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Electronic and magnetic properties of SnS2 monolayer doped with 4d transition metals
Accepted Manuscript Electronic and magnetic properties of SnS2 monolayer doped with 4d transition metals Wen-Zhi Xiao, Gang Xiao, Qing-Yan Rong, Qiao Chen, Ling-Ling Wang PII: DOI: Reference:
S0304-8853(17)30391-8 http://dx.doi.org/10.1016/j.jmmm.2017.04.090 MAGMA 62694
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
8 February 2017 30 April 2017 30 April 2017
Please cite this article as: W-Z. Xiao, G. Xiao, Q-Y. Rong, Q. Chen, L-L. Wang, Electronic and magnetic properties of SnS2 monolayer doped with 4d transition metals, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.04.090
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Electronic and magnetic properties of SnS2 monolayer doped with 4d transition metals Wen-Zhi Xiao,a∗∗ Gang Xiao,a,b Qing-Yan Rong,a Qiao Chen,a Ling-Ling Wang b
Abstract: We investigate the electronic structures and magnetic properties of SnS2 monolayers substitutionally doped with 4-d transition-metal through systematic first principles calculations. The doped complexes exhibit interesting electronic and magnetic behaviors, depending on the interplay between crystal field splitting, Hund’s rule, and 4d levels. The system doped with Y is nonmagnetic metal. Both the Zr- and Pd-doped systems remain nonmagnetic semiconductors. Doping results in half-metallic states for Nb-, Ru-, Rh-, Ag, and Cd doped cases, and magnetic semiconductors for systems with Mo and Tc dopants. In particular, the Nb- and Mo-doped systems display long-ranged ferromagnetic ordering with Curie temperature above room temperature, which are primarily attributable to the double-exchange mechanism, and the p-d/p-p hybridizations, respectively. Moreover, The Mo-doped system has excellent energetic stability and flexible mechanical stability, and also possesses remarkable dynamic and thermal (500 K) stability. Our studies demonstrate that Nb- and Mo-doped SnS2 monolayers are promising candidates for preparing 2D diluted magnetic semiconductors, and hence will be a helpful clue for experimentalists.
School of Science, Hunan Institute of Engineering, Xiangtan 411104, China. E-mail: [email protected] b School of Physics and Electronics, Hunan University, Changsha 410082, China. a
∗
I. Introduction The emergence of graphene[1] has trigged tremendous great attentions in development of a variety of single layer two-dimensional (2D) materials due to their peculiar physics and potential applications in next generation of nanoscale devices[2,3]. Soon after, monolayer MoS2 has been fabricated successfully by exfoliation technique.[4,5] In contrast to the zero band gap of pristine graphene, a single-layer MoS2 sheet is a direct band gap semiconductor[6,7] which makes it a promising material for potential semiconductor devices, such as digital circuits,[8] light-emitting diodes,[9] and diluted magnetic semiconductors[10,11]. Earlier theoretical studies[6,7] predicted a lot of 2D layered materials such as metal chalcogenides, transition metal oxides. Transition metal dichalcogenides consist of hexagonal layers of metal atoms (M) sandwiched between two layers of chalcogen atoms (X) with a MX2 stoichiometry. Among them, the monolayer tin-disulfide (SnS2) also has renewed interest due to its exceptional optical and electronic properties. The freestanding SnS2 single-layers were first time obtained through a scalable liquid exfoliation strategy, and were found to be suitable for visible-light water splitting.[12] Shortly after, based monolayer SnS2 materials Song et al.[13] successfully realized top-gated field-effect transistors and related logic gates. Every recently, monolayer, and bilayer SnS2 have been successfully prepared using mechanical exfoliation.[14] Stirred by these exciting findings, considerable experimental and theoretical efforts have been devoted to controlling the electronic structures and magnetic properties in such unusual materials for a wide range of applications. To control magnetism in 2D crystals has remained a goal in the diluted magnetic semiconductors, [15]
which gives rise to potential applications in spintronics.[16] The field of spintronics
requires an active ferromagnetic ordering temperature, which should ideally be well above room temperature to enable practical applications. It has already been shown that the electronic and magnetic properties of monolayer SnS2 can be modulated by external train, intrinsic defects,[17,18] and doping with 3d transition-metal (TM) atom.[19,20] Theoretical prediction[21] and experimental observation[22] shown that 1
substitution doping of Mg may be the origin of the room temperature ferromagnetism in SnS2 nanoflowers. Although the exact mechanism of Mg-doping for such resultant magnetic order has not been very clear yet, it might be instrumental toward the realization of d 0 ferromagnetism in 2D SnS2 materials. However, as far as we know, there is no report on searching for room temperature ferromagnetism in 2D SnS2 monolayers doped with 4d TM atoms, which is the focus of this work. In this work, we provide a systematic study of the stability, electronic and magnetic properties of 2D doped SnS2 by using the density functional theory (DFT). To better understand and control the magnetic properties of the monolayer SnS2 doped 4d TM atoms (from Y to Cd), the origin of magnetism is systematically analyzed. Our results indicate that the desired long-range ferromagnetism with above room temperature can be achieved by doping with Mo atoms.
II. Computational details We calculate the atomic structure and electronic and magnetic properties within the framework of spin-polarized DFT simulation package (VASP) software
[23]
as implemented in the Vienna ab initio
[24]
. The projector augmented-wave method is
used.[25] A plane-wave basis set with kinetic energy cutoff of 480 eV is used for all the calculations. The exchange-correlation functional is treated using the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) format.
[26]
For
all structural optimization, the convergence criteria for total energy and Hellman–Feynman force are set to 10-4 eV and 0.01 eV/Å, respectively. The vacuum spacing between adjacent monolayers is kept larger than 13 Å. In order to study magnetism induced by substitutional doping of 4d TM atoms, we construct a 4×4×1 SnS2 supercell with one dopant. The Γ-centered 5×5×1 k-mesh scheme and 70 k points along the high symmetry directions are used for electronic density of states (DOS) and band structures, respectively. To determine the stability of these doped sheets, we systemically calculate the binding energy and strain-stress relation, and carry out molecular dynamics simulation. The binding energy, Eb, is calculated by the equation: Eb = Ev + µM - Ed,[10] where, Ev 2
is the total energy of the SnS2 monolayer with one Sn vacnacy, µM is the energy of one M atom in its bulk phase, and Ed is the total energy of the doped SnS2 monolayer with one M dopant. We calculate the in-plane linear elastic constants for Nb- and Mo-doped cases. Since the strain energy is quadratic dependent on the applied strains, the elastic constants can be obtained by calculating the strain-energy relation. More details about the calculation can be found in our previous work [28,29]
[27]
and elsewhere
. The thermal stability is assessed by performing ab initio molecular dynamics
simulations (MD) using the Nosé thermostat model. In the MD simulations, canonical (NVT) ensembles at 500 K with a time step of 1 femtosecond time are adopted.
