Magnetoelectric domain wall dynamics and its implications for magnetoelectric memory

0.1 MB·

by  K. D. Belashchenko

Magnetoelectricdomainwalldynamicsanditsimplicationsformagnetoelectricmemory
K.D.Belashchenko,1 O.Tchernyshyov,2 AlexeyA.Kovalev,1 andO.A.Tretiakov3,4
1)Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,
UniversityofNebraska-Lincoln,Lincoln,Nebraska68588,USA
2)Departmentof Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218,
USA
3)InstituteforMaterialsResearch,TohokuUniversity,Sendai980-8577,Japan
4)School of Natural Sciences, Far Eastern Federal University, Vladivostok 690950,
6 Russia
1
(Dated:22March2016)
0
2 Domainwalldynamicsinamagnetoelectricantiferromagnetisanalyzed,anditsimplicationsformagnetoelectricmem-

r oryapplicationsarediscussed. Cr2O3 isusedintheestimatesofthematerialsparameters. Itisfoundthatthedomain
a
wallmobilityhasamaximumasafunctionoftheelectricfieldduetothegyrotropiccouplinginducedbyit. InCr O
M 2 3
themaximalmobilityof0.1m/(s Oe)isreachedatE 0.06V/nm.Fieldsofthisordermaybetooweaktoovercome
× ≈
the intrinsicdepinningfield, whichis estimatedfor B-dopedCr O . These majordrawbacksfordeviceimplementa-
0 2 3
tion can be overcome by applying a small in-plane shear strain, which blocks the domain wall precession. Domain
2
wallmobilityofabout0.7m/(s Oe)canthenbeachievedat E = 0.2V/nm. Asplit-gateschemeisproposedforthe
×
] domain-wallcontrolledbitelement;itsextensiontomultiple-gatelineararrayscanofferadvantagesinmemorydensity,
i
c programmability,andlogicfunctionality.
s
-
l
r Encoding and manipulation of information by the anti- tiondrivenbythemagnetoelectricpressureF.
t
m ferromagnetic (AFM) order parameter has recently attracted ThemagneticdynamicsinanAFMisqualitativelydifferent
. considerableattention,1–4 andcurrent-inducedswitchingofa fromaferromagnet(FM).15–17Ifthemagnetostaticinteraction
at metallicantiferromagnethasbeendemonstrated.5Devicecon- isneglected,a domainwallinan idealFM with nodamping
m ceptsutilizingamagnetoelectricantiferromagnet(MEAF)as doesnotmoveatall,butratherprecessesintheappliedmag-
the active element are also being actively pursued for appli- netic field. The FM domain wall velocity v in this case is
-
d cationsinnonvolatilememoryandlogic.6–8 Thefundamental proportionaltothesmallGilbertdampingparameterα . The
0
n principleofoperationinvolvesthereversaloftheAFMorder magnetostatic interaction lifts the degeneracy of the Bloch
o
parameterintheMEAFbyappliedvoltageinthepresenceof and Ne´el configurations and blocks the precession, making
c
anexternalmagneticfield,whichisaccompaniedbytherever- v α 1 aslongas v doesnotexceedthe Walker breakdown
[ saloftheboundarymagnetizationoftheMEAF.7,9,10Littleis ve∝loci−0tyvW.20 In contrast, in AFM the Gilbertdampinglim-
2 known,however,aboutthefundamentallimitationsofthisap- its the terminal velocity of the wall. Here we are interested
v
proach. Herewediscusstheswitchingmechanisms,describe inthedynamicsofadomainwallinaMEAF,suchasCr O ,
1 2 3
7 thedynamicsofamovingdomainwall,estimatetherelevant whichisdrivenbytheapplicationoftheelectricandmagnetic
4 metrics,andproposetheschemeofamemorybit. fields. In a finite electric field, a MEAF turns into a nearly-
