Lattice parameter data from the literature have been used to provide a complete description of spontaneous strain variations across each of the six known phase transitions of $WO3$ in the temperature interval 5–1273 K. Analysis of strain/order parameter coupling reveals the character of each phase transition, a unified description of strain across the full temperature range, the relationship between strain and electronic effects, and new insights into the strain gradients likely to be present in each of the different domain walls that develop in four different ferroelastic phases. Tetragonal and orthorhombic shear strains have values of 4%–6% and 2%–3%, respectively, and are dominated by coupling with the order parameter for antiferroelectric-type displacements. Conversely, shear strains, $e4$, $e5$, and $e6$, of up to 2% are controlled by octahedral tilting. Changes in electronic structure and properties have been related back to the susceptibility of $W6+$ to develop cooperative second-order-Jahn–Teller distortions. Proximity to tilt instabilities along with group–subgroup relationships in the $P4/nmm$ parent structure results in two overlapping sequences of structural phase transitions, which differ in the form of their electronic structure. The possibility of a ground state structure in space group $P21/c$ can be rationalized in terms of the efficiency by which different combinations of shearing and tilting of the $WO6$ octahedra can reduce the unit cell volume and would imply that $WO3$ has a re-entrant phase transition. Gradients in up to three order parameters coupled with gradients in strain of up to 12% across ferroelastic domain walls indicate that the different ferroelastic phases of $WO3$ should have domain walls with varied and potentially exotic electronic properties for device applications such as in nanoelectronics and neuromorphic computing.

## I. INTRODUCTION

Tungsten trioxide, $WO3$, has been a material of great societal impact for nearly two centuries, finding purpose in applications as old as pigmentation in paint^{1} and as modern as the active ingredient in electrochromic windows.^{2} It has been the focus of close scientific interest, also, from both structural^{3–13} and theoretical^{14–19} points of view, for the remarkable sequence of structural phase transitions it displays. The crystal structure is a network of corner-linked $WO6$ octahedra, representing a variation of the archetypal perovskite structure $ABO3$ in which the A-cation sites are vacant. Although the cubic phase is never observed, the reported sequences of phase transitions between room temperature and the sublimation point can be understood in terms of distortions from the cubic aristotype structure in space group $Pm3\xafm$ that arise by combinations of different order parameters relating to three basic mechanisms:

Antiferroelectric cation displacements,

Out-of-phase octahedral tilting,

In-phase octahedral tilting.

Howard *et al.*^{12} provided an account of the history and controversy relating to characterization of these transitions at high temperatures. Possible sources of discrepancy between the experimental studies include high sensitivity to stoichiometry, sample preparation method, crystal size, and mechanical treatment.^{20,21} An issue with x-ray diffraction is that scattering from tungsten atoms is much greater than from oxygen atoms, making it difficult to define the oxygen positions precisely.^{21} In addition, multiple twinning in single crystals is common, creating difficulties for single crystal studies. In the particular case of $WO3$, $W6+$ can reduce to metastable $W5+$ stabilized by surfaces,^{22} inside domain walls,^{23} and within the bulk.^{24–27} The favored characterization technique has, therefore, been neutron powder diffraction.^{7–12} As has been shown for other perovskites with multiple phase transitions, including R- and M-point tilting in $SrZrO3$^{28} or tilting + Jahn–Teller distortions + charge ordering in $Pr0.48Ca0.52AlO3$,^{29} one approach is to make use of the high-resolution lattice parameter data which are obtained. These can be used to determine symmetry-adapted strains and, hence, follow the temperature dependence of individual order parameters. The purpose of the present paper is to follow the same approach for $WO3$ by evaluating spontaneous strains associated with each of the known transitions. Group theory and Landau theory are used to define the strain/order parameter relationships.

Two sets of lattice parameters from previous neutron powder diffraction studies have been used for the strain calculations. Data above room temperature were reported by Howard *et al.*^{12} and represent the most definitive determination of the transition sequence, starting from the high temperature tetragonal structure in space group $P4/nmm$, which has antiferroelectric cation displacements only. During cooling, this gives way to a second tetragonal structure, $P4/ncc$, by the development of out-of-phase octahedral tilting. The next transition is from tetragonal $P4/ncc$ to monoclinic $P21/c$ and involves a further out-of-phase octahedral tilt about a second axis. The next transition is to orthorhombic $Pbcn$ and involves the combination of an in-phase tilt about the c-axis and an out-of-phase tilt about the b-axis. The final transition is to the monoclinic structure in space group $P21/n$, which has two out-of-phase tilts about the a- and c-axis and an in-phase tilt about the b-axis. Neutron powder diffraction data below room temperature reported by Salje *et al.*^{8} showed a triclinic structure in space group $P1\xaf$ giving way to a monoclinic structure in space group $Pc$ (formerly referred to as the $\u03f5$-phase^{8}) via a first order transition. In previous studies, the $P1\xaf$ structure had been found to be stable between 290 and 240 K.^{4,30–32}

In comparison with the other structures of $WO3$, the $Pc$ structure has ferroelectric displacements that are not accounted for by the three mechanisms referred to above.^{5,9} However, while experimental data indicated that the crystals are ferroelectric,^{33–35} recent first-principle electronic structure calculations suggest that the ground state structure is in space group $P21/c$.^{17,36} This discrepancy is not resolved here but the strain analysis confirms the closeness of the relationship between the low temperature structure and the $P21/c$ structure observed at high temperatures.

An additional consideration in relation to strain-related properties is the variability and remarkable properties of ferroelastic domain walls in these structures. One focus has been on the conductivity of domain walls in single crystals which, under suitable conditions, can become superconducting below 3 K.^{22,25,37–40} Domain walls arising from a phase transition with only one order parameter, $Q$, have gradients from $+Q$ to $\u2212Q$ across them and $Q=0$ at the midpoint. The associated strain contrast across a ferroelastic wall is twice the symmetry-breaking shear strain. When two order parameters with different symmetries, $Q1$ and $Q2$, vary across a domain wall, the gradients of each do not simply pass through zero. Instead, additional relaxations can lead to different atomic structures locally within the walls, according to whether coupling between the order parameters is biquadratic ($\lambda Q12Q22$) or linear-quadratic ($\lambda Q1Q22$).^{41,42} This aspect of the structure and properties of ferroelastic domain walls in $WO3$ has not yet been explored in detail, but some aspects of the formal order parameter relationships can be set out for each of the known structure types.

## II. STRAIN AND ORDER PARAMETER COUPLING

A full symmetry analysis of octahedral tilting combined with antiferroelectric cation displacements has been given by Howard *et al.* (2002)^{12} based on the group theory software package ISOTROPY.^{43} With respect to the $Pm3\xafm$ parent structure, the three order parameters corresponding to each transition mechanism belong to irreducible representations $M3\u2212$, $R4+$, and $M3+$. The transition from $Pm3\xafm$ to $P4/nmm$ involves a single $M3\u2212$ order parameter component, describing antiferroelectric cation displacements. Upon cooling from the $P4/nmm$ tetragonal phase to room temperature, each subsequent transition can be described in terms of octahedral tilting and a second $M3\u2212$ component. Table I shows the space group, corresponding order parameters, relationships between order parameter components, and tilt systems according to Glazer’s notation^{44,45} for each of the structures considered here.

