Tracing the horizon of tetragonal-to-monoclinic distortion in pressurized trilayer nickelate La₄Ni₃O₁₀

Abstract

The crux of understanding the superconducting mechanism in pressurized Ruddlesden-Popper nickelates hinges on elucidating their structural phases. Under ambient conditions, the trilayer nickelate La₄Ni₃O₁₀ stabilizes in a twinned monoclinic structure with space group P2₁/c. Upon heating, it undergoes a structural transition to the tetragonal I4/mmm phase at T_s  ≈ 1030 K, while a second transition associated with the onset of density-wave (DW) ordering emerges upon cooling below T_DW  ≈ 135 K. Here, from pressure-temperature x-ray diffraction (XRD) on high quality flux-grown single crystals we demonstrate a direct tetragonal-to-monoclinic transition without an intermediate orthorhombic Bmab phase. Ab initio density-functional theory calculations as a function of pressure corroborate the experimental observations. The tetragonal-to-monoclinic transition unfolds as the formation of a two-fold superstructure, as evidenced by the emergence of commensurate superlattice reflections and can be progressively suppressed from 1030 K down to 20 K under 14 GPa. Notably, from XRD we establish the observation of weak incommensurate satellite reflections associated with the DW ordering in flux-grown samples, as previous findings were confined only to crystals grown by the floating-zone technique. This is further reinforced by Raman spectroscopy that reveal the emergence of additional phonon modes below 130 K, concomitant with the onset of the incommensurate DW state.

Understanding the superconducting mechanism in pressurized Ruddlesden-Popper nickelates requires insight into their structural phases. Here, the authors use pressure-temperature XRD and ab initio calculations to reveal a direct tetragonal-to-monoclinic transition in La₄Ni₃O₁₀, highlighting the suppression of this transition under pressure and the presence of intrinsic incommensurate density-wave ordering.

Introduction

Groundbreaking scientific discoveries in Cu-based superconductors, where the superconductivity (SC) is believed to arise from the partially filled Cu 3d electron states in the CuO₂ planes have fundamentally shaped our understanding of high temperature SC over the years^1–3. Nevertheless, the microscopic mechanisms underlying their unconventional superconductivity remain an enigma. In this context, the search for analogous materials continues to be of paramount importance. Rare-earth nickelates have emerged as a fertile platform for exploring correlated electron phenomena. Like the cuprates, they host a rich interplay of quantum phases including charge and spin density waves, antiferromagnetism, and superconductivity offering new avenues to probe the intertwined nature of these competing orders^4–8.

Experimental evidence of SC in the nickelates was first realized on thin films of infinite-layer Nd_1−xSr_xNiO₂^9–11. Interestingly, this structure, composed of NiO₂ planes stacked along the c axis, hosts Ni¹⁺ ions with a 3d⁹ electronic configuration, formally isoelectronic to Cu²⁺ in the cuprates. However, superconductivity has not been observed in bulk crystals^12,13, suggesting a notable influence of the substrate. This breakthrough was shortly followed by the discovery of high temperature SC at T_c = 80 K under hydrostatic pressure in bulk Ruddlesden-Popper bilayer crystal of La₃Ni₂O₇, where the average electronic structure of the Ni is Ni^2.5+ (d^7.5) far from the analogs to the cuprates. Interestingly, the SC is accompanied by a change in the symmetry of the crystal structure^14–19.

Recently, SC up to 30 K was also discovered both in bulk Ruddlesden-Popper trilayer crystals La₄Ni₃O₁₀ and Pr₄Ni₃O₁₀ grown by the floating zone technique and later by the flux method for the La-analogs^20–24. The emergence of superconductivity was evidenced by a drop in the electrical resistance and an anomaly in the magnetic susceptibility around P_c  ≈ 15 GPa. Notably, the reported critical pressure spans a broader range of 5–40 GPa depending on the pressure transmitting medium and the criteria chosen to define T_c²⁰. Pr₄Ni₃O₁₀ at room temperature undergoes a change in symmetry from monoclinic P2₁/c to tetragonal I4/mmm at P_c  ≈ 40 GPa, while for La₄Ni₃O₁₀ the transition occurs at P_c  ≈ 15 GPa ^20–22,24. At ambient pressure the transition towards monoclinic symmetry occurs at T_S  ≈ 1030 K for La₄Ni₃O₁₀^25,26. This suggests that SC emerges once the material has been restored to the higher tetragonal symmetry phase.

In addition under ambient pressure, a density wave (DW) ordering associated with spin and charge manifests in La₄Ni₃O₁₀ at T_DW  ≈ 130 K, characterized by the emergence of incommensurate modulation observed in single crystal by x-ray and neutron diffraction²⁷, followed by other probes^28–30. This DW ordering marked by an anomaly in resistivity and magnetic susceptibility measurements is remarkably fragile as it collapses under the application of a few GPa^20,31, which presumably is the trigger to promote spin fluctuations. This can be attributed to a pairing mechanism for the formation of the Cooper pairs^32–37. However, the pressure-dependence of T_DW remains enigmatic as optical spectroscopy experiments do not corroborate the transport measurements, thereby stating that the DW competes with the SC³⁸. Studies involving temperature dependent powder x-ray diffraction (PXRD) on La₄Ni₃O₁₀ reveals a negative thermal expansion of the b axis that could be potentially linked to structural distortions associated with density-wave ordering^39–41. However, the symmetry in the incommensurate phase and the extent to which it can reshape the 3-dimensional (3-d) lattice remains shrouded in uncertainty as later works do not reproduce or probe the modulation through x-ray diffraction.

Further analysis regarding the modulation is preceded by a lingering matter pertaining to the symmetry of La₄Ni₃O₁₀ at ambient conditions, as it also remains in a state of quandary. Earlier, Zhang et al.⁴⁰ reported the coexistence of a metastable orthorhombic Bmab phase with the monoclinic P2₁/c phase depending on the crystal growth conditions. A follow up work by Li et al.²⁴ propose an experimental phase diagram based on x-ray and transport measurements on single-crystalline La₄Ni₃O_10−δ. However, it is not yet elucidated from a crystallographic point of view. Careful investigation of the diffraction pattern at 11 GPa revealed extremely weak reflections that appear to violate the B-centering, an observation that can be ascribed to the formation of twin domains resulting from symmetry breaking, closely resembling our diffraction pattern. However, their interpretation differs from the present work.

