Large Anomalous Nernst Effect in a Skyrmion Crystal

0.25 MB·

by  Yo Pierre Mizuta

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). N0,whichisgoodreasontoomitσxTy inourcase.
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