Tuning skyrmion Hall effect via engineering of spin-orbit interaction

TL;DR

This paper demonstrates how tuning spin-orbit interaction strength can control and suppress the skyrmion Hall effect, offering new possibilities for spintronic device optimization.

Contribution

It introduces a method to modulate the Magnus force on skyrmions through spin-orbit interaction engineering, enabling suppression of the skyrmion Hall effect.

Findings

Skyrmion Hall effect can be suppressed by tuning SOC strengths.

Emergent magnetic fields influence topological transport.

Different skyrmion types respond uniquely to SOC tuning.

Abstract

We demonstrate that the Magnus force acting on magnetic skyrmions can be efficiently tuned via modulation of the spin-orbit interaction strength. We show that the skyrmion Hall effect, which is a direct consequence of the non-vanishing Magnus force on the magnetic structure can be suppressed in certain limits. Our calculations show that the emergent magnetic fields in the presence of spin-orbit coupling (SOC) renormalize the Lorentz force on itinerant electrons and thus influence the topological transport. In particular, we show that for a N\'eel-type skyrmion and Bloch-type antiskyrmion, the skyrmion Hall effect (SkHE) can vanish by tuning appropriately the strength of Rashba and Dresselhaus SOCs, respectively. Our results open up alternative directions to explore in a bid to overcome the parasitic and undesirable SkHE for spintronic applications.

Equations

$$\hat{\mathcal{H}}=\frac{{\hat{\mathbf{p}}}^2}{2m^{\ast }}+J\mathbf{m}(\mathbf{r},t)\cdot \hat{\mathbf{\sigma }}+{\hat{\mathcal{H}}}_R+{\hat{\mathcal{H}}}_D,$$

$${\mathbf{A}}_s^{\sigma }$$

$${\mathbf{A}}_R^{\sigma }$$

$${\mathbf{A}}_D^{\sigma }$$

$${\mathbf{B}}_{\eta }^{\sigma }=\nabla \times {\mathbf{A}}_{\eta }^{\sigma }=\sigma {\mathrm{B}}_{\eta }{\mathbf{e}}_z,$$

$${\mathrm{B}}_s$$

$${\mathrm{B}}_R$$

$${\mathrm{B}}_D$$

$$⟨{\mathbf{B}}_s{⟩}_{\mathrm{av}}$$

$$⟨{\mathbf{B}}_R{⟩}_{\mathrm{av}}$$

$$⟨{\mathbf{B}}_D{⟩}_{\mathrm{av}}$$

$$\mathcal{H}=\sum_{\bm{i}}\hat{c}_{\bm{i}}^{\dagger}\big{(}\epsilon_{\bm{i}}+J{\bm{m}}_{\bm{i}}\cdot\hat{\bm{\sigma}}\big{)}\hat{c}_{\bm{i}}-\sum_{<{\bm{i}},{\bm{j}}>}\hat{c}_{\bm{i}}^{\dagger}t_{{\bm{i}}{\bm{j}}}\hat{c}_{\bm{j}},$$

$$t_{\mathbf{i}\mathbf{j}}=\{\begin{array}{rlrl}& t_0+i{t}_R{\sigma }_y+i{t}_D{\sigma }_x,& & \mathbf{j}=\mathbf{i}\pm \mathbf{x},\\ & t_0-i{t}_R{\sigma }_x-i{t}_D{\sigma }_y,& & \mathbf{j}=\mathbf{i}\pm \mathbf{y},\end{array}$$

$$I_{\mu }=\frac{e^2}{2\pi \hslash }\sum _{\mu \ne \nu }(T_{\nu \mu }{V}_{\mu }-T_{\mu \nu }{V}_{\nu }),$$

$${\theta }_{\mathrm{TH}}=\frac{V_T-V_B}{V_R-V_L}.$$

Full text

I Introduction

Over the last decade, spintronic research interest has switched towards a novel direction called spin-orbitronics that exploits the relativistic coupling of electron’s spin to its orbit to create new and intriguing effects and materials. Manchon2014 ; Kuschel2015 ; Chen2017 It turns out that the spin-orbit interaction is very crucial in effects related to the efficient conversion of charge to spin current, Sinova2015 ; Hoffmann2015 that is essential for spintronic applications. The former has been largely exploited for the creation of a novel class of topological materials such as chiral domain walls and magnetic skyrmions with enhanced thermal stability, low critical currents and smaller sizes. Therefore, spin-orbit related effects open up promising directions to create, manipulate, and detect spin currents for spintronic applications.

