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Generation and control of charge, spin and valley currents in armchair silicene nanoribbons
Physica E 63 (2014) 81–86
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Generation and control of charge, spin and valley currents in armchair silicene nanoribbons H. Shirkani, M.M. Golshan n Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran
H I G H L I G H T S
G R A P H I C A L
Charge/spin/valley current developed in armchair silicene nanoribbons (ASNs) is reported. Using the symmetries of ASN, the total Hamiltonian is shown to be block-diagonal. The eigenstates and eigenvalues of the total Hamiltonian are then calculated. Consequently, Kubo's formula is used to study charge/spin/valley conductance in ASN. Properties of charge/spin/valley conductance thus obtained are discussed. In particular, topological phase transitions are reported and discussed.
Effects of the width, buckling strength and Fermi energy on the characteristics of charge/spin/valley currents in armchair silicene nanoribbons are reported. Schematic illustration of (a) spin current and (b) valley current. The electronic spin states are denoted by the arrows while the hollow (full) circles designate the valley jK 0 ðKÞ〉 states.
art ic l e i nf o
a b s t r a c t
Article history: Received 22 February 2014 Received in revised form 22 April 2014 Accepted 23 April 2014 Available online 20 May 2014
It is well known that future information technology relies upon electronic states other than charge, through the feasibility of constructing spintronic and valleytronic devices. The main challenge in such applications is the generation and control of the corresponding state currents. The present paper is thus concerned with charge, spin and valley currents which intrinsically develop in armchair silicene nanoribbons (ASNs). In pursuing this consideration the well established Kubo formula is employed. Taking the symmetries of the ASN into account, we analyze the structure of total Hamiltonian and demonstrate how the matrix elements along with the summations involved in the suitably adopted Kubo formula, may be analytically calculated. The results so-obtained along with the corresponding figures reveal that all three currents develop in a step-wise manner. The heights, indicating a jump in the current, and the plateaus, indicating a constant current, are shown to be tunable by adjusting the width and/or the buckled effect. Moreover, we demonstrate that for particular (critical) widths an inversion in band gaps may occur, giving rise to quantum phase transitions. More practical result of this paper, as we show, is the fact that the ASN can generate pure spin or valley current under specific conditions placed on the width, buckled effect and Fermi energy. The material presented in this paper thus provides novel means of generation and control of the charge, spin and valley currents. & 2014 Elsevier B.V. All rights reserved.
Keywords: Silicene Armchair nanoribbon Kubo formula Charge conductance Spin conductance Valley conductance
A B S T R A C T
1. Introduction
n
Corresponding author. E-mail address: [email protected] (M.M. Golshan).
http://dx.doi.org/10.1016/j.physe.2014.04.022 1386-9477/& 2014 Elsevier B.V. All rights reserved.
It is by now well established that any quantum information processing scheme requires entities of limited number of states with least dissipation arising from interactions with the environment [1–3]. Moreover, the states should naturally posses long
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relaxation time and length [4–6] to preserve the information placed on it. In addition, the information processing is carried out through a network of elements that, desirably, are of microscopic scales [7]. A vivid candidate possessing these requirements is the electronic states in nanostructures, in particular, formed with the so-called honeycomb materials [5,6,8,9]. In this regard, graphene [5] and, especially, silicene [6] have attracted much interest. It is well known that the electrons in these materials behave as massless Dirac particles, consequently, possessing long spin diffusion length [6,10], long spin coherency [6,10], highefficiency spin injection [11], etc. [12,13]. Regarding the electronic states, however, there is an important difference between graphene and silicene: in the latter the buckling effect [14,15], where one sublattice is shifted upwards relative to the other, is of great consequences, while in the former such buckling is quite negligible [16–18]. As a result, the buckling effect in silicene breaks the space inversion symmetry, consequently, the behavior of electrons around the Dirac points is distinctly different [14,19]. As we shall show, this fact intrinsically generates the so-called “valley” current. At this point, we might add that in zigzag, but not armchair, graphene nanoribbons such currents have been predicted [2]. It is thus the main aim of the present work to investigate the generation and means of controlling the charge, spin and valley currents in armchair silicene nanoribbons (ASNs) where the space inversion symmetry is naturally broken. The buckling effect in silicene also gives rise to more salient features absent in graphene. Firstly, there exists an intrinsic spin–orbit coupling, much larger than that in graphene [16–18], which manifests itself in the intrinsic generation of quantum spin Hall effect [17,19]. Secondly, it produces a rigid distinction between the Dirac points, the valley states, hence the possibility of intrinsically generating valley currents [14,19]. The former has been suggested for applications in spintronics while the latter has begun to form the concept of “valleytronics” [2]. Lastly, the buckling effect can be controlled by an externally applied electric field, causing the band gaps of controllable nature [20,21]. As we shall demonstrate, this fact leads to quantum topological phase transitions [19,22], with novel applications in quantum information technology [23,24]. In this regard, the stability of the topological phases is of tremendous importance. The stability of topological phases is directly related to the mass term in the Dirac equation [19,20], which in silicene, is much sensitive to the boundaries (bulk, ribbons and dots) and combinations with substrates [20,25]. In what follows, therefore, we consider an ASN, grown on a substrate of silver (111) [26] and report the roles of the externally controllable agents, the width, the electric field and Fermi energy on the behavior of such currents. This particular combination is selected because the intrinsic spin–orbit coupling is relatively large, about 3.9 meV [27,28]. In order to achieve this goal, we employ the well-received Kubo formula [19,33,34] in which summations over the matrix elements of the current operators, with respect to the eigenstates of the total Hamiltonian, appear. Thus, we first introduce the Hamiltonian of an ASN which, upon using the symmetries, can be easily diagonalized yielding the corresponding band structures. In doing so it is noted that the band gaps, for specific states, close at a critical width, resulting in a change in the topological phase [19,22]. It is also demonstrated that by adjusting the width, external electric field and the Fermi energy, pure spin and valley currents are generated. Moreover, as our calculations indicate, the armchair boundaries force the currents to step-wise behavior, whose characteristics are controllable by adjusting these agents. The present paper is organized in the following manner. After a brief introduction of the problem in Section 1, we devote Section 2 to a discussion of the symmetries of the ASN and, thereby, the
calculation of band structures. Since the Kubo formula is the milestone for our calculations, Section 3 is devoted to a brief discussion of this formula, followed by a thorough examination of different conductances in Section 4. Finally, we conclude the paper by summarizing the results in Section 5.
