Effects of biaxial strain on the impurity-induced magnetism in P-doped graphene and N-doped silicene: A first principles study
J. Hernández-Tecorralco, L. Meza-Montes, M. E. Cifuentes-Quintal, R. de Coss
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Effects of biaxial strain on the impurity-inducedmagnetism in P-doped graphene and N-dopedsilicene: A first principles study
J. Hern´andez-Tecorralco , L. Meza-Montes , M. E.Cifuentes-Quintal , and R. de Coss Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apartado PostalJ-48, 72570, Puebla, Puebla, M´exico Departamento de F´ısica Aplicada, Centro de Investigaci´on y de Estudios Avanzadosdel IPN, Apartado Postal 73, Cordemex, 97310, M´erida, Yucat´an, M´exicoE-mail: [email protected]
Abstract.
The effects of biaxial strain on the impurity-induced magnetism in P-dopedgraphene (P-graphene) and N-doped silicene (N-silicene) are studied by means ofspin-polarized density functional calculations, using the supercell approach. Thecalculations were performed for three different supercell sizes 4 ×
4, 5 ×
5, and 6 ×
6, inorder to simulate three different dopant concentrations 3.1, 2.0 and 1.4%, respectively.For both systems, the calculated magnetic moment is 1.0 µ B per impurity atom forthe three studied concentrations. From the analysis of the electronic structure andthe total energy as a function of the magnetization, we show that a Stoner-type modeldescribing the electronic instability of the narrow impurity band accounts for the originof sp -magnetism in P-graphene and N-silicene. Under biaxial strain the impurity banddispersion increases and the magnetic moment gradually decreases, with the consequentcollapse of the magnetization at moderate strain values. Thus, we found that biaxialstrain induces a magnetic quantum phase transition in P-graphene and N-silicene. Keywords: sp magnetism, magnetic phase transition, graphene, silicene, strain, dopingffects of biaxial strain on the impurity-induced magnetism ...
1. INTRODUCTION
Magnetism is one of the most studied phenomena in physics and materials science.The magnetic behavior of the matter is usually attributed to the presence of d- or f- electrons. However, less common is the existence of magnetic materials with only s- and p- electrons. The arise of graphene [1] and related two-dimensional (2D) materials,such as silicene, have aroused great interest due to their outstanding properties andthe expectation to exhibit sp -magnetism under certain conditions. Pristine grapheneand silicene are non-magnetic, however, an alternative to induce magnetism is byintroducing defects. For instance, it has been observed that vacancies [2] or thechemical functionalization using adsorbed [2, 3, 4] or substitutional [5] impurities, areeffective ways to induce magnetism in 2D systems. Thus, the study of impurity-inducedmagnetism in low-dimensional materials is relevant in view of their potential applicationsin spintronics [6, 7] and spin-based quantum information systems [8, 9].The electronic and magnetic properties of doped graphene and silicene withsubstitutional sp -impurities; Al, Si, P, and S, for graphene and B, N, Al, and P forsilicene, have been theoretically studied using first principles calculations based onthe density functional theory [10, 11, 12, 13, 14]. Particularly, it has been reportedthat phosphorus and nitrogen atoms as single substitutional impurities in graphene(P-graphene) and silicene (N-silicene), respectively, present a net magnetic moment.Dai et al. [10, 11] report a net magnetic moment of 1.05 µ B in P-graphene for dopingconcentrations of 1.4 and 3.1 %. The P atom introduces a local curvature in the graphenelattice and they report a metastable non-magnetic state when the P atom is at the plane.Wang et al. [12] obtain similar results at a doping concentration of 2 % with magneticmoment of 1.02 µ B . The authors attribute the origin of the magnetism to the symmetrybreaking of π -electrons in graphene. Furthermore, they show that the spin densitycharge is distributed over the whole lattice. A systematic study of the concentrationeffect in graphene with sp -impurities as Al, Si, P, and S was performed by Denis [13].These results show that for P-graphene the magnetic moment is independent of theconcentration in a range of 0.8-3.1 %. It is important to mention that within the groupof impurities Al, Si, P and S, only the P impurity induces magnetism in graphene.For doped silicene, the effects of the chemical functionalization with B, N, Al and Pimpurities for a doping concentration of 3.1 % were analyzed by Sivek