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Spatial manipulating spin-polarization and tunneling patterns in graphene spirals via periphery structural modification
Accepted Manuscript Spatial manipulating spin-polarization and tunneling patterns in graphene spirals via periphery structural modification Xiaodong Xu, Ruihuan Tian, Qiang Wang, Weiqi Li, Yongyuan Jiang, Xin Zhou, Guiling Zhang, Linhua Liu, Wei Quan Tian PII:
S0008-6223(16)31026-0
DOI:
10.1016/j.carbon.2016.11.052
Reference:
CARBON 11492
To appear in:
Carbon
Received Date: 15 October 2016 Revised Date:
17 November 2016
Accepted Date: 20 November 2016
Please cite this article as: X. Xu, R. Tian, Q. Wang, W. Li, Y. Jiang, X. Zhou, G. Zhang, L. Liu, W.Q. Tian, Spatial manipulating spin-polarization and tunneling patterns in graphene spirals via periphery structural modification, Carbon (2016), doi: 10.1016/j.carbon.2016.11.052. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Spatial Manipulating Spin-polarization and Tunneling Patterns in Graphene Spirals via Periphery Structural Modification Xiaodong Xua, Ruihuan Tiana, Qiang Wangc, Weiqi Lia*, Yongyuan Jianga, Xin Zhoud, Guiling
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Zhange, Linhua Liua,b* and Wei Quan Tianf*
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Graphical Abstract
ACCEPTED MANUSCRIPT Spatial Manipulating Spin-polarization and Tunneling Patterns in Graphene Spirals via Periphery Structural Modification
Guiling Zhange, Linhua Liua,b* and Wei Quan Tianf*
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Xiaodong Xua, Ruihuan Tiana, Qiang Wangc, Weiqi Lia*, Yongyuan Jianga, Xin Zhoud,
Department of Physics, Harbin Institute of Technology, Harbin, 150001, P. R. China
b
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin,
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a
c
Department of Applied Chemistry, College of Science, Nanjing Tech University,
Nanjing, 211816, P. R. China d
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150001, P. R. China
Institute of Theoretical and Simulational Chemistry, Academy of Fundamental and
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Interdisciplinary Sciences, Harbin Institute of Technology, Harbin, 150001, P. R. China e
College of Chemical and Environmental Engineering, Harbin University of Science
College of Chemistry and Chemical Engineering, Chongqing University, Huxi
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f
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and Technology, Harbin 150080, P. R. China
Campus, Chongqing, 401331, P. R. China
∗
Corresponding author
E-mail address: [email protected] (W. Li) or [email protected] (W. –Q. Tian)
ACCEPTED MANUSCRIPT Abstract A new carbon-based morphology, graphene spirals (GSs), possesses interesting electronic features with inter-layer interaction and intra-layer interaction, ascribed to
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its unique intra-electronic coupling states. The spin-polarization and the tunneling patterns of GSs manipulated by the periphery structural modification were investigated in detail with first principle calculations. The spin-polarized edge-states
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and transport properties can be enhanced and modulated by the constructed trigonal
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corners efficiently. Governed by the positions and the numbers of the introduced carbon-hexagons, diverse spin-polarized tunneling states and various edge-state couplings between central spiral structure and electrodes can be achieved. More significantly, the contribution of inter-layer tunneling and intra-layer tunneling can be
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dominated by the topological signatures of GSs. For all spiral conformations, inter-layer tunneling always contributes to the net spin-dependent current. Remarkably, when carbon-hexagons are introduced at some typical positions, the
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complete spiral current along spiral construction is induced by intra-layer tunneling.
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Those features provide a good tunability of spin-polarized couplings and tunneling patterns in GSs for spintronic applications.
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1. Introduction The synergy of geometry, topology, and electromagnetic or optical properties of
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low-dimension materials has become a prevalent theme in physics, especially when its unusual manifestations reveal unexpected effects. In the past decade, numerous researches so far have intensely demonstrated that graphene is one of the best
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candidates to replace the conventional silicon-based devices, owing to its unique
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geometry and excellent physical features [1-4]. However, zero-bandgap hinders the applications of graphene in electronics and optoelectronics. Fortunately, the flexibility of geometry permits it to be cut and synthesized in various shapes and sizes with polytropic edge-states, which provide possibilities to obtain tailored physical features
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for developing multi-functional devices [5-13]. It is well known that the topological electronic characteristics of GNRs (graphene nanoribbons) primarily depend on its periphery structure types, defining ZGNRs (zigzag-GNRs) [14-17] and AGNRs
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(armchair-GNRs) generally [18-21]. In particular, ground-state ZGNRs exhibits
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robust edge-states with ferromagnetic (FM) coupling at each edge individually but antiferromagnetic (AFM) coupling between opposite edges, thus leading to a semiconducting behavior with zero net spin [22]. When extended to finite-size zero-dimensional GNFs (graphene nanoflakes) [23-28] or GQRs (graphene quantum rings) [29-31], graphene is driven by more diverse edge-states to show more unique intra-electronic couplings and transport properties. Additionally, spin-dependent transport properties of graphene and its derivatives have been investigated intensely, 3
ACCEPTED MANUSCRIPT such as spin-filtering [32-35], spin-resolved [36, 37], quantum spin Hall-effect [38-41] and spin-valley Hall-effect [42-44]. Recently, another carbon-based morphology, Graphene spirals (GSs), has
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stimulated the enthusiasm to investigate its topological electronic features and the potential for electronic or spintronic transport [45-50]. GSs as a screw dislocation in graphitic carbons, resembling a Riemann surface, is expected to be fabricated in
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experiment. Because not only intra-layer interaction similar to monolayer graphene
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but also inter-layer interaction similar to graphite are revealed in its intra-system electronic coupling. In experiment, it is possible to design such twisted π-systems by connecting multiple isolated graphene-ring layers or self-assembling particular precursor molecules along spatial spiral trajectory [51, 52]. On the other hand,
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theoretical predictions have revealed that GSs have significant potential applications and present certain unexpected characteristics. Interestingly, the hexagonal-GSs demonstrates an unusual topological signature. With the geometric size extension, the
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band structure shifts from semi-conductor to semi-metallicity at equilibrium state [45].
