Anisotropic damping of the spin fluctuations in doped La2-xSrxCuO4 studied by resonant inelastic x-ray scattering

TL;DR

This study uses resonant inelastic x-ray scattering to analyze spin fluctuations in doped La2-xSrxCuO4, revealing anisotropic damping behavior that varies with doping and momentum, and correlates with superconductivity.

Contribution

It provides the first detailed wavevector-dependent damping analysis of spin fluctuations in doped La2-xSrxCuO4 using RIXS, normalizing data to neutron scattering results.

Findings

Damping parameter increases with doping and peaks near (0.2,0.2).

Wavevector-dependent susceptibility is larger at (1/4,1/4) than at (1/2,0).

Magnetic excitations strongest near (1/2,1/2), relevant for superconductivity.

Abstract

We report high-resolution resonant inelastic x-ray scattering (RIXS) measurements of the collective spin fluctuations in three compositions of the superconducting cuprate system La2-xSrxCuO4. We have mapped out the excitations throughout much of the 2-D (h,k) Brillouin zone. The spin fluctuations in La2-xSrxCuO4 are found to be fairly well-described by a damped harmonic oscillator model, thus our data allows us to determine the full wavevector dependence of the damping parameter. This parameter increases with doping and is largest along the (h, h) line, where it is peaked near (0.2,0.2). We have used a new procedure to determine the absolute wavevector-dependent susceptibility for the doped compositions La2-xSrxCuO4 (x=0.12,0.16) by normalising our data to La2CuO4 measurements made with inelastic neutron scattering (INS). We find that the evolution with doping of the intensity of high-energy excitations measured by RIXS and INS is consistent. For the doped compositions, the wavevector-dependent susceptibility is much larger at (1/4,1/4) than at (1/2,0). It increases rapidly along the (h,h) line towards the antiferromagnetic wavevector of the parent compound (1/2,1/2). Thus, the strongest magnetic excitations, and those predicted to favour superconductive pairing, occur towards the (1/2,1/2) position as observed by INS.

Tables

Table 1. Table 1: Doping dependence of the χ ′ ​ ( 𝐐 ) superscript 𝜒 ′ 𝐐 \chi^{\prime}(\mathbf{Q}) in LSCO as measured with RIXS.

Q	$(1/4,0)$	$(1/4,1/4)$
$x$	${\chi }^{\prime }\left(𝐐\right)({\mu }_B^2{\text{eV}}^{-1}{\text{f.u.}}^{-1}$ )
0	$3.7\pm 0.3$	$5.6\pm 0.6$
0.12	$7.1\pm 0.3$	$9.6\pm 1$
0.16	$7.3\pm 0.8$	$8.0\pm 1$

Equations

$$\ddot{x}+{\omega }_0^{2}x+\gamma \dot{x}=f/m,$$

$${\chi }^{\prime \prime }(\mathbf{Q},\omega )=\frac{{\chi }^{\prime }(\text{Q}){\omega }_0^2(\mathbf{Q})\gamma (\mathbf{Q})\omega }{{\left[{\omega }^2-{\omega }_0^2(\mathbf{Q})\right]}^2+{\omega }^2{\gamma }^2(\mathbf{Q})},$$

$$\omega =\pm {\left[{\omega }_0^2-\left({\gamma }^2/4\right)\right]}^{\frac{1}{2}}=\pm {\omega }_1-\frac{i\gamma }{2}.$$

$${\omega }_{\text{max}}=\frac{1}{6}\sqrt{12{\omega }_0^2-6{\gamma }^2+6\sqrt{{\gamma }^4-4{\gamma }^2{\omega }_0^2+16{\omega }_0^4}}.$$

$${\chi }^{\prime \prime }(\mathbf{Q},\omega )=\frac{{\chi }^{\prime }(\mathbf{Q})\Gamma (\mathbf{Q})\omega }{{\omega }^2+{\Gamma }^2(\mathbf{Q})}.$$

$${\chi }^{\prime }(\mathbf{Q})={\chi }^{\prime }(\mathbf{Q},\omega =0)=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{{\chi }^{\prime \prime }(\mathbf{Q},\omega )}{\omega }d\omega .$$

$$⟨{\chi }^{\prime \text{LSCO}}(\mathbf{Q})⟩={\chi }_{\text{ RIXS}}^{\prime \text{LSCO}}(\mathbf{Q})\times \frac{{\varphi }_{\text{SWT}}^{\text{LCO}}(\mathbf{Q})}{{\varphi }_{\text{RIXS}}^{\text{LCO}}(\mathbf{Q})}.$$

$$V_{\text{eff}}\simeq \frac{3}{2}{U}^2{\chi }^{\prime }({\mathbf{p}}^{\prime }-\mathbf{p}),$$

$${\chi }_{\perp }^{\prime \prime }(\mathbf{Q},\omega )$$

$$=\frac{\pi }{2}{\chi }_{\perp }^{\prime }(\mathbf{Q}){\omega }_0(\mathbf{Q})\delta [\omega \pm {\omega }_0(\mathbf{Q})],$$

$$\hslash {\omega }_0(\text{Q})=2Z_c\sqrt{A_{\mathbf{Q}}^2-B_{\mathbf{Q}}^2},$$

$${\chi }_{\perp }^{\prime }(\mathbf{Q})=Z_d(\mathbf{Q})\frac{g^2{\mu }_B^{2}S}{A_{\mathbf{Q}}+B_{\mathbf{Q}}}.$$

$$Z_{d}(\mathbf{Q})=\begin{cases}\mbox{$Z_{d0}\sin(h\pi)$,}&\mbox{if }h<\frac{1}{4}\\ \mbox{$Z_{d0}\sin({\displaystyle\frac{\pi}{4}})$,}&\mbox{if }h\geq\frac{1}{4}\end{cases},$$

$${\varphi }_{\text{SWT}}^{\text{LCO}}(\mathbf{Q})={\int }_0^{\infty }{\chi }^{\prime \prime }(\mathbf{Q},\omega )d\omega =\frac{\pi }{3}{\chi }_{\perp }^{\prime }(\mathbf{Q}){\omega }_0(\mathbf{Q}).$$

$${\chi }^{\prime \prime }(\mathbf{Q},\omega )=\frac{{\chi }^{\prime }(\mathbf{Q})}{2{\omega }_1(\mathbf{Q})}\left[\frac{{\gamma }^2(\mathbf{Q})}{4}+{\omega }_1^2(\mathbf{Q})\right]\times$$

$$\left\{\frac{\gamma (\mathbf{Q})/2}{{\gamma }^2(\mathbf{Q})/4+[\omega -{\omega }_1(\mathbf{Q}){]}^2}-\frac{\gamma (\mathbf{Q})/2}{{\gamma }^2(\mathbf{Q})/4+[\omega +{\omega }_1(\mathbf{Q}){]}^2}\right\}.$$

$${\varphi }_{\text{RIXS}}^{\text{LCO}}(\mathbf{Q})=\frac{\pi {\chi }^{\prime }(\mathbf{Q}){\omega }_0^2(\mathbf{Q})}{\sqrt{4{\omega }_0^2(\mathbf{Q})-{\gamma }^2(\mathbf{Q})}}.$$

