Structural origins of the infamous"Low Temperature Orthorhombic"to"Low Temperature Tetragonal"phase transition in high-Tc cuprates

We undertake a detailed high-resolution diffraction study of a novel plain band insulator, La$_2$MgO$_4$, which may be viewed as a structural surrogate system of the undoped end-member of the high-T$_c$ superconductors, La$_{2-x-y}$A$^{2+}_x$RE$^{3+}_y$CuO$_{4}$ (A = Ba, Sr, RE= Rare Earth). We find that La$_2$MgO$_4$ exhibits the infamous low-temperature orthorhombic (LTO) to low-temperature tetragonal (LTT) phase transition that has been linked to the suppression of superconductivity in a variety of underdoped cuprates, including the well known La$_{2-x}$Ba$_{x}$CuO$_4$ ($x=0.125$). Furthermore, we find that the LTO-to-LTT phase transition in La$_2$MgO$_4$ occurs for an octahedral tilt angle in the 4 $^{\circ}$ to 5 $^{\circ}$ range, similar to that which has previously been identified as a critical tipping point for superconductivity in these systems. We show that this phase transition, occurring in a system lacking spin correlations and competing electronic states such as charge-density waves and superconductivity, can be understood by simply navigating the density-functional theory ground-state energy landscape as a function of the order parameter amplitude. This result calls for a careful re-investigation of the origins of the phase transitions in high-T$_c$ superconductors based on the hole-doped, $n = 1$ Ruddelsden-Popper lanthanum cuprates.

We undertake a detailed high-resolution diffraction study of a novel plain band insulator, La2MgO4, which may be viewed as a structural surrogate system of the undoped end-member of the high-Tc superconductors, La2-xy A 2+ x RE 3+ y CuO4 (A = Ba, Sr, RE = Rare Earth). We find that La2MgO4 exhibits the infamous low-temperature orthorhombic (LTO) to low-temperature tetragonal (LTT) phase transition that has been linked to the suppression of superconductivity in a variety of underdoped cuprates, including the well known La2-x Bax CuO4 (x = 0.125). Furthermore, we find that the LTO-to-LTT phase transition in La2MgO4 occurs for an octahedral tilt angle in the 4°to 5°range, similar to that which has previously been identified as a critical tipping point for superconductivity in these systems. We show that this phase transition, occurring in a system lacking spin correlations and competing electronic states such as charge-density waves and superconductivity, can be understood by simply navigating the density-functional theory ground-state energy landscape as a function of the order parameter amplitude. This result calls for a careful re-investigation of the origins of the phase transitions in high-Tc superconductors based on the hole-doped, n = 1 Ruddelsden-Popper lanthanum cuprates.
High-temperature superconductors based on La 2x Ba x CuO 4 (LBCO), in which unconventional superconductivity was first reported [1], continue to receive significant attention for their intertwined electronic orders and promise of insights towards roomtemperature superconductivity (SC) [2][3][4]. Nevertheless, a comprehensive understanding of the origin and mechanism behind such phenomena remains elusive [5]. While significant efforts have recently been made to explain the interplay of their competing electronic states [6,7], these fall short of reconciliation with the complex structural behavior concurrently observed [8].
The doping-dependent temperature phase diagram of the canonical system, LBCO, exhibits two pronounced regions of superconductivity, with peaks in the critical transition temperature for superconductivity, T c , at x = 0.095 and 0.155, either side of a pronounced dip for x ∼ 1 8 doping [9]. This suppression of T c is understood to be coincident [10] with a structural phase transition, commencing below 80 K (T LT ) from the low-temperature orthorhombic (LTO) phase (Bmab) to a low-temperature tetragonal (LTT) phase (P 4 2 /ncm) [11]. This phase * m.senn@warwick.ac.uk transition has been linked to the emergence of a chargedensity wave state [12] that is thought to compete with three-dimensional superconductivity [13][14][15]. Since the LTT phase itself is not observed in the undoped endmember, La 2 CuO 4 , and appears otherwise to be localized in the temperature, doping and composition phase diagram at points coincidental with suppression of T c [11,16,17], it has seemed reasonable to assume that this phase arises as a result of an interplay between electronic, spin and lattice degrees of freedom near x ∼ 1 8 chemical doping [10,18].
