Magnetic field induced softening of spin waves and hard-axis order in the Kondo-lattice ferromagnet ${\mathrm{CeAgSb}}_2$

A significant number of Kondo-lattice ferromagnets order perpendicular to the easy magnetization axis dictated by the crystalline electric field. The nature of this phenomenon has attracted considerable attention, but remains poorly understood. In the present paper we use inelastic neutron scattering supported by magnetization and specific heat measurements to study the spin dynamics in the hard-axis ferromagnet $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$. In the zero-field state we observed two sharp magnon modes, which are associated with Ce ordering and extended up to $\approx 3$ meV with a considerable spin gap of 0.6 meV. Application of a magnetic field perpendicular to the moment direction reduces the spectral intensity and suppresses the gap and significantly enhances the low-temperature specific heat at a critical field of $B_{\mathrm{c}}\approx 2.8$ T via a mean-field-like transition. Above the transition, in the field-polarized state, the gap eventually reopens due to the Zeeman effect. We modeled the observed dispersion using linear spin-wave theory taking into account the ground-state ${\mathrm{\Gamma }}_6$ doublet and exchange anisotropy. Our model correctly captures the essential features of the spin dynamics including magnetic dispersion, distribution of the spectral intensity, as well as the field-induced behavior, although several minor features remain obscure. The observed spectra do not show significant broadening due to the finite lifetime of the quasiparticles. Along with a moderate electronic specific heat coefficient $\gamma =46$ mJ/mol ${\mathrm{K}}^2$ this indicates that the Kondo coupling is relatively weak and the Ce moments are well localized. Altogether, our results provide profound insight into the spin dynamics of the hard-axis ferromagnet $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$ and can be used as solid ground for studying magnetic interactions in isostructural compounds including $\mathrm{Ce}\mathrm{Au}{\mathrm{Sb}}_2$, which exhibits nematicity and unusual mesoscale magnetic textures.

Figure 1

(a) Isothermal magnetization $m$, measured for $B\parallel c$ and $B\parallel a$ axes at 1.8 K. (b) Temperature dependence of magnetization measured for $B=0.1\mathrm{T}\parallel c$ and $B\parallel a$ axes. Note the crossing of magnetizations near $T_{\mathrm{C}}=9.6$ K. Inset shows inverse magnetization $1/m$ for both orientations of magnetic field. (c) Temperature dependence of the specific heat divided by temperature in magnetic fields applied along the $a$ axis. (d) Magnetic field dependence of the specific heat at 1.8 K with $B\parallel a$.

Figure 2

Spin-wave dispersion of $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$ in ferromagnetic state at 1.7 K and 0 T measured at CNCS in $\left(HK0\right)$ (a) and $\left(H0L\right)$ (c) scattering planes. The data are integrated by $\pm 0.08$ Å${}^{-1}$ in two orthogonal directions. (b), (d) Magnon spectra for the same path as in panels (a) and (d), respectively, calculated using SpinW software. (e), (f) Sketches of structural “small” and magnetic “large” Brillouin zones for $\left(H0L\right)$ and $\left(0KL\right)$ scattering planes shown by orange dotted and blue solid lines, respectively. The green arrows show the paths in the reciprocal space for the spaghetti plot in panels (a)–(b) and (c)–(d), respectively.

Figure 3

Zero-field magnetic structure and anisotropic exchange interactions in $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$ in order of increasing Ce-Ce bond length. Only Ce ions are shown. (a) $J_1$ and $J_3$ are in-plane interactions along [100] and [110] directions; $J_2$ and $J_4$ are nonequivalent out-of-plane interactions due to the fact that the $z$ coordinate of Ce $0.2388\ne \frac{1}{4}$. (b) $J_5, J_7$, and $J_6$ are out-of-plane and in-plane interactions between Ce in neighboring unit cells. (c) $J_8$ and $J_9$ are interactions along [120] and [331] directions correspondingly.

Figure 4

Spin-wave dispersion measured in $\left(HK0\right)$ scattering plane with magnetic field applied along the $c$ axis using CNCS spectrometer. Left and right parts of the figure show the spaghetti plots along the same $\mathrm{\Gamma }\to M\to X$ path in the reciprocal space at 0 and 2 T as indicated in the panels. The data are integrated by $\pm 0.07$ (r.l.u.) in two orthogonal directions. Black lines shows results of spin-wave modeling. The width of the lines here and in all other figures represents intensities of the magnetic modes.