III. Results and discussions A. Local structure Now, we firstly examine the local structure around the M dopants. Seen from Table I, the first observation is that the bond lengths of dM-S decrease as the 4d TM atom varies from Y to Rh. Then the bond lengths of dM-S go up for the 4d TM after Rh element, suggesting the bonding becomes weaker. The changing tend of bond length is basically the same as that of ionic radius of 4d TM. Checking carefully the bond lengths and bond angles around the 4d TM dopant, we find the local structure centered the 4d TM dopant experiences a breathing relaxation. The 4d TM dopant is six-coordinated with geometry of trigonal bipyramid. For each case, the bond lengths are equal, but the bond angles fall into two kinds, which make the local structure remains the same D3d point-group symmetry as the pristine SnS2 monolayer. It should be noted that the two types of bond angles (∠S-M-S: θ1 and θ2, listed in Table I) are around 90°, and the difference between them is relatively small for all the doped SnS2 monolayers. Therefore, the symmetry of this crystal field approaches an octahedral crystal field with Oh symmetry. Under crystal-field environment, when one of the degenerate 4d bands is partially filled, Jahn-Teller distortions should occur in order to remove the degeneracy and lower the energy. Here, we observe no Jahn-Teller distortion for all the doped systems. 3
B. Electronic structure and magnetic properties Apart from Y, Zr and Pd, all other 4d TM dopants induce spin-polarization and give rise to an integer magnetic moment per dopant, as shown in Table I. Interestingly, the size of magnetic moment changes regularly along the series of 4d TM atoms. From Y to Cd element, the corresponding total magnetic moments are Y: 0, Zr: 0, Nb: 1.0, Mo: 2.0, Tc: 3.0, Ru: 2.0, Rh: 1.0, Pd: 0, Ag: 1.0, and Cd: 2.0 µB. The corresponding spin density distribution in Fig. 1 visualizes the detailed distribution of magnetic moments. In the cases of Nb, Mo, Tc and Ru doped systems, the induced magnetic moment are mainly located on the dopants, attributing to their excess unpaired 4d electrons, whereas the rest resides on the neighboring Sn and S sties. In contrast, the magnetic moment induced by Rh, Ag, and Cd are mainly distributed on the surrounding S atoms rather than the dopants and Sn atoms, and the spin-polarized charge density on S atoms is distinctly characteristic of p orbital. To explore the magnetic stability of the doped systems, we calculate the spin polarization energy, ε=ENSP-ESP, defined as the difference in total energy of the spin-polarized state and non-spin-polarized state. The ε reveals the magnitude of spin-polarization which is also reflected by the spin exchange splitting, as shown in Fig. 2. One can observe the exactly symmetrical DOS of the Y-, Zr-, and Pd-doped systems, implying that they should be nonmagnetic. The Mo-, and Tc doped systems show the largest spin-splitting in the band-gap. The value of ε is much greater than the thermal energy at room temperature, indicating that the local magnetic moments induced by 4d TM are stable. Based on Molecular orbital theory and ligand field theory, we can qualitatively explain the electronic structures and trends in magnetic properties. As motioned above, after structure relaxation, the D3d symmetry is maintained for all dopants. According to group theory, the D3d ligand field splits the neutral Sn vacancy-related defect levels into two non-degenerate a1g (s) and a2u (p z) levels, and a doubly degenerate eu (p x+y) level. The twofold degenerated eu level is lower in energy than the singlet a2u level, but higher than the a 1g level. For a neutral Sn vacancy, anion S dangling bonds (i.e., Sn vacancy-related defect levels) are partially occupied by four unpaired electrons, 4
which will lead to a high-spin state a1↑g eu↑↑ a2↑u with a net moment of 4.0 µB in agreement with theoretical estimation.[19] Upon 4d TM occupying at the Sn site, the dangling bonds of the anion S will couple with the symmetry-adapted 4d and 5s levels of the TM. Under D3d symmetry, the sd orbitals of 4d TM are split into two single a1g [ d z 2 and s, remarked as a1g(d) and a1g(s), respectively] states and fourfold degenerate
eg (dxy, d x2 − y 2 , dxz, and dyz) states which is lower in energy than the two singlet levels. The order of a1g(d) and a1g(s) is mainly dependent on the atomic energy levels of 4d elements. Molecular orbital theory requires that linear combination of atomic orbitals should obey the symmetrical matching principle, the approximate rule of orbital energy, and the maximum overlapping principle. Therefore, the fourfold degenerate eg can couple with the doubly degenerate eu levels of dangling bonds due to the same E symmetry. The a1g(d) and a1g(s) states can couple with the a1g and a2u states because they belonging to the same irreducible A representation. Note that, due to the strong hybridization with SnS2 layer, it is not possible to identify well-defined defect levels with E or A symmetry. The coupling model developed by Raebiger
[30]
and Santos [31]
explained well their calculated results of graphene, and II-VI and III-V compounds modified with 3d TM. We now might as well start the analysis with Nb-doped system. An isolated Nb atom has a 4d45 s1 configuration with one additional valence electron as compared to Sn:5s25p2. Deduced from the electronic band structure in Fig. 3(a) and the PDOS in Fig. 4(a), one can see that the level of a1g(s) is lower than that of a1g(d). The partially filled eu levels of S anion dangling bonds hybridize with two eg(d) electrons, forming a bonding eu state (b-eu). To effectively bond, a eg(d) electron hops to a1g(s) level forming ds hybridization, and then interact with the half-filled a2u and a1g level, resulting in bonding a1g (b-a1g) and a2u (b-a2u) states. In the ds hybridization the hoping lifts energy, but the formation of bond lowers more energy, as a consequence, the system saves energy. As shown in Figs. 3(a) and 4(a), in the spin-down channel, the nonbonding state eg (n-eg) appears about 0.16~0.46 eV, mainly derived from 5
Nb − 4d( xy + xz + yz + x2 ) states. However, in the spin-up channel, the nonbonding state eg crosses the Fermi level, which indicates that the Nb-doped monolayer is essentially half-metallic. Just above the nonbonding state eg, the nonbonding states a1g(d) [na1g(d)] remain unoccupied in both channels. Four out of five electrons of Nb bond to S atoms and fill up b-a1g, b-a2u and b-eu states, leaving one electron fill into nonbonding e state in the spin-up channel. As a result, the Nb-doped system gives rise to magnetic moments of 1.0 µB. As we know that the outer electrons of Mo adopt a 4d45s2 configuration, the valence electrons interact with the dangling bonds in the same manner as Nb does. Two eg(d) electrons of Mo occupy nonbonding fourfold degenerate eg states in the spin-up channel. Consequently, the n-eg level does not cross the Fermi level, so the Mo-doped system is a magnetic semiconductor with a total magnetic moment of 2.0 µB. For Nb-doped SnS2, the exchange splitting of the eg orbitals, ∆ex ≈ 0.30 eV, is
smaller than the crystal field splitting, ∆cf ≈ 0.48 eV [see Fig. (i)]. The spin exchange splitting of Mo-doped case is larger than that of Nb-doped case. We obtain ∆ex ≈ 0.70 eV and ∆cf ≈ 0.3 eV for Mo-doped case. The isolated Y and Zr atoms possess the 4d15s2 and 4d25s2configurations, respectively. For Y, there is lack of one electron for coupling to eu levels, and hence a nonbonding eu state is formed. The nonbonding eu state accommodates one electron which is delocalized in the whole crystal, so that the magnetic moment is quenched. As for the Zr-doped monolayer, all the sd electrons join by bonding, so that the doped system remains the semiconducting properties with a indirect band-gad of 1.22 eV. As shown in Figs. S1 and S2, one can see that the nonbonding and anti-bonding states of Zr-4d occupy the lower levels than that of Y, which attribute to the fact that the 4d orbital energies of the free atoms become deeper as the atomic number increases along the 4d row. For Tc-doped case, the same mechanism works as it does for Mo-doped case. Compared to Mo, the Tc atom has one extra valence electron with a 4d55s2 configuration. Therefore, one electron occupies the nonbonding n-e1g(d) state and the 6
