Table Of ContentSpin-Cherenkov effect in a magnetic nanostrip with interfacial
Dzyaloshinskii-Moriya interaction
Jing Xia,1,2 Xichao Zhang,1,2 Ming Yan,3 Weisheng Zhao,4,5,∗ and Yan Zhou1,2,6,†
1SchoolofElectronicsScienceandEngineering, NanjingUniversity, Nanjing210093, China
2Department of Physics, University of Hong Kong, Hong Kong, China
3Department of Physics, Shanghai University, Shanghai 200444, China
4Fert Beijing Institute, Beihang University, Beijing 100191, China
5SchoolofElectronicandInformationEngineering, BeihangUniversity, Beijing100191, China
6CenterofTheoreticalandComputationalPhysics, UniversityofHongKong, HongKong, China
6 (Dated:April13,2016)
1
Spin-Cherenkoveffectenablesstrongexcitationsofspinwaves(SWs)withnonlinearwavedispersions.
0
TheDzyaloshinskii-Moriyainteraction(DMI)resultsinanisotropyandnonreciprocityofSWspropaga-
2
tion. Inthiswork,westudytheeffectoftheinterfacialDMIonSWCherenkovexcitationsinpermalloy
r thin-filmstripswithintheframeworkofmicromagnetism. Byperformingmicromagneticsimulations,it
p
isshownthatcoherentSWsareexcitedwhenthevelocityofamovingmagneticsourceexceedstheprop-
A
agationvelocityoftheSWs. Moreover,thethresholdvelocityofthemovingmagneticsourcewithfinite
2 DMIcanbereducedcomparedtothecaseofzeroDMI.Ittherebyprovidesapromisingroutetowards
1 efficientSWgenerationandpropagation,withpotentialapplicationsinspintronicandmagnonicdevices.
] PACSnumbers:75.30.Ds,75.78.Cd,85.70.-w,85.75.-d
l
l
a
h TheCherenkovradiationoflightoccurswhenachargedpar-
-
ticle moves faster than the light speed within a medium [1].
s
e ThiseffectisnamedafterPavelAlekseyevichCherenkov,who
m
studied this phenomenon experimentally. Cherenkov radia-
. tion is used frequently in particle identification detectors in
t
a particlephysics[2]. TheCherenkoveffectisanalogoustothe
m sonicboomproducedbyshockwavespropagatingawayfrom
- anaircraft,ifitsspeedexceedsthesoundvelocity. Similarto
d
the Doppler effect observed in different physical system [3–
n
7], the Cherenkov effect belongs to one of the fundamental
o
phenomenainducedbytheradiationofmovingsources. The
c
[ Cherenkov-likeeffectofspinwaves(SWs)hasbeentheoreti-
callystudiedinferromagnets,whichcanbeusedtoexciteco- Figure1. Schematicsofthemicromagneticallymodeledsystem.
2
herentSWswithoutthenecessityofexternalalternatingmag- An external rectangular shape field pulse (H), 12 nm wide in the
v
netic field or current [8, 9]. Recently, the study of the influ- x-directionand100nmlonginthey-direction, isappliedtoa12-
5
5 ence of the antisymmetric exchange interaction, namely the µm-long,100-nm-wide,and10-nm-thickmagneticstripwithamag-
nitude of 10 mT in the z-direction and constant speed of v . The
7 Dzyaloshinskii-Moriya interaction (DMI), on magnetic exci- h
colorscalerepresentstheout-of-planecomponentofthemagnetiza-
7 tations such as domain walls and vortex is one of the hottest
tionm ,whichhasbeenusedthroughoutthispaper.
0 topics in nanomagnetism and spintronics [10–20]. DMI is z
.
1 anantisymmetricinteractioninducedbyspin-orbitalcoupling
0 due to broken inversion symmetry in lattices or at the inter-
6 faceofmagneticfilms[12],whichhasbeenmeasuredforboth nanodiskhasbeenstudiedbymicromagneticsimulationswith
1 magneticinterfaces[21,22]andbulkmaterials[23,24]. DMI interfacialDMI[22].
