Discretized Topological Hall Effect Emerging from Skyrmions in Constricted Geometry

1 MB·English

by  N. Kanazawa

Discretized Topological Hall Effect Emerging from Skyrmions in Constricted
Geometry
N. Kanazawa,1 M. Kubota,2,3,a A. Tsukazaki,4 Y. Kozuka,1 K. S.
Takahashi,2 M. Kawasaki,1,2 M. Ichikawa,1 F. Kagawa,2 and Y. Tokura1,2
1Department ofAppliedPhysicsandQuantum Phase Electronics Center (QPEC),University ofTokyo, Tokyo 113-8656, Japan
2RIKEN Center for Emergent Matter Science (CEMS), Wako, 351-0198, Japan
3Power Electronics R&D Division, ROHM Co., Ltd., Kyoto 615-8585, Japan
4Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
(Dated: January 15, 2015)
5
1 Weinvestigatetheskyrmionformationprocessinnano-structuredFeGeHall-bardevicesbymea-
0
surements of topological Hall effect, which extracts the winding number of a spin texture as an
2
emergentmagnetic field. Step-wiseprofilesoftopological Hall resistivity areobservedin thecourse

n ofvaryingtheappliedmagneticfield,whicharisefrominstantaneouschangesinthemagneticnano-
a structuresuchascreation,annihilation, andjittering motion ofskyrmions. Thediscretechangesin
J topological Hallresistivitydemonstratethequantizednatureofemergentmagnetic fluxinherentin

4 eachskyrmion,whichhadbeenindistinguishableinmany-skyrmionsystemsonamacroscopicscale.
1
PACSnumbers: 72.25.Ba,75.25.-j,85.75.-d

]
l
e
- Recent studies on condensed matters with peculiar creasing recording density facilitate their detection with
r
bandtopology[1–4]haveestablishedthatemergentmag- use of the emergent flux, contrary to conventional case
t
s
netic field arises from the Berry curvature in the mo- where smaller magnetic bits are generally more difficult
.
t mentum space and leads to unique electromagnetic re- to detect: Effective magnetic field observed in Hall volt-
a
m sponses [5–7], such as quantized anomalous Hall effects age increases inversely with the square of skyrmion size
in topological insulators [8–10]. Even for originally non- a , following the relation of B = −φ /a2 , which ide-
- sk eff 0 sk
d topological band (momentum-space) structures, materi- ally reaches approximately 4000 T in a 1-nm skyrmion.
n
als may acquire a topological nature when the electrons Nevertheless, such a remarkable feature of skyrmions re-
o
interact with a specific spin texture in real space. The mains elusive, presumably because a large number of
c
[ magnetic skyrmionin chiralmagnets is anintriguing ex- skyrmions involved in the macroscopic system studied

ample of such topological spin textures, in which con- so far inevitably smears out the quantized nature of the
1
stituent spins point in all directions wrapping a sphere skyrmion [23, 24].
v
0 [11, 12]. Such a winding spin texture whose topology In this Rapid Communication, we demonstrate dis-
9 can be labeled as the integer winding number −1 exerts cretized changes in the emergent magnetic flux arising
2 a quantized emergent magnetic flux of −φ0 = −h/e on from a finite number of skyrmions, by measuring topo-
3
conductionelectronsincaseofstronglimitofspin-charge logicalHallresistivityofnanoscaleFeGeHall-bardevices.
0
. coupling; as a result, topological Hall effect (THE), By measuring the inclination-angle dependence of topo-
1 besides normal and anomalous Hall effects (NHE and logicalHallresistivity,whichworksasatestforskyrmion
0
AHE), can be observed as a hallmark of the skyrmion formation in thin films [25], we can ensure a hysteretic
5
1 formation [13–18]. formationof skyrmions appearing as a hysteresis loop in
v: Inadditiontothetopologicalaspect,theskyrmionpos- Hallresistivity[26]. Inthedeviceswithcircuitlinewidth
i sesses the small-sized particle nature (its typical diame- between50-250nm,thediscretizedprofilesoftopological
X
teris3-200nm),whichpotentiallyenablesmagneticnon- Hall resistivities show up in the hysteresis loops of Hall
r volatilememorieswithultrahighdensity[19,20]. Tofully resistivities where rapid conversions between skyrmions
a
benefitfromsuchnano-sizedcharacteristics,amethodol- withoppositecore-magnetizationsoccur. Further minia-
ogyforidentifyingindividualskyrmionneedstobedevel- turization to 32-nm-wide circuit removes the typical sig-
oped. Although it has been demonstrated that observa- nature of skyrmion formation (a hysteretic topological
tionandevenmanipulationofasingleskyrmionarefeasi- Hall resitivity), which indicates that skyrmions are not
blebymeansofLorentztransmissionelectronmicroscopy able to formina smaller areathan halfthe size ofthem-
[12] and spin-polarized scanning tunneling microscopy selves.
