Graphene, a two-dimensional (2D) layered structure of carbon atoms,^[1,2] due to its unique electronic, mechanical, thermal, and other outstanding properties^[3–7] has attracted a surge of interest recently. The charge carriers in graphene behave like massless Dirac fermions due to the linear dispersion of energy near the Fermi energy level.^[7] The charge mobility in excess of 200000 cm . has been achieved by suspending the single layer grapheme.^[8] In addition to that, one atom thick 2D layered structure of graphene makes it a suitable candidate for field effect transistor (FET) channels in comparison to other materials. Above mentioned features of graphene suggest that it could be the most appropriate material for high speed FETs. Since graphene is a zero band gap semiconductor, it is crucial to open an ample and well-defined band gap to augment the on/off ratio for applications of FETs.

In conjunction to that, dilute magnetic semiconductors (DMSs) and half metals have been under extensive experimental and theoretical research over the last decade.^[9–17] 2D spintronic devices such as spin filters, spin diodes, and spin transistors can be achieved by introducing magnetism in 2D materials.^[18–20] In addition, the long spin diffusion length and rather long spin lifetime make graphene an appropriate material for designing graphene based spintronic devices.^[21] Given the unique structure of graphene, with planar sp² bonding and bonding and with perpendicular p_z orbitals, it is a nonmagnetic semimetal.^[4] Therefore it is very vital to introduce spin polarization in graphene for its applications in future spintronic devices.

In order to broaden the applications of graphene as a magnetic material in novel spintronic devices, various approaches have been utilized to introduce ferromagnetism behavior in monolayer graphene, nanotubes, and silicone.^[21–26] Presently, there are two main approaches widely utilized for substituting transition metal (TM) atoms into graphene to induce magnetic moments in the graphene layer. The first approach is to adsorb TM atoms on graphene sheet. Studies indicate that adsorbed TM atoms bind strongly to graphene sheet^[27] and their migration barrier is low enough to be mobile at room temperature.^[28] Another feasible approach to control the electronic and magnetic properties of graphene is substitutional doping of foreign atoms into graphene containing single vacancy (SV) or divacancy (DV) of carbon atoms. Various studies have been performed to determine the structural, electronic, and magnetic properties of individual TM atoms embedded into SV and DV graphene sheets. These studies indicate that, the individual TM atoms are effective substitutional dopants in graphene and their binding with C atoms is covalent and rather strong.^[28–32] Available methods for introducing impurities into graphene include the synthesis of hybridized grapheme,^[33,34] intercalation,^[35] chemical modification,^[36] low energy ion implantation,^[37] and defect assisted doping by electron beam irradiation.^[38–40] By using first-principles calculations, the effects of 3d and 4d metal atom impurities on electronic and magnetic properties of graphene have also been addressed in the previous studies.^{[30,33,34,41–44]} However, most of the previous work available on 5d TM atom-decorated graphene has been based on their adsorption effect, without considering the effect of substitutional doping of graphene with 5d TM atom impurities.^[45–49] For example, using first-principles calculations, Sun et al.^[50] proposed that magnetism in arsenene can be achieved by doping Ti, V, Cr, Mn, and Fe atoms. The authors also reported that, half metallic states can be achieved in arsenene by substituting Ti and Mn atoms, while spin-polarized semiconducting states can be achieved by V, Cr, and Fe atom doping. In addition, Sun et al.^[51–56] have dedicated a sufficient amount of work to manipulating the electronic and magnetic properties of phosphorene, germanene, and silicene by substituting different impurity atoms. Very recently, Sun et al.^[27] performed a first-principles study on electronic and magnetic behaviors of 5d TM atom substituted SV graphene. The authors reported that TM atom substitution into SV graphene is thermodynamically favorable and induces ferromagnetism behavior in complex structures. That study is focused on TM atom substitution into SV graphene. Thus, could the graphene layer also achieve larger ferromagnetism and display DMS behavior after substituting 5d TM atoms at the DV site?

In this study, we aim to explore the electronic and magnetic properties of 5d TM atom substituted DV graphene. All the 5d TM impurity atoms are tightly bonded with neighboring C atoms in graphene as indicated by robust hybridization. Given their lower binding energies and increased magnetic moments, 5d TM atom-doped DV graphene structures are suitable for utilization in magnetic substrates to induce magnetism in graphene. To the best of our knowledge, the structural, electronic, and magnetic properties of such a system have not been well understood and the study on this topic remains unfinished and scattered. Therefore, a comprehensive theoretical study is timely at present.

