Spin Negative Differential Resistance in Zigzag Graphene

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C: Energy Conversion and Storage; Energy and Charge Transport

Spin Negative Differential Resistance in Zigzag Graphene Nanoribbons with Side-Attached Porphine Molecule Aldilene Saraiva-Souza, Manuel Smeu, José Gadelha da Silva Filho, Eduardo Costa Girão, and Hong Guo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04633 • Publication Date (Web): 25 Jun 2018 Downloaded from http://pubs.acs.org on June 26, 2018

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The Journal of Physical Chemistry

Spin Negative Differential Resistance in Zigzag Graphene Nanoribbons with Side-Attached Porphine Molecule Aldilene Saraiva-Souza,∗,† Manuel Smeu,∗,‡ José Gadelha da Silva Filho,¶ Eduardo Costa Girão,† and Hong Guo§ †Departamento de F´ısica, Universidade Federal do Piau´ı, Teresina, PI, Brazil. ‡Department of Physics, Binghamton University, State University of New York, Binghamton, NY, 13902 ¶Departamento de F´ısica, Universidade Federal do Ceará, P.O. Box 6030, CEP 60455-900, Fortaleza, Ceará, Brazil §Centre for the Physics of Materials and Department of Physics, McGill University, Montreal, QC, Canada. E-mail: [email protected]; [email protected]

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Abstract An isolated roughness at the edge of zigzag graphene nanoribbons (zGNRs) breaks the natural bipartite symmetry in the graphene lattice, which might lead to magnetic frustration. Interestingly, the coupling between delocalized states of the roughness and the extended states of the zGNRs induce quantum phenomena, such as, destructive quantum interference (DQI), which might play an important role to allow or forbid electrons transfer through a system. In this work, the electronic and spin-polarized transport properties of the porphine molecule attached to the 6zGNR via different βand meso- connections are studied using nonequilibrium Green’s function calculations performed within the density functional theory framework (NEGF-DFT). Here, we demonstrated that the porphine/6zGNRs present the FM ground state for all the couplings and the hybridization between porphine and 6zGNR disturbs the bipartite graphene lattice, leading to a spin-frustration. Particularly, the porphine molecule introduces Fano-antiresonances in the transmission spectrum that can be controlled by application of a gate voltage (Vg ) over the energy position, generating a spin filter devices with a high degree of spin polarization of almost 98%. At low bias, the currentvoltage (I-Vb ) characteristics of this system reveal a negative differential resistance (NDR) behavior followed by a flip in spin-current caused by the breaking of symmetry in the scattering region.

Introduction Graphene nanoribbons (GNRs) have gained significant interest owing to their electronic structure that strongly depends on the ribbon width and the edge topology. 1,2 The zigzag GNRs have drawn particular attention because of their peculiar localized states around the zigzag edges, giving rise to the relatively flat bands near the Fermi level. 3–5 Theoretical works have shown that these localized edge states are strongly spin-polarized and coupled with the charge carrier mobility, which makes this material a promising candidate for magnetotransport applications. 6–9 In the ground state, the perfect zGNRs possess ferromagnetic 2


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(FM) ordering along the edge and antiferromagnetic (AF) between the two edges. However, characterization of the edge magnetism in zGNRs is still very limited experimentally due to the huge challenge that top-down techniques face such as the absence of pure sp2 coordination in the edge sites (pedge modifications and impurities) and high hydrogen concentration. On the other hand, a successful bottom-up synthesis reported an important step forward by producing zGNRs either with precise and smooth edges or with topological defects along the edges, which bring the possibility to find magnetic states and confined spin to reality. 10 Theoretical calculations based on density functional theory have revealed that such topological modification breaks the bipartite natural symmetry of the graphene lattice and induces magnetic frustration in the π-bonding as well as in the nearest neighbor atoms. 11,12 Notably, strong efforts have been devoted recently to achieve the ferromagnetic (FM) behavior and nonzero spin conductance in the zGNRs using edge functionalization with single atom or roughness (non-periodic localized defects). 13–16 In such cases, the σ mirror plane that bisects the zGNRs along the periodic direction also vanished due to the broken symmetry, which provides peculiar spin transport properties. For instance, the forbidden transmission energy channels π → π ∗ (π ∗ → π) migth be allowed by resonant tunneling. 6,7,17 Moreover, the coupling between delocalized states of the roughness and the extended states of the zGNRs induces quantum phenomena, including Coulomb blockade, 18,19 resonant tunneling 20 and quantum interference effects (QI). 21 Typically, the holes in T-shaped GNRs and heterocyclic molecules tend to exhibit the QI mechanism and have demonstrated control over the on/off current ratios. 22,23 The current that flows in these systems is determined by the degree of destructive or constructive QI effect between the paths around the hole-shape 24,25 Recently, there was an experimental report on the lateral anchoring of a porphine molecule to the graphene structure through covalent linkages. 26 Several distinct heterocoupling configurations were observed that are influenced by the local structure of the graphene edge and the ability of porphines to engage in lateral bonding schemes involving different numbers of β- and meso- coupling positions.