:
v facilitatesthecreationoftopologicallyprotectedspintextures
Inthispaper,theinfluenceofinterfacialDMIontheSpin-
i inchiralmagneticmaterials, i.e. magneticskyrmions, which
X Cherenkoveffect(SCE)inpermalloystriparestudiedbymi-
arefavorableinformationcarriersinthenext-generationdata
cromagneticsimulations. Pictorialillustrationsofthesetupis
r
storage and spin logic devices. On the other hand, theoret-
a shown in Fig. 1, where a rectangular moving magnetic field
ical [13, 14, 16] and experimental [10, 15, 25] studies have
pulseisappliedtothepermalloystripwithamagnitudeof10
demonstrated that DMI leads to an asymmetrical spin-wave
mT along the +z-direction. Our numerical results show that
dispersion. DMI has also been measured in a wide range of
theinterfacialDMIleadstoareductionofthethresholdveloc-
materials including permalloy [22, 26, 27]. Specifically, the
ityofmovingsource,i.e.theminimumSWphasevelocity,for
asymmetry in the formation of a vortex state in a permalloy
coherentSWexcitationintheabsenceofexternalacmagnetic
field.ThereforeitprovidesapromisingroutetoreducingSCE
threshold to facilitate experimental realization of such effect
∗[email protected] inmagneticmedium.
†[email protected]
2
Results
SW excitation via SCE in permalloy strips. The response
of the magnetization distribution to the pulse velocity v is
h
showninFig.2. Alocalizedmagneticfieldofconstantmag-
nitudetravelingalongthewireaxialdirectionwithvelocityof
v ,isappliedtomimictheinteractingforcewiththemagneti-
h
zation. Themagnetizationdynamicsisstronglydependenton
the pulse velocity v . For D = 0 mJ m−2 and v = 500 m
h h
s−1 and900ms−1 (seeFig.2(a1-a2)),themovingmagnetic
field pulse only causes a distortion of the magnetization dis-
tributiontravelingwiththepulse. ThereisnoSWexcitation
by the moving pulse. As the velocity of the magnetic pulse
increases,SWexcitationsareobservedforv =1050ms−1,
h
1100ms−1, and1200ms−1, asshowninFig.2(a3-a5). At
v =1050ms−1,theSWsproceedingthesourceandlagging
h
thesourcearewell-distinguishedwithdifferentwavelengths.
Upontheapplicationofamovingdcmagneticfieldpulse,the
system reaches a dynamic equilibrium and the excited spin
waves comprise of two branches, giving rise to the SCE as
reportedinRef.9.
For the permalloy nanostrip with interfacial DMI, similar
magnetization dynamics occur as the moving field pulse is
applied. For the case of D = 1 mJ m−2, there is no SW
excitation by the moving magnetic field with the speed of
v = 500 m s−1 and 900 m s−1. The Cherenkov emission
h
ofSWsemergeatv = 1050ms−1, 1100ms−1, and1200
h
ms−1 inthepermalloystripasshowninFig.2(b3-b5). The
SWs excited in permalloy nanostrip are distorted due to the
presence of DMI. The interfacial DMI term for any in-plane
direction u can be expressed as Dz×u [28], which may be
treated as an effective field transverse to the magnetic track,
resulting in the distorted spin waves. As D = 2 mJ m−2,
Figure2. Snapshotsofthez-componentofthemagnetizationin
spin wave is excited when v = 900 m s−1, indicating that
h a magnetic strip in the vicinity of the field pulse traveling at a
thepresenceofinterfacialDMIleadstothedecreasingofthe constantspeed. (a1)D = 0mJm−2 andv = 500ms−1, (a2)
h
threshold velocity of the moving field source for spin wave D = 0 mJ m−2 and v = 900 m s−1, (a3) D = 0 mJ m−2 and
h
excitation. It should be noticed that the SWs with more sig- v =1050ms−1,(a4)D=0mJm−2andv =1100ms−1,(a5)