In magnetic materials with broken inversion symmetry, an atom with strong SOC can mediate an antisymmetric exchange interaction called the Dzyaloshinskii-Moriya iteraction (DMI), that favors the non-collinear alignments of atomic spins. Dzyaloshinsky1958 ; Moriya1960 In such materials, the competition between the DMI and other magnetic interactions notably, the exchange (which favors collinear alignment of atomic spins) is essential for the stabilization of exotic magnetic states such as helimagnets Kishine2015 and magnetic skyrmions. Robler2006 The later have been widely projected as a viable contender for information carriers in the next-generation data storage and spin logic devices due to their small spatial extent, high topological protection, and efficient current-induced manipulation that allows for robust, energy-efficient, and ultra-high density spintronic applications. Nagaosa2013 ; Fert2013 However, the integration of ferromagnetic skyrmions in such applications is hindered by the undesirable SkHE, a transverse motion to the direction of current flow. Litzius2017 ; Jiang2017

To overcome this parasitic effect, several proposals have been put forward such as, the use of antiferromagnetic skyrmions, Barker2016 ; Zhang2016a ; Gobel2017c ; Akosa2018 edge repulsion, Zhang2017 magnetic bilayer-skyrmion, Zhang2016 skyrmionium, Finazzi2013 ; Zheng2017 ; Zhang2018 ; Beg2015 ; Beg2017 ; Zhang2016c antiskyrmions, Huang2017 ; Potkina2019 and via spin current partially polarized along the direction of applied current. Gobel2019b These proposals focus on suppressing the inherent Magnus force in these systems, while little attention has been paid to understanding its source, i.e., the nature of the texture-induced emergent magnetic field. Moreover, since the stabilization of topological magnetic textures such as skyrmions, usually requires strong SOC, it is important to investigate the effect of the latter on these fictitious electromagnetic fields. Indeed, recent studies showed that SOC induces additional fictitious electric field that manifests itself as spin-motive force Kim2012 ; Tatara2013 on the itinerant electrons and gives raise to charge current Tatara2007 and chiral damping. Akosa2018b However, the effect of the SOC-induced emergent magnetic field on the conduction electrons has more or less been overlooked.

In this paper, we provide a theoretical framework that takes into account these fictitious magnetic fields and elucidate their impact on the topological transport inherent to magnetic skyrmions. We demonstrate that the SkHE, which is a direct consequence of a non-vanishing Magnus force on the magnetic structure can be efficiently tuned via modulation of the spin-orbit interaction strength. In particular, we show that for a Néel-type skyrmion and Bloch-type antiskyrmion, the SkHE can be tuned to zero via the modulation of the strength of the Rashba SOC (RSOC) and Dresselhaus (DSOC), respectively. Our results opens up alternative directions to overcome the parasitic and undesirable SkHE in ferromagnetic skyrmions.

II Theoretical Model

It is known that DSOC stabilizes Bloch-type skyrmions, Muhlbauer2009 ; Yu2010 ; Seki2012 while RSOC stabilizes Néel-type skyrmions. Kezsmarki2015 ; Kurumaji2017 However, recent realization of Bloch-type skyrmions in Rashba metals Hayami2018 ; Rowland2016 motivates us here to consider an interplay of both types of SOC in a skyrmionic system described by the Hamiltonian

[TABLE]

where $m^{*}$ is the effective mass of electrons, $\hat{\bm{p}}$ is the momentum operator, $J$ is the constant of exchange interaction between the local moments in the direction of ${\bm{m}}$ and spins of itinerant electrons described by vector of Pauli matrices $\hat{\bm{\sigma}}$. The terms $\mathcal{H}_{R}=\beta_{R}(\sigma_{y}p_{x}-\sigma_{x}p_{y})/\hbar$ and $\mathcal{H}_{D}=\beta_{D}(\sigma_{x}p_{x}-\sigma_{y}p_{y})/\hbar$ describe the RSOC and DSOC, respectively. To render our analysis trackable, we consider an isolated skyrmion (antiskyrmion) with analytical ansatz without loss of generality given in spherical coordinates as {\bm{m}}({\bm{r}},t)=\big{(}\cos\Phi\sin\theta,\sin\Phi\sin\theta,\cos\theta\big{)}, where the azimuthal and radial angles are given as $\Phi(x,y)=q\,\rm{Arg}\,(x,y)+c\pi/2$ and $\theta=\pi(3-p)/2+4\tan^{-1}(e^{r/r_{s}})$, respectively, where $r_{s}$ is the skyrmion radius. In this representation, $p=\pm 1$ is the polarity that defines the orientation of the skyrmion’s core, $q=\pm 1$ is the vorticity, i.e., $q=+1$ for skyrmions and $q=-1$ for antiskyrmions, and $c$ is the heliticity, such that $c=0$ for Néel-type and $c=\pm 1$ for Bloch-type skyrmions (antiskyrmions). Note that the topological charge $Q$ of the magnetic solitons is given by $Q=pq$. Tretiakov2007

