• Domain wall diode based on functionally graded Dzyaloshinskii–Moriya interaction

Topological spin textures have ignited a growing interest in spintronics due to their rich phenomenology,¹1. Topological Structures in Ferroic Materials, edited by J. Seidel ( Springer International Publishing, Switzerland, 2016);Topology in Magnetism, edited by J. Zang , V. Cros , and A. Hoffmann (Springer International Publishing, 2018). as well as novel potential applications. Their nanoscale size and topologically protected stability make them attractive candidates for information carriers in high-density data-storage technologies. For example, domain walls (DWs) and skyrmions in magnetic nanostripes are proposed as key elements of nonvolatile magnetic logic²2. D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005); https://doi.org/10.1126/science.1108813 S. Krause and R. Wiesendanger, Nat. Mater. 15, 493 (2016). https://doi.org/10.1038/nmat4615 and memory³3. P. Xu, K. Xia, C. Gu, L. Tang, H. Yang, and J. Li, Nat. Nanotechnol. 3, 97 (2008); https://doi.org/10.1038/nnano.2008.1 S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008); https://doi.org/10.1126/science.1145799 S. Woo, K. Litzius, B. Krüger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Kläui, and G. S. D. Beach, Nat. Mater. 15, 501 (2016). https://doi.org/10.1038/nmat4593 devices. A diode being the central element of many logical structures is of special interest. Recently, the concepts of ferroelectric domain-wall diode⁴4. J. Whyte and J. Gregg, Nat. Commun. 6, 7361 (2015). https://doi.org/10.1038/ncomms8361 and spin-wave diode⁵5. J. Lan, W. Yu, R. Wu, and J. Xiao, Phys. Rev. X 5, 041049 (2015). https://doi.org/10.1103/physrevx.5.041049 were proposed. Magnetic systems with functionally graded internal material parameters are particularly promising for this use. Recently, spatial engineering of the anisotropy⁶6. J. H. Franken, H. J. M. Swagten, and B. Koopmans, Nat. Nanotechnol. 7, 499 (2012); https://doi.org/10.1038/nnano.2012.111 S. Zhang, A. K. Petford-Long, and C. Phatak, Sci. Rep. 6, 31248 (2016); https://doi.org/10.1038/srep31248 L. Sánchez-Tejerina, E. Martínez, V. Raposo, and Ó. Alejos, AIP Adv. 8, 047302 (2018); https://doi.org/10.1063/1.4993750 C. C. I. Ang, W. Gan, and W. S. Lew, New J. Phys. 21, 043006 (2019). https://doi.org/10.1088/1367-2630/ab1171 and Dzyloshinskii–Moriya interaction (DMI)⁷7. S. A. Díaz and R. E. Troncoso, J. Phys. 28, 426005 (2016); https://doi.org/10.1088/0953-8984/28/42/426005 I.-S. Hong, S.-W. Lee, and K.-J. Lee, Curr. Appl. Phys. 17, 1576 (2017); https://doi.org/10.1016/j.cap.2017.08.024 L. Zhou, R. Qin, Y.-Q. Zheng, and Y. Wang, Front. Phys. 14, 53602 (2019); https://doi.org/10.1007/s11467-019-0897-0 R. M. Menezes, J. Mulkers, C. C. de Souza Silva, and M. V. Milošević, Phys. Rev. B 99, 104409 (2019); https://doi.org/10.1103/PhysRevB.99.104409 D. Toscano, S. Leonel, P. Coura, and F. Sato, J. Magn. Magn. Mater. 