Spin-dependent transport properties in a pyrene–graphene nanoribbon device

Abstract

Based on first-principles density functional theory combined with the nonequilibrium Green's function method, we have investigated the spin-dependent transport properties of a pyrene–zigzag graphene nanoribbon (ZGNR) system. The results show that this system can exhibit high-performance spin filtering, spin rectifying, giant magnetoresistance and negative differential resistance effects, by tuning the magnetization configuration of ZGNR electrodes. By analyzing the spin-resolved transmission spectrum, the local density of states, the transmission pathways, the band structure and symmetry of ZGNR electrodes, as well as the spatial distribution of molecular orbitals within the bias window, we elucidate the mechanism for these intriguing properties. Our results suggest that the pyrene–ZGNR system is a potential candidate for developing high-performance multifunctional spintronic devices.

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1. Introduction

Molecular spintronics, which aims to utilize the spin degree of freedom of an electron in addition to its charge in molecular devices, has recently received tremendous attention.^1–3 Compared with their charge-based counterparts, spintronic molecular devices can achieve high-density information storage and high-speed data processing with low energy consumption. Since its successful preparation in 2004, graphene has been considered as a promising spin-dependent transport material due to its long spin relaxation time and spin diffusion length,^4–7 and high spin injection ratio.^8–10 Especially for its one-dimensional derivatives, zigzag graphene nanoribbons (ZGNRs) are expected to play a critical role in future spintronic molecular devices since they have unique edge states and edge magnetism.^11–13 The ground state of ZGNRs exhibit ferromagnetic (FM) ordering at each edge but antiferromagnetic (AFM) coupling between two opposite edges.¹⁴ It has been shown that this AFM coupling can be tuned into FM coupling by applying a suitable transverse magnetic field,¹⁵ which has triggered intense research on ZGNR-based spintronics.^16–20 Particularly, Bai et al. realized experimentally obvious magnetoresistance effects in ZGNRs under a perpendicular magnetic field even at room temperature.²¹ However, designing ZGNR-based high-performance multifunctional spintronic devices still remains a challenge.

On the other hand, polycyclic aromatic hydrocarbons (PAHs), a group of hydrocarbon compounds consisting of more than two fused aromatic rings, have gained considerable interests because of their connection and application in molecular engineering of electronic device due to their high geometric symmetry and the delocalization of π electrons.^22,23 Pyrene is a peri-condensed PAH molecule and can be viewed as simply a small piece of graphene. Fan et al. studied the electron transport properties of pristine pyrene and boron/nitrogen-doped pyrene sandwiched between gold electrodes, and found negative differential resistance (NDR) effect in boron-doped pyrene molecular devices.²⁴ Here, we design an all-carbon spintronic device with the ZGNR acting as electrodes and a pyrene as the central molecule. The absence of any transition metal can guarantee the long spin relaxation time and spin diffusion length in our device. Our first-principles transport calculations show that the device can present multiple high-performance spin-dependent transport properties, including spin filtering, spin rectifying, giant magnetoresistance (GMR), and NDR effects.

2. Model and method

Fig. 1a shows the designed pyrene–ZGNR all-carbon device, where a pyrene is connected to the most front end of the wedge part of two 6-ZGNR electrodes by a carbon atom. The prefix 6 indicates the number of zigzag carbon chains across the width of ZGNR. Very recently, Prins et al. fabricated successfully similar molecular junctions in which a 9Accm molecule is linked to the wedge parts of two graphene electrodes.²⁵ The device is divided into three regions: the semi-infinite left and right electrodes, and the central scattering region. Each electrode is modeled by a supercell with three repeated ZGNR unit cells along the transport direction. The central scattering region contains not only the pyrene and two anchoring carbon atoms, but also the wedge part and four repeated ZGNR unit cells from each electrode to screen the interaction between the electrode and the central molecule (buffering layers). All the edge carbon atoms in ZGNR are terminated by one hydrogen atom (sp² hybridization type) in order to eliminate the dangling bonds. A 15 Å vacuum slab is adopted to eliminate interactions between periodic images. All the atomic positions in the central scattering region are relaxed fully using quasi-Newton method until the energy and force on each atom reach the tolerance limit of 10⁻⁵ eV and 0.02 eV Å⁻¹, respectively.

Fig. 1 (a) Schematic view of the pyrene–ZGNR transport system. The gray and white balls represent carbon and hydrogen atoms, respectively. The number in figure indicates the bond length (in Å). (b) The spin charge density difference for the P and AP spin configurations under zero bias, where the red and blue colors indicate the α- and β-spin components, respectively, and the isosurface level is taken as ±0.01 |e| Å⁻³.

