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Serial and parallel spin circuits at the molecular scale with two atomic-vacancies in graphene: Amplification of spin-filtering effect
Carbon 154 (2019) 357e362
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Serial and parallel spin circuits at molecular scale with two atomic-vacancies in graphene: Amplification of spin-filtering effect Shizheng Wen a, b, Shiwu Gao b, **, ChiYung Yam b, * a
Jiangsu Province Key Laboratory of Modern Measurement Technology and Intelligent Systems, School of Physics and Electronic Electrical Engineering, Huaiyin Normal University, 223300, Huaian, China b Beijing Computational Science Research Center, Haidian District, 100193, Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2019 Received in revised form 16 July 2019 Accepted 10 August 2019 Available online 10 August 2019
Our previous work demonstrated that graphene devices with monovacancy defects possess spin filtering effect which offers potential applications in spintronics. Here, using first-principles calculations, we further study the spin-dependent electron transport properties and spin filtering efficiency of graphene devices with double vacancies that are arrayed in parallel and serial connections. It is found that devices with vacancies in parallel connection follow classical Kirchhoff circuit law. Both spin-up and spin-down currents are amplified while spin filtering efficiency remains the same as compared to the monovacancy devices. In contrast, amplification of spin filtering efficiency is realized in devices with serially connected double vacancies. In addition, it is shown that the spin current can be flipped reversibly by nanomechanical deformation as we observed in devices with monovacancy. Our findings demonstrate the possibility to engineer the spin filtering effect for vacancy based spintronic devices. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Graphene based spintronics has emerged as a new subject for extensive experimental and theoretical studies due to its unique electronic and mechanical properties [1e11]. While pristine graphene is nonmagnetic, vacancy and defect states can generate local magnetism in graphene due to the breaking of the local sp2 symmetry and the removal of the p bond perpendicular to the graphene plane [10e18]. Such a local magnetism shows sensitive response to variation of local atomic configurations, which is controllable by either shear distortion or vertical manipulation [19e22]. It holds the promise for many potential spintronic applications such as spin filtering and injection, magnetic storage, and information processing [1,7,23]. Single carbon vacancy is the simplest defect state and is present in many graphene fabrication techniques [11,15,24]. Previous theoretical works have been carried out to understand the origin of defect magnetism in graphene [14]. It is understood that the magnetic moment associated with a vacancy is attributed to the localized p states that is close to the Fermi level. In addition, the hybridization of p
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (S. Gao), [email protected] (C. Yam). https://doi.org/10.1016/j.carbon.2019.08.029 0008-6223/© 2019 Elsevier Ltd. All rights reserved.
and s states gives rise to a total magnetic moment of about 1.5 mB for each vacancy in graphene [12,17,25]. Fully and carefully designing the electronic devices with defected-graphene are rare in present exception of the discussion on the zigzag edge graphene nanoribbons (ZGNR) systems [26]. The ZGNRs are predicted with nearly perfect spin polarization [27]. The defects in ZGNR devices impact dramatically effect for spin transport channel due to the symmetry breaking [28e31], whereas preserving edge states [32]. In a recent study, it has been demonstrated that nanomechanical control of magnetic states of graphene vacancy can lead to the spin-filtering effect in prototype graphene devices with single carbon vacancies [33]. While the mechanism of the spin-current flip has been established and understood in previous work, the spin-filtering efficiency (SFE), up to 20% depending on the bias voltage and polarity, achieved so far is not that high compared with other device candidates like for example nanoribbons [26,27,32,34e37]. In light of applications, it is also desirable to consider device structures with multiple vacancy states arranged in possible different arrays in the graphene sheet, where the vacancy-vacancy interaction can change the overall magnetic state of the coupled system [12,38]. How the vacancy-vacancy interaction modifies the magnetic state and transport properties, and in particular whether it is possible to achieve higher SFE using different architectures of multiple
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vacancies remains to be a fundamental issue that has yet to be explored. Previous works have demonstrated that defects play an important role in the properties of graphene. For instance, electron mobility in graphene is strongly influenced by the defect states [10] and a pronounced paramagnetic response is found in monovacancy graphene [25]. To our knowledge, the role of defects in the spin transport properties of graphene has not yet been explored in detail. The primary goal of this work is to investigate how parallel and serial connected double-vacancy devices will affect the spin transport properties and whether classical Kirchhoff's circuit law is still valid at the molecular level. Here, we report on the spindependent electron transport and SFE of a model device with double vacancies that are arrayed in parallel and series connections, using the density functional theory (DFT) combined with nonequilibrium Green's function (NEGF) method [39e41]. It is found that the devices with two atomic-vacancies arranged in parallel connection amplifies the electron currents equally in both the spin-up and spin-down channels, as expected by classical circuit theory, but does not change the SFE in any significant way. In contrast, serially connected double vacancies modify the two spinchannels in different ways (specify later) and thus amplifies the SFE by more than a factor of two due to the suppression of the spindown channel. Such a suppression can be attributed to the vacancy-vacancy interaction and the modulation of the electric potential drop along the transport path. The results clearly elaborate the validity and limitation of the Kirchhoff's circuit law for spintronic transport at the molecular level and provide insights into the design of spin circuits at molecular level and demonstrate a step forward to all-graphene electronics and spintronics. The manuscript is outlined as follows. Computational details and the model employed in this work are given in Section 2. Numerical studies and related discussions are given in Section 3. Finally, we summarize this work in Section 4. 2. Model and computational details Our non-equilibrium transport calculations are performed for a model two-terminal graphene device schematically shown in Fig. 1. This model device is made up of three parts including a central device region, left and right electrode. The central device is modeled by 10 10 graphene supercell with two atomic-vacancies based on our previous work [22,33]. The buffer regions, which connect between the central device and the electrodes, are constructed by extending the outermost atomic layers of central unitcell. In the transport calculations, it is assumed that the semi-infinite electrodes are in thermal equilibrium and maintain the bulk properties. The electrode Hamiltonian therefore possesses translational invariance. This assumption allows us to apply an efficient renormalization technique to calculate the surface Green's function of the semi-infinite electrodes. To screen the influence of the defect states on the electrodes, buffer regions are introduced between the device and the electrodes. The spin-dependent transport calculations were carried out using the TranSIESTA package [40,41]. The Brillouin zone integration is performed using the Monkhorst-Pack sampling scheme with 1 10 1 and 20 10 1 k-points (xy is the plane of graphene with x being the transport direction) for the central device and electrodes, respectively. In addition, a 1 100 1 k-points sampling was used to get smooth well-converged average transmission and current using the TBtrans code [41]. Geometry optimizations of the 10 10 graphene supercell with two atomic-vacancies for different configurations had been performed using the spin-polarized density functional theory (DFT) as implemented in Vienna ab-initio simulation package (VASP) with the projector augmented wave (PAW) pseudopotential [43]. The
Fig. 1. Schematic diagram of the two-terminal graphene device with two atomicvacancy defects, paired by (a) parallel, and (b) series connections. The devices are partitioned into three parts: central device, semi-infinite left and right electrodes. Each dashed square in the central region enclose a 10 10 supercell. (A colour version of this figure can be viewed online.)
