Table Of ContentQuantized Anomalous Hall Effect in Two-Dimensional Ferromagnets
- Quantum Hall Effect from Metal -
Masaru Onoda1∗ and Naoto Nagaosa1,2†
1Correlated Electron Research Center (CERC), National Institute of Advanced
Industrial Science and Technology (AIST), Tsukuba Central 4, Tsukuba 305-8562, Japan
3 2Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
0
(Dated: February 6, 2008)
0
2 We study the effect of disorder on the anomalous Hall effect (AHE) in two-dimensional ferro-
n magnets. The topological nature of AHE leads to the integer quantum Hall effect from a metal,
a i.e., thequantizationof σxy inducedbythelocalization except forthefew extendedstates carrying
J Chernnumber. ExtensivenumericalstudyonamodelrevealsthatPruisken’stwo-parameterscaling
4 theory holds even when the system has no gap with the overlapping multibands and without the
2 uniform magnetic field. Therefore the condition for the quantized AHE is given only by the Hall
conductivityσxy without thequantumcorrection, i.e., |σxy|>e2/(2h).
]
ll PACSnumbers: 72.15.Rn,73.43.-f,75.47.-m,75.70.-i
a
h
- TheoriginoftheanomalousHalleffect(AHE)hasbeen the languageof the effective magnetic field for electrons,
s
e asubjectofextensivecontroversyforalongterm. Oneis it reaches a huge value of the order of ∼ 104 Tesla, i.e.,
m basedonthe bandpicturewiththe relativisticspin-orbit the magnetic cyclotron length is of the order of the lat-
. interaction [1], while the other is due to the impurity ticeconstant,butthenetfluxiszerowhenaveragedover
t
a scatterings [2]. Most of the succeeding theories follows the unit cell. Therefore these two cases belong to quite
m theideathattheAHEoccursduetothescatteringevents different limits although the symmetries of the systems
- modifiedbythespin-orbitinteraction,i.e.,theskewscat- arecommon, i.e., the unitary class withouttime-reversal
d tering or the side jump mechanism [3]. Recently several nor spin-rotationalsymmetry
n
authors recognized the topological nature of the AHE In this paper we report on an extensive numerical
o
discussed in Refs. [4, 5, 6]. In this formalism, the Hall study on two-dimensional (2D) models of AHE includ-
c
[ conductivityσxy isgivenbytheBerryphasecurvaturein ing the disorder potentials. It is found that the topolog-
the momentum (~k-) space integrated over the occupied icalnature ofAHE leads to a dramaticphenomenon, i.e,
1
states [7]. Also there appeared some experimental evi- the ferromagnetic metal turns into an integer quantized
v
2 dences supporting it [8]. Therefore it is very important Hall system by introducing disorder. This is due to the
8 to study the effect of the scatterings by disorder, which topologicalstabilityoftheChernnumberscarriedbythe
4 makes~k ill-defined,toseethetopologicalstabilityofthis extended states which are energetically separatedby the
1 mechanism for AHE. continuumofthe localizedstatesinbetween. Namelythe
0
localized state can not have a finite Chern number, and
3 This issue is closely related to the integer quantum
the integertopologicalnumbercannotchangesmoothly,
0 Hall effect (IQHE) [9] but there are severalessential dif-
i.e., it jumps when it changes. These two facts leads
/ ferences. Usually the topologicalstability which guaran-
t to the protection of the extended state carryinga Chern
a teesthequantizationofsomephysicalquantity,e.g.,σ ,
xy
m numberagainsttheweakdisorder. Thefinite-sizescaling
hasbeendiscussedinthecontextoftheadiabaticcontin-
analysisiscompatiblewiththetwo-parameterRGtheory
- uation [9]. Therefore it appears that the gaps between
d ofPruisken[11], whichpredicts the plateautransitionat
Landau levels in pure system are needed to start with
n |σ | = 0.5e2/h. The critical exponents are consistent
even though the disorder potential eventually buries it. xy
o with that of the IQHE.This problemis not anacademic
c IntheIQHEsystemwithoutdisorder,theperiodicpoten-
one; the recent technology can fabricate very fine thin
: tialisirrelevantbecausethecarrierconcentrationismuch
v films of ferromagnetic metals with large enough single
smaller than unity per atom. Although numerical simu-
i domain. When the coherent length of such a thin film is
X lations[10]uselatticemodels,themainconcernisputon
longer than the thickness, it can be regarded as a multi-
r thelimitofdispersionlessLandaulevelsseparatedbythe
a channel 2D system. These systems can offer a possible
gaps. In the present case, i.e., in ferromagnetic metals,
laboratory to test our theory.
