Critical behavior and evidence of dimensional crossover in quasi-two-dimensional Li$_2$FeSiO$_4$

We report thermal expansion and heat capacity studies on Li$2$FeSiO$4$ single crystals which enable us to investigate the critical behavior in the magnetically quasi-two-dimensional (2D) material. Pronounced $λ$-shaped anomalies at the magnetic ordering temperature $T{\rm N}$ imply significant magneto-elastic coupling. Our analysis of both the thermal expansion and the specific heat data implies the crossover from 2D Ising-like behavior for $|(T-T{\rm N})/T_{\rm N}|>0.3$ to 3D Ising behavior \rev{below $\simeq 1.3\times T_{\rm N}$. The 2D-like behavior is further supported by density functional calculations which show minimal dispersion perpendicular to the crystallographic $ac$ planes of the layered structure, thereby indicating the 2D nature of magnetism at higher temperatures.} Our results extend the available model materials of quasi-2D magnetism to a high-spin $S=2$ system with tetrahedrally coordinated Fe$^{2+}$-ions, thereby illustrating how magnetic order evolves in a 2D Ising-like system with orbital degrees of freedom.

💡 Research Summary

The authors present a comprehensive thermodynamic study of single‑crystalline Li₂FeSiO₄, a layered antiferromagnet containing high‑spin (S = 2) Fe²⁺ ions. Using high‑resolution capacitance dilatometry and heat‑capacity measurements from 2 K to 300 K, they identify sharp λ‑type anomalies at the Néel temperature Tₙ ≈ 17 K in both the linear thermal‑expansion coefficients (αₐ, α_b, α_c) and the magnetic contribution to the specific heat (c_mag). The anomalies are highly anisotropic: the a‑axis expands while the b‑ and c‑axes contract, indicating strong magneto‑elastic coupling and a negative volume anomaly that predicts a pressure‑induced suppression of Tₙ.

To extract the critical behavior, the authors fit the reduced temperature dependence of the volume thermal‑expansion coefficient α_v and c_mag to the standard power‑law form c = A|t|^{-α̃}+B+Et, where t = (T − Tₙ)/Tₙ. In the immediate vicinity of Tₙ (8.3 K < T < 21 K) they obtain critical exponents α̃_te ≈ 0.14 ± 0.03 and α̃_cp ≈ 0.11 ± 0.03, values that match the three‑dimensional Ising universality class (α̃ ≈ 0.11). At higher temperatures (22 K < T < 83 K) the fitted exponents drop to ≈0.02, essentially zero, which is the hallmark of two‑dimensional Ising criticality. This crossover from near‑zero to finite positive α̃ demonstrates a temperature‑driven dimensional crossover: short‑range, quasi‑2D magnetic correlations dominate well above Tₙ, while weak interlayer coupling becomes relevant close to the ordering transition, driving the system into a 3D Ising regime.

Complementary density‑functional theory (DFT) calculations (both LDA‑PW92 and GGA‑PBE) reveal that the Fe 3d bands near the Fermi level are isolated, flat, and largely confined to the ac‑plane. Wannier‑function analysis shows that most Fe 3d orbitals have negligible tails perpendicular to the layers, with only the d_{x²‑y²} orbital displaying a slight interlayer extension. Band dispersion along directions perpendicular to the layers (Γ→X, Z→U, etc.) is essentially flat, whereas in‑plane directions exhibit appreciable dispersion, confirming the quasi‑2D nature of the magnetic exchange pathways. Inclusion of spin‑orbit coupling lifts a small crystal‑field splitting (≈0.8 meV) to a sizable gap (≈30 meV), effectively quenching orbital degeneracy and reinforcing the Ising‑type anisotropy.

The pressure dependence inferred from the negative volume anomaly suggests that hydrostatic pressure reduces Tₙ, consistent with the notion of weak but finite interlayer coupling. The combination of experimental thermodynamics, critical‑exponent analysis, and electronic‑structure calculations positions Li₂FeSiO₄ as a rare example of a high‑spin, quasi‑2D magnet that undergoes a clear dimensional crossover in its critical behavior. This work extends the catalog of model systems for studying Ising physics, provides valuable insight into how orbital degrees of freedom and spin‑orbit interactions influence dimensionality, and offers a template for strain‑engineering magnetic transitions in layered oxides.