2 compensatedferrimagnet.Aswewillseebelow,theexistence
0 We consider the usual case of a collinear MEAF, such as ofasmallmagnetizationhasimportantconsequencesfordo-
1. Cr2O3,withtwomacroscopicallyinequivalentAFMdomains, mainwalldynamicsandhastobetakenintoaccount.
mappedoneontotheotherbytimereversal.Thedrivingforce
0 Werestrictourdiscussiontothelongitudinalmagnetoelec-
6 fortheswitchingofsuchMEAFisthedifferenceF = 2EαˆH tricresponse,inwhichthemagnetizationinducedbytheelec-
1 in thefree energydensitiesofthe two AFM domains, where tricfield isparallelto theAFM orderparameter,irrespective
: αˆ isthemagnetoelectrictensor.11 Thermallyactivatedsingle-
v of its spatial orientation. This is the case for the exchange-
i domainswitchinginvolvesaseveretradeoffbetweenthermal driven mechanism21–23 of magnetoelectric response, which
X stability and switching time — a long-standing problem in
dominatesin Cr O andmanyotherMEAFsattemperatures
r magneticrecordingtechnology.12Inordertosignificantlyre- thatarenottoolo2w.3InCr O theonlynon-zerocomponentof
a 2 3
duce the activation barrier for single-domain switching, the
themagnetoelectrictensorinthisapproximationisα = α ,
zz
appliedfieldsshouldsatisfyαEH K,whereKisthemagne- where z lies along the rhombohedralaxis.22,23 It is akssumed
tocrystallineanisotropyconstant.∼InCr O , whereα . 10 4
2 3 − that the electric field is applied across an epitaxially grown
(Gaussianunits)andK 2 105 erg/cm3,13,14 thiscondition
(0001)film.
requiresEH 1011Oe≈V/c×m.Sincefieldsofthismagnitude
Adding the Berry-phase and magnetoelectric terms to the
∼ ×
areundesirablefordeviceapplications,weareledtoconsider AFM Lagrangian,15–17, we can write the Lagrangiandensity
inhomogeneous switching, which involves nucleation of re-
ofaMEAF,validatlowenergies,as
verse domainsanddomainwall motion. The switching time
isdeterminedbytheslowerofthesetwomechanisms.Nucle- 1
=2ǫ a(n) n˙+ ρn˙ 2 A n2 n n 2ǫ γH n,
ation is a relatively slow thermally-activated process, which L J · 2(cid:16) | | − |∇ | −Kαβ α β(cid:17)− J ·
can be avoided by device engineering, as discussed below. (1)
The switching time is then limited by the domain wall mo- where n is the unit vectorin the directionof the AFM order
1
parameter(staggeredmagnetization)L = (M M )/2, M themovingdomainwallprecesseswiththeangularfrequency
1 2 1
andM thesublatticemagnetizations, = L/(−2γ)theangu- Ω=GX˙/Γ ,andthelinearvelocityofthewallis
2 ΦΦ
J
larmomentumdensityononesublattice,ρtheeffectiveinertia
F
density, A the exchangestiffness, and αβ the magnetocrys- v= . (6)
talline anisotropy tensor.18 Unless noteKd otherwise, it is as- ΓXX +G2/ΓΦΦ
sumedthattheonlynonzerocomponentofthistensoris =
zz Thus, the additional dissipation induced by the gyrotropic
K
<0.Inthefirstterm,ǫ =(M M )/(M +M )=α E/L,
1 2 1 2 couplingreducestheterminalvelocityofthedomainwallby
a−nKd a(n) is the vector potential−of a magnetic monokpole, thefactor1+G2(Γ Γ ) 1.
XX ΦΦ −
n a = n; this term is the Berry-phase contribution from SubstitutingtheexpressionsforΓ ,Γ ,andGinEq.(6),
∇ × XX ΦΦ
the small longitudinalmagnetization M = (M +M )/2 in-
1 2 weobtain
ducedbytheelectricfield.17,19 ThelastterminEq.(1)isthe
magnetoelectricenergydensity;11γisthegyromagneticratio. v= 2ǫ/α0 v , (7)
TheAFMfieldtheoryatE = 0hascharacteristicscalesof 1+(ǫ/α0)2 max
time,length,andpressure:
where v = γH λ /2. The maximum velocity v of the
max z 0 max
domain wall is reached at the optimal electric field strength
t = ρ/ , λ = A/ , ǫ = √A , (2)
0 p K 0 p K 0 K Emax corresponding to ǫ = α0. Interestingly, vmax depends
neither on the magnetoelectric coefficient nor on the Gilbert
which have direct physical meaning. ǫ is the scale of the
0 dampingconstant.