Full text

I Introduction

The origin of high temperature superconductivity (HTS) in doped layered cuprate materials remains a subject of intense interest in both experimental and theoretical research, despite over 30 years of activity. It is widely believed that the magnetic degrees of freedom and in particular spin fluctuations are primarily responsible for superconductive pairing in the cuprates Chubukov et al. (2003); Scalapino (2012); Eschrig (2006); Keimer et al. (2015). In this case, it is important to characterize the collective spin excitations as a function of wavevector, energy, doping and temperature to see how they correlate with the occurrence of superconductivity and compare with theoretical models.

Resonant inelastic x-ray scattering (RIXS) Ament et al. (2011); Braicovich et al. (2009, 2010a); Le Tacon et al. (2011); Ghiringhelli et al. (2012); Dean et al. (2012, 2013a); Dean (2015); Monney et al. (2016); Chaix et al. (2018) and inelastic neutron scattering (INS) Hayden et al. (1991, 1996); Arai et al. (1999); Dai et al. (1999); Coldea et al. (2001); Headings et al. (2010); Lipscombe et al. (2007) are complementary probes which directly yield information about the wavevector and energy of the dynamical structure factor $S(\mathbf{Q},\omega)$ or dynamic susceptibility (response function) $\chi^{\prime\prime}(\mathbf{Q},\omega)$ at high frequencies. The La2-xSrxCuO4 (LSCO) system allows the evolution of $S(\mathbf{Q},\omega)$ to be measured across the phase diagram, from the antiferromagnetic (AF) parent compound La2CuO4 (LCO) through superconducting compositions.

In La2CuO4, the spin waves have their lowest energies at the $\Gamma$, $\mathbf{Q}$=($0,0$) and $M$, $\mathbf{Q}$=($\frac{1}{2},\frac{1}{2}$) positions and $\chi^{\prime\prime}(\mathbf{Q},\omega)$ is small near $\Gamma$ and largest near $M$. INS measurementsHayden et al. (1991); Coldea et al. (2001); Headings et al. (2010) throughout the Brillouin zone have shown that the magnetic excitations can be fairly well-described as spin waves derived from a Heisenberg model with next-nearest neighbour interactions including a ring exchange. As expected, they are strongest near the AF wavevector $\mathbf{Q}$=($\frac{1}{2},\frac{1}{2}$) and show anomalously strong damping at the $X$ or ($\frac{1}{2},0$) position Headings et al. (2010); Dean et al. (2012); Sandvik and Singh (2001).

For superconducting compositions in LSCO, INS shows that the strongest response Hayden et al. (1996); Vignolle et al. (2007); Wakimoto et al. (2007); Lipscombe et al. (2007, 2009) occurs near $\mathbf{Q}$=($\frac{1}{2},\frac{1}{2}$) at low and intermediate energies (0–150 meV), with comparable intensity to the parent antiferromagnet. For optimally doped ($x=0.16$) LSCO, an incommensurate structure is observedVignolle et al. (2007) for $\hbar\omega\lesssim 25$ meV. Above 50 meV the magnetic excitations disperseVignolle et al. (2007); Lipscombe et al. (2007, 2009) away from ($\frac{1}{2},\frac{1}{2}$). At high energies, $\hbar\omega\approx 250$ meV, excitations are observedHayden et al. (1996) on the Brillouin zone boundary at $\mathbf{Q}=(\frac{1}{2},0)$ in LSCO ($x=0.14$) demonstrating the persistence of high energy spin excitations for superconducting compositions. For overdoped compositionsWakimoto et al. (2007); Lipscombe et al. (2007) $x=0.22-0.25$, the lower energy ($\hbar\omega\sim 50$ meV) features observed at optimal doping are suppressed.

Cu $L_{3}$ RIXSBraicovich et al. (2009, 2010a); Dean et al. (2012, 2013a); Dean (2015); Monney et al. (2016); Meyers et al. (2017); Chaix et al. (2018) measurements of the spin fluctuation in LSCO are complementary to INS. They are restricted to a circular region in $(h,k)$ centered on $\Gamma$ [see Fig. 1 (a) and (b) ] but are able to isolate high energy excitations ($\hbar\omega\gtrsim$ 300 meV) more easily. Early RIXS measurements in LSCO Braicovich et al. (2010a) verified the existence of dispersing spin fluctuations. Spin excitations are observedBraicovich et al. (2010a); Dean et al. (2013a); Dean (2015); Monney et al. (2016); Meyers et al. (2017); Chaix et al. (2018) throughout the first AF Brillouin zone including at the boundary [e.g. ($\frac{1}{2},0$) position] where INSHayden et al. (1996) also finds excitations. RIXS studies suggest that these excitations show wavevector-dependent damping Dean et al. (2013a); Monney et al. (2016); Meyers et al. (2017). Spin fluctuations persist to overdoped compositions and evolve relatively slowly with doping Dean et al. (2013a); Monney et al. (2016).

The improved energy resolution of the measurements we have performed allows us to model the nature of the spin fluctuations more precisely. The motivation of this work is to perform a systematic characterisation of the spin fluctuations in LSCO with this enhanced energy resolution including mapping the Q-dependence of the frequency and damping throughout a 2-D portion of the Brillouin zone. We also aim to bridge the techniques of INS and RIXS to establish an estimate of the absolute spin susceptibility.