480, 171 (2019). https://doi.org/10.1016/j.jmmm.2019.02.075 profiles have been suggested as an alternative way of DW and skyrmion guidance and manipulation. The interest in these results is stimulated by the fact that systems with functionally graded material parameters can be fabricated experimentally. For instance, it has been shown that the Bloch DMI can be controlled by the chemical composition⁸8. S. V. Grigoriev, D. Chernyshov, V. A. Dyadkin, V. Dmitriev, S. V. Maleyev, E. V. Moskvin, D. Menzel, J. Schoenes, and H. Eckerlebe, Phys. Rev. Lett. 102, 037204 (2009); https://doi.org/10.1103/PhysRevLett.102.037204 S. V. Grigoriev, D. Chernyshov, V. A. Dyadkin, V. Dmitriev, E. V. Moskvin, D. Lamago, T. Wolf, D. Menzel, J. Schoenes, S. V. Maleyev, and H. Eckerlebe, Phys. Rev. B 81, 012408 (2010); https://doi.org/10.1103/PhysRevB.81.012408 K. Shibata, X. Z. Yu, T. Hara, D. Morikawa, N. Kanazawa, K. Kimoto, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat. Nanotechnol. 8, 723 (2013); https://doi.org/10.1038/nnano.2013.174 S. V. Grigoriev, N. M. Potapova, S.-A. Siegfried, V. A. Dyadkin, E. V. Moskvin, V. Dmitriev, D. Menzel, C. D. Dewhurst, D. Chernyshov, R. A. Sadykov, L. N. Fomicheva, and A. V. Tsvyashchenko, Phys. Rev. Lett. 110, 207201 (2013); https://doi.org/10.1103/PhysRevLett.110.207201 D. Morikawa, K. Shibata, N. Kanazawa, X. Z. Yu, and Y. Tokura, Phys. Rev. B 88, 024408 (2013); https://doi.org/10.1103/PhysRevB.88.024408 S.-A. Siegfried, E. V. Altynbaev, N. M. Chubova, V. Dyadkin, D. Chernyshov, E. V. Moskvin, D. Menzel, A. Heinemann, A. Schreyer, and S. V. Grigoriev, Phys. Rev. B 91, 184406 (2015); https://doi.org/10.1103/physrevb.91.184406 T. Koretsune, N. Nagaosa, and R. Arita, Sci. Rep. 5, 13302 (2015). https://doi.org/10.1038/srep13302 in chiral ferromagnetic materials. On the other hand, the Néel DMI in multilayer thin films can be tuned by the thickness of the Pt layer,⁹9. X. Ma, G. Yu, X. Li, T. Wang, D. Wu, K. S. Olsson, Z. Chu, K. An, J. Q. Xiao, K. L. Wang, and X. Li, Phys. Rev. B 94, 180408 (2016); https://doi.org/10.1103/PhysRevB.94.180408 S. Tacchi, R. E. Troncoso, M. Ahlberg, G. Gubbiotti, M. Madami, J. Akerman, and P. Landeros, Phys. Rev. Lett. 118, 147201 (2017). https://doi.org/10.1103/PhysRevLett.118.147201 the thickness of the ferromagnetic layer,¹⁰10. H. T. Nembach, J. M. Shaw, M. Weiler, E. Jué, and T. J. Silva, Nat. Phys. 11, 825 (2015); https://doi.org/10.1038/nphys3418 M. Belmeguenai, J.-P. Adam, Y. Roussigné, S. Eimer, T. Devolder, J.-V. Kim, S. M. Cherif, A. Stashkevich, and A. Thiaville, Phys. Rev. B 91, 180405 (2015); https://doi.org/10.1103/PhysRevB.91.180405 A. A. Stashkevich, M. Belmeguenai, Y. Roussigné, S. M. Cherif, M. Kostylev, M. Gabor, D. Lacour, C. Tiusan, and M. Hehn, Phys. Rev. B 91, 214409 (2015); https://doi.org/10.1103/physrevb.91.214409 J. M. Lee, C. Jang, B.-C. Min, S.-W. Lee, K.-J. Lee, and J. Chang, Nano Lett. 16, 62 (2015); https://doi.org/10.1021/acs.nanolett.5b02732 M. Kopte, U. K. Rößler, R. Schäfer, T. Kosub, A. Kákay, O. Volkov, H. Fuchs, E. Y. Vedmedenko, F. Radu, O. G. Schmidt, J. Lindner, J. Faßbender, and D. Makarov, arXiv:1706.09322 (2017). electric field,¹¹11. T. Srivastava, M. Schott, R. Juge, V. Křižáková, M. Belmeguenai, Y. Roussigné, A. Bernand-Mantel, L. Ranno, S. Pizzini, S.-M. Chérif, A. Stashkevich, S. Auffret, O. Boulle, G. Gaudin, M. Chshiev, C. Baraduc, and H. Béa, Nano Lett. 18, 4871 (2018); https://doi.org/10.1021/acs.nanolett.8b01502 H. Yang, O. Boulle, V. Cros, A. Fert, and M. Chshiev, Sci. Rep. 8, 12356 (2018). https://doi.org/10.1038/s41598-018-30063-y and by ion irradiation.¹²12. A. Balk, K.-W. Kim, D. Pierce, M. Stiles, J. Unguris, and S. Stavis, Phys. Rev. Lett. 119, 077205 (2017). https://doi.org/10.1103/PhysRevLett.119.077205

