Abstract
Understanding the role played by broken-symmetry states such as charge, spin and orbital orders in the mechanism of emergent properties, such as high-temperature superconductivity, is a major current topic in materials research. That the order may be within one unit cell, such as nematic, was only recently considered theoretically, but its observation in the iron-pnictide and doped cuprate superconductors places it at the forefront of current research. Here, we show that the recently discovered BaTi2Sb2O superconductor and its parent compound BaTi2As2O form a symmetry-breaking nematic ground state that can be naturally explained as an intra-unit-cell nematic charge order with d-wave symmetry, pointing to the ubiquity of the phenomenon. These findings, together with the key structural features in these materials being intermediate between the cuprate and iron-pnictide high-temperature superconducting materials, render the titanium oxypnictides an important new material system to understand the nature of nematic order and its relationship to superconductivity.
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Introduction
Rather than being an anomalous side effect in one or two cuprate systems, broken-symmetry states are now thought to be widespread in strongly correlated electron systems and other complex materials. Extensive study of the manganites1,2, cuprates3,4, iron pnictides5 and a variety of other systems has made it increasingly evident that local and global symmetry breaking in the charge, orbital, lattice and spin degrees of freedom are associated with the appearance of emergent phenomena, such as colossal magnetoresistance and high-temperature superconductivity (HTSC), but the exact relationship is not understood. Historically, the study of such broken-symmetry states has been very challenging. Taking the cuprates as an example, 8 years elapsed from the initial discovery of superconductivity to the first observation of symmetry-broken charge order (stripes) in one system6, another 7 years passed before hints were found in others7,8 and only within the last 3 years has charge order begun to emerge as a possibly ubiquitous feature of the cuprates9,10.
Several possibilities arise when considering the symmetries that can be broken by these states. Most charge-/spin-density waves (C/SDWs) break the translational symmetry of the lattice, folding the Brillouin zone and resulting in superlattice diffraction peaks. On the other hand, orbital ordering, where charge transfers between orbitals centred at the same site, can break the metric rotational symmetry without lowering the translational symmetry. Examples are charge-nematic11 and loop-current12,13 orders in the doped cuprates. In this context, nematic order is defined as one that breaks the rotational point group symmetry while preserving the lattice translational symmetry. The fact that nematic symmetry-broken states have recently been discovered experimentally in both the cuprate11 and iron-based14 superconductors raises the importance and relevance of this observation to HTSC. It is, therefore, critically important to understand the role and ubiquity of symmetry breaking, including intra-unit-cell nematicity, to the superconducting phenomenon.
Standard theoretical treatments of HTSC, such as the effective single-band t–J model, have typically ignored the possibility of intra-unit-cell orders15. When multiple atoms per unit cell are explicitly included in the theory, qualitatively different ground-state solutions may be found16, underscoring the subtlety and importance of accounting correctly for this phenomenon. Hence, finding related but distinct systems that exhibit this phenomenon is expected to shed new light on this critical question.
Very recently, superconductivity was discovered17,18,19 in titanium-oxypnictide compounds, such as ATi2Pn2O (A=Na2, Ba, (SrF)2, (SmO)2; Pn=As, Sb, Bi), which are close structural and chemical cousins to the cuprates and iron-pnictides20,21,22,23,24. In particular, in isovalent BaTi2(Sb1–xBix)2O and aliovalent Ba1–xNaxTi2Sb2O (ref. 25), muon spin rotation and heat capacity measurements point to fully-gapped s-wave superconductivity26,27,28. Interestingly, a number of compounds in this family also show strong anomalies in resistivity and/or magnetic susceptibility that are thought to be signatures of symmetry-breaking charge- or spin-ordered ground states21,24,25,29,30, suggesting that these materials are excellent candidates for studying the interplay between broken-symmetry states and superconductivity. In light of these strong transport anomalies, it is then quite surprising that subsequent experiments have failed to uncover any direct evidence for a conventional spin- or charge-density wave ground state26,27,31, leaving open the question of whether these materials do possess symmetry-broken ground states.
