Photo-induced high-temperature ferromagnetism in YTiO3 – Nature

Optical set-up and MOKE detection

Our MOKE measurements were carried out using the experimental set-up shown in Extended Data Fig. 1. The THz pump pulses were created by the chirped pulse difference frequency generation scheme described in detail in refs. ^31,46. A Ti:sapphire regenerative amplifier (100 fs pulse length, 800 nm centre wavelength, 1 kHz repetition rate) fed two independently tuneable optical parametric amplifiers (OPAs) seeded by a common white light continuum, whose signal outputs produced roughly 70 fs long near-infrared pulses with centre wavelengths between 1,250 and 1,550 nm. A linear chirp was imparted on the OPA outputs by sending them through two transmission grating pairs, after which difference frequency mixing in a roughly 400 μm thick crystal of DAST (4-N,N-dimethylamino-4′-N′-methyl-stilbazolium tosylate) produced the desired THz transient. The centre frequency and bandwidth of the THz pulses were modified by choosing the wavelengths and the chirps of the near-infrared OPA outputs, respectively. Examples of the THz electric field waveforms and their associated spectra are shown in Extended Data Fig. 2.

The generated THz pulses were focused onto a YTiO₃ single crystal mounted in a liquid helium cryostat equipped with a 5 T superconducting magnet. The THz propagation direction and the external magnetic field were both oriented normal to the (001) surface of the sample (that is, parallel to the ferromagnetic c axis). The THz electric field was linearly polarized parallel to the b axis.

A small portion of the 800 nm amplifier output was used for the MOKE detection. These probe pulses were time delayed and focused onto the sample at a small angle (roughly 5°) relative to the sample normal in a polar MOKE geometry. The incident probe was s-polarized perpendicular to the THz polarization to eliminate artefacts from field-driven birefringence. The rotation of the polarization axis (ϕ) of the reflected probe pulses was determined using a standard balanced detection system, consisting of a half-wave plate (HWP), a Wollaston prism and a balanced photodiode. Before each pump-probe measurement, the HWP was adjusted to set the balanced photodiode output to zero for every temperature and magnetic field. The pump-induced polarization rotation changes (Δϕ) and transient reflectance (ΔR) were measured simultaneously from the difference and sum channels of the balanced photodiode, allowing us to rule out isotropic, non-magnetic contributions to the MOKE signal (Extended Data Fig. 3 and Methods section Jones matrix analysis). The time-resolved changes to the MOKE angle following pump excitation, which are shown in the main text, are defined as

where H is the external magnetic field.

Sample preparation and equilibrium measurements

The high-quality, stoichiometric YTiO₃ single crystals were grown by a crucible-free floating zone method in Ar/H₂ = 50/50 flow. A four-mirror type image furnace (CSI) equipped with 1.5 kW halogen lamps was used. The stoichiometry and structure, as well as the thermodynamic, magnetic and optical properties of the sample, have been previously characterized using energy-dispersive X-ray analysis, powder and single crystal X-ray diffraction, thermal gravimetry and/or differential thermal analysis, SQUID magnetometry and spectroscopic ellipsometry. In addition to the information provided here, a detailed description of the sample preparation and characterization can be found in ref. ¹⁷.

MOKE signal calibration

To determine the absolute magnetization of YTiO₃ in the pump-induced state, as reported in Figs. 3b and 4a, we calibrated the MOKE angle on the basis of equilibrium measurements. Without the THz pump impinging on the sample, we measured the static MOKE angle, ϕ_M, as a function of external magnetic field, H. The signal was corrected for a linear background that arises because of the diamagnetic response of the cryostat windows. The resulting static MOKE measurement is shown in Extended Data Fig. 4a), providing the dependence ϕ_M(H). Separately, on the same YTiO₃ single crystal, we carried out measurements of the magnetization as a function of magnetic field, M(H), using a vibrating sample magnetometer (Quantum Design). By correlating the two measurements, we are able to obtain the calibration curve relating the MOKE angle to the magnetization (Extended Data Fig. 4b, which is linear over the meaured field range: (M={beta }_{{rm{M}}}{{phi }}_{{rm{M}}}). The magneto-optical coefficient determined from a fit to the calibration curve is ({beta }_{{rm{M}}},{rm{=; 1.36}}pm 0.05) μ_B mrad⁻¹.

This analysis is applied to the time-resolved MOKE data to obtain the non-equilibrium magnetization in the pump-induced state (({M}_{{rm{pumped}}})) by extrapolating the linear dependence. For a given field,

where Δϕ_M(t) is the pump-induced change in the MOKE angle. As noted in the main text, a positive Δϕ_M corresponds to an increase in M with respect to the equilibrium ferromagnetic magnetization, while a negative Δϕ_M corresponds to a reduction in M.

