Z. JINet al. PHYSICAL REVIEW B102, 054419 (2020) physics deserves further exploration. Fortunately, earlier the-
oretical calculation on the FM skyrmion confirmed that the FM skyrmion profile agrees well with the 360◦ domain-wall formula [37]. This equivalence allows a direct connection between the skyrmion dynamics and magnetic domain-wall dynamics. Certainly, to some extent, the theoretical treatment on the FM skyrmion can be safely transferred to the case of an AFM skyrmion, considering the similarity in Hamiltonian between the FM and AFM skyrmion systems.
On the other hand, it would be even more important to understand the depinning process for an AFM skyrmion under pinning by some defect. Here, we consider the depinning process by spin current, and therefore the depinning current as a function of the system parameters such as damping constant must be well understood. In fact, in our earlier work on the depinning process of an AFM domain wall, it was predicted that the depinning field is remarkably dependent on the damping constant, attributed to the oscillation of the AFM domain wall [38]. This prediction and the equivalence mentioned just above suggests the possibility for a similar dependence regarding the depinning process of an AFM skyrmion under pinning. This is an essential issue to clarify due to its importance in the AFM spintronics.
In this work, we focus on the spin-current-driven dynamics of an AFM skyrmion and present a reliable theoretical treat- ment. Then the predicted dynamic behavior will be compared with the numerical simulations based on the Landau-Lifshitz- Gilbert (LLG) equation. We start from a theoretical treatment of the dynamics of an AFM skyrmion in an ideal AFM lattice (without any pinning defect) and then the motion velocity in dependence of the intrinsic physical parameters of the system will be derived. Subsequently, we extend our treatment to the case with the presence of pinning defect in the lattice, and the depinning field of the skyrmion motion as a function of the damping constant and pinning strength of the defect will be obtained. The whole process of treatment constitutes the theory on the dynamics of AFM skyrmion in the presence of a pinning defect, which is highly valued for potential spintronic applications.
II. MODEL AND METHODS
We consider an ultrathin AFM film on a heavy-metal layer in thexyplane with two magnetic sublattices that have magnetic moments m1 andm2 respectively [39], satisfying condition|m1| = |m2| =Swith spin lengthS. The total mag- netizationmand the unit Néel vector nare defined asm= (m1+m2)/2S and n=(m1−m2)/2S, respectively, which are used to describe the AFM skyrmion dynamics. Taking into account the exchange energy, the anisotropy energy, the interfacial DM interaction, and the constraints |n| =1 and m·n=0, one has the total energy of the system given by [36,40]
H=
dV A0
2 m2+A(∇n)2
−D
nz∇·n−(n·∇)nz
−Kn2z
, (1)
FIG. 1. Schematic illustration of an antiferromagnet-heavy metal bilayer configuration. Top layer: spin configuration along a radial direction of an AFM skyrmion. Bottom layer: a perpendicular spin current (coarse red solid arrow) is induced by a charge current (coarse black solid arrow) and then a spin transfer torque acting on AFM moment is generated.
where A0=8JS2/a2 is the homogeneous exchange constant with AFM interaction J and lattice constant a, A=JS2 is the inhomogeneous exchange constant, D=D0S2/a is the interfacial DM interaction constant with constant D0 in the discrete model, and K=K0S2/a2 is the anisotropy constant along thezaxis with constantK0in the discrete model [41].
Here, the spin-polarized current (spin current) in the perpendicular-to-plane geometry is induced by the spin Hall effect generated by the in-plane charge current in the neigh- boring heavy metal layer, as explicitly depicted in Fig.1. In this case, the AFM dynamics is described by the following two coupled equations [42,43]:
˙n=(γfm−G1m)˙ ×n+γun×(m×p), (2a)
˙
m=(γfn−G2˙n)×n+(γfm−G1m)˙
×m+γun×(n×p), (2b) whereγ is the gyromagnetic ratio, fm = −δH/δm and fn=
−δH/δn are the effective fields, G1 and G2 are the phe- nomenological Gilbert damping parameters, p denotes the unit vector along the electron polarization direction (xaxis, ex),u= jμBθSH/(γeμstz) is the effective field related to the dampinglike spin torque with the Bohr magneton μB, the effective spin-Hall angle θSH, the driving current density j, the electron charge e, the saturation moment μs, and the film thickness tz. Subsequently, the dynamics of the AFM skyrmion can be analytically calculated by solving the two equations.
Furthermore, the velocity and the depinning field of the AFM skyrmion motion are also estimated using the LLG simulations of the discrete model, in order to check the validity of the theoretical treatment. The simulation details and parameter choice are presented in the Appendix.
III. RESULTS AND DISCUSSION A. Velocity for AFM skyrmion motion
In the earlier work [32], it was numerically revealed that the velocity of an AFM skyrmion is proportional to the DM in- teraction for a fixed current density. Here, we can analytically 054419-2
Z. JINet al. PHYSICAL REVIEW B102, 054419 (2020) shown in Fig.2(b)which presents the calculated and simu-
latedvas a function ofDfor u=0.02J/μs. The results are quite consistent with each other (the discrepancy is less than 5%), further confirming the validity of the theory. Moreover, for 32AKπ2D2, a nearly linear relation betweenvandD is obtained, quantitatively explaining the earlier simulations [32].