Crystal structure is the bedrock of every material, governing its electronic structure and all electronic properties, including superconductivity. Accurate determination of the crystalline symmetry is therefore not merely a formality but an essential prerequisite for any theoretical band-structure calculation aimed at uncovering the microscopic mechanisms of superconductivity and in this paper, we seek to resolve the persistent ambiguities surrounding the symmetries reported in this material by constructing a unified crystallographic framework that connects the two phases. Our crystallographic analysis based exclusively on flux-grown crystals, unveils a direct transition from tetragonal (4/mmm) to monoclinic (2/m), with the latter emerging as a 2-fold superstructure of the former. This is further reinforced by ab initio density-functional theory (DFT) calculations which also unequivocally demonstrates the absence of any intermediate phase between the monoclinic and the tetragonal I4/mmm. The absence of an orthorhombic Bmab phase evidenced by minute deviations of the monoclinic angle from 90° and improved fit to the XRD data employing P2₁/c symmetry suggests that superconductivity emerges within the tetragonal symmetry under pressure, consistent with previous reports^20,21,24. In La₄Ni₃O₁₀, a characteristic feature of the tetragonal phase is the absence of tilting of the NiO₆ octahedra along c, unlike in the lower-symmetry phases. The linear Ni–O–Ni bond appears favorable for electron hopping and orbital overlap. Furthermore, tetragonal symmetry preserves the equivalence of the in-plane directions, thereby maintaining the symmetry of the $d_{x^2-y^2}$ electronic states believed to be important for superconductivity in the nickelates. Prior investigations employing ARPES have identified contributions from both the $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals⁴².

We further track the evolution of the structural transition temperature T_S under hydrostatic pressure, observing a reduction from 1030 K at ambient pressure to 20 K under hydrostatic pressures of 14 GPa. Finally, ambient-pressure XRD on flux-grown crystals reveals incommensurate satellite reflections associated with density-wave (DW) ordering below 130 K. This observation confirms the intrinsic nature of the modulation and demonstrates that it is not dependent on the crystal growth method, as previous reports were limited to floating-zone crystals²⁷. These results are consistent with Raman scattering, which shows additional phonon modes around 130 K, indicative of subtle lattice distortions associated with the DW ordering.

Results and discussion

High-temperature powder X-ray diffraction (PXRD) measurements under ambient pressure carried out at beamline BM01 of ESRF in Grenoble, reveal a structural transition from the tetragonal to the monoclinic phase at T_S ≈ 1030 K (See Figs. S1–S2 in the supporting information (SI)⁴³), in close agreement with prior reports^25,26. As in previous studies^27,39 the minute lattice distortion manifested as a small deviation of β from 90° by  ≈ 0.1–0.2° as shown in Fig. S1 in the SI⁴³ can only be understood by employing non-standard crystallographic settings.

Previously, both Nagell et al. and Song et al.^25,26 reported a monoclinic P2₁/a (alternative setting of standard P2₁/c which involves the permutation of the basis vectors a and c) phase with lattice parameters a ≈ b ≈ 5.4 Å and c ≈ 27 Å, treating it as a primitive cell, which effectively doubled the number of atoms and refinement parameters. In our work, we employ the non-standard B2₁/a setting, which uses the same lattice parameters, but the B-centering makes it equivalent to the reduced primitive P2₁/a cell with c ≈ 14 Å. Consequently, the number of atoms and independent refinement parameters remains unchanged compared to the reduced cell. This choice allows a cleaner and more reliable refinement while maintaining equivalence with previously reported structural parameters. In addition, the theoretical calculation by D. Puggioni, et al.⁴⁴ shows that the reported high-temperature phases of La₄Ni₃O₁₀, which have four formula units are unstable at low temperature and transform to a monoclinic P2₁/a phase that can be described using two formula units in agreement with the angle resolved photoemission spectroscopy⁴², which corroborates our interpretation.

Moreover, earlier works do not explicitly describe the transformations between standard monoclinic P2₁/c to non-standard B2₁/a and standard tetragonal I4/mmm to non-standard F4/mmm, thereby inadvertently obscuring symmetry relationships. Here as shown in Fig. 1a, b, we clarify these relationships by explicitly showing that the non-standard tetragonal F4/mmm is related to the standard I4/mmm through a transformation (matrix Q_Tetra) of the basis vectors of the centered cells according to:

$$\left(\begin{array}{c}\hfill {\mathbf{a}}_F\hfill \\ \hfill {\mathbf{b}}_F\hfill \\ \hfill {\mathbf{c}}_F\hfill \end{array}\right)=Q_{Tetra}^{-1,t}\left(\begin{array}{c}\hfill {\mathbf{a}}_I\hfill \\ \hfill {\mathbf{b}}_I\hfill \\ \hfill {\mathbf{c}}_I\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\mathbf{a}}_I\hfill \\ \hfill {\mathbf{b}}_I\hfill \\ \hfill {\mathbf{c}}_I\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\mathbf{a}}_I-{\mathbf{b}}_I\hfill \\ \hfill -{\mathbf{a}}_I-{\mathbf{b}}_I\hfill \\ \hfill -{\mathbf{c}}_I\hfill \end{array}\right)$$ 1

Here, a_F, b_F, c_F are the basis vectors of the F-centered cell, while a_I, b_I, c_I correspond to those of the I-centered cell.

Fig. 1. Crystal structures and phase evolution of La₄Ni₃O₁₀.

a High-pressure tetragonal phase (15 GPa, RT): the F4/mmm unit cell (thick dark lines) superimposed on standard I4/mmm (thin gray lines). b Monoclinic phase (6 GPa, RT): B2₁/a (thick dark lines) on P2₁/c (thin gray lines) to illustrate the transformation (see eqs. (1) and (2)).