The calibration procedure is repeated for each temperature, and it is found that β_M remains constant within the experimental error for temperatures below T_c = 27 K, as shown in Extended Data Fig. 5, and agrees with the value extracted from Extended Data Fig. 4. We note that the observed temperature independence of β_M agrees with magneto-optical studies carried out around the critical region in other ferromagnetic compounds⁴⁷. We take the value of ({beta }_{{rm{M}}},{rm{=; 1.37}}pm 0.09) μ_B mrad⁻¹ determined from the temperature dependence to obtain the non-equilibrium magnetization above T_c.

The experimental values for the total magnetization in the pump-induced state (reported in Figs. 3b and 4a) are obtained from equation (2), where the term ({beta }_{{rm{M}}}{phi }_{{rm{M}}}(H)) is replaced by the equilibrium M(H) determined by vibrating sample magnetometer, which has a relative uncertainty of less than 1% and does not contribute significantly to the uncertainty in ({M}_{{rm{pumped}}}). The error bars on these figures are given by two main contributions: the uncertainty in the determination of β_M (described above) and the uncertainty in the maximum value of Δϕ_M at long times. Due to the slow decay, the maximum saturated value of Δϕ_M was determined by averaging the signal between t = 100 and 200 ps, with the uncertainty given by the standard error of those data points. The maximum value of Δϕ_M can also be determined from the fits to the data (as in the Methods section Time scales of pump-induced magnetization). The extracted values and error bars from the two approaches were found to be equivalent.

Non-magnetic contributions to MOKE signal

Key to our interpretation is the fact that any non-magnetic pump-induced changes to the optical properties of the sample negligibly affect our experimental measurements. To validate this claim, we present here a full analysis of the various contributions to our experimental signal formulated using Jones calculus and argue that we are only sensitive to changes in the magnetization.

Jones matrix analysis

As described in the section Optical setup and MOKE detection, we use a standard balanced detection scheme in which the polarization after the sample is rotated by a HWP and split into orthogonal linear components labelled s and p, which are aligned with the horizontal and vertical axes in the laboratory frame. The intensities of these components I_p and I_s comprise the experimental polarization rotation signal:

where (varDelta I={I}_{s}-{I}_{p}) and ({I}_{{rm{sum}}}={I}_{s}+{I}_{p}).

The general polarization in the s and p basis can be described by the general Jones vector,

The Jones matrix for the HWP, whose primary axis is set at an angle θ with respect to the vertical is,

After the sample, the beam is routed by two metallic mirrors with an angle of incidence (AOI) of 5°. For a general metallic mirror, the Jones matrix can be written as,

In the ideal case, the reflectivity terms r_m and A = 1 and the phase shifts ({kappa }_{{rm{p}}}{rm{=; 0}}) and (varDelta kappa =pi ). For the silver mirrors used in our experiment, r_m ≅ 0.981, A = 0.995, ({kappa }_{{rm{p}}},{rm{=; 0.110}}pi ) and (varDelta kappa =-1.001pi ).

Following ref. ⁴⁸, the YTiO₃ sample is represented by the Fresnel reflection matrix

with ({r}_{ps}=-{r}_{sp}). At normal incidence, ({r}_{ss}={r}_{pp}=r=frac{n+1}{n-1},=,0.395), and the MOKE rotation angle

The refractive index of YTiO₃ at the probe wavelength, n = 2.304, is taken from ref. ¹⁷. The coefficient γ is the magneto-optical constant. The terms multiplying M are combined into a generalized magneto-optical coefficient α_M. In the section Potential magneto-optical effects, we explain how changes in ϕ induced by the pump can be related to changes in the magnetization ΔM through the coefficient α_M. In this section, we describe how the signal that we observe in the real experimental set-up does indeed provide an accurate measure of the intrinsic MOKE angle ϕ.

With a finite AOI of α = 5°, the reflection matrix becomes slightly anisotropic,

where ({r}_{ss}=frac{n{rm{cos }}{alpha }_{n}-{rm{cos }}alpha }{n{rm{cos }}{alpha }_{n}+{rm{cos }}alpha },=,{rm{0.396}}) and (eta =|frac{{r}_{pp}}{{r}_{ss}}|,=,0.993) describes the anisotropy of the Fresnel reflectivities. The internal angle from Snell’s law is ({alpha }_{n}={{rm{sin }}}^{-1}left(frac{1}{n}{rm{sin }}alpha right)approx {2.2}^{^circ }). The apparent rotation angle ({phi }^{{prime} }) is slightly modified from the pure MOKE angle ϕ of equation (8) due to the finite AOI: ({phi }^{{prime} }=cphi ), where (c=frac{{rm{cos }}alpha }{{rm{cos }}(alpha -{alpha }_{n})}). Hence, whereas the finite AOI does slightly modify the reflection parameters, the deviations from the normal incidence case are small.

