Effect of depinning field on determination of angular-momentum-compensation temperature of ferrimagnets

Rare earth (RE)–transition metal (TM) ferrimagnetic amorphous alloys, in which RE and TM moments are coupled antiferromagnetically, have attracted significant attention owing to their fundamental scientific relevance^1–10) as well as their potential for the development of applications such as spintronics devices.^11–15) In a ferrimagnet, the varying temperature dependence of sublattice magnetizations raises the possibility of the existence of a magnetization-compensation temperature T_M at which the resultant magnetization vanishes below the Curie temperature.^3,4,16) In addition, the different Landé g-factors between the RE and TM elements lead to an angular-momentum-compensation temperature T_A at which the net angular momentum becomes zero.^2–4,17,18) As the time evolution of the magnetic state is governed by the commutation relation of the angular momentum, rather than by that of the magnetic moment, Kim et al. demonstrated that fast field-driven antiferromagnetic spin dynamics can be realized in GdFeCo ferrimagnets at T_A,¹⁷⁾ with the maximum domain-wall (DW) velocity occurring at T_A, thereby providing a method for determining T_A. A sharp peak in the DW velocity was demonstrated in the flow regime,¹⁷⁾ but it is unclear whether such a sharp peak occurs in other DW dynamic regimes such as the depinning regime.

In this study, we investigate the temperature dependence of the field-driven DW motion from the depinning regime to the flow regime. We prepared amorphous GdFeCo ferrimagnetic films with the structure 5-nm SiN/30-nm Gd_23.5Fe_66.9Co_9.6/100-nm SiN/Si substrate via co-sputtering. We estimated the compositions of the GdFeCo layer from the relative deposition rates of Gd and FeCo. The films exhibit bulk perpendicular magnetic anisotropy. To detect the anomalous Hall resistance, an e-beam lithography technique was applied to structural devices with a Hall bar geometry having a width of 5 µm and length of 550 µm. We stacked 5-nm Ti/100-nm Au electrodes onto the ends of the wire and Hall bars for current injection and Hall measurement.

To measure the DW velocity v, we employed a real-time DW measurement technique that is described elsewhere.^17–19) We first applied a sufficiently strong magnetic field of −200 mT (−μ₀H_z,sat) to saturate the magnetization along the –z-direction. Subsequently, a constant magnetic field μ₀H_z was applied for driving the DW. To eliminate the nucleation of a domain, we ensured that μ₀H_z was smaller than the coercive field μ₀H_c. Next, we applied a direct current (DC) along the wire to detect the Hall signal; the applied DC was low enough to ignore the spin torques and avoid the Joule heating effect.²⁰⁾ We then injected a current pulse (12 V, 100 ns) through the writing line to nucleate the reversed domain, thereby creating two DWs in the wire. The created DWs moved along the wire because of the presence of μ₀H_z and subsequently passed through the Hall bar. The DW arrival time was detected by monitoring the change in the Hall voltage using an oscilloscope. The DW velocity was calculated using the arrival time and the distance traveled between the writing line and the Hall bar (400 µm). The temperature was controlled in the range of 200 to 300 K by using a low-temperature probe station.

We measured the field-driven DW velocity v over a wide range across the depinning and flow regimes, from 10⁻¹ to 10² m/s.^21–25) Figure 1 shows v as a function of μ₀H_z at a temperature of 280 K. There are two different dynamic regimes: the depinning regime (blue dashed box) and the flow regime (red dashed box). Notably, in the flow regime, where μ₀H_z is considerably greater than μ₀H_dep, the DW exhibits dissipative viscous motion with a linear relationship between v and μ₀H_z, as indicated by the red line showing the best linear fit.

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Figure 2(a) shows v as a function of the temperature T for μ₀H_z = 50, 60, and 70 mT in the flow regime. This plot shows that v exhibits a peak at a certain temperature. Hereinafter, T₀ denotes the temperature corresponding to the maximum value of v. T₀ is a constant irrespective of μ₀H_z (see the orange arrow). This tendency of v with respect to T is consistent with results obtained previously.^17,18) Thus, the present result shows the existence of the angular-momentum-compensation temperature T_A. We then investigated v as a function of T in the depinning regime. Figure 2(b) shows v as a function of T for μ₀H_z = 36, 44, and 68 mT. Apparently, T₀ increases with the decrease of μ₀H_z (see the shift of the orange arrows). Figure 2(c) shows the measured T₀ as a function of μ₀H_z. T₀ monotonically decreases with the increase of μ₀H_z and subsequently approaches T_A as μ₀H_z increases. Here, the red line indicates T_A. To obtain reliable information on T_A, measurements based on field-driven DW motion were performed under strong magnetic fields ($\mu _{0}H_{z} = 50,60,68,70$ mT) that brought the DW into the flow regime. When the magnetic field is low, no DW motion occurs below a certain value of T. At μ₀H_z = 36 mT, there is no DW motion below T = 255 K. A similar behavior is observed at μ₀H_z = 44 mT. According to this result, the DW pinning, which is related to μ₀H_dep, is sensitive to T.^26–29) Therefore, this effect can affect T₀.

