PHYSICAL REVIEW LETTERS 126, 167701 (2021) Hinge Spin Polarization in Magnetic Topological Insulators Revealed by Resistance Switch * Pablo M. Perez-Piskunow Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain Stephan Roche Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain and ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain (Received 28 December 2020; accepted 8 March 2021; published 20 April 2021) We report on the possibility of detecting hinge spin polarization in magnetic topological insulators by resistance measurements. By implementing a three-dimensional model of magnetic topological insulators into a multiterminal device with ferromagnetic contacts near the top surface, local spin features of the chiral edge modes are unveiled. We find local spin polarization at the hinges that inverts the sign between the top and bottom surfaces. At the opposite edge, the topological state with inverted spin polarization propagates in the reverse direction. A large resistance switch between forward and backward propagating states is obtained, driven by the matching between the spin polarized hinges and the ferromagnetic contacts. This feature is general to the ferromagnetic, antiferromagnetic, and canted antiferromagnetic phases, and enables the design of spin-sensitive devices, with the possibility of reversing the hinge spin polarization of the currents. DOI: 10.1103/PhysRevLett.126.167701 Introduction.—The recent discovery of intrinsic mag- magnetism and of topological edge states [22–24]. The netic topological insulator (TI) multilayered MnBi2 Te4 spin texture of topological edge states in both the quantum [1,2] has boosted the expectations for more resilient spin Hall and QAH regimes is usually perpendicular to the quantum anomalous Hall effect (QAHE) [3–9] and observ- material’s surface, limiting the possibility for magnetic- ability of axion insulator states [10–12]. The material sensitive detection or further spin manipulation protocols platforms to realize the quantum anomalous Hall (QAH) [25]. The effective two-dimensional models of these phase can be classified, in a broad sense, in two- and three- materials are often highly symmetric and may overlook dimensional systems. The former includes monolayer the sublattice and spin degree of freedom. However, spin materials and high-symmetry models with spin-orbit textures [26], spin Hall conductivity [27], and local spin coupling and magnetic exchange [13–15]. The latter is polarization [28] provide great insight into the special the case of three-dimensional magnetic TIs, usually real- topological phases that can arise in topological super- ized in thin-film and few-layer systems, including mag- conductors and boundary-obstructed TIs [29,30]. By reduc- netically doped TIs [16,17], proximitized TI surfaces with a ing the symmetry constraints, new spin textures can magnetic insulator [18,19], and the Chern insulator phase develop, such as hidden spin polarization [31] and canted of MnBi2 Te4 [1,11]. The distinction that arises in three- spin textures [32–34]. In the presence of a uniform [35] or dimensional magnetic TIs is that the topological nature alternating [36] Zeeman field, several models of magnetic comes from contributions from two Dirac-like surfaces layers exhibit high-order topological phases and cleavage- that, upon the introduction of a magnetization field dependent hinge modes [35–41]. Thus, a detailed study of throughout the material, become massive with opposite the spin features on a spinful three-dimensional model of effective masses [20,21]. Despite the three-dimensional the QAHE realized in magnetic TI multilayers is missing. nature of magnetic TIs, they are often analyzed near the In this Letter, we use the generic Fu-Kane-Mele (FKM) surface as effective two-dimensional systems. model for three-dimensional topological insulators [42] and However, compared to their two-dimensional counter- introduce exchange terms to describe both ferromagnetic