Space group . | Order parameter components $M3\u2212$ $R4+$ $M3+$ . | Relationships between order parameter components . | Tilts . |
---|---|---|---|

$Pm3\xafm$ | 000 000 000 | a^{o} a^{o}a^{o} | |

P4/nmm | q_{1}00 000 000 | a^{o} a^{o} a^{o} | |

P4/ncc | q_{1}00 q_{4}00 000 | q_{1} ≠ q_{4} | a^{o} a^{o} c^{−} |

P2_{1}/c | q_{1}00 q_{4} q_{5} q_{6} 000 | q_{5} = q_{6} ≠ q_{1} ≠ q_{4} | a^{−} a^{−} c^{−} |

Pbcn | q_{1}0q_{3} q_{4}00 0q_{8}0 | q_{1} ≠ q_{3} ≠ q_{4} ≠ q_{8} | a^{o} b^{+} c^{−} |

P2_{1}/n | q_{1}0q_{3} q_{4} q_{5}0 0q_{8}0 | q_{1} ≠ q_{3} ≠ q_{4} ≠ q_{5} ≠ q_{8} | a^{−} b^{+} c^{−} |

$P1\xaf$ | q_{1}0q_{3} q_{4}q_{5}q_{6} 0q_{8}0 | q_{1} ≠ q_{3} ≠ q_{4} ≠ q_{5} ≠ q_{6} ≠ q_{8} |

Space group . | Order parameter components $M3\u2212$ $R4+$ $M3+$ . | Relationships between order parameter components . | Tilts . |
---|---|---|---|

$Pm3\xafm$ | 000 000 000 | a^{o} a^{o}a^{o} | |

P4/nmm | q_{1}00 000 000 | a^{o} a^{o} a^{o} | |

P4/ncc | q_{1}00 q_{4}00 000 | q_{1} ≠ q_{4} | a^{o} a^{o} c^{−} |

P2_{1}/c | q_{1}00 q_{4} q_{5} q_{6} 000 | q_{5} = q_{6} ≠ q_{1} ≠ q_{4} | a^{−} a^{−} c^{−} |

Pbcn | q_{1}0q_{3} q_{4}00 0q_{8}0 | q_{1} ≠ q_{3} ≠ q_{4} ≠ q_{8} | a^{o} b^{+} c^{−} |

P2_{1}/n | q_{1}0q_{3} q_{4} q_{5}0 0q_{8}0 | q_{1} ≠ q_{3} ≠ q_{4} ≠ q_{5} ≠ q_{8} | a^{−} b^{+} c^{−} |

$P1\xaf$ | q_{1}0q_{3} q_{4}q_{5}q_{6} 0q_{8}0 | q_{1} ≠ q_{3} ≠ q_{4} ≠ q_{5} ≠ q_{6} ≠ q_{8} |

The spontaneous strain tensor contains six components: volume strain, $ea$, has the symmetry of $\Gamma 1+$; tetragonal, $et$, and orthorhombic, $eo$, strains have the symmetry of $\Gamma 3+$; and the remaining three shear strains, $e6$, $e4$, and $e5$, have the symmetry of $\Gamma 5+$. ISOTROPY gives the relationships between strain and the order parameters shown in Table II. A complete Landau expansion, combining these relationships and including elastic energy terms, yields the generalized Landau expansion given as Eq. (A1) in the Appendix. Explicit relationships between strain and order parameter components derived from the equilibrium condition $\u2202G\u2202e=0$ are also given in the Appendix and are used in the strain analysis below.

$M3\u2212$—Antiferroelectric order parameter (q_{1}, q_{2}, q_{3}) | |

$\Gamma 1+$ Volume strain (e_{a}) | $ea(q12+q22+q32)$ |

$\Gamma 3+$ Tetragonal and orthorhombic strain (e_{t}, e_{o}) | $\u22123et(q12\u22120.5q22\u22120.5q32)+[32eo(q22\u2212q32)]$ |

$\Gamma 5+$ Shear strains e_{6}, e_{4}, e_{5} | $e62(q12+q22+q32)+e42(q12+q22+q32)+e52(q12+q22+q32)$ |

$R4+$—Out-of-phase octahedral tilting order parameter (q_{4}, q_{5}, q_{6}) | |

$\Gamma 1+$ Volume strain (e_{a}) | $ea(q42+q52+q62)$ |

$\Gamma 3+$ Tetragonal and orthorhombic strain (e_{t}, e_{o}) | $\u22123et(q42\u22120.5q52\u22120.5q62)+[32eo(q62\u2212q52)]$ |

$\Gamma 5+$ Shear strains e_{6}, e_{4}, e_{5} | e_{6}q_{5}q_{6} + e_{4}q_{4}q_{6} + e_{5}q_{4}q_{5} |

$M3+$—In-phase octahedral tilting order parameter (q_{7}, q_{8}, q_{9}) | |

$\Gamma 1+$ Volume strain (e_{a}) | $ea(q72+q82+q92)$ |

$\Gamma 3+$ Tetragonal and orthorhombic Strain (e_{t}, e_{o}) | $\u22123et(q72\u22120.5q82\u22120.5q92)+[32eo(q82\u2212q92)]$ |

$\Gamma 5+$ Shear strains e_{6}, e_{4}, e_{5} | $e62(q72+q82+q92)+e42(q72+q82+q92)+e52(q72+q82+q92)$ |

$M3\u2212$—Antiferroelectric order parameter (q_{1}, q_{2}, q_{3}) | |

$\Gamma 1+$ Volume strain (e_{a}) | $ea(q12+q22+q32)$ |

$\Gamma 3+$ Tetragonal and orthorhombic strain (e_{t}, e_{o}) | $\u22123et(q12\u22120.5q22\u22120.5q32)+[32eo(q22\u2212q32)]$ |

$\Gamma 5+$ Shear strains e_{6}, e_{4}, e_{5} | $e62(q12+q22+q32)+e42(q12+q22+q32)+e52(q12+q22+q32)$ |

$R4+$—Out-of-phase octahedral tilting order parameter (q_{4}, q_{5}, q_{6}) | |

$\Gamma 1+$ Volume strain (e_{a}) | $ea(q42+q52+q62)$ |

$\Gamma 3+$ Tetragonal and orthorhombic strain (e_{t}, e_{o}) | $\u22123et(q42\u22120.5q52\u22120.5q62)+[32eo(q62\u2212q52)]$ |

$\Gamma 5+$ Shear strains e_{6}, e_{4}, e_{5} | e_{6}q_{5}q_{6} + e_{4}q_{4}q_{6} + e_{5}q_{4}q_{5} |

$M3+$—In-phase octahedral tilting order parameter (q_{7}, q_{8}, q_{9}) | |

$\Gamma 1+$ Volume strain (e_{a}) | $ea(q72+q82+q92)$ |

$\Gamma 3+$ Tetragonal and orthorhombic Strain (e_{t}, e_{o}) | $\u22123et(q72\u22120.5q82\u22120.5q92)+[32eo(q82\u2212q92)]$ |

$\Gamma 5+$ Shear strains e_{6}, e_{4}, e_{5} | $e62(q72+q82+q92)+e42(q72+q82+q92)+e52(q72+q82+q92)$ |

## III. STRAIN ANALYSIS WITH RESPECT TO $Pm3\xafm$ REFERENCE STATE

High-resolution neutron powder diffraction data reported for high temperatures by Howard *et al.*^{12} are displayed in terms of the pseudocubic unit cell in Fig. 1. Data for the “$Pc$” structure below room temperature are neutron powder diffraction results reported in Ref. 8. Inverted commas are used here and below for the space group of this structure in light of the suggestion from recent simulations that the ground state structure should be in space group $P21/c$.^{17,36} Transition temperatures are marked by vertical lines, with the space group of the stable structure given in each temperature interval. Also included are values for lattice parameters of the $P1\xaf$ structure reported in Refs. 4 and 6– 8.