h h
nificantdistortionsareobservedatvh =1050ms−1,1100m D = 0mJm−2 andvh = 1200ms−1,(b1)D = 1mJm−2 and
s−1,and1200ms−1asshowninFig.2(c3-c5).Moreover,the vh = 500ms−1,(b2)D = 1mJm−2 andvh = 900ms−1,(b3)
differencebetweentheSWbranchesproceedingandlagging D = 1mJm−2 andvh = 1050ms−1,(b4)D = 1mJm−2 and
v = 1100 m s−1, (b5) D = 1 mJ m−2 and v = 1200 m s−1,
thesourcebecomesevenmoreobviouswithincreasingDMI h h
(c1)D=2mJm−2andv =500ms−1,(c2)D=2mJm−2and
strengthforagivenpulsevelocity. h
v = 900ms−1,(c3)D = 2mJm−2andv = 1050ms−1,(c4)
Figure 3 shows the numerically determined SW phase ve- Dh= 2mJm−2 andv = 1100ms−1,(c5)hD = 2mJm−2 and
locity vp(k) and group velocity vg(k) for D = 0 mJ m−2. vh =1200ms−1. h
TheinsetofFig.3showstheSWdispersioninthepermalloy
strip, which agrees well with the analytical results for zero
DMI(proportionaltok2). InFig.3,thephasevelocityv and
p are applied and the corresponding responses of the magneti-
groupvelocityv oftheSWsareextractedfromthedispersion
g zationdistributionareshowninFig.2(a1-a5). Thecalculated
relationofv (k) = ω/k andv (k) = dω/dk. Theminimum
p g k are shown with color stars in Fig. 3. As v is below the
ofthephasevelocityoccursaround1000ms−1,meaningthat h
criticalvelocity,thereisnoSWexcitationbythemovingfield
no SW is excited by the moving field pulse in the permalloy pulse. When v = 1050 m s−1, there are two SW modes
h
stripforthepulsevelocitiesbelowtheminimumv . Inother
0 observed in the permalloy strip. The spin waves formed in
words,thevelocityv isthethresholdvelocityofSCE,below
0 front of and behind the pulse exhibit different characteristics
whichthereisnocoherentSWsexcitation. Asthefieldpulse
in Fig. 2(a3), and the corresponding k values are shown in
moves at a velocity larger than v , there are two SW modes
0 Fig.3. TheCherenkovemissionofSWscanalsobeobserved
excited with different group velocity vg but equal phase ve- forthecaseofv =1100ms−1and1200ms−1. Thedisper-
locity vp. The SW packet with larger (cid:126)k (vg > vp) moves sions are in an ehxcellent agreement with the curves of vp(k)
in the front of the source and leaves the one with smaller(cid:126)k andv (k). Thecurvesofv (k)andv (k)extractedfromthe
g p g
(v < v )behind. Thefieldpulseswithdifferentvelocityv numericalSWdispersionrelationω(k),whichisobtainedby
g p h
3
Figure 3. Numerically determined phase velocity v (k) and Figure 4. Numerically determined phase velocity v (k) and
p p
group velocity v (k) of SWs in the magnetic strip which is 12 groupvelocityv (k)ofSWsinthemagneticstrip,withalength
g g
µmlong,100nmwide,and10nmthickasD =0mJm−2. The of 12 µm, a width of 100 nm, and a thickness of 10 nm when
valueofv (k)andv (k)areextractedfromtheSWdispersionrela- D = 2mJm−2. Thevalueofv (k)andv (k)areextractedfrom
p g p g
tionω(k)shownintheinset. v (k)hasaminimumv atk ,where theSWdispersionrelationω(k)shownintheinset.v (k)hasamin-
p 0 0 p
thetwocurvesv (k)andv (k)cross. Thecoloredstarsrepresent imumv atk ,wherethecurvev (k)crossesthecurvev (k). The
p g 0 0 p g
thewavevectorsoftheSWtailsexcitedbythemovingfieldpulse coloredstarsshowthewavevectorsoftheSWtailsexcitedbythe
appliedtothestripatthecorrespondingspeed.Thecoloredhorizon- moving field pulse applied to the strip at the corresponding speed.
talline,indicatingthespeedofthefieldpulse,connectthetwoSW Thecoloredhorizontalline, indicatingthespeedofthefieldpulse,
branches. connectsthetwoSWbranches.
