Current-Driven Dynamics of Magnetic Hopfions

X. S. Wang; A. Qaiumzadeh; A. Brataas

arXiv:1905.09154·cond-mat.mes-hall·November 12, 2020

Current-Driven Dynamics of Magnetic Hopfions

X. S. Wang, A. Qaiumzadeh, A. Brataas

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TL;DR

This paper explores the excitation and current-driven manipulation of three-dimensional magnetic hopfions, revealing their potential as stable information carriers due to their unique topological properties and absence of Hall effects.

Contribution

It demonstrates the existence of metastable Bloch- and Ne9el-type hopfions in ferromagnets and analyzes their dynamics under spin-transfer and spin-Hall torques.

Findings

01

Ne9el-type hopfions can be driven along the current direction by STT and SHT.

02

Bloch-type hopfions can be moved transverse or parallel to current depending on the torque.

03

Hopfions lack a gyrovector, eliminating Hall effects in their motion.

Abstract

Topological magnetic textures have attracted considerable interest since they exhibit new properties and might be useful in information technology. Magnetic hopfions are three-dimensional (3D) spatial variations in the magnetization with a non-trivial Hopf index. We find that in ferromagnetic materials, two types of hopfions, Bloch-type and N\'{e}el-type hopfions, can be excited as metastable states in the presence of bulk and interfacial Dzyaloshinskii-Moriya interactions, respectively. We further investigate how hopfions can be driven by currents via spin-transfer torques (STTs) and spin-Hall torques (SHTs). Distinct from 2D ferromagnetic skyrmions, hopfions have a vanishing gyrovector. Consequently, there are no undesirable Hall effects. N\'{e}el-type hopfions move along the current direction via both STT and SHT, while Bloch-type hopfions can be moved either transverse to the…

Equations57

\mathcal{F}=\int_{V}A_{ex}\bigg{[}\left|\nabla\mathbf{m}\right|^{2}+\mathscr{D}\left(\mathbf{m},\frac{\partial\mathbf{m}}{\partial x_{i}}\right)+K_{b}(1-m_{z}^{2})\\ +BM_{s}(1-m_{z})\bigg{]}\mathrm{d}V+\int_{z=\pm d/2}K_{s}(1-m_{z}^{2})\mathrm{d}S+E_{d},

\mathcal{F}=\int_{V}A_{ex}\bigg{[}\left|\nabla\mathbf{m}\right|^{2}+\mathscr{D}\left(\mathbf{m},\frac{\partial\mathbf{m}}{\partial x_{i}}\right)+K_{b}(1-m_{z}^{2})\\ +BM_{s}(1-m_{z})\bigg{]}\mathrm{d}V+\int_{z=\pm d/2}K_{s}(1-m_{z}^{2})\mathrm{d}S+E_{d},

H = \frac{1}{( 4 π ) ^{2}} \int_{V} F \cdot A d V,

H = \frac{1}{( 4 π ) ^{2}} \int_{V} F \cdot A d V,

G_{z}=\int_{V}F_{z}r\mathrm{d}r\mathrm{d}\phi\mathrm{d}z=-2n\pi\int_{-d/2}^{d/2}\left(\cos\Theta\big{|}^{r=\infty}_{r=0}\right)\mathrm{d}z.

G_{z}=\int_{V}F_{z}r\mathrm{d}r\mathrm{d}\phi\mathrm{d}z=-2n\pi\int_{-d/2}^{d/2}\left(\cos\Theta\big{|}^{r=\infty}_{r=0}\right)\mathrm{d}z.

m_{x} = \frac{4 r ^{'} [ 2 z ^{'} sin ϕ + cos ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{y} = \frac{4 r ^{'} [ - 2 z ^{'} cos ϕ + sin ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{z} = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

m_{x} = \frac{4 r ^{'} [ 2 z ^{'} sin ϕ + cos ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{y} = \frac{4 r ^{'} [ - 2 z ^{'} cos ϕ + sin ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{z} = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

\frac{γ}{M _{s}} T + G \times (v - u) - \tensor D \cdot (α v - β u) = 0,

\frac{γ}{M _{s}} T + G \times (v - u) - \tensor D \cdot (α v - β u) = 0,

τ = \frac{γ ℏ}{e M _{s} d} θ_{SH} m \times [m \times (\hat{J} \times \hat{z})],

τ = \frac{γ ℏ}{e M _{s} d} θ_{SH} m \times [m \times (\hat{J} \times \hat{z})],

H = \frac{1}{( 4 π ) ^{2}} \int F \cdot A d V,

H = \frac{1}{( 4 π ) ^{2}} \int F \cdot A d V,

H^{'} = \frac{1}{( 4 π ) ^{2}} \int F \cdot A^{'} d V = H + \frac{1}{( 4 π ) ^{2}} \int F \cdot \nabla φ d V .

H^{'} = \frac{1}{( 4 π ) ^{2}} \int F \cdot A^{'} d V = H + \frac{1}{( 4 π ) ^{2}} \int F \cdot \nabla φ d V .

\int F \cdot \nabla φ d V = \int \nabla \cdot (φ F) d V - \int φ \nabla \cdot F d V = \oint φ F \cdot d S - 0 = \oint φ F \cdot d S,

\int F \cdot \nabla φ d V = \int \nabla \cdot (φ F) d V - \int φ \nabla \cdot F d V = \oint φ F \cdot d S - 0 = \oint φ F \cdot d S,

F_{r} = m \cdot (\frac{\partial m}{r \partial ϕ} \times \frac{\partial m}{\partial z}) = - n \frac{sin Θ}{r} \frac{\partial Θ}{\partial z},

F_{r} = m \cdot (\frac{\partial m}{r \partial ϕ} \times \frac{\partial m}{\partial z}) = - n \frac{sin Θ}{r} \frac{\partial Θ}{\partial z},

F_{ϕ} = m \cdot (\frac{\partial m}{\partial z} \times \frac{\partial m}{\partial r}) = sin Θ (\frac{\partial Θ}{\partial z} \frac{\partial ΔΦ}{\partial r} - \frac{\partial Θ}{\partial r} \frac{\partial ΔΦ}{\partial z}),

F_{z} = m \cdot (\frac{\partial m}{\partial z} \times \frac{\partial m}{r \partial ϕ}) = n \frac{sin Θ}{r} \frac{\partial Θ}{\partial r} .

