Solid State Communications 150 (2010) 675–679
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Resonant transmission in three-terminal triangle graphene nanojunctions with zigzag edges Yuan Ping Chen ∗ , Yue E. Xie, X.L. Wei, L.Z. Sun, JianXin Zhong Institute for Quantum Engineering and Micro-Nano Energy Technology and Department of Physics, Xiangtan University, Xiangtan 411105, China
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Article history: Received 11 April 2009 Received in revised form 30 November 2009 Accepted 11 December 2009 by E.G. Wang Available online 21 December 2009 Keywords: A. Graphene nanojunction A. Three terminals D. Electron transmission
abstract Three-terminal nanojunctions based on triangle zigzag edged graphene flakes are proposed and their transport properties are studied. In the solid and hollow triangle graphene junctions, there exist different resonant transmissions due to the different electronic states in the two structures. The quasi-bound states in the solid junction are confined in the inner of the triangle flake, while those in the hollow junction are confined at the zigzag edges. In addition, these states are tightly associated with the size of the triangle flake, thus the resonant transmissions in the triangle graphene junctions can be tuned by the structural size and geometry. © 2009 Elsevier Ltd. All rights reserved.
Since the single graphene sheet was fabricated successfully, [1] some lower-dimensional graphene-based microstructures have been obtained one after the other, [2,3] such as graphene nanoribbons (GNRs) and graphene flakes (GFs). These graphene structures own unique energy bands and electronic properties different from those in the other nanostructures of similar size and shape [4–13]. The properties of the quasi-one-dimensional GNRs can range from metallic to semiconducting depending on their widths and edges [7–9]. There are peculiar edge states exist on both sides of the edges of a zigzag edged GNR (ZGNR). While the quasizero-dimensional GFs also show peculiar electronic and magnetic characteristics sensitive to the geometry and size [11–14]. In the (triangular, hexagonal etc.) GFs, triangle flakes have the most persistent shell structure. Based on the GNRs and GFs, various graphene nanojunctions that can be served as nanoelectronics have been proposed [15–20]. The graphene-based nanoelectronics are the important components of graphene nanocircuits [21] and have great potential technological applications in the future. Recently research attention has been focused on the resonant transmission in the graphene nanojunctions. Some studies have indicated that there are exotic resonant phenomena and new physics existing in these structures, due to their unique electronic structures [22–26]. However, recent studies are only concentrated on the resonant transmission in the two-terminal graphene nanostructures. In this paper, we first propose a solid three-terminal graphene nanojunction as shown in Fig. 1(a) where three semi-infinite
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0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.12.019
ZGNRs are connected to a solid triangle zigzag edged GF (ZGF). Electronic states and transport properties of the solid junction are studied. It is found that there exists resonant transmission in the structure. The resonant transport is induced by the quasibound states tightly confined in the center region of the solid ZGF. Whether there are resonant transmissions via quasi-bound states in a graphene junction with a hollow flake? To answer this question, we then study the transport properties of a hollow junction where three ZGNRs are connected to a hollow triangle ZGF as shown in Fig. 1(b). The calculated results indicate that there also exist quasi-bound states in the hollow junction. However the states in the hollow junction are mainly confined at the edges of the hollow ZGF, i.e., are localized edge states. The difference between the electronic states in the solid junction and those in the hollow junction results in different resonant profiles of the two structures. The quasi-bound states in the solid junction have longer lifetimes and higher energies, thus they induce sharp resonant peaks relatively far from the Fermi energy. While the resonant peaks in the conductance of hollow junction are very close to the Fermi energy because they are induced by the localized edge states. The dependence of the quasi-bound states on the size of the two triangle junctions is discussed. The results show that resonant transmissions via quasi-bound states in the graphene junctions are very sensitive to the geometry of the structures. To describe the electron properties of the triangle graphene junctions in Fig. 1, the two devices are both divided into four regions by the dotted lines: three semi-infinite ZGNRs and a triangle ZGF. The widths of the semi-infinite leads and the side length of the flake are labeled by NZ and NF , respectively, while NC represents the side thickness of the hollow flake. It is noted
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Fig. 1. (a) A solid graphene junction where three semi-infinite ZGNRs with widths NZ are connected to a solid triangle ZGF with side length NF . (b) A hollow graphene junction where three semi-infinite ZGNRs with widths NZ are connected to a hollow triangle ZGF with side length NF and side thickness NC .
