Mixed Weyl semimetals and dissipationless magnetization control in insulators by spin-orbit torques

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by  Jan-Philipp Hanke

MixedWeylsemimetalsanddissipationlessmagnetizationcontrolininsulators
Jan-Philipp Hanke,∗ Frank Freimuth, Chengwang Niu, Stefan Blu¨gel, and Yuriy Mokrousov
Peter Gru¨nberg Institut and Institute for Advanced Simulation,
Forschungszentrum Ju¨lich and JARA, 52425 Ju¨lich, Germany
(Dated:January27,2017)
Wepredictthatthestrengthofspin-orbittorquesandtheDzyaloshinskii-Moriyainteractionintopologically
non-trivialmagneticinsulatorscan exceedsignificantlythatofconventionalmetallicmagnets. Inanalogyto
thequantumanomalousHalleffect,weexplainthisextraordinaryresponseinabsenceoflongitudinalcurrents
asahallmarkofmagneticmonopolesintheelectronicstructureofsystemsthatareinterpretedmostnaturally
within the framework of mixed Weyl semimetals. Remarkably, the magnetization switching by anti-damping
7 torques in mixed Weyl semimetals can induce topological phase transitions due to the non-trivial interplay
1 betweenmagnetizationdirectionandmomentum-spacetopology.Theconceptspresentedheremaybeexploited
0 tounderstandandutilizemagnetoelectriccouplingphenomenaininsulatingferromagnetsandantiferromagnets.
2

n
Progressincontrolandmanipulationofthemagnetization composite(k,mˆ)phasespacecombiningcrystalmomentum
a
J in magnetic materials is pivotal for the innovative design of andmagnetizationdirection[18,19]. Bandcrossings,alsore-

7 futurenon-volatile,high-speed,lowpower,andscalablespin- ferredtoasmagneticmonopolesink-space,areknowntoact
2 tronicdevices. Theeffectofspin-orbittorque(SOT)provides asimportantsourcesorsinksofΩkk [20]. Whentransferring
anefficientmeansofmagnetizationcontrolbyelectricalcur- thisconcepttocurrent-inducedtorques,crossingpointsinthe
]
i rents in systems that combine broken spatial inversion sym- compositephasespacecanbeanticipatedtogiverisetoalarge
c metryandspin-orbitinteraction[1–5]. Thesecurrent-induced mixed Berry curvature Ωmˆk, which in turn yields the domi-
s
- torques are believed to play a key role in the practical im- nant microscopic contribution to torkance and spiralization.
l
r plementationofvariousspintronicsconcepts,sincetheywere Thus,materialsprovidingsuchmonopolescanbeexpectedto
t
m demonstrated to mediate the switching of single ferromag- exhibitnotablystrongSOTsandDMI.
netic layers [6, 7] and antiferromagnets [8] via the exchange
.
t ofspinangularmomentumbetweenthecrystallatticeandthe
a
m (staggered)collinearmagnetization. Amongthetwodifferent (a) =0
Z6 =0
contributionstoSOTs,theso-calledanti-dampingtorquesare Z
-
d ofutterimportanceowingtotherobustnessoftheirproperties k1
n x
o withrespecttodetailsofdisorder[5]. kx2
c In a clean sample, the anti-damping SOT T acting on the y kx3
g
[ magnetizationinlinearresponsetotheelectricfieldE isme- r
ne
diated by the so-called torkance tensor τ, i.e., T = τE [9]. E
1
v The Berry phase nature of the anti-damping SOT manifests
0 inthefactthatthetensorelementsτ areproportionaltothe
05 omfixaelldoBcceurrpyiecdursvtaatteusre[9Ω,1miˆj0k],=wheˆiic·h2iInmijco(cid:80)rponoccra(cid:104)∂temˆsduekrniv|∂atkijvueksno(cid:105)f Magn. direction
08 lattice-periodicwavefunctionsuknwithrespecttobothcrys- (b) Momentumky (c)
1. tdaelnmotoemstehnetuitmhCkaratnedsiamnaugnniettvizeacttioorn. Idnitriemcatitoenlymrˆel.atHederteo,theˆei ) 0.2 90� V) 0.2 45�
70 anti-damping SOT is the Dzyaloshinskii-Moriya interaction gy(t 0 60� y(e 0 30�
:1 (wDaMllsI)an[1d1c,h1ir2a]l,sckryurcmiailofnosr[t1h3e–1e6m]e,rwgehnicchecoafncbheiraqluadnotmifiaeidn Ener�0.2 30� nerg 0.2 15�
v bytheso-calledspiralizationtensorDreflectingthechangeof 0.4 0� E� 0�
i � 0.2 0.3 0.4 0.2 0.4 0.6
X thefreeenergyF duetochiralperturbations∂jmˆ according kx kx
(cid:80)
r toF = D eˆ ·(mˆ ×∂ mˆ)[9].
