Thermodynamic properties of the anisotropic frustrated spin-chain compound linarite PbCuSO${}_4$(OH)${}_2$

We present a comprehensive macroscopic thermodynamic study of the quasi-one-dimensional (1D) $s=\frac{1}{2}$ frustrated spin-chain system linarite. Susceptibility, magnetization, specific heat, magnetocaloric effect, magnetostriction, and thermal-expansion measurements were performed to characterize the magnetic phase diagram. In particular, for magnetic fields along the $b$ axis five different magnetic regions have been detected, some of them exhibiting short-range-order effects. The experimental magnetic entropy and magnetization are compared to a theoretical modeling of these quantities using density matrix renormalization group (DMRG) and transfer matrix renormalization group (TMRG) approaches. Within the framework of a purely 1D isotropic model Hamiltonian, only a qualitative agreement between theory and the experimental data can be achieved. Instead, it is demonstrated that a significant symmetric anisotropic exchange of about 10% is necessary to account for the basic experimental observations, including the three-dimensional (3D) saturation field, and which in turn might stabilize a triatic (three-magnon) multipolar phase.

Figure 1

Upper part: The crystallographic structure of PbCuSO${}_4$(OH)${}_2$ consisting of buckled neutral Cu(OH)${}_2$ chains propagating along the crystallographic $b$ direction surrounded by Pb${}^{2+}$ cations and SO${}_4^{2-}$ anions. Lower panel, left: The main exchange paths $J_1$ and $J_2$ [notation in the general anisotropic case for the two intrachain exchange paths shown would be $D_1{J}_1$ and $D_2{J}_2$, see Eq. (1) and the text below] in the basal $bc$ plane as well as the dominant skew interchain coupling $J_{\mathrm{ic}}$. The photographic picture shows one of our mineral specimens from the Grand Reef Mine in Graham County, Arizona.

Figure 2

Susceptibility of PbCuSO${}_4$(OH)${}_2$ for magnetic fields between 0.5 and 7 T parallel to the crystallographic $a_{\perp }$, $b$, and $c$ direction in the temperature range from 1.8 and 10 K. The insets depict the temperature derivative of the product $\chi T$ for selected field values used to determine the transition temperature $T_{\mathrm{N}}$.

Figure 3

(a) Low-temperature susceptibility in different fields for the intermediate-field range of PbCuSO${}_4$(OH)${}_2$ for $H\parallel b$. (b) Field-dependent magnetization of linarite for $H\parallel b$. The steps and hystereses indicate field-induced transitions from the helical ground state to another phase. For clarity the curves are shifted to each other.

Figure 4

Magnetization data $M\left({\mu }_{0}H\right)$ and the derivatives $dM/d\left({\mu }_{0}H\right)$ of PbCuSO${}_4$(OH)${}_2$ for all crystallographic directions as a function of magnetic field in the temperature range between 1.8 and 2.8 K.

Figure 5

High-field magnetization and its field derivative of PbCuSO${}_4$(OH)${}_2$ at low temperatures for $H\parallel b$.

Figure 6

Specific heat of linarite (sample 6) in zero magnetic field. The open circles represent the measured data, the dashed line shows the modeled phononic contribution to the specific heat (for details see text).

Figure 7

Magnetic entropy of PbCuSO${}_4$(OH)${}_2$ in zero magnetic field. The dashed line corresponds to the expected entropy for a spin-$\frac{1}{2}$ system, $R\ln \left(2\right)$, while the solid line indicates the entropy derived from the measured specific-heat data.

Figure 8

Magnetic specific heat of linarite (sample 3) as a function of the magnetic field aligned parallel to $b$. The inset shows data at selected fields on a double-logarithmic scale. The arrow indicates one of the many small anomalies that hint towards another phase transition.

Figure 9

Specific heat of linarite (sample 3) as a function of magnetic fields aligned along $a_{\perp }$ and $c$.

Figure 10

Field scan for the determination of the magnetocaloric effect of linarite for $H\parallel b$ at a starting temperature of 1.476 K. The inset enlarges the feature seen in the magnetocaloric effect in high magnetic fields.

