Vanadium-Containing Planar Heterostructures Based on Topological Insulators

Vanadium-containing heterostructures consisting of an ultrathin magnetic film on the surface of a nonmagnetic topological insulator have been studied theoretically. A method has been demonstrated to control the Dirac point shift in the k space, which is a length measure of an exotic flat band appearing upon the formation of domain walls on the surface of antiferromagnetic topological insulator. The Dirac point shift is inversely proportional to the group velocity of electrons at the Dirac point and is proportional to the degree of localization of the topological state in the magnetic film. The shift is controlled by selecting a substrate with a certain work function. Particular systems have been proposed for the experimental study of flat band features in antiferromagnetic topological insulators.

1 INTRODUCTION

The combination of a nontrivial band topology and the exchange interaction allows the implementation of physical effects such as the quantum anomalous Hall effect [1–9] and the topological magnetoelectric effect [4, 10, 11], which are promising for applications. This combination can lead to new states of matter with unique properties. One of these states is the antiferromagnetic (AFM) topological insulator (TI) phase [5, 12–17]. Depending on the direction of the magnetization vector (parallel or perpendicular to the surface), the band structure of the surface of such systems can include both gapless (similar to three-dimensional topological insulators) and exchange-split topological surface states. Together with the possibility of controlling spin-dependent transport properties, this property makes these materials particularly promising for applications. The first discovered AFM TI is the MnBi₂Te₄ compound whose properties were first predicted theoretially and then were confirmed experimentally [12]. Owing to the magnetization perpendicular to the (0001) plane, the quantum anomalous Hall effect [6, 7] and quantized Hall conductivity in an external magnetic field [7, 18] were observed in thin films and in MnBi₂Te₄-based systems. The discovery of this compound initiated rapid expansion of both theoretical and experimental studies of the MnBi₂Te₄ compound [6, 11, 19–33] and MnBi₂Te₄-based systems, including thin films [5, 14, 34], heterostructures [35, 36], and superlattices [37–40].

It has recently been predicted theoretically [17] that the formation of magnetic inhomogeneities can strongly affect the electronic properties of AFM TIs. In particular, the existence of domain walls on the surface of planar AFM TIs (with the magnetization in the (0001) plane) [26, 41, 42] results in additional specific one-dimensional states, which modify the energy spectrum of the surface [17, 27, 43, 44]. These one-dimensional states connect two Dirac points from different domains and have the form of flat bands with a large effective mass and a high density of states. Flat bands are intensively studied [45–48] and can be manifested in important physical phenomena such as orbital magnetism [49], correlated insulator [50], and superconductivity [49, 51, 52]. It is noteworthy that superconductivity is also observed in nonmagnetic TIs owing to the proximity effect [53–55].

The intensity of effects caused by the flat band is proportional to the length of the flat band, which depends on the electronic and magnetic characteristics of a material. In this context, we demonstrate in this work an efficient method of increasing the length of the flat band by the formation of heterostructures such as the magnetic extension of the TI [56], which is an ultrathin magnetic film on the surface of the TI. Using the results obtained, we also propose systems for the experimental observation of flat bands in planar topological magnets.

2 RESULTS AND DISCUSSION

The band structure of the (0001) surface of planar AFM TIs contains a gapless state (Dirac cone) whose Dirac point is shifted from the $\bar {\Gamma }$ point in the direction perpendicular to the magnetization [17], as in other magnetic TI-based systems with the magnetization in the surface plane [3, 32, 57, 58]. It was shown that the Dirac point shift in these systems is approximately given by the expression [17]

$$\Delta k = j\frac{{{{M}_{0}}}}{v},$$

(1)

where j is the exchange interaction energy, ${{M}_{0}}$ is the magnetization of the surface, and $v$ is the group velocity of electrons at the Dirac point. The length of the flat band formed in the presence of a magnetic inhomogeneity in the form of a domain wall [17, 43, 44] is $2\Delta k$. In this work, we consider one of the possible ways to increase $\Delta k$, namely by varying the group velocity of electrons at the Dirac point $v$ at a fixed magnetization ${{M}_{0}}$. This way allows one to “fine-tune” the electronic structure and can be implemented by designing magnetic heterostructures consisting of an ultrathin magnetic film on the TI substrate. In this way, the magnetization ${{M}_{0}}$ of the surface septuple layers and the exchange interaction energy j will be the same for heterostructures with the same type of magnetic film because these parameters are determined primarily by the magnetic film. The degree of localization of the topological state in the magnetic film also affects the shift.

We consider thin magnetic films that are fragments of recently predicted tetradymite-like interplane planar antiferromagnets VBi₂Se₄ (trivial insulator) and VBi₂Te₄ (AFM TI) (magnetic moments on V atoms lie in the (0001) plane; see Fig. 1а) [17]. Ultrathin films of these compounds consisting of a single structural block with a thickness of seven atomic layers (septuple layer, SL) are two-dimensional ferromagnets because they contain only one V atomic layer and have a semiconductor band structure (see Figs. 1b, 1c). Tetradymite-like TI Bi₂Se₃ and PbBi₂Se₄ were considered as substrates for VBi₂Se₄ and Bi₂Te₃, PbBi₂Te₄, SnBi₂Te₄, and GeBi₂Te₄ were considered as substrates for VBi₂Te₄ (Fig. 1d). These materials were chosen for substrates because they are isostructural with VBi₂Se₄ or VBi₂Te₄, their lattice parameters are close to those of the respective films, and the combination of the work functions of the components of the heterostructures is appropriate (see Table 1) [56, 59].

Fig. 1.

