Coexistence of valley polarization and Chern insulating states in MoS2 monolayers with n-p codoping

The electronic and topological properties of MoS2 monolayers with n-p codoping effect are investigated by using first-principles calculations. Two types of the doped Nb atoms play the roles of the p-type and n-type dopants, respectively. The n-p codoping is found inducing a large valley polarization, associated with the strong magnetization induced by the Nb dopants. Interestingly, the system simultaneously owns a perfect Chern insulating band gap opened exactly at the Fermi level. The nontrivial band gap comes from the lifting of the degeneracy of the dxz and dyz orbitals of Nb2 atoms after the spin-orbit coupling is considered. Our work inspires exciting prospects to tune the novel properties of materials with n-p codoping effects.

adsorption energy reflects the interaction strength between the Nb 2 atom and the two-dimensional sheet. For the pristine MoS 2 ML, the adsorption energy is 2.485 eV, while if the sheet is the Nb 1 doped MoS 2 ML ( Fig. 1(a)), the adsorption energy increases to 3.789 eV. Thus, we infer that the interaction strength of the Nb 2 adatom in the Nb 1 -Nb 2 codoping case is enhanced (by 1.3 eV). This behavior can be ascribed to the strong electrostatic attractive interaction between the n-type (Nb 2 ) and p-type (Nb 1 ) dopants. In the following, we primarily focus on the most stable N configuration for the Nb 1 -Nb 2 codoping case.
The band structure and the densities of states (DOSs) for the MoS 2 ML with Nb 1 -Nb 2 codoping are given in Fig. 2(a) and the upper panel of Fig. 2(b), respectively. The spin-orbit coupling (SOC) is not yet considered. The Nb 1 DOSs distribute over a wide range of energy, similar to those of Mo atoms in the pristine MoS 2 22 , indicating strong bonds formed between Nb 1 and its neighboring S atoms. The DOSs of the Nb 2 are, however, mainly located within the band gap of the pristine MoS 2 . To further comprehend the roles of the two types of Nb atoms, the electronic structures of the pristine MoS 2 ML with solely Nb 1 or Nb 2 are also calculated. For the MoS 2 ML with one Mo atom substituted by one Nb atom in the 3 × 3 supercell, labeled as MoS 2 (Nb 1 ), the Fermi level (E F ) is now located below the top of the valence bands of the pristine MoS 2 (middle panel of Fig. 2(b)), implying the p-type acceptor character of Nb 1 dopant. When one Nb atom is adsorbed on the top of one of the Mo atoms of the pristine MoS 2 ML, marked as MoS 2 (Nb 2 ), the E F of the system is very close to the bottom of the conduction bands of  the pristine MoS 2 (lower panel of Fig. 2(b)). Thus, Nb 2 adatom acts as the n-type donor. The d orbital of Nb 2 atom splits into A (d z2 ), E 1 (d xz , d yz ), and E 2 (d xy , d x2−y2 ) ( Fig. 2(a,b)), due to the C 3v symmetry owned. The band evolution from the pristine MoS 2 ML to the Nb 1 -Nb 2 codoped MoS 2 ML is summarized in Fig. 1(c). For the pristine MoS 2 ML, the E F is within the band gap. For the MoS 2 (Nb 1 ) case, the E F moves downwards to the valence bands of the pristine MoS 2 , while for the MoS 2 (Nb 2 ) case, the E F moves upper to the conduction bands of the pristine MoS 2 , indicating Nb 1 and Nb 2 act as p-type and n-type dopants, respectively. For the MoS 2 (Nb 1 , Nb 2 ) case, the compensation effect of the n-type and p-type doping causes the E F located exactly at the E 1 bands.
Besides the movements of the E F position, the magnetic behaviors of the Nb 1 and Nb 2 atoms in the systems are also different from each other. For example, the MoS 2 (Nb 1 ) is weakly spin polarized with a total magnetic moment of 0.73 μB (the middle panel in Fig. 2(b)), while the total magnetic moment is 5.00 μB for the MoS 2 (Nb 2 ). The strong magnetism of the MoS 2 (Nb 2 ) can be comprehended through strong atomic behavior of Nb 2 atoms, consistent with the case of the 3d transition metal atom adsorption in MoS 2 MLs in previous studies 23,24 . For the Nb n-p codoping case, the total magnetic moment of 4.00 μB, instead of 5.00 μB, is obtained for the most stable N configuration. The decrease of the magnetism in the n-p codoping case can be ascribed to the p-type Nb 1 in the n-p codoping case gaining one unpaired electron from Nb 2 . The local magnetic moments of the Nb 1 and Nb 2 in the n-p codoped system are about 0.05 μB and 2.49 μB, respectively. Thus, the magnetism in the n-p codoped MoS 2 ML is primarily induced by the Nb 2 adatoms. To find whether the above FM structure is the magnetic ground state, the total energies of non-magnetic (NM) and antiferromagnetic (AFM) structures are also calculated. The total energy of the FM ordering per Nb 1 -Nb 2 pair is found to be lower than that of the NM ordering by 355 meV and AFM ordering by 2 meV. Thus, the FM ordering is the most stable magnetic configuration for the Nb codoped system. The Curie temperature (T c ) estimated with the mean-field approximation 25 is about 15.5 K.
