Tunable topological quantum states in three- and two-dimensional materials

We review our theoretical advances in tunable topological quantum states in three- and two-dimensional materials with strong spin-orbital couplings. In three-dimensional systems, we propose a new tunable topological insulator, bismuth-based skutterudites in which topological insulating states can be induced by external strains. The orbitals involved in the topological band-inversion process are the d- and p-orbitals, unlike typical topological insulators such as Bi2Se3 and BiTeI, where only the p-orbitals are involved in the band-inversion process. Owing to the presence of large d-electronic states, the electronic interaction in our proposed topological insulator is much stronger than that in other conventional topological insulators. In two-dimensional systems, we investigated 3d-transition-metal-doped silicene. Using both an analytical model and first-principles Wannier interpolation, we demonstrate that silicene decorated with certain 3d transition metals such as vanadium can sustain a stable quantum anomalous Hall effect. We also predict that the quantum valley Hall effect and electrically tunable topological states could be realized in certain transition-metal-doped silicenes where the energy band inversion occurs. These findings provide realistic materials in which topological states could be arbitrarily controlled.

In two-dimensional (2D) systems, the quantum Hall c The Author(s) 2015. This article is published with open access at link.springer.com effect emerges due to magnetic-field-induced Landau quantization. However, the requirement of an external magnetic field means that it may not be convenient for use in realistic devices. To overcome this inconvenience, researchers proposed the quantum anomalous Hall effect (QAHE) [13] in which the internal magnetism breaks time-reversal symmetry and the spin-orbit coupling (SOC) opens a band gap, resulting in a topologically nontrivial insulating state characterized with the quantized Hall conductivity. Several QAHE materials have been proposed, such as mercury-based quantum wells [14], graphene [15], and transition-metal-doped topological insulators [16,17,20,21]. Recently, the QAHE has been verified experimentally [22]. In this review, we first propose a new tunable topological insulator, bismuth-based skutterudite in which the topological insulating states can be induced by external strains. Then, we demonstrate that silicene decorated with certain 3d transition metals such as vanadium can sustain a stable quantum anomalous Hall effect using both an analytical model and first-principles Wannier interpolation. We also predict the quantum valley Hall effect and electrically tunable topological states could be realized in certain transition-metal-doped silicene. Finally, we summarize our results.

Tunable topological insulator in bismuthbased skutterudites
Pressure and strain have been demonstrated as effective methods for tuning the band structures and even topological properties of materials. For instance, CdSnAs 2 under a 7% decrease in the lattice constant will become topological insulator [25], while a 6% change in the length of c-axis will drive Bi 2 Se 3 from a topological non-trivial phase into a topological trivial phase [26]. Here, we predict a new tunable topological insulator in bismuth-based skutterudites in which the bands involved in the topological band-inversion process are d-and porbitals. This band involvement is distinct from that in conventional topological insulators; for instance, in Bi 2 Se 3 and BiTeI, the bands involved in the topological band-inversion process are only p-orbitals. Due to the presence of large d-electronic states, the electronic interaction in this topological insulator is much stronger than that in other conventional topological insulators. The stability of this new material is verified by binding energy calculation, phonon mode analysis, and finitetemperature molecular dynamics simulations. This new material can provide nearly zero-resistivity signal current for devices and is expected to be applied in spintronics devices.

Crystal structure and optimized lattice parameter
Skutterudites, a class of materials with cage-like crystal structure which have received considerable research interest in recent years, are the origin of several unusual phenomena such as heavy fermion superconductivity [27], exciton-mediated superconducting states [28], and Weyl fermions [29]. To include the strong SOC effects in this system, we investigate skutterudites containing the strong SOC element Bi. In particular, the bismuthbased skutterudite IrBi 3 has space group IM 3, and its crystal structure is shown in Fig. 1. There are eight Ir atoms and twenty-four Bi atoms in a unit cell. Each Ir atom is surrounded by six Bi atoms, and each Bi atom has two Ir nearest neighbors [see Fig. 1(a)]. The structure has space inversion symmetry with the inversion center of (1/2,1/2,1/2). The structure belongs to the bodycentered lattice type, and its primitive cell [ Fig. 1 has half the volume of the unit cell. Figur 1(c) shows the Brillouin zone and high symmetric points with Γ (0,0,0), H (0,1/2,0), N (1/4,1/4,0), and P (1/4,1/4,1/4).
