Hinge Spin Polarization in Magnetic Topological Insulators Revealed by Resistance Switch
HHinge Spin Polarization in Magnetic Topological InsulatorsRevealed 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 andICREA–Instituci´o Catalana de Recerca i Estudis Avan¸cats, 08010 Barcelona, Spain
We report on the possibility to detect hinge spin polarization in magnetic topological insulatorsby resistance measurements. By implementing a three-dimensional model of magnetic topologicalinsulators into a multi-terminal device with ferromagnetic contacts near the top surface, local spinfeatures of the chiral edge modes are unveiled. We find local spin polarization at the hinges thatinverts sign between top and bottom surfaces. At the opposite edge, the topological state withinverted spin polarization propagates in the reverse direction. Large resistance switch betweenforward and backward propagating states is obtained, driven by the matching between the spinpolarized hinges and the ferromagnetic contacts. This feature is general to the ferromagnetic,antiferromagnetic and canted-antiferromagnetic phases, and enables the design of spin-sensitivedevices, with the possibility of reversing the hinge spin polarization of the currents.
Introduction. — The recent discovery of intrinsic mag-netic topological insulator (TI) multilayered MnBi Te [1, 2] has boosted the expectations for more resilientquantum anomalous Hall effect [3–6] and observabilityof axion insulator states [7, 8]. The material platformsto realize the quantum anomalous Hall (QAH) phase canbe classified in two- and three-dimensional systems. Theformer includes monolayer materials with spin-orbit cou-pling and magnetic exchange [9, 10]. The latter is thecase of three-dimensional magnetic TIs, including mag-netically doped TIs [11, 12], proximitized surfaces of a TIwith a magnetic insulator [13, 14], and the Chern insula-tor phase of MnBi Te [1, 8]. The distinction that arisesin three-dimensional magnetic TIs is that the topologicalnature comes from contributions from two Dirac-like sur-faces that, upon the introduction of a magnetization fieldthroughout the material, become massive with oppositeeffective masses [15, 16]. Despite the three-dimensionalnature of magnetic TIs, they are often analyzed near thesurface, as effective two-dimensional systems.However, compared to their two-dimensional counter-parts, three-dimensional magnetic TIs present a higherlevel of complexity that reflects in layer-to-layer mag-netic exchange and termination-dependent surface states,which ultimately dictate the nature and properties of sur-face magnetism and of topological edge states [17, 18].The spin texture of topological edge states in both thequantum spin Hall and quantum anomalous Hall (QAH)regimes is usually perpendicular to the material’s surface,limiting the possibility for magnetic-sensitive detection orfurther spin manipulation protocols. The effective two- ∗ Electronic address: [email protected] dimensional models of these materials are often highlysymmetric and may overlook the sublattice and spin de-gree of freedom. By reducing the symmetry constrains,new spin textures can develop, such as hidden spin polar-ization [19] and canted spin textures [20–22]. In presenceof a uniform [23] or alternating [24] Zeeman field, severalmodels of magnetic layers exhibit high-order topologi-cal phases and cleavage-dependent hinge modes [25–27].Thus, a detailed study of the spin features on a spinfulthree-dimensional model of the QAHE realized in mag-netic TIs multilayers is missing.In this Letter, we use the generic Fu-Kane-Mele (FKM)model for three-dimensional topological insulators [28]and introduce exchange terms to describe both ferro-magnetic (FM) and antiferromagnetic (AFM) multilay-ered TIs. Contrary to ordinary spin- z polarization ofedge states in the QAH regime, the model exhibits anin-plane hinge spin polarization (HSP) which becomesapparent (and observable) in a specific device setup. In-deed, the topological states are characterized by an in-plane HSP perpendicular to both the current flow and thesample magnetization direction. The in-plane polariza-tion reverses sign along the vertical direction, betweenthe top and bottom surfaces. By using efficient quan-tum transport simulation methods [29] implemented intoa three-dimensional multi-terminal device, such peculiarlocal spin polarization is shown to give rise to a giantresistance switching (or spin valve ) triggered upon eitherinverting the magnetization of the sample, varying thepolarization of the magnetic detectors, or reversing thecurrent direction. Such fingerprints of HSP in the QAHregime are rooted in the chiral-like [27, 30] symmetriesof the lattice, and are highly robust to Anderson-type ofenergetic disorder, and to structural edge disorder. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Hamiltonian of the three-dimensional magnetic TI. —The magnetic TI is described by a three dimensional (di-amond cubic lattice) FKM Hamiltonian [28, 31, 32], withmagnetic layers modelled by an exchange coupling termthat well captures the effect of magnetic impurities [33]or magnetic layers [23, 34]. To simulate a multilayerFM or AFM magnetic TI we tune the orientation of themagnetic moments per layer. The FKM lattice vectorsare a = ( / , −√ / / , √ / ) , a = ( , √ / , √ / ) ,and a = (− / , −√ / / , √ / ) ; each unitcell has twosublattices: A with offset, and B with offset d =( , , √ / / ) . The other first neighbors of A sites areat relative positions d q = d − a q for q = , ,
3. The fullHamiltonian reads H = ∑ ⟨ i,j ⟩ ∑ α c † i,α t ij c j,α ; H Z = ∑ i,α,β c † i,α [ m i ⋅ s ] α,β c i,β H SO = i λ SO a ∑ ⟪ i,j ⟫ ∑ α,β c † i,α [ s ⋅ ( d ij × d ij )] α,β c j,β H = H + H SO + H Z . (1)The latin indices go over the lattice sites, and the Greekindices over the spin indices in the s z basis. The Zee-man magnetization vector m i may depend on the layerof the orbital i , and s is a vector of Pauli matrices act-ing on the spin degree of freedom. The parameter λ SO denotes the spin-orbit coupling strength, while t ij de-scribes the first nearest neighbors coupling between sites i and j , and takes four different values labeled t q with q = ... r j − r i = d q . As described in Fu et al. [28], theisotropic case where t q = t defines a multicritical point,but adding anisotropy t q = t for q = , , t > t ,sets the phase to a strong TI, characterized by a non-trivial Z invariant. We tune the parameters in unitsof the in-plane hopping t to the strong TI phase with t = . t and λ SO = . t . The FKM model can be in-terpreted as a stack of coupled Rashba layers, with al-ternating Rashba field [24, 35]. In absence of Zeemanfield the strong TI phase is the three-dimensional real-ization of the Shockley model [35], hosting surface statesthat are sublattice polarized. The magnetic momentsper layer describe the AFM (alternating magnetizationbetween layers m i = ± m ) or FM (constant magnetiza-tion m i = m ) coupling between layers. In a slab ge-ometry perpendicular to the z axis, a Zeeman exchangecoupling field m = . t ˆ z normal to the slab leads toa gap opening of the surface states, and sets the QAHphase described by a non-trivial Chern number [36].We present the main electronic and spin characteristicsof the magnetic topological insulator model in Figure 1.The details of the edge modes vary with the geometric de-sign. For a heterostructure infinite along the y -directionbut finite in both other directions, we obtain the usual (a)(b) (c)(d) FIG. 1. Magnetic TI in the FM phase, m = . t ˆ z . a) Dis-persion relation of a slab geometry infinite in the y direction.The left (right) inset depicts the local spin density of states ⟨ s x ⟩ of the edge state at k y = − . π / a ( k y = . π / a ). The edgestate covers the side wall of the slab and propagates to theright (left). b) Local density of states of a finite square slab.The edge state circulates around the sample, covers the sidesurfaces perpendicular to ˆ x , and propagates along the top orbottom hinges of the side surfaces perpendicular to ˆ y . c) Sideview of transport setup geometry: metallic leads connect tothe whole walls at both ends of the slab (golden color), andferromagnetic leads connect to the lateral walls only near tophinge (red color). d) Top view and reference numbering ofthe leads on the transport setup. linear energy dispersion of topological edge states seenin Fig.1-a). These states cover the whole side surface ofthe stack (wall states) with a very large electronic den-sity at the hinges. Interestingly, the projected local spindensity of the wall states is seen to be dominated by the ⟨ s x ⟩ value near the hinges. The hinge spin polarization (HSP) switches sign between opposite surfaces. Further-more, the HSP changes sign for the back-propagatingstates, located at the opposite walls (see insets). On afinite slab, Fig. 1-b), the nature of the chiral states be-comes richer, with the emergence of hinge states for cer-tain surface cleavage orientations, a property predictedfor M¨obius fermions [23, 24, 27]. The
