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.

Tungsten trioxide, WO3, has been a material of great societal impact for nearly two centuries, finding purpose in applications as old as pigmentation in paint1 and as modern as the active ingredient in electrochromic windows.2 It has been the focus of close scientific interest, also, from both structural3–13 and theoretical14–19 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¯m that arise by combinations of different order parameters relating to three basic mechanisms:

  1. Antiferroelectric cation displacements,

  2. Out-of-phase octahedral tilting,

  3. 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 SrZrO328 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¯ giving way to a monoclinic structure in space group Pc (formerly referred to as the ϵ-phase8) via a first order transition. In previous studies, the P1¯ 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 Q 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 (λQ12Q22) or linear-quadratic (λQ1Q22).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.

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¯m parent structure, the three order parameters corresponding to each transition mechanism belong to irreducible representations M3, R4+, and M3+. The transition from Pm3¯m to P4/nmm involves a single M3 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 component. Table I shows the space group, corresponding order parameters, relationships between order parameter components, and tilt systems according to Glazer’s notation44,45 for each of the structures considered here.

TABLE I.

Order parameter components, their relationships, and tilt systems44,45 for subgroups of the Pm3¯m parent structure. The system of reference axes is taken from ISOTROPY.

Space groupOrder parameter components M3R4+M3+Relationships between order parameter componentsTilts
Pm3¯m 000 000 000  aoaoao 
P4/nmm q100 000 000  aoaoao 
P4/ncc q100 q400 000 q1 ≠ q4 aoaoc 
P21/c q100 q4q5q6 000 q5 = q6 ≠ q1 ≠ q4 aac 
Pbcn q10q3q400 0q8q1 ≠ q3 ≠ q4 ≠ q8 aob+c 
P21/n q10q3q4q50 0q8q1 ≠ q3 ≠ q4 ≠ q5 ≠ q8 ab+c 
P1¯ q10q3q4q5q6 0q8q1 ≠ q3 ≠ q4 ≠ q5 ≠ q6 ≠ q8  
Space groupOrder parameter components M3R4+M3+Relationships between order parameter componentsTilts
Pm3¯m 000 000 000  aoaoao 
P4/nmm q100 000 000  aoaoao 
P4/ncc q100 q400 000 q1 ≠ q4 aoaoc 
P21/c q100 q4q5q6 000 q5 = q6 ≠ q1 ≠ q4 aac 
Pbcn q10q3q400 0q8q1 ≠ q3 ≠ q4 ≠ q8 aob+c 
P21/n q10q3q4q50 0q8q1 ≠ q3 ≠ q4 ≠ q5 ≠ q8 ab+c 
P1¯ q10q3q4q5q6 0q8q1 ≠ q3 ≠ q4 ≠ q5 ≠ q6 ≠ q8  

The spontaneous strain tensor contains six components: volume strain, ea, has the symmetry of Γ1+; tetragonal, et, and orthorhombic, eo, strains have the symmetry of Γ3+; and the remaining three shear strains, e6, e4, and e5, have the symmetry of Γ5+. 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 Ge=0 are also given in the  Appendix and are used in the strain analysis below.

TABLE II.

Strain and order parameter coupling where Γ1+ is a component of the order parameter linked to the volume strain, Γ3+ is the component of the order parameter linked to tetragonal and orthorhombic strains, and Γ5+ corresponds to the final piece of the order parameter linked to shear strains e6, e4, and e5.