2. Armchair silicene nanoribbons: the model The π-electron in silicene is described in a Hilbert space, H ¼ Hc Hp Hs Hv , where Hc , Hp , Hs and Hv are, respectively, the configuration, the pseudospin, ordinary spin and the ! “valley” Hilbert spaces. The elements of Hc are p ¼ iℏ∇, those of Hp are si ði ¼ x; yÞ which act on the lattice sites jAðBÞ〉, while the elements in Hs are si ði ¼ x; y; zÞ acting upon the bases j↑ð↓Þ〉. The valley Hilbert space, Hv , is spanned by the so-called valley states, jKðK 0 Þ〉, the eigenstates of operator τ with eigenvalues þ ð Þ1. Taking the buckled structure and intrinsic spin–orbit coupling of silicene into account, the Hamiltonian of the π-electron, around the Dirac points, is then given by the 8 8 matrix H ¼ vf ðsx px τsy py Þ ΔSO τsz sz þ Δz sz ;
ð1Þ
where the Fermi velocity, vf, is in the order of 10 m=s, Δso is the well-known Kane–Mele intrinsic spin–orbit coupling [29], about ten times larger than the Rashba coupling in silicene [17]. Part of the buckled effect, contained in Δz, can be controlled by the application of a transverse electric field. It is well known that the Hamiltonian of Eq. (1) preserves the time reversal symmetry, but not the space inversion [29,30]. This conclusion is due to the ! ! fact that s is even (odd), s is odd (even) and τ is odd (odd) under time reversal (space inversion) symmetry. Hence, all the terms in Eq. (1) preserve the time reversal symmetry, while the last term does indeed break the space inversion. As we shall see, considerations along these lines lead to controllable charge/spin/valley currents. For the ASN shown in Fig. 1, the eigenstates of Eq. (1), as far as the symmetries allow, are expanded as ( 5
jψ 〉 ¼
½φAK↑ ðyÞjAK↑〉 þ φBK↑ ðyÞjBK↑〉 !! þ φAK↓ ðyÞjAK↓〉 þ φBK↓ ðyÞjBK↓〉ei K r þ ½φAK 0 ↑ ðyÞjAK 0 ↑〉þ φBK 0 ↑ ðyÞjBK 0 ↑〉
!0 !) þ φAK 0 ↓ ðyÞjAK 0 ↓〉 þ φBK 0 ↓ ðyÞjBK 0 ↓〉ei K r eikx x :
ð2Þ
It is evident from Eq. (1) that the operators sz and τ commute with H. As a result, the probability amplitudes, φsz τsz ðyÞ, are the solutions to four 2 2 subblocks ! ! ! φAτsz ðyÞ φAτsz ðyÞ Δz τsz ΔSO vf ℏðkx þ τ∂y Þ φBτsz ðyÞ ¼ εðτ; sz Þ φBτsz ðyÞ ; vf ℏðkx τ∂y Þ Δz þ τsz ΔSO ð3Þ which allows us to reduce the eigenvalue problem to a system of two coupled first order differential equations. It is noted that Eq. (3) is written in the sublattice space for fixed values of spin, sz ¼ 7 1, and Dirac points, τ ¼ 7 1. As Fig. 1 indicates, the probability of finding the π-electron at sublattices should vanish at the boundaries, thus the following conditions are imposed: 0
φAK↑ ð0Þ þ φAK 0 ↓ ð0Þ ¼ 0; φAK↑ ðwÞeiKw þ φAK 0 ↓ ðwÞeiK w ¼ 0 0 φBK↑ ð0Þ þ φBK 0 ↓ ð0Þ ¼ 0; φBK↑ ðwÞeiKw þ φBK 0 ↓ ðwÞeiK w ¼ 0 0 φAK↓ ð0Þ þ φAK 0 ↑ ð0Þ ¼ 0; φAK↓ ðwÞeiKw þ φAK 0 ↑ ðwÞeiK w ¼ 0 0 φBK↓ ð0Þ þ φBK 0 ↑ ð0Þ ¼ 0; φBK↓ ðwÞeiKw þ φBK 0 ↑ ðwÞeiK w ¼ 0;
ð4Þ
where the width of ASN is discrete, taking integral multiples of the lattice constant, w ¼ Na0 (a0 ¼ 3:86 Å in silicene [26]). Accordingly,
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3. Conductances in armchair silicene nanoribbons: Kubo formula The response of a quantum system to an external agent, in a range as wide as mechanical (viscosity [31]), magnetic (magnetic susceptibility [32]) and electric (in particular, conductance [17]), is well described by the celebrated Kubo formula [19,33,34]. Under the action of an external electric field, the response is given by the conductance for which the Kubo formula is cast into
sηij ðωÞ ¼
η
〈αjJ i jα0 〉〈α0 jJ j jα〉 iℏ ∑ ½ f ðEα Þ f ðEα0 Þ ; A α;α0 ðEα Eα0 ÞðEα Eα0 þ ℏωÞ
ð7Þ
η
Fig. 1. Schematic illustration of (a) spin current and (b) valley current. The electronic spin states are denoted by the arrows while the hollow (full) circles designate the valley jK 0 ðKÞ〉 states.
the positive and negative energy (electron–hole) eigenstates of the π-electrons are straightforwardly calculated as ! !0 ! ψ p ðm; kx ; τ; sz Þ〉 ¼ N p eið K Þð K Þ r eikx x eisz km y
sηxy ¼ s ηxy ðK; ↑Þ þ s ηxy ðK; ↓Þ þ s ηxy ðK 0 ; ↑Þ þ s ηxy ðK 0 ; ↓Þ;
ð8Þ
where each partial conductance, s ηxy ðτ; sz Þ, gives rise to the current
!
sz
where sij is an element of the conductance for the current of the observable η, f ðEα Þ is the Fermi–Dirac distribution of an eigenstate jα〉, with energy Eα , and ω is the field frequency. For electrical response, the operator Ji in Eq. (7) is defined as J i ¼ evi ¼ e ∂H=∂pi , where H, the total Hamiltonian, is given in Eq. (1), while η J j ¼ ½η; vj þ =2 corresponds to the current of the observable η. η Physically, the nonzero elements of sij render the current of the observable η (in the present work, charge, spin and valley currents) in the jth direction, under the action of an electric field applied in the ith direction. Since under ordinary conditions the thermal energy is much smaller than other energy scales [16,34], the following calculations are performed at absolute zero temperature. At this limiting case, f(E) takes the values zero or one, depending on whether the state lies above or below the Fermi energy, respectively [34]. For the ASN shown in Fig. 1, and after performing the summations over the spin and valley states in Eq. (7), the conductance of the observable η is decomposed into partial ones as
where N pðnÞ is a normalization factor. The corresponding discrete eigenenergies are
(charge, spin and valley) of the observable η, carried by electrons of definite spin and valley states (see, in particular, Fig. 1). To calculate the explicit form of partial conductances, we note that jcy ¼ 12 ½e; vy þ ¼ evy , jsy ¼ 12 ½ℏsz =2; vy þ ¼ ℏsz vy =2 and jvy ¼ 12 ½τ; vy þ ¼ τvy , for charge, spin and valley currents, respectively. Using this fact along with the eigenstates given in Eq. (5), one can easily determine the matrix elements involved in Eq. (7), thereby, finding the partial conductances as
εðm; kx ; τ; sz Þ ¼ 7 ½v2f ℏ2 ðk2x þ k2m Þ þ ðΔz τsz ΔSO Þ2 1=2 ;
s ηxy ðτ; sz Þ ¼
! !0 ! ψ n ðm; kx ; τ; sz Þ〉 ¼ N n eið K Þð K Þ r eikx x eisz km y
sz vf ℏðkx isz τkm Þ jεj þ Δz τsz ΔSO vf ℏðkx þ isz τkm Þ jεj þ Δz τsz ΔSO
1
! ;
ð5Þ
ð6Þ
where km ¼ mπ =w 4π =3a0 (with m ¼ 0; 1; 2; …Þ. It is important to note that the spin states and valley states at each sublattice site are separated due to the fact that the intrinsic and buckled structure of silicene preserves the time-reversal and breaks the space inversion symmetries, respectively [29]. From Eq. (6) it is also evident that a controllable (through Δz) gap, much larger than that in silicene sheet, opens in ASN. The general behavior of band structure of ASN is illustrated in Fig. 2. In the first row of this figure Eq. (6) is plotted against the longitudinal momentum for a fixed (transverse electric field) Δz ¼ 1:95 meV and different widths and interband states specified by n, while in the second row the width is fixed but Δz varies. The blue (red) graphs in this figure indicate the spin up (down) states. Moreover, it is clear from Eq. (6) that similar graphs, with the replacement of spin states (red and blue graphs), result for the K 0 valley. An important consequence of Eq. (6) and Fig. 2 is the fact that around the KðK 0 Þ point km vanishes at certain critical widths, w ¼ Na0 ¼ 3ma0 =4, with m being an integral multiple of 4. For such ribbon widths the gap closes for the spin up (down) states. The band structure of ASN, given in Eq. (6) and illustrated in Fig. 2, as we shall see in the following section, leads to intrinsic charge/spin/valley currents.