Here we report RIXS measurements on three dopings of LSCO, $x$ = 0, 0.12 and 0.16. We have made use of the high resolution and high intensity of the RIXS spectrometers ID32 at the European Synchrotron Radiation Facility (ESRF) and I21 at the Diamond Light Source (DLS) to map out magnetic spectra over 2-D $(h,k)$ space. We find that, for doped compositions, the magnetic response is fairly well-described by a damped harmonic oscillator line shape. The pole frequency and damping are strongly anisotropic in agreement with previous studies along the $(h,0)$Dean et al. (2013a) and $(h,h)$Meyers et al. (2017) lines, with the strongest damping along the $(h,h)$ line and centred near (0.2, 0.2) for the optimally doped composition. By comparing data on La2CuO4, where the spin waves are well studied, with LSCO, we make quantitative estimates of the wavevector-dependent susceptibility $\chi^{\prime}(\mathbf{Q})$. This quantity is a vital input to theories of the HTS phenomenon Chubukov et al. (2003); Scalapino (2012); Eschrig (2006).

II Experimental details

II.1 Samples

Measurements were performed on single crystal samples of LSCO. Three different compositions were measured: the $x$ = 0 parent compound which displays antiferromagnetism below $T_{N}\simeq$ 320 K Headings et al. (2010) and two hole-doped compoundsCroft et al. (2014); Vignolle et al. (2007), $x$ = $0.12\pm 0.005$ ($T_{c}=29.5$ K) and $x=0.16\pm 0.005$ ($T_{c}=38$ K). The $x$ = 0.16 composition is close to optimal doping for the superconducting phase and $x$ = 0.12 shows charge density wave (CDW) order with short range CDW correlations developing at $T\sim 150$ K and a longer range CDW developing at $T_{\text{CDW}}\simeq$ 75 K Croft et al. (2014). Crystals were grown via the travelling solvent floating zone technique and used in previous neutron Headings et al. (2010); Vignolle et al. (2007); Lipscombe et al. (2007) and x-ray Croft et al. (2014) studies. The crystals were re-cut into posts with typical dimensions $\simeq$ 2 x 1 x 1 mm3. The samples were aligned using Laue x-ray diffraction and cleaved in-situ to expose a clean surface to the beam. The sample used for measurements of the $x$ = 0.12 compound at the ESRF was polished following the procedure in Croft et al. Croft et al. (2014). For the same composition, the elastic peak observed close to the specular condition is approximately 15 times greater in the polished sample compared to the cleaved sample. This makes the low energy excitations at low Q difficult to extract and we therefore only use data from this sample in the map plots. We verified that the lineshape, intensity and energy of the magnetic excitations is the same in both datasets.

II.2 Notation

LSCO undergoes a structural transition to a low-temperature orthorhombic (LTO) phase below $T_{\text{LTO}}\simeq$ 240 K, however, we use the high-temperature tetragonal (HTT) I/4mmm crystal structure notation to allow comparison between the three compounds. In this notation, $a$ = $b\simeq$ 3.8 Å, $c\simeq$ 13.2 Å. The momentum transfer $\mathbf{Q}$ is defined in reciprocal lattice units (r.l.u.) as $\mathbf{Q}=h\mathbf{a^{*}}+k\mathbf{b^{*}}+l\mathbf{c^{*}}$ where $\mathbf{a^{*}}=2\pi/a$ etc. The measured excitations are labelled via their energies $\hbar\omega=c\left|\mathbf{k}\right|-c\left|\mathbf{k}^{\prime}\right|$ and momenta $\mathbf{Q}=\mathbf{k}-\mathbf{k}^{\prime}$, where $\mathbf{k}$ and $\mathbf{k}^{\prime}$ are in initial and final wavevectors.

II.3 Spectrometers

High resolution RIXS spectra were measured at beamline ID32 of the ESRF Braicovich et al. (2012); Brookes et al. (2018) and the I21 RIXS spectrometer at DLSdia . The incoming beam energy was tuned to the Cu L3-edge ($\sim$ 932 eV) with linear horizontal (LH) $\pi$ polarisation. We present LH data from the grazing-out orientation where the single magnon intensity is favoured Braicovich et al. (2010b); Sala et al. (2011). Recent experiments with polarisation analysisPeng et al. (2018); Fumagalli et al. (2019) have established that this configuration is primarily sensitive to magnetic scattering. Samples were mounted on the sample holder in ultra-high vacuum and cooled to $T\simeq 20$ K. Magnetic excitations in cuprates are dispersive predominantly in the $a$-$b$ plane of LSCO, allowing paths to be measured in the ($h,k$) plane by varying the sample orientation, and keeping the scattering angle $2\theta$ fixed at 146*∘* and 149.5*∘* for I21 and ID32 respectively. The scattering geometry is shown in Fig. 1 (a). We assume there is negligible dispersion in the features of interest from variation of $l$, and therefore we focus only on the momentum transferred in the ($h,k$) plane. Spectra were principally measured along the two high-symmetry lines ($h,0$) and ($h,h$) as indicated with red arrows in Fig. 1 (b) with energy resolution $\Delta E$ $\simeq$ 35 meV. The $x$ = 0 and 0.12 measurements were performed at I21 and the $x$ = 0.16 measurements were performed at ID32 and repeated at I21. In both doped compounds, further measurements were performed at ID32 with $\Delta$E $\simeq$ 50 meV on a grid of Q-points evenly distributed throughout a quadrant of the Brillouin zone indicated by the red shaded region in Fig. 1 (b). The energy resolution was established using elastic scattering from a silver paint or carbon tape reference. For I21, a background was measured from either a dark image taken after the collection or by fitting a constant background outside the excitation range, $\leq-0.1$ eV and $\geq$ 5 eV.

II.4 Analysis

II.4.1 Data processing

In order to carry out a quantitative analysis of the data, we follow recent practice Braicovich et al. (2009, 2010a); Dean et al. (2013b); Monney et al. (2016); Meyers et al. (2017); Peng et al. (2018) and assume that the magnetic intensity observed in RIXS is proportional to the spin-spin dynamical structure factor $S(\mathbf{Q},\omega)$ which is used to interpret neutron scattering experiments Lovesey (1986). $S(\mathbf{Q},\omega)$ is, in turn, proportional to $\chi^{\prime\prime}(\mathbf{Q},\omega)$ multiplied by the Bose factor $n(\omega)+1=[1-\exp(-\hbar\omega/k_{B}T)]^{-1}$. Clearly, the scattering processes in RIXS and INS are very different, with the observed RIXS intensity being dependent on the relative orientation of the photon electric field to the Cu $3d$ orbitals as well as the absorption of the x-ray photons within the sample. These factors are known to vary slowly with $\mathbf{Q}$ Ament et al. (2009); Jia et al. (2014), nevertheless, to correct for these effects we initially normalise our raw counts $I_{\mathrm{raw}}$ to the energy-integrated $dd$ excitation intensity obtained from the same spectrum. The intensity of the $dd$ excitations is known to be dependent on the polarisation $\epsilon$ and wavevector $\mathbf{k}$ and can be described by a function $g(\epsilon,\epsilon^{\prime},\mathbf{k},\mathbf{k^{\prime}})$. We denote the measured intensity $I_{\mathrm{RIXS}}$ as $I_{\mathrm{raw}}/g$ where $g=\int g(\epsilon,\epsilon^{\prime},\mathbf{k},\mathbf{k^{\prime}})\,d\omega$ is the integral described above evaluated over the range 1–3 eV.