that the inside edges of the hollow flake are also zigzag edges. The electronic structures of the junctions are described by the nearest-neighbor π -orbital tight-binding model and the hopping energy is represented by parameter t. The lattice Green’s function method is used to calculate the transport properties of the two devices [27]. According to the Green’s function scheme, the total Green’s function of the junction can be expressed as Gt = [E − H0 − Σ1 − Σ2 − Σ3 ]−1 , where H0 is the Hamiltonian of the (solid or hollow) ZGF and Σi is the self-energy of the ith lead. The self-energies can be calculated by the recursive Green’s function method [28]. Once the Green’s function Gt is obtained, one can calculate the conductance G21 from lead 1 to lead 2 and the density of state (DOS) DOS of the structure: G21 =
2e2 h
Ď
Tr [Γ1 Gt Γ2 Gt ],
(1)
DOS = −Im tr [Gt ]/π ,
(2)
Σ1∗(2) ]
where Γ1(2) = i[Σ1(2) − is coupling function of lead 1 or 2. For the two triangle junctions in Fig. 1, the conductances between arbitrary two leads are equal due to the rotative symmetry of the structures. In addition, by solving the equation H Ψ = E Ψ (H = H0 + Σ1 + Σ2 + Σ3 ), one can obtain the eigenenergies E and wave functions Ψ of the eigenstates in the structure. In general, the eigenvalue is a complex expressed as E = (ER , −γ ), in which the real part ER represents the eigenenergy and the imaginary part γ is associated with the lifetime τ of the eigenstate through τ ∼ h¯ /2γ [27]. While the lifetime τ is of the order of the reciprocal of the width of the resonance peak (h¯ /∆E ) in the transmission spectrum. So the smaller γ , the longer the lifetime τ and the narrower the resonance width ∆E in transmission spectrum. While the wave function Ψ can be used to construct electron density probability of the state. In Fig. 2(a), we show the conductance and DOS for the solid graphene junction in Fig. 1(a). It is found that sharp resonant peaks appear symmetrically at the two sides of the Fermi energy (E = 0) in the conductance. Moreover each resonant peak corresponds to a DOS peak. This indicates that there exist resonant transmissions via quasi-bound states in the solid junction. In Fig. 2(b) and (c), the quasi-bound states corresponding to the two resonant peaks (E > 0) closest to the Fermi energy are depicted. The electron probability densities of both states show three-fold rotational symmetry due to the symmetry of the structure. The electrons in the two quasibound states are mainly confined in the center region of the solid ZGF, which is somewhat similar to the case of the bound
states in a triangle cavity. [14] Our calculations indicate that the lifetimes of the states in the solid junction are long. For example, the eigenvalue of the state in Fig. 2(b) is E = (0.345, −0.00261), i.e., the lifetime of the state is τ ∼ 191.571h¯ . So these states induce sharp resonant peaks in the conductance. In addition, one can find that in Fig. 2(a) there is a sharp DOS peak at the Fermi energy which does not correspond to resonant peak. Fig. 2(d) shows the probability densities of the electronic state at E = 0. One can find that the electrons in this state are completely confined at the edges of the triangle ZGF, which exhibits unique electronic characteristics of the zigzag edged graphene nanostructures. Due to very long lifetime (τ ∼ 1016 h¯ ) the state is a quasi-bound state even close to a true bound state. Thus it does not induce resonant peak in conductance. The former discussion indicates that the resonant transmissions in the solid junction are induced by the quasi-bound states mainly confined at the center region of the structure. If the graphene flake is hollow, whether resonant transmissions via quasi-bound states can be still observed in the hollow junction? In Fig. 3(a), the conductance and DOS for the hollow graphene junction in Fig. 1(b) are shown. There also exist resonant peaks in the conductance and each resonant peak corresponds to a DOS peak. However, by comparing Fig. 3(a) with Fig. 2(a), one can find that the conductance profile of the hollow junction is different from that of the solid junction. The resonant peaks in Fig. 2(a) are very sharp and far from the Fermi energy. In the conductance of the hollow junction, however, the resonant peaks distribute closely around the Fermi energy [see Fig. 3(a)]. The difference of transport properties is originated from the difference of electron states in the two structures. To explain this point, in Fig. 3(b) and (c), we depict the quasi-bound states corresponding to the two resonant peaks (E > 0) closest to the Fermi energy in Fig. 3(a). Compared with the states in the solid junction, the states in the hollow junction are mainly confined at the zigzag edges of the hollow ZGF. These localized edge states are similar to the edge states in the perfect ZGNRs, and thus their energies are very close to the Fermi energy. On the other hand, the lifetimes of these edge states are relatively shorter (e.g., the lifetime of the state in Fig. 3(b) is τ ∼ 85.324h¯ ), because they can directly communicate with the edge states in the leads. So the quasi-bound states in the hollow junction cannot induce isolated and sharper resonant peaks. In addition, as a comparison to Fig. 2(d) we show in Fig. 3(d) the electron state at E = 0 in the hollow junction. It is found that the state in Fig. 3(d) are not confined in the hollow ZGF, on the
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b 1
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Fig. 2. (Color online) (a) The conductance (between arbitrary two leads) and DOS as a function of electron energy for the solid graphene junction as shown in Fig. 1(a). (b–d) The density probabilities of electronic states in the solid junction. The quasi-bound states in (b) and (c) correspond to the first and the second resonant peak above the Fermi energy in (a), respectively. (d) The quasi-bound state corresponds to the DOS peak at E = 0 in (a). Other parameters are NZ = 3, NF = 31.
