Magnetism-induced massive Dirac spectra and topological defects in the surface state of Cr-doped Bi₂Se₃-bilayer topological insulators

The surface topography on large scales revealed pyramid-like terraces with steps corresponding to single atomic layers, as described previously [28, 29]. For an averaged, nominal top layer thickness d₁-QL, the local thickness of the top layer could vary up to 1-QL. Atomically resolved topographic images exhibited triangular lattice patterns that were always consistent with that of pure Bi₂Se₃, as exemplified in figure 2(a) and figure 2(b) for (1 + 6)-10% and (5 + 6)-10% samples, respectively. On the other hand, the Fourier transformation (FT) of the surface topography appeared to be dependent on d₁. We found that FT of the (1 + 6)-10% topography showed an expected hexagonal Bragg diffraction pattern for Bi₂Se₃ plus an additional, faint superlattice structure (figure 2(c)), which may be attributed to the underlying Cr-doped Bi₂Se₃. For instance, a periodic substitution of Cr for Bi as exemplified in figure 2(f) for a two-dimensional projection of the two Bi-layers within one-QL yields a FT pattern (figure 2(e)) similar to that in figure 2(c). This superlattice structure corresponds to a local Cr concentration of 1/12. Another similar structure with a local Cr concentration of 1/8 (figure 2(h)) is also feasible within experimental uncertainties of the superlattice constant and its angle relative to the Bi lattice (figure 2(g)). In contrast, the FT in figure 2(d) for the surface topography of a (5 + 6)-10% sample only revealed the hexagonal diffraction pattern of pure Bi₂Se₃ due to the relatively thick d₁ layer. Interestingly, we note that the FT topography of the (1 + 6)-5% samples also agreed with figure 2(d), suggesting random Cr substitutions of Bi for a smaller Cr concentration, which is consistent with the randomly distributed Cr clusters found with STM studies directly on 2% Cr-doped Bi₂Se₃ [30].

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For the zero-field studies, tunneling conductance (dI/dV) versus biased voltage (V = E/e) measurements were carried out on each sample over multiple areas, followed by detailed analysis of the spatially resolved spectral characteristics. While apparent spatial variations were found in all samples, systematic investigations led to several general findings. First, all samples revealed gapless Dirac tunneling spectra at 300 K. Second, with decreasing T there were two distinctly different types of spectral characteristics: For samples with nominal d₁ = 5 and 7, the tunneling spectra remained gapless for all T, except for occasional areas where the actual d₁ values in the nominal d₁ = 5 sample were ∼4. In contrast, for samples with nominal d₁ = 1 and 3, the majority spectra revealed gapped features at T < ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ and the temperature evolution for all samples are exemplified in figures 3(a)–(d). Third, the surface gap Δ (r, T), obtained by the spectral analysis illustrated in figure 3(e), appeared to be spatially inhomogeneous where r denotes the two-dimensional spatial coordinate.

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For a given r, Δ mostly increased with decreasing T except near T_x ∼ (110 ± 10) K where a slight dip appeared, and eventually saturated to a maximum value at T < ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$ ≪ ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ as exemplified by the T evolution of the gap maps and the corresponding gap histograms in the left and middle panels of figures 4(a)–(c) for the (1 + 6)-5%, (1 + 6)-10% and (3 + 6)-10% samples, and also summarized in the right panels for the temperature dependence of the corresponding mean gap $\bar{{\rm{\Delta }}}$(T). Here the mean gap value $\bar{{\rm{\Delta }}}$ at a given T was determined from the peak value of Gaussian fitting to the gap histogram, and the errors were determined from the one sigma linewidth of the Gaussian fitting. Based on the data for$\bar{{\rm{\Delta }}}(T),$ the onset T for the surface gap opening was found to be ${{T}_{{\rm{c}}}}^{2{\rm{D}}}=\left({\rm{240}}\pm {\rm{10}}\right)\;{\rm{K}}$ for x = 10% and ${{T}_{{\rm{c}}}}^{2{\rm{D}}}=\left(210\;\pm \;10\right)\;{\rm{K}}$ for x = 5%, significantly higher than the bulk Curie temperatures ${{T}_{{\rm{c}}}}^{3{\rm{D}}}=30\;\sim \;20\;{\rm{K}}$ obtained from the onset temperature of the anomalous Hall resistance. This finding of ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ ≫ ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$ is in fact consistent with previous reports on other families of 3D-TIs [15–17]. As we shall elaborate further in Discussion, the non-monotonic temperature dependence of the surface gap and the relation of ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ ≫ ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$ may be the result of multiple magnetic interaction components in the bilayer samples, with each component having a different temperature dependence and interaction range.

