A pyroxene-based quantum magnet with multiple magnetization plateaus

Abstract

Pyroxenes (AMX₂O₆) consisting of infinite one-dimensional edge-sharing MO₆ chains and bridging XO₄ tetrahedra are fertile ground for finding quantum materials. Thus, here, we have studied calcium cobalt germanate (CaCoGe₂O₆) and calcium cobalt silicate (CaCoSi₂O₆) crystals in depth. Heat capacity data show that the spins in both compounds are dominantly Ising-like, even after being manipulated by high magnetic fields. On cooling below the Néel temperatures, a sharp field–induced transition in magnetization is observed for CaCoGe₂O₆, while multiple magnetization plateaus beneath the full saturation moment are spotted for CaCoSi₂O₆. Our analysis shows that these contrasting behaviors potentially arise from the different electron configurations of germanium and silicon, in which the 3d orbitals are filled in the former but empty in the latter, enabling electron hopping. Thus, silicate tetrahedra can aid the interchain superexchange pathway between cobalt(II) ion centers, while germanate ones tend to block it during magnetization.

The observation of anisotropic field–induced magnetization plateaus in pyroxenes depends on the choice of bridging tetrahedra.

INTRODUCTION

Solid-state materials are one of the fundamental building blocks of modern society. Many modern technologies have changed in almost unimaginable ways in the past few decades, largely attributed to the discovery of solid-state materials providing sophisticated functionalities (1–8). Pyroxenes, with a general formula AMX₂O_6, are one of the most abundant minerals in Earth’s crust and are well studied in geoscience (9–12). The edge-sharing MO₆ octahedra form infinite one-dimensional (1D) chains that propagate in a zigzag fashion, and the XO₄ tetrahedra bridge adjacent 1D chains to form the structural motif of the lattice. The edge-sharing MO₆ octahedra within each 1D chain naturally lead to shorter distances between adjacent metal centers within the chains. These metal centers communicate with metal centers in the neighboring chains through the bridging tetrahedra. This unique structural motif can have a marked impact on the physical properties. The fruitful choice of elements residing in the centers of the octahedra and tetrahedra provides degrees of freedom for manipulation of these properties. Therefore, this class of materials has attracted enduring interest from the science community because a wide variety of intriguing physical phenomena have been observed, including multiferroicity, magnetoelectric effects, etc. (13–17), and the competing interchain and intrachain interactions within the highly versatile crystal structure of pyroxenes make them a fertile ground for finding quantum materials.

The electronic structure and magnetic properties of pyroxenes containing transition metals have been studied as low-dimensional magnets with 90° bonds via theoretical calculations (18). Despite the fact that an enormous number of studies have focused on pyroxenes bearing the magnetic transition metals Cr, Mn, and Fe (13–17), the analogous Co-containing compounds are incompletely studied (19–25). The Co-containing pyroxenes have been reported to serve as pigments (26, 27) or functional ceramics due to their thermal (28) and dielectric properties (29, 30) and, very recently, have been reported to serve as a playground for Kitaev physics and Ising model realization (31). We here show that on cooling through the Néel temperature T_N of CaCoGe₂O₆ and CaCoSi₂O₆ in powder form, a sharp field–induced transition in magnetization is observed for CaCoGe₂O₆, while multiple magnetization plateaus beneath the full saturation moment are spotted for CaCoSi₂O₆ at reasonable applied fields. A series of CaCoGe_2−xSi_xO₆ (x = 0, 0.5, 1, 1.5, and 2) solid solutions has been synthesized as powders to show the continuous evolution of the observed phenomena in magnetization. To study these phenomena further, we grew macroscopic crystals of CaCoGe₂O₆ and CaCoSi₂O₆ by the floating-zone method. We found that the observed magnetization transitions are highly anisotropic, as expected from the crystal structure. Heat capacity data show that the spins in both compounds are dominantly Ising-like, even after being manipulated by high magnetic fields, which is unusual for Co²⁺ in extended lattices (32, 33). Beyond revealing the enormous orbital contribution to magnetism, our first-principles calculation results further justify the observed discrepancy in anisotropic magnetic saturation between two compounds, which stems from different crystal field hybridizations of Co 3d orbitals.

Magnetization plateaus pinned at fractional values of the full saturation moment during the magnetization of quantum magnets are intriguing and often induced by collective effects (20, 34). The findings in this study ultimately reveal that the choice of bridging tetrahedra (i.e., SiO₄ versus GeO₄) between adjacent 1D CoO₆ chains in a pyroxene-based quantum magnet can locate magnetization plateaus in a measurable field range, potentially attributed to the empty 3d orbitals of Si that enable electron hopping. Hence, this work not only promotes pyroxenes as an excellent platform to study quasi-1D spin chain systems with nontrivial interchain coupling but also opens more research opportunities to further study those observed anisotropic plateaus during the magnetization process.

