Signatures of spin-liquid state in a 3D frustrated lattice compound KSrFe₂(PO₄)₃ with S = 5/2

Geometrically frustrated antiferromagnetic (GFAFM) systems have attracted considerable attention in condensed matter physics due to their exotic magnetic properties, such as the spin-liquid state. In those GFAFM systems, spins are put into a bizarre situation that does not allow a single conventional antiferromagnetic (AFM) ground state because they fail to satisfy all the AFM interactions simultaneously. Moreover, it has been known that the materials with a small spin value of S = ½ bear strong zero-point quantum fluctuation even at T = 0 K.^1,2 Therefore, the spin lattices with geometrically frustrated interaction, with low dimensions and low spin values, have been actively investigated in the search for exotic spin liquid states. A few ideal examples for the two-dimensional (2D) edge-shared triangular lattice and 2D corner-shared triangular (kagome) lattice are κ-(BEDT-TTF)₂Cu₂(CN)₃ and ZnCu₃(OH)₆Cl₂, respectively.^3–7 These materials are promising examples of the quantum spin-liquid (QSL) state with S = 1/2 moments and exhibit strong quantum fluctuations. As quantum fluctuations are known to be dominant in low-dimensional materials and diminish in higher dimensions, it has been challenging to discover a QSL candidate with no static spin ordering in three-dimensional (3D) systems, especially with large spins, such as S = 5/2.

Recently, emergent magnetic behavior has been observed in 3D geometrically frustrated magnets due to the interplay between competing spin interactions and quantum fluctuations. A few examples of 3D GFAFM systems can be found in corned-shared triangle (hyperkagome and trillium) lattices and corned-shared tetrahedral (pyrochlore) lattices; the 3D corner-shared triangular networks include Na₄Ir₃O₈,^8–10 PbCuTe₂O₆,^11–14 Yb₃Ga₅O₁₂,¹⁵ and K₂Ni₂(SiO₄)₃.¹⁶ Na₄Ir₃O₈ has a 3D hyperkagome structure with Ir⁴⁺ ions forming a 3D corner-shared triangular network with z = 4, where z is the number of nearest neighbor (NN) atoms.⁸ Ir⁴⁺ (5d⁵) bears the magnetic moment of J_eff = ½ due to the strong spin–orbit coupling. Initially, magnetic data analysis shows no long-range order (LRO), and the heat capacity follows T² behavior, indicating a formation of a spin-liquid phase with gapless excitations.^8,9 The recent nuclear magnetic resonance (NMR) and neutron scattering studies reveal slowing down of spin fluctuations and a frozen spin state in the system.^17,18 On the other hand, the system PbCuTe₂O₆ has an S = ½ (Cu²⁺), 3D corner-shared triangular lattice.¹¹ Bulk magnetic susceptibility suggests that it is a paramagnetic system with a Curie–Weiss temperature (θ_CW) of −22 K. Neither magnetic ordering nor spin freezing is seen down to 20 mK. Electronic structure calculations suggest that it has a 3D hyperkagome network with additional frustrated couplings. Low-temperature NMR, muon-spin rotation, and inelastic neutron scattering studies have revealed that it is a spin-liquid system with gapless, fractionalized spinon excitations.^12,13 Recently, a ferroelectric ordering with quantum critical behavior has been observed at 1 K due to the strong lattice distortion.¹⁴ 

In search of 3D QSLs, recently, interest has been raised in another type of 3D frustrated system called the trillium lattice, which constitutes a 3D network of corner-shared triangles with z = 6. The trillium lattice is usually found in B20 structures. A few example materials for B20 structures are MnSi, FeGe, and FeSi.^19–23 Due to the lack of inversion symmetry, these trillium lattice systems exhibit skyrmion-type ordering.²⁴ A few more frustrated trillium lattice network systems include CeIrSi,²⁵ demonstrating a spin-ice ground state with Ising interactions, and EuPtX (Ge, Si),²⁶ showing strong magnetic frustration with a finite LRO at 4.1 K. Recently, the field-induced quantum spin liquid behavior has been found in an S = 1 3D frustrated system K₂Ni₂(SO₄)₃. Magnetic susceptibility data follow the Curie–Weiss behavior with θ_CW = −18 K, suggesting AFM interactions. Heat capacity data (C_p) show the second order transition at 1.14 K and the first order transition at 0.74 K. The magnetic field H suppresses these transitions, and C_p data, following T² behavior, suggest the field-induced QSL ground state with gapless spinon excitations. It was claimed that K₂Ni₂(SO₄)₃ is an inter-connected trillium lattice of two Ni²⁺ ions; however, the non-trillium coupling is, indeed, more dominant than trillium interactions, according to the reported electronic structure calculations.¹⁶ The presence of dominant non-trillium couplings could render the system exhibit LRO at finite temperatures.