there are multiple bands overlapping without the gaps
in the density of states. The periodicity of the lattice The essence ofthe AHE is that the Berry phaseof the
remains unchanged, which prohibits the uniform mag- Bloch electron is induced by the spin-orbit interaction
netic field and also gives a large energy dispersion. In in the presence of the magnetization, which is modeled
by the complex transfer integrals [5]. Each band often
gains finite Chern number even though the density of
states has no gap. The minimal model describing this
∗Electronicaddress: [email protected] situation is that proposed by Haldane [12] and its ex-
†Electronicaddress: [email protected] tension. This model is defined on a honeycomb lattice
2
t = 0.2 t, m = 0.4 t, φ = π / 3
1 0 0
at2r atr 2e/ h] 01 (a)
1 [
σ xy −1
atr ahc
3 2 S
O0.5
D
ahc ahc
3 1
0
FIG. 1: Haldane’s model defined on honeycomb lattice [12]. h] 1 (b) u = 0.8 t
OpenandclosedcirclesrepresenttheAandBsublatticesites 2/ 0
e 0
treicsepevcetcivtoerlys.ofThhoentehyiccokmabrrloawttsic~aeh1,ca2n,3datnhdos~aet1r,o2f,3trairaengtlheesluabt-- σ [xy −1
lattice respectively. The dashed lines represent next-nearest-
neighbor hopping. S
O0.5
D
0
containing two atoms in a unit cell (Fig. 1). Using the −4 −2 0 2 4
Fouriertransformationc† =(c† ,c† )ofthespinorrep- E / t0
~k A~k B~k
resentation (c† ,c† ), its Hamiltonian is written as
H = c†H~rc∈A, w~rh∈eBre FIG. 2: Hall conductivity σxy and the density of states for
~k ~k ~k ~k the pure single-layer model with t1 = 0.2t0, m = 0.4t0 and
φ = π/3 (a), and for the pure double-layer model with t1 =
P
H~k = 2t1cosφ cos(~k·~atir)τ0 0.2t0, m=0.4t0, u=0.8t0 and φ=π/3 (b).
i=1,2,3
X
+t0 cos(~k·~ahic)τ1−sin(~k·~ahic)τ2 havealsocalculateduptothesystemsizeM =64around
Xi h i thelowerextendedstate. (Theseadditionaldataareused
+ m+2t1sinφ sin(~k·~atir) τ3. (1) ienxpthoneeannta.)lyTsehsefolerntghtehloocfaaliztuatbieonisletnygptichaallnyd∼it1s0c5ristiitceasl
" #
i
X and the accuracy of data is within a few percent. We
τ0 is the unit matrix and τ1,2,3 are the Pauli matrices. canseethatextendedstatesareisolatedinenergy. They
Hereweassumetheperfectspin-polarizationandusethe mergewitheachotheratacriticalvalue6<Wc <7,and
spinless fermions. The complex next-nearest neighbor disappear, i.e., all states are localized. This behavior is
hoppingintegralt1eiφisintroducedinadditiontothereal the same as that observed in the ordinary IQH system
one t0 between the nearest neighbors. ~at1r,2,3 are the lat- onasquarelatticewithexternalmagneticfield[10]. Itis
tice vectors of triangle sub-lattice, while ~ahc are those notedthatthepairannihilationofextendedstatesalways
1,2,3 occurs between those with the opposite Chern numbers
of honeycomb lattice.
