Table Of ContentS.-Z.LinandA.Saxena,Phys.Rev.B93,060401(R)(2016).
DynamicsofDiracstringsandmonopole-likeexcitationsinchiralmagnetsunderacurrentdrive
Shi-Zeng Lin∗ and Avadh Saxena
TheoreticalDivision, LosAlamosNationalLaboratory, LosAlamos, NewMexico87545, USA
(Dated:February11,2016)
Skyrmionlinesinmetallicchiralmagnetscarryanemergentmagneticfieldexperiencedbytheconduction
electrons. Theinflowandoutflowofthisfieldacrossaclosedsurfaceisnotnecessarilyequal, thusitallows
fortheexistenceofemergentmonopoles. Oneexampleisasegmentofskyrmionlineinsideacrystal,where
a monopole and antimonopole pair is connected by the emergent magnetic flux line. This is a realization of
Dirac string-like excitations. Here we study the dynamics of monopoles in chiral magnets under an electric
current. Weshowthatintheprocessofcreationofskyrmionlines,skyrmionlinesegmentsarefirstcreatedvia
theproliferationofmonopolesandantimonopoles. Thentheselinesegmentsjoinandspanthewholesystem
6 throughtheannihilationofmonopoles.Theskyrmionlinesaredestroyedviatheproliferationofmonopolesand
1 antimonopolesathighcurrents,resultinginachiralliquidphase. Wealsoproposetocreatethemonopolesin
0 acontrolledwaybyapplyinganinhomogenouscurrenttoacrystal. Remarkably,anelectricfieldcomponent
2
inthemagneticfielddirectionproportionaltothecurrentsquaredinthelowcurrentregionisinducedbythe
b motionofdistortedskyrmionlines,inadditiontotheHallandlongitudinalvoltage.Theexistenceofmonopoles
e canbeinferredfromtransportorimagingmeasurements.
F
PACSnumbers:75.10.Hk,75.25.-j,72.25.-b,75.78.-n
0
1
Skyrmioninmagnetsisastablemesoscopictopologicalex- not be applied to insulators. While magnetic monopoles as
]
l citationwherespinswrapthesurfaceofasphereonce.[1,2] elementary particles have never been found experimentally,
e
- Skyrmionshavebeenobservedinmagneticmaterials,suchas quasiparticles resembling monopoles have been identified in
r
t B20chiralmagnetswithoutinversionsymmetry [3–5],mul- several condensed-matter systems, such as spin ice [23] and
s
tiferroics[6,7]andevencentrosymmetriccompounds[8,9]. chiral magnets discussed here as well as Bose-Einstein con-
.
at Thesefindingssuggestthatskyrmionsareubiquitousinmag- densates [24]. A unique feature of monopoles in skyrmion
m netic systems. The typical size of skyrmions is about 10 nm systems is that they can be driven by electric currents. The
and skyrmions form a triangular lattice. Skyrmions can be monopoles in chiral magnets have attracted considerable in-
-
d drivenbyelectriccurrent [10–12],temperaturegradient [13– terestrecently. [25–27]
n
15]andelectricfieldgradient[16,17]. Moreoverthethresh- Inathinfilm,skyrmionbecomesapancake-likeobjectand
o
c old current that makes skyrmions mobile is 5 to 6 orders of nomonopoleisallowedtoappearbecausethesystemisuni-
[ magnitudesmallerthanthatforamagneticdomainwall. [10– form along the direction perpendicular to the film. It was
12]Thusskyrmionsarepromisingcandidatesforspintronics demonstratedtheoreticallythatskyrmionscanbecreateddy-
2
v applicationsandrecentlytheyhaveattractedsignificantatten- namically by applying a strong current to a fully spin po-
6 tion. [18,19] larized state in chiral magnetic thin films. [28] This is be-
5 Inmetals,thespinoftheconductionelectroninteractswith causethecurrentinducesaDopplershiftinthemagnonspec-
3
the localized magnetic moment associated with skyrmions trum [29, 30] and renders the spin polarized state unstable
4
through the Hund’s coupling. In the strong coupling limit, when the spectrum becomes gapless, after which skyrmions
0
. thespinofaconductionelectronchangesadiabaticallywhen are nucleated dynamically. When the current increases fur-
1
itmovesaroundaskyrmionanditgainsaBerryphasewhich ther, skyrmions are destroyed and the system evolves into a
0
5 canbedescribedbyaneffectivemagneticfield. Theemergent chiral liquid with strong spatial and temporal fluctuations in
1 magnetic field is related to the skyrmion topological charge, spinchirality.
