Improving the bias range for spin-filtering by selecting proper electrode materials

Abstract

Using the non-equilibrium Green’s function method combined with density function theory, we investigate the spin transport for carbon chains connected to electrodes of different materials. When a carbon chain is linked to the C–H (C–H₂) bonded edges of H₂–ZGNR–H, the carbon chain displays a net spin polarization with a net magnetic moment of 1.367 μB (−0.935 μB) for the C–H (C–H₂) bonded edge contacts, but the directions of the net magnetic moment are opposite, and the latter system shows a larger spin conductance. Then, we choose N-doped H₂–ZGNR–H as the left electrode, and the right electrode is replaced with a single-capped carbon nanotube, armchair graphene nanoribbon (AGNR), or gold electrode. The conductance and the bias range for perfect spin-filtering of these systems shows obvious differences: the carbon nanotube (Au) system shows weaker conductance, and the AGNR system shows the largest bias range for perfect spin-filtering.

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1. Introduction

Graphene nanoribbons (GNRs) derived from wafer-scale material are attractive candidates for next-generation integrated circuits.^1–4 In particular, the properties of GNRs are strongly dependent on their chirality, like carbon nanotubes.⁵ Two particular types of GNR with armchair and zigzag shaped edges have been extensively studied. Armchair graphene nanoribbons (AGNRs) are predicted to be metallic when n = 3m + 2, where n is the width and m is an integer, and are insulating otherwise.⁶ Zigzag graphene nanoribbons (ZGNRs) are semiconducting with nonzero energy gaps due to the existence of ferromagnetically (FM) ordered edge states at each zigzag edge and an antiferromagnetic (AFM) arrangement of spins between two zigzag edges,⁷ and have peculiar localized electronic states at each edge,^8–11 which extend along the edge direction and decay exponentially into the centre of the ribbon, with the decay rates depending on their momentum. The localized edge states of the nonmagnetic case form a two-fold degenerate flat band at the Fermi energy (E_F), existing in about one-third of the Brillouin zone away from the zone centre.^12,13 Kunstmann et al. consider that magnetic edge states might not exist in real systems as the intrinsic magnetism would not be stable at room temperature in GNRs.¹⁴ Moreover, it has been estimated that the spin correlation length ξ limits the long-range magnetic order to 1 nm at 300 K.¹⁵ A feasible approach to realize graphene-based electronics is to construct the device junctions by connecting GNRs of different widths and orientations,¹⁶ and giant magnetoresistance phenomena can be obtained with the application of an external magnetic field.¹⁷ ZGNRs with asymmetric sp²–sp³ edges, however, exhibit an interesting bipolar magnetic semiconducting behavior, where the top valence bands and bottom conduction bands have opposite spin orientations in the proximity of the Fermi level,¹⁸ and the composition of sp² and sp³ types at the edges of the GNRs can be easily controlled experimentally via the temperature and pressure of H₂ gas.^19,20 On the other hand, doping is a popular method to tune the properties of graphene.^21,22 For instance, p-type or n-type doped graphene can serve as a promising anode for high-power and high-energy lithium ion batteries under high-rate charge and discharge conditions, and the conductance of doped graphene is related to the doping element and position.^23–25 The spin-filtering effect is a particular phenomenon in spintronic devices, which is an interesting topic due to its importance in next-generation electronics systems.²⁶ The range of bias for the spin-filtering effect, an important technical parameter for devices, is narrow in current designed devices. For example, the ZGNR–H/ZGNR–H₂ heterostructure²⁷ shows spin filtering with a bias range of [0, 0.5] V, and the ZGNR–H/ZGNR–O heterostructure²⁸ exhibits perfect spin-filtering when the bias is less than 0.2 V. Carbon atomic chains coupled to ZGNR electrodes,²⁹ which are also high-efficiency thermospin devices,³⁰ display perfect spin-filters under a small bias of [−0.18, 0.18] V. Therefore, it is necessary to design some new devices with larger bias ranges for spin-filtering. As we all know, the magnetism and electronic structure are directly related to the edge states for graphene.³¹ H₂–ZGNR–H has two different edge states, C–H₂ and C–H edges. In this paper, we first investigate the spin transport features of a carbon chain linked to the C–H (C–H₂) bonded edge of H₂–ZGNR–H with N-doping, which shows half-metallicity. A perfect (100%) spin filtering effect can be obtained. Interestingly, the carbon chain displays a net spin polarization with a net magnetic moment of 1.367 μB (−0.935 μB) for the C–H (C–H₂) bonded edge contact, i.e, the directions of the net magnetic moments are anti-parallel, and the latter shows a larger conductance. Then, one electrode is replaced with a single-capped carbon nanotube, armchair graphene nanoribbon (AGNR), or Au (111); the spin filtering effect can still be obtained, but the conductance and the bias range for spin-filtering change obviously.

