Quantitative analysis and optimization of magnetization precession initiated by ultrafast optical pulses

Spintronic devices consisting of materials with large perpendicular magnetic anisotropy (PMA) are promising for the advancement of computer memory, data storage, and spintronics. Due to the time scale of magnetic switching in these devices (∼1 ns),^1–3 it is crucial to understand magnetization dynamics at such short time scales. To understand magnetization dynamics requires knowledge of the magnetic anisotropy and the Gilbert damping (α), which is defined in the Landau-Lifshitz-Gilbert (LLG) equation. While anisotropy can be determined through magnetostatic measurements, extracting α requires measurements that can capture the dynamic magnetization at time scales faster than magnetic switching. To date, the most common method to determine α is frequency-domain measurements of ferromagnetic resonance (FMR).^4,5 By measuring the resonance frequency and linewidth as a function of applied field, FMR probes both the magnetic anisotropy and Gilbert damping.^6–10 As spintronic applications favor materials with large PMA, an all-optical technique, time-resolved magneto-optical Kerr effect (TR-MOKE), has emerged. TR-MOKE is essentially a time-domain FMR technique that can be readily integrated with large external fields to capture high resonance frequencies using optical excitation. Technically, TR-MOKE is limited only by the sampling frequency (∼1 THz) and available external fields. This technique allows materials with large PMA (>10⁶ erg/cm³) to be measured.^11–14 

There are a number of studies reporting TR-MOKE measurements of the Gilbert damping in PMA thin films (e.g., films with large magnetocrystalline anisotropy such as L1₀ FePd or films with interfacial anisotropy including CoFeB).^13–17 While these studies utilized similar polar MOKE measurement techniques, there exists large variations in the choice of both the amplitude range and the angle (θ_H with respect to the sample surface normal z, see Fig. 1) of the external field (H_ext). For example, some literature studies utilized in-plane external fields because of their well-understood frequency dependence,¹⁸ while others applied H_ext at a chosen angle away from the in-plane in order to reduce the impact of inhomogeneous broadening likely caused by a distribution of magnetic anisotropy throughout the sample.¹⁹ Additionally, it has been experimentally observed that the process of applying H_ext at some angle between 0° and 90° from the surface normal is beneficial to increase the TR-MOKE signal amplitude.^17,18 Nevertheless, a systematic study that explores the angular dependence of the TR-MOKE measurement signal is still lacking. In this paper, we will first address this issue by discussing the mechanisms behind the θ_H dependence of the TR-MOKE signal and then predicting the optimal angle of θ_H for conducting TR-MOKE measurements with the improved signal-to-noise (SNR) ratio. The theoretically predicted optimal TR-MOKE operational conditions are further validated by direct experimental studies of a representative sample consisting of a tungsten-seeded CoFeB thin film with large PMA.

Simulations in this work utilize a finite difference approach to solve the LLG equation [Eq. (1)] with an explicit solution for the magnetization vector (M) as a function of time, following the forward Euler method²⁰ 

where M is the magnetization vector with a magnitude of M_s (the saturation magnetization), γ is the gyromagnetic ratio, H_eff is the effective magnetic field, and α is the Gilbert damping parameter. The vector H_eff is determined by taking the gradient of the magnetic free energy density (F) with respect to the magnetization direction (⁠$Heff=−∇MF$⁠). The scalar quantity F is the summation of contributions from the Zeeman energy (resulting from the external magnetic field, H_ext), perpendicular uniaxial magnetic anisotropy (K_u), and the demagnetizing field.

While Eq. (1) is often used to describe magneto-dynamics, it is not conducive to numerical solutions of this ordinary differential equation. To simplify the development procedures of computational algorithms, it is preferable to utilize the Landau-Lifshitz equation [Eq. (2)]²¹ 

In equilibrium, M is parallel to H_eff, and thus, the magnetization does not precess. Once the magnetization is slightly tilted away from the equilibrium direction (θ), it will begin to precess around the equilibrium direction and finally damp towards equilibrium at a rate determined by the magnitude of α (shown in Fig. 1). For the purposes of this work, we use the macrospin approximation, in which all of the parameters in Eq. (1) are independent of the position. This means that the explicit dependence of any parameters on the position (inhomogeneous broadening) and coupling of excitations at different wave-vectors (two-magnon scattering) are ignored. This approximation is justified at high frequencies, at which α is considered to be an intrinsic parameter. In principle, both the inhomogeneous broadening and two-magnon scattering can be made small relative to the intrinsic Gilbert damping by going to a sufficiently high applied field. Just as importantly, the two-magnon contribution depends only weakly on the angle for field orientations that are nearly in plane.²² For this reason, we expect the observations of this paper to apply even in samples with significant two-magnon scattering.

