Dynamic conductance in double-bend silicene nanosystem

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Dynamic conductance in double-bend silicene nanosystem Yun-Lei Sun a,∗ , Chun Zhu b , Jian Chen b , En-Jia Ye b,∗ a b

School of Information and Electrical Engineering, Zhejiang University City College, Hangzhou 310015, China Jiangsu Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, School of Science, Jiangnan University, Wuxi 214122, China

a r t i c l e

i n f o

Article history: Received 11 May 2017 Received in revised form 14 June 2017 Accepted 18 June 2017 Available online xxxx Communicated by M. Wu Keywords: Silicene Ac transport Spin–orbit interaction

a b s t r a c t In this work, dynamic conductance transport properties are theoretically investigated in the zigzag silicene nanosystem with a double-bend structure. We numerically study the dc conductance and ac emittance in the nanosystem based on the tight-binding approach, Green’s function method and ac transport theory, by considering the second-nearest-neighbor spin–orbit interaction (SOI) and external electric field. The numerical results suggest that the nanosystem undergoes a quantum phase transition driven by the relatively strong SOI, which results in a large dc conductance and a vanishing ac emittance around the Dirac point despite the interface scattering. The distribution of the local density of states in the real space reveals that the SOI induces the quantum edge state by establishing transport paths at the edge of the nanosystem. Further investigation indicates that the dynamic conductance related to the quantum edge state are topologically protected from the geometrical size change of the nanosystem. Finally, the nanosystem can be tuned to be a trivial band insulator without any dc or ac response by applying external electric field. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Silicene, a monolayer honeycomb structure of silicon, has recently been synthesized and attracted much attention [1–6], since it has Dirac cones akin to graphene, and shares almost all the striking properties of graphene. Furthermore, due to the large ionic radius of silicon, silicene has a buckled structure, which leads to a relatively strong spin–orbit interaction (SOI), and makes the quantum spin Hall or the topological insulator experimentally accessible [7,8]. A variety of topologically protected states or exotic phenomena, for instance, chiral superconductivity and giant magnetoresistance have also been theoretically predicted in silicene [9,10]. Besides, due to the buckled structure, silicene can generate a staggered sublattice potential under the electric field perpendicular to the monolayer, which drives the silicene to undergo a phase transition to a trivial band insulator [11]. Therefore, silicene possesses the potential for a widely tunable two-dimensional monolayer and future nanoelectronics, since the fundamental properties like band structure and transport properties can be tuned by external field [12–16]. On the other hand, the breakthroughs in silicene experimental fabrication, such as silicene field-effect transistor towards achieving the ultimate in miniaturization of elec-

*

Corresponding authors. E-mail addresses: [email protected] (Y.-L. Sun), [email protected] (E.-J. Ye). http://dx.doi.org/10.1016/j.physleta.2017.06.030 0375-9601/© 2017 Elsevier B.V. All rights reserved.

tronic device [17], promote the investigations of silicene nanocircuits formed by nanoribbon and heterojunction with zigzag or armchair edges [18–21]. The results indicate that some factors, such as quantum topological phase, polarization of spin and valley and external electric field could have influence upon the transport of the silicene nanosystem [22–26]. Besides, the Coulomb interactions have also been studies in the silicene nanosystem, which are demonstrated to have significant influences on the magnetic, topological and thermoelectric properties [6,27,28]. Inspired by the fact that multiterminal graphene nanodevices with different lithographic structures play an important role in graphene-based electronic circuits [29–32], we carry out a theoretical investigation on the dynamic conductance transport properties in the zigzag silicene nanosystem with a double-bend structure in this paper. Double-bend structure is one of the typical nanojunction models, whose structure is significantly different from that of stripe nanoribbon. The dc conductance, ac emittance and local density of states (LDOS) of the nanosystem are numerically calculated by considering the spin–orbit interaction and external electric field, based on the tight binding approach, Green’s function method and ac transport theory. The nanosystem exhibits an anti-resonance behavior due to the interface scattering between the wide and narrow nanoribbons, which results in a dip in the dc conductance and a capacitive-like response in the ac emittance at the Dirac point. After considering the SOI, the nanosystem undergoes a quantum phase transition, which establishes transport paths at the edge of the nanosystem in the real space, and results

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in a large dc conductance and a vanishing ac emittance around the Dirac point. This transport property induced by the relatively strong SOI is topologically protected against the geometrical size change of the nanosystem. In addition, the external electric field will open an energy gap around Fermi level, which destroys the topological quantum state and makes the nanosystem a trivial band insulator without any dc or ac response. 2. Theoretical methods As shown in Fig. 1, the nanosystem is composed of two narrow semi-infinite zigzag silicenen nanoribbons with a width N N and a finite wide one with a length L D and a width N D . The silicene nanoribbons have a low-buckled structure and consist of A-sublattice and B-sublattice with layer separation 2l. Under the electric field E z perpendicular to the layer, silicene structure can generate a staggered sublattice potential. We employ the four-band second-nearest-neighbor tight binding Hamiltonian to depict the electronic structure [8],

H =H1 + H2 + H3

=−t



λSO

†



†

ciα c jα + i √ νi j c iα σαz β c jβ 3 3 i , j α i , j α β

− lE z



(1)

† t zi c i α c i α .