RESULTS

Structural characterization

The crystal structures of the entire series of magenta CaCoGe_2−xSi_xO₆ (x = 0, 0.5, 1, 1.5, and 2) compounds can be indexed by a monoclinic unit cell (space group C2/c), in good agreement with previous reports on the two end members of this series, CaCoGe₂O₆ (25) and CaCoSi₂O₆ (23). The lattice parameters extracted from the Le Bail fitting of the acquired powder x-ray diffraction (p-XRD) patterns at 300 K change continuously, confirming the purity of the as-made polycrystalline powder samples (fig. S1). In general, the unit cell gradually shrinks upon Si-for-Ge substitution as expected, since Ge⁴⁺ (0.53 Å) has a larger effective ionic radius than Si⁴⁺ (0.40 Å) (35). The end members CaCoGe₂O₆ and CaCoSi₂O₆ are of more interest in the present study, and, thus, macroscopic (i.e., a few millimeters on a side) single crystals suitable for anisotropic physical property measurements have been grown by the floating-zone method (fig. S2). Our single-crystal XRD data were collected from small pieces of each compound at ambient temperature. The refined structural parameters are perfectly aligned with those extracted from the abovementioned p-XRD data, as well as the literature (23, 25). Detailed crystallographic data, fitting statistics (table S1), atomic coordinates and equivalent isotropic atomic displacement parameters (table S2), and bond lengths (table S3) are presented in the Supplementary Materials.

The crystal structure is depicted in Fig. 1 (A to C). The 1D Co²⁺ (3d⁷) chains run parallel to c. The Ge and Si reside in the XO₄ tetrahedra. These tetrahedra bridge adjacent infinite edge-sharing CoO₆ 1D chains to form the monoclinic pyroxene crystal structure. The interchain Co-Co distance is dominated by the size of these bridging tetrahedra, with the smallest value found in the all-Si case (d ≈ 6.635 Å) and the largest value found in the all-Ge case (d ≈ 6.772 Å). The intrachain Co-Co distance is less influenced by the size of bridging tetrahedron and only shows a slight contraction in the all-Si case (about 3.101 Å) compared to the all-Ge case (about 3.155 Å). To make sure that all the isotropic properties measured in the present work are reliable, we collected p-XRD data from the crushed and ground crystals of CaCoGe₂O₆ and CaCoSi₂O₆. The powder data were then refined against the unit cell we got from the single-crystal XRD results (Fig. 1, D and E).

Fig. 1. Crystal structure.

(A) Extended monoclinic unit cell of the CaCoGe_2−xSi_xO₆ compounds. (B) Infinite edge-sharing CoO₆ 1D chains run parallel to the c axis, separated by (Ge,Si)O₄ tetrahedra. (C) The zigzag nature of the edge-sharing CoO₆ 1D chain. Le Bail fitting of the p-XRD data collected from crushed and ground crystals of (D) CaCoGe₂O₆ and (E) CaCoSi₂O₆. Insets: Pictures of single crystals on millimeter grids, with the flat crystal surface normal to the [010] axis. a.u., arbitrary units.

Isotropic magnetic properties

The isotropic zero field–cooled (ZFC) temperature-dependent dc magnetization data were collected from the crushed and ground crystals of CaCoGe₂O₆ and CaCoSi₂O₆, under an applied field of H = 0.1 T, and are plotted as magnetic susceptibility χ against temperature T (Fig. 2, A and B). The dc magnetic susceptibility data, over a suitable temperature range (selected as the straight line part of the 1/χ versus T curves, marked in red in Fig. 2, A and B) for each composition, were fitted to the Curie-Weiss law [χ = C/(T − θ) + χ₀], to yield the Curie constants C, Weiss temperatures θ, and effective moments μ_eff. The Curie constants C for both compounds show a marginal difference [Ge, 4.040 (2) cm³·K/mol; Si, 4.002 (4) cm³·K/mol], and, hence, the value of effective moment μ_eff can be calculated [Ge, 5.69 μ_B/formula unit (f.u.); Si, 5.66 μ_B/f.u.]. Both values are larger than the spin-only expectation for S = 3/2 Co²⁺ (high-spin d⁷, 3.87 μ_B/f.u.), revealing that spin-orbit coupling (SOC) makes an enormous contribution to magnetism in both cases, which is expected (36). The Weiss temperatures θ [Ge, −21.2 (1) K; Si, −9.6 (2) K] were also determined. The negative sign of θ suggests that the dominant magnetic coupling between spins is antiferromagnetic (AFM), agreeing with the well-defined paramagnetic-to-AFM transition observed in both cases. Further, the magnitude of θ almost coincides with the Néel temperature T_N (the local maximum in the magnetic susceptibility curve: Ge, 19.8 K; Si, 10.1 K) for both compounds. The Curie-Weiss fitting results of our isotropic magnetic susceptibility data in general agree well with the literature, with the only anomaly being the sign of θ for CaCoGe₂O₆, which can potentially be attributed to differences in sample preparation and fitted temperature range (23, 25). The analogous temperature-dependent data collected from the polycrystalline powder show that the migration of T_N to a lower temperature is a continuous process throughout the whole range of the solid solution (Fig. 2C).