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Moreover, very recently, the successful construction and magnetic characterization of porphyrin molecules bonded to chiral GNRs have been investigated using scanning tunneling microscopy (STM). 27 In particular, it was demonstrated that the magnetic properties of the porphyrin molecule are maintained when it is attached to the GNRs, which makes this system an ideal material to be used as the electrode and in bringing such electronic devices to reality. Thus, this design of structure might offer a distinct electronic transport behavior because the QI mechanism in the ring-shape molecule is preserved in different hybridizations during coupling. Moreover, the electronic and magnetic properties of porphine-based molecules can be effectively modulated. 28–30 In this regard, we use first-principles calculations to study the electronic and spinpolarized transport properties of the typical porphine molecule attached to the 6zGNR via different β- and meso- connections, as can seen in Fig. 1. We found that, in contrast to the pristine zGNRs, the porphine/6zGNRs present the FM ground state for all the couplings. Moreover, the hybridization between porphine and 6zGNR disturbs the bipartite graphene lattice, leading to a spin-frustration. More importantly, our calculations show that the porphine molecule introduces Fano-antiresonances in the transmission function and their energetic position depends on the kind of coupling. We explore the dependence of these Fano-antiresonances under gate voltage (Vg ) with the goal of controlling the spin-polarized transport at zero bias. The results show a spin filter devices with a high degree of spin polarization of almost 98%. The transport properties controlled by a finite source-drain bias were studied for all systems. At low bias, the Fano-antiresonances induced from the porphine molecule reveal a negative differential resistance (NDR) behavior. As the voltage rises, the spin-up current sharply increases and the spin-down fluctuates around 10 µA. This behaivor migth be related with the breaking of symmetry in the scattering region.

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Figure 1: (a) Schematic representation of two-probe system where typical porphine molecule is attached to 6zGNR, where β- (red circle) and meso- (pink circle) positions indicate the two major points available for coupling. The junction is composed of three parts, two leads (left and right) based on 6zGNRs and the scattering region (porphine/6zGNR). Schematic representation of the junctions where porphine is linked by (b) β-, β- (M1), (c) β-, β- (M2), (d) β-, meso-, β- (M3) and (e) β-, β-, meso-, β- (M4) at 6zGNR. Note that the M1 and M2 structures differ by the arragement of H atoms in the middle of the porphine ring. A and B represent the different A/B sublattice carbon atoms in the nanoribbon and the numbers (1,...,11) indicate the atoms on porphine. The atoms are coloured as white for H, blue for N and black for C.

Computational details The structures of the proposed device can be divided into three regions: two electrodes (left and right) and a central scattering region, as schematically shown in Fig. 1. The semiinfinite electrodes are modeled by using a 1 × 1 × 3 six-layer zGNRs supercell (6zGNRs), with periodic boundary conditions along the z-direction, while the central scattering region consists of a porphine molecule attached in different configurations at the edge sites (A and B) of a 6zGNR supercell (59.03 ˚ A in length). In Fig. 1(a), we can see that the porphine

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molecule exhibits two major points available for coupling, i.e., four meso- (pink circle) and eight β- (red circle). According to the distinct heterocoupling configurations, four two-probe devices were modelled, namely: M1 (β-, β-), M2 (β-, β-), M3 (β-, meso-, β-) and M4 (β-, β-, meso-, β-), as schematically shown in Fig. 1(b-e), respectively. Note that the M1 and M2 structures differ by the arragement of H atoms in the middle of the porphine ring. The equilibrium configurations and electronic properties of each structure were investigated using density functional theory (DFT), 31,32 as implemented in the SIESTA code, 33 which provides self-consistent calculations by expanding the Kohn-Sham (KS) orbitals as a linear combination of atomic orbitals (LCAO) for the valence electrons. The PerdewBurke-Ernzerhof approach was employed for the exchange-correlation functional in the spindependent generalized gradient approximation (SGGA). 34 Core electrons were described by Troullier-Martins 35 pseudo-potentials and a polarized double-ζ (DZP) basis was used for the basis set and a cutoff energy of 300 Ry. Atom positions were optimized through a conjugate gradient algorithm until the residual force acting on each atom were less than 10−5 Ry/˚ A. A vacuum region of 10 ˚ A was included along the non-periodic directions (x- and y-axis) to avoid interactions between adjacent images. The Brillouin-zone (BZ) integrations for the central scattering region were performed with 1×1×4 Monkhorst-Pack mesh. 36 The spin-dependent transport properties were investigated using DFT combined with the nonequilibrium Green’s function (NEGF) technique within the Keldysh formalism, 37,38 via the Nanodcal code. 39,40 In these calculations, all atoms are described self-consistently at the same level of theory for the central region as well as the leads. The Brillouin zone samplings were represented by a Monkhorst-Pack grid of 1×1×100 and 1×1×1 k-points for the (left and right) electrodes and central scattering region, respectively. To briefly summarize, the retarded Green’s function is defined as

