Hall eect induced by topologically trivial target skyrmions Tan Dao1 2Sergey S. Pershoguba2and Jiadong Zang2 3 1Department of Physics Harvard University Cambridge MA 02138 USA

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Hall eﬀect induced by topologically trivial target skyrmions

Tan Dao,1, 2 Sergey S. Pershoguba,2and Jiadong Zang2, 3, ∗

1Department of Physics, Harvard University, Cambridge, MA 02138, USA

2Department of Physics and Astronomy, University of New Hampshire, Durham, New Hampshire 03824, USA

3Materials Science Program, University of New Hampshire, Durham, New Hampshire 03824, USA

(Dated: October 21, 2022)

Electrons moving through a noncoplanar magnetic texture acquire a Berry phase, which can be

described as an eﬀective magnetic ﬁeld. This eﬀect is known as the topological Hall eﬀect and has

been observed in topological spin textures. Motivated by recent experimental realizations, here we

study the Hall eﬀect in a nontopological magnetic texture known as a target skyrmion. We start

from a simpliﬁed semiclassical picture and show that the Hall signal is a nonmonotonic function of

both the electronic energy and target skyrmion radius. That observation carries over to the fully

quantum mechanical treatment in a Landauer-B¨uttiker formalism in a mesoscopic setting. Our

conclusion challenges the popular opinion in the community that the Hall eﬀect in such structures

necessarily requires a nonzero skyrmion number.

I. INTRODUCTION

In contrast to the classical Hall eﬀect [1] induced by

a nonzero external magnetic ﬁeld, the anomalous Hall

eﬀect (AHE) represents a component of the Hall signal

proportional to the magnetization of a magnet [2,3].

Therefore it can exist even in the absence of the exter-

nal magnetic ﬁeld. In frustrated chiral magnets, there

is an additional contribution to AHE arising from the

noncoplanar structure of spin magnetic moments [4–

8]. In these materials, electrons acquire a Berry phase,

therefore generating an anomalous velocity which can

be understood as the Lorentz force of an eﬀective mag-

netic ﬁeld in momentum space [2–5]. More interest-

ingly, in the adiabatic regime where itinerant electrons

are strongly coupled to the spins, the Berry curvature

in real space can give rise to the AHE eﬀect [9–11]. This

eﬀect is also known as the topological Hall eﬀect (THE).

As the real space Berry phase is directly related to the

topology of the spin texture, the THE is often used as

evidence of nontrivial spin texture [12–17]. One no-

table example of nontrivial spin texture is the magnetic

skyrmion, whose topological charge (Q = 1) is deﬁned

as the integer winding number of spin S(r)on a unit

sphere [18].

Since the discovery of skyrmions in MnSi[14], the

role of topology and magnetization has increasingly ex-

panded for its potential use in dense and robust data

storage, as well as racetrack memory for quick data

access [19–21]. However, chiral spin textures that are

classiﬁed as topologically trivial have remained under-

explored in theory and experiment. In particular, the

target skyrmion whose texure is a concentric ring of one

skyrmion inside another skyrmion with reversed spins

(see Fig.1a) can be moved linearly with a current, mak-

ing it promising for racetrack memory applications [22–

26]. Currently, target skyrmion are being identiﬁed by

techniques such as magnetic force microscopy, Lorentz

∗Jiadong.Zang@unh.edu

transmission electron miscroscopy, X-ray photoelectron

spectroscopy, and electron hollography, which can not

be integrated to electronic devices [22,25,27]. The

topological Hall eﬀect, however, can be used to quickly

detect presence of spin texture by measuring the trans-

verse voltage of devices. Here we show that the topo-

logical trivial target skyrmion can give rise to the Hall

eﬀect.

We start with a classical model of electrons traveling

in the eﬀective magnetic ﬁeld of the target skyrmion

and show a non-zero transverse current. We then per-

formed quantum transport calculations and observed a

sign change in the non-monotonic Hall angle dependence

of target skyrmion radii. This indicates the inner and

outer skyrmion shells contribute to the Hall angle inde-

pendently. This result is further supported by the cal-

culation of the Hall angle for each shell; showing that

the sum of the Hall angle of each shell is the same as the

target skyrmion’s Hall angle. We also shows that our

results respect the adiabatic approximation. This im-

plies that the Hall eﬀect came from real space topology,

which can be interpreted as the THE.