In this paper, we study the current-induced dynamics of DWs in a chiral ferromagnetic film with functionally graded DMI. We show that the gradient of DMI parameter results in the driving force for DWs similarly to the curvature gradient in wires¹³13. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, Phys. Rev. B 92, 104412 (2015). https://doi.org/10.1103/PhysRevB.92.104412 and stripes.¹⁴14. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, O. V. Pylypovskyi, D. Makarov, and Y. Gaididei, Phys. Rev. B 98, 060409 (2018). https://doi.org/10.1103/PhysRevB.98.060409 Considering the coordinate-dependent DMI parameter, we propose a general approach valid for an arbitrary profile of the DMI parameter distribution. We also show how the competition of DMI-induced driving force and current pumping can be potentially used in applications.

We consider a thin and narrow ferromagnetic stripe whose thickness and width are small enough to ensure the one-dimensional character of changes in the magnetization, and the stripe length significantly exceeds the lateral dimensions. Thus, the magnetization is described by a continuous and normalized function $\mathit{m}=\mathit{M}/M_s=\mathit{m}(x,t)$, where M_s is the saturation magnetization, x-axis is orientated along the stripe, and t denotes time. The magnetization dynamics is described by the Landau–Lifshitz–Gilbert equation with additional Zhang–Li spin-torque terms,¹⁵15. Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998); https://doi.org/10.1103/PhysRevB.57.R3213 S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004); https://doi.org/10.1103/PhysRevLett.93.127204 A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). https://doi.org/10.1209/epl/i2004-10452-6

$$\begin{array}{c}\frac{\partial \mathit{m}}{\partial t}={\omega }_0\hspace{0.17em}\mathit{m}\times \frac{\delta \mathcal{E}}{\delta \mathit{m}}+\alpha \hspace{0.17em}\mathit{m}\times \frac{\partial \mathit{m}}{\partial t}\hfill \\ +\hspace{0.17em}\mathit{m}\times \left[\mathit{m}\times \left(\mathit{u}·\nabla \right)\mathit{m}\right]+\beta \hspace{0.17em}\mathit{m}\times \left(\mathit{u}·\nabla \right)\mathit{m},\end{array}$$

(1)

where ${\omega }_0={\gamma }_{0}K/M_s$ determines the characteristic timescale of the system, with γ₀ being the gyromagnetic ratio. Here, $\mathcal{E}=E/K$ is the normalized total energy of the system, where K > 0 is the easy-axis anisotropy constant; see Fig. 1(a). The driving strength is represented by the quantity $\mathit{u}=\mathit{j}P{\mu }_{\mathrm{B}}/(|e|M_s)$, which is close to the average electron drift velocity in the presence of a current of density $\mathit{j}\left|\right|\widehat{\mathit{x}}$. Here, P is the rate of spin polarization, ${\mu }_{\mathrm{B}}$ is the Bohr magneton, and e is the electron charge. Constants α and β denote the Gilbert damping and the nonadiabatic spin-transfer parameter, respectively.

To write the energy functional $\mathcal{E}$, we consider a simple model, which takes into account only three contributions to the total magnetic energy

$$\mathcal{E}=\mathcal{S}{\int }_{-\infty }^{+\infty }\left[{\ell }^2{\mathcal{E}}_{\text{ex}}+{\mathcal{E}}_{\text{an}}+\frac{D(x)}{K}{\mathcal{E}}_{\text{DM}}\right]\mathrm{d}x,$$