et al. [14], usingthe density functional theory with the local density approximation for the exchange-correlation potential. Their results showed that the nitrogen substitutional impurityinduces a magnetic moment of 0.9 µ B and the system is vibrationally stable. Theauthors also discuss that the contribution of s - and p - states of N gives rise to metallicbands in silicene. Hence, in a similar way that for P-graphene, Sivek et al. [14] foundthat within the group of impurities B, N, Al, and P, only the N impurity inducesmagnetism in silicene. Thus, first-principles calculations predict that P-graphene andN-silicene belong to the group of sp -magnetic systems. However, the details of themechanism that gives rise the magnetism in P-graphene and N-silicene still needs to be ffects of biaxial strain on the impurity-induced magnetism ... sp -magnetism in P-graphene or N-silicene is still lacking. Thus,a systematic and comparative theoretical study of the magnetic properties in P-grapheneand N-silicene is necessary in order to understand the effect of strain for different dopantconcentration values.Therefore, the aim of this work is to contribute to the understanding of the impurity-induced magnetism in P-graphene and N-silicene and how the magnetic moment of thesesystems could be modulated under a positive isotropic deformation (biaxial strain).Here, we present results of first-principles calculations based on the Density FunctionalTheory (DFT) for the structural, electronic and magnetic properties of P-graphene andN-silicene for three different doping concentrations (3.1, 2.0 and 1.4%) in a moderaterange of deformations (0 − sp -magnetism in P-graphene andN-silicene. Secondly, the results for the evolution of the magnetic moment as a functionof the biaxial strain are presented. We find that biaxial strain gradually destabilizes themagnetic state inducing a magnetic to paramagnetic phase transition in these systems.The paper is structured as follows: In Sec. II we describe the computational details ofour calculations. In Sec. IIIA structural results are presented and the magnetic andelectronic properties are discussed in Sec. IIIB. Finally, in Sec. IV we report our mainconclusions. Particularly useful is Appendix A presenting a detailed description of thenarrow impurity band model for ferromagnetism used throughout the paper.
2. Computational details
The DFT calculations were performed within the framework of the plane-wavespseudopotential approach, as implemented in the QUANTUM-ESPRESSO code [18].Core electrons were replaced by ultrasoft pseudopotentials taken from the
PSlibrary s and p electrons, we expect that this functional provides agood description of the magnetic behavior, compared with cases with highly localized d or f orbitals, where usually tends to give a poor description due to an effect ofdelocalization of the wavefunctions. We simulated P (N) substitutional impurities byreplacing one C (Si) atom from the graphene (silicene) pristine lattice. We considered ffects of biaxial strain on the impurity-induced magnetism ... ×
4, 5 × × c ), respectively. For each concentration wecalculate the corresponding ground-state lattice constant a by a direct minimization ofthe electronic total energy. Biaxial tensile strain ε was applied by increasing the latticeconstant as a = (1 + ε ) a . During all the structural calculations the atomic positionswere relaxed until the internal forces were less than 0.01 eV/˚A. In order to simulate anisolated layer, we left at least 15 ˚A of vacuum space between periodic images. Specialattention was paid to the sampling of the Brillouin zone. For structural calculations, weused a 9 × × k − grid with a smearingof 0.005 and 0.002 Ry for graphene and silicene, respectively. This was needed in orderto properly converge the magnetic moment up to 0.01 µ B .
3. Results and discussion
We begin our discussion by analyzing the energetic stability of the impurity on the host.For that, the binding energy for the unstrained ground state was calculated as E B = E system − ( E D − vacancy + E atom ) , (1)where E system is the total energy of the full relaxed doped system, whereas E D − vacancy isthe total energy for a full relaxed vacancy and E atom is the total energy for the isolatedimpurity-atom, in this case P or N. For P-graphene we found for E B values of − . − .
34 eV and − .
42 eV, for the concentrations of 1.4, 2.0, and 3.1 %, respectively,which are in good agreement with the value of − .
31 eV reported by Paˇsti et al. [23]. Inthe case of N-silicene E B is − .
24 eV, − .
25 eV and − .