parts, three-dimensional magnetic TIs present a higher level (FM) and antiferromagnetic (AFM) multilayered TIs. of complexity that reflects in layer-to-layer magnetic Contrary to ordinary spin-z polarization of edge states in exchange and termination-dependent surface states, which the QAH regime, the model exhibits an in-plane hinge spin ultimately dictate the nature and properties of surface polarization (HSP) which becomes apparent (and 0031-9007=21=126(16)=167701(6) 167701-1 © 2021 American Physical Society PHYSICAL REVIEW LETTERS 126, 167701 (2021) observable) in a specific device setup. Indeed, the topo- In absence of a Zeeman field, the strong TI phase is the logical states are characterized by an in-plane HSP three-dimensional realization of the Shockley model [53], perpendicular to both the current flow and the sample hosting sublattice polarized surface states. The magnetic magnetization direction. The in-plane polarization reverses moments per layer describe the AFM (alternating sign along the vertical direction between the top and bottom magnetization between layers mi ¼ m) or FM (constant surfaces. By using efficient quantum transport simulation magnetization mi ¼ m) coupling between layers. In a slab methods [43] implemented into a three-dimensional multi- geometry perpendicular to the z axis, a Zeeman exchange terminal device, such peculiar local spin polarization is coupling field m ¼ 0.05tẑ opens a gap on the surface states shown to give rise to a giant resistance switching (or spin and sets the QAH phase described by a nontrivial Chern valve) triggered upon either inverting the magnetization number [54]. of the sample, varying the polarization of the magnetic We present the main electronic and spin characteristics of detectors, or reversing the current direction. The the magnetic topological insulator model in Fig. 1. The appearance of HSP in the QAH regime is rooted in the details of the edge modes vary with the geometric design. chiral-like [26,40] symmetries of the lattice and on the half- For a heterostructure infinite along the y direction but finite quantization of the topological charge at the surfaces in both other directions, we obtain the usual linear energy [21,44–47]. Therefore, the HSP fingerprints are highly dispersion of topological edge states seen in Fig. 1(a). robust to Anderson energetic disorder, and to structural These states cover the whole side surface of the stack (wall edge disorder. Hamiltonian of the three-dimensional magnetic TI.— (a) The magnetic TI is described by a three-dimensional (diamond cubic lattice) FKM Hamiltonian [42,48,49], with magnetic layers modeled by an exchange coupling term that well captures the effect of magnetic impurities [50] or magnetic layers [35,51]. To simulate a multilayer FM or AFM magnetic TI, we tune the orientation of the magnetic moments per layer. pffiffiffiffiffiffiffi ffi pThe ffiffiffiffiffiffiffiffi FKM latticepffiffiffiffiffiffiffi vectors are ffi pffiffiffiffiffiffiffiffi a1 ¼ ð1=2; − 1=3=2; 2=3 Þ, pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi2 a ¼ ð0; 1=3 ; 2=3 Þ, and a3 ¼ ð−1=2; − 1=3=2; 2=3Þ; each unit cell has pffiffiffiffiffiffiffiffi A with 0 offset, and B with offset two sublattices: d4 ¼ ð0; 0; 3=2=2Þ. The other first neighbors of A sites are at relative positions dq ¼ d4 − aq for q ¼ 1, 2, 3. The full Hamiltonian reads (b) (c) XX † X † H0 ¼ ci;α tij cj;α ; HZ ¼ ci;α ½mi · sα;β ci;β hi;ji α i;α;β 8λSO X X † HSO ¼ i c ½s · ðd1ij × d2ij Þα;β cj;β (d) d)) a2 ⟪i;j⟫ α;β i;α H ¼ H0 þ HSO þ HZ ; ð1Þ with latin indices for lattice sites, and Greek indices for spin value in the sz basis. The Zeeman magnetization vector mi may depend on the layer of the orbital i, and s is a vector of Pauli matrices acting on the spin degree of freedom. The FIG. 1. Magnetic TI in the FM phase, m ¼ 0.05tẑ. parameter λSO denotes the spin-orbit (SO) coupling (a) Dispersion relation of a slab geometry infinite in the y strength, while tij describes