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Large AnomalousNernstEffect inaSkyrmionCrystal Yo Pierre Mizuta∗ GraduateSchoolofNaturalScienceandTechnology, KanazawaUniversity,Kakuma,Kanazawa,920-1192JAPAN Fumiyuki Ishii FacultyofMathematicsand Physics, Kanazawa University, Kakuma, Kanazawa, 920-1192 JAPAN (Dated:January15,2016) Thermoelectricpropertiesofamodelskyrmioncrystalweretheoreticallyinvestigated,anditwasfoundthat 6 itslargeanomalousHallconductivity,correspondingtolargeChernnumbersinducedbyitspeculiarspinstruc- 1 tureleadstoalargetransversethermoelectricvoltagethrough theanomalous Nernsteffect. Thisimpliesthe 0 2 possibilityoffindinggoodthermoelectricmaterialsamongskyrmionsystems,andthusmotivatesourquestsfor thembymeansofthefirst-principlescalculationsaswereemployedhere. n a J I. INTRODUCTION phasetheory3,theemergenceoftopologicaltermsinthecon- 4 tinuum limit (spin variation scale ≫ atomic spacing), is an 1 analogueoftheordinaryHalleffectwiththeexternalB field Thermoelectric(TE)generation,harvestingwasteheatand justreplacedbyspin magneticfield B , whichisthe real- ] turningitintoelectricity,shouldplayanimportantroleinreal- spin i space Berry curvature Ω(R) itself, proportional to a quan- c izingmoreenergy-efficientsocietyandovercomingtheglobal s warming. Nevertheless, it is yet to be widely used, mainly tity called spin scalar chirality χijk ≡ mi · mj × mk re- - flecting the geometrical structure spanned by each spin trio l duetoitsstilllimitedefficiency. Therehavebeenmanystud- r (m , m , m ). Althoughitshouldbebettertotreatσintand t ies pursueing highly efficient TE systems with a large value i j k xy m of figure of merit: Z = σX2/κ, where σ and κ are lon- σxTy onanequalfooting,wewillomitthelatter(topological) X . gitudinalelectricalandthermalconductivityrespectively,and terminthemainpartofthispaperbecausethevalidityofthe t a X = S or N is the Seebeck or Nernst coefficient depend- simple relationBspin ∝ Ω(R)is notclear foroursystem of m rathershortspinvariationscale. Still,theomissioncanbejus- ingon whetherwe use the longitudinalor transversevoltage - for power generation. Here we omitted labels xx or yy in tifiedwithinthisapproximation(SeeAppedixBfordetails). d σ andκbyassuminganisotropicsystem. Amongthose,our n o studyshedslightontheanomalouseffectofelectricalconduc- Among many possible spin textures, what we target here c tionperpendiculartoanelectricfield(anomalousHalleffect, is the so-calledskyrmioncrystal(SkX) phase observedeven [ AHE)1ortoatemperaturegradient(anomalousNernsteffect, nearroom-temperature11,whereskyrmions,particle-likespin 1 ANE)2onTEperformance,focusingonaparticularcontribu- whirls, align on a lattice. The skyrmionic systems, origi- v tiontotheconductivityofAHE(ANE),namelytheso-called nallyexploredinnuclearphysics12, havebeenstudiedexten- 0 intrinsic term σxinyt(αixnyt) expressed as a functional of Berry sively these days in condensed matter physics as well. Typ- 1 curvature3 Ω(k) ≡ ih∂ku|× |∂kui in momentum(k) space ical SkX-hosting materials are some of the transition metal 5 (Seethe middleline ofEq.