Structurally triggered metal-insulator transition in rare-earth nickelates

Rare-earth nickelates form an intriguing series of correlated perovskite oxides. Apart from LaNiO3, they exhibit on cooling a sharp metal-insulator electronic phase transition, a concurrent structural phase transition, and a magnetic phase transition toward an unusual antiferromagnetic spin order. Appealing for various applications, full exploitation of these compounds is still hampered by the lack of global understanding of the interplay between their electronic, structural, and magnetic properties. Here we show from first-principles calculations that the metal-insulator transition of nickelates arises from the softening of an oxygen-breathing distortion, structurally triggered by oxygen-octahedra rotation motions. The origin of such a rare triggered mechanism is traced back in their electronic and magnetic properties, providing a united picture. We further develop a Landau model accounting for the metal-insulator transition evolution in terms of the rare-earth cations and rationalizing how to tune this transition by acting on oxygen rotation motions.


SUPPLEMENTARY NOTE 1. VALIDATION OF THE DFT+U APPROACH
In order to assess the validity of our DFT+U approach and determine the appropriate U parameter, we have considered a wide range of possible values for U (from 0 to 8 eV) and have compared the computed structural, magnetic and electronic properties to experimental data.
The results are summarized below for YNiO 3 considered as a test case. In line with what was reported independently in Ref. [1], it appears that a DFT approach with a moderate U value of 1.5 eV provides for nickelates an unprecedented agreement with experimental data, combining accurate description not only of the structural but also of the magnetic and electronic properties. It therefore oers a robust and ideal framework for the study of the interplay between these properties.

SUPPLEMENTARY NOTE 2. ATOMIC STRUCTURE
In Supplementary Figure 1, we report the relative deviations respect to experimental data at low temperature [2] for the lattice parameters and atomic distortions in the E'-type AFM P 2 1 /n phase of YNiO 3 in terms of the amplitude of the U parameter. The atomic distortions are those with respect to the P m3m phase and are quantied from a symmetryadapted mode analysis performed with AMPLIMODE [3,4]. The labels of the modes that are allowed by symmetry in the P bnm and P 2 1 /n phases and a brief description of the related atomic motions are reported in Supplementary Table 1. In Supplementary Figure 2, we report comparison with experiment data [2] of the absolute amplitudes of the atomic distortions and lattice parameters in the E'-type AFM P 2 1 /n phase of YNiO 3 as computed in DFT with U = 1.5 eV. It conrms that the atomic structure of YNiO 3 is very accurately described in DFT using PBESol and a U parameter of 1.5 eV. Relative deviations respect to experimental data at low temperature [2] for the lattice parameters (a) and atomic distortions (b) in the E'-type AFM P 2 1 /n phase of YNiO    In our calculations, we get a magnetic moment µ = 1.2µ B on the Ni atoms associated to the large oxygen octahedra and µ ≈ 0µ B on the Ni atoms associated to the small octahedra. This is similar with what has been reported in Ref. [1,5] and in line with the d 8 − d 8 L 2 picture [6]. It is also compatible with experimental data as discussed in Ref. [7].
Beyond the fact that DFT calculations with U = 1.5 eV provides the right magnetic ground state, it is interesting to check if it properly accounts for the strength of the magnetic interactions. To that end, we built a simple spin model interactions up to fourth neighbours (6 independent parameters) and tted the parameters on our rst-principles data [8]. As illustrated in Supplementary Figure 4 for YNiO 3 this spin model properly reproduces the energetics of the rst-principles calculations.
Monte-Carlo simulations (using large boxes up to 1728 Ni atoms) from this spin-model [8] (i) conrmed the E'-type ground state and (ii) provided a Neel temperature T N = 154 K, very similar to the mean-eld estimate of 166 K and in close agreement with the experimental value of 150 K for YNiO 3 [9].
This demonstrates that our DFT calculations with U = 1.5 eV reproduces not only the correct E'-type magnetic ground state of nickelates but also properly describes the strength and anisotropy of their magnetic interactions.

SUPPLEMENTARY NOTE 4. ELECTRONIC PROPERTIES
Our DFT calculations with U = 1.5 eV properly accounts for the insulating character of the E'-type AFM P 2 1 /n ground state of YNiO 3 . For the electronic bandgap, we get a value of 0.46 eV in reasonable agreement with the experimental estimate of 0.305 eV [10].
The electronic properties are further discussed in the manuscript. As it appears clearer there, the structural and electronic properties are intimately linked together in nickelates.
Hence, the fact that our simulations describe accurately the structural properties of these compounds strongly suggests that they can also be trusted to investigate their electronic properties.

SUPPLEMENTARY NOTE 5. PHONON DISPERSION CURVES
In Supplementary Figure 5, we report the full phonon dispersion curves of the P m3m phase of YNiO 3 , as calculated for a FM spin ordering at the volume of the P 2 1 /n AFM-E' phase (a pc = 3.728Å). Similar curves have been obtained at the relaxed volume (a 0 = 3.695Å). Interpolation of these phonon dispersion curves relies on the calculation of the interatomic force constants within a 2 × 2 × 2 supercell. Although this might not be totally sucient to get a fully converged interpolation, it provides already a good estimate of the shape of the dispersion curves. It is worth to notice that the frequencies at the highsymmetry points, which are the only ones discussed below and in the manuscript, are not interpolated but calculated explicitly within our approach.
On the one hand, the phonon dispersion curves highlight strong instabilities at the R The parameters α R and α M are assumed to be temperature dependent as while all the other parameters are supposed to be constant.
Other modes allowed by symmetry (see Supplementary Table 1)  The expansion has been limited to 4 th order for all three order parameters including Q B . This is justied by the fact that, from the t of the parameters, the triggered transition appears to be second order . We notice however that explicit treatment of the strain (neglected here) could aect the order of the phase transition as further discussed below.