Here, we show that superconducting BaTi2Sb2O, and its non-superconducting parent compound BaTi2As2O, do indeed undergo a tetragonal-orthorhombic phase transition, corresponding to a C4–C2 symmetry lowering, that occurs at the temperature of the transport anomaly. On the other hand, high-sensitivity electron diffraction measurements failed to detect any superlattice peaks in the bulk at low temperature, indicating that this transition does not break translational symmetry. The low-temperature phase, therefore, constitutes a nematic state. In light of the pronounced upturn in resistivity accompanying this nematic transition, together with the absence of any ordered SDW26, we attribute the nematicity to an intra-unit-cell charge order with d-wave symmetry by charge transfer between neighbouring Ti sites—similar to that between neighbouring oxygen sites in cuprate superconductors—and find that it naturally explains the temperature dependence of the lattice constants. These results establish this family of materials as another playground for studying symmetry-breaking electronic phases and their relationship to superconductivity.
Results
Structural and electronic properties of BaTi2Pn2O
The basic structural unit of BaTi2Pn2O is a planar square net of titanium and oxygen, in analogy with the cuprates (Fig. 1a,b), with the crucial difference that the positions of the metal and oxygen ions are switched between the structures (the complete titanate structure is shown as an inset in Fig. 2b). In Fig. 1, the square net is shown by solid lines along the nearest neighbour bonds, with dashed lines showing the net joining second neighbour ions. This second-nearest-neighbour square net connects oxygen ions in the cuprates, but metal ions in the titanate compounds and also in the iron-based superconductors (Fig. 1c). Thus, in terms of chemistry and structure, the titanate compounds bridge between the ferrous and cuprate superconductors. The Ti 3d orbitals are occupied by one electron per Ti atom, which is found to reside in a nominally 1/4-filled band formed via hybridization of the dxy and orbitals for the Ti(1)/Ti(2) ions (defined in Fig. 1b). The local geometry of the Ti(1) site is shown in Fig. 1d and the arrangement of the d-energy levels is shown in Fig. 1e.
Furthermore, the phase diagram (Fig. 2b) is highly reminiscent of the cuprates and iron-based superconductors, with superconductivity appearing on doping and transport behaviour that is strongly suggestive of a competing electronic transition such as the formation of a CDW or SDW21,24. The transport is metallic at high temperature21, with a positive resistivity slope versus temperature (Fig. 2a). However, on cooling, a pronounced upturn in the resistivity is found for all x in the solid solution BaTi2As1−xSbxO. The feature occurs at a temperature Ta that decreases monotonically from 200 K for x=0 to 50 K for x=1, with superconductivity appearing below ~1 K for the antimony endmember and increasing to 5 K for BaTi2Bi2O (refs 19, 24). Anomalies in the magnetic susceptibility and specific heat are also observed21 at Ta.
Density functional theory (DFT) calculations for BaTi2Sb2O predicted an instability towards a bicollinear SDW formation32,33 or a commensurate CDW ground state driven by an unstable phonon mode that doubles the unit cell by distorting the Ti squares and preserves the tetragonal symmetry34. The possibility of SDW formation in BaTi2(As,Sb)2O, either commensurate or incommensurate, was subsequently ruled out by muon spin relaxation and 121/123Sb nuclear resonance measurements, which show conclusively that no magnetic order develops at any temperature probed26,27,31. On the other hand, a conventional CDW should be evident through an associated structural distortion. However, initial electron and neutron diffraction studies on the Sb endmember26 found no broken symmetry or any signature of superlattice formation at low temperatures, nor was a CDW gap formation observed in angle-resolved photoemission measurements of the nested Fermi surfaces (although a slight depression of the density of states at other momenta was found to correlate in temperature with the resistivity anomaly35). The possibility of an incommensurate CDW was also ruled out by 121/123Sb nuclear resonance measurements31.