(A4) inAppendixA),where|uki silicides/germanides: MnSi13, MnGe, FeGe, or heterostruc- 3 istheperiodicpartofaBlochstate. tures such as monolayerFe on Ir(111)14, in all of which the 0 . SystemshostingAHE/ANEarefoundinmagneticmateri- Dzyaloshinskii-Moriyaterm, a spin-orbitcouplingeffectpe- 1 als,bothnormalsemiconductors4andtopologicalinsulators5. culiartotheirinversion-asymmetriccrystalstructure,playsa 0 6 We previously studied simple models of 2D electron gas in crucialroleintheemergenceofskyrmions. 1 an interface composedof those materials6,7. The anomalous : effect on Seebeck coefficient was fairly large there, but re- v Regardingthe AHE in skyrmionicsystems, phases with a mainedrathersmallcomparedtowhatwillbereportedinthis i quantizedAHC,i.e. σint =C(e2/h)withanintegerC called X paper,whichcanmainlybeattributedtothelimitedmagnitude xy Chern number, have been recently predicted in the models r oftheanomalousHallconductivity(AHC)σint ≤1(Theunit a xy of SkX where the conduction electron spins are exchange- istakentobee2/hfortheAHCin2Dinthispaper)there. coupled to the SkX either strongly15 or weakly16, preceded Therehavebeenseveralideasproposedforobtaininglarge byareportofQAHEinameron(half-skyrmion)crystal17. In AHC, such as the manipulation of massive Dirac cones thispaperwefocusonthecasepickedoutinRef.[15]because (each of them being the source of σint = 1) by control- ofitsparticularlylargeσint,implyingthepossibilityoflarge xy xy lingparameters8,ortheextentionofsystemdimensioninthe ANE as well, thanksto the close relation between AHE and normal-to-planedirection(2Dto3D)9. ANE2,18,andthatisthemainpointweconfirmedinthisstudy. Whatwefocushereisanotherone, namely,thecontrolof real-space spin textureswhich are well knownto induce an- Attheendofthissection,westressthatthecomputational othercontributiontotheAHE/ANE,oftencalledthetopolog- method used in our study is based on first-principles, which ical Hall/Nernst conductivity10 σT (αT ) that also has a ge- canbeappliedtotheexplorationofrealisticmaterialsofSkX xy xy ometrical meaning related to σint(αint). In terms of Berry- etc. fromthesameperspective. xy xy 2 (a) (b) (c) -1.5 σ | σ | xxT=0 xyT=0 V) e µ y ( -2 0 g er n E FIG.1.Aviewofanunitcellof6×6SkX. -2.5 Γ X M Γ 0 10 20 -12 0 16 II. EXPRESSIONSOFTHERMOELECTRICQUANTITIES (e2/h) (e2/h) Theformulaeforthethermoelectriccofficientstobeevalu- atedfollowfromthelinearresponserelationofchargecurrent: FIG.2. (a)BandstructureandFermi-leveldependenceof(b)longi- j=σ˜E+α˜(−∇T),whereEand∇T aretheelectricfieldand tudinaland(c)anomalousHallconductivityof6×6SkX.Theblue temperaturegradientpresentinthesample. dashedlineindicatestheµ0mentionedinthemaintext. Usingtheconductivitytensorsσ˜ =[σ ]andα˜ =[α ],we ij ij obtain IV. COMPUTATIONALPROCEDURE E S +θ N x 0 H 0 Ourcalculationsconsistofthreesteps: (1)Obtaintheelec- S ≡S ≡ =  xx ∂xT 1+θH2 (1) tronic states of target SkX using OpenMX code19, (2) Con-  Ex N0−θHS0 struct maximally localized Wannier functions (MLWF) em- N ≡Sxy ≡ ∂ T = 1+θ2 =−Syx. ployingWannier90 code20 and finally (3) Compute from the  y H obtainedMLWFallthenecessarytransportquantities[tensors σ and α in Eq.(A4)] expressed according to the Boltzmann Here we defined S0 ≡ αxx/σxx,θH ≡ σxy/σxx, N0 ≡ semiclassical transport theory, where we adopted constant- αxy/σxx for a simpler notation. See Eq.