[10]. Actually, two extended states in Fig. 3 originate
The extended model is given by adding another layer
from lower and upper bands which have opposite Chern
with the change t1 → −t1 to the original single-layer
numbers, ±1, respectively.
modelgivenabove. Furthermoreweintroducetheenergy
difference between the layersby shifting the uniform po- Next we analyze the data to obtain a characteris-
tential±u. Thenthe extendedmodelhas the symmetric tic length ξ(E,W), which depends on E and W but
and gapless density of states in contrast to the original not on M, by the scaling hypothesis, λM(E,W)/M =
one. In Figs. 2 are shown the density of states and σ f(ξ(E,W)/M),wheref(x)isascalingfunction[14]. As
xy
for the single-layer (a) and double-layer (b) models, re- for a localized state, ξ(E,W) is interpreted as its local-
spectively. In the single-layer case, σ is quantized to ization length in the thermodynamic limit. Figure 3(c)
xy
bee2/hwhentheFermienergylieswithinthegap,while shows ξ(E,W = 5.0t0) around the lower extended state
itisnotinthedouble-layercasewherethegapcollapses. at E = Ec. From this data, the critical exponent ν
(ξ ∝ |E −E |−ν) is estimated as ν = 2.37±0.05 with
c
Now we introduce the on-site disorderpotentialto the Ec =(−0.69±0.01)t0. Figure 3(d) gives the log-logplot
single-layer model, which is randomly distributed in the of the localization length ξ(E,5.0t0) as a function of the
range [−W/2,W/2], and study the localization problem energy measured from Ec. The fitting is also shown as
in terms of the transfer matrix method [13]. Figure 3(a) a solid line with the slope −ν = −2.37. This value is in
shows the dependence of the renormalized localization reasonableagreementwiththatestimatedintheordinary
length λ /(2M) of a quasi-1D tube with 2M-sites cir- IQH system [15], e.g. ν =2.35±0.03 [16].
M
cumference on the strength of disorder potential W. In The system-size dependence of σ shown in Fig. 3(b)
xy
eachfigure,the lines forM =4,8,16,32areplotted. We represents its scaling property. There are two critical
3
t = 0.2 t, m = 0.4 t, φ = π / 3, W = 0.5 t t = 0.2 t, m = 0.4 t, u = 0.8 t, φ = π / 3
1 0 0 0 1 0 0 0
100 (a) (b) L = 4 1
λ / (2M)M10−1 MM == 48 LLL === 13862 0.5xy [e/ hσ2 λ / (2M)M100 W0 = 5.0 t0 W0 = 5.0 t0
M = 16 ]
W = 0.2 t W = 1.0 t
M = 32 1 0 1 0
0 10−1
−2 0 2 −2 0 2 1
E / t
0
]
h
(c) (d) 2/ 0.5
4 [e
σ xy
ξ 0
n
l 2
−3 −2 −1 0 −3 −2 −1 0
E / t
0
0
−2 −1 0 −2 −1 0 1
E / t0 ln( | E−Ec | / t0 ) FIG.4: Upperpanels: thelocalizationlengthλM ofaquasi-
1D tube where M is the number of A(B) sites on the cir-
cumference. M =4,8,16,32 are plotted. Lower panels : the
FIG. 3: (a) Localization length λM of a quasi-1D tube system-sizedependenceofσxyinthedouble-layersystemwith
where M is the number of A(B) sites on the circumference.