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The inhomogeneous gap distribution may be attributed to multiple reasons. First, the Cr-substitution of Bi may not be uniform due to the size mismatch, which could induce lattice strain and inhomogeneous ferromagnetism. Second, the magnetic moments of Cr ions may not be well aligned along the sample c-axis without an external magnetic field. Given that only c-axis magnetization component can induce a surface gap in 3D-TIs according to equations (1) and (2), varying spin alignments in different magnetic domains would result in varying surface gaps [31]. Third, the sample surface exhibited terrace structures with ∼1-QL thickness variations [28], which could give rise to varying proximity-induced gaps from the RKKY interaction mediated by surface Dirac fermions [18, 22–24].

To better understand how the aforementioned components contribute to the gap inhomogeneity, we compare the surface topography of the bilayer samples with their corresponding gap maps. As exemplified in supplementary figures 2(a)–(b) and discussed further in the supplementary information, we find that the correlation between a typical surface topography map and the corresponding gap map over the same area of a bilayer sample is finite although relatively weak. The weak correlation is primarily due to small height variations over the typical areas of our STM studies. Thus, the primary cause of spatially inhomogeneous gaps in the bilayer systems may be attributed to inhomogeneous Cr distributions and the misalignment of Cr magnetic moments. Experimentally, the latter effect may be minimized by applying a strong external magnetic field along the c-axis, which is the subject of our investigation in the next subsection.

We further investigated the effect of increasing c-axis magnetic field (H) on the gap distribution over the same area of each sample at a constant T. As exemplified in figure 5(a)–(f) for samples of (1 + 6)-5% and (3 + 6)-10% taken at T = 18 K and H = 0, 1.5 T and 3.5 T, the gap maps became increasingly homogeneous and the mean gap value $\bar{{\rm{\Delta }}}$ derived from the histogram also increased slightly with increasing H. This finding suggests that the observed surface gap is consistent with c-axis ferromagnetism induced by Cr-doping and proximity effect, and the characteristic field for saturating the surface state ferromagnetism appears to be larger than that for saturating the bulk ferromagnetism [18], probably due to the helical spin textures of the former. The small but finite residual gap inhomogeneity in high fields may be primarily attributed to spatially inhomogeneous Cr-distributions.

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While the majority of the tunneling spectra in the (1 + 6)-5%, (1 + 6)-10% and (3 + 6)-10% samples revealed gapped characteristics for T < ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ spatially localized and intense conductance peaks were occasionally observed along the borders of gapless and gapped regions, as exemplified in figures 6(a)–(d) for a (1 + 6)-5% sample and in figure 6(f) for a (1 + 6)-10% sample. These long-lived minority spectra either consisted of a single sharp conductance peak at a small negative energy E = E₋ near the Dirac point E_D; or comprised of double conductance peaks (figure 6(b)) at E = E₋ and E = E₊, where E₊ is near the Fermi energy E_F = 0. These double-peak spectral characteristics were consistent with theoretical predictions for magnetic impurity resonances [25]. Further, the numbers of both single- and double-peak impurity resonances at H = 0 were found to increase rapidly near ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ (figure 6(f)). In contrast, all resonances disappeared under a large c-axis magnetic field at low T when gapless regions diminished.

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We attribute the sharp impurity resonances to isolated Cr-impurities that were far away from neighboring Cr ions and probably had partially diffused from the d₂-layer into the interfacial region, because they were only found in zero-field along the borders between gapless and gapped regions, and then disappeared under a large c-axis magnetic field when gapless regions diminished and long-range ferromagnetism was established. The strongly non-monotonic temperature dependence of the number of impurity resonances at H = 0 (see figure 6(f)) may be understood as the result of weakening surface ferromagnetism near ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ so that more Cr-impurities became decoupled and acted like isolated impurities. The strong spatial localization and long lifetime of these magnetic impurity resonances at H = 0 may be attributed to topological protection of the surface state in 3D-TIs when the Dirac energy E_D is relatively close to E_F, similar to the case for non-magnetic impurities [25, 32].

