Three-dimensional Models of Topological Insulator Films: Dirac Cone Engineering and Spin Texture Robustness
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Three-dimensional Models of Topological Insulator Films: DiracCone Engineering and Spin Texture Robustness
David Soriano ∗ , Frank Ortmann , ∗ and Stephan Roche , CIN2 (ICN-CSIC) and Universitat Aut`onoma de Barcelona,Catalan Institute of Nanotechnology,Campus de la UAB, 08193 Bellaterra (Barcelona), Spain ICREA, Instituci´o Catalana de Recerca i Estudis Avan¸cats, 08010 Barcelona, Spain (Dated: June 26, 2018)
Abstract
We have designed three-dimensional models of topological insulator thin films, showing a tun-ability of the odd number of Dirac cones on opposite surfaces driven by the atomic-scale geometryat the boundaries. This enables creation of a single Dirac cone at the Γ-point as well as possiblesuppression of quantum tunneling between Dirac states at opposite surfaces (and gap formation),when opposite surfaces are geometrically differentiated. The spin texture of surface states wasfound to change from a spin-momentum-locking symmetry to a progressive loss in surface spinpolarization upon the introduction of bulk disorder, related to the penetration of boundary statesinside the bulk. These findings illustrate the richness of the Dirac physics emerging in thin filmsof topological insulators and may prove utile for engineering Dirac cones and for quantifying bulkdisorder in materials with ultraclean surfaces. ∗ These authors contributed equally. ntroduction.- The nascent field of Topological Insulators (TI) sparked by the seminalpaper of Kane and Mele [1], together with the prediction of three-dimensional structures forTI [2], and the subsequent experimental discoveries of two-dimensional HgCdTe quantumwells[3] and three-dimensional TI (3D-TI) materials [4–8] has thrusted these fascinating ma-terials to the forefront of modern condensed matter physics [9–11]. Topological insulatorsare governed by strong spin-orbit coupling and special crystalline symmetries that yieldan insulating bulk phase complemented by highly robust, gapless Dirac boundary states,revealed through spin-resolved ARPES profiles, or through peculiar Landau levels finger-prints in scanning tunneling spectroscopy (STM) measurements [12–14]. However, despitethe success in identifying these chiral surface states by photoemission and STM, the natureof surface transport in 3D TI lacks experimental characterization. This is because in all thematerials studied to date, residual conduction through bulk states is irremediably driven byunintentional doping introduced by the electrical gates and contacts[15, 16].The aforementioned boundary states are described by Dirac-cone physics, similarly tothe case of low-energy excitations in graphene [17], but with Dirac cones appearing in oddnumbers. The robustness of the physics of these chiral states, with respect to the thicknessof a TI film, deserves particular attention. Indeed when TI are reduced to thin films,quantum tunneling between Dirac states at opposite surfaces can eventually occur, yieldinggap formation, as recently shown for Bi Se [18] or freestanding thin Sb films [19]. However,and surprisingly, specific interactions between film and substrate prevent gap formation[19], a feature which could be of considerable interest for spintronic applications, but whichremains poorly understood.Understanding the effects of disorder on quantum transport of massless Dirac fermions isa challenging but fundamental task. For a single scattering event, the spin (or pseudospin,for graphene) quantum degree of freedom may lead to partial or full suppression of back-ward reflection when the charge crosses a local tunneling barrier (referred to as the Kleintunneling mechanism [23–25]). Additionally, quantum interferences between propagatingtrajectories may lead to an increase in the semiclassical conductivity monitored by the π Berry phase (weak antilocalization) [20–22]. These mechanisms prevent the transition to astrong Anderson localization regime and vanishing conductivity; their dependence on thenature and strength of disorder demands in-depth scrutiny.All