Fig. 2. Isotropic magnetization.

The temperature-dependent and field-dependent dc magnetization data collected from (A, B, D, and E) the crushed and ground crystals of CaCoGe₂O₆ and CaCoSi₂O₆, as well as (C and F) the polycrystalline powder form of the CaCoGe_2−xSi_xO₆ (x = 0, 0.5, 1, 1.5, and 2) series.

The field-dependent magnetization data were collected from the crushed and ground crystals of CaCoGe₂O₆ and CaCoSi₂O₆. At temperatures higher than T_N, the magnetization (M) versus field (H) curves are linear and pass through the origin, while upon cooling to below T_N, they exhibit features that indicate either field-induced spin-state transitions or spin reorientations (Fig. 2, D and E). For CaCoGe₂O₆, at T = 1.8 K, the M versus H curve shows a sharp kink under an external applied field of around 5.5 T. The isotherm shows no signs of saturation up to H = 9 T, and the largest magnetic moment achieved is approximately 2 μ_B/f.u. (Fig. 2D). In contrast, CaCoSi₂O₆ has a subtle magnetization plateau observed in the low-field regime. At T = 1.8 K, this subtle magnetization plateau arises when a relatively small field (0 ≤ H/T ≤ 1.2) is applied, and then a flattened saturation plateau at about 2.5 μ_B/f.u. starts to occur under a magnetic field of about 5.5 T (Fig. 2E). The analogous field-dependent magnetization data collected from the CaCoGe_2−xSi_xO₆ (x = 0, 0.5, 1, 1.5, and 2) series further illustrate the evolution of the field-induced magnetization plateaus/transitions in this system (Fig. 2F).

Anisotropic magnetic properties

The magnetization plateaus observed in the low-temperature field-dependent magnetization data, collected from the crushed and ground crystals of CaCoGe₂O₆ and CaCoSi₂O₆, were deemed to be worth further investigation. Low-temperature neutron diffraction data have been collected by others for both CaCoGe₂O₆ (25) and CaCoSi₂O₆ (23) powders. The long-range AFM unit cells adopt a similar arrangement, with spins aligned parallel to each other within the infinite edge-sharing CoO₆ 1D chain and antiparallel to those in adjacent chains. Spins are found to align in the ac plane (23, 25); hence, they should behave differently upon an external magnetic field applied in or out of plane. This could yield an intriguing anisotropy during the magnetization process.

By using Laue alignment, we identified that the flat crystal surface is normal to the [010] axis (fig. S3); thus, the naturally cleaved crystal surface is confirmed to be the ac plane that contains in-plane Co²⁺ spins. Hence, the anisotropic magnetization data were collected from these crystals by applying the external field parallel and perpendicular to the [010] axis (Fig. 3). After noting how the infinite edge-sharing CoO₆ 1D chains propagate along the c axis and how the adjacent chains are bridged together by XO₄ tetrahedra along the a axis, together with the zero-field AFM magnetic unit cells determined in the literature (23, 25), the anisotropic magnetization data are less daunting to interpret.

Fig. 3. Anisotropic magnetization.

The (A and B) temperature-dependent and (C to F) field-dependent dc magnetization data collected from crystals of CaCoGe₂O₆ and CaCoSi₂O₆.