(2)

where $\mathcal{S}$ is the stripe cross sectional area. The first term in (2) is the exchange energy density ${\mathcal{E}}_{\text{ex}}=\sum _{i=x,y,z}{({\partial }_i\mathit{m})}^2$. The competition between exchange and anisotropy results in the magnetic length $\ell =\sqrt{A/K}$, which determines a length scale of the system; here, A is the exchange constant and K is the easy-axis anisotropy constant. The second term in (2) corresponds to the biaxial anisotropy contribution ${\mathcal{E}}_{\text{an}}=1-m_z^2+\epsilon \hspace{0.17em}{m}_y^2$, where $\epsilon =K_p/K$, with $K_p>0$ being the easy-plane anisotropy coefficient. The easy-axis is perpendicular to the stripe plane (xy-plane), while easy-plane coincides with the xz-plane; see Fig. 1. Such kind of anisotropy is effectively induced by the magnetostatic interaction in the thin stripes, with the perpendicular easy-axis magnetocrystalline anisotropy.¹⁶16. A. Aharoni, J. Appl. Phys. 83, 3432 (1998); https://doi.org/10.1063/1.367113 D. G. Porter and M. J. Donahue, J. Appl. Phys. 95, 6729 (2004); https://doi.org/10.1063/1.1688673D. G. Porter and M. J. Donahue, Spin Dynamics in Confined Magnetic Structures III, Topics in Applied Physics Vol. 101, edited by B. Hillebrands and A. Thiaville ( Springer, Berlin, 2006). For thin and narrow stripes, the approximation of the shape anisotropy is also used for inhomogeneous magnetization states,¹⁷17. Y. Gaididei, A. Goussev, V. P. Kravchuk, O. V. Pylypovskyi, J. M. Robbins, D. Sheka, V. Slastikov, and S. Vasylkevych, J. Phys. A 50, 385401 (2017). https://doi.org/10.1088/1751-8121/aa8179 in particular, DWs.^14,1814. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, O. V. Pylypovskyi, D. Makarov, and Y. Gaididei, Phys. Rev. B 98, 060409 (2018). https://doi.org/10.1103/PhysRevB.98.06040918. A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferré, Europhys. Lett. 78, 57007 (2007). https://doi.org/10.1209/0295-5075/78/57007 The last term in (2) corresponds to the Néel DMI ${\mathcal{E}}_{\text{DMI}}^{\mathrm{N}}=m_z\mathbf{\nabla }·\mathit{m}-\mathit{m}·\mathbf{\nabla }{m}_z$, and D(x) describes the spatial profile of the DMI strength. This type of DMI is taken in the form typical for ultrathin films,^19,2019. A. Bogdanov and U. Rößler, Phys. Rev. Lett. 87, 037203 (2001). https://doi.org/10.1103/PhysRevLett.87.03720320. A. Thiaville, S. Rohart, É. Jué, V. Cros, and A. Fert, Europhys. Lett. 100, 57002 (2012). https://doi.org/10.1209/0295-5075/100/57002 bilayers,²¹21. H. Yang, A. Thiaville, S. Rohart, A. Fert, and M. Chshiev, Phys. Rev. Lett. 115, 267210 (2015). https://doi.org/10.1103/PhysRevLett.115.267210 or materials belonging to the C_nv crystallographic group.²²22. A. O. Leonov, T. L. Monchesky, N. Romming, A. Kubetzka, A. N. Bogdanov, and R. Wiesendanger, New J. Phys. 18, 065003 (2016). https://doi.org/10.1088/1367-2630/18/6/065003

Since $\left|\mathit{m}\right|=1$, it is convenient to proceed to the angular representation $\mathit{m}={\mathit{e}}_x\hspace{0.17em}\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\varphi +{\mathit{e}}_y\hspace{0.17em}\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\varphi +{\mathit{e}}_z\hspace{0.17em}\text{cos}\hspace{0.17em}\theta$, where $\theta =\theta (x)$ and $\varphi =\varphi (x)$ are the magnetic angles. In terms of angular parametrization, the energy density in (2) has the following form:

$$\begin{array}{c}\mathcal{E}={\left({\theta }^{\prime }\right)}^2+{\left({\varphi }^{\prime }\right)}^2{\text{sin}}^2\theta +{\text{sin}}^2\theta \left(1+\epsilon \hspace{0.17em}{\text{sin}}^2\varphi \right)\hfill \\ +\hspace{0.17em}d\left(\theta \prime \hspace{0.17em}\text{cos}\hspace{0.17em}\varphi -\frac{\varphi \prime }{2}\text{sin}\hspace{0.17em}2\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\varphi \right).\hfill \end{array}$$

(3)

Here, and below, prime denotes the derivative with respect to the dimensionless coordinate $\xi =x/\ell$, and $d(\xi )=D/\sqrt{AK}$ is a dimensionless DMI parameter.

