Introduction

The current rectification in a diode is fundamental in electronics and the spin current rectification is equally important in spintronics. In the diffusive transport regime, various charge and spin diodes have been proposed1,2,3,4. Since the pioneer work done by Aviram and Ratner on molecular rectifier5, rectification systems in atomistic scale have been extensively studied due to the scientific and technologic interests in quantum transport and molecular electronics6,7. Ballistic spin diodes have also been proposed, including two quantum wire electrodes coupled by a tunnel barrier in antiparallel magnetic configuration8 and molecular wires bridging magnetic and nonmagnetic electrodes9. Recently, large current rectification ratio has been detected in longitudinally asymmetric but transversely symmetric diblock dipyrimidinyldiphenyl molecules connected10,11. Meanwhile, a device composed of organic magnetic molecules sandwiched by gold electrodes10 has been expected to obtain a spin rectification ratio over 100. A half-metallic behavior in one dimensional infinite chromium porphyrin array has been predicted11.

Carbon based electronics and spintronics have been attracting a huge amount of interest especially after the discovery of graphene12,13,14,15,16,17,18,19. Graphene has advantageous electronic properties and can be easily patterned into designed nanostructures and nanodevices such as nanoribbons and field effect transistors19,20,21,22,23,24. On the other hand, atomically perfect narrow graphene nanoribbons (GNRs) can be fabricated by bottom-up methods like self-assembly25,26. In addition, the spin current can be injected into graphene from ferromagnetic electrodes at room temperature27,28. Particularly, the zigzag graphene nanoribbons (ZGNRs) have been the focus of many studies due to its tunable band gap and intrinsic edge magnetism29,30 and giant magneto-resistance devices of graphene nanoribbon were predicted and fabricated31,32. Various methods were proposed to enhance the charge and spin rectification as well as the spin polarization in ZGNR devices33,34,35,36. A current rectification ratio up to 6000 is predicted in diarylethene molecule sandwiched between a ZGNR and an armchair graphene nanoribbon37. The spin rectification ratio up to 100 in bent graphene nanoribbons has been predicted based on ab initio simulation38.

Recently, carbon atomic chains (CACs) have been successfully fabricated from graphene with the help of a high energy electron beam39,40,41 and its transport properties were also characterized experimentally42. The advancement of such technology is significant since CACs may be used as transport channels or on-chip interconnects for graphene-based devices, nanoelectronic or spintronic nanodevices29,43,44,45. Ab initio simulations have indicated that the CACs between ZGNRs exhibit similar electron transport properties as those between metal electrodes46,47,48 and can be employed as spin filters or spin valves49,50. It is predicted that in C doped BN atomic wire connected between gold electrodes via S atoms51, a current rectification ratio can be enhanced up to 130. In the current work, we elucidate the importance of geometry symmetry match among CACs and the two electrodes in enhancing the rectification ratio in atomistic systems. We demonstrate how to realize a high performance spin filter with high spin rectification ratios as well as giant current rectification ratio in transversely symmetric ZGNR-CAC systems.

Computational Models and Methods

We consider device systems where a CAC is connected between two transversely symmetric electrodes of pristine or doped 6-ZGNRs shown in Figures 1(a). A 7-atom CAC is employed in the simulation since ballistic transport maintains in the chains with an odd number of atoms52 and we have verified that our results/conclusions are not sensitive to the length of the chain. Three-membered carbonic rings are designed to link between the CACs and the ZGNRs to keep the geometry symmetry of the system. The two-probe device is partitioned into the left (L) and right (R) electrodes and the central scattering region where a buffer layer of 6-ZGNR is put at each ends of the CAC to ensure a negligible contact effect on the electrostatic potential in the electrodes. The edges of the electrode ZGNRs are assigned in ferromagnetic states27,31 and the left electrode is in either antiparallel (FMa) or parallel (FMp) magnetization configurations to the right electrode.