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Under axial tensile strain, the band structure of triangular-GSs convert between semi-conductor with FM-coupling edge-states and metallicity with nonmagnetic states [46]. Remarkably, Xu et al. have simulated the graphene spiral as a quantum inductance with “bond currents” concept and predicted the spatial distribution of magnetic field in GSs, induced by applied electrical field [48]. However, the behavior of electron tunneling in GSs still remains non-transparent. In this work, based on DFT (density functional theory) calculation, series of GSs 4
ACCEPTED MANUSCRIPT are investigated to illuminate the electron tunneling behaviors with spin-polarization. The simulated systems composed of central single spiral and metallic graphene electrodes are designed as two-probe conformations to insure that the carriers can be
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transported from source to drain (see Fig. 1(a)). The electron tunneling behaviors and the spin-polarized characters in those π-conjunction systems can be well manipulated by the periphery structural modification through introducing carbon-hexagons at the
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periphery of central spiral moiety of origin conformation (labeled as ). Two
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transport patterns, inter-layer tunneling and intra-layer tunneling, are demonstrated among GSs conformations. All spiral systems are defined as , where m donates the number of the introduced carbon-hexagons and n is defined to distinguish the
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isomers by the positions of carbon-hexagons.
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Fig. 1. (a) The two-probe conformation of original GS ( ) is divided into central spiral region and electrode regions. The top panel is side view and the bottom panel is
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top view. Z indicates the transport direction. The yellow shadows represent the
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electrode components and the vertical distance between electrode-layers is fixed to H=3.38Å. The atoms are colored as green for C and white for H. (b) The equilibrium spin-dependent transmission spectrum of the original GS with the energy region [-0.5eV, 0.5eV] and the SP-LDOS around the selected transmission peaks. (c) The spatial distribution of spin-polarized electron density calculated by ∇ρ = ρ↑ − ↓ . The isosurface value is set to 0.01Å-3 for all conformations.
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2. Results and Discussion 2.1.
Spin-polarized transport properties of
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The geometric construction of the original GS ( ) as two-probe conformation is presented in Fig. 1(a). The vertical distance between electrode-layers is fixed to 3.38Å for all spiral conformations which is optimized precisely for . In Fig. 1(b), the
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equilibrium transmission spectrum splits apart into two kinds of spin-polarized states,
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where the spin-up resonant tunneling peak locates below the Fermi level contributed by valence bands and the spin-down peak locates above the Fermi level contributed by the conduction bands. From the spin-polarized local electron density of states (SP-LDOS) around the selected peaks, the spin-polarized character of is induced
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by the edge-states on electrodes, while the central spiral moiety makes no contribution. Additionally, the strength of spin-up peak surpasses spin-down peak, indicating that the spin-up current is larger than spin-down current at lower applied biases. With the
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spin-polarized interaction, the electron density on the edge of electrodes reveals a
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ground FM-coupling (see Fig. 1(c)). On the spiral moiety, however, there is no distribution of spin-polarized electron. Under non-equilibrium condition, the current-voltage (I-V) characteristics were
investigated. With finite applied source-drain biases, the chemical potential of both electrodes is shifted to open bias windows. Consequently, the resonance tunneling occurs in a well-matched energy level in those bias windows, thereby inducing the enhancement of the tunneling current. Meanwhile, the strength and the position of the 7
ACCEPTED MANUSCRIPT resonant states could be modulated by the applied bias as well. Fig. 2(a) shows the spin-dependent tunneling current as a function of homologous applied bias (I-V curves). Due to the symmetric geometry, the spin-dependent I-V curves possess the
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same character under positive and negative biases. The spin-up tunneling current still keeps larger than spin-down tunneling current at lower biases range [0V, ±0.7V]. This feature can be explained from the evolution of transmission spectra in the bias
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windows. At this bias region, the portion of spin-up transmission spectra entering bias
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windows has higher amplitude than that of spin-down, resulting in stronger enhancement in spin-up tunneling current integral. When the bias exceeds ±0.7V, the magnitude of spin-up and spin-down current have a reversal change, dominated by a larger portion of spin-down transmission spectra entering the bias windows below
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Fermi level.
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Fig. 2. (a) The spin-dependent I-V curves and the evolution of the homologous spin-dependent transmission spectra of . The green lines indicate the bias widows and the black arrows indicate the direction from negative bias to positive bias for all conformations. (b-c) The patterns of local transmission pathways for spin-up and
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spin-down states respectively. The green arrows represent the transport direction and the red cross-mark suggests no contribution for intra-layer tunneling for all conformations. The side views of the transmission pathways are inset in the black
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ellipses for all conformations.
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To explore how the carriers travel in this spiral π-conjunction system, the spin-dependent local transmission pathways is calculated in the considered energy range (see Fig. 2(b-c)). The distribution of arrows represents the local transmission pathways and the colors indicate the transport direction. The local pathways of both spin-states on the central spiral moiety support backward transport for the interlayer tunneling. Namely, there is no generation of spiral current along the spiral moiety from source to drain. From the side view of the pathways distribution, the inter-layer 9
ACCEPTED MANUSCRIPT tunneling primarily contributes to the net current for both spin-polarized states. The reason for the intense inter-layer tunneling is that there is a wide overlap of scattering states between layers to provide a primary channel for the electron transport,
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meanwhile the central spiral moiety without robust edge-states cannot open a feasible
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way for intra-layer tunneling.
Fig. 3. The two-probe conformations and the spatial distribution of the spin-polarized
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electron density manipulated by the single-introduced carbon-hexagon. (a-c) The
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carbon-hexagon is introduced at different positions with various colors to construct three isomers, labeled as (n=1, 2, 3) respectively. (d-f) The homologous spatial distribution of the spin-polarized electron density.
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2.2.
Spin-polarized transport properties of When a single carbon-hexagon is introduced to construct a trigonal corner at the
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periphery of central spiral moiety, three isomers can be constructed, (n=1, 2, 3, see Fig. 3(a-c)). Compared with , the single-introduced carbon-hexagon systems exhibit a robust spin-polarized interaction. From the spatial distribution of the
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spin-polarized electron density (see Fig. 3(d-f)), the constructed trigonal corner exerts
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a key role to stimulate the redistribution of spin-polarized edge-states. Governed by the positions of the constructed trigonal corner, the conformations have diverse couplings of edge-states between electrodes and central spiral moiety. Basically, similar to GNFs, the edge-state couplings obey the so-called Lieb’s theorem regarding
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to the total spin of the Hubbard model in bipartite sublattices, A and B [53-56]. The constructed trigonal corner has extra atoms appear in one sublattice, thereby breaking the balance, NA≠NB. Consequently, the emergence of magnetism is induced by the
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edge imbalance and topological frustration of π-bonds. Generally, the edge spin shows
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FM-coupling, where the neighboring edges are at angles of 60° to each other, whereas AFM-coupling at angle of 120° [55]. In those three conformations, the periphery of the constructed trigonal corner, where the atoms belong to the equivalent sublattice, has the same FM-coupling. For , the periphery of trigonal corner and R-electrode manifests FM-coupling. However, AFM-coupling appears between the trigonal corner and the R-electrode in , where the neighboring edges are at 120° and the atoms belong to non-equivalent sublattice. Additionally, the trigonal corner cannot induce 11
ACCEPTED MANUSCRIPT the redistribution of spin-polarized electron density on L-electrode in and , owing to the weak continuity of edge-state couplings. Analogously, in , the coupling between the trigonal corner and the electrodes is very weak as well. As
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analyzed above, the spin-polarized character can be well-manipulated by the position
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of carbon-hexagon, suggesting the great potential applications in spintronics.
