An Appraisal of the Understanding Pressure Effects on Structural, Optical, and Magnetic Properties of CsMnF₄ and Other 3dⁿ Compounds

1. Introduction

CsMnF₄ is a foundational example of a cooperative Jahn-Teller (JT) system, whose structural distortions mediate its distinctive optical and magnetic properties. While the effects of pressure on this material have been extensively studied [1,2,3,4,5,6,7,8,9,10,11,12], its high-pressure structural evolution and electronic ground state remain subjects of active debate [1,7]. A recent publication [1] has intensified this controversy by presenting theoretical calculations that contradict interpretations from prior experimental work. That study, however, omits several key experimental findings on CsMnF₄ and other Mn³⁺ fluorides under pressure [2,3,4,5,6,7,8,9,10,11]. This perspective aims to critically reexamine the structural, optical, and magnetic properties of CsMnF₄ under pressure, specifically to address the interpretation in Ref. [1] that conflict with the established literature. A primary focus is the disputed assignment of the high-spin to low-spin transition observed in CsMnF₄ near 37 GPa [7].

At ambient pressure, CsMnF₄ crystallizes in a tetragonal structure (space group P4/n), forming a layered perovskite. The MnF₆^3- octahedra are subject to a cooperative JT distortion, resulting in a locally elongated complex with a D_2h symmetry. The crystal structure is characterized by an antiferrodistortive arrangement where the in-plane equatorial ligands of one MnF₆^3- unit act as the axial ligands for two adjacent complexes (Figure 1). The in-plane Mn-F bond distances are 2 × 2.168 Å and 2 × 1.817 Å, while the out-of-plane distances are 2 × 1.854 Å [12]. This specific connectivity, with Mn-F-Mn bond angles clase to 180º, favors ferromagnetic superexchange within the layers, successfully explaining the emergence of long-range ferromagnetic order below Tc = 19 K [2,3,4,5].

This paper provides an overview that contrasts the findings of Ref. [1] with the body of existing research. We will first delineate the fundamental Jahn-Teller distortions in this system before critically assessing the high-pressure behavior, with particular emphasis on the debated spin-state transition [1,7]. Our goal is to reconcile the theoretical predictions with experimental evidence and clarify the current understanding of this important material.

2. Results and Discussions

The main scientific concerns to the theoretical work about the structural, electronic and magnetic properties of CsMnF₄ published elsewhere [1] are summarized in the six following sections.

2.1. Structure of CsMnF₄

The authors theoretically describe the structural evolution of CsMnF₄ in terms of a tetragonal structure P4/n (0-40 GPa), which experiences a structural phase transition at 37.5 GPa to another tetragonal phase (P4) that is stable up to at least 50 GPa. In this structural evolution, the ground state of Mn³⁺ is high spin (S = 2). The point is that there are three publications [2,3,4] dealing with the structural evolution of CsMnF₄ with pressure by x-ray diffraction (XRD): energy dispersive XRD at the Daresbury Synchrotron, UK [2,3], and angle dispersive XRD at the European Synchrotron Radiation Facility (ESRF), France [4]. These independent works concluded that the tetragonal structure P4/n of CsMnF₄ is unstable above 2 GPa, transitioning to a monoclinic structure stable up to at least 16 GPa [4] (Figure 1). Energy dispersive XRD data [2,3] conclude that a tetragonal to orthorhombic phase transition at 1.4 GPa followed by a monoclinic transition above 6 GPa (see Orthorhombic to Monoclinic section of Ref. 3). None of these publications were discussed in [1] in relation to the experimentally observed structural evolution –Refs. 2 and 4 were not cited– and therefore, this low-pressure structural transition calls into question all results derived from a tetragonal structure above 2 GPa. Although Ref. 3 was cited in the article, it literally states that “No signs of a structural phase transition around 1.4 GPa, early suggested by Moron et al. [22], have been found in the optical measurements on CsMnF₄” (page 13233 in Ref. 1). Besides the recognition of a structural phase transition around 1.4 GPa earlier reported by Moron et al. [3], the authors ignored two other publications [2,4] not cited in Ref. 1, which confirm the tetragonal to monoclinic structural phase transition (around 2 GPa in Ref. 4). In addition to the XRD results, reference 4 also reports precise optical data in the 0-16 GPa range, which provides clear evidence of the influence of the phase transition at 2 GPa in the optical spectra of CsMnF₄ (Figure 1).

