Interplay of octahedral rotations and breathing distortions in charge ordering perovskite oxides

Interplay of octahedral rotations and breathing distortions in charge ordering perovskite oxides


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 valence-state 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.

61.50.Ks, 31.15.xh, 71.30.+h

I 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 corner-connected 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 three-dimensional irreducible representations (irreps), and , of the high-symmetry 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 (low-symmetry) can also produce electronic metal–insulator (MI) transitions. Perovskite oxides with -site cations in (), () and () electronic configurations are particularly susceptible, because the low-energy electronic structure is dictated by the octahedral crystal-field split antibonding orbitals—the atomic-like -states that are spatially directed at the coordinating oxygen ligands. MI-transitions which occur simultaneously with lattice distortions are common in low-dimensional materials, e.g. Peierls systems.Canadell et al. (2012) In three-dimensional perovskite oxides, however, the most familiar electronic transitions with concomitant changes in the –O bond lengths and octahedral rotations result from cooperative first-order Jahn-Teller 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 Jahn-Teller distortions are described by irreps , and , and their interaction with octahedral rotations are well-established.Carpenter and Howard (2009a); ?

Unusually high-valence 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 on-site 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 so-called charge ordering (CO) lowers the potential energy of the crystal and gaps the Fermi surface. 1

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, 2 largely because of the change in ionic radii, i.e., the effective radius of is larger than .

Figure 1: (Color online.) Illustration of the effect of charge disproportionation (CDP) on the local O octahedral site equivalence. Before the CDP transition all octahedra are equivalent (left); afterwards the O octahedra disproportionate into non-equivalent sites with the simplest two-site case shown here: charge transfer between -sites causes one octahedron to dilate (site 1) and the other octahedron to contract (site 2), resembling a “breathing” mode of the O perovskite building blocks.

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 MI-transitions, however, is not as well understood Matsuno et al. (2002); Saha-Dasgupta et al. (2005) as the role of Jahn-Teller 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 octahedron-derived 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.3 Using this information, we then illustrate how to quantify relative contributions of octahedral breathing distortions and rotations across phase boundaries in prototypical nickelate and bismuthate perovskites. This rigorous mapping of the unit-cell level structural distortions into a symmetry-adapted basis enables us to disentangle the role that the atomic structure plays in directing the macroscopic electronic metal–insulator transitions. Finally, we show that a synergistic combination of group-theoretical analysis with statistical analysis makes it possible to understand the complex interplay pervasive in mixed-metal perovskite oxides.

Ii Cooperative Breathing Distortions

The octahedral breathing distortions, which create two unique crystallographic -sites, must tile in three-dimensions to maintain the corner-connectivity 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.

Figure 2: (Color online.) Schematic octahedral representations of the two cooperative structural breathing distortions that coexists with charge ordering in O perovskites. In (a) the breathing distortion is two-dimensional, irrep , and produces a columnar ordering of the two cations reducing the symmetry to . In (b) a three-dimensional ordering produces a checkerboard arrangement of the -cations and is described by the irrep ( symmetry).

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. 4 This distortion is associated with the active three-dimensional irrep with order parameter and manifests as a zone edge lattice instability of the cubic phase. The order parameter (OP) describes a vector in the irrep space and corresponds to specific directions along which the physical distortion may be induced. For the irrep, the OP can have three general components (), where the values , and correspond to amplitudes of the two-dimensional breathing distortion along each Cartesian direction , and , respectively. Here we consider only the case where , i.e. a restricted one-dimensional space. The possible directions for the OP correspond to , and or columnar arrangements of the CBD along the -, - and -directions, respectively, so that the octahedra distort in the same sense along the given direction. As a result, the symmetry is reduced to tetragonal, (space group no. 123), and the -site Wyckoff position of the cubic aristotype is split as , doubling the number of perovskite formula units (f.u.) in the primitive cell (Table 1).