A_{r} = - (1 + cos Θ) \frac{\partial ΔΦ}{\partial r},

A_{r} = - (1 + cos Θ) \frac{\partial ΔΦ}{\partial r},

A_{ϕ} = \frac{n}{r} (1 - cos Θ),

A_{z} = - (1 + cos Θ) \frac{\partial ΔΦ}{\partial z} .

H = \frac{1}{( 4 π ) ^{2}} \int F \cdot A d V = \frac{n}{4 π} \int_{- \infty}^{+ \infty} \int_{0}^{+ \infty} sin Θ (\frac{\partial Θ}{\partial z} \frac{\partial ΔΦ}{\partial r} - \frac{\partial Θ}{\partial r} \frac{\partial ΔΦ}{\partial z}) d r d z .

H = \frac{1}{( 4 π ) ^{2}} \int F \cdot A d V = \frac{n}{4 π} \int_{- \infty}^{+ \infty} \int_{0}^{+ \infty} sin Θ (\frac{\partial Θ}{\partial z} \frac{\partial ΔΦ}{\partial r} - \frac{\partial Θ}{\partial r} \frac{\partial ΔΦ}{\partial z}) d r d z .

F (x) = i \sum (\nabla\nabla - \nabla^{2}) g (x, x_{i}) c_{i} .

F (x) = i \sum (\nabla\nabla - \nabla^{2}) g (x, x_{i}) c_{i} .

A (x) = - i \sum (c_{i} \times \nabla) g (x, x_{i}),

A (x) = - i \sum (c_{i} \times \nabla) g (x, x_{i}),

m_{x} = \frac{4 [ 2 x z - y ( x ^{2} + y ^{2} + z ^{2} - 1 ) ]}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{x} = \frac{4 [ 2 x z - y ( x ^{2} + y ^{2} + z ^{2} - 1 ) ]}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{y} = \frac{4 [ 2 y z + x ( x ^{2} + y ^{2} + z ^{2} - 1 ) ]}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{z} = 1 - \frac{8 ( x ^{2} + y ^{2} )}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{x} = \frac{4 [ - 2 x z - y ( x ^{2} + y ^{2} + z ^{2} - 1 ) ]}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{x} = \frac{4 [ - 2 x z - y ( x ^{2} + y ^{2} + z ^{2} - 1 ) ]}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{y} = \frac{4 [ - 2 y z + x ( x ^{2} + y ^{2} + z ^{2} - 1 ) ]}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}},

m_{z} = 1 - \frac{8 ( x ^{2} + y ^{2} )}{( 1 + x ^{2} + y ^{2} + z ^{2} ) ^{2}} .

m_{x} = \frac{4 r [ - 2 cos ϕ z - sin ϕ ( r ^{2} + z ^{2} - 1 ) ]}{( 1 + r ^{2} + z ^{2} ) ^{2}},

m_{x} = \frac{4 r [ - 2 cos ϕ z - sin ϕ ( r ^{2} + z ^{2} - 1 ) ]}{( 1 + r ^{2} + z ^{2} ) ^{2}},

m_{y} = \frac{4 r [ - 2 sin ϕ z + cos ϕ ( r ^{2} + z ^{2} - 1 ) ]}{( 1 + r ^{2} + z ^{2} ) ^{2}},

m_{z} = 1 - \frac{8 r ^{2}}{( 1 + r ^{2} + z ^{2} ) ^{2}} .

r \to r^{'} = \frac{e ^{r / w_{R}} - 1}{e ^{R / w_{R}} - 1}, z \to z^{'} = \frac{∣ z ∣}{z} \frac{e ^{∣ z ∣/ w_{h}} - 1}{e ^{h / w_{h}} - 1},

r \to r^{'} = \frac{e ^{r / w_{R}} - 1}{e ^{R / w_{R}} - 1}, z \to z^{'} = \frac{∣ z ∣}{z} \frac{e ^{∣ z ∣/ w_{h}} - 1}{e ^{h / w_{h}} - 1},

r \to r^{'} = (\frac{e ^{r / w_{R}} - 1}{e ^{R / w_{R}} - 1})^{- 1} = \frac{e ^{R / w_{R}} - 1}{e ^{r / w_{R}} - 1},

r \to r^{'} = (\frac{e ^{r / w_{R}} - 1}{e ^{R / w_{R}} - 1})^{- 1} = \frac{e ^{R / w_{R}} - 1}{e ^{r / w_{R}} - 1},

cos Θ = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

cos Θ = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

Φ = ϕ + arctan (- \frac{1 + r ^{'} ^{2} + z ^{'} ^{2}}{2 z ^{'}}) .

m_{x} = \frac{4 r ^{'} [ - 2 z ^{'} \frac{x}{x ^{2} + y ^{2}} - \frac{y}{x ^{2} + y ^{2}} ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

m_{x} = \frac{4 r ^{'} [ - 2 z ^{'} \frac{x}{x ^{2} + y ^{2}} - \frac{y}{x ^{2} + y ^{2}} ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

m_{y} = \frac{4 r ^{'} [ - 2 z ^{'} \frac{y}{x ^{2} + y ^{2}} + \frac{x}{x ^{2} + y ^{2}} ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

m_{z} = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}},

m_{x} = \frac{4 r ^{'} [ 2 z ^{'} sin ϕ + cos ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{y} = \frac{4 r ^{'} [ - 2 z ^{'} cos ϕ + sin ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{z} = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}} .

m_{x} = \frac{4 r ^{'} [ 2 z ^{'} sin ϕ + cos ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{y} = \frac{4 r ^{'} [ - 2 z ^{'} cos ϕ + sin ϕ ( r ^{'} ^{2} + z ^{'} ^{2} - 1 ) ]}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}}, m_{z} = 1 - \frac{8 r ^{'} ^{2}}{( 1 + r ^{'} ^{2} + z ^{'} ^{2} ) ^{2}} .