The geometric optimization and subsequent spin-dependent transport calculations are both carried out by the Atomistix Toolkit (ATK) program package based on nonequilibrium Green's functions (NEGF) and density functional theory (DFT).^26–29 In this work, the spin-dependent generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) parameterization of correlation energy is used for the exchange–correlation functional.³⁰ The core electrons are described by the Troullier–Martins norm-conserving pseudopotentials³¹ and the valence electronic orbitals are expanded in a double-ξ plus polarization (DZP) basis set for all atoms. The energy cutoff for the real space grid is 200 Ry and the size of mesh grid in k space for electrode parts is 1 × 1 × 100. Under a given bias V_b, the spin-dependent current through the central scattering region is calculated by the Landauer–Büttiker formula:³²

here, e and h are the electron charge and Planck's constant, respectively. The spin index σ is α (spin up) or β (spin down). f_L/R = 1/{1 + exp[(E − μ_L/R)/k_BT]} is the Fermi–Dirac distribution function of the left/right electrode. μ_L/R = E_F ± eV_b/2 is the electrochemical potential of the left/right electrode, E_F is the Fermi level of the electrodes which has been set to zero for practical purposes. T_σ is the spin-resolved transmission function defined as:

T_σ = Tr[Γ_LσG^RσΓ_RσG^Aσ]. (2)

where G^Rσ/Aσ is the retarded/advanced Green's function of the central region with spin index σ and Γ_Lσ/Rσ is the coupling matrix resulting from the spin-dependent coupling between the central scattering region and left/right electrode. The energy region, [μ_L, μ_R], in which the transmission functions contribute the current, is referred to as the bias window.

3. Results and discussion

Experimentally, the magnetization of the left and right electrodes can be set to parallel (P) or antiparallel (AP) spin configuration by adjusting the applied magnetic field on two electrodes. Therefore, both two spin configurations are adopted to investigate the spin-dependent transport properties of our proposed all-carbon device. Fig. 1b shows the spin charge density difference between α- and β-spin states for the P and AP spin configurations under zero bias, where the red and blue colors indicate the α- and β-spin components, respectively. Clearly, these spin-polarized states present strong edge effect, i.e., they are mainly localized on the edge carbon atoms of ZGNR electrodes. The pyrene (with two anchoring carbon atoms) is also spin-polarized with a small net spin magnetism 0.22 μ_B in the P spin configuration, while the one is zero in the AP spin configuration.

Fig. 2a shows the spin-resolved current–voltage (I–V) curves for the P spin configuration. The distinct feature is that the α-spin current (I_α) is significantly larger than the β-spin one (I_β). Especially the I_β is nearly zero in the whole bias range. This means that the α-spin channel is always opened while the β-spin channel is blocked almost completely. Then similar to the conventional spin filtering devices, our proposed all-carbon device in the P spin configuration can also exhibit an spin filtering effect in both bias polarities. To evaluate this behavior, we define the bias-dependent spin polarization (SP) as SP = [(I_α − I_β)/(I_α + I_β)] × 100%, as shown in Fig. 2b. At zero bias, when all the currents vanish, we obtain the SP from the formula SP = [(T_α(E_F) − T_β(E_F))/(T_α(E_F) + T_β(E_F))] × 100%, where T_α(E_F) and T_β(E_F) are the α- and β-spin transmission coefficients at the Fermi level, respectively. As we can see, the device in the P spin configuration presents a near 100% SP in a wide bias range from −1.35 to 1.35 V. To explain the spin-dependent transport behaviors, the spin-resolved transmission spectra as a function of the electron energy E and bias V_b are plotted in Fig. 2c and d, where the region between two dotted white lines is the bias window. It is clear that there are always evident α-spin transmission peak in the bias window (Fig. 2c), resulting in a large I_α according to the formula (1). By contrast, the β-spin transmission peak keeps outside the bias window all along, leading to the strong suppression of I_β in the whole bias range. Therefore, a near perfect spin filtering effect occurs in both bias polarities for the P spin configuration. To further elucidate the significant difference between two spin channels, we plot the zero-bias spin-resolved local density of states (LDOS) and transmission pathways³³ at the E_F, as shown in Fig. 3 and 4, respectively. Clearly, the LDOS is delocalized on the whole device for the α-spin (Fig. 3a), showing the α-spin channel is opened substantially. It is also confirmed by the corresponding transmission pathways plot (Fig. 4a), which shows the presence of continuous pathways within the device. On the contrary, the LDOS is highly localized on two electrodes and no distribution on the pyrene for the β-spin (Fig. 3b), indicating the β-spin channel is blocked completely. Accordingly, there is no continuous transmission pathways (Fig. 4b).