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Fig. 4. (a-c) The equilibrium spin-dependent transmission spectra of (n=1, 2, 3) and the SP-LDOS around the selected peaks. (d-f) The spin-dependent I-V curves. The inserted maps at the bottom are the evolution of the homologous spin-dependent transmission spectra. In comparison with , the equilibrium spin-dependent transmission is enhanced obviously, stimulated by the conformation of the trigonal corner with robust spin-polarization (see Fig. 4(a-c)). Moreover, manipulated by the position of the 12
ACCEPTED MANUSCRIPT introduced carbon-hexagon, the resonant tunneling states display certain differences. It is well-known that the electronic properties of systems primarily depend on the geometric structure, thus the introduced carbon-hexagon not only enhances the
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spin-polarized interaction but also alters the energy levels of the spiral systems. That is the reason for the splitting of the transmission peaks. The resonant tunneling states are contributed by series of spin-dependent scattering eigenstates which open
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channels for the spin-dependent carriers transportation. From the SP-LDOS around
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the selected peaks, the scattering patterns of the spin-polarized electrons are observed clearly. The trigonal corner gathers abundant spin-polarized electrons around the tunneling peaks, indicating the significant impact of the introduced carbon-hexagon on the transport characters. Additionally, due to the higher intensity of transmission
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spectrum close to Fermi level, displays a better conductivity under lower biases. From the variation of the transmission spectra, the effect of the position of the
applications.
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introduced carbon-hexagon provides more possibilities for multi-functional spintronic
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The spin-dependent I-V curves of the single-introduced carbon-hexagon conformations are presented in the right panel of Fig. 4. Transparently, the tunneling current is much more enhanced in contrast with . For and , the I-V curves present an obvious rectification feature for both spin-polarized states, deriving from the asymmetric geometry which induces the differences of the electrons tunneling paths between polarities. This feature can be well demonstrated by the evolution of the transmission spectra in the expanding bias windows. As presented in Fig. 4(d), 13
ACCEPTED MANUSCRIPT there is an asymmetric distribution of the spin-dependent transmission spectra of in the expanding bias windows. The negative applied biases drive more portion of transmission spectra entering the bias windows, leading to the large enhancement in
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the net tunneling current. Asymmetric distribution also occurs in , while the transmission spectra entering the bias windows are driven by positive biases (see Fig. 4(e)). Consequently, as revealed by the evolution of the transmission spectra, and
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have a rectification feature of current but in opposite directions. For , the
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spin-dependent transmission spectra exhibit a symmetric distribution in the expanding bias windows. There is no rectification consequently, but when the bias exceeds ±0.4V, the spin-polarized character of system is switched off (see Fig. 4(f)). Namely, the spin-polarization merely occurs under lower bias. The bidirectional rectification and
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the on-off spin-polarization phenomenon suggest that the single-introduced carbon-hexagon spiral systems could be designed as multi-functional spintronic
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devices, such as spin-switch and spin-filter.
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Fig. 5. The spin-dependent local transmission pathways of (n=1, 2, 3). From the spin-dependent local transmission pathways (see Fig. 5), the behaviors
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of the tunneling current modulated by the single-introduced carbon-hexagon can be well-understood. For both spin-polarized states of , the net tunneling current is mainly contributed by inter-layer tunneling, while the central spiral bridge makes no
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obvious contribution to intra-layer tunneling. The local pathways largely localize on
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the trigonal corner which forms more overlap of wave functions in the interlayer, further prompting direct inter-layer tunneling in comparison with . In , the spin-up pathways also localize on the trigonal corner, to some extent, driving the current scattering to the spiral bridge in backward direction, while the spin-down pathways form a tunneling current along the periphery of the trigonal corner in forward direction. And the inter-layer tunneling still occupies a larger proportion. Interestingly, for both spin-dependent pathways of , there appears an obvious 15
ACCEPTED MANUSCRIPT integrated spiral current of intra-layer tunneling along the periphery of the central spiral construction, which makes a primary contribution to the net tunneling current. From the side view, there still exists minor inter-layer tunneling component.
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Compared with , the introduced carbon-hexagon plays a significant role to enhance the transport capacity of the spiral conformations and even induces the generation of intra-layer tunneling along the periphery of spiral central moiety, such
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Spin-polarized transport properties of
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2.3.
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as .
Fig. 6. The two-probe conformations (a-c) and the spatial distribution of the spin-polarized electron density (d-f) of (n=1, 2, 3). The positions of the introduced carbon-hexagons are labeled with different colors. 16
ACCEPTED MANUSCRIPT From the previous analysis, it is known that the trigonal corner can induce robust spin-polarized interaction and may open intra-layer tunneling pattern for carriers transporting along periphery of the spiral structure. Then, based on , two
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carbon-hexagons are introduced to construct double trigonal corners, thus forming three isomers, (n=1, 2, 3, see Fig. 6(a-c)). As is anticipated, the robust spin-polarized interaction is stimulated in those spiral systems, as shown in Fig. 6(d-f). with
the
single-introduced
carbon-hexagon
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Compared
conformations,
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double-introduced carbon-hexagons drive the localization of the spin-polarized electron density to expand to entire spiral moiety, thereby improving the continuity of the edge-state coupling. Owing to the symmetric geometry, the spin-polarized electron density in (see Fig. 6(d)) presents a symmetric distribution, forming a completely
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continuous magnetic edge-state couplings. In (see Fig. 6(e)), there is a reduction of spin-polarization on the central spiral moiety, caused by the interaction between trigonal corners with same spin direction but in non-equivalent sublattices. Due to the
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weak continuity of edge-states coupling the spin-polarized electron density on
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cannot spread to entire system (see Fig. 6(f)).