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Full text

Current-Driven Dynamics of Magnetic Hopfions

X. S. Wang

A. Qaiumzadeh

A. Brataas

Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

Abstract

Topological magnetic textures have attracted considerable interest since they exhibit new properties and might be useful in information technology. Magnetic hopfions are three-dimensional (3D) spatial variations in the magnetization with a non-trivial Hopf index. We find that in ferromagnetic materials, two types of hopfions, Bloch-type and Néel-type hopfions, can be excited as metastable states in the presence of bulk and interfacial Dzyaloshinskii-Moriya interactions, respectively. We further investigate how hopfions can be driven by currents via spin-transfer torques (STTs) and spin-Hall torques (SHTs). Distinct from 2D ferromagnetic skyrmions, hopfions have a vanishing gyrovector. Consequently, there are no undesirable Hall effects. Néel-type hopfions move along the current direction via both STT and SHE, while Bloch-type hopfions move either transverse to the current direction via SHT or parallel to the current direction via STT. Our findings open the door to utilizing hopfions as information carriers.

Topological solitons are of fundamental interest in nonlinear field theories. Additionally, their magnetic realizations are promising candidates as information carriers in the next generation of data storage and processing devices soliton1 ; book . Low-dimensional topological soliton-like textures in ferromagnetic (FM) and antiferromagnetic (AFM) materials, such as 1D magnetic domain walls DW1 ; DW2 ; DW3 ; DW4 , 2D magnetic vortices VT1 ; VT2 , and 2D magnetic skyrmions SK1 ; SK2 ; SK3 ; SK4 ; SK5 ; SKAQ1 ; SKAQ2 ; SK6 ; SK7 , have been extensively studied in recent years.

The existence of 3D topological solitons with string-like properties has been proposed by Ludvig D. Faddeev Faddeev as a limit of the Skyrme model Skyrme . These 3D topological solitons are known as Faddeev-Hopf knots FHK or hopfions, which are classified by a topological charge called the Hopf index Hp1 . Hopfions have been discussed in many physical systems, such as gauge theories Faddeev ; Hp2 , cosmic strings Hp3 , ferromagnets VR (as a special case of dynamical vortex rings), low-temperature bosonic systems Egor1 ; Egor2 ; Hp4 , fluids Hp5 , and liquid crystals Hp6 ; Hp7 ; news . Recently, stable magnetic hopfions were numerically predicted in finite-size noncentrosymmetric FM systems with Dzyaloshinskii-Moriya interaction (DMI) DMI1 ; DMI2 and interfacial perpendicular magnetic anisotropy (PMA) magHp1 ; magHp2 ; magHp3 or higher-order exchange interaction Bluegel2019 . However, 3D topological solitons such as hopfions in magnetic systems are still underexplored compared to well-studied 1D and 2D solitons.

In this Letter, we show that, in addition to interfacial PMA, a bulk PMA assists in stabilizing a localized hopfion that can exist in nanostrips, in contrast to the boundary-confined hopfions in nanodisks proposed in previous studies magHp1 ; magHp2 . In addition to the Bloch-type hopfions studied previously magHp1 ; magHp2 ; magHp3 , which can be stable in the presence of bulk DMI DMI1 ; DMI2 ; DMI3 , we identify another type of hopfion, Néel-type hopfions, which can be stable in the presence of interfacial DMI AQDMI . We also introduce an ansatz that can accurately describe the hopfion profile. We then study the current-driven dynamics of ferromagnetic hopfions in nanostrips. Although the hopfions are topologically nontrivial, their gyrovectors vanish. This is in contrast to magnetic skyrmions, whose nontrivial topology induces an unwanted “skyrmion Hall effect” Hall1 ; Hall2 ; Hall3 and hinders the device applications Yoo2017 ; Device1 ; Device2 . As a result, hopfions move along the current via spin-transfer torques (STTs) ZhangLi . Spin Hall torques (SHT) SHT also cause Néel-type hopfions to move along the current, while Bloch-type hopfions move transverse to the current. Hopfions may be superior to skyrmions as information carriers in racetrack memories since their current-induced motion is more straightforward.

We consider a magnetic film of thickness $d$ with interfacial PMA at the top and bottom surfaces as well as bulk PMA in the bulk. The zero-temperature micromagnetic free energy of the system reads

[TABLE]

where $A_{ex}$ is the exchange constant; $\mathscr{D}$ is the DMI energy density functional, which depends on the symmetry of the system; $K_{b}$ and $K_{s}$ are the bulk PMA and the interfacial PMA, respectively; $B$ is a perpendicular magnetic field; $M_{s}$ is the saturation magnetization; and $E_{d}$ is the demagnetization energy. In bulk noncentrosymmetric materials such as FeGe and MnSi, the DMI is bulk-like $\mathscr{D}=D_{b}\mathbf{m}\cdot(\nabla\times\mathbf{m})$ , where $D_{b}$ is the bulk DMI strength in units of J/m2 SK2 . In inversion-symmetry-broken films such as $\text{Pt/Co/AlO}_{\text{x}}$ , the DMI is interfacial-like $\mathscr{D}=D_{i}\left[(\hat{\mathbf{z}}\cdot\mathbf{m})\nabla\cdot\mathbf{m}-(\mathbf{m}\cdot\nabla)(\hat{\mathbf{z}}\cdot\mathbf{m})\right]$ , where $\hat{\mathbf{z}}$ is the direction normal to the film and $D_{i}$ is the interfacial DMI strength in units of J/m2 SK1 ; AQDMI . Because the hopfions are non-isomorphic maps from $\mathbb{R}^{3}\cup\{\infty\}$ to $\mathbb{S}^{2}$ , the topological invariant of hopfions, known as the Hopf index $H$ , differs from the skyrmion number. This index is defined as

[TABLE]

where $F_{i}=\varepsilon_{ijk}\mathbf{m}\cdot\left(\partial_{j}\mathbf{m}\times\partial_{k}\mathbf{m}\right)/2$ , in which $i,j,k=\{x,y,z\}$ and $\varepsilon$ is the Levi-Civita tensor, and $\mathbf{A}$ is a vector potential, which satisfies $\nabla\times\mathbf{A}=\mathbf{F}$ HPnumber . The components of $\mathbf{F}$ are solid angle densities in different coordinate planes. $\mathbf{F}$ can be understood as the gyrovector density Thiele , emergent magnetic field SZhang2009 , or topological charge SK2 .