M3—Antiferroelectric order parameter (q1, q2, q3
Γ1+ Volume strain (eaea(q12+q22+q32) 
Γ3+ Tetragonal and orthorhombic strain (et, eo3et(q120.5q220.5q32)+[32eo(q22q32)] 
Γ5+ Shear strains e6, e4, e5 e62(q12+q22+q32)+e42(q12+q22+q32)+e52(q12+q22+q32) 
R4+—Out-of-phase octahedral tilting order parameter (q4, q5, q6
Γ1+ Volume strain (eaea(q42+q52+q62) 
Γ3+ Tetragonal and orthorhombic strain (et, eo3et(q420.5q520.5q62)+[32eo(q62q52)] 
Γ5+ Shear strains e6, e4, e5 e6q5q6 + e4q4q6 + e5q4q5 
M3+—In-phase octahedral tilting order parameter (q7, q8, q9
Γ1+ Volume strain (eaea(q72+q82+q92) 
Γ3+ Tetragonal and orthorhombic Strain (et, eo3et(q720.5q820.5q92)+[32eo(q82q92)] 
Γ5+ Shear strains e6, e4, e5 e62(q72+q82+q92)+e42(q72+q82+q92)+e52(q72+q82+q92) 
M3—Antiferroelectric order parameter (q1, q2, q3
Γ1+ Volume strain (eaea(q12+q22+q32) 
Γ3+ Tetragonal and orthorhombic strain (et, eo3et(q120.5q220.5q32)+[32eo(q22q32)] 
Γ5+ Shear strains e6, e4, e5 e62(q12+q22+q32)+e42(q12+q22+q32)+e52(q12+q22+q32) 
R4+—Out-of-phase octahedral tilting order parameter (q4, q5, q6
Γ1+ Volume strain (eaea(q42+q52+q62) 
Γ3+ Tetragonal and orthorhombic strain (et, eo3et(q420.5q520.5q62)+[32eo(q62q52)] 
Γ5+ Shear strains e6, e4, e5 e6q5q6 + e4q4q6 + e5q4q5 
M3+—In-phase octahedral tilting order parameter (q7, q8, q9
Γ1+ Volume strain (eaea(q72+q82+q92) 
Γ3+ Tetragonal and orthorhombic Strain (et, eo3et(q720.5q820.5q92)+[32eo(q82q92)] 
Γ5+ Shear strains e6, e4, e5 e62(q72+q82+q92)+e42(q72+q82+q92)+e52(q72+q82+q92) 

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¯ structure reported in Refs. 4 and 6– 8.

FIG. 1.

Lattice parameters of WO3 expressed in terms of the pseudocubic unit cell: from Ref. 12 for heating through the temperature interval 298–1273 K, Ref. 8 for heating through the temperature interval 5–250 K, and Refs. 4 and 6– 8 for the P1¯ structure between 240 and 298 K. Vertical dotted lines are estimated transition temperatures: 230 K (placed between hysteretic limits), 295 K (placed between hysteretic limits), 611±1 K (from fitting of e54 vs T), 1000 K (placed within two phase field), 1047±1 K (from fitting of e54 vs T), and 1171±1 K (from fitting of a first order transition).

FIG. 1.

Lattice parameters of WO3 expressed in terms of the pseudocubic unit cell: from Ref. 12 for heating through the temperature interval 298–1273 K, Ref. 8 for heating through the temperature interval 5–250 K, and Refs. 4 and 6– 8 for the P1¯ structure between 240 and 298 K. Vertical dotted lines are estimated transition temperatures: 230 K (placed between hysteretic limits), 295 K (placed between hysteretic limits), 611±1 K (from fitting of e54 vs T), 1000 K (placed within two phase field), 1047±1 K (from fitting of e54 vs T), and 1171±1 K (from fitting of a first order transition).

Close modal

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

e1=asinγaoaoaaoao,
(1)
e2=baoao,
(2)
e3=csinαsinβaoaocaoao,
(3)
e4=ccosαaocosα,
(4)
e5=csinαcosβaocosβ,
(5)
e6=acosγaocosγ.
(6)

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. β 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 α=βγ90°, giving e4=e5e60. 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 [q12, 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¯mP4/nmm transition. WO3 is already a highly sheared perovskite structure before any of the subsequent phase transitions occur.

FIG. 2.

Spontaneous shear strains calculated from lattice parameter data shown in Fig. 1 and the high temperature cubic phase Pm3¯m as a reference state. Vertical lines mark the same transition temperatures as shown in Fig. 1.

FIG. 2.

Spontaneous shear strains calculated from lattice parameter data shown in Fig. 1 and the high temperature cubic phase Pm3¯m as a reference state. Vertical lines mark the same transition temperatures as shown in Fig. 1.

Close modal

The transition from tetragonal P4/nmm to tetragonal P4/ncc at 1173 K gives q40, q5=q6=0 (R-point tilts about the c-axis). It is co-elastic46 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=q60. 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 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¯m and Pbcn structures. Strain values for the P1¯ 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¯ 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.

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 λeaqi2 and, hence, ea scales with qi2. 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.

FIG. 3.

