Electronic properties of Bi₂Se₃ dopped by 3d transition metal (Mn, Fe, Co, or Ni) ions

The first-principles calculations are preformed using the projector augmented-wave potentials [39] implemented in the Vienna ab initio simulation package code [40–42]. The calculations are made within the local density aproximation (LDA) [43, 44] and the generalized gradient approximation (GGA) in the Perdew, Burke, and Ernzerhof (PBE) parametrization [45]. First, the crystal structure is optimized in the conventional unit cell including three primitive cells (cf. figure 1, where the primitive cell is marked by gray shadow) with the SOC. As a break of the optimization loop, we take the condition with an energy difference of 10⁻⁴ eV and 10⁻⁶ eV for ionic and electronic degrees of freedom. During the crystal structure optimization loop, ionic positions, cell volume, and cell shape are allowed to relax [46]. Van der Waals (vdW) corrections are included using the Grimme scheme (DFT-D2) [47]. In the case of the doped system, the 3 × 3 × 1 supercell (containing 27 primitive cells) is used with one TM impurity substituting Bi site. It corresponds to the concentration of magnetic impurities of about 1.85%. For the summation over the reciprocal space, one uses 12 × 12 × 3 and 4 × 4 × 3 k–point Γ–centered grids in the Monkhorst–Pack scheme [48], in the case of the conventional cell (for the clean system) and the supercell (for the doped system), respectively. The energy cutoff for the plane-wave expansion is set to 350 eV. The crystal symmetry is analyzed using FindSym [49] and SpgLib [50], while the momentum space analysis is done with using SeeK-path tools [51].

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The Bi₂Se₃ family of compounds crystallizes with the $R\bar{3}m$ symmetry (space group no. 166) [38]. As shown in figure 1, the system has a layered structure with the quintuple layer (QL) as a basic unit. Each QL consists of five atomic layers formed by Se(1)–Bi–Se(2)–Bi–Se(1) atoms. Within the QL, Se atoms take two nonequivalent positions. Se(1) atomic layer is adjacent only to one Bi layer. Contrary to this, Se(2) atomic layer located inside the QL is surrounded by two Bi layers. Inside the QL, strong bonding between molecular p-like orbitals is produced along the five-atom linear chain [52]. The interaction between two Se(1) layers belonging to two different QLs layers is of the vdW type.

In the case of Bi₂Se₃, lattice parameters and gap values are very sensitive to used pseudopotentials [53]. For instance, some hybrid pseudopotentials lead to underestimation of gap values, while GGA PBE gives an incorrect indirect gap [54]. In this compound, interactions of vdW type play a crucial role [55, 56]. Previous studies showed, that the fullest consistency of experimental and theoretical results can be obtained from LDA + GW calculations with the relaxed atomic positions [9, 57, 58]. In addition, theoretical studies show existence of the strong correlation between the value of the topological gap and the value of lattice constant c [19].

A comparison of our results in the case of LDA and GGA PBE pseudopotentials is shown in table 1 and figure 2. In both cases, BSs and DOS have qualitatively similar form. However, the LDA calculations reproduce both the experimental parameters and the band gap of the clean Bi₂Se₃ (i.e., without substitutions) more accurately. Therefore, in the following parts of the manuscript, we present the results obtained solely by using the LDA pseudopotentials.

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In TM-doped Bi₂Se₃, the optimization of the crystal structure leads to the lowering of the supercell symmetry to the P3m1 (space group no. 156), due to the distortion of atoms located near the substituted TM atom. Additionally, distance between the TM atom and the nearest Se atoms in the doped system decreases in the series Mn, Fe, Co, Ni and, simultaneously, it is smaller than the distance between Bi–Se atoms in one QL of the clean system, due to smaller atomic radius of considered TMs atoms than Bi. We present explicitly values of these distances in figure 3. As we can see, the TM–Se(1) distance (to the vdW gap site) is more suppressed than the TM–Se(2) bond (directed to the QL center). In the clean system, these relaxed distances are equal 2.903 Å and 3.102 Å, respectively. Experimental data for the clean system indicates 2.850 Å and 3.074 Å, respectively. Moreover, the strong bonding between TM and Se atoms is visible in the charge density. This shows a strong local contraction of the bonds, which has been also observed in similar TM-doped structures [59–61]. This contraction enhances the covalent character of the TM–Se bonds, which can have important consequences for the electronic and magnetic properties of the TM-doped Bi₂Se₃ [59].