Figures 1(a) and 1(b) show the typical magnetization profiles of Bloch-type and Néel-type hopfions, respectively, obtained by numerical simulations. We consider a 16-nm-thick film with $A_{ex}=0.16$ pJ m*-1* and $M_{s}=1.51\times 10^{5}$ A m*-1*, representing MnSi parameters magHp1 . No external field is applied. The Bloch-type (Néel-type) hopfions are favorable in bulk (interfacial) DMI systems. In Fig. 1(a), we use $K_{s}=0.5$ mJ m*-2*, $K_{b}=41$ kJ m*-3*, and $D_{b}=0.115$ mJ m*-2*, while in Fig. 1(b), we use $K_{s}=0.5$ mJ m*-2*, $K_{b}=20$ kJ m*-3*, and $D_{i}=0.115$ mJ m*-2* (these parameters are also used in the study of current-driven dynamics below). The simulations are mainly performed using mumax3 mumax at zero temperature (additional details of the simulations can be found in the Supplemental Materials SM ). We compute that the Hopf indices are 0.96 (Bloch) and 0.95 (Néel) by numerical integration of Eq. (2) SM . The two types of hopfions are topologically equivalent but behave differently in the presence of SHT, which we will discuss later. The upper and lower panels are the midplane cross-sections in the $xy$ -plane and $xz$ -plane. The magnetization profile in each $xy$ -midplane cross-sections is Bloch-type (a) or Néel-type (b) skyrmionium or the target skyrmion TSK1 ; TSK2 , while the $xz$ -midplane cross-section shows a pair of vortices with opposite chirality. The right ( $x>0$ ) $xz$ -midplane contains a vortex (antivortex) with chirality $+1$ ( $-1$ ) for an $H=+1$ ( $H=-1$ ) hopfion. Outside the hopfions and at the center of the hopfions, the magnetization is along the $z$ direction, and the donut-shape transition region is chiral (for Bloch-type hopfions) or hedgehog-like (for Néel-type hopfions). Figure 1(c) and (d) show the corresponding preimages (constant- $\mathbf{m}$ curves in real space) of Fig. 1(a) and (b). The preimages link with each other once, which is consistent with the Hopf index calculation, justifying the hopfion nature of the textures in (a) and (b).

Different from the hopfions observed in previous studies magHp1 ; magHp2 ; magHp3 that are confined in small magnetic disks, the introduction of a finite bulk PMA causes the hopfions in our work to be metastable, localized objects that can exist in long strips with a hopfion radius $R$ , defined as the radius of the preimage $\mathbf{m}=(0,0,-1)$ . Thus, these hopfions can be candidates of information carriers, and devices such as hopfion racetrack memories can be designed DW2 ; device1. Moreover, unlike skyrmions, although the topology of a hopfion is nontrivial, the gyrovector $\mathbf{G}=\int\mathbf{F}\mathrm{d}V$ of a hopfion vanishes. Consequently, the main drawback of a FM skyrmion racetrack memory, the skyrmion Hall effect, is absent in the hopfion racetrack memory. In addition to the numerical verification, the vanishing gyrovector of a hopfion can be understood as follows. Consider a film that is isotropic in the $xy$ plane. The hopfion profile centered at a certain location can be expressed via $\Theta(r,\phi,z)$ , $\Phi(r,\phi,z)$ , where $(r,\phi,z)$ are cylindrical spatial coordinates, and $\Theta$ , $\Phi$ are the polar and azimuthal angles of the magnetization. Because of the isotropy in the $xy$ plane, it is natural to assume that $\Theta$ is independent of $\phi$ , and $\Phi(r,\phi,z)=\Delta\Phi(r,z)+n\phi$ , where $n$ is an integer and $\Delta\Phi$ is a function independent of $\phi$ . These assumptions are well justified by our numerical results. Thus, in cylindrical coordinates, $F_{z}=n\frac{\sin\Theta}{r}\frac{\mathrm{d}\Theta}{\mathrm{d}r}$ . We can rewrite $G_{z}=\int F_{z}\mathrm{d}V$ as

[TABLE]

Since in a hopfion the magnetization directions are the same at both the periphery ( $r=\infty$ ) and the center ( $r=0$ ), $G_{z}$ vanishes. Since the two vortices in any $xz$ (or $yz$ ) midplane cross-section have opposite chirality, as shown in the lower panels of Fig. 1(a)(b), the integration of $F_{x}$ (or $F_{y}$ ) over the volume gives a vanishing contribution to $G_{x}$ (or $G_{y}$ ). The components of $\mathbf{G}$ are invariant under continuous deformation SK2 ; therefore, $\mathbf{G}=0$ applies to all the hopfions.

The magnetic hopfions discussed in previous studies magHp1 ; magHp2 ; magHp3 were Bloch-like. In the following, we mainly focus on Néel-type hopfions. Although the analytical expression of the hopfion profile is unknown, we find an ansatz that describes the $H=+1$ Néel-type hopfion profile very well:

[TABLE]

where $r^{\prime}=\frac{e^{R/w_{R}}-1}{e^{r/w_{R}}-1}$ , $z^{\prime}=\frac{z}{|z|}\frac{e^{|z|/w_{h}}-1}{e^{h/w_{h}}-1}$ . $R$ , $w_{R}$ , $h$ and $w_{h}$ are lengths parametrizing the hopfion profile. $R$ is the hopfion radius, defined from $m_{z}(r=R,z=0)=-1$ . $h$ is the hopfion height describing the extent of the hopfion in the out-of-plane direction, defined from $m_{z}(r=R,z=h)=1/9$ . $w_{R}$ and $w_{h}$ are hopfion wall widths in the radial and out-of-plane directions respectively, describing the length scale of the magnetization variation from $m_{z}=+1$ to $m_{z}=-1$ SK5 . The ansatz (4) is based on the well-known ansatz FHK augmented by a non-linear rescaling of $r$ and $z$ SK5 and can also describe Bloch-type hopfions and $H=-1$ hopfions after simple transformations SM . Figure 2(a) shows a comparison of $m_{z}$ between the above ansatz and the numerical data along the $x$ direction for $y=z=0$ (bottom axis) and along the $z$ direction for $r=R$ (top axis), with $R=8.3$ nm, $w_{R}=5.6$ nm, $h=6.3$ nm and $h_{w}=1.6$ nm obtained from fitting. The comparison gives good agreement (more comparisons can be found in the Supplemental Materials SM ). The numerical data along the $z$ direction are slightly asymmetric with respect to $z=0$ , which is because of the asymmetric bulk magnetic charge. If the dipolar interaction is turned off, or if the hopfion is a Bloch-type hopfion, this asymmetry will vanish.