†

Here, the first term H 1 represents the normal nearest hopping † ciα

interaction, where t = 1.6 eV is the hopping energy, (c i α ) is the creation (annihilation) operator with spin polarization α at site i. The second term H 2 represents the second-nearest-neighbor effective SOI with λSO = 0.2t, where νi j = +1 (νi j = −1) if the second-nearest-neighbor hopping is anticlockwise (clockwise) with respect to the positive z axis [33,34]. σ z is the z component of Pauli matrix. Here the Rashba spin–orbit coupling is neglected, whose effects are relatively small compared to those of the second term [15]. The third term H 3 is the staggered sublattice potential term originating from the external electric field E z . The silicene nanoribbon was demonstrated to undergo a topological phase transition from a topological insulator to a trivial band insulator when E z = E cr , with E cr = λSO /l [11,16]. Here we consider a weakly correlated electron system, where the Coulomb interactions are simply neglected, to focus on the tuning effect of SOI and external electric field on the dynamic conductance in the title nanosystem. The Coulomb interactions are demonstrated to play an important role in the edge magnetism and thermoelectric properties in graphene-like nanosystems [6,27,28,35,36]. Nevertheless, the nonmagnetic one-electron Hamiltonian can be an effective model to study quantum phase transition and electron transport properties in silicene nanosystem [7,8]. The silicene nanosystem can be divided into three parts for calculation convenience, the center device region (D), the left lead (L) and right lead (R) as shown in Fig. 1. The two leads with semiinfinite and periodic structure consist of repeating zigzag super-cell with the width N N , which is indicated by the dashed green box. Then, the Hamiltonian of the system can be written as

HL H = ⎣ H DL 0

HLD HD HRD

L ( R ) is the self-energy of the left (right) lead. And the recursive decimation method was employed to calculate the self-energies of the leads [37]. can be calculated by The local partial density of states



dn(α , r, β)/dE = Re δα β G D α G D + i G D β G D α G D

iα

⎡

Fig. 1. Schematic structure of the double-bend nanosystem model with side view of the silicene nanoribbon super-cell at the left. The nanosystem, composed of two narrow semi-infinite zigzag silicenen nanoribbons with a width N N and a finite wide one with a length L D and a width N D , can be divided into three parts, the center device region (D), the left lead (L) and right lead (R). The electric field E z can be applied perpendicular to the layer with a separation of 2l.

⎤

0 HDR ⎦ , HR

(2)

where H D , H L (H R ), and H D L (H D R ) represent the center device, left (right) semi-infinite electrode, and the coupling between them, respectively. And Green’s function of the silicene system can be written as

 −1  G D ( E ) = ( E + i0+ ) − H D −  L −  R ,

(3)

rr

/2π [38],

which represents the scattering process of carriers incident from contact β , reach site r and exit through contact α . By summing of the incident channels or outgoing channels, dn(r, β)/dE = †

†

(G D β G D )rr /2π and dn(α , r)/dE = (G D α G D )rr /2π are defined to be the injectivity and emissivity [39]. Summing all the channel indexes, one can get the local density of state (LDOS)

dn(r) dE

= −π −1 Im(G D )rr .

(4)

Dc conductance and linear ac response of the silicene system are obtained from the ac transport theory. Among them, dynamic conductance with a low frequency ω is written as [39,40]

G α β (ω) = G α β (0) − i ωe 2 E α β + O (ω2 ),

(5)

where G α β (0) is dc conductance, and E α β is emittance, which represents ac response of the system. Here the index α (β) is the label for left

(right) contact. From the definition of dynamic conductance I α = β G α β V β , the current conservation and gauge invariance are

satisfied, since α G α β (ω) = 0 and β G α β (ω) = 0. According to Landauer–Büttiker formalism, the dc conductance is [41]

G α β (0) =

2e 2 h

T ( E ),

(6)

where h and e are Planck’s constant and electron charge, respectively. T ( E ) is the electron transmission parameter, which

 can be †

calculated by Green’s function as T ( E ) = Tr R G D L G D , where

 † L ( R ) = i L ( R ) − L ( R ) is the imaginary part of the self-energies. In terms of the ac transport theory, emittance is given by