Let us first analyze the static magnetization distribution determined by the minimum of the energy (2). Minimization of (2) with associated energy density (3) with respect to $\varphi$ results in a solution²³23. Here, we consider the spatial distribution of the DMI strength without the change of sign, i.e., $d\ge 0$. $\text{cos}\hspace{0.17em}\varphi =\text{cos}\hspace{0.17em}{\varphi }_0=\mathcal{C}=\pm 1$. Here, $\mathcal{C}=\pm 1$ is a helicity parameter, which determines the orientation of the longitudinal magnetization component m_x. This means that vectors m lie within the xz-plane. The corresponding function θ is determined by a driven pendulum equation (see the supplementary material),

$$\theta ″-\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\theta =-\frac{d\prime }{2}\text{cos}\hspace{0.17em}{\varphi }_0.$$

(4)

Equation (4) is analogous to one, which determines the DW structure in flat curved wires¹³13. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, Phys. Rev. B 92, 104412 (2015). https://doi.org/10.1103/PhysRevB.92.104412 and stripes,¹⁴14. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, O. V. Pylypovskyi, D. Makarov, and Y. Gaididei, Phys. Rev. B 98, 060409 (2018). https://doi.org/10.1103/PhysRevB.98.060409 where curvature results in a coordinate-dependent effective DMI.

For the case $d\prime \equiv 0$, Eq. (4) has a well-known DW solution $\text{cos}\hspace{0.17em}\theta =-p\hspace{0.17em}\tanh [(\xi -q)/\Delta ]$, where q is the DW position, $p=\pm 1$ being the topological charge ($p=+1$: kink, p = −1: antikink), and Δ is the DW width. Here, q and Δ are the dimensionless quantities measured in units of $\ell$. For the case $d\prime \ne 0$, an additional driving force appears similarly to the case discussed in Refs. 1313. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, Phys. Rev. B 92, 104412 (2015). https://doi.org/10.1103/PhysRevB.92.104412 and 1414. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, O. V. Pylypovskyi, D. Makarov, and Y. Gaididei, Phys. Rev. B 98, 060409 (2018). https://doi.org/10.1103/PhysRevB.98.060409. In the following, we consider a case of the spatial distribution of the DMI strength, with $d\prime (\pm \infty )=0$, which allows the boundary conditions $\text{cos}\hspace{0.17em}\theta (\pm \infty )=\mp p$. We restrict ourselves to the case $d<4/\pi$ and consider $d\prime$ as a small perturbation, which does not significantly modify the profile of the DW and its width Δ. Therefore, to analyze the DW properties, we use the collective variable approach based on the q–$\Phi$ model,^24,2524. J. C. Slonczewski, AIP Conference Proceedings 5, 170 (1972). https://doi.org/10.1063/1.369941625. A. P. Malozemoff and J. C. Slonzewski, Magnetic Domain Walls in Bubble Materials ( Academic Press, New York, 1979).

$$\text{cos}\hspace{0.17em}\theta =-p\hspace{0.17em}\tanh \frac{\xi -q(t)}{\Delta },\hspace{1em}\varphi =\Phi (t).$$

(5)

Here, $\{q,\Phi \}$ are the time-dependent conjugat collective variables, which determine the DW position and phase, respectively. The DW width Δ is assumed to be a slaved variable,¹⁶16. A. Aharoni, J. Appl. Phys. 83, 3432 (1998); https://doi.org/10.1063/1.367113 D. G. Porter and M. J. Donahue, J. Appl. Phys. 95, 6729 (2004); https://doi.org/10.1063/1.1688673D. G. Porter and M. J. Donahue, Spin Dynamics in Confined Magnetic Structures III, Topics in Applied Physics Vol. 101, edited by B. Hillebrands and A. Thiaville ( Springer, Berlin, 2006). i.e., $\Delta (t)=\Delta [\Phi (t),q(t)]$.

