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Bipolar magnetic semiconductor in silicene nanoribbons
Accepted Manuscript Bipolar magnetic semiconductor in silicene nanoribbons Rouhollah Farghadan PII: DOI: Reference:
S0304-8853(16)31808-X http://dx.doi.org/10.1016/j.jmmm.2017.04.016 MAGMA 62619
To appear in:
Journal of Magnetism and Magnetic Materials
Please cite this article as: R. Farghadan, Bipolar magnetic semiconductor in silicene nanoribbons, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.04.016
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.
Bipolar magnetic semiconductor in silicene nanoribbons Rouhollah Farghadan1, ∗ 1
Department of Physics, University of Kashan, Kashan,87317-51167, Iran (Dated: April 8, 2017)
A theoretical study was presented on generation of spin polarization in silicene nanoribbons using the singleband tight-binding approximation and the non-equilibrium Green’s function formalism. We focused on the effect of electric and exchange magnetic fields on the spin-filter capabilities of zigzag-edge silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. The results show that a robust bipolar magnetic semiconductor with controllable spin-flip and spin-conserved gaps can be obtained when exchange magnetic and electric field strengths are both larger than the intrinsic spin-orbit interaction. Therefore, zigzag silicene nanoribbons could act as bipolar and perfect spin filter devices with a large spin-polarized current and a reversible spin polarization in the vicinity of the Fermi energy. We also investigated the effect of edge roughness and found that the bipolar magnetic semiconductor features are robust against edge disorder in silicene nanoribbon junctions. These results may be useful in multifunctional spin devices based on silicene nanoribbons. PACS numbers:
I. INTRODUCTION
Combination of silicon, as the mainstream of semiconductors, with magnetic materials could widely improve the spinbased electronics1,2 . Silicene, the two dimensional honeycomb structure of silicon, is a good candidate for nanoelectronic and spintronic devices, because it has a high carrier mobility, large spin coherence lengths, and a relatively large intrinsic spin-orbit gap, and because it is relatively easier to induce band gap in silicene than in graphene, and it is possibly compatible with the contemporary industry3,4 . Spin Hall effect5 and anomalous Hall effect due to a relatively large spin-orbit interaction6 and a giant magnetoresistance7 have been investigated in a monolayer silicene. Silicene nanoribbons (SiNRs) are narrow stripes of silicene that have been synthesized with a very large aspect ratio8 . Silicene nanoribbons have significant electronic and transport properties suitable for nanoelectronic and spintronic devices9–11 . Moreover, a silicene field effect transistor has been fabricated at room temperature and showed a Diraclike ambipolar charge transport similar to graphene in ambient conditions9,10 . A high efficiency spin-filter device based on silicene nanoribbon with a ferroelectric polymer has also been reported11 . Lattice structures of silicene are very similar to those for graphene, but silicene structure is low-buckled and composed of two vertically moved sublattices unlike graphene3,4 . Due to its buckled structure, silicene could overcome graphene limitations for application in nanoelectronics. The combination of the perpendicular electric field and the low-buckled structure of silicene generates a controllable band gap and spin-polarized states. Some theoretical studies have proposed various designs to control and produce a spin-polarized current with an external electric field based on silicene nanostructures6,12–14 . Various theoretical mechanisms have also been investigated in order to generate a spin-polarized current in silicene nanoribbon. Local magnetic exchange field on one edge or on both edges of a silicene nanoribbon could produce a spin polarized current14,15 . A giant magnetoresistance originating from the edge magnetism in zigzag silicene nanoribbons (ZSiNR) has been the-
oretically predicted16 . Moreover, a large magnetoresistance has been reported in edge-doped SiNRs with various spin configurations17 . Edge states are crucial for applications of nanoribbons in spintronics. In zigzag-edge SiNRs, transport properties are dominated by edge states. Moreover, edge defects significantly affect the electronic, magnetic and transport properties of silicene nanostructures, similar to graphene18–22 . Various types of defects have been experimentally observed in nanoribbons, such as the Klein edge23 and pentagon-heptagon reconstructions24 . The effect of extended vacancies on silicon stability has been also investigated25 . Interestingly, Li et al. found a new class of spintronic materials as ”bipolar magnetic semiconductor” (BM S) in semi-hydrogenated single-walled carbon nanotubes26,27 . The robust BM S was predicted based on the intrinsic edge magnetism and designing in graphene nanoribbons28 . Moreover, silicene nanoribbons under asymmetry Klein defects, edge hydrogenation, and functionalized silicene sheet by nitrophenyl diazonium show BM S features21,29,30 . In this paper by using the non-equilibrium Green’s function formalism, we investigated spin-filter capabilities of a ZSiNR in the presence of electric and exchange magnetic fields. By tuning electric and exchange magnetic fields, we found the robust BM S in perfect