Similarly, the non-standard monoclinic B2₁/a is related to standard P2₁/c by:

$$\left(\begin{array}{c}\hfill {\mathbf{a}}_B\hfill \\ \hfill {\mathbf{b}}_B\hfill \\ \hfill {\mathbf{c}}_B\hfill \end{array}\right)=Q_{Mono}^{-1,t}\left(\begin{array}{c}\hfill {\mathbf{a}}_P\hfill \\ \hfill {\mathbf{b}}_P\hfill \\ \hfill {\mathbf{c}}_P\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\mathbf{a}}_P\hfill \\ \hfill {\mathbf{b}}_P\hfill \\ \hfill {\mathbf{c}}_P\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{\mathbf{c}}_P\hfill \\ \hfill {\mathbf{b}}_P\hfill \\ \hfill 2\left({\mathbf{a}}_P\right)+{\mathbf{c}}_P\hfill \end{array}\right)$$ 2

Here, a_B, b_B, c_B correspond to the basis vectors of the B-centered cell, while a_P, b_P, c_P denote the basis vectors of the primitive cell. See supporting information (SI)⁴³ for details. We thus employ the use of the non-standard F4/mmm for the tetragonal and B2₁/a for the monoclinic phases to emphasize the importance of maintaining a consistent crystallographic framework in view of connecting both and to avoid the impression of a large monoclinic distortion that arises in the standard setting P2₁/c with β  ≈ 100°, thus enabling a more transparent and meaningful description of how the distortion evolves under temperature and pressure. To map the evolution of this structural transition by pressure, single-crystal X-ray diffraction (SXRD) under pressure was performed at the beamline ID15b of the ESRF. Application of hydrostatic pressure suppresses the transition, as elucidated by the phase diagram shown in Fig. 2, which reveals a pronounced reduction of the transition temperature to 20 K under an applied pressure of 14 GPa.

Fig. 2. Pressure–temperature phase diagram showing the tetragonal-to-monoclinic transition; dashed line indicates the phase boundary between 4/mmm and 2/m symmetries.

Bragg reflections at RT and 80 K illustrate the evolution from 4/mmm to 2/m, with broadening and anisotropy in the monoclinic phase. Complementary ab initio calculations predict P_c ≈ 16 GPa at 0 K, in agreement with experiments.

Establishing the critical pressure and temperature points for the phase diagram not only required rigorous scrutiny of reflections in the hk0 plane as shown in Fig. 3, but also tracking the appearance of the 2-fold superstructure and the evolution of the structural parameters as indicated in Figs. 4, 5 and 6. The reflections, nearly circular in the high-pressure tetragonal phase, become progressively broadened and anisotropic as the transition sets in, eventually splitting into distinct components at lower pressures. Purple dashed squares indicate the onset of the broadening associated with the transition towards monoclinic from tetragonal in Fig. 3. This bifurcation of the reflections becomes particularly pronounced at low temperatures 80 and 20 K, showing that the evolution of the distortion in the structure is significantly tuned by both pressure and temperature.

Fig. 3. Reconstructed reciprocal-space hk0 planes for the non-standard tetragonal F4/mmm and monoclinic B2₁/a settings (a ≈ 5.41 Å, b ≈ 5.43 Å, c ≈ 27 Å, β ≠ 90^∘), measured at 300, 80, and 20 K.

Upon reduction of pressure, broadening and splitting of the reflections reveal a 4/mmm → 2/m structural transition accompanied by pseudo-merohedral twinning. Purple dashed squares indicate the onset of the transition, where the reflections first become anisotropically broadened. Both the anisotropy and the bifurcation exhibit a pronounced temperature dependence.

Fig. 4. Selective sections of the diffraction pattern at three pressure points showing reflections associated with the superstructure and twinning.

a Excerpts of the 0kl plane; reflections ($0\overline{5}\overline{4}$) and ($0\overline{5}4$) (white circles) correspond to a two-fold superstructure. b $hk\overline{1}$ plane showing weak reflections below 11.3 GPa (yellow circles) associated with loss of four-fold symmetry; white rectangles denote new reflections that are allowed by the B-centering and sky-blue triangles mark domain-wall related scattering.

Fig. 5. Temperature-pressure dependence of the apical Ni-O-Ni bond angle.

a, b In the tetragonal phase the bond angle of Ni1-O2-Ni2 is 180° at 15 GPa (RT), while in the monoclinic phase at 6 GPa (RT) the bond angle of Ni1-O2-Ni2 is 171.1°. c Evolution of the octahedral tilt as a function of pressure at 300 and 80 K. Dashed lines are a guide to the eye. Estimated standard deviations (esds) of the bond angles were calculated from the structural refinement using Jana2006⁷⁷.

Fig. 6. Evolution of the lattice parameters under temperature and pressure.

In the tetragonal phase (4/mmm) (green-shaded region), the lattice parameters correspond to the non-standard F-centered setting of I4/mmm and for the monoclinic phase (2/m) (pink-shaded region) we use the B2₁/a non-standard space group in order to preserve the same setting of the lattice which allows for a comparison. Estimated standard deviations (esds) computed from the refinement are smaller than the symbols which can be understood (a–d) Lattice parameters, distortion and the volume within the pressure range of 0–16 GPa for temperatures at 300, 80 and 20 K from the numerical values listed in Table S5 in the SI ⁴³. Fits of a Birch-Murnaghan equation of state (EOS) to the volume-pressure data has resulted in V₀ = 825(1) ų and K₀ = 155(4) GPa at 300 K, V₀ = 818(1) ų and K₀ = 182(4) GPa at 80 K, and V₀ = 815(1) ų and K₀ = 185(2) GPa at 20 K. See the SI for details⁴³.