(A4) in Appendix relaxation-time approximation with a fixed value of τ = A for the specific form of the conductivity tensors σ and α 0.1ps21. weconsiderhere.Throughoutthispaper,sinceour2Dsystem Instep(1),ones-characternumericalpseudo-atomicorbital has no periodicity in z direction, we discuss σij ≡ σi2jD ≡ withcutoffradiusof7bohrwasassignedtoeachHatom.The d×σ3D,whichisindependentofthefilmthicknessdandhas present calculation for a non-collinear magnetic system was ij thedimensionofe2/hwhoseunitcanbeΩ−1,insteadofreal realizedbyapplyingaspin-constrainingmethod22inthenon- conductivityσ3D. collineardensityfunctionaltheory23.Step(1)yielded6×6=36 ij non-spin-degenerateoccupiedbandsandtheequalnumberof NotethattheSeebeckcoefficientS estimatedwithoutcon- 0 unoccupiedones,amongwhichonlytheformer36bandswere sidering Berry curvature is obtained by setting θ = 0 and H usedinstep(2)toconstructMLWFandinterpolatedwiththem N =0inEq. (1). 0 tocalculatetheconductivities. Instep(3),twomoduleswere usedinWannier90: berrymodulebasedontheformalismin Ref.[24]tocomputeσintandboltzwannmoduleintroducedin xy Ref.[25] to compute σ and S , in both of which the sam- xx 0 III. MODEL plingforintegrationswasperformedon50×50k-points.Be- sidesthese,numericalintergrationswerecarriedouttoevalu- ateα fromσ viaEq.(A4). WeconsideramagneticSkXonatwodimensionalsquare xy xy Theaboveprocedurewastested26,referringtoRef.[15]. lattice, withits unitcelloflatticeconstant2λ = 19.8A˚ con- taining6×6spins,thustheatomiclatticespacinga = 2λ/6 being 3.3A˚. The spin configuration is equivalent to the one V. RESULTSANDDISCUSSIONS studied in Ref.[15], i.e., the spherical coordinates of spin m(r )locatedatsiteiaresetasθ =π(1−r /λ)forr <λ i i i i andθ = 0forr > λ, alongwithφ = tan−1(y /x )+C, A. Electronicstructureandconductivities i i i i i whereC isanarbitraryconstant.Inordertosimulatethesim- plestcase,weassumedeachspinisthatofahydrogenatom. First we show the obtained band structure of the 36 oc- ThespinmodulationinthesystemisshowninFig.1. cupied states in Fig.2(a). We notice there that each band is 3 wellisolated from(nottouchingwith)each other,exceptfor (a) thefourbandsaroundthemiddleenergyrange[-2.0,-1.9]eV, 200 s S whichweshallhereafterrefertoas“centralbands”. Wealso nt N steenedthtoatcsoonmveerngeeigtohwboarridngMbapnodins,ti(n0c.l5u,di0n.g5)tπhe/(c2eλn)t.erAowneasy, coefficie µ [V/K] 0 fromMpoint,weseesomedegeneratepoints. Regardingthe E T -200 dispersions, quite nice symmetry with respect to the energy µ0 ≃ −1.93eVinthecenterbandsalsodeservescloseatten- S θ N 8 tion. Further analyses are needed in order to understandthe (b) 0 H 0 200 secretsbehindtheseinterestingfeatures. nts 4 andNAexHt,CleattuTso=bs0eKrvedehpoewndthoenlothnegibtaunddinfialllcinognd(Fuecrtimviityleσvexlx) coefficie µ [V/K] 0 0 θH in Fig.2(b)and(c)respectively. Theσxx(µ)|T=0 hasa large TE -200 -4 peakinthecentralbandsregion,ascanbeexpectedfromthe obviously large density of states there. The σ ’s roughly -8 xx symmetricvariationwithrespecttoµ0 (recognizedwhenav- 0.9 (c) epstraratugecdetduirnoet.vueitrivseolmyefrospmretahdeedabeonveer-gmy)enctoiounldedhasyvme mbeeetnricanbtaicnid- er factor 2W/(K m)] 0.6 PPFFNS w 3 The AHC, on the other hand, shows good anti-symmetric o -0 0.3 P 1 behavior with respect to µ , which is quite understandable [ 0 from the combined consideration of, again, the symmetric 0 -2.8 -2.4 -2 -1.6 -1.2 band structure and the assumption that each of the isolated Chemical potential (eV) (otherthan thecentral)bandsisanalogousto a Landaulevel formedbyexternalmagneticfieldwhichcontributesAHC=1 (e2/h) in the quantum Hall effect. The maximum absolute value of AHC is approximately 16, reached just below the FIG.3. Chemicalpotentialdependence ofthethermoelectricquan- central bands, while the second largest value of about 12 is titiesof6×6SkXatT =300K:(a)SandN,(b)S0, N0(leftaxis) located just above them. These behaviors result in the dras- andθH(rightaxis)[seeEq.