Figure 1
figure 1

(a) Schematic illustration of the pristine device system and its partitions in the calculation. The big dark (small light) spheres indicate the C (H) atoms. The left electrode is assigned magnetized-up. For a transport channel, the electronic orbitals of the four C atoms in the contact area (shadow) have a symmetric form as shown in the inset. (b) The energy bands of pristine 6-ZGNRs. The black solid (red dotted) curves are for spin up (down) bands and the thick (thin) curves for π* (π) states. (c) and (d) the electronic wave function isosurfaces of the spin-up (π*) and the spin-down (π) bands near the Fermi energy, respectively. (e) The linear spin-up and spin-down conductance spectra in the FMa configuration. (f) The available (arrows) and prohibited (crossed arrows) electron transport channel from states in electrodes L to those in electrode R. (g) and (h) the linear spin-up and spin-down conductance spectra in FMp configuration. (i) and (j) The spin-up and spin-down local density of states respectively, at the Fermi energy in the FMp configuration. The three bond lengths from the end to the center of the seven-atom CAC are 1.354, 1.290 and 1.312 Å, respectively. The bond lengths in the contact between CAC and GNRs electrodes are all 1.447 Å.

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In the doped ZGNRs, the edge C atoms are replaced by dopant atoms of element B or N. The dangling bonds of the edge atoms are terminated by H atoms. In Table 1 we present the formation energies per primitive cell Eform = EnH-ZGNR-(EZGNR + nEH2/2) of pristine, B-doped and N-doped 6-ZGNRs with each edge atom terminated by n H atoms, n = 1 or 2. Here EnH-ZGNR is the total energy per primitive cell of ZGNR terminated by n H atoms, EZGNR is the total energy per primitive cell of ZGNR without H atom and EH2 is the total energy of a H2 molecule. Pristine ZGNRs prefer 2H termination (sp3) while doped ZGNRs prefer 1H termination (sp2). Since 1H termination is also feasible in pristine ZGNRs by controlling the chemical potential of hydrogen via temperature and pressure of H2 gas53,54 or by the hydrogen plasma etching method55, here we assume 1H termination in all the cases.

Table 1 Formation energies per primitive cell of ZGNRs in which each edge atom is terminated by one H atom or two H atoms
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The geometry optimization for each system is performed firstly by the Vienna ab initio simulation package (VASP)56,57 and then by the Atomistix Toolkits (ATK)59,60 until the forces on each atom is less than 0.02 eV/Å. The structure parameters are given in Fig. S1. Secondly, the electron transports are calculated by the nonequilibrium Green's functions method (NEGF) combined with ab initio density functional theory (DFT) implemented in the ATK package58,59. We use the Perdew-Zunger parametrization of the local spin density approximation (LSDA) for the exchange correlation functionals, which leads to the same conclusion as the spin dependent generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization (SGGA-PBE). The wave functions are expanded in a basis set of double-ζ orbitals plus one polarization orbital (DZP), which can preserve an accurate description of π and π* bonds. A 15Å thick vacuum layer is adopted to separate the ribbons in neighbouring supercells which is enough to suppress the of the coupling between them. In the transport calculation, the energy cutoff is 2000 eV, the mesh grid in the k-space is of size 1 × 1 × 100 and an electronic temperature of 300 K is used in the technique of the real-axis integration for the nonequilibrium Green's functions.

The current (Iσ) of spin σ through the device is evaluated by the Landauer-Büttiker formula: . Here Tσ(E) is the transmission spectrum and f(E−μX) is the Fermi distribution function with the Fermi energy μX in electrode X (L or R). Under a voltage bias Vb, the current flows from the left to the right electrode and we assume μL = −eVb/2 and μR = eVb/2. The integration is over the transport window in energy at zero temperature and mainly the transmission spectra in the transport window contribute to the current even at finite temperature. The linear conductance of the device reads .

Results and discussion

Our calculation results show that the pristine 6-ZGNRs are metallic at the ferromagnetic state and have an energy band structure near the Fermi energy as shown in Figure 1(b) when being assumed up-magnetized. The corresponding states are composed of py orbitals and are antisymmetric with respect to the x-z plane. In addition, the wave functions of the π* bands are antisymmetric and those of the π band symmetric with respect to the y-z plane as shown in Figure 1(c) and (d), respectively. The spin up/down π* band crosses with the spin up/down π band near 0.2 eV above/below the Fermi energy at k = 0.4 × 2π/a with a = 2.461Å the lattice constant of the ZGNR. The two bands then twist with each other and become almost degenerate at higher k when the corresponding wave functions are confined to the edges of the ZGNR. Interestingly, the two spin up bands (black solid) decrease in energy at k > 0.4 × 2π/a while the spin down ones (red dotted) become flat.

