Interplay of electronic and spin degrees in ferromagnetic SrRuO$_3$: anomalous softening of magnon gap and stiffness

TL;DR

This study investigates how electronic and spin interactions in ferromagnetic SrRuO$_3$ influence magnon properties, revealing anomalous softening of magnon gap and stiffness linked to Weyl points and the anomalous Hall effect.

Contribution

It demonstrates the direct influence of Weyl points and the anomalous Hall effect on spin dynamics in metallic ferromagnets, a novel insight into their interplay.

Findings

Magnon modes exhibit broadening due to strong interaction with charge carriers.

Magnon gap and stiffness soften upon cooling in the ferromagnetic phase.

The temperature dependence of these effects correlates with the anomalous Hall effect.

Abstract

The magnon dispersion of ferromagnetic SrRuO$_3$ was studied by inelastic neutron scattering experiments on single crystals as function of temperature. Even at low temperature the magnon modes exhibit substantial broadening pointing to strong interaction with charge carriers. We find an anomalous temperature dependence of both the magnon gap and the magnon stiffness, which soften upon cooling in the ferromagnetic phase. Both effects trace the temperature dependence of the anomalous Hall effect. We argue that these results show that Weyl points and the anomalous Hall effect can directly influence the spin dynamics in metallic ferromagnets.

Equations

$${\mathbf{m}}^{W,i}=\frac{\hslash {\mu }_B}{2e}{\mathbf{g}}^i({\mathbf{v}}^i{)}^{-1}{\mathbf{j}}^{W,i}$$

$$S$$

$$-\kappa (T)(m_x^2+m_y^2)-A(T)\left((\nabla m_x{)}^2+(\nabla m_y{)}^2\right)+\dots$$

$$\Delta =\frac{\kappa (T)}{\alpha (T)},D=\frac{A(T)}{\alpha (T)}$$

$$\alpha (T)\approx \frac{1}{g{\mu }_{B}m(T)}+c(T){\sigma }_{xy}^a(T)$$

$$\Delta (T)\approx \frac{a_{\Delta }m(T)/m_0}{1+b(m(T)/m_0)({\sigma }_{xy}(T)/{\sigma }_0)}$$

$$D(T)\approx \frac{a_{D}m(T)/m_0}{1+b(m(T)/m_0)({\sigma }_{xy}(T)/{\sigma }_0)}$$

$$\alpha =-\underset{\omega \to 0,\mathbf{q}\to 0}{\lim }\frac{({\chi }_0^{-1}(\mathbf{q},\omega ){)}_{xy}}{i\omega }$$

$$\mathbf{m}\approx {\mathbf{m}}_0+\frac{\hslash {\mu }_B}{2e}\sum _i{\mathbf{g}}^i({\mathbf{v}}^i{)}^{-1}({\mathbf{j}}^{W,i}-{\mathbf{j}}^{W,\bar{i}})$$

$$\alpha (T)\approx \frac{1}{g{\mu }_{B}m(T)}+c(T){\sigma }_{xy}^a(T)$$

Full text

SrRuO3 Randall and Ward (1959); Callaghan et al. (1966); Koster et al. (2012) with the 4d ion Ru4+ is a prime candidate to observe the impact of strong SOC in a ferromagnetic metal. It crystallizes in the cubic perovskite structure but undergoes two structural phase transitions at 975 and 800 K into an orthorhombic phase (space group Pnma) associated with rotations of the RuO6 octahedra Randall and Ward (1959); Jones et al. (1989); Chakoumakos et al. (1998). The ferromagnetic transition occurs at 165 K in single crystals and at low temperature a magnetization of $\sim$1.6$\mu_{B}$ is observed at 6 T Kunkemöller et al. (2016). The electric resistivity is linear at moderate temperatures and breaks the Ioffe-Regel limit, but it drops in the ferromagnetic phase with a clear kink at the ferromagnetic $T_{c}$ Allen et al. (1996); Klein et al. (1996) attaining very low residual resistivity values of only 3 $\mu\Omega cm$ in high-quality single crystals Kunkemöller et al. (2016, 2017). This coupling of ferromagnetic excitations and charge transport inspired the proposal of spin-triplet pairing in the superconducting sister compound Sr2RuO4 Rice1995 , in which quasiferromagnetic excitations could only recently be observed in neutron experiments Steffens2019 . The anomalous Hall effect in SrRuO3 shows a peculiar temperature dependence undergoing a sign change Izumi et al. (1997); Fang et al. (2003); Kats et al. (2004); Haham et al. (2011); Koster et al. (2012); Itoh et al. (2016). Fang et al. Fang et al. (2003) proposed that the anomalous Hall effect in SrRuO3 arises from the impact of magnetic monopoles in momentum space associated with Weyl points. The magnetic exchange splitting combined with the impact of SOC causes Weyl points in the band structureFang et al. (2003); Itoh et al. (2016), which DFT calculations predict to occur near the Fermi level Chen et al. (2013).