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The results of our calculations show that the supercell volume of the doped system is smaller than that in the clean case (cf. table 2). One should note that this behavior was reported previously in the case of the Mg doped system [62]. In the case of the Sb substitution, the volume initially increases for low doping and then decreases for high doping concentration [63]. If Bi₂Se₃ is doped by TMs, magnetic moment appears in the system which is localized mainly on TM ions. Due to hybridization with the rest of the states, we observe smaller magnetic moments than that for the corresponding free atoms. The fractional magnetic moments are discussed in section 2.4 in more detail.

Band structure and density of states. Calculated BS and DOS are shown in figure 4. Projection of states into the TM substituted atoms are marked by blue dots and blue background area, in the case of BS and DOS, respectively. In every case, we observe many well-localized dispersion-less states of the TM atoms. However, only for Co and Ni substitutions, we observe their localization around the Fermi level, e.g., in the form of clear peak at E = 0 eV in DOS. Typically, the system becomes an electron-doped and the Fermi level is shifted to the bottom of the CB [64]. Similar behavior was observed experimentally in the case of MnBi₂Se₄/Bi₂Se₃ heterostructure [65], i.e., a magnetic monolayer deposited into the TI. A shift of Fermi level is associated with a change of the energy levels around ∼0.5 eV [66], which is bigger than the related value of insulating gap. Contrary to this, in the case of Mn-doped system, the Fermi level is located below the top of the valence band (VB) of the clean system.

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The band structures reveals differences between the TM and the QL atoms. First, energy levels of the bands of the base system (gray lines in figure 4) exhibit weak momentum dependence in the z-direction due to QL structure. An impact of the TM is indicated by a size of the blue dots in figure 4. In each case, we notice a 'wide' energy window of the TM states below the Fermi level, what is related to strong hybridization of the d-orbital energy levels (of TM) with p-orbital states. As opposed to this, the TM states above or near the Fermi level are well-localized in 'narrow' energy window in a form of dispersionless bands. Generally, the shift of the energy between levels below and above the Fermi level is relatively concordant with the splitting due to the field associated with the magnetic moment of the TM atom (cf. table 2). From this, independently of the SOC, in the studied system, one expects the separation of electrons in spin-↑ and spin-↓ subspaces in the energy domain.

Similarly as in the clean system, the valence and CBs consist of Bi and Se p orbitals [52]. All other orbitals are located far away from the Fermi level (in the energy domain). In the clean system, these p orbitals are responsible for a creation of strong σ-bonding between Bi–Se atoms [52]. Introduced substituted TM atom modifies the bonding in the frame of one QL, what is very well visible in modifications of the charge distribution (figure 5).

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Charge and spin redistribution. The changes in the charge distribution induced by the substituted TM atom can be specified more precisely using the differences of charge and spin densities. The modification of the charge density in Bi₂Se₃ by substitution of TM atom, can be shown by the difference between the charge densities of the (initial) doped system and both separated components [67], namely:

$$\begin{equation}{\Delta}\rho ={\rho }_{{\text{TM}}_{x}{\mathrm{B}\mathrm{i}}_{2-x}{\mathrm{S}\mathrm{e}}_{3}}-\left({\rho }_{{\text{TM}}_{x}/\text{vac}}+{\rho }_{\text{vac}/{\mathrm{B}\mathrm{i}}_{2-x}{\mathrm{S}\mathrm{e}}_{3}}\right),\end{equation} \tag{ 1 }$$