To determine the polarization measured from our balanced detection set-up, we can propagate the incident polarization through the Jones matrix of each element. If the incident polarization is perfectly s-polarized,

then:

For measurements of the pump-induced MOKE angle (varDelta {phi }_{{rm{M}}}^{{rm{sig}}}(H,t)), we first apply the external field H, then select the HWP angle θ to compensate for the static field-induced rotation ({phi }_{0}^{{rm{sig}}}(H)). This is accomplished by finding the angle ({theta }_{{rm{bal}}}) for which the difference signal ΔI is zero, or equivalently, when the two outgoing polarization components E_p and E_s are equal. Here, the balancing condition is given by,

Then, fixing the HWP setting at this angle and allowing for pump-induced changes in the magnetization, which would yield a time-dependent change in the intrinsic MOKE angle, (varDelta {phi }_{{rm{M}}}(H,t)=varDelta ({alpha }_{{rm{M}}},M)), the resulting pump-induced signal is

The approximation holds for small angles ϕ₀ and Δϕ_M, which is certainly satisfied in our experiment, where the maximum we observe is less than 1 mrad or 0.06°. Hence, from equation (12), with a perfectly s-polarized incident beam, the detected time-resolved changes in the MOKE angle are proportional to the intrinsic time-resolved MOKE angle coming from pump-induced changes to the magnetization of YTiO₃ with a mismatch of less than 1%. Notice, importantly, that as the difference of the two polarization states is normalized by their sum in our signal, the reflectivity and (non-magnetic) pump-induced changes thereof drop out of the expression for the detected signal and do not affect it at all under these ‘ideal’ experimental conditions. The only errors result from the imperfections of the mirrors used after the sample.

We can also investigate additional errors that might result from a slight misalignment of the incident polarization from perfect s polarization. We can imagine that the incident polarization is rotated from the ideal s polarization by a small angle δ,

Following the same procedure as above, we get for the static case,

The algebraic forms of the subsequent equations for the ({theta }_{{rm{bal}}}) and (varDelta {phi }_{{rm{M}}}^{{rm{sig}}}) are cumbersome, so instead we numerically analyse the extracted signal. Fixing ({phi }_{0}(H)=1) mrad (twice the value measured statically at 2 T) to provide an upper bound, we compute the resulting pump-induced MOKE signal (varDelta {phi }_{{rm{M}}}^{{rm{sig}}}) for varying δ and compare to the actual intrinsic value of Δϕ_M from the sample (Extended Data Fig. 6).

In all cases, (varDelta {phi }_{{rm{M}}}^{{rm{sig}}}) is linearly proportional to Δϕ with small deviations only visible for large δ. It is unlikely that a misalignment of more than a few degrees in the probe polarization would appear in our experiment; for realistic values of δ < 5°, the error between the measured and actual value of Δϕ_M is 1% or less.

We note that, even in this geometry with a misaligned probe polarization, any pump-induced changes in the overall sample reflectivity (here, given by Δr_ss) are again cancelled out due to the signal normalization. However, pump-induced changes to the anisotropic Fresnel factors η and c could still influence the signal. These effects turn out to be extremely small and well within the uncertainty in the experimental measurements. To see this, one can determine the pump-induced change in the refractive index from the measured change in the reflectivity Δr/r and the Fresnel equations,

From Extended Data Fig. 3, we see that the magnitude of Δr/r is on the order of 3 × 10⁻³ at its largest. Plugging in for n and α from above, we can get an upper bound of (|varDelta n|approx 0.006). The resulting pump-induced changes in η and c would then be,

With these deviations, the total error in our measured MOKE signal is only around 0.77% and the contribution specifically from Δη and Δc is extremely small: on the order of 0.001%. Therefore, after a thorough analysis and estimate of the errors of our experimental set-up, we conclude that non-magnetic effects arising from imperfect optics and pump-induced changes in reflectivity do not affect our experimental signal.