: BE =(cid:126)c(cid:15) n·(∂ n×∂ n)/(2e),andisextremelystrongdueto The creation and destruction of skyrmions by current can
v ijk j k
i thesmallsizeoftheskyrmion. Herenisaunitvectorrepre- also occur in 3D, where the emergent monopoles play an
X
sentingthedirectionoflocalizedspinand(cid:15) isthetotallyan- important role. A skyrmion line percolating the whole sys-
ijk
r tisymmetricmatrixwithi, j, k = x, y, z. Askyrmionlinein tem(connectingsurfacesofthesystem)cannotbecreatedin-
a
a3Dmetalthuscanberegardedasanemergentmagneticflux stantly.Insteadsegmentsofskyrmionlinesarecreated.Atthe
line with quantum flux Φ = hc/e. It was demonstrated ex- endsoftheselines,∇·BE =±ρ ,i.e. thereisamonopoleand
0 m
perimentallythattwoskyrmionlinescanmergeandbecomea antimonopolepairconnectedbyaskyrmionline,asschemat-
singleline.Thisimpliesanemergentmagneticmonopolewith ically shown in Fig. 1 (a). This is a realization of Dirac
fluxΦ atthemergingpoint,whichcanbeverifiedbyintegrat- stringincondensedmattersystems. Thesesegmentsthenjoin
0
ing BE over a closed surface around the merging point. [20] together to form skyrmion lines spanning the whole system
A monopole and antimonopole pair can also be created by throughtheannihilationofmonopolesandantimonopoles. In
thermalfluctuations. [21]Theemergentmagneticfluxofthe thecourseofskyrmiondestruction,skyrmionlinesbreakinto
skyrmionandhencethemonopoleisaneffectivedescription segments and monopole and antimonopole pairs are created
for the conduction electrons [22], thus these concepts can- dynamicallybycurrent. Thesephysicalpicturesareborneout
2
(a) (b) (c) (d)
z
H
a y
J x
FIG.1.(coloronline)(a)SchematicviewofaskyrmionsegmentasaDiracstring-likeexcitation,wherethereisamonopoleandantimonopole
pairatthe twoends. (b)Straight skyrmionlinelattice. (c)Snapshotofspinconfiguration afterthemagnonicinstabilityofspin polarized
state.Skyrmionsegmentsarecreateddynamicallyassociatedwithmonopoleandantimonopoleexcitations.Themonopoleandantimonopole
annihilateordisappearatthesurfaces,throughthemergingoftheskyrmionsegments.Finallythesystemreachesastatewithstraightskyrmion
linelattice,andtheymoveasawhole.(d)Chiralliquidatahighcurrentwhenskyrmionlinesaredestroyed.Coloratthreesurfacesrepresents
n (redforn =1andblueforn =−1)andthevectorfieldatthetopsurfacedenotesthen andn components.
z z z x y
byournumericalsimulationsbelow. 40×40×40 (J /D)3. The system is discretized into a cu-
ex
Weconsidera3Dchiralmagnetdescribedbythefollowing bicmeshwithperiodicboundaryconditionsinmostcases. To
Hamiltonianinthecontinuumlimitbecausetheskyrmionsize countthemonopole/antimonopolenumber,wespliteachsur-
ismuchbiggerthantheunderlyinglatticeparameter[1,2,31– face of a unit cubic cell into two triangles, and calculate the
33] solid angle Ω subtended by the three spins n in a triangle,
i i
(cid:90) (cid:20)J (cid:21) tan(Ωi/2)=n1·(n2×n3)[1+n1·n2+n2·n3+n3·n1]−1. [20]
H = dr3 ex(∇n)2−Ha·n +HDM, (1) The monopole number of this unit cell is obtained by sum-
2
ming the solid angle for these 12 triangles on the surface of
(cid:90) the unit cubic cell, i.e. n = (4π)−1(cid:80)12 Ω, which is quan-
(cid:104) (cid:16) (cid:17)(cid:105) M i=1 i
H = D dr3 n ∂ n −n ∂ n +n ∂ n −∂ n , (2) tized,n =±1, 0.
DM x y z y x z z x y y x M
Wefirststudythenucleationofskyrmionsintheferromag-
where Jex is the exchange coupling and HDM is the netic state due to the magnon instability induced by current.
