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

TL;DR

This paper investigates critical behavior and dimensional crossover in the magnetically quasi-2D Li$_2$FeSiO$_4$ using thermal expansion and heat capacity on single crystals, complemented by density-functional theory that reveals strong in-plane confinement of Fe $3d$ states. The experimental analysis shows lambda anomalies at $T_N$ and a crossover from 2D-Ising-like to 3D-Ising universality as $T$ approaches $T_N$, supported by a consistent 3D-Ising critical exponent near Tc and 2D-like behavior at higher temperatures. DFT and Wannier-downfolding demonstrate minimal interlayer dispersion and orbital contributions, with SOC lifting orbital degeneracy and producing a crystal-field splitting that stabilizes the 2D magnetic character. Overall, the work extends quasi-2D magnetism models to high-spin $S=2$ Fe$^{2+}$ systems and clarifies how dimensional crossover and magneto-elastic coupling emerge in layered oxides, with implications for strain tuning and orbital physics in 2D magnets.

Abstract

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.

Figures (8)

• Figure 1: Schematics of the crystal and magnetic structure of Li$_2$FeSiO$_4$ . FeO$_4$ tetrahedra are shown in beige, with Fe$^{2+}$ ions located inside. Blue and green arrows indicate the orientations of the antiparallel magnetic moments. Small black spheres represent oxygen. The sketch was made with the VESTA software Vesta using structural data from CCDC 1859157 Hergett2019b.
• Figure 2: (a) Linear and volume thermal expansion coefficients $\alpha_i$ ($i = a,b,c, \rm{vol}$) along the three crystallographic axes and for the volume (solid black line). (b) Magnetic specific heat $c_p^{\rm mag}$ obtained by correcting the measured specific heat by the lattice contribution from measurements of the non-magnetic analog material Li$_2$ZnSiO$_4$Hergett2025. The vertical dashed line indicates $T_{\rm N}$. Black triangles mark the dimensional crossover region as discussed in Sec. \ref{['ch:critical']}.
• Figure 3: (a,b) Semi-logarithmic plots of the volume thermal expansion coefficient $\alpha_{\rm v}$ and the magnetic specific heat $c_p^{\rm mag}$ vs. the reduced temperature $\left|t\right| = \left| (T - T_{\rm N})/T_{\rm N} \right|$ for $T_{\rm N}$ = 17.14 K and $T_{\rm N}$ = 17.11 K, respectively. Blue dots (black dots) mark data for $T < T_{\rm N}$ ($T > T_{\rm N}$). Lines are fits to the data using Eq. \ref{['eq:critical']} for different values of the critical exponent $\tilde{\alpha}^{\rm cp,te}$ (see text). (c,d) Experimental data and obtained fit functions. Triangles mark the crossover region.
• Figure 4: (a) Paramagnetic band structure of Li$_2$FeSiO$_4$ as computed with FPLO FPLO1999 using the PW92 functional. (b) Projected density of states for the Fe $3d$, O 2$p$, Li 2$s$ and Si 3$p$ orbitals.
• Figure 5: Isosurface plots of the Wannier functions localized on the iron shown in cut through the crystal structure parallel to the $ab$-plane. The Isosurface is chosen such that 90% of the electronic density is contained within. The subplots correspond to the (a) $3d_{{xy}}$, (b) ${3d}_{{yz}}$, (c) ${3d}_{{z}^2}$, (d) ${3d}_{{xz}}$, and (e) ${3d}_{{x}^2-{y}^2}$-like Wannier functions.