The measured spectra show that the pressure shift rate of the ⁵B₁ → ⁵A₁ transition energy at 1.89 eV changes from -42 meV/GPa to +2 meV/GPa at 2 GPa, coinciding with the tetragonal to monoclinic phase transition [4] (Figure 1). It must be emphasized that Refs. 2-4 are the only ones dealing with the crystal structure of CsMnF₄ under high pressure conditions, and were not considered, or cited [2,3,4] in the reported work [1], although one of the authors cited Ref. 4 in a previous publication [6]. Therefore, the structural phase transition detected by XRD by two different research groups, at two different synchrotron facilities, between 1-2 GPa was neither considered in the article¹ nor detected by using the authors’ DFT methodology. Furthermore, the structural evolution of CsMnF₄ with increasing pressure reported in reference 4 coincides with the structural evolution in the series AMnF₄ (A: Cs, Rb, Tl, K, Na) with decreasing the unit cell volume [2,3,4,5] as it can be seen in Figure 2 and Figure 3.

2.2. Pressure Dependence of the Optical and Electronic Properties

The pressure dependence of the optical and electronic properties derived theoretically from the tetragonal phase [1] are unable to explain the experimental results by optical spectroscopy published previously [4,7,8,9]. CsMnF₄ in the tetragonal phase is a uniaxial crystal instead of a biaxial crystal attained in the monoclinic structure (P > 2 GPa). Furthermore, the structure of the Mn³⁺ one-electron d-orbitals obtained theoretically [1], predicts that the energy of the first absorption band associated with the ⁵B₁ →⁵A₁ (denoted by the one-electron orbital transition 3y²-r²→x²-z² in Figure 4 of Ref. 1), decreases continuously from 1.92 eV to 1.43 eV in the 0-40 GPa range, while experimentally it changes from 1.89 to 1.81 eV in the 0-2 GPa range (tetragonal phase) but blueshifts by only 0.03 eV in the 2-16 GPa range (monoclinic phase) [4,7] giving a total shift of +0.07 eV at 37 GPa [4,8].

It means that the band shifts from 1.89 eV at ambient pressure to 1.88 eV at 37 GPa, which is two orders of magnitude lower than the shift predicted theoretically of -0.49 eV in the same pressure range (Figure 1). Neither the magnitude of the pressure band shift, nor the two opposite pressure rates below and above 2 GPa were accounted for theoretically [1]. References 4 and 8 were not cited in Ref. 1, and their experimental results could have guided the authors to search the actual structural evolution of CsMnF₄ as well as its associated properties. It must be noted that the pressure-induced blueshift of the first absorption band experimentally observed in the CsMnF₄ monoclinic phase above 2 GPa is also observed in NaMnF₄ in the 0-6 GPa range, which has a monoclinic structure at ambient conditions [8,9] (Figure 3). It seems that the preservation of the tetragonal structure in the whole explored pressure range¹ is unable to explain the observed optical properties.

2.3. Optical Band Assignment of Mn³⁺ Fluorides

The band assignment of the crystal-field spectra related to d-d transitions in CsMnF₄ as well as the high-spin (HS) to low-spin (LS) transition detected by optical spectroscopy at 37 GPa reported elsewhere [7] (Figure 4), the spectra of which are reproduced in Figure 4 of Ref. 1, are questioned in the article [1]. Although no alternative interpretation to the assignment of Ref. 7 is given in the article [1], it must be noted that the optical absorption band assignment of Mn³⁺ fluorides is based on spectroscopic results reported elsewhere [9]. This work, which is not cited in Ref. 1, deals with single-crystal polarized optical absorption spectroscopy and the temperature dependence of the various bands appearing in the optical spectra. Based on their transition energy, spectral shape and oscillator strength, as well as their temperature dependence (Figure 5, Figure 6 and Figure 7), it was concluded that there are two types of bands in the optical spectra: three broad bands associated with single-electron transitions within the d-orbitals of Mn³⁺ in an almost D_4h local environment, and two spin-flip narrow peaks arising from the 3d⁴ electronic configuration of Mn³⁺ (Figure 5).

It is experimentally shown that the former ones are electric-dipole spin-allowed transitions, whose oscillator strength increases with increasing temperature by coupling to odd parity vibrations within the (MnF₆)^3- complex. The spin-flip transitions are shown to be electric-dipole spin-forbidden transitions within a single Mn³⁺ ion but are activated by the spin-effective mechanism in exchange-coupled systems via the Mn-F-Mn superexchange pathway [9] (Figure 7). The temperature dependence of their oscillator strength demonstrates the spin-flip nature of these narrow peaks observed in the optical spectra of exchange-coupled systems [7,8,9]. In addition, these spin-flip transitions can be easily identified in the optical spectra because, unlike the spin-allowed broad bands whose energies are strongly dependent on the (MnF₆)^3- octahedral distortion, their energy is not as sensitive to the distortion [7,8,9] (Figure 5). Spin-flip transitions are within 0.05 eV at the same position in all series of manganese (III) fluorides: E_SP1 = 2.380 eV and E_SP2 = 2.873 eV in 2D CsMnF₄ [7], 2.397 and 2.890 eV in 2D NaMnF₄ [8,9], 2.380 and 2.860 eV in 2D TlMnF₄ (2D fluorides) [8] and, 2.418 and 2.914 eV in the 1D Tl₂MnF₅.H₂O [9].