generalized gradient approximation (GGA) in Perdew-BurkeErnzerholf form for the exchange-correlation energy was employed with an energy cutoff of 600 eV [44]. To simulate the shear distortion, a torsional deformation angle of Dq ¼ 1 is applied to the device [22]. It is formerly shown that under the shear distortion, the residue strain in the buffer region and electrodes has no effect on the spin-polarization properties of pristine graphene systems [33]. We had considered the two atomic-vacancies either on the same (AA) or on different (AB) sublattices of graphene with various models. To minimize vacancy-vacancy interactions, the distance of between the vacancies are chosen to be at least 8 Å within the same supercell [42]. The SIESTA package had also been used to check the electronic properties of the structures as the transport properties were carried out by TranSIESTA [41,45e48]. For SIESTA simulations, the local spin density approximation (LSDA) was used for exchange-correlation potentials and singlezeta polarized (SZP) numerical atomic orbital basis sets were chosen for expanding the valence states. Atomic cores were described by non-local norm conserving Troullier-Martins pseudopotentials [49]. An energy cutoff of 120 Ry was used to define the real-space grid for numerical integration of the electron density. An electronic temperature of 20 K was used for the smearing of the electronic occupations using the Methfessel and Paxton method [50]. The computational parameters were checked carefully for convergence. Basically, all the results obtained with VASP optimizations are doubled checked and reproduced in the SIESTA calculation, before subsequent transport calculation. Under finite bias voltage Vb , the spin-dependent current Is passing through the device via spin channel s is given by the Landauer-Büttiker formula for resonance tunneling,
e Is ðVb Þ ¼ ∬ Ts;k ðε; Vb Þ½fL ðε mL Þ fR ðε mR Þdkdε; h
(1)
where fL=R is the Fermi-Dirac distribution function and mL=R is the
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electrochemical potential of the left (L) and right (R) electrodes, respectively. Ts;k ðε; Vb Þ is the transmission coefficient for spin channel s, as given by
Ts;k ðε; Vb Þ ¼ Tr½GRs;k Gs;k GLs;k Gys;k ;
(2)
where GL=Rs;k is the coupling functions measuring the tunneling interaction between of the device and electrodes L and R, respectively. Gs;k is the one-electron Green's function of the central device. 3. Results and discussion Fig. 2 (a) and (b) shows the spin-dependent IeV curves for the device with defects arrayed in serial connection and in parallel connection, respectively. The two vacancies are situated on the same sublattices of graphene, which favors a ferromagnetic coupling according to the Lieb's theorem and ab initio DFT calculations [51,52]. The IeV curves of single-atom vacancies (SV) connected in a classical circuit are also plotted in Fig. 2 (a) and (b) for comparison. For two atomic-vacancies in parallel connection, the
Fig. 2. Spin-polarized IeV characteristics of the simulated devices with two atomicvacancies with serial (a) and parallel (b) connections. Solid and dotted lines correspond to spin-up and spin-down currents. The currents of SV connected in classical circuits are also plotted for comparison. (c) Spin filtering efficiency as a function of applied voltage. (A colour version of this figure can be viewed online.)