More recently Itoh et al. argued that the Weyl points not only induce the peculiar temperature dependence of the anomalous Hall effect but also affect the spin dynamics. At each Weyl point, index $i$, two bands cross and the single-particle current $j^{W,i}_{n}=e\sum_{m}v^{i}_{nm}{\boldsymbol{\sigma}}^{m}$ and magnetization $m^{W,i}_{n}=\sum_{m}g^{i}_{nm}\mu_{B}\frac{\hbar{\boldsymbol{\sigma}}^{m}}{2}$ are proportional to each other Burkov and Balents (2011)

[TABLE]

where ${\boldsymbol{g}}^{i}$ is the $g$-tensor, $n,m=x,y,z$ and ${\boldsymbol{v}}^{i}$ the tensor of Fermi velocities characterizing the Weyl point $i$. The intimate relation of current, magnetization and Berry phases will be tested by our paper.

Inelastic neutron scattering (INS) experiments in reference Itoh et al. (2016) indeed find a temperature dependence of the magnon gap resembling that of the anomalous Hall effect while the magnon stiffness was claimed to exhibit a normal temperature dependence, i.e. a hardening upon cooling. However, these measurements were performed with powder samples that do not give direct access to the parameters of the magnon dispersion. Here we report on INS, magnetization and anomalous Hall effect measurements on single-crystalline SrRuO3. We confirm a close similarity between the temperature dependencies of the magnon gap and the anomalous Hall conductivity, but the magnon gap differs from the powder experiment by almost a factor two. Furthermore, the magnon stiffness also softens upon cooling in the ferromagnetic phase. We argue that this unusual temperature dependence originates from the coupling of current and magnetization thus confirming the impact of the anomalous Hall effect on magnetization dynamics.

Large single crystals of SrRuO3 were grown by the floating-zone technique and characterized by resistivity and magnetization measurements Kunkemöller et al. (2016). For the INS experiments six crystals with a total mass of $\sim$6g were coaligned in the [100]/[011] scattering plane in respect to the pseudocubic lattice with $a_{c}\sim 3.92$ Å. INS experiments were performed on cold triple-axis spectrometers (TAS) 4F at the Laboratoire Léon Brillouin and PANDA at the Meier-Leibnitz Zentrum, and on the thermal TAS 2T at the Laboratoire Léon Brillouin. The anomalous Hall effect was measured on a rectangular sample with edge lengths of $2.2\times 1.64\times 0.193$ mm3 along the cubic directions [1$\overline{1}$0], [001], and [110], respectively. We applied the electrical current $I_{x}$ (typically 5 mA) along [1$\overline{1}$0] (orthorhombic $a$), the external magnetic field along [110] (the easy axis, orthorhombic $c$, up to $\pm 7$ T) and measured the longitudinal ($U_{x}||I_{x}$) and the transverse voltage $U_{y}||[001]$ (orthorhombic $b$). Using a SQUID magnetometer, we also measured the magnetization $M$ of this sample for the same field direction in order to precisely determine the normal and anomalous Hall effects. Further experimental details are given in the supplemental material.sup