and rough silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. Therefore a large spin-polarized current with a perfect and reversible spin polarization and controllable spin-flip and spinconserved gaps can be obtained in ZSiNRs. We also investigated the effect of extended vacancies along zigzag edges on the spin-filter capabilities of silicene junctions. The results show that the edge roughness can decrease the magnitude of charge and spin-polarized current, but the BM S features are robust against edge disorder in silicene nanoribbon junctions. The paper is organized as follows. In Sec. II, the Hamiltonian of silicene junction is formulated in the presence of electric and exchange magnetic fields and a theoretical model describing the spin-polarized transport is then derived using the non-equilibrium Green’s function formalism. In Sec. III, the results are presented and discussed. Section IV contains
conclusions. II. MODEL AND METHOD
The tight-binding Hamiltonian of a ZSiNR in the presence of the intrinsic spin-orbit interaction, and electric and exchange magnetic fields, can be written as6 : ˆσ = H
X λ z −t c†iα cjα + i √ νij c†iα σαβ cjβ 3 3α <>,αβ X † X +Mz ciα σz ciα + elEz µi c†iα ciα (1) X
iα
iα
In this expression, c†iα ( ciα ) creates (annihilates) an electron at site α ; and t is the tight binding parameter which will be set to t = 1.6 eV 6 < ij > and << ij >> denote the summation over all nearest and next-nearest neighbor hopping sites, respectively. The second term describes the effective spin-orbit coupling with coefficient of λ=3.9meV 6 . Here, σ is the Pauli matrix. νij =dj × di , where di and dj are two bonds connecting next-nearest neighbor atoms. In this expression, Mz , is the uniform exchange magnetic field with a strength of Mz . The fourth term is the staggered sublattice potential due to the low-buckled structure of silicene, Ez is external electric field perpendicular to the silicene plane, where µi = ±1 for the ˚ 6 . Note that, the strength of intrinsic A(B) site, and l=0.23A and extrinsic Rashba spin-orbit interactions are smaller than the intrinsic spin-orbit interaction, so we ignored them in our calculations. According to the Landauer-Buttiker formalism the spin-dependent conductance can be written as31 : 2
Gσ (ε) =
e ˆ σ (ε)Γ ˆ R (ε)G ˆ † (ε)]. ˆ L (ε)G T r[Γ σ h
(2)
ˆ σ is the spin-dependent Green’s function of the Where G ˆσ − Σ ˆ S (ε) − ˆ σ (ε) = [εIˆ− H junction and can be defined as: G −1 ˆ ΣD (ε)] . Using the self-energy matrices due to the connecˆ tion of electrodes to the channel Σ(ε), the coupling matrices 32 ˆ ˆ ˆ . Γ(ε) can be expressed as Γ(ε) = −2Im[Σ(ε)] Moreover, Pz (ε) spin polarization in z direction can be defined as:
Pz (ε) =
Gσ (ε) − G−σ (ε)) × 100 Gσ (ε) + G−σ (ε)
(3)
III. RESULTS AND DISCUSSION
This paper focuses mainly on the effect of electric and exchange magnetic fields on spin-filter features of silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. The exchange magnetic field can be originated from the proximity with a ferromagnetic layer or ferroelectric polymers and can induce spin splitting between spin-up and spindown bands. The value of exchange magnetic field in silicene is still unknown and we have used a different value, which
FIG. 1: The band structure of the zigzag-edge silicene nanoribbon for Mz = 0 and elEz = .10t (a), Mz = .10t and Ez = 0. The strength of intrinsic spin-orbit interaction is chosen λ = 3.9meV . The ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
is much smaller than the one reported in graphene9,33,34 . Recently, Wei et. al show that graphene coupling to a magnetic insulator (EuS) as an efficient control for local spin generation could induce a substantial exchange magnetic field with the potential to reach hundreds of tesla, which can produce the Zeeman splitting energy of magnitude 2 meV 35 . The electric field strength in experimental observation is about Ez = 10 V /nm36 . Fig. 1 shows the band structures of an ideal ZSiNR with 24 zigzag chains and 48 silicon atoms in the unit cell, in the presence of electric and exchange magnetic fields and the intrinsic spin-orbit interaction. In all calculations, we investigated low energy states around the fermi energy and around the K=±π/a points, and also set λ=3.9meV . The electric field could break the degeneracy between spin-up and spindown subbands and induces a gap energy about 2lEz . Note that, for elEz > λ the gap energy can be opened, similar to Ref.6 . As shown in Fig 1(a), the bands possess a spin inversion symmetry with respect to k axis, i.e., E↑ (k)=E↓ (−k). This spin splitting of band structures could not solely produce a spin-polarized current in silicene nanoribbon junctions. The effect of uniform exchange magnetic field on the band structure of a silicene nanoribbon is presented in Fig. 1(b). Clearly, a uniform exchange magnetic field in z direction breaks the spin inversion symmetry with respect to the Brillouin zone boundary. When a uniform exchange magnetic field is applied, the spin inversion symmetry occurs with respect to the
FIG. 2: The band structure of the zigzag-edge silicene nanoribbon in the presence of perpendicular electric field with elEz = .10t for Mz = .01t (a), Mz = .04t (b), Mz = .08t (c), Mz = .10t (d). The ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
