We report a theoretical investigation of the impact of spin-transfer torque on magnetic head-to-head domain walls of Fe and Ni_{80}Fe_{20} (Permalloy Py^{TM}) nanowires exchange coupled to a two-sublattice uniaxial antiferromagnetic substrate. Our results indicate that provided the interface exchange interaction is large enough, the domain walls pin to interface defects consisting of steps perpendicular to the easy antiferromagnetic axis, separating terraces with opposite effective interface exchange fields. We also found that the dipolar and interface energies lead to narrow V-shaped domain walls and that the spin-transfer torque effects are restricted to the domain wall. Depinning walls from the step defect at the interface require polarized spin current densities of the order of 10^{7} A/cm^{2} for both materials.

## I. INTRODUCTION

There is currently considerable interest in studies of magnetic domain walls of ferromagnetic nanosized metal rings^{1} and micrometer long ferromagnetic wires of nanometer range widths and thicknesses.^{2} Owing to the significant impact of the magnetostatic energy on the magnetic patterns of these nanosized systems, there is plenty of room to tailor the domain wall patterns in these confined geometries to suit key applications such as memory cells^{2} or magnetic logic devices.^{3}

The interest in studying magnetic domain walls (DW) in constrained geometries is largely motivated by the promise of a high-performance nonvolatile memory device, the Racetrack Memory (RM) device, with data storage in a sequence of DW confined in micrometer long flat wires.^{2}

Due to the strong shape anisotropy, long and narrow rectangular wires, made of soft ferromagnetic (FM) materials, are prone to magnetization orientation along the wire edge. Such wires may turn into possible platforms to hold sequences of DWs, provided a periodic sequence of magnetic DW pinning centers might be created along the wire.

Recent investigations of magnetic domain walls in ferromagnetic wires have, most commonly, made use of protrusions and constrictions to modify the DW patterns and set pinning barriers to DW motion.^{2,4–7} Choosing an antiferromagnetic (AFM) substrate with uniaxial anisotropy easy axis direction along the FM wire edge is an alternative strategy to produce a stable magnetic DW sequence in the FM/AFM bilayer. The magnetic stability of this bilayer benefits from the substrate uniaxial anisotropy field, the FM wire shape anisotropy field, and the interface exchange coupling effective field, all of which are oriented along the wire edge.

We have recently reported a theoretical investigation of external magnetic field effects in vicinal FM/AFM wires. We have shown that the critical field strength to release the DW sequence is of the same order of magnitude as the effective interface exchange field.^{8}

This paper reports a theoretical study of the impact of spin-polarized current on interface pinned DW. We consider a single interface defect consisting of a step perpendicular to the easy antiferromagnetic axis, separating terraces with opposite AFM sublattices.

We assume the AFM substrate has a strong uniaxial anisotropy and consider the AFM substrate frozen in the antiferromagnetic state. As seen in Fig. 1(a), at the step defect, there is a change in the direction of the substrate magnetic moments, pointing along the ± *x* − axis direction as indicated schematically by the yellow arrows. Owing to this magnetic pattern of the AFM top surface terraces, there is a corresponding pattern of the effective interface exchange field acting on the FM, which produces a pair of FM domains separated by a head-to-head domain wall.

## II. THEORETICAL MODEL

We consider ferromagnetic (FM(*L*_{x}, *L*_{y}, *t*)) nanowires exchange coupled to a two-sublattice AFM substrate, where FM stands either for Fe or Py^{TM}, and *L*_{x}, *L*_{y}, and *t*)) give the wire dimensions, where *t* is the wire thickness. The AFM substrate spins are oriented along the longitudinal wire direction, − *x* direction, from − *L*_{x}/2 to 0, and + *x* direction, from 0 to *L*_{x}/2, so that the interface exchange field favors the formation of a head-to-head magnetic DW, as shown in Fig. 1(a).

Notice that we assume that the AFM substrate is frozen in the antiferromagnetic order, and the interface exchange coupling is represented by an effective field *H*_{int}, along the *x* − axis direction.