To explore exotic properties in the class of 3D complicated network systems, we have investigated a compound KSrFe₂(PO₄)₃ that has a similar structure to K₂Ni₂(SO₄)₃ but has a spin S = 5/2. Quite interestingly, no long-range order (LRO) is found down to 0.19 K although it has relatively strong AFM correlation with θ_CW = −70 K. Magnetic heat capacity follows the power law T ^α (α ∼ 2.3) behavior, suggesting that our titled compound is a possible candidate of forming a spin-liquid state with gapless excitations in the family of 3D frustrated spin network with S = 5/2.

Polycrystalline samples of KSrFe₂(PO₄)₃ were synthesized using the conventional solid-state heating method at 1000 °C.²⁷ The non-magnetic compound KSrSc₂(PO₄)₃²⁸ with the same structure as KSrFe₂(PO₄)₃ was prepared at 1100 °C. Polycrystalline powder samples were used to measure x-ray diffraction (XRD), magnetization, and heat capacity measurements. The magnetic and heat capacity measurements on polycrystalline samples were carried out using the Physical Properties Measurement System (PPMS, Quantum Design). The low-temperature heat capacity was measured on a tiny pellet using a CF-150 dilution fridge (Leiden cryogenics). The Rietveld refinement of the XRD profile has been done using FULLPROF SUITE software [see Fig. 1(a)]. The extracted lattice parameters are a = 9.780 Å and χ² = 2.30, R_p = 20.2%, R_wp = 19.2%, and R_exp = 12.63%. As reported previously,²⁷ there is nearly 20% of site exchange between K and Sr atoms.

The compound KSrFe₂(PO₄)₃ holds a cubic crystal system with space group P2₁3 (space group number 198). The 3D framework is built by Fe1O₆ and Fe2O₆ octahedral units, PO₄ tetrahedral units, K, and Sr atoms, as shown in Fig. 1(b). The Fe1O6 octahedral unit is slightly distorted than the Fe2O₆ octahedral unit. The magnetic interactions are exchanged among Fe1 and Fe2 atoms through PO₄ units. The first, second, and third NN couplings (J₁, J₂, and J₃) are between Fe1 and Fe2 atoms. The combination of first NN and second NN couplings forms a J₁–J₂ anisotropic diamond lattice, known to be a non-frustrated system. After adding the third NN coupling (J₃), the system still remains non-frustrated, as shown in Fig. 2(a). On the other hand, the fourth and fifth NN couplings (J₄ and J₅) are through Fe1–Fe1 and Fe2–Fe2 bonds, respectively (see Table I). The couplings J₄ and J₅ alone built up two independent trillium lattices, respectively [see Fig. 2(b)], which are highly frustrated 3D triangular networks. Overall, the compound KSrFe₂(PO₄)₃ has the combination of all exchange couplings J₁, J₂, J₃, J₄, and J₅ and stands for a complicated 3D frustrated network system, as shown in Fig. 2(c). The super-super-exchange interaction between Fe³⁺ ions is through the O–P–O path. The details of various magnetic couplings are listed in Table I. The magnetic exchange energies were extracted from DFT calculations, which will be discussed in Sec. IV.

