Lattice dynamics of KAgF3 perovskite, unique 1D antiferromagnet

Theoretical DFT calculations using GGA+U and HSE06 frameworks enabled vibrational mode assignment and partial (atomic) phonon DOS determination in KAgF3 perovskite, a low-dimensional magnetic fluoroargentate(II). Twelve bands in the spectra of KAgF3 were assigned to either IR active or Raman active modes, reaching very good correlation with experimental values (R2>0.997). Low-temperature Raman measurements indicate that the intriguing spin-Peierls-like phase transition at 230 K is an order-disorder transition and it does not strongly impact the vibrational structure of the material.


Introduction
Fluoroargentates(II) attract interest because of their noticeable similarities with isoelectronic copper(II) oxides; many interesting physical phenomena occur in the latter. One of them is low dimensional magnetism, which plays an important role in superconducting materials. KAgF3, which adopts a distorted perovskite structure [1], is an example of such low-dimensional magnetic system. Magnetic susceptibility measurements indicate substantial magnetic anisotropy in this compound, with strong antiferromagnetic ordering along the crystallographic b axis, and weak ferromagnetic one in the ac plane (Pnma setting). KAgF3 can be regarded as quasi-1D antiferromagnet in which superexchange constant along kinked Ag-F-Ag chains, J1D, is about 20 times larger than in the remaining two dimensions [2,3]. Importantly, standard Generalized Gradient Approximation (GGA) approach fails to reproduce antiferromagnetic ground state of KAgF3. Due to considerable correlation of silver valence electrons on d orbitals it is necessary to introduce Mott-Hubbard correction within the GGA+U framework [2]. The GGA+U approach permits to reproduce experimental J values for both strong intra-chain and much weaker intra-sheet interactions [3,4]. The unique character of KAgF3 stems from the fact that the intrachain antiferromagnetic interactions are immensely strong, of the order of 100 meV, thus comparable to those for copper(II) oxides, as well as from substantial magnetic anisotropy.
Interestingly, KAgF3 exhibits an intriguing phase transition at 230 K which resembles a spin-Peierls transition; it has been preliminarily assigned to structural order/disorder-type (Pnma/Pcma) transition based on X-ray diffraction studies [2]. Moreover, this compound shows a complex magnetic ordering at temperatures below 66 K [4] which is a subject to ongoing investigation. Analysis of X-ray diffraction experiments suggests that phase transition present at ca. 230 K might be of order/disorder type. The low temperature ordered polymorph crystallizes in Pnma space group while the high-temperature disordered one in the Pcma one, exhibiting tilting of [AgF4] 2-units around the a and c vectors [2]. Most characteristic structural features (tilting of octahedra, Jahn-Teller effect), as well as magnetic features (antiferromagnetic 1D character) are retained during this transition.
Lattice dynamics studies of related AAgF3 systems have been scarce despite the fact that they host interesting physical phenomena. To this date there are no spectroscopic data published on KAgF3. The goal of the current work is to gain insight into lattice dynamics (phonons) of KAgF3 with both experimental and theoretical tools, assign the bands appearing in the Raman scattering and infrared absorption spectra, as well as further elucidate the nature of the 230 K phase transition. Understanding the impact of disorder on the lattice dynamics is also of interest here.

Experimental
KAgF3 has been prepared using published synthesis route yielding high purity product (equimolar mixture of KF and AgF2 was heated in nickel reactor, under F2 atmosphere, for four days at 300 °C) [2]. Infra-red measurements were carried out on Bruker Vertex 80V vacuum spectrometer using powdered samples placed on HDPE windows. Raman spectra were obtained on Horiba Jobin Yvon LabRam-HR Raman micro-spectrometer with 632.8 nm He-Ne laser exciting beam. As with other Ag II compounds, used laser beam had to be of very low power, in order to minimize thermal (or photo-) decomposition [5,6]. In this work we have used power of circa 0.3 mW. KAgF3 can also easily decompose in contact with the atmosphere, so the sample was enclosed inside a sealed quartz capillary. In case of lowtemperature Raman measurements, the capillary was placed in a home-made flow cryostat, described in more detail below, and measured using Horiba Jobin Yvon T64000 Raman spectrometer with 514.5 nm Ar-Kr laser exciting beam, with on-sample power lower than 0.5 mW.
Low-temperature setup was prepared in the following way: 5N Ar gas bottle was connected to FEP coil placed in a 3 L dewar ( Figure 2). The amount of LN2 in the dewar and the flow rate of cool Ar gas were chosen to ensure a near constant temperature in the vicinity of the sample (a maximum of 1-2 K drift was observed during measurements). The cryostat itself was made from FEP tube with aluminum disks. Holes drilled in the aluminum disks enabled gas flow through the measurement chamber, and retained sample in place. The main body of the cryostat, made from FEP tube, had two holes drilled in on opposing sides of the tube -the first one, about 3 mm in diameter to let in the excitation line, the second one, about 1 mm in diameter, to let in a thermocouple. The temperature inside the cryostat was measured using a thermocouple positioned about 1 mm from the capillary with enclosed sample. To ensure that no condensation formed in the hole that excitation line was coming in, a heat gun was used to blow the newly formed ice crystals away from the cryostat. Minimum stable temperature reached in this setup was equal to 123 K, way below the 230 K order/disorder-type phase transition but above the temperature of magnetic ordering (66 K), and we report here spectra measured at 123 K as well as at room temperature.