[21, 22], identifying individual skyrmions by electrical Hall-bar devices of various circuit-line-widths (32 nm
means remains as a challenge for developing skyrmion- - 250 nm and 10 µm) were fabricated from 40-nm-thick
basedmagnetic memories. In this context, we canenvis- FeGe epitaxial thin film by using electron-beam lithog-
age that the quantized emergent magnetic flux inherent raphy method. A top-view image of a 50-nm-wide de-
in the skyrmion may play a crucial role in identifying a vice is examplified in Fig. 1(a). The electrical leads for
single skyrmion. Mostnotably,smaller skyrmionsforin- current and voltage were made to have the same width
2
(w); skyrmion formations in the intersectional areas of magnetic state is induced, the difference between Hall
w×w were probed by measurements of the topological resistivities at low and high (θ =30◦) angles reads
Hall effect [27]. The FeGe film was grown on a highly-
∆ρ =ρ (θ)−ρ (30◦)
resistive Si (111) substrate (ρ > 1000 Ω cm) by codepo- yx yx yx
sition of Fe and Ge at a substrate temperature of 325 =[ρN (θ)+ρA (θ)+ρT (θ)]−[ρN (30◦)+ρA (30◦)]
yx yx yx yx yx
◦
C by molecular beam epitaxy. A very low concentra- =(R0B⊥+RsM⊥+R0Beffcosθ)−(R0B⊥+RsM⊥)
tion of impurity phases was verified by high-angle X-ray
=R B cosθ.
0 eff
diffraction (XRD) as detailed in Supplemental Material
[27]. Here we note that the electrical conduction at 2 Otherwise (H⊥ >Hc),
K is not the ballistic one because the residual resistivity
ranges between 50-150 µΩ cm [27]. We do not have to ∆ρyx =[ρNyx(θ)+ρAyx(θ)]−[ρNyx(30◦)+ρAyx(30◦)]
take mesoscopic effect into account. =(R0B⊥+RsMscosθ)−(R0B⊥+RsMscos30◦)
We first confirm skyrmion formation in our FeGe thin =R M (cosθ−cos30◦).
s s
film by THE. Topological Hall resistivity ρT associated
yx
Here R and R are normal and anomalous Hall coeffi-
with formation of a non-coplanar magnetic structure is 0 s
usually defined as an additionalcomponent to the terms cients, and M⊥ and Ms the vertical component of mag-
netization M and the saturated magnetization, respec-
varyinglinearlywithmagneticfieldH andmagnetization
M, i.e., the normal and anomalous Hall resistivity ρN tively. Although the extracted ∆ρyx inevitably includes
and ρA , respectively. However, those two componenytxs anomalousHallcomponentRsMs(cosθ−cos30◦)athigh
(ρN aynxdρA )mayshowH-andM-nonlineardependence fields, ∆ρyx for H⊥ < Hc is almost equivalent to ρTyx.
yx yx The rapid shrinkage of ρT is again highlighted in Fig.