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currents are increased by nearly a factor of two for both the spin-up and spin-down channels with a relationship of I sparallel x2I ssv . Such an increase in both spin channels does not significantly change the SFE for voltages up to 0.7 V, as shown in Fig. 2(c). As discussed in our previous works, the spin-dependent current results from the polarization of the p bands near the Fermi level in the central device. Such a polarization persists in two vacancies with ferromagnetic coupling. Detailed analysis indicates that the total magnetic moments and the DOS of the p bands near the Fermi level for double vacancies in parallel connection are roughly doubled compared to those of the monovacancy. It implies that the vacancyvacancy interaction in a model device with 10 10 supercell has minor effect on the SFE. This conclusion is in line with the classical Kirchhoff circuit law. More results for different vacancies configurations are shown in Figs. S1eS4. It is interesting to notice that classical circuit law is valid even at such molecular scale. While device with parallelly connected vacancies follows the Kirchhoff law, it behaves differently when the vacancies are connected serially. As shown in Fig. 2(b), the overall IeV characteristics of the serially connected double vacancies looks similar to the SV case. However, the currents are substantially reduced, especially in the spin-down channel. The reduction of the current is less than half in magnitude compared to the SV case. So the classical law breaks down for serial connection. To understand the different behaviors of parallel and serial devices, we further analyze the projected density of states (PDOS) and the transmission coefficients of the graphene devices. Fig. 3 shows the transmission coefficient (T) as a function of energy at different applied bias for parallel (a-c) and serial connections (d-f). For the parallel circuit, the transmission spectra for the parallel connection is similar to monovacancy case studied before [33]. Its main feature is the development of a triangle structure at finite bias in the energy window of (-eV/2, eV/2), which results from the local density of states (DOS) of the central device and their coupling to the two graphene electrodes. For the serial connection, right panels, more complex structures emerge in the transmission spectra especially in the negative energy region, and the spin-down current (reddotted lines) is more suppressed compared with the spin-up channel. The suppression of the spin-down current for serial connection is mainly responsible for the amplification of SFE shown in Fig. 2(c). To gain insight into the observed SFE enhancement, we analyzed the DOS projected on the two unsaturated carbon atoms near the two vacancies (atom 1 and atom 2, respectively in Fig. 1), which determines the magnetic vacancy states and thus the spinpolarized current through the whole device. For the parallel connection, the energy distributions of the projected DOS (PDOS) for the two atoms are degenerate at all energies and all bias voltages, as illustrated in Fig. 4(aec). The number of channels involved in the central device are thus nearly doubled, which explains the enhanced currents in both channels. Since the two defect sites in parallel configuration are located at the same distance with respect to the electrodes, they experience the same voltage drop across the junction and this explains the degeneracy of their PDOS. As the bias increases, the PDOS of the two atoms shift downwards in both spinup and spin-down channels, which follows the same trend as that of monovacancy studied before [33]. As shown in Fig. S2, the shaded region encloses the p bands which clearly shows the spin polarization. We further present in Fig. S3 the wave functions of the corresponding states that contribute to the magnetic moments. In our simulations of electron transport, polarization of spin currents is mainly contributed by the p bands near the Fermi level since only low bias voltages are considered. For the serial connection, the PDOS of the two atoms shift in opposite directions relative to the Fermi energy when a finite bias is
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Fig. 3. Spin-polarized transmission coefficient at various bias voltages for devices with double vacancies in (aec) parallel and (def) serial connections. (A colour version of this figure can be viewed online.)
Fig. 4. The DOS projected on the two carbon atoms (atom 1 and atom 2 marked in Fig. 1) at various bias voltages for the double vacancy devices with (aec) parallel and (def) serial connections. The solid and dotted lines represent the spin-up and spin-down DOS, respectively. A smearing of 0.01 eV is used for energy broadening. (A colour version of this figure can be viewed online.)
applied. The PDOS of atom 1 is shifted upward for both spin-up and spin-down channels, while for atom 2, these peaks are moving downwards as shown in Fig. 4(def). This opposite shift of PDOS between atom 1 and 2 results from the change of potential energy induced by the applied bias. It therefore offsets the spectral overlap between the two vacancy states, which in turn reduces transmission spectra and thus the current through the device. Such an opposite shift of defect states, which is modulated by the potential energy drop along the tunneling path, results in destructive interference in the transmission spectra. Such a destructive interference is particularly obvious in the spin-down channels. Fig. S7 shows the spin-polarized local DOS of the defect state near Fermi level for a
10 10 supercell with double vacancies in serial connection. It can be clearly seen that the spin-down defect states are more localized compared to the spin-up states. It is therefore expected that the spin-down states have a weaker coupling to the electrodes and this results in a lower transmission. It explains why a large current reduction occurs in the spin-down channels, and thus a significant enhancement of the SFE. In Fig. 5, the energies of the spin-polarized defect states on the two atoms are displayed as a function of voltage for the serial device. Overall, it is observed that the energies of the two defect sites are positively correlated with the applied bias voltage for both spinup and spin-down channels, although there are deviations at
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Fermi level is also shown in Fig. S5 which plots the transmission coefficients for different applied bias voltages. The maximum SFE is 17.2% and 23.9% for the deformed devices with monovacancy and double vacancies, respectively. Our results show that by introducing serially connected vacancies, the spin filtering effect of devices can be amplified and a maximum change of SFE by up to 51% can be achieved by applying a shear distortion. In Fig. S6, we show also the results when the two vacancies are situated on the different sublattices of graphene. It is observed that the configuration favors an antiferromagnetic state and results in insignificant spin current. First-principles calculations however show that the transition barrier from antiferromagnetic to ferromagnetic state is low [52,53]. In practice, it is therefore believed that a spin-filtering graphene device can be readily prepared in ferromagnetic state by applying an external magnetic field.
4. Conclusions Fig. 5. Energy of the defect states as a function of applied bias for the device with double vacancies in serial connection. Red lines and blue lines correspond to atom 1 and atom 2, respectively (as marked in Fig. 1). Solid lines: spin-up and dotted lines: spin-down. (A colour version of this figure can be viewed online.)