a ij ij i j
Optimizing the efficiency of magnetization switching in Figure1.(Coloronline)(a)Schematicevolutionoftwoenergybands
inthecomplexphasespaceofcrystalmomentumandmagnetization
spintronic devices by current-induced SOTs relies crucially
direction,wherethecolorsofthebandsindicatedifferentk . Ifk
on the knowledge of the microscopic origin of most promi- x y
istuned,theelectronicstructuredisplaysamonopole,whichiscor-
nent contributions to the electric-field response. To promote
relatedwithachangeinthemixedChernnumberZ. Suchcrossing
the understanding, it is rewarding to draw an analogy be- pointsareobservedin(b)themodelofmagneticallydopedgraphene
tween the anti-damping SOT as given by Ωmˆk and the in- withhoppingt,and(c)thefunctionalizedbismuthfilm,wherecol-
ij
trinsicanomalousHalleffectasdeterminedbytheBerrycur- orsindicatethemagnetizationdirectionmˆ = (sinθ,0,cosθ). The
ΩvamtˆukreaΩrekijkco=mp2oImne(cid:80)ntsoncocf(cid:104)∂akigueknner|∂alkjcuukrvna(cid:105)tu[1re7]t.eBnsootrhΩΩkiknatnhde s(bh)o,wanndmθo=nop4o3l◦esanadriske=at(θ0.=489,00◦.1a9n)dinkin=ter(n0a.l2u9na2iπxts,f0o.r41(ca2)πy.)for
2
Inthefieldoftopologicalcondensedmatter[21,22],there- (a) �
l ) 0
cent advances in the realization of quantum anomalous Hall, /h K (c)
or, Cherninsulatorshavebeenstriking[23,24]. Thesemag- tso,�nl t 2(e�1
netic materials are characterized by a quantized value of the y
a x 2 -30 0
anomalousHallconductivityandanintegernon-zerovalueof ��
theChernnumberink-space,C =1/(2π)(cid:82) Ωkxykdkxdky. On (b) � K0 K � ) (d)
theotherhand,topologicalsemimetalshaverecentlyattracted ea0.1 K
1 (
great attention due to their exceptional properties stemming x
from monopoles in momentum space. Among these lat- t) 0 ⌧y 0 12
( 0
ter systems, magnetic Weyl semimetals host gapless low- rgy�1 uc) 10 (e)
energy excitations with linear dispersion in the vicinity of ne0.4 a/ 5 K
non-degenerate band crossings at generic k-points [25–28], E 0 mt 0
which are sources of Ωkk. Their conventional description 0 (
in terms of the Weyl Hamiltonian can be formally extended �0.4 -2 Dyx�5 -300 0
0 0.1 0.2 0.8 0.4 0.0 0.4
to the case of what we call the mixed Weyl semimetal as de- k (2⇡/a ) � E�nergy (t)
y y
scribedbyH =v k σ +v k σ +v θσ ,whereθisthe
W x x x y y y θ z
anglethatthemagnetizationmˆ =(sinθ,0,cosθ)makeswith Figure 2. (Color online) (a) Sketch of the tight-binding model.
the z-axis. As illustrated in Fig. 1a, mixed Weyl semimet- (b) Top: Band structure of magnetically doped graphene with out-
alsfeaturemonopolesinthecompositephasespaceofkand of-plane magnetization and t = 0.3t, λ = 0.1t, λ = 0.4t.