especially in multi-band metals [25, 28]. Here we adopt yx
1(d). As magnetization at the core of the skyrmion is
anotherprecisemethodforevaluatingTHEbasedonthe
anti-parallel to the field [11, 12], skyrmions with posi-
H-direction-sensitive stability of skyrmions in thin films
tive (negative) core-magnetization is stabilized at neg-
[25]. According to neutron scattering experiments on
ative (positive) magnetic field. Hereafter, we refer to
skyrmionic materials [11], the plane of skyrmion lattice,
skyrmionswithpositive(negative)core-magnetizationas
where the three magnetic modulation vectors Q lie, is
core-up (down) skyrmions, which produce positive (neg-
strictly perpendicular to the magnetic field. Namely, en-
ative) emergent magnetic fields. The hysteresis loop of
ergygainfrominteractionsamongthe spins onthe same
ρT demonstrates that core-up or core-down skyrmions
plane perpendicular to H is crucial for stabilization of yx
are stabilized even at zero magnetic field depending on
skyrmions. When H is tilted from the normal vector of
a magnetic-fieldhistory,as discussedlater. Possibilityof
the thin film plane n, Qs have out-of-plane components,
other interpretations of the hysteretic Hall resistivity is
whichmeans thatspins near the surfaces lose their part-
discussedintheSupplementalMaterial[27];nevertheless,
nerspinstointeractwith[Fig. 1(b)]. Theskyrmionstate
the skyrmion formationis the most plausible scenarioto
is thus easily destabilized with H tilted from n in thin
explain the hysteretic behavior.
films, and this feature should become more significantin
To pursue the quantized nature of emergent flux, we
thinner films compared to their skyrmion sizes.
fabricated Hall-bar micro-devices and found that such a
Figure 1(c) shows magnetic-field dependence of Hall
hysteresis loop representing skyrmion formation remain
resistivity at various inclination angles θ at 2 K in a 10-
robust down to a 50-nm-wide device. We show the line-
µm-wide Hall-bar device. We observe a loop in Hall re-
width dependence of Hall resistivity in Fig. 2. Devices
◦
sistivity at θ = 0 , which is recognized to be induced
withwidercircuitlinesthan50nmshowthecharacteris-
by residual skyrmions of metastable state dependent on
tichysteresisloopsrepresentingskyrmionformation[29].
magnetic-field history, as reported by Huang and Chien
By contrast, Hall resistivity in a 32-nm-wide circuit al-
[26]. The hysteresis loop dramatically shrinks with a
mosttracesthemagnetizationcurve[27]. Thecollapseof
slight change in the inclination angle: the loop size be-
the hysteresis loop with narrowingthe circuit line repre-
◦
comes less than the half just tilting at θ =4 andnearly
sentsthatthecriticalsizefortheformationofskyrmions
◦
vanishes above θ = 10 . This sudden reduction of the
of FeGe lies between 32 nm and 50 nm [30]; this criti-
ρ loop, which is highly unlikely for the magnetization-
yx cal size is to be compared with the helimagnetic period
related anomalous Hall effect in ordinary ferromagnets, (≈70nm)orthe latticeconstant(≈80nm)ofskyrmion
represents the destruction of two-dimensional skyrmion
crystalofFeGebulkcrystal. Weassumethatahelix-like
structures as expected.
magnetic structurewith no topologicalchargeis realized
WeextracttopologicalHallcomponentinFig. 1(d)by in the 32-nm device as theoretically indicated by Du et
subtracting ρ at a high angle of θ = 30◦ consisting of al.[24].
yx
normal and anomalous Hall components from low-angle Takingacloselookatthehysteresisloopsinthenano-
data[27]. IftheverticalcomponentofH tothefilmplane scale circuits, we notice that Hall resistivity shows dis-
(H⊥) is lower than the critical field Hc, where the ferro- continuous changes with field variation, which indicates
3
(a)! (b)! B! the discrete change in emergent magnetic field. Figure 3
I! showsmagnifieddrawingsofhysteresisloops(insets)and
θ!
difference betweenHallresistivitiesin increasingandde-
Q!
[112] 500 nm! 70 nm! 40 nm! Q! creasing field processes, ∆ρTyx (main panels), in various-
[111] [110] || I size devices; ∆ρTyx is plotted as the normalized value by
0.2 its maximum value in the figure. Since the conventional
(c)! 0.04 (d)!