22 eV, for the concentrations of1.4, 2.0, and 3.1%, respectively. Thus, the calculated values for the binding energy showthat the substitutional impurity of P(N) is energetically stable in graphene (silicene).With respect to the structural properties, it is important to remember that pristinegraphene is a flat crystal whereas pristine silicene has a buckled structure. Thus, becauseof the different structural character and the contrasting size of the impurity atom ineach case, a different structural behavior for P-graphene and N-silicene is anticipated.The lattice structure for P-graphene in the ground (unstrained) state shows significantdistortions owing to the presence of substitutional P, which has an atomic size largerthan the carbon atom. The impurity causes a distorted tetrahedral-type structure witha bond angle of θ CPC =100 ◦ and P-C bond length of 1.76 ˚A which is larger than theC-C bond in pristine graphene (1.42 ˚A). For N-silicene, because N atom is smaller insize than Si, as substitutional impurity it has a N-Si bond length smaller than Si-Si,given by 1.83 ˚A and 2.27 ˚A, respectively. The difference in distances causes a bondangle θ SiNSi =119.4 ◦ , which is close to the typical value of sp hybridization whereasfor pristine silicene the θ SiSiSi =116.2 ◦ is usually attributed to sp - sp hybridization.The main structural difference is localized around the impurity, for P-graphene the P ffects of biaxial strain on the impurity-induced magnetism ... (a) (b)(c) (d) P−Graphene B ond l eng t h P − C ( Å ) N−Silicene B ond l eng t h N − S i ( Å ) B ond ang l e C − P − C ( deg ) e (%) 100104108112116120 0 1 2 3 4 5 6 7 8 9 10 B ond ang l e S i − N − S i ( deg ) e (%) B L BA Figure 1.
Bond length and bond angle for P-graphene(a, c) and N-silicene (b, d) asa function of deformation ε , at the three studied concentrations (1.4, 2.0, and 3.1%). atom is out of the plane respect to the flat graphitic lattice whereas for N-silicene the Natom is in the same plane with their three first nearest neighbors of a deformed bluckedstructure. That is the reason for the different evolution of the structural parameters asa function of the biaxial strain for each system as it can be seen in Fig. 1.For both systems, the evolution of the bond length (BL) and the bond angle (BA)is almost independent of the concentration, but under biaxial deformations ( ε ), anappreciable difference occurs between P-graphene and N-silicene as we discuss below.In the case of P-graphene we can observe two regimes (see Fig. 1a), for 0 to 5% ofstrain the BL is almost constant while P is completely out of the plane, followed by areduction and then a linear increment from 5 to 10%. From 0 to 5% the BA increaseswith deformation and from 5 to 10% the angle becomes constant θ = 120 ◦ (see Fig. 1c),indicating that the system becomes flat as pristine graphene for deformations largerthan 5%. For N-silicene, in Fig. 1b we can see that the BL has a monotonic increasewith strain, although around of 9% it seems to reach a maximum. With respect to theBA in N-silicene under strain, interestingly, we can see in Fig. 1d) that biaxial straindoes not affect the θ SiNSi , remaining almost constant in the whole range of deformations0-10%. This means that for the N atom is more favorable to be almost aligned to theirfirst nearest neighbors. This behavior is an effect of the strong hybridization betweenthe N and Si orbitals, avoiding a buckled configuration around N. ffects of biaxial strain on the impurity-induced magnetism ... In carbon-based magnetism, it has been proposed that the origin of sp -magnetism canbe explained by the Stoner theory adapted to the case of a narrow impurity band[24], where a paramagnetic system becomes unstable with respect to the ferromagneticcase when a high density of states exists at the Fermi level. The Stoner criterion forthe existence of ferromagnetism is IN ( E F ) >
1, where I is the Stoner parameter and N ( E F ) is the density of states (DOS) of the paramagnetic case at Fermi level. It hasbeen suggested that this criterion is useful to describe the emergence of magnetismin systems with sp -electrons, particularly in graphene nanostructures with vacancies orwith adsorbed hydrogen atoms [2, 4, 25, 26]. The cases of substituted graphene and evensilicene are not the exception. However, for these substitutional cases, the role of thespecific impurity is important, otherwise, other sp -impurities could induce magnetismin graphene and silicene. An alternative to the Stoner model for the ferromagnetismoccurring in a narrow impurity band can be found in Appendix A, which is based onthe proposal of Edwards and Katsnelson [24] and the work of Gruber et al. [27]In Fig. 2 the paramagnetic electronic band structure of P-graphene (left) and theN-silicene (right) are shown at the unstrained state for the three different values ofconcentration ( c = 1.4, 2.0 and 3.1%). As it is well known, the band structure ofgraphene and silicene shows a linear dispersion around the Fermi level in the K point,the so-called Dirac cones. From Fig. 2, it is observed that for concentrations of 2.0 and3.1% this linearity dissapears with doping, and in fact we find a band gap opening anda narrow impurity band at the Fermi level (band in red color). For the concentration of1.4% the band structure shows differences with respect to the other two concentrationsdue to band folding effect by the use of the supercells [28]. For this concentration,the K point of the unit cell is folded to the Γ point of the supercell as Figs. 2a and2b show. Besides, the band gap opening does not occur, but the impurity band