Furthermore, we show that the emergence of the monoclinic (2/m) phase in La₄Ni₃O₁₀ cannot be interpreted as a trivial distortion of the tetragonal lattice. As shown in Figure 4a, reflections such as $\left(0\overline{5}\overline{4}\right)$ and $\left(0\overline{5}4\right)$ (white circles) that appear below 11.3 GPa correspond to a 2-fold monoclinic superstructure. When transformed into the standard I-centered ‘pseudo-tetragonal’ lattice (a = b ≈ 3.7 Å, c ≈ 27 Å), these reflections can alternatively be indexed as $\left(2.52.5\overline{4}\right)$ and (2.5 2.5 4), corresponding to positions ${\mathbf{q}}^1=\left(\frac{1}{2},\frac{1}{2},0\right)$ and ${\mathbf{q}}^2=\left(\frac{1}{2},-\frac{1}{2},0\right)$ in this pseudo-tetragonal description. These vectors are not independent modulation wavevectors but rather reflect a different indexing convention for the same monoclinic superlattice reflections, highlighting the 2-fold nature of the monoclinic structure relative to the parent tetragonal lattice. Rather than invoking higher-dimensional (3 + d − dimensional) superspace formalism⁴⁵, where the intensities of the main Bragg reflections are kept distinct from the superlattice peaks, we reproduce the solution of the monoclinic structure in 3-dimensions by indexing and integrating as a superstructure, echoing earlier works^{20,21,24,26,27}. Figure 4b reveals the emergence of new reflections that deviate from the expected centering conditions at pressures below 11.3 GPa, consistent with the recent observations of Li et al.²⁴. Reflections such as $\left(\overline{3}2\overline{1}\right)$ and $\left(\overline{1}2\overline{1}\right)$ highlighted by white rectangles are fully compatible with the B-centering. In contrast, those marked by yellow circles like $\left(\overline{4}3\overline{1}\right)$, $\left(\overline{4}1\overline{1}\right)$, and $\left(\overline{2}3\overline{1}\right)$ originate from a twin domain rotated by 90° about the c-axis. Sky-blue triangles denote reflections arising from domain-wall scattering, which correlates with the pronounced strengthening of diffuse lines (Fig. 4a) in the monoclinic phase.

Figure 5a–c shows the deviation of the Ni1-O2-Ni2 bond angle from 180° that corresponds to the tilt of the NiO₆ octahedra along c upon entering the 2/m phase, thereby highlighting the effect of the phase transition on an atomic level. Figure 5 also illustrates that La₄Ni₃O₁₀ consists of three layers of Ni planes, with each Ni atom surrounded by six oxygen to form an octahedral site. The inner layer is sandwiched between two outer layers. The outer layers of different blocks are stacked with a rock salt layer of LaO. It is worth noting that the octahedral coordination is strongly distorted. Each Ni atom in the inner layer has two apical oxygen (i.e., along the c-axis in the pseudo-tetragonal description of the structure), which are close together, and four oxygen in the basal plane, which are further apart. The octahedra in the outer layers differ. Indeed, the apical O close to the rock salt layer (O4) and the four O in the basal plane (O3) form a square-based pyramid, and the second apical O (O2) shared with the inner layer Ni is far away. At 15 GPa, the octahedra is perfect. As it was already noticed by the previous work, at low pressure, the NiO₆ blocks are tilted from the pseudo c-axis.

Evolution of the lattice parameters are shown in Fig. 6a, b. The distortion toward monoclinic symmetry is minimal, particularly at 300 K where it is negligible. Much like the structural transition at ambient pressure outlined in the SI⁴³ in Fig. S1, it is also noted that in Fig. 6b there is a jump in the value of β at the onset of the 2/m phase, which could be attributed to a weakly first-order phase transition. The loss of symmetry from 4/mmm to 2/m generates four twin domains whose relation is explained by Fig. S3 in the SI⁴³. Essentially, these domains are described by distinct twin laws, and their presence is evident from reflection splitting in the diffraction data as shown in Fig. 3. The degree of overlap between the domains referred to as the ’twin obliquity’ is partial and indicates pseudo-merohedral twinning as observed from Fig. 3. Detailed information pertaining to the concept of twinning by phase transformation can be obtained from Parsons⁴⁶. The splitting appears to increase at 80 and 20 K, which contributes to the increase of the lattice distortion as defined by (b − a)/(b + a) at lower temperatures as depicted in Fig. 6c. Lattice parameters and Volume as displayed in 6d decreases monotonously with increasing pressure and the equation of state (EOS) fit depicting its compressibility is in good agreement with²⁰. Details of the EOS fit are shown in Fig. S4 in the SI⁴³. Recent work by Li et al.²⁴, based on measurements at the ID27 beamline of ESRF on flux-grown crystals, proposes the presence of an intermediate phase with Bmab symmetry under 11 GPa at room temperature. In contrast, our experiments were performed on flux-grown crystals synthesized using the same technique as Li et al. at beamline ID15b of ESRF, whose capabilities are analogous to ID27. While Li et al. conducted measurements only at room temperature, where β is nearly 90°, we carried out low-temperature XRD measurements under pressure. Under these conditions at temperatures of 80 and 20 K, we clearly observe a deviation of β from 90°, ruling out orthorhombic symmetry. At room temperature and around 10.5–11 GPa, where β is nearly 90° (see Fig. 6b), the structure may be interpreted as orthorhombic Bmab.

However, a full structural refinement at 10.5 GPa, near the 11 GPa pressure point reported by Li et al.²⁴, indicates that the monoclinic B2₁/a model, compared with the orthorhombic Bmab model, results in an improvement of R_F(obs) by approximately 1% and a substantial reduction in the residual electron density by about 50% (see Table S4 in the SI⁴³). This preference for the monoclinic phase is further reinforced by the Hamilton-R test^47,48, which demonstrates that the B2₁/a model provides a statistically significant better fit to the diffraction data, with a probability of less than 1% that this result arises by chance, confirming that the monoclinic structure more accurately represents the crystal under these conditions. Table S7 in the SI⁴³ also shows similar tests at 230 and 80 K under ambient pressure, where the B2₁/a model shows a significantly better fit to the SXRD data than obtained with Bmab symmetry, while the monoclinic lattice distortion is more pronounced. The minor difference between our refinement and that of Li et al.²⁴ arises from the choice of criterion of observability. While they adopt a more permissive 2σ threshold, we employ a 3σ cutoff, resulting in a reduced number of observed reflections in our refinement. This choice has no effect on the weighted R factors and does not lead to any significant change in the refined structural parameters within experimental uncertainty. It should also be noted that the less than ideal goodness-of-fit values (g.o.f.  ≈ 4, see Table S5 in the SI⁴³) at higher pressures may arise due to residual strain and de-twinning effects under high-pressure conditions as the crystal evolves toward the tetragonal phase. These results indicate that the transition from tetragonal (4/mmm) to monoclinic (2/m) proceeds directly without stabilizing an intermediate orthorhombic phase. Moreover, the coexisting metastable orthorhombic Bmab phase reported by Zhang et al.⁴⁰ on crystals grown by high-pressure floating zone technique appears to be absent in flux-grown crystals.

