Longitudinal magnetic dynamics and dimensional crossover in the quasi-one-dimensional spin-$\frac{1}{2}$ Heisenberg antiferromagnet ${\mathrm{KCuF}}_3$

The spin dynamics of coupled spin-$\frac{1}{2}$ antiferromagnetic Heisenberg chains is predicted to exhibit a longitudinal mode at low energies and temperatures below the Néel temperature. This mode is a dimensional crossover effect and reveals the presence of a limited amount of long-range antiferromagnetic order coexisting with quantum fluctuations. In this paper the existence of such a mode is confirmed in the model material ${\mathrm{KCuF}}_3$ using polarized and unpolarized inelastic neutron scattering, and the longitudinal polarization of the mode is definitively established. The line shape is broadened, suggesting a reduced lifetime due to decay into spin waves. In addition, the data show evidence of continuum scattering with a lower edge greater than the longitudinal mode energy. A detailed comparison is made with theoretical predictions and experimental work on other model materials.

Figure 1

Predicted $T=0$ low-energy magnetic response near the $(0,0,L)$ 1D antiferromagnetic wave vector in ${\mathrm{KCuF}}_3$. The darkness of the shading gives the intensity of the scattering. The theories illustrated are (a) the Muller ansatz; (b) spin-wave theory up to second order, showing one-magnon signal (black lines) and two-magnon scattering (shading); (c) predictions of Essler et al. (Ref. 29) and Schulz (Ref. 18) showing transverse spin waves and the longitudinal mode at low energies, and continuum at higher energies.

Figure 2

(Color online) The magnetic ordering in ${\mathrm{KCuF}}_3$ and the experimental setup. Below the Néel temperature in zero field the spin moments of ${\mathrm{KCuF}}_3$ lie in the $\mathit{a}\text{–}\mathit{b}$ plane and are canted slightly away from the [1,1,0] direction, giving a total of eight possible antiferromagnetic domains [see panel (a)]. If however a magnetic field is applied along the $\mathit{b}$ direction, the spin moments lie parallel to the $\mathbf{a}$ axis for fields greater than $0.8\mathrm{T}$ [see panel (b)]. In the experiment the neutrons are polarized parallel to the $\mathit{b}$ axis. Components of the magnetic ordering that are parallel to the polarization $\left(\mathit{P}\right)$ direction are seen in the NSF channel, while components perpendicular to $\mathit{P}$ are observed in the SF channel [see panel (c)]. In zero field the multidomain structure means that magnetic Bragg peak intensity is observed in both the NSF and SF channels; however, the single domain achieved with $\mathit{B}=0.8\mathrm{T}∕∕\mathit{b}$ means that this signal is observed solely in the SF channel. Longitudinal magnetic excitations (parallel to ordering direction) can also be separated from transverse magnetic excitations (perpendicular to ordering direction) using this technique.

Figure 3

(Color online) Constant-energy and constant-wave vector scans through the predicted longitudinal mode position using unpolarized neutron scattering. (a) Constant-wave vector scan through $(0,0,-1.5)$ below $T_N$ at $T<11\mathrm{K}$ (circles) and at $T=200\mathrm{K}$ (open squares); the data are given per unit monitor $(\sim 1\mathrm{s})$. The high-temperature data are used to identify the phonons (gray shading) and the solid lines are guides to the eye. (b) Magnetic signal for $T<11\mathrm{K}$, where the phonon background (adjusted for thermal populations factors) has been subtracted from the low-temperature data. The scan consists of transverse magnetic signal (gray shading) and a lump of scattering at $16\mathrm{meV}$ close to the expected longitudinal mode position. (c) Constant-energy scan at $16\mathrm{meV}$; again, the gray shading gives the scattering expected from the transverse spin waves. In parts (b) and (c) the dashed line is the spin-wave intensity while the solid line is the fitted intensity of both the spin waves and the longitudinal mode convolved with the instrumental resolution.