Temperature-dependent magnetic susceptibility χ(T) measurements were carried out on polycrystalline samples of KSrFe₂(PO₄)₃ down to 2 K at H = 10 kOe, as shown in Fig. 3(a). No conventional magnetic LRO is observed in the χ(T) data down to 2 K. A minor splitting between zero-field-cooled (ZFC) and field-cooled (FC) susceptibility data is noted at 3.5 K in the low fields, as shown in the inset of Fig. 3(a). However, this splitting disappears at a small H = 5 kOe. It could be due to the site exchange between K and Sr atoms, as also noted in many geometrically frustrated magnets, such as Na₄Ir₃O₈⁸ and LiGaCr₄S₈,²⁹ due to the disorder. The inverse χ(T) data follow the Curie–Weiss behavior at high temperatures, as shown in Fig. 3(b). The data fit to the expression 1/χ = 1/[χ_dia + χ_vv + C/(T − θ_CW)]. The extracted θ_CW is about −70 K, suggesting dominant AFM couplings. The calculated diamagnetic susceptibility (χ_dia) from the individual ions of K⁺, Sr²⁺, Fe³⁺, P⁵⁺, and O²⁻ is −9.75 ⨯ 10⁻⁵ cm³/mol.³⁰ The yielded Van Vleck susceptibility χ_vv from the fit is 1 ⨯ 10⁻⁵ cm³/mol-Fe. The calculated effective magnetic moment from the Curie constant (C) = 4.43 cm³ K/mol-Fe is about μ_eff = 5.95 μ_B, which is consistent with the expected value for S = 5/2 of the Fe³⁺. The χ(T) data show field-independent behavior down to 2 K in the magnetic fields of up to 50 kOe [see Fig. 3(c)]. Interestingly, the log–log scale of χ(T) data shows nearly temperature-independent behavior at low temperatures, as seen in the gapless spin-liquid materials, such as PbCuTe₂O₆, EtMe₃Sb[Pd(dmit)₂]₂, and κ-H₃(Cat-EDT-TTF)₂.^12,31,32 The H-dependence of magnetization M(H) was also measured at 1.9 K by varying H up to 160 kOe, as shown in Fig. 3(d). The magnetization did not saturate and reached only 1.42 μ_B even at 160 kOe, indicating the presence of AFM correlations in this system. The absence of hysteresis indicates the antiferromagnetic nature of the samples.

Temperature-dependent heat capacity C_p(T) measurements were done down to 0.19 K at H = 0 and 30 kOe on KSrFe₂(PO₄)₃ samples. C_p(T) for the nonmagnetic analog KSrSc₂(PO₄)₃ is also measured down to 2 K at 0 kOe, as shown in Fig. 4(a). The C_p(T) data of KSrSc₂(PO₄)₃ are well-matched with its C_p(T) data of KSrFe₂(PO₄)₃ at high temperatures. It deviates at low temperatures due to the presence of magnetic contribution of KSrFe₂(PO₄)₃. Following the analysis on KSrSc₂(PO₄)₃,²⁸ the C_p(T) data fit the polynomial AT³ + BT⁵ in the range from 1.8 to 7 K. It is extrapolated down to 0.19 K. The magnetic heat capacity C_m(T) is obtained after subtracting C_p(T) of KSrSc₂(PO₄)₃ from that of KSrFe₂(PO₄)₃. The C_m(T) data do not show any sharp peak down to 0.19 K, suggesting the absence of magnetic ordering. In particular, C_m(T) does not have an anomaly at 3.5 K, where ZFC-FC magnetic susceptibility shows the bifurcation, which rules out the possibility of magnetic LRO. However, the possibility of spin-glass behavior and its percentage of the frozen spin state are to be further investigated through the detailed Muon spin rotation measurements. The frustration parameter (f), the ratio of Curie–Weiss temperature (θ_CW) to Néel temperature (T_N), is larger than 370. C_m(T) data show a broad maximum at 9 K, indicating the presence of short-range spin correlations [see Fig. 4(b)] as observed in most of the geometrically frustrated magnets. The C_m data are compared with the T² and T³ power-law fits. The C_m(T) data fall between T² and T³ behavior, meaning that the power is between 2 and 3. To understand further, we have fitted the C_m data with the power-law, T^α, over the temperature range from 0.19 to 1.7 K, resulting in power-law behavior with a coefficient of α ≈ 2.3 [see Fig. 4(c)]. In general, the C_m(T) data follow the T³ behavior for conventional AFM LRO systems due to 3D magnon excitations, which, thus, rule out the presence of AFM LRO in our system. Conversely, the sublinear (T^2/3) dependent behavior is expected for the 2D QSL with spinon Fermi surface. Table II shows the experimental 3D spin-liquid systems following the nearly quadratic behavior. The peculiar power-law behavior of C_m(T) in our system could be due to the spin-liquid nature of the system. Furthermore, detailed theoretical and experimental studies are needed to understand the precise nature of excitations. Despite the application of H = 110 kOe, no field dependence was observed in this compound, suggesting that the low-energy excitations are robust in this spin-liquid candidate.⁸ 

