1. Introduction

Nowadays, the continuous shrinking of conventional semiconductors is facing great challenges [1]. That is, people’s demand for chip integration is increasing, but the chip integration and miniaturization is constrained because of the quantum interference effects. Then people’s attention has shifted to molecular size electronics from silicon based semiconductor devices. Among the molecular electronics, spintronics, using electron spin as a degree of freedom, make spin a vehicle for energy and information transfer [2,3]. In spintronics, if a certain number of spin-up electrons flow in one direction and an equal number of spin-down electrons flow in the opposite direction, when the charge current disappears, leading to a pure spin current with no Joule heat and ultra-low power characteristics [4]. These properties make spintronic devices promising for fast, low-energy and highly integrated applications in quantum computing, information storage and processing [5,6]. So the search for a perfect candidate material to realize pure spin current is of great importance for spintronics. Till now, tremendous [7–13] advances have been found in graphene-based related devices in recent years, making it the most promising material in spintronics. Ideal graphene nanoribbons usually fall into two categories: armchair graphene nanoribbons (AGNRs) and ZGNRs. In the ground state of ZGNR, the magnetization directions of the two edges are antiparallel, that is antiferromagnetic state (AFM), and a direct band gap exists in the band structure [14–16]. Ferromagnetic (FM) state can be obtained at slightly higher energies via either a magnetic field or a transverse electric field [15], and parallel magnetic structures are electrically conductive. In addition to intrinsic graphene, some interesting properties have been obtained by joining two or more adjacent nanoribbons in different ways to form nanojunctions [17,18]. Due to the presence of its spin alternation rule, the application of non-intrinsic graphene nanoribbons in spintronics is greatly extended [17].

The discovery of photogalvanic effect (PGE) [19–21] provided a new way of generating spin currents. The PGE effect is that when light is irradiated into the central region of the optoelectronic device, electrons in the valence band are excited into the conduction band and these excited electrons flow with unequal probability into the two leads, which in turn produce a net DC. There has been a great deal of work related to the generation of photocurrents via PGE [22–24]. Xie et al. proposed a two-dimensional (2D) spin-battery system that delivers pure spin current without accompanying an charge current to the outside world at zero bias [23]. Tao et al. generated pure spin current with photogalvanic effect (PGE) by designing devices with spatial inversion symmetry based on 2D spin semiconducting materials [24]. In particular, pure spin currents have been obtained in configuration using a hexagon as connector to connect two ground state ZGNRs, while no pure spin current in the tetragon connnected configurations [25]. Therefore, the following question naturally arises: if the ground state of the ZGNR is changed by external magnetic field, will the configuration using hexagons as connectors still express pure spin currents? The answer to this question will have important implications for the application of magnetic field-tuned nanodevices in spintronics.

In this work, we investigate the spin-dependent photocurrent in spatial antisymmetric configuration via modulation of magnetic fields. To meet the structural requirements of pure spin current, the configuration is constructed based on three ZGNRs where two hexagons are used as connectors. The results show that the same spatial inversion symmetric configuration produces different photocurrent phenomena by changing the direction of magnetic moment via magnetic field modulation. Pure spin currents obtained in the structure at the ground state without magnetic fields and at the state with the left and right leads in anti-parallel magnetic fields. These pure spin currents are independent of photon energy and polarization angle. In contrast, no pure spin current is presented in the structure when the left and right leads in parallel magnetic fields. The above results show the realization of pure spin current needs not only the spatial antisymmetry of the device geometry, but also the spatial inverse antisymmetry of spin density.

2. Simulation model and calculation details

We constructed a two-probe photovoltaic device model using three ZGNRs connected by two carbon hexagonal connectors. All the ZGNRs consist six zigzag carbon chains along the $x$-axis (labeled as 6-ZGNRs). The structure consists of the left and right leads and the central region, which extends to infinity in the leads direction ($z$=$\pm \infty$). The central region consists of three nanoribbons, referred to as the left nanoribbon (‘L’), the middle nanoribbon (‘M’ ) and the right nanoribbon (‘R’), as shown in Fig. 1. A hexagon is attached to each of the two opposite corners of the ‘M’ ribbon (top right and bottom left) to act as a link between it and the ‘L’ or ‘R’ ribbon. It is known that device structures formed by joining nanoribbons together using hexagonal connectors have been synthesized experimentally using mechanical etching techniques [26], making it possible to construct our model into a real device. We refer to this structure as “2-C6". Figure 1 shows a top view of the structure, with the red dotted circles showing the hexagonal connection points.