E αβ =

dN α β dE

− D αβ , 

(7)

where dN α β /dE = dr 3 dn(α , r, β)/dE is the internal bare charge  response, and D α β = dr 3 [dn(α , r)/dE ]u β (r) is external induced charge response. The sign of emittance indicates different ac responses of the system to the applied voltage. The characteristic potential u β (r) measures the induced potential landscape in the scattering process. It satisfies equation as u β (r) =

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Fig. 2. Dc conductance (in unit of G 0 = e 2 /h), density of states (DOS) and ac emittance (the transmissive part E RL and reflective part E LL ) as a function of electron energy in the double-bend silicene nanosystem, with considering nearest-neighbor hopping term H 1 (a)/(d), nearest-neighbor hopping and second-nearest-neighbor SOI terms H 1 + H 2 (b)/(e) and nearest-neighbor hopping, second-nearest-neighbor SOI and on-site external electric field terms H 1 + H 2 + H 3 (c)/(f).

dn(r, β)/dE [dn(r)/dE]−1 [42], with the Thomas–Fermi screening model (r, r ) = δ(r − r )dn(r)/dE applied to the Poisson equation [43]. 3. Results and discussion Fig. 2 shows the dc conductance and ac emittance of the nanosystem with N D = 14, L D = 6 and N N = 6, by considering nearest-neighbor hopping term H 1 (a)/(d), nearest-neighbor hopping and second-nearest-neighbor SOI terms H 1 + H 2 (b)/(e), nearest-neighbor hopping, second-nearest-neighbor SOI and onsite external electric field terms H 1 + H 2 + H 3 (c)/(f), respectively. All the conductances and emittances exhibit a symmetrical behavior about the Fermi level, due to the electron–hole symmetry. Meanwhile, because of the interface scattering between the wide nanoribbon and the narrow ones inside the device, dc conductances are all confined by the perfect-step dotted lines in Fig. 2(a), (b) and (c). These dotted lines are the dc conductances of the corresponding uniform zigzag nanoribbons with a width of N N , which implies that the dc transport is primarily confined by the narrow nanoribbons. In Fig. 2(a), there appears some oscillations in the dc conductance spectrum because of the interface scattering. In particular, a significant dip emerges at the Dirac point due to the anti-resonant effect as shown in the inset of Fig. 2(a), where the density of states (DOS) shows a large peak. Accordingly, the ac emittance changes the signs, suggesting inductive-like or capacitive-like behaviors as shown in Fig. 2(d). At the Dirac point as shown in the inset of Fig. 2(d), the ac emittance shows a peak with a negative value, indicating a capacitive-like ac response. After taking the SOI into account, the dc conductance is shown in Fig. 2(b). One can see that the peak-like behavior of DOS, especially at the Dirac point, is suppressed substantially. Interestingly, there are some resonance states induced by the SOI around the Dirac point with the dc conductance dip offset. As previously reported, it is the SOI that induces a quantum phase transition to a quantum spin Hall insulator in the silicene nanosystem [16]. It establishes topologically protected edge state despite the geometrical shape. As a result, new transport channel opens, and the dc conductance is enlarged around the Dirac point, though the DOS is relatively small. Moreover, the ac emittance is accordingly suppressed to be a vanishing value at the Dirac point as shown in