Values of individual strain components depend on the pseudocubic lattice parameters according to

Linear strain components $e1$, $e2$, and $e3$ are defined as being parallel to X, Y, and Z axes of the cubic reference state, respectively, and yield values of the symmetry-adapted shear strains $et$ and $eo$ according to Eqs. (A3) and (A4). $ao$ is the lattice parameter of the cubic reference state but, because this structure is not stable at any temperature below the sublimation point, $ao=V13$, where $V$ is the volume of the pseudocubic cell, is used as an approximation that allows shear strains to be determined without significant loss of precision. This does not allow any determination of volume strains, however. $\beta \u2217$ is the reciprocal lattice angle of the monoclinic structures. Of the remaining shear strains, $e5$ has non-zero values in the case of the $P21/n$ structure ($e4=e6=0$). In the pseudocubic setting of the $P21/c$ structure, the cell angles are $\alpha =\beta \u2260\gamma \u226090\xb0$, giving $e4=e5\u2260e6\u22600$. Shear strains calculated using the lattice parameter data in Fig. 1 are shown in Fig. 2. The single most significant feature of these is the size of the tetragonal shear strain, $et$ [$\u221dq12$, Eq. (A7)] across the full temperature range. At 4%–6%, this is an order of magnitude greater than would be expected in the case of octahedral tilting transitions and reflects large distortions of the structure associated with the antiferroelectric displacements of the $Pm3\xafm\u2192P4/nmm$ transition. $WO3$ is already a highly sheared perovskite structure before any of the subsequent phase transitions occur.

The transition from tetragonal $P4/nmm$ to tetragonal $P4/ncc$ at 1173 K gives $q4\u22600$, $q5=q6=0$ (R-point tilts about the c-axis). It is co-elastic^{46} and involves increases in the value of $et$ by up to 0.0023. According to Eq. (A11), the variation of $et$ scales with $q42$ if $q1$ (the antiferroelectric order parameter) remains approximately constant. The tetragonal $P4/ncc$ to monoclinic $P21/c$ transition at 1073 K is ferroelastic. Additional R-point tilts of octahedra about an axis which would correspond to the [110] of the cubic reference structure give $q5=q6\u22600$. This is not accompanied by any overt change in $et$ but, instead, gives rise to values of shear strains $|e4|(=|e5|)$ and $|e6|$ up to 0.008. According to Eqs. (A17) and (A18), the variation of $e6$ scales with the square of the tilt order parameter, $q52$, and the variation of $e4$ scales with $q4q5$. This strain is relatively large in comparison with pure tilt systems because the tilting is of octahedra which are already distorted.

From the obvious discontinuities in all strain parameters and the interval of two-phase co-existence, the monoclinic $P21/c$ to orthorhombic $Pbcn$ transition at 993 K is clearly first order in character, as required by the fact that the two space groups do not have a group–subgroup relationship.^{12} The $Pbcn$ structure is ferroelastic with respect to the cubic reference structure. It is notable for having tetragonal and orthorhombic shear strains that, at 4% and 3%, are substantially greater than would be observed for pure tilting transitions. Since both $et$ and $eo$ include the influence of a new component of the $M3\u2212$ order parameter [$q3$ in Table I, Eqs. (A22) and (A23)], it is clear that significant changes in the antiferroelectric displacements accompany the different R-point and new M-point tilts.

The transition from orthorhombic $Pbcn$ to monoclinic $P21/n$ at 623 K involves the addition of R-point tilt component $q5$. The influence of strains coupled to the additional degree of freedom is seen as small variations in the large values of $et$ and $eo$ [Fig. 2, Eqs. (A27) and (A28)] but is most clear in the variation of $e5$, which is expected to scale with $q4q5$ to good approximation [Eq. (A30)]. Relatively large values of shear strains $|e5|$ and $|eo|$ up to 2% and 3%, respectively, are again due to tilting of deformed octahedra. The $P21/n$ structure is ferroelastic with respect to both the $Pm3\xafm$ and $Pbcn$ structures. Strain values for the $P1\xaf$ structure are sparse in the sense that lattice parameters have not yet been collected through its full stability field. The limited available data reveal that $et$ and $eo$ closely match or are slightly smaller than values for the $P21/n$ structure, while values of $e5$ and $e6$ closely match those of $e5$ for the $P21/n$ structure. The development of $e4$ is discontinuous and the transition, to another ferroelastic state, is clearly first order. In principle, it occurs by the appearance of the third R-point tilt order parameter, $q6$ (Table I).

The transition from $P1\xaf$ to “$Pc$” does not involve a group–subgroup relationship and is discontinuous. The most striking feature of the strain variations is the way in which $e4(=e5)$ and $e6$ are clearly extensions of trends for the temperature dependence of the same strain components established in the stability field of the $P21/c$ structure at high temperatures.

## IV. VOLUME STRAINS

The volume strain, $ea$, provides a convenient measure of how order parameters evolve with temperature because it has the symmetry properties of the identity representation. The lowest order coupling term allowed by symmetry has the form $\u2211\lambda eaqi2$ and, hence, $ea$ scales with $\u2211qi2$. In the present case, there are no lattice parameter data available for the parent cubic structure, so it is only possible to determine volume strains, $Vs$, with respect to the $P4/nmm$ structure, as shown in Fig. 3.

Following the approach used in Refs. 47–50, for example, the function $Vo=A+B\Theta scoth\u2061\Theta so/T$ has been fit to unit cell volume data in the temperature interval 1213–1273 K [Fig. 3(a)], where $Vo$ is the unit cell volume of the $P4/nmm$ structure as reference extrapolated into the stability fields of the lower symmetry structures. This has the correct form of evolution, $dV/dT\u21920$, as $T\u21920$ K. On the basis of experience from other perovskites,^{28,29} $\Theta so$ was set at 150 K. Defining $Vs$ as $(V\u2212Vo)/Vo$ gives the variations shown in Fig. 3(b), which reveals a repeating pattern of negative volume strains associated with each transition, analogous to the strain variations seen through successive phase transitions in the mineral lawsonite, $CaAl2Si2O7(OH)2.H2O$.^{51}