In this respect, the d-orbital electronic structure derived theoretically for CsMnF₄ in the article,¹ does not agree with the measured spectra [7] (Figure 4), which are reproduced in Figure 4 of Ref. 1. The spectra show three broad bands associated with single-electron transitions from 3z²-r², xy, and the nearly degenerate (xz, yz) to x²-y², in order of increasing energy (note that the local z-axis in Ref. 7 refers to the long Mn-F bond in CsMnF₄ of the nearly D_4h (MnF₆)^3- complex). In the article [1], the xy, xz, and yz d-orbitals have different energies with respect to the x²-y² level: 2.2, 2.5 and 2.8 eV (Figure 7 of Ref. 1). Except for the first absorption band 3z²-r² → x²-y² at about 1.9 eV, the other three transition energies cannot give rise to a two-broad-band structure in the optical spectra as observed experimentally (Figure 5), but to three almost equally spaced broad bands, which would probably give rise to a structureless broad band instead of the two well-resolved bands observed experimentally [7]. Furthermore, the calculations performed in the article deal with one-electron d-orbitals and not with the states arising from the d⁴ electronic configuration, and thus the observed spin-flip transitions are missed in the reported calculations [1]. This is an important point as the authors should know that the spin-flip transitions observed in the spectra of CsMnF₄ (Refs. 4,7) are crucial to explain the spin transition at 37 GPa on the basis of optical spectroscopy [7] (Figure 4 and Figure 5).

$$f\left(T\right)={\displaystyle \frac{2f\left(0\right)}{(2S-1)}}\left[S\left(U^2-1\right)+{\displaystyle \frac{4SU\left(S+1\right)UV}{2V}}\right]$$

(1)

Where f(0) is the 0 K oscillator strength, U=cothV-1/V, V=2JS(S+1)/k_BT , S = 2 and J (< 0) is the intrachain exchange constant [9] (Left side). Temperature dependence of the oscillator strength for the ⁵B_1g→⁵A_1g broad band (1.45 eV) in π and σ polarizations. Solid lines are fits to $f\left(T\right)=f\left(0\right) \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\hslash {\omega }_u/{2k}_{B}T\right)$ (vibronic mechanism, parity-forbidden electric-dipole transitions activated by odd parity vibrations ${\omega }_u$). The different temperature dependences and polarization reveal distinct activation mechanisms: exchange for spin-flip peaks and vibronic for spin-allowed broad bands (Right side). Adapted with permission from Reference 9. Copyright 1994 American Chemical Society

2.4. Magnetic Properties of CsMnF₄

Regarding the magnetic properties, it is theoretically found that CsMnF₄ undergoes an abrupt change from ferromagnetic to antiferromagnetic at 10 GPa within the tetragonal phase [1]. However, there is a misunderstanding with the magnetic measurements under high pressure conditions reported elsewhere [11]. This paper reports a singular magnetic behavior above 2 GPa with a reduction of the Curie temperature and a sharp decrease of the saturation magnetization of CsMnF₄ above 3.3 GPa, which the authors explain in terms of a progressive transition from ferromagnetic to antiferromagnetic with a permanent ferromagnetic component, which the authors attribute to canting antiferromagnetism [11]. This behavior is overlooked in the article [1] and therefore, properties observed in the monoclinic phase are not predicted in the tetragonal phase. In fact, this behavior has been explained along the AMnF₄ series of layered perovskites, where the only tetragonal member of the series is the ferromagnetic CsMnF₄, the remaining compounds having a monoclinic structure are all antiferromagnetic [2,5,11,12]. Moreover, Refs. 2 and 3 show that the AFM Neel temperature correlates with the Mn-F-Mn tilting angle in the monoclinic phase, a figure which cannot be reproduced in the same way using a tetragonal phase due to structural constrains.

