Strain–Phonon Cooperation as a Necessary Ingredient to Understand the Jahn–Teller Effect in Solids

Spatial degeneracy is the cause of the complex electronic, geometrical, and magnetic structures found in a number of materials whose more representative example is KCuF3. In the literature the properties of this lattice are usually explained through the Kugel–-Khomskii model, based on superexchange interactions. Here we provide rigorous theoretical and computational arguments against this view proving that structural and magnetic properties essentially arise from electron–vibration (vibronic) interactions. Moreover, based on the work of Öpik and Pryce, we show that the coupling between lattice (homogeneous strain) and motif (phonons) distortions is essential to understand the main stable configurations of the lattice. Using this information, we predict a new low-energy phase in KCuF3 that could strongly alter its properties and provide guidance on how to stabilize it through strain engineering.


June 12, 2024 1 Computational details
The many-body model of the cooperative Jahn-Teller effect in solids with locally degenerate centers proposed in this letter is supported by first principles methods in the framework of Density Functional Theory (DFT).Assuming periodic boundary conditions, the calculations are carried our using the VASP and CRYSTAL17 codes.The Vienna Ab initio Simulation Package (VASP) 1,2 utilizes a plane wave approach to represent the Bloch orbitals.To explore the interplay of electron and vibrations and their influence on the magnetic characteristics of the system, it becomes imperative to address electron correlation beyond the conventional scope of DFT.Liechtenstein's LDA+U implementation (with U=7.5 eV and J=0.9 eV) 3 and the highly standardized hybrid functional HSE06 4 with a 25% of Hartree-Fock (HF) exchange, both based on the Generalized Gradient Approximation (GGA) approach of Perdew-Burke-Ernzerhof (PBE).The results are also compared with other hybrid functionals as PBE0. 5 This approach enhances the description of these systems as insulators by correcting the intrinsic self-interaction error of DFT.To encompass the full spectrum of geometry distortions and magnetic configurations, we conducted simulations of the cubic Pm-3m and P4/mmm phases of KCuF3 within a tetragonal √ 2 × √ 2 × 2 supercell, aligning with the configuration employed to characterize the extensively distorted ground state exhibiting I4/mcm symmetry.
The convergence threshold for the electronic self-consistency loop was set at 1 × 10 −6 eV, with atomic positions undergoing relaxation via the conjugate gradient algorithm until forces per atom reached below 0.03 eV/ Å.Initial orbital occupancies were obtained by a Gaussian smearing with the default value σ = 0.05.The Brillouin zone of the reciprocal space was sampled with a 2 × 2 × 2 mesh, centered at the Γ point.Even though it may seem not dense enough, it strikes a balance between accuracy and computational cost for the highly demanding hybrid-functional calculations.Additionally, LDA+U calculations were replicated using a much finer mesh of 8 × 8 × 6, and the results were entirely equivalent.The representation of valence electrons involved utilizing a plane-wave basis set with an energy cutoff of 520 eV, whereas core electrons were characterized using the Projector Augmented Wave method (PAW) 6 in conjunction with pseudopotentials. 7Specifically, the PAW potentials incorporated 17 and 9 valence electrons for Cu and K cations respectively, 7 for F and Cl, and 6 for O.The 3D representation of the spin density represented in Figure 3 for both stable phases of KCuF 3 , obtained from VASP, was extracted from the CHGCAR file utilizing the Python tool vaspkit, 8 and then visualized using the graphical software vesta. 9 For comparison purposes, all calculations were executed concurrently utilizing the CRYS-TAL17 package. 10In this software, crystalline orbitals are expanded via a linear combination of Bloch functions, articulated in terms of local functions, which are, in turn, Gaussian type functions.These local functions, available in CRYSTAL website, 11 are defined employing high-quality triple-ζ polarized basis sets, which were developed by Peintinger et al. 12 The hybrid functionals B1WC 13 (including 16% of exact HF exchange), PW1PW 14 (20% HF exchange), HSE06 4 (25% HF exchange) and PBE0 15 (25% HF exchange) were used.To integrate over the first Brillouin zone, an 8 × 8 × 8 grid was utilized.TOLINTEG parameters for real space integrals of the electronic density were set to 9, 9, 9, 9, and 18.The convergence criterion for the energy was established at 10 −8 Hartree.

Definition of distortion modes
The definition of the distortions applied in the KCuF 3 lattice are represented in Figure S1.For the tetragonal strain mode η Γθ , the distortion is performed by changing the lattice parameters by a quantity η, that allows changing all local octahedra in the cubic perovskite as dictated by the normalized coordinate Q θ of a local Jahn-Teller tetragonal e g mode 16 (see Figure S2).That is, this local Q θ mode in a molecule transforms into the strained η Γθ

S3
mode in the Γ point for a solid.The atomic fractional coordinates remain fixed during all the process as corresponds to the application of homogenous strain.On the other hand, for the phonon modes, the lattice parameters remain fixed, while the atomic positions change following the normalized coordinates of a Jahn-Teller tetragonal/orthorhombic e g mode for the Q Rθ /Q Rε modes respectively.As it is a phonon in R, both Q Rθ and Q Rε components of the e g phonon alternates Q θ /Q ε and -Q θ /-Q ε local coordinates for neighboring active centers (see Figure S1).This scheme allows characterizing the strain/phonon distortions using a unique criteria (the local JT coordinate) that has been followed to represent Fig. 2 in the main text of the manuscript.S1 and S2.These results correspond to geometry optimizations of the different phases represented in Figure 2. The energy of each phase is given with respect to the absolute minimum.For both codes, the calculations have been carried out taken symmetry intro account.

Figure S1 :
Figure S1: Represetation of the strain (green) and phonon (red) distortions applied in the KCuF 3 lattice.Both of them are based on the local distortions of the original Jahn-Teller effect introduced by Bersuker.16

Figure S2 :
Figure S2: Illustration of the two degenerate components of an e g mode: Q θ (tetragonal) and Q ε (orthorhombic) in an octahedral complex.Quantity a represents the absolute displacement of a single atom.The description of the normalized coordinate of each mode is described at the bottom of the figure.Notation from Bersuker's Jahn-Teller Theory 16