Fig. 1 shows the INS data obtained by constant energy scans across the magnon dispersion on cold and thermal TAS. The peaks arising from the magnon on both sides of the (100) Bragg points are clearly visible and allow for a reliable determination of the dispersion. In order to quantitatively analyze this data we calculate the folding of the magnon dispersion with the resolution of the cold and thermal TAS using the ResLib program package Res . The lines superposed to the constant energy scans refer to this folding procedure with only a few global fit parameters. For small momenta, the magnon dispersion can approximately be described by $E_{0}({{\boldsymbol{q}}})\approx\Delta+D\,{{\boldsymbol{q}}}^{2}$, with the anisotropy gap $\Delta$ and the magnon stiffness $D$. Magnetic anisotropy is sizeable in SrRuO3 as it can be inferred from the macroscopic magnetic anisotropy Kanbayasi (1976); Cao et al. (1997); Kunkemöller et al. (2016) and the shape memory effect Kunkemöller et al. (2017) and from an optical study Langner et al. (2009). The small orthorhombic distortion of SrRuO3 induces a tiny anisotropy in $D$, see below. Note, that throughout the paper we use reduced reciprocal lattice units with respect to the pseudocubic cell, $\frac{2\pi}{a_{c}}$, but the stiffness is typically given in units of meVÅ2. In order to describe the measured scan profiles we need to assume a width of the magnon modes that amounts to 40% of their energy. Part from this broadening can stem from the twinning of the crystals superposing different direction of the orthorhombic lattice, but due to the small orthorhombic distortion this broadening should be of the order of a few % only. Magnons in SrRuO3 thus exhibit strong scattering most likely due to the coupling to electrons, see also the kink in resistivity at $T_{c}$ Allen et al. (1996); Klein et al. (1996), and due to the presence of sizable SOC. The dispersion, which for small $|{\boldsymbol{q}}|$ values is quadratic, is presented in Fig. 1(c). A consistent description of the data at low temperatures is obtained yielding $D$=87(2) meVÅ2 and $\Delta$=0.94(3) meV and an energy width of $0.4\cdot E_{0}(q)$. Note that the gap $\Delta$ is almost a factor two smaller than the result obtained from the powder experiment, and also the stiffness $D$, considerably differs Itoh et al. (2016).

With the better resolution of the cold TAS 4F we scanned across the magnon gap at the zone center, see Fig. 2. Again the gap can be directly read from the raw data in contrast to the previous powder INS experiment.Itoh et al. (2016) Our single-crystal INS result further agrees with the optical study Langner et al. (2009) and with the extrapolation of the anisotropic magnetization Kunkemöller et al. (2017). The easy axis of SrRuO3 is found along the orthorhombic $c$ direction in Pnma notation Kunkemöller et al. (2017) and therefore two anisotropy gaps can be expected for the orthorhombic system, i.e. for rotating the magnetic moments towards $a$ and $b$ directions. Our data indicate little splitting for the two gaps that were examined with an untwined smaller crystal on PANDA. This crystal was first mechanically detwinned Kunkemöller et al. (2016) and then mounted with its cubic [01$\bar{1}$] direction parallel to a magnetic field in a vertical field cryostat. After applying a magnetic field of 3 T a fully monodomain crystal was obtained Kunkemöller et al. (2017) with the orthorhombic $c$ axis, the magnetic easy axis, parallel to cubic [01$\bar{1}$], and thus $a\|$[011] and $b\|$[100]. The gaps measured at the scattering vectors $\bf{Q}$=(100) and (011) correspond thus to the rotation of the moments towards $a$ and $b$, respectively. In agreement with the macroscopic analysis in reference Kunkemöller et al. (2017) the $a$ direction is only slightly softer than $b$. The anomalies of the gap temperature dependence are qualitatively confirmed by these monodomain measurements, but the temperature dependence of the averaged gap profits from higher statistics. From the constant $\mathbf{Q}$ scans at the zone center we deduce the temperature dependence of the anisotropy gap by fitting a gaussian peak, see Fig. 2 (b). In only qualitative agreement with the powder INS there is a rather anomalous softening and rehardening of the gap upon cooling deep in the ferromagnetic phase, while the closing of the gap upon heating above the Curie temperature corresponds to the expected behavior.

Fig. 3 summarizes the temperature dependence of the magnon stiffness. We recorded constant energy scans at 8 meV between 13 and 280 K that were analyzed by fitting the $D$ value through the folding of the resolution with the dispersion. The characteristic two peak structure remains visible even well above the Curie temperature, which underlines the persistence of ferromagnetic correlations. This scattering agrees with the expectation for a nearly ferromagnetic metal, which still exhibits a paramagnon signal very similar to that of ferromagnetic material with broad magnons Moriya (1985). The fitted $q$ positions can be directly transformed into temperature dependent stiffness constants, $D$, by taking the temperature dependent anisotropy gaps into account, Fig. 3 (b). The magnon stiffness clearly softens upon cooling well below the Curie temperature, while the magnetization increases. In Fig. 3 (c) we compare the stiffness to the spontaneous magnetization, because in a usual system one expects the two quantities to scale. In contrast in SrRuO3 the stiffness softens upon cooling well below Tc. Scaling the magnetization, $M_{s}$, and $D$ by the low-temperature values, one recognizes that at 0.8 times $T_{c}$ the stiffness almost twice as large than what follows from $D\propto m$.