Analysis of volume strains with respect to the P4/nmm structure as a reference state. (a) Unit cell volume data from Refs. 4, 6– 8, and 12, expressed in terms of the pseudocubic unit cell. The solid curve is a fit of the expression Vo=A+BΘs(cothΘso/T) to data for the P4/nmm structure in the temperature interval 1213–1273 K: A=53.74 and B=0.00113 with Θso fixed at 150 K. (b) Volume strains calculated as Vs=(VVo)/Vo from the data in (a). Colored curves represent a description of each phase transition as being tricritical, q4(TcT) with transition temperatures Tc, on the basis that volume strains scale with q2. Blue and red curves are for Vs2T, i.e., for transitions to P4/ncc and Pbcn structures directly from the P4/nmm structure. The brown curve is for the square of the difference in Vs between the blue curve and data for the P21/c structure, representing the transition P4/nccP21/c. The green curve is for the square of the difference in Vs between the red curve and data for the P21/n structure, representing the transition PbcnP21/n. (c) Vs for P4/nmm and P4/ncc structures, with a fit for the P4/ncc structure to represent a first order transition at Ttr=Tc+2.5 K. The dotted line shows a discontinuity in Vs of Vso=0.0014 at T=Ttr.

FIG. 3.

Analysis of volume strains with respect to the P4/nmm structure as a reference state. (a) Unit cell volume data from Refs. 4, 6– 8, and 12, expressed in terms of the pseudocubic unit cell. The solid curve is a fit of the expression Vo=A+BΘs(cothΘso/T) to data for the P4/nmm structure in the temperature interval 1213–1273 K: A=53.74 and B=0.00113 with Θso fixed at 150 K. (b) Volume strains calculated as Vs=(VVo)/Vo from the data in (a). Colored curves represent a description of each phase transition as being tricritical, q4(TcT) with transition temperatures Tc, on the basis that volume strains scale with q2. Blue and red curves are for Vs2T, i.e., for transitions to P4/ncc and Pbcn structures directly from the P4/nmm structure. The brown curve is for the square of the difference in Vs between the blue curve and data for the P21/c structure, representing the transition P4/nccP21/c. The green curve is for the square of the difference in Vs between the red curve and data for the P21/n structure, representing the transition PbcnP21/n. (c) Vs for P4/nmm and P4/ncc structures, with a fit for the P4/ncc structure to represent a first order transition at Ttr=Tc+2.5 K. The dotted line shows a discontinuity in Vs of Vso=0.0014 at T=Ttr.

Close modal

Following the approach used in Refs. 47–50, for example, the function Vo=A+BΘscothΘso/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/dT0, as T0 K. On the basis of experience from other perovskites,28,29Θso was set at 150 K. Defining Vs as (VVo)/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(TcT), for the P4/nmmP4/ncc transition. The brown curve is a fit of Vs2 vs T for the P4/nccP21/c transition, where Vs for P4/nccP21/c is the difference between Vs and values on the blue curve extrapolated below the P4/nccP21/c transition point (Tc=1044±4K, 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, PbcnP4/nmm, with Tc=1116±1 K. The green curve is a fit of Vs2 vs T for the PbcnP21/n transition, where Vs for PbcnP21/n is the difference between Vs and values on the red curve extrapolated below the PbcnP21/n transition point (Tc=619±4 K, from the fit). Although fits to the data in Fig. 3(b) provide a good representation of the P4/nccP21/c and PbcnP21/n transitions as being tricritical, the P4/nmmP4/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.13Figure 3(c) shows a fit of the Landau solution for a first order transition,

Vs=23Vso{1+[134(TTcTtrTc)]}12,
(7)

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=0.0014. Values of Vs calculated in the same way for the P1¯ 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 1% at the transition from the P1¯ 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.

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.

Values of the tetragonal shear strain, et, arising from the Pm3¯mP4/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 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.

When expressed with respect to space group P4/nmm, the order parameter for P4/nmmP4/ncc transforms as Z1 and has one component, corresponding to R4+(a,0,0) of space group Pm3¯m. In the absence of M3 distortions, it would be equivalent to the Pm3¯mI4/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 0.003 and 0.001, respectively, comparable in sign and magnitude with the strains observed at the equivalent pure tilting transition at 1367 K in SrZrO328 and at 1578 K in CaTiO3.54 The transitions in SrZrO3 and CaTiO3 are also close to tricritical in character.

FIG. 4.