other two fill the nonbonding n-eg state, and hence the developing magnetic moment reaches a peak value of 3.0 µB in the 4d TM series [see Fig. 5(a) and Fig 6(a)]. The exchange splitting of the eg orbitals, ∆ex, also reaches a maximum of ~0.98 eV which is larger than the crystal field splitting, ∆cf, of ~0.27 eV. However, we find that the magnetic moment and exchange splitting drop down along the heavy 4d (Ru, Rh, Ag) elements. A crossover from high-spin state to low spin state happens around the Ru-doped case. Under the D3d crystal field, the free atomic configuration of Ru: 4d75s1 splits into three levels, as shown in Fig. 5(b) and Fig 6(b). Unlike the scene for the light 4d elements (Y, Zr, Nb, Mo, and Tc), the a1g(s) state now is the highest in energy. According to Hund’s rule, the nonbonding eg spin-up state should have been filled by four electrons, giving rise to a magnetic moment of 4.0 µB. As mentioned above, the 4d orbital energies of the free atoms move to lower energies as the atomic number increases along the 4d row. Therefore, the electrons in lower-energy state trend to create internal electron pair, which lie deeper levels and exhibit chemical inertness. For Ru-doped case, if two electrons form an electron pair, the nonbonding eg spin-up state should be filled up by two electrons instead of four electrons. As a result, the effective magnetic moment is 2.0 µB, but not 4.0 µB. Similar ideas can be applied to the later heavy 4d elements. The calculated band structures for metals after Ru are presented in Fig. S3 and Fig. 7, and the PDOS are shown in Fig. S4 and Fig. 8. By observing the band structures, one can see the relatively localized character of the deep-level electrons gradually becomes evident, particularly in the cases of Pd, Ag, and Cd. The strong atomic character and localization of these states appear in the range of -3.8 ~ -4.2 eV, -5.0 ~ -5.6 eV, and -7.0 ~ -8.0 eV for Pd, Ag, and Cd, respectively. Assuming that 4, 6, 8, 8, and 10 electrons form electron pairs for atoms from Rh to Cd, the values of magnetic moments for heavy 4d TM are understood according to the schemes of the electronic levels shown in Fig. S3 and Fig. 7. Similar to the case of Nb-doped case, one d electron of Ag hops to 5s orbital with A-symmetry, so that it well hybridize with the dangling bonds with same symmetry. In the competition between hoping and hybridizing, the hybridization saves more much energy for the system. Finally, for Ag 7
and Cd doping, the nonbonding eu states are half-filled (Fig. 7), as distinguished from the case before Pd doping where the nonbonding eg or a1g(d) levels accommodate the unpaired electrons. The corresponding PDOS and electronic bands (Figs. 7 and 8) reflect that the partially occupied eu states are dominantly composed of the relatively localized sulfur p states near Femi level. What is worth mentioning is that the non-bonding eu states are equally composed of 4-d and S-3p orbitals due to strong p-d hybridization. The systems doped with Ru, Rh, and Ag show acceptor levels and exhibits half-metal behavior. We find that the band gap of the Pd-doped case is the same as the value of 1.57 eV[32] obtained with PBE functional for pristine SnS2 monolayer. C. Stability
To estimate the energetic stability of implantation of foreign atoms, we calculate the binding energies Eb, as presented in Table I. For better comparison, the binding energy of Sn self-substitution is calculated to be 6.14 eV. Binding energies for transition metals vary in the range of 3.3 ~ 9.1 eV. Using the Bader charge analysis,[33] we find the binding energies almost follow the same order as the charge transfer (∆ρM ) from M atom to S atoms (see Table I). Except for Pd, the Eb basically tend to decrease with the increasing atomic number of M element. Zr appears the maximum Eb, because all the valence electrons of Zr bond to S. Pd possess relatively large Eb, which can be easily understood since for this element there is no nonbonding state 4d electron. Ag has the minimum Eb, because this element cannot provide enough valence electrons to effectively hybridize with sp electrons of S. Therefore, Cd has a larger binding energy than that of Ag. There are three competing factors (i.e., effective ionic radius, effective hybridization, and nonbonding electron) that affect on the Eb of the 4d TM. Comparing to Sn, the Eb show that Y, Zr, Nb, Mo and Pd is easy to be doped in SnS2 matrix, while the Ag and Cd is difficult to be implanted into SnS2 matrix. It should be noted that the Eb is not the sole criterion for identifying the energetic stability because the µM depends on the specific reservoir of element M. Here the shorter bond lengths of dM-S suggesting more stable structures. 8
In particular, we further check the thermal stability of Nb- and Mo-doped SnS2 monolayers based on finite temperature MD calculations, due to their half-metallic and magnetic semiconductoring properties. Figure S5 shows the fluctuations of the temperature and total energy as a function of simulation time. As expected, after the 8 ps MD simulations, the Nb- and Mo-doped SnS2 monolayers show no observable distortion of structure. Therefore, both the Nb- and Mo-doped SnS2 monolayers are thermally stable up to at least 500 K. To verify the dynamic stability, the phonon calculations are also performed by using the Phonopy package
[34]
based on finite
displacement method. The phonon dispersion plot (Figure S6) shows no imaginary frequency value for Mo-doped monolayer and thus it can be considered to be a dynamically stable structure. However, the presence of large imaginary modes in Nb-doped structures confirms that the Nb-doped SnS2 structures are dynamically instable. It is necessary to examine the mechanical stablity of Nb- and Mo-doped SnS2 monolayers. Due to D3d symmetry there are only two independent elastic constants, namely, C11 and C12. We also can obtain the in-plane elastic constant C66 [C66 = (C11 -C12 ) / 2] . The main criteria for mechanical stability is C11 > |C12| and C66 > 0.[35] The parabolic fitting of the strain-energy curves are present in Fig. S7. The test results for MoS2 monolayer are C11=134.1 N/m, C12=33 N/m, which are in excellent agreement with the available data (C11=132.7 N/m, C12=33 N/m).[36] The calculated elastic constants for SnS2 monolayer are C11=72.2 N/m, C12=18.0 N/m. For Nb-doped case, the computed elastic constants are C11=69.2, C12=18.3 N/m. For Mo-doped case, the elastic constants are C11=67.2, C12=20.1 N/m. Thereby, all the positive elastic constants fulfill the mechanical stability criteria for Nb- and Mo-doped SnS2 monolayers. It should be noted that the criterion between dynamic stability and mechanical stability is not completely equivalent. The mechanical stability cannot guarantee the dynamic stability. For example, the Soft Mode was found in cubic ZrO2, but the elastic constants are positive values. Another example, the OsC with NiAs-type structure shows positive elastic constants and imaginary frequencies. In the 9
2 2 linear-elastic regime, we can derive the in-plane Young modulus Y = (C11 ) / C11 − C12
and the Poisson’s ratioν = C12 / C11 . Compared to 2D MoS2, the small Young modulus of pristine and doped SnS2, which are less than half of that of MoS2, demonstrate well mechanical flexibility. D. Magnetic coupling and Curie temperature
Having established energetic, thermal, and mechanical stability of Nb- and Mo-doped SnS2 monolayers, we further explore their magnetic ground states. To determine the magnetic exchange interactions, we construct a large 6×6×1 supercell and a 3×6×1 supercell [see Fig. 1(a)]. We study three configurations, with two 4d TM atoms being placed in various positions. We define them as C12, C34 and C56, where the subscript means the Sn site that will be occupied by 4d TM atom, as marked in Fig. 1(a). Due to periodicity, in C12, C34 and C56 configurations each 4d TM atom has 2, 3, and 6 nearest neighboring TM atoms, respectively. Mainly considering the energetic and magnetic stability, here we focus on Nb-, Mo-, Rh-, and Ru-doped cases (especially
Mo-doped).
For
each
configuration,
ferromagnetic
(FM)
and
anti-ferromagnetic (AFM) alignments of spins on 4d TM atoms are considered. The total-energy difference (∆Eex = EAFM -EFM) between these two alignments is an important index of magnetic exchange interaction and is tabulated in Table II. The total energy calculations suggest that the magnetic coupling between the two 4d TM atoms prefer FM states with lower total energies for both Nb- and Mo-doped cases. However, the AFM ground states are energetically favorable in Ru- and Rh-doped (apart from C56) cases. The importance in dilute magnetic semiconductors is the appearance of room-temperature and long-ranged ferromagnetism or half-metallic property for technological applications in prospective spintronic devices. Based on the mean-field theory and Heisenberg model, the Curie temperature (TC) for the Nb- and Mo-doped systems is calculated (Table II). The exchange coupling parameter J can obtain from the ∆Eex, which reflects the magnetic ground state and the TC in the 2D doped systems. Here, we map the ∆Eex onto the Heisenberg model Hamiltonian:[37] 10
H = ∑ J 0 Si ⋅ S j i, j
where Si represents spin on lattice sites, J0 is the exchange coupling parameter that
gives the nearest neighbor interaction. J0 is also proportional to the ∆Eex. Taking C12 configuration as an example, in the AFM state, there are two nearest neighbors with anti-aligned spins and hence the magnetic energy is EAFM=2J0S2. In the FM state, there are two nearest neighbors with aligned spins, so that the magnetic energy is EFM=-2J0S 2. One can obtain the value of J0 by the relation
1 2
∆Eex = 4J 0 S 2 for the C12
configuration. Similarly, the ∆Eex is 12J0S2 and 16J0S 2 for C34 and C56 configurations, respectively. The evaluated exchange parameter J0 is listed in Table II. There exist several approximations to determine the TC from the J0 value. Here, we will use the mean field approximation estimate the magnetic properties of 2D doped SnS2 monolayers. The TC can be obtained by determining the bifurcation critical point of the ensemble-average spin. Considering all possible spin configurations of Nb-doped case, the partition function can be expressed as follows: Z=