To make progress in understanding any entanglement between structural symmetries, superconductivity and charge density waves, it is important to understand the precise symmetries and structural mechanisms implicated in their behavior. Indeed, even in the undoped end-member, La 2 CuO 4 , the true structure and its physical origin remains a topic of discussion [19][20][21]. To gain insight into this question, we synthesize a novel surrogate of the end-member, La 2 MgO 4 , in which the possibility of electronic contributions to the phase behavior is eliminated.
Polycrystalline La 2 MgO 4 was prepared via high pressure synthesis from stoichiometric amounts of La 2 O 3 (Alfa Aesar, 99.999%, dried by heating at 1273 K for 12 hours) and MgO (Acros, 99.99%). The starting ma-arXiv:2111.11888v1 [cond-mat.supr-con] 23 Nov 2021 terials were mixed thoroughly, sealed in a gold capsule and heated at 1273 K at 6 GPa for 30 minutes in a DIA-type cubic anvil high pressure apparatus. The sample was quenched to room temperature after the heating program and the pressure released slowly. La 2 MgO 4 adopts a Ruddlesden-Popper (RP) n = 1 structure consisting of single MgO 6 perovskite slabs separated by LaO rock-salt layers (Fig. 1). As Mg 2+ has almost the same ionic radius as Cu 2+ (0.72Å vs. 0.73Å, respectively) [22], it may be viewed as a structural surrogate to the end-member of the RP n = 1 superconducting cuprate La 2 CuO 4 , but with the key difference being that necessarily lacks the electron-electron correlations from which competing emergent phenomena may arise.
To ascertain the temperature-dependent structural phase diagram, synchrotron X-ray powder diffraction measurements were conducted on beamline 19A of the Taiwan Photon Source, National Synchrotron Radiation Research Center (TPS, NSRRC) with a 20 keV and 16 keV incident beam (λ = 0.61994Å and λ = 0.77495Å) for the low-and high-temperature sweeps, respectively, using a MYTHEN detector. The La 2 MgO 4 sample was sealed in a 0.1 mm borosilicate capillary for low temperature and 0.1 mm quartz capillary for high-temperature experiments, each kept spinning during data collection for better averaging. A Cryostream 800 Plus (Oxford Cryosystems) was utilized for data collection from 100 K to 450 K with intervals of 50 K (omitting 400 K). A hot air gas blower (FMB Oxford) was utilized for data collection from 300 K to 1000 K, at intervals of 10 K from 300 K to 370 K, 50 K from 400 K to 500 K, and 10 K from 510 K to 1000 K. All measurements were taken upon warming. Rietveld refinement of the data was performed in Topas 6 [23] using the symmetry-adapted displacement formalism [24], as implemented in ISODISTORT [25]. All refinements were performed in the (a+b, a-b, c) supercell with respect to the I4/mmm, HTT aristotype phase, and in the common subgroup P ccn with appropriate constraints imposed on the symmetry-adapted displacements and lattice parameters to reproduce the symmetry of the HTT (F 4/mmm in the aforementioned supercell), LTO or LTT phase. For each phase, the scale factor, isotropic thermal displacement parameters (for each atom type), Stephens strain parameters [26], and atomic positions as described above were fitted, along with impurity phases as detailed in the Supplemental Information (SI) [27]. For the temperature range 300 K -340 K of the high-temperature sweep, we observed clear phase coexistence of the LTO and LTT phases (Fig. 2), although a model reliable enough for structural discussion could not be obtained until 370 K. Further details and representative fits are given in the SI.