(a) Strain variations of the P4/ncc structure, as determined with respect to reference parameters of the P4/nmm structure extrapolated into the stability field of the P4/ncc structure. The transition occurs at 1170 K and is weakly first order. (b) Variations of shear strains, eo, and cosβ (e5), as defined for the P21/c structure with respect to the P4/ncc structure in their conventional settings. (c) eo and Vs for the P21/c structure, as determined with respect to the P4/ncc structure, scale with cosβ2 (e52), confirming the relationships e52eoVsQ2 expected for a pseudoproper ferroelastic transition. (d) cosβ4 for the P21/c structure, in its conventional setting, varies linearly with temperature, indicating that the P4/nccP21/c transition is close to tricritical in character, with Tc=1047±1 K. (e) Variations of eo4 in the stability field of the Pbcn structure and of Vs2, where Vs is the volume strain of the Pbcn structure with respect to the P4/nmm structure, go linearly to zero at 1113±2 K and 1115±2 K, respectively, consistent with tricritical character for a (metastable) P4/nmmPbcn transition with eo2VsQ2. (f) Variations of shear and volume strains through the stability field of the P21/n structure. The linear fit to cosβ4 (e54Q4) extrapolates to zero at Tc=611±1 K and the fit to Vs2, where Vs is the volume strain of the P21/n structure with respect to the Pbcn structure, extrapolates to zero at 612±2 K.

FIG. 4.

(a) Strain variations of the P4/ncc structure, as determined with respect to reference parameters of the P4/nmm structure extrapolated into the stability field of the P4/ncc structure. The transition occurs at 1170 K and is weakly first order. (b) Variations of shear strains, eo, and cosβ (e5), as defined for the P21/c structure with respect to the P4/ncc structure in their conventional settings. (c) eo and Vs for the P21/c structure, as determined with respect to the P4/ncc structure, scale with cosβ2 (e52), confirming the relationships e52eoVsQ2 expected for a pseudoproper ferroelastic transition. (d) cosβ4 for the P21/c structure, in its conventional setting, varies linearly with temperature, indicating that the P4/nccP21/c transition is close to tricritical in character, with Tc=1047±1 K. (e) Variations of eo4 in the stability field of the Pbcn structure and of Vs2, where Vs is the volume strain of the Pbcn structure with respect to the P4/nmm structure, go linearly to zero at 1113±2 K and 1115±2 K, respectively, consistent with tricritical character for a (metastable) P4/nmmPbcn transition with eo2VsQ2. (f) Variations of shear and volume strains through the stability field of the P21/n structure. The linear fit to cosβ4 (e54Q4) extrapolates to zero at Tc=611±1 K and the fit to Vs2, where Vs is the volume strain of the P21/n structure with respect to the Pbcn structure, extrapolates to zero at 612±2 K.

Close modal

The R-point of space group Pm3¯m becomes folded back to the Γ-point in P4/ncc. As a result, the additional R-point tilt gives rise to the P4/nccP21/c transition and has a one component order parameter, Q, which transforms as Γ5+(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=(e1e2)=(ab)ao,e5=ccocosβcosβ, where β=(180β) 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 e5Q,eoQ2. The volume strain, Vs, determined with respect to the P4/ncc structure is expected to scale as VsQ2. Figure 4(b) shows the variations of eo and cosβ 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(Q4) varies linearly with temperature, consistent with tricritical character and Tc=1047±1 K.

The P21/cPbcn 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/nccPbcn obscured by the stability field of the P21/c structure and a metastable transition P4/nmmPbcn 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 component, q3, appears in both cases. In group theory terms, M3 of space group Pm3¯m becomes M1 of space group P4/ncc and the active representation for P4/nccPbcn is M1(a,0). M3 and R4+ of space group Pm3¯m become M1 and Z1 of space group P4/nmm so the two order parameters needed for P4/nmmPbcn are M1(a,0) and Z1. In both cases, A1(0,a), corresponding to M3+ of Pm3¯m, 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, (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/nmmPbcn transition could be driven by the addition of q3 coupled with the R-point tilt and the P4/nccPbcn 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(ab)/(a+b) (to good approximation) with respect to the Pm3¯m 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 order parameter component q3, with tilting (q8) as secondary.