Potential magneto-optical effects

On the basis of the Jones matrix discussion above, we established that our measured signal describes the intrinsic MOKE angle of the sample. One additional assumption in our analysis is that pump-induced changes in β_M (that is, the magneto-optical constant) are negligible. To verify this assumption, we consider the different contributions to the pump-induced change of the MOKE angle^49,50,

where ({alpha }_{{rm{M}}}={beta }_{{rm{M}}}^{-1}) is the magneto-optical coefficient from equation (8). Equation (19b) can be written in terms of the relative changes ((delta {phi }_{{rm{M}}}=frac{varDelta {phi }_{{rm{M}}}}{{phi }_{{rm{M}}}}), (delta {alpha }_{{rm{M}}}=frac{varDelta {alpha }_{{rm{M}}}}{{alpha }_{{rm{M}}}}) and (delta M=frac{varDelta M}{M})), as

Both ϕ_M and α_M are complex quantities: ({phi }_{{rm{M}}}^{{prime} }={rm{Re}}({phi }_{{rm{M}}})) is the MOKE rotation and ({phi }_{{rm{M}}}^{{primeprime} }={rm{I}}{rm{m}}({phi }_{{rm{M}}})) is the MOKE ellipticity. Assuming the dynamics associated with the real and imaginary parts of α_M follow each other, (delta {alpha }_{{rm{M}}}^{{prime} }=kdelta {alpha }_{{rm{M}}}^{{primeprime} }), one can isolate the relative change in the magnetization⁵⁰,

In Extended Data Fig. 7, we compare the pump-induced MOKE rotation to the ellipticity, measured by replacing the HWP in the detection with a quarter-wave plate. We find that the two signals lie on top of each other within our experimental error, indicating that (delta {phi }_{{rm{M}}}^{{prime} }approx delta {phi }_{{rm{M}}}^{{primeprime} }). Then, equation (21) simplifies to (delta Mapprox delta {phi }_{{rm{M}}}^{{prime} }), or, expanding,

That is, the dynamics we observe can be attributed to true magnetization dynamics.

In addition, no probing volume correction is needed because the penetration depth at the probe wavelength (({delta }_{{rm{probe}},800{rm{nm}}}) = 178 nm) is much shorter than those at all of the pump wavelengths used in the experiment (({delta }_{{rm{pump}},4{rm{THz}}}) = 3.5 μm, ({delta }_{{rm{pump}},9{rm{THz}}}) = 6.6 μm, ({delta }_{{rm{pump}},17{rm{THz}}}) = 600 nm). The penetration depths were extracted from our measured infrared spectra (Fig. 2a and ref. ⁵¹).

Time scales of pump-induced magnetization

The time-resolved data presented in the main text focus on the early to intermediate time response of the pump-induced state. On the time scales of those measurements (Δt < 200 ps), the signal grows following pump excitation then remains relatively constant, allowing one to analyse the saturated magnetic behaviour. The saturation of the pump-induced response over hundreds of picoseconds points to the existence of a metastable non-equilibrium magnetic state, which persists for much longer than the coherent structural response induced by the resonant phonon excitation (typically tens of picoseconds, Extended Data Fig. 8). To figure out the lifetime of the metastable state, we also carried out time-resolved MOKE measurements over longer time scales, up to roughly 1 ns. A representative measurement taken with 9 THz pump excitation at low temperature (T = 10 K) is shown in Extended Data Fig. 9. We observe a sharp rise and slow decay of the pump-induced MOKE angle Δϕ_M, which can be fit by a decaying exponential model:

The time constants σ and τ represent the rise time to reach saturation and the lifetime of the non-equilibrium state, respectively, and A is the saturation value of the MOKE angle. We obtain fitted values of σ = 30 ± 2 ps and τ = 3.8 ± 0.3 ns. The several-nanosecond lifetime demonstrates the metastability of the pump-induced phase, as this time scale is much longer than any external time scale of the system.

To help us understand the mechanism leading to the formation of the non-equilibrium magnetic state, we compare the MOKE signal rise time to the lifetime of the driven phonon as a function of temperature (Extended Data Fig. 8). The time scales vary on the basis of the mode being excited. At low temperatures, the rise time is roughly 30 ps for the 4 THz mode, 20 ps for the 9 THz mode and 10 ps for the 17 THz mode. The lifetimes of the phonons at T = 13 K, determined from equilibrium vibrational spectra, are 24, 16 and 8, respectively, remaining relatively constant crossing through T_c. These values provide a lower bound assuming the spectral features are homogeneously broadened; if inhomogeneous broadening plays a role, the decoherence time of the driven phonon could be longer. From these comparisons, we conclude that the rise time to reach the saturated, long-lived magnetization state is roughly equivalent to the lifetime of the driven phonon for all pump excitations studied. The fact that the lattice is in a coherently driven state throughout most of the transition points to a non-thermal, phonon-mediated mechanism underlying the dynamics of the pump-induced magnetization. A detailed discussion of the proposed mechanism, which is based on the coupling between coherently driven phonons, the orbital state, and the associated magnetic order, is presented in the Supplementary Information.