types of non-magnetic disorders, including structural imperfections (e.g. vacancies),2urface contaminants, or doping with chemical impurities[26–29], preserve time-reversal sym-metry and are expected to weakly affect TI transport physics. In contrast, magnetic impuri-ties (which break time reversal invariance) can develop net magnetic moments inducing localmagnetic ordering, spin-dependent scattering or gap formation [30–35]. Henk and cowork-ers [36] recently reported on the robustness of Dirac states upon moderate Mn doping of aBi Te surface layer. However, they observed complicated spin textures for both undopedas well as Mn-doped Bi Te , which exhibited layer-dependent spin reversal and spin vor-tices. Since the topological protection of Dirac states is inherently driven by the non-trivialtopology of bulk electronic wavefunctions, surface and bulk disorders are actually expectedto tailor spin polarization features. However, these effects and their relation to the eventualAnderson localization of Dirac fermions have yet to be quantified.In this Letter, we describe 3D models of TI thin films and show that Dirac-cone charac-teristics on opposite surfaces of the film can be tuned upon differentiation of atomic-scalesurface terminations. Reducing the film thickness to several bulk layers leads to a loss of low-energy Dirac physics, owing to quantum tunneling between chiral states lying at oppositesurfaces. In striking contrast, when atomic-scale bottom and top surfaces are geometricallydifferentiated, Dirac cones develop either at the Γ-point (single Dirac cone) or at M-points(triple Dirac cones) and remain uncoupled down to a few bulk-layers. Our findings areconsistent with recent experimental observations [19] and open the way to controlled engi-neering of thin 3D-TI with highly robust chiral states. Furthermore, upon analyzing the spintextures of Dirac states on surfaces of thick TI films as a function of the strength of non-magnetic bulk disorder, we found that disorder leads to steady randomization of polarizationproperties and to suppression of certain spin-momentum locking symmetries. Model.-
To describe the 3D-TI films, we used the Fu–Kane–Mele (FKM) Hamiltonianwhich is defined on a diamond lattice with a single orbital per site. [37] This is a three-dimensional generalization of the model proposed by Kane and Mele to study the quantumspin Hall (QSH) effect in two-dimensional honeycomb lattices in the presence of spin-orbitcoupling (SOC) [1, 38] H = t X h ij i c † i c j + i (8 λ SO /a ) X hh ij ii c † i s · ( d ij × d ij ) c j . (1)The first term denotes the hopping term ( t >
0) between nearest neighboring orbitals, while3he second describes the spin-orbit interaction (SOI) given by a spin-dependent complexterm connecting second neighbors i and j in the diamond structure through vectors d ij and d ij along first-neighbor bonds (see Fig.1(a)). λ SO is the SOI strength, a is the cubic cellsize and s = ( σ x , σ y , σ z ) is elaborated from the Pauli matrices. From the diamond bulkHamiltonian we create slabs with varying number of layers (up to 48) and (111) surfaceorientation.An important feature of the FKM model is that it enables the description of either aweak or a strong topological insulator depending on the value of the hopping t ′ along the(111) direction (see Fig.1(c)). When t ′ < t , a weak topological insulating phase is generatedwhose physics resembles that of stacked bilayer bismuth, where each layer is in a 2D QSHstate [39]. This phase is characterized by an even number of Dirac points in the surfaceBrillouin zone (SBZ). Alternatively, if t ′ > t , the system is driven into a strong topologicalinsulating (STI) phase, with an odd number of Dirac cones centered at the M-points in theSBZ. Such a phase has been found for instance in Bi − x Sb x [5].Here we focus on the STI phase, which seems more relevant in light of recent experimentalworks [4–6, 18, 40–43], and show that a gap can form at the M points (by removing theoutermost layers of a diamond slab) and create a new surface Dirac cone at the Γ- point[9] showing a band inversion at k = 0. This extends the applicability of the FKM modelto strong topological insulators such as Bi Se , Bi Te or Sb Te and could enable furthermodeling of some ternary [44, 45] and non-centrosymmetric [46] compounds. Dirac cone engineering by atomic-scale surface geometry differentiation.-