Figure 4(e) shows, further, that eo4 scales with (TcT)/Tc, Tc=1113±2 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/nmmPbcn, 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, eo4Vs2Q4(TcT)/Tc, where Q is a driving order parameter. However, this implies that the metastable transition is pseudoproper ferroelastic in character, eoQ, which is not consistent with the expected improper ferroelastic behavior, eoQ2. For this to be true, the symmetry of Q would need to be Γ4+ 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 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/nmmPbcn would be close to tricritical in character. Discontinuities in et and eo at 1020 K (Fig. 2) indicate that P4/nccPbcn would be the first order.

The PbcnP21/n transition occurs by the addition of a second R-point tilt, q5 (Table I). The R-point of space group Pm3¯m folds back to become the Γ-point of space group Pbcn so that the order parameter has one component and transforms as Γ3+ of space group Pbcn. This is referred to here, again generically, as Q. The new symmetry-breaking shear strain is e5 (cosβ) which couples bilinearly with Q, leading to the expectation e5Q. Figure 4(f) shows linear variations of cosβ4 and Vs2 with temperature, where Vs is the volume strain with respect to the Pbcn structure. The strain relationships are, therefore, e54Vs2Q4(TcT)/Tc, i.e., pseudoproper ferroelastic character, tricritical evolution, and overall behavior closely analogous to that of the P4/nccP21/c transition. This temperature dependence for the shear strain was previously reported by Locherer et al.55 

There is no single data set for the variations of lattice parameters through the stability field of the P1¯ 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, βγ91° in the triclinic structure which lies on or close to an extension of the variation of the angle β in the monoclinic structure. The abrupt change in the value of α from 90° to 89.2° signifies that the transition from the P21/n structure is first order in character. An alternative choice of crystallographic axes would give αβγ91°, 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 q4q5q6. 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.

Significant discontinuities in the volume strain (Fig. 3) and all the observed shear strains (Fig. 2) indicate that the P1¯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, α (=β), and γ 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. cosβ2 is shown as a function of temperature for the two temperature intervals, 5–250 K and 1003–1053 K, using the β angle of the conventional monoclinic setting. This is a good approximation for e52 and scales with Q2 for the transition P4/nccP21/c. The curve through the combined data in Fig. 5(a) is a solution to GQ=0 for the standard Landau 246 expansion, including the saturation temperature, Θs,

G=12AΘs(cothΘsTcothΘsTc)Q2+14BQ4+16CQ6.
(8)

With Q2 replaced by cosβ2, the fit parameters are B/A=3.02×105±1.1×104 K, C/A=7.35×108±1.3×107 K, Θs = 123±4 K, Tc=1057±1 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.

FIG. 5.

(a) The curve is a solution of GQ=0 for Eq. (8) fit to the data for cosβ2 of the “Pc” and P21/c structures, where β is the unit cell angle in the conventional monoclinic setting. The quality of the fit is permissive of the structure being the same in both temperature intervals and the P4/nccP21/c transition as being between second order and tricritical in character, rather than tricritical as implied by the fit in Fig. 4(d) for the high temperature structure alone. (b) Variation of cosα (e4) with respect to cosγ (e6) for the high and low temperature stability fields of the P21/c structure. The straight line marks |e4|=|e5|=|e6| as would occur for q4=q5=q6.

FIG. 5.

(a) The curve is a solution of GQ=0 for Eq. (8) fit to the data for cosβ2 of the “Pc” and P21/c structures, where β is the unit cell angle in the conventional monoclinic setting. The quality of the fit is permissive of the structure being the same in both temperature intervals and the P4/nccP21/c transition as being between second order and tricritical in character, rather than tricritical as implied by the fit in Fig. 4(d) for the high temperature structure alone. (b) Variation of cosα (e4) with respect to cosγ (e6) for the high and low temperature stability fields of the P21/c structure. The straight line marks |e4|=|e5|=|e6| as would occur for q4=q5=q6.

Close modal

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 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¯mR3¯c 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 cosα(e4=e5) with respect to cosγ(e6) 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.

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¯m 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.

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 ordering59 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, (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/nmmP4/nccP21/c, involves changes in R-point tilting. The second, P4/nmmPbcnP21/nP1¯, involves almost the same changes in R-point tilting but with additional shearing due to the second M3 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 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 Γ-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¯m(0,0,0)P4mm(0,0,a)Amm2(a,a,0)R3m(a,a,a), where a signifies values for non-zero components of the Γ4 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¯mP4mm transition is weakly first order61 (see, also, Refs. 62 and 63). Amm2 and R3m are subgroups of Pm3¯m 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¯m(0,0,0)I4/mcm(a,0,0)Imma(a,a,0)R3¯c(a,a,a). This occurs as far as the Imma stability field in SrZrO3, for example, where the Pm3¯mI4/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)(a,0,0)(a,b,b) or (0,0,0)(a,0,0)(a,b,0)(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 2% for the sequence P4/nmmPbcnP21/nP1¯ and 3.5% for the sequence P4/nmmP4/nccP21/c. Even without including some additional volume strain accompanying the transition to P4/nmm from the parent Pm3¯m 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.