Substituting the Ansatz (5) into (3) and performing integration over the ξ coordinate, we obtain the energy of a DW in the stripe in the form

$$\frac{\mathcal{E}}{2\mathcal{S}\ell }=\frac{1}{\Delta }+\Delta \left(1+\epsilon \hspace{0.17em}{\text{sin}}^2\Phi \right)+p\frac{d(q)}{2}\pi \hspace{0.17em}\text{cos}\hspace{0.17em}\Phi ,$$

(6)

where the condition $\Delta \hspace{0.17em}d\prime \ll 1$ was imposed when integrating (2). The structure of the energy (6) has similar form as DW energy in stripes with constant DMI strength^26,2726. O. A. Tretiakov and A. Abanov, Phys. Rev. Lett. 105, 157201 (2010). https://doi.org/10.1103/PhysRevLett.105.15720127. O. Boulle, S. Rohart, L. D. Buda-Prejbeanu, E. Jué, I. M. Miron, S. Pizzini, J. Vogel, G. Gaudin, and A. Thiaville, Phys. Rev. Lett. 111, 217203 (2013). https://doi.org/10.1103/PhysRevLett.111.217203 and in a curved biaxial stripe.¹⁴14. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, O. V. Pylypovskyi, D. Makarov, and Y. Gaididei, Phys. Rev. B 98, 060409 (2018). https://doi.org/10.1103/PhysRevB.98.060409 The first two terms on the right-hand side in (6) determine the competition of the isotropic exchange and anisotropy contributions, while the third term originates from the DMI and demonstrates the coupling between the DMI strength d, DW topological charge p, and helicity $\mathcal{C}$; i.e., DMI energy is minimized when $\mathcal{C}=-\text{sgn}(pd)$. The coupling between DMI and DW topology determined by p and $\mathcal{C}$ was previously indicated in Ref. 2828. M. Heide, G. Bihlmayer, and S. Blügel, Phys. Rev. B 78, 140403 (2008). https://doi.org/10.1103/PhysRevB.78.140403. In the following, a DW, which corresponds to the global minimum of the energy (6) in the parametric space $\{q,\Phi ,\Delta \}$, is called stable. Under the condition $\left|d\right|/\epsilon <4/\pi$, energy (6) also has a local minimum, which corresponds to a DW with the opposite helicity $-\mathcal{C}$. In the following, this DW is called metastable.

In terms of collective variables, the equations of motion (1) take the form (see the supplementary material)

$$\begin{array}{c}\frac{\alpha }{\Delta }\dot{q}+p\dot{\Phi }=-p\frac{\pi }{2}\frac{\partial d(q)}{\partial q}\text{cos}\hspace{0.17em}\Phi +\frac{\beta }{\Delta }\tilde{u},\\ p\dot{q}-\alpha \Delta \dot{\Phi }=-p\frac{\pi }{2}d(q)\hspace{0.17em}\text{sin}\hspace{0.17em}\Phi +\epsilon \Delta \hspace{0.17em}\text{sin}\hspace{0.17em}2\Phi +p\tilde{u},\hfill \end{array}$$

(7)

where overdots indicate the derivative with respect to the dimensionless time $\tau =t{\omega }_0$, and $\tilde{u}=u/({\omega }_0\ell )$ is a dimensionless current. The DW width is $\Delta (\tau )=\Delta [\Phi (\tau )]=1/\sqrt{1+\epsilon \hspace{0.17em}{\text{sin}}^2\Phi }$. The behavior of the DW width is discussed in detail in the supplementary material.

For the case $\tilde{u}=0$, Eq. (7) coincide with the equations of motion for DW in a curved biaxial stripe presented in Ref. 1414. K. V. Yershov, V. P. Kravchuk, D. D. Sheka, O. V. Pylypovskyi, D. Makarov, and Y. Gaididei, Phys. Rev. B 98, 060409 (2018). https://doi.org/10.1103/PhysRevB.98.060409, with the curvature gradient replaced by the gradient of the DMI strength. The DMI-induced driving force can suppress the action of the pumping by the spin-polarized current or can reinforce it; see Fig. 2. In other words, the metastable (stable) DW has to overcome the energy barrier when it enters the region with larger (smaller) $\left|d\right|$. If the applied current is small enough, then such a DW is pinned in position $[d(q_{\text{pin}})-d(q_0)]/[q_{\text{pin}}-q_0]=2p\hspace{0.17em}\mathcal{C}\beta \tilde{u}/\pi$. For small currents, the phase of the pinned DW does not significantly deviate from its equilibrium value: ${\Phi }_{\text{pin}}\approx \{0,\pi \}$. The average DW velocity as a function of current for the DMI profile $d=d_0[\tanh (\xi /w)+1]/2$ with amplitude d₀ and width w is presented in Fig. 2(a); also see the supplementary material movies for the corresponding DW dynamics. Zero averaged velocity²⁹29. The velocity of DW is calculated as $\overline{V}=T^{-1}{\int }_0^T\dot{q}(\tau )\mathrm{d}\tau$, where $\dot{q}(\tau )$ is extracted from numerical simulations and $T\gg 1$ is a time of simulation. corresponds to the case of pinning.