We further remark that our finding of atomically sharp impurity resonances due to isolated Cr ions differs from direct STM studies of bare 2% Cr-doped Bi₂Se₃ [30]: in the latter case, direct atomic imaging revealed that Cr substituted Bi in clusters so that the resulting spectroscopy did not behave like isolated magnetic impurities embedded in the surface state of pure Bi₂Se₃. In contrast, our identification of isolated Cr-impurities in the bilayer systems was solely based on spatially resolved spectroscopic evidences without directly imaging the Cr impurities, because Cr ions could not diffuse all the way to the top surface and were still buried by the undoped Bi₂Se₃ layer. Therefore, STM imaging of all bilayer samples always revealed the atomic structure of pure Bi₂Se₃, as exemplified in figures 2(a) and (b). Additionally, we have found no evidences for the formation of Cr clusters at the interface of our bilayer samples: The STEM studies [18] of typical bilayer samples revealed atomically sharp interfaces without any cluster-induced variations in the c-axis lattice spacing; our bilayer samples also exhibited well defined 3D ferromagnetism below an appreciable ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$ (20 ∼ 30 K), which was consistent with more dispersed and uniform distributions of Cr in order to achieve long-range ferromagnetic order. The latter situation was in stark contrast to the absence of long-range ferromagnetism down to 1.5 K in the 2% Cr-doped Bi₂Se₃ sample that exhibited randomly distributed surface Cr-clusters [30]. In fact, a recent report based on direct STM studies on a similar TI system (Bi_0.1Sb_0.9)_1.92Cr_0.08Te₃ (with ${{T}_{{\rm{c}}}}^{3{\rm{D}}}\sim 18\;{\rm{K}}$ and 4% Cr-doping) revealed completely dispersed distributions of Cr and inhomogeneous gaps without evidences for cluster formation [33]. These findings are consistent with what we have inferred from our experimental studies of the (Bi_1−xCr_x)₂Se₃/Bi₂Se₃ bilayer systems.

Our experimental results have demonstrated that the dependence of the spectral characteristics on T, H, x and d₁ for the bilayer samples are all consistent with the scenario that a surface-state gap was induced by the proximity effect of predominantly c-axis ferromagnetism in the Cr-doped bottom layer. The appearance (absence) of gapped tunneling spectra below ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ for bilayer samples with d₁ ≤ 3-QL (d₁ ≥ 5-QL) suggests that the proximity effect due to c-axis magnetic correlation is limited to a critical thickness d_c ∼ 4-QL.

In our investigation of the Cr-doped bilayer system, it is important to realize that the undoped Bi₂Se₃ layer should not be considered as an isolated ultrathin film on a 'substrate' of Cr-doped Bi₂Se₃. This realization is because of the continuous and seamless growth of undoped Bi₂Se₃ layer on top of the isostructure, Cr-doped Bi₂Se₃ layer [18], leading to an effective total thickness of (d₁ + d₂)-QL. In contrast, for ultra-thin TI films on dissimilar substrates, an energy gap can open up due to coupling between the top and bottom topological surface states, and a Rashba-like coupling with further energy splitting in momentum can also be expected due to the asymmetric chemical potentials between the surface of the thin film and its interface with a dissimilar substrate [34]. This Rashba-like coupling has indeed been confirmed by ARPES and STS studies for 2-QL to 5-QL Bi₂Se₃ on various dissimilar substrates [35, 36]. In particular, the energy gap induced by the Rashba-like coupling for ultra-thin TI films on dissimilar substrates does not exhibit any discernible temperature or magnetic field dependence, which is in sharp contrast to the behavior of the proximity-magnetism induced surface gaps described in this work.