Fermi energy (E=0), i.e. E↑ (k)=-E↓ (k). As a result, electric and uniform exchange magnetic fields cannot produce a spin-polarized current solely. So in order to enhance the spin-polarized features in silicene nanoribbons, we consider the effects of electric and exchange magnetic fields simultaneously. In order to understand the mutual effects of electric and exchange magnetic fields on electronic structure of silicene nanoribbons, we plotted the band structures in Fig. 2. We considered various strengths of exchange magnetic field with a constant electric field strength that is perpendicular to the surface of silicene nanoribbon. It is demonstrated that taking into account the effects of electric and exchange magnetic fields simultaneously produces an asymmetry in the spin-dependent band structure. In the rest of the paper, we show that by tuning the strengths of these fields, the spin asymmetry and the spin-polarized current can both be enhanced in ZSiNRs. Fig. 2(a) shows the band structure for an exchange magnetic field that is relatively weaker than the strength of electric field and is equal to the intrinsic spin-orbit interaction (Mz =λ). The results show that even a weak exchange magnetic field breaks the spin-inversion symmetry of the band structure in the presence of external electric field and the intrinsic spin-orbit interaction. It is interesting that the spindependent band structures show a little spin-splitting when the intrinsic spin-orbit interaction, electric field, and exchange magnetic field are present simultaneously. Therefore, the bro-
ken spin-inversion symmetry may causes a spin-polarized current in the silicene junction. Furthermore, we considered three different strengths of exchange magnetic field that are more than the strength of the intrinsic spin-orbit interaction, in order to emphasize on spin splitting effects between spin-up and spin-down bands. As the strength of exchange magnetic field (Mz =.04t) increases, the two spin bands are completely separated from each other, see Fig. 2 (b). Interestingly, the band structures show that ZSiNRs in the presence of electric and exchange magnetic fields exhibit an interesting bipolar magnetic semiconductor (BM S), where the spin orientation of top valence bands and bottom conduction bands flips around the Fermi energy. The bands in the vicinity of Fermi energy possess an opposite spin polarization. Thus, the BM S can be found in ZSiNR when exchange magnetic and electric field strengths are both very large compared to the intrinsic spin-orbit interaction and the value of elEz is more than the strength of exchange magnetic fields. Interestingly, we found that ZSiNRs are BM S that is robust to ribbon width, indicating the potential application of ZSiNRs in spin devices, similar to hydrogenated armchair silicene nanoribbons with transvers electric field modulations29 . This new class of spintronic materials can be defined by three energy parameters (∆1, ∆2, ∆3), as shown in Fig. 2(b). ”∆1” indicates the spin-flip gap between top valence and bottom conduction bands. ”∆1 + ∆2” and ”∆1 + ∆3” are the spin-conserved gaps between two spin-up
FIG. 3: Evolution of energy gap in a ZSiNR in the presence of intrinsic spin-orbit interaction. (a) The energy gap of the ZSiNR as a function of the exchange magnetic field strength in the presence of electric field with elEz = .10t. (b) The energy gap of the ZSiNR as a function of the electric field strength in the presence of exchange magnetic field with Mz = .02t.
and spin-down bands around the Fermi energy. Also due to the inversion symmetry for spin-up and spin-down electrons with respect to the energy axis, the values of ∆2 and ∆3 are equal, so ZSiNRs have two equal spin-conserved gaps. Band structures depicted in Fig. 2 show that as the exchange field strength increases the spin flip-gap decreases. Moreover, when the strengths of exchange magnetic fields and electric field become equal to each other, the spin flip-gap vanishes and ZSiNR exhibits a metallic behavior at the Fermi energy. In fact, in the energy interval of ( | E(k) |<λ), spin up and spin-down bands are degenerate. Therefore, the spin-flip gap vanishes but the spin-conserved gap persists and ZSiNR shows a bipolar spin polarization behavior in the interval energy of λ < | E(k) |
FIG. 4: The spin-dependent conductance of a ZSiNR with 12 zigzag chains and 480 atoms for various exchange magnetic field strengths Mz = .04t (a), Mz = .08t (b). The spin polarization for Mz = .04t, Mz = .08t (c). The strength of electric field is chosen elEz = .10t and the ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
exchange magnetic field shows that as the exchange magnetic strength increases the magnitude of spin-flip gap decreases, but for a constant electric field strength the spin-conserved gap is unchanged, so the sum of the spin-flip gap and ∆1 must be constant. The variation of spin-flip gap versus the exchange magnetic field strength shows the spin-filter capabilities of ZSiNR junctions. In order to clarify the effect of electric and exchange magnetic fields on the spin-flip and spin-conserved gaps, we plotted in Fig. 3 the variation of energy gap as a function of electric and exchange magnetic field strengths. The results show that the variation of electric and exchange magnetic field strengths could change the value of spin-flip gap, but the magnitude of the spin-conserved gap is only proportional to the electric field strength. We plotted in Fig. 3(a) the variation of energy gap as a function of exchange magnetic field for a constant electric field.