The behavior of DW velocity presented in Fig. 2(a) demonstrates that functionally graded DMI allows filtering of DWs of a certain type; i.e., we have built a DW diode in a planar nanostripe.³⁰30. The principle of operation of our diode is based on the creation of a potential barrier for DWs of a certain helicity. This makes it similar to the conventional semiconductor diodes. Depinning of DWs takes place when current $\tilde{u}$ exceeds some critical value ${\tilde{u}}_c\approx \pi \hspace{0.17em}{\text{max}}_q[d\prime (q)]/(2\beta )$; see Fig. 2(d).

Next, we study the linear dynamics of the DW in the vicinity of the DW pinning position. With this purpose, we introduce small deviations as $q(\tau )=q_{\text{pin}}+\tilde{q}(\tau )$ and $\Phi (\tau )={\Phi }_{\text{pin}}+\tilde{\Phi }(\tau )$. The equations of motion (7) linearized with respect to the deviations read as $(1+{\alpha }^2)\dot{\mathit{\psi }}=\widehat{\mathcal{M}}\mathit{\psi }$, where $\mathit{\psi }={(\tilde{q},\tilde{\Phi })}^{\mathrm{T}}$ and matrix $\widehat{\mathcal{M}}=\widehat{\mathcal{M}}(\tilde{u})$ depends on current (see the supplementary material). For the case of low damping and low current, we have decaying oscillations with frequency

$$\Omega \approx \frac{\pi }{2}\sqrt{d″(q_{\text{pin}})\left[p\hspace{0.17em}\mathcal{C}\epsilon \frac{4}{\pi }-d\left(q_{\text{pin}}\right)\right]}.$$

(8)

The frequency Ω as a function of the applied current is plotted in Figs. 2(b) and 2(d). Depinning of the DW takes place for cases (i) $d″(q_{\text{pin}})=0$ [see Fig. 2(b)] or (ii) $|d(q_{\text{pin}})|/\epsilon >4/\pi$. In the latter case, the metastable DW experiences the phase flip $\Phi \to \Phi +\pi$ and transforms into the stable DW. The limiting case of $d=\text{const}$ does not produce any pinning due to the zero gradient of the DMI strength.

Now, we consider the case with zero gradient of the DMI strength, i.e. $d=d_0=\text{const}$. For this case, Eqs. (7) have a solution for the traveling-wave regime, with $q=V\tau$ and $\Phi =\text{const}$ (the case $d_0=0$ is discussed in Ref. 1515. Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998); https://doi.org/10.1103/PhysRevB.57.R3213 S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004); https://doi.org/10.1103/PhysRevLett.93.127204 A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). https://doi.org/10.1209/epl/i2004-10452-6). The corresponding DW velocity and phase (in the small current approximation) are

$$V=\frac{\beta }{\alpha }\tilde{u},\hspace{1em}\Phi \approx {\Phi }_0+\frac{2p}{4\epsilon -p\hspace{0.17em}\mathcal{C}\pi d_0}\left(\frac{\beta -\alpha }{\alpha }\right)\tilde{u}.$$

(9)

It is necessary to mention that the DW velocity (9) is independent of the DMI parameter and coincides with the DW velocity reported in Ref. 1515. Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998); https://doi.org/10.1103/PhysRevB.57.R3213 S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004); https://doi.org/10.1103/PhysRevLett.93.127204 A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). https://doi.org/10.1209/epl/i2004-10452-6; see Fig. 2(a). The traveling-wave solution (9) exists for the currents $\tilde{u}<\left|{\tilde{u}}_{\mathrm{W}}\right|$, where

$${\tilde{u}}_{\mathrm{W}}\approx {\tilde{u}}_{\mathrm{W}}^0\pm \frac{\alpha }{|\alpha -\beta |}\left(\frac{\pi }{2\sqrt{2}}\right)d_0,\hspace{1em}{\tilde{u}}_{\mathrm{W}}^0=\frac{\alpha }{|\alpha -\beta |}\epsilon .$$