contrary it directly extends to the leads, i.e., it is not a quasi-bound state. To illustrate the evolution process from the quasi-bound states in solid junction to those in hollow junctions, in Fig. 4(a) and (b), two middle quantum states between the two states in Figs. 2(b) and 3(b) are shown. Seen from Fig. 4(a) which exhibits a quasibound state in a hollow junction with a very small triangle hole, the electrons in the state are tightly confined at the inner zigzag edges while the outer edge states does not form. When the inner triangle hole becomes wider, the side thickness NC of the hollow junction decreases, accordingly the coupling between the inner and outer edges of the junction is strengthened. As a result outer edge states begin to form. So one can find from Fig. 4(b) that some electrons are confined at the outer edges besides the inner edge states. With the further decrease of NC , the coupling of inner and outer edges is further strengthened and thus inner and outer edge states are formed at the same time, as shown the quasi-bound states in Fig. 3(b). Fig. 4(c) shows the relationship between the energy of the quasi-bound state in the triangle junction and the side width NC of the junction. Here we only consider the quasi-bound state closest to the Fermi energy and E > 0. One can find that the energy shift to the Fermi energy fast as the side width deceases. The energy of the quasi-bound state in the solid junction is highest. A very small triangle hole in the graphene junction will result in the fast decrease of the energy. As the state in the hollow junction evolves into a quasi-bound state including both inner and outer edge states, the energy of state is close to the Fermi energy. From these results, one can conclude that the resonant transmissions via quasi-bound states in the triangle graphene junctions are very sensitive to the structural geometry.
In Fig. 5, we show the dependence of the quasi-bound states in the solid or hollow junctions on the side length NF of the triangle flakes. As the case of Fig. 4(c), we only consider the quasibound state closest to the Fermi surface and E > 0. The solid triangles represent the energies of states in the solid junction, while the hollow triangles represent those in the hollow junction. Although both energies decrease as the side length increases, the latter decreases much faster than the former. This implies that, for a hollow ZGF with wider side, many localized edge states around the Fermi surface are expected to exist in it. Accordingly many resonant peaks closer to the Fermi energy will appear in the conductance of the hollow junction with large size. On the other hand, the inset of Fig. 5 shows that the lifetimes of quasi-bound states in the two junctions also change with the variation of the side length. Especially to the states in solid junction, their lifetimes dramatically increase with the side length. In another words, with the size increase of the solid junction the state will gradually evolve into a true bound state. As to the states in hollow junction, their lifetimes do not increase obviously with the side length because they mainly localize at the edges rather than in the inner of the structure. In conclusion, we propose two types of three-terminal resonant transmission nanostructures based on ZGNRs and a triangle ZGF. For the solid triangle junction, the resonant transmissions are induced by the quasi-bound states confined in the inner of the solid ZGF, and the resonant peaks in the conductance are relatively far from the Fermi energy. While for the hollow triangle junction, the quasi-bound states are mainly confined in the zigzag edges of the hollow ZGF. The resonant peaks induced by these edge states are much closer to the Fermi energy. In addition, the energies and also
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a
b 2
DOS
G(2e2/h)
1
E(t)
c
d
Fig. 3. (Color online) (a) The conductance (between arbitrary two leads) and DOS as a function of electron energy for the hollow graphene junction as shown in Fig. 1(b). (b–d) The density probabilities of electronic states in the hollow junction. The quasi-bound states in (b) and (c) correspond to the first and the second resonant peak above the Fermi energy in (a), respectively. (d) The electronic state corresponds to the DOS peak at E = 0 in (a). Other parameters are NZ = 3, NF = 31 and NC = 3.
a
b
E(t)
c
NC Fig. 4. (Color online) The electron probability density of the quasi-bound state closest to the Fermi energy in the hollow graphene junction with (a) NC = 5 and (b) NC = 4. Other parameters are NZ = 3, NF = 31. (c) The energy (E > 0) of the quasi-bound state (closest to the Fermi energy) in the hollow junction with NZ = 3 as a function of the side thickness NC .
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References
Fig. 5. The energy (E > 0) and lifetime (inset) of the quasi-bound state closest to the Fermi energy as a function of the side length NF . The solid triangles represent the energies or lifetimes (inset) of states in the solid junction, while the hollow triangles represent the energies or lifetimes (inset) of states in the hollow junction. Other parameters are NZ = 3 and NC = 3.
the lifetimes of the quasi-bound states in the two structures are tightly associated with the side length of the triangle flake. Thus the resonant transmissions in the three-terminal junctions can be tuned by the structural geometry and size. Acknowledgements This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education (No. 200805301001), the Open Fund based on innovation platform of Hunan colleges and universities (No. 09K034) and Scientific Research Fund of the Hunan Provincial Education Department (No. 09C956).
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