∑ S = 12 , − 12
eγ J 0 S / kB T = 2 cosh(γ J 0 12 < S > / kBT)
where, the γ is the number of nearest neighbors of each dopant. The ensemble-average spin of each site can be obtained by the following equation: < S >=
1 Z
∑
S = 12 , − 21
Seγ J 0 S < S > / kB T =
1 tanh(γ J 0 12 < S > / kBT) 2
Therefore, it is easy to find that the critical point locates at γ J 0 ( 14 ) / kBT=1 . The details in calculations are present in Supplementary Material. The possible values of S are 1, 1/2, -1/2, and -1, because the calculated magnetic moment is 2.0 µB, for the Mo-doped system. Similarly, the critical point is found to locate at γ J 0 ( 58 ) / kBT=1 . According to these relations, the corresponding TC is listed in Table II for Nb- and Mo-doped systems. Generally, the mean field approximation (MFA) tends to overestimate the TC, due to the neglect of spin fluctuations. For several 2D materials, the TC are overestimated by 30%-60% by MFA relative to those more precise values estimated by Monte Carlo simulations.[38,39] Therefore, although MFA gives an upper estimate 11
of Tc, we believe that the room temperature ferromagnetism is expected to be achieved in Mo-doped SnS2 monolayers. Next, it is necessary to understand what is responsible for the fundamental mechanism of the observed strong magnetism in Nb- and Mo-doped SnS2 systems. We attribute the FM coupling between the spins on Nb atoms to Zener’s double exchange interactions,[16,40] which requires the presence of delocalized carriers that can hop between the localized spins and lower the kinetic energy of the system by aligning the localized spins ferromagnetically. It is evident in the DOS for Nb-doped systems [see Fig. 9(a)] that a peak straddles the Fermi level in up-spin channel while leaving the down-spin channel semiconducting. This suggests that the effective coupling between spins on neighboring Nb atoms arises from the double-exchange mechanism mediated by the delocalized spins on S and Mo atoms that ferromagnetically align to the Nb spins, as illustrated by the spin density distribution in Fig. 10(a). From electronic density of states of Mo-doped systems [see Fig. 9(b)], we note that the impurity states do not cross the Fermi level in up-spin channel, thus other exchange mechanisms, rather than double-exchange, may be responsible for the ferromagnetic ordering in Mo-doped systems. Examining the spin density distribution in Fig. 10(b), we find that the spins on the six nearest neighboring S of Mo is aligned antiparallel to the spin at Mo. The AFM coupling between the localized Mo 4d states and the p states of the S atoms originates from p-d hybridization [see Fig. 9(b)], which also was observed in the Mn-doped two-dimensional dichalcogenides.[11, 41] Similarly, the AFM p-p exchange interaction between the nearest neighboring S and the farther Sn and S in turn drives FM coupling of Mo to farther atoms. As a consequence, the long-ranged FM coupling between Mo atoms is established by the coupling chain Mo↑-S↓-(Sn, S)↑-(S, Sn)↑-S↓-Mo↑, as being observed in Fig. 10(b). Additionally, we find that the Tc-doped system shows an AFM magnetic ground state, and the Ru-doped system exhibits FM and AFM couplings, depending on the distance between the Ru atoms. 12
IV. SUMMARY In summary, the stability, electronic structures and magnetic properties of SnS2 monolayer doped with 4d transition metal by means of density functional theory. There is no magnetism for Y, Zr, and Pd dopings, while interesting magnetic behaviors are observed for Nb, Mo, Tc, Ru, Rh, Ag, and Cd dopings. Based on molecular orbital theory and ligand field theory, we qualitatively explain the electronic structures and trends in magnetic properties. Especially, the Nb-doped system is a half-metallic ferromagnetic material, in which the exchange interaction is primarily governed by the double-exchange mechanism. The Mo-doped system is a magnetic semiconductor with a Curie temperature above room temperature. Furthermore, it has good energetic stability and flexible mechanical stability and also possesses remarkable dynamic and thermal (500 K) stability. The magnetic ground state originates from the strong S:p-Mo:d hybridization and p-p couplings for Mo-doped case. This study gives potential ways for fabricating diluted magnetic semiconductors based on Nb- and Mo-doped SnS2 monolayers.
ACKNOWLEDGEMENTS This work was supported by the Hunan Provincial Natural Science Foundation under grant Nos. 2016JJ6028 and 2017JJ3049, and the Scientific Research Fund of Hunan Provincial Education Department (Nos. 16A046 and 16C0391). We acknowledge the computational support provided by the computing platform of Network Information Centre of Hunan Institute of Engineering.
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15
Table I Calculated values of binding energy Eb (eV), bond length of M-S dM-S (Å), bond angle θ1/θ2 , spin polarization energy ε (meV), total magnetic moment Mtotal (unit in µB), magnetic moments on M atom Mm and on its nearest neighboring S MS, and the charge transfer ∆ρ (e) from M to S for each doped system. M Y Zr Nb Mo Tc Ru Rh Pd Ag Cd Sn
Eb 8.613 9.039 6.999 5.959 5.725 5.361 5.761 7.345 3.392 3.953 6.141
dM-S 2.736 2.602 2.516 2.463 2.440 2.400 2.392 2.422 2.541 2.667 2.597
θ1/θ2 88.0/92.0 89.4/90.6 89.7/90.3 89.5/90.5 89.7/90.3 89.2/90.8 89.2/90.8 89.2/90.8 89.9/90.1 89.3/90.7 89.2/90.8
ε 0 0 46 513 516 200 31 0 39 73
Mtotal 0 0 1 2 3 2 1 0 1 2 0
Mm
MS
0.523 1.384 2.069 1.105 0.298
0.025 0.034 0.033 0.034 0.024
0.186 0.069
0.049 0.095
∆ρM 1.9408 2.1396 1.8395 1.5023 1.1925 0.8721 0.6102 0.5139 0.5837 0.8638 1.5212
Table II. The calculated values of the M–M distance (d), FM stabilization energies (∆EFM), exchange coupling constant (J0), total magnetic moment (Mtotal) for ground state, and magnetic coupling for each configure (Cij`) of two-M-atom doped SnS2 monolayers. Type
Config.
d (Å)
∆EFM (meV)
J0 (meV)
Mtotal (µB)
Nb
C12 C34 C56 C12 C34 C56 C12 C34 C56 C12 C34 C56
11.09 12.83 11.09 11.09 12.81 11.09 11.08 12.83 11.09 11.09 12.81 11.09
100 7 102 572 652 425 -100 -551 -18 -285 -38 62
100 4.7 51 143 108.7 53.1 -11.1 -40.8 -1.0 -71.3 -6.3 7.8
2.00 2.00 2.00 4.00 4.00 4.00 0.00 0.00 0.00 0 0.00 4.00
Mo
Tc
Ru
16
Coupling
TC (K)
FM FM FM FM FM FM AFM AFM AFM AFM AFM FM
291 20 444 1038 1183 1156
Captions
Figure 1. (a) Geometry of models for the SnS2 monolayer. The prism and parallelogram shown by solid black lines in (a) indicate a (6×6×1) and a (3×6×1) supercell, respectively. The Sn atom sites marked by numbers 1–6 will be substituted by TM; Spin-density isosurfaces for SnS2 monolayer containing one (b) Nb, (c) Mo, (d) Tc, (e) Ru, (f), Rh, (g) Ag, and (h) Cd impurity, respectively. The up-spin density is indicated by blue color with an isosurface of 0.005 e/Å3. (i) Schematic of the crystal-field splitting, (∆cf) of Nb-4d orbitals due to the D3d symmetry and the exchange splitting of the eg orbitals (∆ex ).
Figure 2. Calculated spin-polarized TDOS of M-doped SnS2 monolayer structures.
Figure 3. The spin resolved electronic band structures and schematic representations of the coupling between Sn vacancy-related defect states and symmetry-adapted 4d5s orbitals of (a) Nb and (b) Mo under crystal field with D3d symmetry. The arrows of ↑ and ↓ represent up-spin and down-spin channels, respectively. The red open circles and blue regular hexagons indicate, respectively, the contribution from 4d z 2 state with A symmetry and the rest of the 4d orbitals with E symmetries. The symbol sizes denote the weights of the specific site. The atomic configuration for Nb and Mo is 4d45s1and 4d45s2, respectively.
Figure 4. The spin resolved partial density of states of SnS2 monolayers doped with (a) Nb and (b) Mo atoms. The Fermi level is indicated by the dashed line.
Figure 5. The spin resolved electronic band structures and schematic representations of the coupling between Sn vacancy-related defect states and symmetry-adapted 4d5s orbitals (a) Tc and (b) Ru under crystal field with D3d symmetry. The green “↑↓” denotes a pair of deep-level electrons which are chemically inert.
17
Figure 6. The spin resolved partial density of states of SnS2 monolayers doped with (a) Tc and (b) Ru atoms. The Fermi level is indicated by the dashed line.
Figure 7. The spin resolved electronic band structures and schematic representations of the coupling between Sn vacancy-related defect states and symmetry-adapted 4d5s orbitals (a) Ag and (b) Cd under crystal field with D3d symmetry. The green stars denote the components of S-3px and S-3py.
Figure 8. Partial DOSs of (a) Ag-doped and (b) Cd-doped SnS2 systems. The Fermi level is indicated by the dashed line.
Figure 9. The total DOS and partial DOS for the (a) Nb- and (b) Mo-doped SnS2 for both C12 and C56 configurations. The Fermi level is indicated by the dashed line.
Figure 10. Calculated spin-density isosurfaces of configurations C12 and C56 for (a) Nb- and (b) Mo-doped systems. The blue and red surfaces represent up-spin and down-spin components, respectively, with an isosurface of 0.0012 e/ Å3.