Interestingly, the spin-flip gap is about 2elEz − λ − 2Mz regardless of the nanoribbon width, which is in agreement with the variation of spin-flip gap depicted in Fig. 3(b), and the spin-conserved gap is constant. Furthermore, as the electric field strength increases for a constant exchange magnetic field the spin-conserved gap increases linearly (see Fig. 3b)). Furthermore, the spin-conserved gap is about 2elEz − λ regardless of the exchange magnetic field strength. Therefore, the ZSiNR exhibits a robust BM S feature with controllable spin-flip and spin-conserved gaps. Generally, BM S feature is created when exchange magnetic and electric field strengths are both larger than the intrinsic spin-orbit interaction and the value of elEz is slightly more than the strength of exchange magnetic fields, i.e. elEz 1 Mz > λ. In order to study the spin-filter features of silicene nanoribbon junctions in the presence of electric and exchange magnetic fields, we sandwiched zigzag silicene nanoribbon slices with 12 zigzag chains and 480 atoms between two zigzag silicene nanoribbons. The chemical potential of the semi-infinite electrode as the source is set equal to elEz − Mz . Thus by using silicene nanoribbons, we proposed bipolar spin-filter devices. Fig. 3 shows that the out-of-plane electric field opens a gap energy, so causes the conductance vanishes for | E(k) |
FIG. 5: (a) Schematic view of a ZSiNR with extended vacancies. (b) and (c) show the spin-dependent conductance and spin polarization as a function of energy for the rough ZSiNR with extended vacancies. The dashed lines show the conductance and spin polarization for the perfect ZSiNR without extended vacancies. The ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
termined by ∆1 gap, and the zero spin polarization is equal to the ∆1 gap, which is in agreement with the band structure calculations shown in Fig. 2. Generally, in the case of elEz 1 Mz > λ, the interplay between elEz and Mz preserves BM S features, but the width of spin energy gap significantly changes. We also investigated the effect of edge roughness on the spin polarization in ZSiNRs in the presence of external electric and exchange magnetic fields. An imperfect silicene flake with a finite length and 12 zigzag chains (Fig. 5(a)) is sandwiched between two perfect semi-infinite zigzag silicene nanoribbons as electrodes. In this calculation, the length of the finite zigzag nanoribbon is chosen as L = 20a where ˚ is the silicene lattice constant, and the strengths a = 3.86A of the exchange magnetic and electric fields are chosen as .04t and .10t, respectively. ZSiNRs with edge modifications with extended vacancies are investigated in Fig 5. Fig. 5(a) introduces extended vacancies along the zigzag edges of the nanoribbon. Extended vacancies are similar to single vacancies, except that now more than one atom are removed along the zigzag edges. The conductance curve shows that the extended vacancies could extremely reduce the charge and spin-
polarized currents in the rough silicene junction (see Fig 5(b)). For more clarification, the conductance function of a perfect silicene nanoribbon is also plotted in Fig. 5(b). However, perfect spin-polarized currents with a reversible spin polarization can be found in rough silicene nanoribbons. Moreover, due to the effect of edge roughness, electron and hole show an asymmetry spin polarization with respect to the energy axis. The edge roughness breaks the spin-inversion symmetry with respect to the energy axis. So the spin energy gap parameters of ∆2 and ∆3 have different values. As can be seen in Fig. 5, the spin energy gap determined by ∆2 and ∆3 for electron and hole have different widths in the conductance and spin polarization functions. Interestingly, in addition to the bipolar spin polarization in the vicinity of the Fermi energy, the spin polarization in Fig 5(c) shows a high spin polarization in some energy ranges far from the Fermi energy. Therefore, in some energy ranges the spin polarization may be enhanced by the edge disorder in a relatively large external electric and magnetic fields. In summary, robust BM S features can be found even with extended vacancies in ZSiNR when elEz 1 Mz > λ. In comparison with Ref. 6, we obtained a new electronic phase for silicene nanoribbon in the presence of electric and magnetic fields when Mz and elEz are both very larger than λ, and elEz is slightly more than Mz , i.e. elEz 1 Mz > λ. In detail, our results show that by modulation of external fields two different spin gaps can be opened near the Fermi energy. Therefore, these changes induce the bipolar magnetic semiconductor behavior in silicene nanoribbons. Throughout the study, we ignored the effect of temperature scattering; these factors affect the tight binding parameter and the band struc-
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IV. CONCLUSION
Using the single-band tight-binding approximation, we investigated the spin-filter capabilities of a zigzag silicene nanoribbons in the presence of electric and exchange magnetic fields. By tuning electric and exchange magnetic fields, we found a robust bipolar magnetic semiconductor in perfect and rough silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. Zigzag silicene nanoribbons exhibit BM S behavior when Mz and elEz are both very larger than λ, and elEz is slightly more than Mz , i.e. elEz 1 Mz > λ. Generally, a perfect and reversible spin polarization can be found in silicene nanoribbon junctions with the spin-flip gap and spin-conserved gaps tunable by exchange magnetic and external electric fields, i.e. spin-flip gap = 2elEz − λ − 2Mz and spin-conserved gap = 2elEz − λ. These properties could provide a feasible solution for controlling the direction of spin-polarized current in ZSiNRs. Furthermore, our proposed bipolar spin-filter device based on ZSiNRs can operate as multifunctional spin devices. We found robust BM S features in a rough silicene with a large spin-polarized current and a reversible spin polarization in the vicinity of the Fermi energy, similar to perfect silicene nanoribbons.