We describe the magnetic structure using the micromagnetic theory. We consider cubic simulation cells with the edge of *d* = 2.5 nm, and the magnetic energy density is given by:

where the first term is the intrinsic exchange energy, and *A* is the exchange stiffness. The second term is the Zeeman energy, the third is the interface exchange energy, and the last is the dipolar interaction energy. *M*_{S} is the saturation magnetization, $m\u0302i$ is the direction of the magnetic moment of the *i*-th cell, and *n*_{ij} is the distance between the cells *i* and *j* in units of cell size *d*. For the Ni_{80}Fe_{20} magnetic parameters, we use saturation magnetization *M*_{S} = 0.8 × 10^{5} A/m, and exchange stiffness *A* = 1.3 × 10^{−11} J/m.^{9} For the iron parameters, we use *M*_{S} = 1.7 × 10^{5} A/m, *A* = 2.5 × 10^{−11} J/m, and the anisotropy constant *K* = 4.7 × 10^{4} J/m^{3}.^{9–11}

We use micromagnetic simulation and the model by Zhang and Li,^{12} in order to model the interaction between a uniform electric current density *J* and a spatially varying magnetization. Therefore, the equation of motion of the magnetization of the *i*-th cell is given by:^{12}

where $Mi\u20d7=MSmi\u0302$, *γ* is the gyromagnetic ratio, and *α* is the Gilbert damping parameter. The effective gyromagnetic ratio *γ*′ = *γ*/(1 + *η*), and the effective Gilbert damping parameter *α*′ = (*α* + *ξη*)/(1 + *η*), where *η* = *n*_{0}/(*M*_{S}(1 + *ξ*^{2})), *n*_{0} is the local equilibrium spin density whose direction is parallel to the magnetization, and *ξ* is the non-adiabatic spin-transfer parameter.^{12} We defined the global effective magnetic field $h\u20d7i$ by:

where $H\u20d7eff(i)=\u22121MS\u2202E\u2202m\u0302i$ is the effective field, derived from the magnetic energy density *E* in Eq. (1), and $h\u20d7J=1\gamma \u2032(1+\eta )bJmi\u0302\xd7(j\u0302e.\u2207)mi\u0302+cJ(j\u0302e.\u2207)mi\u0302$ describe the spin-polarized current effective magnetic field *h*_{J}, with contributions from the adiabatic^{13} and non-adiabatic^{12} spin-transfer torque.

According to Zhang and Li,^{12} for a typical ferromagnet (Ni, Co, Fe, and their alloys), *n*_{0}/*M*_{s} ≈ 0.01, thus we have used *η* = 0.01, *ξ* = 0.01, and the damping parameter *α* = 0.01.^{12}

We assume the current density is in the *x*-direction, $j\u20d7e=Ji\u0302$. *b*_{J} = *PJμ*_{B}/*eM*_{S}(1 + *ξ*^{2}), and *c*_{J} = *ξb*_{J}, where *e* is the electron charge, *J* is the current density, *μ*_{B} is the Bohr magneton, and *P* is the spin current polarization of the ferromagnet. For Fe and Py^{TM}, we use *P* = 0.5 and *P* = 0.7, respectively.^{13}

We use the static micromagnetic simulation to get the magnetic equilibrium configuration, $Mi\u20d7\xd7h\u20d7(i)\u22480$. For each value of the external field strength, the equilibrium configuration is found by seeking a set of directions of the moments in all cells ($m\u0302i$, *i* = 1, …, *N*), which makes the torque smaller than 10^{−26} J in any of the cells.^{14–16}

We calculate the magnetic phases starting from *J* = 0. For each current density value, the self-consistent procedure is initialized with the magnetic state corresponding to the equilibrium state of the previous value of the current density. Proceeding this way, we find the metastable equilibrium state nearest to the preceding one, as appropriate to modeling the release of the magnetic DW from the pinning center at the step edge.

## III. RESULTS AND DISCUSSIONS

We have investigated the impact of spin-polarized currents on domain walls of Fe and Py^{TM} on narrow wires exchange coupled to an antiferromagnetic substrate.

We have focused on the process of depinning the DW from interface step defects that produced a localized change of the effective interface exchange field and, as a result, favors the nucleation of domains magnetized in opposite directions, separated by a head-to-head DW.

We have considered Fe(250 nm, 25 nm, t) and Py^{TM}(250 nm, 25 nm, t) nanowires with thicknesses (t) ranging from 10 nm to 17.5 nm, and interface exchange field strengths (*H*_{int}) ranging from 150 Oe to 300 Oe.

In all cases, we have found that in the absence of spin-polarized current, the DW sits at the step defect (as seen in Fig. 1(a)), and the spin-polarized current shifts the wall position in the direction opposite do the current density vector, $J\u20d7$, (as seen in Fig. 1(b)).