2. Model and method

In our models, as shown in Fig. 1, the ribbon width of H₂–ZGNR–H is characterized by the number of zigzag-shaped C chains, N_z, along the direction perpendicular to the nanoribbon axis; here N_z = 6, namely H₂–6ZGNR–H. For N-doped H₂–ZGNR–H, where the N atom is doped near the C–H bonded edge, the nanoribbon is transformed to half-metallicity irrespective of the ribbon width.³² M1 and M2 are considered for the carbon chain linked to the C–H₂ or C–H edge of H₂–6ZGNR–H, where the carbon chain consists of seven carbon atoms. Then, by varying the right electrode materials and keeping the left electrode material (H₂–ZGNR–H) the same based on M2, we gain models M3, M4, and M5,where the right electrodes are a single-capped carbon nanotube, AGNR, and Au, respectively. Each device is composed of the left electrode, scattering region (the device region), and right electrode, marked by L, C, and R, respectively.

Fig. 1 The geometric structures of M1–M5. The blue spheres denote doping N atoms, the gray (white) spheres denote the C (H) atoms, and L, R, and C mean the left and right electrodes, and the central scattering region. The gold spheres denote Au atoms.

Geometry optimizations and calculations of electronic structure are performed by using spin-polarized density functional theory (DFT) combined with the non-equilibrium Green’s function (NEGF) method as implemented in the Atomistix ToolKit11.8 (ATK11.8).³² We employ Troullier–Martins norm-conserving pseudopotential to present the atom core and linear combinations of atom orbitals to expand the valence state of electrons. The local spin density approximation (LSDA) is used as the exchange–correlation functional. Real space grid techniques are used with an energy cutoff of 150 Ry, as a required cutoff energy in numerical integrations, and the solution of the Poisson equation using fast Fourier transform (FFT). The k-point sampling is 1, 1, and 100 in the x, y and z directions, respectively, where z is the electronic transport direction. Open boundary conditions are used to describe the electronic and transport properties of the nanojunctions. The geometrical structures used are optimized until all residual forces on each atom are smaller than 0.05 eV Å⁻¹ under the periodic boundary condition. The wave functions of the C and H atoms are expanded with a single-zeta polarized (SZP) basis set. The current, I_σ, in the systems as a function of the applied external bias, V_b, can be calculated from the Landauer-like formula,³³

where σ = ↑ (spin up) and ↓ (spin down), T_σ(E, V_b) is the bias-dependent transmission coefficient, f_l/r(E, V_b) is the Fermi–Dirac distribution function of the left and right electrodes; as a result, the electrochemical potentials correspond to μ_l(V_b) = μ_l(0) − eV_b/2 and μ_r(V_b) = μ_r(0) + eV_b/2, when the external bias is V_b. Considering the fact that the Fermi level is set to zero, the region of the energy integral window [μ_l(V_b), μ_r(V_b)] can be written as [−V_b/2, V_b/2].