Figure 5

Spin-wave dispersion measured in $\left(H0L\right)$ scattering plane with magnetic field applied along the $b$ axis using LET spectrometer. Panels (a), (b), and (c) show the data collected at 0, 1, and 3 T. The data are integrated by $\pm 0.07$ (r.l.u.) in two orthogonal directions. The color scale is identical for each panel. Black lines show results of spin-wave modeling. (d) Orange and blue lines show sketches of the structural and magnetic Brillouin zones; the green arrows show the path of spaghetti plots.

Figure 6

CEF excitations extracted from high-energy INS data measured using LET spectrometer with $E_{\mathrm{i}}=14.9$ meV. (a), (b) Powder-averaged INS signals of $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$ measured at $B=0$ T and different temperatures, as indicated in panels. Note that these patterns differ from the real powder signal, because the integration was performed within the $\left(H0L\right)$ scattering plane, rather than full reciprocal space. Bright vertical excitation at high-$\left|\mathbf{Q}\right|$ is due to acoustic phonons emanating from (200) structural peak. (c) Field dependence of magnetic excitations at $T=1.7$ K and $B=0,1,3$ T. The curves were obtained by integration of the INS signal at low $\mathbf{Q}$ within $H=0\pm -1.5;K=0\pm 0.1;L=0\pm 1.5$ r.l.u. Two shaded peaks at $\approx 2$ and $\approx 6$ meV show magnon and CEF excitations, respectively. The data are shifted vertically for clarity.

Figure 7

Spin-wave dispersion measured in $\left(H0L\right)$ scattering plane with magnetic field $B=6$ T applied along the $b$ axis using CNCS spectrometer. The data are integrated by $\pm 0.07$ (r.l.u.) in two orthogonal directions. Black lines shows results of spin-wave modeling. (a) Dispersion along $\left(003\right)\to \left(002\right)\to \left(\frac{1}{2}02\right)$ direction. (b) Two energy-momentum slices along the equivalent directions of the Brillouin zone $\left(10\frac{5}{2}\right)\to \left(00\frac{5}{2}\right)$ and $\left(00\frac{3}{2}\right)\to \left(10\frac{3}{2}\right)$ clearly show the absence of the mode crossing. (c) Orange and blue lines show sketches of the structural and magnetic Brillouin zones; the green and purple arrows show the paths for panels (a) and (b), respectively.

Figure 8

INS spectra of $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$ measured using FLEXX spectrometer at Gamma point $\mathbf{Q}=\left(101\right)$. Temperature was fixed to $T=1.7$ K and magnetic fields are indicated in legends. Panels (a) and (b) show the data collected below and above the critical field $B_{\mathrm{c}}$, respectively. Data in both panels are vertically shifted for clarity.

Figure 9

Spin gap at $\mathrm{\Gamma }$ point calculated from the INS spectra (colored dots) and temperature of spin-reorientation transition deduced from thermodynamic measurements (gray dots) plotted as a function of magnetic field. Red and blue lines show the spin gap calculated with Eqs. (3), (4), and (5) for $g=3$ and $g=3.43$. Black line is a fit of the transition temperature with $T_{\mathrm{c}}\propto \sqrt{B_{\mathrm{c}}^2-B^2}$. The temperature and energy axes are shown on the same scale $E=k_{\mathrm{B}}T$.

Figure 10

INS spectra of $\mathrm{Ce}\mathrm{Ag}{\mathrm{Sb}}_2$ measured using CNCS spectrometer at $T=1.7$ K and $B=0$ T obtained by integration of the signal around $\mathrm{\Gamma }\left(101\right)$ point, $H=1\pm \mathrm{\Delta }Q,K=0\pm \mathrm{\Delta }Q,L=1\pm \mathrm{\Delta }Q$ with different $\mathrm{\Delta }Q$ as indicated in the legend. Green lines show model calculations using SpinW software (on top of an empirical linear background), which take into account $Q$ integration and energy-dependent experimental resolution of CNCS instrument. The data are shifted vertically for clarity.

Figure 11

INS spectra calculated by RPA and INS. The color plots were obtained using SpinW software for the $S=1/2$ model and are similar to Figs. 2 and 2. Solid lines show low-energy magnon dispersion calculated for Hamiltonian (D1) using RPA with McPhase software.