For TR-MOKE measurements, a “pump” laser pulse increases the temperature at an ultrafast time scale (∼400 fs determined from the pump pulse duration), which causes a thermal demagnetization (a decrease in M_s resulting from the increase in temperature).^23,24 This thermal demagnetization temporarily shifts the equilibrium direction initiating magnetization precession, which is continued even when M_s has recovered to its original state. Here, the demagnetization process is treated as a step decrease in M_s that lasts for 2.5 ps before an instant recovery to its initial value. All signal analysis discussed in this work follows the recovery of M_s.

For polar MOKE measurements, the Kerr rotation is proportional to the projected magnetization in the z-direction (M_z, the through-plane magnetization).²⁵ The evolution of M_z in time during precession will appear as a decaying sinusoid as captured by TR-MOKE measurements, i.e., $Mz(t)∝sin(2πft+φ)exp(−t/τ)$ with f, φ, and τ being the angular resonance frequency, the phase term, and the relaxation time of spin precession, respectively. The amplitude of the precession will greatly depend on the magnitude and angle of the external applied field, which is directly related to the SNR of TR-MOKE signals. By analyzing the precession as a function of the field (H_ext) and angle (θ_H), the precession amplitude (ΔM_z) can be extracted. Figure 2(a) shows the predicted ΔM_z as a function of the time delay between pump excitation and probe sensing, which can be treated as a direct simulation of TR-MOKE signals. Figure 2(b) depicts the θ_H-dependent ΔM_z normalized to the maximum ΔM_z for θ_H = 90° for two representative regions of magnetic field, H_ext > H_k,eff and H_ext < H_k,eff. Here, H_k,eff denotes the effective anisotropy field of the sample that is related to K_u through H_k,eff = 2K_u/M_s-4πM_s and can be readily determined from VSM measurements. Tracking this signal amplitude as a function of θ_H reveals that the precession (and thus the signal) will be maximized for a certain θ_H as shown in Fig. 2(b). Maximizing the oscillation implies that it will be beneficial to maximize the “magnetic torque” term (M × H_eff, which prefers a large angle between M and H_eff), but it is also important to include that TR-MOKE measures the projection of the magnetization along the z-direction (which prefers θ = 90°). Consequently, the value of θ_H,MAX requires weighing inputs from both the magnetic torque and the z-direction projection of magnetization.

Depending on the field ratio (H_ext/H_k,eff), the angular dependence of magnitude will drastically change. For H_ext < H_k,eff, the magnetization will be in equilibrium between the perpendicular direction and the in-plane direction (0° ≤ θ ≤ 90°). Maximizing the magnetic torque and projection in the z-direction in these cases will cause H_ext applied in-plane (θ_H = 90°) to be the optimal setup [shown by the black line in Fig. 2(b)]. Once H_ext exceeds H_k,eff, the Stoner-Wohlfarth minimum energy model predicts that the magnetization will approach the direction of the external field but never perfectly align with H_ext (except for the extreme cases of θ_H = 0° or 90°).²⁶ When θ_H = 0 or 90°, these two directions will excite no magnetic torque and therefore no magnetic precession will occur, as indicated by the amplitude minima at these extreme cases. For the intermediate range of θ_H in between 0° and 90°, the two effects for optimizing the signal (maximizing torque and maximizing projection) will compete, leading to an amplitude maximum at an angle that depends on the field ratio of H_ext/H_k,eff. The dependence of the M_z amplitude on θ_H can be readily obtained by dividing ΔM_z in Fig. 2(b) by sinθ with θ being the equilibrium angle.

Figure 3 shows a contour plot of the dependence of the normalized ΔM_z representing the spin precession amplitude as a function of both H_ext and θ_H. The highest amplitude of precession will occur near H_k,eff when the field is applied in the film plane. If the external field is greater than H_k,eff, it is beneficial to conduct the measurement at an angle that is out of the film plane. To better illustrate this trend, the dotted red line in Fig. 3 indicates the angle of the maximum signal (θ_H,MAX) at specified field ratios. Based on these results, measurement conditions can be optimized to maximize the precession signal based on the field ratio. For example, if the maximum strength of the magnetic field is 2H_k,eff, then it would be beneficial to set θ_H > 70°. Furthermore, measurements conducted at a constant H_ext but with varying magnetic field angles should not necessarily choose the highest possible H_ext to achieve the optimal SNR.