In Fig. 2, we show the magnetization’ *x*-component as a function of the current density *J* for Fe(250 nm, 25 nm, *t*) nanowires, with thicknesses of *t* = 10, 12.5, 15, and 17.5 nm, for *H*_{int} = 150 Oe. As shown, for *J* = 0, the magnetization is zero. This corresponds to two domains with the magnetization in opposite direction, as shown in Fig. 3(a). Furthermore, for each value of thickness, there is a threshold current density (*J**) for which the domain wall is released from the pinning site, and the domain parallel to the current density flips out to the direction opposite to $J\u20d7$ and the magnetization saturates (*M*_{x}/*M*_{S}=−1).

In Fig. 3(a), the panel shows the magnetization profile for Fe(250 nm, 25 nm, 10 nm) nanowire at the interface plane. We selected a fraction of the wire plane near the DW center. Notice that the magnetic DW width has a non-homogeneous structure due to the dipolar interaction. It is wider at the upper side of the wire (*y* = 25 nm). Also, we can see in Fig. 3(a) that the out-of-plane magnetization component (*M*_{z}(*x*, *y*)) at the magnetic DW center, at the lower side of the wire (for *y* = 2.5 nm), is very small, *M*_{z}(0.0, 2.5 nm) = −0.04*M*_{S}, while for *y* = 25 nm is larger, *M*_{z}(0.0, 25 nm) = −0.25*M*_{S}.

As seen in Fig. 3(b), increasing the magnitude of *J*, the magnetic DW position shifts from *x* = 0 in the direction opposite to the current density vector. For example, for *J* = 33 × 10^{7} A/m^{2}, point (b) in Fig. 2, the DW center shifts to *x* = −27.5 nm. Apart from the DW shift, we also see changes in the magnetization pattern at the displaced position.

The changes are not apparent in the magnetization profile projection in the surface plane. However, there are clear changes in the magnetization component along the wire normal. As shown in Fig. 3(b), at the magnetic DW center, *M*_{z}( − 27.5 nm, 2.5 nm) = 0.4*M*_{S} and *M*_{z}( − 27.5 nm, 25 nm) = −0.24*M*_{S}. At the magnetic DW center, the tenfold increase of *M*_{z}, from *M*_{z}(0.0, 2.5 nm) = −0.04*M*_{S} to *M*_{z}( − 27.5 nm, 2.5 nm) = 0.4*M*_{S}, highlights the distortions of the magnetic DW as it moves from the step defect.

In Fig. 4, we present *h*_{J} pattern, at the interface plane, corresponding to the magnetization profile shown in Fig. 3(b), where the magnetic DW is displaced from the step defect, in *x* = 0, to − 27.5 nm. We highlight that the magnetic field (*h*_{J}) only exists in the region where there is magnetization variation, i.e., near the magnetic DW or at the ends of the wire, where its strength is negligible. Therefore, we zoomed in on the magnetic DW region from − 70 nm to 20 nm. In Fig. 4(a), the color barcodes show the magnitude, in kOe, and in Fig. 4(b), the color barcodes show the out-of-plane angle, in degrees, of *h*_{J}. In both panels of Fig. 4, the *h*_{J} strength in the dark shadow region is negligible.

As shown in Fig. 4(a), at the center of the magnetic DW, around *x* = −27.5 nm, there is an area where *h*_{J} is stronger, *h*_{J} is around 0.5 kOe. In this region, the magnetic DW is the narrowest. Therefore, the gradient of *M*_{x} is large to accomplish a variation of *π*.

In Fig. 4(b), we show that, at the magnetic DW center, the *h*_{J} field has a robust out-of-plane component from the adiabatic spin-transfer torque term. However, it has a slight negative *x*-component from the non-adiabatic spin-transfer torque term, which leads to the displacement of the magnetic DW in the opposite direction of the current density.

The energy barrier between the magnetic DW state and the uniform state is proportional to the interface exchange energy. The interface energy density may change by changes in geometric parameters, such as the nanowire dimensions (*L*_{x} and *L*_{y}) and the nanowire thickness (*t*).

In Table I, and Fig. 2, we show the values of *J** for Fe(250 nm, 25 nm, *t*) and Py^{TM}(250 nm, 25 nm, *t*) nanowires, with thicknesses of *t* = 10, 12.5, 15, and 17.5 nm, and an interface field strength of *H*_{int} = 150 Oe. It is clear that the critical current density value *J** decreases as the thickness increases. This is intuitive and well known since the FM/AFM interface exchange interaction is an interface effect and must turn wicker as the FM thickness increases. In a thicker nanowire, the effectiveness of the interface effects is weaker. Thus, a thicker nanowire requires a smaller critical value of the current density, as shown in Table I.