18
E
-2
-2
-3
-3
-4
-4
-5
-5
4
4
3
3
2
2
Energy(eV)
1
1
M o-up
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
Μ
M odow n
Γ
Κ
Μ
Μ
Γ
S0304-8853(17)30391-8 http://dx.doi.org/10.1016/j.jmmm.2017.04.090 MAGMA 62694
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
8 February 2017 30 April 2017 30 April 2017
Please cite this article as: W-Z. Xiao, G. Xiao, Q-Y. Rong, Q. Chen, L-L. Wang, Electronic and magnetic properties of SnS2 monolayer doped with 4d transition metals, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.04.090
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Electronic and magnetic properties of SnS2 monolayer doped with 4d transition metals Wen-Zhi Xiao,a∗∗ Gang Xiao,a,b Qing-Yan Rong,a Qiao Chen,a Ling-Ling Wang b
Abstract: We investigate the electronic structures and magnetic properties of SnS2 monolayers substitutionally doped with 4-d transition-metal through systematic first principles calculations. The doped complexes exhibit interesting electronic and magnetic behaviors, depending on the interplay between crystal field splitting, Hund’s rule, and 4d levels. The system doped with Y is nonmagnetic metal. Both the Zr- and Pd-doped systems remain nonmagnetic semiconductors. Doping results in half-metallic states for Nb-, Ru-, Rh-, Ag, and Cd doped cases, and magnetic semiconductors for systems with Mo and Tc dopants. In particular, the Nb- and Mo-doped systems display long-ranged ferromagnetic ordering with Curie temperature above room temperature, which are primarily attributable to the double-exchange mechanism, and the p-d/p-p hybridizations, respectively. Moreover, The Mo-doped system has excellent energetic stability and flexible mechanical stability, and also possesses remarkable dynamic and thermal (500 K) stability. Our studies demonstrate that Nb- and Mo-doped SnS2 monolayers are promising candidates for preparing 2D diluted magnetic semiconductors, and hence will be a helpful clue for experimentalists.
School of Science, Hunan Institute of Engineering, Xiangtan 411104, China. E-mail: [email protected] b School of Physics and Electronics, Hunan University, Changsha 410082, China. a
∗
I. Introduction The emergence of graphene[1] has trigged tremendous great attentions in development of a variety of single layer two-dimensional (2D) materials due to their peculiar physics and potential applications in next generation of nanoscale devices[2,3]. Soon after, monolayer MoS2 has been fabricated successfully by exfoliation technique.[4,5] In contrast to the zero band gap of pristine graphene, a single-layer MoS2 sheet is a direct band gap semiconductor[6,7] which makes it a promising material for potential semiconductor devices, such as digital circuits,[8] light-emitting diodes,[9] and diluted magnetic semiconductors[10,11]. Earlier theoretical studies[6,7] predicted a lot of 2D layered materials such as metal chalcogenides, transition metal oxides. Transition metal dichalcogenides consist of hexagonal layers of metal atoms (M) sandwiched between two layers of chalcogen atoms (X) with a MX2 stoichiometry. Among them, the monolayer tin-disulfide (SnS2) also has renewed interest due to its exceptional optical and electronic properties. The freestanding SnS2 single-layers were first time obtained through a scalable liquid exfoliation strategy, and were found to be suitable for visible-light water splitting.[12] Shortly after, based monolayer SnS2 materials Song et al.[13] successfully realized top-gated field-effect transistors and related logic gates. Every recently, monolayer, and bilayer SnS2 have been successfully prepared using mechanical exfoliation.[14] Stirred by these exciting findings, considerable experimental and theoretical efforts have been devoted to controlling the electronic structures and magnetic properties in such unusual materials for a wide range of applications. To control magnetism in 2D crystals has remained a goal in the diluted magnetic semiconductors, [15]
which gives rise to potential applications in spintronics.[16] The field of spintronics
requires an active ferromagnetic ordering temperature, which should ideally be well above room temperature to enable practical applications. It has already been shown that the electronic and magnetic properties of monolayer SnS2 can be modulated by external train, intrinsic defects,[17,18] and doping with 3d transition-metal (TM) atom.[19,20] Theoretical prediction[21] and experimental observation[22] shown that 1
substitution doping of Mg may be the origin of the room temperature ferromagnetism in SnS2 nanoflowers. Although the exact mechanism of Mg-doping for such resultant magnetic order has not been very clear yet, it might be instrumental toward the realization of d 0 ferromagnetism in 2D SnS2 materials. However, as far as we know, there is no report on searching for room temperature ferromagnetism in 2D SnS2 monolayers doped with 4d TM atoms, which is the focus of this work. In this work, we provide a systematic study of the stability, electronic and magnetic properties of 2D doped SnS2 by using the density functional theory (DFT). To better understand and control the magnetic properties of the monolayer SnS2 doped 4d TM atoms (from Y to Cd), the origin of magnetism is systematically analyzed. Our results indicate that the desired long-range ferromagnetism with above room temperature can be achieved by doping with Mo atoms.
II. Computational details We calculate the atomic structure and electronic and magnetic properties within the framework of spin-polarized DFT simulation package (VASP) software
[23]
as implemented in the Vienna ab initio
[24]
. The projector augmented-wave method is
used.[25] A plane-wave basis set with kinetic energy cutoff of 480 eV is used for all the calculations. The exchange-correlation functional is treated using the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) format.
[26]
For
all structural optimization, the convergence criteria for total energy and Hellman–Feynman force are set to 10-4 eV and 0.01 eV/Å, respectively. The vacuum spacing between adjacent monolayers is kept larger than 13 Å. In order to study magnetism induced by substitutional doping of 4d TM atoms, we construct a 4×4×1 SnS2 supercell with one dopant. The Γ-centered 5×5×1 k-mesh scheme and 70 k points along the high symmetry directions are used for electronic density of states (DOS) and band structures, respectively. To determine the stability of these doped sheets, we systemically calculate the binding energy and strain-stress relation, and carry out molecular dynamics simulation. The binding energy, Eb, is calculated by the equation: Eb = Ev + µM - Ed,[10] where, Ev 2
is the total energy of the SnS2 monolayer with one Sn vacnacy, µM is the energy of one M atom in its bulk phase, and Ed is the total energy of the doped SnS2 monolayer with one M dopant. We calculate the in-plane linear elastic constants for Nb- and Mo-doped cases. Since the strain energy is quadratic dependent on the applied strains, the elastic constants can be obtained by calculating the strain-energy relation. More details about the calculation can be found in our previous work [28,29]
[27]
and elsewhere
. The thermal stability is assessed by performing ab initio molecular dynamics
simulations (MD) using the Nosé thermostat model. In the MD simulations, canonical (NVT) ensembles at 500 K with a time step of 1 femtosecond time are adopted.
