Observation of Unprecedented Fractional Magnetization Plateaus in a New Shastry-Sutherland Ising Compound

TL;DR

Er$_2$Be$_2$GeO$_7$ realizes an Ising Shastry–Sutherland lattice (SSL) with Er$^{3+}$ dimers and exhibits fractional magnetization plateaus at $m=1/4$ and $m=1/2$ for $H\parallel[001]$, defying the canonical $m=1/3$ SSL prediction. A subtle orthorhombic distortion introduces intra-dimer exchange anisotropy, realized as an anisotropic Shastry–Sutherland Ising model (ASSLIM) with $Cmm2$ symmetry, which reproduces the observed plateau sequence and field-induced magnetic peaks. Analyses combining crystal-field theory, neutron scattering, and CTMRG indicate strong Ising anisotropy with a dominant $c$-axis moment and reveal the role of longer-range interactions and quantum fluctuations in lifting degeneracy and eliminating residual entropy. The work establishes Er$_2$Be$_2$GeO$_7$ as a platform for exploring frustrated magnetism and demonstrates how lattice distortions can stabilize unexpected plateau states in rare-earth SSL systems, with accessible energy scales enabling precise experimental tests.

Abstract

Geometrically frustrated magnetic systems, such as those based on the Shastry-Sutherland lattice (SSL), offer a rich playground for exploring unconventional magnetic states. The delicate balance between competing interactions in these systems leads to the emergence of novel phases. We present the characterization of Er2Be2GeO7, an SSL compound with Er3+ ions forming orthogonal dimers separated by non-magnetic layers whose structure is invariant under the P-421m space group. Neutron scattering reveals an antiferromagnetic dimer structure at zero field, typical of Ising spins on that lattice and consistent with the anisotropic magnetization observed. However, magnetization measurements exhibit fractional plateaus at 1/4 and 1/2 of saturation, in contrast to the expected 1/3 plateau of the SSL Ising model. By comparing the energy of candidate states with ground-state lower bounds we show that this behavior requires spatially anisotropic interactions, leading to an anisotropic Shastry-Sutherland Ising Model (ASSLIM) symmetric under the Cmm2 space group. This anisotropy is consistent with the small orthorhombic distortion observed with single-crystal neutron diffraction. The other properties, including thermodynamics, which have been investigated theoretically using tensor networks, point to small residual interactions, potentially due to further couplings and quantum fluctuations. This study highlights Er2Be2GeO7 as a promising platform for investigating exotic magnetic phenomena.

Figures (18)