So far, the dependence of the AFM skyrmion velocity on these internal and external parameters is clarified, allowing one to estimate the velocity easily and to understand the physics clearly. As a matter of fact, the dynamics of the AFM skyrmion has attracted attention for many years, but a detailed formulation for its velocity has remained ambiguous.
B. Pinning and depinning of AFM skyrmion
Subsequently, we investigate the AFM skyrmion dynamics in a system with pinning defect, in particular the pinning and depinning behaviors. Without loss of generality, the defect is introduced by a local variation in the magnetic anisotropy [36,48,49], expressed byK∗=K{1−λexp[−(r−rd)2/R2d]} [26], whereλ denotes the defect pinning strength, rd is the position of the defect center, andRd is the defect size.
For our calculation, parameters K=0.8S2/a2 and D= 0.78S2/aare selected to generate a skyrmion with small size to suppress the defect induced skyrmion distortion. Thus, for Rs<Rd, one may reasonably assume the defect potential to be a parabolic one [38,50]:
V(r)= 1
2λ0|r−rd|2 (|r−rd|<Rp)
1
2λ0R2p (|r−rd|Rp) , (13) whereλ0=cλλis the pinning strength prefactor related to the defect, andRpis the radius of the potential well.
For simplicity, the skyrmion is assumed to be initially at the defect centerrd. It is noted that the skyrmion will be captured by the defect in the low current region. By applying the Thiele approach and considering Eq. (13), the equation of motion for the skyrmion positionqis obtained:
¨
q+2εωq˙+ω2q+C=0, (14) where 2ε=γA0G2/ω,ω2=γcλλ/Dxx, and C=
−γ2A0Ixyu/Dxx. Equation (14) clearly describes the damping oscillation of the AFM skyrmion. Specifically, for G2<2ω/γA0, we have the solution representing an underdamped oscillation:
q(τ)=e−εωτ(C1cosωpτ+C2sinωpτ)− C
ω2, (15) whereC1 andC2 are parameters to be determined by initial conditions,ωp=[ω2−(γA0G2)2/4]1/2 is the oscillating an- gular frequency, andωp≈ωdue toω2(γA0G2)2/4. It is noted thatG1is much smaller thanG2 [42], which has been safely neglected in the derivation of Eq. (15).
As a matter of fact, the underdamped oscillation has been confirmed by the LLG simulations, as shown in Fig. 3(a), which presents the simulatedq(τ) curves for variousG2. For a fixed G2, the skyrmion oscillates around its equilibrium position with an attenuating amplitude. It is noted that at the equilibrium position, the driving torque is well cancelled out by the retarding torque caused by the defect, and the position
FIG. 3. The skyrmion position as a function of time (a) for variousG2forλ=0.4 and (b) for variousλforG2=0.03aunder u=0.012J/μs.
hardly depends onG2. Interestingly, the oscillation magnitude is highly related to the damping constant, which significantly affects the pinning and depinning of the skyrmion. Actually, one may define the maximum displacement of the skyrmion from its initial position as |q|max, which is approximately given by
|q|max=e−γA0G2arctan(C2/C1)2ωp
C12+C22− C
ω2. (16) The simulated |q|max for variousG2 are summarized in Fig.4(a), in which the excellent fitting of the simulated data based on Eq. (16) confirms the validity of the theory.
When a large current is applied, the AFM skyrmion will be depinned from the defect, benefiting from the strong driving torque. The critical depinning field could be theoretically estimated based on the condition that the maximum displace- ment of the skyrmion equals the radius of the potential well,
|q|max=Rp. Substituting the condition into Eq. (16), one
FIG. 4. Numerical (empty circles) and analytical (solid line) calculated for (a), (c)|q|maxand (b), (d)udepinas functions of (a), (b)G2forλ=0.2 and (c), (d)λforG2=0.01a.
054419-4
obtains the critical depinning field:
udepin=(Rp−e−γA0G2arctan(C2/C1)2ωp
×
C12+C22)ω2Dxx
γ2A0Ixy. (17)
The simulated udepin as a function of G2 are plotted in Fig.4(b), which reveals a significant dependence ofudepinon G2 in its small region. Specifically,udepin gradually increases with G2 until the large G2 region where udepin becomes saturated, similar to the depinning of the AFM domain wall in notched nanostructures [38]. Importantly, the simulated results are quite consistent with Eq. (17), revealing that the oscillation of the AFM skyrmion plays an important role in its depinning, regardless of the defect type.