In CACs where the C atoms are bonded to each other via the sp hybridization, the wave functions of states near the Fermi energy are composed of carbon px or py orbitals. The px (py) orbitals are symmetric with the x (y) axis and antisymmetric with respect to the plane perpendicular to the x (y) axis. When a CAC is connected to a symmetric ZGNR electrode along its central line as shown in Figure 1(a), those electrons passing through the device are required to have a wave function symmetric with respect to the y-z plane and antisymmetric with respect to the x-z plane as indicated in the inset.

The conductance spectra in the linear response region are presented in Figure 1(e) and (g) for the FMa and FMp configurations of the electrode magnetizations, respectively. In the FMa configuration, for each spin there is an energy difference about 0.4 eV between the cross points of π and π* bands in the two electrodes as shown in Figure 1(e) and (f). Because the electrons in the π band cannot pass through the CAC due to the transverse symmetry mismatch, the spin-up and spin-down conductance spectra show similar step forms. The step edges are located at 0.2 eV above the Fermi energy, meanwhile, a sharp conductance peak appears next to the edges corresponding to the flat part of the energy bands. Note that a conductance dip follows the conductance peak and results in a conductance oscillation near the edges. This is an interference result of the transverse confinement mismatch between the wide ZGNRs and the narrow CAC. In the FMp configuration, the energy bands of the two electrodes are aligned with each other as shown in Figure 1(g) and (h). The spin-down conductance spectrum appears the same as that in the FMa configuration, while the step edge of the spin-up one shifts down to −0.2 eV below the Fermi energy and the sharp peak next to the edge shifts to lower energy but the range becomes wider, resulting in a perfect spin transport window about 0.4 eV around the Fermi energy. It means the system can work as a spin filter. Based on spin up and spin down local density of states (LDOS) at the Fermi energy presented in Figure 1(i) and (j), it is seen that the spin up electrons in the edge states of the electrodes can flow through the CAC. Meanwhile, the spin-down LDOS in the CAC at the Fermi energy is almost zero and there is no transport channel for the spin-down electrons to pass through the device.

Furthermore, we study the electronic conductance in this CAC system. The calculated spin-up (black) and spin-down (red) I-V curves in the FMa configuration are presented in Figure 2(a). In the linear regime, the device is OFF for both spin currents due to the symmetry mismatch between the electrodes as illustrated in Figure 1(f). Under a bias Vb> 0, the band structure of the left (right) electrode shift to higher (lower) energies. The step edge of the spin-up conductance spectrum shifts to higher energies accordingly and always remains outside the transport window as shown in Figure 2(c). Therefore, the spin-up transport channel is OFF. On the other hand, the step edge of the spin-down conductance spectrum shift to lower energy and moves into the transport window at Vb = 0.2 V [see Figure 2(d)]. The spin-down transport channel is then ON and the current increases quickly with Vb until saturates at Vb = 1.0 V. On the contrary, the spin-up transport channel becomes ON when Vb < −0.2 V and the current saturates at Vb = −1.0 V while spin-down transport channel remains OFF.

Figure 2
figure 2

(a) I-V curves of spin-up (solid) and down (dotted) components for a CAC device with pristine 6-ZGNR electrodes in the FMa configuration. (b) The corresponding spin-up (triangles) and down (dots) rectification ratios versus Vb. (c) and (d) 3D plot of the spin-up and down transmission spectra on the E-Vb plane, respectively. (e) The same as (a) in the FMp configuration. (f) The spin filtering polarization versus Vb in the FMp configuration. (g) and (h) The same as (c) and (d), respectively, in the FMp configuration.

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As discussed above, the device works as a high performance spin diode as well as a perfect spin filter. When the applied bias reverses, both the direction and the magnitude of the spin current remain unchanged. According to the definition of spin dependent rectification ratio, RRσ = [Iσ(V)−Iσ(−V)]/|Min{Iσ(V), Iσ(−V)}|, we plot the rectification ratio versus the bias amplitude for each spin in Figure 2(b). The rectification ratio becomes higher than 104 for Vb > 0.2 V. The rectifying behavior in the ZGNR-CAC structures is similar to that in ZGNRs32,34 but with much higher rectification ratio.