ispresent. It is interesting to note that the dispersion of the impurity band is larger inN-silicene with respect to P-graphene. Nevertheless, in both systems the dispersion ofthe impurity band increases with the concentration. A further characterization of theimpurity band was done and we find that for both systems the Fermi level is at half-filling and that the maximum occupancy is 1.0. Additionally, the value of the impuritybandwidth ( W imp ) for the concentration of 2.0% in P-graphene and N-silicene is 65and 84 meV, respectively. This very narrow impurity-band is able to produce a sharppeak in the Density of States (DOS) at Fermi level. Hence, an electronic instabilityin the paramagnetic state is expected from the high value of N ( E F ), in particular aband-splitting induced by spin polarization, generating a net magnetic moment in thesystem.In order to have more physical insight on this instability, we have performed first-principles calculations for the total-energy as a function of the magnetic moment inthe supercell using the Fixed Spin Moment (FSM) method [29]. In Fig. 3, we showthe calculated values (symbols) of the total-energy as a function of the spin magnetic ffects of biaxial strain on the impurity-induced magnetism ... −1.2−0.60.00.61.2 G M K G (a) . % (b)(c) (d)(e) (f) E ne r g y ( e V ) P−Graphene −1.2−0.60.00.61.2 G M K G . % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G c = . % E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G E ne r g y ( e V ) N−Silicene −0.6−0.30.00.30.6 G M K G E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G E ne r g y ( e V ) Figure 2.
Electronic band structure for P-graphene and N-silicene in the paramagneticstate for the three studied concentrations. The origin of the energy scale has been setat the Fermi level ( E F ). The narrow impurity band is emphasized in red. moment, E ( M ), for P-graphene and N-silicene for c = 2 . E ( M ) indicates that the moststable state is the full polarized state with a spin magnetic moment of 1.0 µ B /cell,corresponding to strong ferromagnetism [30, 31].From the model for ferromagnetism in a narrow impurity band described inAppendix A, we have that in the case of half-filling, maximum occupancy N = 1,and assuming the rigid band splitting, the total energy as a function of the magneticmoment is thus given by E ( M ) = E + M W imp − U ) , (2)where E corresponds to the reference energy, for instance the total energy of theparamagnetic state, and U to the Coulomb type interaction. Thus, the condition forspontaneous magnetization is U/W imp > W imp whereobtained previously from the paramagnetic band structure, 65 meV for P-graphene and84 meV for N-silicene at c = 2 . U , we have fitted(2) to the calculated values of E ( M ), solid line in Fig. 3, resulting in U = 250 meVfor P-graphene and U = 172 meV for N-silicene. Therefore, U/W imp is 3.8 and 2.0 forP-graphene and N-silicene, respectively, fulfilling the Stoner-type criterion
U/W imp > ffects of biaxial strain on the impurity-induced magnetism ... −50−40−30−20−10 0 100.0 0.2 0.4 0.6 0.8 1.0 E ne r g y ( m e V ) Spin magnetic moment ( m B /cell) P−GrapheneModelN−SiliceneModel
Figure 3.
Energy as a function of the magnetic moment for P-graphene and N-siliceneat c = 2%, calculated using the fixed-spin moment method (symbols). The solid linecorresponds to the fit of E ( M ) given by eq. (2) to the calculated values (symbols). moment using a Stoner-type model, it is clear that the value of U in these systemsis small. Consequently, the spontaneous polarization in P-graphene and N-silicene isdriven by the very small value of W imp .The electronic band structure and DOS for the spin-polarized state for P-grapheneand N-silicene are presented in Figures 4 and 5, respectively. The spin-up states areindicated in red color and the spin-down states in blue color. For reference, also wehave included the paramagnetic bands in grey color. The largest spin-splitting is for theimpurity band, but the host bands also show an important spin-splitting, indicating thatthe spin-polarization is not only localized at the impurity atom. It is also interesting tonote that the impurity bandwidth for spin-down is larger than for the spin-up, indicatingthat the spin-polarization of the impurity band is not a rigid-band splitting. Theimpurity band splitting is a result of the exchange interaction between the electronsin the narrow impurity band. Thus, in order to characterize the exchange interaction,we have calculated the spin-splitting of the impurity band (∆ s ) by taking a weightedaverage in the first Brillouin zone. The values for ∆ s in P-graphene and N-silicene are267 meV and 137 meV, respectively.The analysis of the density of states revealed that the high DOS at Fermi levelin Figures 4 and 5, which gives rise to magnetism comes from the hybridization ofthe orbitals at the impurity atom and the orbitals at the atoms which belong to thesublattice adjacent to the impurity. For P-graphene the localized state at the Fermilevel have contributions of s and p z orbitals from P, and from p z orbital of C atoms.In the case of N-silicene, the main contributions to the peak at the DOS that causesthe instability come from the s and p z orbitals from N, s and p z from Si, and a smallcontribution p x and p y , which makes sense due to the buckled structure of silicene. ffects of biaxial strain on the impurity-induced magnetism ... −1.2−0.60.00.61.2 G M K G . % E ne r g y ( e V ) P−Graphene −1.2−0.60.00.61.2 G M K G . % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G c = . % E ne r g y ( e V ) DOS spin−upspin−dwPM
DOS DOS
Figure 4.