G = (E + S − H − ΣL (E) − ΣR (E))−1 ,

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(1)

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where E + = limη→0 [E + iη] is the energy plus an infinitesimal imaginary part iη. H is the Hamiltonian, and S is the corresponding overlap matrix obtained from a conventional DFT calculation on the central scattering region. ΣL/R are the self-energies that account for the effects of each electrode on the central scattering region. Each consists of two parts: the energy level shift is given by the real part as ∆L/R (E) = ReΣL/R (E), and the level broadening is given by the imaginary part: ΓL/R (E) = i(ΣL/R − Σ†L/R ).

(2)

The spin resolved transmission coefficient 37 is calculated by the trace over the matrix product of the coupling matrices ΓL/R,σ 41 and the (G/G† ) Green’s function of the central region. The transmission around the Fermi energy at zero bias is [Tσ (E, Vb = 0)], Tσ (E, Vb ) = Tr(ΓR,σ Gσ ΓL,σ G† ),

(3)

which represents the probability that an electron having spin σ and with a given energy E transmits from the left electrode, through the central region, into the right electrode. Under finite bias voltage (Vb ), the spin resolved current was obtained by integrating the transmission function in the bias window [−Vb/2 , Vb/2 ], e Iσ (E, Vb ) = − h

Z

µR

Tσ [f (E − µL ) − f (E − µR )]dE,

(4)

µL

where h is the Plank’s constant, e the the electron charge, eV = µL − µR the electrochemical potential difference between left and right electrodes and f is the Fermi distribution,

fL/R (E, µ) =

1 exp[(E − µL/R )/kT ] + 1

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.

(5)


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The spin filter efficiency 42,43 (SFE) at the Fermi level is defined as

SFE =

|Tup (EF ) − Tdown (EF )| × 100% Tup (EF ) + Tdown (EF )

(6)

and represents the excess transmission of one spin type over the other as a percentage of the total transmission. We calculated the SFE achieved at zero bias under different gate voltages (Vg ) for all systems. The gate effect is simulated by setting a gate-induced electrostatic boundary condition for the Hartree potential when solving the Poisson equation in the NEGF-DFT self-consistent procedure.

Results and discussion The kind of coupling between porphine/6zGNR is extremely relevant to the transport behavior through these molecular junction since a minor chemical modification in the system might dramatically affect the electronic transmission. For this reason, we first examine the geometries and how the porphine molecule will affect the electronic structure of 6zGNR when it is attached on the edge of the nanoribbon. Previous theoretical studies showed that, in the pristine zGNRs, the two nonequivalent carbon atoms that form the A and B bipartite sublattices are occupied by the spin-up and spin-down electrons exhibiting an anti-ferromagnetic ordering in the ground state, which means that each edge is represented exclusively by one carbon sublattice. 44,45 At the coupling between porphine and 6zGNR, the atom on the A sub-lattice possesses three neighboring atoms from the B sub-lattice, being analogous to a topological defect coming from the pentagonal ring at the edge of the zigzag backbone. This arrangement disturbs the bipartite graphene lattice and breaks translational symmetry with respect to the σ mirror plane of the zGNRs. 6,7,10,12 For the 6zGNR not affected by porphine, the optimized C–C and C–H bond lengths were fairly uniform, about 1.42 ˚ A and 1.08 ˚ A, respectively. In the coupling between the molecule and ribbon edge, the bond lengths are computed and found to be in good agreement 8