III. Results and discussions A. Local structure Now, we firstly examine the local structure around the M dopants. Seen from Table I, the first observation is that the bond lengths of dM-S decrease as the 4d TM atom varies from Y to Rh. Then the bond lengths of dM-S go up for the 4d TM after Rh element, suggesting the bonding becomes weaker. The changing tend of bond length is basically the same as that of ionic radius of 4d TM. Checking carefully the bond lengths and bond angles around the 4d TM dopant, we find the local structure centered the 4d TM dopant experiences a breathing relaxation. The 4d TM dopant is six-coordinated with geometry of trigonal bipyramid. For each case, the bond lengths are equal, but the bond angles fall into two kinds, which make the local structure remains the same D3d point-group symmetry as the pristine SnS2 monolayer. It should be noted that the two types of bond angles (∠S-M-S: θ1 and θ2, listed in Table I) are around 90°, and the difference between them is relatively small for all the doped SnS2 monolayers. Therefore, the symmetry of this crystal field approaches an octahedral crystal field with Oh symmetry. Under crystal-field environment, when one of the degenerate 4d bands is partially filled, Jahn-Teller distortions should occur in order to remove the degeneracy and lower the energy. Here, we observe no Jahn-Teller distortion for all the doped systems. 3
B. Electronic structure and magnetic properties Apart from Y, Zr and Pd, all other 4d TM dopants induce spin-polarization and give rise to an integer magnetic moment per dopant, as shown in Table I. Interestingly, the size of magnetic moment changes regularly along the series of 4d TM atoms. From Y to Cd element, the corresponding total magnetic moments are Y: 0, Zr: 0, Nb: 1.0, Mo: 2.0, Tc: 3.0, Ru: 2.0, Rh: 1.0, Pd: 0, Ag: 1.0, and Cd: 2.0 µB. The corresponding spin density distribution in Fig. 1 visualizes the detailed distribution of magnetic moments. In the cases of Nb, Mo, Tc and Ru doped systems, the induced magnetic moment are mainly located on the dopants, attributing to their excess unpaired 4d electrons, whereas the rest resides on the neighboring Sn and S sties. In contrast, the magnetic moment induced by Rh, Ag, and Cd are mainly distributed on the surrounding S atoms rather than the dopants and Sn atoms, and the spin-polarized charge density on S atoms is distinctly characteristic of p orbital. To explore the magnetic stability of the doped systems, we calculate the spin polarization energy, ε=ENSP-ESP, defined as the difference in total energy of the spin-polarized state and non-spin-polarized state. The ε reveals the magnitude of spin-polarization which is also reflected by the spin exchange splitting, as shown in Fig. 2. One can observe the exactly symmetrical DOS of the Y-, Zr-, and Pd-doped systems, implying that they should be nonmagnetic. The Mo-, and Tc doped systems show the largest spin-splitting in the band-gap. The value of ε is much greater than the thermal energy at room temperature, indicating that the local magnetic moments induced by 4d TM are stable. Based on Molecular orbital theory and ligand field theory, we can qualitatively explain the electronic structures and trends in magnetic properties. As motioned above, after structure relaxation, the D3d symmetry is maintained for all dopants. According to group theory, the D3d ligand field splits the neutral Sn vacancy-related defect levels into two non-degenerate a1g (s) and a2u (p z) levels, and a doubly degenerate eu (p x+y) level. The twofold degenerated eu level is lower in energy than the singlet a2u level, but higher than the a 1g level. For a neutral Sn vacancy, anion S dangling bonds (i.e., Sn vacancy-related defect levels) are partially occupied by four unpaired electrons, 4
which will lead to a high-spin state a1↑g eu↑↑ a2↑u with a net moment of 4.0 µB in agreement with theoretical estimation.[19] Upon 4d TM occupying at the Sn site, the dangling bonds of the anion S will couple with the symmetry-adapted 4d and 5s levels of the TM. Under D3d symmetry, the sd orbitals of 4d TM are split into two single a1g [ d z 2 and s, remarked as a1g(d) and a1g(s), respectively] states and fourfold degenerate
eg (dxy, d x2 − y 2 , dxz, and dyz) states which is lower in energy than the two singlet levels. The order of a1g(d) and a1g(s) is mainly dependent on the atomic energy levels of 4d elements. Molecular orbital theory requires that linear combination of atomic orbitals should obey the symmetrical matching principle, the approximate rule of orbital energy, and the maximum overlapping principle. Therefore, the fourfold degenerate eg can couple with the doubly degenerate eu levels of dangling bonds due to the same E symmetry. The a1g(d) and a1g(s) states can couple with the a1g and a2u states because they belonging to the same irreducible A representation. Note that, due to the strong hybridization with SnS2 layer, it is not possible to identify well-defined defect levels with E or A symmetry. The coupling model developed by Raebiger
[30]
and Santos [31]
explained well their calculated results of graphene, and II-VI and III-V compounds modified with 3d TM. We now might as well start the analysis with Nb-doped system. An isolated Nb atom has a 4d45 s1 configuration with one additional valence electron as compared to Sn:5s25p2. Deduced from the electronic band structure in Fig. 3(a) and the PDOS in Fig. 4(a), one can see that the level of a1g(s) is lower than that of a1g(d). The partially filled eu levels of S anion dangling bonds hybridize with two eg(d) electrons, forming a bonding eu state (b-eu). To effectively bond, a eg(d) electron hops to a1g(s) level forming ds hybridization, and then interact with the half-filled a2u and a1g level, resulting in bonding a1g (b-a1g) and a2u (b-a2u) states. In the ds hybridization the hoping lifts energy, but the formation of bond lowers more energy, as a consequence, the system saves energy. As shown in Figs. 3(a) and 4(a), in the spin-down channel, the nonbonding state eg (n-eg) appears about 0.16~0.46 eV, mainly derived from 5
Nb − 4d( xy + xz + yz + x2 ) states. However, in the spin-up channel, the nonbonding state eg crosses the Fermi level, which indicates that the Nb-doped monolayer is essentially half-metallic. Just above the nonbonding state eg, the nonbonding states a1g(d) [na1g(d)] remain unoccupied in both channels. Four out of five electrons of Nb bond to S atoms and fill up b-a1g, b-a2u and b-eu states, leaving one electron fill into nonbonding e state in the spin-up channel. As a result, the Nb-doped system gives rise to magnetic moments of 1.0 µB. As we know that the outer electrons of Mo adopt a 4d45s2 configuration, the valence electrons interact with the dangling bonds in the same manner as Nb does. Two eg(d) electrons of Mo occupy nonbonding fourfold degenerate eg states in the spin-up channel. Consequently, the n-eg level does not cross the Fermi level, so the Mo-doped system is a magnetic semiconductor with a total magnetic moment of 2.0 µB. For Nb-doped SnS2, the exchange splitting of the eg orbitals, ∆ex ≈ 0.30 eV, is
smaller than the crystal field splitting, ∆cf ≈ 0.48 eV [see Fig. (i)]. The spin exchange splitting of Mo-doped case is larger than that of Nb-doped case. We obtain ∆ex ≈ 0.70 eV and ∆cf ≈ 0.3 eV for Mo-doped case. The isolated Y and Zr atoms possess the 4d15s2 and 4d25s2configurations, respectively. For Y, there is lack of one electron for coupling to eu levels, and hence a nonbonding eu state is formed. The nonbonding eu state accommodates one electron which is delocalized in the whole crystal, so that the magnetic moment is quenched. As for the Zr-doped monolayer, all the sd electrons join by bonding, so that the doped system remains the semiconducting properties with a indirect band-gad of 1.22 eV. As shown in Figs. S1 and S2, one can see that the nonbonding and anti-bonding states of Zr-4d occupy the lower levels than that of Y, which attribute to the fact that the 4d orbital energies of the free atoms become deeper as the atomic number increases along the 4d row. For Tc-doped case, the same mechanism works as it does for Mo-doped case. Compared to Mo, the Tc atom has one extra valence electron with a 4d55s2 configuration. Therefore, one electron occupies the nonbonding n-e1g(d) state and the 6