so nl
θ, which are sources of the general curvature Ω. In anal- Bottom: Top of valence band (red circles) and bottom of conduc-
tion band (blue squares) in the (k ,θ)-space. The bold integers
ogy to conventional Weyl semimetals [25], we can charac- x
denote the mixed Chern number Z in the insulating regions, and
terize the topology and detect magnetic monopoles by mon-
a =3a/2. (c)–(e)EnergydependenceoftheanomalousHallcon-
itoring the flux of the mixed Berry curvature through planes duyctivity σ = Ce2/h = e2/(2πh)(cid:82) Ωkkdk dk , the torkance
xy xy x y
oZf c=ons1t/a(n2tπk)y(cid:82)aΩsmygˆxikvdenθdbkyx,thFeigi.nt1eag.erAmisxigednifiCchaenrtnenluecmtrbice-r nτyexti,zaantidonth.eInsspeirtsalsihzoatwiotnheDcyoxr,rerespspoencdtiinvgelmy,ofmoreanntuomut--sopfa-cpeladniestmribaug--
field response near monopoles in mixed Weyl semimetals tionssummedoveralloccupiedstatesinthevicinityoftheK-point.
would be invaluable in paving the road towards dissipation-
lessmagnetizationcontrolbySOTs[29].
to these effects. We point out intriguing perspectives for the
In this Letter, using model arguments and first-principles
electric-field control of magnetization in absence of longitu-
calculations [30–35], we investigate the origin and size of
dinalchargecurrentsinthesesystems,whichmayultimately
anti-damping SOTs and DMI in representative topologi-
lead to an ultralow power consumption of future spintronic
cally non-trivial magnetic insulators: (i) magnetically doped
devicesexploitingSOTs.
graphene,and(ii)afunctionalizedbismuthfilm. Wedemon-
Magnetically doped graphene.— We begin with a tight-
strate that these systems can be naturally interpreted as re-
bindingmodelofmagneticallydopedgraphene[30,36]:
alizations of mixed Weyl semimetals in the composite phase
space of k and mˆ. To characterize the anti-damping SOTs, (cid:88) (cid:88)
H =−t c† c +it eˆ ·(σ×d )c† c
weevaluatewithinlinearreponsethetorkance[9] iα jα so z ij iα jβ
(cid:104)ij(cid:105)α (cid:104)ij(cid:105)αβ
(3)
occ (cid:88) (cid:88)
2e (cid:88)(cid:2) (cid:3) +λ (mˆ ·σ)c† c −λ (mˆ ·σ)c† c ,
τij = N eˆi· mˆ ×Im(cid:104)∂mˆukn|∂kjukn(cid:105) , (1) iα iβ nl iα jβ
k iαβ (cid:104)ij(cid:105)αβ
kn
where N is the number of k-points, and e > 0 denotes the whichissketchedinFig.2a. Here,c† (c )denotesthecre-
k iα iα
elementarypositivecharge. Similarly, thespiralization[9]is ation(annihilation)ofanelectronwithspinαatsitei,(cid:104)...(cid:105)re-
obtainedas strictsthesumstonearestneighbors,theunitvectord points
ij
fromj toi,andσisthevectorofPaulimatrices. Besidesthe
occ
D = eˆi ·(cid:88)(cid:2)mˆ ×Im(cid:104)∂ u |h |∂ u (cid:105)(cid:3) , (2) usual hopping with amplitude t, the first line in Eq. (3) con-
ij N V mˆ kn kn kj kn tainstheRashbaspin-orbitcouplingofstrengtht originating
k kn so
inthesurfacepotentialgradientofthesubstrate. Theremain-
where h = H + E − 2E , H is the lattice-periodic ingtermsinEq.(3)aretheexchangeenergyduetothelocal
kn k kn F k
Hamiltonian with eigenenergies E , E is the Fermi level, (λ)andnon-local(λ )exchangeinteractionbetweenspinand
kn F nl
and V is the unit cell volume. Remarkably, we find that the magnetization. Dependingonmˆ,thenon-localexchangede-
magnitude of SOTs and DMI in the considered family of in- scribesahoppingprocessduringwhichthespincanflip.