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with previous experimental values. 26 As shown in Table 1, the CA –CB bond lengths in the 6zGNR are about 1.44 ˚ A (M1 and M3) and 1.43 ˚ A (M2 and M4), which is very close to the experimental value (1.43±0.08 ˚ A) for porphine molecule coulpled in the zigzag edge of the graphene structure via the β-,β-position. The CA –CA distance is the graphene lattice constant, an impotant parameter related to the coupling configuration between the porphine and nanoribbon. For the M1 and M2 systems the calculated values are about 2.39 ˚ A. This same distance is slightly greater in the M3 and M4 systems with values of 2.51 ˚ A and 2.48 ˚ A, respectively. These predictions are slightly different from the experimental value of 2.48±0.15 ˚ A (porphine/graphene via β-,meso-,β-position). 26 ˚) between carbons atoms in different couTable 1: Optimized bond lengths (A plings of porphine/6zGNR. The experimental values of porphine/graphene are included in parentheses for comparison. 26

CA −CB CA −CA C1 −C2 ; C3 −C4 C2 −C3 C3 −C6 C6 −C8 C8 −C9 C8 −C7 ; C9 −C10 CA −C2 CA −C3 CA −C6 ; CA −C8 CB −C3

M1 M2 M3 1.44 (1.43±0.08) 1.43 (1.43±0.08) 1.44 2.39 2.39 2.51 (1.43±0.08) 1.45 1.44 1.44 1.43 (1.46±0.15) 1.44 (1.46±0.15) 1.41 − − 2.50 (2.81±0.18) − − 2.50 (2.81±0.18) − − 1.41 − − 1.42 1.43 1.44 − 1.43 1.44 1.44 − − 1.44 − − −

M4 1.43 2.48 1.43 1.41 2.46 − 1.41 1.45 − 1.42 1.42 2.46

In the porphine molecule, the bond lengths between two neighboring carbon atoms is relatively shorted compared to the experimental values. For instance, the C1 –C2 bond lengths are 1.45 ˚ A (M1), 1.44 ˚ A (M2 and M3) and 1.43 ˚ A (M4) while the experimental value is about 1.46±0.15 ˚ A for porphine/graphene in β-,β-position. 26 This reduction of bond length respect to experimental values is expected for organic compounds. 42 The bond length at the coupling of porphine/6zGNR is defined as the average distance between carbon atoms at the connection points (CA –Cx ; where x= 2, 3, 6 and 8). We found this value to be around 9



1.43 ˚ A (M1), 1.44 ˚ A (M2 and M3) and 1.42 ˚ A (M4), which is quite close to the length of the C–C bond in the pristine graphene sheet. 46 The magnetic ground state is calculated for all the systems by performing the total energy difference between AFM and FM ordering (∆E = EAFM − EFM ). The ∆E for M1 (0.3 eV) and M3 (0.43 eV) are higher than those of the M2 (0.06 eV) and M4 (0.04 eV) systems. Thus energy difference suggests that these structures, unlike the zGNRs, display a stable ferromagnetic (FM) spin configuration. Moreover, it is important to note that these energy differences are very high, which means that even the lowest energy value (M4 ≈ 0.04 eV) is larger than the room-temperature thermal energy (kB T ) around 25 meV, making it possible to discriminate experimentally. Therefore, only the ferromagnetic ordering will be explored in this work.

Figure 2: (a-d) Plots of spin density isosurfaces for M1, M2, M3 and M4, respectively. Note that the different couplings change the ordering and the extent of the magnetization in the porphine molecule. The colors blue and red correspond to spin-up and spin-down (orbital contour value is 0.002) The high value of ∆E is directly related to strong magnetic moment. Figure 2 presents the spatial distribution of the charge difference between spin-up and spin-down for all systems. The coupling of porphine/6zGNR breaks the bipartite character of the lattice and the FM ordering remains unaffected along the edge outside the coupling. We found that the total magnetic moment per unit cell is dependent on the coupling, with values around 10 µB (M1, 10