Figure 11

(a) Magnetostriction of linarite at various temperatures as a function of magnetic field. (b) The thermal expansion of linarite for various magnetic fields as a function of temperature. The insets depict the field and temperature derivatives $\beta$ and $\alpha$, respectively. Here the peaks indicate the transition into the long-range ordered ground state.

Figure 12

Magnetic phase diagram of PbCuSO${}_4$(OH)${}_2$ for $H\parallel a_{\perp }$ and $c$ normalized to $H_{\mathrm{sat}}$ (upper panel) and for $H\parallel b$ (lower panel).[36]

Figure 13

Low-temperature $T$ dependence of the magnetic entropy for an 1D isotropic $J_1$-$J_2$ chain for $\alpha =0.36$. The temperature is measured in units of $|J_1|$. Inset: The same as in the main figure on a larger temperature scale comparable with $J_1$. The behavior for $T\to 0$ has been extrapolated linearly to $T=0$ using the lowest available numerical TMRG data (in between $T=0.006$ and 0.012) as suggested by the adopted scenario of interacting spinons (see text).

Figure 14

Upper panel: Temperature dependence of the magnetic entropy for a 1D isotropic $J_1$-$J_2$ chain as compared with the measured total entropy including the lattice contribution. Lower panel: The phenomenological lattice contribution resulting from a subtraction of the theoretical 1D contribution shown in the upper panel from the measured total one as compared with that from a harmonic-lattice model explained in the text. Since the behavior of the theoretical curve for $T\to 0$ has been extrapolated linearly to $T=0$ (see also the note in the caption of Fig. 13), the difference becomes artificially negative in the region with magnetic ordering at $T<T_{\mathrm{N}}\approx 2.8$ K where the 1D model naturally fails. Inset: Difference between the calculations and the above-mentioned harmonic model with one Debye spectrum and two Einstein modes.

Figure 15

Influence of the interchain coupling $J_{\mathrm{ic}}$ and the easy-axis exchange (spin) anisotropy $D_1$ of the ferromagnetic (FM) inchain NN-coupling $J_1$ on the ground state of a system of coupled anisotropic $J_1$-$J_2$ spin chains [cf. Eq. (1)] for an intrachain frustration rate $\alpha =-J_2/J_1=0.36$. (a) Zero-field plot of the interchain coupling $J_{\mathrm{ic}}$ vs easy-axis anisotropy $D_1$ for various fixed pitch angles $\varphi$ (given in degrees at the left side of each curve). The FM ground-state phase (i.e., $\varphi =0$), present for large enough $J_{\mathrm{ic}}$, is shown in the light blue upper part of the figure. The NNN-coupling $J_2$ is isotropic (i.e., $D_2=1$). Note that the red curve corresponds to the observed pitch for linarite. (b) Character of the lowest excitations above the FM state for large external fields above the saturation field applied in the easy-axis (b) direction. These two figures, which have been slightly modified for clarity here, are taken from Nishimoto et al.[102]

Figure 16

Magnetization at finite temperature for an effective single chain (1D) $J_1$-$J_2$ model for the frustration ratio $\alpha =-J_2/J_1=0.365$ as compared with the experimental data for linarite for $H\parallel b$. $H_{c3}$ is a fit parameter in order to get a reasonable description at low fields. $H_{c3}$ corresponds approximately to the inflection points of the experimental magnetization curves shown in Figs. 4 and 5.

Figure 17

Temperature (in units of $|J_1|$) dependence of the magnetic specific heat of a single chain within the $J_1$-$J_2$ model. (a) Complete diagonalization-based calculations for periodic rings with $N=22$ sites for different dimensionless magnetic fields $h=gH/|J_1|$. (b) The same as in (a) for TMRG calculations. (c) $T$ dependence of the magnetic specific heat at zero magnetic fields but with symmetric anisotropic exchange included. $D_1$ $>1$ means easy-axis anisotropy for $J_1$, see Eq. (1). $D_1=1$ corresponds to the isotropic limit of the $J_1$-$J_2$ model as shown in (b), too.