Ωµ (cm)ρyx-000...011 042oo0,,o 26, 3oo,,0 o Ωµ (cm)∆ρyx--0000....00004202 06ooR,, S22Mo0S,o (4coos, θ - cos30o)! sHcaiussasrlovblceeci,haowtmaevhdpiioocwrhnsietsnh(hstotsewh,eeis.Sehlui.yt,ptsρplteNyelxerhemyasseinstndeltoraρeolAyspxiMs,rebadpetorehernasiovaetilnotsfrsoh[rot2hw7te]h.hg)ey,esnM∆tueρi-rnTyHexe-
-0.2 -1.0 0.0 1.0 -1.0 0.0 1.0 contributionfromthe topologicalHalleffect. Incontrast
µ0Hcosθ (T) µ0Hcosθ (T) to the smooth variation of ∆ρT in the 10-µm device
yx
[Fig. 3(a)], ∆ρT in nano-circuits of w = 50-250 nm ex-
yx
FIG. 1. (color online). (a) Scanning electron microscope im-
hibit step-wise profiles [Figs. 3(b)-3(f)]. The noise level
age of a Hall-effect measurement circuit with line width of
is one order of magnitude smaller than the smallest step
w = 50 nm. (b) Schematic illustrations of the skyrmion-
height, 5 nΩ cm, indicating that the discretizedstep can
lattice state in a 40-nm-thick FeGe thin film. The magnetic
modulationperiodintheFeGethinfilmisassumedtobethe not be electrical noise of our experimental system [27].
same length of 70 nm as the bulk sample. Magnetic-field de- Constricted geometry comparable to the size of a few
pendence of (c) Hall resistivity ρyx and (d) ∆ρyx at various skyrmions emphasizes skyrmions’ individuality, so that
inclination angles at 2 K in a 10-µm-wide circuit. (See text we successfully observed the quantized nature of emer-
for the definition of ∆ρyx.) A sudden shrinkage of hysteresis gentmagneticfieldasdiscretechangesintopologicalHall
loops in panels (c) and (d) represents declines in skyrmion
resistivity mainly originating from creation/annihilation
numbers.
of skyrmions in changing the applied magnetic field.
Incidentally, we found that the step heights display
considerablevariationandare not equalto integer times
0.2 0.3 of unit topological Hall resistivity ρT = −R φ /w2,
(a)! 0.2 (b)! corresponding to creation/annihilationyxof one sk0yrm0 ion.
m) 0.1 0.1 This indicates additional contributions to the discrete
c
Ω 0.0 0.0 changesinemergentmagneticfield,whichweattributeto
µ
( x -0.1 discontinuousmotionsofskyrmionsviatrappingbyorre-
ρy -0.1
-0.2 leasingfromimpurityordefectsitesinthe courseoffield
w = 250 nm w = 200 nm
-0.2 -0.3 changes: thisbearsanalogytotheBarkhauseneffect,i.e.,
-2 -1 0 1 2 -2 -1 0 1 2
0.2 discontinuous changes in magnetization via the similar
(c)! 0.1 (d)! motion of ferromagnetic domains. The possible fluctua-
m) 0.1 tionofskyrmions’positions,especiallyskyrmionsputting
c
Ω 0.0 0.0 their feet on and off the verge of the probed square area
µ
( x oftheHalldevice,mayproduce“halfway”discretetopo-
ρy -0.1 -0.1 logical Hall effect. For examples, the fluctuation of ρT
w = 120 nm w = 100 nm yx
-0.2 around µ0H = 0.2-0.3 T in 120-nm-wide device [Fig.
-2 -1 0 1 2 -2 -1 0 1 2 3(d)] and dip structure of ρT around µ H = 0-0.1 T
0.3 0.2 yx 0
0.2 (e)! (f)! in 100-nm-widedevice [Fig. 3(e)] canbe assignedto dis-
m) 0.1 continuous changes in skyrmion position. Here we note
c 0.1
Ω 0.0 0.0 that the steady flow of skyrmions via the spin-transfer
µ
(x -0.1 torque,whichmayalsocontributetothevariationinthe
ρy -0.1 step heights, probably does not occur in present study
-0.2
w = 50 nm w = 32 nm consideringtheenhancedcriticalcurrentdensityinnano-
-0.3 -0.2
-2 -1 0 1 2 -2 -1 0 1 2 structures [27, 31].