The spectra were aligned to the elastic reference and the exact zero-energy position was established by fitting an elastic peak with a Gaussian function. The aligned spectra were modelled within a range –80 to 800 meV. As well as the spin excitations, we fit an elastic peak and low-energy excitations, which are interpreted as phonons, using Gaussian functions. Electron-hole excitations and broadened $dd$ excitations contribute to the low-energy RIXS scattering for doped compositions Dean et al. (2013a). This contribution was modelled with a linear function which was fixed for all spectra of the same composition. The gradient of the linear function was found by fitting the spectra at low Q. In the insulating parent compound this contribution was not required. However, a broad continuum of multimagnon excitations is resolvable at $\sim$ 400–600 meV. This was modelled with a Gaussian function.

The spectra were not deconvolved to take account of the instrument energy resolution $\simeq 35$ meV. The most noticeable effect of this was in the determination of $\gamma$ and $\Gamma$ values (see Sec. II.4.2). We estimate that our fitted values are increased by 5% in the worse case.

II.4.2 Damped harmonic oscillator model

A damped harmonic oscillator (DHO) model may be used to describe a given spin-wave mode with wave vector Q. This approach has recently been taken in a number of RIXS studies Dean et al. (2013a); Monney et al. (2016); Lamsal and Montfrooij (2016); Peng et al. (2018). The analogous mechanical DHO equation is Chaikin and Lubensky (1995)

[TABLE]

where $\omega_{0}$ is the frequency of the undamped mode and $\gamma$ is the damping parameter. In our case, both of these are Q-dependent, thus $\omega_{0}=\omega_{0}(\mathbf{Q)}$ and $\gamma=\gamma(\mathbf{Q)}$.

The imaginary part of the DHO response function for a given wavevector can be written as,

[TABLE]

where $\chi(\mathbf{Q})\equiv\chi^{\prime}(\mathbf{Q})\equiv\chi^{\prime}(\mathbf{Q},\omega=0)$ is the real part of the zero frequency susceptibility. The solution of Eq. 1 can be represented by two poles with complex frequencies:

[TABLE]

If $\omega_{0}^{2}\geq\gamma^{2}/4$, $\omega_{1}$ is real and the frequency of the pole. The solutions (response) correspond to damped oscillations in time. If $\omega_{0}^{2}\leq\gamma^{2}/4$, $\omega_{1}$ is imaginary and the system is overdamped. We may introduce a third frequency, $\omega_{\textrm{max}}$, defined as the frequency at the peak in $\chi^{\prime\prime}(\omega)$. This can be shown to be

[TABLE]

Using the DHO function (Eqn. 2) to analyse all of the data allows a consistent model to be applied to the underdamped and overdamped regimes. This is useful when comparing excitations from undoped and doped compositions. In particular, $\gamma/2>\omega_{0}$ is allowed in this model, however, beyond critical damping, $\gamma/2=\omega_{0}$, the shape of the response function evolves relatively slowly. Further, the fitted values of $\gamma$ and $\omega_{0}$ become correlated. This is the case at small $|\mathbf{Q}|$. In the limit of large dampingChaikin and Lubensky (1995) $\gamma/\omega_{0}\rightarrow\infty$, $\chi^{\prime\prime}(\mathbf{Q},\omega)$ can be approximated by the overdamped harmonic oscillator (ODHO) Lorentzian form,

[TABLE]

Eqn. 5 only has two parameters, $\chi^{\prime}$ and the relaxation rate $\Gamma=\omega_{0}^{2}/\gamma$. We found it convenient to use Eqn. 5 in some of the overdamped region. Thus, the grey region in Figs. 7 and 8 indicate the low Q regime where Eqn. 5 is used to fit the data.

III Results

III.1 RIXS spectra of La2-xSrxCuO4

Fig. 1 (c) shows example spectra from each composition at Q = (0.25,0). The low-energy magnetic spectrum of the parent ($x=0$) compound (bottom), is dominated by resolution-limited spin-wave excitations. The magnetic excitations in the doped $x=0.12$ (middle) and $x=0.16$ (top) compositions are considerably broader as noted in previous studiesBraicovich et al. (2010a); Ghiringhelli et al. (2012); Dean et al. (2012, 2013a); Dean (2015); Monney et al. (2016). The dd excitations occur in the energy range 1–3 eV. These are considerably broadened and shifted to lower energy in the doped compositions. The spectra are consistent with published lower resolution data Dean et al. (2013a).

Figs. 3 and 4 show examples of our RIXS data. Spectra such as those in Figs. 3 and 4 are collected together into intensity maps plotted as a function of $\mathbf{Q}$ and energy in Fig. 2. Thus Fig. 2 gives an overall picture of the excitations observed in the present study. The strongest feature in Fig. 2(a) is the magnon which disperses to an energy $\sim 355\pm 34$ meV along $(h,0)$ in agreement with previous studiesHeadings et al. (2010); Dean et al. (2012). The magnetic excitations are much broader in energy for doped compositions as shown in Figs. 2(c)-(f). Phonons can be seen in the La2CuO4 spectra below 100 meV, for example in Fig. 4(c) and also visible in the map plots in Fig. 2. In Fig. 2(c), for $x=0.12$ and $0.16$ a particularly strong phonon branch can be seen below 100 meV along $(h,0)$ near $h=0.3$. This indicates coupling to charge excitations. In the $x=0.12$ composition, CDW order is seen near $h=0.23$. Similar behaviourChaix et al. (2017) is seen in Bi2Sr2CaCu2O8+δ.