Having established proximity magnetism as the physical origin of the surface gap in the bilayer system, we consider next the primary components that contribute to the surface ferromagnetism in the undoped Bi₂Se₃ layer of d₁ ≤ 3-QL. A natural component is the dipole fields associated with the magnetic moments of Cr ions. Additionally, RKKY-like magnetic interactions mediated by the bulk carriers and the surface Dirac fermions are also feasible contributions to the surface ferromagnetism. Therefore, we may express the effective ferromagnetic coupling constant J_eff in terms of the sum of these three components, J_eff = (J_dipole + J_bulk) + J_Dirac, where (J_dipole + J_bulk) ≡ J_3D may be considered as a 3D component and J_Dirac ≡ J_2D a 2D component. We expect J_3D to vary slowly with d₁ because it primarily represents the bulk magnetic properties of the Cr-doped Bi₂Se₃. In contrast, J_2D ∝ exp (−2d₁/d_c) is strongly dependent on d₁ because it involves the wavefunction overlaps between the surface Dirac fermions of the undoped Bi₂Se₃ layer and those of the d-electrons in Cr-doped Bi₂Se₃ layer [18, 22–24].

Concerning the temperature dependence of J_eff, we expect J_3D to dominate at low T because of the strong confinement of Dirac fermions to the surface layer and therefore negligible RKKY interaction between surface Dirac fermions and bulk d-electrons. On the other hand, J_3D is sensitive to 3D long-range order of magnetic moments and so should diminish significantly above ${{T}_{{\rm{c}}}}^{3{\rm{D}}}.$ In contrast, finite temperature can enhance wavefunction overlaps between the surface Dirac fermions and the interfacial d-electrons. Therefore, we expect J_2D to dominate at elevated temperatures, and have further elaborated our rationale for this temperature dependence in supplementary information. Moreover, the long Fermi wavelength of surface Dirac fermions could result in much enhanced RKKY interaction in the bilayer system so that J_2D > J_3D [18, 22–24], which may account for the empirical observation of ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ ≫ ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$.

Based on the scenario described above and noting that the proximity-induced surface gap is given by Δ = (J_effM) where M is the surface magnetization that increases monotonically below ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ we find that with decreasing T, the surface gap Δ (T < ${{T}_{{\rm{c}}}}^{2{\rm{D}}})$ can indeed exhibit a non-monotonic T-dependence that is consistent with our experimental findings depicted in figure 4. As qualitatively exemplified in supplementary figure 3, Δ (T) first increases with decreasing T for T < ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ and then exhibits a 'dip' at T_x where ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ > T_x > ${{T}_{{\rm{c}}}}^{3{\rm{D}}},$ and T_x represents a dimensional crossover temperature below which J_2D becomes negligible. While we do not have sufficient information to model Δ(T) quantitatively, the qualitative agreement of supplementary figure 3 with experimental results in figure 4 suggests that our proposed scenario for proximity magnetism in the bilayer system may provide useful guide for future investigation.

Next, we comment on the magnitude of the proximity-induced surface gaps in Bi₂Se₃, which were found to be comparable or even larger than the bulk gap (∼0.3 eV) of Bi₂Se₃. While one may question whether it is feasible for a magnetism-induced surface gap to exceed its bulk gap of a TI, we point out that the bulk gap ∼0.3 eV of Bi₂Se₃ is an indirect gap; the direct gap at the Γ-point where the Dirac cone resides is in fact >0.5 eV based on bandstructure calculations [37]. In this context, the largest mean gap values (0.4 eV ∼ 0.5 eV) found in our work are well within the range of the bulk gap. We further note that the gap values determined from STS studies would be dominated by the local gap of the surface layer, whereas the gap value determined from ARPES would be sensitive to the smallest gap over an extended surface area. Therefore, the surface gap determined from STS tends to be larger than that determined from ARPES for an inhomogeneous sample. Indeed, ARPES studies on Bi_1.8Cr_0.2Se₃ have revealed a surface gap of ∼0.2 eV [18], which is slightly smaller than but still in reasonable agreement with our finding of Δ = (0.25 ± 0.09) eV for a (1 + 6)-10% sample at H = 0 and T = 79 K, as shown in figure 4(b).