In addition, we also plot the I-V curve for the FMp configuration in Figure 2(e). For spin-up electrons, the device is metallic in the linear transport regime and the spin-up current increases monotonically with the bias until it saturates at Vb = 1.0 V. Meanwhile, spin-down current keeps OFF. The spin filter polarization (SFP = [(Iup−Idown)/(Iup + Idown)] × 100%) approaches to 100% in the studied bias region [−1, 1] V as shown in Figure 2(f). In Figures 2(g) and (h) for spin-up and spin-down, respectively, we present the corresponding conductance spectrum under bias to illustrate the physical mechanism behind. Part of the spin-up π* bands in both electrodes is inside the transport windows and the spin-up transport is ON. In contrast, the spin-down π* band in electrode R shifts down with the increase of bias and is always localized outside of the transport windows. The spin-down channel is then OFF.

Previous experiments have suggested that substitutional doping on the edges of graphene nanoribbons may greatly enhance the current rectification ratio and a ratio up to 16 has been measured in armchair nanoribbon24. Here we show that a giant current rectification ratio can be realized in a CAC device with edge C atoms in the left electrode substituted by B atoms as schemed in the inset of Figure 3(a). Its simulated I-V curves for both up and down spins in the FMp configuration are plotted in Figure 3(a) and the results are similar to that in the FMa configuration due to the magnetism suppression from B doped electrode. The device is ON within linear I-V curves under negative bias and the current is spin polarized at low bias. In contrast, the device is OFF under positive bias where the current becomes very small and decreases with the bias. As a result, the rectification ratio increases quickly with the bias as shown in Figure 3(b). The spin-up and spin-down rectification ratios can reach over 108 and 106 respectively; while, the average current rectification ratio is over 106 under bias |Vb| > 0.4 V.

Figure 3
figure 3

(a) I-V curves of spin-up (solid) and down (dotted) components in a device with left electrode doped by B atoms on both edges as schemed in the inset. The big black, big brown and small grey spheres represent the C, B and H atoms, respectively. (b) Spin-up and down rectification ratios versus Vb. (c) Atomic magnetic moment μ in B-doped (dots) and pristine (squares) 6-ZGNRs for atoms indexed from lower (index 0) to higher (index 13) edge. (d) Energy bands of B-doped 6-ZGNRs near the Fermi energy. (e) and (f) The spin-up and down transmission spectra, respectively.

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In Figure 3(c), we plot the transverse distribution of the atomic magnetic moment across pristine (squares) and B-doped (dots) 6-ZGNRs. The doping suppresses the edge magnetic moment55 and the 4 energy bands near the Fermi level in B-doped 6-ZGNRs become almost spin degenerate as plotted in Figure 3(d). Bands 1 and 2 are degenerate except at small k and their wave functions are symmetric with respect to the x-z plane, showing σ bonding characteristics (see Figure S2(a)–(b)). Bands 3 and 4 originate from py orbitals but the wave functions in band 3 are antisymmetric with respect to the y-z plane (see Figure S2(c)–(d)). So electrons near the Fermi energy in bands 1, 2 and 3 cannot reach to the right electrode of pristine 6-ZGNR due to the symmetry mismatch of wave functions. Because the top of Band 4 is slightly (~23 meV) below the Fermi energy and locates ouside of the transport window [−eVb/2,eVb/2] in the linear regime, the device is OFF when Vb ≥ 0 as shown in Figure 3(a). When Vb < 0, Band 4 shifts to higher energies with the Fermi energy in the left electrode and overlaps with the spin-up π* band in the right electrode. The spin-up transmission spectrum fills almost uniformly the transport windows of negative bias as shown in Figure 3(f). The spin-up current shows a very small threshold voltage and then increases linearly with the bias. On the other hand, the spin-down π* band in the right electrode enters the transport windows only when bias Vb < −0.2 V. The spin-down transmission spectrum has a shift to higher energy and negative bias with a bigger maximum compared to its spin-up counterpart. The spin-down current has a bigger threshold voltage and become almost the same at Vb = −1V.