Next, we numerically calculate the Hopf index $H$ and the layer-averaged gyrovector $G_{z}/d$ by integrating over a cylinder of height $d$ and radius $R_{0}$ (symbols), and we compare the numerical results with the analytical result calculated using the ansatz (4) (solid lines), as shown in Fig. 2(b). As $R_{0}$ increases, $H$ converges toward 1, and $G_{z}/d$ converges toward 0. Note that the $R_{0}$ used here is smaller than the sample size of our numerical simulation such that the edge structures are discarded. Below, we use this ansatz to discuss the current-driven dynamics of the hopfions, and we compare the results with numerical simulations.

Disregarding deformations, the motion of a hopfion, as a rigid body, is governed by Thiele’s equation Thiele ; VT2 :

[TABLE]

where $\gamma$ is the gyromagnetic ratio; $\alpha$ is the Gilbert damping; $\beta$ is the STT non-adiabaticity ZhangLi ; $\mathbf{v}$ is the velocity of the hopfion; $\mathbf{u}=-\mu_{B}p\mathbf{J}/[eM_{s}(1+\beta^{2})]$ is a vector with dimension of velocity proportional to the current density $\mathbf{J}$ , in which $p$ is the spin polarization and $e$ is the electron charge; $\mathbf{G}$ is the above-mentioned gyrovector; and $\tensor{\mathcal{D}}$ is the dissipation tensor defined as $\mathcal{D}_{ij}=\int\partial_{i}\mathbf{m}\cdot\partial_{j}{\mathbf{m}}\mathrm{d}V$ . $\mathbf{T}$ is the force on the hopfion, expressed as $T_{i}=-\frac{\partial\int\mathcal{F}\mathrm{d}V}{\partial X_{i}}-\int\frac{\partial\mathbf{m}}{\partial x_{i}}\cdot(\mathbf{m}\times\boldsymbol{\tau})\mathrm{d}V$ , where $\mathcal{F}$ is the free-energy functional (1), $X_{i}$ is the center position of the hopfion, and $\boldsymbol{\tau}$ represents non-conservative torques other than STT such as the SHT. In our model, all the material parameters are spatially homogeneous; therefore, the first term in $\mathbf{T}$ is 0. Since $\mathbf{G}=0$ , the hopfions move along the applied current via STT with velocity $\mathbf{v}=\frac{\beta}{\alpha}\mathbf{u}$ . Figure 3(a) shows the trajectory during a period of 15 ns of the Néel-type hopfion driven by STT under $J=10^{11}$ A m*-2*, with $p=0.12$ (a typical value for Co Copolar ), $\alpha=0.05$ and $\beta=0.1$ , obtained by numerically solving the Landau-Lifshiz-Gilbert (LLG) equation LLG with STT ZhangLi ; mumax . The strip is 128 nm-wide in the $y$ direction, and periodic boundary conditions are used in the $x$ direction. The trajectory is almost along the $x$ direction after moving for 15 ns. The small deviation may come from the discretization and the deformation of the hopfion. Figure 3(b) shows the longitudinal component of the hopfion velocity $v_{x}$ versus the applied current density $J$ . The numerical data from LLG simulations (black squares) are in good agreement with the analytical formula $v=\frac{\beta}{\alpha}u$ (black line). Above $J=2\times 10^{11}$ A m*-2*, the hopfion becomes distorted, and at even higher currents $J=5\times 10^{11}$ A m*-2*, the hopfion is destroyed. In contrast to the threshold current for the annihilation of FM skyrmions, this limitation on the current is not intrinsic and can be improved by material engineering. For the Bloch-type hopfion in Fig. 1(a), similar results are obtained.

Recently, spin-orbit torques (SOTs) have attracted attention for driving magnetic textures because of their possibly higher angular momentum transfer efficiency SOTReview . SOTs arise from a variety of origins such as interfacial Rashba spin-orbit coupling Manchon , spin-Hall-effect-induced spin currents from adjacent heavy metal layers SHT , and the intrinsic SOT in magnetic materials Hals . The field-like component of the torque Manchon ; Hals can be regarded as a uniform magnetic field on the system. Since a hopfion is a localized object in a domain, a uniform magnetic field deforms (or even destroys) the hopfion without exerting a net force on it. We consider the antidamping-like SHT SHT ; ZY ,

[TABLE]

which is usually the dominant SOT for a heavy metal/magnet system where $\theta_{\text{SH}}$ is the spin Hall angle. Consider a current applied along the $x$ direction. The SHT is then $\boldsymbol{\tau}=\tau_{0}\mathbf{m}\times(\mathbf{m}\times\hat{\mathbf{y}})$ , where $\tau_{0}$ denotes the prefactors in (6). Using the ansatz (4) with $R$ , $w_{R}$ , $h$ and $w_{h}$ obtained by fitting the numerical data, we can calculate the force $\mathbf{T}$ and dissipation tensor $\tensor{\mathcal{D}}$ . According to the polarity of the hopfion profile, the force on a Bloch-type hopfion is along the $y$ direction, while the force on a Néel-type hopfion is along the $x$ direction, similar to the skyrmion or target skyrmion Hall2 ; TSK1 . Thus, only Néel hopfions move along the current under SHT, while the Bloch hopfions move transverse to the current and are blocked by the edge of the racetrack. Because of the isotropy in the $xy$ plane, $\tensor{\mathcal{D}}$ is diagonal, with $\mathcal{D}_{xx}=\mathcal{D}_{yy}\equiv\mathcal{D}$ . Thus, we have $v_{x}=\frac{T}{\alpha\mathcal{D}}$ for Néel hopfions. The trajectory of the Néel-type hopfion during 15 ns driven by SHT under $J=10^{11}$ A m*-2* and $\theta_{\text{SH}}=0.05$ (a typical value for Pt SHT ) obtained from the LLG simulation is shown in Fig. 3(c). The damping is assumed to be $\alpha=0.05$ . The Néel hopfion propagates along the wire. The longitudinal velocity component $v_{x}$ under different current densities is plotted in Fig. 3(b) by red circles. The analytical formula (red line) agrees well with the numerical data. Note that the values of $T$ and $\mathcal{D}$ depend on the hopfion profile. Since the ansatz introduced gives very good agreement with the numerical results, it may be useful in other investigations on hopfions.