III Analytical Results

It is well established that when itinerant electrons traverse a smooth and slowly varying magnetic texture, ${\bm{m}}({\bm{r}},t)$, there is a reorientation of their spins to follow the direction of local magnetization. This process gives raise to fictitious electromagnetic fields that act on the itinerant electrons. The emergent electrodynamics resulting from the system described by Eq. (1) is derived following the standard approach, i.e., the exchange term is diagonalized via a unitary transformation $\hat{U}={\bm{n}}\cdot\hat{\bm{\sigma}}$, where {\bm{n}}=\big{(}\cos\Phi\sin\theta/2,\sin\Phi\sin\theta/2,\cos\theta/2\big{)} in the spin-space. Tatara1994a ; Tatara1994b In the adiabatic limit, itinerant electrons in the transformed frame are subjected to a uniform ferromagnetic state and weakly coupled to the spin gauge fields with the two contributions: (i) due to the magnetic texture ${\bm{A}}_{s}^{\sigma}$, and (ii) as a result of an interplay between SOC and the magnetic texture ${\bm{A}}_{so}^{\sigma}={\bm{A}}_{R}^{\sigma}+{\bm{A}}_{D}^{\sigma}$, where ${\bm{A}}_{R(D)}^{\sigma}$ is the RSOC (DSOC)-induced spin gauge field given by Akosa2018b

[TABLE]

where $i=x,y$, and $\lambda_{R(D)}=\hbar^{2}/2m^{*}\beta_{R(D)}$ is the characteristic length scale of the RSOC (DSOC). Kim2013 While the previous studies have focused on the effect of the SOC-induced emergent electric field on the itinerant electrons (spin-motive force), here, we focus on the effect of the corresponding emergent magnetic field on the itinerant electrons (Lorentz force). In the strong exchange limit, these fields are calculated from the corresponding spin gauge fields using

[TABLE]

where $\eta=s,R,D$, and the field components are

[TABLE]

It turns out that ${\bm{B}}_{s}$ and ${\bm{B}}_{R}$ in Eq. (4) act in the opposite directions for Néel-type skyrmions as illustrated in Fig. 1. The direct consequence of this fact is that the electrons traversing an array of Néel-type skyrmions in the positive x-direction, experience two opposite fictitious magnetic fields: ${\bm{B}}_{s}$ in the positive z-direction that tends to deflect the electrons in the positive y-direction and ${\bm{B}}_{R}$ in the negative z-direction that tends to deflect the electrons in the negative y-direction. Since ${\bm{B}}_{R}$ has two free parameters $r_{s}$ and $\lambda_{R}$, by tuning them, one can realize a condition when ${\bm{B}}_{R}$ completely cancels out ${\bm{B}}_{s}$. In this case, the transverse Lorentz force acting on electrons transversing the skyrmions is completely suppressed, or equivalently the Magnus force on the magnetic structure is vanishing.