 

Fig. 1. Photovoltaic junction model of “2-C6". $\mathbf {A}$ is the electromagnetic vector potential, $\vec {e}_1=\hat {z}$ and $\vec {e}_2=-\hat {x}$ are the two polarization vectors that determine the direction of incidence of the polarized light, which is determined by $\vec {e}_1\times \vec {e}_2$. Angle $\theta$ between $\vec {e}_1$ and $\mathbf {A}$ is defined as the angle of polarization. The red spiral arrows indicate the polarized light applied, which is only applied to the atoms in the ‘M’ ribbon. The brown and pink atoms are carbon and hydrogen atoms respectively, the pink and green shaded parts of the structure indicate the left and right leads respectively, and the red dot $\mathbf {O}$ indicates the center of inversion symmetry.

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Our calculations used density generalized function theory (DFT) combined with the first principle of non-equilibrium Green’s functions and implemented in Atomistix Toolkit transport package [27]. In terms of the parameters set for the calculations, the calculations use local atomic orbitals of the form of norm-conserving pseudopotentials and the basis set DZP (double zeta polarized basis) for all atoms, the cut-off energy is 50 Ry, and the exchange-correlation potential is treated at the level of the generalized gradient approximation (GGA) in the form of PerdewBurkeErnzerhof (PBE) [28]. For the lead self-consistency and electron transport calculations of this optoelectronic device, the inverse space $\mathbf {K}$ points are set to 1$\times$1$\times$100 and 1$\times$1$\times$1, respectively. The density matrix and the Hamiltonian convergence criterion are both set to $1{\times }10^{-4}$a.u. With the transport direction along the $z$-axis and the vacuum direction in the $x$ and $y$-axis, a vacuum of 12 Å was chosen in $x$ and $y$ directions to eliminate interactions between adjacent images. To avoid edge reconstruction, all edge-hanging suspension bonds were passivated with hydrogen atoms. The structure has been fully relaxed using the quasi-Newton method until the force tolerance reaches 0.02 $eV/$ Å. In addition, such structures can be obtained through etching in experiments. In fact, even more complex graphene nanoribbons connected by carbon tetragons have been experimentally prepared [17]. When the ‘M’ ribbon in the central region is illuminated by linearly polarized light with a photon energy of $\hbar \omega$ and a polarization angle of $\theta$, photocurrent is generated. The photocurrent from the central region to each lead is calculated in a post-processing way. Firstly, the self-consistent Hamiltonians of the leads and the central region are obtained by DFT or DFT-NEGF calculations, without taking into account the electron–photon interaction. With these self-consistent Hamiltonians, the retarded (advanced) Green’s function $G^{r}_{0}$($G^{a}_{0}$) of the central region is calculated. Then the electron–photon interaction $\frac {q}{m}p\cdot {A}$ (where $p$ is the electron momentum and ${A}$ is the vector potential) is included in the calculation of the lesser (greater) Green’s function of the central region by $G^{<(>)}_{ph}=G^{r}_{0}[\sum _{ph}^{<(>)}+\sum _{L}^{<(>)}+\sum _{R}^{<(>)}]G_{0}^{a}$, where $\sum _{\alpha }^{r/a}$ is the self-energy of the semi-infinite lead $\alpha$($\alpha =L, R$) and $\sum _{ph}^{<(>)}$ is the self-energy from the electron–photon interaction considered in the first-order Born approximation [29]. The photocurrent flowing from the central region to lead $\alpha$ can be expressed as [30]

The effective transmission function is

In Eq. (2), the parameters $\alpha (=L,R)$ denote the lead and $s(=\uparrow,\downarrow )$ the spin indicator. $\Gamma _{\alpha }$ is a linewidth function describing the strength of the coupling between lead $\alpha$ and the central region. $f_\alpha$ is the Fermi-Dirac distribution function for the electrons in lead $\alpha$. $G_{ph}^{>}/G_{ph}^{<}$ is the larger/smaller Green’s function for the central region with electron-photon interaction, which is considered in the first-order Born approximation for the electron-photon interactions [30]. More details on the calculation of the photocurrent can be found by consulting the relevant references [29].