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M4) and 11 µB (M2, M3). In Fig. 2(a-b), as we expected for the M1 and M2 systems, the magnetic moment at the molecule is 0.0 µB in both case. This absence of states is caused by the coupling between the two nearest neighbor carbon atoms C1 −C2 (belonging to the same sub-lattice, e.g., B sub-lattice) at the molecule and the CA atoms in the edge of 6zGNR, that generate a competing interaction where each state tends to populate equally both spin components, leading to a spin-frustration. Note that inner hydrogen atoms perpendicular to the z−direction (M2 system) reduce the spin-frustration and the carbon atoms at the edge of the ribbon are also populated. For the M3 and M4 systems, the porphine possesses a local magnetic moment of ca. -2.5 and 4.0 µB , respectively. Once a spin density appears on the molecule (see Fig. 2(c and d) top view), there are no carbon bonds that belong to the same sub-lattice at the coupling. However, it is impossible to assign the A or B sites at the porphine molecule. Moreover, the exchange interactions in each system might be influenced by the topological frustration of the π carbon bonds C2 −C3 and C8 −C9 , leading to the opposite signs. The above results are quantitatively consistent with previous works. 10,47 Next, we discuss how the topological frustration influences the electronic properties of 6zGNRs. In Fig. 3 (right panel) the calculated spin polarized band structure shows an interesting behavior for all systems, especially at the Fermi energy where the dispersive bands that become flat at the high symmetric point clearly cross the Fermi energy. In all cases, both spin channels exhibit a metallic behavior, except for the M4 system in which the spin-down channel is a semiconductor. The PDOS (right panel) shows that these bands consist of states primarly derived from 6zGNR having strong interaction with the localized states on the porphine molecule (shaded region). Immediately below EF two almost flat bands appear in the spin-up channel, while the spin-down bands display only one nearly flat band for the M1, M2, and M3 systems (see Fig. 3(a-c), right panel). The PDOS (right panel) shows that these states mainly arise from the porphine molecule. Furthermore, it also shows that the states above the Fermi level (E − EF > 0.2 eV) are almost solely from the 6zGNR.

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Figure 3: (a-d) Spin polarized band structures (left panel) and projected density of states (PDOS) (right panel) for M1, M2, M3 and M4 systems, respectively. The contributions from porphine for each system corresponds to the shaded region in the PDOS. The dashed/solid lines indicate spin-up/spin-down contributions. The wave functions for the states around the Fermi energy for all systems at Γ-point can be seen Figure S1 in Supporting Information. The M4 system was found to be semiconducting for the spin-down channel. The porphine molecule introduces only one significant flat band below and two flat bands above the Fermi level for the spin-up and spin-down channels, respectively. The asymmetric feature of the spin-polarized band structure in all systems is due to the hybridization between porphine/6zGNR that no longer keeps the σ mirror plane of the pristine zGNR along the y-direction. These bands around the Fermi energy are derived from edge-states as can be seen from the wave function analysis at the Γ-point in Supporting Information Fig. S1. In the zigzag GNRs with an even (symmetric) number of carbon atoms in the sublattice across the lateral direction, the transport properties around the Fermi energy are dominated 12


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by the perfect match between the two dispersive bands that become partially flat near the Fermi level (π → π or π ∗ → π ∗ ). 17 In particular, the corresponding spin dependent transmission spectra under zero bias for the FM configuration preserve a metallic behavior, displaying non-zero electronic states at the Fermi energy. 48,49 For porphine/6zGNR, the hybridization can break the symmetry of the nanoribbon lattice along the edge, leading to new available electronic states around the Fermi energy. Such electronic configuration might allow the forbidden hopping between π and π ∗ subbands in some energy range, revealing a distinct transport feature. In that case, the transport properties are better understood when the spin transmission spectra (Tσ E)) are compared with their projected density of states (PDOS) as a function of the electron energy in each scattering region. The spin-resolved transmission spectra (Tσ (E)) of all four models are plotted in Figure 4(a-h) (top panel). A common feature in these curves is the presence of transmission dips observed below the Fermi level, caused by destructive quantum interference (DQI) effects. The M4 junction, in particular, also exhibit some dips in the energy range above the Fermi level (see Figure 4(g and h)). These DQI effects appear for both spin orientations, but are slightly separated in energy. Note that the Tσ (E)’s are almost spin-degenerate above the energy value 0.50 eV, no matter how the molecule is attached to the 6zGNR. However, this degeneracy is completely broken around the fermi level, since the spin-up (blue lines) transmission dips tend to be relatively wide and deep, whereas the spin-down (red lines) dips tend to be more narrow and shallow. In order to understand the nature of the DQI effects, we have calculated the projected density of states (PDOS) for all junctions. In Fig. 4(a-h) (bottom panel), we can see that each system has nearly the same contribution of 6zGNR electronic states to the total DOS, i.e., a rather wide peak around -0.40 eV for spin-up and 0.30 eV for spin-down, leading to the higher transmission coefficient (T = 2G0 ) below and above the Fermi level, respectively. For the porphine molecule (shaded plot) there are three broad peaks which correlate well with the energy values of the transmission dips. In particular, the transmission dip in the