Concerning the physical feasibility of a relatively high ${{T}_{{\rm{c}}}}^{2{\rm{D}}},$ we note that the energy splitting of the double-peak spectrum associated with isolated magnetic impurities in the surface state is comparable to the effective ferromagnetic coupling constant J_eff [21]. Noting that the mean-field Curie temperature may be estimated from J_eff via the relation k_B${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ ∼ S (S + 1) (J_eff/3) [38] and that J_eff ∼ 0.1 eV as determined from the measured energy splitting of the double resonances, we obtain ${{T}_{{\rm{c}}}}^{2{\rm{D}}}\sim {\rm{250}}\;{\rm{K}}$ by assuming S = 1/2. While this rough estimate does not include the screening effect from Dirac fermions, it is still comparable to our STS measurements of ${{T}_{{\rm{c}}}}^{2{\rm{D}}}\sim {\rm{240}}\;{\rm{K}}$ for x = 10% and ${{T}_{{\rm{c}}}}^{2{\rm{D}}}\sim {\rm{210}}\;{\rm{K}}$ for x = 5% for the observed surface ferromagnetism, and is much higher than the bulk ${{T}_{{\rm{c}}}}^{3{\rm{D}}}\sim {\rm{25}}\;{\rm{K}}.$ The disparity of ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ and ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$ may be attributed to the different microscopic mechanisms for mediating ferromagnetism via the surface-state Dirac fermions versus via the bulk-state carriers, particularly given the diverging Fermi wavelength of the Dirac fermions when the Fermi energy approaches the Dirac point [18, 22–24]. While quantitative details of the microscopic mechanism responsible for ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ ≫ ${{T}_{{\rm{c}}}}^{3{\rm{D}}}$ require further investigation, the relatively high ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ values are promising for realistic spintronic applications, particularly if our findings may be generalized to more homogeneously Cr-doped 3D-TIs such as (Bi_1−xCr_x)₂Te₃ [18].

The appearance of both single- and double-peak magnetic impurity resonances is suggestive of two types of topological defects associated with an isolated magnetic impurity: The single-peak resonance may be attributed to a helical Dirac spin texture coupling to a magnetic moment pointing along the c-axis, whereas the double-peak resonance is associated with two opposite chiral spin textures of Dirac fermions coupling to an in-plane magnetic moment [21], as illustrated in figure 6(e). Our assignment of the single-peak resonances in this work to the helical Dirac spin texture coupled with isolated c-axis magnetic moments can be justified by their strong and non-monotonic temperature dependence near ${{T}_{{\rm{c}}}}^{2{\rm{D}}}$ (figure 6(f)), and also by their suppression under large c-axis magnetic fields. In contrast, our previous studies of non-magnetic impurity resonances in pure Bi₂Se₃ MBE-grown thin films [32] did not find any dependence of their occurrences on either the temperature or the applied magnetic field, implying that the physical origin of the single-peak resonances in pure Bi₂Se₃ is fundamentally different from that of the single-peak resonances observed in the bilayer systems.

Our finding of occurrences of impurity resonances only along the borders of gapped and gapless regions is consistent with suppressed long-range magnetic order along the boundaries between the c-axis oriented (gapped) and in-plane oriented (gapless) magnetic domains [31]. Thus, proximity-induced magnetization in the top Bi₂Se₃ layer is also much suppressed for regions above the boundaries of c-axis and in-plane magnetic domains [31], and Cr-ions located in these regions were more likely to decouple from long-range magnetization and became effectively isolated magnetic impurities, particularly with the assistance of thermal fluctuations at sufficiently high temperatures (figure 6(f)).

Finally, we note that the spin textures associated with an in-plane magnetic impurity may be considered as a stable 'topological bit' with two levels associated with the two opposite spin chirality. In principle, for a completely isolated topological bit, the two-level states may be tuned by a local c-axis magnetic field. Further, for E_F → E_D these topological bits are long-lived as the result of topological protection. On the other hand, tuning E_F away from E_D will result in increasing coupling among spatially separated topological bits [21], which may be viewed as inducing an effective entanglement of wavefunctions among these topological bits. The feasibility of tuning the two levels within a topological bit by an external magnetic field and the coupling among spatially separated bits by gating the Fermi level may present an interesting opportunity for applying these topological bits to quantum information technology.