The blue curve shown through the data for $Vs$ of the $P4/ncc$ structure is a fit of $Vs2$ vs T and corresponds to tricritical character, $q4\u221d(Tc\u2212T)$, for the $P4/nmm$–$P4/ncc$ transition. The brown curve is a fit of $Vs2$ vs $T$ for the $P4/ncc$–$P21/c$ transition, where $Vs$ for $P4/ncc\u2212P21/c$ is the difference between $Vs$ and values on the blue curve extrapolated below the $P4/ncc$–$P21/c$ transition point ($Tc=1044\xb14K$, from the fit). It signifies that this transition is also close to tricritical in character. The red curve is a fit to the data as $Vs2$ vs T and represents a third transition that is close to tricritical. In this case, the transition would be metastable, $Pbcn$–$P4/nmm$, with $Tc=1116\xb11$ K. The green curve is a fit of $Vs2$ vs $T$ for the $Pbcn$–$P21/n$ transition, where $Vs$ for $Pbcn\u2212P21/n$ is the difference between $Vs$ and values on the red curve extrapolated below the $Pbcn$–$P21/n$ transition point ($Tc=619\xb14$ K, from the fit). Although fits to the data in Fig. 3(b) provide a good representation of the $P4/ncc$–$P21/c$ and $Pbcn$–$P21/n$ transitions as being tricritical, the $P4/nmm$–$P4/ncc$ transition is better represented as being weakly first order.^{10} First order character is confirmed also by the form of the associated anomaly in specific heat as a steep spike, together with hysteresis of the transition temperature between heating and cooling.^{13} Figure 3(c) shows a fit of the Landau solution for a first order transition,

where $Ttr$ is the transition temperature, $Tc$ is the critical temperature, and $Vso$ is the magnitude of the discontinuity in $Vs$ at $Ttr$ (proportional to $qo2$, where $qo$ is the discontinuity in the order parameter).

The curve shown through the data (obtained in a heating sequence) has $Ttr=1171$ K, $Tc=1168.5$ K, $Vso=\u22120.0014$. Values of $Vs$ calculated in the same way for the $P1\xaf$ structure, using unit cell volumes reported in Refs. 4 and 6– 8 for different samples at or just below room temperature, plot slightly below the trend of values for the $P21/n$ structure. There is then an abrupt reduction by $\u223c$1% at the transition from the $P1\xaf$ structure to the low temperature monoclinic structure. It is well established that both transitions are first order in character, with significant hysteresis between heating and cooling.^{4,30,32,52,53} The trend of $Vs$ values for the “$Pc$” structure appears to be a continuation of the trend established at higher temperatures in the stability field of the $P21/c$ structure.

## V. EVOLUTION OF ORDER PARAMETER COMPONENTS

Observed strain variations are now used to provide an indirect measure of how individual order parameters evolve at each transition between successive structures in the full sequence.

### A. $Pm3\xafm$ → *P4/nmm*

Values of the tetragonal shear strain, $et$, arising from the $Pm3\xafm\u2192P4/nmm$ transition are in the vicinity of 6%, which is larger than for most ferroelastic oxides. They do not vary significantly with temperature in the stability field of the $P4/nmm$ structure, consistent with the temperature of the transition to the cubic structure being substantially higher than the sublimation temperature. Given that coupling between different order parameters weakens with increasing divergence of their critical temperatures, it is expected that coupling of the $M3\u2212$ order parameter with the $R4+$ and $M3+$ order parameters will be weak. In other words, the octahedral tilting transitions occur in a highly sheared structure with, to first approximation, little influence of tilting on the antiferroelectric displacements.

### B. *P4/nmm → **P*4/*ncc*

*P4/nmm →*

*P*4/

*ncc*

When expressed with respect to space group $P4/nmm$, the order parameter for $P4/nmm\u2192P4/ncc$ transforms as $Z1\u2212$ and has one component, corresponding to $R4+$ $(a,0,0)$ of space group $Pm3\xafm$. In the absence of $M3\u2212$ distortions, it would be equivalent to the $Pm3\xafm\u2192I4/mcm$ transition typical of many perovskites. Figure 4(a) shows the values for the only non-zero strains, $e1$ and $e3$, in the stability field of the $P4/ncc$ structure, as determined with respect to the $P4/nmm$ structure using values of $ao$ and $co$ obtained by extrapolation of linear fits to $a$ and $c$ above the transition point. Elongation of $c$ and shrinkage in the $ab$ plane gives $e1$ and $e3$ up to $\u22120.003$ and 0.001, respectively, comparable in sign and magnitude with the strains observed at the equivalent pure tilting transition at 1367 K in $SrZrO3$^{28} and at 1578 K in $CaTiO3$.^{54} The transitions in $SrZrO3$ and $CaTiO3$ are also close to tricritical in character.

### C. *P4/ncc → **P*2_{1}/*c*

*P4/ncc →*

*P*2

_{1}/

*c*

The R-point of space group $Pm3\xafm$ becomes folded back to the $\Gamma $-point in $P4/ncc$. As a result, the additional R-point tilt gives rise to the $P4/ncc\u2192P21/c$ transition and has a one component order parameter, $Q$, which transforms as $\Gamma 5+$ $(a,0)$ of space group $P4/ncc$. The transition is pseudoproper ferroelastic and there are two non-zero shear strains, $eo$ and $e5$, given by $eo=(e1\u2212e2)=(a\u2212b)ao,e5=\u2212ccocos\u2061\beta \u2217\u2248\u2212cos\u2061\beta \u2217$, where $\beta \u2217=(180\u2212\beta )$ for the $P21/c$ structure in its conventional setting. The reference parameter, $ao$, was taken as $(a+b)/2$, without serious loss of precision for values of $eo$. These strains are expected to scale as $e5\u221dQ,eo\u221dQ2$. The volume strain, $Vs$, determined with respect to the $P4/ncc$ structure is expected to scale as $Vs\u221dQ2$. Figure 4(b) shows the variations of $eo$ and $\u2212cos\u2061\beta \u2217$ with temperature. Figure 4(c) displays the expected linear variations of $eo$ and $Vs$ with respect to $e52$, confirming pseudoproper properties. Figure 4(d) shows that $e54$ $(\u221dQ4)$ varies linearly with temperature, consistent with tricritical character and $Tc=1047\xb11$ K.

### D. *P2*_{1}/c → *Pbcn*

*P2*

_{1}/c →*Pbcn*

The $P21/c\u2192Pbcn$ transition is necessarily first order due to the absence of a group–subgroup relationship. However, $Pbcn$ is a subgroup of both $P4/ncc$ and $P4/nmm$ so there must be a metastable transition $P4/ncc\u2192Pbcn$ obscured by the stability field of the $P21/c$ structure and a metastable transition $P4/nmm\u2192Pbcn$ obscured by the stability fields of the $P21/c$ and/or $P4/ncc$ structures. At first glance, the $Pbcn$ structure can be understood to develop from the $P4/ncc$ structure by addition of an M-point tilt component, $q8$, and from the $P4/nmm$ structure by addition of both R-point and M-point tilt components, $q4$ and $q8$ (Table I). However, an additional $M3\u2212$ component, $q3$, appears in both cases. In group theory terms, $M3\u2212$ of space group $Pm3\xafm$ becomes $M1$ of space group $P4/ncc$ and the active representation for $P4/ncc\u2192Pbcn$ is $M1$ $(a,0)$. $M3\u2212$ and $R4+$ of space group $Pm3\xafm$ become $M1$ and $Z1\u2212$ of space group $P4/nmm$ so the two order parameters needed for $P4/nmm\u2192Pbcn$ are $M1$ $(a,0)$ and $Z1\u2212$. In both cases, $A1$ $(0,a)$, corresponding to $M3+$ of $Pm3\xafm$, appears as an additonal order parameter. ISOTROPY shows that both transitions are expected to be improper ferroelastic. The question then arises as to which of these drives the transitions and which are secondary effects arising from symmetry being broken by the other combinations.