domain wall energy per unit area. The magnon dispersion
Usingthevalueγ=1.76 107s 1/Gandareasonablefield
−
ω(k) = qω20+s2k2 has a gap ω0 = 1/t0 and velocity s = Hz =100Oe,wefindvmax ×10.6m/s. Assumingtheswitch-
≈
λ /t . In Cr O , ω = 0.68 meV,24 hence t 1 ps. The ablebitsizeof50nm,weestimatetheswitchingtimeofabout
0 0 2 3 0 0
magnonvelocityiss=12km/s.24Thelengthpa≈rameterλ = 5 ns. Note that the maximal MEAF domain wall mobility
0
st0setsthescaleofthedomainwallwidthd. InCr2O3wefind vmax/Hz ≈ 0.1 m/(s×Oe)is 2–3ordersof magnitudesmaller
λ =12nmandd =πλ 38nm. inthisregimecomparedtoferromagnetslikepermalloy.25
0 0
The effective Lagrangi≈an for low-energy domain wall dy- TheGilbertdampingconstantcanbedeterminedfromthe
namicsisobtainedbyinsertingthedomainwallprofile relationT = ρ/(2α0 ), whereT is therelaxationtime.17 To
J
estimateT inCr O ,weusethewidthoftheAFMresonance
2 3
x X ∆H =900Oe,14whichtranslatesinto∆ω=1.6 1010s 1and
cosθ(x)=tanh λ− , φ(x)=Φ, (3) T = 1/∆ω 60ps. Usingthevalue K = 2 1×05 erg/c−m3,14
0 ≈ ×
we findthe inertia densityρ = 2Kt2 4 10 19 g/cm. The
parameterizedbythecollectivevariablesX andΦ,inEq.(1) 0 ≈ × −
valueof isobtainedfromthelocalmagneticmoment292.76
andtakingtheintegraloverallspace. FortheMEAFdomain J
µ and volume Ω 50 Å3 per formula unit. Putting these
B
wallthisleadsto estimatestogether,w≈eobtainα 2 10 4.
0 −
≈ ×
L= 1MX˙2+ 1IΦ˙2+GX˙Φ V(X,Φ), (4) CrTOhe,rwelhaetiroenwǫe u=sedαt0hethpeenakgivvaelsueEαmax ≈10604,Vw/hµimchiins
2 2 − rea2ch3ed at 260 K. The magnetoelectricpkres≈sure−correspond-
where M = 2ρ/λ0 andI = 2ρλ0 arethemassandmomentof ing to E = Emax and Hz = 100 Oe is Fmax = 2α0LHz 40
inertia per unit area of the wall, V is the potential energyof erg/cm3.Toputthisvalueinperspective,wenotethatinf≈erro-
the wall, which in a uniaxialAFM has no dependenceof Φ, magneticironthemagneticfieldof100Oeexertsapressureof
andG = 4ǫ is the gyrotropicterm couplingthe motion of about3 105erg/cm3onthedomainwalls. The“loss”offour
thewalltoitJsprecession,whichisproportionaltoE. orderso×fmagnitudein MEAFisdueto thesmallmagnitude
Theequationsofmotionforthecollectivecoordinatesare ofthemagneticmomentinducedbytheelectric field. Alter-
natively, onecan say that a 100 Oe coercivityin a MEAFat
MX¨ =−GΦ˙ −ΓXXX˙ +F, E ∼Emaxisequivalent,assumingsimilarmaterialquality,toa
IΦ¨ =GX˙ Γ Φ˙ +τ, (5) 10mOecoercivityiniron.Thus,itisclearthatreasonablyfast
ΦΦ
− switchingofaMEAFwithuniaxialanisotropyrequiressam-
where Γ = 4α /λ and Γ = 4α λ are the viscous ples of very high quality, unless the temperature is close to
XX 0 0 ΦΦ 0 0
J J
dragcoefficientsproportionalto the Gilbertdampingparam- theNe´elpointT wherethedomainwallwidthdivergesand
N
eter α , and F = ∂V/∂X = 2α E H = 2ǫLH . Thetorque thecoercivitybecomessmalleveninlow-qualitysamples.In-
0 z z z
τ = ∂V/∂Φvanis−hesinthecaskeofuniaxialanisotropy. We deed, isothermal MEAF switching has so far been observed
−
willfirstconsiderthecaseτ = 0andthenaddresstheroleof onlyclosetoT .7
N
brokenaxialsymmetry. In the presence of lattice imperfections, switching is pos-
AtG =0wehaveaconventionalAFMdomainwall,which sibleifthemagnetoelectricpressureF appliedtothedomain
behaves as a massive particle subject to viscous drag, and wallexceedsthedepinningpressureF . SinceT =307Kof
c N
whose angular collective variable Φ is completelypassive.17 Cr O istoolowforpassively-cooledcomputerapplications,
2 3
However, the gyrotropic coupling G induced by the elec- it needsto be either dopedor strained to increase its T . In
N
tric field generates precession of the moving domain wall, particular,borondopingontheCrsublatticehasbeenshown
which generates additional dissipation. In the steady state to raise T significantly.30,31 Random substitutionaldisorder
N
2
inadopedmaterialleadstoanintrinsicpinningpotentialand F,asshowninFig.1.
nonzero coercivity. Let us estimate the effective depinning
pressureforthisrepresentativecase.