2L×2Llattice points. Thenumbersofsamplesaveraged are
M = 4,8,16,32 are plotted. (b) System-size dependence of
81920,20480,5120 for L = 4,8,16 respectively. The color of
σxyinthesingle-layersystemwith2L×2Llatticepoints. The each lines is for thesame system-size as in Fig. 3(a) and (b).
numbersof samples averaged are 81920,20480,5120,1280 for
L=4,8,16,32respectively. Theerrorsareonestandarddevi-
ation. (c)Logplotand(d)Log-logplotofLocalizationlength
at which λ /2M does not show M-dependence. This
ξ around the lower extended state at Ec = (−0.69±0.01)t0 M
The solid line is a fittingresult with theslope −ν =−2.37. meansthatthe extendedstatessurvivesthere asinFigs.
3 at least up to W1 =1.0t0.
We next present the system-size dependence of σ ,
xy
points where there is no size dependence and which sep- as shown in the lower panels of Fig. 4. There appears
aratethetwoenergyregionswiththeoppositesizedepen- two critical points for the transitions σxy : 0 ↔ 1
dences. σxy atthese criticalpointstakesthe valueabout and σxy : −1 ↔ 1. From the particle-hole symme-
0.5e2/h. This is consistent with the analysis in terms of try, σxy(−E) = −σxy(E) is concluded. Therefore, there
the effective field theory for the ordinary IQH system in appear three critical points in the whole energy region.
the weak-localization region [11], and strongly suggests Because the single-layer model has at most two critical-
that the critical properties of this transition are same as points,thedouble-layermodelcouldhavemaximumfour.
those of the plateau transition σ : 0 ↔ 1. The energy One may wonder why there appear only three critical-
xy
ofthesecriticalpointscoincidewiththatwherethelocal- points in the present case. This is because the middle
izationlength divergesin Figs.3(a) and(c). This means one is composed of two extended states which originally
that the extended states with Chern number ±1 exist contribute to different critical points but carry the same
there, and σ in the thermodynamic limit (M → ∞) Chern number, i.e. −1. These extended states merge
xy
stays quantized to be e2/h between these two energies (at least in the present numerical accuracy) but never
[broken line in Fig. 3(b)]. pair-annihilate,because the composite ofthese extended
Now we shall consider the double-layer model. The states carry the non-zero Chern number −2. In other
IQHE never occurs in this system with the above set of words,theconservationlawofthetopologicalchargepre-
parameters unless the effect of disorder is taken into ac- vents the localization.
count. Weconsiderthenontrivialcaseinwhichthereare σxy at the lower critical-point takes the value ∼=
scattering events both within and between these layers. 0.5e2/h. This value is again consistent with the anal-
In this case, there is no gap between the initial and final ysis by Pruiskin and coworkers [11]. However, σ at
xy
states of the elastic scattering, and it is possible that all the middle critical-point is zero. This critical behavior
the states are localized once the disorder is introduced. seems to violate the prediction by the analysis in [11].
However as shown below, the extended states and the However, recent numerical studies for the ordinary IQH
Chern number carried by them are stable against the system reveals the new type of critical phenomena, i.e.
weak disorder. Here we define the strengthofintra-layer the direct transitions σ : 0 ↔ n (n > 1) [10], which
xy
scatteringasW0 andrepresentthestrengthofinter-layer were experimentally observed in advance [17]. The crit-
scattering by W1. As seen in the upper panels of Fig. 4, ical property around the middle point is considered to
there occur two energies, i.e., E ∼= −1.5t0 and E = 0, belong to the same class as σxy : 0 ↔ 2. Although the
4
samplesizeisnotlargeenoughinthedouble-layermodel, glected. In IQHE system, the lnT dependence of σ is
xx
the size-dependence of σ is consistent with the quan- observed[18]andis attributed to the quantumCoulomb
xy
tizedplateaushownbythebrokenlineinthelowerpanels correction[19]. However,thequantizedσ intheground
xy
of Fig. 4. state is well described by the noninteracting electron
It is not difficult to generalize the non-linear sigma model. The situation is similar here for the quantized
modelapproachforthelocalizationproblemtothecaseof AHE system. In the thin film of Fe, lnT-dependence of
multi-component model without time-reversal nor spin- σ isobservedwhile notforσ [20], whichis explained
xx xy
rotationalsymmetry. This“components”meansorbitals, by the quantum Coulomb correction combined with the
spins, and channels in the multilayer cases altogether. skew scattering mechanism [21].