the nearest neighbors coupling direction. The left (right) inset depicts the local spin density of between sites i and j, and takes different values tq with states hsx i of the edge state at ky ¼ −0.1π=a (ky ¼ 0.1π=a). The q ¼ 1; …; 4 depending on the direction rj − ri ¼ dq . As edge state covers the sidewall of the slab and propagates to the described in Fu et al. [42], the isotropic case tq ¼ t defines right (left). (b) Local density of states of a finite square slab. The edge state circulates around the sample, covers the side surfaces a multicritical point. Adding anisotropy tq ¼ t for q ¼ 1, 2, perpendicular to x̂, and propagates along the top or bottom hinges 3 and t4 > t, sets the phase to a strong TI characterized by a of the side surfaces perpendicular to ŷ. (c) Side view of transport nontrivial Z 2 invariant. We tune the parameters to the setup geometry: metallic leads connect to the whole walls at both strong TI phase with t4 ¼ 1.4t and λSO ¼ 0.1t [52]. The ends of the slab (golden color), and ferromagnetic leads connect FKM model can be interpreted as a stack of coupled to the sidewalls only near top hinge (red color). (d) Top view and Rashba layers, with an alternating Rashba field [36,53]. reference numbering of the leads on the transport setup. 167701-2 PHYSICAL REVIEW LETTERS 126, 167701 (2021) states) with a very large electronic density at the hinges. Interestingly, the projected local spin density of the wall states is seen to be dominated by the hsx i value near the hinges. The HSP switches sign between opposite surfaces. Furthermore, the HSP changes sign for the back-propagating states, located at the opposite walls (see insets). On a finite slab, Fig. 1(b), the nature of the chiral states becomes richer, with the emergence of hinge states for certain surface cleavage orientations, a property predicted for Möbius fermions [35,36,40]. The Möbius fermion phase depends on the ferromagnetic interlayer exchange and appears in the FM and canted AFM phase on crystalline canting directions. Conversely, the HSP is robust and appears in all phases, that is, FM, AFM, and canted AFM, irrespective of the canting angle, as long as there is a z component of the net magnetization. Next, we FIG. 2. Transport simulations of a FM slab (m ¼ 0.02tẑ) explore the possible fingerprints of such anomalous spin between metallic leads and ferromagnetic leads with spin down features on quantum transport in the QAH regime. ðsx ; ↓Þ polarization. The shaded regions depict the standard error Multiterminal spin transport simulations.—To analyze of considering ten Anderson disorder realizations of strength the spin transport in the QAH regime, we use the Kwant d ¼ 0.04t. A four-terminal device allows us to measure the software package [43] to build the three-dimensional model distinct resistance profiles. The two-terminal resistance setup and implement a multiterminal device configuration, between leads L0 and L1 is depicted in the left inset and the blue curve is the resistance profile. The right inset shows the resistance shown in Figs. 1(c) and 1(d). We perform charge transport setup between leads L1 and L2 , with resistance profile in red. In simulations of a central scattering region connected with the former case, the ferromagnetic lead polarization matches the metallic and ferromagnetic leads. The interplay between the top HSP, and in the latter case, the local spin polarization is states available for transport in the leads and in the opposite. scattering region has a central role. The leads L1 and L3 are the metallic leads (golden color). They are fully To test the robustness of the 2T resistance switching contacting the left and right sides of the slab (all spin effect, we introduce different types of disorder. First, we projections). The ferromagnetic leads L0 and L2 (red color) consider the impact of structural disorder—vacancies near located on the sides only contact the upper part of the the sidewalls of the slab. This disorder