Figure 4

(Color online) Temperature dependence of the magnetic order and excitations measured with unpolarized neutrons. (a) Intensity of the (0,0,1.5) magnetic Bragg peak as the crystal is heated through its Néel temperature $T_N=39\mathrm{K}$. The line through the data is a guide to the eye. (b) Magnetic excitations at (0,0,1.5) for the temperatures $6.15\mathrm{K}$ (filled circles), $40.6\mathrm{K}$ (open squares), and $70.5\mathrm{K}$ (filled diamonds). (c) Constant-energy scan below the longitudinal mode energy at $10\mathrm{meV}$. In parts (b) and (c) the lines through the $T=6.15\mathrm{K}$ data are fits of the transverse spin waves and broadened longitudinal mode (Ref. 26) convolved with the spectrometer resolution. The lines through the $T=70.5\mathrm{K}$ data are fits of the field theory expression for an ideal, one-dimensional, spin-$\frac{1}{2}$ Heisenberg antiferromagnet at $70\mathrm{K}$ that has been corrected for interchain effects by a random phase approximation (Ref. 28) and convolved with the resolution.

Figure 5

(Color) Wave vector–energy intensity maps over the longitudinal mode region measured using unpolarized neutrons. Relative scattering intensity is indicated by color as defined in the color bars. (a) Raw data at $T<11\mathrm{K}$. The most intense scattering is in the V-shaped spin-wave branches dispersing from the antiferromagnetic zone center. The longitudinal mode at $16\mathrm{meV}$ is readily visible near $(0,0,-1.5)$, but the data are contaminated by the $19\mathrm{meV}$ phonon. (b) The phonon scattering modeled using the high-temperature data corrected for the thermal population factor. (c) Low-temperature data with the phonon scattering subtracted out. (d) Theoretical scattering intensity.

Figure 6

(Color online) Magnetic Bragg peak intensity measured with polarized neutrons. A magnetic field applied along the $\mathit{b}$ axis is used to pole the crystal into a single magnetic domain where all the spins point along the $\mathbf{a}$ axis. The intensities of the magnetic Bragg peaks vary as the spin direction changes. The intensities of four magnetic Bragg peaks are plotted in part (c) for both SF and NSF channels. Saturation is achieved for fields greater than $0.8\mathrm{T}$. The angle $\theta$ that the spin moments make with the $\mathbf{b}$ axis is given in (a) and the flipping ratio is also deduced and is plotted in (b).

Figure 7

(Color online) Constant-wave vector scans collected at $\mathit{Q}=[0,0,1.5]$ using polarized neutron scattering data corrected for flipping ratio. The data were collected at $T=6\mathrm{K}$ (filled circles), $T=200\mathrm{K}$ (open squares), and $T=6\mathrm{K}$ with analyzer turned by 15 deg from its optimum position (filled diamonds) to give a measure of the general instrumental background. Longitudinal magnetic scattering is observed in the spin-flip channel [part (a)] while transverse magnetic scattering plus phonons are observed in the non-spin-flip channel [part (b)]. The lines through the data are guides to the eye.

Figure 8

(Color online) Constant-energy scans collected at $E=16\mathrm{meV}$ using polarized neutron scattering data corrected for flipping ratio. As in Fig. 6 the data were collected at $T=6\mathrm{K}$ (filled circles), $T=200\mathrm{K}$ (open squares), and $T=6\mathrm{K}$ with analyzer turned by 15 deg (filled diamonds). Longitudinal magnetic scattering is observed in the spin-flip channel [part (a)] while transverse magnetic scattering plus phonons are observed in the non-spin-flip channel [part (b)]. Again, the lines through the data are guides to the eye.

Figure 9

(Color online) Polarized neutron scattering data corrected for flipping ratio and with the analyzer turned, incoherent, and phonon backgrounds subtracted. (a) Constant-wave vector scan collected at $\mathit{Q}=[0,0,1.5]$ and $T=6\mathrm{K}$ and measured in the SF channel. The longitudinal mode and the longitudinal continuum are fitted simultaneously to the data; the solid line gives the longitudinal mode part of this fit while the dashed line gives the longitudinal continuum part. In an alternative model the scattering observed at $15\mathrm{meV}$ is due to the lower edge of the longitudinal continuum; the dotted line is a simulation of this. (b) Constant-wave vector scan collected at $T=6\mathrm{K}$ in the NSF channel. The transverse spin waves and transverse continuum have been fitted to the data and the solid line is the spin-wave part of this fit while the dashed line is the transverse continuum part. (c) Constant-energy scans at $T=6\mathrm{K}$ in both SF and NSF channels. The solid lines through the data are the fitted longitudinal mode and spin waves. The asymmetry in the spin-wave peaks on either side of [0,0,1.5] is a consequence of instrumental resolution. In all cases the resolution function has been convolved with the fitted line shape.