After the structure is fully relaxed [30,31], the calculated total free energy (solid line) as a function of lattice parameter is shown in Fig. 1(d). We find that the optimized lattice parameter of the primitive cell is 8.493Å, corresponding to the position of free energy minimum. This value is 6% larger than the experimental lattice parameter of IrSb 3 [32], which is realistic considering that the Bi atom has a larger atomic radius than the Sb atom.
2.2 Binding energy calculation, phonon mode analysis, and finite-temperature molecular dynamics simulations To verify the stability of the new material, we perform binding energy calculations, phonon mode analysis, and finite-temperature molecular dynamics (FTMD) simulations. The binding energy is calculated by where E IrBi 3 denotes the free energy of IrBi 3 per primitive cell, E Ir and E Bi are the free energy of crystalline Ir and Bi per atom, respectively, while n Ir and n Bi are the number of Ir and Bi atoms in the IrBi 3 primitive cell, respectively. By simple calculation [33], E b is found to be equal to −3.65 eV per primitive cell. The negative value of the binding energy suggests a stable state of IrBi 3 . The phonon dispersion and phonon density of states (DOS) for IrBi 3 at zero strain is shown in Fig. 2. In the phonon DOS subfigure, the black solid line corresponds  to the total phonon density of states, while the green and red shaded areas correspond to the states of the Ir and Bi atoms, respectively. The phonon states in the low energy range are mostly composed of states from Bi atoms, indicating that Bi atoms in IrBi 3 vibrate more easily than Ir atoms. The phonon dispersion and phonon DOS show no imaginary frequencies, indicating that IrBi 3 is stable.
To further verify the dynamical stability of the material, we perform finite-temperature molecular dynamics simulations at the temperature of 300 K for room temperature and at 30 K for low temperature. In the simulations, a 2 × 2 × 2 supercell containing 256 atoms is used. The length of the time-step is chosen as 5 fs, and simulations with 1000 steps are executed. We observe that the atoms move the equilibrium positions back and forth, and the extent of such motion under 300 K is larger than that under 30 K. However, no structural collapse occurs throughout the simulations, which can also be observed from the free-energy curves as a function of the time-step as shown in Fig. 3. Moreover, the crystal structure always remains nearly the same as the initial crystal structure. In fact, as shown in the inset of Fig. 3, the crystal structure corresponding to the last free energy maximum in the T = 300 K case (right) shows no significant structural differences than the initial crystal structure (left). The lattice relaxation, binding energy calculation, and Finite temperature molecular dynamics. Free energies as functions of time-step at temperature T = 30 K (blue curve) and T = 300 K (red curve). The slight shift of the free energy curves corresponds to the oscillations of each atom around their equilibrium position. The absence of sharp changes in such curves indicates that no structural phase-transition happens throughout the whole simulation process. The initial crystal structure (denoted by the orange circle on the free energy curve) is plotted in inset (a). The crystal structure corresponding to the last free energy maximum (denoted by the green circle on the free energy curve) is shown in inset (b) as a comparison. It can be seen that, the latter still shows no significant structural differences as compared with the initial crystal structure. Reproduced from Ref. [6], Copyright c 2014 Nature Publishing Group.
