Destruction of long-range magnetic order in an external magnetic field and the associated spin dynamics in ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ and ${\mathrm{Cu}}_2{\mathrm{AlBO}}_5$ ludwigites

The quantum spin systems ${\mathrm{Cu}}_2{M}^{\prime }{\mathrm{BO}}_5$ $(M^{\prime }=\mathrm{Al},\mathrm{Ga})$ with the ludwigite crystal structure consist of a structurally ordered ${\mathrm{Cu}}^{2+}$ sublattice in the form of three-leg ladders, interpenetrated by a structurally disordered sublattice with a statistically random site occupation by magnetic ${\mathrm{Cu}}^{2+}$ and nonmagnetic ${\mathrm{Ga}}^{3+}$ or ${\mathrm{Al}}^{3+}$ ions. A microscopic analysis based on density-functional-theory calculations for ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ reveals a frustrated quasi-two-dimensional spin model featuring five inequivalent antiferromagnetic exchanges. A broad low-temperature ${}^{11}\mathrm{B}$ nuclear magnetic resonance points to a considerable spin disorder in the system. In zero magnetic field, antiferromagnetic order sets in below $T_{\text{N}}\approx 4.1$ K and $\sim 2.4$ K for the Ga and Al compounds, respectively. From neutron diffraction, we find that the magnetic propagation vector in ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ is commensurate and lies on the Brillouin-zone boundary in the $\left(H0L\right)$ plane, ${\mathbf{q}}_{\text{m}}=(0.45,0,-0.7)$, corresponding to a complex noncollinear long-range ordered structure with a large magnetic unit cell. Muon spin relaxation is monotonic, consisting of a fast static component typical for complex noncollinear spin systems and a slow dynamic component originating from the relaxation on low-energy spin fluctuations. Gapless spin dynamics in the form of a diffuse quasielastic peak is also evidenced by inelastic neutron scattering. Most remarkably, application of a magnetic field above 1 T destroys the static long-range order, which is manifested in the gradual broadening of the magnetic Bragg peaks. We argue that such a crossover from a magnetically long-range ordered state to a spin-glass regime may result from orphan spins on the structurally disordered magnetic sublattice, which are polarized in magnetic field and thus act as a tuning knob for field-controlled magnetic disorder.

Figure 1

Crystal structure of ${\mathrm{Cu}}_{2}M{\mathrm{BO}}_5$ ludwigites, where $M$ is either Ga [11] or Al [12]. The magnetic ${\mathrm{Cu}}^{2+}$ ions occupy four inequivalent positions in distorted octahedral coordinations, which form edge-sharing zigzag walls perpendicular to the $b$ axis. The chains of edge-sharing octahedra surrounding the $M\left(1\right)$ and $M\left(2\right)$ sites with 100% occupation by the ${\mathrm{Cu}}^{2+}$ ions in neighboring zigzag walls (blue) are connected triplewise by corner sharing into three-leg ladders running along the $a$ axis. They are separated by the structurally disordered three-leg ladders ($3\times \infty$ ribbons) of edge-sharing octahedra (green) surrounding the $M\left(3\right)$ and $M\left(4\right)$ sites that are statistically occupied by magnetic ${\mathrm{Cu}}^{2+}$ and nonmagnetic $M^{3+}$ ions. Solid lines mark the unit cell. The visualization was done in VESTA [21].

Figure 2

Sketch of the crystal structure with the hypothetical magnetic order that one might expect in the simplified three-leg ladder model suggested by the GKA rules, but which is not observed in reality. Nearest-neighbor interactions between the ${\mathrm{Cu}}^{2+}$ ions are shown by solid double lines (AFM) and dashed lines (FM), respectively. Hatched circles show the fractionally occupied metal sites. Dotted ellipses highlight the AFM trimers. To simplify the drawing, spin chains along the $a$ axis are shown for the top row of trimers only.

Figure 3

(a) Oriented ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ single crystal on an aluminum sample holder. (b) X-ray and (c) neutron Laue diffraction measurements for different crystal directions at room temperature.

Figure 4

Neutron diffraction data, measured at the E2 single-crystal diffractometer at HZB on one of the largest single crystals of the ${\mathrm{Cu}}_2{\mathrm{Ga}}^{11}{\mathrm{BO}}_5$ ludwigite. Magnetic Bragg peaks at the AFM propagation vector ${\mathbf{q}}_{\text{m}}=(0.45,0,-0.7)$ and the equivalent position $\left(10\overline{2}\right)-{\mathbf{q}}_{\text{m}}$ are shown with white arrows. Note that both peaks fall on the intersection of the Brillouin-zone boundary (dashed lines) with the straight dash-dotted line connecting the (002) and $\left(10\overline{4}\right)$ reciprocal-lattice vectors.

Figure 5

A set of equidistant planes in direct space, showing the pitch of the AFM spin-spiral structure in ${\mathrm{Cu}}_2{\mathrm{Ga}}^{11}{\mathrm{BO}}_5$. The planes of equal phase are orthogonal to the magnetic propagation vector ${\mathbf{q}}_{\text{m}}$ and are spaced by $2\pi /|{\mathbf{q}}_{\text{m}}|$. In the crystal structure, only the distorted octahedra around the magnetic Cu ions are shown for simplicity. The visualization was done in VESTA [21].