Besides, we also study the N edge-doped effect and the results are given in Figure 4. As shown in Figure 4(a), the device is OFF for spin-down electrons when Vb > 0 and the negative differential resistance appears for spin-up electrons for Vb∈[0.2, 0.5] V with a peak to valley ratio of 4.54. The resulting SFP is almost 100% and the device can work as a perfect spin filter effect when Vb> 0. While, when Vb < 0, the spin-down current shows semiconducting with a threshold voltage around 0.1 V but the spin-up one appears metallic. At this moment, it exhibits unique half-metallic. Both spins show linear I-V dependence at Vb < 0. Due to the block of spin-down current at Vb > 0, the device exhibit a giant rectification effect with a ratio around 104 for spin down electrons as shown in Figure 4(b). The current rectification ratio vanishes at Vb = 0 and may reach 10 at Vb = 1 V. Similar to that in the B doped case, the atomic magnetic moment μ in the N edge-doped 6-ZGNR is also suppressed as shown in Figure 4(c) and the left electrode appears nonmagnetic with spin degenerate energy bands as illustrated in Figure 4(d). There are three bands (1, 2 and 3) near the Fermi energy and all of them are composed of pz orbitals. Bands 1 and 3 show π* characteristics and their wavefunctions are symmetric with respect to the y-z plane (see Figure S2(e) and (g)). Band 2 is a result of π bonding and doesn't contribute to the electronic transport (see Figure S2(f)). Because only electrons in π* can pass through the CAC due to the symmetry selection rule, the spin-up conductance spectrum at Vb = 0 originates from Band 1 near the Fermi energy as shown in Figure 4(e) and the spin-down conductance spectrum from band 3 as shown in Figure 4(f). With the variation of Vb, the energy bands of the two electrodes shift against each other in energy and the conductance spectra change accordingly as illustrated in Figure 4(e) and (f).

Figure 4
figure 4

(a) I-V curves of spin-up (solid) and down (dotted) components in a device with left electrode doped by N atoms on both edges as schemed in the inset. The big black, big blue and small grey spheres represent the C, N and H atoms, respectively. (b) Spin-up and down rectification ratios versus Vb. (c) Atomic magnetic moment μ in N-doped (dots) and pristine (squares) 6-ZGNRs for atoms indexed from lower (index 0) to higher (index 13) edge. (d) Energy bands of N-doped 6-ZGNRs near the Fermi energy. (e) and (f) The spin-up and down transmission spectra, respectively.

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Finally, we study a P/N junction composed of B doped left electrode and N doped right electrodes as schemed in the inset in Figure 5(a) together with calculated I-V curves. The device is OFF under positive bias with a current about 0.1 μA at Vb = 1.0 V. However, this value is much larger that (~10−6 μA at Vb = 1.0 V) in Figure 3(a) for the system with only one electrode doped by B atoms. As a result, the rectification ratio as shown in Figure 5(b) is much lower with the maximum spin-up (down) rectification ratio 80 (47) at 0.3 V. Interestingly, the top of Band 4 in the left electrode (Figure 3(d)) matches in energy with the top of band 3 in the right electrode (Figure 4(d)) at Vb = −0.25 V. The current reaches a maximum and then a NDR effect occurs in the bias range [−0.5, −0.25] V.

Figure 5
figure 5

(a) I-V curves of spin-up (solid) and down (dotted) components in a CAC device with left B-doped and right B-doped electrodes as schemed in the inset (b) Spin-up and down rectification ratios versus Vb. (c) and (d) The spin-up and down transmission spectra, respectively.

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Summary

Based on first-principle simulations, we have shown that transversely symmetric ZGNR-CAC-ZGNR structures are promising materials for high-performance charge and spin rectifier. Compared to symmetric pure ZGNRs, the insertion of the CAC in these structures strengthens the symmetry restriction and greatly enhances the rectification ratios. Structures with electrodes of pristine ZGNR can work as spin filter and their rectification performance strongly depends on the magnetization configurations. In the FMa configuration, the spin currents do not change when the bias inverses. On the contrary, the spin current flips when the bias inverses in the FMp configuration. When the edge C atoms in one of the two ZGNR electrodes are substituted by B atoms, the magnetization of this electrode is suppressed and symmetry matched wave functions exist only in one energy band below the Fermi energy. The electrons can pass through the structures only when this electrode is negatively biased and the current rectification ratio may reach over 106. In addition, the current is spin polarized under low bias and becomes unpolarized as the bias increases.