Kinetics of Domain Structure in Co/Pt/Co Ultrathin Films with Ferromagnetic Interlayer Exchange Interaction: Dependence on Interlayer Thickness

: We studied the in-field evolution of the domain structure in ultrathin Co(0.6 nm)/Pt(t)/Co(0.6) nm trilayers with perpendicular magnetic anisotropy for 5 nm < t < 6 nm using polar Kerr microscopy. The critical interlayer thickness t cr = 5.3 nm was found to separate two principal patterns of domain be-havior including interlayer correlations and motility of the domain walls. It is shown that magnetization in both Co layers is coupled with strong ferromagnetic interaction for small Pt thickness ( t < t cr ), while this coupling is weak for thicker ( t > t cr ) Pt layers. Nonlinear dependence of the wall displacement on the field value is observed. The established final position of domain walls after relaxation depends on the Pt layer thickness. It is determined by balance of the interlayer exchange and energy gain due to the field. The mechanism of wall stabilization is considered in the case of independent wall motion. In the region with weak coupling, dependence of the interlayer interaction energy on Pt thickness was measured.


Introduction
Magnetic thin film structures with perpendicular magnetic anisotropy (PMA) possess unique physical properties [1][2][3]. They can be used as model materials to investigate the formation and evolution of specific domain structures [4][5][6][7][8], thermoactivated domain wall creep [9], magnetoresistance [10], anomalous Hall effect [10,11], and spin-wave dynamics [12]. The list of possible applications includes high-density magnetic memory devices [13,14], magnetic sensors for medicine [15], and superfast elements for spintronics [16]. Switching of magnetization in such materials involves nucleation of new domains and motion of domain walls (DWs) [4,8,[17][18][19][20][21]. These processes are well examined in the case of continuous ultrathin ferromagnetic (FM) Co films with PMA [17][18][19]22]. Meanwhile, they can be quite different in the case of a superlattice where PMA layers are separated by nonmagnetic (NM) layers [23][24][25]. Depending on the number and thickness of both FM and NM layers, different magnetic phases can be observed, including stable interlayer frustration at the domain walls. Switching of such complicated objects in multiple layers can be highly inhomogeneous, and its mechanism is difficult to understand.
More distinct are similar phenomena in trilayers where two FM layers are still coupled across the NM layer by exchange or magnetostatic interaction J [1, [3][4][5][6][7][8][9]26,27]. Typical for exchange coupling is its dependence on the thickness of the NM layer. It was shown theoretically [28,29] and experimentally [26,30,31] that this dependence is oscillatory. Studying magnetoresistance and the Kerr effect in a wedge-shaped trilayer Co/Au(111)/Co, the authors of [31] showed that the mutual orientation of magnetization in FM layers changed from parallel to antiparallel and back when the Au thickness was varied monotonically; that is, interlayer exchange evolved from ferromagnetic to antiferromagnetic and vice versa. However, the mechanism of the magnetization reversal was not revealed. Nucleation and motion of the domain walls were observed for both exchange ( [4][5][6][7][8][9] Co/Pt/Co) and Magnetism 2022, 2 187 magnetostatic ( [27] CoPt/NiO/CoPt) interactions between FM layers. Dependence of the DW velocity on the applied field was investigated both theoretically and experimentally in separate FM layers of different thickness and in a coupled trilayer system [5][6][7][8][9]. It was shown that despite the DWs having different mobility in each separate layer, they proceeded together with the same speed in the case of ferromagnetic coupling between the FM layers. Thickness-dependent coupling parameter J can be determined by measuring the minor hysteresis loops using the magneto-optic Kerr effect [8] or using ferromagnetic resonance studies [26]. Experiments with similar Co/Pt/Co trilayers where the NM Pt layer was wedge-shaped revealed a transition of the domain nucleation pattern and domain wall motion depending on the Pt thickness and, hence, on the strength of interlayer coupling [4,8]. When the thickness of the Pt layer was above some critical value, nucleation of new domains and motion of domain walls in both FM layers were independent and uncorrelated, while the processes were synchronous in the case of thinner Pt interlayers. These features should be determined by kinetics and relaxation of the domain structure depending on the interlayer interaction. Detailed analysis of this problem is the purpose of the present paper.