Dzyaloshinskii-Moriya (DM) interaction due to the absence The magnon dispersion in the fully spin polarized state is
of inversion symmetry. [34–36] Here we consider that the givenby[28]
DMvectorDisalongthebonddirectioninthe x-yplaneand (cid:126)γ γ(1+iα)(cid:16) (cid:17)
the component along the z direction is zero, D = D = D Ω(k)= J·k+ H +J k2 . (4)
andD = 0. [37]Thefieldisalongthezdirectioxn,H y= H zˆ 2e α2+1 a ex
z a a
withaunitvectorzˆ,andtheskyrmionlinesarealignedinthe The term J · k accounts for the magnonic Doppler shift.
same direction. In 3D systems, skyrmions are stable only in The magnon gap for J = 0 is due to the external
a small phase region in magnetic field-temperature parame- fiel√d and it becomes gapless at a threshold current Jm =
ter space [3, 38]. Here we consider a metastable skyrmion 4e HaJex/[(cid:126)(α2+1)],signalinganinstability. Theaveraged
linelatticeatzerotemperaturewhichcanbeobtainedthrough skyrmion and monopole density in the stationary state as a
fieldcooling[20].ThedynamicsofspinsfollowstheLandau- function of current is displayed in Fig. 2 (a). Right after
Lifshitz-Gilbertequation theinstability, skyrmionsegmentsarenucleatedaccompany-
ing the creation of monopoles and antimonopoles, see Fig.
(cid:126)γ
∂tn= 2e(J·∇)n−γn×Heff +αn×∂tn, (3) 1as(ac)f.unInctiFoing.of2ti(mb)e,atrheesshkoywrmniwonheanndthme oinnsotapboilleitydeinsstirtiiges-
where γ is the gyromagnetic ratio, α is the Gilbert damping gered.Shortskyrmionsegmentsappearwithahighmonopole
constant,Heff ≡−δH/δnistheeffectivemagneticfieldandJ density. During the relaxation, the length of skyrmion seg-
isthespinpolarizedcurrent. Thecurrentisinthe xdirection ment increases accompanying an increase in skyrmion den-
inoursimulations. sity. Oncethesesegmentsmeet,monopoleandantimonopole
In simulations, we calculate the skyrmion density in the annihilate resulting in longer skyrmion segments. This pro-
x-y plane ρ(r) = n · (∂ n × ∂ n)/(4π), or the BE compo- cess repeats until the skyrmion lines span the whole system
x y z
nent. The emergent electric field induced by skyrmion mo- whennomonopoleandantimonopoleisleft. Meanwhilethe
tion is EE = (cid:126)n · (∇n × ∂n)/(2e). The system size is skyrmion lines become straight to minimize the line tension.
t
3
(a) 0.010 (b) (c) 0.008 0.004(d)
22Skyrmion density [D/J]ex0000....000000002468 33Monopole Density [D/J]ex1111100000-----118642022Skyrmion density [D/J]ex0000....000000001234 33Monopole density [D/J]ex1100--54 22Skyrmion density [D/J]ex00..000067 33Monopole density [D/J]ex000...000000123
- -0.000 - 0.000
0.0000 1 2 3 4 5 610-14 600 800 1000 1200 140100-6 0.0050 100 200 300 400
J [2De/ ] t [J /(γD2)] t [J /(γD2)]
ex ex
FIG.2. (coloronline)(a)Skyrmionandmonopoledensityversuscurrentinthestationarystatefor H = 0.8 D2/J . At J = 3.84 De/(cid:126),
a ex
skyrmionsegmentsarecreatedaftertheinstabilityoftheferromagneticstate. (b)Timedependenceofskyrmionandmonopoledensitywhen
theinstabilityistriggeredatJ=3.7De/(cid:126)att=0forH =0.6D2/J .Monopolesandantimonopolesarecreatedaftertheinstabilitybutthey
a ex
finallydisappearinthestationarystate,resultinginastraightskyrmionlinelattice. (c)Dynamicsofskyrmiondestructionatahighcurrent
J=10.0De/(cid:126)andH =0.5D2/J ,wheremonopolesandantimonopolesarenucleated.(d)Distributionofmonopolesandantimonopolesin
a ex
thechiralliquidphaseatJ=8.64De/(cid:126)andH =0.8D2/J .