The dynamics of small-amplitude, long-wavelength oscillations of the magnetization in a ferromagnet polarized in the $z$-direction can be described by the action Itoh et al. (2016)

[TABLE]

up to higher order corrections and damping terms. From the corresponding Euler-Lagrange equations one obtains

[TABLE]

In the absence of spin-orbit coupling $\alpha$ is exactly given by $\alpha_{0}=\frac{1}{2m(T)\mu_{B}}$ where $m(T)$ is the magnetisation density. Taking SOC and the Weyl points into account $\alpha(T)$ is determined not only by $m(T)$ but also by the anomalous Hall effect Itoh et al. (2016); sup

[TABLE]

This equation arises because the time-dependent magnetization induces currents close to the Weyl points, see Eq. (1) and sup . Due to the Berry curvature of the Bloch bands, both a transverse current and a transverse magnetization are generated, modifying $\alpha(T)$.

Neglecting the $T$ dependence of $\kappa(T)$, $c(T)$ and $A(T)$, we obtain from Eq. (3) and (8) for the spin gap $\Delta$ Itoh et al. (2016) and the stiffness $D$ the same temperature dependence parametrized by

[TABLE]

where $m_{0}=m(T=0)$ and $\sigma_{0}=\frac{e^{2}}{2\pi\hbar a_{c}}$.

In order to verify the scaling between the anomalous Hall effect and the two characteristic parameters of the magnon dispersion in SrRuO3 we also measured the magnetization and anomalous Hall effect on a single crystal. The magnetization obtained by extrapolating field-dependent magnetization curves to H=0 T is shown in Fig. 3(c) and can be described by a stretched power law $m(T)=m_{0}(1-(\frac{T}{T_{c}})^{a})^{\beta}$ with the parameters $m_{0}$=1.69$\mu_{B}$, $a$=1.27 and $\beta$=0.304. The field-dependent magnetization was inserted in the analysis of the anomalous Hall effect measurement performed on the same sample. The comparison of our and previous Hall conductivity results is shown in Fig. 4(ab). Due to the larger thickness of the single-crystalline sample the Hall voltage is smaller, but it offers the advantage of more precise magnetization data. Our single-crystal data of the Hall conductivity $\sigma_{xy}$ agree with previous single-crystal data but only qualitatively with powder and thin film data Itoh et al. (2016); not . There is a sign change in $\sigma_{xy}$ slightly below the ferromagnetic transition. Differences may stem from the twinning and different orientations of the distorted orthorhombic lattices in the powder and thin-film experiments and from differences in sample quality. The temperature dependence of the anomalous Hall effect was fitted by a polynomial and then inserted in equation (5) to describe the anomalous softening of the magnon gap in the ferromagnetic phase, see Fig. 4(b). This analysis well reproduces the main feature with $a_{\Delta}$ = 1.66 meV and $b$ = 4.98. The same temperature dependence of the anomalous Hall effect was used to describe the softening of the stiffness compared to a normal behavior proportional to the magnetization, $a_{D}$ = 160.90 meVÅ2. Again good agreement is obtained, see Fig. 4(c). Taking into account that Eq. (5) completely neglects corrections arising, e.g., from the $T$ and magnetization dependence of $K(T)$, $J(T)$, $c(T)$ and from the broadening of the spin-waves, the semi-quantitative agreement is satisfactory. It clearly supports a common origin of the unusual softening of the spin-gap and the spin-stiffness towards lower temperatures, explained by the coupling of magnetization and current and by the $T$-dependence of the anomalous Hall effect.

At the phase transition, where $m(T)$ vanishes, Eq. (5) predicts that $D(T)$ vanishes also. However, in this temperature regime the broad spin-waves smoothly transform into paramagnon scattering Moriya (1985) with a very similar shape and thus a drop of $D(T)$ cannot be extracted from the measured neutron scattering data.

In conclusion we have studied the magnetization, anomalous Hall effect and magnon dispersion in SrRuO3 using high-quality single crystals. The magnon modes exhibit sizeable broadening revealing strong scattering by most likely charge carriers. The magnon gap and the magnon stiffness do not follow the temperature dependence of the spontaneous magnetization. Both quantities soften upon cooling over a large temperature range in the ferromagnetic phase and at least the magnon gap passes through a minimum. These findings can be well explained by the effect of Weyl points situated close to the Fermi level. Such Weyl points possess a well-established impact on the anomalous Hall effect and cause an additional term in the magnetic Hamiltonian. The latter leads to a reduction of both the magnon gap and the stiffness, which scales with the anomalous Hall effect. Our data perfectly agree with this scaling between the anomalous Hall effect and the magnon dispersion.