certain bias. Such a deviation could be due to the vacancy-vacancy interaction, and the quantum interference effect between the two atomic-vacancies. Finally, we investigate the spin current flip effect found in our previous work. A shear distortion with torsion deformation angle of Dq ¼ 1 is applied to the devices with serially connected vacancies. The IeV characteristics for bias voltage from 0.0 to 0.7 V are displayed in Fig. 6 (a) for devices with and without the deformation. Similar to the monovacancy case [33], the magnetic moment of the system is changed from 2.8 mB to 1.8 mB and a spin current flip is observed upon the applied shear distortion. The spin current flip can be attributed to the change of spin polarization of the p bands near the Fermi level. Fig. 6(b) shows a comparison the SFE between device with monovacancy and that with serially connected vacancies. For both devices, spin up current is larger than the spin down current. Compared to the monovacancy device, the maximum SFE increases from 8.5% to 27.4% when an extra vacancy is introduced. When a shear distortion is applied to the system, magnetic transition in the devices occurs due to the reversal of spin polarization of the p band of the vacancy states and this leads to the flip of spin-polarized current. The magnetic transition of states near
In summary, we have studied electron transport through model devices with double vacancy defects of graphene, which are arranged in parallel and serial connections. As in the case of monovacancy device, the localized magnetic states lead to spin-polarized currents. In the parallel case, the current response follows the classical Kirchhoff's circuit law, with current increased by a factor of 2 in both spin-up and spin-down channels. However, the SFE does not change significantly. Whereas for the serial connections, the electrostatic potential drop along the transport path offsets spectral distribution between the two defect states, leading to destructive interference effect in the tunneling current. Such a reduction is asymmetric between the spin-up and spin-down channels. Therefore, serial connected vacancies can lead to substantial increase in spin-filtering effect. It should be noted that the SFE in these systems is an intrinsic and internal property due to the defect states of the graphene. In realistic applications, it is expected that the SFE can be much enhanced when the devices are in contact with ferromagnetic electrodes, due to the injection of spinpolarized current externally. In addition, the spin current flip effect in graphene devices with monovacancy is also retained for two-atomic vacancies graphene with nanomechanical distortion. This spin current flip is attributed to the transition between a high and a low magnetic moment state and reaches a maximum SFE of 50%. We believe these results are interesting in the fundamental understanding and can lead to potential applications in graphene spintronics.
Fig. 6. (a) Spin-polarized IeV characteristics of the simulated devices with (black lines) and without (red lines) deformation. (b) Spin filtering efficiency as a function of applied voltage for devices with monovacancy (red lines) and serially connected vacancies (black lines). Solid lines: devices without deformation; dotted lines: devices with a deformation of Dq ¼ 1. (A colour version of this figure can be viewed online.)
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Author contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes The authors declare no competing financial interest. Acknowledgments We acknowledge the financial support from the Science Challenge Project (TZ2018004), National Key R&D Program of China (No. 2017YFA0303400), the National Natural Science Foundation of China (21673017 (C.Y.), U1530401 (C.Y. and S.G.), 21403081 (S.W.)) and the computational resource TH2-JK of Beijing Computational Sciences Research Center (CSRC). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.carbon.2019.08.029. References [1] W. Han, R.K. Kawakami, M. Gmitra, J. Fabian, Graphene spintronics, Nat. Nanotechnol. 9 (10) (2014) 794e807. [2] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, et al., Electric field effect in atomically thin carbon films, Science 306 (5696) (2004) 666e669. [3] P. Avouris, Z. Chen, V. Perebeinos, Carbon-based electronics, Nat. Nanotechnol. 2 (10) (2007) 605e615. [4] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (1) (2009) 109e162. [5] J. Li, S. Sanz, M. Corso, D.J. Choi, D. Pena, T. Frederiksen, et al., Single spin localization and manipulation in graphene open-shell nanostructures, Nat. Commun. 10 (1) (2019) 200. [6] X. Li, X. Wu, Two-dimensional monolayer designs for spintronics applications, Wiley Interdisciplin. Rev.: Computat. Mol. Sci. 6 (4) (2016) 441e455. [7] J. Wu, W. Pisula, K. Müllen, Graphenes as potential material for electronics, Chem. Rev. 107 (3) (2007) 718e747. [8] V. Karpan, G. Giovannetti, P. Khomyakov, M. Talanana, A. Starikov, M. Zwierzycki, et al., Graphite and graphene as perfect spin filters, Phys. Rev. Lett. 99 (17) (2007) 176602. [9] N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Electronic spin transport and spin precession in single graphene layers at room temperature, Nature 448 (7153) (2007) 571e574. [10] J.H. Chen, L. Li, W.G. Cullen, E.D. Williams, M.S. Fuhrer, Tunable Kondo effect in graphene with defects, Nat. Phys. 7 (7) (2011) 535e538. [11] J.J. Chen, H.C. Wu, D.P. Yu, Z.M. Liao, Magnetic moments in graphene with vacancies, Nanoscale 6 (15) (2014) 8814e8821. [12] B. F, K. J, K. AV, Structural defects in graphene, ACS Nano 5 (1) (2011) 26. [13] A.V. Savin, Y.S. Kivshar, Localized defect modes in graphene, Phys. Rev. B 88 (12) (2013) 125417e125424. [14] O.V. Yazyev, L. Helm, Defect-induced magnetism in graphene, Phys. Rev. B 75 (12) (2007) 125408. [15] A. Hashimoto, K. Suenaga, A. Gloter, K. Urita, S. Iijima, Direct evidence for atomic defects in graphene layers, Nature 430 (2004) 870. [16] J.C. Meyer, C. Kisielowski, R. Erni, M.D. Rossell, M.F. Crommie, A. Zettl, Direct imaging of lattice atoms and topological defects in graphene membranes, Nano Lett. 8 (11) (2008) 3582. [17] J.J. Palacios, F. Yndur ain, Critical analysis of vacancy-induced magnetism in monolayer and bilayer graphene, Phys. Rev. B 85 (24) (2012) 245443e245450. [18] N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Electronic spin transport and spin precession in single graphene layers at room temperature, Nature 448 (7153) (2007) 571. [19] B. Uchoa, V.N. Kotov, N.M.R. Peres, A.H. Castro Neto, Localized magnetic states in graphene, Phys. Rev. Lett. 101 (2) (2008), 026805-026808. [20] B. Soodchomshom, Spin currents in graphene under tension, Phys. B Condens. Matter 406 (3) (2011) 614e619. [21] F. Zhai, L. Yang, Strain-tunable spin transport in ferromagnetic graphene junctions, Appl. Phys. Lett. 98 (6) (2011), 062101. [22] F. Gao, S.W. Gao, Controlling magnetic transition of monovacancy graphene
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Serial and parallel spin circuits at molecular scale with two atomic-vacancies in graphene: Amplification of spin-filtering effect Shizheng Wen a, b, Shiwu Gao b, **, ChiYung Yam b, * a
Jiangsu Province Key Laboratory of Modern Measurement Technology and Intelligent Systems, School of Physics and Electronic Electrical Engineering, Huaiyin Normal University, 223300, Huaian, China b Beijing Computational Science Research Center, Haidian District, 100193, Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2019 Received in revised form 16 July 2019 Accepted 10 August 2019 Available online 10 August 2019