The C 3v symmetry owned by the system makes the E 1 (d xz , d yz ) and E 2 (d xy , d x2−y2 ) bands both be degenerate at the Γ point with quadratic non-Dirac band dispersions before the SOC is included (Fig. 2(a)). Under this case, the system could be called as a spin-gapless semiconductor (SGS) [26][27][28] for the bands at the E F are 100% spin polarized. When the SOC is included, the doubly degenerate energy points at the Γ point are lifted and band gaps are opened, as shown in Fig. 2(c,d). Additionally, the degeneracy of the tops of the valence bands of the pristine MoS 2 (around −1.2 eV in Fig. 2(a)) is also lifted (Fig. 2(c)). E 1 is exactly located at the E F without the SOC ( Fig. 2(a)) and a global band gap of 15.0 meV is opened by the SOC in the system (Fig. 2(d)). We also check the results by using the HSE06 functional 29 . The calculations show that the main feature keeps undistributed with the HSE06 functional, i.e., the E 1 bands are still located around the E F and a global band gap is also opened. The opened band gap is 36.4 meV, larger than the result obtained from the metaGGA method, due to the stronger exchange-correlation interactions predicted for the Nb d xz and d yz orbitals in the HSE06 functional. The real value of the band gap may be determined by future experimental measurements. Hence, the n-p codoped MoS 2 ML is an insulator. As shown in Fig. 1(c), if there is no Nb 1 in the system, the E 1 bands are totally occupied by electrons. It is, thus, the Nb n-p codoping that gives rise to the E 1 bands located exactly at the E F .
Due to the strong magnetism of the Nb 2 adatoms, legible spin polarization of about 0.2 eV is induced at the tops of the valence bands of MoS 2 (around −1.2 eV in Fig. 2(a)). This magnetism together with the broken space-inversion symmetry in the system may lead to valley polarization effect 30,31 . However, the use of the 3 × 3 supercell in the calculations causes band folding. To explore the valley polarization of the n-p codoped MoS 2 ML, the bands shown in Fig. 2(a,c) are unfolded onto the k-points in the Brillouin zone (BZ) of the 1 × 1 primitive cell ( Fig. 1(b)) by using a k-projection method [32][33][34] . Figure 3(a,b) show the unfolded bands for the Nb n-p codoped MoS 2 ML without SOC. The obvious spin polarization in the valence bands of the MoS 2 is also outstanding. The bands at K p and K′ p for both the spin-up and spin-down components are degenerate, whose sketches are displayed in the (i) case of Fig. 3(d). Thus, no valley polarization appears in the bands.
When the SOC is included, a valley polarization of 125 meV can be observed, defined as ∆ = − When the HSE06 functional is employed, an almost same value of the valley polarization is obtained, due to the small exchange-correlation interaction for the Mo d orbitals mainly contributing to the valley bands in the system. Only in the unfolded bands (Fig. 3(c)), the valley polarization can be distinguished. How the tops of the valence bands of the MoS 2 in Fig. 2(c) correspond to the K p or K′ p points in the MoS 2 primitive cell is also illustrated. Figure 3(d) shows the schematic diagrams of the valley polarization formed in the Nb n-p codoped MoS 2 ML. Reference 12 has found that the magnitude of the valley polarization is limited by the smaller spin splitting www.nature.com/scientificreports www.nature.com/scientificreports/ arising from the SOC or the exchange field. Therefore, the valley polarization obtained (125 meV) should be limited by the smaller SOC strength other than the spin polarization (200 meV). The valley polarization may be enhanced further with the increase of the SOC, by such as substituting the atoms with heavier elements. The obtained valley polarization is actually an intrinsic attribute owned by the system and no magnetic field or optical pumping is necessary to realize the effect. This valley polarization makes the Nb n-p codoped MoS 2 ML can be called as a ferrovalley material. If the spin orientations of the Nb atoms are reversed, the band features at K and K′ are exchanged, similar to the case in ref. 35 . Thus, the ferrovalley property still exists in the system, with the same valley polarization strength, but an opposite sign. Since the total energies of the two systems are the same, the two ferrovalley states can be regarded as energy degenerate states.