a ex
The monopole density decreases exponentially and the typi- ease of discussion, let us consider the case that current only
caltimescaleforthisprocessisabout300 J /(γD2). Inthe flowsatthebottomsurfaceofthecrystal. Theskyrmionlines
ex
end,straightskyrmionlinelatticedrivenbycurrentmovesas near the surface are driven by the Lorentz force due to the
awhole,seeFig. 1(b). current, Jδ(z), and start to move. This causes distortion of
Theskyrmiondensityincreasesstepwisewithcurrent. For the lines and drags the skyrmion lines inside the crystal to
alowskyrmiondensity,thesystemisclosetoaferromagnetic move due to the elastic energy of the skyrmion lines [see
state and one would expect a similar magnonic instability as Fig. 3(b)]. Forasmallcurrent,theelasticforcecanbalance
thatinEq. (4). For J closeto J ,theskyrmiondensityislow the different Lorentz forces acting on the surface and mid-
m
and the magnetic instability persists at the same current. In dle of the skyrmion line, and the skyrmion lines move as a
this case skyrmion segments do not relax into straight lines. whole. For an elastic skyrmion line lattice, the equation of
Whentheskyrmiondensitybecomeshigh,theinstabilitycur- motionforthelatticecanbedescribedbyadisplacementvec-
rentincreases. Themagnonicinstabilitydisappearswhenthe toru(z)=[u (z), u (z), 0][39,40]
x y
skyrmionlatticeisreachedatahighskyrmiondensityforan
intermediate J. This accounts for the stepwise increase of 4π(cid:2)ηv(z)+v(z)×zˆ(cid:3)= Φ0zˆ×Jδ(z)+ (cid:96)∂2u(z), (5)
γ c 2 z
skyrmiondensity. Inthesesteps,nomonopoleexistsbecause
theskyrmionlinelatticespansthewholesystem. wherev = [∂u , ∂u , 0]istheskyrmionvelocity, andη (cid:28)
t x t y
At a high current, the skyrmion density decreases sharply 1 is the dimensionless viscosity due to the Gilbert damping
indicating destruction of skyrmions [see Fig. 2 (a)]. The for localized spins and the Ohmic dissipation by conduction
system evolves into a chiral liquid displayed in Fig. 1 (d), electrons.Here(cid:96)isthetiltmodulusoftheskyrmionlatticeand
wherespinchiralityfluctuatesstronglyinspaceandtime. In it becomes skyrmion self-energy per unit length for a single
thecourseofskyrmiondestruction,skyrmionlinesbreakinto line. From Eq. (5), the velocity component parallel to the
segmentswithamonopoleandanantimonopoleatends(sim- currentisv = −γ(cid:126)J/[2e(1+η2)L ]andthatperpendicularto
(cid:107) z
ilar to the Dirac strings). In other words, skyrmion lines are the current is v = γ(cid:126)ηJ/[2e(1+η2)L ] with L the system
⊥ z z
destroyedduetotheproliferationofmonopole-antimonopole sizealongthezdirection. Forasmallcurrent,theequationof
pairsinthisdynamicphasetransition. Thetimeevolutionof motionisthesameasthatforasingleskyrmioninthinfilms.
monopole and skyrmion density is presented in Fig. 2 (c). Thedisplacementfieldisu∝ Jz2.
Skyrmion density decreases while the monopole density in- However for a strong current, the skyrmion line breaks.
creases. In the chiral liquid, both densities fluctuate justi- The skyrmion segments at the surface and inside the crystal
fying the liquid nature of the final state. These fluctuations travelwithdifferentvelocities. Thisdecoupledmotionofthe
areinduceddynamicallybycurrent. AsshowninFig. 2(d), skyrmionsegmentsispossiblebecausethemonopoleandan-
themonopolesandantimonopolesaredistributedrandomlyin timonopole pairs are allowed for the emergent magnetic flux
space, and can be regarded as a monopole gas. The dynam- lineassociatedwiththeskyrmion.