The anisotropic dc magnetic susceptibility data of CaCoGe₂O₆ and CaCoSi₂O₆ both show a substantial discrepancy between two field directions (Fig. 3, A and B), with details of the Curie-Weiss fitting provided in the Supplementary Materials (fig. S4). The anisotropic field–dependent magnetization data thus reveal the impact on magnetism brought by the difference in electron configuration of Si and Ge, which are in the superexchange pathway. For the CaCoGe₂O₆ crystal (Fig. 3, C and E), after deconvoluting the isotropic field–dependent magnetization data, it is clearly seen that an AFM-to-ferromagnetic transition has been induced by H = 7.3 T when the field is applied perpendicular to the [010] axis. The proper “square-S-shape” M versus H curve accompanied by a moment of nearly 3 μ_B/f.u. suggests that the system has reached a full (~100%) ferromagnetic saturation (Fig. 3E). Therefore, the spins in every other infinite CoO₆ 1D chain may have been flipped by the applied field and, thus, are all aligned parallel to each other along the field direction. It is necessary to emphasize that we do not rule out the possibility that the M versus H curve with H // [010] (Fig. 3C) will reach saturation at fields greater than 9 T. For the CaCoSi₂O₆ crystal (Fig. 3, D and F), both analogous isotherms collected reach a flattened saturation plateau of ~2.5 μ_B/f.u. at H ≈ 5.5 T (~83% of the expected full saturation moment) with potential further steps in the M versus H curve beyond the range of field measured to reach a full ferromagnetic saturation similar to the Ge case. Although the value of full saturation moment M_s for Co²⁺ varies from case to case in the literature, M_s ≈ 3 μ_B/Co²⁺ is reported for CoGeO₃, a compound that is analogous to ours (20). Thus, we tentatively state that M_s ≈ 3 μ_B/Co²⁺ for our materials based on the current data, although we may observe further steps in the MH isotherms under extremely high fields and the study of which will be of future interest.

In addition, the previously mentioned subtle magnetization plateau observed in the low-field regime of the isotropic M versus H data collected from CaCoSi₂O₆ (Fig. 2E) is now deconvoluted. The presence of nonlinked 1D CoO₆ chains along the [010] direction appreciably dilutes the contribution of the interaction that causes this subtle plateau, as evidenced by its visual absence in the M versus H curve collected with H // [010] (Fig. 3D). It is more important that this subtle magnetization plateau is not observed in the Ge case at all. Therefore, it is valid to postulate that this interaction may be due to the fact that Si resides in the bridging tetrahedra. Hence, this subtle magnetization plateau (~7% of the expected full saturation moment) may only be unveiled if the magnetic interaction is dominated by the superexchange pathway that accommodates an extensive amount of Si (e.g., the interchain coupling pathway in CaCoSi₂O₆). Given that the interchain Co-Co distance only varies ~2% between these two end members, which is too subtle to cause any qualitative changes in magnetism, the electron configuration of Si and Ge can be used to comprehend this phenomenon (Fig. 4A). The critical difference is that Si⁴⁺ has empty 3d orbitals that are available for electron hopping in the superexchange pathway, while Ge⁴⁺ has a filled 3d shell so it acts like a “shield”; thus, the communication between spins in the former is more effective than in the latter. That probably also explains why, in general, CaCoSi₂O₆ requires a much lower field to induce plateaus in its magnetization compared to CaCoGe₂O₆ (Fig. 4B). In summary, we observe that the choice of bridging tetrahedra between adjacent 1D CoO₆ chains can markedly affect magnetization plateaus at finite fields in an anisotropic way. We argue that this is potentially because SiO₄ can aid the interchain superexchange pathway between Co²⁺ centers, while GeO₄ tends to block it during the magnetization process. This occurs as a consequence of the different electron configurations of Si and Ge.

Fig. 4. Electron configuration.

The schematic illustration for (A) the electron configuration of Si and Ge and (B) their impact on the manipulation of interchain AFM coupling by magnetic field.