(Color online) (a) Crystal structure of the bulk of V-containing compounds by an example of VBi₂Te₄. (b, c) Band structures of isolated VBi₂Se₄ and VBi₂Te₄ septuple layers. (d) Crystal structure of the considered heterostructures by an example of VBi₂Te₄/Bi₂Te₃. Blue arrows indicate magnetic moments.

Table 1. Ratios of the lattice parameters ${{a}_{i}}$, band gaps $\Delta {{E}_{i}}$, and work functions ${{\Phi }_{i}}$ for the free magnetic septuple layer ($i = 1$) and substrate ($i = 2$) and the Dirac point shift $\Delta k$ and the group velocity $v$ at the Dirac point for the considered heterostructures

As seen in Fig. 2, the energy spectra of pure substrates include a gapless surface state (Dirac cone) characteristic of TIs. The deposition of the magnetic V-containing SL on the surface of the TI modifies the shape of the cone owing to the Dirac point shift both in the energy and in the $k$ space. The Dirac point in the heterostructures under investigation is shifted upward in energy, which is due to the relations between the work functions and band gaps of the V‑containing SL and substrate (see Fig. 2 and Table 1), similar to nonmagnetic TI-based he-terostructures [59]. The band gaps of the magnetic film and substrate in the VBi₂Te₄/Bi₂Te₃, VBi₂Te₄/PbBi₂Te₄, and VBi₂Te₄/GeBi₂Te₄ heterostructures are partially overlapped. The bulk gap of the substrate in other heterostructures is completely built in the corresponding gap of the magnetic film. Such a combination of the band gaps ensures a significant hybridization of the surface state of the substrate with the states of the V-containing magnetic film, which results in the upward energy shift of the Dirac point [59]. Simultaneously, the magnetization induces an additional Dirac point shift $\Delta k$ from the center of the two-dimensional Brillouin zone ($\bar {\Gamma }$ point) in the direction perpendicular to the magnetization of the surface (see Table 1).

Fig. 2.

(Color online) Band structures of the considered heterostructures near the Fermi level: (from left to right) the band structure of the substrate surface, the band structure of the heterostructure, the band structure of the heterostructure near the Dirac point (DP), the schematic positions of the band gaps (the bottom of the conduction band (CB) and the top of the valence band (VB)) of the thin film and substrate with respect to each other, and the depth distribution of the partial charge density at the Dirac point of the surface state (for the heterostructure); the fraction of the charge density localized in the magnetic septuple layer is also given in percent.

According to Fig. 2 and Table 1, the Dirac point shift from the $\bar {\Gamma }$ point in the considered heterostructures occurs as described above: the Dirac point shift $\Delta k$ increases with a decrease in the Dirac velocity $v$. Furthermore, the $\Delta k$ value in the spectra of the VBi₂Te₄/PbBi₂Te₄, VBi₂Te₄/Bi₂Te₃, and VBi₂Te₄/ SnBi₂Te₄ heterostructures is larger than that in the spectrum of the VBi₂Te₄ AFM TI surface. An additional reason for such an effect can be the fact that only the surface V-containing SL is magnetized in the heterostructures under consideration. On the contrary, all SLs are magnetized in VBi₂Te₄, and neighboring SLs are antiferromagnetically ordered. Although the surface state is localized predominantly in the surface SL, its wavefunction extends into the deeper (subsurface) SL. As a result, the exchange interaction with the surface state in the considered heterostructures is stronger than that in VBi₂Te$_{4}$ because it is not weakened by antiferromagnetic interlayer ordering. The shift $\Delta k$ also depends on the degree of localization of the topological state in the outer magnetic block. As seen in Fig. 2, the shift in the heterostructures under investigation with the VBi₂Te₄ magnetic film is maximal in VBi₂Te₄/PbBi₂Te₄ and is minimal in VBi₂Te₄/GeBi₂Te₄. Shifts $\Delta k$ are comparable with the experimentally observed flat bands in graphene systems [60–64]; consequently, the studied heterostructures can be considered as potential candidates for experimental studies of flat bands in planar topological magnets.

3 CONCLUSIONS

To summarize, our ab initio calculations of the electronic structure have demonstrated the possibility of controlling the Dirac point position in heterostructures of the magnetic extension of topological insulators consisting of a thin V-containing magnetic film and a nonmagnetic topological insulator substrate. It has been shown that the Dirac point shift from the center of the Brillouin zone caused by the in-plane magnetization (and, correspondingly, by the length of the flat band on domain walls) strongly depends on the group velocity of electrons at the Dirac point and the degree of localization of the topological state in the magnetic film. The revealed features and made predictions can stimulate the experimental realization of these materials. The considered heterostructures are promising candidates for growth and experimental studies to observe flat bands on domain walls in planar topological magnets.

4 CALCULATION METHODS

The calculations were performed using the projector augmented wave (PAW) method [65] implemented in the VASP [66–68]. Exchange correlation effects were taken into account in the generalized gradient approximation in the Perdew–Burke–Ernzerhof form [69]. To accurately describe the van der Waals interaction, we used the DFT-D3 method [70]. The V 3d states were described in the GGA + U approach [71] within the Dudarev scheme [72]. The parameter U calculated by the linear response method [73] was found to be 4.8 and 4.7 eV for the heterostructures with VBi₂Se₄ and VBi₂Te₄, respectively.

The heterostructures under study were simulated within the model of repeating films with vacuum gaps of 12 Å. The heterostructures consisted of the TI substrate with a thickness of 45 (for heterostructures with Bi₂Se₃ and Bi₂Te₃) or 49 (for heterostructures with PbBi₂Se₄, PbBi₂Te₄, GeBi₂Te₄, and SnBi₂Te₄) atomic layers and two VBi₂Se₄ or VBi₂Te₄ septuple layers (one on each of the surfaces of the substrate). The interplane distances were optimized so that forces acting on atoms did not exceed 10^–2 eV/Å.