II. TARGET SKYRMION CONFIGURATION

To set the stage, we discuss details of a target-

skyrmion texture in this section. We consider a two-

dimensional ferromagnet described by a magnetization

vector S(r) normalized to unity |S(r)|= 1. We may use

the following parametrization of the target skyrmion,

S(r) = 



sin φ(r) sin θ(r)

cos φ(r) sin θ(r)

cos θ(r)

,

where the polar and azimuthal angles

φ(r) = tan−1y

x−π/2

θ(r) = 2π−8 tan−1exp 4r

r0 (1)

arXiv:2210.11459v1 [cond-mat.mes-hall] 20 Oct 2022

8.1-3

-6.8-4

(a)

(b)

8.1E-03

-6.8E-04

0.0

(b)

-1.0

1.0

FIG. 1. a) Target skyrmion spins conﬁguration. The ar-

rows indicate the spins rotation, and the colorbar indicates

Szmagnitude. b) The eﬀective magnetic ﬁeld of a target

skyrmion.

are position-rdependent. The speciﬁc choice of the an-

gle (1) is motivated by the necessity to eliminate spuri-

ous electronic scattering at the target skyrmion bound-

ary [22,28].

Assuming that we are working in the adiabatic

regime, we may apply the real space Berry phase picture

by evaluating the eﬀective magnetic ﬁeld

Bz(r) = 1

2S(r)·[∂xS(r)×∂yS(r)] (2)

The eﬀective magnetic ﬁeld distribution of the target

skyrmion is shown in Fig. 1b. The inner shell’s ﬁeld

is positive and greatest at the center of the target

skyrmion due to a larger gradient of the spins at around

the center spin. The outer shell’s ﬁeld is negative and is

much weaker. The total magnetic ﬁeld added up to zero

which is consistent with Q= 0 of the target skyrmion

[22,23]. We use the eﬀective ﬁeld of the target skyrmion

for our classical numerical calculations.

III. CLASSICAL MODEL

As a warm-up, we analyze classical scattering of elec-

trons of the target skyrmion. We approximate the

target-skyrmion as a two-shell structure shown in the

Fig. 2. The inner (i) circular shell has a radius ri, and

the outer (o) annular region has radius ro. The eﬀec-

tive magnetic ﬁelds have opposite signs Bi= + ˆ

zBi

FIG. 2. (a) Approximate representation of a target

skyrmion. The magnetic ﬁelds in the inner (i) circular and

outer (o) annular regions have opposite signs [Fig. 1(b)]. The

opposite Lorentz force bends the corresponding cyclotron

trajectories in the corresponding regions in the opposite di-

rection. The scattering angle Φ(ρ) is given by Eq. (5).

and Bo=−ˆ

zBo, which we assume constant within the

respective regions. The speciﬁc values are constrained

by the condition that the total magnetic ﬂux piercing

through the structure vanishes, i.e.

Biπr2

i=Boπr2

o−r2

i.(3)

In the presence of the eﬀective magnetic ﬁelds Biand

Bo, the particles moves along the cyclotron trajectories

with radii

Ri=p

and Ro=p

(4)

determined by the magnetic ﬁelds in the respective re-

gions Biand Boas well as the momentum p=√2mE.

A representative electronic trajectory is shown in Fig. 2.

By matching the circular trajectories in the two regions,

we ﬁnd the scattering angle dependence

Φ(ρ) = 2 Re (acos "Ro+ρ

pR2

o+r2

o+ 2ρRo#

−acos "2R2

o+ 2ρRo+r2

o−r2

2RopR2

o+r2

o+ 2ρRo#(5)

−acos "2RiRo−2ρRo+r2

i−r2

2RopR2

i+r2

i−2ρRi+ (r2

i−r2

o)Ri/Ro#).

on the impact parameter ρ.

To get a feeling of Eq. (5), let us give the low-energy

(high-ﬁeld) limiting cases of Ro/ro1

Φ(ρ)≈2 Re acos ρ

+Ro

ro1−ρ2

o,(6)

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摘要：

HalleectinducedbytopologicallytrivialtargetskyrmionsTanDao,1,2SergeyS.Pershoguba,2andJiadongZang2,3,1DepartmentofPhysics,HarvardUniversity,Cambridge,MA02138,USA2DepartmentofPhysicsandAstronomy,UniversityofNewHampshire,Durham,NewHampshire03824,USA3MaterialsScienceProgram,UniversityofNewHampshire,Du...

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Hall eect induced by topologically trivial target skyrmions Tan Dao1 2Sergey S. Pershoguba2and Jiadong Zang2 3 1Department of Physics Harvard University Cambridge MA 02138 USA

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