In addition to the high-symmetry direction measurements shown in Figs. 2-3, a full quadrant of the Brillouin zone was examined by mapping ($h,k$) in the $x$ = 0.16 and 0.12 compounds. Approximately 90 spectra were collected at the ID32 beamline, distributed throughout the zone with spacing 0.05 (r.l.u.). The RIXS intensity is plotted as a function of ($h,k$) for several energy slices and for $x=0.12$ in Fig. 5 where areas of high-intensity correspond to the spin-excitation intensity. These measurements were performed with lower resolution ($\Delta E\simeq$ 50 meV). The plots are smoothed by averaging neighbouring points within $|\Delta\mathbf{Q}|$ = 0.05 r.l.u.. At low energies, the maximum in the RIXS intensity appears at low Q and is approximately symmetrically distributed around $\Gamma$. As the energy increases, peaks develop along $(h,0)$ and $(0,k)$ and move to larger $h$ and $k$. It is interesting to note that quite similar behaviour is observedHeadings et al. (2010) in La2CuO4, where $(h,k)$ maps measured with INS show a peak in the intensity at ($\frac{1}{2},0$) for energies above about 320 meV. The maps show that for doped LSCO the magnetic spectral weight persists to higher energies near ($\frac{1}{2},0$) than in other parts of the Brillouin zone. This observation is consistent with previous workBraicovich et al. (2010a); Dean et al. (2013a); Dean (2015); Monney et al. (2016); Meyers et al. (2017).

III.2 DHO fitting

Figs. 3 and 4 show fits of the damped harmonic oscillator (DHO) model (Sec. II.4) together with phonon peaks and background to the data. The 35 meV resolution of the instrument allows the phonons and elastic peaks to be separated from the DHO response. For example, in La2CuO4 the frequencies are approximately wavevector independent with energies $\sim 45$ and 90 meV which are attributed to CuO bond-bending and bond-stretching modes respectively Pintschovius (2005); Reznik (2012); Devereaux et al. (2016). As can be seen from the figures, the DHO model generally describes the magnetic excitations well. The measured spectra are shown in black with the total fitted function indicated in red with constituent functions below. The parameters $\omega_{0}$ and $\gamma/2$ extracted from DHO fits are plotted for Q = ($h,0$) and ($h,h$) in Fig. 7 for each compound. Eqn. 5 is used to fit the small Q (grey) regime and the resulting relaxation rate $\Gamma$ is shown. Hole doping the parent compound increases $\gamma/2$. In the doped compounds, it can be comparable to $\omega_{0}$. The damping is anisotropic in wavevectorDean et al. (2013a); Dean (2015); Monney et al. (2016); Meyers et al. (2017), that is $\gamma/2$ is larger along ($h,h$) than along ($h,0$). Our data also reveals that the anisotropy of the damping does not reflect the antiferromagnetic Brillouin zone as $\gamma/2$ peaks at approximately ($0.2,0.2$) [rather than ($\frac{1}{4},\frac{1}{4}$)] along ($h,h$). This effect can be seen both for $x=0.12$ and $x=0.16$ Fig. 7(d,f).

We also fit the lower resolution ($\Delta E\simeq$ 50 meV) spectra from the grid in $(h,k)$. The results of fitting this data to the DHO model are summarised in Fig. 6. The damping $\gamma/2$ is again seen to be largest in the region near ($0.2,0.2$) for both doped compositions. The over-damped region where $\omega_{0}^{2}<\gamma^{2}/4$ and $\omega_{1}$ is imaginary, is indicated in Fig. 6 (d,i) as $\omega_{1}=0$.

It should be noted that normalising the data to an integration over the range of the $dd$ excitations does not account for the energy-dependence of the self-absorption. The lineshape of the excitation is therefore altered by the strong absorption of the scattered photons at low energy. We calculated that the damping parameters $\gamma$ is reduced by approximately 24% as a result of accounting for the energy-dependent self-absorption sup . This reduction decreases slightly with Q, and therefore the key result, that $\gamma/2$ is peaked in Q away from $(\frac{1}{4},\frac{1}{4})$ is unaffected.

III.3 Estimate of the absolute wavevector-dependent susceptibility

Fitting our RIXS data to the DHO response function in Eqn. 2, allows the wavevector-dependent susceptibility $\chi^{\prime}(\mathbf{Q})$ to be estimated, where

[TABLE]

In this section, we estimate $\chi^{\prime}(\mathbf{Q})$ in the superconductors we have investigated by using the parent antiferromagnet La2CuO4 as a reference. This estimation assumes that the DHO response contains only magnetic contributions which is somewhat justified by recent polarisation analysis Peng et al. (2018); Fumagalli et al. (2019). The data of Peng et al. suggest that approximately 82% of spectral weight is magnetic in the region of the magnetic excitations, 150–600 meV. In our analysis, the 18% charge contribution is partially accounted for in the background and multimagnon fits but any remaining charge contribution may lead to an overestimation of $\chi^{\prime}(\mathbf{Q})$.

The analysis discussed so far has relied on normalisation to $g$, an integration over the region of the $dd$ excitations, to take account of angle-dependent effects on the RIXS intensity. This does not affect the determination of excitation energies or damping coefficients. However, this procedure does not account for the difference in absorption between photons scattered from the magnetic excitations, which are close to the resonance, and the $dd$-excitations which are significantly away from the Cu absorption peak (with a width of about 0.4 eV) and are therefore less likely to be absorbed. In order to correct for these effects, we use the measured spin wave RIXS intensity of the parent compound as a reference. INS measurements show that the magnetic excitations of La2CuO4 are fairly well-described by linear spin wave theory (SWT) with some correctionsHeadings et al. (2010) near ($\frac{1}{2},0$). Thus the underlying $S(\mathbf{Q},\omega)$ is known in this case.

Ament et al. Ament et al. (2011) point out that under certain theoretical approximations, the absolute RIXS cross-section can be split into a prefactor $f(\boldsymbol{\epsilon},\boldsymbol{\epsilon}^{\prime},\mathbf{k},\mathbf{k}^{\prime})$ multiplied by a dynamic structure factor $S(\mathbf{Q},\omega)$, where the polarisations of the initial and final photons are $\boldsymbol{\epsilon}$ and $\boldsymbol{\epsilon}^{\prime}$. We note that the exact circumstances when the RIXS response is proportional to $S(\mathbf{Q},\omega)$ is still an active subject of investigation Ament et al. (2011); Jia et al. (2014), however, we will use this approximation in our analysis. Here we propose a simple estimate to remove the effects of $f(\boldsymbol{\epsilon},\boldsymbol{\epsilon}^{\prime},\mathbf{k},\mathbf{k}^{\prime})$ from $S(\mathbf{Q},\omega)$ for doped LSCO. We assume $f(\boldsymbol{\epsilon},\boldsymbol{\epsilon}^{\prime},\mathbf{k},\mathbf{k}^{\prime})$ is the same for doped and undoped compounds. For each $(\mathbf{k},\mathbf{k}^{\prime})$ we first normalise (divide) the raw RIXS spectra by $g$ to yield $I_{\textrm{RIXS}}$ (see Sec. II.4) and find $\chi^{\prime}_{\textrm{RIXS}}$ by fitting to the DHO model. We then multiply $\chi^{\prime}_{\textrm{RIXS}}$ for LSCO by the spin-wave response of LCO determined from INS Headings et al. (2010) divided by the measured RIXS response of LCO, to estimate the dynamic susceptibility of the doped superconductor in absolute units:

[TABLE]

$\phi^{\textrm{LCO}}_{\textrm{SWT}}$ is the energy integrated spin-wave pole weight, determined from a fit of linear SWT to INS data and $\phi^{\textrm{LCO}}_{\textrm{RIXS}}$ is the integrated pole weight of fitted RIXS data, details of this are given in appendix A. In practice, we fit the LCO spectra and then use Eqns. 13 and 16 to evaluate $\phi^{\textrm{LCO}}_{\textrm{SWT}}$ and $\phi^{\textrm{LCO}}_{\textrm{RIXS}}$. Eqn. 7 assumes that the factors $f$ and $g$ are the same in doped and undoped compositions and therefore cancel in the normalisation procedure. We have verified that this is approximately the case in our samples.

Figs. 7 and 8 (a) and (b) show the parameters $\gamma(\mathbf{Q})$, $\omega_{0}(\mathbf{Q})$ and $\chi^{\prime}(\mathbf{Q})$ extracted from fits of Eq. 2 as a function of Q along ($h,0$) and ($h,h$) for the three compounds. For LCO, $\chi^{\prime\prime}(\textbf{Q},\omega)$ is a sum of the single and multimagnon contributions. For LSCO a single response function is used. The resulting $\chi^{\prime}_{\text{RIXS}}(\mathbf{Q})$ due to the magnon pole is shown in Fig. 8 (a) and (b) with a cubic polynomial fit indicated with a solid blue line. The susceptibilities $\chi^{\prime}_{\text{RIXS}}(\mathbf{Q})$ in Fig. 8(a) and (b) contain the effects of the $f$ factor and self absorption mentioned above. In Fig. 8 (c) and (d) we correct for these effects and estimate the absolute $\chi^{\prime}(\mathbf{Q})$ using Eqns. 7, 13 and 16 together with the cubic polynomial fit of $\chi^{\prime}_{\text{RIXS}}(\mathbf{Q})$ to La2CuO4 in Fig. 8(a,b).

By definition, the corrected susceptibility for the parent compound La2CuO4 becomes that of the SWT model described in Appendix A plus additional spectral weight due to the multimagnon excitations observed with RIXS. For all three compositions investigated, $\chi^{\prime}(\mathbf{Q})$ increases as we move along $(h,h)$ towards $(\frac{1}{2},\frac{1}{2})$, where INS finds the strongest spin fluctuations. The magnitude of $\chi^{\prime}(\mathbf{Q})$ is generally larger for the doped compositions $x=0.12,0.16$ than in the parent (see Table 1), this effect is also present when the data is normalised via the $dd$ excitations so does not seem to be an artefact arising from the spin wave normalisation. The increase arises when spectral weight in $\chi^{\prime\prime}(\mathbf{Q},\omega)$ is moved to lower energy and gives a larger contribution to $\chi^{\prime}(\mathbf{Q})$ because of the $1/\omega$ factor in Eqn. 6. For example, if, for a particular $\mathbf{Q}$, a spin wave keeps the same integrated intensity in $\chi^{\prime\prime}(\mathbf{Q},\omega)$ and is broadened in $\omega$, then $\chi^{\prime}(\mathbf{Q})$ can increase. Inspection of Fig. 4 shows that this indeed happens. The modelled excitations are shown in Fig. 9 where $\chi^{\prime\prime}(\mathbf{Q},\omega)$ is calculated from Eqn. 2 with the fitted parameters [$\omega_{0}(\mathbf{Q})$, $\gamma(\mathbf{Q})$, $\chi^{\prime}(\mathbf{Q})$] shown in Figs. 7 and 8 (c, d).

IV Discussion

IV.1 Theoretical Models

Our investigation of the magnetic excitations in cuprates is motivated by spin-fluctuation mediated theories of high temperature superconductivityScalapino (2012) and to gain a fundamental understanding of metallic transition metal oxides. The Hubbard model (in its one or three band variants) is generally considered to be a good starting point. Calculations based on the Hubbard modelScalapino (2012) show that the wavevector-dependent pairing interaction $V_{\text{eff}}$ is approximately Scalapino (1995, 2012)

[TABLE]

where $\mathbf{p}^{\prime}$ and $\mathbf{p}$ are the wavevectors of the two electrons making up a Cooper pair and $U$ is the Hubbard on-site interaction. RIXS measurements of the magnetic excitations over a wide energy range allow the opportunity to determine $\chi^{\prime}(\mathbf{q})$. This can be used as an input to theory or a test of models of the excitations. Numerical studies of the two-dimensional Hubbard model, applied to cuprates, qualitatively reproduce Huang et al. (2017) the slowly-evolving high-energy magnetic excitations which are observed by INS and RIXS experiments, but calculations are restricted to relatively small lattices. Other approaches based on renormalised itinerant quasiparticles James et al. (2012); Eremin et al. (2013); Dean et al. (2014); Monney et al. (2016) with various types of approximation provide a basis for a phenomenological understanding of the physical properties and allow finer structure in wavevector and energy to be predicted. In general, we expect the magnetic excitations and $\chi^{\prime\prime}(\mathbf{Q},\omega)$ to be different around $(0,0)$ and $(\frac{1}{2},\frac{1}{2})$ and the dispersion of the excitations not to be symmetric around $(\frac{1}{4},\frac{1}{4})$.

IV.2 Wavevector dependence of the response

The high-energy magnetic excitations in the parent compound La2CuO4 are anisotropic in two ways. Firstly the single magnon energy varies between points on the antiferromagnetic Brillouin zone boundary with $(\frac{1}{2},0)$ having a higher energy than $(\frac{1}{4},\frac{1}{4})$. Secondly, the single magnon excitation is strongly and anomalously damped at the $(\frac{1}{2},0)$ position. This variation in the magnon energy can be understood in terms of an expansion of the single band Hubbard modelColdea et al. (2001); MacDonald et al. (1990) which gives rise to second nearest neighbour and cyclic exchange interactions. While the anisotropy of the damping in La2CuO4 may be understood in terms of the unbinding of magnons into spinonsHeadings et al. (2010); Dalla Piazza et al. (2015). This is a generic propertyHeadings et al. (2010); Dalla Piazza et al. (2015) of $S=1/2$ square lattice antiferromagnets.