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 (λeQ) or quadratically (λeQ2) with the driving order parameter.46,68,69 All non-symmetry breaking strains couple as λeQ2 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/nccP21/c, PbcnP21/n, P21/nP1¯), and perhaps the metastable P4/nmmPbcn transition, are pseudoproper ferroelastic. Linear coupling of the symmetry-breaking shear strains with the zone center driving order parameter, λeQ, will have increased their transition temperatures by renormalization of the critical temperature in each case.

The fundamental link between structural and electronic effects in WO3 was identified through the observation that the “PcP1¯ 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 3% due to coupling with the q3 component of the M3 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 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 RVO3 and RMnO3 perovskites (R = lanthanide cations57) than to shear strains of up to 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 Γ-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¯ transition point. Even though tilting results in shear strains e4, e5, and e6 having values up to 2%, their influence is clearly much smaller than is associated with shear strains et and eo arising from coupling with the M3 order parameter. Relationships between structure and electronic properties are the focus of several computational studies14–16,19 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 1.5% of the A-site, for example, Ref. 77.

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 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¯m structure and it is, therefore, not possible to generate ferroelastic domain walls by cooling through the Pm3¯mP4/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 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/nmmP4/ncc transition is co-elastic, so ferroelastic twin walls do not develop.

3. Domain walls in the P21/c structure

The P4/nccP21/c transition is ferroelastic, so classical pairs of orthogonal ferroelastic twin walls should appear below the transition point, 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 2%, and a secondary strain contrast of 2eo, up to 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 λq4q5, 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/cPbcn 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/nccPbcn or P4/nmmPbcn. 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 6%. Relaxation of three order parameters could occur within the walls.

5. Domain walls in the P21/n structure

The PbcnP21/n transition would generate a set of orthogonal ferroelastic domain walls with shear strain contrast across of them of 2e5, i.e., up to 3%, additional to those established as a consequence of the prior P4/nmmPbcn transition. Relaxations within the walls would occur by coupling between the two separate R4+ components, q4 and q5, and of both these with M3 and M3+ components.

6. Domain walls in the P1¯ structure

The transition P21/nP1¯ would generate a third set of ferroelastic domain walls. Strain contrasts in e4 and e6 of up to 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 and M3+ components. Experimentally, the P1¯ 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¯ 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 CuCl 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.

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 of space group Pm3¯m. 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.

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.

The authors have no conflicts to disclose.

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

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

G=12a1(TTc1)(q12+q22+q32)+12a2(TTc2)(q42+q52+q62)+12a3(TTc3)(q72+q82+q92)+14b1(q12+q22+q32)2+14b1(q14+q24+q34)+14b2(q42+q52+q62)2+14b2(q44+q54+q64)+14b3(q72+q82+q92)2+14b3(q74+q84+q94)+16c1(q12+q22+q32)3+16c1(q1q2q3)2+16c1(q12+q22+q32)(q14+q24+q34)+16c2(q42+q52+q62)3+16c2(q4q5q6)2+16c2(q42+q52+q62)(q44+q54+q64)+16c3(q72+q82+q92)3+16c3(q7q8q9)2+16c3(q72+q82+q92)(q74+q84+q94)+λ1ea(q12+q22+q32)+λ2ea(q42+q52+q62)+λ3ea(q72+q82+q92)+λ4[3et(q120.5q220.5q32)+[32eo(q22q32)]]+λ5[3et(q420.5q520.5q62)+[32eo(q62q52)]]+λ6[3et(q720.5q820.5q92)+[32eo(q82q92)]]+λ7[e62(q12+q22+q32)+e42(q12+q22+q32)+e52(q12+q22+q32)]+λ8[e6q5q6+e4q4q6+e5q4q5]+λ9[e62(q72+q82+q92)+e42(q72+q82+q92)+e52(q72+q82+q92)]+14(C11°C12°)(eo2+et2)+16(C11°2C12°)ea2+12C44o(e42+e52+e62).
(A1)

q1q3 are components of the M3 order parameter; q4q6 are components of the R4+ order parameter; q7q9 are components of the M3+ order parameter, a1a3, b1b3, and c1c3 are Landau coefficients; Tc1, Tc2, and Tc3 are critical temperatures; λ1λ9 are coupling coefficients; C11°, C12°, 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

ea=e1+e2+e3,
(A2)
et=2e3e1e23,
(A3)
eo=e1e2.
(A4)