! Z ℏ3 v4f e ðkx þ iτsz km Þ2 1 jN p j2 jN n j2 ητsz ∑ dkx A ðjεðm; kx ; τ; sz Þj þ Δz τ sz ΔSO Þ2 m
! 1 ðkx iτsz km Þ2 2 1 : ε ðm; kx ; τ; sz Þ ðjεðm; kx ; τ; sz Þj þ Δz τsz ΔSO Þ2
ð9Þ
In what follows we employ Eq. (8) along with Eq. (9) to investigate charge, spin and valley conductances in an ASN.
4. Intrinsic charge, spin and valley conductances In the previous section, the structural properties of an ASN, viz.,
π-electronic eigenstates, were used to present an explicit expression for different conductances. We are now in a position to employ the resulting Kubo formula, Eq. (9), along with Eq. (8), to investigate the naturally occurring charge, spin and valley conductances. In the following three subsections we present the effects of Fermi energy, the buckled term (Δz in Eq. (1)) and the width of the ribbon on each of these conductances, respectively. For clarity, we also point out that the positive or negative conductances relate the currents to the axes defined in Fig. 1. As a matter of comparison with graphene [14,29], we emphasize that here the buckled effect is so strong that the space inversion
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Fig. 2. Schematic representation of the band structure, around the K point in armchair silicene nanoribbons. The first row is drawn for different widths and fixed buckled term value. In the second row the width is held fixed at the critical value but the buckled term strength is varied. The colors (grayscale) identify the spin states; spin up: blue (dark gray) and spin down: red (light gray). Similar graphs, with the colors (grayscales) interchanged, result around the other Dirac point. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
symmetry is completely broken, even for armchair boundaries [2,20], leading to new features in different conductances. 4.1. Intrinsic charge conductance As given in the previous section, the operator responsible for charge current is evy which is odd under time reversal but even under space inversion. Therefore, around the Dirac points, and Fermi energy falling into the gap, the charge current is expected to vanish. As Fermi energy exceeds the gap, on the other hand, one expects that the charge current appears. Moreover, at the particular nanoribbon width for which the gap closes (see the concluding remarks of Section 2), the charge conductance is nonvanishing for any Fermi energy. Due to the fact that the π-electron is confined to a limited region of space (silicene nanoribbon) the charge conductance, when it exists, would have a step-wise behavior. These expectations are verified in Fig. 3a and b, where the result of Eqs. (8) and (9) for the charge conductance are depicted. In Fig. 3a the charge conductance is plotted against the Fermi energy for three typical values of Δz ¼ 0 o ΔSO , 3:9 meV ¼ ΔSO and 7:8 meV 4 ΔSO , for the critical width of 30a0. The behavior of charge conductance versus Δz (the silicene buckled effect), on the other hand, is presented in Fig. 3b for the same critical width as in Fig. 3a, for three Fermi energies of 0.5, 2 and 2.5 meV. From Fig. 3b it is further observed that by adjusting Δz the ASN changes from an insulator phase to a Hall insulator [35]. Although not indicated in these figures, but obvious from Fig. 2 and the discussion right before it, when the charge current exists, the ASN turns into a spin/valley polarized Hall insulator. We shall further elaborate on this point in the following two subsections. 4.2. Intrinsic spin conductance It has been well established that for systems preserving the time reversal symmetry but not the space inversion, spin currents can occur at the edges, turning the material from an insulator to a topological insulator [19,22,30]. In the case of silicene, near the
Dirac points, the Hamiltonian of Eq. (1) and the spin current operator, ℏsz vy =2 (see the preceding section), are both even under time reversal. The space inversion symmetry, however, is broken due to the buckled effect, the last term of Eq. (1). Because of these physical facts it is expected that, under specific circumstances, spin currents form at the edges of ASN. It is then evident that the formation of topological insulator phase, edge-currents, depends upon where the Fermi energy lies, how strong the buckled term is and the width of the ribbon. Furthermore, the spin conductance (current) exhibits step-wise behavior again due to the confinement of the π-electrons inside the ribbon. The foregoing conclusions are best confirmed from Figs. 4 and 5. In Fig. 4 the “edge” spin conductance as functions of Fermi energy, for different widths (Fig. 4a) and buckled strengths (Fig. 4b) is depicted, while Fig. 5 is devoted to the same conductance as functions of the widths (Fig. 5a) and buckled strengths (Fig. 5b) for fixed Fermi energies. We conclude this subsection by remarking that for Fermi energy placed in the gap and the critical ribbon width, a change from trivial insulator phase to the topological insulator [19,22] one occurs at Δz ¼ ΔSO (in particular note Fig. 5b). As the Fermi energy exceeds the gap, pure spin currents cease to exist; it is accompanied by charge and/or valley currents. This point is most clear from Fig. 1.
4.3. Intrinsic valley conductance Under the space inversion symmetry the two valley states are degenerate and hence, in spite of the fact that the valley current operator, τv, is odd, the valley conductance vanishes in armchair graphene ribbons [2]. In ASN, on the other hand, the buckled effect destroys this symmetry and an intrinsic valley current occurs [14,19]. Here again, as discussed in the previous subsections, the band structure, where the Fermi energy lies and the strength of buckled effect determine whether the valley current is pure. Moreover, the inversion of band structure at Δz ¼ ΔSO causes the signs in Eq. (8) to alternate for valley conductance relative to that of spin conductance.