other two fill the nonbonding n-eg state, and hence the developing magnetic moment reaches a peak value of 3.0 µB in the 4d TM series [see Fig. 5(a) and Fig 6(a)]. The exchange splitting of the eg orbitals, ∆ex, also reaches a maximum of ~0.98 eV which is larger than the crystal field splitting, ∆cf, of ~0.27 eV. However, we find that the magnetic moment and exchange splitting drop down along the heavy 4d (Ru, Rh, Ag) elements. A crossover from high-spin state to low spin state happens around the Ru-doped case. Under the D3d crystal field, the free atomic configuration of Ru: 4d75s1 splits into three levels, as shown in Fig. 5(b) and Fig 6(b). Unlike the scene for the light 4d elements (Y, Zr, Nb, Mo, and Tc), the a1g(s) state now is the highest in energy. According to Hund’s rule, the nonbonding eg spin-up state should have been filled by four electrons, giving rise to a magnetic moment of 4.0 µB. As mentioned above, the 4d orbital energies of the free atoms move to lower energies as the atomic number increases along the 4d row. Therefore, the electrons in lower-energy state trend to create internal electron pair, which lie deeper levels and exhibit chemical inertness. For Ru-doped case, if two electrons form an electron pair, the nonbonding eg spin-up state should be filled up by two electrons instead of four electrons. As a result, the effective magnetic moment is 2.0 µB, but not 4.0 µB. Similar ideas can be applied to the later heavy 4d elements. The calculated band structures for metals after Ru are presented in Fig. S3 and Fig. 7, and the PDOS are shown in Fig. S4 and Fig. 8. By observing the band structures, one can see the relatively localized character of the deep-level electrons gradually becomes evident, particularly in the cases of Pd, Ag, and Cd. The strong atomic character and localization of these states appear in the range of -3.8 ~ -4.2 eV, -5.0 ~ -5.6 eV, and -7.0 ~ -8.0 eV for Pd, Ag, and Cd, respectively. Assuming that 4, 6, 8, 8, and 10 electrons form electron pairs for atoms from Rh to Cd, the values of magnetic moments for heavy 4d TM are understood according to the schemes of the electronic levels shown in Fig. S3 and Fig. 7. Similar to the case of Nb-doped case, one d electron of Ag hops to 5s orbital with A-symmetry, so that it well hybridize with the dangling bonds with same symmetry. In the competition between hoping and hybridizing, the hybridization saves more much energy for the system. Finally, for Ag 7
and Cd doping, the nonbonding eu states are half-filled (Fig. 7), as distinguished from the case before Pd doping where the nonbonding eg or a1g(d) levels accommodate the unpaired electrons. The corresponding PDOS and electronic bands (Figs. 7 and 8) reflect that the partially occupied eu states are dominantly composed of the relatively localized sulfur p states near Femi level. What is worth mentioning is that the non-bonding eu states are equally composed of 4-d and S-3p orbitals due to strong p-d hybridization. The systems doped with Ru, Rh, and Ag show acceptor levels and exhibits half-metal behavior. We find that the band gap of the Pd-doped case is the same as the value of 1.57 eV[32] obtained with PBE functional for pristine SnS2 monolayer. C. Stability
To estimate the energetic stability of implantation of foreign atoms, we calculate the binding energies Eb, as presented in Table I. For better comparison, the binding energy of Sn self-substitution is calculated to be 6.14 eV. Binding energies for transition metals vary in the range of 3.3 ~ 9.1 eV. Using the Bader charge analysis,[33] we find the binding energies almost follow the same order as the charge transfer (∆ρM ) from M atom to S atoms (see Table I). Except for Pd, the Eb basically tend to decrease with the increasing atomic number of M element. Zr appears the maximum Eb, because all the valence electrons of Zr bond to S. Pd possess relatively large Eb, which can be easily understood since for this element there is no nonbonding state 4d electron. Ag has the minimum Eb, because this element cannot provide enough valence electrons to effectively hybridize with sp electrons of S. Therefore, Cd has a larger binding energy than that of Ag. There are three competing factors (i.e., effective ionic radius, effective hybridization, and nonbonding electron) that affect on the Eb of the 4d TM. Comparing to Sn, the Eb show that Y, Zr, Nb, Mo and Pd is easy to be doped in SnS2 matrix, while the Ag and Cd is difficult to be implanted into SnS2 matrix. It should be noted that the Eb is not the sole criterion for identifying the energetic stability because the µM depends on the specific reservoir of element M. Here the shorter bond lengths of dM-S suggesting more stable structures. 8
In particular, we further check the thermal stability of Nb- and Mo-doped SnS2 monolayers based on finite temperature MD calculations, due to their half-metallic and magnetic semiconductoring properties. Figure S5 shows the fluctuations of the temperature and total energy as a function of simulation time. As expected, after the 8 ps MD simulations, the Nb- and Mo-doped SnS2 monolayers show no observable distortion of structure. Therefore, both the Nb- and Mo-doped SnS2 monolayers are thermally stable up to at least 500 K. To verify the dynamic stability, the phonon calculations are also performed by using the Phonopy package
[34]
based on finite
displacement method. The phonon dispersion plot (Figure S6) shows no imaginary frequency value for Mo-doped monolayer and thus it can be considered to be a dynamically stable structure. However, the presence of large imaginary modes in Nb-doped structures confirms that the Nb-doped SnS2 structures are dynamically instable. It is necessary to examine the mechanical stablity of Nb- and Mo-doped SnS2 monolayers. Due to D3d symmetry there are only two independent elastic constants, namely, C11 and C12. We also can obtain the in-plane elastic constant C66 [C66 = (C11 -C12 ) / 2] . The main criteria for mechanical stability is C11 > |C12| and C66 > 0.[35] The parabolic fitting of the strain-energy curves are present in Fig. S7. The test results for MoS2 monolayer are C11=134.1 N/m, C12=33 N/m, which are in excellent agreement with the available data (C11=132.7 N/m, C12=33 N/m).[36] The calculated elastic constants for SnS2 monolayer are C11=72.2 N/m, C12=18.0 N/m. For Nb-doped case, the computed elastic constants are C11=69.2, C12=18.3 N/m. For Mo-doped case, the elastic constants are C11=67.2, C12=20.1 N/m. Thereby, all the positive elastic constants fulfill the mechanical stability criteria for Nb- and Mo-doped SnS2 monolayers. It should be noted that the criterion between dynamic stability and mechanical stability is not completely equivalent. The mechanical stability cannot guarantee the dynamic stability. For example, the Soft Mode was found in cubic ZrO2, but the elastic constants are positive values. Another example, the OsC with NiAs-type structure shows positive elastic constants and imaginary frequencies. In the 9
2 2 linear-elastic regime, we can derive the in-plane Young modulus Y = (C11 ) / C11 − C12
and the Poisson’s ratioν = C12 / C11 . Compared to 2D MoS2, the small Young modulus of pristine and doped SnS2, which are less than half of that of MoS2, demonstrate well mechanical flexibility. D. Magnetic coupling and Curie temperature
Having established energetic, thermal, and mechanical stability of Nb- and Mo-doped SnS2 monolayers, we further explore their magnetic ground states. To determine the magnetic exchange interactions, we construct a large 6×6×1 supercell and a 3×6×1 supercell [see Fig. 1(a)]. We study three configurations, with two 4d TM atoms being placed in various positions. We define them as C12, C34 and C56, where the subscript means the Sn site that will be occupied by 4d TM atom, as marked in Fig. 1(a). Due to periodicity, in C12, C34 and C56 configurations each 4d TM atom has 2, 3, and 6 nearest neighboring TM atoms, respectively. Mainly considering the energetic and magnetic stability, here we focus on Nb-, Mo-, Rh-, and Ru-doped cases (especially
Mo-doped).
For
each
configuration,
ferromagnetic
(FM)
and
anti-ferromagnetic (AFM) alignments of spins on 4d TM atoms are considered. The total-energy difference (∆Eex = EAFM -EFM) between these two alignments is an important index of magnetic exchange interaction and is tabulated in Table II. The total energy calculations suggest that the magnetic coupling between the two 4d TM atoms prefer FM states with lower total energies for both Nb- and Mo-doped cases. However, the AFM ground states are energetically favorable in Ru- and Rh-doped (apart from C56) cases. The importance in dilute magnetic semiconductors is the appearance of room-temperature and long-ranged ferromagnetism or half-metallic property for technological applications in prospective spintronic devices. Based on the mean-field theory and Heisenberg model, the Curie temperature (TC) for the Nb- and Mo-doped systems is calculated (Table II). The exchange coupling parameter J can obtain from the ∆Eex, which reflects the magnetic ground state and the TC in the 2D doped systems. Here, we map the ∆Eex onto the Heisenberg model Hamiltonian:[37] 10
H = ∑ J 0 Si ⋅ S j i, j
where Si represents spin on lattice sites, J0 is the exchange coupling parameter that
gives the nearest neighbor interaction. J0 is also proportional to the ∆Eex. Taking C12 configuration as an example, in the AFM state, there are two nearest neighbors with anti-aligned spins and hence the magnetic energy is EAFM=2J0S2. In the FM state, there are two nearest neighbors with aligned spins, so that the magnetic energy is EFM=-2J0S 2. One can obtain the value of J0 by the relation
1 2
∆Eex = 4J 0 S 2 for the C12
configuration. Similarly, the ∆Eex is 12J0S2 and 16J0S 2 for C34 and C56 configurations, respectively. The evaluated exchange parameter J0 is listed in Table II. There exist several approximations to determine the TC from the J0 value. Here, we will use the mean field approximation estimate the magnetic properties of 2D doped SnS2 monolayers. The TC can be obtained by determining the bifurcation critical point of the ensemble-average spin
