Magnetic storage and magnetic memory have recently shifted towards the use of magnetic thin films with large perpendicular magnetic anisotropy (PMA) to simultaneously satisfy the requirements in storage density and thermal stability. Understanding the magnetic switching process and its dependence on the Gilbert damping (*α*) of materials with large PMA is crucial for developing low-power consumption, fast-switching, and high-thermal stability devices. The need to quantify *α* of materials with large PMA has resulted in the development of the all-optical ultrafast Time-Resolved Magneto-optical Kerr Effect (TR-MOKE) technique. While TR-MOKE has demonstrated its capability of capturing magnetization dynamics of materials with large PMA, a quantitative analysis regarding the operational optimization of this emerging technique is still lacking. In this paper, we discuss the dependence of the TR-MOKE signal on the magnitude and angle of the applied field, by utilizing a numerical algorithm based on the Landau-Lifshitz-Gilbert equation. The optimized operational conditions that produce the largest TR-MOKE signals are predicted. As an experimental verification, we conduct TR-MOKE measurements on a representative sample of a tungsten-seeded CoFeB PMA thin film to show the excellent agreement of the model prediction with measurements. Our analysis results in a better understanding of the external field influence on the magnetization precession processes. The results of this work can also provide guidance on selecting operational conditions of the TR-MOKE technique to achieve optimal signal-to-noise ratios and thus more accurate measurements of magnetization dynamics.

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 (>10^{6} erg/cm^{3}) 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 L1_{0} 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 method^{20}

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 ($Heff=\u2212\u2207MF$). 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., $Mz(t)\u221dsin(2\pi ft+\phi )exp(\u2212t/\tau )$ 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 (Δ

*M*) can be extracted. Figure 2(a) shows the predicted Δ

_{z}*M*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

_{z}*θ*

_{H}-dependent Δ

*M*normalized to the maximum Δ

_{z}*M*for

_{z}*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}= 2

*K*

_{u}/

*M*

_{s}-4π

*M*

_{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 Δ*M*_{z} in Fig. 2(b) by sin*θ* with *θ* being the equilibrium angle.

Figure 3 shows a contour plot of the dependence of the normalized Δ*M*_{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 2*H*_{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 Δ*M*_{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 $\Delta Ms\u226aMs$, so that the angular displacement of the magnetization during the pulse is proportional to the torque

where $\phi \u0302$ 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 |Δ*M _{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 $\theta K(t)=A+B\u2009exp(\u2212t/C)+D\u2009sin(2\pi ft+\phi )exp(\u2212t/\tau )$ and subtracting the thermal background, as denoted by the decaying exponential $A+B\u2009exp(\u2212t/C)$. 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.