Computational details
Lattice dynamics was modelled using GGA+U framework using Lichtenstein formalism [7] with values of U = 5.0 eV and J = 1.0 eV for silver atoms in order to account for strong electron correlation. Perdew-Burke-Ernzerhof functional revised for solids (PBEsol) [8] and projector-augumented-wave method [9,10] was used, as implemented in VASP 5.2.12 code [11][12][13]. The cut-off energy of the plane wave basis set was equal to 800 eV with a self-consistent-field convergence criterion of 1·10 -7 eV. The kpoint mesh spacing used in calculations was equal to 0.16 Å -1 . Ordered, low temperature, Pnma structure of KAgF3 was chosen as a starting model for the calculations [2]. Antiferromagnetic model was assumed in agreement with Zhang et al. (AFM along the chains, FM in planes) [3].
Phonon dispersion curves and phonon density of states were calculated for 2 x 2 x 2 supercell with the use of PHONON software package [14,15]. Hellmann-Feynman forces were evaluated using series of atomic displacements (0.0018-0.0025 Å) for which single point energies were calculated with settings mentioned above and harmonic approximation. Furthermore, obtained Γ-point frequencies were compared with values obtained using HSE06 functional [16]. For that comparison the finite differences method was used, as implemented in VASP core. These calculations were carried for KAgF3 unit cell, using less dense k-point mesh spacing of 0.25 Å -1 to reduce the calculation cost. This implementation yields frequencies in the centre of the first Brillouin zone (G point).  Infrared and Raman spectra were measured from instrument lower limit up to 1000 cm -1 . FIR measurement resulted in 8 separate bands, while Raman measurement showed 7 bands. Low temperature Raman measurement exhibits slight temperature shift towards higher wavenumbers. Density functional framework was used to gain in-depth understanding of present spectral features. To set the grounds for analysis of the experimental spectra, we begin discussion with group theory analysis. According to Group theory analysis there are 60 normal vibrational modes for KAgF3 in Pnma symmetry (Γacoustic = B1u + B2u + B3u; Γoptic = 7Ag + 8Au + 5B1g + 9B1u + 7B2g + 7B2u + 5B3g + 9B3u).
Calculated phonon dispersion curves and phonon density of states are shown in Figure 5. No imaginary modes are seen; hence, the low-temperature structure is confirmed to be dynamically stable. Partial (atomic) phonon density of states ( Figure 5, right) yields information about atoms involved in oscillations. Up to ca. 175 cm -1 all atoms contribute comparably to normal modes. The characteristic region of vibrations of potassium atoms is within the energy range 0 -225 cm -1 . Silver atoms contribute within slightly broader energy range 0 -260 cm -1 as well as to the highest energy modes between 400 and 450 cm -1 . Fluorine atoms contribute considerably to all phonon modes of the KAgF3. Their contribution dominates the phonon DOS within the energy range 150 -250 cm -1 , which is the characteristic energy region of the F-Ag-F bending vibrations, and the entire higher energy phonon DOS in the range 300 -450 cm -1 , which is the characteristic region of the Ag-F stretching vibrations.
The phonon bands belonging to F-Ag-F bending and low-energy lattice modes (below 250 cm -1 ) show relatively low dispersion. On the other hand, phonon bands stemming from diverse Ag-F stretching modes exhibit substantial dispersion indicating considerable coupling of the stretching vibrations along both the direction of the propagation of the AFM chains (corresponding to Y point in the 1 st Brillouin zone) and within the [AgF2] planes (X and Z points in the 1 st Brillouin zone).
The eigenvectors for selected normal modes assigned to measured IR and Raman spectra are shown in Figure 6, while the complete list of theoretically predicted Γ-point frequencies is contained in Electronic Supplementary Material.
The measured room-temperature far infra-red absorption spectrum of KAgF3 ( Figure 3) consists of eight bands, of which seven was successfully assigned using DFT+U framework while considering both selection rules and calculated absolute values of wavenumbers; we found out that the best match between theoretical and experimental wavenumbers is obtained if the former ones are scaled by the factor of 1.05 (Table 1 and Figure 3). The highest frequency band is placed at 448 cm -1 and was assigned to B2u stretching within [AgF2] planes in ac plane. The 377 cm -1 shoulder cannot be assigned to any fundamental mode; it likely originates either from 126 cm -1 + 252 cm -1 combination mode (A1u) or from an unknown impurity. In a lower frequency range, observed bands are rather broad and not fully resolved. One feature centred at 200 cm -1 , may be assigned to [AgF2] plane buckling vibrations (B2u).