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A new electronic phase for silicene nanoribbon in the presence of electric and magnetic fields Bipolar magnetic semiconductor with controllable spin-flip and spin-conserved gaps in silicene Robust bipolar magnetic semiconductor features in a rough silicene. Perfect and reversible spin polarization in silicene nanoribbon junctions
S0304-8853(16)31808-X http://dx.doi.org/10.1016/j.jmmm.2017.04.016 MAGMA 62619
To appear in:
Journal of Magnetism and Magnetic Materials
Please cite this article as: R. Farghadan, Bipolar magnetic semiconductor in silicene nanoribbons, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.04.016
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.
Bipolar magnetic semiconductor in silicene nanoribbons Rouhollah Farghadan1, ∗ 1
Department of Physics, University of Kashan, Kashan,87317-51167, Iran (Dated: April 8, 2017)
A theoretical study was presented on generation of spin polarization in silicene nanoribbons using the singleband tight-binding approximation and the non-equilibrium Green’s function formalism. We focused on the effect of electric and exchange magnetic fields on the spin-filter capabilities of zigzag-edge silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. The results show that a robust bipolar magnetic semiconductor with controllable spin-flip and spin-conserved gaps can be obtained when exchange magnetic and electric field strengths are both larger than the intrinsic spin-orbit interaction. Therefore, zigzag silicene nanoribbons could act as bipolar and perfect spin filter devices with a large spin-polarized current and a reversible spin polarization in the vicinity of the Fermi energy. We also investigated the effect of edge roughness and found that the bipolar magnetic semiconductor features are robust against edge disorder in silicene nanoribbon junctions. These results may be useful in multifunctional spin devices based on silicene nanoribbons. PACS numbers:
I. INTRODUCTION
Combination of silicon, as the mainstream of semiconductors, with magnetic materials could widely improve the spinbased electronics1,2 . Silicene, the two dimensional honeycomb structure of silicon, is a good candidate for nanoelectronic and spintronic devices, because it has a high carrier mobility, large spin coherence lengths, and a relatively large intrinsic spin-orbit gap, and because it is relatively easier to induce band gap in silicene than in graphene, and it is possibly compatible with the contemporary industry3,4 . Spin Hall effect5 and anomalous Hall effect due to a relatively large spin-orbit interaction6 and a giant magnetoresistance7 have been investigated in a monolayer silicene. Silicene nanoribbons (SiNRs) are narrow stripes of silicene that have been synthesized with a very large aspect ratio8 . Silicene nanoribbons have significant electronic and transport properties suitable for nanoelectronic and spintronic devices9–11 . Moreover, a silicene field effect transistor has been fabricated at room temperature and showed a Diraclike ambipolar charge transport similar to graphene in ambient conditions9,10 . A high efficiency spin-filter device based on silicene nanoribbon with a ferroelectric polymer has also been reported11 . Lattice structures of silicene are very similar to those for graphene, but silicene structure is low-buckled and composed of two vertically moved sublattices unlike graphene3,4 . Due to its buckled structure, silicene could overcome graphene limitations for application in nanoelectronics. The combination of the perpendicular electric field and the low-buckled structure of silicene generates a controllable band gap and spin-polarized states. Some theoretical studies have proposed various designs to control and produce a spin-polarized current with an external electric field based on silicene nanostructures6,12–14 . Various theoretical mechanisms have also been investigated in order to generate a spin-polarized current in silicene nanoribbon. Local magnetic exchange field on one edge or on both edges of a silicene nanoribbon could produce a spin polarized current14,15 . A giant magnetoresistance originating from the edge magnetism in zigzag silicene nanoribbons (ZSiNR) has been the-
oretically predicted16 . Moreover, a large magnetoresistance has been reported in edge-doped SiNRs with various spin configurations17 . Edge states are crucial for applications of nanoribbons in spintronics. In zigzag-edge SiNRs, transport properties are dominated by edge states. Moreover, edge defects significantly affect the electronic, magnetic and transport properties of silicene nanostructures, similar to graphene18–22 . Various types of defects have been experimentally observed in nanoribbons, such as the Klein edge23 and pentagon-heptagon reconstructions24 . The effect of extended vacancies on silicon stability has been also investigated25 . Interestingly, Li et al. found a new class of spintronic materials as ”bipolar magnetic semiconductor” (BM S) in semi-hydrogenated single-walled carbon nanotubes26,27 . The robust BM S was predicted based on the intrinsic edge magnetism and designing in graphene nanoribbons28 . Moreover, silicene nanoribbons under asymmetry Klein defects, edge hydrogenation, and functionalized silicene sheet by nitrophenyl diazonium show BM S features21,29,30 . In this paper by using the non-equilibrium Green’s function formalism, we investigated spin-filter capabilities of a ZSiNR in the presence