The second CBD consists of O octahedra which are tiled in a three-dimensional checkerboard arrangement [Fig. 2(b)]. The distortion is described by a one-dimensional irrep and occurs as a zone-corner 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
(1) 0 0 0
(2) 0
O(1) 0
O(2) 0 0
(no. 225)
Atom Wyck. Site
(1) 0 0 0
O 0 0
Table 1: Crystallographic data including the occupied Wyckoff positions (Wyck. Site) for the cooperative breathing distortions (CBD) available to O perovskites (Figure 2) in the absence of O rotations. Atom positions are given relative to the ideal cubic symmetry such that the relevant CBD imposed on the oxygen positions is indicated by . The value of controls the amplitude of the –O contraction (elongation) and typically scales with the amount of inter-site charge transfer (). The change in cell size is given relative to the pseudo-cubic (pc) lattice constant .

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 three-dimensional 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 second-order 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 well-defined 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 cell-doubling 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 non-zero 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, in-phase rotations (neighboring octahedra along a Cartesian axis rotate in the same direction), or out-of-phase 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 5 (a,0,0)6 (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)7 (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 8 (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: Possible crystallographic space groups, octahedral rotation patterns and unit cell relationships for O perovskites exhibiting rotations of octahedra given by irreps and with order parameter directions given in parentheses () with the CBD or . The lattice vectors and origin shifts are given with respect to the high symmetry 5-atom structure ( cation at the origin).

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 three-dimensional 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 out-of-phase 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 out-of-phase 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
no change no change
Table 3: The change in space group which occurs when the different octahedral CBD patterns are superposed with the 15 simple octahedral tilt systems.

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.

Figure 3: (Color online.) Representation of the rotation patterns that are compatible with each (or both, given by the intersection of the) CBD without requiring a further symmetry reduction.

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 out-of-phase rotations about each axis, or equivalently a single out-of-phase rotation about the three-fold axis. The trigonal symmetry is incompatible with a distortion that would require a loss of the three-fold 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 bond-disproportionation. 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.)

Figure 4: Group—sub-group relationship between and . Irreps within responsible for transitions are shown. In addition, we note that and would also transform . Dotted line indicates that the phase transition is not allowed to be continuous within the confines of Landau theory.

Here we describe the effect of superposition of the common orthorhombic rotation pattern () obtained from irreps and , which gives the six-dimensional order parameter with two unique directions as

The superposition of the three-dimensional CBD and the rotation pattern gives a seven-dimensional order parameter space, i.e., (), which we contract to an effective four-dimensional 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 out-of-phase 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 rare-earth 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); Blanca-Romero 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 non-thermodynamic phases in thin film geometries. Knowledge of quantitative structure–property octahedral distortion relationships are required to accelerate materials discoveries.

Distortion-mode decomposition analysis is an alternative approachCampbell et al. (2006); Carpenter and Howard (2009a, b); Perez-Mato et al. (2010) (widely practiced and followed in the crystallography literature) to study displaciveDove (1997) phase transitions in perovskites. It involves describing a distorted (low-symmetry) structure as arising from a (high-symmetry) parent structure with one or more static symmetry-breaking structural distortions.Perez-Mato et al. (2010) In the undistorted parent structure, each symmetry breaking distortion-mode has zero amplitude. The low-symmetry phase, however, will have finite amplitudes for each irrep compatible with the symmetry breaking. Said another way, the low-symmetry 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 symmetry-adapted 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 distortion-mode 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 distortion-mode analysis relies solely on crystal structural data, which enables both bulk and thin film stabilized structures with identical compositions to be evaluated on equal-footing. 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 distortion-modes to form the basis for the quantitative description of octahedral distortions and CBD on material properties. Using bulk NiO, where is a rare-earth element, and BaKBiO perovskite compounds as prototypical charge-ordering materials, we decompose available low-symmetry structural data into symmetry-adapted structural distortion modes. We then evaluate and correlate the amplitudes of the distortion-modes to macroscopic materials behavior to uncover trends linking octahedral distortions to the structural and physical properties. We use the group-theory program isodistort Campbell et al. (2006) for distortion-mode 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 R-script in the supplementary materials section available at

v.1 RNiO Nickelates

Rare-earth perovskite nickelates, NiO, where Y, Ho, Er, Dy, Lu, Pr, or Nd, exhibit non-trivial changes in structure and physical properties, including sharp first-order temperature-driven MI-transitions, unusual antiferromagnetic order in the ground state, and site- or bond-centered 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 9 (Figure 4), where the Ni cation no longer maintains a unique uniform valence on all sites, and disproportionates as