3. Results and discussion

At first, we discuss the edge-state effects for the carbon chain linked to the C–H₂ or C–H edge of H₂–6ZGNR–H, namely M1 and M2. The I–V curves at various biases are shown in Fig. 2(a), and the distinct features are as follows: the current for the α-spin state is completely suppressed, while the current for the β-spin component is obvious larger than that for the α-spin state, and the corresponding spin polarization, |(I_α − I_β)/(I_α + I_β)|, reaches 100% when the bias is below 0.9 V. Moreover, the negative differential resistance (NDR) behaviors can also be obtained, which are described by a decrease in current through the system with a steady increase in the applied bias. We can also see that the current in M1 is smaller than that of M2, and the NDR phenomenon is also weak. Interestingly, from Fig. 2(b) we can see that the net spin magnetism exists in the carbon chains with an alternate emergence of the stronger β-spin and weaker α-spin in M1, with a net magnetic moment of 1.367 μB. In contrast, the carbon chain almost exhibits an alternate emergence of the stronger α-spin and weaker β-spin in M2 with a net magnetic moment of −0.935 μB, as seen in Fig. 2(c), which means that the direction of the two net magnetic moments is opposite for M1 and M2. The polarization in the A sublattice is much stronger than that in the B sublattice, and the spin direction of the A and B sublattices is antiparallel, indicating negative exchange interactions between the nearest neighbor atoms. The stronger magnetism can be found at the dihydrogen, and a weak magnetism can be observed on the carbon atom with dihydrogen termination, while the atoms next to them feature a stronger spin polarization because it decays much more slowly towards the center due to the different localizations in the edge states. It can be seen that the local magnetization at the doping sites and in their vicinity is strongly suppressed, giving rise to a decrease in the net magnetic moment of the whole system, since nitrogen has one electron more than carbon, which is injected into the device after its substitutional doping. Thus the Fermi energy is pushed up slightly into the conduction band, and some α-spin states are occupied, thereby causing a decrease in the net magnetic moment of the device. The current is determined from the integral area of the transmission curve within the bias window. In Fig. 3(a) and (b), we show the spin-dependent transmission spectra for M1 and M2 at 0.5 V, respectively. The main transmission peaks include the β-spin HOMO (highest occupied molecular orbital) peak and β-spin LUMO (lowest unoccupied molecular orbital) peak near the Fermi level within the bias windows, so the α-spin current is nearly zero, and spin-filter effects can be achieved. To provide an insight into the spin-filter behavior, we investigate the spatially resolved local density of states (LDOS) at the Fermi level under 0.5 V, shown in Fig. 3(c) and (d). The LDOS is localized at the right electrode and is partly distributed on the carbon chain for M1, but is highly delocalized throughout the scattering region and possesses significant values on the carbon chain. It is evident that the LDOS for M1 is much higher than that of M2. We further calculate the electron transmission pathways for the energy point of the transmission peak, which is an analysis option that splits the transmission coefficient into local bond contributions. The pathways have the property that if the system is divided into 2 parts (A, B), then the pathways across the boundary between A and B sum up to the total transmission coefficient: . Usually, the primary use for the pathways is to figure out where (and how) the current propagates; the more arrows, the more transmission.^34,35 Fig. 3(e) shows the transmission pathways of the β-spin at the Fermi energy under 0.5 V for M1, we can see that the local currents (from the left to the right) are nearly zero in the devices, as the electrons of the left electrode can not reach the right one. In contrast, for M2 (seen in Fig. 3(f)) stronger local currents can transmit from the left electrode to the right one via the carbon chain.

Fig. 2 (a) The spin-dependent I–V curves for M1 and M2; (b) and (c) spin density for M1 and M2 under zero bias. The magenta and cyan colors stand for the β-spin and α-spin components, respectively. The isosurface level is taken as 0.01 e Å⁻³.

Fig. 3 (a) The transmission spectrum, (c) the LDOS and (e) local currents transmission pathway at the Fermi energy at 0.5 V for M1; (b), (d) and (f) the same for M2. The blue dotted lines denote the chemical potentials of the left and right electrodes, and the Fermi level is set to zero.

Next, the right electrode is replaced with a single-capped carbon nanotube (the cap is derived from half of a C60 molecule) based on M2, and this model is denoted as M3. The I–V curves at various biases are shown in Fig. 4(a): the current for the β-spin component is obviously larger than that for the α-spin state, which is completely suppressed, and the spin polarization reaches 100% between zero and 1.0 V bias. However, the current of M3 shows an obviously smaller value than that of M1 and M2. An isosurface plot of the spin charge density difference of the α- and β-spin states for M3 is shown in the inset of Fig. 4(a), and the net spin magnetic moment in the carbon chain with an alternate emergence of the stronger α-spin is −1.046 μB. To find the origin of the weak current, the electrostatic potential³⁶ distribution is shown in Fig. 4(b): the potential profile is asymmetric on the interface between the carbon atom chain and the two electrodes, and two tunnel barriers at both interfaces are formed, where the right barrier is higher than the left one, which makes the electron movement difficult under bias. In Fig. 4(c), we give the β-spin transmission spectrum and molecular energy level in the scattering region at 0.5 V for M3. In addition, we also plot the MPSH of the HOMO and LUMO for the β-spin within the bias window. As can be seen, the main transmission peaks, including the β-spin HOMO and LUMO peaks, are near the Fermi level within the bias windows. The magnitude of the transmission coefficients is related to the number of the molecular orbital and the degree of delocalization of the molecular orbital. It can be found that there are two molecular orbitals appearing in the bias window indicated between the black dashed lines, and the spatial distributions of the molecular projected self-consistent Hamiltonian (MPSH) show that the HOMO and LUMO are localized, which leads to a weaker electronic transmission capability. The HOMO state is distributed largely on the left electrode and the carbon chain, and nearly no distribution is found on the right electrode. In contrast, the LUMO state shows much distribution on the right electrode and the carbon chain, and no distribution on the graphene electrode.