Neutron powder diffraction measurements
In the absence of evidence for a long-range ordered CDW, we undertook a neutron diffraction and total scattering measurement on BaTi2Sb2O with a dense set of temperature points to search for evidence for a possible short-range ordered CDW36,37. We also extended the investigation to the previously unstudied BaTi2As2O endmember. Unexpectedly, we found a long-range structural phase transition at Ta. Room temperature measurements of BaTi2As2O confirm the tetragonal P4/mmm space-group symmetry previously reported from X-ray diffraction21. However, on cooling through Ta, we observe a distinct splitting of the (200)/(020) and (201)/(021) Bragg peaks, representing the first observation of a symmetry lowering at the same temperature as the resistivity anomaly in BaTi2As2O. This is shown in Fig. 3, which compares the high- and low-temperature Bragg peaks in panel (a) and displays their temperature evolution in panel (b). The (200) peak at 2.02 Å begins to broaden below 200 K, coinciding with Ta, and appears to split at the lowest temperatures. Similarly, the (201) peak at 1.94 Å displays apparent splitting as the temperature is lowered. These observations demonstrate that BaTi2As2O undergoes a long-range ordered structural phase change at Ta, lowering its symmetry from tetragonal to orthorhombic.
To investigate the structural transition in greater detail, we performed Le Bail38 refinements at all temperatures. We used the parent P4/mmm model for T≥200 K. The simplest possible symmetry-breaking distortion mode of the parent P4/mmm structure consistent with the observed peak splitting is a mode that breaks the degeneracy of the a- and b-axes without otherwise shifting atoms within the unit cell, resulting in a space-group symmetry of Pmmm, which was used for T<200 K. We also explored other candidate orthorhombic structures with lower symmetry, but since these structures do not improve the fit quality within the resolution limitations of the current data, Pmmm is the most appropriate choice. We display the results of these refinements in Fig. 4. As seen in panel (b), the tetragonal a axis clearly splits below T~200 K, with a maximum orthorhombic splitting of ~0.01 Å. The orthorhombicity parameter η=2 × (a–b)/(a+b) is shown in the inset of panel (a), indicating a maximum orthorhombicity of ~0.22%. Panel (a) also displays the temperature dependence of the c axis parameter, which exhibits an upturn below the structural transition deviating from the linear thermal contraction trend seen for T>200 K. This same type of c axis response also accompanies long-range ordered stripe formation in the nickelates36. Additional details regarding Rietveld refinement of BaTi2As2O can be found in Supplementary Note 1. The superconducting BaTi2Sb2O shows qualitatively the same behaviour, albeit with an amplitude decreased by a factor of ~5, with an orthorhombic splitting (0.05%) and a small but observable c axis upturn appearing on cooling through Ta=50 K. The undistorted P4/mmm model can be used with moderate success at all temperatures, but the Pmmm model yields a better fit below 50 K (see Supplementary Note 2). Pair distribution function analysis is consistent with these observations for both compounds and can be found in Supplementary Note 3.
These results offer compelling evidence that the observed structural response is intimately related to the transport anomaly and may be driven by a broken symmetry of the electronic system forming at that temperature. The small distortion amplitude in BaTi2Sb2O explains why this long-range structural phase change escaped notice in previous neutron diffraction measurements26.
Electron diffraction measurements
Since CDW formation is implicated, we made a special effort to search for the appearance of weak superlattice peaks associated with a finite CDW wavevector in electron diffraction (ED) patterns below Ta. The original study on the antimony endmember failed to observe superlattice peaks26. Here, we concentrated on the arsenic endmember where the structural distortion is five times larger, and the ED patterns were heavily overexposed to search for any weak response at intensities close to background. Despite these efforts, the ED patterns taken along the [001] and [011] directions revealed no superlattice peaks in the bulk at low temperature, as shown in Fig. 5. However, in a very small fraction of the sample in the immediate vicinity of grain boundaries, weak superlattice peaks with Q=(1/2, 0, 0) are observed at low temperature. This non-bulk behaviour is explored further in Supplementary Note 4.
Discussion
A picture emerges of a C4–C2 symmetry breaking occurring with an accompanying strong upturn in resistivity, but with no corresponding CDW superlattice peaks appearing. The resistivity upturn is larger than would be expected as a passive response of the electronic system to the structural transition, borne out by a standard DFT calculation with the observed orthorhombicity parameter η=0.22%, which showed that merely 0.0003 electrons are transferred from Ti(1) to Ti(2). Therefore, in common with earlier discussion18,21, we propose that the structural transition is a response to an instability of the electronic system. The earlier muon spin relaxation results26,27, together with the ED measurements, allow us to rule out the existing proposals of SDW formation32,33 or a phonon-driven CDW34. Instead, an intra-unit-cell charge-nematic electronic symmetry breaking is implicated, similar to that proposed for doped cuprates39.