applyingalocalizedacfieldandextractingthewavelengthof whereas v = 944 m s−1 for D = 1 mJ m−2. The criti-
0
the excited spin waves, can predict the Cherenkov excitation cal velocity further drops to 887 m s−1 as D increases to 2
ofSWsprecisely. mJ m−2, indicating that the Cherenkov emission of SWs in
permalloy strip with finite DMI can be excited more easily
Influence of DMI on SCE in permalloy. Figure 4 shows thanthecaseofzeroDMI.TheSWdispersionrelationswith-
the numerically determined phase velocity v (k) and group outDMI(D =0mJm−2)isaparabolicfunctionofSWvec-
p
velocity v (k) with D = 2 mJ m−2, where v (k) and v (k) tork whenk islarge[9,13,29,30]. InRefs.14and25, the
g p g
arecalculatedfromtheSWdispersionrelationω(k)shownin SWdispersionrelationisgivenanalyticallywhenDMIisin-
theinset. SimilartothecaseofD =0mJm−2,thereexistsa cluded,indicatingthattheinterfacialDMIresultsintheasym-
criticalvelocityv fortheCherenkovexcitationofSWs. The metric dispersion. Such asymmetric dependence due to the
0
Cherenkov excitation of SWs can be observed for v > v interfacialDMIdependsonthespinwavevectork andequi-
h 0
(887 m s−1). We also investigate the evolution of the SW librium magnetization distributions. Different with the typi-
branches by varying v for D = 2 mJ m−2. At v = 1050 cal Damon-Eshbach spin waves studied in Refs. 14 and 25,
h h
ms−1,theCherenkovexcitationofSWsisobservedwithtwo theSWsinourcasepropagatealongthe±x-directionswhile
SW modes formed in the permalloy strip. The calculated k theequilibriummagnetizationisalongthe+x-direction,cor-
markedwithcolorstarsagreesverywellwiththenumerically responding to the back-volume mode. The SW dispersion is
(cid:112)
determined curve of v (k). The curves of v (k) and v (k) given by ω = γ ω2−(kD∗m )2 when the DMI is con-
p p g 0 0 z0
extracted from the SW dispersion relation ω(k) can predict sidered. HereD∗ = 2D andω istheangularfrequencyin
theCherenkovexcitationofSWspreciselywiththepresence theabsenceofDMI.mµ0MSisthez0-componentofthemagneti-
z0
offiniteDMI. zation. TheSWdispersionrelationremainssymmetric,when
Figure 5 shows the minimum velocity v0 of the Spin- the interfacial DMI is included. The SCE threshold can be
Cherenkov excitation for D = 0 mJ m−2, 1 mJ m−2, 2 mJ calculatedby dω = ω,whichdecreaseswithDMI[9].
dk k
m−2 and 3 mJ m−2, as well as the corresponding dispersion
relation. Onlythebranchesofpositivekareshown. Thecrit-
ical velocity for the SCE decreases linearly with increasing Discussion
DMI in the investigated DMI range. As shown in the inset TheCherenkovemissionofspinwaveshasbeennumerically
of Fig. 5, the dispersion relation for D = 1 mJ m−2 devi- studied by considering the effect of DMI in permalloy strip.
ates from the case of D = 0 mJ m−2, resulting in a smaller TheresonantSpin-Cherenkoveffectcanbeexcitedastheve-
critical velocity v . v = 1000 m s−1 for D = 0 mJ m−2 locityofmovingmagneticfieldpulseexceedsacertainthresh-
0 0
4
whereMisthemagnetization,H istheeffectivefield,γ is
eff 0
theGilbertgyromagneticration,andαistheGilbertdamping
coefficient. Theeffectivefieldreadsasfollows
∂E
H =−µ−1 . (2)
eff 0 ∂M
TheaverageenergydensityE asafunctionofMisgivenby
M (n·M)2
E =A[∇( )]2−K −µ M ·H
M M2 0
S S
µ
− 0M ·H (M)+E , (3)
2 d DM
whereAandK aretheexchangeandanisotropyenergycon-
stants,respectively. HandH (M)aretheappliedandmagne-
d
tostaticself-interactionfieldswhileM =|M(r)|isthespon-
S
taneousmagnetization. TheE istheenergydensityofthe
DM
interfacialDMIoftheform[21,31,36,37]
Figure5.Theminimumvelocityv fortheSpin-Cherenkovexci-
0
tationasD =0mJm−2,1mJm−2,2mJm−2,and3mJm−2. E = D (M ∂Mx +M ∂My
Theinsetsshowsthecorrespondingdispersionrelations, whichare DM M2 z ∂x z ∂y
S
obtainedbyapplyingalocalizedacmagneticfieldandextractingthe ∂M ∂M
wavelengthoftheexcitedspinwaves. −Mx ∂xz −My ∂yz), (4)
wheretheM ,M ,M arethecomponentsofthemagnetiza-
x y z
oldvelocity. TheSpin-Cherenkoveffectthresholdcanbere- tionMandDistheinterfacialDMIconstant. Thefiveterms
ducedinthepresenceoffiniteDMI.Byfurthertuningthema- at the right side of equation (3) correspond to the exchange
terialparametersandgeometries,resonantspinwavescanbe energy,theanisotropyenergy,theappliedfield(Zeeman)en-
excitedthroughSpin-Cherenkoveffectatmuchlowerthresh- ergy, the magnetostatic(demagnetization) energyand thein-
oldvelocityofthemovingdcfieldsource. Ontheotherhand, terfacialDMIenergy,respectively.