Spin-polarized electronic band structure for P-graphene at the three studiedconcentrations. The red and blue lines correspond to the spin-up and spin-down,respectively. For reference, the paramagnetic case is included in grey color.
As a first approximation to the analysis of the local magnetic moments distribution,we analyzed the L¨owdin charges for each atom. On the top of Fig. 6 we have showniso-surfaces of the spin electronic density charge ( ρ ↑ − ρ ↓ ) whereas the local magneticmoment distributions are plotted at the bottom. The red color corresponds to majorityspin density and blue (green) color to minority spin density for P-graphene (N-silicene).For the bottom of Fig. 6, the color represents the contribution to the local magneticmoment from σ ( s + p x + p y ) orbitals in red and π ( p z ) orbitals in blue. The spin densitycharge plots are presented in an extended supercell which involves a supercell with itsfirst six neighbors. We can observe a spin distribution spread on all of the lattice,where the impurity and their first atomic three neighbors have a majority spin density.Interestingly, each kind of spin density is majority on each sublattice. The dashedblack line is used to define a radial distance with respect to the impurity. Insidethis circumference, we plot the local magnetic moments as a function of the radialdistance and for each distance, we have performed a sum according to the number ofthe atoms at that distance. It is clear that the local magnetic moment distribution isnot homogeneous on the lattice, but the pattern is the same in both P-graphene andN-silicene. Interestingly, only a small fraction of the total magnetic moment is locatedat the impurity atom. ffects of biaxial strain on the impurity-induced magnetism ... −0.6−0.30.00.30.6 G M K G . % E ne r g y ( e V ) N−Silicene −0.6−0.30.00.30.6 G M K G . % E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G c = . % E ne r g y ( e V ) DOS spin−upspin−dwPM
DOS DOS
Figure 5.
Spin-polarized electronic band structure for N-silicene at the three studiedconcentrations. The red and blue lines correspond to the spin-up and spin-down,respectively. For reference, the paramagnetic case is included in grey color.
Thus, from the present analysis we can now answer the question; why P-grapheneand N-silicene are magnetic? According to our observations, there are two fundamentalreasons: i ) the impurity have an electronic configuration with one more electron thanthe host material, and ii ) the impurity induces a very narrow band at Fermi level.After the analysis and characterization of the electronic structure and the spin-magnetic moment for unstrained P-graphene and N-silicene, in Fig. 7, we show theevolution of the electronic structure with strain for P-graphene and N-silicene at c = 2 .
0% in the paramagnetic state. The band structure along the high symmetrypaths of the first Brillouin zone in P-graphene shows a narrow band around the Fermilevel, except in the K point where a parabolic dispersion is observed. As the strainincreases, this effect is more pronounced. The electronic behavior in N-silicene underisotropic deformation shows a different evolution. The nearly flat impurity band aroundthe Fermi level along the high symmetry paths with a small parabolic dispersion aroundthe K point observed for ε = 0%, is strongly distorted under biaxial strain. As thedeformation is increased the flat character disappears and shows parabolic dispersion ffects of biaxial strain on the impurity-induced magnetism ... −0.20.00.20.40.6 0 2 4 6 8 (a) (b)(c) (d) C ( , ) C(2,6) C ( , ) C ( , ) C(5,3)C(6−7,9) C ( , ) C ( , ) P Lo c a l m agne t i c m o m en t ( m B ) r (Å) ps P−graphene −0.20.00.20.40.6 0 3 6 9 12 S i ( , ) S i ( , ) S i ( , ) Si(5,3)Si(6−7,9) S i ( , ) S i ( , ) N Si(2,6) r (Å) ps N−silicene
Figure 6.