Our data show how the anisotropies of the parent compound persist into the doped compositions and are qualitatively consistent with previous studiesDean et al. (2013a); Monney et al. (2016); Meyers et al. (2017). However, the higher energy resolution of the present study ($\Delta E\approx 35$ meV as compared to $\Delta E\gtrsim 100$ meV in previous workDean et al. (2013a); Monney et al. (2016); Meyers et al. (2017)) allows us to separate the magnetic excitations from lower energy features. In Fig. 7 we see that the frequency of the undamped mode $\omega_{0}(\mathbf{Q})$ extracted from the DHO model shows similar dispersions along $(h,0)$ and $(h,h)$ in the doped $x=0.12$ and $x=0.16$ compositions as in the parent $x=0$. At $\mathbf{Q}=(\frac{1}{2},0)$, $\hbar\omega_{0}$ increases with doping from $356\pm 45$ meV $(x=0)$ to $396\pm 54$ meV ($x=0.16$), while at $\mathbf{Q}=(\frac{1}{4},\frac{1}{4})$ it increases from $298\pm 27$  meV to $313\pm 30$ meV.

A new result from this work is the extent of the variation of $\gamma(\textbf{Q})$ and $\omega_{0}(\mathbf{Q})$ across the Brillouin zone in doped LSCO. Significantly, the damping is seen to increase in the underdoped compound, $x$ = 0.12 and again in the optimally-doped material, $x$ = 0.16. From the damping maps shown in Fig. 6 (b) and (g) it can be seen that the enhanced damping is most prominent close to the ($h,h$) direction. It is notable that the maxima in $\gamma(\textbf{Q})$ and $\omega_{0}(\mathbf{Q})$ along $(h,h)$ are actually near ($0.2,0.2$) rather than at ($\frac{1}{4},\frac{1}{4}$). Our $(h,k)$ maps of the fitted parameters in Fig. 6 show that $\gamma(\mathbf{Q})$ actually shows a local maximum around this point. These features appear to be qualitatively present in theoretical calculations based on itinerant quasiparticle such as those in Refs. Dean et al., 2014; Monney et al., 2016 and presumably arise from (nesting) features in the underlying quasiparticle band structure. The general damping anisotropy between $(h,0)$ and $(h,h)$ for the doped compositions has also been described by theories based on determinantal quantum Monte Carlo (DQMC) Huang et al. (2017).

The normalisation procedure described in Sec. III.3 allows us to obtain the estimates of $\chi^{\prime}(\mathbf{Q})$ in Fig. 8. Values of $\chi^{\prime}$ at representative wavevectors are shown in Table 1. A striking feature of the analysis is that it shows that there is a large anisotropy in $\chi^{\prime}(\mathbf{Q})$ at the antiferromagnetic Brillouin zone boundary. In particular, $\chi^{\prime}(\mathbf{Q})$ is about 4 times larger at $(\frac{1}{4},\frac{1}{4})$ than at $(\frac{1}{2},0)$. This arises because of the smaller $\omega_{0}(\mathbf{Q})$ at $(\frac{1}{4},\frac{1}{4})$ (see Figs. 4 and 7) which shifts spectral weight to lower energy.

A maximum on $\chi^{\prime}(\mathbf{Q})$ along ($h,0$) is seen for all compositions. This may derive from the combination of two effects present in the parent antiferromagnetic state. Firstly, linear spin-wave theory of a square lattice $S=1/2$ antiferromagnet (see Appendix A) predicts that $\chi^{\prime}(\mathbf{Q})$ increases from $(0,0)$ to $(\frac{1}{2},0)$. Secondly, square lattice $S=1/2$ antiferromagnets such as La2CuO4Headings et al. (2010) and CFDTDalla Piazza et al. (2015) show anomalous broadening and weakening of their magnetic excitations near $(\frac{1}{2},0)$ and thus a dip in $\chi^{\prime}(\mathbf{Q})$ at this position. This is not predicted in the pure SWT model and has been understood in terms of the unbinding of magnons into spinon pairsDalla Piazza et al. (2015). Our results suggest that these effects persist for doped compositions.

Also of interest is the fact that $\chi^{\prime}(\mathbf{Q})$ increases monotonically along $(h,h)$ from $\Gamma$ to $M$. The increase is consistent with the fact that the magnetic response is strongest in the antiferromagnetic Brillouin zone centred on $M$. This is expected because of the residual antiferromagnetic exchange interactions and is qualitatively consistent with INS measurementsHayden et al. (1996); Vignolle et al. (2007); Wakimoto et al. (2007); Lipscombe et al. (2007, 2009). Thus, to our knowledge, Fig. 8 (c), (d) are the first attempts to determine $\chi^{\prime}(\mathbf{Q})$ in absolute units based on integrals of the magnetic response over a wide energy range. It should be noted that theoretical calculations based on the Hubbard modelScalapino (1995); Huang et al. (2017) show that spin fluctuation in the $M$ zone contribute most to pairing in spin-fluctuation mediated theories of HTC. Fig. 9 shows the total modelled excitation for all compositions. The wavevector dependence of the susceptibility and damping is clearly shown.

In Fig. 10 we compare slices with calculations from the DQMC calculations of Huang et al.Huang et al. (2017). The DQMC calculations reproduce qualitatively some of the features of our data such as the increase in the strength of $\chi^{\prime\prime}(\mathbf{Q},\omega)$ moving towards $(\frac{1}{2},\frac{1}{2})$. However, the RIXS spectra are generally much sharper and show a stronger wavevector dependence.