(L_{x}, L_{y}, t) (nm)
. | Fe (A/cm^{2})
. | Py^{TM} (A/cm^{2})
. |
---|---|---|

(250, 25, 10.0) | 34 × 10^{7} | 25 × 10^{7} |

(250, 25, 12.5) | 21 × 10^{7} | 17 × 10^{7} |

(250, 25, 15.0) | 9 × 10^{7} | 12 × 10^{7} |

(250, 25, 17.5) | 3 × 10^{7} | 7 × 10^{7} |

(L_{x}, L_{y}, t) (nm)
. | Fe (A/cm^{2})
. | Py^{TM} (A/cm^{2})
. |
---|---|---|

(250, 25, 10.0) | 34 × 10^{7} | 25 × 10^{7} |

(250, 25, 12.5) | 21 × 10^{7} | 17 × 10^{7} |

(250, 25, 15.0) | 9 × 10^{7} | 12 × 10^{7} |

(250, 25, 17.5) | 3 × 10^{7} | 7 × 10^{7} |

Notice, in particular, that Fe(250 nm, 25 nm, 15 nm) and Fe(250 nm, 25 nm, 17.5 nm) have critical current densities values of 9 × 10^{7}A/cm^{2} and 3 × 10^{7}A/cm^{2}.

On the other hand, it is intuitive to expect that, since the stability of the DW originates in the effect of the interface energy, the value of the critical current density to release the DW from the defect pinning site should increase with the interface exchange field strength. As shown in Table II, the value of the critical current density increase with the interface exchange field strength.

H_{int} (Oe)
. | Fe(250 nm, 25 nm, 10.0 nm) . | Py^{TM}(250 nm, 25 nm, 10.0 nm)
. |
---|---|---|

300 | 80 × 10^{7} A/cm^{2} | 35 × 10^{7} A/cm^{2} |

200 | 49 × 10^{7} A/cm^{2} | 25 × 10^{7} A/cm^{2} |

150 | 34 × 10^{7} A/cm^{2} | 20 × 10^{7} A/cm^{2} |

80 | 12 × 10^{7} A/cm^{2} | 8 × 10^{7} A/cm^{2} |

H_{int} (Oe)
. | Fe(250 nm, 25 nm, 10.0 nm) . | Py^{TM}(250 nm, 25 nm, 10.0 nm)
. |
---|---|---|

300 | 80 × 10^{7} A/cm^{2} | 35 × 10^{7} A/cm^{2} |

200 | 49 × 10^{7} A/cm^{2} | 25 × 10^{7} A/cm^{2} |

150 | 34 × 10^{7} A/cm^{2} | 20 × 10^{7} A/cm^{2} |

80 | 12 × 10^{7} A/cm^{2} | 8 × 10^{7} A/cm^{2} |

## IV. CONCLUSIONS

We have considered ferromagnetic (Fe ou Py^{TM}) nanowires on an antiferromagnetic substrate. An interface step defect introduces a variation in the interface exchange field pattern and is responsible for forming magnetic DW in the ferromagnetic nanowires. We investigated the release of these magnetic DW by the spin-polarized current.

We have shown that in the absence of current, for *J* = 0, the magnetic DW center sits on the step defect at *x* = 0. However, in the presence of a spin-polarized current (*J* ≠ 0), the magnetic DW shifts from the step defect in the direction opposite to the current.

Our results show, as expected that the threshold value of the current density (*J**) that releases the magnetic domain wall increases with the strength of the interface exchange field. Furthermore, for thicker nanowires, *J** is smaller.

We note that our present results should be appropriate for estimating the effects of spin-transfer torque on magnetic DW sequences, which may be stabilized in a vicinal FM/AFM bilayer wire.^{8} Furthermore, there are advantages in using the FM/AFM bilayer interface exchange coupling as the DW pinning mechanism, as compared to using notches and other geometrical wire constrictions.

In the absence of current (*J* =0), with the domain wall sitting at the step edge, as shown in Fig. 1, with the FM magnetization oriented along the direction of the effective exchange field at each terrace, the value of the pinning energy is proportional to the area of each terrace.^{8} Thus, as in a vicinal surface,^{17} the vicinal angle controls the terrace’s length (*L*_{x}/2), and one may control the value of the terrace area and the interface exchange pinning energy by choosing the length of the terraces. Furthermore, since the pinning energy is an interface effect, it is inversely proportional to the FM film thickness. Thus, one must be able to tailor the pinning energy strength reproducibly.

## ACKNOWLEDGMENTS

The authors acknowledge the financial support from CNPq, CAPES and FAPERN.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.