phonon mode analysis together with the FTMD simulations provide an authentic test for the stability of the bismuth-based skutterudite IrBi 3 . Figure 4 shows the band structures, where the black and blue lines represent the generalized gradient approximation (GGA) and GGA+U band structures, respectively. As shown in Fig. 4(a), IrBi 3 resides in the normal metal state before exerting pressure with its bands crossing the Fermi level E F several times. Moreover, the conduction band minimum and valence band maximum degenerate so that the material behaves as a semi-metal that has a zero energy gap, similar to graphene and CeOs 4 As 12 [10]. This degeneracy at Γ is protected by the cubic symmetry of crystal, which cannot be eliminated by small changes of the lattice constant as we have verified. To shift the degeneracy at Γ, we need to break the symmetry. A simple method is to add an anisotropy similar to the method used for CdSnAs 2 [25]. Here, we simply impose an additional 2% suppression on the c-axis of the primitive cell without changing the length of a-and b-axis, which imposes an anisotropy on the system. Though the anisotropy does not change the parities of each band, it opens a gap at the Fermi level, shifting the system to the insulating state [see Such band-inversion character is further verified by the modified Becke-Johnson (mBJ) potential, which could predict an accurate band gap and band order [34][35][36]. The highly dispersive band below the Fermi level is primarily due to the p-orbitals of Bi atoms, and it has little weight from the Ir atoms in k-points far away from the Γ point. However, in the vicinity of Γ point, the weight of the Ir atoms in that band increases rapidly and becomes the dominating orbital component, showing an apparent band inversion. To further confirm the topological properties in such conditions, we calculate the Z 2 topological quantum number. After both applying an isotropic strain  Table 1 Parities of top-most isolated valence bands at eight time-reversal invariant momenta. Positive parity is denoted by + while negative denoted by −. Products of the occupied bands at each time-reversal invariant momentum are listed in the right-most column. As is shown, the product of parities of occupied bands contributes a −1 at (0,0,0) while +1 at the seven other time-reversal invariant momenta, resulting in ν 0 = 1, ν 1 = ν 2 = ν 3 = 0. Reproduced from Ref. [6], Copyright c 2014 Nature Publishing Group.

TRIM
Parity Product of the parities and further applying a uni-axial strain along c-axis, the system still possesses spatial inversion symmetry. Thus, the Fu-Kane method [7] can be utilized to calculate the Z 2 number of our system. The index for strong topological insulators v 0 is expressed as  Table 1, where the deeper states (those states lower than -9.5 eV in Fig. 5) that were separated far from top-most isolated valence bands are ignored because they do not change the system band topology. As shown, the product of the parities of occupied bands contributes −1 at Γ and contributes +1 at the seven other time-reversal invariant momenta. As a result, the Z 2 quantum number is ν 0 = 1, ν 1 = ν 2 = ν 3 = 0, which corresponds to a strong topological insulator.
2.4 Partial-density of states and the d-p orbital dominating property  Figure 5(a) shows the atomic-resolved DOS in which the green curve represents the states of Ir and the red curve represents the states of Bi. The DOS of both types of atoms is clearly large, indicating that both types of atoms make a significant contribution to the total DOS. Figures 5(b) and (c) show the orbital-resolved DOS of the Ir and Bi atoms, respectively. Green, blue and red curves represent s-, p-and d-orbitals, respectively. One character of the material introduced here is the large proportion of d-states near E F . Further, we calculate the projected band structures of the d-orbitals (see Fig. 6) in the local coordinates of the Bi octahedral. The orange, violet, red, green, and blue colors in Fig. 6 represent the d z 2 , d x 2 −y 2 , d xy , d yz , and d xz orbitals, respectively. The radii of the circles are proportional to the weights of corresponding orbitals. The t 2g orbitals (including the d xy , d yz , and d xz orbitals) reside far below the Fermi level and are fully occupied. On the other hand, the lowest three conduction bands are primarily contributed by the e g orbitals (including the d z 2 and d x 2 −y 2 orbitals). More specifically, the d x 2 −y 2 orbital makes an even larger contribution than the d z 2 orbital for the lowest conduction band. The large pro-portion of d-states near E F is distinctively different from conventional TI materials. For example, states near E F primarily contain s-p electrons in HgTe and p-electrons in Bi 2 Se 3 . The large proportion of d-states near E F indicates that the electrons in the proposed material process strong electronic correlations and are more localized [37][38][39][40][41][42][43][44][45] than those in other TI materials. These strong electronic correlations make this material a useful platform for investigating the effect of correlations on the topology as well as a candidate for realizing quantum information devices based on correlations. The localization will enhance the effective mass of bulk electrons, and hence reduce the bulk contribution to the local current at finite temperatures, enhancing the spin-binding property, which is helpful for fabricating spintronics devices with higher stability.