is detrimental to the device near the top hinge [55]. They carry electrons with formation of well-defined HSP, which only occurs for wall only one spin polarization: ðsx ; ↓Þ. In this way, these states at crystalline edges. Nevertheless, it is relevant for contacts couple with the edge state in the region of maximal predictions on experiments, since the side walls of material local spin polarization. samples have edge disorder. We find that the HSP effect The expected resistance measurements for the QAHE survives to structural disorder, up to %5 vacancies [52]. are shown in the inset of Fig. 2. We use the notation Rij;kl , Next, we simulate Anderson disorder by adding an on-site for the resistance measured from passing current between energy dχ, where χ is a random variable with normal terminals i and j and measuring the voltage drop between distribution on ½−0.5; 0.5Þ. We find robustness of the HSP terminals k and l. The two-terminal (2T) resistance up to d much larger than the magnetization strength. In Rkl;kl is noted R2T;kl . The typical values of Hall resistance Fig. 2, we use d ¼ 2jmj ¼ 0.04t, and average the resistance Rxy ¼ R13;20 and the longitudinal resistance Rxx ¼ R01;23 curve over ten disorder realizations. We see that spin of a QAH insulator [56,57] take the quantized values transport measurements can still distinguish the peculiar Rxy ¼ ð1=CÞh=e2 , where C is the Chern number, and spin texture of the edge states. vanishing Rxx inside the gap. The two-terminal resistance The fact that Anderson disorder and structural disorder R2T ¼ h=e2 is also quantized for perfect tunneling show the resistance switch is crucial in establishing the between the leads and the scattering region [58]. Such robustness of our results. The limit mismatching case, is the case of the matching ferromagnetic lead. The where the edge state and ferromagnetic lead are completely matching or mismatching between the spin current carried decoupled from the transport setup, results in voltage by the leads and the spin polarization of the edge states probes that have zero transmission probability to any other gives rise to a remarkable resistance switch, as seen in leads, leaving a floating probe with an arbitrary value of the Fig. 2. The 2T resistance in the matching case is quantized chemical potential and the voltage [59]. However, in our inside the topological gap, while in the mismatching case, the ferromagnetic leads are not fully disconnected case the resistance increases by more than 1 order of when the spins do not match, rather they are weakly magnitude. connected. Even though the value of the 2T resistance is 167701-3 PHYSICAL REVIEW LETTERS 126, 167701 (2021) (a) (a) (b) (b) FIG. 4. Magnetic TI slab with the same parameters as in Fig. 3. HSP of the edge state at ky ¼ −0.02π=a and positive energy (propagating in the ŷ direction) for a wide range of magnetization angles ðθ; ϕÞ. The arrows show the components of hsi projected FIG. 3. 2T resistance at eF ¼ 0 of a magnetic TI slab in the FM on the top-left (-right) hinges of an infinite slab along the y axis. phase, with magnetization jmj ¼ 0.02t (see right inset). In (a), the The direction of the arrows gives the component in the two-terminal resistance is measured between leads L1 and L2 ; in ðhsx i; hsz iÞ plane, and the color of the arrows is the component (b), the resistance is measured between leads L0 and L1 . The plots in hsy i. (a) Propagating edge states on the top-left hinge are only in (a) and (b) are almost identical by reflecting around ϕ ¼ 90°, present when the magnetization has a positive projection along z, where the magnetization component along the z axis changes thus, ϕ ≲ 90°. (b) Propagating edge states on the top-right hinge sign. The mismatching configurations ϕ ¼ 0° in (a), and are only present when the magnetization has a negative projection ϕ ¼ 180° in (b) show the largest resistance that slowly decreases along z, thus, ϕ ≳ 90°. when rotating the magnetization