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Fig. 7. (a-c) The equilibrium spin-dependent transmission spectra and the homologous
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SP-LDOS at selected peaks of (n=1, 2, 3). (d-f) The spin-dependent I-V curves and the evolution of the homologous spin-dependent transmission spectra.
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For , there appear several additional peaks in the spin-dependent transmission spectra at zero bias, which open extra channels for electron tunneling
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(see Fig 7(a-c)). Meanwhile, the spin-polarized electrons also largely localize on the trigonal corners. Consequently, the double-introduced carbon-hexagons induce the generation of more spin-polarized electronic states on center spiral moiety so as to further enhance the strength of the net tunneling current under applied biases in contrast to (see Fig.7(d-f)). Similar to , the interesting on-off spin-polarization phenomenon also occurs in . It is confirmed by the evolution of the spin-dependent transmission spectra, where the spectra split into two kinds of spin-polarized states 18
ACCEPTED MANUSCRIPT within the range [-0.4V, 0.4V], while the splitting disappears with the bias beyond ±0.4V. For and , due to asymmetric geometry, the I-V curves present a rectification phenomenon as well. Moreover, the comparison of the I-V curves and
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rectification rate for all spiral conformations is shown in Fig. S1 and Fig. S2 (see
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SM).
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Fig. 8. The spin-dependent local transmission pathways of (n=1, 2, 3) To clearly understand the effects of the double-introduced carbon-hexagons on
the tunneling behavior, the local transmission pathways were analyzed in detail as well, as shown in Fig. 8. From the side views, the inter-layer tunneling still has major contribution to the net tunneling current. For , the spiral current of intra-layer tunneling is generated and flows along the periphery of the two trigonal corners for both spin-polarized states. However, there is no obvious generation of complete spiral 19
ACCEPTED MANUSCRIPT current along the periphery in and . Spin-polarized transport properties of
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2.4.
Fig. 9. (a) The two-probe conformation for . (b) The homologous spatial distribution of the spin-polarized electron density. (c) The spin-dependent local
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transmission pathways for spin-up and spin-down states. (d) The equilibrium spin-dependent transmission spectrum. (e) The spin-dependent I-V curves.
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The spin-dependent transport properties and magnetic character of the triple-introduced carbon-hexagons spiral conformation (see Fig. 9(a)) were
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investigated as well. In Fig. 9(b), the spin-polarized electron density distributes over the spiral system, while a reduction of spin-polarization emerges on the overlap, according to Pauli exclusion principle. Correspondingly, the spin-dependent local transmission pathways are driven to spread over the spiral moiety, but in opposite directions for spin-up and spin-down states respectively (see Fig.9(c)). The spin-up pathways in forward direction along the spiral track make contributions to the intra-layer tunneling current, while the spin-down pathways in backward direction 20
ACCEPTED MANUSCRIPT open a spiral path for inter-layer tunneling. From the side view, the inter-layer tunneling for both spin-polarized states still has significant contribution to the net tunneling current. In the equilibrium spin-dependent transmission spectra, there is a
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sharp dip near Fermi level obviously, suggesting a destructive quantum interference (DQI) [57] (see Fig. 9(c)). This interesting phenomenon has been observed in experiment and predicted to control the current through single molecular junctions
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[58]. Additionally, the spin-polarized net tunneling current is also enhanced in
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contrast with (see Fig. 9(e)). 3. Conclusion
Based on first principle calculations, the spin-polarized edge-state couplings and transport properties of series of GS conformations were simulated by altering the
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positions and numbers of introduced carbon-hexagons. Due to that the constructed trigonal corners make A and B sublattice to be imbalanced, the spiral systems are stimulated to generate robust spin-polarized edge-states. With the numbers of
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introduced carbon-hexagons increasing, the continuity of the edge-states couplings is
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enhanced significantly. Moreover, different introduced positions will induce various edge-state couplings between spiral moiety and electrodes. Meanwhile, the spin-dependent transport capacity is enhanced significantly as well. There exist two patterns of electron tunneling, inter-layer tunneling and intra-layer tunneling. The inter-layer tunneling makes contributions in all spiral conformations. When the trigonal corners are constructed at some typical positions, like and , there will appear a complete spiral current of intra-layer tunneling. Furthermore, if the 21
ACCEPTED MANUSCRIPT symmetric geometry is broken by the introduced carbon-hexagons, the I-V curves demonstrate a rectification phenomenon. For and , there is an interesting on-off feature of spin-polarization. Those unique characteristics provide great
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potentials for GSs as multi-functional spintronic devices. 4. Computational Details
In calculation processes, the geometry optimization of all spiral systems is
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executed firstly by employing DFT implemented in the Vienna ab initio simulation
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package (VASP) [59-62] with the projector-augmented wave (PAW) pseudopotential and plane-wave basis set [63, 64]. The cut-off energy is 500eV. To take the weak interaction between electrode-layers into consideration, a Grimme DFT-D2 semiempirical dispersion-correction method is utilized. The maximum absolute force
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on each atoms is less than 0.01eV/Å. A vacuum slab is set with 20 Å to avoid interaction between periodic images.
The two-probe conformation with further geometry optimization is established to
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calculate the transport properties by using NEGF+DFT approach, which is
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implemented in ATK package [65, 66]. Double-zeta plus polarization (DZP) basis sets are utilized for all atoms and the real-space mesh cut-off is 150Ry. The temperature is 300K and a Monkhorst-Pack 1×1×150 is used for both of electrodes and central scattering region. The spin-dependent tunneling current is simulated using the Landauer-Büttiker formula [67]: Iσ ( E , Vb ) =
e µL Tσ ( E , Vb )[ f L ( E , Vb ) − f R ( E , Vb )]dE h ∫µR
(1)
Where e is the electron charge, h is the Planck constant, and Tσ is the transmission 22
ACCEPTED MANUSCRIPT probability at a given bias Vb with spin σ. fL(R)(E,Vb) is the Fermi-Dirac distribution function for the left(L)/right(R) electrode with the chemical potential of µL(R)(Vb)=E±eVb/2. r a ∑ ( E , V b ) G σ ( E , V b )] R σ
(2)
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r T σ ( E , V b ) = T r[Im ∑ ( E , V b ) G σr ( E , V b ) Im Lσ
r
r
Lσ
Rσ
Where Gr/Ga is the retarded/advanced Green’s function matrix, and ∑ /∑ ( E,Vb ) is the
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retarded self-energy matrix for the left/right electrode. Finally, spin-dependent Generallized-gradient-approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) form
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[68] to the exchange-correction functional is used throughout this work. Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11574062, 51473042, 21373112), the Open Project of State Key Laboratory of Structure
and
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Supramolecular
materials
(JLU)
(sklssm201620),
and the Fundamental Research Funds for the Central Universities and Program for Inn
201620).
