Colossal enhancement of spin-chirality-related Hall effect by thermal fluctuation
CColossal enhancement of spin-chirality-related Hall effect by thermal fluctuation
Yasuyuki Kato and Hiroaki Ishizuka
Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, JAPAN (Dated: September 4, 2019)The effect of thermal fluctuation on the spin-chirality-induced anomalous Hall effect in itinerantmagnets is theoretically studied. Considering a triangular-lattice model as an example, we findthat a multiple-spin scattering induced by the fluctuating spins increases the Hall conductivity ata finite temperature. The temperature dependence of anomalous Hall conductivity is evaluated bya combination of an unbiased Monte Carlo simulation and a perturbation theory. Our results showthat the Hall conductivity can increase up to 10 times the ground state value; we discuss that thisis a consequence of a skew scattering contribution. This enhancement shows the thermal fluctuationsignificantly affects the spin-chirality-related Hall effect. Our results are potentially relevant to thethermal enhancement of anomalous Hall effect often seen in experiments. I. INTRODUCTION
Anomalous Hall effect (AHE) has been one of the cen-tral topics in the study of quantum transport phenom-ena [1]. Continuous study over more than a century haverevealed that AHE shows rich properties which attractsthe interest not only from basic science but also from ap-plications (e.g., high-accuracy Hall-effect sensors) [2, 3].Microscopically, the mechanism of the AHE is often clas-sified into two groups: Intrinsic mechanism related to theBerry curvature of the electronic bands [4] and the ex-trinsic mechanism due to impurity scattering [5–7]. Thedifference in the microscopic origin is often reflected inthe behaviors of the AHE. For instance, the intrinsicAHE reflects singular structures in the Berry curvature.This gives rise to non-monotonic temperature ( T ) [8] andfield [9] dependence of the anomalous Hall conductivity σ AHE . On the other hand, the extrinsic AHE by magneticscatterings shows a peaklike enhancement of the Hall re-sistivity at a certain T which characterizes the underlyingphysics such as the magnetic transition [10] and coher-ence [11–13] T s. These rich features of the AHE havebeen intensively studied in both theory and experiment,and are also useful in identifying the physics behind thephenomena.Among various studies, a recent breakthrough was thediscovery of AHE related to scalar spin chirality whichis often called topological Hall effect (THE). Scalar spinchirality is a quantity defined by the scalar triple prod-uct of magnetic moments S · ( S × S ), where S i is alocal magnetic moment [Fig. 1(a)]). This quantity is ameasure of the non-coplanar nature of spin texture be-cause the spin chirality is zero whenever the three spinslie in a same plane. It was pointed out that the spinsproduce a fictitious magnetic field b when the three ad-jacent spins have a finite scalar spin chirality, resultingin an AHE [14–16] [Fig. 1(a)]. Alternatively, it is in-terpreted as an AHE due to the magnetic scattering bymultiple scatterers [17, 18]. The spin-chirality-relatedmechanism is studied in various materials, such as per-ovskite [19–22] and pyrochlore [16] oxides, chiral mag-nets [23–26], triangular oxides [27–31], and kagome anti-ferromagnets [32, 33]. The THE in these materials are of- ten investigated by the magnetic field dependence, whichare consistent with the theoretical predictions [16, 23, 24].In contrast, the T dependence of the THE in thenon-coplanar magnetic states is less understood. TheHall conductivity is expected to decrease with increasingtemperature in magnets with non-coplanar magnetic or-ders because the scalar spin chirality decreases [Curve Bin Fig. 1(b)]. In experiment, however, many materialsshow an increase of the Hall conductivity with increas-ing T [25, 30, 31] [Curve A of Fig. 1(b)]; some materialsshow the maximum slightly above the magnetic transi-tion temperature T c [30, 31]. This is in contrast to theknown theories, where the maximum is expected to bebelow [10] or much [11] higher than T c . To the best ofour knowledge, no theoretical understanding on the T dependence is reached so far.In this work, we theoretically study the enhancement ofHall conductivity ( σ THE ) by the thermal fluctuation fo-cusing on the fluctuation-induced skew scattering. As anexample, we consider a triangular lattice model with four-sublattice non-coplanar order called 3 Q order [Figs. 1(c)and 1(d)]. The T dependence of σ THE is calculated com-bining a Monte Carlo (MC) simulation and a large-sizenumerical calculation using Kubo formula. We find that σ THE increases with increasing T , sometimes up to 10 times compared with the ground state. The scan over thecarrier density n el (0 . ≤ n el ≤ . T . II. MODEL
In this study, we consider a classical Heisenberg spinmodel on a triangular lattice as an example of short-range non-coplanar magnetic order [34]. The Hamilto- a r X i v : . [ c ond - m a t . s t r- e l ] S e p b
1] direc-tions ( D > B a term represents the Zeemancoupling between the a -sublattice spins and an externalmagnetic field B a (cid:107) [111] ( B a > B χ term rep-resents a coupling between a fictitious field B χ > χ t because of which χ > T range. With thesethree terms, the low- T chiral phase is replaced by a four-sublattice long-range magnetic ordered phase [Figs. 1(c)and 1(d)]. In this state, the four spins on each sublatticein Fig. 1(c) points along different directions [Fig. 1(d)],forming a non-coplanar magnetic texture. III. RESULTS
Monte Carlo simulation — The finite T properties of this N = 24 N = 32 N = 64 N = 128 S y
T/K
T/K
024 0 1 2 3 N = 24 N = 32 N = 64 N = 128
024 0 1 2 3 N = 24 N = 32 N = 64 N = 128
024 0 1 2 3 N = 24 N = 32 N = 64 N = 128
024 0 1 2 3 N = 24 N = 32 N = 64 N = 128 FIG. 2. Results of MC simulations of the model (1):(a) specific heat C and (b) normalized spin structure factor S ( π, π ) /N and scalar spin chirality χ . (c) A spin configura-tion in MC simulation at T /K = 0 .