Our previous work demonstrated that graphene devices with monovacancy defects possess spin filtering effect which offers potential applications in spintronics. Here, using first-principles calculations, we further study the spin-dependent electron transport properties and spin filtering efficiency of graphene devices with double vacancies that are arrayed in parallel and serial connections. It is found that devices with vacancies in parallel connection follow classical Kirchhoff circuit law. Both spin-up and spin-down currents are amplified while spin filtering efficiency remains the same as compared to the monovacancy devices. In contrast, amplification of spin filtering efficiency is realized in devices with serially connected double vacancies. In addition, it is shown that the spin current can be flipped reversibly by nanomechanical deformation as we observed in devices with monovacancy. Our findings demonstrate the possibility to engineer the spin filtering effect for vacancy based spintronic devices. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Graphene based spintronics has emerged as a new subject for extensive experimental and theoretical studies due to its unique electronic and mechanical properties [1e11]. While pristine graphene is nonmagnetic, vacancy and defect states can generate local magnetism in graphene due to the breaking of the local sp2 symmetry and the removal of the p bond perpendicular to the graphene plane [10e18]. Such a local magnetism shows sensitive response to variation of local atomic configurations, which is controllable by either shear distortion or vertical manipulation [19e22]. It holds the promise for many potential spintronic applications such as spin filtering and injection, magnetic storage, and information processing [1,7,23]. Single carbon vacancy is the simplest defect state and is present in many graphene fabrication techniques [11,15,24]. Previous theoretical works have been carried out to understand the origin of defect magnetism in graphene [14]. It is understood that the magnetic moment associated with a vacancy is attributed to the localized p states that is close to the Fermi level. In addition, the hybridization of p
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (S. Gao), [email protected] (C. Yam). https://doi.org/10.1016/j.carbon.2019.08.029 0008-6223/© 2019 Elsevier Ltd. All rights reserved.
and s states gives rise to a total magnetic moment of about 1.5 mB for each vacancy in graphene [12,17,25]. Fully and carefully designing the electronic devices with defected-graphene are rare in present exception of the discussion on the zigzag edge graphene nanoribbons (ZGNR) systems [26]. The ZGNRs are predicted with nearly perfect spin polarization [27]. The defects in ZGNR devices impact dramatically effect for spin transport channel due to the symmetry breaking [28e31], whereas preserving edge states [32]. In a recent study, it has been demonstrated that nanomechanical control of magnetic states of graphene vacancy can lead to the spin-filtering effect in prototype graphene devices with single carbon vacancies [33]. While the mechanism of the spin-current flip has been established and understood in previous work, the spin-filtering efficiency (SFE), up to 20% depending on the bias voltage and polarity, achieved so far is not that high compared with other device candidates like for example nanoribbons [26,27,32,34e37]. In light of applications, it is also desirable to consider device structures with multiple vacancy states arranged in possible different arrays in the graphene sheet, where the vacancy-vacancy interaction can change the overall magnetic state of the coupled system [12,38]. How the vacancy-vacancy interaction modifies the magnetic state and transport properties, and in particular whether it is possible to achieve higher SFE using different architectures of multiple
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vacancies remains to be a fundamental issue that has yet to be explored. Previous works have demonstrated that defects play an important role in the properties of graphene. For instance, electron mobility in graphene is strongly influenced by the defect states [10] and a pronounced paramagnetic response is found in monovacancy graphene [25]. To our knowledge, the role of defects in the spin transport properties of graphene has not yet been explored in detail. The primary goal of this work is to investigate how parallel and serial connected double-vacancy devices will affect the spin transport properties and whether classical Kirchhoff's circuit law is still valid at the molecular level. Here, we report on the spindependent electron transport and SFE of a model device with double vacancies that are arrayed in parallel and series connections, using the density functional theory (DFT) combined with nonequilibrium Green's function (NEGF) method [39e41]. It is found that the devices with two atomic-vacancies arranged in parallel connection amplifies the electron currents equally in both the spin-up and spin-down channels, as expected by classical circuit theory, but does not change the SFE in any significant way. In contrast, serially connected double vacancies modify the two spinchannels in different ways (specify later) and thus amplifies the SFE by more than a factor of two due to the suppression of the spindown channel. Such a suppression can be attributed to the vacancy-vacancy interaction and the modulation of the electric potential drop along the transport path. The results clearly elaborate the validity and limitation of the Kirchhoff's circuit law for spintronic transport at the molecular level and provide insights into the design of spin circuits at molecular level and demonstrate a step forward to all-graphene electronics and spintronics. The manuscript is outlined as follows. Computational details and the model employed in this work are given in Section 2. Numerical studies and related discussions are given in Section 3. Finally, we summarize this work in Section 4. 2. Model and computational details Our non-equilibrium transport calculations are performed for a model two-terminal graphene device schematically shown in Fig. 1. This model device is made up of three parts including a central device region, left and right electrode. The central device is modeled by 10 10 graphene supercell with two atomic-vacancies based on our previous work [22,33]. The buffer regions, which connect between the central device and the electrodes, are constructed by extending the outermost atomic layers of central unitcell. In the transport calculations, it is assumed that the semi-infinite electrodes are in thermal equilibrium and maintain the bulk properties. The electrode Hamiltonian therefore possesses translational invariance. This assumption allows us to apply an efficient renormalization technique to calculate the surface Green's function of the semi-infinite electrodes. To screen the influence of the defect states on the electrodes, buffer regions are introduced between the device and the electrodes. The spin-dependent transport calculations were carried out using the TranSIESTA package [40,41]. The Brillouin zone integration is performed using the Monkhorst-Pack sampling scheme with 1 10 1 and 20 10 1 k-points (xy is the plane of graphene with x being the transport direction) for the central device and electrodes, respectively. In addition, a 1 100 1 k-points sampling was used to get smooth well-converged average transmission and current using the TBtrans code [41]. Geometry optimizations of the 10 10 graphene supercell with two atomic-vacancies for different configurations had been performed using the spin-polarized density functional theory (DFT) as implemented in Vienna ab-initio simulation package (VASP) with the projector augmented wave (PAW) pseudopotential [43]. The
Fig. 1. Schematic diagram of the two-terminal graphene device with two atomicvacancy defects, paired by (a) parallel, and (b) series connections. The devices are partitioned into three parts: central device, semi-infinite left and right electrodes. Each dashed square in the central region enclose a 10 10 supercell. (A colour version of this figure can be viewed online.)