To further assist in the design of TR-MOKE measurements with optimal SNR, we note that there is a simple means to estimate the amplitude of the TR-MOKE signal based on the ansatz that the magnetization during the pulse is modified by an amount ΔM_s, so that the effective field during the pulse is

which accounts for the demagnetizing field due to the non-equilibrium magnetization. We make the simplifying assumption that this field is constant during the pulse and that $ΔMs≪Ms$⁠, so that the angular displacement of the magnetization during the pulse is proportional to the torque

where $φ̂$ is a unit vector in the x-y plane of Fig. 1. After the pulse, the magnetization precesses about the equilibrium effective field on the trajectory shown in Fig. 1, and the amplitude |ΔM_z| of the modulation of the z-component of the magnetization is then proportional to sin θ, so that

We recall that θ is the angle of the equilibrium effective field relative to the z-axis in Fig. 1. It is possible to express θ in terms of the angle θ_H minimizing the free energy

with respect to θ, yielding the compact expression

for the amplitude θ_K of the TR-MOKE signal. It is then easy to calculate the angle θ_H,MAX at which θ_K is maximized for each value of H_ext. This defines the contour shown as the dashed red line in Fig. 3. The result is shown in Fig. 4, in which a comparison is made to the full simulation.

To verify the model prediction for the maximum TR-MOKE signal amplitude, we conducted a series of measurements on a W/CoFeB/MgO thin-film sample with PMA. The CoFeB sample was post-annealed at 300 °C. The magnetic properties of this sample were found to be H_k,eff = 6.1 kOe as determined from VSM and α = 0.018 as measured by TR-MOKE.¹⁶ A schematic of the sample stack is shown as an inset of Fig. 5(b). The TR-MOKE setup used for these measurements involves an ultrafast Ti:Sapphire laser system to initiate and capture the magnetization precession. Additional information regarding this ultrafast system can be found elsewhere.^16,24 During measurements, we considered four different amplitudes of H_ext (4, 6, 8, and 10 kOe) to show the θ_H dependence of TR-MOKE signals covering both the low- and high-field ratio regimes. To avoid blocking the laser with the magnetic poles, the range of θ_H was confined from 80° to 90° with a 2° interval. For the amplitude range (4–10 kOe) and angle range (80°–90°) of the external field described here, the measured resonance frequencies are less than 25 GHz. Additional TR-MOKE data at higher frequencies of up to ∼50 GHz and details regarding the sample preparation and structural and magnetic property characterization are provided in Ref. 16.

For ease of comparison, we subtracted the thermal background from the raw TR-MOKE measurement data to obtain the pure precession signal, which oscillates sinusoidally at a decaying rate related to the Gilbert damping. As shown in Fig. 5(a), this involves fitting the data to the equation $θK(t)=A+B exp(−t/C)+D sin(2πft+φ)exp(−t/τ)$ and subtracting the thermal background, as denoted by the decaying exponential $A+B exp(−t/C)$⁠. The amplitude is then calculated using the same method as shown in Fig. 2(a). Figures 5(b)–5(e) summarize the comparison of the normalized oscillation amplitudes from both measurements (red symbols) and model prediction (black lines) for all four values of H_ext. The data and theoretical curves are normalized to the highest amplitude for a given H_ext.

Comparisons between the trends of predicted simulations and measurement results show excellent agreement. As expected, the signal amplitude increases monotonically with increasing angle for H_ext < H_k,eff and has an angle of the maximum signal for H_ext > H_k,eff. These measurements can even capture the predicted peak of amplitude at nearly the same θ_H for fields near H_k,eff. For the 6 kOe measurements, there is a slight deviation in the amount of decay in signal strength for decreasing θ_H (simulations predict a slower decrease). This is most likely due to the inhomogeneity resulting from a nonuniform distribution of the values of H_k,eff in the sample, which leads to a deviation from theory near H_k,eff. While the θ_H in the setup used in this experiment was limited, these results verify the excellent agreement between simulation and measurement.

In conclusion, we have developed a numerical approach to simulate the dynamic response of magnetization to a thermally induced demagnetization process. This approach identifies the optimal angle of the external field for the maximum magnetization precession signal in TR-MOKE measurements by balancing the projection of the dynamic magnetization and the magnetic torque. To verify the theoretical prediction, we have conducted TR-MOKE measurements on a W/CoFeB/MgO sample with perpendicular magnetic anisotropy for multiple external fields and field angles. The measurement results demonstrate that the dependence of the TR-MOKE signal magnitude on the external field can be well captured by the theoretical prediction. Our study provides a better understanding of how the external field influences the magnetization precession signals obtained in TR-MOKE measurements and thus facilitates the design and optimization of measurement conditions for maximizing SNR and improving accuracy.

This work was supported by C-SPIN (Award No. 2013-MA-2381), one of the six centers of STARnet, a Semiconductor Research Corporation Program, sponsored by MARCO and DARPA.