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vicinity of the Fermi level (vertical dashed line) might be due to Fano-antiresonance effects, which result from the interference of the localized states (here, the porphine orbitals) with the continuum states of the ribbon. 50–52 A more direct approach to confirm the Fano-antiresonance effect is by projecting the scattering states associated with the dips around the Fermi energy. For spin-up, these scattering states are shown in the top panel of Fig. 4(i). The scattering wave function of the M1 device is delocalized throughout the left-hand side and further extended to the bottom atoms of the porphine molecule covalently bonded to the ribbon structure. A similar behavior was found for the M2 and M4 systems, but in these cases, the states are also delocalized over the entire porphine molecule. For the M3 system, the scattering states penetrate through the central region and then vanish gradually on the right hand side, but are mainly concentrated on the porphine molecule. In contrast to the M3 model, the projected scattering wave functions of the M1, M2 and M4 systems do not extend through the right side, which is in accordance with the transmission dips observed in these three models. Likewise, the spindown scattering states are strongly confined to the left hand side, as well as in the porphine molecule. The pattern of the scattering wave-functions in both spin configurations indicates that the Fano-antiresonance effect appears when the resonant channel from the porphine molecule aligns with non-resonant states from the 6zGNR. Furthermore, the energy position and the depth of the transmission dips are strongly related with the coupling between the porphine and the 6zGNR. The DQI effect has a high probability to be detected experimentally in the vicinity of the Fermi level. More importantly, this phenomenon plays a crucial role in the conduction regime through the molecular devices. 53,54 However, the ability to control the negative peak and drive it to the Fermi energy is a big challenge that can be overcome by using a gate voltage (Vg ). The application of the Vg might control the position of the energy levels of the system relative to the EF and tune the conductance properties to enable a large on-off ratio. 55 Thus, we use the gate bias to continuously tune the Fano-antiresonances (around

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Figure 4: Spin resolved transmission spectra (top panel) and the projected density of states (PDOS) at 6zGNR and porphirin molecule for two-probe systems: (a, b) M1, (c, d) M2, (e ,f) M3 and (g, h) M4. (i) The scattering states wave function corresponding to selected negative peaks around EF indicated by vertical lines (isovalue = 0.25 au). The blue/red curves correspond to the spin-up/spin-down contribution. EF ) in all molecular junctions. The positive and negative Vg is applied across the scattering region (6zGNR/porphirin) along the y-direction ranging from -5.0 to +5.0 V, as illustrated in Fig. 1(a). The gate effect is included by effective potential, therefore all atoms in the central region are determined selfconsistently from first principles. Figure 5(a-h) shows the Tσ (E)’s for M1, M2, M3 and M4 systems under gate voltage. For the spin-up (left panel), the transmission spectra of the M1 and M2 systems exhibit a small shift in the Fano-antiresonance dip to more negative energies with varying the Vg . In addition, the sharp DQIs observed in the M1 and M2 devices are kept exponentially small (T (E) ≈ 10−6 ). For the M3 system, the transmission dip increases by three orders of magnitude (T (E) ≈ 10−3 ) at Vg = −5.0 V and practically disappears for Vg = +5.0 V. 15



Meanwhile, the transmission dip of the M4 nanojunction shifts to negative and positive energies upon increasing and decreasing the bias voltage Vg , respectively. The spin-down transmission spectra for all models are shown in the right panel of the 5(a-h). In all systems, the DQI effects are shifted towards negative energies as the gate voltage increases from -5.0 V to +5.0 V. The general trend of the transmission dips under gate voltage for both spin channels are shown in Figure 5(i-l). As one can see, for all studied junctions the spin-up (blue lines) and the spin-down (red lines) transmission dips exhibit small shifts in the energy as a function of the gate voltage. As a consequence, the transmission coefficients at the Fermi level for both spin directions becomes quite different from each other, depending on the applied gate voltage. This behavior might be used to control the spin filter efficiency (SFE) through the 6zGNR/porphine systems. To quantify the difference between the two spin channels in transmission spectra at the Fermi energy, the SFE (according Eq. 6) was calculated as a function of Vg . For the M1 junction, the transmission dip in both spin directions is restricted to the energy below the Fermi level in all gate range, see the Fig. 5(i). In fact, as the gate voltage increases the spin-up transmission dip experienced a slight fluctuation around -0.05 eV, leading to a small transmission coeficient value at EF . For the spin-down case, the Fano-antiresonance dip shifts away the Fermi level as a function of the Vg . Since the spin-down transmission is higly transparent at the Fermi level, we found that the spin filtering is quite high in the range of 75% to 95%. In Fig. 5(j) and (k), we see that the spin-down transmission dip for the M2 and M3 junctions crosses the EF for the gate voltage of -4.0 V and -2.0 V, respectively. On the other hand, the spin-up plot show that the electron can flow readily in both cases by T (E) ≈ 1. Thus, the maximum spin filter efficiency for M2 and M3 junctions was calculated to be 98% and 88%. The Fig. 5(l) for the M4 system , the transmission dips for both spin components shows a quasi-linear decrease in energy with increasing the gate voltage. In contrast with