Unlike the damping constant, the defect strength λ also affects the equilibrium position of the skyrmion, as revealed in Eq. (15). For example, for a smallλ, the skyrmion moves to an equilibrium position far away from its initial position to make the torque from the defect cancel out the driving torque, as shown in Fig.3(b), which presents the simulatedq(τ) curves for variousλ. In this case, the oscillation magnitude hardly depends onλ for the small damping constant, allowing one to fit the simulated |q|max by |q|max=C∗−CDxx/cλλγ where the constantC∗ represents the first term on the right side of Eq.16. The updated equation of |q|max coincides well with the simulated results, as shown in Fig.4(c), which presents the LLG-simulated |q|max as a function of λ for G2 =0.01a. Similarly, the equation ofudepincould be updated toudepin=(Rp−C∗)cλλ/IxyγA0, revealing a linear relation betweenudepin andλ. This relation has been verified by the fitting of the simulatedudepin as a function ofλ, as shown in Fig.4(d).
C. Brief discussion
In Sec.III A, the dependence of the skyrmion velocity on the internal parameters including the exchange interaction and DM interaction is predicted, based on the fact that the AFM skyrmion profiles agree with the 360◦ domain-wall formula.
In Sec.III B, the critical depinning field of the AFM skyrmion is discussed, based on Thiele’s theory, and its dependence on the damping constant and pinning strength is derived. The theory predictions are very consistent with the simulation results based on the LLG equation, confirming the validity of the theory.
On one hand, the deformation of the AFM skyrmion in shape could be induced during the motion or by the inter- action with the defect. For simplicity, the deformation [32]
is completely ignored in our derivation of the dynamics, resulting in the small deviation between the simulations and analytical calculations. It is noted that the skyrmion size significantly depends on the exchange and DM interactions which could be also changed in the defect region. Their effects on the depinning of the skyrmion have been numerically investigated in earlier numerical simulations [36], which is beyond the scope of this theory. Moreover, the defect potential could be complex when the defect size is smaller than the skyrmion size. Thus, the present theory could work well for the skyrmion with a small and stable size [51].
On the other hand, the spin current, as an example, is used to drive the motion of the AFM skyrmion in this work. How- ever, the theory could be easily transferred to other proposed schemes for driving the AFM skyrmion motion through re- calculating the effective field. More interestingly, a parabolic potential is also expected for other types of defects such as notches [38,52], and the derived formula for calculating the depinning field could also work in notched nanostructures, which deserves to be checked in future simulations and/or experiments [53].
Thus, the present theory not only strengthens the earlier conclusions, but more importantly helps one to understand the physical picture behind the numerical simulations, providing useful information for future application design.
IV. CONCLUSION
In conclusion, we have studied theoretically the spin- current-induced dynamics of the AFM skyrmion, and have confirmed the validity of the theory through a detailed com- parison between the theory and numerical simulations based on the LLG equation. The dependence of the skyrmion veloc- ity on the intrinsic physical parameters are uncovered, allow- ing a quantitative and clear understanding of the dynamics.
Moreover, the depinning field of the AFM skyrmion depend- ing on the damping constant and the defect’s pinning strength is derived theoretically, demonstrating the effect of the time- dependent oscillation of the skyrmion on the depinning. Thus, the present work helps one to understand clearly the velocity and defect depinning of the AFM skyrmion, benefiting future experiment designs and AFM spintronics applications.
ACKNOWLEDGMENTS
We sincerely appreciate the insightful discussions with X.
Zhang, H. Yuan, and L. Shen. The work was supported by the Natural Science Foundation of China (Grants No. 51971096 and No. 51721001), the Science and Technology program of Guangzhou (No. 2019050001), the Natural Science Founda- tion of Guangdong Province (Grant No. 2019A1515011028), the Science and Technology Planning Project of Guangzhou in China (Grant No. 201904010019), and Special Funds for the Cultivation of Guangdong College Students Scientific and Technological Innovation (Grant No. pdjh2020a0148).
APPENDIX: NUMERICAL SIMULATIONS OF THE ATOMISTIC SPIN MODEL
In order to check the validity of the theory, we also perform the numerical simulations of the discrete model. Here, the two-dimensional Hamiltonian of the atomistic spin model is given by
H =J
i,j
Si·Sj
−D0
i
(Si×Si+x·ˆy−Si×Si+y·ˆx)
−K0(Siz)2, (A1) where the first term is the exchange interaction with J=1 between the nearest-neighbor spins, the second term is the DM interaction, and the last term is the anisotropy energy.
Z. JINet al. PHYSICAL REVIEW B102, 054419 (2020) The dynamics of the AFM skyrmion driven by the current is
investigated by solving the LLG equation
∂Si
∂τ = −γSi×Hi+αSi×∂Si
∂τ +γuSi×(Si×p), (A2) whereαis the damping constant andHi= −μs−1∂H/∂Siis the effective field. Without loss of generality,u=0.02J/μs, K0 =0.8Jandα=0.03 are selected. Generally, we use the
fourth-order Runge-Kutta method to solve the LLG equation on a 50×50 square lattice. The position of the skyrmionqis estimated by
q=
[xn·(∂xn×∂yn)]dxdy
[n·(∂xn×∂yn)]dxdy. (A3) Then, the velocity is numerically calculated byv=dq/dτ with timeτ.
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