∑ S = 12 , − 12
eγ J 0 S
where, the γ is the number of nearest neighbors of each dopant. The ensemble-average spin
1 Z
∑
S = 12 , − 21
Seγ J 0 S < S > / kB T =
1 tanh(γ J 0 12 < S > / kBT) 2
Therefore, it is easy to find that the critical point locates at γ J 0 ( 14 ) / kBT=1 . The details in calculations are present in Supplementary Material. The possible values of S are 1, 1/2, -1/2, and -1, because the calculated magnetic moment is 2.0 µB, for the Mo-doped system. Similarly, the critical point is found to locate at γ J 0 ( 58 ) / kBT=1 . According to these relations, the corresponding TC is listed in Table II for Nb- and Mo-doped systems. Generally, the mean field approximation (MFA) tends to overestimate the TC, due to the neglect of spin fluctuations. For several 2D materials, the TC are overestimated by 30%-60% by MFA relative to those more precise values estimated by Monte Carlo simulations.[38,39] Therefore, although MFA gives an upper estimate 11
of Tc, we believe that the room temperature ferromagnetism is expected to be achieved in Mo-doped SnS2 monolayers. Next, it is necessary to understand what is responsible for the fundamental mechanism of the observed strong magnetism in Nb- and Mo-doped SnS2 systems. We attribute the FM coupling between the spins on Nb atoms to Zener’s double exchange interactions,[16,40] which requires the presence of delocalized carriers that can hop between the localized spins and lower the kinetic energy of the system by aligning the localized spins ferromagnetically. It is evident in the DOS for Nb-doped systems [see Fig. 9(a)] that a peak straddles the Fermi level in up-spin channel while leaving the down-spin channel semiconducting. This suggests that the effective coupling between spins on neighboring Nb atoms arises from the double-exchange mechanism mediated by the delocalized spins on S and Mo atoms that ferromagnetically align to the Nb spins, as illustrated by the spin density distribution in Fig. 10(a). From electronic density of states of Mo-doped systems [see Fig. 9(b)], we note that the impurity states do not cross the Fermi level in up-spin channel, thus other exchange mechanisms, rather than double-exchange, may be responsible for the ferromagnetic ordering in Mo-doped systems. Examining the spin density distribution in Fig. 10(b), we find that the spins on the six nearest neighboring S of Mo is aligned antiparallel to the spin at Mo. The AFM coupling between the localized Mo 4d states and the p states of the S atoms originates from p-d hybridization [see Fig. 9(b)], which also was observed in the Mn-doped two-dimensional dichalcogenides.[11, 41] Similarly, the AFM p-p exchange interaction between the nearest neighboring S and the farther Sn and S in turn drives FM coupling of Mo to farther atoms. As a consequence, the long-ranged FM coupling between Mo atoms is established by the coupling chain Mo↑-S↓-(Sn, S)↑-(S, Sn)↑-S↓-Mo↑, as being observed in Fig. 10(b). Additionally, we find that the Tc-doped system shows an AFM magnetic ground state, and the Ru-doped system exhibits FM and AFM couplings, depending on the distance between the Ru atoms. 12
IV. SUMMARY In summary, the stability, electronic structures and magnetic properties of SnS2 monolayer doped with 4d transition metal by means of density functional theory. There is no magnetism for Y, Zr, and Pd dopings, while interesting magnetic behaviors are observed for Nb, Mo, Tc, Ru, Rh, Ag, and Cd dopings. Based on molecular orbital theory and ligand field theory, we qualitatively explain the electronic structures and trends in magnetic properties. Especially, the Nb-doped system is a half-metallic ferromagnetic material, in which the exchange interaction is primarily governed by the double-exchange mechanism. The Mo-doped system is a magnetic semiconductor with a Curie temperature above room temperature. Furthermore, it has good energetic stability and flexible mechanical stability and also possesses remarkable dynamic and thermal (500 K) stability. The magnetic ground state originates from the strong S:p-Mo:d hybridization and p-p couplings for Mo-doped case. This study gives potential ways for fabricating diluted magnetic semiconductors based on Nb- and Mo-doped SnS2 monolayers.
ACKNOWLEDGEMENTS This work was supported by the Hunan Provincial Natural Science Foundation under grant Nos. 2016JJ6028 and 2017JJ3049, and the Scientific Research Fund of Hunan Provincial Education Department (Nos. 16A046 and 16C0391). We acknowledge the computational support provided by the computing platform of Network Information Centre of Hunan Institute of Engineering.
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15
Table I Calculated values of binding energy Eb (eV), bond length of M-S dM-S (Å), bond angle θ1/θ2 , spin polarization energy ε (meV), total magnetic moment Mtotal (unit in µB), magnetic moments on M atom Mm and on its nearest neighboring S MS, and the charge transfer ∆ρ (e) from M to S for each doped system. M Y Zr Nb Mo Tc Ru Rh Pd Ag Cd Sn
Eb 8.613 9.039 6.999 5.959 5.725 5.361 5.761 7.345 3.392 3.953 6.141
dM-S 2.736 2.602 2.516 2.463 2.440 2.400 2.392 2.422 2.541 2.667 2.597
θ1/θ2 88.0/92.0 89.4/90.6 89.7/90.3 89.5/90.5 89.7/90.3 89.2/90.8 89.2/90.8 89.2/90.8 89.9/90.1 89.3/90.7 89.2/90.8
ε 0 0 46 513 516 200 31 0 39 73
Mtotal 0 0 1 2 3 2 1 0 1 2 0
Mm
MS
0.523 1.384 2.069 1.105 0.298
0.025 0.034 0.033 0.034 0.024
0.186 0.069
0.049 0.095
∆ρM 1.9408 2.1396 1.8395 1.5023 1.1925 0.8721 0.6102 0.5139 0.5837 0.8638 1.5212
Table II. The calculated values of the M–M distance (d), FM stabilization energies (∆EFM), exchange coupling constant (J0), total magnetic moment (Mtotal) for ground state, and magnetic coupling for each configure (Cij`) of two-M-atom doped SnS2 monolayers. Type
Config.
d (Å)
∆EFM (meV)
J0 (meV)
Mtotal (µB)
Nb
C12 C34 C56 C12 C34 C56 C12 C34 C56 C12 C34 C56
11.09 12.83 11.09 11.09 12.81 11.09 11.08 12.83 11.09 11.09 12.81 11.09
100 7 102 572 652 425 -100 -551 -18 -285 -38 62
100 4.7 51 143 108.7 53.1 -11.1 -40.8 -1.0 -71.3 -6.3 7.8
2.00 2.00 2.00 4.00 4.00 4.00 0.00 0.00 0.00 0 0.00 4.00
Mo
Tc
Ru
16
Coupling
TC (K)
FM FM FM FM FM FM AFM AFM AFM AFM AFM FM
291 20 444 1038 1183 1156
Captions
Figure 1. (a) Geometry of models for the SnS2 monolayer. The prism and parallelogram shown by solid black lines in (a) indicate a (6×6×1) and a (3×6×1) supercell, respectively. The Sn atom sites marked by numbers 1–6 will be substituted by TM; Spin-density isosurfaces for SnS2 monolayer containing one (b) Nb, (c) Mo, (d) Tc, (e) Ru, (f), Rh, (g) Ag, and (h) Cd impurity, respectively. The up-spin density is indicated by blue color with an isosurface of 0.005 e/Å3. (i) Schematic of the crystal-field splitting, (∆cf) of Nb-4d orbitals due to the D3d symmetry and the exchange splitting of the eg orbitals (∆ex ).
Figure 2. Calculated spin-polarized TDOS of M-doped SnS2 monolayer structures.
Figure 3. The spin resolved electronic band structures and schematic representations of the coupling between Sn vacancy-related defect states and symmetry-adapted 4d5s orbitals of (a) Nb and (b) Mo under crystal field with D3d symmetry. The arrows of ↑ and ↓ represent up-spin and down-spin channels, respectively. The red open circles and blue regular hexagons indicate, respectively, the contribution from 4d z 2 state with A symmetry and the rest of the 4d orbitals with E symmetries. The symbol sizes denote the weights of the specific site. The atomic configuration for Nb and Mo is 4d45s1and 4d45s2, respectively.
Figure 4. The spin resolved partial density of states of SnS2 monolayers doped with (a) Nb and (b) Mo atoms. The Fermi level is indicated by the dashed line.
Figure 5. The spin resolved electronic band structures and schematic representations of the coupling between Sn vacancy-related defect states and symmetry-adapted 4d5s orbitals (a) Tc and (b) Ru under crystal field with D3d symmetry. The green “↑↓” denotes a pair of deep-level electrons which are chemically inert.
17
Figure 6. The spin resolved partial density of states of SnS2 monolayers doped with (a) Tc and (b) Ru atoms. The Fermi level is indicated by the dashed line.
Figure 7. The spin resolved electronic band structures and schematic representations of the coupling between Sn vacancy-related defect states and symmetry-adapted 4d5s orbitals (a) Ag and (b) Cd under crystal field with D3d symmetry. The green stars denote the components of S-3px and S-3py.
Figure 8. Partial DOSs of (a) Ag-doped and (b) Cd-doped SnS2 systems. The Fermi level is indicated by the dashed line.
Figure 9. The total DOS and partial DOS for the (a) Nb- and (b) Mo-doped SnS2 for both C12 and C56 configurations. The Fermi level is indicated by the dashed line.
Figure 10. Calculated spin-density isosurfaces of configurations C12 and C56 for (a) Nb- and (b) Mo-doped systems. The blue and red surfaces represent up-spin and down-spin components, respectively, with an isosurface of 0.0012 e/ Å3.
18
E
-2
-2
-3
-3
-4
-4
-5
-5
4
4
3
3
2
2
Energy(eV)
1
1
M o-up
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
Μ
M odow n
Γ
Κ
Μ
Μ
Γ