Fragile 3D Order in V$_{1-x}$Mo$_x$O$_2$
Matthew A. Davenport Matthew J. Krogstad Logan M. Whitt Chaowei Hu Tyra C. Douglas Ni Ni Stephan Rosenkranz Raymond Osborn Jared M. Allred





Department of Chemistry and Biochemistry, The University of Alabama, Tuscaloosa, AL 35487, USA

Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA

Department of Chemistry and Biochemistry, The University of Alabama, Tuscaloosa, AL 35487, USA

Department of Physics and Astronomy and California NanoSystems Institute, University of California, Los Angeles, Los Angeles CA 90095, USA

Department of Chemistry and Biochemistry, The University of Alabama, Tuscaloosa, AL 35487, USA

Department of Physics and Astronomy and California NanoSystems Institute, University of California, Los Angeles, Los Angeles CA 90095, USA

Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA

Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA

[]jmallred@ua.edu



Department of Chemistry and Biochemistry, The University of Alabama, Tuscaloosa, AL 35487, USA





The metal-to-insulator transition (MIT) in rutile VO_2 has proven uniquely difficult to characterise because of the complex interplay between electron correlations and atomic structure. A number of approaches have been explored since its discovery, relying on some combination of Mott localization, Peierls distortions, superexchange, and orbital ordering to explain the structural instability at the MIT. Here we report the discovery of the sudden collapse of three-dimensional order in the low-temperature phase of V_1-xMo_xO_2 at x=0.17 and the emergence of a novel frustrated two-dimensional order at x=0.19, with only a slight change in electronic properties. Single crystal diffuse x-ray scattering reveals that this transition from the 3D M1 phase to a 2D variant of the M2 phase results in long-range structural correlations along symmetry-equivalent (11L) planes of the tetragonal rutile structure, yet extremely short-range correlations transverse to these planes. These findings emphasise the fragility of structural order in VO_2, the role of geometric frustration, and the need for improved electronic structure models.


Fragile 3D Order in V_1-xMo_xO_2
    Jared M. Allred
    December 30, 2023
================================



§ MAIN


High temperature (ht) VO_2 is metallic due to the d^1 V^4+ species, which somehow localizes into a paramagnetic state in low-temperature (lt) semiconducting VO_2.  At this time, electron-electron correlations are considered by many to be fundamental to this localization process, possibly in combination with a Peierls instability<cit.>. Others still favour the primacy of a pure structural instability, suggesting previous calculations underestimated the effect of bond covalency, entropy, and/or orbital ordering<cit.>. As such, 60 years after its discovery,<cit.> the current literature represents many different approaches and conclusions.
 
What has become clear is that the structural instability is more complex than initially perceived, and it is not fully understood. The rutile structure of metallic VO_2 (R) consists of two chains (A and B) of edge-sharing octahedra running along the c axis that are related by a 4_2 screw axis. The screw axis means that the A and B chains are not mutually equivalent by translation. The structure is shown looking across the chains in Fig. <ref>a and down the chains in Fig. <ref> b. Also shown are the M1 and M2 phases, which are both lt phases. Below the MIT, all of the metal atoms dimerise and buckle (A chain in the [1,1,0] direction, B chain in the [-1,1,0] direction), yielding the M1 phase. The M2 phase forms upon addition of Cr^3+, Al^3+, and Ga^3+, and is quite similar to the M1 phase, except only half of the metal atoms dimerise (e.g. the A chain).<cit.> The remaining B chain metal sites show the buckling. 

Simple Coulombic arguments can be used to explain the connection between dimerisation and buckling, and is illustrated in Fig. <ref>c. When a M-M bond forms along a given chain, the c-axis displacement of one M atom (d_z) couples to the ab displacement of another atom (d_xy) at [1/2,1/2,1/2] through the shared oxygen atom (Fig. <ref>c). This propagates in a network in the (110) planes, with two types of V atoms—those with only d_z and those with only d_xy. This network is entirely 2D, as described. In the M2 structure, 3D ordering occurs when the distortions propagate symmetrically along the a axis and antisymmetrically along the b axis (or vice versa). The M1 phase can be seen as simply a superposition of two, orthogonal M2 order parameters. The fact that half the atoms are dimerised, but the electronic properties remain semiconducting, is strong evidence that the MIT is not pure Peierls.<cit.> The M2 phase has been observed in pure VO_2 when strained or in a thin film,<cit.> but not in the bulk material.<cit.>