Heat capacity

Heat capacity data were collected from sintered pellets of the crushed and ground crystals of CaCoGe₂O₆ and CaCoSi₂O₆, under various external applied fields (μ₀H = 0, 3, 6, and 9 T), with details of background fitting (fig. S5) provided in Supplementary Text. The total heat capacity C_total data (Fig. 5, A and B) and the extracted magnetic contribution C_mag/T (Fig. 5, C and D) are plotted against temperature for both CaCoGe₂O₆ and CaCoSi₂O₆. The sharp peaks in the 0-T curves (Ge, ~20 K; Si, ~10 K) match well with the phase transitions in the temperature-dependent magnetization data. In the 3-T curves, the peaks are suppressed in magnitude, while their shapes remain almost intact, aligning perfectly with the AFM nature of this long-range magnetic ordering. However, the heat capacity data become more complicated under higher external magnetic fields (μ₀H = 6 and 9 T), since these fields induce the plateaus observed in the magnetization data, hence severely interfering with the otherwise perfect AFM long-range ordering under low fields. For CaCoGe₂O₆, although the transition peaks in the 6- and 9-T curves have been suppressed extensively, the peak width only gets moderately broadened, accompanied by the sequential shift of peak position to a lower temperature in minor steps. In contrast, for CaCoSi₂O₆, the peaks in the 6- and 9-T curves are severely deformed, and the anomalies tend to move up in temperature, especially in the 9-T data. Thus, under high magnetic fields, the spins in CaCoGe₂O₆ tend to retain the original AFM arrangement or undergo a proper AFM-to-ferromagnetic transition (nearly 3 μ_B/f.u., full saturation for S = 3/2 Co²⁺), depending on the direction of field applied (Fig. 3, C and E). However, although the spins in CaCoSi₂O₆ communicate more effectively through the bridging SiO₄ tetrahedra, maximum saturated magnetization plateaus reached are only ~2.5 μ_B/f.u. at 9 T in both field directions (Fig. 3, D and F), which indicates that a fully ordered spin arrangement cannot be achieved within our measurable field range. In summary, the heat capacity of CaCoGe₂O₆ shows qualitatively different features compared to CaCoSi₂O₆ under high fields and suggests that spins in the former tend to form longer-range ordering than in the latter, which coincides with their magnetization data. The magnetic entropy change ΔS_mag can be used to diagnose whether the spins in the magnetic system are inclining toward the Ising model or the Heisenberg model. Our experimental data indicate that ΔS_mag reaches a saturation value of ~5.41 J/mol/K for CaCoGe₂O₆ (Fig. 5E) and ~5.16 J/mol/K for CaCoGe₂O₆ (Fig. 5F), respectively. The Heisenberg spin prediction for an S = 3/2 (Co²⁺ high-spin d⁷) system should be Rln(2S + 1) = 11.53 J/mol/K, while the Ising spin prediction of Rln(2) = 5.76 J/mol/K is clearly closer to our experimental values (Ge, ~93.9%; Si, ~89.6%). Thus, we can tentatively conclude that the magnetic entropy released at the magnetic phase transition is associated with the bulk long-range ordering of spins in both compounds that are dominantly Ising-like, even after manipulated by high magnetic fields, which is unusual for Co²⁺ in extended lattices. Specific geometric arrangement of Co²⁺ (3d⁷) is often required, so that the interplay of crystal electric field and SOC could lead to an effective S = ½ ground state. Although Ising models have been adopted in some other compounds containing octahedral Co²⁺ 1D chains or 2D layers, the preserve of Ising-like behavior under high magnetic fields has been rarely spotted.

Fig. 5. Heat capacity.

(A and B) the total heat capacity C_total, (C and D) the magnetic contribution C_mag/T, and (E and F) the magnetic entropy S_mag plotted against temperature for CaCoGe₂O₆ and CaCoSi₂O₆.