With respect to the order parameter components shown in Table I, a transition driven by $M3\u2212$, ($q1,0,q3$), would give a structure in space group $Ibam$. Addition of the R-point tilt with components ($q4,0,0$) gives the structure in space group $Pbcn$ but also gives $M3+$ ($0,q8,0$) as a secondary order parameter. In other words, the $P4/nmm\u2192Pbcn$ transition could be driven by the addition of $q3$ coupled with the R-point tilt and the $P4/ncc\u2192Pbcn$ transition could be driven by the addition of $q3$ alone. The M-point tilt would be the secondary effect, developing simply as a consequence of the symmetry change rather than being responsible for driving it. Shear strain $|eo|=2(a\u2212b)/(a+b)$ (to good approximation) with respect to the $Pm3\xafm$ structure depends on $q3$ and $q8$, but not on $q4$ [Eq. (A23)]. The magnitude of $eo$ is in the range 2%–3% and the transition is accompanied by a reduction in $et$ of 1% (Fig. 2). Both these shear strains are substantially larger than would be expected from a typical M-point octahedral tilting transition in perovskites, suggesting that the transition is driven by the new $M3\u2212$ order parameter component $q3$, with tilting ($q8$) as secondary.

Figure 4(e) shows, further, that $eo4$ scales with $(Tc\u2212T)/Tc$, $Tc=1113\xb12$ K. Since $eo$ is also the symmetry-breaking shear strain with respect to $P4/nmm$ as a reference structure, $Tc$ marks the transition point for a metastable transition $P4/nmm\u2192Pbcn$, which is pseudoproper ferroelastic and tricritical in character. The same temperature dependence is observed for $Vs2$, where $Vs$ is the volume strain with respect to the $P4/nmm$ structure from Fig. 3(b). The observed relationships are, therefore, $eo4\u221dVs2\u221dQ4\u221d(Tc\u2212T)/Tc$, where *Q* is a driving order parameter. However, this implies that the metastable transition is pseudoproper ferroelastic in character, $eo\u221dQ$, which is not consistent with the expected improper ferroelastic behavior, $eo\u221dQ2$. For this to be true, the symmetry of $Q$ would need to be $\Gamma 4+$ and would need to represent a different transition mechanism, such as a previously unrecognized electronic instability.

What emerges is a view that the $q3$ component of $M3\u2212$ is more important than tilting in stabilizing the $Pbcn$ structure, corresponding to a dominant role for differently ordered antiferroelectric displacements of cations in comparison with $q1$. Speculation of an additional zone center instability remains to be tested, such as by following the temperature dependence of single crystal elastic moduli that have readily distinguishable patterns of evolution for pseudoproper and improper ferroelastic transitions. In any case, $P4/nmm\u2192Pbcn$ would be close to tricritical in character. Discontinuities in $et$ and $eo$ at 1020 K (Fig. 2) indicate that $P4/ncc\u2192Pbcn$ would be the first order.

### E. *Pbcn → **P*2_{1}/*n*

*Pbcn →*

*P*2

_{1}/

*n*

The $Pbcn\u2192P21/n$ transition occurs by the addition of a second R-point tilt, $q5$ (Table I). The R-point of space group $Pm3\xafm$ folds back to become the $\Gamma $-point of space group $Pbcn$ so that the order parameter has one component and transforms as $\Gamma 3+$ of space group $Pbcn$. This is referred to here, again generically, as $Q$. The new symmetry-breaking shear strain is $e5$ ($\u2248\u2212cos\u2061\beta \u2217$) which couples bilinearly with $Q$, leading to the expectation $e5\u221dQ$. Figure 4(f) shows linear variations of $cos\u2061\beta \u22174$ and $Vs2$ with temperature, where $Vs$ is the volume strain with respect to the $Pbcn$ structure. The strain relationships are, therefore, $e54\u221dVs2\u221dQ4\u221d(Tc\u2212T)/Tc$, i.e., pseudoproper ferroelastic character, tricritical evolution, and overall behavior closely analogous to that of the $P4/ncc\u2192P21/c$ transition. This temperature dependence for the shear strain was previously reported by Locherer *et al.*^{55}

### F. $P21$/n → *P*$1\xaf$

There is no single data set for the variations of lattice parameters through the stability field of the $P1\xaf$ structure, but the combined data from separate samples in Fig. 1 show values of $a$, $b$, and $c$, which follow closely the trend established in the stability field of the $P21/n$ structure. In addition, $\beta \u2248\gamma \u224891\xb0$ in the triclinic structure which lies on or close to an extension of the variation of the angle $\beta $ in the monoclinic structure. The abrupt change in the value of $\alpha $ from $90\xb0$ to $89.2\xb0$ signifies that the transition from the $P21/n$ structure is first order in character. An alternative choice of crystallographic axes would give $\alpha \u2248\beta \u2248\gamma \u224891\xb0$, suggesting that octahedral tilting gives a distortion that is close to rhombohedral. The structure can, therefore, be understood as developing from the $P21/n$ structure by the addition of an R-point tilt such that $q4\u2248q5\u2248q6$. With respect to $P21/n$ as the parent structure, the transition is pseudoproper ferroelastic, with $e4$ and $e6$ as the new symmetry-breaking shear strains.

### G. $P1\xaf$ → “*Pc***”**

*Pc*

Significant discontinuities in the volume strain (Fig. 3) and all the observed shear strains (Fig. 2) indicate that the $P1\xaf\u2192$ “$Pc$” is unambiguously first order, whatever space group might be chosen for the low temperature structure. On the other hand, variations of all the lattice parameters and strains calculated from them clearly fall along the extension of trends established in the stability field of the $P21/c$ structure. This is most obvious for values of $b$, $\alpha $ ($=\beta $), and $\gamma $ of the pseudocubic unit cell in Fig. 1, $e4$ ($=e5$) and $e6$ in Fig. 2, and $V$ in Fig. 3(a). On this basis, the correct space group for the structure would be $P21/c$ rather than $Pc$, unless the strains associated with loss of the $21$ axis are substantially smaller than any of the strains coupled to other order parameters. Figure 5(a) provides a more quantitative test of whether the “$Pc$” structure can be treated simply as the $P21/c$ structure reappearing at a re-entrant transition. $cos\u2061\beta \u22172$ is shown as a function of temperature for the two temperature intervals, 5–250 K and 1003–1053 K, using the $\beta $ angle of the conventional monoclinic setting. This is a good approximation for $e52$ and scales with $Q2$ for the transition $P4/ncc\u2192P21/c$. The curve through the combined data in Fig. 5(a) is a solution to $\u2202G\u2202Q=0$ for the standard Landau 246 expansion, including the saturation temperature, $\Theta s$,

With $Q2$ replaced by $cos\u2061\beta \u22172$, the fit parameters are $B/A=3.02\xd7105\xb11.1\xd7104$ K, $C/A=7.35\xd7108\xb11.3\xd7107$ K, $\Theta s$ = $123\xb14$ K, $Tc=1057\xb11$ K. The curve provides what seems to be a single fit to the data in both temperature intervals, implying that they can be described in terms of a single instability at 1057 K with evolution of an order parameter that is between second order and tricritical in character.