Several attempts have been made to more fully describe the connection between the M1 and M2 phases in tandem, including an all electron embedded DMFT model by Brito et al.<cit.>. Another approach was taken by Lovorn and Sarker, who instead used an Ashkin-Teller model, with displacements taking the place of spins, which resulted in several novel predictions<cit.>. They found that in order to make the 2D displacement network shown in Fig. <ref>c fully three-dimensional, inclusion of second-order terms to the Hamiltonian are required. These favoured weak cooperative (i.e. ferroelectric) displacements in the [100] and [010] directions. This interaction is thus geometrically frustrated, and the 3D-M2 ordering is only stabilized by next-nearest neighbour interactions. However, there is still no model that can predict whether the M1 or M2 phase is more stable under a given set of conditions, nor why the M1 phase so often devolves into the M2 phase.

We have found an entirely new type of ordering when suppressing the M1 phase. In V_1-xMo_xO_2, the d^2 Mo^4+ suppresses the structural transition temperature monotonically<cit.>. Previously, V_1-xMo_xO_2 was not known to show the M2 phase. Instead, Mo^4+ (a d^2 species) reportedly suppresses the structural instability to zero Kelvin by ∼ x=0.3 and enhances the metallicity and Sommerfeld constant, γ. The rutile phase was believed to persist at least up to x=0.5, before giving way to the M1 ground state observed in MoO_2 at some higher Mo concentration<cit.>. Here we show instead that between x=0.17 and x=0.19 the structural instability suddenly loses periodicity along the [110] axis while maintaining essentially long range M2-like order within the (110) plane. This “2D-M2” ordering was revealed using X-ray three-dimensional difference pair distribution function (3D-ΔPDF). 

3D-ΔPDF is a recently developed method, which builds substantially on the powder-PDF method<cit.>. Using single crystals, reciprocal space is reconstructed in 3D, which allows the Bragg peaks to be removed, leaving only diffuse scattering, using the “punch-and-fill” method<cit.>. The resulting PDF map only contains information about the difference (hence the Δ) between short range and long-range ordering, resulting in positive and negative peaks. The positive peaks are real-space vectors of increased electron density compared to the average long-range structure, and the negative peaks are lost electron density. From this, features such as atomic displacements are visually apparent. 

Besides using x-ray diffraction to analyse the diffuse scattering, the electronic transport properties were also measured. This revealed that below their transitions the new 2D-M2 phase and the regular M1 phase just below the critical composition (x = 0.17) are both metallic and are virtually indistinguishable in their bulk transport. This shows that the M1 and 2D-M2 ordering types are similar electronically, despite the large differences in structure, and that addition of sufficient electrons to VO_2 does suppress the MIT, but not the underlying electronic instability. This does not appear consistent with a lattice-instability-only model, but is consistent with a Mott-Peierls one.




§ RESULTS


Total X-ray scattering was measured while rotating a single crystal of V_1-xMo_xO_2 over 360°, which was then used to reconstruct three-dimensional data in reciprocal space coordinates.  Cuts from this reconstruction are shown on a log scale in Figure <ref>. Intense, sharp rods of scattering were observed below 150 K in x = 0.19 (19 % Mo) at l = n/2 planes, where n is an integer. Fig. <ref>a-e show the temperature dependence of the (hk1/2)^* plane in 19% Mo. The scattering is mostly constant below 140 K, with small changes in intensity and linewidth, down to the minimum measured temperature, 30 K. The detail view from a (hk3/2)^* slice (h), shows that the scattering rods actually have a periodic bending to them. At 150 K (Figure <ref>c), the rods have weakened and broadened substantially, and by 160 K only weak diffuse scattering is visible, persisting to room temperature. 