(1)],(c)PowerfactorscorrespondingtoS ticalchangeofAHCaroundcentralbandsasisprominentin andN.ThebluedottedlineforPFNin(c)isdrawntoguidetheeye. Fig.2(c). This very character has a strong effect on the TE propertiesofthesystem,aswillbeseenbelow. θ . H OnthetransversepartN,whatwenoticeinFig.3(a,b)and their interpretationsare the followingtwo points (i) and (ii): B. Filling-dependenceofthermoelectricproperties (i) Aroundµ , it showsa largepeak N ≃ N ≃ 180µV/K. 0 0 Thisisbecauseofθ (µ )≃0[SeeEq.(1)]andlargeσ′ (µ ) Wewillproceedtoourmainsubjects,theTEquantitiesof H 0 xy 0 (theprimeindicatingthederivativewithrespecttoµatT =0) thesystem,especiallyfocusingontheirelectronfillingdepen- asaresultofthelargeanti-symmetryofσ aroundµ ,which dence,whichweassumeto beparametrizedbythe chemical xy 0 affectsN approximatelythroughtheMott’srelation[Eq.(2), potentialµ within the rigid bandapproximation. In all what latertobe0discussedquantitavely]N ∝ σ′ /σ . (ii)Away is reported in this paper, the room temperature T = 300K 0 xy xx from µ , N is strongly suppressed compared to N due to is assumed27. The Seebeck S and Nernst N coefficients, 0 0 large θ , similarly to the relation between S and S . This the terms S , θ , N constituting them [Eq.(1)], and the H 0 0 H 0 is different from the situation where N is the far dominant power factors associated with each of S and N, are shown 0 contribution to N, as reported for example in Ref.[28]. In inFig.3(a),(b)and(c),respectively. view of the relation between S and N, it is instructive that As to the longitudinal S(µ), it is largely affected by the largeN appearsthankstothelargeasymmetryofσ ,atthe anomalouseffectwhenθ (µ)isfinite(almostthroughoutthe xy H same filling where S diminishesdue to the complete loss of plottedµrange)WerecognizeinFig.3(a,b)thefollowingthree asymmetry in σ , showing that σ could be an additional points (i)-(iii) including their explanation based on what we xx xy freedominseekingbetterthermoelectricity,whenthematerial have seen in Fig.2: (i) S(µ ) ≃ 0 and S (µ ) ≃ 0, result- 0 0 0 isproperlydesigned. ing from the combination of the above-mentioned behavior of σ | and σ | , whose symmetrical centers came xx T=0 xy T=0 across with each other aroundµ . (ii) Just below and above 0 µ , S showsthe peaks(the higherone reaching≈ 80µV/K) C. Largestpredictedthermoelectricvoltages 0 of the sign opposite to that of S despite rather small Hall 0 angle θ there. This is thanks to the large N that satisfies In order to better capture quantitatively the above- H 0 |θ N | > |S |. (iii) Between the central and edge energy mentioned connection between conductivities and TE coef- H 0 0 range, S is stronglysuppressed comparedto S dueto large ficients that leads to large values of the latter, which are 0 4 seeminglyattractiveforTEapplications,letuscheckwhether Themainlimitationofourresultsliesinthelackofknowl- the