Theoretical calculations

In (37), we developed a single-ion model with proper crystal field extracted from the density functional theory calculations to describe the complicated spin-orbit moments caused by the interplay among the crystal field, Hund’s rule coupling, and the SOC. For such a model, the single-ion magnetic susceptibility tensor can be obtained by the exact diagonalization of the single-ion many-body Hamiltonian, from which three unique crystalline directions can be determined from the singular value decomposition as the “magnetic principal axes.” The external magnetic field along these directions will be parallel to its induced magnetization with the ratio between the two being the corresponding eigenvalues of the susceptibility tensor. In the present study, we applied the above method to CaCoGe₂O₆ and CaCoSi₂O₆. Our results indicate that for both CaCoGe₂O₆ and CaCoSi₂O₆, the principal axis $\tilde{y}$ is along the crystallographic b axis, while $\tilde{x}$ and $\tilde{z}$ lie in the ac plane. We further calculated the Curie-Weiss effective magnetic moments along different principal axes, as shown in Table 1, with CaCoGe₂O₆ exhibiting moments of 5.07, 3.19, and 2.44 μ_B along $\tilde{x}$, $\tilde{y}$, and $\tilde{z}$, respectively. In comparison, CaCoSi₂O₆ displays magnetic moments of 4.27, 4.32, and 1.95 μ_B along the corresponding axes. Our theoretical results obtained from the single-ion model indicate three important points. First, the large experimental isotropic effective magnetic moments can be well explained from our simulations. On the basis of the calculated magnetic moments along different principal axes, we can yield values of 6.47 μ_B (CaCoGe₂O₆) and 6.38 μ_B (CaCoSi₂O₆) for theoretical isotropic effective magnetic moments. They show an acceptable deviation (~13 to 14%) compared to those values obtained from experimental data (CaCoGe₂O₆, 5.69 μ_B; CaCoSi₂O₆, 5.66 μ_B). Second, both the size and anisotropy of the saturation magnetic moments in these two materials can be explained by the single-ion physics. As shown in Table 1, the Curie-Weiss saturation magnetic moment of CaCoGe₂O₆ manifests substantial anisotropy, with a large value of 2.93 μ_B along $\tilde{x}$, a moderate value of 1.84 μ_B along $\tilde{y}$, and a relatively small value of 1.41 μ_B along $\tilde{z}$. These results are in good agreement with experimental anisotropic magnetization data shown in Fig. 3 (C and E). The 1.8-K MH isotherm in blue (Fig. 3C) does not show any signs of saturation, with the maximum magnetization reached ~1.3 μ_B at 9 T, which is much smaller compared to the analogous data in orange (~3 μ_B; Fig. 3E), consistent with the calculated values listed in Table 1. In contrast, for CaCoSi₂O₆, the theoretical values of saturation magnetic moment of $\tilde{x}$ (2.47 μ_B) and $\tilde{y}$ (2.49 μ_B) are only marginally different, which coincides with the fact that the 1.8-K MH isotherms in blue and orange saturate at almost the same plateau (~2.5 μ_B) in Fig. 3 (D and F). This discrepancy in the saturation magnetic moment between two materials stems from the different crystal field hybridization of d_xz and d_yz orbitals (17 meV for CaCoGe₂O₆ and 3 meV for CaCoSi₂O₆; referring to fig. S6), which results from varying degrees of distortion in the CoO₆ octahedra. Third, in both materials, the magnetic moments contain considerably large orbital contribution, namely, 1.30 μ_B (26%) for $\tilde{x}$ and 0.64 μ_B (20%) for $\tilde{y}$ of CaCoGe₂O₆ and 0.95 μ_B (22%) for $\tilde{x}$ and 0.97 μ_B (22%) for $\tilde{y}$ of CaCoSi₂O₆, which are in good agreements with the experimental data. The large orbital component in the magnetic moment can be interpreted by the domination of Hunds’ coupling over the octahedral crystal field splitting, leading to large portion of ${\mathrm{t}}_{2\mathrm{g}}^5{\mathrm{e}}_{\mathrm{g}}^2$ rather than ${\mathrm{t}}_{2\mathrm{g}}^6{\mathrm{e}}_{\mathrm{g}}^1$ configurations in the many-body ground state. For the spin state with ${\mathrm{t}}_{2\mathrm{g}}^5{\mathrm{e}}_{\mathrm{g}}^2$ configuration, the formation of magnetic moment can be understood from LS coupling scheme with total orbital angular moment L = 1 and spin angular moment S = 3/2, which contains large portion of orbital magnetic moments.

Table 1. Calculated effective magnetic moments.

Materials	Magnetic principal axis	Total effective magnetic moments (μB)	Saturation magnetic moments (μB)	Orbital effective magnetic moments (μB)	Spin effective magnetic moments (μB)	Percentage of orbital component
CaCoGe2O6	$\tilde{x}$	5.07	2.93	1.30	3.78	26%
	$\tilde{y}$	3.19	1.84	0.64	2.55	20%
	$\tilde{z}$	2.44	1.41	0.35	2.10	14%
CaCoSi2O6	$\tilde{x}$	4.27	2.47	0.95	3.32	22%
	$\tilde{y}$	4.32	2.49	0.97	3.35	22%
	$\tilde{z}$	1.95	1.13	0.12	1.83	6%

DISCUSSION

In this work, the Co-bearing pyroxenes CaCoGe₂O₆ and CaCoSi₂O₆, as well as their solid solutions, have been investigated both theoretically and experimentally. Their crystal structures extracted from our XRD data are in good agreement with those reported in the literature (23, 25). The Curie-Weiss fits of the isotropic temperature-dependent magnetization data of both compounds indicate the local maxima of susceptibility curves as the onset of AFM ordering, consistent with the neutron powder diffraction data (23, 25). The calculated effective moments reveal substantial contributions due to SOC. Upon cooling down below T_N, field-dependent magnetization data unveil a sharp field–induced transition for CaCoGe₂O₆ and multiple magnetization plateaus beneath the full saturation moment for CaCoSi₂O₆. These magnetization anomalies are highly anisotropic. Heat capacity data indicate that the spins in both compounds are dominantly Ising-like, even after being manipulated by high applied magnetic fields.