Moreover, the insulating ground-state 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 distortion-modes 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); García-Muñoz et al. (2009); Munoz et al. (2009) into the orthonormal symmetry-modes. 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 Jahn-Teller
(c) in-phase rotation (d) out-of-phase tilting
(e) -type Jahn-Teller (f) out-of-phase rotation
(g) bending/buckling mode (h) in-phase tilting
Figure 5: (Color online.) Illustration of the symmetry-adapted orthonormal distortion modes found in the low-symmetry NiO and BaKBiO perovskites. The undistorted octahedra are shown in panel (a) for comparison. Octahedral tilting distortions refer to the rotation of the octahedral units (we use ‘tilting’ and ‘rotation’ interchangeably). The out-of-phase distortions are differentiated using arrows (lines and dashes) indicating the direction of cooperative atomic displacements. (b) describes the -type Jahn-Teller mode, where two bonds shrink and two elongate; (c) describes an in-phase rotation mode; (d) describes an out-of-phase tilting mode, which could also act as an in-phase tilting mode depending on the order parameter direction (e) describes the -type Jahn-Teller mode, where four bonds contract and two expand, and vice versa; (f) describes an out-of-phase rotation mode; (g) describes an out-of-phase bending mode; (h) describes an in-phase tilting mode accompanied by cation displacements, which are not shown for clarity.

The structure decomposes into eight algebraically independent symmetry-modes 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
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
Table 4: Summary of the amplitudes of each irrep (notation given with respect to the parent space-group) in the structure (units in Å). Std. Err. = , where is the standard deviation and (=7) is the sample size, gives an estimate of the sampling error. The total distortion amplitude is given in the column indicated by .

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 zone-boundary 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 distortion-modes 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 anti-parallel 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 symmetry-mode description of the structural distortions in nickelates that identifies bulk NdNiO compound as anomalous in the series. Combining temperature-dependent 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 Jahn-Teller 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 non-polar distortions through anharmonic interactionsPerez-Mato 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 O-O isotope effect Medarde et al. (1998) indicate the presence of dynamic Jahn-Teller polarons; however, the interpretation of orbital-ordering has lacked conclusive support from diffraction experiments Catalan (2008). More recently, ultra-thin 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 Jahn-Teller 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 Jahn-Teller modes, described by irreps and , as recently proposed in Ref. Tung et al., 2013. Alternatively, orbital-ordering could emerge in nickelates, e.g., LaNiO where the Jahn-Teller modes are prohibited by symmetry in the bulk ground-state structure, through quantum confinement effects in the limits of ultra-thin films Chakhalian et al. (2011a); Freeland et al. (2011); although in this case, strain-induced Jahn-Teller distortions would likely result in larger energetic penalties.

Statistical Analysis

We now evaluate the statistical correlation between the amplitude of the symmetry-adapted 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 rare-earth nickelate in our set is described using eight distortion-modes, , 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.

Figure 6: (Color online.) Correlation heat map capturing the degree of linear relationship between unit cell level structural distortion modes and macroscopic transition temperatures, , and in NiO perovskites. Dark blue (red) indicates a strong positive (negative) correlation and white indicates no statistical correlation between between the two variables. Units for the irreps and temperatures are given in Å and Kelvin (K), respectively.

A strong positive correlation is found between irreps that describe distortions to the NiO octahedra: (a planar Jahn-Teller mode), , , , and modes. Even though the distortion-modes 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 high-temperature, reinforcing the concept that the orthorhombic distortions are largely responsible for the bandwidth-controlled 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 Jahn-Teller mode or the tilting mode do not contribute significantly to . In fact, we find they are anti-correlated 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 MI-transition.

We now shift our attention to the Type-E antiferromagnetic ordering. Unlike the transport behavior, where the