1. Each point representsspin orientation. Arrows are guide for eyes. (d) Spin chiralityof fluctuating spins δχ . model is calculated by MC simulations using a standardsingle-spin-flip Metropolis algorithm [38]. Figure 2 showsthe results of MC simulations with H spin . The specificheat C in Fig. 2(a) shows a peak at T c /K (cid:39)
2, and thenormalized structure factor S ( π, π ) /N = (cid:104) [( (cid:80) i ∈ a,c S i ) − ( (cid:80) i ∈ b,d S i )] (cid:105) /N , [ N (= L ) is the number of spins and L is the size of the system.] becomes nonzero below T c reflecting a phase transition to a magnetic order phase.In the lowest T , S ( π, π ) /N approaches S Q /N = 1 / χ approaches χ Q = 4 / (3 √
3) as shown in Fig. 2(b).Figure 2(c) shows a spin configuration obtained in simu-lation at the sufficiently low T . These results consistentlyshow that the ground state is the 3 Q order and the phasetransition is continuous. We also find that the overall be-havior of the above quantities are sufficiently convergedwith L ≥
24 with some finite size effect close to T c . Thebehavior of C , S ( π, π ) /N , and χ as well as the observedfinite size effect indicates that the phase transition is con-tinuous.We note that the scalar spin chirality remains positivein the entire T range, even above T c [Fig. 2(b)]. Thenonzero χ comes from the local correlation of fluctuatingspins under the B χ field in Eq. (1), which acts as the“magnetic field” for χ . As a measure of the chirality dueto the fluctuating spins, we use δχ = χχ Q − (cid:20) S ( π, π ) S Q (cid:21) . (2)In contrast, δχ shows a different T dependence. Fig-ure 2(d) shows δχ increases with increasing T and showsa cusp like peak at T c . The magnetic scattering by fluctu-ating spins produces anomalous Hall effect proportionalto the scalar spin chirality [17, 18]. Therefore, the fluctu-ating spins may produce a non-monotonic T dependenceof σ THE . Anomalous Hall conductivity — To study the T depen-dence of σ THE , we consider itinerant electrons coupled tothe spins in H spin . The electrons are coupled to the local-ized spins via Hund’s coupling, i.e., we consider a Kondolattice model on the triangular lattice. The Hamiltonianreads: H KL = − t (cid:88) (cid:104) i,j (cid:105) ,s ( c † is c js +h . c . ) − J (cid:88) i,s,s (cid:48) S i · ( c † is σ ss (cid:48) c is (cid:48) ) , (3)where σ = ( σ x , σ y , σ z ) are the vector of Pauli matri-ces, and c † is ( c is ) is a creation (annihilation) operator ofitinerant electron at site i with spin s . The first termrepresents the kinetic energy term of itinerant electrons,and the second term the Hund’s coupling. We assumethat the coupling is relatively weak ( J = t/ T el /t = 0 . σ THE is calculated by Kubo formula usingspin configurations generated by the MC simulation [38–43].Figures 3(b,c) show σ THE ( T ) at n el = 0.3 and 0.7 asexamples. Different lines are for different choices of L and L φ ; we find only small finite size effect after taking -0.8-0.40.00.40.80.0 0.5 1.0 1.5 2.0 T / T c = 10 T / T c = 5 T / T c = 2.5 T = 0 T / T c = 1/2 T / T c = 1 T / T c = 3/2
100 10 1 1011 12 2 2 T / T c
3) Perturbation theory
1. However, σ THE monotonically increases withincreasing T , reaching σ THE ∼ . e /h at T ∼ T c ; this isapproximately 30 times larger than σ THE ( T /T c = 0 . σ THE innearly all choices of n el . Figure 3(c) shows the resultsfor n el = 0 .
