Magnetic field dependence of low-energy magnons, anisotropic heat conduction, and spontaneous relaxation of magnetic domains in the cubic helimagnet $\mathrm{Zn}{\mathrm{Cr}}_2{\mathrm{Se}}_4$

Anisotropic low-temperature properties of the cubic spinel helimagnet $\mathrm{Zn}{\mathrm{Cr}}_2{\mathrm{Se}}_4$ in the single-domain spin-spiral state are investigated by a combination of neutron scattering, thermal conductivity, ultrasound velocity, and dilatometry measurements. In an applied magnetic field, neutron spectroscopy shows a complex and nonmonotonic evolution of the spin-wave spectrum across the quantum-critical point that separates the spin-spiral phase from the field-polarized ferromagnetic phase at high fields. A tiny spin gap of the pseudo-Goldstone magnon mode, observed at wave vectors that are structurally equivalent but orthogonal to the propagation vector of the spin helix, vanishes at this quantum critical point, restoring the cubic symmetry in the magnetic subsystem. The anisotropy imposed by the spin helix has only a minor influence on the lattice structure and sound velocity but has a much stronger effect on the heat conductivities measured parallel and perpendicular to the magnetic propagation vector. The thermal transport is anisotropic at $T\lesssim 2\mathrm{K}$, highly sensitive to an external magnetic field, and likely results directly from magnonic heat conduction. We also report long-time thermal relaxation phenomena, revealed by capacitive dilatometry, which are due to magnetic domain motion related to the destruction of the single-domain magnetic state, initially stabilized in the sample by the application and removal of magnetic field. Our results can be generalized to a broad class of helimagnetic materials in which a discrete lattice symmetry is spontaneously broken by the magnetic order.

Figure 1

(a) Thermal-expansion measurements in longitudinal and transverse magnetic fields above the domain-selection transition. All curves are normalized to the same $l_0$ value at $T=30\mathrm{K}$ and $B=0$. (b) Zero-field thermal expansion, measured during zero-field cooling (ZFC) and after field-cooling in a longitudinal (FC 0 T $\parallel$) or transverse (FC 0 T $\perp$) field of 7 T. The measurement steps (i)–(iii) are explained in the text. (c), (d) Magnetostriction measurements at several constant temperatures for longitudinal and transverse field directions, respectively.

Figure 2

(a) The change in the sound velocity of the longitudinal acoustic mode, $\mathrm{\Delta }v=v-v_0$, normalized to the value $v_0$ at 30 K, in the single-domain (SD, solid lines) and multidomain (MD, dashed lines) states. The top and bottom curves correspond to the ultrasound wave vector ${\mathbf{k}}_{\text{s}}$ perpendicular and parallel to the propagation vector of the spin helix, ${\mathbf{q}}_{\text{h}}$, respectively. The inset shows a similar temperature dependence, measured in the field-polarized FM phase at $B=7\mathrm{T}$ for ${\mathbf{k}}_{\text{s}}\perp \mathbf{B}$ (top curve) and ${\mathbf{k}}_{\text{s}}\parallel \mathbf{B}$ (bottom curve), for comparison. (b) Magnetic-field dependence of the relative change in the sound velocity, $\mathrm{\Delta }v/v_0$, normalized to the respective $v_0$ values at 12 T. The measurements are done at a constant temperature with the ultrasound wave vector parallel and perpendicular to the magnetic-field direction. Note the tenfold difference in the ranges of the vertical scales between (a) and (b).

Figure 3

Time dependence of the relative change in the sample length $\mathrm{\Delta }l/l_0$, measured during and after a rapid linear sweep of magnetic field from 7 T to zero in the transverse and longitudinal directions. The time dependence of magnetic field is plotted at the bottom. At $t<0$, the sample temperature is stabilized at $T=8$ or 12 K and a downward field sweep with 1 T/min starts. After the time $t_0=420\mathrm{s}$, when the field reaches zero, slow relaxation toward the multidomain state can be described by a stretched exponential, Eq. (1), with $\beta =0.5$ and the average decay constants ${\tau }_{\text{8}\text{K}}=980\mathrm{s}$ and ${\tau }_{\text{12}\text{K}}=190\mathrm{s}$ (thin white lines).

