Magnetic excitations in the metallic single-layer ruthenates Ca${}_{2-x}$Sr${}_x$RuO${}_4$ studied by inelastic neutron scattering

By inelastic neutron scattering, we have analyzed the magnetic correlations in the paramagnetic metallic region of the series Ca${}_{2-x}$Sr${}_x$RuO${}_4$, $0.2⩽x⩽0.62$. We find different contributions that correspond to two-dimensional ferromagnetic fluctuations and to fluctuations at incommensurate wave vectors $Q_1^{\mathrm{IC}}=(0.11,0,0)$, $Q_2^{\mathrm{IC}}=(0.26,0,0)$, and $Q_{\alpha \beta }^{\mathrm{IC}}=(0.3,0.3,0)$. These components constitute the measured response as a function of the Sr concentration $x$, of the magnetic field, and of the temperature. A generic model is applicable to metallic Ca${}_{2-x}$Sr${}_x$RuO${}_4$ close to the Mott transition, in spite of their strongly varying physical properties. The amplitude, characteristic energy, and width of the incommensurate components vary only slightly as functions of $x$, but the ferromagnetic component depends sensitively on concentration, temperature, and magnetic field. While ferromagnetic fluctuations are very strong in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$ with a low characteristic energy of 0.2 meV at $T=1.5$ K, they are strongly suppressed in Ca${}_{1.8}$Sr${}_{0.2}$RuO${}_4$, but reappear upon the application of a magnetic field, and form a magnon mode above the metamagnetic transition. The inelastic neutron scattering results document how the competition between ferromagnetic and incommensurate antiferromagnetic instabilities governs the physics of this system.

Figure 1

Magnetic scattering in Ca${}_{1.8}$Sr${}_{0.2}$RuO${}_4$ at $T=2$ K and $\Delta E=4$ meV. (a) Intensity map over several Brillouin zones at a constant value of $L=1.4$. A smooth background has been subtracted. Data was taken on IN20 using the flat-cone setup. (b) Magnetic scattering in one Brillouin zone, obtained from the data in (a) by projecting to the range $H,K=[-0.5,0.5]$, thereby averaging the scattering from different Brillouin zones (after correction for the Ru magnetic form factor). (c) Sketch of the (2D) Brillouin zone and the positions of the different signals. (d) Intensity along a line $Q=(H,0,1.4)$, obtained by integration of the 2D data set in (b). (e) The same for a diagonal cut through the Brillouin zone through $(0,0,1.4)$. The data in all parts of the figure are normalized to a monitor count rate that corresponds to $~$1 min counting time per point.

Figure 2

Incommensurate signals in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$. (a) and (b) Diagonal scans across $Q_{\alpha \beta }^{\mathrm{IC}}$; the thin dotted line is the scattering-angle-dependent background. (c) and (d) Transverse scans (background subtracted) across $(1,0,0)$, crossing the positions $Q_1^{\mathrm{IC}}$ and $Q_2^{\mathrm{IC}}$ at two temperatures. Scans were taken with an energy transfer of 4 meV on the 1T spectrometer.

Figure 3

Magnetic scattering in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$ taken around the FM wave vector $(0,0,1.6)$ on the 4F spectrometer. (a) Constant energy scans at different temperatures. Shifted by two units each. (b) Signal at (0,0,1.6) as a function of temperature. (c) Signal as a function of energy at different temperatures (shifted by 100${\mu }_B^2/\mathrm{eV}$ each). Lines are fits to a single-relaxor function (see text).

Figure 4

Analysis of FM scattering in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$. (a) Characteristic energy, as obtained from the fits in Fig. 3. (b) Value of the real part of the macroscopic susceptibility, calibrated to absolute units, obtained from the same fits. (Inset: Inverse of the same values.)

Figure 5

Magnetic scattering in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$ ($T=0.05$ K, shifted by 20 counts each). The model used to fit the data in (a) consists of a ferromagnetic contribution and two incommensurate antiferromagneticlike excitations at $Q_1^{\mathrm{IC}}$ and $Q_2^{\mathrm{IC}}$. These components are given by Eqs. (2)--(4) and are displayed separately in (b) and (c). The ferromagnetic one has to be modified to account for the measured finite energy at the zone center. Data taken on the IN14 spectrometer.

Figure 6

Ferromagnetic fluctuations in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$: Different data sets taken at constant temperature or energy transfer (same as in Fig. 3). The quantity $\omega {\chi }^{\text{'}\text{'}}$ only depends on $\omega /(T+\Theta )$. The line is the function $f$ for the parameter values discussed in the text.

Figure 7

Magnetic scattering in Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$ as a function of temperature and energy (background subtracted, shifted by 300 counts each). All constant energy scans along $Q=(1,K,0)$ were performed on the thermal triple-axis spectrometer 1T. The signal changes its shape qualitatively owing to the interplay of ferromagnetic and incommensurate contributions. At high energy and low temperature the incommensurate signals dominate, whereas at low energy and high temperature the centered peak in these scans arises from the ferromagnetic component.

Figure 8

Ferromagnetic scattering in Ca${}_{1.8}$Sr${}_{0.2}$RuO${}_4$. (a) and (b) Scans across $Q^{\mathrm{FM}}$ at different energy transfers and temperatures. (c) Energy dependence at $Q^{\mathrm{FM}}$ and fits with a relaxor function (shifted by 20 counts and 300${\mu }_B^2/\mathrm{eV}$, respectively). (d) Characteristic energy $\Gamma$ of the signal at $Q^{\mathrm{FM}}$. (e) Additional ferromagnetic contribution at $T=65$ K: The blue line is the fit to the corresponding scan at $T=2$ K. For demonstration purposes, the green curve shows a Gaussian broadening of the blue curve, which can produce a maximum at $K=0$, but only for parameters that fail to describe the full scan. The red line is the sum of the blue $T=2$ K curve and a Gaussian peak at $K=0$.

Figure 9

Effect of a magnetic field on the ferromagnetic fluctuations in (a) Ca${}_{1.38}$Sr${}_{0.62}$RuO${}_4$ and (b) Ca${}_{1.8}$Sr${}_{0.2}$RuO${}_4$. The temperature is 1.5 K and the background has been subtracted in both cases (but the scales are not normalized to each other). Lines correspond to a single relaxor ($B=0$, with the characteristic energies discussed in the text) and to Lorentzian functions ($B=10$ T), respectively. The 0 T data in (b) have been measured with low statistics only, in order to fix the intensity scale relative to the high-field data under the same experimental conditions. To determine their spectral shape and energy scale, we use the information from the previously discussed zero-field data. The data were taken on the (a) IN12 and (b) Panda spectrometers.

Figure 10

Magnetic excitations in Ca${}_{1.8}$Sr${}_{0.2}$RuO${}_4$ at $B=10$ T ($B\parallel c$ and $T=1.5$ K). (a) Transverse scans through $(1,0,0)$ at different energy transfers (shifted by 7.5 counts each). The lines are fits to the model described in the text. (b) Dispersion of the magnon, extracted from fits to the scans in (a) with symmetric Gaussian peaks. Red data points are taken from Ref. 31. (c) Colour plot of the data in (a) (background subtracted).