of electric and exchange magnetic fields. By tuning electric and exchange magnetic fields, we found the robust BM S in perfect and rough silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. Therefore a large spin-polarized current with a perfect and reversible spin polarization and controllable spin-flip and spinconserved gaps can be obtained in ZSiNRs. We also investigated the effect of extended vacancies along zigzag edges on the spin-filter capabilities of silicene junctions. The results show that the edge roughness can decrease the magnitude of charge and spin-polarized current, but the BM S features are robust against edge disorder in silicene nanoribbon junctions. The paper is organized as follows. In Sec. II, the Hamiltonian of silicene junction is formulated in the presence of electric and exchange magnetic fields and a theoretical model describing the spin-polarized transport is then derived using the non-equilibrium Green’s function formalism. In Sec. III, the results are presented and discussed. Section IV contains
conclusions. II. MODEL AND METHOD
The tight-binding Hamiltonian of a ZSiNR in the presence of the intrinsic spin-orbit interaction, and electric and exchange magnetic fields, can be written as6 : ˆσ = H
X λ z −t c†iα cjα + i √ νij c†iα σαβ cjβ 3 3
iα
iα
In this expression, c†iα ( ciα ) creates (annihilates) an electron at site α ; and t is the tight binding parameter which will be set to t = 1.6 eV 6 < ij > and << ij >> denote the summation over all nearest and next-nearest neighbor hopping sites, respectively. The second term describes the effective spin-orbit coupling with coefficient of λ=3.9meV 6 . Here, σ is the Pauli matrix. νij =dj × di , where di and dj are two bonds connecting next-nearest neighbor atoms. In this expression, Mz , is the uniform exchange magnetic field with a strength of Mz . The fourth term is the staggered sublattice potential due to the low-buckled structure of silicene, Ez is external electric field perpendicular to the silicene plane, where µi = ±1 for the ˚ 6 . Note that, the strength of intrinsic A(B) site, and l=0.23A and extrinsic Rashba spin-orbit interactions are smaller than the intrinsic spin-orbit interaction, so we ignored them in our calculations. According to the Landauer-Buttiker formalism the spin-dependent conductance can be written as31 : 2
Gσ (ε) =
e ˆ σ (ε)Γ ˆ R (ε)G ˆ † (ε)]. ˆ L (ε)G T r[Γ σ h
(2)
ˆ σ is the spin-dependent Green’s function of the Where G ˆσ − Σ ˆ S (ε) − ˆ σ (ε) = [εIˆ− H junction and can be defined as: G −1 ˆ ΣD (ε)] . Using the self-energy matrices due to the connecˆ tion of electrodes to the channel Σ(ε), the coupling matrices 32 ˆ ˆ ˆ . Γ(ε) can be expressed as Γ(ε) = −2Im[Σ(ε)] Moreover, Pz (ε) spin polarization in z direction can be defined as:
Pz (ε) =
Gσ (ε) − G−σ (ε)) × 100 Gσ (ε) + G−σ (ε)
(3)
III. RESULTS AND DISCUSSION
This paper focuses mainly on the effect of electric and exchange magnetic fields on spin-filter features of silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. The exchange magnetic field can be originated from the proximity with a ferromagnetic layer or ferroelectric polymers and can induce spin splitting between spin-up and spindown bands. The value of exchange magnetic field in silicene is still unknown and we have used a different value, which
FIG. 1: The band structure of the zigzag-edge silicene nanoribbon for Mz = 0 and elEz = .10t (a), Mz = .10t and Ez = 0. The strength of intrinsic spin-orbit interaction is chosen λ = 3.9meV . The ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
is much smaller than the one reported in graphene9,33,34 . Recently, Wei et. al show that graphene coupling to a magnetic insulator (EuS) as an efficient control for local spin generation could induce a substantial exchange magnetic field with the potential to reach hundreds of tesla, which can produce the Zeeman splitting energy of magnitude 2 meV 35 . The electric field strength in experimental observation is about Ez = 10 V /nm36 . Fig. 1 shows the band structures of an ideal ZSiNR with 24 zigzag chains and 48 silicon atoms in the unit cell, in the presence of electric and exchange magnetic fields and the intrinsic spin-orbit interaction. In all calculations, we investigated low energy states around the fermi energy and around the K=±π/a points, and also set λ=3.9meV . The electric field could break the degeneracy between spin-up and spindown subbands and induces a gap energy about 2lEz . Note that, for elEz > λ the gap energy can be opened, similar to Ref.6 . As shown in Fig 1(a), the bands possess a spin inversion symmetry with respect to k axis, i.e., E↑ (k)=E↓ (−k). This spin splitting of band structures could not solely produce a spin-polarized current in silicene nanoribbon junctions. The effect of uniform exchange magnetic field on the band structure of a silicene nanoribbon is presented in Fig. 1(b). Clearly, a uniform exchange magnetic field in z direction breaks the spin inversion symmetry with respect to the Brillouin zone boundary. When a uniform exchange magnetic field is applied, the spin inversion symmetry occurs with respect to the
FIG. 2: The band structure of the zigzag-edge silicene nanoribbon in the presence of perpendicular electric field with elEz = .10t for Mz = .01t (a), Mz = .04t (b), Mz = .08t (c), Mz = .10t (d). The ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