Experimental protocols
The new strain-induced topological insulator IrBi 3 could be grown using the Bridgman method, which has been successfully used for synthesizing the similar materials CoP 3 [46] and RuSb 3 [47]. The crystal growth should be conducted in a sealed quartz ampoule. The iridium and bismuth should be coated by graphite and then introduced into the quartz ampoule. Similar to the case of RhSb 3 [48], a temperature gradient of about 50 • C/cm should be maintained at the growth interface. To remove the excess bismuth in the as-grown crystal, post-annealing should be performed [49]. After the synthesis, the crystal structure of the new material could be characterized by X-ray diffraction using monochromatic Cu Kα radiation [50]. Then, the strains could be generated by a pair of diamond anvils [51], which could generate strong pressure above 200 GPa [52]. Moreover, to detect the real-time pressure strength, the ruby fluorescence method [51,53] could be utilized. To verify the Fig. 6 Orbital-projected band structures. The orange, violet, red, green and blue colors in subfigures represent the d z 2 , d x 2 −y 2 and dxy, dyz and dxz orbitals respectively. The radii of circles are proportional to the weights of corresponding orbitals. The Fermi level is set to be zero energy. It can be seen that, the t 2g orbitals (including the dxy, dyz, and dxz orbitals) reside far below the Fermi level and are fully occupied. While, the lowest three conduction bands are mainly contributed by the eg orbitals (including the d z 2 and d x 2 −y 2 orbitals). More specifically, the d x 2 −y 2 orbital makes an even larger contribution than the d z 2 orbital for the lowest conduction band. Reproduced from Ref. [6], Copyright c 2014 Nature Publishing Group.
topological property of the material, angle-resolved photoemission spectroscopy (ARPES) measurements [9] or the transport measurements [54,55] could be performed. Similar to Bi 2 Se 3 , the observation of the spin-Hall current [56] and the non-equally spaced Landau levels [57] in IrBi 3 will be signatures of the Dirac fermions in the surface of the topological insulator [1].

Tunable quantum anomalous Hall effect and valley Hall effect in transition-metal-doped silicene
For considering the tunable topological topological states in 2D systems, we take transition-metal-doped silicene as an example. Silicene is closely analogous to graphene [58][59][60] in the sense that it consists of a single layer of Si atoms arranged in a low buckled honeycomb lattice. Moreover, its low energy physics can be described by Dirac-type energy-momentum dispersion akin to that in graphene [61]. Therefore, silicene inherits many intriguing properties such as the expected Dirac fermions and quantum spin Hall effect [11]. The additional buckling degree in the silicene monolayer results in a relatively large (1.55 meV) SOC-induced gap [11], which is absent in graphene. A number of recently reported unusual quantum phenomena [62][63][64][65][66] together with its natural compatibility with the current silicon-based microelectronics industry make silicene a promising candidate for future nano-electronics applications. In this regard, it is highly valuable if magnetism or a sizable band gap or both can be established in nonmagnetic silicene for realizing QAHE with dissipation-less edge states protected by topology.
In this section, we introduce the underlying topological nontrivial states of silicene via systematic investigation of the adsorption of 3d transition metals (TMs). We demonstrate that 3d TMs strongly bonded with silicene and that TM-silicene systems are strongly magnetic. With the combination of tight-binding (TB) model analysis and first-principles Wannier interpolation, show that the vanadium-doped silicene hosts a stable QAHE that survives the strong correlation effects of the adatom. This system can also be half-metallic [67] if the Fermi level is properly tuned. Further, a close study of the TB model in the band-inverted regime gives rise to another topologically nontrivial state, which supports the quantum valley Hall effect (QVHE) [68]. We predict that the resulting QAHE and QVHE can be tuned directly using an external electrical field, which is rather appealing for future nanoelectronics and spintronics applications.