The Pm3¯mP4/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 q10, which reduces Eq. (A1) to

G=12a1(TTc1)(q12)+14b1(q14)+14b1(q14)+16c1(q16)+16c1(q16)+λ1ea(q12)+λ4[3et(q12)]+λ7[e62(q12)+e42(q12)+e52(q12)]+14(C11°C12°)(eo2+et2)+16(C11°+2C12°)ea2+12C44o(e42+e52+e62).
(A5)

Under equilibrium conditions, the following apply: Gea=Get=Geo=Ge6=Ge4=Ge5=0. These yield relationships

ea=3λ1q12C11°+2C12°,
(A6)
et=23λ4q12(C11°C12°),
(A7)
eo=e6=e4=e5=0.
(A8)

For the Pm3¯mP4/ncc transition, q2=q3=q5=q6=q7=q8=q9=0 and q1q40. Equation (A1) reduces to

G=12a1(TTc1)(q12)+12a2(TTc2)(q42)+14b1(q14)+14b1(q14)+14b2(q44)+14b2(q44)+16c1(q16)+16c1(q16)+16c2(q46)+16c2(q46)+λ1ea(q12)+λ2ea(q42)+λ4[3et(q12)]+λ5[3et(q42)]+λ7[e62(q12)+e42(q12)+e52(q12)]+14(C11°C12°)(eo2+et2)+16(C11°+2C12°)ea2+12C44o(e42+e52+e62).
(A9)

Applying the same equilibrium conditions as before yields

ea=3(λ1q12+λ2q42)(C11°+2C12°),
(A10)
et=23(λ4q12+λ5q42)(C11°C12°),
(A11)
eo=e6=e4=e5=0.
(A12)

For the Pm3¯mP21/c transition, q2=q3=q7=q8=q9=0 and q5=q6q1q40, which reduces Eq. (A1) to

G=12a1(TTc1)(q12)+12a2(TTc2)(q42+2q52)+14b1(q14)+14b1(q14)+14b2(q42+2q52)2+14b2(q44+2q54)+16c1(q16)+16c1(q16)+16c2(q42+2q52)3+16c2(q4q52)2+16c2(q42+2q52)(q44+2q54)+λ1ea(q12)+λ2ea(q42+2q52)+λ4[3et(q12)]+λ5[3et(q42q52)]+λ7[e62(q12)+e42(q12)+e52(q12)]+λ8[e6q52+e4q4q5+e5q4q5]+14(C11°C12°)(eo2+et2)+16(C11°+2C12°)ea2+12C44o(e42+e52+e62).
(A13)

Applying the same equilibrium conditions as before yields

ea=3(λ1q12+λ2(q42+2q52))(C11°+2C12°),
(A14)
et=23(λ4q12+λ5(q42q52))(C11°C12°),
(A15)
eo=0,
(A16)
e6=λ8q522λ7q12+C44o,
(A17)
e4=λ8q4q52λ7q12+C44o,
(A18)
e5=λ8q4q52λ7q12+C44o.
(A19)

In the case of the Pm3¯mPbcn transition, q2=q5=q6=q7=q9=0 and q1q3q4q80, which reduces Eq. (A1) to

G=12a1(TTc1)(q12+q32)+12a2(TTc2)(q42)+12a3(TTc3)(q82)+14b1(q12+q32)2+14b1(q14+q34)+14b2(q44)+14b2(q44)+14b3(q84)+14b3(q84)+16c1(q12+q32)3+16c1(q12+q32)(q14+q34)+16c2(q46)+16c2(q46)+16c3(q86)+16c3(q86)+λ1ea(q12+q32)+λ2ea(q42)+λ3ea(q82)+λ4[3et(q120.5q32)[32eo(q32)]]+λ5[3et(q42)]+λ6[3et(0.5q82)+[32eo(q82)]]+λ7[e62(q12+q32)+e42(q12+q32)+e52(q12+q32)]+λ9[e62(q82)+e42(q82)+e52(q82)]+14(C11°C12°)(eo2+et2)+16(C11°+2C12°)ea2+12C44o(e42+e52+e62).
(A20)