Beyond revealing the enormous orbital contribution to magnetism, our first-principles calculation results further justify the observed discrepancy in anisotropic magnetic saturation between two compounds, which stems from different crystal field hybridizations of Co 3d orbitals. Furthermore, despite the fact that our single-ion model manages to capture some substantial aspects when comparing these two compounds, additional phenomena such as the plateau observed only in the direction perpendicular to [010] under low magnetic fields for CaCoSi₂O₆, as well as the discrepancy in magnetic fields required to induce any saturation between these two compounds, cannot be adequately addressed if only the local interactions are taken into account. Hence, it ultimately proves that exchange interactions beyond our single-ion model, which are strongly affected by the bridging tetrahedra (SiO₄ versus GeO₄), play an important role during magnetization.

In summary, we conclude that the choice of bridging tetrahedra (i.e., SiO₄ versus GeO₄) between adjacent 1D CoO₆ chains in a pyroxene-based quantum magnet can locate magnetization plateaus in a measurable field range, potentially attributed to the empty 3d orbitals of Si that enable electron hopping. Our research makes pyroxenes an excellent platform to study the dynamically intertwined lattice, orbital, charge, and spin degrees of freedom in the quantum regime, as a result of their highly versatile crystal structure and fruitful choice of elements. Further investigations of these materials including but not limited to various spectroscopic measurements (neutron, Raman, etc.) and magnetization/heat capacity under extreme high fields will be of future interest.

MATERIALS AND METHODS

Materials synthesis

The powders of magenta CaCoGe_2−xSi_xO₆ (x = 0, 0.5, 1, 1.5, and 2) were made through conventional solid-state synthesis. Stoichiometric amounts of CaCO₃ (Alfa Aesar; 99.999%), CoO (Sigma-Aldrich; 99%), SiO₂ (Sigma-Aldrich; 99.5%), and GeO₂ (Alfa Aesar; 99.999%) were thoroughly ground together using an agate mortar and pestle and then transferred into a dense alumina crucible. The reaction mixture was first slowly (ramp rate = 1°C/min) heated to 1000°C in air, held overnight to decompose the carbonate, and then annealed in air at 1100° to 1200°C (ramp rate = 3°C/min) for two to three periods of 48 hours with intermittent grindings. The compounds with a high Si content required SiO₂ in a fine particle size to facilitate the product formation.

Macroscopic single crystals of CaCoSi₂O₆ and CaCoGe₂O₆ were grown via the floating-zone method. The precursor was ground using an agate mortar and pestle for 10 min to achieve a fine and uniform polycrystalline state to facilitate rod making. The powder was then molded into a long thin balloon and pressed under 60 MPa in a hydraulic press. The rods were sintered in a high-temperature furnace at 1150° and 1250°C for 8 to 12 hours for the Si and Ge compounds, respectively. The optical floating zone furnace (Crystal Systems, HP-10000) has four halogen lamps. We used an airflow of 2 liters/min under 1-atm pressure. After a stable molten zone was formed, the feed rod and seed rod rotated in a reverse direction, each at 25 rpm. Since the material was quite easy to crystalize, a flat crystal surface formed at a low rotation speed disconnected the molten zone. Thus, a high rotating speed was necessary to maintain a uniform molten zone. The growth rate was kept at 5 mm/hour until the end of growth. Scanning electron microscopy images of as-grown crystals were collected using a Quanta 200 FEG environmental scanning electron microscopy.

Powder-XRD and single-crystal XRD measurements and refinements

The solid-state ceramic reaction progress was monitored using laboratory p-XRD data collected at room temperature on a Bruker D8 FOCUS diffractometer (Cu Kα) over the 2θ range between 5° and 70°. Once the reactions were deemed finished, laboratory p-XRD data with much better statistical significance, covering a 2θ range between 5° and 110°, were collected from each sample. The Le Bail fitting of the acquired p-XRD patterns was conducted via the TOPAS software.

Single-crystal XRD was carried out using a Bruker D8 Quest Eco diffractometer equipped with a Photon III CPAD detector and monochromated Mo Kα radiation (λ = 0.71073 Å). Frames were integrated and corrected for Lorentz and polarization effects using SAINT 8.40b and were corrected for absorption effects using SADABS V2016/2 (38). Space groups were assigned on the basis of systematic absences, the E-statistics, agreement factors for equivalent reflections, and the successful refinement of the structures. The structures were solved by direct methods, expanded through successive difference Fourier maps using SHELXT, and refined against all data using the SHELXL-2018 (39, 40) software package as implemented in Olex2 (41, 42). Weighted R factors, R_w, and all goodness-of-fit indicators are based on F². Thermal displacement parameters for all atoms were refined anisotropically. The crystal structures were visualized by the VESTA program.

