Thermal transport of the frustrated spin-chain mineral linarite: Magnetic heat transport and strong spin-phonon scattering

The mineral linarite $\left[\mathrm{Pb}\mathrm{Cu}\mathrm{S}{\mathrm{O}}_4{\left(\mathrm{O}\mathrm{H}\right)}_2\right]$ forms a monoclinic structure where a sequence of $\mathrm{Cu}{\left(\mathrm{O}\mathrm{H}\right)}_2$ units forms a spin-$\frac{1}{2}$ chain. Competing ferromagnetic nearest-neighbor ($J_1$) and antiferromagnetic next-nearest-neighbor interactions ($J_2$) in this quasi-one-dimensional spin structure imply magnetic frustration and lead to magnetic ordering below $T_N=2.8\mathrm{K}$ in a multiferroic elliptical spin-spiral ground state. Upon the application of a magnetic field along the spin-chain direction, distinct magnetically ordered phases can be induced. We studied the thermal conductivity $\kappa$ in this material across the magnetic phase diagram as well as in the paramagnetic regime in the temperature ranges 0.07–1 K and 9–300 K. We found that in linarite the heat is carried mainly by phonons but shows a peculiar nonmonotonic behavior in field. In particular, $\kappa$ is highly suppressed at the magnetic phase boundaries, indicative of strong scattering of the phonons off critical magnetic fluctuations. Even at temperatures far above the magnetically ordered phases, the phononic thermal conductivity is reduced due to scattering off magnetic fluctuations. The mean free path due to spin-phonon scattering ($l_{\text{spin-phonon}}$) was determined as a function of temperature. A power law behavior was observed mainly above $500\mathrm{m}\mathrm{K}$ indicating the thermal activation of spin fluctuations. In the critical regime close to the saturation field, $l_{\text{spin-phonon}}$ shows a $1/T$ dependence. Furthermore, a magnon thermal transport channel was verified in the helical magnetic phase. We estimate a magnon mean free path which corresponds to about 1000 lattice spacings.

Figure 1

Magnetic phase diagram of $\mathrm{Pb}\mathrm{Cu}\mathrm{S}{\mathrm{O}}_4{\left(\mathrm{O}\mathrm{H}\right)}_2$ for $\mathbf{H}\left|\right|\mathbf{b}$ adapted from [21]. The Roman numerals label the magnetic phases: helical incommensurate phase (I), hysteretic commensurate region (II), coexistence phase (III), collinear Néel antiferromagnetic order (IV), and spin-density wave phase (V).

Figure 2

Thermal conductivity for $T>9\mathrm{K}$ with magnetic field applied along the spin-chain direction ($\mathbf{H}\left|\right|\mathbf{b}$).

Figure 3

Temperature dependence of the low-temperature thermal conductivity of sample 1 with the heat current and magnetic field oriented along the $b$ direction for the magnetic field ranges (a) 0–4 T, (b) 4–9 T, and (c) 9–16 T. In (d) the open circles depict the magnetic contribution to the thermal conductivity at zero field obtained from subtracting the phononic background in the polarized phase at 16 T from the zero-field curve ${\kappa }_{\text{mag}}^{0\text{T}}=\kappa \left(0\text{T}\right)-\kappa \left(16\text{T}\right)$; the solid line is a guide to the eye.

Figure 4

Temperature dependence of the low-temperature thermal conductivity of sample 2 with the heat current perpendicular and the magnetic field parallel to the $b$ direction for the magnetic field ranges (a) 0–9 T and (b) 9–16 T.

Figure 5

Thermal conductivity as function of magnetic field where in (a) both the heat current and the magnetic field are applied along $b$ in sample 1. Panel (b) shows the thermal conductivity $\kappa /T$ of sample 2 as a function of magnetic field where the heat current $j$ is applied perpendicular to $b$ and the magnetic field parallel to $b$. The dashed lines correspond to the phase boundaries at $T\approx 250$ mK as referred to in [14]. The dotted lines refer to the phononic background which is defined in the fully polarized phase.

Figure 6

Relative field dependence of thermal conductivity in sample 1 ($\mathbf{j},\mathbf{H}\left|\right|\mathbf{b}$), where $\kappa$ is normalized to the value at $15\mathrm{T}$ in the polarized phase. The dashed lines refer to the phase boundaries at $250\mathrm{m}\mathrm{K}$ as referred to in [14].

Figure 7

(a) Phonon mean free path $l_{\text{total}}$ as function of temperature in sample 1; (b) spin-phonon mean free path $l_{\text{spin-phonon}}$, which describes the limit of phonon propagation due to spin-phonon scattering. Dashed and solid black lines are guides to the eye and illustrate the power law behavior. The respective exponents are shown in Table 1. The left scales are normalized to the width of the sample.

Figure 8

Magnon mean free path of the magnon excitations at ${\mu }_{0}H=0\mathrm{T}$ estimated for a one-dimensional system. The right scale is normalized to the distance of magnetic ions.

Figure 9

Magnon mean free path of the magnon excitations at ${\mu }_{0}H=0\mathrm{T}$ estimated using the kinetic model with the experimental magnetic specific heat of linarite. The right scale is normalized to the distance of magnetic ions.

Figure 10

Zero-field thermal conductivity of sample 1 fitted with a power law ($\kappa =C_0·T^{\alpha }$), where $\alpha =2.50$ for $T<0.4\mathrm{K}$ (solid blue line) and $\alpha =2.04$ for $0.4\mathrm{K}<T<1\mathrm{K}$ (dotted black line).

Figure 11

Thermal conductivity at $16\mathrm{T}$ of sample 1 fitted with a power law ($\kappa =C_0·T^{\alpha }$), where $\alpha =2.42$ for $T<0.4\mathrm{K}$ (solid blue line) and $\alpha =2.10$ for $0.4\mathrm{K}<T<1\mathrm{K}$ (dotted black line).