largest value S and N seen in the central energy edgeon the processesthroughwhich electronsare scattered. max max rangeinFig.3(a)canberoughlyestimatedviathewell-known Even if the single-τ approximation is valid, the magnitude Mott’sformula,whichsays, of τ matters a lot. This is clearly manifested in the relation N(µ ) ∝ τ−1. For examplewe obtain N(µ ) ≈ 20µV/K N N α =−π2kB2Tσ′ (µ)| , (2) ifwesupposeτ =1ps,althoughtheNernstcoefficientofthis ij 3 e ij T=0 sizeisstillmuchlargerthanthesofarreportedvaluesandis practicallyvaluable30. where the prime in σ′ means the derivative with regard to xy the Fermi energy (chemical potenatial µ at T = 0). We writethechemicalpotentialvaluesthatrespectivelygiveS max VI. SUMMARY and N as µ and µ , both of which are close to µ but max S N 0 slightly different from each other. Finding that S (µ ) ≈ 0 S We computed the thermoelectric (TE) coefficients in a 0, θ (µ ) ≈ 1, S reduces to S(µ ) ≃ N (µ )/2 = H S S 0 S σ′ /2σ . Similarly found is that N(µ ) ≃ N (µ ) be- modelofsquareskyrmioncrystal,wherespin-scalar-chirality- xy xx N 0 N driven anomalous Hall/Nernst effects directly expressed in cause of S (µ ) ≈ 0 ≈ θ (µ ). A rough estimation of 0 N H N σ′ (µ )| can be obtained by linearly-approximatingthe termsofBerrycurvatureinmomentumspace, werefoundto xy S T=0 drasticaldropofAHC∆σ (µ)≈−28(e2/h)Ω−1intheen- giveriseto,notonlyaverydifferentbehaviorinSeebeckco- ergyrange∆µ ≈ 0.13eVxiynFig.2(c). Thisgivesσ′ (µ ) ≃ efficient from in the case of conventional TE materials, but xy S alsoasurprisinglylargeNernstcoefficient. Thefuturetaskis ∆σxy ≈2.2×102(e2/h)eV−1. Theotherquantityweneedis ∆µ tounderstandthephysicsuniquetosuchsystemsthatleadsto σxx(µS),whichwasfoundtobeabout7.7(e2/h).Combining the prominentTE effect, and to find or design good TE ma- these,adimensionlessfactorofx=kBT/(σxx/σx′y)≃0.73 terialsamongthisclassofmaterialsusingthefirst-principles is found at T = 300K (kT ≃ 0.026eV). Finally our eval- methodusedhere,whichwillalsobeofgreatpracticalimpor- uation arrives at Nmax = N(µN) ≃ N(µS) ≃ N0(µS) ≃ tanceforenergy-savingapplications. −π32keBx ≈ 2×102µV/K and thereforeSmax = S(µS) ≈ 1 × 102µV/K. These values are in fairly good accordance withtheintegration-derivedvalues,henceclarifyingthatT = ACKNOWLEDGMENTS 300Kislowenough(incomparisontotheFermilevel)forthe Mott’sformulatobevalidatleastforroughestimation. TheauthorsthankK.Hamamotoforprovidinguswithde- At this point we compare the maximum value of αxy ≈ tailedinformationonthemodelheandcoworkershadstudied 10−8 A/K correspondingto the above-estimatedNmax, with in Ref.[15]. This work was partially supported by Grants- a value reasonably expected to be maximum within the low in-Aid on Scientific Research under Grant Nos. 25790007, temperature approximation for the two-band Dirac-Zeeman 25390008and15H01015fromJapanSocietyforthePromo- (DZ) model we studied before7. The low-T approximated tion of Scienceand by the MEXT HPCI Strategic Program. αxy in Eq.(4) of Ref.