Experiment
The sample used in our experiments was a Pt(10 nm)/Co(0.6 nm)/Pt(t)/Co(0.6 nm)/Pt(3 nm) structure synthesized by magnetron sputtering on a 5 × 6 mm oxidized silicon substrate at room temperature. Thickness of the Pt layer increased uniformly from 5 to 6 nm ( Figure 1). The hysteresis loops were obtained by measuring the intensity of the Kerr effect from local (0.3 × 0.3) mm areas of the sample. Domain structure and DW motion were studied by magneto-optic (MO) microscopy based on the Kerr effect. To improve the magneto-optic contrast, we registered preliminary images of the saturated sample in a strong positive and negative magnetic field. The average of these two magneto-optic pictures was subtracted from every experimental image of the transient domain structure.
Magnetism 2022, 2, FOR PEER REVIEW 2 varied monotonically; that is, interlayer exchange evolved from ferromagnetic to antiferromagnetic and vice versa. However, the mechanism of the magnetization reversal was not revealed. Nucleation and motion of the domain walls were observed for both exchange ( [4][5][6][7][8][9] Co/Pt/Co) and magnetostatic ( [27] CoPt/NiO/CoPt) interactions between FM layers. Dependence of the DW velocity on the applied field was investigated both theoretically and experimentally in separate FM layers of different thickness and in a coupled trilayer system [5][6][7][8][9]. It was shown that despite the DWs having different mobility in each separate layer, they proceeded together with the same speed in the case of ferromagnetic coupling between the FM layers. Thickness-dependent coupling parameter J can be determined by measuring the minor hysteresis loops using the magneto-optic Kerr effect [8] or using ferromagnetic resonance studies [26]. Experiments with similar Co/Pt/Co trilayers where the NM Pt layer was wedge-shaped revealed a transition of the domain nucleation pattern and domain wall motion depending on the Pt thickness and, hence, on the strength of interlayer coupling [4,8]. When the thickness of the Pt layer was above some critical value, nucleation of new domains and motion of domain walls in both FM layers were independent and uncorrelated, while the processes were synchronous in the case of thinner Pt interlayers. These features should be determined by kinetics and relaxation of the domain structure depending on the interlayer interaction. Detailed analysis of this problem is the purpose of the present paper.

Experiment
The sample used in our experiments was a Pt(10 nm)/Co(0.6 nm)/Pt(t)/Co(0.6 nm)/Pt(3 nm) structure synthesized by magnetron sputtering on a 5 × 6 mm oxidized silicon substrate at room temperature. Thickness of the Pt layer increased uniformly from 5 to 6 nm ( Figure 1). The hysteresis loops were obtained by measuring the intensity of the Kerr effect from local (0.3 × 0.3) mm areas of the sample. Domain structure and DW motion were studied by magneto-optic (MO) microscopy based on the Kerr effect. To improve the magneto-optic contrast, we registered preliminary images of the saturated sample in a strong positive and negative magnetic field. The average of these two magneto-optic pictures was subtracted from every experimental image of the transient domain structure.

Results and Discussion
Contrary to [5][6][7][8] where FM layers in Co/Pt/Co had different thickness, we performed a direct experimental investigation with equal thickness of FM layers but with a spatially varying NM layer. Magnetization of both Co layers was the same. Similar to [4], kinetics of the magnetization reversal was different at opposite sides of the sample. Application of a number of field pulses (4 ms, ±100 Oe) resulted in the formation of a complicated domain structure. Four principal configurations of mutual orientation of magnetization in both FM layers could be extracted from magneto-optic Kerr images (Figure 1a). Further magnetic treatment using a 10 Hz AC magnetic field with amplitude close to coercivity allowed obtaining a solitary domain wall at the position x 0 where the antiferromagnetic interaction between FM layers was replaced with a ferromagnetic one (Figure 1b). Examination of the Kerr intensity along the wedge confirmed that the magnetization in FM layers was parallel (that is, the interlayer coupling was ferromagnetic) everywhere for x > x 0 . In the field H = +800 Oe normal to the surface, both FM layers were saturated (phase I). Elimination of this field and application of a field with opposite sign caused nucleation of new domains and motion of domain walls.