Fermi energy (E=0), i.e. E↑ (k)=-E↓ (k). As a result, electric and uniform exchange magnetic fields cannot produce a spin-polarized current solely. So in order to enhance the spin-polarized features in silicene nanoribbons, we consider the effects of electric and exchange magnetic fields simultaneously. In order to understand the mutual effects of electric and exchange magnetic fields on electronic structure of silicene nanoribbons, we plotted the band structures in Fig. 2. We considered various strengths of exchange magnetic field with a constant electric field strength that is perpendicular to the surface of silicene nanoribbon. It is demonstrated that taking into account the effects of electric and exchange magnetic fields simultaneously produces an asymmetry in the spin-dependent band structure. In the rest of the paper, we show that by tuning the strengths of these fields, the spin asymmetry and the spin-polarized current can both be enhanced in ZSiNRs. Fig. 2(a) shows the band structure for an exchange magnetic field that is relatively weaker than the strength of electric field and is equal to the intrinsic spin-orbit interaction (Mz =λ). The results show that even a weak exchange magnetic field breaks the spin-inversion symmetry of the band structure in the presence of external electric field and the intrinsic spin-orbit interaction. It is interesting that the spindependent band structures show a little spin-splitting when the intrinsic spin-orbit interaction, electric field, and exchange magnetic field are present simultaneously. Therefore, the bro-
ken spin-inversion symmetry may causes a spin-polarized current in the silicene junction. Furthermore, we considered three different strengths of exchange magnetic field that are more than the strength of the intrinsic spin-orbit interaction, in order to emphasize on spin splitting effects between spin-up and spin-down bands. As the strength of exchange magnetic field (Mz =.04t) increases, the two spin bands are completely separated from each other, see Fig. 2 (b). Interestingly, the band structures show that ZSiNRs in the presence of electric and exchange magnetic fields exhibit an interesting bipolar magnetic semiconductor (BM S), where the spin orientation of top valence bands and bottom conduction bands flips around the Fermi energy. The bands in the vicinity of Fermi energy possess an opposite spin polarization. Thus, the BM S can be found in ZSiNR when exchange magnetic and electric field strengths are both very large compared to the intrinsic spin-orbit interaction and the value of elEz is more than the strength of exchange magnetic fields. Interestingly, we found that ZSiNRs are BM S that is robust to ribbon width, indicating the potential application of ZSiNRs in spin devices, similar to hydrogenated armchair silicene nanoribbons with transvers electric field modulations29 . This new class of spintronic materials can be defined by three energy parameters (∆1, ∆2, ∆3), as shown in Fig. 2(b). ”∆1” indicates the spin-flip gap between top valence and bottom conduction bands. ”∆1 + ∆2” and ”∆1 + ∆3” are the spin-conserved gaps between two spin-up
FIG. 3: Evolution of energy gap in a ZSiNR in the presence of intrinsic spin-orbit interaction. (a) The energy gap of the ZSiNR as a function of the exchange magnetic field strength in the presence of electric field with elEz = .10t. (b) The energy gap of the ZSiNR as a function of the electric field strength in the presence of exchange magnetic field with Mz = .02t.
and spin-down bands around the Fermi energy. Also due to the inversion symmetry for spin-up and spin-down electrons with respect to the energy axis, the values of ∆2 and ∆3 are equal, so ZSiNRs have two equal spin-conserved gaps. Band structures depicted in Fig. 2 show that as the exchange field strength increases the spin flip-gap decreases. Moreover, when the strengths of exchange magnetic fields and electric field become equal to each other, the spin flip-gap vanishes and ZSiNR exhibits a metallic behavior at the Fermi energy. In fact, in the energy interval of ( | E(k) |<λ), spin up and spin-down bands are degenerate. Therefore, the spin-flip gap vanishes but the spin-conserved gap persists and ZSiNR shows a bipolar spin polarization behavior in the interval energy of λ < | E(k) |
FIG. 4: The spin-dependent conductance of a ZSiNR with 12 zigzag chains and 480 atoms for various exchange magnetic field strengths Mz = .04t (a), Mz = .08t (b). The spin polarization for Mz = .04t, Mz = .08t (c). The strength of electric field is chosen elEz = .10t and the ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
exchange magnetic field shows that as the exchange magnetic strength increases the magnitude of spin-flip gap decreases, but for a constant electric field strength the spin-conserved gap is unchanged, so the sum of the spin-flip gap and ∆1 must be constant. The variation of spin-flip gap versus the exchange magnetic field strength shows the spin-filter capabilities of ZSiNR junctions. In order to clarify the effect of electric and exchange magnetic fields on the spin-flip and spin-conserved gaps, we plotted in Fig. 3 the variation of energy gap as a function of electric and exchange magnetic field strengths. The results show that the variation of electric and exchange magnetic field strengths could change the value of spin-flip gap, but the magnitude of the spin-conserved gap is only proportional to the electric field strength. We plotted in Fig. 3(a) the variation of energy gap as a function of exchange magnetic field for a constant electric field.