Tiny deviations of axial angles from 90° are not uncommon and have been reported in other systems, such as the CDW compound SrAl₄⁴⁹, incommensurately modulated Rb₂ZnCl₄⁵⁰, antiferromagnetic Eu₂Sb₃⁵¹, the incommensurate phase of ferroelectric KNbO₃⁵², and the superstructure phase of CoSn₂⁵³. The refined atomic coordinates are provided in Tables S1 and S2 the SI⁴³. Table S5 in the SI⁴³ summarizes the crystallographic information. The lattice parameters indicate that the monoclinic distortion is minute, rendering the lattice pseudo-tetragonal.

The subtlety of the direct tetragonal-to-monoclinic phase transition realized from XRD can also be understood from Fig. 7, where we present the results of an ab initio density-functional theory (DFT) crystal structure relaxation using the PBE (Perdew-Burke-Ernzerhof)⁵⁴ generalized-gradient approximation (GGA). The calculations at all pressures were performed within the least symmetric monoclinic spacegroup 14 in the P2₁/c setting, which is a subgroup of both orthorhombic Bmab and tetragonal I4/mmm, namely P2₁/c ⊂ Bmab ⊂ Fmmm ⊂ I4/mmm. Therefore, our crystal relaxation is free to span all these four structures and can capture all transitions between these phases. The figure illustrates the evolution with pressure of the three order parameters associated to these transitions, namely: the orthorhombic distortion factor o taken as the basal distortion o = (b − a)/(a + b), associated with the orthorhombic-tetragonal Fmmm → I4/mmm phase transition; the NiO octahedra tilt angle θ = 180 − bond angle of Ni1-O2-Ni2 (see Fig. 5c), associated with the Bmab → Fmmm phase transition; the ’monoclinicity’ (monoclinic distortion) β − 90 degrees, associated with the monoclinic-orthorhombic P2₁/c → Bmab phase transition. Here, β is the angle between the a and c vectors in the P2₁/c monoclinic cell. After applying the P2₁/c → B2₁/a transformation matrix, β is  ≈ 90°, compared to  ≈ 100° in the primitive P2₁/c setting, consistent with the experimental analysis.

Fig. 7. Density-functional theory GGA PBE crystal relaxation as a function of applied external pressure.

a Blue dots and lines: NiO octahedra tilt angle θ = 180 − Ni1-O2-Ni2 in degrees. b Black dots and lines: Monoclinic distortion β − 90 in degrees. c Red dots and lines: Basal distortion (b − a)/(a + b) scaled by 10³. Critical pressure P_c ≈ 16 GPa. Both horizontal and vertical error bars are smaller than the dot size when not visible. Dashed lines refer to fits, whereas dotted lines are a guide to the eyes. Green-shaded region marks the tetragonal phase, while the pink-region denotes the monoclinic phase.

Figure 7 unequivocally shows that, for ab initio DFT PBE, the system undergoes only one phase transition, from the monoclinic P2₁/c directly to the tetragonal I4/mmm. There is no room for intermediate structures: both the orthorhombic Bmab found by ref. ²⁴, and also a hypothetical orthorhombic Fmmm without octahedra distortions, are excluded. Indeed, all three order parameters nullify at the same critical pressure found to be P_c = 16.47 ± 0.04 GPa. The reported error reflects the spread between the three independent P_c values obtained from fits to each order parameter and does not include the error associated with the PBE approximation.

It is well known that the PBE approximation overestimates lattice parameters by at least 1%, and the error increases for example on the c parameter of layered hexagonal crystals. At T = 0 K and P = 0 GPa our DFT PBE relaxation provided a = 5.43 Å, b = 5.51 Å, and (in the B2₁/a setting) c = 27.89 Å with β = 90.42 deg, and finally θ = 17.25 deg. With respect to the experiment at T = 20 K, P = 0 GPa, we see that here the PBE overestimation is slightly below 1% on a, slightly above 1% on b, and remarkably only 0.3% on c. There is no common consensus on the PBE error on lattice angle parameters. Here we see that PBE and XRD at least agree on the order of magnitude. It can be expected that the systematic overestimation on lattice parameters leads also to overestimation of critical pressures. Estimating the error on pressure as the value here required to shrink the b parameter by 1%, yields ΔP = 3 GPa as the most pessimistic bound; keeping into account also the other lattice parameters leads to a more realistic estimate of ΔP = 1 GPa. Regardless, while the quantitative result on P_c might be affected by a large PBE error, the qualitative result on the non-existence of intermediate phases is much less questionable.

In Fig. 7 we show by dashed lines our best fits on the three last points closest to the critical pressure. We have used a standard scaling $~{\left(P_c-P\right)}^{\alpha }$ with α = 1/2 both for the octahedra tilt angle θ and for the monoclinic β angle, while a linear fit has been used for the basal distortion b − a/b+a. The square-root scaling is closely respected by the θ parameter down to P = 0. The θ scaling and magnitude seems in agreement with the experiment (see Fig. 5c at 80 K). On the other hand, we observe a quick breakdown of the scaling for the monoclinic β parameter. Letting the critical coefficient free in the fit yields α = 0.4 but without any improvement on the low pressure side. Remarkably, the monoclinic angle β presents a maximum at ~4 GPa signaling a departure from a simpler mean-field description. It would be interesting to check the existence of such maximum in XRD data, but the lowest level of monoclinicity β ~ 10⁻¹ deg, close to the experimental accuracy, will make it difficult. Finally, we also remark that the basal distortion o is linear down to 9 GPa, and then we observe a departure from linearity again in the region where the system presents its maximum monoclinicity. Overall, the theoretical results do not conform to a simple mean-field description of the phase transition.