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Fig. 3. Charge conductance, in units of e2 =ℏ, for the critical width, w ¼ 30a0 . Part (a) (grayscale): versus the Fermi energy for different buckled effect parameters of Δz ¼ 0 o ΔSO , black (black), Δz ¼ 3:9 meV ¼ ΔSO , red (light gray) and Δz ¼ 7:8 meV 4 ΔSO , blue (dark gray). Part (b) (grayscale): as functions of buckled effect strength for different Fermi energies, εf ¼ 0:5 meV, black (black), εf ¼ 2:0 meV, red (light gray) and εf ¼ 2:5 meV, blue (dark gray). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 4. Spin conductance, in units of e=4π, versus Fermi energy for different values of the width and buckled effect parameter: (a) (grayscale) fixed Δz ¼ ΔSO =2 ¼ 1:95 meV for different ribbon's widths of w ¼ 20a0 , blue (dark gray), w ¼ 30a0 , red (light gray) and w ¼ 40a0 , black (black). (b) (grayscale) at the critical width, w ¼ 30a0 , for different buckled effect parameters of Δz ¼ ΔSO =2 ¼ 1:95 meV, blue (dark gray), Δz ¼ ΔSO ¼ 3:9 meV, red (light gray) and Δz ¼ 2ΔSO ¼ 7:8 meV, black (black). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 5. Spin conductance, in units of e=4π, as functions of the ribbon's width, w, and the buckled parameter, Δz. In (a) it is observed that for Δz 4 ΔSO , as indicated by blue dots (dark gray, if grayscaled), the spin current vanishes at the critical widths, 3na0, for zero Fermi energy. It is also noted that for w≳100a0 the spin conductance approaches that of a silicene sheet, namely, zero. In (b) the spin conductance is presented as a function of the buckled effect parameter, at the critical width, w ¼ 30a0 , for three typical Fermi energies of 0.0 meV, blue, 2.0 meV, black and 2.5 meV, red. Part (b) clearly indicates a change of topological phase. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 6. Valley conductance, in units of e=ℏ, for the critical width, w ¼ 30a0 . Part (a) (grayscale): versus the Fermi energy for different buckled effect parameters of Δz ¼ ΔSO =2 ¼ 1:95 meV, blue (dark gray), Δz ¼ ΔSO ¼ 3:9 meV , red (light gray) and Δz ¼ 2ΔSO ¼ 7:8 meV, black (black). Part (b): as functions of buckled effect strength for different Fermi energies, εf ¼ 0:0 meV, blue, εf ¼ 2:0 meV, black and εf ¼ 2:5 meV, red. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
It is then expected that the valley states flow in a direction opposite to that of spin, if existed. This point is also indicated in Fig. 1. Expressed differently, when the Fermi energy is inside the gap and Δz 4 ΔSO a pure valley current intrinsically develops. The general behavior of the valley conductance, along the same line of reasoning as in the preceding subsections, is similar to the spin conductance. To complete the discussion, in Fig. 6a we present the valley conductance versus the Fermi energy for different values of the buckled term, while in Fig. 6b the same is illustrated as functions of Δz for different Fermi energies. To emphasize on the role of the width, Fig. 6 is drawn for the critical one, w ¼ 30a0 . 5. Conclusion In the present work we report the characteristics of charge, spin and valley intrinsic conductances in an armchair silicene nanoribbon (ASN). We utilize the Kubo formula, appropriately adopted, to study effects of Fermi energy, buckled term and the ribbon's width on each of such conductances. This is done by a thorough consideration of the symmetries which, consequently, leads to conserved quantities. We then employ these facts to determine the band structure of ASN, thereby, introducing a method to calculate the summations appearing in the Kubo formula. Effects of these controlling agents on the conductances are best observed from figures, which we also present. Although a thorough discussion of the results, along with physical reasons, are given in Section 4, in what follows we highlight the more important findings of the paper. All three conductances (currents) behave step-wisely, due to the confinement of π-electrons in the ASN. The height (a jump in the current) and the plateaus (constant current) of the steps may be most effectively manipulated by the ribbon's width. There exists a critical width for which the band gaps can invert, resulting in a quantum phase change of the ribbon. In particular, by adjusting the buckled effect one can force the change in phase to be of a topological type. When the Fermi energy lies in the gap, which is controlled by the width and buckled effect, the charge current vanishes, while either spin current or valley current develops. When these conditions are met the corresponding currents are pure, otherwise, they accompany each other. In short, the material presented in this paper provides a deeper understanding of different conductances, including the valley, in ASN and, thereby, novel means of controlling such currents.
Acknowledgments This work has been in part supported by a grant from the Research Council of Shiraz University, under the contract 92-GR-SC-82. References [1] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnar, M. L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488. [2] B. Trauzettel, P. Recher, Physics 4 (2011) 25. [3] T. Kampfrath, M. Battiato, P. Maldonado, G. Eilers, J. Notzold, S. Mahrlein, V. Zbarsky, F. Freimuth, Y. Mokrousov, S. Blugel, M. Wolf, I. Radu, P. M. Oppeneer, M. Munzenberg, Nat. Nanotechnol. 8 (2013) 256. [4] S. Sanvito, Chem. Soc. Rev. 40 (2011) 3336. [5] M.H.D. Guimaraes, A. Veligura, P.J. Zomer, T. Maassen, I.J. Vera-Marun, N. Tombros, B.J. van Wees, Nano Lett. 12 (2012) 3512. [6] Y. Wang, J. Zheng, Z. Ni, R. Fei, Q. Liu, R. Quhe, C. Xu, J. Zhou, Z. Gao, J. Lu, Nano 07 (2012) 1250037. [7] H.J. Kimble, Nature 453 (2008) 1023. [8] D. Xiao, G.B. Liu, W. Feng, X. Xu, W. Yao, Phys. Rev. Lett. 108 (2012) 196802. [9] S. Wu, J.S. Ross, G. Liu, G. Aivazian, A. Jones, Z. Fei, W. Zhu, D. Xiao, W. Yao, D. Cobden, X. Xu, Nat. Phys. 9 (2013) 149. [10] B. Bishnoi, B. Ghosh, RSC Adv. 3 (2013) 26153. [11] B. Dlubak, M.-B. Martin, C. Deranlot, B. Servet, S. Xavier, R. Mattana, M. Sprinkle, C. Berger, W.A. DeHeer, F. Petroff, A. Anane, P. Seneor, A. Fert, Nat. Phys. 8 (2012) 557. [12] C. Jozsa, M. Popinciuc, N. Tombros, H.T. Jonkman, B.J. van Wees, Phys. Rev. B 79 (2009) 081402. [13] S. Cho, Y.F. Chen, M.S. Fuhrer, Appl. Phys. Lett. 91 (2007) 123105. [14] C.J. Tabert, E.J. Nicol, Phys. Rev. B 87 (2013) 235426. [15] C.J. Tabert, E.J. Nicol, Phys. Rev. B 88 (2013) 085434. [16] A. Dyrdal, V.K. Dugaev, J. Barna´s, Phys. Rev. B 80 (2009) 155444. [17] A. Dyrdal, J. Barna´s, Phys. Status Solidi Rap. Res. Lett. 6 (2012) 340. [18] H. Shirkani, F. Amiri, M.M. Golshan, Superlattices Microstruct. 64 (2013) 418. [19] M. Tahir, A. Manchon, K. Sabeeh, U. Schwingenschlogl, Appl. Phys. Lett. 102 (2013) 162412. [20] M. Ezawa, Phys. Rev. Lett. 109 (2012) 055502. [21] N.D. Drummond, V. Zolyomi, V.I. Falko, Phys. Rev. B 85 (2012) 075423. [22] M. Ezawa, Appl. Phys. Lett. 102 (2013) 172103. [23] J. Alicea, Nat. Nanotechnol. 8 (2013) 623. [24] H.P. Paudel, M.N. Leuenberger, Phys. Rev. B 88 (2013) 085316. [25] H. Peng, K. Lai, D. Kong, S. Meister, Y. Chen, X.-L. Qi, S.-C. Zhang, Z.-X. Shen, Y. Cui, Nat. Mater. 9 (2010) 225. [26] P. Vogt, P. DePadova, C. Quaresima, J. Avila, E. Frantzeskakis, M.C. Asensio, A. Resta, B. Ealet, G. LeLay, Phys. Rev. Lett. 108 (2012) 155501. [27] C.C. Liu, H. Jiang, Y. Yao, Phys. Rev. B 84 (2011) 195430. [28] M. Ezawa, New J. Phys. 14 (2012) 033003. [29] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95 (2005) 226801. [30] L. Fu, C.L. Kane, Phys. Rev. B 76 (2007) 045302. [31] B. Bradlyn, M. Goldstein, N. Read, Phys. Rev. B 86 (2012) 245309. [32] D. Manske, I. Eremin, K.H. Bennemann, Phys. Rev. B 63 (2001) 054517. [33] R. Kubo, H. Hasegawa, N. Hashitsume, J. Phys. Soc. Jpn. 14 (1959) 56. [34] E.I. Rashba, Phys. Rev. B 70 (2004) 201309. [35] M. Hilke, D. Shahar, S.H. Song, D.C. Tsui, Y.H. Xie, D. Monroe, Nature 395 (1998) 675.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Generation and control of charge, spin and valley currents in armchair silicene nanoribbons H. Shirkani, M.M. Golshan n Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran
H I G H L I G H T S
G R A P H I C A L
Charge/spin/valley current developed in armchair silicene nanoribbons (ASNs) is reported. Using the symmetries of ASN, the total Hamiltonian is shown to be block-diagonal. The eigenstates and eigenvalues of the total Hamiltonian are then calculated. Consequently, Kubo's formula is used to study charge/spin/valley conductance in ASN. Properties of charge/spin/valley conductance thus obtained are discussed. In particular, topological phase transitions are reported and discussed.
Effects of the width, buckling strength and Fermi energy on the characteristics of charge/spin/valley currents in armchair silicene nanoribbons are reported. Schematic illustration of (a) spin current and (b) valley current. The electronic spin states are denoted by the arrows while the hollow (full) circles designate the valley jK 0 ðKÞ〉 states.
art ic l e i nf o
a b s t r a c t
Article history: Received 22 February 2014 Received in revised form 22 April 2014 Accepted 23 April 2014 Available online 20 May 2014
It is well known that future information technology relies upon electronic states other than charge, through the feasibility of constructing spintronic and valleytronic devices. The main challenge in such applications is the generation and control of the corresponding state currents. The present paper is thus concerned with charge, spin and valley currents which intrinsically develop in armchair silicene nanoribbons (ASNs). In pursuing this consideration the well established Kubo formula is employed. Taking the symmetries of the ASN into account, we analyze the structure of total Hamiltonian and demonstrate how the matrix elements along with the summations involved in the suitably adopted Kubo formula, may be analytically calculated. The results so-obtained along with the corresponding figures reveal that all three currents develop in a step-wise manner. The heights, indicating a jump in the current, and the plateaus, indicating a constant current, are shown to be tunable by adjusting the width and/or the buckled effect. Moreover, we demonstrate that for particular (critical) widths an inversion in band gaps may occur, giving rise to quantum phase transitions. More practical result of this paper, as we show, is the fact that the ASN can generate pure spin or valley current under specific conditions placed on the width, buckled effect and Fermi energy. The material presented in this paper thus provides novel means of generation and control of the charge, spin and valley currents. & 2014 Elsevier B.V. All rights reserved.
Keywords: Silicene Armchair nanoribbon Kubo formula Charge conductance Spin conductance Valley conductance
A B S T R A C T
1. Introduction
n
Corresponding author. E-mail address: [email protected] (M.M. Golshan).
http://dx.doi.org/10.1016/j.physe.2014.04.022 1386-9477/& 2014 Elsevier B.V. All rights reserved.
It is by now well established that any quantum information processing scheme requires entities of limited number of states with least dissipation arising from interactions with the environment [1–3]. Moreover, the states should naturally posses long
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relaxation time and length [4–6] to preserve the information placed on it. In addition, the information processing is carried out through a network of elements that, desirably, are of microscopic scales [7]. A vivid candidate possessing these requirements is the electronic states in nanostructures, in particular, formed with the so-called honeycomb materials [5,6,8,9]. In this regard, graphene [5] and, especially, silicene [6] have attracted much interest. It is well known that the electrons in these materials behave as massless Dirac particles, consequently, possessing long spin diffusion length [6,10], long spin coherency [6,10], highefficiency spin injection [11], etc. [12,13]. Regarding the electronic states, however, there is an important difference between graphene and silicene: in the latter the buckling effect [14,15], where one sublattice is shifted upwards relative to the other, is of great consequences, while in the former such buckling is quite negligible [16–18]. As a result, the buckling effect in silicene breaks the space inversion symmetry, consequently, the behavior of electrons around the Dirac points is distinctly different [14,19]. As we shall show, this fact intrinsically generates the so-called “valley” current. At this point, we might add that in zigzag, but not armchair, graphene nanoribbons such currents have been predicted [2]. It is thus the main aim of the present work to investigate the generation and means of controlling the charge, spin and valley currents in armchair silicene nanoribbons (ASNs) where the space inversion symmetry is naturally broken. The buckling effect in silicene also gives rise to more salient features absent in graphene. Firstly, there exists an intrinsic spin–orbit coupling, much larger than that in graphene [16–18], which manifests itself in the intrinsic generation of quantum spin Hall effect [17,19]. Secondly, it produces a rigid distinction between the Dirac points, the valley states, hence the possibility of intrinsically generating valley currents [14,19]. The former has been suggested for applications in spintronics while the latter has begun to form the concept of “valleytronics” [2]. Lastly, the buckling effect can be controlled by an externally applied electric field, causing the band gaps of controllable nature [20,21]. As we shall demonstrate, this fact leads to quantum topological phase transitions [19,22], with novel applications in quantum information technology [23,24]. In this regard, the stability of the topological phases is of tremendous importance. The stability of topological phases is directly related to the mass term in the Dirac equation [19,20], which in silicene, is much sensitive to the boundaries (bulk, ribbons and dots) and combinations with substrates [20,25]. In what follows, therefore, we consider an ASN, grown on a substrate of silver (111) [26] and report the roles of the externally controllable agents, the width, the electric field and Fermi energy on the behavior of such currents. This particular combination is selected because the intrinsic spin–orbit coupling is relatively large, about 3.9 meV [27,28]. In order to achieve this goal, we employ the well-received Kubo formula [19,33,34] in which summations over the matrix elements of the current operators, with respect to the eigenstates of the total Hamiltonian, appear. Thus, we first introduce the Hamiltonian of an ASN which, upon using the symmetries, can be easily diagonalized yielding the corresponding band structures. In doing so it is noted that the band gaps, for specific states, close at a critical width, resulting in a change in the topological phase [19,22]. It is also demonstrated that by adjusting the width, external electric field and the Fermi energy, pure spin and valley currents are generated. Moreover, as our calculations indicate, the armchair boundaries force the currents to step-wise behavior, whose characteristics are controllable by adjusting these agents. The present paper is organized in the following manner. After a brief introduction of the problem in Section 1, we devote Section 2 to a discussion of the symmetries of the ASN and, thereby, the
calculation of band structures. Since the Kubo formula is the milestone for our calculations, Section 3 is devoted to a brief discussion of this formula, followed by a thorough examination of different conductances in Section 4. Finally, we conclude the paper by summarizing the results in Section 5.