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Figure 5: Spin resolved transmission spectra under the influence of an external gate voltage (Vg ) around the Fermi energy for (a, b) M1, (c, d) M2, (e, f) M3 and (g, h) M4 systems. (i-l) Position in energy of Fano-antiresonances and the spin filter efficiency (SFE) as a function of the gate voltage (Vg ). The blue/red lines correspond to the spin-up/spin-down contribution. the other junctions, the SFE increases as the bias voltage increases from -5.0 V to +4.0 V reaching 100% of the spin-up polarization and then decreasing upon further increase of Vg . The Fano-antiresonance can have a significant impact on the magnitude of the predicted current through the systems. Figure 6 shows the spin polarized current versus bias voltage ∆Vb = VL − VR for all juntions applied from 0.0 to 1.5 V. The total current is obtained by adding the currents for the spin-up and spin-down states. In Fig. 6(a) and (b), one can see that for both M1 and M2 systems the current rises sharply, followed by a critical decrease with an additional bias voltage, then increases for greater Vb values, revealing a strong negative differential resistance (NDR) effect. In addition, a slight NDR effect was also verified for the M3 and M4 junction, as shown in Fig. 6(c) and (d). The contribution of spin-up and spin-down states to the total current shows that the transport properties are highly dependent on the spin configuration and several insights can be gained from the I-Vb 17



curves. The current features of the M1, M2 and M3 molecular junctions showed qualitatively similar behavior, i.e., for low bias voltage, the spin-down current is higher than the spin-up current, however, increasing the bias voltage the spin current exhibits a flip in this behavior. For instance, the M1 system has a maximum current peak of Ip ≈ 6 µA (11 µA) for the spin-up (spin-down) channel at a bias voltage of 0.3 V. As the bias voltage increases, the current reaches a valley of Iv ≈ 1.1 µA at 0.4 V for both spin configurations. Note that, below this critical point, the spin-down current is greater than the spin-up one. However, beyond 0.6 V, the current of the spin-up channel continuously increases while the spin-down current keeps a nearly constant value. For M2 and M3 junctions, this flip in the current can be seen only above 1.0 V and 0.4 V, respectively. Interestingly, the M4 junction is the only case where the current flip is not observed.

Figure 6: The spin-resolved I-Vb characteristics for (a) M1, (b) M2, (c) M3 and (d) M4 junctions. The blue and red lines indicates the spin-up and spin-down contribution, respectively. The NDR effect is characterized by a local change in the transmission inside the bias window as the bias voltage increases. 56–58 To better understand the physical mechanism, 18


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we plotted the spin-resolved transmission spectra at three different Vb values (see Fig. 7). Since all junctions displayed a similar trend, we will focus on the M1 system. At zero bias voltage, the transmission around the EF is dominated by the Fano-antiresonances at -0.11 eV and -0.03 eV for spin-up and spin-down component, respectively (Fig. 4(a)). At low bias (Vb = 0.3 V) the spin-resolved transmission curves resemble closely those observed at zero bias. As we can see in Fig. 7(a), the transmission dips for both spin orientations are shifted to lower energies, the computed spin-down transmission dip is out of the transport window (−Vb /2, Vb /2), while the spin-up Fano-antiresonance is inside. This is consistent with the observation that the spin-down current is greater than spin-up one at this applied bias voltage. Such reduction in spin-up transmission spectrum is related to the PDOS peak from the porphine molecule at -0.9 eV, as shown in the bottom panel. The right tail of the PDOS peak at -0.15 eV from de porphine molecule indicates that a small increase in Vb might turn off the spin-down transmission spectrum. As expected, a further increase in the bias voltage to Vb = 0.4 V (Fig. 7(b)) leads to several transmission dips for both spin channels into the bias window and the total current reduces significantly. Fig. 7(c) shows the transmission spectra at Vb = 1.0 V. It can be found that the system displays a high transmission coefficient for the spin-up channel in the positive energy region. While, at the negative energy region, there are several transmission dips for both spin configurations. Thereby the spin-up electron flows easily through the system reaching 20 µA when compared with the spin-down current that saturates around 10 µA. To demonstrate the dramatic effect of the porphine molecule on the transport properties of 6zGNR, Fig. 7(d) shows the scattering states at relevant energy points (black cicle) inside and outside the bias window of Vb = 0.3 V. From a physical point of view, a scattering state is an eigenstate of the Hamiltonian calculated in real space that indicates the path available to electron flow through the device at certain energies. The isosurface for the spinup transmission dip (I: -0.9 eV, T (E) ≈ 8 × 10−6 ) shows that there is a high concentration