Fig. 2 (a) The calculated spin-resolved I–V curves for the P spin configuration. (b) The corresponding bias-dependent spin polarization curve. (c) and (d) The spin-resolved transmission spectra as a function of the electron energy E and bias V_b. The region between two dotted white lines is the bias window.

Fig. 3 Under zero bias, the LDOS for (a) α- and (b) β-spin, respectively, at the E_F. The isosurface level is taken as ±0.005/(ų eV).

Fig. 4 Under zero bias, the transmission pathways for (a) α- and (b) β-spin, respectively, at the E_F.

Fig. 5a shows the spin-resolved I–V curves for the AP spin configuration. The most striking feature is the unidirectional characteristics of both I_α and I_β, i.e., the I_α/I_β can only flow through the device when the negative/positive bias exceeds −0.25/0.25 V. This indicates that the α-spin channel is only opened under negative bias, while it is completely opposite for the β-spin channel. Therefore, the device in the AP spin configuration can exhibit bipolar spin filtering and spin rectifying behaviors. The corresponding bias-dependent spin polarization curve is plotted in Fig. 5b. As we can see, except for the small bias range, the SP reaches near ±100% (especially in the bias ranges of [−1.35, −0.1 V] and [0.1, 1.35 V]). To further quantify the observed bipolar spin rectifying behavior, we define the bias-dependent spin rectification ratio (SRR) as SRR = |I_β(+V_b)/I_β(−V_b)| (for β-spin) or |I_α(−V_b)/I_α(+V_b)| (for α-spin), respectively. As shown in Fig. 5c, the SRR surpasses 10⁴ within a large bias range of [0.45, 1.25 V], and the maximum SRR reaches up to 3.87 × 10⁵ at 0.7 V, which is much larger than those in previous reports.^17,34 Besides, since the spin-polarized currents are negligible within small bias range in the AP spin configuration, one can expect a GMR effect when the spin magnetization of two electrodes switches between P and AP spin configurations. As a figure of merit, the magnetoresistance ratio (MRR) is defined as MRR = [(I_P − I_AP)/I_AP] × 100%, where I_P and I_AP are the total currents of P and AP spin configurations, respectively. At zero bias, when all the currents vanish, we obtain the MRR from the formula MRR = [(T_P(E_F) − T_AP(E_F))/T_AP(E_F)] × 100%, where T_P(E_F) and T_AP(E_F) are the total transmission coefficients of P and AP spin configurations at the E_F, respectively. As shown in Fig. 5d, the MRR exceeds 10² within the bias range of [−0.5, 0.5 V], and the maximum MRR reaches up to 7.90 × 10⁶ at zero bias, which is far larger than those in previous reports.^18,19 All those spin-related effects can be understood by the corresponding spin-resolved transmission spectra. As shown in Fig. 5e, the α-spin transmission peak only runs into the negative bias window (V_b < −0.25 V), while none appears in the whole positive bias window. By contrast, as shown in Fig. 5f, the variation of β-spin transmission peak is just opposite to the α-spin one. As a result, bipolar spin filtering and spin rectifying effects emerge in the AP spin configuration. And the GMR effect originates undoubtedly from the existence of threshold bias for transmission peak entering into the bias window. Moreover, from Fig. 2a and 5a, one can also observe obvious NDR effect, namely, the I_α of P spin configuration and the I_α/I_β of AP spin configuration begin to drop when the bias exceeds ±0.6 V and −1.0/+1.0 V, respectively. The NDR effect has found many applications in the field of semiconductor physics, including digital applications,^35,36 amplification,³⁷ and oscillators,³⁸ since it was first observed.³⁹ The NDR behavior is usually ascribed to the strength reduction and position shift of transmission peak,^40,41 as shown in Fig. 2c and 5e and 5f. Consequently, our proposed all-carbon device can present multiple high-performance spin-dependent transport properties.

Fig. 5 (a) The calculated spin-resolved I–V curves for the AP spin configuration. (b)–(d) The corresponding bias-dependent spin polarization, spin rectification ratio, and magnetoresistance ratio curves. (e) and (f) The spin-resolved transmission spectra as a function of the electron energy E and bias V_b. The region between two dotted white lines is the bias window.