sulators can reach gigantic values exceeding by far that of First,bymonitoringtheevolutionofthemixedChernnum-
conventional metallic ferromagnets. We show that emergent ber Z we demonstrate that the above model hosts a mixed
magnetic monopoles as sources of a large mixed Berry cur- Weylsemimetalstate. Indeed,asshowninFig.2b,thetopo-
vatureinthecompositephasespacearepivotalingivingrise logicalindexZ changesfrom−2to0atacertainvalueofk ,
y
3
(f) H 2
(a) h) 1 (c) K Bi h) K0 (h)
C 2e/ 0 2e/ 1 K
( 1 (
Ir �xy �2 -1000 0 �xy 0 0 100
� (g) � K0 M K �
0 ea)0 0.2 (d) K 1 ea)0 1 (i) K0
(b) ( ) ( K
ergy(eV)��21 Va/uc)⌧0yx 01.010 (e) 0 K8 Energy(eV�0.210 ⌧Va/uc)yx01098.050 K0 0 10(j)
En 3 (me 10 0 500 0 -1 0 -1 (me 0 K
� � KM � Dyx� 2.7 2.6 2.5 �0.20 0.5 1 Dyx�90 00.410000.2 0.0 0.2
� E�nergy (e�V) ky (2⇡/a) � E�nergy (eV)
Figure3.(Coloronline)(a)CrystalstructureofgraphenedecoratedbyIradatomsin2×2geometry.(b)First-principlesbandstructureofthe
systemforanout-of-planemagnetization.Thetopologicallynon-trivialgapabout2.6eVbelowtheFermilevelishighlighted.(c)–(e)Energy
dependenceofanomalousHallconductivityσ ,torkanceτ ,andspiralizationD ,respectively,foranout-of-planemagnetization.Vertical
xy yx yx
linesindicatetheenergyuptowhichstatesaresummedupforthemicroscopicdistributionsinmomentumspaceneartheK-point, shown
in the insets. (f) Crystal structure of the semihydrogenated Bi(111) bilayer. (g) Top: First-principles band structure of the system for an
out-of-planemagnetization,wheretheregionofthetopologicallynon-trivialbandgapishighlighted. Bottom: Evolutionoftopofvalence
band(redcircles)andbottomofconductionband(bluesquares)inthe(k ,θ)-space.BoldintegersdenotethemixedChernnumberZ,anda
x
isthein-planelatticeconstant.(h)–(j)Energydependenceofσ ,τ ,andD formˆ outoftheplane.
xy yx yx
indicating thus the presence of a band crossing in compos- haviorofthespiralizationD withinthegap,changingfrom
yx
ite phase space that carries a topological charge of +2. One 8mta/ucto−6mta/ucasshowninFig.2e,where“uc”refers
of these monopoles appears near the K-point off any high- tothein-planeunitcellcontainingtwoatoms. Wefurtherex-
symmetrylineifthemagnetizationisorientedin-planealong plicitly confirm the origin of the large SOT and DMI in the
the x-direction (see Fig. 1b). The emergence of the quan- monopoles by removing the Weyl points from the electronic
tumanomalousHalleffect[36],Fig.2c,overawiderangeof structure via, e.g., including an intrinsic spin-orbit coupling
magnetization directions can be understood as a direct con- term in Eq. (3), in which case the electric-field response in
sequenceofthemagneticmonopolesactingassourcesofthe termsoftorkanceandspiralizationisstronglysuppressed.
curvatureΩkk. Correspondingly,formˆ outoftheplane,the Toprovidearealisticmanifestationofthemodelconsider-
system is a quantum anomalous Hall insulator. Moreover, ationsabove,westudyfromabinitiothesystemofgraphene
large values of the mixed curvature Ωmˆk in the vicinity of decorated by 5d transition-metal adatoms. This system is
themonopolearevisibleinthemomentum-spacedistributions known to realize the quantum anomalous Hall effect due to
oftorkanceandspiralizationintheinsetsofFigs.2dand2e, thespin-orbitmediatedcomplexhybridizationofgraphenep
respectively. For an out-of-plane magnetization, the primary stateswithdstatesofthetransitionmetal[39]. Inthecaseof
microscopiccontributiontotheeffectsarisesfromanavoided Iradatomsdepositedongraphene(cf.Fig.3a),aglobalband
crossing along ΓK – a residue of the Weyl point in (k,θ)- gap of 85meV opens about 2.6eV below the Fermi level if
space. Sincetheexpressions(1)and(2)differentiatethewave themagnetizationisorientedout-of-plane,Fig.3b. Asvisible
function only with respect to one of the momentum coordi- inFig.3c,thesystematthisenergyisaCherninsulatorwith
nates,thesymmetrybetweenkxandky inthedistributionsof C = −2, where the main contribution to the Chern number
torkanceandspiralizationisbrokennaturally. comesfromanavoidedbandcrossingalongKM. Themag-
As a consequence of the monopole-driven momentum- nitude of the torkance in the gap amounts to τ = 0.1ea
yx 0
spacedistribution,theenergydependenceofthetorkanceτ , (with a being Bohr’s radius), Fig. 3d, and the spiralization
yx 0
Fig.2d,displaysadecentmagnitudeof0.1eaintheinsulating D ranges from −1meVa /uc to −10meVa /uc, Fig. 3e,
yx 0 0
region(withabeingtheinteratomicdistance),andstayscon- values that are the same order of magnitude as those ob-
stantthroughoutthebandgap. IncontrasttotheChernnum- tained in metallic magnetic heterostructures [5, 9] and non-
bersC andZ,thetorkanceτ is,however,notguaranteedto centrosymmetric bulk magnets [16]. Since the details of the
yx
bequantizedtoarobustvalue,i.e.,theheightofthetorkance electronicstructurecaninfluencethevalueofthetorkancein
plateau in Fig. 2d is sensitive to fine details of the electronic the gap, e.g., upon replacing Ir with other 5d transition met-
structure such as magnetization direction and model param- als such as W, the magnitude of the SOT and DMI can be
eters. Because of their intimate relation in the Berry phase enhanced drastically in the gapped regions of corresponding
theory[9,37,38],theplateauintorkanceimpliesalinearbe- materials,accordingtoourcalculations.