Equations (A17)–(A19) show that a rhombohedral distortion, $e4=e5=e6$, would arise from the R-point tilting if $q4=q5=q6$. The sign of these shear strains, as defined with respect to the parent cubic structure, depends on whether the cell angles are chosen to be acute or obtuse. Values of $|e4|(=|e5|)$ and $|e6|$ are almost the same (at 2.2%) throughout the low temperature stability field of the “$Pc$” structure, implying that this condition is nearly met. Thus, the structure can be understood, to good approximation, as having rhombohedral (R-point) tilting of distorted octahedra. $M3\u2212$ distortions have the effect of causing the associated strains to be much greater than for equivalent rhombohedral tilting in a perovskite with regular octahedra. For example, the value of $|e4|$ is 0.002 at 4 K for the second order $Pm3\xafm\u2192R3\xafc$ transition in $LaAlO3$, where $Tc=817$ K.^{56,57} The fit in Fig. 5(a) is permissive of the low temperature monoclinic structure being the same as the high temperature $P21/c$ structure. Another test of this is provided by examining the interdependence of $e4$ and $e6$, as defined with respect to the cubic reference structure. Figure 5(b) shows how the variation of $cos\u2061\alpha (\u223ce4=e5)$ with respect to $cos\u2061\gamma $ $(\u223ce6)$ tends toward or stays close to the rhombohedral condition $|e4|=|e5|=|e6|$ with falling temperature across both stability fields. In the high temperature stability field of the $P21/c$ structure, $e4$ and $e6$ go continuously to zero, which is possible only if $q5$ goes to zero while the value of $q4$ from the parent $P4/ncc$ structure remains finite. However, the fact that data for the low temperature structure fall on the other side of the $|e4|=|e6|$ line from data for the high temperature structure perhaps provides the only hint that there is a difference between them such as the low temperature structure having additional strain due to lowering of the symmetry to $Pc$.

## VI. DISCUSSION

Analysis of spontaneous strains in this formal manner has yielded details of how the three different structural (multicomponent) order parameters combine to determine the sequence of multiple transitions in $WO3$. The benefit of using the Landau theory with the $Pm3\xafm$ structure for the reference state is that a unified description of the complete strain coupling behavior throughout the full temperature interval from 5 to 1273 K is obtained. It lends some support for the ground state as being in space group $P21/c$, identifies the character of each transition, relates specific strain characteristics to electronic effects, and allows some understanding of strain gradients likely to be present through different domain walls in four different ferroelastic phases.

### A. Two transition sequences from group–subgroup relationships

Group theoretical treatments have been effective in identifying structure hierarchies and transformation sequences in other perovskites with three discrete instabilities, such as combinations of two tilt systems with ferroelectric displacements,^{58} with cation ordering^{59} or with cooperative Jahn–Teller distortions.^{60} The three order parameters might all vary on the same timescale and, hence, be active in influencing the variations of each other but this is not necessarily always the case. For example, cation ordering can break cubic symmetry but generally will not change on the same timescale as variations in the tilt order parameters. Analogously, if the tilting transitions have high critical temperatures, Jahn–Teller and related charge ordering transitions at low temperatures would occur in crystals that already have more or less fixed tilt angles, such as in the case of ($Pr$,$Ca$)-manganites.^{29} It appears at first glance that $WO3$ belongs to the class in which one order parameter is effectively passive because the critical temperature for $M3\u2212$, $(q1,0,0)$ displacements is substantially higher than the critical temperatures for $R4+$ and $M3+$ tilting. The tilting transitions occur in crystals, which have a large tetragonal distortion where there is a strong coupling between $et$ and a saturated value of the order parameter component $q1$. Variations in $et$ as a function of temperature seen in Fig. 2 can then be understood in terms of additional terms in Eqs. (A11), (A15), (A22), and (A27) as specifying the contributions of coupling with tilt order parameters for each structure, rather than variations in $q1$.

Closer inspection reveals that there are two transition sequences in the tetragonal parent structure. The first, $P4/nmm\u2192P4/ncc\u2192P21/c$, involves changes in R-point tilting. The second, $P4/nmm\u2192Pbcn\u2192P21/n\u2192P1\xaf$, involves almost the same changes in R-point tilting but with additional shearing due to the second $M3\u2212$ component, $q3$. The development of an M-point tilt in the second sequence can be understood as a secondary effect, permitted by the loss of symmetry due to the combination of non-zero $M3\u2212$ and $R4+$ components. It is not resolved whether $q3$ effectively acts as the driving order parameter or whether, as hinted at by the variation of $eo$ in the stability field of the $Pbcn$ structure, there is an additional $\Gamma $-point instability.

It is informative to contrast these sequences with $BaTiO3$, the classic cubic perovskite in which multiple phase transitions arise from different components of a single order parameter. With falling temperature, $BaTiO3$ undergoes successive transitions $Pm3\xafm(0,0,0)\u2192P4mm(0,0,a)\u2192Amm2(a,a,0)\u2192R3m(a,a,a)$, where $a$ signifies values for non-zero components of the $\Gamma 4\u2212$ order parameter. Each of the three low temperature structures can have an equilibrium field of stability if the fourth order coefficient of the Landau expansion is negative, i.e., if the $Pm3\xafm\u2192P4mm$ transition is weakly first order^{61} (see, also, Refs. 62 and 63). $Amm2$ and $R3m$ are subgroups of $Pm3\xafm$ but not of $P4mm$ or $Amm2$, respectively, so that spontaneous strains calculated for each structure all follow trajectories through first order transitions toward a common transition temperature (Fig. 2 of Ref. 64). The equivalent R-point tilting sequence would be $Pm3\xafm(0,0,0)\u2192I4/mcm(a,0,0)\u2192Imma(a,a,0)\u2192R3\xafc(a,a,a)$. This occurs as far as the $Imma$ stability field in $SrZrO3$, for example, where the $Pm3\xafm\u2192I4/mcm$ transition is also close to tricritical.^{65} Tetragonal shear strains of the $I4/mcm$ and $Imma$ structures extrapolate to zero at the same temperature because there is only one intrinsic instability. The effect of prior broken symmetry in $WO3$ is that the R-point tilt systems with $(a,a,0)$ and $(a,a,a)$ are not accessible and are replaced by $(a,b,0)$, $(a,b,b)$, or $(a,b,c)$. As an immediate consequence, there is a group–subgroup relationship between the successive structures and each transition in the sequences $(0,0,0)\u2192(a,0,0)\u2192(a,b,b)$ or $(0,0,0)\u2192(a,0,0)\u2192(a,b,0)\u2192(a,b,c)$ has a separate critical temperature. As has been shown previously,^{10–12} each transition is close to tricritical in character.

If the ground state structure is in space group $P21/c$ instead of $Pc$, $WO3$ provides a relatively rare example of a system with a re-entrant transition. In general, the ground state is likely to be the structure with the highest density and there may be a simple geometric argument to account for it in this case. Figure 3 shows that volume strains reach $\u223c\u22122$% for the sequence $P4/nmm\u2192Pbcn\u2192P21/n\u2192P1\xaf$ and $\u223c\u22123.5$% for the sequence $P4/nmm\u2192P4/ncc\u2192P21/c$. Even without including some additional volume strain accompanying the transition to $P4/nmm$ from the parent $Pm3\xafm$ structure, this is almost an order of magnitude larger than typical volume strains accompanying tilting transitions in $SrZrO3$, $(Ca,Sr)TiO3$, or $BaCeO3$.^{28,54,66,67} It appears that the most efficient way of filling vacant space in the $WO3$ structures ultimately involves R-point tilting of octahedra, which have distortions relating to $q1$ rather than by combined R- and M-point tilting of octahedra with distortions relating to both $q1$ and $q3$.