Physical property measurements

The magnetization data were collected using the VSM option of a Quantum Design Physical Property Measurement System. Temperature-dependent magnetization (M) data were collected under an applied external field (H) of 0.1 T. Magnetic susceptibility is defined as M/μ₀H. Field-dependent isothermal magnetization data between applied fields of +9 and −9 T were collected at various temperatures. Isotropic magnetization data were collected from crushed and ground FZ-grown crystals, while anisotropic magnetization data were collected with the applied external field perpendicular or parallel to the relatively flat surface of the FZ-grown crystals. Heat capacity was measured using the standard relaxation method in the Physical Property Measurement System over the temperature range of 1.8 to 50 K under an applied external field of 0, 3, 6, and 9 T, with the 0-T data extended to 150 K for the background phonon fitting.

Laue alignment for crystal orientations

Laue pattern images were produced using the HUBER Laue Imaging Plate x-ray diffractometer with a 1500-W Cu target. Samples were mounted parallel to beam axis at a distance of 10 mm from a collimator tube with an aperture of 0.8 mm. The fine-focus x-ray was operated at 40 mA and 20 kV in air at room temperature, and samples were measured for a typical exposure time of 1 to 3 min—longer times proving necessary for higher-quality patterns. These diffraction patterns were then transferred onto the OrientExpress software and oriented perpendicular with the [010] axis given lattice parameters derived from single-crystal XRD measurements.

Theoretical calculations

Local moments including spin and orbital components are crucial ingredients of the complicated magnetism in both CaCoGe₂O₆ and CaCoSi₂O₆. After extracting the crystal field and SOC terms from first principles, a single-ion model is derived to understand the formation mechanism of local moments of Co²⁺ ion in both materials, which reads

$${\widehat{H}}_{\text{atom}}={\widehat{H}}_{\text{kanamori}}+{\widehat{H}}_{\text{cf}}+{\widehat{H}}_{\text{soc}}$$

where the first term encodes both the Coulomb interaction (U) and Hund’s coupling (J) and the last two terms denote the local crystal field interaction and SOC, respectively. The Vienna ab initio simulation package (43, 44) is used to perform the first-principles calculations with the choice of the Perdew-Burke-Ernzerhof generalized gradient approximation for the exchange-correlation potential (43). An energy cutoff of 550 eV for the plane wave basis and a fine grid of 9 × 9 × 9 for k-point are adopted. Then, a tight-binding model is constructed with the help of Wannier90 software, which quantifies the crystal field interaction (∆_Oct for octahedral splitting between t_2g and e_g orbitals, ∆₁ for further splitting between d_xz/d_yz and d_xy due to the symmetry breaking from C₃ to C₂ symmetry) and SOC (λ_soc for the strength of SOC). Because the little group of the Co²⁺ ion is C₂ with the two-fold axis along the crystallographic unique b axis, we select the local axes x and y to lie in the O-Co-O plane, with the constraint that x + y equals b. For CaCoGe₂O₆, ∆_Oct ≈ 830 meV, ∆₁ ≈ 33 meV, and λ_soc ≈ 66 meV, while ∆_Oct ≈ 910 meV, ∆₁ ≈ 59 meV, and λ_soc ≈ 67 meV for CaCoSi₂O₆. A typical value of 5 eV for the Hubbard U and 0.8 eV for the Hunds’ coupling J is set for the many-body interactions. Following the method we proposed in a previous paper (37), the lowest Kramers’ doubly degenerate eigenstates (pseudo-spin space) are obtained by solving the many-body atomic Hamiltonian within the Fock subspace of seven electrons, which is about 25 and 20 meV below the first excited states for CaCoGe₂O₆ and CaCoSi₂O₆, respectively. The theoretical discussion is based on the calculation within the aforementioned pseudo-spin subspace.

For convenience, we define the isotropic effective magnetic moment as

$${\mathrm{\mu }}_{\text{eff}}^{\text{iso}}=\sqrt{{\mathrm{\mu }}_{\text{eff},\tilde{x}}^2+{\mathrm{\mu }}_{\text{eff},\tilde{y}}^2+{\mathrm{\mu }}_{\text{eff},\tilde{z}}^2}$$

where ${\mathrm{\mu }}_{\text{eff},\tilde{x}}$, ${\mathrm{\mu }}_{\text{eff},\tilde{y}}$, and ${\mathrm{\mu }}_{\text{eff},\tilde{z}}$ are the calculated effective magnetic moment along each principal axis.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S7

Tables S1 to S3