Note that hopfions can also be stabilized in AFM systems, where the staggered Néel field forms a hopfion profile SM ; unpub .

The Néel-type hopfions should be realizable in experiments SM ; multilayer1 ; multilayer2 ; multilayer3; CoAni . In device application, a hopfion can be created by applying a spin-polarized current or a localized magnetic field through a ring-shaped nanocontact SM ; TSK1 ; SK3 ; SKAQ1 ; SKAQ2 ; heat . A strong out-of-plane magnetic field can eliminate a hopfion. The creation and elimination of hopfions will be studied in detail in future. Since the hopfions have finite magnetic moment, any existing techniques that can detect local magnetic moment are also capable to detect hopfions magHp1 ; elec ; NV . The hopfions that we found remain geometrically confined by the thickness of the film with the help of strong PMA. Indeed, in the presence of DMI, Derrick’s theorem Derrick , which prohibits the existence of 3D solitons in infinite conventional (non-chiral) magnets, is no longer valid Bog1995 . Whether it is possible to stabilize hopfions in 3D chiral magnets without confinement is still an open question for further investigations. Our study also implies that magnetic systems represent a fertile playground for research on nonlinear 3D topological solitons.

In conclusion, we identified a new type of hopfion, the Néel-type hopfion, and studied the current-driven dynamics of hopfions. In FM systems, despite the nontrivial topology, neither Bloch- nor Néel-type hopfions exhibit Hall effects and propagate along external currents via spin transfer torque. The SHT only drives the Néel-type hopfions to move along the current. Hopfions have the potential to be efficient information carriers.

Acknowledgements.

The research leading to these results was supported by the European Research Council via Advanced Grant No. 669442, “Insulatronics,” and by the Research Council of Norway through its Centres of Excellence funding scheme, Project No. 262633, “QuSpin.” X.S.W. acknowledges the support from the Natural Science Foundation of China (Grant No. 11804045) and the China Postdoctoral Science Foundation (Grant No. 2017M612932 and 2018T110957).

I Supplemental Materials

I.1 Simulation Details

Most of the simulations are performed using the mumax3 package mumax1 . Some of the results of the static hopfion profile are double-checked by the oommf package oommf . All the calculates are performed at zero temperature. The mesh size is 0.5 nm $\times$ 0.5 nm $\times$ 0.5 nm. The surface pinning is modeled by imposing a very strong PMA $K=10^{6}$ J m*-3* on two additional layers attached to the top and bottom surfaces. This corresponds to a surface anisotropy $K_{s}=0.5$ mJ m*-2* by multiplying the mesh size.

For the static hopfion profile, the conjugate gradient method CG is used to minimize the total energy with an error toleration of $10^{-5}$ . The sample size is 128 nm $\times$ 128 nm $\times$ 16 nm, as shown in Fig. S1. To be consistent with the current-driven dynamical simulations, periodical boundary conditions are imposed along the $x$ direction to mimic a long strip along the $x$ direction. Two sets of initial magnetizations are used. One magnetization is a ring of $\mathbf{m}=(0,0,-1)$ at 25 nm $\leq r\leq$ 40 nm and $|z|<5$ nm inside a uniform domain of $\mathbf{m}=(0,0,1)$ . The other magnetization is a profile of a well-known ansatz that will be discussed below. Both sets of initial magnetizations give the same results. For the current-driven dynamics, the RK45 method is used for the temporal integration of the Landau-Lifshitz-Gilbert (LLG) equation LLG . The spin-transfer torque is in the Zhang-Li form ZhangLi2 ; mumax1 .

Exchange constant $A_{ex}=0.16$ pJ m*-1* and saturation magnetization $M_{s}=1.51\times 10^{5}$ A m*-1*, are used throughout the paper, representing MnSi parameters. Other material parameters used are $K_{s}=0.5$ mJ m*-2*, $K_{b}=41$ kJ m*-3* and $D_{b}=0.115$ mJ m*-2* [for the Bloch-type hopfion shown in Fig. 1(a)], and $K_{s}=0.5$ mJ m*-2*, $K_{b}=20$ kJ m*-3*, and $D_{i}=0.115$ mJ m*-2* [for the Néel-type hopfion shown in Fig. 1(b) as well as Fig. 2 and Fig. 3.]

I.2 Calculation of Hopf Index

**Analytical discussions-**As mentioned in the main text, for an infinite system, the Hopf index is defined as

[TABLE]

where $F_{i}=\frac{1}{2}\varepsilon_{ijk}\mathbf{m}\cdot\left(\partial_{j}\mathbf{m}\times\partial_{k}\mathbf{m}\right)$ , in which $i,j,k=\{x,y,z\}$ and $\varepsilon$ is the Levi-Civita tensor, and $\mathbf{A}$ is a vector potential satisfying $\nabla\times\mathbf{A}=\mathbf{F}$ .

We now demonstrate that the Hopf index is well-defined for an infinite system. Straightforward derivation shows that $\mathbf{F}$ is divergenceless ( $\nabla\cdot\mathbf{F}=0$ ) when $|\mathbf{m}|=$ constant such that the vector potential $\mathbf{A}$ exists. However, obviously, $\mathbf{A}$ is not unique. For any continuous function $\varphi(\mathbf{r})$ , $\mathbf{A}^{\prime}=\mathbf{A}+\nabla\varphi$ is also a vector potential. The corresponding Hopf index is

[TABLE]

The integral in the extra term can be rewritten as

[TABLE]

where Gauss’s theorem has been used and $\oint$ means the integration over the surface of the volume. In an infinite system, the surface is infinitely far away, and the $\mathbf{m}$ field should be homogenous such that, on the surface, $\mathbf{F}$ is 0, and the integral $\oint\varphi\mathbf{F}\cdot d\mathbf{S}$ vanishes. Thus, we have $H^{\prime}=H$ , meaning that the Hopf index is well-defined independent of the choice of $\mathbf{A}$ .