The structural, electronic, and magnetic properties of 5d TM atom-doped DV monolayer graphene were investigated using first-principles density functional theory (DFT) method. The DFT method has already proven to be one of the most accurate methods for the computation of the electronic structure of solids.^[57–62] All the calculations were performed in spin-polarized mode. The projector augmented wave (PAW) potentials^[63] with Perdew–Burke–Ernzerhof (PBE) functional^[64] were utilized by Vienna ab-initio simulation package (VASP).^[65] A kinetic energy cutoff of 500 eV was used for wave function expansion. Our structure model consists of a 5 × 4 monolayer graphene supercell with a vacuum layer of 15 Å in the Z-direction to eliminate the interaction between adjacent layers. The Brillouin zone (BZ) was sampled using a 7 × 7 × 1 Γ-centered k-point mesh. All the structures were fully optimized until the Hellmann–Feynman forces were less than 0.01 eV/Å and the total change in energy was less than 10⁻⁶ eV. A Gaussian smearing method was utilized to deal with the partial occupancies. All our calculations were performed at temperature T = 0 K. Bader analysis was used to calculate the charge transfer.^[66,67] Due to a controversial difference between GGA PBE functional predictions and high level GW and BSE results for graphene, we tested both LDA and PBE functionals to determine the band gap sensitivity. We found that, the band gap is not very sensitive to the functional choice. Both functional parameters predicted similar band gap values with a minimal variation of approximately 0.02 eV in the band gap results.

In this section, the band structure and density of states (DOS) plots are investigated for 5d TM atom-doped DV monolayer graphene structures. For band structure calculations, 20 points are collected along each high symmetry line using path Γ–M–K–Γ in the irreducible Brillouin zone to obtain the band structure with a very fine grid. Total and projected density of states (PDOS) are calculated for all 5d TM atom-doped structures using an 11 × 11 × 1 Γ centered Brillouin-zone sampling and the energy eigenvalues are smeared with Gaussians of width 0.02 eV. The band structure and DOS plots were calculated in spin-polarized mode.

Figures 6(a)–6(g) show the spin polarized band structure diagrams of 5d TM (Hf, Ta, W, Re, Os, Ir, and Pt) atoms incorporated into DV monolayer graphene. In order to determine the effect of 5d TM atom doping on the band structure of graphene, we show the spin un-polarized band diagram of pure graphene in Fig. 6(a). The pure graphene band structure diagram demonstrates a zero band gap semiconductor behavior, which is consistent with the previous results available.^[1–4,7] The band structure diagram of pure graphene depicts that the valence and conduction bands are straddling each other at the Dirac point, depicted by π and bands fusing at the high symmetric k-point thereby making graphene a zero band gap semiconductor. Fermi energy level is presented by a dotted purple line and its value is set to 0 eV as shown in the band structure diagrams.

Fig. 6.

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	Fig. 6. (color online) Spin polarized band structure diagrams for all 5d TM atom-doped DV monolayer graphene 5 × 4 supercell structures. The black and red lines represent the spin up and spin down bands, respectively.

From the obtained results of band structure diagrams for 5d TM atom-doped DV graphene structures given in Figs. 6(a)–6(g), we can predict that after TM atom doping into DV graphene, the linear dispersion of energy at the Dirac point is maintained for all the structures but a finite band gap appears at the high symmetric K-point. As described in the charge density difference diagrams, the impurity TM atoms have excess charge compared to monolayer graphene, thus TM atom doping can be considered as donor impurity doping. Since the impurity atoms have excess charge, it causes the shifting of the Fermi level into the conduction band and the Dirac cone into the valence band along with producing a band gap at high symmetric K-point as shown in the band structure diagrams in Figs. 6(a)–6(g). An important factor to note here is that, the spin up and spin down states in these bands are polarized, which indicates the existence of magnetic moments (we will further elaborate the origins of magnetism in DOS plots below). In case of Hf, Ta, and W atom substitution, the Fermi level moves up above the valence band maximum (VBM) by approximately 0.4 eV, 0.2 eV, and 0.3 eV, respectively and a band gap of 0.5 eV, 0.2 eV, and 0.4 eV appears at the high symmetric K-point in their spin up channel. In the spin down channel, Fermi level moves up above the VBM by approximately 0.35 eV, 0.05 eV, and 0.1 eV respectively and a band gap of 0.5 eV, 0.15 eV, and 0.2 eV appears at the high symmetric K-point as shown in Figs. 6(a)–6(c), respectively. Os substituted DV graphene structure shows direct band gap semiconductor behavior in the spin up band channel with a band gap of approximately 0.4 eV. Since Ir and Pt substituted DV graphene structures produce 0.217 and 0.00 magnetic moments, therefore there is no spin polarization in their spin up and spin down channels. Interestingly, Ir and Pt substituted DV graphene structures show indirect band gap semiconductor behavior with a band gap of approximately 0.2 eV and 0.1 eV, respectively. These results are consistent with the previous reports available.^[27–30]