where ${\rho }_{{\text{TM}}_{x}{\mathrm{B}\mathrm{i}}_{2-x}{\mathrm{S}\mathrm{e}}_{3}}$, ${\rho }_{{\text{TM}}_{x}}/\text{vac}$, and ${\rho }_{\text{vac}/{\mathrm{B}\mathrm{i}}_{2-x}{\mathrm{S}\mathrm{e}}_{3}}$ correspond to charge density calculated for the doped system, separated TM atom, and the system without substituted atoms (i.e., the system in the presence of vacancy [68]), respectively. The redistribution of the spin density can be defined analogously. Modifications of the charge distribution due to the TM substitution are shown in figure 5. In both cases strongest modification of charge and spin (not shown here) densities are observed along strong bonds between TM and Se atoms. Here, it should be noticed that Se(1) atoms play more important role than Se(2) atoms in the frame of one QL. The redistribution of charge and spin inherits of the C₃ symmetry of the system. This behavior can be observed indirectly in the STM mapping of the Bi₂Se₃ surface [31–33]. In this case, when vacancy or impurity is included to the system, modification of the charge is strongly observed along σ bonding. As a consequence characteristic charge density defects with C₃ symmetry on the surface is observed [31–33].

Finally, one notes that the magnetic moment of the system is strongly localized on the TM atom (cf. table 2). Thus, only a small modification (in a range of 1 mμ_B Å^{− 3}) of the spin density around atoms near the TM impurity is observed.

Electron localization function. The influence of the TM on the character of the electronic states around the substituted atom can be studied by the ELF [69–71], which can take values from 0 to 1. Large values of ELF (i.e., close to 1) is associated with a high localization (small mobility) of electron at a given place. Small values (i.e., around 0) correspond to delocalized (itinerant) electrons.

Similar as in the case of the charge density, the modifications of the ELF can be calculated as [cf. equation (1)]:

$$\begin{align}\hfill {\Delta}\text{ELF}& ={\text{ELF}}_{{\text{TM}}_{x}{\mathrm{B}\mathrm{i}}_{2-x}{\mathrm{S}\mathrm{e}}_{3}}\hfill \\ \hfill & \quad -\left({\text{ELF}}_{{\text{TM}}_{x}/\text{vac}}+{\text{ELF}}_{\text{vac}/{\mathrm{B}\mathrm{i}}_{2-x}{\mathrm{S}\mathrm{e}}_{3}}\right).\hfill \end{align} \tag{ 2 }$$

For instance, we show the ELF modification in the case of the Fe dopant (figure 6). The base system (i.e., clean Bi₂Se₃) is TI, which implies large value of ${\text{ELF}}_{{\text{Bi}}_{2}{\mathrm{S}\mathrm{e}}_{3}}$ (nearly 1 in whole volume of the system). Introduction of the TM atom leads to decrease of the localization electrons around this atom. This behavior is compliant with experimental data, where the influence of magnetic impurity Fe on single crystals Bi₂Se₃ was investigated in the high magnetic fields by Shubnikov–de Haas (SdH) effect [72]. Based on the SdH effect and the Hall effect measurements, it was established that the doping of Bi₂Se₃ single crystals by Fe leads to an increase of the concentration of free electrons.

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In this subsection, we begin with discussing the impact of the impurity on the band structures in the general case. The schematic representation of the band structure of the doped system is presented in figure 7. In the case of the clean bulk system, we observed existence of the topological gap (cf. figure 2), between the VB and the CB (represented by the black parabolas with the gray background in figure 7). Due to topological nature of the Bi₂Se₃, in a real system with the 'edge', one observes forming of the metallic surface states in the form of the Dirac cone (the blue lines in figure 7) [10, 12–18]. Doping of the system leads to the introduction of the impurity levels (ILs) to the energy spectrum of the system (the red line in figure 7).