Fig.1(d-f) showthe band structures of the three different diamond films obtained by varying surface ter-minations. We have fixed t = 1, t ′ = 1 . λ SO = 1 /
8, and have used a unit cell with N = 48 sites (one site per layer) which corresponds to a slab thickness of about 19 a (where a = a/ √ IG. 1. (a-c) Thin slabs showing the atomic structure of the top and bottom surfaces for theT1–T1, T1–T2 and T2–T2 slab geometries. (d-f) Band structure along the path Γ - M - M -M - Γ for the three slab geometries shown in (a-c). In the T1–T2 case the bands are no longerdegenerated due to inversion symmetry breaking. (g-i) Charge density plots corresponding to thedegenerate valence bands | Ψ V B ( k ) | and | Ψ ′ V B ( k ) | obtained for each slab geometry. | Ψ V B ( k ) | and | Ψ ′ V B ( k ) | (Fig.1(g)), calculated along the path Γ–M –M –M –Γ.Removing the uppermost layer from the top surface (while keeping the second-neighborhopping within the layer underneath) one generates another termination labeled T2 (seeFig.1(b)). The corresponding slab geometry with differing terminations is labeled T1–T25nd its band structure exhibits four Dirac cones along the path Γ–M –M –M –Γ (Fig.1(e)).Although the number of Dirac cones is even the system remains in the STI phase becausethe number of Dirac points on each surface remains odd. In fact, the three Dirac cones lo-cated at the M points are related to the surface with T1 termination, while the single Diraccone (emerging at the Γ point) is localized at the T2 terminated surface, as evidenced bythe valence band charge density plot (Fig.1(h)). In such a T1–T2 slab geometry, the inver-sion symmetry is broken and the valence and conduction bands are no longer degenerated.However, the topological states in both surfaces can be continuously transformed from oneto another by tuning the wave vector. This fact has important implications regarding filmthickness, as we explain below.A third possible structure is obtained by also removing one layer from the bottom surface,which leads to a T2–T2 slab geometry (Fig.1(c)). The electronic structure of this slabincludes a single Dirac cone at the Γ point in the SBZ of each surface (Fig.1(f)) and resemblesthe typical band structure of topological insulators like Bi Se [6] or Bi Te .[40] The valenceband charge density plot shows that the states at the Dirac cones are localized at oppositesurfaces (Fig.1(i)).We now compute the spin texture on each surface for the three slab geometries by eval-uating the expectation value of the spin operator h ˆs i of the corresponding surface valenceband state Ψ V B ( k ) projected onto the surface sites i S ( k ) = X i ∈ surfτ,τ ′ h Ψ V B ( k ) | i, τ ih i, τ | ˆs | i, τ ′ ih i, τ ′ | Ψ V B ( k ) i (2)where τ, τ ′ are spin indices.In Fig.2, we superimpose the spin textures (restricted to E/t = − . ± .
1, see dot-ted line on Fig.1(d-f)) on the valence band energy where Dirac points correspond to thebrightest areas ( E = 0). The top and bottom figures correspond to the top and bottomsurfaces, respectively. Blue and red arrows respectively correspond to negative and posi-tive z -components (out-of-plane) of the spin. For the three studied surface terminations,the states around the Dirac points show an out-of-plane helical spin texture preservingtime-reversal symmetry. It is noteworthy that in the T1–T1 and T2–T2 cases, the spinpolarizations in the two surfaces are related by inversion symmetry and exhibit a vortexor a spin reversal texture. This is not the case for the T1–T2 case, in which the inversion6 IG. 2. (Color). Spin texture on top and bottom surfaces (top and bottom pannels respectively)for the three different slabs studied (a,d) T1–T1, (b,e) T1–T2 and (c,f) T2–T2. The gray scaleindicates the valence band energy where the Dirac cones are located at the bright spots. symmetry is broken and the spin texture is centered around the Γ point (at the top surface)and the three M points (at the bottom surface).