To further clarify the above interesting phenomena, we analyze the overlap of spin-resolved band structure of electrodes (Fig. 6) and the spatial distribution of spin-resolved molecular projected self-consistent Hamiltonian (MPSH) eigenstates⁴² of molecular orbitals (MOs) within the bias window (Fig. 7) at different bias. For the P spin configuration, the bands of two electrodes have the identical structures at zero bias (not shown here), and this constructive matching is the precondition of perfect transmission channels. When the positive/negative bias is applied, the bands of left electrode shift downwards/upwards, while those of right electrode shift upwards/downwards. At 0.6 V (Fig. 6a), there are two α-spin (α_298 and α_299) and three β-spin MOs (β_296, β_297, and β_298) within the bias window (the region between two dotted blue lines). However, MOs α_298 and β_298 can not contribute the spin-polarized currents, since they overlap with the π* band of left electrode and the π band of right electrode, which have completely opposite parity with respect to the mirror plane σ, as shown by the corresponding isosurface plots of Bloch wave functions in Fig. 8. As shown in Fig. 7a, MO α_299 is delocalized on the whole device, while MO β_296 (β_297) is localized on the right electrode (left electrode and pyrene). Therefore, the α-spin channel is opened with a strong and broad α-spin transmission peak within the bias window, while the β-spin channel is closed entirely. This leads to the perfect spin filtering behavior. When the positive bias rises further, for example, at 1.5 V (Fig. 6b), more MOs enter into the bias window. To be specific, MOs α_297–α_300 and β_297–β_302 are shut down due to the symmetry mismatching of bands (π* → π).^43,44 Besides, as shown in Fig. 7a, MOs α_302, α_303 and β_296 are highly localized orbitals, having no contribution to the transport. Although MOs α_301 and α_304 are delocalized orbitals, their delocalization degree is obviously reduced compared with the case of α_299 at 0.6 V. As a result, the strength of α-spin transmission peak within the bias window is weakened, and the NDR effect occurs accordingly. For the AP spin configuration, the bands of α-spin and β-spin are exchanged for two electrodes at zero bias (not shown here), resulting in the transmission gap around the E_F (Fig. 5e and f), which lays the foundation for the GMR effect at small bias. Due to the opposite shift of bands under different bias polarities, as shown in Fig. 6c–f, all α-spin MOs under positive bias and all β-spin MOs under negative bias are closed due to the symmetry mismatching of bands (π* → π for α-spin and π → π*for β-spin). For this reason, the device in the AP spin configuration exhibit bipolar spin filtering and spin rectifying behaviors. Moreover, at 1.0 V, as shown in Fig. 7b, there are two delocalized MOs (β_299 and β_300), which give rise to the strong and broad β-spin transmission peak within the bias window (Fig. 6c). When the positive bias goes up again, for example, at 1.5 V (Fig. 6d), although more MOs enter into the bias window, there are only two somewhat delocalized MOs (β_301 and β_304), which leads to the damping of β-spin transmission peak within the bias window and generates the NDR behavior. Here, in Fig. 7, we only plot those MPSH eigenstates satisfying the symmetry matching of electrode bands at positive bias (the case at negative bias can be elucidated similarly).

Fig. 6 The spin-resolved band structure of left electrode (left panel) and right electrode (right panel), and the transmission spectrum (middle panel) at finite bias. (a) and (b) correspond to the case of P spin configuration at 0.6 and 1.5 V. (c)–(f) correspond to the case of AP spin configuration at 1.0, 1.5, −1.0 and −1.5 V, respectively. The region between two dotted blue lines is the bias window. The positions of molecular orbitals within the bias window are labeled with short black bars for α-spin and short red bars for β-spin.

Fig. 7 The spatial distribution of spin-resolved MPSH molecular orbitals within the bias window at finite bias. (a) corresponds to the case of P spin configuration at 0.6 and 1.5 V. (b) corresponds to the case of AP spin configuration at 1.0 and 1.5 V, respectively. The isosurface level is taken as ±0.04/Å^3/2.

Fig. 8 The spin-resolved isosurface plots of Bloch wave functions calculated for the Γ point. The different colors indicate the phase of wave functions. Evidently, the π and π* bands have odd and even parity with respect to the mirror plane σ.

4. Conclusions

In conclusion, we have designed a pyrene–ZGNR all-carbon device and investigated its spin-dependent transport properties by using the first-principles DFT + NEGF method. Our results show that this system can exhibit multiple high-performance spin-related properties, including spin filtering, spin rectifying, GMR and NDR effects, by modulating the applied magnetic field on electrodes. The physical mechanisms of these intriguing properties are analyzed by the spin-resolved transmission spectrum, the LDOS, the transmission pathways, the band structure and symmetry of ZGNR electrodes, and the spatial distribution of molecular orbitals within the bias window. Our findings suggest that the pyrene–ZGNR system is a potential candidate for developing the high-performance multifunctional spintronic devices.

Acknowledgements

This work is jointly supported by the National Natural Science Foundation of China (Grant No. 11104115), the Science Foundation of Middle-aged and Young Scientist of Shandong Province of China (Grant No. BS2013DX036).

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