On the other hand, our ab initio DFT PBE calculation of the crystal structure carried within the P2₁/c structure cannot state anything about the possibility of a density-wave ordering associated with either charge or spin. We could in principle check the possibility of a commensurate DW by carrying the calculation within the supercell at the corresponding superperiodicity, but this is impossible in the case of an incommensurate DW.

Despite our extensive temperature-dependent high-pressure SXRD measurements down to 20 K, no signatures of incommensurate modulation associated with density wave (DW) ordering were detected, contrary to what was observed by Zhang et al.²⁷ under ambient pressure. To eliminate the possibility that weak satellite reflections were obscured by background scattering from the diamond anvil cell or residual pressure effects, we conducted low-temperature SXRD experiments under ambient-pressure at P24 beamline DESY, Hamburg using the same batch of single-crystals. Although the main Bragg reflections were over-saturated, these measurements below 130 K likewise revealed no incommensurate satellites in any region of reciprocal space as shown in Figure S5 in the SI⁴³, thereby suggesting the weak nature of any incommensurate modulation. Crystallographic data for the data collected in DESY are provided in Tables S3 and S6 in the SI⁴³.

To conclusively verify their presence, we performed preliminary measurements at the ID28 beamline of ESRF at 140 and 80 K under ambient pressures, where the incommensurate satellites could be clearly observed, presumably due to the higher brilliance enabled by the multi-bend achromat lattice design of the fourth-generation synchrotron as shown in Fig. 8. The weak intensity of the satellites with average intensities of approximately 10⁻⁴ relative to the principal Bragg reflections indicates that the modulation is likely weak and the underlying three-dimensional lattice remains essentially undistorted, as commonly observed in other incommensurate systems such as Mo₂S₃, CuV₂S₄, EuAl₄, SrAl₄, Sm₂Ru₃Ge₅, and Gd₂Os₃Si₅^{49,50,55–59}.

Fig. 8. Reconstructed reciprocal lattice planes (1 k l) at temperatures of 140 and 80 K (B-centered setting).

a (1 k l) for X-ray diffraction data at 140 K showing the absence of satellite reflections. b (1 k l) for X-ray diffraction data at 80 K, showing satellite reflections with q ≅ 0.76b* + 1c*) as indicated by blue arrows.

The present work shows first observation of incommensurate satellite reflections in flux-grown crystals, whereas prior work by Zhang et al.²⁷ was performed on crystals grown using the floating-zone method. This demonstrates that the manifestation of the weak satellites is impervious to the crystal growth technique and can therefore be regarded as an intrinsic property of La₄Ni₃O₁₀. Notably, following Zhang et al.’s²⁷ work, subsequent investigations have not consistently reproduced the reported low-temperature single-crystal XRD results. The difficulty in detecting the satellites may reflect a combination of their weak intensity and experimental limitations. We now turn to the spectroscopic signatures, where the observations from X-ray diffraction are fully consistent with Raman spectroscopy. The DW-driven distortions is understood through the emergence of additional phonon modes, thereby confirming that the DW-driven lattice distortion is a robust intrinsic feature of the system.

From the space group P2₁/c (no. 14, standard setting), at room temperature, the Wyckoff positions are identified to be 8 4e and 1 2a. Thus, the number of phonon modes is expected to be: Γ = 24A_g+24B_g+27A_u+27B_u. The Raman-active phonons are 24 A_g and 24 B_g, which are active in parallel and crossed polarizations, respectively, when the Poynting vector is perpendicular to the (bc) plane (here we use the standard setting with b as unique axis). The Raman activity is reported in table 1. All these modes have single degeneracy (See also Figs. S8–S10 in the SI⁴³).

Table 1.

The Raman observable symmetries for a given scattering orientation of light with respect to the global symmetry of the lattice of La₄Ni₃O₁₀

Porto notation	P21/c
X(YY)$\overline{\mathrm{X}}$	24 Ag
X(YZ)$\overline{\mathrm{X}}$	24 Bg

X,Y and Z are taken in the standard setting of the space group P2₁/c. The Poynting vector is parallel to X, while the polarization Y and Z are in the (bc) plane but not aligned with a particular axis. Y is defined to be perpendicular to Z.

As shown Fig. 9a, b, 15 modes in parallel and 19 in crossed polarization configurations are identified by polarized Raman spectroscopy at 170 K (so above the transition temperature), consistent with the reported space group P2₁/c. New modes appear at temperatures between 120 and 150 K, as show Fig. 9a, b. At 26 K, a total of 31 modes in parallel configuration and 20 ones in crossed polarization configurations are measured. This evaluation of the number of modes appearing at low temperature has been done in a conservative way (see Fig. S6 in the SI⁴³), so these are minimum numbers of modes. When we adopt the spectral fitting approach (see the SI⁴³), a total of 38 modes in parallel and 23 ones in cross configurations of polarization of light are measured at 26 K.

Fig. 9. Temperature-dependence of Raman spectra showing the emergence of new Raman modes associated with the DW ordering.

a, b Spectra as a function of temperature in yy and yz configurations of polarizations of light. The modes that emerge across the transition at ~130 K are highlighted with diamond (yz) and stars (yy). Their fits at 26 K are shown by plain black lines. The normal-state modes that strongly change with temperature are shown with triangles. The green dot mode is a common-behavior phonon modes for comparison. Their behavior is reported in Fig. S11 of the SI⁴³. c Phonon energy as a function of temperatures for the new modes and the modes with peculiar behaviors. The gray region indicates the transition temperature range.