OF
HIT
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ovation Research of Science in Harbin Institute of Technology
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DOI:
10.1016/j.carbon.2016.11.052
Reference:
CARBON 11492
To appear in:
Carbon
Received Date: 15 October 2016 Revised Date:
17 November 2016
Accepted Date: 20 November 2016
Please cite this article as: X. Xu, R. Tian, Q. Wang, W. Li, Y. Jiang, X. Zhou, G. Zhang, L. Liu, W.Q. Tian, Spatial manipulating spin-polarization and tunneling patterns in graphene spirals via periphery structural modification, Carbon (2016), doi: 10.1016/j.carbon.2016.11.052. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Spatial Manipulating Spin-polarization and Tunneling Patterns in Graphene Spirals via Periphery Structural Modification Xiaodong Xua, Ruihuan Tiana, Qiang Wangc, Weiqi Lia*, Yongyuan Jianga, Xin Zhoud, Guiling
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Zhange, Linhua Liua,b* and Wei Quan Tianf*
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Graphical Abstract
ACCEPTED MANUSCRIPT Spatial Manipulating Spin-polarization and Tunneling Patterns in Graphene Spirals via Periphery Structural Modification
Guiling Zhange, Linhua Liua,b* and Wei Quan Tianf*
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Xiaodong Xua, Ruihuan Tiana, Qiang Wangc, Weiqi Lia*, Yongyuan Jianga, Xin Zhoud,
Department of Physics, Harbin Institute of Technology, Harbin, 150001, P. R. China
b
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin,
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a
c
Department of Applied Chemistry, College of Science, Nanjing Tech University,
Nanjing, 211816, P. R. China d
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150001, P. R. China
Institute of Theoretical and Simulational Chemistry, Academy of Fundamental and
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Interdisciplinary Sciences, Harbin Institute of Technology, Harbin, 150001, P. R. China e
College of Chemical and Environmental Engineering, Harbin University of Science
College of Chemistry and Chemical Engineering, Chongqing University, Huxi
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f
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and Technology, Harbin 150080, P. R. China
Campus, Chongqing, 401331, P. R. China
∗
Corresponding author
E-mail address: [email protected] (W. Li) or [email protected] (W. –Q. Tian)
ACCEPTED MANUSCRIPT Abstract A new carbon-based morphology, graphene spirals (GSs), possesses interesting electronic features with inter-layer interaction and intra-layer interaction, ascribed to
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its unique intra-electronic coupling states. The spin-polarization and the tunneling patterns of GSs manipulated by the periphery structural modification were investigated in detail with first principle calculations. The spin-polarized edge-states
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and transport properties can be enhanced and modulated by the constructed trigonal
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corners efficiently. Governed by the positions and the numbers of the introduced carbon-hexagons, diverse spin-polarized tunneling states and various edge-state couplings between central spiral structure and electrodes can be achieved. More significantly, the contribution of inter-layer tunneling and intra-layer tunneling can be
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dominated by the topological signatures of GSs. For all spiral conformations, inter-layer tunneling always contributes to the net spin-dependent current. Remarkably, when carbon-hexagons are introduced at some typical positions, the
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complete spiral current along spiral construction is induced by intra-layer tunneling.
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Those features provide a good tunability of spin-polarized couplings and tunneling patterns in GSs for spintronic applications.
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1. Introduction The synergy of geometry, topology, and electromagnetic or optical properties of
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low-dimension materials has become a prevalent theme in physics, especially when its unusual manifestations reveal unexpected effects. In the past decade, numerous researches so far have intensely demonstrated that graphene is one of the best
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candidates to replace the conventional silicon-based devices, owing to its unique
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geometry and excellent physical features [1-4]. However, zero-bandgap hinders the applications of graphene in electronics and optoelectronics. Fortunately, the flexibility of geometry permits it to be cut and synthesized in various shapes and sizes with polytropic edge-states, which provide possibilities to obtain tailored physical features
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for developing multi-functional devices [5-13]. It is well known that the topological electronic characteristics of GNRs (graphene nanoribbons) primarily depend on its periphery structure types, defining ZGNRs (zigzag-GNRs) [14-17] and AGNRs
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(armchair-GNRs) generally [18-21]. In particular, ground-state ZGNRs exhibits
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robust edge-states with ferromagnetic (FM) coupling at each edge individually but antiferromagnetic (AFM) coupling between opposite edges, thus leading to a semiconducting behavior with zero net spin [22]. When extended to finite-size zero-dimensional GNFs (graphene nanoflakes) [23-28] or GQRs (graphene quantum rings) [29-31], graphene is driven by more diverse edge-states to show more unique intra-electronic couplings and transport properties. Additionally, spin-dependent transport properties of graphene and its derivatives have been investigated intensely, 3
ACCEPTED MANUSCRIPT such as spin-filtering [32-35], spin-resolved [36, 37], quantum spin Hall-effect [38-41] and spin-valley Hall-effect [42-44]. Recently, another carbon-based morphology, Graphene spirals (GSs), has
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stimulated the enthusiasm to investigate its topological electronic features and the potential for electronic or spintronic transport [45-50]. GSs as a screw dislocation in graphitic carbons, resembling a Riemann surface, is expected to be fabricated in
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experiment. Because not only intra-layer interaction similar to monolayer graphene
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but also inter-layer interaction similar to graphite are revealed in its intra-system electronic coupling. In experiment, it is possible to design such twisted π-systems by connecting multiple isolated graphene-ring layers or self-assembling particular precursor molecules along spatial spiral trajectory [51, 52]. On the other hand,
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theoretical predictions have revealed that GSs have significant potential applications and present certain unexpected characteristics. Interestingly, the hexagonal-GSs demonstrates an unusual topological signature. With the geometric size extension, the
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band structure shifts from semi-conductor to semi-metallicity at equilibrium state [45].