### B. Strain coupling and thermodynamic properties

A classical consequence of strain coupling is that individual strains contribute differently to the thermodynamic character of a phase transition, depending on whether they couple linearly $(\lambda eQ)$ or quadratically $(\lambda eQ2)$ with the driving order parameter.^{46,68,69} All non-symmetry breaking strains couple as $\lambda eQ2$ and result in renormalization of the fourth order Landau coefficient such that it tends toward zero (tricritical) or becomes negative (first order). Coupling with generally large volume strains will have contributed to the tendency for the transitions with group–subgroup transitions to be tricritical or first order. Three of the transitions $(P4/ncc\u2192P21/c$, $Pbcn\u2192P21/n$, $P21/n\u2192P1\xaf)$, and perhaps the metastable $P4/nmm\u2192Pbcn$ transition, are pseudoproper ferroelastic. Linear coupling of the symmetry-breaking shear strains with the zone center driving order parameter, $\lambda eQ$, will have increased their transition temperatures by renormalization of the critical temperature in each case.

### C. Strain coupling and electronic properties

The fundamental link between structural and electronic effects in $WO3$ was identified through the observation that the “$Pc$” $\u2192P1\xaf$ transition is accompanied by a large and abrupt reduction in DC and AC electrical resistivity.^{4,30,32,53,70} The structural transition is, in effect, from an insulator to a semi-conductor. From the formal analysis of shear strains in Fig. 2, it is associated specifically with an abrupt distortion of the $WO6$ octahedra, corresponding to the change in $|eo|$ from zero to $\u223c$3% due to coupling with the $q3$ component of the $M3\u2212$ order parameter.

On the basis of equations given in the Appendix, the shear strains can be divided into two groups: $et$ and $eo$ are dominated by contributions of coupling with the $M3\u2212$ order parameter for antiferroelectric-type displacements, while $e4$, $e5$, and $e6$ arise predominantly from R-point tilting. Values of $et$ and $eo$ fall in the ranges 4%–6% and 2%–3%, respectively (Fig. 2) and are, thus, closer in magnitude to shear strains accompanying Jahn–Teller distortions in *R*$VO3$ and *R*$MnO3$ perovskites (*R* = lanthanide cations^{57}) than to shear strains of up to $\u223c$1% which typically accompany tilting. They define two antiferroelectric ordering processes which are effectively electronic instabilities, the first occurring at some temperature above the sublimation point, described by $q1$, and the second arising at a much lower temperature, described by $q3$, with or without the hypothetical $\Gamma $-point instability. $W6+$ is not Jahn–Teller active but is susceptible to second-order-Jahn–Teller distortions,^{71} and there is an obvious analogy also with the large shear strains that accompany martensitic transitions driven by a band Jahn–Teller distortion in Heusler compounds.^{72–75}

Anomalies in electrical conductivity accompany all the phase transitions at high temperatures,^{4,76} but their magnitude is smaller than seen at the “$Pc$”–$P1\xaf$ transition point. Even though tilting results in shear strains $e4$, $e5$, and $e6$ having values up to $\u223c$2%, their influence is clearly much smaller than is associated with shear strains $et$ and $eo$ arising from coupling with the $M3\u2212$ order parameter. Relationships between structure and electronic properties are the focus of several computational studies^{14–16,19} and will lead to tuning of the electrical properties of bulk and thin film samples through the effect of dopant cations or by imposing a particular strain by the choice of substrate.^{19} The effect of dopant cations on the A cation site should be particularly marked because they will act strongly to hold the structure open. It is known from other studies that the strain fields around replacement cations in the perovskite structure start to overlap through an entire crystal once their substitution reaches $\u223c$1.5% of the A-site, for example, Ref. 77.

### D. Ferroelastic domain walls

Because of the additional degrees of freedom allowed for structural relaxation in domain walls which have gradients in more than one order parameter, it must be expected that ferroelastic domain walls in $WO3$ could have local variations in structure that differ from any of the formal structure types listed in Table I. For perovskites that become ferroelastic due to octahedral tilting with shear strains of up to 1%, the total strain contrast across individual domain walls will be up to 2%. Given the much larger shear strains and the interdependence between electronic properties and shearing of the perovskite structure, it is inevitable that ferroelastic domain walls in $WO3$ will also have electronic properties that are distinct from bulk properties within the domains. These will be superimposed on bulk properties, which may already be exotic due to the 6% tetragonal shear strain of the $P4/nmm$ structure that persists through all subsequent transitions.

Volume strains associated with all the transitions are negative, with the result that local structure within the walls should be relatively expanded by up to $\u223c$3.5%. This has implications for chemical diffusion pathways of impurities and dopant cations as well as providing an additional degree of freedom with respect to structural relaxations that might occur within the walls.

#### 1. Domain walls in the P4/nmm structure

Crystals cannot be grown in the stability field of the $Pm3\xafm$ structure and it is, therefore, not possible to generate ferroelastic domain walls by cooling through the $Pm3\xafm\u2192P4/nmm$ transition. Individual walls could, in principle, be present as growth defects in crystals grown at lower temperatures, however, and would have a shear strain contrast of $\u223c$12% across them. These would be subject to additional local relaxations in all the subsequent transitions but have not yet been identified in experimental samples.

#### 2. Domain walls in the P4/ncc structure

The $P4/nmm\u2192P4/ncc$ transition is co-elastic, so ferroelastic twin walls do not develop.

#### 3. Domain walls in the P2_{1}/c structure

The $P4/ncc\u2192P21/c$ transition is ferroelastic, so classical pairs of orthogonal ferroelastic twin walls should appear below the transition point, $\u223c$1150 K. As defined with respect to the axes of the $P4/ncc$ structure as parent phase, the primary strain contrast would be $2e5$, i.e., up to $\u223c$2%, and a secondary strain contrast of $2eo$, up to $\u223c$1%. In principle, the symmetry of the structure at the center of the walls would be $P4/ncc$ but coupling of the two $R4+$ octahedral components $q4$ and $q5$ $(=q6)$, possibly as $\lambda q4q5$, could cause additional relaxations to a different local structure. Any twin walls inherited from the as-grown crystals would also experience additional local relaxations by tilting.

#### 4. Domain walls in the Pbcn structure

The $P21/c\u2192Pbcn$ transition is first order in character due to the lack of a group–subgroup relationship, but the orthorhombic structure develops in a crystal with monoclinic distortions that will influence the configuration of domain walls arising, in effect, from a symmetry change $P4/ncc\u2192Pbcn$ or $P4/nmm\u2192Pbcn$. The suggestion in Sec. V D that the real parent structure has $P4/nmm$ symmetry is supported by first-principle calculations showing the local structure of domain walls in $Pbcn$ crystals to have $P4/nmm$ symmetry.^{40} In this case, the strain contrast across a new set of ferroelastic domain walls would be $2eo$, up to $\u223c$6%. Relaxation of three order parameters could occur within the walls.

#### 5. Domain walls in the P2_{1}/n structure

The $Pbcn\u2192P21/n$ transition would generate a set of orthogonal ferroelastic domain walls with shear strain contrast across of them of $2e5$, i.e., up to $\u223c$3%, additional to those established as a consequence of the prior $P4/nmm\u2192Pbcn$ transition. Relaxations within the walls would occur by coupling between the two separate $R4+$ components, $q4$ and $q5$, and of both these with $M3\u2212$ and $M3+$ components.