To gain more insight into the physics originating from the interplay of the different contributions to emergent magnetic fields given by Eq. (4), we consider a discrete square system of size $101\times 101a_{0}^{2}$, with the equilibrium skyrmion radius $r_{s}=12a_{0}$, where $a_{0}=0.3\mbox{ nm}$ is the lattice constant. Furthermore, we take the effective mass of electrons $m^{*}=0.4m_{0}$, where $m_{0}$ is the bare mass of electrons, and the RSOC strength $\beta_{R}=2.5\times 10^{-11}\mbox{eV}\mbox{ m}$ (equivalent to $\lambda_{R}=3.8\mbox{ nm}$). Then we calculate the corresponding magnetic fields for different vorticities and helicities. In Fig. 2, the results for Néel-type skyrmions $(q=+1)$ and antiskyrmions $(q=-1)$ are shown with the magnetization profiles given by those presented in Figs. 2(a) and (d), respectively. We see that for Néel-type skyrmions, indeed, the fictitious magnetic fields ${\bm{B}}_{s}$ [c.f. Fig. 2(b)] and ${\bm{B}}_{R}$ [c.f. Fig. 2(c)] act in opposite directions such that, it is possible to achieve a current-driven motion without SkHE by tuning the strength of the RSOC. However, in the case of a Néel antiskyrmion as shown in Figs. 2(e) and (f), even though the transversing electrons experience these fictitious magnetic fields, the effect of ${\bm{B}}_{R}$ on its trajectory cancels out by symmetry [c.f. Fig. 2(f)]. As such, only ${\bm{B}}_{s}$ influences its trajectory leading to SkHE for current-driven motion.

Similar arguments can be used to explain the characteristics of Bloch-type skyrmions and antiskyrmions as shown in Fig. 3, with the magnetization profiles as depicted in Figs. 3(a) and (d), respectively. It turns out that unlike in the Néel-type case, the SOC-induced fictitious magnetic field ${\bm{B}}_{D}$, does influence the trajectory of electrons traversing Bloch-type skyrmions, since the latter cancels out by symmetry [c.f. Fig. 3(c)]. As such the trajectory of traversing electrons are detected by ${\bm{B}}_{s}$ [c.f. Fig. 3(b)] leading to SkHE for current-driven motion. However, in the case of Bloch-type antiskyrmions, ${\bm{B}}_{s}$ [c.f. Fig. 3(e)] and ${\bm{B}}_{D}$ [c.f. Fig. 3(f)] act in the opposite directions and as a result, by tuning the strength of DSOC and/or the $r_{s}$ via material engineering, it is possible to achieve a SkHE-free current-driven motion of Bloch antiskyrmions.

Therefore, our analysis shows that, depending on the vorticity and chirality of ferromagnetic solitons, it is possible to achieve a current-driven motion of the latter without SkHE via the engineering of the spin-orbit interaction in the system. Although the heuristic analysis presented above, captures the important physics, In what follows, we provide a more rigorous argument based on the average emergent magnetic field that electrons traversing magnetic skyrmions experience. From Eq. (4), the average fictitious magnetic fields are calculated as

[TABLE]

We immediately infer from Eq. (5) that up to the linear order in SOC, the average SOC-emergent magnetic field: (i) vanishes for Néel antiskyrmions [c.f. $q=-1$, in Eq. (5b)] and Bloch skyrmions [c.f. $q=+1$, in Eq. (5c)] (ii) is finite and is opposite to ${\bm{B}}_{s}$ for Néel skyrmions [c.f. $q=+1$, in Eq. (5b)] and Bloch antiskyrmions [c.f. $q=-1$, in Eq. (5c)]. As a result, since the topological Hall effect of itinerant electrons as the traverse magnetic skyrmions is detected by these average fictitious magnetic fields, we recover the conclusions discussed in our heuristic analysis above.

Interestingly, these average additional fictitious magnetic fields are proportional to $\beta_{R,D}/r_{s}$, and since the DMI has a subtle dependence on $r_{s}$, Kiselev2011 ; Rohart2013 but is proportional to the strength of the SOC Kikuchi2016 ; Koretsune2018 , one expects that the former should be at least dependent on the strength of the DMI in these systems. This theoretical prediction provides an interesting direction to explore to tune and potentially completely overcome this undesirable SkHE for spintronic applications. An interesting extension, which is however out of the scope of the present work, would be to directly investigate this effect via micromagnetic simulations. This is achievable via for example, taking into account the effect of the spin torques induced by the fictitious magnetic fields in Eq. (4). Indeed, previous studies have incorporated the textured-induced magnetic and electric fields and shown that this gives raise to the so-called topological torque and topological damping that directly influence the mobility of skyrmions. Bisig2016 ; Akosa2017

IV Numerical Results

We corroborate our analytical predictions via numerical calculations of the THE of electrons as they traverse an isolated skyrmion in the presence of RSOC and DSOC. Our considerations are based on a two-dimensional tight-binding model on a square lattice, described by the Hamiltonian