2. Armchair silicene nanoribbons: the model The π-electron in silicene is described in a Hilbert space, H ¼ Hc Hp Hs Hv , where Hc , Hp , Hs and Hv are, respectively, the configuration, the pseudospin, ordinary spin and the ! “valley” Hilbert spaces. The elements of Hc are p ¼ iℏ∇, those of Hp are si ði ¼ x; yÞ which act on the lattice sites jAðBÞ〉, while the elements in Hs are si ði ¼ x; y; zÞ acting upon the bases j↑ð↓Þ〉. The valley Hilbert space, Hv , is spanned by the so-called valley states, jKðK 0 Þ〉, the eigenstates of operator τ with eigenvalues þ ð Þ1. Taking the buckled structure and intrinsic spin–orbit coupling of silicene into account, the Hamiltonian of the π-electron, around the Dirac points, is then given by the 8 8 matrix H ¼ vf ðsx px τsy py Þ ΔSO τsz sz þ Δz sz ;
ð1Þ
where the Fermi velocity, vf, is in the order of 10 m=s, Δso is the well-known Kane–Mele intrinsic spin–orbit coupling [29], about ten times larger than the Rashba coupling in silicene [17]. Part of the buckled effect, contained in Δz, can be controlled by the application of a transverse electric field. It is well known that the Hamiltonian of Eq. (1) preserves the time reversal symmetry, but not the space inversion [29,30]. This conclusion is due to the ! ! fact that s is even (odd), s is odd (even) and τ is odd (odd) under time reversal (space inversion) symmetry. Hence, all the terms in Eq. (1) preserve the time reversal symmetry, while the last term does indeed break the space inversion. As we shall see, considerations along these lines lead to controllable charge/spin/valley currents. For the ASN shown in Fig. 1, the eigenstates of Eq. (1), as far as the symmetries allow, are expanded as ( 5
jψ 〉 ¼
½φAK↑ ðyÞjAK↑〉 þ φBK↑ ðyÞjBK↑〉 !! þ φAK↓ ðyÞjAK↓〉 þ φBK↓ ðyÞjBK↓〉ei K r þ ½φAK 0 ↑ ðyÞjAK 0 ↑〉þ φBK 0 ↑ ðyÞjBK 0 ↑〉
!0 !) þ φAK 0 ↓ ðyÞjAK 0 ↓〉 þ φBK 0 ↓ ðyÞjBK 0 ↓〉ei K r eikx x :
ð2Þ
It is evident from Eq. (1) that the operators sz and τ commute with H. As a result, the probability amplitudes, φsz τsz ðyÞ, are the solutions to four 2 2 subblocks ! ! ! φAτsz ðyÞ φAτsz ðyÞ Δz τsz ΔSO vf ℏðkx þ τ∂y Þ φBτsz ðyÞ ¼ εðτ; sz Þ φBτsz ðyÞ ; vf ℏðkx τ∂y Þ Δz þ τsz ΔSO ð3Þ which allows us to reduce the eigenvalue problem to a system of two coupled first order differential equations. It is noted that Eq. (3) is written in the sublattice space for fixed values of spin, sz ¼ 7 1, and Dirac points, τ ¼ 7 1. As Fig. 1 indicates, the probability of finding the π-electron at sublattices should vanish at the boundaries, thus the following conditions are imposed: 0
φAK↑ ð0Þ þ φAK 0 ↓ ð0Þ ¼ 0; φAK↑ ðwÞeiKw þ φAK 0 ↓ ðwÞeiK w ¼ 0 0 φBK↑ ð0Þ þ φBK 0 ↓ ð0Þ ¼ 0; φBK↑ ðwÞeiKw þ φBK 0 ↓ ðwÞeiK w ¼ 0 0 φAK↓ ð0Þ þ φAK 0 ↑ ð0Þ ¼ 0; φAK↓ ðwÞeiKw þ φAK 0 ↑ ðwÞeiK w ¼ 0 0 φBK↓ ð0Þ þ φBK 0 ↑ ð0Þ ¼ 0; φBK↓ ðwÞeiKw þ φBK 0 ↑ ðwÞeiK w ¼ 0;
ð4Þ
where the width of ASN is discrete, taking integral multiples of the lattice constant, w ¼ Na0 (a0 ¼ 3:86 Å in silicene [26]). Accordingly,
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3. Conductances in armchair silicene nanoribbons: Kubo formula The response of a quantum system to an external agent, in a range as wide as mechanical (viscosity [31]), magnetic (magnetic susceptibility [32]) and electric (in particular, conductance [17]), is well described by the celebrated Kubo formula [19,33,34]. Under the action of an external electric field, the response is given by the conductance for which the Kubo formula is cast into
sηij ðωÞ ¼
η
〈αjJ i jα0 〉〈α0 jJ j jα〉 iℏ ∑ ½ f ðEα Þ f ðEα0 Þ ; A α;α0 ðEα Eα0 ÞðEα Eα0 þ ℏωÞ
ð7Þ
η
Fig. 1. Schematic illustration of (a) spin current and (b) valley current. The electronic spin states are denoted by the arrows while the hollow (full) circles designate the valley jK 0 ðKÞ〉 states.