The new modes appearing below  ~ 120 K are interpreted as phonons, naturally originating from a transition, either a purely structural distortion or alternatively an electronic one which backfolds the phonon modes originally at finite $\vec{Q}$ into the Γ point of the Brillouin zone, reminiscent of the observation of stripe order in hole-doped nickelate compounds^60,61. Each mode as presented in Fig. 10 clearly follows a strict Raman selection rules, being either active in crossed or in parallel polarization configurations.

Fig. 10. Raman spectra of La₄Ni₃O₁₀ with polarizations in the (bc) plane (standard-setting P2₁/c) and in both yy and yz configurations, at 26 K.

Clearly identified phonons are reported as ticks (red : yz, black: yy). The longer and thicker ticks identify the modes emerging below 120–140 K.

The number of modes in parallel polarization as well as the total number of modes exceeds the authorized ones in the P2₁/c room temperature space group. Thus we can conclude to a lowering of symmetry below ~120 K. Recent results by Gim et al.⁶² presented the phonon modes up to 120 meV (= 970 cm⁻¹). They do not report any new mode at low temperature whereas we observe many new ones in this energy range. Other recent publications^63,64 reported the appearance of new modes at low temperature. Interestingly, Suthar et al. observed a total of 48 modes at low temperature (24 in each polarization configurations), so it is still consistent with the room temperature space group. Our results, while being quite similar in many aspects (behavior of peculiar phonons, energies of phonon modes as described in the SI⁴³), show more than 48 modes at low temperature, thus ruling out the P2₁/c as the low temperature space group. In Table S8 in the SI⁴³, we report a full list of our detected phonon modes. Generally, the samples’ quality and exact composition seems to have important impact on the Raman responses, pointing to this characterization technique as a sensible one.

From our results in Fig. 9c, the transition temperature to this lower symmetry space group is at  ~ 130 K. Besides, certain modes detected at room temperature (large colored symbols in Fig. 9a) show anomalous temperature dependence upon cooling, also in correlation with the emergence of these new modes (see Fig. S11 in SI⁴³). We naturally interpret this transition temperature as the onset of the Density Wave, which is reported to be at about 135 K by Zhang et al.²⁷.

To conclude, we report that the monoclinic phase consistently manifests as a twinned two-fold superstructure below 1030 K which is easily observed by XRD. The monoclinic distortion is exceedingly small, thereby maintaining an overall pseudo-tetragonal crystallographic framework. Pressurizing the material from ambient conditions to 14 GPa leads to a gradual suppression of the distortion over a broad temperature range from 1030 to 20 K, reflecting its delicate energetic balance. Guided by ab initio calculations and pressure-temperature dependent XRD measurements, we show that the tetragonal-to-monoclinic phase transition is subtle and direct, without an intermediate orthorhombic Bmab phase. By adopting a consistent crystallographic description, we are able to elucidate how the symmetries of the two phases evolve and are interconnected, thereby resolving the long-standing ambiguities among the tetragonal, orthorhombic and monoclinic structures. Crucially from XRD, we present the first observation of incommensurate satellite reflections in flux-grown crystals associated with DW ordering, thereby establishing that the associated lattice modulation is an intrinsic property of the material and not contingent on the method of synthesis. The results corroborate the Raman-active modes indicative of DW-driven lattice distortions. Determination of the symmetry of the incommensurately modulated phase requires a comprehensive investigation employing the superspace approach to disentangle the atomic modulation and uncover the degrees of freedom underlying the density-wave ordering.

Methods

Crystal growth

There are two methods to synthesize single crystals of La₄Ni₃O₁₀. One is using the floating zone technique with an image furnace under oxygen pressure²⁷ and the other one is by the flux method⁶⁵. We used the latter technique resulting in flattened cuboid black crystals of La₄Ni₃O₁₀, with long edges of approximately 50–100 μm. La₂O₃ and Ni were ground together in stoichiometric ratio, then thoroughly mixed with K₂CO₃ flux in the molar ratio Ni:K₂CO₃ 1:35. The mixture was placed in an alumina crucible with a lid to prevent K₂CO₃ evaporation, then heated in a box furnace at 1050 °C for 48 h, cooled slowly to 975 °C at 0.4 °C/h, and cooled with the furnace inertia to room temperature. The remaining flux was then washed away with distilled water. Powder X-ray diffraction data were collected with a Bruker D8-Endeavor with Cu Kα radiation. The pattern of the pulverized reaction product shows La₄Ni₃O₁₀ as the majority phase, along with NiO and La₂NiO₄ as side products of the reaction.

Scanning electron microscope and electron probe micro-analysis

Morphology of grains were investigated with a FESEM Zeiss Ultra+ scanning electron microscope. Energy dispersive X-ray analysis on selected crystals was carried out using a Bruker SDD detector and Spirit software. EPMA analysis were performed on a Jeol - 8800A electron microprobe equipped with five wavelength-dispersive spectrometers. Three of them were used to determine the chemical composition of grains: PETJ for Lanthanum (Lα), LiFH for Nickel (Kα), and LDE1H (W/Si) for oxygen (Kα). Analysis were made at 15 keV accelerating voltage, with a 30 nA probe current with 2 μm beam diameter. Similar conditions (counting times, current, samples and standards together in the chamber) were respected. KTiOPO₄ for Oxygen 40.4%, LaB₆ for Lanthanum and pure Nickel were used as standards. Concentrations were calculated using a Φ(ρz) procedure. Both techniques confirm the La:Ni 4:3 atomic ratio.

Temperature dependent powder XRD

High temperature synchrotron powder XRD data were measured at BM01 (Swiss-Norwegian Beamlines, SNBL) at ESRF in Grenoble, France, at a wavelength of λ = 0.71913 Å and using a 2D PILATUS 2M detector⁶⁶. The carefully ground powder sample was filled into a 0.1 mm diameter quartz capillary and heated with a resistive capillary heater⁶⁷. Data were recorded every 2 K from 290 to 1223 K in heating and cooling and Rietveld refinements were performed with the Fullprof software⁶⁸.