For a rotationally symmetric system, it is natural to assume that the hopfion profile following $\Theta$ is independent of $\phi$ , and $\Phi(r,\phi,z)=\Delta\Phi(r,z)+n\phi$ , where $n$ is an integer. This form means that the polar angle (or $z$ component) of $\mathbf{m}$ is independent of $\phi$ , and when transversing a whole circle centered at the origin in real space ( $\phi$ changes from 0 to $2\pi$ ), the azimuthal angle $\Phi$ of $\mathbf{m}$ uniformly rotates by $2n\pi$ . With this assumption, we can write the $\mathbf{F}$ field in cylindrical coordinates as

[TABLE]

The vector potential $\mathbf{A}$ is

[TABLE]

Then, the Hopf index is

[TABLE]

Thus, the Hopf index equals the whirling number $n$ along the $\phi$ direction multiplied by the skyrmion number at the $rz$ half plane HPindex .

Numerical evaluation of Hopf index- As discussed above, the Hopf index is well-defined when $\oint\left(\varphi\mathbf{F}\right)\cdot d\mathbf{S}=0$ is satisfied. To numerically evaluate the Hopf index, we first cut off the nonhomogeneous edge such that $\mathbf{m}$ is homogeneous on the surface of the sample to ensure that the Hopf index is well-defined (see Fig. S1). We then employ two methods to calculate the Hopf index: a real space method and a Fourier space method.

In real space, we first numerically calculate $\mathbf{F}$ from $\mathbf{m}$ utilizing the standard central finite difference method. Then, we employ a radial basis function (RBF) interpolation to ensure that $\mathbf{F}$ is divergenceless RBF . We use a Gaussian function $g(\mathbf{x}_{i},\mathbf{x}_{j})=e^{-\epsilon\left(|\mathbf{x}_{i}-\mathbf{x}_{j}|^{2}\right)}$ as the RBF with control parameter $\epsilon=1$ . Here, $i$ and $j$ label two grid points, and $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ denote the positions of $i$ and $j$ . The interpolated field is

[TABLE]

After obtaining the RBF coefficients $\mathbf{c}_{i}$ , $\mathbf{A}$ can be directly calculated:

[TABLE]

where the Coulomb gauge is used. Then, the standard numerical integration is performed to calculate $H=\frac{1}{\left(4\pi\right)^{2}}\int\mathbf{F}\cdot\mathbf{A}\mathrm{d}V$ .

The Fourier space method has been introduced in Ref. magHp22 . The relative difference between the two methods is less than 5%. In the main text, we show the Fourier space result.

I.3 Construction and Verification of the Ansatz for Hopfion Profile

We start from the well-known ansatz for hopfions ansatz12 :

[TABLE]

which describes an $H=-1$ Bloch-type hopfion with $\mathbf{m}$ upward at the center and infinity. Note that in some references, the definitions of Hopf index differ by a sign. Here, we use the definition mentioned in the main text. The radius is 1 because at $r^{2}=x^{2}+y^{2}=1$ , $\mathbf{m}=(0,0,-1)$ . The magnetization rotates counterclockwise (clockwise) at $r>1$ ( $r<1$ ) in the top view. An $H=+1$ hopfion ansatz can be obtained simply by inverting the sign of the first term in the numerators of $m_{x}$ and $m_{y}$ :

[TABLE]

In cylindrical coordinates:

[TABLE]

This operation can invert the Hopf index and retain the rotational sense, which is preferred for $D_{b}>0$ . For $D_{b}<0$ , we let $m_{x}\rightarrow-m_{x}$ and $m_{y}\rightarrow-m_{y}$ . For a realistic hopfion with a given radius $R$ and height, it is natural to consider a linear rescaling , where $h$ describes the extent of the hopfion in the $z$ direction. This profile is used as the initial condition, with $R=20$ nm and $h=10$ nm. However, this ansatz cannot describe the numerical data well. In Fig. S2(a), we show the numerical magnetization profile along $x$ at $y=z=0$ (symbols), and the above ansatz (dashed blue line) with $R=7.8$ nm obtained from the numerical data for the Bloch-type hopfion is shown in Fig. 1(a) of the main text. Obviously, the numerical data show a much faster decay away from $x=R$ than the polynomial decay of the ansatz. Inspired by work on skyrmion profiles wxs2 , we introduce another length scale to describe how fast $\mathbf{m}$ decays to $(0,0,1)$ . Since the $z$ direction is special because of the surface PMA, we introduce $w_{R}$ and $w_{h}$ for the $xy$ plane and $z$ direction, respectively. We try a monotonic nonlinear rescaling

[TABLE]

and use the resulting ansatz to fit the numerical result to determine the parameters $R$ , $w_{R}$ , $h$ , and $w_{h}$ . $R$ and $w_{R}$ are determined by fitting the radial profile $m_{z}(r)$ at $z=0$ . $h$ and $w_{h}$ are determined by fitting the profile $m_{z}(z)$ along $z$ at $r=R$ . The result of this nonlinear rescaling (solid red line) is also compared with the numerical data in Fig. S2(a). The agreement is obversely better. For a larger hopfion (which can be obtained by using a smaller $K_{b}$ ), the agreement of our ansatz is even better.

For an $H=+1$ hopfion, with $z\neq 0$ , the position of the minima of $m_{z}(r)$ moves outward as $|z|$ increases according to the ansatz, which is consistent with the numerical result, as shown in Fig. S2(b). In contrast, for an $H=-1$ hopfion, the minima of $m_{z}(r)$ moves inward, as shown in Fig. S2(c). To describe this, we further invert the rescaling of $r$ as

[TABLE]

which maps $r=0$ to $r^{\prime}=\infty$ and vice versa. In cylindrical coordinates for space and spherical coordinates for $\mathbf{m}$ , the ansatz can be written as

[TABLE]

The Hopf index can be calculated using Eq. (S10). Because of the inverse rescaling of $r$ , the Hopf index becomes $1$ . To retrieve the rotational sense, we further let $m_{x}\rightarrow-m_{x}$ , $m_{y}\rightarrow-m_{y}$ , or $\Phi\rightarrow\Phi+\pi$ . The resultant ansatz gives good agreement for $z\neq 0$ , as shown in Fig. S2(b) by the solid lines. Although the ansatz cannot quantitatively overlap with the numerical data, the tendency of the minima position is correct.