the positive and negative energy (electron–hole) eigenstates of the π-electrons are straightforwardly calculated as ! !0 ! ψ p ðm; kx ; τ; sz Þ〉 ¼ N p eið K Þð K Þ r eikx x eisz km y
sηxy ¼ s ηxy ðK; ↑Þ þ s ηxy ðK; ↓Þ þ s ηxy ðK 0 ; ↑Þ þ s ηxy ðK 0 ; ↓Þ;
ð8Þ
where each partial conductance, s ηxy ðτ; sz Þ, gives rise to the current
!
sz
where sij is an element of the conductance for the current of the observable η, f ðEα Þ is the Fermi–Dirac distribution of an eigenstate jα〉, with energy Eα , and ω is the field frequency. For electrical response, the operator Ji in Eq. (7) is defined as J i ¼ evi ¼ e ∂H=∂pi , where H, the total Hamiltonian, is given in Eq. (1), while η J j ¼ ½η; vj þ =2 corresponds to the current of the observable η. η Physically, the nonzero elements of sij render the current of the observable η (in the present work, charge, spin and valley currents) in the jth direction, under the action of an electric field applied in the ith direction. Since under ordinary conditions the thermal energy is much smaller than other energy scales [16,34], the following calculations are performed at absolute zero temperature. At this limiting case, f(E) takes the values zero or one, depending on whether the state lies above or below the Fermi energy, respectively [34]. For the ASN shown in Fig. 1, and after performing the summations over the spin and valley states in Eq. (7), the conductance of the observable η is decomposed into partial ones as
where N pðnÞ is a normalization factor. The corresponding discrete eigenenergies are
(charge, spin and valley) of the observable η, carried by electrons of definite spin and valley states (see, in particular, Fig. 1). To calculate the explicit form of partial conductances, we note that jcy ¼ 12 ½e; vy þ ¼ evy , jsy ¼ 12 ½ℏsz =2; vy þ ¼ ℏsz vy =2 and jvy ¼ 12 ½τ; vy þ ¼ τvy , for charge, spin and valley currents, respectively. Using this fact along with the eigenstates given in Eq. (5), one can easily determine the matrix elements involved in Eq. (7), thereby, finding the partial conductances as
εðm; kx ; τ; sz Þ ¼ 7 ½v2f ℏ2 ðk2x þ k2m Þ þ ðΔz τsz ΔSO Þ2 1=2 ;
s ηxy ðτ; sz Þ ¼
! !0 ! ψ n ðm; kx ; τ; sz Þ〉 ¼ N n eið K Þð K Þ r eikx x eisz km y
sz vf ℏðkx isz τkm Þ jεj þ Δz τsz ΔSO vf ℏðkx þ isz τkm Þ jεj þ Δz τsz ΔSO
1
! ;
ð5Þ
ð6Þ
where km ¼ mπ =w 4π =3a0 (with m ¼ 0; 1; 2; …Þ. It is important to note that the spin states and valley states at each sublattice site are separated due to the fact that the intrinsic and buckled structure of silicene preserves the time-reversal and breaks the space inversion symmetries, respectively [29]. From Eq. (6) it is also evident that a controllable (through Δz) gap, much larger than that in silicene sheet, opens in ASN. The general behavior of band structure of ASN is illustrated in Fig. 2. In the first row of this figure Eq. (6) is plotted against the longitudinal momentum for a fixed (transverse electric field) Δz ¼ 1:95 meV and different widths and interband states specified by n, while in the second row the width is fixed but Δz varies. The blue (red) graphs in this figure indicate the spin up (down) states. Moreover, it is clear from Eq. (6) that similar graphs, with the replacement of spin states (red and blue graphs), result for the K 0 valley. An important consequence of Eq. (6) and Fig. 2 is the fact that around the KðK 0 Þ point km vanishes at certain critical widths, w ¼ Na0 ¼ 3ma0 =4, with m being an integral multiple of 4. For such ribbon widths the gap closes for the spin up (down) states. The band structure of ASN, given in Eq. (6) and illustrated in Fig. 2, as we shall see in the following section, leads to intrinsic charge/spin/valley currents.
! Z ℏ3 v4f e ðkx þ iτsz km Þ2 1 jN p j2 jN n j2 ητsz ∑ dkx A ðjεðm; kx ; τ; sz Þj þ Δz τ sz ΔSO Þ2 m
! 1 ðkx iτsz km Þ2 2 1 : ε ðm; kx ; τ; sz Þ ðjεðm; kx ; τ; sz Þj þ Δz τsz ΔSO Þ2
ð9Þ
In what follows we employ Eq. (8) along with Eq. (9) to investigate charge, spin and valley conductances in an ASN.
4. Intrinsic charge, spin and valley conductances In the previous section, the structural properties of an ASN, viz.,
π-electronic eigenstates, were used to present an explicit expression for different conductances. We are now in a position to employ the resulting Kubo formula, Eq. (9), along with Eq. (8), to investigate the naturally occurring charge, spin and valley conductances. In the following three subsections we present the effects of Fermi energy, the buckled term (Δz in Eq. (1)) and the width of the ribbon on each of these conductances, respectively. For clarity, we also point out that the positive or negative conductances relate the currents to the axes defined in Fig. 1. As a matter of comparison with graphene [14,29], we emphasize that here the buckled effect is so strong that the space inversion
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Fig. 2. Schematic representation of the band structure, around the K point in armchair silicene nanoribbons. The first row is drawn for different widths and fixed buckled term value. In the second row the width is held fixed at the critical value but the buckled term strength is varied. The colors (grayscale) identify the spin states; spin up: blue (dark gray) and spin down: red (light gray). Similar graphs, with the colors (grayscales) interchanged, result around the other Dirac point. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
symmetry is completely broken, even for armchair boundaries [2,20], leading to new features in different conductances. 4.1. Intrinsic charge conductance As given in the previous section, the operator responsible for charge current is evy which is odd under time reversal but even under space inversion. Therefore, around the Dirac points, and Fermi energy falling into the gap, the charge current is expected to vanish. As Fermi energy exceeds the gap, on the other hand, one expects that the charge current appears. Moreover, at the particular nanoribbon width for which the gap closes (see the concluding remarks of Section 2), the charge conductance is nonvanishing for any Fermi energy. Due to the fact that the π-electron is confined to a limited region of space (silicene nanoribbon) the charge conductance, when it exists, would have a step-wise behavior. These expectations are verified in Fig. 3a and b, where the result of Eqs. (8) and (9) for the charge conductance are depicted. In Fig. 3a the charge conductance is plotted against the Fermi energy for three typical values of Δz ¼ 0 o ΔSO , 3:9 meV ¼ ΔSO and 7:8 meV 4 ΔSO , for the critical width of 30a0. The behavior of charge conductance versus Δz (the silicene buckled effect), on the other hand, is presented in Fig. 3b for the same critical width as in Fig. 3a, for three Fermi energies of 0.5, 2 and 2.5 meV. From Fig. 3b it is further observed that by adjusting Δz the ASN changes from an insulator phase to a Hall insulator [35]. Although not indicated in these figures, but obvious from Fig. 2 and the discussion right before it, when the charge current exists, the ASN turns into a spin/valley polarized Hall insulator. We shall further elaborate on this point in the following two subsections. 4.2. Intrinsic spin conductance It has been well established that for systems preserving the time reversal symmetry but not the space inversion, spin currents can occur at the edges, turning the material from an insulator to a topological insulator [19,22,30]. In the case of silicene, near the
Dirac points, the Hamiltonian of Eq. (1) and the spin current operator, ℏsz vy =2 (see the preceding section), are both even under time reversal. The space inversion symmetry, however, is broken due to the buckled effect, the last term of Eq. (1). Because of these physical facts it is expected that, under specific circumstances, spin currents form at the edges of ASN. It is then evident that the formation of topological insulator phase, edge-currents, depends upon where the Fermi energy lies, how strong the buckled term is and the width of the ribbon. Furthermore, the spin conductance (current) exhibits step-wise behavior again due to the confinement of the π-electrons inside the ribbon. The foregoing conclusions are best confirmed from Figs. 4 and 5. In Fig. 4 the “edge” spin conductance as functions of Fermi energy, for different widths (Fig. 4a) and buckled strengths (Fig. 4b) is depicted, while Fig. 5 is devoted to the same conductance as functions of the widths (Fig. 5a) and buckled strengths (Fig. 5b) for fixed Fermi energies. We conclude this subsection by remarking that for Fermi energy placed in the gap and the critical ribbon width, a change from trivial insulator phase to the topological insulator [19,22] one occurs at Δz ¼ ΔSO (in particular note Fig. 5b). As the Fermi energy exceeds the gap, pure spin currents cease to exist; it is accompanied by charge and/or valley currents. This point is most clear from Fig. 1.