2.5. High-Spin to Low-Spin Transition at 37 GPa

The change in the optical spectra (Figure 4 of Ref. 1; or Figure 2 of Ref. 7) associated with a HS→LS crossover transition at 37 GPa and room temperature in the original paper [7] (Figure 4) is now questioned and associated with a tetragonal to tetragonal phase transition P4/n→P4 at 37.5 GPa, both phases being HS (S=2). However, this interpretation lacks the spectral evolution at the crossover transition point. There are three main reasons for retaining the HS to LS transition: 1) The three broad bands observed below 30 GPa, associated with the strong octahedral distortion due mainly to the E⊗e JT effect, transform into a structureless broad band characteristic of a nearly octahedral coordination. The T⊗e JT effect in the LS ³T₁ ground state (O_h) is a factor of 4 weaker than the E⊗e HS ⁵E ground state strongly distorted by the JT effect [7,13]; its stabilization energy in LS is an order of magnitude smaller than in HS. 2) The spin-flip transitions E_SP1 and E_SP2, which are characteristic of a HS ground state, disappear completely above 37 GPa. According to the Tanabe-Sugano diagrams for d⁴ ions [13,14], these transitions are absent in a LS ground state (³T₁), making them a precise spectroscopic probe for layered AMnF₄ (A: Cs, Na, Tl) perovskites in HS [7,9] (Figure 4, Figure 5 and Figure 6). 3) The LS broad band peaking at 2.5 eV (Figure 4) coincides with the centroid of the spin-allowed crystal-field bands at the spin crossover transition (S= 2→1), which corresponds to a crystal-field splitting 10Dq of 27B (C/B = 4.6), where B and C are the Racah parameters for (MnF₆)³⁻ [7,13,14]. On the other hand, there is an argument given in Ref. 1 to support a structural transition at 37 GPa instead of a HS→LS crossover transition, which is not properly justified. The authors state that the 10Dq required to stabilize the LS state should be higher than 3 eV, without any further justification. The value of 10Dq derived from the TS diagram at the transition point requires accurate knowledge of the Racah parameter B, which is known to be B = 0.097 eV (780 cm^-1) in (MnF₆)^3- at ambient conditions [10]. This value can also be obtained from the spin-flip transition energies, since the first observed spin-flip transition depends on pressure as E_SP2 = 2.397 – 0.0018 P (in eV and GPa units, respectively) [7,9]. From the TS diagram [13,14] (Figure 4) its energy varies as 24.6 B in HS up to the spin crossover point. This means that B slightly decreases from 0.097 eV at ambient pressure to about 0.094 eV at 37 GPa, which is considered to be the spin crossover pressure as stated elsewhere [7]. This means that 10Dq should be about 2.54 eV –lower than 3 eV–, and considering that the single-electron t_2g⁴ e_g⁰ → t_2g³ e_g¹ transition actually gives rise to several spin-allowed transitions within the d⁴ electronic configuration, from the LS ³T₁ ground state to the ³Γ_i excited states, whose energies spread over the 0.4 eV range: 2.39 eV (³T₁), 2.48 eV (³T₂), 2.65 eV (³A₂), and 2.76 eV (³A₂) [13,14]. All these transitions give rise to a broad band whose centroid peaks at about 2.54 eV, in agreement with the observed spectra of CsMnF₄ above 37 GPa in LS [7] (Figure 4). 4) The argument given to rule out spin transition about the enthalpy difference between HS and LS states in tetragonal symmetry is not supported by any information or evidence even in the Supporting Information [1]. It is well known that the estimation of a spin crossover pressure from ab initio calculations is subtle and delicate since it requires a good knowledge of the actual HS and LS structures and the calculations are very sensitive to parameters such as the Hubbard U among others [15,16]. It is therefore scientifically irrelevant to state that the enthalpies for LS and HS are a few tenths of eV, based on an inadequate HS structure and an unknown LS structure, without providing details of the performed calculations. However, optical spectroscopy offers a more suitable means of identifying spin crossover phenomena. This technique can effectively distinguish between the distinct spectra associated with each spin state, irrespective of the specific crystal structures involved in the spin transition. This has been well established for Fe²⁺ compounds and extensively documented elsewhere [17,18,19].

2.6. The Jahn-Teller Effect: Theorem, Theory, and Molecular Distortion

Finally, some concepts about the JT effect need to be clarified in order to avoid misunderstandings among scientists working with systems that exhibit JT distortions. This follows from sentences (i-iii) on page 13234 of Ref. 1, which state: (i) “The existence of a JT effect requires a degenerate electronic state in the initial geometry”, (ii) “CsMnF₄ is a layered compound where layers are perpendicular to the crystal c axis (Figure 1). Accordingly, one would expect that the axis of the MnF₆³⁻ unit perpendicular to the layer plane plays a singular role, a fact seemingly not consistent with the JT assumption.”, and (iii) “The local equilibrium geometry for MnF₆³⁻ in CsMnF₄ is not tetragonal. Indeed, even assuming Y as the main axis (Figure 1) the symmetry would be at most orthorhombic because R_X − R_Z = 0.037 Å. Accordingly, one should expect four and not only three d−d transitions with ΔS = 0 for CsMnF₄.”