When the Pt layer was thin, the switching proceeded via coherent nucleation of new domains (phase III) in both FM layers (Figure 2a). DW motion was also coupled in both layers. Figure  The intensity of the total MO signal in the three phases described above was I I , I II , and I III . (c,f) magneto-optic hysteresis loops for t < t cr (a-c) and t > t cr (d-f).
When t was higher than this critical value, both the DW image and the MO profile were different. The new domains arose at different fields in the top FM layer and the lower one, and their domain walls moved independently. As a rule, the process began in one Co layer (magnetically soft layer with magnetization M S ), and, when it was reversed completely (phase II), nucleation and DW motion began in the second layer (magnetically hard layer with magnetization M H ) (Figure 2d). To distinguish which layer was switched first, we analyzed the corresponding MO profile (Figure 2e). The Kerr intensity was obviously weaker for the bottom FM layer.
The intensity of the total MO signal in the three phases described above was I I , I II , and I III .
I I = I 1↑ + I 2↑ , I II = I 1↑ + I 2↓ , The separate intensity of each layer I 1 and I 2 magnetized up or down was determined by the Malus law.
where I 0 is the intensity of incident light polarized linearly, β is the rotation of the polarization plane when two polarizers are slightly uncrossed, and ϕ 1 and ϕ 2 are angles of rotation induced by magnetization M 1 and M 2 of two FM layers.
The intensity provided by each layer could be estimated from the MO profile across DW (Figure 2e).
I II − I I = I 0 Sin(2β) Sin(2ϕ 2 ) ≈ D 2ϕ 2 , where D = I 0 sin(2β) is a constant value, and sin(2ϕ 1,2 ) at small ϕ is approximately 2ϕ 1,2 . Therefore, the difference in MO intensity across domain walls (Figure 2d) was determined by the rotation angle in a corresponding layer. The measured values of I II − I I and I III − I II were 50 and 42, respectively. This means that both nucleation of new domains and DW motion started in the upper Co layer where the MO signal was strongest. The difference in coercivity of lower (hard) and upper (soft) Co layers allowed us to directly observe incoherent nucleation and separated domain walls, DW S and DW H , as well as motion in these soft and hard layers, respectively. The local MO hysteresis loop in this area of the sample is not rectangular (Figure 2f) exhibiting consecutive magnetization reversal. The typical separate steps often observed in magnetic multilayers [1,9,10,31] were not well resolved as the difference in coercivity was rather small. The measured values of the start field in both layers (Figure 2d) were H c1 = 24 Oe and H c2 = 30 Oe. The measured intensity difference (I III − I I ) was the same for both thin (Figure 2b) and thick (Figure 2e) Pt layers as the total thickness of the FM layer remained constant.