To confirm the absence of magnetic LRO, neutron diffraction (ND) experiments were performed on the polycrystalline samples of KSrFe₂(PO₄)₃ down to 2 K. The measurements were carried out using the neutron powder diffractometer PD-I (λ = 1.094 Å) at various temperatures from 6 to 300 K. The ND data 2 and 20 K were obtained at another neutron powder diffractometer PD-II (λ = 1.244 300 Å) with three linear position-sensitive detectors at the Dhruva reactor, Bhabha Atomic Research Center, India. Rietveld refinement on ND data was performed using the FULLPROF software package using the inputs from the structural parameters obtained from the powder XRD data. Neither the additional magnetic Bragg peaks nor enhancement in the intensity of the fundamental nuclear Bragg peaks from 300 to 2 K was found (see Fig. 5). The measured ND pattern could be fitted by considering only the nuclear phase. In addition, no significant difference was found upon high-temperature ND data at 300 K being subtracted from that at 6 K. This suggests that the broad maximum in the heat capacity may stem from short-range correlations. Similarly, no significant difference was seen in the subtracted ND data at 20 K from that at 2 K, ruling out the presence of magnetic LRO down to 2 K.

The electronic structure calculations were performed using a full-potential linearized muffin-tin orbital method as implemented in the RSPt code.^40–36 For the exchange-correlation functional, we have employed a generalized gradient approximation + Hubbard U (GGA+U) approach. The Brillouin zone integration is done using the thermal smearing method with an 8 × 8 × 8 k-mesh. For charge density and potential angular decomposition inside the muffin tin sphere, maximum angular momentum is considered to be l_max = 8. To account for electronic correlation within the rotationally invariant GGA+U approach,³⁷ we have considered coulomb interactions U = 4.0 eV³⁸ and Hund coupling J_H = 0.8 eV for Fe-3d states.

To begin with, we have checked the relative stability of a few possible magnetic states within the GGA+U approach and found that an AFM state where the NN Fe spins are oppositely aligned comes out to be the lowest in energy. The computed total and partial density of states for Fe-d and O-p states are displayed in Fig. 6. The total density of states [Fig. 6(a)] clearly shows that the ground state is insulating with a gap of 2.2 eV. From the projected density of states [Figs. 6(b) and 6(c)], we find that the majority Fe-t_2g and e_g, states are completely occupied, while the corresponding minority states are completely empty. According to Hund’s rule, such distribution of Fe-states is expected for Fe³⁺ (d⁵). The gap primarily arises between the majority and minority d-states, which are exchange split due to the formation of local moments and the correlation effect of the Fe-3d subshell. The magnetic moment on the Fe-site is estimated to be 4.3 µ_B, which is smaller than the moment expected from the nominal d⁵ configuration. This also implies strong Fe-d and O-p hybridization in this system, as evident from the spectral distribution of the O-p states [Fig. 6(c)], which appears in a broader energy range between −4 eV and Fermi energy. We will see below that such strong d-p hybridization will facilitate a super-super-exchange mechanism for the magnetic interactions between the Fe-spins, which are even quite far from each other.

Furthermore, we have attempted to understand the nature of magnetism and employed the magnetic force theorem to extract magnetic exchange energies.^45–41 The computed inter-site exchange interactions are displayed in Table I. Surprisingly, the fifth NN coupling (J₅) is dominant among all five exchange couplings although the bond distance of J₅ is larger than those of other couplings, which forms the trillium lattice by the Fe2 atoms [see Fig. 2(b)]. As per the structure, the fourth NN coupling J₄ also forms another trillium lattice by Fe1 atoms. The magnitude of J₄ is slightly smaller than that of J₅, which could be due to the distortion in Fe1O₆ being slightly larger than that of Fe2O₆ environments. In addition to these J₄ and J₅ couplings, there are strong and compatible AFM couplings J₁, J₂, and J₃. All these spin interactions make a complicated 3D spin system.

