Magnetic phase diagram of the frustrated spin chain compound linarite ${\mathrm{PbCuSO}}_4{\left(\mathrm{OH}\right)}_2$ as seen by neutron diffraction and ${}^1\mathrm{H}$-NMR

We report on a detailed neutron diffraction and ${}^1\mathrm{H}$-NMR study on the frustrated spin-1/2 chain material linarite, ${\mathrm{PbCuSO}}_4{\left(\mathrm{OH}\right)}_2$, where competing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions lead to frustration. From the magnetic Bragg peak intensity studied down to 60 mK, the magnetic moment per Cu atom is obtained within the whole magnetic phase diagram for $H\parallel b$ axis. Further, we establish the detailed configurations of the shift of the SDW propagation vector in phase V with field and temperature. Finally, combining our neutron diffraction results with those from a low-temperature/high-field NMR study, we find an even more complex phase diagram close to the quasisaturation field suggesting that bound two-magnon excitations are the lowest energy excitations close to and in the quasisaturation regime. Qualitatively and semiquantitatively, we relate such behavior to $XYZ$ exchange anisotropy and contributions from the Dzyaloshinsky-Moriya interaction to affect the magnetic properties of linarite.

Figure 1

Scans in reciprocal space along $(0k1$/2) with varying $k$ recorded for several temperatures in a field of 7.5 T at the instrument D10, ILL. The phase boundary IV–V can be identified by the change in the appearance of the magnetic Bragg peaks. The peaks are broadened due to a slight mosaic spread within the linarite crystal. The data for $T>0.1\mathrm{K}$ are shifted along the vertical axis for clarity. As example, for the scans at 0.1 and 1.0 K, a Gaussian fit curve is added to the data.

Figure 2

Detector images of the magnetic Bragg peaks recorded at E2 for selected temperatures in an external magnetic field of 6 T $\parallel b$ axis. The direction out of the scattering plane was scaled as $k$ in units of r.l.u. The change in the appearance of the magnetic Bragg peaks represents the phase borderline IV–V. The dashed red lines indicate $k=0,\pm 0.1$ r.l.u.

Figure 3

The contour plot of the magnetic moment $m$ per Cu atom for the different magnetic phases. On passing the phase boundaries I–V and IV-V there is an abrupt drop of the magnetic moment. The dashed grey and blue lines denote the scans from the Figs. 1 and 2. In the black-and-white striped low-temperature area, no neutron diffraction scans were carried out. The phase borderlines were added to the contour plot by copying them from the phase diagram in Ref. [39].

Figure 4

Hysteretic behavior of (a) the commensurate (0 0 1/2) peak and (b) the incommensurate (0 0.186 1/2) peak in region II at 50 mK for increasing (red)/decreasing (black) magnetic field; solid lines are guides to the eye.

Figure 5

First row: cuts through the magnetic moment map in Fig. 3 at various fields. At the phase boundary IV–V, a steplike decrease of the magnetic moment is observed for all magnetic fields. The background color in each plot corresponds to the coloring of the magnetic phases in Fig. 12. The notation “phase V (1), (2), (3)” is defined later in the text and described in Fig. 12. The phase boundaries were drawn according to the phase diagram from the thermodynamic measurements on linarite [6, 35, 39]. In the diagram for the 8-T data, the error bars for the magnetic moment values in phase V represent the typical error size for the phase V magnetic moments. The magnetic moment error in phases I and IV is of the order of the data point size. Second row: cuts through magnetic moment map at various temperatures. Note that the cut at 1.5 K passes through phase III where both reflections at (0 0 1/2) and (0 0.186 1/2) are present.

Figure 6

Temperature dependence of the SDW propagation vector component $k_y$ in magnetic fields $H\parallel b$ axis. The temperature behavior of $k_y$ separates phase V into three different regions: (1) the low magnetic field region where $k_y$ decreases when increasing temperature, (2) the intermediate-field region with $k_y$ increasing when increasing temperature, and (3) the high-field region where there is no significant temperature dependence of $k_y$ [55]. The upper limit of the $k_y$ scale in (a) and (b) is $k_y=0.186$ r.l.u. The dashed lines in (b) are guides to the eye.