7. Similar to Fig. 3(b), σ THE increases withincreasing T and decreases above T c ; the curve shows amaximum around T ∼ T c with a kink slightly below it.This trend also appears for n el > σ THE [38].The increase of σ THE implies the enhancement is re-lated to the fluctuation effect. Indeed, the T dependenceof σ THE is in contrast to that of χ , which decreases mono-tonically with increasing T [Fig. 2(b)]. Therefore, theenhancement is different from what is expected in the in-trinsic THE mechanism. On the other hand, the increaseof σ THE below T c and the maximum around T c are coinci-dent with the T dependence of δχ . Furthermore, at somefilling e.g., n el = 0 . σ THE shows a cusp at T c resembling δχ . These features imply the enhancement isrelated to the spin chirality of fluctuating spins δχ , pre-sumably related to the skew scattering mechanism [18]Our results in Fig. 3(a) also find that the thermal ef-fect is larger when the Fermi level is close to the bandedge, i.e., n el ∼ ∼
2. Figure 3(d) shows the ratio σ THE ( T ) /σ THE ( T = 0) at T /T c =1/2, 1, and 3/2. Asshown in the figure, σ THE ( T /T c = 1 / , , /
2) is typ-ically 2–10 times larger than σ THE ( T = 0). On theother hand, the enhancement at the band edges are muchlarger, sometimes up to 10 times of that at T = 0.The n el ∼ ∼ n el ∼ n el ∼ ξk F (cid:28) k F and ξ are the Fermi wavenumber and the cor-relation length for χ t , respectively. Therefore, the skewscattering theory in Ref. [18] applies to this case. The ξk F (cid:28) High-temperature region — We next turn to the n el de-pendence of σ THE at a high- T region well above T c . Theresults of MC simulation are shown in Fig. 3(e). The re-sults show a qualitatively different n el dependence com-pared to the T = 0 result; σ THE for n el < n el > T regime whilethe trend is opposite in the low T near T = 0. This con-trasting trend at a high- T is explained by the relaxation-time (electron lifetime) dependence of skew scatteringmechanism. To see the density of state [ ρ ( µ )] depen-dence, we evaluated σ THE using a perturbation methodin Ref. [17]. In the T (cid:29) T c region, the fluctuation contri-bution is expected to be the only contribution to the Halleffect. Also, the correlation length of the spins becomesvery short in this region. Therefore, we only take intoaccount the contribution from the nearest-neighbor spincorrelation. With these approximations, the conductivityreads [17]: σ THE = − e J τ πN (cid:88) ( ijk ) ∈ t (cid:15) αβγ (cid:104) S α r k S β r i S γ r j (cid:105)× I x ( r j − r k ) I ( r k − r i ) I y ( r i − r j ) , (4)where I a ( r ) ≡ τN (cid:80) k v a k e i k · r ε k +1 / (4 τ ) , and v k ( ε k ) is the ve-locity (energy) of electrons with momentum k ( v k = 1).Here, the sum of ( ijk ) is limited to the three spinsforming the triangles t [Fig. 1(c)]. The electron life-time τ is evaluated using the first Born approximation, τ − ( µ ) = 2 πJ ρ ( µ ). Here, we neglected the spin-spincorrelation for the evaluation of τ .The result of Eq. (4) is shown in Fig. 3(e). The pertur-bation theory semi-quantitatively reproduces the overalltrend of numerical results. The similarity between thenumerical results and the perturbation suggests that theHall effect is related to the skew scattering by the fluc-tuating spins in the high- T regime; in the perturbationtheory, larger skew scattering contribution to σ THE is expected when τ ∝ ρ ( µ ) is larger [2, 17, 18], and indeed ρ ( µ ) for n el < n el > IV. SUMMARY AND CONCLUDINGREMARKS
To summarize, in this work, we studied the effect of thethermal fluctuation to the spin-chirality-related anoma-lous Hall effect. By an unbiased numerical simulation, wefind the Hall conductivity σ THE increases with increas-ing temperature, sometimes approximately 10 times theground state value. Detailed analysis on the temperatureand electron-density dependence shows the enhancementis consistent with the skew scattering mechanism pro-posed recently [18]; the thermal enhancement is largerwhen the Fermi level is close to the band edge, and isalso related to the density of states. These results showa significant effect of the thermal fluctuation to the Halleffect induced by non-coplanar magnetic orders.In contrast to our results, the skew scattering mech-anism was also discussed in relation to the sign changeof σ THE close to the critical temperature in chiral mag-nets with long-period magnetic orders (e.g., MnGe) [18].This is a decidedly different behavior from the currentcase where the skew scattering enhances the Hall effect.Presumably, a key difference is the size of the magneticstructure, i.e., the characteristic wave number k ∗ is large(small) in the 3 Q order (magnetic skyrmion crystals). Inthe skew scattering mechanism [18], the scattering ampli-tude is proportional to sin θ where θ is the angle betweenthe in-comming and out-going electrons, namely, largerangle scattering is important. In addition to the skewscattering, the small angle scattering is also induced bythe intrinsic topological Hall effect (THE) when k ∗ issmall. In other words, from the scattering theory view-point, the scattering channels for the skew scattering andintrinsic THE are different for small k ∗ . In contrast, sincethe magnetic unit cell of 3 Q order has only four sites ( k ∗ is large), both the skew scattering and intrinsic THE in-duce a large angle scattering. Our results presented hereshows that the magnetic fluctuation plays a non-trivialand crucial role in magnets with such a short period or-der. ACKNOWLEDGMENTS
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