3

Fig. 2(e). The transport properties with large dc conductance and vanishing ac emittance at the Dirac point could be closely related to the topologically protected edge state, which is demonstrated in various silicene nanosystems [16,25]. One can tune the band character and transport properties of the nanosystem by applying external electric field E z , since silicene has a buckled lattice. The external electric field E z can lead to spin splitting of the edge states. Theoretical studies on the band structure and spin Hall conductivity in silicene indicate that a relatively large external electric field (E z > E cr ) will drive a phase transition from topological insulator to trivial band insulator, by opening energy gaps for spin-up and spin-down at different points in the k space [16,44]. In Fig. 2(c), an energy gap emerges after considering the on-site electric potential energy (E z > E cr ). As seen, zero dc conductance and DOS locate around the Fermi level. Consistently, the ac emittance, as shown in Fig. 2(f), is also gapped by the external electric field, resulting in a zero ac emittance around the Fermi level. The insulating transport indicates that the external electric field opens an energy gap in the silicene nanosystem, and drives the nanosystem undergo a phase transition to a trivial band insulator. The spatial-resolved local density of states (LDOS) of the double-bend nanosystem is plotted in Fig. 3, to further illustrate the behaviors of the electronic transport at the Dirac point (E = 0) in the real space. In Fig. 3(a), LDOS distributes dispersedly at the upside and downside margin of silicene nanosystem with a large value, consistent with the peak of DOS in Fig. 2(a). However, because of the interface scattering, LDOS vanishes at the interface between the narrow ribbons and the wide one. As a result, dc conductance exhibits a dip and ac response shows a capacitivelike behavior at the Dirac point, though DOS has a large value. After considering SOI in Fig. 3(b), the LDOS obviously concentrates at the edge of the nanosystem with a smaller value, even at the interface of the narrow ribbons and the wide one, suggesting a topological edge state transport regardless of the geometrical shape of the nanosystem. Concretely, the SOI induces a quantum topological phase by establishing transport paths at the edge of the nanosystem in the real space, as indicated by the green arrows, while the bulk states of the nanosystem correspond to a band insulator. In other words, the SOI opens an energy gap at the Dirac point K and K  in the bulk electronic spectrum, while the edge state bands cross over the energy gap. The topological edge state enlarges the dc conductance, and results in a vanishing ac response in this nanosystem. Nevertheless, once applying external electric field, energy gaps of edge state open for both spin orientations. Thus, LDOS vanishes completely to zero in the whole nanosystem as shown in Fig. 3(c), since it undergoes a phase transition from topological edge state to trivial band insulator without any dc and ac response. As discussed above, the SOI indeed induces a quantum topological edge state, which protects the dc conductance from the interface scattering inside the device and results in a vanishing ac response. Here we further investigate the ac transport by calculating the ac response of the nanosystem, with the change of the SOI strength λSO and the geometrical size at the Dirac point (E = 0). Fig. 4(a) shows the bare and induced charge responses of the nanosystem with N D = 14, L D = 6 and N N = 6, as a function of the SOI strength λSO (E z = 0). When the SOI is very small (λSO → 0), the induced charge response is dominant and the nanosystem responds capacitivelikely, consistent with the results in Fig. 2(d). As the SOI increases (λSO > 0.08t), the bare and induced charge responses tend to be equal, resulting in a vanishing ac emittance as shown in Fig. 4(c). It is the relatively strong SOI in silicene that leads to a vanishing ac response at the Dirac point. Fig. 4(b) shows the size-dependent ac response of the nanosystem with the geometrical size increasing proportionally L D = N N =

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Fig. 3. The distribution of LDOS at the Dirac point in the double-bend nanosystem, with considering nearest-neighbor hopping term H 1 (a), nearest-neighbor hopping and second-nearest-neighbor SOI terms H 1 + H 2 (b) and nearest-neighbor hopping, second-nearest-neighbor SOI and on-site external electric field terms H 1 + H 2 + H 3 (c).

on the effect of the Coulomb interactions on the topologically protected dynamic conductance in the silicene nanosystem is desired. Acknowledgements The authors would like to acknowledge financial support from the National Natural Science Foundation of China (Grant Nos. 11604293, 11447206 and 11504137), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20140131). This work was also supported by the Zhejiang University City College Scientific Research Foundation (No. J-16017) and the Key Lab of Information Processing and Intelligent System of Hangzhou. References

Fig. 4. The bare (dN RL /dE) and induced charge (D RL ) responses/ac emittance (the transmissive part E RL and reflective part E LL ) change as a function of the strength of SOI λSO (a)/(c), the geometrical size (b)/(d) in the double-bend nanosystem. 3 N (λSO = 0.2t, E z = 0). As shown, the bare and induced charge 7 D responses merge together along with the size changing. Accordingly, the ac emittance keeps vanishing zero with the size changing as seen in Fig. 4(d). All these results imply that the strong SOI induces a topological quantum state with equal contributions of the bare charge response and induced one, regardless of the size variation. We could conclude that the ac transport is topologically protected by the quantum state, which is induced by the SOI in the silicene nanosystem, from the interface scattering and the change of geometrical size.

4. Conclusion To conclude, we study the dynamic conductance transport properties in the double-bend nanosystem, by considering the spin–orbit interaction and external electric field. The numerical results suggest that the nanosystem undergoes a quantum phase transition driven by the SOI, which results in a large dc conductance and a vanishing ac emittance around the Dirac point despite the interface scattering between the narrow and wide nanoribbons. LDOS distribution in the real space reveals that the SOI induces the quantum edge state by establishing transport paths at the edge of the nanosystem. Moreover, further investigation indicates that the dynamic conductance related to the quantum edge state are topologically protected against the geometrical size change of the nanosystem. Besides, the external electric field E z can open an energy gap around the Dirac point in the nanosystem, and destroy the topological quantum state. Thus the transport properties of the nanosystem can be tuned by external electric field to be insulating transport without any dc or ac response. Further investigation

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