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Properties of the bands formed by the ILs depends on, e.g., the type of the impurity and its concentration. The type of the impurity atoms plays an important role in the shifting of the Fermi level (schematically indicated by the green dashed line in figure 7). The single impurity leads to an appearance of discrete energy levels, i.e., flat bands, while increasing concentration of the impurities leads to forming of dispersive bands. A theoretical study of the Bi₂Se₃ doped with Sn in the frame of the Korringa–Kohn–Rostoker method [73], shows that the change of the doped atom concentration from 0.05% to 1% leads to modification ILs from separate narrow peak to dispersion band with the width of about 0.25 eV. A similar effect can be expected in the case of the TM atom doping. In our case, increasing number of impurities can be associated with emergence of the sublattice of the TM atoms. However, distances between them should be relatively large (of the order of several lattice constants). In lower concentration, the mediation between dopant atoms should be carried out via itinerant electrons of the base system. Increasing concentration should enhance second mechanism of dispersive band forming, i.e., increasing overlap between orbitals of the TM atoms within the sublattice. Additionally, above some critical concentration, a realization of the long-range FM order can be expected due to the Ruderman–Kittel–Kasuya–Yosida (RKKY) mechanism [74]. This effect was theoretically investigated, e.g., in the case of the Bi₂Se₃ doped with Mn, for concentrations above 3.2%. However, experimental data from the Mn doped Bi₂Se₃ epitaxial layer, suggest existence of long-range FM order even for smaller concentration of Mn (around concentration of 1.0%) [75].

In the case of the atoms with a high (low) electron concentration, the Fermi level can be shifted to the conduction (valence) band [cf. figures 7(a) and (d)]. One can expect a situation, in which ILs are located around the topological gap above or below the Fermi level [figures 7(b) and (c), respectively].

Here, we should mention a role of the hybridization of the doped atom and atomic levels of the base system. Dependently on a position of the energy levels of the TM doped atom, we observe different behavior of its DOS contribution (cf. figure 4). States around the topological gap are narrow, but states located deep in the bands are more smeared. This can be associated with the hybridization of the TM orbitals with the orbitals of base system. Exact analyses of the partial electronic DOS [76] show that the VB is composed mostly of p states of Se, while CB is composed of p states of Se and Bi as well as s states of Se. The d states of TM are hybridized with p orbitals of the neighboring Se atoms, which suppress their localization. If the ILs lie around the topological gap, the strong peaks appear in DOSs and become more delocalized as the dopant concentration increases. The situation is similar to ordinary semiconductors [77–79], in which the TM ILs can form clear peaks only if they are weakly hybridized with the host states of the same symmetry.

Doping of the Bi₂Se₃ by the magnetic atoms, leads not only to the Fermi energy level shift but also to breaking of the time reversal symmetry. Thus, the gap is opened at the Dirac point (so-called Dirac gap) [21, 37]. Moreover, the spin texture of the surface states is modified [80]. This effects have consequences within the surface electric transport measurements [14], and will be discussed in the next paragraphs.

The Fermi level in real system. As we mentioned in the Introduction, the Bi₂Se₃ crystal exhibits metallic character due to intrinsic defects [34, 35], which have their donor levels above the CB minimum as in figure 7(a). This is in contrast to the theoretical results, where the TI nature of this compound is well visible (cf. section 2.2 and figure 2). In the simplest case, the shift of the Fermi level to the topological gap region can be obtained by doping of the system by nonmagnetic atoms. For example, in the case of the 0.25% doping by Ca atoms, the Fermi level was shifted to the Dirac point [13, 14]. Similar effect was observed in the case of 1% Mg doping [17]. Further increase of the doped atom concentration leads to the shift of Fermi level to the CB [20]. Also some more complicated crystal structures are obtained, e.g., (Bi_1−x Sb_x )₂(Se_1−y Te_y )₃ [36, 81] and other related doped compounds [82, 83] to compensate all donors in the system and to lower the Fermi level.

A shift of the Fermi level to lower energies can be also carried out by doping the system by magnetic atoms. This shift strongly depends on a type of doped atoms. For example, doping by Fe opens the Dirac gap, while the Fermi level is located in the CB. Replacing Fe by Mn atoms leads to a shift of the Fermi level to the lower energies [21, 37]. ARPES experiments for the Mn doped system show the location of the Fermi level above the Dirac gap (more precisely, the Fermi level crosses the surface states from the CB) for Mn concentration smaller than 0.5% [37]. Then, the Mn energy levels are visible in the DOS at the top of the VB [37]. This type of electron structure modification is the most interesting when it leads to the location of the Fermi level around the Dirac point and consequently to the realization of the quantum topological transport. Experimental data suggests that this can be realized for Mn concentration of 1.0% [21].