To verify the continuity of the ansatz, we write it in Cartesian coordinates as

[TABLE]

where $r^{\prime}=\left(\frac{e^{\sqrt{x^{2}+y^{2}}/w_{R}}-1}{e^{R/w_{R}}-1}\right)^{\pm 1}$ . When $xyz\neq 0$ , the ansatz has no singularity. At $z=0$ , we have $\lim_{z\rightarrow 0}z^{\prime}=0$ and $\partial_{z}z^{\prime}\big{|}_{0+}=\partial_{z}z^{\prime}\big{|}_{0-}$ . At $x=y=z=0$ , it is easy to verify $\lim_{x,y,z\rightarrow 0}\mathbf{m}=(0,0,1)$ , and all the first-order derivatives $\partial_{i}m_{j}$ ( $i,j\in x,y,z$ ) are continuous. Since the highest order of derivative in the energy functional is 1, the ansatz is well-defined in the whole space.

To obtain a Néel-type hopfion, we can locally rotate the $\mathbf{m}$ by 90 degrees, as shown in the main text:

[TABLE]

Because the interfacial DMI only exists in the $xy$ plane and because there is no chiral interaction in the $z$ direction, the minima in $m_{z}(r)$ remains in the same position. Therefore, either $r^{\prime}=frac{e^{R/w_{R}}-1}{e^{r/w_{R}}-1}$ or $r^{\prime}=\frac{e^{r/w_{R}}-1}{e^{R/w_{R}}-1}$ can be used as the ansatz. We choose $r^{\prime}=frac{e^{R/w_{R}}-1}{e^{r/w_{R}}-1}$ because it gives better results in the calculation of $\tensor{D}$ and $\mathbf{T}$ [Fig. S2(d)].

The midplane cross-sections in the $xy$ plane and $yz$ plane are compared in Fig. S3(a-d) for $H=+1$ Bloch-type, $H=-1$ Bloch-type, $H=+1$ Néel-type, and $H=-1$ Néel-type hopfions. The 2D cross-sections also give fairly good agreement with the numerical results. The senses of the rotation of the spins are retrieved in all the cross-sections, while the shapes of the textures in $yz$ cross-sections are not as good as those in the $xy$ cross-sections. Nevertheless, the ansatz can well describe the topological properties and current-driven dynamics of the hopfions.

I.4 Antiferromagnetic Hopfions

We consider that $A_{ex}=-0.16$ pJ m*-1*, $D_{b}=0.115$ mJ m*-2*, $K_{b}=16$ kJ m*-3*, and the other parameters to be the same as those in Fig. 1(a). The ground state is an out-of-plane AFM Néel state. Because the dipolar field is negligible in an antiferromagnet, to speed up the simulation, we turn off the dipole-dipole interaction. For a numerical cell labelled by $(i,j,k)$ , if $i+j+l$ is even (sublattice 1), we impose the ansatz [Eq. (S19-S22)] with the collective coordinates $R=15$ nm, $w_{R}=5$ nm, $h=8$ nm and $h_{w}=5$ nm. If $i+j+k$ is odd (sublattice 2), we impose the opposite direction. After relaxation, we obtain an antiferromagnetic Bloch-type hopfion. In Fig. S4(a), the mid-plane cross-sections in the $xy$ and $xz$ planes are shown for each sublattice. If we use $K_{b}=5$ kJ m*-3* and $D_{i}=-0.115$ mJ m*-2* instead, a Néel-type AFM hopfion is obtained, as shown in Fig. S4(b).

I.5 Discussions about Experimental Realization, Writing, and Reading of Magnetic Hopfions

Candidate materials- The possible material for realization of Bloch-type hopfions has been discussed in Ref.maghopf1 . For Néel-type hopfions proposed in our paper, inversion-symmetry-breaking multilayer systems multilayer1 ; multilayer2 may be good candidates for experimental realization. Inversion-symmetry-breaking multilayer stacks such as Pt/Co/AlOx has been shown to possess interfacial DMI and support magnetic skyrmions. The interlayer coupling is RYYK-type and can be tuned to be ferromagnetic. The anisotropy of the ferromagnetic layers (usually Co or Co alloys) can be engineered in the wide range, from negative PMA (in-plane anisotropy) to very strong PMA CoAni . Thus, the weak bulk PMA and strong surface PMA are possible in these systems. In this case, the spin Hall torque is much larger than that calculated in the main text because the torque is exerted on layers with substantially smaller thickness $d$ . Thus, the expected hopfion velocity can be much larger.

Writing of hopfion bits- To use hopfions as information carriers, we need to create or eliminate a hopfion to write binary “1” or “0”. We expect that Hopfions can be created from a single domain by reversing the spins in a torus in analogous ways as for skyrmions. The reversal can be achieved by applying a field pulse field1 ; field2 or a current pulse Roland ; TSK1 , and may be assisted by local heating heat . The field or current pulse can be applied through a ring-shaped nanocontact with radius closed to the hopfion radius, with the field direction or spin-polarization direction opposite to the magnetization direction of the domain. Since the top and bottom surfaces are pinned by the strong PMA while the bulk PMA is much weaker, a moderate field or current pulse can reverse the magnetization only in a torus. After relaxation, the structure is able to evolve to a hopfion state. A hopfion can be eliminated simply by applying a uniform magnetic field that is strong enough to overcome the energy barrier between the hopfion state and the single-domain state. The optimal pulse duration and field/current intensity for hopfion creation depend strongly on the material parameters, and extensive numerical calculations are needed. We intend to study the writing and erasing of hopfions in details elsewhere.

Reading of hopfion bits- Although hopfions are local solitons in a uniform domain, they have non-zero net magnetic moments. Thus, any existing techniques that can detect local magnetic moments can be used to read the hopfion information. For example, the possible Lorentz TEM image has been shown in Ref. maghopf1 . Other techniques like magnetic force microscopy (MFM) also works for reading of hopfions. More practically, the all-electric detection elec based on the non-collinear magnetoresistance effects is also possible. The use of NV-centers should be able to detect the stray fields from the Hopfions NV .

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