The relevant results presented in this letter are expressed in terms of the photo response function (normalized current)

The unit of $a_{0}^{2}$/photon, where $a_{0}^{2}$ represents the Bohr radius. $I_{\alpha,s}^{ph}$ and $e$ represent the photocurrent and electron charge defined in Eq. (1), respectively, and $I_{\omega }$ is defined as the number of photons per unit area per unit time. Thus, the charge and spin currents can be defined as

Here, $R_{\uparrow }$ and $R_{\downarrow }$ denote the spin-up and spin-down light responses, respectively.

3. Results and discussion

Localized edge magnetism are known to exist at the edged carbon atoms of all ZGNRs and any adjacent carbon atoms at the edges in ZGNR belong to different sub-lattices and that the carbon atoms of the different (same) lattice have the opposite (same) magnetic moment direction. The use of carbon hexagons as connectors between two ZGNRS allows for opposite magnetic moments at adjacent edges of adjacent nanoribbons based on the magnetic moments of two adjacent atoms in opposite directions, as shown in Fig. 2. In order to illustrate the effect of magnetic field on the production of light-induced pure spin current in carbon hexagonal-connected ZGNRs, three representative magnetic structures are considered via modulating the magnetization direction of the ZGNRs. The three representative magnetic structures considered are, the ground states (labeled as ‘S1’, as shown in Fig. 2(a)), the antiparallel state with ‘L’ and ‘R’ ribbons in antiparallel magnetic fields (labeled as ‘S2’, as shown in Fig. 2(b)) and the parallel state with parallel magnetic fields in the ‘L’ and ‘R’ ribbons (labeled as ‘S3’, as shown in Fig. 2(c)). The three structurues satisfy the spin alternation rule of maintaining opposite magnetic moment at adjacent edges of multiple ZGNRs.

 

Fig. 2. Schematic diagram of the magnetic moment distribution at the ZGNR edges for the three representative magnetic structures constrcucted by "2-C6": (a) AFM-AFM-AFM (from left to right), (c) FM-AFM-FM and (c) FM-FM-FM, respectively. Arrows indicate the direction of the edge magnetic moments of the three ZGNRs connected by carbon hexagons. Red, yellow and blue rectangles indicate the left, middle and right ZGNR, respectively.

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In the following, we present the light-induced photocurrents in the three representative magnetic structures of the "2-C6" in Fig. 3. For ’S1’, the photocurrents in both spin channels are zero at photon energies below 0.62 eV because the band gap of the ground state (AFM) 6-ZGNR is 0.62 eV [25] which determines the minimum photon energy for photoexcitation, then own small values when the photon energies exceeds 0.62 eV and start to have a sufficiently large value in a certain range of photon energies exceeds 0.75 eV, and the two spin components are exactly equal in magnitude but opposite in sign, as can be seen in Fig. 3(a). The photocurrents in the two different spin channels have opposite signs means that the flow of electrons in the two spin channels are in opposite directions, so pure spin current with zero charge current but non-zero spin current can be obtained in ’S1’. In addition to varying the photon energy, pure spin current can be also obtained by varying the light polarization angle $\theta$. Figure 3(b) shows the photocurrents of ‘S1’ as a function of the polarization angle $\theta$ at the selected photon energy of $E$=0.85 eV. It can be seen from Fig. 3(b) that the sum of the spin currents in both spin directions is always zero, so the pure spin currents are independent of the light polarization angle. We also obtain rather great pure spin current in ‘S2’, as shown in Fig. 3(c-d). However, in ‘S3’, the photocurrent in both spin channels can be negelected at the considered photon energy range and at any photon polarization angle (see Fig. 3(e-f)), so no pure spin currents can be realized in ‘S3’.

 

Fig. 3. (a, c, e) The spin-dependent photocurrents as a function of linear photon energy at polarization angle $\theta = 0^{\circ }$ for the three representative magnetic configurations ‘S1’, ‘S2’,‘S3’, respectively; (b, d, f) the corresponding photocurrent as a function of linear polarization angle at photon energy 0.85eV for the magnetic configurations as in (a, c, e). “$R_{\uparrow }$", “$R_{\downarrow }$" and “$I_{c}$" denote the photocurrent contributed by the spin-up and spin-down photocurrent respectively and the sum of them.