We now identify the possible topological behaviors of the band gap opened at the E F (Fig. 2(d)). The bands are fitted by using maximally localized Wannier functions (MLWFs) method [36][37][38] (Fig. 4(a)). The Berry curvatures can then be calculated. Two peaks of the Berry curvatures appear around the Γ point ( Fig. 4(b)), ascribed to the parabolic band dispersions of E 1 (Fig. 2(a)) without SOC 39 . The calculated Chern number of 1 gives a direct evidence for the existence of Chern insulating state in the system. Figure 4(c) displays the band structure of a one-dimensional Nb n-p codoped MoS 2 nanoribbon, calculated by using the tight-binding model constructed from the MLWFs. The two edge states (red curves in Fig. 4(c)) connecting the conduction and valence bands exist inside the SOC induced band gap, representing one chiral dissipationless conducting channel existing on each side of the nanoribbon sample. Besides, the edge states should be 100% spin-polarized and half-metallic since the two-fold E 1 bands in Fig. 2(a) are spin up 40,41 , availing the applications of the Nb n-p codoped MoS 2 ML system in spintronic devices. The quantized Hall conductivity of e 2 /h at E F (Fig. 4(d)) matches well with the nontrivial band gap contributed by E 1 orbitals, while the Hall conductivity of −e 2 /h at −0.8 eV appears due to the E 2 orbitals. The forming mechanism of these topological states can be ascribed to the SOC-induced band splitting of the degenerate quadratic non-Dirac bands at the Γ point [39][40][41] . Thus, the large ferrovalley effect and the Chern insulating state achieved in the Nb n-p codoped MoS 2 monolayer are contributed from the bands at the K (K′) and Γ points in the momentum space, respectively.
The electronic and topological properties of the Nb codoped systems with different Nb doping concentrations are also studied. If the Nb doping concentration is increased, such as one pair of Nb 1 -Nb 2 doped in one 2 × 2 supercell, the band structure is found being changed much and becomes metallic, ascribed to the strong interaction between the different Nb pairs. If the Nb doping concentration is lowered, for example, one pair of Nb 1 -Nb 2 doped in one 4 × 4 supercell, the valley polarization and the nontrivial bands can still be clearly observed, although global band gaps may not be opened. Therefore, the Nb doping concentration should be carefully examined to give perfect Chern insulting states and large valley polarization in the MoS 2 MLs.
In experiments, the n-p codoping has been realized in some materials, such as dilute magnetic semiconductors of ZnMnAlO and ZnCoAlO 42 . Based on these techniques, the n-p codoped MoS 2 ML may be fabricated by following two steps. First, synthesize the p-type doped MoS 2 ML with Nb 1 atoms substituting the Mo atoms via e-beam evaporation plus chemical vapor deposition (CVD), as implemented in ref. 16 or chemical vapor transport www.nature.com/scientificreports www.nature.com/scientificreports/ (CVT) in refs. 17,18 . Second, deposit some Nb atoms onto the surface of the prepared p-type MoS 2 ML. Due to the electrostatic attraction between the n-type and p-type dopants, these deposited Nb atoms (Nb 2 ) tend to be located on the top of the Nb 1 atoms, as the obtained adsorption energies indicate. In this way, the Nb n-p codoped system might be achieved in experiments.

conclusion
The electronic states of the Nb n-p codoped MoS 2 monolayer are studied with ab initio calculations. Two Nb atoms (Nb 1 and Nb 2 ) serve as p-type and n-type dopants, respectively. Large valley polarization is predicted, caused by the induced magnetism together with the broken space-inversion symmetry and the SOC interaction in the MoS 2 system. The Nb n-p codoped MoS 2 monolayer is also a Chern insulator, whose edge can conduct the pure spin-up current without energy dissipation. The coexistence of the large valley polarization and Chern insulting states provides encouraging routes in applying the codoped two-dimensional materials in the fabrication of valleytronic, microelectronic, and spintronic devices.

Methods
Since experimental studies have shown that the Mo atoms in MoS 2 thin films can be substituted by Nb atoms as efficient acceptors [16][17][18] , we build the MoS 2 ML with n-p codoping by first substituting one of the Mo atoms in the 3 × 3 supercell with an Nb atom, marked as Nb 1 in Fig. 1(a). Another Nb atom (Nb 2 ) is then considered to adsorb at the surface of the MoS 2 ML. Totally four typical adsorption sites are explored for the Nb 2 atom: H (Hollow), M (Mo-top), N (Nb 1 -top), and S (S-top) ( Fig. 1(a)) 43 . Some other adsorption sites, far away from the Nb 1 atom, are also considered, which are, however, found being unfavorable in energy, compared with the above corresponding sites. The calculations of the electronic structures are performed by using first-principles methods 44 . The exchange-correlation interaction is described with the metaGGA with SCAN form 45 . The energy cutoff is set as 450 eV. Monkhorst-Pack k-point meshes with 5 × 5 × 1 are adopted for structural relaxation and electronic structure calculations. The experimental lattice constant of 3.16 Å 46 is used and the vacuum space along the z direction is set about 20 Å. The test calculations show that the Nb dopants can expand slightly the lattice structure. Since the energy bands are found insensitive to the lattice constant, the experimental lattice is employed in the subsequent calculations. The convergences of the total energy and Hellmann-Feynman forces are set to be 10 −6 eV and 0.01 eV/Å, respectively.