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Under axial tensile strain, the band structure of triangular-GSs convert between semi-conductor with FM-coupling edge-states and metallicity with nonmagnetic states [46]. Remarkably, Xu et al. have simulated the graphene spiral as a quantum inductance with “bond currents” concept and predicted the spatial distribution of magnetic field in GSs, induced by applied electrical field [48]. However, the behavior of electron tunneling in GSs still remains non-transparent. In this work, based on DFT (density functional theory) calculation, series of GSs 4
ACCEPTED MANUSCRIPT are investigated to illuminate the electron tunneling behaviors with spin-polarization. The simulated systems composed of central single spiral and metallic graphene electrodes are designed as two-probe conformations to insure that the carriers can be
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transported from source to drain (see Fig. 1(a)). The electron tunneling behaviors and the spin-polarized characters in those π-conjunction systems can be well manipulated by the periphery structural modification through introducing carbon-hexagons at the
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periphery of central spiral moiety of origin conformation (labeled as ). Two
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transport patterns, inter-layer tunneling and intra-layer tunneling, are demonstrated among GSs conformations. All spiral systems are defined as , where m donates the number of the introduced carbon-hexagons and n is defined to distinguish the
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isomers by the positions of carbon-hexagons.
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Fig. 1. (a) The two-probe conformation of original GS ( ) is divided into central spiral region and electrode regions. The top panel is side view and the bottom panel is
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top view. Z indicates the transport direction. The yellow shadows represent the
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electrode components and the vertical distance between electrode-layers is fixed to H=3.38Å. The atoms are colored as green for C and white for H. (b) The equilibrium spin-dependent transmission spectrum of the original GS with the energy region [-0.5eV, 0.5eV] and the SP-LDOS around the selected transmission peaks. (c) The spatial distribution of spin-polarized electron density calculated by ∇ρ = ρ↑ − ↓ . The isosurface value is set to 0.01Å-3 for all conformations.
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2. Results and Discussion 2.1.
Spin-polarized transport properties of
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The geometric construction of the original GS ( ) as two-probe conformation is presented in Fig. 1(a). The vertical distance between electrode-layers is fixed to 3.38Å for all spiral conformations which is optimized precisely for . In Fig. 1(b), the
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equilibrium transmission spectrum splits apart into two kinds of spin-polarized states,
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where the spin-up resonant tunneling peak locates below the Fermi level contributed by valence bands and the spin-down peak locates above the Fermi level contributed by the conduction bands. From the spin-polarized local electron density of states (SP-LDOS) around the selected peaks, the spin-polarized character of is induced
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by the edge-states on electrodes, while the central spiral moiety makes no contribution. Additionally, the strength of spin-up peak surpasses spin-down peak, indicating that the spin-up current is larger than spin-down current at lower applied biases. With the
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spin-polarized interaction, the electron density on the edge of electrodes reveals a
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ground FM-coupling (see Fig. 1(c)). On the spiral moiety, however, there is no distribution of spin-polarized electron. Under non-equilibrium condition, the current-voltage (I-V) characteristics were
investigated. With finite applied source-drain biases, the chemical potential of both electrodes is shifted to open bias windows. Consequently, the resonance tunneling occurs in a well-matched energy level in those bias windows, thereby inducing the enhancement of the tunneling current. Meanwhile, the strength and the position of the 7
ACCEPTED MANUSCRIPT resonant states could be modulated by the applied bias as well. Fig. 2(a) shows the spin-dependent tunneling current as a function of homologous applied bias (I-V curves). Due to the symmetric geometry, the spin-dependent I-V curves possess the
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same character under positive and negative biases. The spin-up tunneling current still keeps larger than spin-down tunneling current at lower biases range [0V, ±0.7V]. This feature can be explained from the evolution of transmission spectra in the bias
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windows. At this bias region, the portion of spin-up transmission spectra entering bias
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windows has higher amplitude than that of spin-down, resulting in stronger enhancement in spin-up tunneling current integral. When the bias exceeds ±0.7V, the magnitude of spin-up and spin-down current have a reversal change, dominated by a larger portion of spin-down transmission spectra entering the bias windows below
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Fermi level.
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Fig. 2. (a) The spin-dependent I-V curves and the evolution of the homologous spin-dependent transmission spectra of . The green lines indicate the bias widows and the black arrows indicate the direction from negative bias to positive bias for all conformations. (b-c) The patterns of local transmission pathways for spin-up and
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spin-down states respectively. The green arrows represent the transport direction and the red cross-mark suggests no contribution for intra-layer tunneling for all conformations. The side views of the transmission pathways are inset in the black
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ellipses for all conformations.
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To explore how the carriers travel in this spiral π-conjunction system, the spin-dependent local transmission pathways is calculated in the considered energy range (see Fig. 2(b-c)). The distribution of arrows represents the local transmission pathways and the colors indicate the transport direction. The local pathways of both spin-states on the central spiral moiety support backward transport for the interlayer tunneling. Namely, there is no generation of spiral current along the spiral moiety from source to drain. From the side view of the pathways distribution, the inter-layer 9
ACCEPTED MANUSCRIPT tunneling primarily contributes to the net current for both spin-polarized states. The reason for the intense inter-layer tunneling is that there is a wide overlap of scattering states between layers to provide a primary channel for the electron transport,
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meanwhile the central spiral moiety without robust edge-states cannot open a feasible
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way for intra-layer tunneling.
Fig. 3. The two-probe conformations and the spatial distribution of the spin-polarized
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electron density manipulated by the single-introduced carbon-hexagon. (a-c) The
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carbon-hexagon is introduced at different positions with various colors to construct three isomers, labeled as (n=1, 2, 3) respectively. (d-f) The homologous spatial distribution of the spin-polarized electron density.
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2.2.
Spin-polarized transport properties of When a single carbon-hexagon is introduced to construct a trigonal corner at the
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periphery of central spiral moiety, three isomers can be constructed, (n=1, 2, 3, see Fig. 3(a-c)). Compared with , the single-introduced carbon-hexagon systems exhibit a robust spin-polarized interaction. From the spatial distribution of the
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spin-polarized electron density (see Fig. 3(d-f)), the constructed trigonal corner exerts
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a key role to stimulate the redistribution of spin-polarized edge-states. Governed by the positions of the constructed trigonal corner, the conformations have diverse couplings of edge-states between electrodes and central spiral moiety. Basically, similar to GNFs, the edge-state couplings obey the so-called Lieb’s theorem regarding
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to the total spin of the Hubbard model in bipartite sublattices, A and B [53-56]. The constructed trigonal corner has extra atoms appear in one sublattice, thereby breaking the balance, NA≠NB. Consequently, the emergence of magnetism is induced by the
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edge imbalance and topological frustration of π-bonds. Generally, the edge spin shows
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FM-coupling, where the neighboring edges are at angles of 60° to each other, whereas AFM-coupling at angle of 120° [55]. In those three conformations, the periphery of the constructed trigonal corner, where the atoms belong to the equivalent sublattice, has the same FM-coupling. For , the periphery of trigonal corner and R-electrode manifests FM-coupling. However, AFM-coupling appears between the trigonal corner and the R-electrode in , where the neighboring edges are at 120° and the atoms belong to non-equivalent sublattice. Additionally, the trigonal corner cannot induce 11
ACCEPTED MANUSCRIPT the redistribution of spin-polarized electron density on L-electrode in and , owing to the weak continuity of edge-state couplings. Analogously, in , the coupling between the trigonal corner and the electrodes is very weak as well. As
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analyzed above, the spin-polarized character can be well-manipulated by the position
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of carbon-hexagon, suggesting the great potential applications in spintronics.