[7] can be rewritten as a 2D quantity Computations in this research were performed using super- αxy ≃ (πke/12h)(kT/∆)/µ˜2, where ∆ is the Zeeman gap computersatISSP,UniversityofTokyo. and µ˜ ≡ µ/∆ ≥ 1 for the electron-dopedcase. Assuming rather small µ˜ = 2, and choosing(kT/∆) = 0.3 to loosely satisfy the low-T criterion kT ≪ µ − ∆ = ∆, we obtain AppendixA:Expressionsofconductivitytensors α ≈ 10−10A/K fortheDZ model,whichis bytwo orders xy ofmagnitudesmallerthaninthe6×6SkX.Althoughsome Wederivebelowthesemiclassicalformulaeforconductiv- larger α can be achieved beyond low-T range, still larger xy ity tensors. The starting point is the expression for charge valuesin the SkX should be ascribed to the AHC morethan currentjobtainedinRef.[2],whichreads, tentimeslargerthanthatoftheDZmodel. Finally, since N ≃ 180µV/K is a good enough value max dk as a TE material, in order to further investigatethe practical j=e g(r,k)vk Z (2π)2 performance of the present system, we plot the powerfactor PF ≡ σ S2 and PF ≡ σ N2 in Fig.3(c). Note that, dk εk−µ forSthe evaxlxuationofpowNerfactxoxrs, unlikein the discussions +e∇r×kBT(r)Z (2π)2Ω(k)log(cid:20)1+exp(cid:18)−kBT(r)(cid:19)(cid:21), ofTEquantitiesuptonow,weneedtoknowσ3D. Therefore (A1) xx weassumedtheSkXfilmthicknessof10nm. Themaximum ofPFN ≈10−3W/(K2m)iscomparabletothevaluesofpos- wheree, g(r,k), T(r), vk, εk, Ω(k), andµ standforthe sibleoxideTE candidatessuchasNaCo O andZnO29. Al- electron’s charge (e < 0), distribution function, local tem- 2 4 though the thermal conductivity κ was beyond the scope of perature,(velocity,energyandk-spaceBerrycurvature)ofan thisstudy,wejustmakearoughestimatehere: Assumingthe electronwithwavenumberkandchemicalpotential,respec- thermalconductivityκoftheorderof1W/(Km)correspond- tively. ingtogoodTEmaterials29,thepresentcaserealizesZ T of The second term is a correction that appearswhen spatial N theorderof0.3atT =300K. inhomogeneity[T(r)inthepresentcase]exists. Simplifying 5 the secondterm followingRef.[2] andsubstitutingthe set of AppendixB:CommentonthetopologicalHallcontribution equationsofmotionofaperturbed(byEfield)Blochelectron Weroughlyestimatethemagnitudeofchangeinourresult vk = 1∂εk −k˙ ×Ω(k), k˙ =eE (A2) to be broughtabout by considering σT omitted in the main ¯h ∂k xy partofthepaper. whichwasderivedinRef.[31],forvk andtheformofdistri- TheexpressionforσT intheDrudemodeltreatmentis, xy bution ω τ ∇T ∂f σT = s σ , (B1) g(r,k)=τkvk· eE+(εk−µ) − − , xy 1+(ωsτ)2 xx (cid:20) (cid:18) T (cid:19)(cid:18) ∂ε(cid:19)(cid:21) (A3) where ω ≡ eB /m with spin-scalar-chirality-induced obtained as the solution of Boltzmann transport equation s spin e field B ≡ (h¯/2e)mˆ · (∂ mˆ ×∂ mˆ) determined by the withinrelaxationtime(τk)approximation,forg(r,k),weob- spin x y unitvectorofthe backgroundspin texturemˆ ≡ m/|m|and tain electronmassm . e ∂f Substituting the present SkX form of mˆ(r), we σ =e2τ dkv (k)2 − ,  xx Z x (cid:18) ∂ε(cid:19) obtain the spatially-inhomogeneous Bspin(r) =  σxy =−e¯h2 Z dkΩz(k)=−σyx, (A4) si(tih¯sz/eu2λnei)-ti(-nπcde/elrlpλea)nvsdeiernanπgte(fld1uxv−aolufreh/λ/Be),s1p5i.nwAh=ipcphr(ohxg/iimvee)a/st(inπagλ2bs)kyyttrhamaktiinoigns 1 ε−µ ∂f roughly 103 T in our case of λ ≈ 1nm, we obtain the  αij = e Zfodrεσiij=(ε)x|To=r0y,Tj =(cid:18)−x ∂oεr(cid:19)y, ωafrssesτquum≈eendc2yth0oroffouωrgshthoe=utstc(haeitsBtepsrapinpinge/r-.mreTleahxe≈aretifo2onr×et,itm1h0ee1c4oosfn−τt1riba=untdi0o.tn1hpuossf wherevx(k) ≡ ¯h−1∂ε(k)/∂kx istheelectron’sgroupveloc- σxTy ≈σxx/20addslessthan0.1toθH andlessthan0.1S0to ity,andweassumedaconstantrelaxationtime(τk =τ). 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