Figure 4

The decay constant $\tau$ defined by Eq. (1) as a function of temperature. Error bars are comparable to the size of data points. The solid line is a fit with Eq. (3).

Figure 5

The neutron TOF spectrum of $\mathrm{Zn}{\mathrm{Cr}}_2{\mathrm{Se}}_4$, measured in magnetic fields of 0, 3, 6, and 8 T, applied along the [001] direction. The top row of panels (a)–(d) shows the corresponding constant-energy cuts in the $\left(H0L\right)$ plane, integrated within $\pm 0.03\mathrm{meV}$ around 0.17 meV energy transfer. The bottom row (e)–(h) shows energy-momentum cuts along the radial directions connecting the $\mathrm{\Gamma }$ point with the local minimum of the dispersion, as shown with dashed lines in (a)–(d). Integration ranges in momentum directions orthogonal to the plane of the figure are approximately $\pm 0.05{\AA }^{-1}$. The vertical dotted lines in (a) and (b) indicate the direction of energy-momentum cuts in Fig. 6.

Figure 6

Energy-momentum cuts along the $q_z$ direction in the $B=0$ and 3 T data sets, passing through the minima in the spin-wave dispersion, as indicated in Figs. 5 and 5, respectively, with vertical dotted lines. The integration range in $q_x$ and $q_y$ momentum directions is approximately $\pm 0.05{\AA }^{-1}$.

Figure 7

Magnetic-field dependence of the INS spectrum at ($q_{\text{h}}$ 0 0), measured as a function of magnetic field applied along the [001] direction. The color map shows INS intensity measured on a TAS instrument (dark-blue color corresponds to high intensity, white to the background level). Open circles show fitted positions of the peak, resulting from Gaussian fits. Red squares show fits of additional TAS data, taken with tighter collimation to improve the resolution (the corresponding raw data are not shown). Orange star symbols correspond to the spin-gap values, extracted from the TOF data (see Fig. 6) at the local minimum of the dispersion, which was not accessible in our TAS experiment.

Figure 8

Anisotropy of the thermal conductivity with respect to the direction of the magnetic ordering vector. (a) Temperature dependence of ${\kappa }_{\perp }/T$ ($\circ$) and ${\kappa }_{\parallel }/T$ ($•$) at $B=0$ and 5 T (main panel) and at two intermediate fields of 1.5 and 3 T (inset). The dotted line in the inset shows the 3 T data set reproduced from Ref. [22], which we recalculated into $\kappa \left(T\right)/T$ and normalized by a constant factor to match with our data in the high-temperature region. (b) Temperature dependence of the relative thermal-conductivity anisotropy, $({\kappa }_{\perp }-{\kappa }_{\parallel })/{\kappa }_{\parallel }$, at different magnetic fields. The inset shows the low-$T$ part of the same data without normalization, i.e., ${\kappa }_{\perp }-{\kappa }_{\parallel }$, on a linear temperature scale. (c) Magnetic-field dependence of ${\kappa }_{\perp }$ ($\circ$) and ${\kappa }_{\parallel }$ ($•$) for three different temperatures. (d) Magnetic-field dependence of the thermal-conductivity anisotropy, ${\kappa }_{\perp }-{\kappa }_{\parallel }$, resulting from the pairwise subtraction of the data in panel (c).

Figure 9

Schematic of planar boundaries between helimagnetic domains of opposite chirality: (a), (b) at $B=0$ and (c), (d) within the conical phase. The red dashed boxes in (b) and (d) highlight energetically disfavored maximal deviations from FM nearest-neighbor spin orientations.