While the bottom FM layer showed the same picture of magnetization reversal for every position along the sample, switching of the top layer depended on the interlayer Pt thickness when t > t cr . Figure 3a shows the consecutive displacement of the DW toward the thinner side of the Pt wedge when a sequence of H = 50 Oe, τ = 4 ms magnetic field pulses was applied. After each single pulse, the domain wall shifted to a new position passing the distance ∆x i = x i+1 − x i , where i is the number of pulses. Total displacement of the wall is shown in Figure 3b. Figure 3a shows that, when the wall approached the position where t = t cr , its displacement after each pulse became smaller before finally (here, after the 25th pulse) reaching zero, such that the wall reached its equilibrium position for a given amplitude of the field pulse (Figure 3b). tance between the two walls was much larger than the wall thickness; thus, their motion was likely independent. The two wall displacements x1 and x2 became comparable only near tcr (insert III in Figure 4). Dependence xi(H) was obviously nonlinear, tending to stop in the vicinity of the value corresponding to t = tcr. When the upper DW reached the part of the sample where t ≤ tcr, it did not move despite a further increase in the field amplitude until the lower wall joined it. Subsequently, both walls moved together as a single object.   Figure 4 shows the dependence of the DW position on the amplitude of the field after some transient series of pulses (5)(6)(7)(8)(9). Duration τ = 4 ms was chosen such that the upper DW could move ahead of the bottom one as shown in Figure 4 (inserts I and II). The distance between the two walls was much larger than the wall thickness; thus, their motion was likely independent. The two wall displacements x 1 and x 2 became comparable only near t cr (insert III in Figure 4). Dependence x i (H) was obviously nonlinear, tending to stop in the vicinity of the value corresponding to t = t cr . When the upper DW reached the part of the sample where t ≤ t cr , it did not move despite a further increase in the field amplitude until the lower wall joined it. Subsequently, both walls moved together as a single object.  The observed monotonic decrease in the DW displacement with every field pulse when it moved from the thicker interlayer side to the thinner one assumed the presence of an additional effective field HJ(x). This additional field was antiparallel to the driving one, and its value increased with a decrease in interlayer thickness t. The free energy necessary to move the wall in the x-direction can be written as where b is the thickness of the FM layer, Y is the length of DW along the y-axis, and H and Hc1 are the driving field and coercivity of the upper layer, respectively. DW stopped at The observed monotonic decrease in the DW displacement with every field pulse when it moved from the thicker interlayer side to the thinner one assumed the presence of an additional effective field H J (x). This additional field was antiparallel to the driving one, and its value increased with a decrease in interlayer thickness t. The free energy necessary to move the wall in the x-direction can be written as where b is the thickness of the FM layer, Y is the length of DW along the y-axis, and H and H c1 are the driving field and coercivity of the upper layer, respectively. DW stopped at some new position x (Figure 4) when it reached the new equilibrium state ∆W(x) = 0. That is, or By subtracting coercivity from the driving field and taking into account the relation in Equation (6), one can determine dependence H J (x). It is shown in Figure 5a (circles). The initial left point x 0 where the interlayer exchange field changed its sign (J = 0) was determined separately by moving the upper DW with an AC magnetic field around this point with the same field amplitude H ≈ H c .
The measured dependence HJ(x) fits a third-degree polynomial (shown by the line in Figure 5a), HJ = K (19.76 X 3 − 110.8 X 2 + 208.19 X − 130.32), (13) where K = 1 Oe and dimensionless variable X = x/1 mm. Equations (12) and (13) allow deducing dependence J(x) from the experimental data. The interlayer thickness varies linearly with x with a factor of C = −∆t/Δx = −0.2 nm/mm, t = t0 + Cx, where ∆t is the maximal change in interlayer thickness in the interval under study (1 nm), and t0 is the maximal interlayer thickness (6 nm); hence, dependence J(x) can be easily recalculated to J(t). The final dependence J(t) is shown in Figure 5b. Thus, we obtained the detailed dependence J(t) in the FM/NM/FM wedge heterostructure in the area with ferromagnetic interlayer interaction, J(t) > 0. It is shown that the energy density increased for thinner Pt layers. Furthermore, this rise was steeper when thickness approached tcr. The effective magnetic field H(t) due to exchange interaction between layers also increased with the decrease in t (i.e., increase in x). This is the reason for the asymmetric domain nucleation and domain wall motion observed in [4]. According to Equation (6), the wall moves when H − Hj > Hc. Otherwise, the walls stop, in agreement with Figure 4. However, the quick rise in J(t) and Hj(t) when the wall moved toward a thinner Pt area resulted in a stronger counteraction of Hj. Hence, one needs to increasingly enhance driving field H while the wall travels progressively smaller distance. Finally, the individual motion of the wall becomes suppressed when its displacement exceeds xcr. Application of an even stronger H leads to joined motion of the top and bottom domain walls, as observed in [4].