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Fig. 4. (a-c) The equilibrium spin-dependent transmission spectra of (n=1, 2, 3) and the SP-LDOS around the selected peaks. (d-f) The spin-dependent I-V curves. The inserted maps at the bottom are the evolution of the homologous spin-dependent transmission spectra. In comparison with , the equilibrium spin-dependent transmission is enhanced obviously, stimulated by the conformation of the trigonal corner with robust spin-polarization (see Fig. 4(a-c)). Moreover, manipulated by the position of the 12
ACCEPTED MANUSCRIPT introduced carbon-hexagon, the resonant tunneling states display certain differences. It is well-known that the electronic properties of systems primarily depend on the geometric structure, thus the introduced carbon-hexagon not only enhances the
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spin-polarized interaction but also alters the energy levels of the spiral systems. That is the reason for the splitting of the transmission peaks. The resonant tunneling states are contributed by series of spin-dependent scattering eigenstates which open
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channels for the spin-dependent carriers transportation. From the SP-LDOS around
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the selected peaks, the scattering patterns of the spin-polarized electrons are observed clearly. The trigonal corner gathers abundant spin-polarized electrons around the tunneling peaks, indicating the significant impact of the introduced carbon-hexagon on the transport characters. Additionally, due to the higher intensity of transmission
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spectrum close to Fermi level, displays a better conductivity under lower biases. From the variation of the transmission spectra, the effect of the position of the
applications.
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introduced carbon-hexagon provides more possibilities for multi-functional spintronic
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The spin-dependent I-V curves of the single-introduced carbon-hexagon conformations are presented in the right panel of Fig. 4. Transparently, the tunneling current is much more enhanced in contrast with . For and , the I-V curves present an obvious rectification feature for both spin-polarized states, deriving from the asymmetric geometry which induces the differences of the electrons tunneling paths between polarities. This feature can be well demonstrated by the evolution of the transmission spectra in the expanding bias windows. As presented in Fig. 4(d), 13
ACCEPTED MANUSCRIPT there is an asymmetric distribution of the spin-dependent transmission spectra of in the expanding bias windows. The negative applied biases drive more portion of transmission spectra entering the bias windows, leading to the large enhancement in
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the net tunneling current. Asymmetric distribution also occurs in , while the transmission spectra entering the bias windows are driven by positive biases (see Fig. 4(e)). Consequently, as revealed by the evolution of the transmission spectra, and
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have a rectification feature of current but in opposite directions. For , the
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spin-dependent transmission spectra exhibit a symmetric distribution in the expanding bias windows. There is no rectification consequently, but when the bias exceeds ±0.4V, the spin-polarized character of system is switched off (see Fig. 4(f)). Namely, the spin-polarization merely occurs under lower bias. The bidirectional rectification and
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the on-off spin-polarization phenomenon suggest that the single-introduced carbon-hexagon spiral systems could be designed as multi-functional spintronic
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devices, such as spin-switch and spin-filter.
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Fig. 5. The spin-dependent local transmission pathways of (n=1, 2, 3). From the spin-dependent local transmission pathways (see Fig. 5), the behaviors
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of the tunneling current modulated by the single-introduced carbon-hexagon can be well-understood. For both spin-polarized states of , the net tunneling current is mainly contributed by inter-layer tunneling, while the central spiral bridge makes no
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obvious contribution to intra-layer tunneling. The local pathways largely localize on
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the trigonal corner which forms more overlap of wave functions in the interlayer, further prompting direct inter-layer tunneling in comparison with . In , the spin-up pathways also localize on the trigonal corner, to some extent, driving the current scattering to the spiral bridge in backward direction, while the spin-down pathways form a tunneling current along the periphery of the trigonal corner in forward direction. And the inter-layer tunneling still occupies a larger proportion. Interestingly, for both spin-dependent pathways of , there appears an obvious 15
ACCEPTED MANUSCRIPT integrated spiral current of intra-layer tunneling along the periphery of the central spiral construction, which makes a primary contribution to the net tunneling current. From the side view, there still exists minor inter-layer tunneling component.
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Compared with , the introduced carbon-hexagon plays a significant role to enhance the transport capacity of the spiral conformations and even induces the generation of intra-layer tunneling along the periphery of spiral central moiety, such
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Spin-polarized transport properties of
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2.3.
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as .
Fig. 6. The two-probe conformations (a-c) and the spatial distribution of the spin-polarized electron density (d-f) of (n=1, 2, 3). The positions of the introduced carbon-hexagons are labeled with different colors. 16
ACCEPTED MANUSCRIPT From the previous analysis, it is known that the trigonal corner can induce robust spin-polarized interaction and may open intra-layer tunneling pattern for carriers transporting along periphery of the spiral structure. Then, based on , two
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carbon-hexagons are introduced to construct double trigonal corners, thus forming three isomers, (n=1, 2, 3, see Fig. 6(a-c)). As is anticipated, the robust spin-polarized interaction is stimulated in those spiral systems, as shown in Fig. 6(d-f). with
the
single-introduced
carbon-hexagon
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Compared
conformations,
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double-introduced carbon-hexagons drive the localization of the spin-polarized electron density to expand to entire spiral moiety, thereby improving the continuity of the edge-state coupling. Owing to the symmetric geometry, the spin-polarized electron density in (see Fig. 6(d)) presents a symmetric distribution, forming a completely
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continuous magnetic edge-state couplings. In (see Fig. 6(e)), there is a reduction of spin-polarization on the central spiral moiety, caused by the interaction between trigonal corners with same spin direction but in non-equivalent sublattices. Due to the
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weak continuity of edge-states coupling the spin-polarized electron density on
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cannot spread to entire system (see Fig. 6(f)).
17
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ACCEPTED MANUSCRIPT
Fig. 7. (a-c) The equilibrium spin-dependent transmission spectra and the homologous
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SP-LDOS at selected peaks of (n=1, 2, 3). (d-f) The spin-dependent I-V curves and the evolution of the homologous spin-dependent transmission spectra.