Spin polarized TDOS and PDOS for all the TM atom-doped DV graphene structures are analyzed to understand the different effects after TM atom substitution into DV graphene. TDOS and PDOS on the doped TM atoms, C atom p orbital, and C atom s orbitals for Hf, Ta, W, Re, Os, Ir, and Pt doped DV monolayer graphene are shown in Figs. 7(a)–7(g), respectively. Fermi energy level is marked by the thin grey line in the PDOS plots appearing at the 0 eV energy level. Some surface impurity states appear near the top of the valence band and near the bottom of the conduction band, which are attributed to the hybridization between d orbitals of TM atoms and p orbitals of C atoms. These impurity states are also shown in band structure diagrams at the Fermi energy as presented in Figs. 6(a)–6(g). Since we know that Hf, Ta, W, Re, and Os doped DV graphene show 0.75 , 1.65 , 1.78 , 0.51 , and 1.5 magnetic moments, respectively, hence their orbitals are spin polarized as shown in PDOS plots for given structures. However Ir and Pt atom-doped graphene structures have almost 0 magnetic moment, hence no polarization in their orbitals is observed. Upon detailed analysis of PDOS plots, we can predict that, five different orbitals ( , , , , and ) of 5d TM atoms give rise to magnetic moments of TM atom-doped DV graphene structures. In case of 5d TM atom-doped DV graphene, and orbitals do not hybridize. However, slight hybridization occurs between , , and orbitals of TM atoms and p orbitals of C atoms as described in Figs. 7(a)–7(g). An important factor to note here is that the band gap is sensitively dependent on the type of impurity atom, which offers a sensible approach to tune the band structure of monolayer graphene. In addition to that, the band gaps differ between their spin up and spin down channels, which suggests that the 5d TM cluster-doped graphene structures can exhibit metallic or dilute magnetic semiconductor behaviors depending upon the applied external magnetic field.

Fig. 7.

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	Fig. 7. (color online) TDOS and PDOS plots for 5d TM atom-doped DV monolayer graphene complexes. Fermi level is denoted by the vertical solid grey line at 0 eV.

The structural, electronic, and magnetic behaviors of 5d TM atom-doped DV monolayer graphene structures were investigated by means of first-principles calculations based on density functional theory (DFT) method. It is observed that 5d TM atom substitution into DV monolayer graphene can produce ferromagnetism coupling in graphene. Spin densities and charge density difference for all 5d TM atom-doped graphene structures were calculated. Significant ferromagnetic couplings were observed between TM atom and neighboring C atoms on the graphene layer in all 5d TM atom-doped DV graphene structures. The direction of charge transfer is always from 5d TM impurity atom to the monolayer graphene. We calculated the band structure and partial density of states in order to understand the effect of 5d TM substitution into DV monolayer graphene. It is observed that the band gap is sensitively dependent upon the dopant atoms, which suggests a viable approach to tune the electronic band structure of monolayer graphene. In addition, the band gap in the spin up channel is different from that in the spin down channel, which suggests that 5d TM atom-doped graphene complexes carry potential applications for the development of spintronic devices. Detailed analysis of PDOS plots indicate that in all cases of 5d TM atom-doped DV graphene structures, five different orbitals, namely, , , , , and , of 5d TM atoms give rise to magnetic moments in monolayer graphene. From the results given above, we can conclude that some of the 5d TM atom-doped DV graphene structures introduce band gaps and magnetic moments converting semimetal graphene into metallic/dilute magnetic semiconductor structures. Therefore, we can predict that 5d TM atoms-doped DV graphene complexes are suitable for future applications in nanoelectronics and spintronic devices.