M¨obius fermionsphase depends on the ferromagnetic interlayer exchange,and appears in the FM and canted-AFM phase on crys-talline canting directions. Conversely, the HSP is robustand appears in all phases, that is: FM, AFM, and canted-AFM, irrespective of the canting angle, as long as there isa z -component of the net magnetization. We next explorethe possible fingerprints of such anomalous spin featureson quantum transport in the QAH regime. Multi-terminal spin transport simulations. — To anal-yse the spin transport in the QAH regime, we usethe Kwant software package [29] to build the three-dimensional model, and implement a multi-terminal de-vice configuration, shown in Figs. 1 c), and d). We per-form charge transport simulations of a central scatteringregion connected with metallic and ferromagnetic leads.The interplay between the states available for transportin the leads and in the scattering region has a centralrole. The leads L and L are the metallic leads (goldencolor). They are fully contacting the left and right sidesof the slab (all spin projections). The ferromagnetic leads L and L (red color) located on the sides only contactthe upper part of the device near the top hinge. Theycarry electrons with only one spin polarization: ( s x , ↓) .In this way, these contacts couple with the edge state inthe region of maximal local spin polarization.The expected resistance measurements for the QAHEare shown on the inset of Fig. 2. We use the notation R ij,kl , for the resistance measured from passing currentbetween terminals i and j , and measuring the voltagedrop between terminals k and l . The two-terminals (2T)resistance R kl,kl is noted R T,kl . The typical values ofHall resistance R xy = R , and the longitudinal resis-tance R xx = R , of a QAH insulator [37, 38] take thequantized values of R xy = C h / e , where C is the Chernnumber, and vanishing R xx inside the gap. The two-terminal resistance R T = h / e is also quantized in thecase of perfect tunneling between the leads and the scat-tering region. Such is the case of the matching ferro-magnetic lead. The matching or mismatching betweenthe spin current carried by the leads and the spin po-larization of the edge states gives rise to a remarkableresistance switch, as seen in Fig. 2. The 2T resistance inthe matching case is quantized inside the topological gap,while in the mismatching case the resistance increases bymore than one order of magnitude.To test the robustness of the 2T resistance switchingeffect, we introduce different types of disorder sources.First, we consider the impact of structural disorder de-scribed by a random distribution of a certain density ofvacancies near the side walls of the slab that makes thescattering region. The impact of this disorder is detri-mental to the formation of well-defined HSP, which onlyoccur for wall-states at crystalline edges Nevertheless, isrelevant for predictions on experiments, since the sidewalls of material samples have edge disorder. We findthat the HSP effect survives to structural disorder, upto %5 vacancies [39]. Next, we simulate Anderson disor-der by adding an onsite energy dχ , where χ is a randomvariable with normal distribution on [− . , . ) . We find FIG. 2. Transport simulations of a FM slab ( m = . t ˆ z ) be-tween metallic leads and ferromagnetic leads with spin down ( s x , ↓) polarization. The shaded regions depict the stan-dard error of considering 10 Anderson disorder realizationsof strength d = . t . A 4-terminals device allows to measurethe distinct resistance profiles. The two-terminal resistancesetup between leads L and L is depicted on the left insetand the blue curve is the resistance profile. The right insetshows the resistance setup between leads L and L , with re-sistance profile in red. In the former case the ferromagneticlead polarization matches the top HSP, and in the latter caseand in the second case the local spin polarization is opposite. robustness of the HSP up to d much larger than the mag-netization strength. In Fig. 2, we use d = ∣ m ∣ = . t ,and average the resistance curve over 10 disorder realiza-tions. We see that spin transport measurements can stilldistinguish the peculiar spin texture of the edge states.The fact that Anderson disorder and structural disor-der show the resistance switch is crucial in establishingthe robustness of our results. There is a limit case wherethe edge state is fully polarized in a large region nearthe top hinge, and the ferromagnetic leads with oppo-site spin are completely decoupled from the edge state,and thus, from the transport setup. In this case, thevoltage probes may have zero transmission probabilityto any other leads, leaving a floating probe with an arbi-trary value of the chemical potential and the voltage [40].However, in our case the ferromagnetic leads are not fullydisconnected when the spins do not match, rather theyare weakly connected. Even though the value of the 2Tresistance is sensitive to the details of the weak coupling,seen on the large standard error in Fig. 2, the trend isclear. In a QAH thin-film contacted on its lateral sideswith ferromagnetic leads, we can selectively get, eitherfull transmission, or blocking of the edge state transport.Such phenomenon is sensitive to the direction of the mag-netization of the ferromagnetic leads, the direction of thecurrent, and the net magnetization of the sample.Another experimentally relevant analysis is to explorethe resistance switch for different directions of the mag- FIG. 3. 