From 950 K, the aristotypical n = 1 RP symmetry is observed with space group F 4/mmm in the supercell setting, with any splitting of the (200) diffraction profile clearly absent (Fig. 2(a)). Below 950 K, a second-order phase transition occurs, evidenced by a continuous evolution of the lattice parameters (Fig. 3). This phase is well fit by the Bmab (LTO) phase ( Fig. 1) that is nominally the ground-state structure of La 2 CuO 4 [28]. The diffraction profile of the (200) peak shows a continuous splitting into well-resolved (200) and (020) reflections. This phase transition may be described as an out-of-phase rotation of the MgO 6 octahedra about the a-axis. Following the conventions of ISODISTORT [25], the order parameter (OP) associated with this tilting may be labeled as transforming as the irreducible representation (irrep) X + 3 (a;0), while the tilt system of the perovskite slabs can be labeled in Glazer notation as a − a − c 0 with respect to the lattice vectors of the aristotypical HTT cell.
Below 350 K, a third distinct phase is observed with tetragonal symmetry and fitting well to the LTT phase that has been much discussed in the underdoped cuprates, most notably in LBCO (x ≈ 1 8 ). This corresponds now to alternating tilting of the octahedra about the [110] and [110] axes in the perovskite layers, centered at z and z + 1 2 , and may be described by an OP transforming as the irrep X + 3 (a;a) (a − a 0 c 0 ).Upon warm-   ing through this transition, a clear coexistence between the LTT and LTO phases is observed (Fig. 2(e)). Along with a discontinuous jump in the lattice parameters and refined magnitude of the OP (Fig. 3), this is indicative of first-order behavior. Since this phase transition has invariably been found to be concomitant with the suppression of superconductivity and the formation of CDWs, the novel observation of the LTT phase in a system that is a plain band insulator, i.e., lacking the possibility of strong electronic correlations, allows opportunity for clear insight into the structural contributions to the mechanism of the LTO-to-LTT phase transition, isolated from the influences of competing electronic phenomena.
To obtain further insight into the origin of the LTOto-LTT phase transition in La 2 MgO 4 , we perform singlepoint energy calculations using density-functional theory (DFT) to map out an energy landscape spanning the X + 3 (a;0) and X + 3 (a;a) OPs. The symmetry in all cases can be described within the P ccn space group, which is a common subgroup of both the LTT and LTO phases and encompasses intermediate tilt propagation vectors that 3 OP mode amplitude is the total mode amplitude Ap value as defined by ISODISTORT [25]. A discussion of the errors in this plot can be found in the SI.
describe the so-called low-temperature less orthorhombic (LTLO) structure. Figure 4(a) shows the energy landscape associated to an interpolation of the strain, X + 3 OP magnitude and direction, and the Γ + 1 OP magnitude between the relaxed HTT, LTO and LTT phases (see SI). These single-point energy calculations reveal no metastable, local minima for any intermediate P ccn phases between the LTO and LTT structures. This is further supported by full structural relaxations of intermediate P ccn structures which invariably relax to the LTT structure. Since the P ccn phase has lower symmetry than the LTO and LTT phases, we would equally expect its vibrational entropy to be lower. The lower entropy of this phase also makes a stable P ccn phase unexpected with increasing temperature. The LTO-to-LTT phase transition can then not occur via a continuous rotation of the OP that passes through the P ccn phase, meaning it must be first-order. This is consistent with the first-order nature of the phase transition evidenced experimentally (i.e., the pronounced phase coexistence; Fig. 2(e)).
