Overheated Topological Hall Effect: How Possible Artifacts Emerge?
OOverheated Topological Hall Effect: How Possible Artifacts Emerge?
Liang Wu a) and Yujun Zhang Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming Yunnan,650093 China. Institute of High Energy Physics, Chinese Academy of Science, Yuquan Road 19B, Shijingshan District, Beijing,100049 China (Dated: 2 September 2020)
Topological Hall effect (THE) originates from the real-space Berry phase that an electron gains when its spin followsthe spatially varying non-trivial magnetization textures, such as skyrmions. Such topologically protected magnetizationtextures can provide great potential for information storage and processing, which spurs a new wave of THE research.Since directly imaging the skyrmions or detecting the magnetic diffraction of skyrmion lattice are more challengingthan conducting Hall measurements, THE has been widely used to attest the presence of skyrmions. However, as thekey feature of THE, the bump/dip in Hall signal is not a sufficient proof of THE [Phys. Rev. B. ,180408(2018) andPhys. Rev. B. , 214440(2018)]. Here, we use empirical numerical modeling to demonstrate all possible THE-likesignals that two anomalous Hall effect (AHE) signals with opposite signs can superpose. We accentuate that similar Hallsignals observed in experiments require scrupulous re-examination to claim the advent of THE and related skyrmions.In addition, the origin of two-channel AHE in several representative examples are also been analyzed.The concept of Berry phase has been deeply rooted in themodern condensed matter physics, which reveals the role ofmomentum-space topology in various observable novel ef-fects. The quintessential examples are those notable membersin the Hall family, such as quantum Hall effect and anoma-lous Hall effect (AHE) . Recently, the topology of the non-trivial spin texture in real space was found to contribute a non-vanishing Berry phase as well, whose effect on the transversemotion of electrons was dubbed as topological Hall effect(THE) . To date, the widely-accepted criterion of the pres-ence of THE is the additional bumps/dips along with the AHE.However, this would be no longer sufficient when there existmultiple conduction channels with opposite AHE signs . Inthis letter, numerical modeling was utilized to extend and gen-eralize the results of reference [3-4] and simulate all possiblesituations of two superposed AHE signals with a comprehen-sive summary of the previously published results. We under-line that THE-like Hall signals should be taken more seriouslywhen claiming the existence of THE and related skyrmions. A. Numerical modeling
THE-like Hall signal can be mimicked by superposition oftwo AHE signals with opposite signs. Similar to the magneti-zation versus field hysteresis, an empirical model of R IAHE = M I tanh ( ω I ( H − H Ic )) , can be used to model the AHE signals,where M I , ω I and H Ic stand for the AHE magnitude, slopeparameter at coercive field and coercive field. An additionalAHE channel can be represented as R IIAHE = M II tanh ( ω II ( H − H IIc )) . Thus the overall AHE can be presented by the follow-ing equation, a) Electronic mail: [email protected] R totAHE = M I R IAHE tanh ( ω I ( H − H Ic ))+ M II R IIAHE tanh ( ω II ( H − H IIc )) (1)To simplify the discussion, we set M I > M II < | M I | = | M II | and ω I = ω II , (b) | M I | = | M II | and ω I (cid:54) = ω II ,(c) | M I | (cid:54) = | M II | and ω I = ω II , (d) | M I | (cid:54) = | M II | and ω I (cid:54) = ω II ,by varying the relative magnitude of their coercive fields H c ,as shown in Fig. 1. Without loss of generality, here we set | M I | > || M II | for | M I | (cid:54) = | M II | , which indicates the magnitudeof AHE for channel I is larger than that of channel II. Whilewe set ω I < ω II for ω I (cid:54) = ω II , which means channel II has asteeper transition than channel I near H c .As immediately seen in Fig. 1, the THE-like bumps/dipscan generally be generated in all cases. From experimen-tal point of view, Hall signals similar to a , b , c , andc are widely observed and explained by references andmany of them are listed in Table I. Special cases like d can be found in references , representing the situations ofnegligible H c (eg. superparamagnetic or ferromagentic nearCurie temperature), which can be understood by reference .While similar putative THE like d has been found recentlyin LaMnO /SrIrO heterostructure . In addtion, we also pre-dict a novel Hall signal similar to b , representing the situa-tion with difference only in ω of the two channels, which hasnot been experimentally observed yet.To summarize, the hysteresis behaviors of the previouslypublished THE results can generally be imitated by two su-perposed AHE hysteresis as shown Fig. 1 . To offer a deeperunderstanding, we next discuss the possible origin of the two-channel AHE signals. a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p 𝑀 ! = M !! 𝜔 ! = 𝜔 !! 𝐻 "! 𝐻 "!! (a) a a a a a a a a a a 𝑀 ! = M !! 𝜔 ! ≠ 𝜔 !! 𝐻 "! 𝐻 "!! (b) b b b b b b b b b b 𝑀 ! ≠ M !! 𝜔 ! = 𝜔 !! 𝐻 "! 𝐻 "!! (c) c c c c c c c c c c 𝑀 ! ≠ M !! 𝜔 ! ≠ 𝜔 !! 𝐻 "! 𝐻 "!! (d) d d d d d d d d d d FIG. 1.