SUPPLEMENTARY NOTE 8. FIT FROM DFT
Parameters of our Landau-type model have been tted on rst-principles results. At rst, we focused on YNiO 3 .
We considered in our calculations a xed cubic P m3m cell at a volume similar to that of the P 2 1 /n AFM-E' ground-state (a pc = 3.728Å), which corresponds to imposing a negative The calculations have been performed with a FM spin order which does not break any symmetry. We checked explicitly that the key physical features and conclusions (cooperative bi-quadratic coupling between rotations and breathing and triggered mechanism) remain similar for dierent AFM spin orders. The results remain even very similar in a non-magnetic (NM) calculation (with or without U correction) although, in that case, the amplitude of rotations required to destabilise B OC is slightly larger (≈ 160%); this last result illustrates that electronic Hund's rule energy, although playing a role, is not driving alone the appearance of B OC as sometimes suggested [11].
The parameters of the Landau model at 0 K have then been extracted from DFT data as follows.
• λ RM was tted to reproduce the energy of a relaxed P bnm-like phase (full atomic relaxation while keeping the cubic cell xed). From this, we renormalize the natural competition between R xy and M z by including implicitly the stabilising eect of X −

,
R − 4 and M + 3 modes. We notice that in all compounds, R xy and M z compete with each other and should yield λ RM > 0. However, because of the renormalization due to the implicit presence of the other modes, λ RM becomes negative for large cations (i.e. X − 5 helps stabilizing the P bnm phase consistently with the discussion in Ref. [12]. • α B was tted on the single well associated to B OC ( Figure 1).
• λ BR and λ BM were tted from the change of curvature of the well of B OC when freezing 100% of Q R and Q M respectively ( Figure 2).
• β B was tted to reproduce the right amplitude of B OC in the ground state of the model and it was checked that the result still properly describes the single well associated to B OC .
Within the model, the amplitude for the atomic distortion are renormalised to the one obtained from DFT calculation for the YNiO 3 ground state. This means 1 for rotation, tilts and breathing mode correspond to the amplitude of these modes in a cubic box with lattice parameters coresponding to 3.728 Å.
We applied the same procedure to GdNiO 3 and SmNiO 3 . All the computed parameters are summarized in Table II.
As illustrated in Supplementary Figure 7, all the parameters have an almost linear dependence in terms of the tolerance factor t. So, in our model, we assumed such a linear dependence to determine the value of the parameters at arbitrary t.
At the energy minimum, we should have : The solutions for that, other than Q M = 0 and Q R = 0, are : Introducing this in Supplementary Eq. (4) we get : where The MIT is linked to the appearance of the B OC . This will appear at a temperature T MI at which α B = 0. This critical temperature is given by : Furthermore, supposing a linear dependence for all the coecients with respect to the tolerance factor, we get a generic expression : where t is the tolerance factor and a, b, c, d, e, f and g are a combination of model parameters.
Using the coecients determined from DFT calculations in the previous Section, we can predict the evolution of T MI as a function of the tolerance factor as illusrated in Figure 2b, blue line.
Independently, we can also t the experimental data point using Supplementary Eq. 13.
Making such a t, while excluding Nd and Pr compounds, we get the dashed blue line in (T MI = T N ), the MIT is rather abrupt and hysteretic and unanimously considered as being rst order [13]. The magnetic transition that takes place at the same temperature is also rst order [14]. For small cations (T MI > T N ), the MIT is less hysteretic and sometimes considered as evolving to second-order. Some studies seem however to show that it stays rst-order [15,16], while the less hysteretic behavior could be related to the fact that kinetics are better at higher temperatures [13]. For these compounds the magnetic transition is second-order.
As previously mentioned, the MIT is predicted to be second-order within our very simple model. As highlighted in Supplementary Table 2 and SmNiO 3 respectively) corresponding therefore to a second-order transition.
Yet, we have to stress that our approach does not allow us to address the order of the transition conclusively. First, our model is built at xed cubic cell and does not include strain relaxation. Explicit treatment of the latter will further renormalize the 4 th -order term and might potentially make it negative, so eventually changing the order of the transition. Second, at a more fundamental level, even if our DFT+U results suggest that Q B undergoes a second-order transition, that does not rule out the possibility that thermal eects eectively render a rst-order transformation driven by temperature. The ferroelectric phase transitions of BaTiO 3 , a well studied case, are a concrete example of this [17], and also illustrate the critical role of strains to enhance the discontinuous character of the transformation [18]. Hence, discussing the character of the transition from rst-principles would require explicitly statistical simulations that fall beyond of the scope of this work.  [14]: they suggest indeed that, contrary to other cases, there is a possible interplay between electronic and magnetic degrees of freedom when T MI = T N and that any further model of the TMI should address that fact. Our manuscript explicitly addresses that point and we believe that it convincingly answers their questioning.

SUPPLEMENTARY NOTE 11. ELECTRONIC BAND STRUCTURES
In Supplementary Figure 8, we report the electronic dispersion curves of YNiO 3 with a FM spin order, along a more exhaustive path of the Brillouin zone of the P bnm or P 2 1 /n 20-atom cell. The majority spins are in colors while the minority spins are in light grey. The latter have been omitted for clarity in the main manuscript. We notice that the cubic phase is essentially non magnetic (up and down spin bands nearly degenerate) and magnetism starts to develop with the rotations.
In Supplementary Figure 9, we report similarly the electronic dispersion curves of YNiO 3 but with an AFM-A spin order. This gure is very similar to the previous one, demonstrating that our results are not dependent of the specic choice of spin order. a. b. c. d. e.
f. The graph connects high-symmetry points in the Brillouin zone of the P bnm or P 2 1 /n 20-atom cell.