for a series of NiO compounds, our analysis reveals that they (the modes describing the distortions) are statistically dependent and coupled. These modes largely describe bond angle distortions and are positively correlated with the electronic transition . The conventional route to describe variations in primarily focus on tolerance factor and average Ni–O–Ni bond angle, whereby bending of the Ni–O–Ni angle further from the ideal case of 180 decreases the bandwidth, promoting the insulating state over the metallic stateTorrance et al. (1992); Obradors et al. (1993); Catalan (2008). While we recover this behavior, we also identify the unique displacement patterns that geometrically sum to give the aggregate bond angle: irreps , , , , and cooperatively act to bend the Ni–O–Ni angle. These five irreps fully describe the crystal structure relative to the cubic phase found in the metallic nickelates at hightemperature, reinforcing the concept that the orthorhombic distortions are largely responsible for the bandwidthcontrolled transport behavior in nickelates, and hence prepare the electronic system for the MIT.
Intriguingly, the CBD distortion, which is the usual signature for CO has only a moderate effect on , indicating that the dominant structural route to engineer the electronic transition may not be solely through isotropic bond length distortions. Moreover, anisotropic bond distortions obtained with the type JahnTeller mode or the tilting mode do not contribute significantly to . In fact, we find they are anticorrelated with the electronic transition temperature (Figure 6). Our conclusion to this point is that the geometry (tilt pattern) of the oxygen framework structure is the important atomic scale feature governing the MItransition.
We now shift our attention to the TypeE