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Figure 7: (a, b and c) Spin resolved transmission spectra for M1 system under Vb =0.3, 0.4 and 1.0 V, respectively. (d) Scattering states at transmisson points inside (I and II) and outside (III) bias window for Vb =0.3V. The vertical dashed lines in indicate the bias windows. The dashed blue (solid red) lines correspond to the spin-up (spin-down) contribution. of states on the porphine molecule (Top and End view), while a high amplitude of states appears at the carbon atoms closest to the coupling region between porphine and 6zGNR. For spin-down energy point named II (-0.9 eV, T (E) = 1G0 ) (Top view), the high amplitude of states on 6zGNR reveals that the electron transmits quite well from left to the right electrode. In that case, the porphine molecule is unrecognized in terms of transmission (End view). Furthermore, the scattering states recall the shape of the wave function in the antisymmetric distribution due to the perfect matching of π → π subbands in the left and right 6zGNR electrodes. It’s well known that the bias voltage not only shifts the chemical potential of the electrodes but also the position of the resonant levels from the scattering region allowing new conduction states for the electronic transport. Interestingly, for all porphine/6zGNR junctions the transmission spectra do not show an obvious behavior under the applied voltage,

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since the mirror plane symmetry no longer exists in those systems (see discussion regarding Fig. 3), allowing for electron hopping between π and π ∗ states. For instance, the break of symmetry in the scattering region induced for the porphine molecule is shown in the transmission point outside the bias window (III: 0.35 eV, T (E) ≈ 0.92), where the scattering wave function associated with this point has a strong contribution to the states crossing the central scattering region from the left to right 6zGNR electrode. Moreover, it is expected that similar transport properties would be obtained when different symmetric ribbon widths are coupled to the porphine molecule. On the other hand, antisymmetric ZGNRs coupled to porphyrins may present I-Vb current curves that are completely different from those observed in the symmetrical one. Hence, from the I-Vb characteristics, one can clearly see that the Fano-antiresonance generated from the coupling between the porphine and the 6zGNR are directly related to the low bias NDR effect and those systems are ideally suited to the development of reduced power consumption devices.

Summary The electronic and spin-polarized transport properties of the typical porphine molecule attached to the 6zGNR via different β- and meso- connections were investigated using nonequilibrium Green’s function technique combined with density functional theory. The porphine/6zGNRs present the FM ground state for all the couplings and the hybridization between porphine and 6zGNR disturbs the bipartite graphene lattice, inducing new available electronic states around the Fermi energy which provide a distinct transport feature. Particularly, a Fano-antiresonance effect appears when the resonant channel from the porphine molecule aligns with non-resonant states from the 6zGNR. By applying a gate voltage crossing the device, the Fano-antiresonance can be controlled to bring it into and out of alignment with the Fermi level, leading to a control of the spin filter efficiency of this system up to 98%. Finally, the calculated current-voltage (I − Vb ) characteristics of this system show that

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the transport properties are highly dependent on the spin configuration. At low bias, the current-voltage characteristics of this system reveal a negative differential resistance (NDR) behavior, where spin-down is greater than spin-up current. With increases in Vb , the spin currents are flipped and greater spin-up current values are observed due to the break of symmetry into the systems. We believe that the devices based on porphine/6zGNRs hold great promising applications as active components for controlling the spin-transport properties in spintronic devices

Supporting Information Available The evolution of band structures for M1, M2, M3 and M4 systems in Figure 3 and it’s correspondents square wave function at the Γ-point for the spin-up and spin-down contribution around the Fermi energy. The atomic coordinates and parameter of calculations used for all structures in the work. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement A.S.-S. acknowledges the Brazilian agency CAPES for the postdoctoral program fellowship (process 1510765). J. G. Silva Filho acknowledges the Brazilian agency CAPES for the postdoctoral program fellowship (process 1746352) and support from PRONEX (PR2-010100006.01.00/15). E.C.G. acknowledges support from CNPq (Process No. 473714/2013-2 and Process No. 306378/2014-0). We gratefully acknowledge FQRTN-Team funding and financial support by NSERC of Canada (H.G.). We also thank CalculQuebec and Compute Canada for computation facilities.

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Spin Negative Differential Resistance in Zigzag Graphene

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