[TABLE]

where $\epsilon_{\bm{i}}$ and $\hat{c}_{\bm{i}}^{\dagger}(\hat{c}_{\bm{i}})$ are the onsite energy and the spinor creation (annihilation) operators on site ${\bm{i}}=({i_{x},i_{y}})$, respectively. $J$ is the exchange energy that couples the spin of electrons $\hat{\bm{\sigma}}$ to local magnetization ${\bm{m}}_{\bm{i}}$, and $t_{{\bm{i}}{\bm{j}}}$ is the nearest-neighbor hopping that incorporates the spin-orbit interaction and is given by

[TABLE]

Here $t_{0}$ is the hopping in the absence of SOC, $t_{R(D)}=\beta_{R(D)}/2a_{0}$ and $a_{0}$ is the lattice constant. We note that at low band filling, there is direct correspondence between the continuous and discrete Hamiltonians given in Eq. (1) and Eq. (6), respectively for $t_{0}=\hbar^{2}/2m^{*}a_{0}^{2}$ Nikolic2005 . An isolated skyrmion of radius $r_{s}=10a_{0}$ in embedded in a ferromagnetic background to which four ferromagnetic leads are attached as depicted in Fig. 4. We employ the Landauer-Büttiker formalism Buttiker1986 to investigate the coherent charge transport in our system which we subject to a longitudinal bias voltage across the left (L) and right (R) leads and measure the transverse responds via the top (T) and bottom (B) leads. The terminal current of the $\mu$-lead is calculated as

[TABLE]

where $V_{\mu}$ is the voltage at the $\mu$-lead and $T_{\nu\mu}$ is the transmission coefficient for electrons from the $\mu$-lead to the $\nu$-lead, which is calculated via the use of the KWANT software package Groth2014 . The terminal voltages are calculated following Ref. Ndiaye2017, , from which the THE is quantified via the topological Hall angle defined as

[TABLE]

We consider the strong exchange limit with $J=5t_{0}$, and investigate the dependence of $\theta_{\rm TH}$ on the strength of RSOC ($t_{R}$) for Néel-type and DSOC ($t_{D}$) for Bloch-type skyrmions and antiskyrmions. Furthermore, to rule out any possibility that our results stems from size-effect, we perform a systematic study with different system sizes $L\times L$, for $L=101a_{0}$, $161a_{0}$, $181a_{0}$ and an optimal (to ensure smooth enough magnetization variation from system to leads Akosa2017 ) skyrmion radius of $10a_{0}$.

Our numerical results as shown in Fig. 5, confirm the physics underscored by our analytical derivations. Furthermore, the green arrow in Figs. 5 (a) and (b) shows that the value of $t_{R}$ for Néel skyrmion and $t_{D}$ for Bloch antiskyrmion at which the THE vanishes (green arrow) is independent on the system size as such, we rule out the possibilities that the observe results are artifact from size effect.

V Conclusions

Magnetic skyrmions are hugely considered as a contender for information carriers in future spintronic applications. However, the parasitic SkHE constitutes a technological challenge for the integration of the former in such applications. Several theoretical proposals, which focus on suppressing the Magnus force that gives raise to this detrimental SkHE, have been put forward. In this study, we focus on exploring the possibilities of overcoming the SkHE via tuning spin-orbit interactions that are inherent in skyrmionic systems. Starting from the emergent electrodynamics in the latter in the presence of SOC, we demonstrate that the additional SOC-induced emergent fictitious magnetic fields can be used to tune the SkHE. Our calculations show that by tuning the strength of RSOC in Néel skyrmions or DSOC in Bloch antiskyrmions, it is possible to achieve a current-driven motion without SkHE in these systems. Our findings open up promising perspective on overcoming the SkHE.

This work was supported by Grant-in-Aid for Exploratory Research (No. 16K13853) and Grant-in-Aid for Scientific Research (B) (No. 17H02929) from Japan Society for the Promotion of Science (JSPS). H. L. acknowledges support from HeNan University (Grant No. CJ3050A0240050) and National Natural Science Foundation of China (Grants No. 11804078). O. A. T. acknowledges support by the Cooperative Research Project Program at the Research Institute of Electrical Communication, Tohoku University and by Ministry of Science and Higher Education of the Russian Federation in the framework of Increase Competitiveness Program of NUST ”MISiS” (No. K2-2019-006), implemented by a governmental decree dated 16th of March 2013, N 211. C. A. A. thanks A. Abbout for useful discussions.