According to a standard single crystal x-ray diffraction structure solution, the 17% Mo crystal exhibits the normal structural phase transition to the M1 phase at 168 K (See Supplemental for more details).  Figure <ref>f and g show that the high temperature scattering for this composition is essentially equivalent to 19% Mo, and that Bragg peaks are observed at [h/2,0,l/2] below the transition. A crystal with 27% Mo (i) shows the same network of rods as 19% Mo, except with a reduced intensity and correlation length.

The cross-section of the scattering rods in the [hh0]^* and [00l]^* axes can be fit to a Pseudo-Voigt function. The correlation length, ξ was calculated from the Lorentzian line width component using the expression ξ = 1/L where L is the line width in reciprocal lattice coordinates (Å^-1). Fitting the T dependence of ξ to a power law ξ=A(1-T/T_s)^β gives a transition temperature of 151.0(3) K and a critical exponent of β = 0.109(18) according to the [hh0]^* cross-section. The resulting ξ_xx and ξ_z are plotted in Fig. <ref>a as red and blue circles, respectively. The maximum intensity of the scattering rods occurs around 120 K. The structural properties are compared to the electronic ones using electronic transport, which is shown for the x = 0.17 and 0.19 in Fig. <ref>b. These two compounds are metallic in both ht and lt phases with the resistivity in the mΩ-cm range. The temperature-dependence of the resistivity is reminiscent of a charge-density-wave (CDW) metal. That is, upon cooling, the structural phase transition comes with a sudden increase in resistivity of a factor between 3 to 8, suggesting the formation of charge gaps.






Cuts from the 3D-ΔPDF are shown in Fig. <ref>a-f. The 17% Mo phase again serves as a standard for usual 3D M1 ordering (Fig. <ref>a), with long-range correlations along all crystallographic axes. This is in contrast to the unusual two-dimensional scattering observed in 19% Mo (Fig. <ref>b). In the latter, the strongest correlations fall along (110) and (110) planes, as expected from the scattering rod orientations. The two crossed planes must be independent scattering from different regions that are twinned on the nanoscale. That is, some regions have strong correlations in the (110) direction and others have strong (110) correlations, but not both. Otherwise, the cooperation of both correlation directions should give a 3D correlation, e.g. sharp Bragg peaks. Within the planes of high intensity (Fig. <ref>c-f), there are intense positive and negative peaks at integer points, [u,u,w], corresponding to correlations commensurate with the unit cell. There is also a quadrupolar feature of intensity centred at [u/2,u/2,w/2] positions (e.g. vectors connecting cell corners to the body centres, and vice versa). Weaker dipolar features are also observed around the points [u/3,u/3,w], [-u/3,-u/3,w], [u/6,u/6,w/2], and [-u/6,-u/6,w/2].

Beyond the principal {110} planes, there are weaker planes of correlations that alternate with essentially zero intensity along the adjacent diagonals, except the first off-diagonal plane, which does show significant intensity (about one fifth the intensity of principal diagonal sites).






 §.§ Interpretation and modelling of the 3D-ΔPDF


The oscillation of positive/negative intensity at each integer point in the 3D-ΔPDF shows the periodicity of local correlations. By definition, each integer site connects sites that are translationally equivalent in the crystalline (i.e. rutile) model. Negative (blue) correlation implies broken translation symmetry at that vector, while a positive (red) correlation means the symmetry is maintained. It is clear that the local planar cell is doubled on each axis, but also centred, giving a new cell with double the area. 