4.3. Intrinsic valley conductance Under the space inversion symmetry the two valley states are degenerate and hence, in spite of the fact that the valley current operator, τv, is odd, the valley conductance vanishes in armchair graphene ribbons [2]. In ASN, on the other hand, the buckled effect destroys this symmetry and an intrinsic valley current occurs [14,19]. Here again, as discussed in the previous subsections, the band structure, where the Fermi energy lies and the strength of buckled effect determine whether the valley current is pure. Moreover, the inversion of band structure at Δz ¼ ΔSO causes the signs in Eq. (8) to alternate for valley conductance relative to that of spin conductance.
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Fig. 3. Charge conductance, in units of e2 =ℏ, for the critical width, w ¼ 30a0 . Part (a) (grayscale): versus the Fermi energy for different buckled effect parameters of Δz ¼ 0 o ΔSO , black (black), Δz ¼ 3:9 meV ¼ ΔSO , red (light gray) and Δz ¼ 7:8 meV 4 ΔSO , blue (dark gray). Part (b) (grayscale): as functions of buckled effect strength for different Fermi energies, εf ¼ 0:5 meV, black (black), εf ¼ 2:0 meV, red (light gray) and εf ¼ 2:5 meV, blue (dark gray). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 4. Spin conductance, in units of e=4π, versus Fermi energy for different values of the width and buckled effect parameter: (a) (grayscale) fixed Δz ¼ ΔSO =2 ¼ 1:95 meV for different ribbon's widths of w ¼ 20a0 , blue (dark gray), w ¼ 30a0 , red (light gray) and w ¼ 40a0 , black (black). (b) (grayscale) at the critical width, w ¼ 30a0 , for different buckled effect parameters of Δz ¼ ΔSO =2 ¼ 1:95 meV, blue (dark gray), Δz ¼ ΔSO ¼ 3:9 meV, red (light gray) and Δz ¼ 2ΔSO ¼ 7:8 meV, black (black). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Fig. 5. Spin conductance, in units of e=4π, as functions of the ribbon's width, w, and the buckled parameter, Δz. In (a) it is observed that for Δz 4 ΔSO , as indicated by blue dots (dark gray, if grayscaled), the spin current vanishes at the critical widths, 3na0, for zero Fermi energy. It is also noted that for w≳100a0 the spin conductance approaches that of a silicene sheet, namely, zero. In (b) the spin conductance is presented as a function of the buckled effect parameter, at the critical width, w ¼ 30a0 , for three typical Fermi energies of 0.0 meV, blue, 2.0 meV, black and 2.5 meV, red. Part (b) clearly indicates a change of topological phase. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 6. Valley conductance, in units of e=ℏ, for the critical width, w ¼ 30a0 . Part (a) (grayscale): versus the Fermi energy for different buckled effect parameters of Δz ¼ ΔSO =2 ¼ 1:95 meV, blue (dark gray), Δz ¼ ΔSO ¼ 3:9 meV , red (light gray) and Δz ¼ 2ΔSO ¼ 7:8 meV, black (black). Part (b): as functions of buckled effect strength for different Fermi energies, εf ¼ 0:0 meV, blue, εf ¼ 2:0 meV, black and εf ¼ 2:5 meV, red. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
It is then expected that the valley states flow in a direction opposite to that of spin, if existed. This point is also indicated in Fig. 1. Expressed differently, when the Fermi energy is inside the gap and Δz 4 ΔSO a pure valley current intrinsically develops. The general behavior of the valley conductance, along the same line of reasoning as in the preceding subsections, is similar to the spin conductance. To complete the discussion, in Fig. 6a we present the valley conductance versus the Fermi energy for different values of the buckled term, while in Fig. 6b the same is illustrated as functions of Δz for different Fermi energies. To emphasize on the role of the width, Fig. 6 is drawn for the critical one, w ¼ 30a0 . 5. Conclusion In the present work we report the characteristics of charge, spin and valley intrinsic conductances in an armchair silicene nanoribbon (ASN). We utilize the Kubo formula, appropriately adopted, to study effects of Fermi energy, buckled term and the ribbon's width on each of such conductances. This is done by a thorough consideration of the symmetries which, consequently, leads to conserved quantities. We then employ these facts to determine the band structure of ASN, thereby, introducing a method to calculate the summations appearing in the Kubo formula. Effects of these controlling agents on the conductances are best observed from figures, which we also present. Although a thorough discussion of the results, along with physical reasons, are given in Section 4, in what follows we highlight the more important findings of the paper. All three conductances (currents) behave step-wisely, due to the confinement of π-electrons in the ASN. The height (a jump in the current) and the plateaus (constant current) of the steps may be most effectively manipulated by the ribbon's width. There exists a critical width for which the band gaps can invert, resulting in a quantum phase change of the ribbon. In particular, by adjusting the buckled effect one can force the change in phase to be of a topological type. When the Fermi energy lies in the gap, which is controlled by the width and buckled effect, the charge current vanishes, while either spin current or valley current develops. When these conditions are met the corresponding currents are pure, otherwise, they accompany each other. In short, the material presented in this paper provides a deeper understanding of different conductances, including the valley, in ASN and, thereby, novel means of controlling such currents.
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