Temperature and pressure dependent single crystal XRD

High-pressure experiments were performed at ID15b⁶⁹ ESRF Grenoble France, using membrane-driven diamond-anvil cells (DACs) with 500 μm diamond culets employing a radiation of a wavelength of λ = 0.4099 Å. Diffracted intensity was collected on a EIGER 9M area detector during continuous rotation of the crystal about the ω axis, in frames of 0.5 deg rotation and 1 s exposure time. Stainless steel gasket were used for the pressure chambers. The pressure transmitting medium used was Helium, loaded at 1.2 kbar, to ensure high hydrostatic pressure conditions up to the highest pressure reached in this study. The pressure was measured using the R1-line emission of a ruby ball placed close to the sample using the International Practical Pressure Scale IPPS-Ruby2020 equation of state⁷⁰. The ruby signal is measured before and after each measurement in order to control the pressure drift during acquisitions. The recorded pressure is set at the average of these two pressure values and the uncertainty is set as the half of the difference between these two values. It is typically found smaller than the symbol size used for the figures in this paper. The homogeneity of the pressure in the DAC was followed from both the width and the splitting between the R1 and R2 ruby lines^71,72.

Under ambient pressure temperature-dependent single-crystal x-ray diffraction (SXRD) was measured at station EH1 of beamline P24 of PETRA-III extension at DESY in Hamburg, Germany, employing radiation of a wavelength of λ = 0.5002 Å. Diffracted X rays were detected by a Pilatus 1M CdTe detector. The temperature of the sample was regulated with a CRYOCOOL open-flow cryostat, employing helium as cryogenic gas. Crystal of 50 μm was selected for the SXRD experiment at beamline P24. Diffracted intensity was collected on the detector during continuous rotation of the crystal about the ϕ axis, in frames of 0.1 deg rotation and 0.1 s exposure time. Each run of data collection comprises a 10 times repeated measurement of 3600 frames, corresponding to a total rotation of the crystal by 360 deg, repeated 10 times. These data were binned to 360 frames of 1 deg of rotation and 10 seconds exposure time employing the software addrunscbf and combcbf⁷³. Due to the anisotropic shape of the reflection the EVAL15 suite⁷⁴ could not be used successfully. Instead, data processing has been done with the EVAL14⁷⁵ to integrate the data. In EVAL14 the concept of ray tracing is not used, as EVAL14 defines a box, where inside box there is a reflection and outside the box no reflection. The border of the box is used for evaluating the background level. Integration now is simply adding up intensities of pixels inside the box. SADABS⁷⁶ has been used for the absorption correction for all the data sets. JANA2006⁷⁷ was used for the structure refinements at 230 and 80 K. Preliminary single crystal XRD measurements were also performed on ID28 in ESRF using the Pilatus 1M area Si-detector without any attenuation of the beam at 140 K and 80 K close to the wavelengths of DESY, where incommensurate satellites are observed at 80 K.

Raman scattering

Raman measurements were performed with a 532 nm solid-state laser with an incident laser power between 0.5–2 mW. A Trivista 777 spectrometer equipped with ultra-low noise, cryogenically-cooled PyLon CCDs was used in single-stage configurations for which we have access to Raman signal down to 70 cm⁻¹. A cryo-free cryostat⁷⁸ was used to perform measurements down to 2 K (26 K with laser heating included).

Ab initio density-functional theory calculation

The density-functional theory (DFT) crystal relaxation has been performed using the ABINIT code in the the PBE⁵⁴ generalized-gradient approximation (GGA) with pseudopotentials from the PsuedoDojo table. As explained in the main text, the calculation has been carried out in the lowest symmetry monoclinic P2₁/c elementary and largest cell able to encompass all crystal structures up to the I4/mmm. Within this cell the total energy was found converged within 1 mHa using a plane-waves cutoff of 100 Ryd and a Brillouin zone k-point sampling of 2 × 4 × 4 shifted by 1/2 × 1/2 × 1/2 (as explained in the crystallographic part, within the P2₁/c setting the Ruddlesden-Popper (RP) stacking axis is oriented along a and is doubled). The relaxation was stopped when the forces and stresses reduced below a threshold of only 5 × 10⁻⁶ Ha/Bohr: this was an absolutely critical parameter of the calculation to get the right phase diagram. With just only one order of magnitude more, 5 × 10⁻⁵ Ha/Bohr, the relaxation could have stopped in the tetragonal (or even in one of the orthorhombic) at low pressure, when the minimum of the total energy was on the P2₁/c, and vice-versa at highest pressure, depending on the starting crystal structure. It was not even a question of relative minima: just only tiny forces/stresses between energetically very close structures. Finally, the errors on lattice parameters have been estimated by calculating the stiffness tensor from the data out of the last four relaxation steps, and using its inverse, the compliance tensor, applied to the residual stress to evaluate the residual strain which is then taken as the lattice error estimate. We did the same to estimate the error on the octahedra tilt angle by relying on elastic constants associated to the internal atomic positions.

Author contributions

E.P. and A.H.-A. synthesized the single crystals. V.O. carried out the DFT calculations. X-ray diffraction experiments were performed by S.R., E.P., G.G., M.D., O.P., A.P., D.V., D.C., L.N., C.P., J.B., A.B., S.vS., P.T., and P.R. EPMA was conducted by M.Q. and S.P. SXRD data analysis was carried out by S.R., while PXRD analysis was done by E.P. Raman measurements and analysis were done by Y.G. and M.A-M. The manuscript was written by S.R., Y.G., V.O., E.P., M.A-M. and P.R. with inputs from all authors. The project was initiated by P.T. and P.R. and the work was supervised by P.R. Fundings were acquired by A.P., P.T., M.A-M., and P.R.

Peer review

Peer review information

Communication Materials thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Funding

We thank the support from the Agence Nationale de la Recherche under the project SUPERNICKEL (Grant No.ANR-21-CE30-0041-04). M.-A.M. and Y. G. thank the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 865826).

Data availability

Data is available upon reasonable request from the authors.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s43246-026-01297-5.