generalized gradient approximation (GGA) in Perdew-BurkeErnzerholf form for the exchange-correlation energy was employed with an energy cutoff of 600 eV [44]. To simulate the shear distortion, a torsional deformation angle of Dq ¼ 1 is applied to the device [22]. It is formerly shown that under the shear distortion, the residue strain in the buffer region and electrodes has no effect on the spin-polarization properties of pristine graphene systems [33]. We had considered the two atomic-vacancies either on the same (AA) or on different (AB) sublattices of graphene with various models. To minimize vacancy-vacancy interactions, the distance of between the vacancies are chosen to be at least 8 Å within the same supercell [42]. The SIESTA package had also been used to check the electronic properties of the structures as the transport properties were carried out by TranSIESTA [41,45e48]. For SIESTA simulations, the local spin density approximation (LSDA) was used for exchange-correlation potentials and singlezeta polarized (SZP) numerical atomic orbital basis sets were chosen for expanding the valence states. Atomic cores were described by non-local norm conserving Troullier-Martins pseudopotentials [49]. An energy cutoff of 120 Ry was used to define the real-space grid for numerical integration of the electron density. An electronic temperature of 20 K was used for the smearing of the electronic occupations using the Methfessel and Paxton method [50]. The computational parameters were checked carefully for convergence. Basically, all the results obtained with VASP optimizations are doubled checked and reproduced in the SIESTA calculation, before subsequent transport calculation. Under finite bias voltage Vb , the spin-dependent current Is passing through the device via spin channel s is given by the Landauer-Büttiker formula for resonance tunneling,
e Is ðVb Þ ¼ ∬ Ts;k ðε; Vb Þ½fL ðε mL Þ fR ðε mR Þdkdε; h
(1)
where fL=R is the Fermi-Dirac distribution function and mL=R is the
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electrochemical potential of the left (L) and right (R) electrodes, respectively. Ts;k ðε; Vb Þ is the transmission coefficient for spin channel s, as given by
Ts;k ðε; Vb Þ ¼ Tr½GRs;k Gs;k GLs;k Gys;k ;
(2)
where GL=Rs;k is the coupling functions measuring the tunneling interaction between of the device and electrodes L and R, respectively. Gs;k is the one-electron Green's function of the central device. 3. Results and discussion Fig. 2 (a) and (b) shows the spin-dependent IeV curves for the device with defects arrayed in serial connection and in parallel connection, respectively. The two vacancies are situated on the same sublattices of graphene, which favors a ferromagnetic coupling according to the Lieb's theorem and ab initio DFT calculations [51,52]. The IeV curves of single-atom vacancies (SV) connected in a classical circuit are also plotted in Fig. 2 (a) and (b) for comparison. For two atomic-vacancies in parallel connection, the
Fig. 2. Spin-polarized IeV characteristics of the simulated devices with two atomicvacancies with serial (a) and parallel (b) connections. Solid and dotted lines correspond to spin-up and spin-down currents. The currents of SV connected in classical circuits are also plotted for comparison. (c) Spin filtering efficiency as a function of applied voltage. (A colour version of this figure can be viewed online.)