Because of these three sentences, it is convenient to distinguish between the JT theorem, the JT theory, and the JT distortion when dealing with the JT effect. The JT theorem states that a nonlinear polyatomic molecule with an orbital degenerate ground state is unstable and must distort to a lower energy configuration, or literally “All nonlinear configurations are therefore unstable for an orbital degenerate electronic state” [20]. Based on the JT theorem, the JT theory goes further and explores the distortions of the molecule under different structural perturbations. In particular, for highly distorted E⊗e JT systems such as octahedral MX₆ complexes with M: Cr²⁺, Cu²⁺, Mn³⁺, and X: Cl^-, F^-, most of which exhibit elongated octahedral distortions [21,22,23,24,25], the JT theory shows that the equilibrium geometry caused by a tetragonal symmetry perturbation of the octahedron corresponds to the JT distorted geometries of the octahedron (three degenerate minima associated with tetragonally elongated octahedra along the x, y and z axes) slightly modified by the axial perturbation [21,22,23,24,25] (Appendix A). The JT theory shows that the equilibrium coordination geometry of the JT ion varies from elongated tetragonal along the tensile perturbation direction, to two equivalent minima corresponding to the JT elongated octahedra with a slight orthorhombic distortion for a compressive D_4h perturbation of the site, to a tetragonally compressed geometry if the compressive perturbation is large enough (Appendix A). In general, a compressive D_4h perturbation results in two equivalent elongated octahedra associated with the two JT distorted geometries with the long axes perpendicular to the D_4h axis of the compressed site. The two short M-X bonds are generally different; their difference, which is zero for an O_h site (three equivalent D_4h elongated geometries), increases with the D_4h compressive perturbation of the site, yielding local JT distortions of D_2h symmetry. Thus, E⊗e JT theory shows that slight tetragonal distortions of the initial octahedral symmetry lead to equilibrium geometries corresponding to the JT distortions obtained in O_h slightly perturbed by the D_4h compressive strain of the site. Therefore, JT theory shows that it is possible to have equilibrium geometries of the MX₆ complex corresponding to JT distortions when the JT ion is placed in a D_4h compressed –near O_h– symmetry, resulting in two out of three JT equivalent D_2h elongated equilibrium geometries with their elongated M-X bonds perpendicular to each other and both perpendicular to the D_4h axis of the compressive perturbation with an initial non-degenerate ground state [22,23,24,25].

This implies a reduction in local symmetry relative to the D₄_h symmetry of the site, although the average of the two orthorhombic local symmetries retains the D₄_h symmetry of the site (Appendix A). This result contradicts the authors' assertion that the local symmetry of Mn³⁺ or Cu²⁺ in a compressed D₄_h symmetry must remain unchanged, contrary to the JT theory.

Other structural perturbations such as non-centrosymmetric strains may be present, but they have no effect on the JT distortion since such distortion modes are not coupled to the parent-octahedral degenerate electronic e_g levels in first-order perturbation.

3. Conclusions

In conclusion, JT theory shows that JT ions placed in O_h-perturbed low-symmetry sites, where electronic degeneracy is left, exhibit JT distortions different from the initial site symmetry in which it is placed, making the exclusion of such distortions as not being JT distortions doubtful. The authors in Ref. [1] conclude that reasons for not considering the formation of low-symmetry complexes in Cu²⁺ and Mn³⁺ as a JT effect are that “They are also behind the so-called plasticity property of compounds of Cu²⁺ and Mn³⁺ ions”, citing reference 84 (Reference 26 herein). In this context, the term plasticity appears to be a euphemism, as Ref. 26, Chapter E is entitled “The Jahn-Teller and pseudo-Jahn-Teller origin of the plasticity of Cu^II coordination sphere and distortion isomerism”.

Finally, it is crucial to emphasize that translating the behavior of a complex like (MnF₆)³⁻ into any fluoride lattice, such as CsMnF₄, is plausible. This is because the phenomenon is inherently local, and the JT model parameters can effectively mimic the real structural conditions of the complex within the lattice, including crystal anisotropy, cooperative effects, and electron-lattice coupling. This approach, often referred to as cluster model, significantly simplifies the problem and provides a general framework for understanding distortions induced by the JT or pseudoJT effect [27].