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The difference directions of magnetic moment of edge carbon atoms in the representative structures ‘S1’, ‘S2’,‘S3’ induce the above great different photocurrents. Since the spin-polarized photocurrents are directly related to the spin-resolved density of states of the system, which are mainly contributed by the carbon atoms on the zigzag edges in ZGNRs. The underline physical origin can be given via the spin-dependent partial density of states (PDOS) of the edge carbon atoms of the ‘L’, ‘M’ and ‘R’ regions combined with the schematic diagrams of how the electrons in the valence band of the M region excited to the conduction band and flowing to the left or right region to produce photocurrents. For case ‘S1’, the spin-up and spin down PDOSs in the M ribbon have two identical spikesas at the electron energies −0.48 and 0.32 eV as can be seen in Fig. 4(b). Simultaneously, there are also PDOS spikes at the corresponding energies in the L and R ribbons, and the PDOS peaks of the L ribbon own the same values as that of the R ribbon peak at the same electron energy positions, however, their spin-up and spin down PDOSs are swapped, as shown in Fig. 4(a, c). Based on these spikes (energy levels), we draw a schematic diagram of the photocurrent generation process, as shown in Fig. 4(d). When the phonon energy of the irradiated light exceeds 0.8eV, the spin down electron in the valence band of the M ribbon with an energy of −0.48 eV is excited to the conduction band of the M ribbon with an energy of 0.32 eV and moves further towards the L ribbon due to spin matching. Similarly, equivalent spin up electrons in the M ribbon is excited to 0.32 eV and moves further towards the R ribbon. So pure spin photocurrents are obtained in case ’S1’ and the spin-up and spin down photocurrent peaks happen at the photon energy exceeds $\sim$0.80 eV (see Fig. 3(a)). Similar analysis can be made for case ‘S2’, as can be seen from Fig. 5.

 

Fig. 4. (a)-(c) The partial density of states (PDOS) of the edge carbon atoms of the L, M and R ribbon in case ‘S1’, respectively. (d) schematic diagram of the photocurrent generation process in S1. The spin-up and spin-down electron flows are indicated by the blue and red arrow spirals, respectively. The black vertical dashed line indicates the mirror plane. The red and blue dashed (solid) lines indicate the incident (out) process, and the black solid line with arrows indicates the flow direction of the excited electrons during photoexcitation

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Fig. 5. The same as Fig. 4, but for the case ‘S2’.

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Why can the photocurrent in both spin channels be nearly neglected in case ‘S3’? We can obtain the reason again from the PDOS of the edge carbon atoms of L, M and R ribbon and the schematic diagram of photocurrent generation process as shown in Fig. 6. In the M ribbon, as can be seen from Fig. 6(b), the spin-up and spin-down PDOS have a spike at photon energy −0.40eV below the Fermi energy level; while above the Fermi energy level, the spin-up and spin-down DOS have a spike at photon energies of 0.24eV and 0.28eV, respectively. In both L and R ribbons, as can be known from Fig. 6(a, c), there is a spin-down PDOS spike at the photon energy 0.28eV in the conduction bands and a spin-up PDOS spike at −0.40eV in the valance bands, respectively. The PDOS peak of the L ribbon is exactly at the same photon energy position as that of the R ribbon. Based on the photon energy position of these PDOS spikes,a schematic of the photocurrent generation process is presented in Fig. 6(d). Under light illumination, the spin-down electrons at the energy level −0.40 eV in the valence band of the M ribbon are excited to the energy level 0.28 eV of the conduction band and move further towards the L and R ribbons according the spin matching. Coincidentally, there is a same spin-down PDOS spike at the photon energy 0.28eV in the conduction bands in the L and R ribbons, so the probabilities of spin-down electrons moving to the left and right are the same, leading to zero photocurrent. Similarly but not the same, equivalent spin-up electrons at the −0.40 eV energy level in the valence band of the M ribbon are excited to the 0.24 eV energy level of the conduction band and move further towards the L and R ribbons due to spin matching. There is no PDOS peak at the photon energy 0.24eV in the conduction bands in the L and R ribbons, the probabilities of spin-up electrons moving to the left and right are very low but still the same, and also the spin-up electrons make no contribution to the photocurrent. So the photocurrent in both spin channels can be nearly negelected in case ‘S3’.