Vectors at half-integer points in the plot must correspond to interchain metal-metal (M-M) correlations. The z component of the displacement around the body-centre sites and the two strong lobes above and below each integer vector are clear evidence that there are strong dimerising correlations between some metal sites. The [x,x,0] component of the quadrupole feature shows that at least one site has a d_xy displacement. The fact that there is no orthogonal shift in the cross-section of the same feature shown in Fig. <ref>b, shows that the dimerisation is dissimilar to the M1 structure. In the M1 phase, d_xy causes buckling between atoms in the neighboring dimer along the diagonal (Fig. <ref>c), so they do not lie along the same z axis as they do in this 3D-ΔPDF map. Instead, they should be displaced off of the diagonal, which is obviously not the case in Fig. <ref>. This relationship looks much closer to the M2 structure, where the dimerised d_z atoms contain no d_xy, and the undimerised set do. This would explain why every pair in the 3D-ΔPDF map is offset laterally as a whole. This is consistent with the dipolar features at [u/3,u/3,w] and [u/6,u/6,w/2] sites, which correspond to M-O and O-O interatomic vectors. They show a commensurate shift that is in agreement with the M2 interpretation. In this sense, the (110) cut of the 3D-ΔPDF map can be interpreted as a 2D cut of the M2 phase on the (110)_R (Fig. <ref>c). This structural model will be referred to as the `2D-M2' model. The correlations are very strong, and remain measurable in the 3D-ΔPDF at least to 25 unit cell diagonals (about 20 nm), but die off within just a few unit cells ( 1 nm) in the orthogonal axis. The 3D-ΔPDF map was used as an alternate method for estimating ξ, by fitting the PDF amplitudes to a back-to-back exponential decay function<cit.>. The results are plotted in Fig. <ref>a (triangular symbols) along with the fits from scattering rod widths. Notably, the [x,x,0] length scale agrees very closely for both methods, while the [0,0,z] seems to be underestimated by the pseudo-Voigt fits.

In order to test this interpretation, we constructed metal-only 2D-M2 models for both the X-ray scattering and 3D-ΔPDF<cit.>. Fig. <ref> shows the results of these simulations (right) compared to observations (left). The match between experiment and model appears successful, given that there was no actual parameter fitting involved; the observed data was simply used for scaling. It is clear that the constructed 2D-M2 model naturally conforms to the observation, even down to the relative intensities of the scattering rods. Note that one extra parameter, ferroelectric correlations along the [100] and [010] axes, was required in the diffuse scattering simulations to reproduce the wave-like bending of the scattering rods. The weak ferroelectric correlations are frustrated by the main in-plane correlation. This will be discussed further below. The calculated 3D-ΔPDF map reproduces the integer lattice vectors features and the body centred quadrupolar features. The weaker dipolar features attributed to oxygen atoms are missing, as expected. 




§ DISCUSSION


The constructed models validate the 2D-M2 interpretation of the data. Combining this with the other structural and electronic properties we have measured, we can state that the 2D-M2 is a different thermodynamic phase from both the long-range M1 and R phases. The transition from 2D-M2 to R is a first-order, metal-to-metal phase transition. Compositionally, the structural instability in V_1-xMo_xO_2 changes from M1 to 2D-M2 ordering quite discontinuously between 0.17 <  x ≤ 0.188. The transition is very similar to the observed V_1-xCr_xO_2 phase behaviour, except that here the M2-like phase only has medium-range order in two dimensions and very short-range order in the third dimension. Another difference is that Cr-substitution immediately (x≤ 0.015) suppresses M1 in favour of M2. For Mo-substitution, the M1 structure is robust up to x=0.18, before suddenly disappearing. 

This fragility implies that long-range order of displacements is somehow independent between the orthogonal sets of {110} planes, and that x=0.19 corresponds to the point where two order parameters diverge. This is reminiscent to the `embedded 2D crystal model' proposed by Lovorn and Sarker<cit.>. In accordance with the Lovorn-Sarker Ashkin-Teller displacements model, the critical composition near x=0.19 is when order parameters become m_A≠ 0≠ m_B. They predicted that in this case, interactions between the ⟨ 110⟩ and ⟨ 100 ⟩ directions are geometrically frustrated and prevent ordering along both directions simultaneously. Instead, regions with strongly coupled displacements in either [110] or [110] planes form. This is exactly what we required to explain the wavy feature in our diffraction data. Another feature of the geometric frustration is that only the odd numbered pairs of parallel are frustrated, while the even-numbered neighbours still have some correlations. It is also consistent with the observed 3D-ΔPDF map. It is important that the disordered dimension in the 2D-M2 phase would be symmetrically equivalent to one of the ordered ones in the ht-R phase. This is in contrast to most low-dimensional disordered phases, such as clays or layered chalcogenides, where weak structural interactions are what lead to scattering rods<cit.>). To our knowledge an ordered state like this has never been observed experimentally in rutile or any other extended solid.