Numerical modeling of the superposition of two opposite AHE signals.
The doted, dashed and solid lines represent the positive,negative and total Hall channels, respectively. The colors indicate the directions of field sweep (red: increase, blue:decrease). The arrowsindicate the increasing of H c . (a) | M I | = | M II | and ω I = ω II , (b) | M I | = | M II | and ω I (cid:54) = ω II , (c) | M I | (cid:54) = | M II | and ω I = ω II , (d) | M I | (cid:54) = | M II | and ω I (cid:54) = ω II . B. The origin of two-channel AHE
Since the AHE is both sensitive to the intrinsic electronicstructure near the Fermi level and extrinsic sample inhomo-geneity, the coexistence of two opposite AHE channels couldoccur in many materials and heterostructures. As for the in-trinsic properties, the multiple Weyl nodes with opposite signsof Berry curvature can induce the bump/dip in the resultantHall signal, such as (Bi − x Mn x ) Se and EuTiO . On theother hand, as thickness, temperature, doping, electrical gat-ing etc. can induce the AHE sign reversal, an inhomogeneoussample with a corresponding broadening sign reversal rangecould naturally give rise to a Hall signal mimicking THE fea-tures. In addition, heterostructures may contain opposite AHEsignals at different interfaces to induce a THE-like Hall.For instance, SrRuO shows a sign reversal at a criticalthickness of 4 unit cell (u.c.). Even one u.c. thickness fluc-tuation can blend the positive AHE from 4 u.c. and negativeAHE from 5 u.c. (or thicker regions) to induce the bump inHall signal in a nominal 4 or 5 u.c. sample (See Fig. 2a). SrRuO also possesses a temperature dependent AHE sign re-versal. We can assume there exists two spatial regions withdifferent critical temperatures, T and T (T
C. Approaches to distinguish THE and two-channel AHE
From the aforementioned discussion, other than the Hallmeasurements, additional methods are necessary to distin-guish THE and two-channel AHE. To predicate a real THE,reliable methods to directly image the spin textures likeskyrmions are neutron scattering in momentum space andlorentz electron microscopy techniques in real space andspin-resolved scanning tunnelling microscopy . Other mag-netic domain resolved methods, such as magnetic force mi-croscopy (MFM), magneto-optical Kerr effect (MOKE), pho-toemission electron microscopy (PEEM), can hardly distin-guish the topologically trivial magnetic bubble and non-trivialspin textures. The trivial magentic bubbles have zerowinding number, which will make no contribution to THE. Onthe other hand, if a material can be proved absence of AHEsign reversal, a THE-like Hall should be real and originatefrom non-trivial spin textures.To witness the two-channel AHE is to identify possiblesample inhomogeneity. For instance, MFM can be used to de-tect the relative magnitude of magnetization. As for SrRuO , since the 4 u.c. SrRuO shows a weaker magnetism than thatof the 5 u.c. or thicker SrRuO , different thickness can bemapped by MFM to dementrate the thickness fluctuation inSrRuO . Another hint to dertermine the sample inhomogene-ity is the two-step transition MH or magnetoresistance (MR)loops in a inhomogenous magentic sample . It is worth high-lighting that the bump/dip induced by two-channel oppoisteAHE is highly sensitive to the their ω (slope parameter) and H c . However, on the contrary, a two-step transition of MH(MR) will not be distinguishable when the differences be-tween of their ω and H c are small as the two-channel MH(MR) are of the same sign (see Fig. 2c). In addition, the uni-versal scaling relation between anomalous Hall conductivity σ AHE and longitudinal conductivity σ xx can also used to ver-ify the nature of the bump/dip in Hall signal. This is becausea real THE is a result of real-space spin topology, which theo-retically will not affact the scaling relation solely determinedby the AHE component at a high field. Otherwise, the super-posed overall AHE of two-channel opposite signal is expectedto deviate the scaling relation, which can be seen when the op-posite signal is of comparable magnitude, featured by a smalloverall AHE with large bump/dip . D. Conclussion
As a newly prominent field, dedicated efforts have been de-voted to the study of THE. However, the authenticity of THEhas long been overlooked. We argue that different forms ofTHE can be imitated with tricky two-channel opposite AHE,which is a natural result in inhomogeneous AHE sign reversalsystems. Besides the observation of bump/dip in Hall mea-surements, additional characterizations are demanded to claimthe advent of THE and related non-trivial spin textures.
Data Availability Statement
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