(10)

Here, “+” sign corresponds to the stable DW, while “−” sign corresponds to the metastable DW, and ${\tilde{u}}_{\mathrm{W}}^0$ is the Walker current for the DMI-free biaxial ferromagnetic system.¹⁵15. Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998); https://doi.org/10.1103/PhysRevB.57.R3213 S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004); https://doi.org/10.1103/PhysRevLett.93.127204 A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005). https://doi.org/10.1209/epl/i2004-10452-6 Estimation (10) is obtained under the assumption $d_0\ll \epsilon \ll 1$. From (10), it follows that the DMI results in the shift of the Walker current in the biaxial stripe similarly to the case of field-driven DW.³¹31. V. P. Kravchuk, J. Magn. Magn. Mater. 367, 9 (2014). https://doi.org/10.1016/j.jmmm.2014.04.073 The value of the Walker current for metastable DW is smaller as compared to that of the stable one and it defines the current of the DW phase flip; i.e., for ${\tilde{u}}_{\mathrm{W}}^{\mathrm{m}\mathrm{s}\mathrm{t}}<\tilde{u}<{\tilde{u}}_{\mathrm{W}}$, we have traveling-wave motion with a single flip of the phase with $\Phi \to \Phi +\pi$ [see Figs. 3(b) and 3(c)]. One should note that for the case $\beta =\alpha$, both DWs move in a traveling-wave regime without any flip of the phase $\Phi$, i.e., ${\tilde{u}}_{\mathrm{W}}(\beta =\alpha )\to \infty$. The average DW velocity as a function of current is presented in Fig. 3.

Let us estimate the effective mass of the DW.³²32. W. Döring, Z. Naturforsch. 3, 373 (1948). https://doi.org/10.1515/zna-1948-0701 To this end, we consider a no driving case (u = 0) with vanishing damping. In this case, a small deviation $\phi =\Phi -{\Phi }_0$ of the DW phase from its equilibrium value results in the traveling-wave DW motion with the velocity $V\approx [2p\epsilon -\mathcal{C}\pi d_0/2]\phi$. Using the latter relation and energy expression (6), one can estimate the energy of the moving DW as $\mathcal{E}/(\mathcal{S}\ell )\approx {\mathcal{E}}_0/(\mathcal{S}\ell )+\mu V^2/2$, where ${\mathcal{E}}_0$ is the energy of a stationary DW, and the quantity $\mu =4/(4\epsilon -p\hspace{0.17em}\mathcal{C}\pi d_0)$ can be interpreted as the effective mass of the DW. For the DMI-free case ($d_0=0$), the effective mass μ coincides with the Döring mass.³²32. W. Döring, Z. Naturforsch. 3, 373 (1948). https://doi.org/10.1515/zna-1948-0701 The DMI results in the modification of the DW mass; i.e., for stable (metastable) DW, the mass μ decreases (increases) with the DMI strength.

We have demonstrated that the presence of the biaxial anisotropy ($\epsilon >0$) and DMI ($\left|d\right|/\epsilon <4/\pi$) allows the existence of DWs with different combinations of topological charge and helicity ($p\hspace{0.17em}\mathcal{C}=\pm 1$). One of these DWs becomes energetically preferable (stable DW); i.e., it minimizes the DMI energy ${\mathcal{E}}_{\text{DM}}\propto p\hspace{0.17em}\mathcal{C}d$. By engineering the profile of the DMI strength, the stable DW will move to the area with a larger DMI strength and will be pinned for the opposite direction, while behavior for the metastable DW is vice versa. This effect can be used for the fabrication of a DW diode, traps, and ratchets in a planar stripe. These can be potentially used for the development of logic devices. We show that the gradient of functionally graded DMI in magnetic systems results in the driving force for DWs. The competition between the DMI driving force and pumping by the current determines the behavior of DW dynamics. The intrinsic DMI results in the shift of the Walker current (10), which allows us to increase the maximal velocity of traveling-wave motion for the DW; DMI modifies the DW Döring mass μ. This shows that functionally graded materials open new possibilities in the manipulation of DWs. We expect that similar pinning effects can appear for magnetic skyrmions as well.