Adsorption and magnetism analysis
We use a 4 × 4 supercell of silicene to model the interaction between 3d TMs (Sc, Ti, V, Cr, Mn, Fe, Co, Ni) and silicene. Because silicene has a buckled geometry, we consider three high symmetry adsorption sites, namely the hollow (H) site at the center of a hexagon and two top sites denoted as T A and T B , corresponding to the top Si atoms belonging to the A and B sublattice, respectively (see Fig. 7). The simulations have been performed separately within the GGA [69] and GGA+U [71,72] framework to evaluate the effect of on-site Coulomb interactions among the 3d electrons of the adatoms on the equilibrium structure and the magnetic properties of the TM-silicene system, respectively. We first focus on the GGA case. Of the three adsorption sites, all 3d TMs energetically favor the H site, which is 0.02 eV -0.60 eV and 0.20 eV-0.80 eV higher in adsorption energy than the T A and T B site, respectively. Here, ΔE = E s +E ads −E s−ads is the adsorption energy within GGA, and E s , E ads , and E s−ads represent the energies of the 4 × 4 pristine silicene, single adatom, and siliceneadatom system, respectively. The bonding between 3d TMs and silicene is strongly covalent as manifested by a much larger ΔE ranging from 2.44 eV to 4.75 eV. The unusual large ΔE compared with that in graphene [73,74] could be related to the covalently more active sp 3like orbitals of silicene, which are a result of the unique buckled geometry.
Most TM-doped silicenes exhibit magnetism with sizable magnetic moments ranging from ∼ 1 μ B to ∼ 5 μ B , with the exception of Ni-doped silicene, which is similar to the graphene case [73,74]. A relatively large magnetic moment is key to the realization of QAHE in silicene, which we discuss later. We also note that when some TMs (Sc, Ti, Cr) adsorb on the H site, the DOS show peaks at the Fermi level, indicating that these systems could be magnetically unstable and may undergo Jahn-Teller distortion to lower total energy. In the case of Sc-silicene, we artificially move one of three Si atoms nearest to Sc to break the C 3 rotational symmetry. After relaxation, the three nearest Si atoms to Sc, which originally arranged themselves as a coplanar, regular triangle (bond length d Si−Sc equal to 2.62Å) is now distorted to an isosceles triangle (d Si−Sc with bond lengths of 2.63Å, 2.63Å, and 3.12Å) by pushing an Si atom away from the upper sublattice plane by 1.24Å. The distorted Sc-silicene system becomes more stable than C 3v symmetric system because the total energy is lowered by 0.1 eV. Similar to the Sc-doped silicene case, we could expect the Jahn-Teller distortion to further stabilize the Ti-silicene and Cr-silicene systems; nevertheless, the distortion for these two systems was rather weak.
The resulting magnetic moments and possible aforementioned Jahn-Teller distortion can be understood by symmetry considerations. When TMs are deposited on high symmetry sites of silicene (H, T A , T B ), the 3d subshell of the adatom splits into three groups under the C 3v symmetric crystal field of system: the 3d 3z 2 −r 2 state corresponding to the A 1 symmetry group, the twofold degenerate E 1 group consisting of the 3d xz and 3d yz states, and the E 2 group consisting of the 3d xy and 3d x 2 −y 2 states. Therefore, the three groups of 3d states hybridize with the π orbitals of silicene weakly or strongly according to the different symmetrical properties in similar way as that in benzene [75] and graphene [76]. Because 3d orbitals are anti-bonding states, the energy order of the orbitals are usually (E 2 ) < (A 1 ) < (E 1 ). In general, (E 2 ) and (A 1 ) are close to each other due to their similar hybridization strength with π orbitals. After incorporating spin polarization, the three groups of states split, according to different splitting energy, into 10 spinpolarized orbitals. Meanwhile, the outer 4s electrons of the adatoms experience relatively large electrostatic interaction from the π manifold of silicene, compared to that from the 3d shells, due to the spherical symmetry and delocalized nature of the π manifold, which enables charge transfer from the 4s to 3d shells. Thus, the crystal field splitting and spin splitting, together with the occupation number, primarily dominate the electronic structure of adsorbed TM ions.
The spin splitting for Sc in Sc-doped silicene is around 0.2 eV, which is smaller than ligand field splitting between E 2 and A 1 . In total, there are three electrons occupying 3d orbitals (see Fig. 8). As a result, two of these three electrons occupy the majority E 2 orbitals, while the remaining electron occupies the doubly degenerate minority E 2 orbitals, leading to 1 μ B magnetic moment and potential Jahn-Teller distortion. Owing to the relatively large splitting of A 1 (1 eV), the majority A 1 orbital in Ti-doped silicene is occupied before the doubly degenerate minority E 2 orbitals as indicated in the inset of Fig.  8, showing the projected density of states (PDOS) of Ti from GGA calculations. This leads to the peaks at the Fermi level and the magnetic moment of 2 μ B . For V, the spin splitting is much larger, and a high spin state with 5 μ B moment state is realized, which is essential to the realization of QAHE in silicene. The other cases can be understood with similar arguments.