 §.§ The MIT in VO_2

Having found an existing theoretical model explaining a key aspect of our structural one, we next consider how these new observations fit into the controversial MIT in VO_2. The structure of the 2D-M2 phase elucidates how discrete structural units coalesce to create long-range order, and more importantly, it identifies the strongest interactions. Lovorn and Sarker described the relative Hamiltonian parameters associated with onset of geometric frustration<cit.>, but it is hard to connect this to a real compositional phase diagram. We posit that molybdenum's extra electrons are likely important in the stabilization of the 2D-M2 phase, so it is possible that a special band-filling is involved.

Special orbital/band filling is a key part of many interpretations invoking the Mott instability<cit.>. Said Mott-Peierls models, have emphasized that the orbital filling is driven by the Mott instability, and that the structural distortions follow. That is, the electronic properties are robust against changes in the structure or disorder<cit.>. Here we found that with Mo substitution, the metallic properties are retained in the lt structures near the critical composition x = 0.19. This is true for both the M1 phase in x = 0.17 and the 2D-M2 phase in x = 0.19 (Figure <ref>b); indeed, we as yet see no way to distinguish them through their transport properties. This suggests that the electronic ordering is fundamentally the same in both cases, and it is only the the structural manifestation of those electronic effects that changes. We also note that the existence of a metallic M1 phase ground state has not been observed adjacent to VO_2, and is itself evidence in favour of a Mott-based scenario. It has also been observed that μ _eff and γ both increase with Mo composition, indicating heightened electronic correlations<cit.>.



 §.§ Comparison the 2D-2M phase to important non-rutile systems

The lamellar structure of the 2D-M2 phase is reminiscent of the stripe order observed in both the cuprate and iron-based high-T_C superconducting families. In the underdoped cuprates, a transition from tetragonal to orthorhombic symmetry is accompanied by a striped CDW ground state, which then develops into more complex charge and spin interactions at higher doping. In the prototypical iron arsenides phases, AFe_2As_2, the striped spin density wave ground state also lowers the tetragonal symmetry to orthorhombic, and is connected to a much more complex, symmetry-breaking electronic ground state<cit.>. Despite the differences in physics—cuprates are almost certainly Mott-like, while said iron arsenides are more weakly correlated—both show complex physics through broken C4 symmetry.

Rutile is also tetragonal, yet it must lack C4 symmetry due to the staggered placement of atoms in the 1D chains. Instead, tetragonal symmetry is maintained only by the C4_2 axes that relates the two chain orientations. It is the translation that is inherent to the screw axis that actually decouples the two order parameters m_A and m_B, and thus is the origin of the fragile order in V_1-xMo_xO_2. How would stronger or weaker coupling between the principal components affect the electronic phase diagrams? Are further electronic ground states, such as superconductivity, accessible in proximity to broken symmetry? It would be telling to determine the affect of uniaxial strain on the electronic properties of V_1-xMo_xO_2 in the vicinity of x=0.19.



§ CONCLUSION

We have uncovered a new type of ground state in V_1-xMo_xO_2, 2D-M2, governed by largely 2D ordering of atomic displacements. We have shown that the structure is driven by geometric frustration, and that the properties of this phase support a Mott-Peierls mechanism as its origin. The real structure would have been impossible to describe using conventional crystallographic techniques, showing the power of the new 3D-ΔPDF method. A full electronic structure treatment of the 2D-M2 remains to be determined. The extremely sharp diffraction features indicate that most of the correlations derive from static order, but there may be a dynamic component, which could be revealed using inelastic x-ray/neutron scattering. The 2D-M2 phase weakens with increasing Mo content, but does exist at least up to x = 0.3. A systematic study of the structural parameters in detail over a wide compositional range may reveal how the fundamental order parameters evolve, which may help explain what conditions are required for the 2D-M2 state to manifest. This could result in a more complete physical model for VO_2 and the underlying physics universal to all open shell rutile phases. 