 

Fig. 6. The same as Fig. 4, but for the case ‘S3’.

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Furture, from the flow directions of spin-up and spin-down electrons, the properties of photocurrents generated in cases ‘S1’, ‘S2’ and ‘S3’ can be uderstood via the distribution of spin density in real space. As can be seen from Fig. 7(a, b), cases ‘S1’ and ‘S2’ both have spin density inversion antisymmetry. The spin density has an inverse antisymmetric property means that the electrons in one spin channel experience exactly the same situation in one direction as the electrons in the other spin channel experience in the opposite direction, while the electrons in the same spin channel experience a different situation along the opposite direction. Thus, we always get photocurrents with opposite spin directions and equal magnitude in the magnetic configuration of S1 and S2, resulting in pure spin currents with zero charge currents and non-zero spin currents. The distribution of spin density in‘S3’ does not own the spatial inversion antisymmetry, instead it is Cs symmetric (see Fig. 7(c)), so the probabilities of light-irradiated electrons (Whether it’s spin up electrons or spin-down electrons) in the M ribbon moving to the left and right are the same, giving rise to the nearly zero photocurrent. Here, as the channel region of the designed device is M, the light only illuminates M. If we want to irradiate some regions in L and R, it is also possible [31]. As long as the spin density of the illuminated region satisfies the spatial inversion anti-symmetry, the above conclusions remain valid. In addition, the generation of the pure spin current is dependent on the spatial inversion anti-symmetry of spin density, so the perfect zigzag-edged structure is a prerequisite for ensuring this spatial inversion anti-symmetry. However, in reality, the presence of impurities or defects in the nanoribbons may affect the observation of pure spin current, depending on whether perfect (or at least with low defect concentration) graphene nanoribbons can be prepared experimentally. With the development of high-precision nanofabrication technology, this problem will be solved.

 

Fig. 7. Real space distribution of spin density for: (a) “S1" ; (b) “S2" ; (c) “S3"

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4. Conclusion

In summary, we studied the effect of magnetic structure on the photocurrent properties through first-principles calculations. Specifically, we implemented our ideas in a structure constructed by three ZGNRs connected by two carbon hexagonal connectors (2-C6) and used magnetic fields to modulate the direction of magnetic momentum of three ZGNR regions. Three representative magnetic structures: AFM-AFM-AFM (from left to right, ’S1’), FM-AFM-FM (‘S2’) and FM-FM-FM (‘S3’), were considered via modulating the magnetization direction of the edge carbon atoms in the ZGNRs. Pure spin currents with zero charge currents and non-zero spin currents were obtained in magnetic structures in zero magnetic field (‘S1’) and anti-parallel magnetic field (‘S2’), but not in the magnetic structure in parallel magnetic field (‘S3’). In addition, the obtained pure spin currents in ‘S1’ and ‘S2’ do not depend on photon energy and polarization angle, so they are robust. The above phenonmena were explained by the PDOSs of the edge carbon atoms of the three regions and the schematic diagrams of the photocurrent generation processes. The underline reason for obtaining the robust pure spin currents is attributed to the fact that the spin density of both structures possesses spatially inverse antisymmetric character. While the distribution of spin density in‘S3’ does not own the spatial inversion antisymmetry, it is Cs symmetric, so the probabilities of light-irradiated electrons (spin up and spin-down electrons) in the M ribbon moving to the left and right are the same, giving rise to the nearly zero photocurrent and also no pure spin current, which again proves that the realization of pure spin photocurrent needs not only the spatial inversion symmetry of the structure, but also the spatial inversion anti-symmetry of the spin density. Our findings can provide a solution for obtaining pure spin currents in ZGNR constructed devices via magnetic field modulation, which have important implications for future applications of spintronics and can furture extended to other two-dimentional materials.

Author Contributions

Y.H. Zhou and X.H. ZHeng proposed the initial idea. X.F. Shang and Y.J. Li performed quantum transport calculations and got the results. All authors discussed the results and contributed to the preparation of the manuscript.