Considering that the strong correlation effect of 3d electrons apparently affect the description of adsorption, we consider the GGA+U case. After ionic relaxation, the adsorption geometry of the adatom-silicene system strongly changed compared with GGA case. Moreover, most 3d TMs still favor the H site (except for Mn, which now energetically favors the T A site). The geometry change of the adatom-silicene systems is a direct consequence of the on-site Coulomb interactions among 3d electrons, which modify the electron distribution in 3d and 4s shells of adatoms and π orbitals of silicene as shown from the changes in the magnetic moment.

Topological states from Chern number analysis
In this section, we review the QAHE in the absence of external field via doping with 3d TMs and predict the electrically tunable topological states. As shown in Refs. [74,77], the QAHE could be realized via doping certain 3d or 5d adatoms on the hollow site of graphene. In Fedoped graphene [74], the QAHE gaps occur around the Dirac K points of the Brillouin zone, and the low-energy physics can be described by a Hamiltonian for graphene in the presence of the extrinsic Rashba SOC (λ ext R ) and exchange field (M ) [78], introduced solely by the adatom. However, in silicene, when depositing 3d TMs on the stable adsorption site or when applying a perpendicular external electric field, the induced inequality of the AB sublattice potential (Δ) necessarily occurs and competes with the magnetization [62,63]. Furthermore, the QAHE in silicene lives only in certain ranges of parameter space [62]. Moreover, owing to the low buckled structure, there exists so-called intrinsic Rashba SOC (λ int R ) [70]. The interplay between the two types of Rashba SOC (λ ext R and λ int R ) will lead to an electrically tunable topological phase transition, as we will demonstrate later. Below, we identify conditions for the realization of QAHE in silicene based on an effective Hamiltonian [70] by introducing a staggered AB sublattice potential in addition to SOC (λ ext R and λ so ) and the exchange field (M ), which is similar to the one used in Ref. [62].
The Hamiltonian of this system is H Here, the basis is chosen as {A, B} ⊗ {↑, ↓}, and H ± ef f are the total Hamiltonians for the two inequivalent Dirac points K (+) and −K (−). Moreover, H ± s are the low-energy effective Hamiltonians for the quantum spin Hall insulator silicene, and H ± d include all effects introduced by the 3d dopants, including the effective spin-dependent magnetic field M , site-dependent staggered potential Δ, and the resulting extrinsic Rashba SOC λ ext R . The Pauli matrices τ and σ act separately on pseudospin (sublattice) and spin space, respectively, ε ef f is the ε 1 − λ 2nd term in Ref. [11], v F and a represent the Fermi velocity and the lattice constant, respectively, and λ so stands for the effective SOC.
We can derive the band structures around each valley  (K and −K) in the Brillouin zone by diagonalizing the above Hamiltonian. The interplay between the exchange field (M ) and staggered potential (Δ) (see Fig. 9) causes the energy bands with opposite spin to intersect, which is essential to the QAHE in Fe-doped graphene when M/Δ > 1. The situation here is different from that in Ref. [78], where Δ = 0 and where two degenerate points always exist around the Dirac point as long as M = 0.