4
Functionalized bismuth film.— While the SOT and DMI for example, by doping or applying strain. Such versatility
in magnetically doped graphene are comparable in magni- could be particularly valuable for the stabilization of chiral
tudetothoseinconventionalmetallicmagnets,nowwefocus magneticstructuressuchasskyrmionsininsulatingferromag-
on a semihydrogenated Bi(111) bilayer with an out-of-plane nets. Inthelattercase,verylargevaluesoftheanti-damping
magnetization,Fig.3f,anothermixedWeylsemimetalbutfor SOT arising in these systems would open exciting perspec-
whichgiganticeffectscanberealized. Thesystemisavalley- tives in manipulation and dynamical properties of chiral ob-
polarizedquantumanomalousHallinsulator[40]withasize- jectsassociatedwithminimalenergyconsumptionbymagne-
able magnetic moment of 1.0µ per unit cell, and exhibits toelectric coupling effects. Generally, we would like to re-
B
a large band gap of 0.18eV at the Fermi energy as well as a markthatmagneticmonopolesinthecompositephasespace,
distinctasymmetrybetweenthevalleysK andK(cid:48),Fig.3g. which we discuss here, do not only govern the electric-field
AnalyzingtheevolutionofthemixedChernnumberZ asa response in insulating magnets but are also relevant in met-
functionofk inFig.3g,wedetecttwomagneticmonopoles als, where they appear on the background of metallic bands.
y
ofoppositechargethatemergeatthetransitionpointsbetween Ultimately,inanalogyto(non-quantized)anomalousHallef-
thetopologicallydistinctphaseswithZ =−1andZ =0.Al- fect in metals, this makes the analysis of SOT and DMI in
ternatively,thesecrossingpointsandthemonopolechargesin metallicsystemsverycomplexowingtocompetingcontribu-
thecompositephasespacecouldbeidentifiedbymonitoring tionstotheseeffectsfromvariousbandspresentattheFermi
the variation of the momentum-space Chern number C with energy. Inaddition,theelectric-fieldstrengthinmetalsistyp-
magnetizationdirection. Thesemonopoles,actingassources icallymuchsmaller,limitingthusthereachablemagnitudeof
of the general curvature Ω, occur at generic points near the responsephenomenaascomparedtoinsulators.
valley K for θ = 43◦ (see Fig. 1c) and in the vicinity of
At the end, we comment on the relevance of the physics
theK(cid:48)-pointforθ = 137◦,respectively. Foranout-of-plane
discussed here for antiferromagnets (AFMs) that satisfy the
magnetization,thecomplexnatureoftheelectronicstructure
combined symmetry of time reversal and spatial inversion.