In the current case, a charge redistribution between the on-site orbital states at the Fermi level, to dxy, does not break the rotational symmetry, so can be ruled out. Instead, a simple but novel intra-unit-cell charge order (IUC-CO) naturally explains the observed phenomenology. A transfer of charge from Ti(1) to Ti(2) (see Fig. 1b) lowers the rotational symmetry of the Ti2O plaquette locally from C4 to C2, with the effect on the overall lattice symmetry depending on the ordering pattern of the distinct Ti ions in neighbouring unit cells. Repeating the symmetry-lowered plaquette uniformly along the a and b directions results in no change of the unit cell, but breaks the metric symmetry from C4 in P4/mmm to C2 in Pmmm, as observed experimentally. This arrangement of charges can accordingly be described as a nematic IUC-CO. Our data are therefore consistent with the formation of this type of charge order on cooling through Ta. Such a model is also consistent with the breaking of C4 symmetry at the Pn site that has been observed from 121/123Sb nuclear resonance measurements31.
This nematic IUC-CO is energetically favoured on Coulombic grounds if the on-site Hubbard energy U is sufficiently small, which is a reasonable assumption as the system is a metal rather than a Mott insulator. The small Hubbard U arises from significant screening due to the solvation effect of the high polarizabilities of the As3− and Sb3− anions, which are an order of magnitude larger than that of O2−. Moreover, the As and Sb ions reside on a lower symmetry site in the Pmmm structure, which enhances the effects of their polarizabilities. This physics was proposed in an earlier study of the iron-pnictide superconductors40. Returning to the present system, the transfer of a charge of δ from Ti(1) to Ti(2) will result in a lowering of the electrostatic energy, V(1–δ)(1+δ)=V(1–δ2), where V is the screened Coulombic interaction between Ti sites and is positive. The result is charge order with a d-wave symmetry41 with the sign of the modulated charge density varying as −+−+ around the plaquette. From plaquette to plaquette, the orientation of the axis of the distortion can be parallel or perpendicular, forming a ferro- or anti-ferro- type ordering, which would preserve or break translational symmetry, respectively. The former is consistent with the experimental observations in this material. We suggest that the rather rigid face-shared octahedral topology in each layer favours the uniform ferro- over the anti-ferro- ordering. It is noteworthy that this V is the counterpart of the Coulumbic repulsion Vpp between neighbouring oxygen atoms in the CuO2 plane, which was shown to drive IUC nematic order in nonstoichiometric doped cuprates42. Hence, the present results obtained on these stoichiometric materials (thus having less ambiguity such as disorder effects) yield insight into the origin of IUC nematic order in the cuprates.
This nematic IUC-CO naturally explains the observed changes in the a- and c-lattice parameters. The transfer of charge from the Ti(1) orbital to Ti(2) results in increased electrostatic repulsion between the charge-rich orbitals extending along the a axis, breaking the tetragonal degeneracy of the a and b axes and leading to the observed orthorhombic distortion. Furthermore, a uniform stacking of the IUC-CO in each layer can also explain the response of the c-lattice parameter, which expands upon entering the charge-ordered state (Fig. 4a). This lattice expansion may be attributed in part to increased electrostatic repulsion between inter-layer Ti ions from the transferred charge. The net energy contribution is V′[(1+δ)2+(1–δ)2]=V′(2+2δ2), where V′ is the inter-layer screened Coulombic interaction. This acts in addition to other elastic energy contributions.
The structural effects observed on cooling are much smaller in BaTi2Sb2O than BaTi2As2O, suggesting that the IUC-CO is relatively suppressed both in amplitude and temperature. This may be a result of the larger unit cell in the Sb compound, due to its larger Sb3− ionic diameter (2.25 versus 1.98 Å for As3−; ref. 18), resulting in a smaller V.