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To investigate the T-dependent μ₀H_dep, we measured v as a function of μ₀H_z for various values of T. Figures 3(a)–3(e) show v–μ₀H_z curves for T = 210, 220, 237, 260, and 280 K, respectively. μ₀H_dep decreases with the increase of T (see the colored arrows). Recently, Nishimura et al. reported that the μ₀H_dep of a GdFeCo ferrimagnet strongly depends on the magnetic properties in accordance with the relationship $\mu _{0}H_{\text{dep}} \propto \sqrt{\mu _{0}H_{\text{K}}/M_{\text{S}}}$,²⁶⁾ where μ₀H_K is the effective magnetic anisotropy field, and M_S is the saturation magnetization. This implies that the primary origin of the T-dependent μ₀H_dep can be attributed to the variations in μ₀H_K and M_S. To eliminate the effect of T on μ₀H_dep, v is plotted as a function of the reduced coordinate μ₀(H_z − H_dep) for various values of T in Fig. 4(a), from which we can obtain v as a function of T at a fixed μ₀(H_z − H_dep), as shown in Fig. 4(b). This clearly indicates that the DW velocity has a sharp peak at almost the same temperature regardless of μ₀(H_z − H_dep) (see the orange arrow). Figure 4(c) shows T₀ as a function of μ₀(H_z − H_dep), where T₀ is independent of μ₀(H_z − H_dep) (see the red line).

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In the flow regime, where the magnetic field is sufficiently large, v is ideally proportional to μ₀H_z.²¹⁾ However, many reports show finite x-axis interception, which generally arises from imperfections of the sample or complexities of the internal DW structure.^30–32) Although we do not know the exact physical meaning of the reduced coordinate, by introducing the reduced coordinate, we empirically determine that the DW velocity peak is observed at T_A even in the depinning regime.

It is worthwhile to discuss the physical mechanism underlying the maximum DW velocity at T_A. In the flow regime above the Walker field, DW shows precessional motion in general, leading to the Walker breakdown. However, this precessional motion is suppressed at T_A because of the zero net angular momentum, resulting in the suppression of the Walker breakdown. Therefore, DW velocity is maximized at T_A.^17,18) However, the maximum DW velocity at T_A in the depinning regime remains debated. As the physics in the flow regime is related to the suppression of the precessional DW motion, we attempt to estimate the Walker field μ₀H_W, which can be defined as μ₀H_W = μ₀αM_S|N_y − N_x|/2,³³⁾ where μ₀ is the permeability, α is the Gilbert damping parameter, and N_x and N_y are the demagnetizing factors along the x- and y-directions, respectively. According to a previous study,²⁾ α is in the range of 0.08–0.2. M_S was determined to be in the range of 10⁴–10⁵ A/m by using a superconducting quantum interference device.²⁶⁾ N_x and N_y are defined as N_x ≈ t/(t + w) and N_y ≈ t/(t + Δ), respectively.^33,34) Here, w is the strip width, t is the GdFeCo thickness, and Δ is the DW width parameter, which can be expressed as $\Delta = \sqrt{A/K_{\text{eff}}}$, where A is the exchange stiffness constant, and K_eff is the effective magnetic anisotropy energy. To estimate the limiting value of μ₀H_W, we chose the maximum values of each parameter, i.e., α = 0.2, M_S = 10⁵ A/m, and Δ = 10 nm. The limiting value μ₀H_W is estimated to be 10.7 mT, which is smaller than μ₀H_dep. Furthermore, this value is consistent with a recent experimental result.³⁵⁾ We observed only precessional flow motion in all our experiments. Therefore, a universal value of T_A from the depinning regime to the flow regime can be realized via the suppression of the precessional motion in both the regimes.

In conclusion, we investigated the magnetic DW motion of a ferrimagnetic system near the angular-momentum-compensation temperature. Two dynamic regimes — the depinning and flow regimes — were explored by varying the magnetic field. The DW velocity exhibited a peak at T_A in the flow regime, whereas the temperature corresponding to the peak DW velocity increased monotonically with the decrease of μ₀H_z in the depinning regime, making it impossible to determine T_A. Therefore, to obtain reliable T_A values even in the depinning regime, we measured the depinning field for a wide range of temperatures and replotted the DW velocity as a function of μ₀H_z − μ₀H_dep, finding that the DW velocity peak is observed at T_A even in the depinning regime. Thus, our finding suggests that the identification of the depinning field is important for determining T_A.

Acknowledgments