µH (T) µH (T)
0 0
Hallresistivitydataenableustoenvisagetheskyrmion
formationprocessinthenano-structuredHallbars. Here
FIG. 2. (color online). (a)-(f) Magnetic-field dependence
we take, for example, the case of w = 250 nm wide de-
of Hall resistivity for FeGe nano-scale circuits with vari-
vice,whichcanaccommodateapproximately9skyrmions
ous widths ranging from 250 nm to 32 nm. We subtracted
the H-symmetric component using the equation ρyx(±H) = at the overlap of current and voltage leads w×w. Fig-
±[Vy(H)−Vy(−H)]t. ures 4(a) and 4(b) are the conceivable development of
2I numbers of core-up and -down skyrmions in increasing
and decreasing field processes, respectively. In increas-
4
TNormalized ∆ρyx 100000......086420 (wa )=! 10 µm!Ωµ ( cm)ρyx--00000.....00000420240.00.10.20.30.40.5100000......086420 w(b =)! 250 nm!Ωµ ( cm)ρyx---0000000.......000000064202460.00.10.20.30.40.5 mber118620 (aUS)kp! IncCreaosnincDSg ekofiweplndtuaΩµ cm)l d00ia..01gr(ac)m! s!
TNormalized ∆ρyx 100000......08642000..00w(c )=! 002..1100 n00m..22!Ωµ ( cm)ρyx-0000....01000505000...0330.10.2000...4430.400.05..55100000......08642000..00(d)!00..11 00..22!Ωµ ( cm)ρyx--00000000......000000w246420..033.0 =0.1 10.002H2. .0(404T.3) 0n.4m000.5..55 number118642020 (bUS-1)kp!.0 De0c.0reasingDS1 kof.i0ewlnd Ω cm)(ρ yx-000...011 (d-1)!.0 0.0w = 2510NA. 0nHHmEE
Tormalized ∆ρyx 10000.....08642 (e)! !Ωµ ( cm)ρyx---0000000.......000000064202460w.0 0=.1 01.2000.3 0n.4m0.510000.....08642 (wf) != 50 nm!Ωµ ( cm)ρyx--00000.....00000840480.00.10.2H 0(T.3)0.40.5 nu 420 -1.0 µ0H0. 0(T) 1.0 µ(ρ yx -0.1 -1.0 µ0H0. 0(T) 1T.0HE
N 0.0 0.0
0.0 0.1 0µ.20H (0T.)3 0.4 0.5 0.0 0.1 0µ.20H (0T.)3 0.4 0.5 (e)! 250 n +m!z!
m
FIG. 3. (color online). (a)-(f) Magnetic-field dependence i! ii! iii! iv! ! -!
of difference between Hall resistiveities in increasing and de-
creasing field processes, which is denoted by ∆ρT . Normal-
yx
ized ∆ρT by its maximum value is presented. Insets are
yx
magnified images of each ρyx at low fields. v! vi! vii! viii! ix!
FIG. 4. (color online). Conceptual diagrams for skyrmion
formation and corresponding Hall resistivities in a 250-nm-
ing magnetic field from a large negative field (< −Hc) widenano-scalecircuit. Variationsofnumbersofcore-upand
[Fig. 4(a)],the magnetic state is anticipatedto tracethe core-down skyrmions as functions of magnetic field in (a) in-
successive stages as illustrated in Fig. 4(e): first, core- creasingand(b)decreasingfieldprocesses. Corresponding(c)
Hallresistivityand(d)itsdecompositionintoρN (NHE),ρA
up skyrmions gradually pop up [panel (ii) of Fig. 4(e)] yx yx
(AHE), and ρT (THE). (e) Series of schematic illustrations
in the background of ferromagnetic state of pointing- yx
representing an expected development of magnetic structure
downmoments(i);next,high-densitycore-upskyrmions,
as a function of magnetic field.