Figure 7

(a) NMR spectra for different magnetic fields applied along the $b$ axis as a function of the relative field ${\mu }_{0}H-2\pi {\nu }_0/\gamma$. (b) NMR spectra for fields at 9.6 T and above represented on a different scale. The background signal around ${\mu }_{0}H-2\pi {\nu }_0/\gamma =0\mathrm{T}$ is expected to vary slowly with field and was determined separately only for ${\mu }_{0}H=9.9\mathrm{T}$. The two other peaks are associated with the hydrogen sites H(4) and H(5) [38].

Figure 8

NMR shift and FWHM as a function of the applied magnetic field $H\parallel b$ axis. The full and open symbols represent the H(4) and H(5) sites, respectively. The lines are guides to the eye and indicate the (quasi)saturation field at ${\mu }_0{H}_{\mathrm{sat}}=9.64\pm 0.10\mathrm{T}$. The light grey area indicates the field interval where the occurrence of a multipolar state is possible. At 9.1 T, the broad NMR spectrum has been observed. The measurements were carried out at 160 mK except for the 8.5 T data which were taken at 500 mK and serve as reference.

Figure 9

Normalized magnetization recovery $m/m_{\infty }$ of the hydrogen nuclei H(4) as a function of time $t$ after its suppression by $\pi /2$ pulses at ${\mu }_{0}H=9.57\mathrm{T}$ and different temperatures. The dashed and solid lines are fits to Eqs. (7) and (8), respectively. The inset shows the fit parameter $\beta$ as a function of temperature. The solid line in the inset is a guide to the eye.

Figure 10

(a) Spin-lattice relaxation rate $1/T_1$ as a function of temperature $T$ and magnetic field $H\parallel b$ axis. The lines are guides to the eye. (b) Relaxation rate $1/T_1$ as a function of the inverse temperature $1/T$. The lines are linear fits to Eq. (6) in the temperature range 300 mK $<T<1$ K. The inset shows the field dependence of the gap $\mathrm{\Delta }$ extracted from the fits. The blue and red dashed lines correspond to the expected behavior for a one-magnon gap $\left(p=1\right)$ and a bound two-magnon gap $\left(p=2\right)$, respectively. The solid black line in the inset is a linear fit to our experimental data and gives $d\mathrm{\Delta }/d{\mu }_{0}H=(2.6\pm 0.5)$ K/T.

Figure 11

(a) Cluster used in our DMRG calculations. (b) Calculated magnetization curve by DMRG method with $(J_1^x,J_1^y,J_1^z)=(-91.1,-86.6,-88.4),J_2=28.3\left(J_2/\left|{\tilde{J}}_1\right|\sim 0.32\right)$, and $J_{\mathrm{ic}}=2.7$. (c) Integrated weight of $S(q_x,\pi )$ as a function of field $\left(H\parallel b\right)$, where the results using anisotropic and isotropic NN exchange couplings are compared. For the isotropic case, we set $J_1^x=J_1^y=J_1^z=-88.4\mathrm{K}$. [(d)–(g)] Field dependence of static spin structure factor $S(q_x,\pi )$.

Figure 12

Updated magnetic phase diagram of linarite for $H\parallel b$ axis. The red diamonds represent the $(T,{\mu }_{0}H)$ points where a change of the magnetic behavior is observed in the neutron diffraction experiments. The solid black lines representing the phase borderlines were drawn from the thermodynamic measurements. The green dotted lines highlight the magnetic fields where a change of the behavior of the incommensurability vector $\vec{q}=\left(0k_{y}0.5\right)$ in phase V has been observed in neutron diffraction measurements (see Fig. 6). These regions are therefore named phase V (1), (2) and (3).

Figure 13

Enlarged high-field section of the updated magnetic phase diagram of linarite for $H\parallel b$ axis; labels as in Fig. 12. The blue stars indicate the points within the magnetic phase diagram where the NMR measurements from Fig. 7 have been carried out. The solid black lines representing the phase borderlines were drawn from the thermodynamic measurements. The gray line indicates the lower border for a possible nematic state as observed by NMR.