While our DFT calculations do not explicitly capture the effect of temperature, we may still gain valuable insights into the sequence of phase transitions in La 2 MgO 4 . The energy landscape for a given direction and magnitude of the X + 3 (a;b) OP with simultaneous interpolation of the Γ + 1 OP and strain field between the HTT, relaxed LTO and relaxed LTT phases, calculated relative to the non-standard, F 4/mmm, HTT phase (see SI for further details). (b) Plot of the the energy landscape along the X + 3 (a;0) and X + 3 (a;a) OP directions at the calculated relaxed values of strain and Γ + 1 OP for the LTO and LTT phases, respectively, relative to the HTT aristotype (see SI). All mode amplitudes are given in Ap values as defined by ISODISTORT [25]. The black data points denote the positions of the lowest-energy structures along each OP direction.
the magnitude of the X + 3 OP along the (a;0) and (a;a) OP directions for strain and Γ + 1 magnitudes fixed at the calculated relaxed structures for LTO and LTT, respectively. This reveals that there is a clear departure of the LTO and LTT phases from the harmonic approximation (which requires that all OP directions be iso-energetic) at an amplitude of around A p = 0.35Å. Such an amplitude lies within the discontinuous region between the LTO and LTT phases, at around 300 K to 350 K in our diffraction study (Fig. 3(d)).
The origin of the experimentally observed LTT-to-LTO phase transition in La 2 MgO 4 may then be understood within the quasi-harmonic approximation in which the thermal lattice expansion renormalizes the OP amplitude, as we have found in our previous work on RP n = 1 systems in which we included the effects of thermal expansion [29]. The increase in amplitude of the OP upon cooling leads to a scenario where the LTT structure becomes more favorable from an enthalpic perspective. Any entropic contributions to the Gibbs free energy that favor LTO at higher temperatures are evidently then quickly overwhelmed. The observation of such a cross-over between LTO and LTT superstructures in our surrogate system at a tilt angle between 4°and 5°may indicate why a similar value has been identified as a critical octahedral tilt angle for superconductivity in the family of compounds, La 2xy A 2+ x RE 3+ y CuO 4 (A = Ba, Sr, RE = Rare Earth) [30], where superconductivity is presumed to be the preserve of the LTO phase.
Finally, we turn our attention to La 2 CuO 4 where we employ DFT with an on-site Hubbard U on the Cu-site of 9 eV (SI), an approach which has been demonstrated to be effective in approximating the electronic ground state of La 2 CuO 4 [31,32]. While our DFT+U calculations for La 2 MgO 4 find the LTT phase to be lower in energy than the LTO phase by ∼ 11.2 meV/(formula unit), for La 2 CuO 4 this energy difference is of the order of the typical accuracy of DFT. Prior calculations on La 2 CuO 4 using the SCAN exchange-correlation functional also found the two phases to be very close in energy [33]. These results are consistent with the experimental observations that the LTO-to-LTT phase transition is not observed upon cooling for La 2 CuO 4 [19][20][21]28] but is evidenced here as occurring at significantly elevated temperatures in La 2 MgO 4 . The small energy scale associated with variation in the ionic radii and changes in spin and electron correlations that occur as the La 2 CuO 4 system is doped towards x = 1 8 may be sufficient to affect small changes in the amplitude of the X + 3 OP, in turn triggering or indeed inhibiting the transition from LTO to LTT in these systems. This resulting change in structural symmetry is what has often been correlated with the suppression of superconductivity in these systems.
In conclusion, we have shown that the infamous lowtemperature tetragonal (LTT) to low-temperature orthorhombic (LTO) phase transition, observed in many underdoped lanthanum cuprates, may be reproduced in the surrogate plain band insulator, La 2 MgO 4 , which is presumably devoid of electron-electron correlation inherent to the competing superconducting and CDW states. The transition is shown to be triggered simply by the change in order parameter magnitude associated with the tilt of the BO 6 octahedra (B = Mg or Cu). In the doped La 2 CuO 4 , small changes in magnetic and electronic correlations (experimentally or in silico) will couple to the order parameter magnitude, consequently selecting either LTO or LTT as the ground state, which presumably themselves favor superconductivity or CDW formation, respectively. It is noteworthy that the tilt angle at the relevant OP magnitude lies close to the critical value of tilting that has been associated to the suppression of superconductivity in the doped lanthanum cuprates.