IV.3 Comparison to INS

RIXS and INS provide complementary views of the collective spin excitations in the cuprates Wakimoto et al. (2007). However, INS measurements of the high-energy magnetic excitations are difficult because the background increases when high incident energies are used. Nevertheless, some data does exist for La2-xSrxCuO4. An early studyHayden et al. (1996) on La1.86Sr0.14CuO4 revealed magnetic excitations up to 260 meV. In particular, excitations were observed at $\textbf{Q}=(\frac{3}{2},0)$ which is equivalent to the $\textbf{Q}=(\frac{1}{2},0)$ position investigated here with RIXS. Our RIXS normalisation procedure (Sec. III.3) allows us to estimate $\chi^{\prime}(\mathbf{Q})=1.8\pm 0.6\mu_{B}^{2}$ eV*-1* f.u.-1 in LSCO $x$ = 0.16 at $(\frac{1}{2},0)$ based on an integration of the spectrum up to about 800 meV. Integrating the INS data in Ref. Hayden et al., 1996 up to 260 meV we obtain $\chi^{\prime}(\mathbf{Q})\approx$ 0.5 $\mu_{B}^{2}$ eV*-1* f.u.-1. Thus, if it were possible to perform neutron scattering experiments over a wider energy range the integration of INS data may produce a comparable value for $\chi^{\prime}(\mathbf{Q})$ at $\textbf{Q}=(\frac{1}{2},0)$. The approximate agreement is satisfying, however further work is required to develop the comparison of the two probes of collective magnetic excitations.

The INS study in Ref. Hayden et al., 1996 [Fig. 4(d)] also estimated $\chi^{\prime}(\mathbf{Q})$ along the line $(h,h)$ for La1.86Sr0.14CuO4. Unfortunately, the energy integration was only carried out over the range $0\leq\hbar\omega\leq 150$ meV. However, the increase in $\chi^{\prime}(\mathbf{Q})$ in the doped compound in the range $0.25\leq h\leq 0.34$ [Fig. 8(d), present paper] is also seen with INS. The absolute values of $\chi^{\prime}(\mathbf{Q})$ measured with neutrons are of the same order of magnitude but less than those reported in the present RIXS study presumably because the INS study integrates only up to 150 meV.

V Summary and Conclusions

We have made high-resolution RIXS measurements of the collective magnetic excitations for three compositions of the superconducting cuprate system La2-xSrxCuO4. Specifically, we have mapped out the excitations throughout the 2-D $(h,k)$ Brillouin zone to the extent that is possible at the Cu-$L$ edge. In addition, we have attempted to determine the wavevector-dependent susceptibility of the doped compositions La2-xSrxCuO${}_{4}(x=0.12,0.16)$ by normalising data to the parent compound. This procedure allows comparison with INS measurements. We find that the evolution of the intensity of high-energy ($\hbar\omega\gtrsim 200$ meV) excitations measured by RIXS and INS is consistent.

The high-energy spin fluctuations in La2-xSrxCuO4 are fairly well-described by a damped harmonic oscillator model. The DHO damping parameter increases with doping and is largest along the ($h,h$) line although it is not peaked at the high symmetry point $(\frac{1}{4},\frac{1}{4})$. While the pole frequency is peaked at $(\frac{1}{2},0)$ for doped and undoped compositions, for the doped compositions, the wavevector-dependent susceptibility $\chi^{\prime}(\mathbf{Q})$ is much larger at $(\frac{1}{4},\frac{1}{4})$ than at $(\frac{1}{2},0)$. Both of these positions are on the antiferromagnetic zone boundary of the parent compound. The wavevector-dependent susceptibility increases rapidly along the $(h,h)$ line towards the antiferromagnetic wavevector of the parent compound $(\frac{1}{2},\frac{1}{2})$. Thus the strongest magnetic excitations and those predicted to favour superconductive pairing occur towards the $(\frac{1}{2},\frac{1}{2})$ position. Our quantitative determination of the wavevector-dependent susceptibility will be useful in testing magnetic mediated theories of high-temperature superconductivityChubukov et al. (2003); Scalapino (2012).

Appendix A Linear spin-wave theory calculations

The magnetic excitations can be modelled in LCO with classical linear spin-wave theory. We consider the case of a $S=1/2$ square lattice antiferromagnet with nearest- and next-nearest exchange interactions. The susceptibility transverse to the ordered moment $\chi^{\prime\prime}_{\perp}(\mathbf{Q},\omega)$ due to one-magnon creation is given by:

[TABLE]

where

[TABLE]

and

[TABLE]

The amplitude factors $A_{\mathbf{Q}}$ and $B_{\mathbf{Q}}$ are givenColdea et al. (2001) by $A_{\mathbf{Q}}=J-J_{c}/2-(J^{\prime}-J_{c}/4)(1-v_{h}v_{k})-J^{\prime\prime}(1-(v_{2h}+v_{2k})/2$), $B_{\mathbf{Q}}=(J-J_{c}/2)(v_{h}+v_{k})/2$, where $v_{x}=\cos(2\pi x)$ and $x\mapsto h$ or $k$. $Z_{d}$ and $Z_{c}$ are renormalisation constants which take account of quantum fluctuations in the AF ground state.

Headings et al. Headings et al. (2010) have made INS measurements of the spin waves in La2CuO4 and fitted the model described by Eqns. 9-11. They find $J$ = 143, $J^{\prime}$ = $J^{\prime\prime}$ = 2.9 and $J_{c}$ = 58 meV, assuming $Z_{c}=1.18$. The wavevector dependence of $Z_{d}(\mathbf{Q})$ is also determined from the INS data,

[TABLE]

where $Z_{d0}$ = 0.4. In order to compare the INS and RIXS measurements, we assume that RIXS is equally sensitive to the three components of the susceptibility and compute the average susceptibility $\chi=\frac{1}{3}(\chi_{xx}+\chi_{yy}+\chi_{zz})=\frac{2}{3}\chi_{\perp}$. The energy integrated intensity of the spin wave pole $\phi_{\textrm{SWT}}^{\textrm{LCO}}(\mathbf{Q})$ is then:

[TABLE]

We derive a comparable measure of the energy integrated spin-wave pole measured with RIXS by rewriting Eqn. 2 for LCO (in the limit $\omega_{0}\geq\gamma/2$) as,

[TABLE]

Integrating over the positive energy pole, we obtain the measured pole intensity from the fitted parameters $\omega(\textrm{Q}),\gamma(\textrm{Q})$ and $\chi^{\prime}(\textrm{Q})$:

[TABLE]

Acknowledgements.

The authors acknowledge funding and support from the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Condensed Matter Physics (CDT-CMP), Grant No. EP/L015544/1 as well as Grant EP/R011141/1. We acknowledge Diamond Light Source for time on Beamline I21 under proposals SP18469 and SP18512 and the European Synchrotron Radiation Facility for time on Beamline ID32 under proposal HC/2696. We would like to thank G. B. G. Stenning and D. W. Nye for help on the Laue instrument in the Materials Characterisation Laboratory at the ISIS Neutron and Muon Source.