Microscopic parameters of the van der Waals CrSBr antiferromagnet from microwave absorption experiments

Microwave absorption experiments employing a phase-sensitive external resistive detection are performed for a topical van der Waals antiferromagnet CrSBr. The magnetic-field dependence of two resonance modes is measured in an applied field parallel to the three principal crystallographic directions, revealing anisotropies, magnetic transitions, and a strong magnon-magnon coupling in this material. To account for the observed results, we formulate a microscopic spin model with a bi-axial single-ion anisotropy and interplane exchange. Theoretical calculations give an excellent description of full magnon spectra enabling us to precisely determine microscopic interaction parameters for CrSBr.

Figure 1

(a) Schematic representation of the experimental microwave absorption setup. (b) “In-plane” and (c) “out-of-plane” magnetic-field $B$ configurations and crystal structure. Photos show the actual sample environment. (d) Representative raw data of magnetic-field sweep-induced resonances in different magnetic-field configurations. Upper panel: resonances observed for different magnon modes at a single microwave frequency. Relative change of the photoresponse amplitude $\mathrm{\Delta }R$ with respect to its “off-resonant” value $\mathrm{\Delta }{R}_0$ (solid black lines); relative change of the phase $\mathrm{\Phi }$ with respect to its “off-resonant” value ${\mathrm{\Phi }}_0$ (dashed blue lines). Lower panel: a single magnon mode resonance followed vs magnetic field for different microwave frequencies.

Figure 2

Absorption spectra for a magnetic field $B$ along the $b$- [left panel (a)], $a$- [center panel (b)], and $c$- [right panel (c)] axis. Color plots of the phase variation of the thermometer photoresistance as a function of microwave frequency and magnetic field $B$. The phase difference with respect to its median (“off-resonance”) value is color-coded from 0 (white) to $-8$% (dark blue). Fits (red dashed lines) obtained with our microscopic model (see the text for details).

Figure 3

Absorption spectrum for a magnetic field $B$ tilted by an angle $\psi =5.6^{\circ }$ from the $a$-axis in the ($a,b$) plane. Color plot of the phase variation of the thermometer photoresistance as a function of microwave frequency and $B$. The phase difference with respect to its median (“off-resonance”) value is color-coded from 0 (white) to $-8$% (dark blue). Theoretical modes (red dashed lines) are obtained with the (fixed) “global parameters” of our microscopic model. Inset: coupling strength $\mathrm{\Delta }f/2$ for different temperatures.

Figure 4

Microwave power dependence of the magnon excitations. Left panel (a): relative change (arbitrary Y-scale) of the phase of the thermometer photoresistance $\mathrm{\Phi }$ with respect to its “off-resonant” value ${\mathrm{\Phi }}_0$ as a function of the magnetic field $B$ (in this example, applied along the $a$-axis), for different microwave output powers (from 7.5 to 13 dBm). The top (bottom) panel displays resonances of the upper (lower) energy mode, and microwave powers are indicated above each traces. Right panel (b): resulting center fields (top panel) and linewidths (full width at half-maximum, bottom panel) of the resonant signals, with the horizontal dotted lines corresponding to the average values. Red (blue) dots correspond to the upper (lower) energy mode.

Figure 5

Determination of the zero-field magnon energies with frequency sweeps. Normalized relative change of the thermometer photoresistance $\mathrm{\Delta }R$ (solid gray line) as a function of the microwave frequency. The normalization is made by dividing the $B=0$ frequency sweep by a frequency sweep taken at a different finite magnetic field (here $B=3$ T). Solid green and red lines are Lorentzian fits of the resonances, from which the resonant frequency $f_R$ is extracted. Inset: temperature dependence of the zero-field resonant frequencies for each mode. Red (green) dots correspond to upper (lower) frequency mode.

Figure 6

Waterfall plots of relative variations of the thermometer phase photoresponse for a magnetic field $B$ applied along the $b$-axis (arbitrary Y-scale). Left panel: selected data on the entire studied range. Right panel: zoom-in data closer to the $B=0$ magnon energies.

Figure 7

Waterfall plots of relative variations of the thermometer phase photoresponse for a magnetic field $B$ applied along the $a$-axis. Left panel: selected data on the entire studied range (arbitrary Y-scale). Right panel: zoom-in data closer to the $B=0$ magnon energies.

Figure 8

Waterfall plots of relative variations of the thermometer phase photoresponse for a magnetic field $B$ applied along the $c$-axis. Left panel: selected data on the entire studied range (arbitrary Y-scale). Right panel: zoom-in data closer to the $B=0$ magnon energies.

Figure 9

Absorption spectrum for a magnetic field $B$ along the $b$-axis. Color plot of the phase variation of the thermometer photoresistance (same data as the main text) as a function of microwave frequency and magnetic field $B$ along the $b$-axis, for $T=5.4\mathrm{K}$. Data at a higher temperature of $T=19\mathrm{K}$ are plotted as orange open circles.

Figure 10

Absorption spectra for a magnetic field $B$ tilted by an angle $\psi =5.6^{\circ }$ from the $a$-axis in the ($a,b$) plane for different temperatures. Color plot of the phase variation of the thermometer photoresistance as a function of microwave frequency and $B$. The phase difference with respect to its median value (which corresponds to the “off-resonance” value) is color-coded from zero (white) up to $-8\%$ (dark blue).

Figure 11

Absorption spectra for a magnetic field $B$ along the $b$ (left panel), $a$ (center panel), and $c$ (right panel) axis. Color plot of the phase variation of the thermometer photoresistance as a function of microwave frequency and magnetic field $B$. The phase difference with respect to its median value (which corresponds to the “off-resonance” value) is color-coded from zero (white) up to its measured maximum value (dark blue). Fits (red dashed lines) obtained with our microscopic model and a single set ($J_{\perp }=0.069, D=0.396$, and $E=0.207$) of “global” parameters (see the text for details).