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currents are increased by nearly a factor of two for both the spin-up and spin-down channels with a relationship of I sparallel x2I ssv . Such an increase in both spin channels does not significantly change the SFE for voltages up to 0.7 V, as shown in Fig. 2(c). As discussed in our previous works, the spin-dependent current results from the polarization of the p bands near the Fermi level in the central device. Such a polarization persists in two vacancies with ferromagnetic coupling. Detailed analysis indicates that the total magnetic moments and the DOS of the p bands near the Fermi level for double vacancies in parallel connection are roughly doubled compared to those of the monovacancy. It implies that the vacancyvacancy interaction in a model device with 10 10 supercell has minor effect on the SFE. This conclusion is in line with the classical Kirchhoff circuit law. More results for different vacancies configurations are shown in Figs. S1eS4. It is interesting to notice that classical circuit law is valid even at such molecular scale. While device with parallelly connected vacancies follows the Kirchhoff law, it behaves differently when the vacancies are connected serially. As shown in Fig. 2(b), the overall IeV characteristics of the serially connected double vacancies looks similar to the SV case. However, the currents are substantially reduced, especially in the spin-down channel. The reduction of the current is less than half in magnitude compared to the SV case. So the classical law breaks down for serial connection. To understand the different behaviors of parallel and serial devices, we further analyze the projected density of states (PDOS) and the transmission coefficients of the graphene devices. Fig. 3 shows the transmission coefficient (T) as a function of energy at different applied bias for parallel (a-c) and serial connections (d-f). For the parallel circuit, the transmission spectra for the parallel connection is similar to monovacancy case studied before [33]. Its main feature is the development of a triangle structure at finite bias in the energy window of (-eV/2, eV/2), which results from the local density of states (DOS) of the central device and their coupling to the two graphene electrodes. For the serial connection, right panels, more complex structures emerge in the transmission spectra especially in the negative energy region, and the spin-down current (reddotted lines) is more suppressed compared with the spin-up channel. The suppression of the spin-down current for serial connection is mainly responsible for the amplification of SFE shown in Fig. 2(c). To gain insight into the observed SFE enhancement, we analyzed the DOS projected on the two unsaturated carbon atoms near the two vacancies (atom 1 and atom 2, respectively in Fig. 1), which determines the magnetic vacancy states and thus the spinpolarized current through the whole device. For the parallel connection, the energy distributions of the projected DOS (PDOS) for the two atoms are degenerate at all energies and all bias voltages, as illustrated in Fig. 4(aec). The number of channels involved in the central device are thus nearly doubled, which explains the enhanced currents in both channels. Since the two defect sites in parallel configuration are located at the same distance with respect to the electrodes, they experience the same voltage drop across the junction and this explains the degeneracy of their PDOS. As the bias increases, the PDOS of the two atoms shift downwards in both spinup and spin-down channels, which follows the same trend as that of monovacancy studied before [33]. As shown in Fig. S2, the shaded region encloses the p bands which clearly shows the spin polarization. We further present in Fig. S3 the wave functions of the corresponding states that contribute to the magnetic moments. In our simulations of electron transport, polarization of spin currents is mainly contributed by the p bands near the Fermi level since only low bias voltages are considered. For the serial connection, the PDOS of the two atoms shift in opposite directions relative to the Fermi energy when a finite bias is
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Fig. 3. Spin-polarized transmission coefficient at various bias voltages for devices with double vacancies in (aec) parallel and (def) serial connections. (A colour version of this figure can be viewed online.)
Fig. 4. The DOS projected on the two carbon atoms (atom 1 and atom 2 marked in Fig. 1) at various bias voltages for the double vacancy devices with (aec) parallel and (def) serial connections. The solid and dotted lines represent the spin-up and spin-down DOS, respectively. A smearing of 0.01 eV is used for energy broadening. (A colour version of this figure can be viewed online.)
applied. The PDOS of atom 1 is shifted upward for both spin-up and spin-down channels, while for atom 2, these peaks are moving downwards as shown in Fig. 4(def). This opposite shift of PDOS between atom 1 and 2 results from the change of potential energy induced by the applied bias. It therefore offsets the spectral overlap between the two vacancy states, which in turn reduces transmission spectra and thus the current through the device. Such an opposite shift of defect states, which is modulated by the potential energy drop along the tunneling path, results in destructive interference in the transmission spectra. Such a destructive interference is particularly obvious in the spin-down channels. Fig. S7 shows the spin-polarized local DOS of the defect state near Fermi level for a
10 10 supercell with double vacancies in serial connection. It can be clearly seen that the spin-down defect states are more localized compared to the spin-up states. It is therefore expected that the spin-down states have a weaker coupling to the electrodes and this results in a lower transmission. It explains why a large current reduction occurs in the spin-down channels, and thus a significant enhancement of the SFE. In Fig. 5, the energies of the spin-polarized defect states on the two atoms are displayed as a function of voltage for the serial device. Overall, it is observed that the energies of the two defect sites are positively correlated with the applied bias voltage for both spinup and spin-down channels, although there are deviations at
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Fermi level is also shown in Fig. S5 which plots the transmission coefficients for different applied bias voltages. The maximum SFE is 17.2% and 23.9% for the deformed devices with monovacancy and double vacancies, respectively. Our results show that by introducing serially connected vacancies, the spin filtering effect of devices can be amplified and a maximum change of SFE by up to 51% can be achieved by applying a shear distortion. In Fig. S6, we show also the results when the two vacancies are situated on the different sublattices of graphene. It is observed that the configuration favors an antiferromagnetic state and results in insignificant spin current. First-principles calculations however show that the transition barrier from antiferromagnetic to ferromagnetic state is low [52,53]. In practice, it is therefore believed that a spin-filtering graphene device can be readily prepared in ferromagnetic state by applying an external magnetic field.
4. Conclusions Fig. 5. Energy of the defect states as a function of applied bias for the device with double vacancies in serial connection. Red lines and blue lines correspond to atom 1 and atom 2, respectively (as marked in Fig. 1). Solid lines: spin-up and dotted lines: spin-down. (A colour version of this figure can be viewed online.)