Figure 6

Magnetic-field and temperature dependence of the $(0.55,0,-1.3)$ magnetic Bragg peak from the E2 data. (a), (c) Neutron diffraction data, measured at various temperatures in zero magnetic field and at 2 T, respectively. (b), (d) The corresponding $T$ dependence of the magnetic Bragg intensity (after background subtraction) at zero field and at 2 T. (e) Magnetic-field dependence of the intensity profile across the magnetic Bragg peak at the base temperature, $T=1.5$ K. (f) Magnetic-field dependence of the full width at half maximum of the peaks shown in panel (e). All the data points in panels (b), (d), and (f) were obtained from Gaussian fits that are shown in panels (a), (c), and (d), respectively, with solid lines.

Figure 7

Suppression of the magnetic Bragg peaks in ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ by an external magnetic field. (a) Elastic neutron-scattering intensity along the $(H02-6H)$ line in momentum space, measured in zero field and in the field of 5 and 10 T, applied along the $b$ axis. (b) Magnetic-field dependence of the peak intensity, measured on top of the two Bragg reflections at $(0.45,0,-0.7)$ and $(0.55,0,-1.3)$.

Figure 8

Inelastic neutron scattering data collected at the Panda spectrometer at the base temperature of 1.8 K. (a) Constant-energy scans along the $(H02-6H)$ line in momentum space, measured in zero magnetic field at several energies. The solid lines are fits with two Gaussian peaks of different amplitudes, symmetrically offset from the commensurate $\left(\frac{1}{2}0\overline{1}\right)$ wave vector, plus a linear background (shown by a dashed line). (b) Comparison of the $\mathbf{Q}$ scans at an energy transfer of 0.4 meV, measured in zero field and in a 10 T field applied along the $b$ axis. (c) Corresponding scans in the orthogonal $\mathbf{Q}$ direction, $(H01.2H-1.96)$. (d) Energy dependence of the background-subtracted magnetic intensity in zero and 10 T magnetic field. Solid lines are guides to the eyes.

Figure 9

(a) Field-sweep ${}^{11}\mathrm{B}$ NMR spectra of in ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ collected at 33 MHz at various temperatures. (b) Temperature dependence of the spin-lattice relaxation rates at three different external fields. The dashed lines are guides to the eyes.

Figure 10

(a), (b) Zero-field $\mu \mathrm{SR}$ time spectra measured at various temperatures for ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ and ${\mathrm{Cu}}_2{\mathrm{AlBO}}_5$. The solid lines are fits to Eq. (2). (c) $\mu \mathrm{SR}$ measurements of the magnetic volume fraction in ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ as a function of temperature, obtained in a weak transverse magnetic field of 15 mT. The solid lines are fits to Eq. (3). (d) $\mu \mathrm{SR}$ time spectra measured in longitudinal fields between zero and 750 mT at the base temperature of 1.6 K. The lines are fits to Eq. (2).

Figure 11

Comparison of the initial muon depolarization in the time window below 1 $\mu \mathrm{s}$ at the base temperature for the ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ and ${\mathrm{Cu}}_2{\mathrm{AlBO}}_5$ samples. The solid lines are fits to Eq. (2), with the individual terms corresponding to the static, dynamic, and background contributions shown with the dashed, dash-dotted, and dotted lines, respectively.

Figure 12

Temperature dependencies of the (a) static and (b) dynamic depolarization rates in zero-field $\mu \mathrm{SR}$ measurements of ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ and ${\mathrm{Cu}}_2{\mathrm{AlBO}}_5$, obtained from fitting Eq. (2) to the data in Fig. 10 as described in the text. (c) Temperature dependence of the magnetic volume fraction, obtained by fitting the transverse-field data in Fig. 10 to Eq. (3). (d) Static part of the muon asymmetry ($A_{\text{st}}$) as a function of longitudinal magnetic field, obtained by fitting the data in Fig. 10 to Eq. (2). The dotted lines are guides to the eyes.

Figure 13

(a) Spin model of ${\mathrm{Cu}}_2{\mathrm{GaBO}}_5$ comprising four exchanges—$J_{\text{t}}$, $J_{\text{c}}$, $J_{\text{tt}}$, and $J_{\text{ct}}$—that operate within the (102) planes, and the interplane exchange $J_{\perp }$. Spins of the front layer are depicted as orange (shaded) spheres, spins of the second plane are shown with open circles. All exchanges are antiferromagnetic. (b) Intratrimer $J_{\text{t}}$ and intertrimer $J_{\text{tt}}$ exchanges form eight-membered loops that are not frustrated. (c) Frustration is induced by the combination of $J_{\text{t}}$ and $J_{\text{tt}}$ with the intrachain exchange $J_{\text{c}}$ and the exchange $J_{\text{ct}}$ coupling trimers to chains, forming five (odd)-membered rings.