For simplicity, we assume that B dopants modify the ex-
changeinteractionlocallybutdonotstronglyaffectthemag-
netocrystallineanisotropy. AccordingtoRef.30, borondop-
ingenhancestheexchangecouplingfortheCratomsthathave
aBneighborbyafactorof2–3.TheconcentrationofBatoms
is n = 3x/Ω, where x is the B-for-O substitution concentra-
tion. Therefore,wemakea crudeestimatethattheexchange
stiffness A is enhancedby a factorof2 inregionsofvolume
2Ω,whoseconcentrationisn. FIG.1. Averagedomainwallvelocityvasafunctionof E/E at
max
Leta∗betheradiusofaspherewithvolume2Ω.Theforce K⊥ =0(dottedblueline),4Fmax(dashedgreen),and16Fmax(solid).
acting on the domain wall from the vicinity of one B atom
is f (a∗/λ0)3A. The typical pinning force on a portion of In the presence of & 900 erg/cm3, the fields E 0.2
the d∼omain wall of size R2 then becomes f nλ R2f2. V/nm and H 100 OKe⊥result in v 70 m/s and F ≈140
pin ∼ p 0 ≈ ≈ ≈
The typical correlation length for the domain wall bending erg/cm3. Under these conditions the switching time of a
displacementis the Larkin lengthR ,32–35 which is foundby nanoscalebitcanbewellbelowananosecond,whilethemag-
c
equating f tothetypicalelasticforce f u√A produced netoelectric pressure F exceeds the intrinsic depinning field
pin el
∼ K
by the domain wall, where u λ corresponds to the situ- ofB-dopedCr O byanorderofmagnitude. Clearly,theim-
0 2 3
ation in which the domain wal∼l deforms weakly. This gives positionof in-planeanisotropyofferscompellingadvantages
Rc ∼ qλn0Af2K. Thedepinningthresholdcanthenbeestimated forItdiesviincteeraepsptilnicgattioonnostbeythimatpthroevdinogmsawinitwchaallbmiliotybialintdycspaneebde.
asF A/R2 =nλ A(2Ω/λ3)2. Usingx=0.03andA 10 6
c ∼ c 0 0 ∼ − changedbyordersof magnitudebyimposinga non-zeroK
ethrge/mcmag,nweetofielnedctrFicc p∼res1s0ureerga/tcHm3=, w1h0i0chOiesacnodmEpa=rabElematxo, tiunrtehoefsMtroEnAg-Feldeocmtriaci-nfiweladllredgyinmaemǫic≫scaαn0.beTdhiirsepcetlcyuclihaercfkeea⊥d-
as we have estimated above. Other imperfections may fur- experimentally.
therincreaseFc. Thus,asexpectedfromthecomparisonwith Devicesbased on MEAF switchingoffer a distinctadvan-
typicalferromagnets,evenweakpinningassociatedwith ho- tagein termsofenergyefficiency. Energydissipatedwhena
mogeneousdopingcanimpedeMEAFswitching. Thissensi- bit is switched is E = 2α E H V = FV, where V is the
dis z z
tivity to lattice disorder, along with the low upper bound on switched volume. This is thek energy difference between the
the domain wall mobility, present serious challenges for the twoAFMdomainstatesofthebit. Takingtheswitchingvol-
implementationofmagnetoelectricdevices. umetobeacubewitha50nmedge,weestimateE 10 14
dis −
Wewillnowshowthatbothoftheselimitationscanbeover- ergforthe fieldmagnitudeschosenabove. Thiscorre∼sponds
come by introducing a relatively small in-plane anisotropy to an upper limit on the intrinsic power consumption of 1
component yy = in addition to the axial component mW/Gbit, assumingthateach bitis switched everynanosec-
zz = .KSuchKin⊥-plane anisotropy can be induced by ond.Clearly,energydissipationinamagnetoelectricmemory
K −K
applying a small in-plane shear strain to the magnetoelec- devicewouldbedominatedbylossesintheexternalcircuitry.