Spin density charge (top) and the π and σ contributions to the localmagnetic moments as a function of radial distance from the impurity atom (bottom)for P-graphene and N-silicene at c = 2%. The two indexes at each C or Si atomposition represents the order of nearest neighbor with respect to the impurity and thenumber of nearest neighbors. around the K and Γ points. Thus, we find that in both systems the dispersion of theimpurity band increases with the biaxial strain.In Fig. 8 we show the behavior of W imp as a function of the biaxial strainfor P-graphene and N-silicene for the three studied concentrations, as obtained fromthe paramagnetic band structure. For P-graphene, we can see that W imp follows alinear behavior with the applied strain in the range 0-5% of deformations for the threeconcentrations, with a large step at ε ∼ W imp with the biaxial strain is strongly dependent on the concentration. For instance, for c = 3 .
1% the impurity bandwidth increases monotonically with the deformation, whilefor c = 1 .
4% the W imp shows a minimum around 4% of deformation. According tothe narrow impurity band model discussed above, these results anticipate the loss ofmagnetism in both systems, but the behavior of the magnetic moment with the biaxialstrain and the concentration dependence for each system will be different.In Fig. 9, the spin-polarized electronic band structure and DOS for P-graphene andN-silicene at c = 2 .
0% under biaxial strain are presented. In both systems we find thatthe impurity band dispersion increases with the strain for both spin channels, inducingoverlapping of the spin-up and spin-down bands, as can be seen in the DOS. At thesame time, the spin-splitting of the impurity band decreases. For this concentration, wecan see that in both systems the spin-splitting vanishes at ε = 6%, indicating that themagnetism has been lost.Interestingly, for P-graphene at ε = 6% we recover a linear dispersion aroundK point as in graphene pristine (Fig. 9d), however, the Fermi level is located above ffects of biaxial strain on the impurity-induced magnetism ... −1.2−0.60.00.61.2 G M K G (a) ε = % E ne r g y ( e V ) P−Graphene −1.2−0.60.00.61.2 G M K G (b) ε = % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G (c) ε = % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G (d) ε = % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G (e) ε = % E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G (f) E ne r g y ( e V ) N−Silicene −0.6−0.30.00.30.6 G M K G (g) E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G (g) E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G (h) E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G (i) E ne r g y ( e V ) Figure 7.
Electronic band structure for P-graphene and N-silicene at c = 2 .
0% in theparamagnetic state as a function of the strain. The origin of the energy scale has beenset at the Fermi level ( E F ). The impurity band is emphasized in red color. the Dirac point corresponding to electron doped graphene. For this deformation, asmentioned before, the system recovers its flatness, showing the close relation betweenstructural and electronic properties. The electronic structure of N-silicene underisotropic deformation (Fig. 2e-h) shows a different evolution with respect to P-graphene.In this case, the impurity band at ε = 0% shows a parabolic dispersion around theK point and a nearly flat character along the high symmetry paths. However, withdeformation the evolution of the impurity band leads to a parabolic nature around Kand Γ points, with hole character at K and electron character at Γ. We notice that forN-silicene under strain, in contrast to P-graphene, we do not recover the Dirac cones(see Fig. 9h). Thus, the electronic character of N-silicene under biaxial strain in thenon-magnetic state will be of a normal metal.The evolution of the magnetic moment as a function of the strain for eachconcentration is shown in Fig. 10. For the unstrained case both systems are magnetic ffects of biaxial strain on the impurity-induced magnetism ... W i m p ( e V ) e (%) 1.4 %2.0 %3.1 %0.00.10.20.30.40.50.60.70.8 0 1 2 3 4 5 6 7 8 9 10N−silicene W i m p ( e V ) e (%) Figure 8.
Evolution of the impurity bandwidth ( W imp ) as a function of the strain ( ε )for P-graphene and N-silicene at the three studied concentrations (1.4, 2.0, and 3.1%)in the paramagnetic state. with a net magnetic moment of 1.0 µ B / cell regardless of concentration. Under strain, themagnetic moment changes from 1.0 to 0 µ B / cell, indicating that a magnetic transitionappears when a biaxial strain is applied. Although P-graphene presents almost similartransition with the doped concentration, N-silicene has a strong dependence on thedoping concentration. Nevertheless, a common feature is that in both systems we havea range of deformations starting from ε = 0% where the magnetic moment remainconstant ( M = 1 . < M < .