To identify the microscopic driving force of this nematic instability, we present a symmetry-based zero-order analysis in which only the leading energy scales are retained. Ti atoms reside at the centre of a distorted octahedron with oxygen at the apices and pnictide atoms around the equatorial plane, as shown in Fig. 1d for Ti(1). The Ti 3d-energy levels are illustrated in Fig. 1e, similar to the case of nearly isostructural (LaO)2CoSe2O (ref. 43). The nominal electron occupation is one electron per Ti atom, which would have been assigned to the locally lowest-lying and orbitals on the Ti(1) and Ti(2) ions, respectively. However, the σ-bonding between the Ti(1) and Ti(2) dxy orbitals forms a relatively wide band that overlaps the locally lowest level. Therefore, the minimum model for the titanium oxypnictides involves the two orbitals: and dxy on Ti(1) and and dxy on Ti(2). Since the dxy orbital has the same impact on both the a and b direction, the C4–C2 symmetry lowering around the central oxygen atom is mainly determined by the charge imbalance between the quasi-one-dimensional band on Ti(1) and on Ti(2) as a result of the Stoner instability42. No doubt this mechanism will be complicated by hybridization and other issues, but this symmetry-based analysis provides the appropriate realistic starting point. Since the proposed nematic intra-unit-cell charge order elegantly explains all the observed structural effects, it is anticipated to be the electronic ground state of BaTi2Pn2O.
Methods
Neutron powder diffraction
Powder specimens of BaTi2As2O and BaTi2Sb2O were prepared via conventional solid state reaction methods. Details of the synthesis are provided in a previous study26. Time-of-flight neutron total scattering experiments were performed at the Neutron Powder Diffractometer at Los Alamos Neutron Science Center (LANSCE) at Los Alamos National Laboratory. Data were collected using a closed-cycle He refrigerator at temperatures ranging from 10–300 K in steps of 10 K near the structural transition and 25 K away from the transition over a wide range of momentum transfer Q. Le Bail38 fits to the intensity profiles were performed with GSAS44 on the EXPGUI platform45. Pair distribution function (PDF) profiles were obtained by Fourier transforming the measured total scattering intensity up to a maximum momentum transfer of Qmax=24 Å−1 using established protocols46,47 as implemented in the programme PDFgetN48. The Le Bail fits were used to extract lattice parameters and space-group symmetry, the PDF fits to extract atomic displacement parameters. Symmetry mode analysis using the programme ISODISTORT49 was conducted to identify candidate distorted structures.
Electron diffraction and DFT calculations
Electron diffraction patterns were recorded using a JEOL ARM 200CF transmission electron microscope (TEM), operated at 200 keV, at Brookhaven National Laboratory. The TEM samples were prepared by crushing powder specimens into thin flakes transparent to the electron beam, which were supported by a lacey carbon copper grid. DFT calculations were performed within the generalized gradient approximation implemented in the Wien2k software package50.
Additional information
How to cite this article: Frandsen, B. A. et al. Intra-unit-cell nematic charge order in the titanium-oxypnictide family of superconductors. Nat. Commun. 5:5761 doi: 10.1038/ncomms6761 (2014).
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Acknowledgements
Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract No. DE-AC02-98CH10886. Work at Columbia University was supported by the U.S. National Science Foundation (NSF) Partnership for International Research and Education (PIRE) Super-PIRE project (grant OISE-0968226). Y.J.U. also acknowledges support from NSF DMR-1105961, the Japan Atomic Energy Agency Reimei project, and the Friends of Todai Inc. The work at Kyoto University was supported by the FIRST program, Japan Society of the Promotion of Science (JSPS). Neutron scattering experiments were carried out on NPDF at LANSCE, funded by DOE Office of Basic Energy Sciences. LANL is operated by Los Alamos National Security LLC under DOE Contract No. DE-AC52-06NA25396.
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Y.J.U., S.J.L.B. and B.A.F. initiated this work. Y.N. and H.K. carried out sample preparation and characterization. E.S.B. carried out the neutron diffraction measurements and analysed the data with assistance from B.A.F. H.H. performed the electron diffraction measurements with help from Y.Z. W.-G.Y. proposed the charge-ordering model and provided theoretical support. B.A.F., W.-G.Y. and S.J.L.B. wrote the paper, with input from all the authors.
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Frandsen, B., Bozin, E., Hu, H. et al. Intra-unit-cell nematic charge order in the titanium-oxypnictide family of superconductors. Nat Commun 5, 5761 (2014). https://doi.org/10.1038/ncomms6761
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DOI: https://doi.org/10.1038/ncomms6761
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