Either the extrinsic or the intrinsic Rashba SOC would induce an insulating state when starting from the case M/Δ > 1. To identify the topological properties of the resulting insulating state, we perform the Chern number (C) analysis [79], where C can be obtained by integrating over the first Brillouin zone (BZ): Here, Ω (k) is the usual Berry curvature of all occupied states [80]: where f n is the Fermi-Dirac distribution function for band n, ψ nk is the Bloch function of the eigenenergy nk , and v x and v y are the velocity operators. The anomalous Hall conductivity is readily given by σ xy = (e 2 /h)C. Interestingly, the extrinsic Rashba SOC (λ ext R ) gives the insulating state with C = +2, while the intrinsic Rashba SOC (λ int R ) leads to that with C = −2. One may expect  that a different Chern number state can be realized by properly tuning the two types of Rashba SOC in experiments. Figure 10(a) indicates that this is indeed the case: C can take integer values of +2, 0, −2 with different combinations of λ int R and λ ext R . Careful study shows that the tunable C originates from the different response of bulk gap to the two types of Rashba SOC (λ int R and λ ext R ) around K and −K. For example, when increasing λ ext R while keeping λ int R fixed at 0.03t (t = 1.6 eV is the nearest neighbor hopping parameter [70]) as shown in Fig. 10 Fig. 10(h)]. Consequently, the system can be in the QAHE phase (with C being +2 or −2) or the QVHE phase (with C = 0 and C K = −C −K = −1) depending on the value of λ ext R , which is controllable through an external gate voltage.
However, the effective SOC (λ so ) further breaks the particle-hole symmetry of the above Hamiltonian, causing the energy bands shift up (at valley K) or down (at valley −K) relative to the Fermi level while leaving the topological charge of each valley unchanged. Hence, as long as the shifting is small, the system is still insulating, and the above discussion of topological transition remains valid.

Verification of quantum anomalous Hall effect in 3d-transition-metal-doped silicene
We now introduce the verification of topologically nontrivial phases using first-principles calculations. As an example, in Ti-, V-, Cr-, or Mn-doped silicene, the opposite spin subbands cross near the Fermi level due to the relatively large magnetization. This behavior closely resembles the band inversion case (M/Δ > 1) in the TB model above [see Fig. 9(e)]. The spin-up and spin-down subbands near the Fermi level are also gapped [Figs. 11(a) and (c)], which makes these systems candidates for half-metallic materials if the Fermi level is tuned properly. We take V-silicene as a prototype because this system is insulating with an energy gap around 6 meV [ Fig.  11(b)] when only the SOC is considered. Moreover, the SOC-induced band gap in V-silicene is stable regardless of the inclusion of the effective U .
As discussed above, the V-silicene system could be in one of three topologically nontrivial states [QAHE(+2), QVHE(0), QAHE(-2)] when only SOC is considered. Thus, we need to obtain an accurate Berry curvature distribution and Chern number at the first-principles level in V-doped silicene [81,82]. As indicated in Fig. 11(d), the band structure produced by Wannier interpolation is consistent with that from first-principles calculations. In Fig. 12, the Berry curvature [Ω z (k)] in k-space is explicitly shown. As we may observe, the most nonzero values (positive) of the Berry curvature are distributed near the Dirac K points, forming small circles, where the avoided crossing occurs [ Fig. 11(d)]. Moreover, we find that the Chern number of all occupied bands equals an integer value of +2, which indicates that the V-doped silicene is in the QAHE(+2) phase, by the integral over BZ. The tight-binding parameters corresponding to this case [Fig. 11(d)] are estimated as M =75 meV, Δ=21 meV, λ so =5 meV, λ int = 8 meV, and λ ext = 1.5 meV. Note that, here, λ ext is induced solely by vanadium adatom.
To introduce the topological phase transition in the V-silicene system, we can apply an additional external electric field to increase λ ext R . According to the phase diagram presented in Fig. 10, we can give an estimation of the magnitude of the external electric field. The value of λ ext R required for the system to enter the QVHE(0) state is about 1.55 meV. Consequently, an extra 0.05 meV λ ext R should be supplied by an external electric field E z , which is about (12.5 V)/(300 nm) according to a rough estimation [70]. Further, we need to apply E z of about (225 V)/(300 nm) to generate an additional 0.9 meV λ ext R (total λ ext R is about 2.4 meV) for the system to enter the QAHE(-2) phase.

Conclusions
In summary, we have reviewed our recent progress in tunable topological quantum states in three-dimensional and two-dimensional materials with strong spin-orbital couplings. By applying proper strain, we demonstrate the bismuth-based skutterudite could be tuned into a topological insulator with d-p topological band inver-sion. Moreover, by 3d-transition-metal doping, silicene can sustain the quantum anomalous Hall effect and quantum valley Hall effect. These findings provide evidence for realistic materials in which the topological states could be arbitrarily controlled.