0) and a third range where the system is non-magnetic ( M = 0),after reaching the critical deformation where M → U does not change with strain. This consideration comes froma further analysis of the spin-splitting for the impurity band and the magnetic momentas a function of the strain, which shows a practically constant behaviour of U at differentdeformations. In this case both systems are at half-filling and the maximum occupancyis N = 1, therefore the full polarized state remains as long as W imp < U . This conditionis fulfilled in the first range where the spin magnetic moment is 1.0 µ B /cell with thesystem in the strong ferromagnetic state. In Fig. 8, we can see that for P-graphenethe impurity bandwidth W imp increases linearly with strain independently of the dopingconcentration, with practically the same slop up to ε ∼ ffects of biaxial strain on the impurity-induced magnetism ... −1.2−0.60.00.61.2 G M K G (a) ε = % E ne r g y ( e V ) P−Graphene −1.2−0.60.00.61.2 G M K G (b) ε = % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G (c) ε = % E ne r g y ( e V ) −1.2−0.60.00.61.2 G M K G (d) ε = % E ne r g y ( e V ) DOS spin−upspin−dw
DOSDOSDOS −0.6−0.30.00.30.6 G M K G (e) E ne r g y ( e V ) N−Silicene −0.6−0.30.00.30.6 G M K G (f) E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G (g) E ne r g y ( e V ) −0.6−0.30.00.30.6 G M K G (h) E ne r g y ( e V ) DOSDOSDOSDOS
Figure 9.
Spin-polarized electronic band structure for P-graphene and N-silicene at c = 2 .
0% as a function of the strain. The origin of the energy scale has been set at theFermi level ( E F ). where the system is in the weak ferromagnetic state, characterized by an unsaturatedspin-up band and a magnetic moment M < . µ B /cell. In the case of N-silicene, thepicture is the same, however the details of the transition are strongly dependent on thedoping concentration as a result of the different behavior of W imp with strain (see Fig.8).For the analysis of N-silicene, we will start with the highest concentration c = 3 . W imp increases almost linearly with the biaxial strain. Thus, in this particularcase the transition of the magnetic moment is similar to P-graphene. For N-silicene at c = 2 .
0% the value of W imp remain almost constant up to ε ∼ c = 1 .
4% we can see in Fig. 8 that W imp decreasesin the range of 0-4% for the strain, reaching a minimum value at ε = 4%, and thenincrease linearly with a very small slope. In this way, the strong ferromagnetic statein N-silicene at c = 1 .
4% extends up to ε = 4% and the wide transition region up to ε = 8 . ffects of biaxial strain on the impurity-induced magnetism ... S p i n m agne t i c m o m en t ( m B / c e ll ) P−Graphene
N−Silicene e (%) Figure 10.
Spin magnetic moment as a function of biaxial strain for P-graphene andN-silicene at the three studied concentrations (1.4, 2.0, and 3.1%). the case of N-silicene with the doping concentration, a common feature was recognizedand characterized, emerging the picture depicted in Fig. 11. So, we have that for lowstrain values the system remains in the strong ferromagnetic (SF) state, followed by atransition region corresponding to a weak ferromagnetic (WF) state with a impurityband partially polarized, and finally the non-magnetic region after reaching the criticalstrain where the ground state corresponds to a paramagnetic (PM) state. This picture,was confirmed performing fixed spin moment calculations of the total energy as afunction of the spin magnetic moment, for three values of biaxial deformations 0, 4.0 and7.0% corresponding to SF, WF, and PM states, respectively. The results are presentedin the inset of Fig. 11. The calculated plots for E ( M ) are in close agreement with thegeneral model of itinerant electrons in a narrow band [31] .
4. Conclusions
For P-graphene and N-silicene the origin of the magnetism is result of a partially filledvery narrow band at Fermi level induced by the impurity. This narrow impurity bandcauses an electronic instability which favours a magnetic state, which is characterizedby magnetic moment of 1.0 µ B / cell corresponding to a full-polarized impurity band.We found that a Stoner-type model describing the electronic instability of the narrowimpurity band accounts for the origin of sp -magnetism in P-graphene and N-silicene.The evolution of the spin magnetic moment as a function of the biaxial strain is mainlygoverned by the behavior of the impurity bandwidth with the applied strain. Thus, thebiaxial strain evolves the systems from a strong ferromagnetic state to a paramagnetic ffects of biaxial strain on the impurity-induced magnetism ... P−Graphene
SF WF PM S p i n m agne t i c m o m en t ( m B / c e ll ) e (%) −40−2002040 E ( m e V ) M ( m B /cell) SF (0%)WF (4%)PM (7%)
Figure 11.