angle. forward propagating edge states on the top half of an sensitive to the details of the weak coupling, seen on the infinite slab in the ŷ direction, and finite in the xz plane, see large standard error in Fig. 2, the trend is clear. In a QAH insets of Fig. 4. At momentum k ¼ −0.02π=a we select the thin film contacted on its lateral sides with ferromagnetic positive eigenvalue inside the topological gap, similar to the leads, we can selectively get either full transmission or states shown in the insets of Fig. 1(a). Panels (a) and (b) of blocking of the edge state transport. Such a phenomenon is Fig. 4 show finite length arrows that indicate the spin sensitive to the direction of the magnetization of the density and components in the ðhsx i; hsz iÞ plane of the ferromagnetic leads, the direction of the current, and the forward propagating state, while the color represents the net magnetization of the sample. mostly null hsy i component. A vanishing arrow length (a Another experimentally relevant analysis is to explore point in the plot) indicates that there is no net spin density at the resistance switch for different directions of the mag- that region enclosing that hinge [60]. When the system is in netization m of the slab in the FM phase. Figure 3 shows the topological phase (ϕ≁90°), there is electronic density in two measures of 2T resistance, as in Fig. 2 at the charge one edge or the other, and spin density near the hinge neutrality point, for different directions of the Zeeman (a finite arrow). Accordingly, we see that panel 4(b) exchange field ðcos θ cos ϕx̂ þ sin θ cos ϕŷ þ sin ϕẑÞ (see complements perfectly panel 4(a). In both cases, the right inset). At low ϕ angles (m pointing mostly toward HSP direction changes with the magnetization angle, þẑ), the configuration R2T;1;2 in 3(a) shows large resis- giving a notch to control the matching or mismatching tance, while R2T;0;1 in 3(b) is close to the quantized value cases in a transport setup. h=e2 . When sweeping the magnetization to the inverse Conclusions.—We have demonstrated that the edge direction (toward −ẑ) at ϕ180°, the roles of 3(a) and 3(b) states in thin-film ferromagnetic and antiferromagnetic reverse, giving a clear signature of the highly spin-polar- TIs host HSP, spin polarized states at the hinges, which ized hinges and of the spin-dependent matching and leads to a large resistance switch. The HSP of the edge mismatching with the ferromagnetic leads. In the middle states is in plane, but the sign depends on the propagation of both extremes, where ϕ ¼ 90°, the magnetization lies in direction and the magnetization of the sample. For a the plane of the slab and does not open a gap on the top and crystalline edge direction, the local spin polarization bottom surfaces. At intermediate angles, we note that the reverses across the vertical direction. Thus, the HSP inverts resistance switch is more robust for θ ¼ 90°, where m tilts across the vertical direction, and switches sign for toward ŷ, the transport direction and edge direction that the the opposite current direction. Carefully engineering ferro- FM leads contact. magnetic contact leads in a transport setup allows us to The HSP of the edge states is a good proxy for predicting obtain a giant resistance (spin valve effect) upon reversing the switch in resistances that is measured in the device the current direction or, conversely, tuning the total shown in Fig. 3. We obtain the spin projection of the magnetization of the sample. The hsx i component of the 167701-4 PHYSICAL REVIEW LETTERS 126, 167701 (2021) spin direction in Figs. 4(a) and 4(b) can be directly [17] S. Qi, Z. Qiao, X. Deng, E. D. Cubuk, H. Chen, W. Zhu, E. translated to the resistance values found in Figs. 3(a) Kaxiras, S. B. Zhang, X. Xu, and Z. Zhang, Phys. Rev. Lett. and 3(b). This highlights that the resistance switching 117, 056804 (2016). mechanism, once established, can be used to gain insight [18] Y. Hou, J. Kim, and R. Wu, Sci. Adv. 5, eaaw1874 (2019). [19] M. Mogi, T. Nakajima, V. Ukleev, A. Tsukazaki, R. about the magnetization of the sample. Yoshimi, M. Kawamura, K. S. Takahashi, T. Hanashima, Finally, we observe the fact that FM and AFM K. Kakurai, T.