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For , there appear several additional peaks in the spin-dependent transmission spectra at zero bias, which open extra channels for electron tunneling
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(see Fig 7(a-c)). Meanwhile, the spin-polarized electrons also largely localize on the trigonal corners. Consequently, the double-introduced carbon-hexagons induce the generation of more spin-polarized electronic states on center spiral moiety so as to further enhance the strength of the net tunneling current under applied biases in contrast to (see Fig.7(d-f)). Similar to , the interesting on-off spin-polarization phenomenon also occurs in . It is confirmed by the evolution of the spin-dependent transmission spectra, where the spectra split into two kinds of spin-polarized states 18
ACCEPTED MANUSCRIPT within the range [-0.4V, 0.4V], while the splitting disappears with the bias beyond ±0.4V. For and , due to asymmetric geometry, the I-V curves present a rectification phenomenon as well. Moreover, the comparison of the I-V curves and
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rectification rate for all spiral conformations is shown in Fig. S1 and Fig. S2 (see
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SM).
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Fig. 8. The spin-dependent local transmission pathways of (n=1, 2, 3) To clearly understand the effects of the double-introduced carbon-hexagons on
the tunneling behavior, the local transmission pathways were analyzed in detail as well, as shown in Fig. 8. From the side views, the inter-layer tunneling still has major contribution to the net tunneling current. For , the spiral current of intra-layer tunneling is generated and flows along the periphery of the two trigonal corners for both spin-polarized states. However, there is no obvious generation of complete spiral 19
ACCEPTED MANUSCRIPT current along the periphery in and . Spin-polarized transport properties of
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2.4.
Fig. 9. (a) The two-probe conformation for . (b) The homologous spatial distribution of the spin-polarized electron density. (c) The spin-dependent local
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transmission pathways for spin-up and spin-down states. (d) The equilibrium spin-dependent transmission spectrum. (e) The spin-dependent I-V curves.
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The spin-dependent transport properties and magnetic character of the triple-introduced carbon-hexagons spiral conformation (see Fig. 9(a)) were
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investigated as well. In Fig. 9(b), the spin-polarized electron density distributes over the spiral system, while a reduction of spin-polarization emerges on the overlap, according to Pauli exclusion principle. Correspondingly, the spin-dependent local transmission pathways are driven to spread over the spiral moiety, but in opposite directions for spin-up and spin-down states respectively (see Fig.9(c)). The spin-up pathways in forward direction along the spiral track make contributions to the intra-layer tunneling current, while the spin-down pathways in backward direction 20
ACCEPTED MANUSCRIPT open a spiral path for inter-layer tunneling. From the side view, the inter-layer tunneling for both spin-polarized states still has significant contribution to the net tunneling current. In the equilibrium spin-dependent transmission spectra, there is a
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sharp dip near Fermi level obviously, suggesting a destructive quantum interference (DQI) [57] (see Fig. 9(c)). This interesting phenomenon has been observed in experiment and predicted to control the current through single molecular junctions
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[58]. Additionally, the spin-polarized net tunneling current is also enhanced in
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contrast with (see Fig. 9(e)). 3. Conclusion
Based on first principle calculations, the spin-polarized edge-state couplings and transport properties of series of GS conformations were simulated by altering the
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positions and numbers of introduced carbon-hexagons. Due to that the constructed trigonal corners make A and B sublattice to be imbalanced, the spiral systems are stimulated to generate robust spin-polarized edge-states. With the numbers of
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introduced carbon-hexagons increasing, the continuity of the edge-states couplings is
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enhanced significantly. Moreover, different introduced positions will induce various edge-state couplings between spiral moiety and electrodes. Meanwhile, the spin-dependent transport capacity is enhanced significantly as well. There exist two patterns of electron tunneling, inter-layer tunneling and intra-layer tunneling. The inter-layer tunneling makes contributions in all spiral conformations. When the trigonal corners are constructed at some typical positions, like and , there will appear a complete spiral current of intra-layer tunneling. Furthermore, if the 21
ACCEPTED MANUSCRIPT symmetric geometry is broken by the introduced carbon-hexagons, the I-V curves demonstrate a rectification phenomenon. For and , there is an interesting on-off feature of spin-polarization. Those unique characteristics provide great
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potentials for GSs as multi-functional spintronic devices. 4. Computational Details
In calculation processes, the geometry optimization of all spiral systems is
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executed firstly by employing DFT implemented in the Vienna ab initio simulation
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package (VASP) [59-62] with the projector-augmented wave (PAW) pseudopotential and plane-wave basis set [63, 64]. The cut-off energy is 500eV. To take the weak interaction between electrode-layers into consideration, a Grimme DFT-D2 semiempirical dispersion-correction method is utilized. The maximum absolute force
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on each atoms is less than 0.01eV/Å. A vacuum slab is set with 20 Å to avoid interaction between periodic images.
The two-probe conformation with further geometry optimization is established to
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calculate the transport properties by using NEGF+DFT approach, which is
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implemented in ATK package [65, 66]. Double-zeta plus polarization (DZP) basis sets are utilized for all atoms and the real-space mesh cut-off is 150Ry. The temperature is 300K and a Monkhorst-Pack 1×1×150 is used for both of electrodes and central scattering region. The spin-dependent tunneling current is simulated using the Landauer-Büttiker formula [67]: Iσ ( E , Vb ) =
e µL Tσ ( E , Vb )[ f L ( E , Vb ) − f R ( E , Vb )]dE h ∫µR
(1)
Where e is the electron charge, h is the Planck constant, and Tσ is the transmission 22
ACCEPTED MANUSCRIPT probability at a given bias Vb with spin σ. fL(R)(E,Vb) is the Fermi-Dirac distribution function for the left(L)/right(R) electrode with the chemical potential of µL(R)(Vb)=E±eVb/2. r a ∑ ( E , V b ) G σ ( E , V b )] R σ
(2)
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r T σ ( E , V b ) = T r[Im ∑ ( E , V b ) G σr ( E , V b ) Im Lσ
r
r
Lσ
Rσ
Where Gr/Ga is the retarded/advanced Green’s function matrix, and ∑ /∑ ( E,Vb ) is the
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retarded self-energy matrix for the left/right electrode. Finally, spin-dependent Generallized-gradient-approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) form
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[68] to the exchange-correction functional is used throughout this work. Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11574062, 51473042, 21373112), the Open Project of State Key Laboratory of Structure
and
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Supramolecular
materials
(JLU)
(sklssm201620),
and the Fundamental Research Funds for the Central Universities and Program for Inn
201620).
OF
HIT
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