Applying the same equilibrium conditions as before yields

ea=3(λ1(q12+q32)+λ2q42+λ3q82)C11°+2C12°,
(A21)
et=23[λ4(q120.5q32)+λ5q420.5λ6q82]C11°C12°,
(A22)
eo=3(λ4q32λ6q82)C11°C12°,
(A23)
e6=e4=e5=0.
(A24)

For the Pm3¯mP21/n transition, q2=q6=q7=q9=0 and q1q3q4q5q80, which reduces Eq. (A1) to

G=12a1(TTc1)(q12+q32)+12a2(TTc2)(q42+q52)+12a3(TTc3)(q82)+14b1(q12+q32)2+14b1(q14+q34)+14b2(q42+q52)2+14b2(q44+q54)+14b3(q82)2+14b3(q84)+16c1(q12+q32)3+16c1(q12+q32)(q14+q34)+16c2(q42+q52)3+16c2(q42+q52)(q44+q54)+16c3(q82)3+16c3(q86)+λ1ea(q12+q32)+λ2ea(q42+q52)+λ3ea(q82)+λ4[3et(q120.5q32)+[32eo(q32)]]+λ5[3et(q420.5q52)+[32eo(q52)]]+λ6[3et(0.5q82)+[32eo(q82q92)]]+λ7[e62(q12+q32)+e42(q12+q32)+e52(q12+q32)]+λ9[e62(q82)+e42(q82)+e52(q82)]+14(C11°C12°)(eo2+et2)+16(C11°2C12°)ea2+12C44o(e42+e52+e62).
(A25)

Applying equilibrium conditions yields

ea=[λ1(q12+q32)+λ2(q42+q52)+λ8q82]C11°+2C12°,
(A26)
et=23[λ4(q120.5q32)+λ5(q420.5q52)0.5λ6q82]C11°C12°,
(A27)
eo=3(λ4q32+λ5q52λ6q82)C11°C12°,
(A28)
e6=e4=0,
(A29)
e5=λ8q4q52[λ7(q12+q32)+λ9q82+12C44o].
(A30)

For the Pm3¯mP1¯ transition, q2=q7=q9=0 and q1q3q4q5q6q80, which reduces Eq. (A1) to

G=12a1(TTc1)(q12+q32)+12a2(TTc2)(q42+q52+q62)+12a3(TTc3)(q82)+14b1(q12+q32)2+14b1(q14+q34)+14b2(q42+q52+q62)2+14b2(q44+q54+q64)+14b3(q84)+14b3(q84)+16c1(q12+q32)3+16c1(q12+q32)(q14+q34)+16c2(q42+q52+q62)3+16c2(q4q5q6)2+16c2(q42+q52+q62)(q44+q54+q64)+16c3(q86)+16c3(q86)+λ1ea(q12+q32)+λ2ea(q42+q52+q62)+λ3ea(q82)+λ4[3et(q120.5q32)+[32eo(q32)]]+λ53et(q420.5q520.5q62)+[32eo(q62q52)]+λ6[3et(0.5q82)+[32eo(q82)]]+λ7[e62(q12+q32)+e42(q12+q32)+e52(q12+q32)]+λ8[e6q5q6+e4q4q6+e5q4q5]+λ9[e62(q82)+e42(q82)+e52(q82)]+14(C11°C12°)(eo2+et2)+16(C11°2C12°)ea2+12C44o(e42+e52+e62).
(A31)

Applying equilibrium conditions yields

ea=3[λ1(q12+q32)+λ2(q42+q52+q62)+λ3q82]C11°C12°,
(A32)
et=23[λ4(q120.5q32)+λ5(q420.5q520.5q62)0.5λ6q82]C11°C12°,
(A33)
eo=3[λ4q32λ5(q62q52)λ6q82]C11°C12°,
(A34)
e6=λ8q5q62[λ7(q12+q32)+λ9q82+12C44o],
(A35)
e4=λ8q4q62[λ7(q12+q32)+λ9q82+12C44o],
(A36)
e5=λ8q4q52[λ7(q12+q32)+λ9q82+12C44o].
(A37)
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