Electron spin resonance in spiral antiferromagnet linarite: Theory and experiment

We present a combined experimental and theoretical investigation of the low-frequency ESR dynamics in the ordered phases of magnetic mineral linarite. This material consists of weakly coupled spin-1/2 chains of copper ions with frustrated ferro- and antiferromagnetic interactions. In zero magnetic field, linarite orders into a spiral structure and exhibits a peculiar magnetic phase diagram sensitive to the field orientation. The resonance frequencies and their field dependence are analyzed combining microscopic and macroscopic theoretical approaches and precise values of magnetic anisotropy constants are obtained. We conclude that possible realization of exotic multipolar quantum states in this material is greatly influenced by the biaxial anisotropy.

Figure 1

Upper panel: Schematic crystal structure of linarite showing the positions of magnetic copper ions (small red circles). The spiral magnetic structure in zero field is illustrated by gray ellipses. The orthogonal triad $\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}}$ marks the principal axes of the anisotropy tensor. The hard axis $\widehat{\mathbf{z}}$ is perpendicular to the spiral plane, whereas the intermediate axis $\widehat{\mathbf{y}}\parallel \mathbf{b}$. Bottom panel: Magnetic structures appearing for different orientations of an external field for $T=1.3\mathrm{K}$.

Figure 2

The frequency-field diagram for $\mathbf{H}\parallel \mathbf{z}$ measured at $T=1.3\mathrm{K}$. The ESR resonances are marked by solid symbols, empty squares correspond to phase transitions observed as anomalies on $P_{\text{tr}}\left(H\right)$ scans. The dotted line shows the transition field into the fan state. Solid lines are theoretical curves $\nu \left(H\right)$ computed for the spiral state. Dash-dotted line is the paramagnetic $\nu \left(H\right)$. Bottom panel: Examples of $P_{\text{tr}}\left(H\right)$ scans measured at $T=1.3\mathrm{K}$.

Figure 3

Upper panel: The frequency-field diagram for $\mathbf{H}\parallel \mathbf{y}$. Resonances corresponding to absorption lines measured at different temperatures are shown by symbols of different colors (blue squares $T=0.5\mathrm{K}$, red squares $T=1.3\mathrm{K}$, green rectangles $T=1.8\mathrm{K}$). Empty squares are measured transition fields. The vertical dotted lines show transition fields from the bulk measurements [22] obtained for $T=0.5\mathrm{K}$ (${\mu }_0{H}_C=2,7\mathrm{T}$, ${\mu }_0{H}_F=8\mathrm{T}$), $T=1.3\mathrm{K}$ (${\mu }_0{H}_C=2,7\mathrm{T}$, ${\mu }_0{H}_F=6.7\mathrm{T}$), and $T=1.8\mathrm{K}$ (${\mu }_0{H}_C=2.8\mathrm{T}$, ${\mu }_0{H}_F=6\mathrm{T}$). Solid black and purple lines show theoretical curves $\nu \left(H\right)$ computed for spiral and canted antiferromagnetic phases. Solid line is the paramagnetic mode. Bottom panel: Examples of $P_{\text{tr}}\left(H\right)$. Upper mode was measured at $T=1.8\mathrm{K}$, other modes at $T=1.3\mathrm{K}$.

Figure 4

Upper panel: The temperature evolution of the power absorption scans $P_{\text{tr}}\left(H\right)$ measured at $\nu =77.72$ GHz for $\mathbf{H}\parallel \mathbf{y}$. The roman numbers correspond to the notations of branches in Figs. 1 and 3. Bottom panel: Temperature dependence of the lowest zero-field gap ${\mathrm{\Delta }}_1$.

Figure 5

Upper panel: The frequency-field diagram for $\mathbf{H}\parallel \mathbf{x}$. The solid lines show theoretical $\nu \left(H\right)$ computed for the spiral phase before and after the spin-flop transition. Dashed lines indicate the transition fields. Dash-dotted line gives the paramagnetic $\nu \left(H\right)$. Bottom panel: Examples of $P_{\text{tr}}\left(H\right)$ recorded at $T=1.3\mathrm{K}$.

Figure 6

Angular dependence in the $xz$ plane of the resonance field for $\nu =80.6$ GHz, solid squares, and the phase transition fields, empty symbols, measured at $T=1.3\mathrm{K}$. The solid line is a theoretical curve computed for the spiral magnetic structure. Dotted lines are guides for the eye.

Figure 7

Angular dependence in the $yz$ plane of the resonance field for $\nu =80.6$ GHz, solid squares, and the phase transition fields, empty symbols, measured at $T=1.3\mathrm{K}$. The solid lines are theoretical curves computed for the spiral magnetic structure. Dotted lines are guides for the eye.