Tinhgi.sFapolplorowaicnhgdtoheesdneortivaastsiuomneinthReefif.ni[t1e1g],awpeatotbhteaisntatrhte- Usually AHE is estimated by ρH ∼= −ρ2xxσxy, where
ρ , ρ and σ are measured as quantities in 3D. In
Lagrangian: H xx xy
good metals, ρ is very small at low temperatures, and
xx
L[{Ql}] = − 1[g−1]ll′TrQlQl′ hfoerneciet|iσsxpyo|ssisibllaertgheaatltthheouqguhan|ρtiHze|disAvHeEryissmreaallli.zeTdheevreen-
2
l,l′ in the conventional metallic ferromagnets such as Fe or
X
f Ni, when the thin film is fabricated. Actually, when we
+Trln E−Hˆ +iηs+ iQlIl , (2) virtually consider thin film of n-layer systems, the 2D
" Xl # |σxy| at TC/2 is estimated as ∼ 0.59ne2/h for Fe [22],
∼ 0.47ne2/h for Ni [22], and ∼ 0.20ne2/h for SrRuO3
where [g]ll′ is the scattering strength between compo- (from the first article in Ref. [8]). Therefore, the con-
nents l and l′, and [Il]l′l′′ = δl′lδl′′l. Tr (ln) is the trace dition |σ | > 0.5e2/h is not so difficult to achieve in
xy
(logarithm) of matrix with functional index (~r) and dis-
the thin films of metallic ferromagnets. Extrapolating
crete indices (p,a,l), where p = ± corresponds to the
the lnT behavior of σ experimentally observed [20],
xx
advanced and retarded fields respectively, and a runs
the crossovertemperature Tcross from weak to strong lo-
over replicas. Tr O is the abbreviation for d~r TrO(~r) calization is estimated as Tcross ∼= T0e−Aσ0eh2, where T0 a
where O(~r) is a matrix with p and a indices. The non-
R reference temperature of the order of 10K,σ0 the Drude
linearsigmamofdelistheeffectivemodelforthemassless
conductivityatT0,andAisasample-dependentscaling-
Goldstone modes. In order to extract these modes, the
parametrizationQl =TlPlTl−1 isuseful. Fromthe above eσx0pho/n(Aenet2)ofisthleesosrtdhearnof∼un10it,y.wTehhearveefotrhee,icfhtahnecme itnoimoba-l
Lagrangian, it is clear that inter-component scatterings
[g−1]ll′ (l 6= l′) lock the out-of-phase modes Tl 6= Tl′ simerevnetathlleyqreuaalniztaizbelde atenmompearlaotuusreH.alCloenffseicdteriinngthtehaetxpσer-
(l 6= l′), and therefore the effective model for massless 2 xy
havetobelargerthan0.5e /h,thisconditionmeansthat
in-phase modes, i.e., T = T, reduces to the model iden-
tical to that in Ref. [11l]. The coefficients of the stiffness theratioσ0/σxy shouldbesmallerthan∼10. Thisnovel
quantizedHallstatewouldprovethemostdramaticcon-
and topological terms for these modes coincide with σ
xx sequence of the topological nature of the AHE.
and σ respectively. It is noted that these σ and σ
xy xx xy
contain the contributions from all components, i.e., all The authors would like to thank Y. Tokura and
orbitals,spinsandchannels. Thenthescalingofσ and A. Asamitsu for useful discussion. M. O. is supported
xy
σ remainsthesameasgiveninRef.[11]. Thissupports byDomestic ResearchFellowshipfromJapanSocietyfor
xx
the finite-size scaling study given above. thePromotionofScience. N.N.issupportedbytheMin-
Inrealsystems,theCoulombinteractioncannotbene- istryofEducation,Science,CultureandSportsofJapan.
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