TI-film thickness and robustness of Dirac physics.-
Fig.3 gives the electronic structuresof slabs with varying film thickness. For the T1–T1 terminated structure, a sizable gapalready opens at all M points for slabs with twelve layers. However, reducing the thicknessdown to four layers (Fig.3 (d)) provides insulating surface states, since gap values are above7 M M M -2-101 E ne r g y ( t ) -2-1012 E ne r g y ( t ) T1-T1 Γ M M M Γ T2-T2 Γ M M M Γ T1-T2
FIG. 3. Band structure of slabs of various thicknesses (layers L) and surface terminations madefrom the STI phase as explained in the text. The surface terminations of upper and lower surfacesare T1–T1 (a,d), T1–T2 (b,e), and T2–T2 (c,f). one (in t unit). For the T2–T2 case, the twelve-layer slab evidences a small gap at Γwhich is further widened upon reduction of film thickness (Fig.3 (c,f)). Note that a similarsituation has been encountered in recent experiments. [18, 50] For four layers, a gap ofapproximately one (in t unit) develops, similarly to the T1–T1 termination. Turning tothe mixed (T1–T2) termination, one might expect a similar trend. However, the behavioris completely different (see Fig.3 (b,e)): gapless surface states are insensitive to quantumtunneling and gap formation is suppressed regardless of film thickness. These results supportthe interpretation of recent experiments by Bian and coworkers [19], who also reported theabsence of a gap opening for thin TI-films. In that case strong interfacial bonding to thesubstrate prevents gap opening, in contrast to freestanding TI-films. Bulk disorder effects on spin textures.-
We investigate the changes in the spin texture8
IG. 4. (Color). Spin polarization of T2–T2 slab (12 layers) with disorder. (a,b) norm distribu-tion of spin vectors close to Γ. (c) Spin texture of clean (semi-transparent) and disordered slab(opaque). (d) Squared valence band wave-function near the Γ-point for different disorder and in-verse participation ratio (IPR) for increasing lateral slab size (inset). The l − scaling behavior isshown in dashed lines for comparison. upon introduction of bulk disorder. It is indeed very instructive to determine the extentto which the topological protection of surface states is reduced in the presence of bulkdisorder with increasing strength. To facilitate comparison with experiments, we focus onthe geometry which induces single Γ-centered Dirac points (T2–T2 surface configuration).For each k-point close to Γ, the spin vectors S ( k ) are computed (using Eq.2 but neglectingsurface projection) for the valence band. The spin textures are plotted for the clean case inFig.4(c) (semi-transparent arrows) where the in-plane projection is indicated by the vectorlength and the out-of plane component by the color index. For all k-points in the vicinityof Γ, the total length is one (in units of ~ / ε i ), with ε i selected at random inthe interval [ − W/ , W/ ×
3) unit cells (UC). Complementary datafor different supercells are discussed in the supplementary material[52].Upon varying the disorder strength W (Fig.4(a,b)), we find that as long as W ≤ W ≃
2, spinpolarization starts being randomized with a reduced norm (for W ≥
4) which eventuallyvanishes in the strong disorder limit. An illustration of the randomization and loss of spin-polarization for W = 4 and 9UC is given in Fig.4(c) (opaque arrows) in comparison to theclean case.To unveil the mechanisms leading to the randomization of the spin-polarization, we plotthe absolute square of the valence band wavefunction | Ψ V B ( k ) | near the Γ-point ( k = Γ + δ )along the z -axis perpendicular to the surface (Fig.4(d)). For W = 0 the valence bandwavefunction is mainly localized at the surface, but as the disorder strength is tuned from W = 4 to W = 10, it further spreads over bulk layers. The observed penetration depth ofelectronic states progressively increases with W , changing the nature of wavefunctions from(quasi) two-dimensional confined states at the surface to more three-dimensional real-spaceextended states promoted by bulk disorder. For W=10, the wavefunction is seen to be spreadall over the system. However we also observe that inter-surface coupling mediated by bulkdisorder is a minor effect. In fact, from a comparison of spin polarization histograms fordifferent slab thicknesses (12-layer, 22-layer and 46-layer slabs)[52] we conclude that inter-surface coupling between Dirac cones is negligible for spin randomization at thicknessesdown to 12 layers.To deepen the analysis, we compute and analyze the scaling behavior of the inverseparticipation ratio defined as IPR = P i | Ψ V B | / ([ P i | Ψ V B | ] ). The IPR is a commonmeasure for the localization nature of electronic states (see e.g. Brndiar et al [51]). Inabsence of disorder, the IPR is predicted to scale as l − d (where l is system length, d the spacedimension) being a fingerprint of truly extended states and a metallic regime, whereas the10nderson localization regime (in the strong disorder limit) manifests in a length-independentIPR value (with IPR ∼ ξ − d , ξ the localization length). Fig.4-d (inset) shows IPR forincreasing lateral slab sizes l and for W = 0, 4 and 10. It is found that IPR ∼ l − for W = 0, in agreement with extended wavefunctions at the surface (only weak unavoidablespreading to nearest bulk layers is observed for | Ψ V B | , Fig.4-d (main frame)). By increasingthe bulk disorder strength from W = 4 to W = 10, the IPR are seen to vary in absolutevalue in a non-monotonic fashion while maintaining the IPR( l ) ∼ l − scaling behavior,with no sign of saturation for the considered system sizes. This scaling analysis excludesshort localization lengths and Anderson insulating regime for bulk disorder strengths whichhowever significantly suppress surface spin polarization. Conclusion.-
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