Infrared spectra and experimental spin–phonon couplings

We used synchrotron-based spectroscopic ellipsometry to accurately determine the infrared phonon spectra of YTiO₃, as described in ref. ⁵¹. The ellipsometric measurements in the frequency range from 9 to 85 meV (70 to 690 cm⁻¹) used synchrotron edge radiation of the 2.5 GeV electron storage ring at the IR1 beamline of the Karlsruhe Research Accelerator at the Karlsruhe Institute of Technology, Germany, and were performed using a home-built ellipsometer in combination with a Bruker IFS 66v/S Fourier-transform infrared spectrometer. The ellipsometric parameters (varPsi ) and Δ, measured at an AOI of 15°, define the complex ratio ({r}_{p}/{r}_{s}={rm{tan }}(varPsi ){{rm{e}}}^{ivarDelta }), where r_p and r_s are the complex Fresnel coefficients for light polarized parallel and perpendicular to the plane of incidence, respectively. For anisotropic samples, a direct analytical inversion of the ellipsometric parameters into the diagonal components of the complex dielectric tensor ε_xx, ε_yy and ε_zz is not possible, and a numerical regression procedure is required. To determine the dielectric function of YTiO₃, we measured on the ac and bc surfaces cut from the same crystal, with a or b axes aligned either parallel or perpendicular to the plane of incidence, respectively. A nonlinear fitting procedure was applied to extract point by point the complex dielectric response throughout the covered spectral range. Figure 2a in the main text shows the true b axis complex dielectric response ε = ε_yy extracted from the raw ellipsometry spectra (varPsi (omega )) and Δ(ω). The transverse optical phonon modes appear as peaks in Im(ε).

To study the changes in the parameters of the phonon modes with temperature, we used a simplified approach. The complex pseudo-dielectric function ε* in Extended Data Fig. 10a is derived by a direct inversion of (varPsi ) and Δ measured on the bc plane with the b axis in the plane of incidence, assuming semi-infinite bulk isotropic behaviour of the crystal. Several features of the phonon modes are found to change with temperature, namely the amplitudes, linewidths and frequencies of the modes. Here, we focus on the frequency shifts with temperature. As YTiO₃ is magnetic, there are two main contributions to the shift of the phonon frequency Δω, which can be written as

where (langle {S}_{i}{S}_{j}rangle ) is the nearest-neighbour spin correlation function. The first term on the right-hand side is the contribution due to lattice anharmonicity. The second term arises due to spin–phonon coupling, which, for infrared-active modes, enters to the lowest order into the lattice potential as ({E}_{{rm{spin-ph}}}=omega lambda langle {S}_{i}{S}_{j}rangle {Q}^{2}), where λ is the spin–phonon coupling constant and Q is the phonon amplitude. The frequency shift associated with this term is (varDelta {omega }_{{rm{spin-ph}}}approx lambda langle {S}_{i}{S}_{j}rangle ). To isolate the spin–phonon contribution, we subtract off the anharmonic background, which can be determined by fitting the data at high temperatures to equation (3.8) in ref. ⁵². Note that in reality, we use the frequency shift of the 7 THz phonon as the background function, as it can be described well by the anharmonic model throughout the entire temperature range, indicating negligible spin–phonon coupling for this mode. The spin–phonon frequency shift for the other modes then becomes (varDelta {omega }_{k,{rm{spin-ph}}}(T)=varDelta omega (T)-{g}_{k}varDelta {omega }_{7{rm{THz}}}), where ({g}_{k}) is determined by the condition (varDelta {omega }_{k,{rm{spin-ph}}}approx 0) for T > 150 K, which is easily satisfied for all modes with (mathrm{0.45 < }{g}_{k}mathrm{ < 1.1}).)

The temperature dependent spin–phonon frequency shifts for the three phonons driven in our pump-probe experiment are plotted in Extended Data Fig. 10b. At low temperatures, (langle {S}_{i}{S}_{j}rangle ) reaches a maximum so that the spin–phonon coupling constant λ is given by (varDelta {omega }_{{rm{spin-ph}}}). One can see from this analysis that the sign and magnitude of λ differ between the three modes. Furthermore, (varDelta {omega }_{{rm{spin-ph}}}) is non-zero all the way up to more than 100 K, indicating that (langle {S}_{i}{S}_{j}rangle ) is still finite at these temperatures well above T_c. This high-temperature fluctuating spin order provides a potential basis for the non-equilibrium magnetic state we report in the main text.