-h. Arima, M. Kawasaki, and Y. Tokura, Phys. topological insulators are able to host maximally spin Rev. Lett. 123, 016804 (2019). polarized currents along crystalline hinges opens new [20] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. avenues to implement disruptive proposals using axion Fang, Science 329, 61 (2010). and magnetic TIs to manipulate dislocation, hinge, and [21] R.-L. Chu, J. Shi, and S.-Q. Shen, Phys. Rev. B 84, 085312 edge states [37], with the additional value of spin polari- (2011). zation features. [22] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nat. Rev. Phys. 1, 126 (2019). We thank Sergio O. Valenzuela, David Soriano, [23] X. Wu et al., Phys. Rev. X 10, 031013 (2020). and Aron W. Cummings for fruitful discussions. We [24] D. Nevola, H. X. Li, J.-Q. Yan, R. G. Moore, H.-N. Lee, H. acknowledge the European Union Horizon 2020 research Miao, and P. D. Johnson, Phys. Rev. Lett. 125, 117205 and innovation programme under Grant Agreement (2020). No. 824140 (TOCHA, H2020-FETPROACT-01-2018). [25] For tunable spin polarization in the QAHE see R.-X. Zhang, ICN2 is funded by the CERCA Programme/Generalitat H.-C. Hsu, and C.-X. Liu, Phys. Rev. B 93, 235315 (2016). de Catalunya, and is supported by the Severo Ochoa [26] A 2D model including sublattice and spin can yield in-plane program from Spanish MINECO (Grant No. SEV- spin polarization, see J. Wu, J. Liu, and X.-J. Liu, Phys. Rev. 2017-0706). Lett. 113, 136403 (2014). [27] M. Costa, G. R. Schleder, C. M. Acosta, A. C. M. Padilha, F. Cerasoli, M. B. Nardelli, and A. Fazzio, arXiv:2006.07270. [28] K. Plekhanov, N. Müller, Y. Volpez, D. M. Kennes, H. * Schoeller, D. Loss, and J. Klinovaja, Phys. Rev. B 103, pablo.perez.piskunow@gmail.com L041401 (2021). [1] M. M. Otrokov et al., Nature (London) 576, 416 (2019). [29] A study of the quantum spin Hall effect in magnetic [2] I. I. Klimovskikh et al., npj Quantum Mater. 5, 54 (2020). topological insulators indicates the presence of hinged [3] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). [4] C. Z. Chang et al., Science 340, 167 (2013). quantum spin Hall states, characterized by a nontrivial [5] Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. spin-Chern number, see Y.-r. Ding, D.-h. Xu, C.-z. Chen, Chen, and Y. Zhang, Science 367, 895 (2020). and X. C. Xie, Phys. Rev. B 101, 041404(R) (2020); [6] T. Hirahara et al., Nat. Commun. 11, 4821 (2020). However similar, our model cannot be characterized by [7] D. A. Estyunin, I. I. Klimovskikh, A. M. Shikin, E. F. the spin-Chern number due to the large SO coupling and Schwier, M. M. Otrokov, A. Kimura, S. Kumar, S. O. nonconservation of the spin, see E. Prodan, Phys. Rev. B 80, Filnov, Z. S. Aliev, M. B. Babanly, and E. V. Chulkov, 125327 (2009); and D. Monaco and L. Ulčakar, Phys. Rev. APL Mater. 8, 021105 (2020). B 102, 125138 (2020). Instead, our model is characterized [8] S. Wimmer et al., arXiv:2011.07052. by a nonzero Chern number. [9] V. Bonbien, F. Zhuo, A. Salimath, O. Ly, A. Abbout, and A. [30] E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, Manchon, arXiv:2102.01632. arXiv:1908.00011, [Phys. Rev. X, (to be published)]. [10] D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, [31] X. Zhang, Q. Liu, J.-W. Luo, A. J. Freeman, and A. Zunger, Phys. Rev. Lett. 122, 206401 (2019). Nat. Phys. 10, 387 (2014). [11] C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. [32] L.-k. Shi and J. C. W. Song, Phys. Rev. B 99, 035403 Zhang, and Y. Wang, Nat. Mater. 19, 522 (2020). (2019). [12] F. Giustino et al., J. Phys. Mater. 3, 042006 (2021). [33] M. Vila, C.-H. Hsu, J. H. Garcia, L. A. Benítez, X. [13] Z. Qiao, S. A. Yang, W. Feng, W.