ical phase transitions associated with the skyrmion creation The above picture is confirmed by our simulations. We
anddestructionareofthefirst-orderbecausethereisastrong consider a T-shaped crystal shown in the inset of Fig. 3
hysteresis.[28]Bothtransitionscanbedetectedfromcurrent- (a). The current is injected through the bottom of the crys-
voltagecharacteristicsbecausetheemergentelectricfieldde- tal. Theskyrmionlinesproduceanemergentelectromagnetic
pendsontheskyrmiondensity. fieldwhichaffectsthemotionofconductionelectrons. How-
We next discuss the controlled creation of monopoles and ever, as shown in the Supplement Materials [41], the emer-
antimonopoles by driving part of the skyrmion lines. For gent electric field is much smaller than the real electric field
4
(a) 0.05 (b) (c) 6.0x10-5 (d)
EE at the top monopole
32d [D/(2eJ)]γex00..0034 J EEExEyExEy aaattt ttthhheee tbbooopttttoomm 33density [D/J]ex3.0x100.0-5 antimonopole
ctric fiel 00..0012 onopole -3.0x10-5
Ele M
0.00 -6.0x10-5
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
J [2De/ ] J [2De/ ]
FIG.3.(coloronline)(a)Emergentelectricfieldsatthetopandbottomsurfacesofthecrystal.Atasmallcurrent,EE attheoppositesurfaces
isthesameindicatinganelasticmotionofskyrmionlines.Beyondathresholdcurrent,EE atthebottomsurfaceunderacurrentdriveismuch
biggerthanthetopone,implyingthebreakingofskyrmionlines.Insetisasketchshowinghowtoinjectcurrentatthebottomsurfaceandthe
calculatedcurrentdistribution[41]. (b)Snapshotofspinconfigurationafterthebreakingofskyrmionlines. (c)Monopoleandantimonopole
densityversuscurrent. (d)Distributionofmonopolesandantimonopolesafterthesegmentationofskyrmionlines. Inthiscalculation, we
have used open boundary conditions in the z direction. Here H = 0.5 D2/J and J = 0.26 De/(cid:126) for (b) and (d). The system size is
a ex
L =2L =2L =100J /D.
z x y ex
inducedbytheinjectedcurrent,andweneglecttheemergent component. When the skyrmion lines are distorted accord-
electric field in the calculation of current distribution. The ing to Eq. (5), BE acquires an in-plane component, BE =
(cid:112)
currentdistributionisobtainedbysolving∇2V =0,subjectto BE[∂ u , ∂ u , 1]/ (∂ u )2+(∂ u )2+1andEE isnonzero,
z x z y z x z y z
theproperboundaryconditions,withVbeingthevoltage[41]. as shown in Fig. 4 (a). The induced electric field EE ∝ J2
z
AsdisplayedinFig. 3(a),belowathresholdcurrent,theelec- increases nonlinearly with current in the elastic region and
tricfieldatthetopandbottomarethesame,indicatingmotion it suddenly decreases when the skyrmion line breaks, after
of skyrmion lines as a whole. Beyond the threshold current, which skyrmion segments become mostly aligned with the
theskyrmionsegmentsatthebottomsurfacemoveatahigher magneticfield.Inthechiralliquidphase,EEfluctuatesaround
z
velocity. Atthesametime,monopolesandantimonopolesare zero due to the fluctuations of the density of monopoles and
created which are confined at the bottom surface [see Figs. antimonopoles;seeFig. 4(b). ThereforeEE providesaclear
z
3 (c) and (d)]. At a large current, the skyrmion density near transportsignatureoftheexistenceofmonopoles. Weremark
thebottomsurfacedecreasesduetothedynamicaldestruction that the nonuniform current distribution can also produce a
of skyrmions. This causes a rapid increase of antimonopole local real electric field E along the magnetic field direction.
z
density. Meanwhile the skyrmion density in the current-free However E averagedoverthecrystalvanishesbecausethere
z
region does not change, which results in a higher density of isnonetcurrentinthemagneticfielddirection.
antimonopolesthanthatofmonopoles,seeFig. 3(c). Theexistenceofmonopolescanalsobeinferredfromimag-
The electric field along the magnetic field direction EE ing measurements, such as magnetic force microscope [20].
z
is particularly interesting. Let us first consider EE due to Forinstance,onemayfirstapplyastrongcurrenttodrivethe
z
the stretching of skyrmion lines by a surface current. The system into the chiral liquid phase. Then the current is re-
electric field is given by EE = v × BE/c. For a straight moved and the system can be trapped in a metastable state
skyrmion line, EE = 0 because BE = BEzˆ only has the z with monopoles. By visualizing the skyrmion density at the
z
opposite surface, one can extract the information about the
monopolesinsidethecrystal.
0.002 0.06
(a) (b ) If we start with a skyrmion line lattice in the ground state
and drive it, the magnon instability due to the Doppler shift
2eJ)]ex 0.03 is absent. At a high current, we still have the destruction of
3/(20.001 0.00 skyrmionlinesinnonequilibriumsimilartothoseinFig. 2(a)
g(cid:1)D viatheproliferationofmonopolesandantimonopoles.