Evolution of the spin-magnetic moment as a function of the strain forP-graphene at c = 2% indicating the regions for strong-ferromagnetism (SF), weak-ferromagnetism (WF) and paramagnetic (PM) behaviors. In the inset the calculated E ( M ) using the fixed-spin moment method (symbols), for deformations of 0.0, 4.0,and 7.0%, corresponding to SF, WF, and PM states, respectively. state, with a transition region corresponding to a weak ferromagnetic state characterizedby a partially polarized impurity band. Furthermore, it has been demonstrated thatwith strain it is possible to modulate the spin magnetic moment and induce a magneticquantum phase transition. Consequently, the control and manipulation of the magneticproperties in these two-dimensional magnetic systems are technologically attractive forthe potential applications in spintronics and spin-based quantum computation systems,using strain engineering as an effective way to modulate the magnetic properties in2D-materials.
5. Acknowledgments
The authors thankfully acknowledge the computer resources, technical expertise andsupport provided by the Laboratorio Nacional de Superc´omputo del Sureste de M´exico.J.H.T. acknowledges a student fellowship from the Consejo Nacional de Ciencia yTecnolog´ıa (Conacyt, M´exico) and VIEP-BUAP. M.E.C.Q. gratefully acknowledges aposdoctoral fellowship from Conacyt-M´exico. This research was supported by Conacyt-M´exico under grant No. 288344.
Appendix A. Model of narrow impurity band ferromagnetism
According to the paramagnetic band structures at zero deformation, a narrow impurityband at Fermi level is present in P-graphene and N-silicene. Thus, the origin ofmagnetism could be attributed to this band because the exchange interaction inducesa spin-splitting. Edwards and Katsnelson [24] restated the conventional Stoner criteria ffects of biaxial strain on the impurity-induced magnetism ... IN ( E F ) > N ( E F ) = n imp /W imp , being n imp the impurity density is and W imp the width of the impurity rectangular band. On the other hand, Gruber et al.[27] using a mean-field version of the Hubbard like interaction describe the band energyof the electronic states for the impurity band for each spin channel, as follows: E = E ↑ band + E ↓ band + U n ↑ n ↓ , (A.1)where the terms E band within the tight-binding approximation are E ↑ band = Z E F E ↑ min ( E − E ↑ c ) N ↑ ( E ) dE , (A.2) E ↓ band = Z E F E ↓ min ( E − E ↓ c ) N ↓ ( E ) dE , (A.3)here E ↑ c and E ↓ c are the center of the band for each spin channel which suffers an energyshift in the spin-polarized case with respect to the paramagnetic case, in order to havea non-zero magnetic moment. The E F , E ↑ min and E ↓ min are the Fermi level and the lowerlimits of each band. The third term in (A.1), corresponds to a Coulomb type interaction U suppressing double occupancy and n ↑ ( n ↓ ) is the number of the spin-up (spin-down)electrons in the band. The number of electrons in each spin band can be obtained from n ↑ = Z E F E ↑ min N ↑ ( E ) dE , (A.4) n ↓ = Z E F E ↓ min N ↓ ( E ) dE . (A.5)The total number of electrons Z is given by Z = n ↑ + n ↓ , (A.6)and the magnetic moment MM = n ↑ − n ↓ . (A.7)Now, according to the Friedel’s model [30], we assume a DOS with rectangularshape for each spin channel. In Fig. A1 we show the paramagnetic and spin-polarizedcase according to this model. Thus, the density of states N ( E ) for each spin channel isconstant and is given by N ↑ ( E ) = N W ↑ , (A.8) N ↓ ( E ) = N W ↓ , (A.9)where N is maximum occupancy of the band and W ↑ ( W ↓ ) is the bandwidth for eachspin-up (spin-down). ffects of biaxial strain on the impurity-induced magnetism ... D en s i t y o f s t a t e s D en s i t y o f s t a t e s N › (E)N fl (E)N › (E)N fl (E) EE PMspin−upspin−down E F E F M = 0M „ WWW › W fl Figure A1.
Density of states of the impurity band for the paramagnetic (top) andspin polarized (bottom) states in the rectangular model.
With (A.2) and (A.8) we rewrite the band energy for the spin-up band E ↑ band = Z E F E ↑ min ( E − E ↑ c ) N W ↑ dE , (A.10) E ↑ band = N W ↑ " ( E − E ↑ c ) E F E ↑ min . (A.11)The lower limit corresponds to E ↑ min = E ↑ c − ( W ↑ /
2) and the Fermi energy isobtained from (A.4) using (A.8) n ↑ = (cid:20) N W ↑ E (cid:21) E F E ↑ min = N W ↑ " E F − E ↑ c − W ↑ ! , (A.12) E F = n ↑ W ↑ N + E ↑ c − W ↑ . (A.13)With E ↑ min = E ↑ c − ( W ↑ /