To understand the origin of the highly frustrated magnetism and spin liquid features in the complicated 3D spin network, one needs to look into the details of structural features and magnetic coupling strengths. It is clear from Fig. 2(a) that the system is non-frustrated upon the magnetic couplings of J₁, J₂, and J₃ only being considered. Hence, we need to look at other couplings, J₄ and J₅, to trace the origin of magnetic frustration. Figure 2(b) clearly shows that J₄ and J₅ interactions built two uniform equilateral triangular networks (3D perfect trillium lattices). From the DFT electronic structure calculations, it is noted that J₅ is a dominant AFM coupling, which is the starting point to explain the geometrical frustration in this compound as the J₁–J₂ anisotropic diamond lattice is not frustrated even with the inclusion of the J₃ coupling. However, the isolated trillium lattice model is insufficient to describe the geometrical frustration of this compound. There exist two trillium lattices formed by J₄ and J₅ couplings, respectively. The connections between two trillium lattices are made by the three interactions of J₁, J₂, and J₃. These are AFM of whose strengths are almost comparable to those of J₄ and J₅. Importantly, these AFM inter-trillium connections again produce the triangular-type of interactions between J₄ and J₅ triangles [see Fig. 2(c)]. These connections make the J₄–J₂ and J₅–J₃ anisotropic tetrahedral environments, which are another type of frustrating motifs [see Fig. 2(c)]. Overall, the compound forms a more complicated 3D frustrated spin network than the trillium lattice. The formation of a complex 3D frustrated network with competing and compatible magnetic interactions (J₁, J₂, J₃, J₄, and J₅) is likely the primary source of high frustration and the related gapless spin liquid feature in the compound.

Since the crystal structure and space group of KSrFe₂(PO₄)₃ are the same as those of K₂Ni₂(SO₄)₃, it will be worthwhile to compare the structural and magnetic features of the two compounds. There are a few differences noticed, leading to different magnetic ground states. While K₂Ni₂(SO₄)₃ shows magnetic LRO at zero magnetic fields, the titled compound KSrFe₂(PO₄)₃ shows the spin-liquid feature. In the case of K₂Ni₂(SO₄)₃, one of the interactions (J₂) is, indeed, ferromagnetic; the combination of AFM and FM couplings likely reduces the levels of magnetic frustration. In addition, one of the inter-trillium couplings (J₃) in the compound K₂Ni₂(SO₄)₃ is nearly three times more dominant than the intra-trillium couplings.¹⁶ Unlike K₂Ni₂(SO₄)₃, all the magnetic couplings in the KSrFe₂(PO₄)₃ are AFM and comparable with each other, which may enhance the levels of magnetic frustration. The elevated levels of magnetic frustration in the compatible magnetic interactions could lead to the robust spin-disorder in KSrFe₂(PO₄)₃. Moreover, the system has a site disorder between K and Sr atoms and thus, induces, some randomness in magnetic exchange couplings. The randomness could destabilize the LRO and help the system stay in a spin-liquid ground state.⁴² Our experimental and theoretical results, thus, reveal that the titled compound is a suitable candidate for forming the spin-liquid state with gapless excitations.

We have investigated thermodynamic properties and spin interactions by electronic structure calculations in the compound KSrFe₂(PO₄)₃, which forms a 3D frustrated lattice with S = 5/2 magnetic moments. The absence of a conventional magnetic LRO is confirmed down to 0.19 K, despite the relatively sizable Curie–Weiss temperature θ_CW = −70 K. The power-law T^2.3 behavior of the magnetic heat capacity C_m(T) at lower temperatures, followed by a broad maximum at 9 K, supports the formation of a gapless ground state. The appearance of geometrical spin frustration is understood as due to the complicated frustrated AFM exchange couplings, rendering the system remains in an unusual spin-liquid state, at least down to 0.19 K. Our results suggest that the compound KSrFe₂(PO₄)₃ is an appealing candidate for forming a gapless spin-liquid state among the family of S = 5/2, 3D spin networks.

B.K. acknowledges the support from DST, India. The work at SNU was supported by the NRF (Grant No. 2019R1A2C2090648) and the Ministry of Education (Grant No. 2021R1A6C101B418). E.K. acknowledges the financial support from the labex PALM for the QuantumPyroMan project (Project No. ANR10-LABX-0039-PALM).

The authors have no conflicts to disclose.

K. Boya and K. Nam contributed equally to this work.

K. Boya: Conceptualization (equal); Investigation (equal); Methodology (equal). K. Nam: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal). K. Kargeti: Investigation (equal); Software (equal); Validation (equal). A. Jain: Data curation (equal); Formal analysis (equal). R. Kumar: Data curation (equal); Formal analysis (equal). S. K. Panda: Software (equal); Supervision (equal). S. M. Yusuf: Data curation (equal); Formal analysis (equal). P. L. Paulose: Data curation (equal); Formal analysis (equal). U. K. Voma: Data curation (equal). E. Kermarrec: Data curation (equal); Formal analysis (equal); Validation (equal). Kee Hoon Kim: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Supervision (equal). B. Koteswararao: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.