itisalsofeasibletoincreasetheinteractionregionofthemov- The typical material parameters of permalloy, µ M = 1
0 S
ingdcfieldsourceonthemagneticstripinrealexperiments. T, exchange constant A = 1.3 × 10−11 J m−1, and zero
Indeed, wehaveinvestigatedtheSpin-Cherenkoveffectwith anisotropy are adopted [8, 9, 22, 26]. Considering that the
differentsizesofthemovingdcfieldsource(seeSupplemen- valueoftheeffectiveinterfacialDMIconstantinthePy/Ptbi-
taryFigure1),aswellaswithdifferentthicknessesofthemag- layershasbeenestimatedtobewithintherangeof1.0 ∼ 2.2
neticstrip(seeSupplementaryFigure2).TheSpin-Cherenkov mJ m−2 in Refs. 27 and 38, the interfacial DMI constant D
effect in finite-DMI system might be interesting for funda- is varied from 0 to 3 mJ m−2 in this paper. A rectangular
mental physics and also promising for potential applications shapefieldpulseisappliedto12-µm-long,100-nm-wide,and
in spintronic and magnonic devices, easing the experimental 10-nm-thickmagneticstripwithamagnitudeof10mTinthe
complexityanddifficultyofapplyinganexternalacmagnetic +z-directionanda12nmwidthinthex-direction,asshown
fieldorcurrentforresonantSWexcitations. in Fig. 1. The moving field pulse can be realized, for exam-
ple, withalaserbeamscanningoverthesurfaceofmagnetic
thinfilms[39]. Theresultswithdifferentwidthsofthemov-
Methods ing field pulse are shown in Supplementary Figure 1. The
results with different thicknesses of the magnetic strip are
Modeling and simulation. The micromagnetic simu-
shown in Supplementary Figure 2. For simplicity, the uni-
lations are performed using the Object Oriented Micro-
form field along the film thickness is assumed. All samples
Magnetic Framework (OOMMF) software including the
arediscretizedintocellsof3nm×5nm×5nminthesimu-
interface-induced Dzyaloshinskii-Moriya interaction (DMI)
lation. Gilbertdampingcoefficientαissettobe0.02andthe
extension module [19–21, 31–33]. The three-dimensional
value for Gilbert gyromagnetic ratio γ equals 2.211 × 105
time-dependent magnetization dynamics is controlled by the 0
mA−1s−1. Initially,themagnetizationorientsalongthe+x-
Landau-Lifshitz-Gilbert (LLG) ordinary differential equa-
directionduetotheshapeanisotropy.Theabsorbingboundary
tion[34–36]
condition has been implemented at both ends of the nanos-
trip, which effectively avoids the spurious spin wave reflec-
dM α dM
=−γ M ×H + (M × ), (1) tions[40].
dt 0 eff M dt
S
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ITS/203/14), the RGC-GRF under Grant HKU 17210014, project. X.Z.carriedoutthenumericalsimulations. J.X.per-
and University Grants Committee of Hong Kong (Contract formed the theoretical analysis with the input from M.Y. All
No. AoE/P-04/08). W.S.Z. acknowledges the support by authorsdiscussedtheresultsandpreparedthemanuscript.
the projects from the Chinese Postdoctoral Science Founda-
tion (No. 2015M570024), National Natural Science Founda- AdditionalInformation
tion of China (Projects No. 61501013, No. 61471015 and Supplementary information accompanies this paper at
No. 61571023), Beijing Municipal Commission of Science http://www.nature.com/srep
and Technology (Grant No. D15110300320000), and the In-
ternational Collaboration Project (No. 2015DFE12880) from
Correspondence and requests for materials should be ad-
theMinistryofScienceandTechnologyofChina.X.Z.thanks
dressedtoW.Z.andY.Z.
M.J.Donahueforusefuldiscussions.
AuthorContributions CompetingFinancialInterests
Y.Z.conceivedtheproblem. Y.Z.andW.S.Z.coordinatedthe Theauthorsdeclarenocompetingfinancialinterests.