Fig. 4 (a) The spin-dependent I–V curves for M3 (the inset is the spin density), (b) the electrostatic potential distribution, and (c) β-spin molecular energy level spectra and transmission curves in the scattering region for M3. The dashed black lines represent the bias window. The Fermi level is set at zero. The left region of (c) gives the MPSH of the HOMO and LUMO.

The right electrode is replaced by armchair graphene with width n = 11, based on M2, and this model is denoted as M4. The I–V curves at various biases are shown in Fig. 5(a); the obvious spin polarization can be seen in the bias range [0, 1.2] V, where the α-spin current is completely suppressed. This is less than the β-spin component, which has a large magnitude, and the spin polarization reaches 100% in the bias range [0, 1] V, and 86% in the bias range [1.1, 1.2] V. The isosurface plot of the spin charge density difference of the α- and β-spin states for M4 is shown in the inset of Fig. 5(a), and the net spin magnetic moment of the carbon chain is −1.137 μB. To explain the spin polarization, we give the band structure for the left and right electrodes, with transmission at 0.8 V, shown in Fig. 5(b). The horizontal dashed lines denote the chemical potentials of the left and right electrodes. The energy bands are shifted downward and upward for the left and right electrodes under positive bias, respectively. Within the bias window, the α-spin bands for both the left and right electrodes have no overlaps, and no α-spin current; the β-spin band of the left electrode overlaps with that of the right electrode, and therefore the β-spin transmission peaks can be obtained. For contrast, we also give the plot at the same bias for M2, shown in Fig. 5(c). Only one narrow β-spin transmission peak appears in the window for such a small degree of overlap of the β-spin band between left and right electrode, which leads to a weaker current compared with M4. Then, with the increase in bias, the energy bands are shifted further downward and upward for the left and right electrodes, the overlaps will decrease and the currents also become weaker and weaker accordingly.

Fig. 5 (a) The spin-dependent I–V curves for M4 (the inset is the spin density); band structure for the left electrode, transmission spectrum, and band structure for the right electrode at the FM state under 0.8 V bias (b) for M4 and (c) for M2, where the dashed black lines represent the bias window.

Finally, the right electrode is replaced by semi-infinite Au (111) based on M2, where the surface consists of 3 × 3 Au atoms, from which we obtained M5. The I–V curves at various biases are shown in Fig. 6(a); the obvious spin polarization only occurs in the bias range [0, 0.8] V, and the β-spin currents show a small magnitude similar to M3. The net spin magnetic moment of the carbon chain is −1.124 μB, which shown in the inset of Fig. 6(a). To understand the origin of the weak current, we plot the transmission spectrum and the electrostatic potential distribution at zero bias, shown in Fig. 6(b). One obvious barrier appears at the interface between the carbon chain and the Au electrode, which hinders the movement of electrons. The α-spin transmission coefficient is zero near the Fermi energy, but one obvious β-spin transmission peak emerges at the Fermi energy, and the coefficient is around 0.15, which leads to the weak β-spin current under low bias.

Fig. 6 (a) The spin-dependent I–V curves (the inset is spin density) and (b) transmission spectrum under zero bias for M5 (the inset is the electrostatic potential distribution).

4. Conclusion

In summary, the spin transport of a carbon chain connected to electrodes of different materials is investigated using the non-equilibrium Green’s function method combined with density function theory. We first investigate the spin transport features of a carbon chain linked to the C–H (C–H₂) bonded edge of H₂–ZGNR–H. Interestingly, the carbon chain displays a net spin polarization with a net magnetic moment of 1.367 μB (−0.935 μB) for the C–H (C–H₂) bonded edge contact, but the directions of the net magnetic moments are opposite, and the latter device has a larger spin conductance. Then, we choose N-doped H₂–ZGNR–H as the left electrode, and the right electrode is replaced with a single-capped carbon nanotube, armchair graphene nanoribbon (AGNR), and gold electrode. The conductance and the bias range for perfect spin-filtering of these systems show obvious differences, with the carbon nanotube (Au) system showing weaker conductance, and the AGNR system showing the largest bias range for perfect spin-filtering.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos 61371065 and 61201080), the Hunan Provincial Natural Science Foundation of China (Grant no. 2015JJ3002), the Construct Program of the Key Discipline in Hunan Province, and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

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