Two shoulders at 295 and 252 cm -1 are associated with similar plane buckling, as well as bending of the [AgF + ] chain (B3u and B1u, respectively). The 136 cm -1 band of medium strength is also assigned to chain bending (B2u), but its two shoulders originate from potassium and fluoride lattice modes (both B2u, at 110 -1 and 59 cm -1 , respectively).
In the measured room-temperature Raman spectrum seven bands are clearly visible. Two bands at 444 cm -1 and 359 cm -1 were assigned to Ag-F stretching within chain-forming [AgF6] octahedra and within [AgF2] planes (B2g and Ag, respectively). Weak band at 196 cm -1 is associated with [AgF2] intra sheet bending vibrations (B2g). The lowest lying 126 cm -1 and 101 cm -1 features correspond, respectively, to B1g and/or a nearby Ag, and B2g lattice modes involving potassium cations. There are two more bands in the spectral region above 800 cm -1 (not shown); the shoulder at 899 cm -1 is supposedly the first overtone of the IR-active B2u mode at 448 cm -1 ; the origin of the 949 cm -1 band currently remains unknown. The assignment presented above is substantially strengthened by the fact that HSE06 hybrid DFT calculations yield identical band assignment without necessity to scale wavenumbers by the 1.05 factor.
The measured low-temperature Raman spectra are compared with the room-temperature one in Figure 4. Both spectra are characteristic of the same four broad main features, while in the lowtemperature spectrum these features are shifted to slightly higher energies. All bands that are present at low temperature, are observable also at RT, and general shape of the spectrum remains unchanged. There is some phonon stiffening observed at low temperature, with the 444 cm -1 band now appearing at 464 cm -1 , 359 cm -1 one at 380 cm -1 , 196 cm -1 one at 217 cm -1 , 126 cm -1 one at 127 cm -1 and 101 cm -1 one at 109 cm -1 .
In published work, the disordered room temperature Pcma structure is derived by dividing Pnma unit cell in half, and introducing disorder of fluoride anions. The fluorine anions are placed alternately along the [AgF + ] chains with an occupancy of 0.5 [2]. As this type of phase transition does not substantially alter the network of covalent bonds, the thermal shift observed in Raman spectra can be satisfactorily explained by thermal compressibility of KAgF3 lattice. With decreasing temperature Ag-F bond lengths also decrease, causing energy of associated stretching and bending phonons to rise. The fact that the room temperature vibrational spectra of KAgF3 may be successfully assigned using the lowtemperature Pnma structure serves as a confirmation of the X-ray diffraction-based hypothesis described above. This is additionally confirmed by comparing Raman spectrum obtained at room temperature and at 123 K ( Figure 4, Table 2).  sh -shoulder, w -weak, m -medium, s -strong, vs -very strong, R -Raman active mode, IR -infrared active mode, a -mode with assignment to two nearby modes)

Conclusions
The vibrational spectra of KAgF3 perovskite, a unique 1D antiferromagnet, have successfully been assigned with the help from theoretical DFT calculations. The calculated frequencies are in excellent agreement with experimental values when applying a 1.05 scaling factor for DFT+U calculations, and without any scaling for hybrid DFT ones. Phonon dispersion curves show that the low-temperature Pnma structure is dynamically stable, as no phonon softening is present. Comparison of low temperature and room temperature Raman measurements proves that order-disorder phase transition in 230 K has no major effect on the crystal structure and vibrational characteristics of the compound. To achieve complete understanding of the vibrational structure of KAgF3 additional studies using e.g. Inelastic Neutron Scattering are needed which could determine the energies of these phonons, which have so far eluded experiment, and enable comparison with theoretical values. Table S1: List of all theoretically predicted normal modes, Figure S1: Visualization of theoretically predicted normal modes in order of decreasing wavenumber.