in momentum space manifests in the quantization of C to
SOTsinsuchantiferromagnetsareintimatelylinkedwiththe
+1, Fig. 3h, which is primarily due to the pronounced pos-
physics of Dirac fermions, which are doubly-degenerate ele-
itivecontributionsnearK. Calculationsoftheenergydepen-
mentary excitations with linear dispersion [43, 44]. In these
dence of the torkance and spiralization in the system, shown
systems, the reliable switching of the staggered magnetiza-
inFigs.3iand3j,revealtheextraordinarymagnitudesofthese
tion by means of current-induced torques has been demon-
phenomena of the order of 1.1ea for τ and 50meVa /uc
0 yx 0 stratedveryrecently[8]. Inanalogytotheconceptofmixed
forD ,exceedingbyfarthetypicalmagnitudesoftheseef-
yx Weyl semimetals presented here, we expect that the notion
fectsinmagneticmetallicmaterials[5,9,16]. Asinthecase
of mixed Dirac semimetals in a combined phase space of
of graphene, the SOT and DMI in this material are strongly
crystal momentum and direction of the staggered magneti-
suppressediftheFermienergyissettoliewithinthetopolog-
zation vector will prove fruitful in understanding the micro-
icallytrivialgapsthatdonotexhibitthemixedWeylpoints.
scopicoriginofSOTsininsulatingantiferromagnets. Follow-
Remarkably,themagnetizationswitchingviaanti-damping ing the very same interpretation that we formulated here for
torques in mixed Weyl semimetals can be utilized to induce ferromagnets,monopolesintheelectronicstructureofAFMs
topologicalphasetransitionsfromaCherninsulatortoatriv- can be anticipated to constitute prominent sources or sinks
ialmagneticinsulatormediatedbythecomplexinterplaybe- of the corresponding general non-Abelian Berry curvature,
tween magnetization direction and momentum-space topol- whose mixed band-diagonal components correspond to the
ogy in these systems. In the case of the functionalized bis- sublattice-dependentanti-dampingSOT,inanalogytothespin
muth film, for instance, the material is a trivial magnetic in- Berry curvature for quantum spin Hall insulators and Dirac
sulator with a band gap of 0.25eV if the magnetization is semimetals [45–47]. Correspondingly, exploiting the princi-
oriented parallel to the film plane. Nevertheless, the result- ples of electronic-structure engineering for topological prop-
ing anti-damping torkance τyx = 0.5ea0 is still very large, ertiesdependingonthestaggeredmagnetizationcouldresult
and also the DMI varies from Dyx = 105meVa0/uc to in an advanced understanding and utilization of pronounced
−60meVa0/uc within the gap [30]. Overall, combining an magnetoelectricresponseininsulatingAFMs.
exceptionalelectric-fieldresponsewithalargebandgap, the
We gratefully acknowledge computing time on the super-
H-functionalizedbismuthfilmservesasadistinctrepresenta-
computers JUQUEEN and JURECA at Ju¨lich Supercomput-
tiveofaclassofmixedWeylsemimetalmaterials,whichlay
ing Center as well as at the JARA-HPC cluster of RWTH
outextremelypromisingvistasinroom-temperatureapplica-
Aachen, and funding under the HGF-YIG programme VH-
tionsofmagnetoelectriccouplingphenomenafordissipation-
NG-513andSPP1538ofDFG.
lessmagnetizationcontrol–asubjectwhichiscurrentlyunder
extensivescrutiny(see,e.g.,Refs.[29,41,42]).
Intheexamplesthatweconsideredhere,theDMIchanges
overawiderangeofvaluesthroughoutthebulkbandgap,im-
plyingthatproperelectronic-structureengineeringenablesus ∗ [email protected]
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1
Supplementalmaterialsfor
“MixedWeylsemimetalsanddissipationlessmagnetizationcontrolininsulators”
METHODS representationoftheHamiltonian(3)onthebipartitelatticeof
graphene by introducing four orthonormal basis states |Nα(cid:105)
First-principles calculations.— Using the full-potential thatdescribeelectronswithspinα = {↑,↓}onthesublattice
linearized augmented plane-wave code FLEUR [35], we per- N ={A,B}. UsingFouriertransformations,wetransformed
formed self-consistent density functional theory calculations this matrix to a representation H(k) in momentum space,
of the electronic structure of (i) graphene decorated with whichwassubsequentlydiagonalizedateveryk-pointtoac-
Ir adatoms in 2 × 2-geometry, and (ii) a semihydrogenated cess the electronic and topological properties of the system.