• Figure 1: Crystal structure and magnetic properties of Er_2Be_2GeO_7. (a) Schematic representation of Er_2Be_2GeO_7 crystal structure, highlighting the 2D Er^3+ network with nearest and next nearest neighbor Er-Er bond lengths of 3.31 and 3.92, respectively, and interlayer separation of 4.72. (b) Isothermal magnetization $M_{[001]}$ measurements at various temperatures, exhibiting saturation to $7.39~\mu_B$ at 7 T. (c) Isothermal magnetization $M_{[010]}$ at various temperatures, showing a tendency to saturate around $4.5~\mu_B$. (d) Low-temperature magnetic susceptibility ($\chi_{[001]}$) showcasing an antiferromagnetic transition near 0.85 K with $H$=10 Oe applied in [001] direction. (e) Low-temperature magnetic susceptibility ($\chi_{[010]}$) revealing a ferromagnetic transition around 0.85 K when $H$=10 Oe applied in [010] direction. The missing data points from 1.5 K and 2.5 K are due to the use of different Helium-3 and Helium-4 setups. The inset presents a zoomed-in view of $M_{[010]}(H)$ at 0.3 K. Within the -0.2 T to +0.2 T range, a sigmoid-like curve is observed, altering $M_{[010]}$ from -1.5 $\mu_B$ to +1.5 $\mu_B$ over a mere $\pm0.07$ T field range. No hysteresis is observed during field sweeps. (f) Zoomed-in isothermal magnetization $M_{[001]}(H)$ at 0.3 K, illustrating the emergence of fractional magnetization plateaus in presence of the applied magnetic field, with prominent 1/4 and 1/2 plateaus. The right axis displays the derivative of the magnetization, $\Delta M/\Delta H$. The peaks of the $\Delta M/\Delta H$ curve are used to identify the critical fields associated with the beginning and end of each plateau.
• Figure 2: Field-dependent $C_m(T)$ of Er_2Be_2GeO_7 single crystal for $\mathbf{H} \parallel$ [001]. (a--d) $C_m(T)$ grouped by fields corresponding to plateau phases, illustrating transitions and peak evolution. (e) Phase diagram in the $(\mathbf{H},T)$ plane as revealed by a color plot of $C_m$. The green lines connect the phase transitions detected in the $m=0$ and $m=1/4$ plateaus, while the green circle emphasizes an isolated critical point at the boundary between the $m=1/4$ and $m=1/2$ plateaus. The arrows mark the positions of the first-order transitions between the plateaus as deduced from magnetization at low temperature. (f) $S_m$ calculated by $\int_{0.4~\mathrm{K}}^{T_{\mathrm{max}}} (C_m/T) \, dT$, showing similar entropy loss across plateau phases and exceeding $R\ln(2)$ above 3.5 K, indicating contributions from a low-lying CEF level.
• Figure 3: CEF excitations of Er_2Be_2GeO_7 obtained from inelastic neutron scattering. (a, b) Representative INS spectrum at 6 K using incident energies $E_i$ of 30 meV and 60 meV, respectively, collected at SEQUOIA. The non-magnetic Lu_2Be_2GeO_7 spectrum is subtracted for background, removing phonon contributions. (c,d) Constant $Q$ cuts of the experimental data spanning $\Delta E$ from 0 to 25 meV with $E_i=30$ meV and from 0 to 55 meV with $E_i=60$ meV. The red lines represent the CEF fits to the experimental data. The inset in (d) and (e) shows the CEF fits to the isothermal magnetization and magnetic susceptibilities for $[001]$ and $[010]$ directions.
• Figure 4: Magnetic structure characterization at 0 T in the ordered state of Er_2Be_2GeO_7. (a) NPD spectrum at 100 K, illustrating a successful refinement within the space group $P\overline{4}2_1m$ with a low $R_{wp}$ of 5.02%. Points around $2\theta= (61.4^\circ-62.3^\circ, 72.0^\circ-73.8^\circ, 113.1^\circ-115.9^\circ )$ were excluded from the refinement due to prominent Al can peaks. (b) NPD spectra at 0.3 K, showcasing the emergence of magnetic Bragg peaks atop the nuclear Bragg peaks. Fit from the proposed magnetic model is shown in red and agrees well with the data achieving a low $R_M$ value of 2.17. (c) Schematic representation of magnetic structures of Er_2Be_2GeO_7, showing the magnetic unit cell outlined in black, with Er atoms labeled 1, 2, 3, and 4, and corresponding moments are listed in table in panel (d). Each dimer pair has an antiferromagnetic arrangement of spins with a slight canting along the dimer bond. (d) Magnetic moments (in $\mu_B$) and fractional atomic coordinates of Er atoms. The lattice constants are $a=b=$7.377 and $c=$4.777.
• Figure 5: Single-crystal neutron diffraction results for the $(h, k, 0)$ plane obtained by integrating over the elastic energy range $-0.05 < \Delta E < 0.05$ meV and along $l$ with $-0.1 < l < 0.1$. The Brillouin zones are outlined with white dashed lines for reference. (a) Diffraction data measured at $T = 12$ K in the paramagnetic phase revealing weak intensities at the forbidden $(1,0,0)$ and $(0,1,0)$ reflections. (b) Structural distortion in the $Cmm2$ space group, described by a basis change $\mathbf{a}' = \mathbf{a} + \mathbf{b}$, $\mathbf{b}' = -\mathbf{a} + \mathbf{b}$, $\mathbf{c}' = \mathbf{c}$, and an origin shift $\left(0, \tfrac{1}{2}, 0\right)$. Here, one dimer sublattice (purple) expands while the other sublattice (magenta) contracts, as indicated by white arrows. This distortion results in anisotropic inter- and intra-dimer bond lengths. (c-f) Magnetic diffraction data collected at $T = 60$ mK, obtained by subtracting the paramagnetic dataset shown in (a). Panels depict different magnetization phases: (c) $m=0$ phase at $H=0$ T, (d) $m=1/4$ phase at $H=0.275$ T, (e) $m=1/2$ phase at $H=0.45$ T, and (f) $m=1$ phase at $H=7$ T.