#### 6. Domain walls in the $P1\xaf$ structure

The transition $P21/n\u2192P1\xaf$ would generate a third set of ferroelastic domain walls. Strain contrasts in $e4$ and $e6$ of up to $\u223c$4% would exist across them and possible relaxations would include coupling that would occur between three R-point tilts and all three of these with the $M3\u2212$ and $M3+$ components. Experimentally, the $P1\xaf$ structure is seen under the optical microscope to be riddled with twin structures, often inherited from the high temperature transitions.

#### 7. Domain walls in the “Pc” structure

If the space group of this structure is $P21/c$, the domain walls should have characteristics similar to those described above for the same structure at high temperatures, though the magnitudes of the shear strain contrast across individual walls would be greater. Optical micrographs in Refs. 52 and 78 show that, even though there is no group–subgroup relationship, transition from the $P1\xaf$ structure is accompanied by a marked increase in the density of domain walls. If the assignment to space group $Pc$ is correct and the structure is ferroelectric, the domain walls could have gradients in both ferroelectric and ferroelastic order parameters—as occurs, for example, in $Cu\u2212Cl$ boracite.^{79} Catalan (2014)^{80} has suggested that domain walls which are both ferroelectric and ferroelastic will be thinner than pure ferroelastic walls but thicker than pure ferroelectric walls. This, in turn, has implications for their mobility since thin walls are more likely to be pinned by point defects than thick walls.^{81} Another possibility that remains to be explored is that ferroelectric displacements occur only within the domain walls.

## VII. CONCLUSIONS

Functional properties of $WO3$ emerge essentially from the strong interdependence of atomic and electronic structures. At the heart of this is the susceptibility of $W6+$ to cooperative second-order-Jahn–Teller distortions which can be described, formally, as large shear strains accompanying structural instabilities associated with irrep $M3\u2212$ of space group $Pm3\xafm$. The additional proximity to tilt instabilities leads to two discrete but overlapping sequences of structural phase transitions. One can be considered to be the development of a sequence of R-point tilts in crystals, which have a large tetragonal distortion due to a zone boundary instability at some temperature above the sublimation point. The second is a sequence of essentially the same sequence of tilting transitions in crystals which acquire an orthorhombic distortion due to coupling with a different orientation of antiferroelectric displacements, maybe with an additional zone center electronic instability. M-point tilts appear to be a secondary relaxation, which appears only because it is allowed by symmetry in crystals with both tetragonal and orthorhombic distortions.

Exotic properties of domain walls are expected in $WO3$ due to the combined influence of large strain gradients and order parameter coupling. Gradients in up to three order parameters can exist through selected individual walls and it is inevitable that coupling between these will lead to new structural relaxations within the walls that do not exist in homogeneous domains of any of the other known structures. Because of the strong relationship between shear strains and electronic properties, the steep shear strain gradients must also give rise to wide variations in electronic properties. The key to tuning these as functional properties in device applications using either bulk samples or thin films will be to impose preferred strain states. For example, it should be possible to choose substrates that impose tetragonal or orthorhombic shear strains to pre-determine the structure of the thin film and the nature of twin walls within it.

Formal strain analysis with respect to the cubic parent structure has drawn attention to the closeness of the relationship between the structures of the low temperature monoclinic structure and the $P21/c$ structure at high temperatures. Re-entrant phase transitions are rare and will depend on some unusual feature to reverse the typical sequence of structures in which symmetry is lowered as temperature reduces.^{82} In the case of $WO3$, the issue may simply be a matter of how distorted octahedra can most efficiently fill empty space in the perovskite structure by tilting—with R-point tilting of octahedra deformed by $q1$ leading to a higher density than R- and M-point tilting of octahedra deformed by $q1+q3$. If the ground state is correctly assigned to $Pc$, small additional ferroelectric displacements would add to the stabilization of the structure with the lowest molar volume.

The present analysis provides a clear view of the different types of domain walls that might develop in $WO3$. Materials that are of interest for domain wall nanoelectronics especially depend on having controllable domain wall properties and configurations. Strain/order parameter coupling in conjunction with Landau theory, therefore, provides critical insights for tailoring domain wall nanoelectronics in the wider context of materials with multiple instabilities.

## ACKNOWLEDGMENTS

The project has received funding from the EU’s Horizon 2020 programme under the Marie Sklodowska-Curie Grant Agreement No. 861153. This work was supported by the Engineering and Physical Sciences Research Council under Grant No. EP/P024904/1 to E.K.H.S. and M.A.C.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: EXPRESSIONS FOR STRAIN AND ORDER PARAMETER COUPLING

The complete Landau expansion obtained from ISOTROPY for analysis of the spontaneous strain variations is

$q1\u2212q3$ are components of the $M3\u2212$ order parameter; $q4\u2212q6$ are components of the $R4+$ order parameter; $q7\u2212q9$ are components of the $M3+$ order parameter, $a1\u2212a3$, $b1\u2212b3$, and $c1\u2212c3$ are Landau coefficients; $Tc1$, $Tc2$, and $Tc3$ are critical temperatures; $\lambda 1\u2212\lambda 9$ are coupling coefficients; $C11\xb0$, $C12\xb0$, and $C44o$ are elastic constants of the parent cubic structure; and $e6$, $e4$, and $e5$ are components of the spontaneous strain tensor. The symmetry-adapted strains $eo$, $ea$, and $et$ are defined in terms of linear components of the strain tensor, $e1$, $e2$, and $e3$ as

The $Pm3\xafm\u2192P4/nmm$ transition would occur above the sublimation temperature and is not observed. In the $P4/nmm$ structure, $q2=q3=q4=q5=q6=q7=q8=q9=0$ and $q1\u22600$, which reduces Eq. (A1) to

Under equilibrium conditions, the following apply: $\u2202G\u2202ea=\u2202G\u2202et=\u2202G\u2202eo=\u2202G\u2202e6=\u2202G\u2202e4=\u2202G\u2202e5=0$. These yield relationships

For the $Pm3\xafm\u2192P4/ncc$ transition, $q2=q3=q5=q6=q7=q8=q9=0$ and $q1\u2260q4\u22600$. Equation (A1) reduces to

Applying the same equilibrium conditions as before yields

For the $Pm3\xafm\u2192P21/c$ transition, $q2=q3=q7=q8=q9=0$ and $q5=q6\u2260q1\u2260q4\u22600$, which reduces Eq. (A1) to

Applying the same equilibrium conditions as before yields

In the case of the $Pm3\xafm\u2192Pbcn$ transition, $q2=q5=q6=q7=q9=0$ and $q1\u2260q3\u2260q4\u2260q8\u22600$, which reduces Eq. (A1) to

Applying the same equilibrium conditions as before yields

For the $Pm3\xafm\u2192P21/n$ transition, $q2=q6=q7=q9=0$ and $q1\u2260q3\u2260q4\u2260q5\u2260q8\u22600$, which reduces Eq. (A1) to

Applying equilibrium conditions yields

For the $Pm3\xafm\u2192P1\xaf$ transition, $q2=q7=q9=0$ and $q1\u2260q3\u2260q4\u2260q5\u2260q6\u2260q8\u22600$, which reduces Eq. (A1) to

Applying equilibrium conditions yields

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