certain bias. Such a deviation could be due to the vacancy-vacancy interaction, and the quantum interference effect between the two atomic-vacancies. Finally, we investigate the spin current flip effect found in our previous work. A shear distortion with torsion deformation angle of Dq ¼ 1 is applied to the devices with serially connected vacancies. The IeV characteristics for bias voltage from 0.0 to 0.7 V are displayed in Fig. 6 (a) for devices with and without the deformation. Similar to the monovacancy case [33], the magnetic moment of the system is changed from 2.8 mB to 1.8 mB and a spin current flip is observed upon the applied shear distortion. The spin current flip can be attributed to the change of spin polarization of the p bands near the Fermi level. Fig. 6(b) shows a comparison the SFE between device with monovacancy and that with serially connected vacancies. For both devices, spin up current is larger than the spin down current. Compared to the monovacancy device, the maximum SFE increases from 8.5% to 27.4% when an extra vacancy is introduced. When a shear distortion is applied to the system, magnetic transition in the devices occurs due to the reversal of spin polarization of the p band of the vacancy states and this leads to the flip of spin-polarized current. The magnetic transition of states near
In summary, we have studied electron transport through model devices with double vacancy defects of graphene, which are arranged in parallel and serial connections. As in the case of monovacancy device, the localized magnetic states lead to spin-polarized currents. In the parallel case, the current response follows the classical Kirchhoff's circuit law, with current increased by a factor of 2 in both spin-up and spin-down channels. However, the SFE does not change significantly. Whereas for the serial connections, the electrostatic potential drop along the transport path offsets spectral distribution between the two defect states, leading to destructive interference effect in the tunneling current. Such a reduction is asymmetric between the spin-up and spin-down channels. Therefore, serial connected vacancies can lead to substantial increase in spin-filtering effect. It should be noted that the SFE in these systems is an intrinsic and internal property due to the defect states of the graphene. In realistic applications, it is expected that the SFE can be much enhanced when the devices are in contact with ferromagnetic electrodes, due to the injection of spinpolarized current externally. In addition, the spin current flip effect in graphene devices with monovacancy is also retained for two-atomic vacancies graphene with nanomechanical distortion. This spin current flip is attributed to the transition between a high and a low magnetic moment state and reaches a maximum SFE of 50%. We believe these results are interesting in the fundamental understanding and can lead to potential applications in graphene spintronics.
Fig. 6. (a) Spin-polarized IeV characteristics of the simulated devices with (black lines) and without (red lines) deformation. (b) Spin filtering efficiency as a function of applied voltage for devices with monovacancy (red lines) and serially connected vacancies (black lines). Solid lines: devices without deformation; dotted lines: devices with a deformation of Dq ¼ 1. (A colour version of this figure can be viewed online.)
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Author contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes The authors declare no competing financial interest. Acknowledgments We acknowledge the financial support from the Science Challenge Project (TZ2018004), National Key R&D Program of China (No. 2017YFA0303400), the National Natural Science Foundation of China (21673017 (C.Y.), U1530401 (C.Y. and S.G.), 21403081 (S.W.)) and the computational resource TH2-JK of Beijing Computational Sciences Research Center (CSRC). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.carbon.2019.08.029. References [1] W. Han, R.K. Kawakami, M. Gmitra, J. Fabian, Graphene spintronics, Nat. Nanotechnol. 9 (10) (2014) 794e807. [2] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, et al., Electric field effect in atomically thin carbon films, Science 306 (5696) (2004) 666e669. [3] P. Avouris, Z. Chen, V. Perebeinos, Carbon-based electronics, Nat. Nanotechnol. 2 (10) (2007) 605e615. [4] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (1) (2009) 109e162. [5] J. Li, S. Sanz, M. Corso, D.J. Choi, D. Pena, T. Frederiksen, et al., Single spin localization and manipulation in graphene open-shell nanostructures, Nat. Commun. 10 (1) (2019) 200. [6] X. Li, X. Wu, Two-dimensional monolayer designs for spintronics applications, Wiley Interdisciplin. Rev.: Computat. Mol. Sci. 6 (4) (2016) 441e455. [7] J. Wu, W. Pisula, K. Müllen, Graphenes as potential material for electronics, Chem. Rev. 107 (3) (2007) 718e747. [8] V. Karpan, G. Giovannetti, P. Khomyakov, M. Talanana, A. Starikov, M. Zwierzycki, et al., Graphite and graphene as perfect spin filters, Phys. Rev. Lett. 99 (17) (2007) 176602. [9] N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Electronic spin transport and spin precession in single graphene layers at room temperature, Nature 448 (7153) (2007) 571e574. [10] J.H. Chen, L. Li, W.G. Cullen, E.D. Williams, M.S. Fuhrer, Tunable Kondo effect in graphene with defects, Nat. Phys. 7 (7) (2011) 535e538. [11] J.J. Chen, H.C. Wu, D.P. Yu, Z.M. Liao, Magnetic moments in graphene with vacancies, Nanoscale 6 (15) (2014) 8814e8821. [12] B. F, K. J, K. AV, Structural defects in graphene, ACS Nano 5 (1) (2011) 26. [13] A.V. Savin, Y.S. Kivshar, Localized defect modes in graphene, Phys. Rev. B 88 (12) (2013) 125417e125424. [14] O.V. Yazyev, L. Helm, Defect-induced magnetism in graphene, Phys. Rev. B 75 (12) (2007) 125408. [15] A. Hashimoto, K. Suenaga, A. Gloter, K. Urita, S. Iijima, Direct evidence for atomic defects in graphene layers, Nature 430 (2004) 870. [16] J.C. Meyer, C. Kisielowski, R. Erni, M.D. Rossell, M.F. Crommie, A. Zettl, Direct imaging of lattice atoms and topological defects in graphene membranes, Nano Lett. 8 (11) (2008) 3582. [17] J.J. Palacios, F. Yndur ain, Critical analysis of vacancy-induced magnetism in monolayer and bilayer graphene, Phys. Rev. B 85 (24) (2012) 245443e245450. [18] N. Tombros, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Electronic spin transport and spin precession in single graphene layers at room temperature, Nature 448 (7153) (2007) 571. [19] B. Uchoa, V.N. Kotov, N.M.R. Peres, A.H. Castro Neto, Localized magnetic states in graphene, Phys. Rev. Lett. 101 (2) (2008), 026805-026808. [20] B. Soodchomshom, Spin currents in graphene under tension, Phys. B Condens. Matter 406 (3) (2011) 614e619. [21] F. Zhai, L. Yang, Strain-tunable spin transport in ferromagnetic graphene junctions, Appl. Phys. Lett. 98 (6) (2011), 062101. [22] F. Gao, S.W. Gao, Controlling magnetic transition of monovacancy graphene
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