The above-mentioned results are predominantly in the compliance with our ab initio theoretical results (see figure 8 for details of the electronic band structures near the Fermi level). Doping of the system by Mn leads to the shift of the Fermi level to the VB [figure 8(a)], while in all other cases the Fermi level is located in the CB. Comparing the band structures obtained for the TM doped systems [panels (a)–(d) of figure 8] with the reference band structure of the clean Bi₂Se₃ [figure 8(e)], one notices that the band degeneracy is removed. This is a consequence of two effects: (i) modification of the crystal structure by doped atoms (i.e., a change of the distances between atoms) and (ii) removal of the spin-degeneracy due to the time reversal symmetry breaking by magnetic TM dopants. However, the location of the Fermi level depends also on the impurity atom concentration.

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Here, we would like to discuss the role of the doping on the value of the magnetic moment of the substituted TM atoms. From the results collected in table 2, we can conclude that the magnetic moments are fractional and thus the calculated charge states of the TM ions are between 2+ and 3+ (except for Ni ions, in which the magnetic moment indicates the charge state between 3+ and 4+ ). Taking into account that the real samples exhibit conductivity of n-type, we expect that they act as acceptors and exist in 2+ charge state. Indeed, Mn²⁺ signal with magnetic moment of 5  μ_B in Bi₂Se₃ was observed by EPR measurements [84]. The Mn²⁺ level is located within the VB [see figure 8(a)] and thus compensates nonintentional donors and shift the Fermi energy below the CB. Assuming that the main effect of increasing Mn concentration on the electron structure is broadening of the Mn-induced bands, the Fermi level should be shifted down to the Mn level (so to the maximum of the VB) as long as all donors are not compensated. For higher concentrations, both Mn²⁺ (with maximal magnetic moment of 5 μ_B) and Mn³⁺ (with 4 μ_B magnetic moment) can coexist in the sample.

Experimental observation. An increase of the doped atom concentration can also lead to realization of the FM order, which is induced by the spin–spin interaction mediated by the surface states [21, 28]. In fact, the magnetic impurities lead to the effect similar to the external magnetic field, where negative magnetoresistance (MR) is observed [85]. In the absence of magnetic doping, the negative MR is due to the weak localization effect coexisting with the weak anti-localization effect (positive MR) under a low magnetic field [83, 86–88]. The transition between weak localization and weak anti-localization is demonstrated as a gap opening at the Dirac point of surface states in the quantum diffusive regime. This phenomena was reported in the case of the Cr doped TI [89]. This type of experiments, realized in the TI nanoflake devices, suggest the occurence of the surface dominated transport in low temperatures [90]. Here, doping is an effective way to manipulate the magneto-transport properties of the TI [88]. Also in the MR measurements, results strongly depend on the Fermi level location. For example, in the Fe doped material, the experiments show exactly a location of the Fermi level in the CB [72], what is in concordance with results presented in this work. Morover, in this case the negative MR was reported [91]. In Bi₂Te₃, an increase of the Fe doping leads to several transition on the phase diagram, from a paramagnetic TI, through a magnetic band insulator, to a valence bond glass TI [92]. Similar effect is expected in the Bi₂Se₃ [91].

Finally, an interesting phenomena associated with the topological properties of the magnetically doped 3D TI is the realization of the quantum anomalous Hall effect (QAHE) [93, 94]. When one spin channel is suppressed due to time reversal symmetry breaking, the other spin channel should exhibit QAHE in the form of the dissipationless spin chiral edge mode. The time reversal symmetry can be broken due to magnetic doping, and thus the external magnetic field is not necessary [95, 96]. When the Fermi level is located around the Dirac point, the QAHE is expected. Such conditions are provided by Mn²⁺ or Fe³⁺, see figure 8. However, the presence of native donors makes it impossible to realize Fe³⁺ for any Fe concentration. Indeed, experimental realization of this phenomena was reported in the Mn [97, 98], Cu [99], or Cr [100], but not in Fe doped TI.

One should also notice that the mentioned above experimental results were obtained in the case of the quasi-2D system, i.e., in a form of nanoflakes or nanodevices. In such situations, the role of the magnetic doping is expected to be more important than in real 3D TI.