-K. Tse, J. Ding, Y. Yao, J. Waintal, S. Valenzuela, V. M. Pereira, and S. Roche, Wang, and Q. Niu, Phys. Rev. B 82, 161414(R) (2010). arXiv:2007.02053. [14] W.-K. Tse, Z. Qiao, Y. Yao, A. H. MacDonald, and Q. Niu, [34] J. H. Garcia, M. Vila, C.-H. Hsu, X. Waintal, V. M. Pereira, Phys. Rev. B 83, 155447 (2011). and S. Roche, Phys. Rev. Lett. 125, 256603 (2020). [15] An alternative route to realize the QAHE is to include strong [35] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 110, interactions, leading to orbital magnetization, see F. Zhang, 046404 (2013). J. Jung, G. A. Fiete, Q. Niu, and A. H. MacDonald, Phys. [36] K. Plekhanov, F. Ronetti, D. Loss, and J. Klinovaja, Phys. Rev. Lett. 106, 156801 (2011); W. Chen and J. L. Lado, Rev. Research 2, 013083 (2020). Phys. Rev. Lett. 122, 016803 (2019); M. Serlin, C. L. [37] N. Varnava and D. Vanderbilt, Phys. Rev. B 98, 245117 Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, (2018). T. Taniguchi, L. Balents, and A. F. Young, Science 367, 900 [38] L. Trifunovic and P. W. Brouwer, Phys. Status Solidi (b) (2020). 258, 2000090 (2021). [16] G. Xu, J. Wang, C. Felser, X.-L. Qi, and S.-C. Zhang, Nano [39] Y. Tanaka, R. Takahashi, T. Zhang, and S. Murakami, Phys. Lett. 15, 2019 (2015). Rev. Research 2, 043274 (2020). 167701-5 PHYSICAL REVIEW LETTERS 126, 167701 (2021) [40] R.-X. Zhang, F. Wu, and S. Das Sarma, Phys. Rev. Lett. 124, [53] S. S. Pershoguba and V. M. Yakovenko, Phys. Rev. B 86, 136407 (2020). 075304 (2012). [41] For a similar model describing magnons, see A. Mook, S. A. [54] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. Rev. Condens. Díaz, J. Klinovaja, and D. Loss, arXiv:2010.04142. Matter Phys. 7, 301 (2016). [42] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, [55] A similar device can be envisioned to measure the axion 106803 (2007). insulator phase R. Chen, S. Li, H.-P. Sun, H.-Z. Lu, and [43] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, X. C. Xie, arXiv:2005.14074. New J. Phys. 16, 063065 (2014). [56] M. Büttiker, Phys. Rev. B 38, 9375 (1988). [44] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, [57] M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986). 195424 (2008). [58] In the case of all normal (metallic) leads, we obtain the [45] M. Gu, J. Li, H. Sun, Y. Zhao, C. Liu, J. Liu, H. Lu, and Q. typical quantized values of Rxx ¼ 0, Rxy ¼ h=e2 , and Liu, arXiv:2005.13943. R2T;ij ¼ h=e2 for all pairs of leads i, j. This is due to the [46] R. Lu et al., Phys. Rev. X 11, 011039 (2021). fact that metallic leads carry current with any spin projec- [47] We verified the half-quantized topological charge at the tion, thus, metallic leads always realize the matching case. surfaces using the method described in D. Varjas, M. [59] In the Ladauer-Bttiker formalism, after setting one reference Fruchart, A. R. Akhmerov, and P. M. Perez-Piskunow, Phys. probe L0 with an arbitrary voltage value V 0 , we eliminate Rev. Research 2, 013229 (2020). the rows and columns of L0 from the transmission matrix to [48] L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007). make it nonsingular. However, if there is a disconnected [49] D. Soriano, F. Ortmann, and S. Roche, Phys. Rev. Lett. 109, voltage probe Ld , there is one extra row and column with 266805 (2012). zero values corresponding to Ld , and this matrix is again [50] Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang, Phys. singular. The value that the voltage can take in this probe V d Rev. Lett. 102, 156603 (2009). is arbitrary, and we must eliminate the corresponding [51] J. Wang, B. Lian, H. Zhang, and S.-C. Zhang, Phys. Rev. column and row to solve the system. Lett. 111, 086803 (2013). [60] The norm of the spin components is not conserved, since it [52] See the Supplemental Material at http://link.aps.org/ can be averaged out within the region where it is computed. supplemental/10.1103/PhysRevLett.126.167701 for more Therefore, the arrows may vanish without a significant hsy i details on the model and structural disorder simulations. component. 167701-6