tric crystal, for example, by using a piezoelectric element, As we argued above, fast memory operation should be
an anisotropicsubstrate, or anisotropic thermalexpansionin based on domain wall-mediated switching. Therefore, it is
a patterned structure. The physics of domain wall motion necessary to design the architecture of a bit in such a way
at , 0 is similar to Walker breakdown in ferromagnets, that the domain wall is not annihilated at the surface as the
whKer⊥e the anisotropy with respect to Φ appears due to the bitisswitched. Onewaytoachievethisisthroughtheuseof
magnetostaticinteraction.20 a multiple-gate scheme, as shown in Fig. 2. In this scheme,
In the equations of motion (5) we now have, after inte- additional“setgates” are used to initialize andmaintaintwo
grating out the domain wall profile (3), a nonzero torque differentAFM states at the edgesof the active magnetoelec-
τ = λ0 sin2Φ per unit area. There is a steady-state so- tric layer, which are labeled + and in Fig. 2, therebytrap-
lution−wiKth⊥Φ˙ = 0andv = F/ΓXX, aslongasv < vW, where ping the domain wall inside the dev−ice. The set gates need
vW/vmax = 2( /Fmax)1/2 is analogousto the Walker break- to be activated only during the write operation, along with
downvelocityK.20⊥For example, in order to achieve vW 100 the control gate. Positive or negative voltage applied to the
m/s,weneedtohave &900erg/cm3,whichisthree∼orders control gate selects the AFM domain state in the switched
ofmagnitudesmallerKth⊥an . Itislikelythat ofthisorder areaanddrivesthedomainwallbetweenthepositionsshown
canbeachievedwithafairKlysmallin-planeshKe⊥arstrain. by the dashed lines. This scheme is somewhat reminiscent
Below the Walker breakdown the domain wall velocity is ofthespin-transfertorquedomainwalldevice.27 Thecontrol
linear in E: v/v = 2E/E . At F > Γ v the in-plane gatecanalsoprovidethememoryreadfunctionbyemploying
max max XX W
anisotropycannolongersuppressdomainwallprecession,so a FM layer, coupled, via the boundary magnetization of the
thatitsvelocitybecomesoscillatory.Theaveragevelocityhas MEAF,toitsAFMdomainstate,andaspinvalve,orasimilar
a cuspat F = Γ v and declineswith a furtherincrease in magnetoresistive element, grown on top of it. Alternatively,
XX W
3
theAFMdomainstatecanbedetectedthroughtheanomalous cussions and KITP for hospitality. KB acknowledges sup-
Halleffectinathinnon-magneticcontrolgate.28 portfromtheNanoelectronicsResearchCorporation(NERC),
a wholly-owned subsidiary of the Semiconductor Research
Corporation (SRC), through the Center for Nanoferroic De-
vices (CNFD), an SRC-NRI Nanoelectronics Research Ini-
tiative Center, under Task ID 2398.001. KB and AK ac-
knowledgesupportfromthe NSF throughthe NebraskaMa-
FIG. 2. Two logical states of asplit-gate magnetoelectric memory terials Research Science and Engineering Center (MRSEC)
bit. Central region: MEAF layer. Arrows: AFM order parameter (GrantNo.DMR-1420645). AKacknowledgessupportfrom
L. Thecontinuousbottomelectrodeisgrounded. Theleftandright theDOEEarlyCareerAwardDE-SC0014189. OTacknowl-
setgateslabeled“+”and“–”areactivatedduringthewriteoperation, edges support from the DOE Office of Basic Energy Sci-
enforcingfixedAFMdomainstatesunderneaththesegates.Thestate
ences,DivisionofMaterialsSciencesandEngineeringunder
ofthebitisrecordedbythevoltageappliedtothecentralcontrolgate.
Award DE-FG02-08ER46544. OAT acknowledges support
Thepermanentmagneticfieldisappliedvertically(notshown).
bytheGrants-in-AidforScientificResearch(Nos.25800184,
25247056and 15H01009)from the MEXT, Japan and Spin-
Sincethedomainwallshouldfitinsidethebit, itswidthd
Net. Thisresearchwas also supportedin partbyNSF under
setsalimitationforthedownwardscalingofthelengthofthe
GrantNo.PHY11-25915.
MEAFelement. The width ofthis element, however,can be
significantly smaller. To facilitate downscaling, the domain
wall width d can be reducedby increasingthe magnetocrys-
tallineanisotropyoftheMEAF.Forexample,itisknownthat
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