2T resistance at e F = ∣ m ∣ = . t (see right inset).In a) the two-terminal resistance is measured between leads L and L ; in b) the resistance is measured between leads L and L . The plots in a) and b) are almost identical by re-flecting around φ = ○ , where the magnetization componentalong the z -axis changes sign. The mismatching configura-tions φ = ○ in a), and φ = ○ in b) shows the largest resis-tance, that slowly decreases when rotating the magnetizationangle. netization m of the slab in the FM phase. Figure 3 showstwo measures of 2T resistance, as in Fig. 2 at the chargeneutrality point, for different directions of the Zeeman ex-change field ( cos θ cos φ ˆ x + sin θ cos φ ˆ y + sin φ ˆ z ) (see rightinset). At low φ angles ( m pointing mostly towards + ˆ z )the configuration R T, , in a) shows large resistance,while the values of R T, , in b) are close to the quan-tized value h / e . When sweeping the magnetization tothe inverse direction (towards − ˆ z ) at φ ○ , the roles ofa) and b) reverse, giving a clear signature of the highlyspin-polarized hinges and of the spin-dependent match-ing and mismatching with the ferromagnetic leads. Inthe middle of both extremes where φ = ○ , the magne-tization lies in the plane of the slab and does not open agap on the top and bottom surfaces. At intermediate an-gles, we note that the resistance switch is more robust for θ = ○ , where m tilts towards ˆ y , the transport directionand edge direction that the FM leads contact.The HSP of the edge states is a good proxy to predictthe switch in resistances that is measured in the deviceshown in Fig. 3. We obtain the spin projection of theforward propagating edge states on the top half of an in-finite slab in the ˆ y direction, and finite in the xz plane,see insets of Fig. 4. At momentum k = − . π / a we selectthe positive eigenvalue inside the topological gap, similarto the states shown in the insets of Fig. 1-a). Panels a)and b) of Fig. 4 show finite length arrows that indicatethe spin density and components in the (⟨ s x ⟩ , ⟨ s z ⟩) planeof the forward propagating state, while the color repre- FIG. 4. Magnetic TI slab with the same parameters as inFig. 3. HSP of the edge state at k y = − . π / a and posi-tive energy (propagating in the ˆ y direction) for a wide rangeof magnetization angles ( θ, φ ) . The arrows show the com-ponents of ⟨ s ⟩ projected on the top- left (-right) hinges of aninfinite slab along the y -axis. The direction of the arrows givesthe component in the (⟨ s x ⟩ , ⟨ s z ⟩) plane, and the color of thearrows is the component in ⟨ s y ⟩ . a) Propagating edge stateson the top-left hinge are only present when the magnetizationhas a positive projection along z , thus φ ≲ ○ . b) Propagatingedge states on the top-right hinge are only present when themagnetization has a negative projection along z , thus φ ≳ ○ . sents the mostly null ⟨ s y ⟩ component. A vanishing arrowlength (a point in the plot) indicates that there is nonet spin density at that region enclosing that hinge [41].When the system is in the topological phase, that is, φ away from ∼ ○ , there is electronic density in one edge or the other, and spin density near the hinge (a finite arrow).Accordingly, we see that panel b) complements perfectlypanel a). In both cases the HSP direction changes withthe magnetization angle, giving a notch to control thematching or mismatching cases in a transport setup. Conclusions. — We have demonstrated that the edgestates in thin-film ferromagnetic and antiferromagneticTIs host HSP, spin polarized states at the hinges, whichleads to a large resistance switch. The HSP of the edgestates is in-plane, but the sign depends on the propaga-tion direction and the magnetization of the sample. Fora crystalline edge direction, the local spin polarizationreverses across the vertical direction. Thus, the HSP in-verts across the vertical direction, and switches sign forthe opposite current direction. Carefully engineering fer-romagnetic contact leads in a transport setup, allows usto obtain a giant resistance (spin valve effect) upon re-versing the current direction or, conversely, tuning thetotal magnetization of the sample. The ⟨ s x ⟩ componentof the spin direction in Fig. 4 a) and b) can be directlytranslated to the resistance values found in Fig. 3 a) andb). 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Therefore, the arrows may vanish without a sig-nificant ⟨ s y ⟩⟩