Effective field H J (x) preventing DW motion was caused by an interlayer exchange interaction J(x) between FM Co layers, taking into account the enhancement of energy when the moving DW passed the surface S where S = Yx is the surface swept by the wall during its motion. Equations (7) and (9) result to Separating the variables yields Finally, taking the derivative, one gets dependence J(x), The measured dependence H J (x) fits a third-degree polynomial (shown by the line in Figure 5a), where K = 1 Oe and dimensionless variable X = x/1 mm. Equations (12) and (13) allow deducing dependence J(x) from the experimental data. The interlayer thickness varies linearly with x with a factor of C = −∆t/∆x = −0.2 nm/mm, t = t 0 + Cx, where ∆t is the maximal change in interlayer thickness in the interval under study (1 nm), and t 0 is the maximal interlayer thickness (6 nm); hence, dependence J(x) can be easily recalculated to J(t). The final dependence J(t) is shown in Figure 5b. Thus, we obtained the detailed dependence J(t) in the FM/NM/FM wedge heterostructure in the area with ferromagnetic interlayer interaction, J(t) > 0. It is shown that the energy density increased for thinner Pt layers. Furthermore, this rise was steeper when thickness approached t cr . The effective magnetic field H(t) due to exchange interaction between layers also increased with the decrease in t (i.e., increase in x). This is the reason for the asymmetric domain nucleation and domain wall motion observed in [4]. According to Equation (6), the wall moves when H − H j > H c . Otherwise, the walls stop, in agreement with Figure 4. However, the quick rise in J(t) and H j (t) when the wall moved toward a thinner Pt area resulted in a stronger counteraction of H j . Hence, one needs to increasingly enhance driving field H while the wall travels progressively smaller distance. Finally, the individual motion of the wall becomes suppressed when its displacement exceeds x cr . Application of an even stronger H leads to joined motion of the top and bottom domain walls, as observed in [4].
The origin of the observed distinction in the switching scenarios at t > t cr and t < t cr is the relationship of the exchange energy J(t) and energy W C2 = 2MH C2 b necessary to overcome coercivity in the bottom FM layer. J(t) < W c2 when t > t cr (or x < x cr ). In this case, the upper domain wall can move separately as the interlayer exchange cannot overcome pinning of the wall in the bottom layer. If J(t) > W c2 (t < t cr , x > x cr ), the interlayer exchange is capable of pulling the bottom wall, and both walls travel together.

Conclusions
We performed measurements of the field-driven domain wall displacement along the FM/NM/FM wedge heterostructure using the magneto-optic Kerr effect. This procedure allowed us to extract the dependence of the effective field on the interlayer thickness H j (t). Hence, we obtained the detailed dependence J(t) in the area with ferromagnetic interlayer interaction. It was shown that H j (t) and J(t) increased significantly with the decrease in Pt thickness. The obtained values of 0 erg/cm 2 < J < 5·10 −2 erg/cm 2 with~5.6 nm > t > t cr = 5.3 nm are in good agreement with the results reported by other authors [8,31]. We studied the transition between the two switching scenarios of the FM/NM/FM wedge for t < t cr and t > t cr observed in [4,8]. It is shown that the nonlinear growth of interlayer exchange requires progressively increasing the driving field to push the domain wall toward the thin platinum layer side. The separated motion of the wall in the top FM layer becomes suppressed, and, for a higher field, the walls in both layers move together, as observed in [4]. Distinction between the two regimes of the wall motion is determined by the relationship between the interlayer exchange field and pinning in the bottom layer, i.e., when J(t) becomes equal to or greater than energy W C2 , as determined by the coercivity H C2 of the bottom layer.