E [z -0.03
E Tosummarize,weshowthattheemergentmonopolesplay
0.0000.0 0.1J [2 D e /(cid:1)]0.2 0.3-0.0165000 160t0 [0J /(gD 21)7]000 18000 an important role in the skyrmion line creation and destruc-
ex tionprocessinthepresenceofanelectriccurrent. Thecurrent
densityrequiredforthemonopoleproliferationisoftheorder
FIG. 4. (color online) (a) Emergent electric field along the mag-
neticfielddirectionEE whenskyrmionlinesarestretchedbyacur- of1012A/m2fortypicalmaterialparameters. Themonopoles
z
rent at the bottom surface in a T-shaped crystal in Fig. 3 (a). (b) canalsobecreatedinacontrolledwaybyapplyinganinho-
FluctuationsofEE asafunctionoftimeinthechiralliquidphaseat mogeneous current to the crystal. Especially we predict an
z
J=10.0De/(cid:126).HereHa=0.5D2/Jex. inducedelectricfieldcomponentalongthemagneticfielddi-
5
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thank Cristian D. Batista for helpful suggestions and a crit-
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6
wherea isthelinearsizeofskyrmionandvistheskyrmion
s
velocity, which can be estimated using the Thiele’s equation
ofmotionforskyrmions[39]
a3J
v≈ , (7)
2e
withathelatticeconstantforthespinsystem. TheratioofEE
totherealelectricfieldE=ρJwithρbeingtheresistivityis
V+ V- EE ha3
≈ . (8)
E 2e2a2ρ
s
For MnSi, we have a ≈ 10 nm, a ≈ 0.45 nm and ρ ≈ 3×
s
FIG.5. (coloronline)T-shapedcrystalandtheresultingcurrentdis- 10−7 Ω·m [43], thus we estimate EE/E ≈ 0.03. Therefore,
tribution. Thevectorfielddenotesthe xandycomponentsofJand we can neglect the skyrmion contribution in the calculations
colorrepresentstheintensity.
oftheelectriccurrentdistributioninthecrystalbelow.
The current distribution is governed by the equation for
voltageV
CalculationofthecurrentdistributioninaT-shapedcrystal
∇2V =0. (9)
WeconsideraT-shapedcrystalsketchedinFig. 5. Acur-
rent is applied into the crystal through two electrodes at the The boundary condition in the region without electrodes is
bottom, V+ and V−. The presence of skyrmion affects the thatthespatialderivativeofthevoltagenormaltothebound-
motion of conduction electrons and changes the conductiv- aryvanishes,∂V = 0. Fortheboundaryincontactwithelec-
l
ityofthecrystal. Theskyrmionmodifiestheconductivityin trodes, the boundary condition is V = V+ and V = V− = 0.
twodifferentways. Theskyrmionproducesaneffectivemag- The resulting current distribution is J = −σ∇V, with σ be-
neticfieldactingontheconductionelectronsandchangesthe ing the conductivity. We solve Eq. (9) numerically for the
conductivity. Meanwhile when the skyrmions start to move, geometryshowninFig. 5.
the motion of skyrmions produces an emergent electric field Thecurrentmainlyflowsinthebottomregionofthecrys-
whichalsomodifiestheconductivity. Forthetransversecon- tal. Thecurrentdistributiondoesnotdependontheintensity
ductivity,theformercontributionisknownasthetopological ofthecurrentbecausewecanobtainthedesiredcurrentinten-
Hallconductivityandthelatterastheskyrmionflowconduc- sitysimplybyrescalingV+.Theregionwherecurrentextends
tivity. ForMnSi,bothcontributionsduetotheskyrmionsare intheverticaldirectionisofthesameorderasthelateraldi-
found to be small compared to the background conductivity mension of the crystal at the top. The nonuniform electric
from the conduction electrons in experiments. [12, 42] Be- current distribution yields a local electric field in the verti-
low we provide an explicit estimate of the emergent electric cal direction E . However, the averaged E over the crystal
(cid:82) z (cid:82) z
field induced by skyrmion motion and compare it to the real vanishes, dr3E = dr3J /σ = 0, because there is no net
z z
electricfield. Theemergentelectricfieldis[22] electriccurrentintheverticaldirection.
Knowing the current distribution in the crystal, we then
hv
EE ≈ , (6) solvetheLandau-Lifshitz-Gilbertequationnumerically.
ea2
s
