Atomically-sharp magnetic soliton in the square-net lattice EuRhAl$_{4}$Si$_{2}$

Abstract

Topological spin textures are hallmark manifestations of competing interactions in magnetic matter. Their effective description by nonlinear field theories reflects an energetic frustration that destabilizes uniform order while selecting finite-size, topologically nontrivial configurations as stationary states. Among the most extreme realizations are atomically-sharp domain wall excitations, namely one-dimensional (1D) magnetic solitons, which represent the ultimate scaling limit of magnetic textures. Such solitons may emerge in magnetic systems where effective exchange interactions compete directly with uniaxial magnetic anisotropy. Here we show that the square-net rare earth compound EuRhAl$_{4}$Si$_{2}$ realizes a very susceptible regime where the magnetic anisotropy competes with highly frustrated exchange interactions stabilizing a rare ferrimagnetic $\uparrow\uparrow\downarrow$ state that, under applied magnetic field, supports the formation of atomically-sharp soliton defects. We confirm the bulk response of the 1D magnetic solitons via magnetization and electrical transport measurements. We establish both the zero- and in-field $\uparrow\uparrow\downarrow$ order via neutron diffraction, while magnetic force microscopy visualizes its real-space evolution into a stripe-like array. To elucidate the microscopic origin of the soliton, we relate the Ruderman-Kittel-Kasuya-Yosida (RKKY)-driven exchange interactions and the magnetic anisotropy through density functional theory, and we construct an effective 1D $J_{1}$-$J_{2}$-$K$ model whose atomistic spin dynamics simulations reproduce the observed soliton states as a function of external field. Our results demonstrate that EuRhAl$_{4}$Si$_{2}$ hosts atomically-sharp, field-driven 1D magnetic solitons, providing a new platform for studying 1D topological excitations at the atomic length scale.

Figures (4)

• Figure 1: Magnetization and soliton defects in EuRhAl$_4$Si$_{2}$. (a) Magnetization isotherm for EuRhAl$_4$Si$_{2}$ taken at T = $1.8$ K and $H \parallel c$ and the corresponding differential magnetic susceptibility $\text{d}M/\text{d}H$ (right axis, dashed line). The gray box indicates the region of interest. Inset shows the side and top view of the EuRhAl$_4$Si$_{2}$ crystal structure. (b) Enlarged view of the $M = M_\text{sat}/3 \cdot (1 \pm 1/100)$ plateaus and the corresponding spin arrangement of the Eu moments (inset). (c) Resistivity $\rho_c$ measured for the FIB-fabricated microstructure measured at T = $1.8$ K (left solid black line) and the corresponding derivative d$\rho_c$/dH (right axis, dashed line). (d) Resistivity for $\rho_c$ (black line) and $\rho_a$ (blue line). The inset shows the FIB-fabricated microstructured device on EuRhAl$_4$Si$_{2}$ single crystal. (e) Phase-diagram of the 1D frustrated $J_{1}$-$J_{2}$-$K$ Heisenberg model (inset effective lattice model). At the isotropic case (magnetic anisotropy $K=0$, left panel), there are three phases: ferromagnetic (FM), antiferromagnetic (AFM) and the non-collinear spin-spiral (SS) phase. The orange dotted line corresponds to the period-3 SS phase. Shown in the orange box (right panel) is the phase change along the dotted line as function of $K$, with the Fan state, up-up-down state up to the strong anisotropic limit. (f) Schematic of the soliton defect, the structure is not to scale. Bottom step: Time-reversal domains with domain wall soliton consisting of down-down-up spins (below). Middle step: 1D representation of the up-up-down order with q = $1/3$, Top step: additional formation of soliton defects (above).
• Figure 2: Up-up-down magnetic structure of EuRhAl$_{4}$Si$_{2}$(a) Neutron diffraction patterns in the (HK0) scattering plane measured at $5$ K and $0$ T. Magnetic peaks are found at $\mathbf{k} = 0$, $\mathbf{k} = (1/3, 0 ,0)$ and $\mathbf{k} = (0, 1/3, 0)$. The latter two wave vectors are results of a- and b- direction magnetic twin domains. (b) Rocking curves of $(2/3, 0, 0)$ magnetic peak at zero field. Colors (blue to red) indicate temperatures from $5\,K$ to $12.5$ K. (c) Rocking curves of $(1, 0 ,0)$ Bragg peak measured at zero field. The intensity at $15$ K is from nuclear scattering, and the intensity at $5\,K$ is from both nuclear and magnetic scattering. (d) Up-up-down magnetic structure of EuRhAl$_{4}$Si$_{2}$ in the $3\!\times\! 1\times\! 1$ magnetic unit cell. (e) The temperature dependence of $(1, 1/3, 0)$ and $(1, 0 ,0)$ magnetic peaks at zero field. The solid lines represent critical exponent fitting. (f) The field hysteresis of $(2/3, 0 ,0)$ and $(1, 0 ,0)$ magnetic peaks measured at $1.5$ K. The intensity difference of $(2/3, 0 ,0)$ before and after magnetic field history shows magnetic twin-domain redistribution.
• Figure 3: Magnetic field dependent MFM images of EuRhAl$_{4}$Si$_{2}$ at $4.2\,$K in the magnetically ordered state. (a) Magnetization M as a function of magnetic field $\mu_{0}H$ (H $\parallel$ c. T = $4.2\,$K). (b) $0\,$T MFM image displaying the stripe-like $\uparrow \uparrow \downarrow$ and $\downarrow \downarrow \uparrow$ domains. The net magnetization and the MFM phase contrast switch as it crosses from one domain (blue) to the next (yellow). (c) $0.3\,$T (bottom step, inset (c) MFM image, the majority of the spins are $\uparrow \uparrow \downarrow$ with a smaller phase contrast from the domain wall soliton $\downarrow \downarrow \uparrow$ as shown by the dashed boxed spins. (d) At the middle plateau (inset), $0.8\,$T, the MFM contrast disappears. This state corresponds to a uniform $\uparrow \uparrow \downarrow$ domain. (e) At $1.4\,$T, the top step, the periodic weak stripes reappear with opposite contrast from (c). Most spins are in the $\uparrow \uparrow \downarrow$-state with the domain wall soliton formed by $\uparrow \uparrow \uparrow$. Scanned areas are $10\mu$m$\times10\mu$m. Black lines illustrate the schematic MFM phase contrast and red arrows show their corresponding spin configurations.
• Figure 4: Magnetic interactions and atomistic spin dynamics. (a) Real-space isotropic exchange interactions ($J_{ij}$) among the Eu magnetic atoms as function of the interatomic distance $R_{ij}$ given in units of the in-plane lattice constant $a$. (b) DFT spin-spiral calculations of EuRhAl$_{4}$Si$_{2}$ for $q$ along the X-$\Gamma$ (blue) and $\Gamma$-M (red) directions. Total energies for colinear ($\uparrow \downarrow$, $\uparrow \uparrow \downarrow$) states along the two directions are indicated with crosses. (c) Magnetic moment per atom averaged over 3-atom unit cell, $M$ (red), and susceptibility $\chi$ (blue), obtained from a Monte Carlo simulation at $T = 0.1\,$K for a quasi-one-dimensional system with exchange parameters as described in the text. Shown is the transition from a spin-spiral ($H=0$, $M =0$) via a $\uparrow \uparrow \downarrow$ state ($M=7/3$) to a FM state ($M=7$) as function of magnetic field. The small steps indicate the formation of magnetic defects as indicated by the spin configurations.