possibly skyrmion lattice state, is realized and remains
aroundzerofieldasthemetastablestate(iii);asthemag-
netic field increase from zero to a positive value, core-up
skyrmions are destabilized to rapidly disappear coalesc- fields above H and magnetization M, respectively. The
c
ing with neighboring ones (iv); and then form a convo- overall profile of ρT bears a remarkable resemblance to
yx
lutedhelicalstructure(v);thehelicalstructurefragments the extracted ρT of the 10-µm device [Fig. 1(d)], which
yx
to formcore-downskyrmions(vi), which then crystallize
supports our model. The total Hall resistivity as simu-
(vii) and subsequently begin to disappear (viii); finally,
lated [Fig. 4(c)] also shows a good agreement with the
spins are fully polarized upward to form the ferromag-
experimental results [Fig. 2(a)]. The discretized feature
netic state (ix). of ρT especially shows up in parts of the hysteresis loop
yx
Magnetic-field profile of Hall resistivity can be simu- where the rapid switch between core-up and core-down
lated,asshowninFigs. 4(c)and4(d),fromthe assumed skyrmions occurs, which we focus on in Fig. 3.
skyrmion formation [Figs. 4(a) and 4(b)]. A unit of
In conclusion, we observed the discretized topologi-
topologicalHallresistivityproducedbyasingleskyrmion
cal Hall effect in nano-structured FeGe Hall-bar devices
is determined from ∆ρT : Since the maximum value of
yx with their sizes of one or a few skyrmions. The dis-
∆ρTyx represent the difference between topological Hall continuouschangesin topologicalHallresistivity,arising
resistivitiesproducedby close-packedcore-upand-down from discontinuous motion, creation and annihilation of
skyrmions, the unit of ρTyx is ∆ρTyx divided by twice the skyrmions, represent the quantized nature of the emer-
maximum capacity (i.e., 2×9 skyrmions) of the overlap gentmagneticfieldinherentineachskyrmion,whichgets
area (w×w). In the present example, the unit of ρTyx is buried in many other skyrmions residing in a large-size
derivedasapproximately5nΩ cmfrom∆ρT /(2×9). We sample used in prior experimental studies. The present
yx
showthesimulatedρT inFig. 4(d)alongwithNHEand study shows that the emergent magnetic field produc-
yx
AHE evaluated from the slope of Hall resistivity at high ing the topological Hall effect can be utilized to detect
5
single skyrmion in future skyrmion-based nanodevices. [18] C.Franz,F.Freimuth,A.Bauer,R.Ritz,C.Schnarr,C.
Skyrmion formations in FeGe films and nano-devices re- Duvinage,T.Adams,S.Blu¨gel,A.Rosch,Y.Mokrousov,
quirefurthervalidationbyreal-spaceobservations,which and C. Pfleiderer, Phys.Rev.Lett. 112, 186601 (2014).
[19] A. Fert, V. Cros, and J. Sampaio, Nature Nanotech. 8,
are left as future challenges.
152 (2013).
The authors thank T. Arima, W. Koshibae, and T. [20] N. Nagaosa, and Y. Tokura, Nature Nanotech. 8, 899
Yokouchi for enlightening discussions. This work was (2013).
supported by Young Scientists(Start-up) No. 26886005, [21] S.Heinze,K.vonBergmann,M.Menzel,J.Brede,A.Ku-
Scientific Research(S)Nos. 24224009and 24226002,and betzka, R. Wiesendanger, G. Bihlmayer, and S. Blu¨gel,
the Funding ProgramofWorld-LeadingInnovativeR&D Nature Phys.7, 713 (2011).
[22] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B.
onScience andTechnology(FIRST program)on“Quan-
Wolter,K.vonBergmann,A.Kubetzka,andR.Wiesen-
tum Science on Strong Correlation” initiated by the danger, Science 341, 636 (2013).
Council for Science and Technology Policy, Japan.
[23] Here we note that thereare experimentalreports study-
ingfinite-sizeeffectonskyrmionformation byreal-space
observation [X. Z. Yu,J. P. DeGrave,Y. Hara, T. Hara,
S. Jin, and Y. Tokura, NanoLett. 13, 3755 (2013).] and
resistivity measurements [H. F. Du, J. P. DeGrave, F.
a PresentAddress: TechnologyandBusinessDevelopment Xue, D. Liang, W. Ning, J. Y. Yang, M. L. Tian, Y. H.
Unit,MurataManufacturingCo.,Ltd.,Nagaokakyo,Ky- Zhang, andS.Jin, NanoLett.14, 2026 (2014).] inMnSi
oto 617-8555, Japan. nanowires.However,thewirewidthwasstilltoolargeto
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