Interestingly, the spin-flip gap is about 2elEz − λ − 2Mz regardless of the nanoribbon width, which is in agreement with the variation of spin-flip gap depicted in Fig. 3(b), and the spin-conserved gap is constant. Furthermore, as the electric field strength increases for a constant exchange magnetic field the spin-conserved gap increases linearly (see Fig. 3b)). Furthermore, the spin-conserved gap is about 2elEz − λ regardless of the exchange magnetic field strength. Therefore, the ZSiNR exhibits a robust BM S feature with controllable spin-flip and spin-conserved gaps. Generally, BM S feature is created when exchange magnetic and electric field strengths are both larger than the intrinsic spin-orbit interaction and the value of elEz is slightly more than the strength of exchange magnetic fields, i.e. elEz 1 Mz > λ. In order to study the spin-filter features of silicene nanoribbon junctions in the presence of electric and exchange magnetic fields, we sandwiched zigzag silicene nanoribbon slices with 12 zigzag chains and 480 atoms between two zigzag silicene nanoribbons. The chemical potential of the semi-infinite electrode as the source is set equal to elEz − Mz . Thus by using silicene nanoribbons, we proposed bipolar spin-filter devices. Fig. 3 shows that the out-of-plane electric field opens a gap energy, so causes the conductance vanishes for | E(k) |
FIG. 5: (a) Schematic view of a ZSiNR with extended vacancies. (b) and (c) show the spin-dependent conductance and spin polarization as a function of energy for the rough ZSiNR with extended vacancies. The dashed lines show the conductance and spin polarization for the perfect ZSiNR without extended vacancies. The ↑ and ↓ signs correspond to the majority and minority spin electrons, respectively.
termined by ∆1 gap, and the zero spin polarization is equal to the ∆1 gap, which is in agreement with the band structure calculations shown in Fig. 2. Generally, in the case of elEz 1 Mz > λ, the interplay between elEz and Mz preserves BM S features, but the width of spin energy gap significantly changes. We also investigated the effect of edge roughness on the spin polarization in ZSiNRs in the presence of external electric and exchange magnetic fields. An imperfect silicene flake with a finite length and 12 zigzag chains (Fig. 5(a)) is sandwiched between two perfect semi-infinite zigzag silicene nanoribbons as electrodes. In this calculation, the length of the finite zigzag nanoribbon is chosen as L = 20a where ˚ is the silicene lattice constant, and the strengths a = 3.86A of the exchange magnetic and electric fields are chosen as .04t and .10t, respectively. ZSiNRs with edge modifications with extended vacancies are investigated in Fig 5. Fig. 5(a) introduces extended vacancies along the zigzag edges of the nanoribbon. Extended vacancies are similar to single vacancies, except that now more than one atom are removed along the zigzag edges. The conductance curve shows that the extended vacancies could extremely reduce the charge and spin-
polarized currents in the rough silicene junction (see Fig 5(b)). For more clarification, the conductance function of a perfect silicene nanoribbon is also plotted in Fig. 5(b). However, perfect spin-polarized currents with a reversible spin polarization can be found in rough silicene nanoribbons. Moreover, due to the effect of edge roughness, electron and hole show an asymmetry spin polarization with respect to the energy axis. The edge roughness breaks the spin-inversion symmetry with respect to the energy axis. So the spin energy gap parameters of ∆2 and ∆3 have different values. As can be seen in Fig. 5, the spin energy gap determined by ∆2 and ∆3 for electron and hole have different widths in the conductance and spin polarization functions. Interestingly, in addition to the bipolar spin polarization in the vicinity of the Fermi energy, the spin polarization in Fig 5(c) shows a high spin polarization in some energy ranges far from the Fermi energy. Therefore, in some energy ranges the spin polarization may be enhanced by the edge disorder in a relatively large external electric and magnetic fields. In summary, robust BM S features can be found even with extended vacancies in ZSiNR when elEz 1 Mz > λ. In comparison with Ref. 6, we obtained a new electronic phase for silicene nanoribbon in the presence of electric and magnetic fields when Mz and elEz are both very larger than λ, and elEz is slightly more than Mz , i.e. elEz 1 Mz > λ. In detail, our results show that by modulation of external fields two different spin gaps can be opened near the Fermi energy. Therefore, these changes induce the bipolar magnetic semiconductor behavior in silicene nanoribbons. Throughout the study, we ignored the effect of temperature scattering; these factors affect the tight binding parameter and the band struc-
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IV. CONCLUSION
Using the single-band tight-binding approximation, we investigated the spin-filter capabilities of a zigzag silicene nanoribbons in the presence of electric and exchange magnetic fields. By tuning electric and exchange magnetic fields, we found a robust bipolar magnetic semiconductor in perfect and rough silicene nanoribbons in the presence of the intrinsic spin-orbit interaction. Zigzag silicene nanoribbons exhibit BM S behavior when Mz and elEz are both very larger than λ, and elEz is slightly more than Mz , i.e. elEz 1 Mz > λ. Generally, a perfect and reversible spin polarization can be found in silicene nanoribbon junctions with the spin-flip gap and spin-conserved gaps tunable by exchange magnetic and external electric fields, i.e. spin-flip gap = 2elEz − λ − 2Mz and spin-conserved gap = 2elEz − λ. These properties could provide a feasible solution for controlling the direction of spin-polarized current in ZSiNRs. Furthermore, our proposed bipolar spin-filter device based on ZSiNRs can operate as multifunctional spin devices. We found robust BM S features in a rough silicene with a large spin-polarized current and a reversible spin polarization in the vicinity of the Fermi energy, similar to perfect silicene nanoribbons.
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A new electronic phase for silicene nanoribbon in the presence of electric and magnetic fields Bipolar magnetic semiconductor with controllable spin-flip and spin-conserved gaps in silicene Robust bipolar magnetic semiconductor features in a rough silicene. Perfect and reversible spin polarization in silicene nanoribbon junctions