§ METHODS




 §.§ Crystal Growth

Large single crystal samples of V_1-xMo_xO_2 have been synthesised for 0 <  x ≤ 0.60 using a two-part synthesis. First, a mixture of small single crystals and polycrystalline V_1-xMo_xO_2 batches were synthesised using VO_2 (Materion, 99.5%), V_2O_3 (Alfa Aesar, 99.7%), MoO_3 (Alfa Aesar, 99.95%), and MoO_2 starting materials. MoO_2 was obtained by passing H_2 gas over MoO_3 for 24 hours at 565 ℃. Stoichiometric V:Mo:O ratios of the starting materials were placed into evacuated quartz tubes of 1 cm diameter and 23 cm length along with 0.15 g of MoCl_3 to act as a transport agent in a chemical vapour transport (CVT) synthesis. The sealed tubes were placed into a multi-zone furnace with the starting materials placed in the hot zone, which was heated to 950 ℃ over 8 hours, held at 950 ℃ for 6 days and then cooled to room temperature (RT) over 8 hours. The opposite end of the tube was the cold zone, which was heated to 850 ℃ over 10 hours, held at 850 ℃ for 6 days, and then cooled to RT over 9 hours, yielding small single crystals of V_1-xMo_xO_2. This was then ground into a fine powder and placed into evacuated quartz tubes of 1 cm diameter and 23 cm length with 0.15 g of TeCl_4 transport agent. These tubes were treated to the same heating schedule as the first step of the synthesis, this time yielding large rod shaped single crystals (maximum 5 mm×0.5 mm×0.5 mm) of V_1-xMo_xO_2. The crystals typically had x values comparable to the input value, although in some cases lower x crystals were found in one zone, and MoO_2 formed in the other. 


 §.§ Electrical Resistivity

Electrical Resistivity was determined using the four-probe technique on a Quantum Design Dynacool physical property measurement system (PPMS). 50 µm platinum wire contacts were adhered to the crystals using silver paint.


 §.§ X-ray Total Scattering Experiment

Total scattering experiments were performed at the Advanced Photon Source (APS) at Argonne National Lab (ANL) on sector 6-ID-D. Incident beam energy was 87 keV. Samples were mounted on Kapton capillaries using either GE varnish or Duco cement. The detector was a Dectris Pilatus CdTe 2M detector, held at a distance of 650 mm, with a threshold detection limit set to 43 keV. Goniometer geometry allowed free rotation of ϕ, limited ω and fixed χ at 90°. Every temperature scan had three 360°ϕ scans, at ω = 0, -15, +15. The second and third sub-scans also included a shift of the detector of 5 mm x 5 mm and 10 x 10 mm. The detector and ω shifts were to correct for gaps in the detector and blooming artefacts. The data were processed using nexpy, and transformed using the crystal coordinate transformation work-flow (CCTW). 


 §.§ 3D-ΔPDF

A 3D-ΔPDF map was created using a modification of the punch and fill method developed by Weber et al.<cit.>, by taking a Fourier transform of a symmetrised reciprocal space map of the diffuse scattering only. More details can be found in the supplemental information.





§ END NOTES


Correspondence and requests for materials should be addressed to J.M.A. (email: jmallred@ua.edu).



 §.§ Acknowledgements

The work was primarily supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division. The work at the University of Alabama was supported under award DE-SC0018174. X-ray experiments were performed at the Advanced Photon Source, which is supported by the Office of Basic Energy Sciences under Contract No. DE-AC02- 06CH11357. Work at UCLA was supported by NSF DMREF program under the award NSF DMREF project DMREF-1629457. During the diffraction experiments, sample environment maintenance and data collection were both aided by Dr. Doug Robinson.



 §.§ Contributions

The experiment was designed and planned by M.A.D. and J.M.A.. Sample synthesis and structural characterisation was performed by M.A.D with support from L.M.W and T.C.D. Total scattering experiments at ANL were performed by M.A.D., L.M.W., M.J.K., S.R., R.O., and J.M.A.. R.O. and M.J.K. provided extra support in the data acquisition and interface steps. Data analysis was performed by M.J.K., M.A.D, R.O. and J.M.A.. Anisotropic resistivity measurements were performed by M.A.D., C.H. and N.N.. The manuscript was written by J.M.A and M.A.D, with support from M.J.K and R.O. 



 §.§ Declarations


The authors declare that they have no competing financial interests.