Bi(111)bilayer. Thestructuralandcomputationalparameters The model parameters tso = 0.3t, λ = 0.1t, and λnl = 0.4t
of Refs. [39] and [40] were assumed in the respective cases. wereemployedinthiswork. Wechosethemagnetizationdi-
Starting from the converged charge density, the Kohn-Sham rection as mˆ = (cosθ,0,sinθ) for a direct comparison be-
equationsweresolvedonanequidistantmeshof8×8k-points tweenthemodelandthefirst-principlescalculations.
for8differentmagnetizationdirectionsmˆ =(sinθ,0,cosθ),
where the angle θ covers the unit circle once. Based on the
SEMIHYDROGENATEDBISMUTHFILM:ANISOTROPY
resulting wave-function information in the composite phase
WITHMAGNETIZATIONDIRECTION
space,weconstructedasinglesetofhigher-dimensionalWan-
nierfunctions(HDWFs)[31]foreachofthesystemsbyem-
In Fig. S1, we show the dependence of anomalous Hall
ploying our extension of the WANNIER90 code [32]. In case
conductivity, torkance, and spiralization on the magnetiza-
(i), we generated 82 HDWFs out of 120 energy bands with
tion direction mˆ = (sinθ,0,cosθ) in the semihydrogenated
the frozen window up to 3.3eV above the Fermi level, and
bismuth film. For general magnetization directions, both
inthecase(ii),weextractedfrom28bands14HDWFsfora
torkance and spiralization display also small non-zero com-
frozenwindowthatextendsto2.1eVabovetheFermienergy.
ponents τ and D , respectively, since the shape of these
We used the Wannier interpolation [33, 34] that we gen- yy yy
responsetensorsisdictatedbythecrystalsymmetriesandnot
eralized to treat crystal momentum and magnetization direc-
due to Onsager’s reciprocity relations. When the Weyl point
tion on an equal footing [31] in order to evaluate the Berry
curvatures Ωkk and Ωmˆk. Taking into account the above emerges in the electronic structure at θ = 43◦, the system
undergoes a topological phase transition from a Chern insu-
parametrizationofthemagnetizationdirectionbyθ,wewere
latortoatrivialmagneticinsulator,whichisaccompaniedby
thusabletoaccessefficientlytheanomalousHallconductivity
a jump in σ and a similar drop of τ . As apparent from
σ ,thetorkanceτ ,andthespiralizationD : xy ij
ij yj yj
Fig.S1,thetorkanceτ isstillremarkablyprominentinthe
yx
σij = eh221π (cid:90) 2Im(cid:88)occ (cid:28)∂∂ukkn(cid:12)(cid:12)(cid:12)(cid:12)∂∂ukkn(cid:29)dkxdky, (S1) regimeofthetrivialinsulator,i.e.,forθ >43◦.
i j
n
σ (e2/h) τ (ea ) D (meVa /uc)
τyj =e(cid:90) 2Im(cid:88)occ (cid:28)∂∂uθkn(cid:12)(cid:12)(cid:12)(cid:12)∂∂ukkn(cid:29)dkxdky, (S2) 0◦ 0xy 1 0 ij0.5 0 1 −5i0j 0 0 50
j
n
Dyj = V1 (cid:90) Im(cid:88)oncc (cid:28)∂∂uθkn(cid:12)(cid:12)(cid:12)(cid:12)hkn(cid:12)(cid:12)(cid:12)(cid:12)∂∂ukkjn(cid:29)dkxdky, (S3) ctionθ 30◦ τyx Dyx
e
r
di
with the same definitions as in the main text. Convergence
n. 60◦
ofthesequantitieswasachievedusing1024×1024k-points ag τyy
in the Brillouin zone. We obtained the mixed Chern number M D
yy
Z(k ) = 1/(2π)(cid:82) 2Im(cid:80)occ(cid:104)∂ u |∂ u (cid:105)dθdk by inte-
y n θ kn kx kn x 90◦
gratingthemixedBerrycurvatureonauniformmeshof1024
kx-valuesand512anglesθin[0,2π). FigureS1. TheanomalousHallconductivityσxy,thetorkanceτij,
Tight-binding model.— To arrive at the Hamiltonian (3) andthespiralizationD attheactualFermilevelinthesemihydro-
ij
ofthemaintext,themodelinRef.[36]hasbeengeneralized genated bismuth film as a function of the magnetization direction
to account for arbitrary magnetization directions mˆ and the mˆ = (sinθ,0,cosθ). The Weyl point in the electronic structure
non-local exchange interaction. We obtained a 4×4-matrix emergesatθ=43◦.