Valley-And-Spin-Resonant-Tunneling-Current-In-Ferromagnetic-Nonmagnetic-Ferromagnetic-Silicene-Junction.pdf

Valley and spin resonant tunneling current in ferromagnetic/nonmagnetic/ ferromagnetic silicene junction , Yaser Hajati and Zeinab Rashidian Citation: AIP Advances 6, 025307 (2016); doi: 10.1063/1.4942043 View online: http://dx.doi.org/10.1063/1.4942043 View Table of Contents: http://aip.scitation.org/toc/adv/6/2 Published by the American Institute of Physics AIP ADVANCES 6, 025307 (2016) Valley and spin resonant tunneling current in ferromagnetic/nonmagnetic/ferromagnetic silicene junction Yaser Hajati1,a and Zeinab Rashidian2 Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran Department of Physics, Faculty of Science, University of Lorestan, Lorestan, Iran (Received 26 October 2015; accepted February 2016; published online 10 February 2016) We study the transport properties in a ferromagnetic/nonmagnetic/ferromagnetic (FNF) silicene junction in which an electrostatic gate potential, U, is attached to the nonmagnetic region We show that the electrostatic gate potential U is a useful probe to control the band structure, quasi-bound states in the nonmagnetic barrier as well as the transport properties of the FNF silicene junction In particular, by introducing the electrostatic gate potential, both the spin and valley conductances of the junction show an oscillatory behavior The amplitude and frequency of such oscillations can be controlled by U As an important result, we found that by increasing U, the second characteristic of the Klein tunneling is satisfied as a result of the quasiparticles chirality which can penetrate through a potential barrier Moreover, it is found that for special values of U, the junction shows a gap in the spin and valley-resolve conductance and the amplitude of this gap is only controlled by the on-site potential difference, ∆z Our findings of high controllability of the spin and valley transport in such a FNF silicene junction may improve the performance of nano-electronics and spintronics devices C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4942043] I INTRODUCTION Very recently, silicene a two-dimensional (2D) honeycomb lattice of silicon atoms has attracted enormous attention in the field of physics due to its unusual electronic properties and potential applications in electronics and spintronics devices.1–6 In silicene the low-energy excitations are governed by the Dirac equation near the K and K ′ points.7,8 In contrast to graphene,9 silicene has a large spin-orbit coupling and due to the low-buckled geometry, its energy gap can be further tuned by an external electric field perpendicular to the silicene sheet.10–12 The large spin-orbit interaction of silicene couples the spin and valley degrees of freedom and it plays an important role in the valley and spin transport The low-buckled geometry of silicene with strong atomic intrinsic spin-orbit interactions leads to a gap of 1.55 meV between the conduction and valence bands.7,8 Also opening the energy gap leads to the topological phase transition in silicene by applying electric field and the conductance can be controlled by the gate voltage.12 So, this characterization distinguishes silicene from other 2D-materials such as gapped graphene and MoS2, because its gap is controllable by an external electric field.10,11,13–17 The valley physics aims to control the valley transport of electrons in 2D-materials.18–20 The valley-valve and valley-filter effects were originally proposed in graphene nanoribbons with zigzag edge.18,20 The valleytronics is protected by the suppression of intervalley scattering via a potential step and it is controllable by local application of a gate potential The two valleys are inequivalent Dirac points in the Brillouin zone, K and K ′ and they are degenerate in energy and related to the time reversal symmetry Therefore the valley degree of freedom is similar a Author to whom correspondence should be addressed Electronic mail: yaserhajati@gmail.com 2158-3226/2016/6(2)/025307/13 6, 025307-1 © Author(s) 2016 025307-2 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) to the spin degree of freedom and it provides another probe to control electron Due to the strong spin-valley coupling and the tunability of the spin-splitting band gap by an external electric field in silicone, it is worth studying the valleytronics in silicene and comparing the results with those in graphene in which the spin-orbit interaction is weak Recently, the transport properties of silicene-based tunneling junction have attracted much attention Yokoyama showed that the current through the normal/ferromagnetic/normal (NFN) silicene junction is valley and spin polarized due to the coupling between the valley and spin degrees of freedom and it can be tuned by an external electric field.21 Also remarkable spin/valley polarization can be accessed through the spinor relying resonant tunneling mechanism in the normal/ferromagnetic/normal multiple silicene junction by aligning the spin and valley-resolve confined states in magnetic well.22 It is shown that the presence of the ferromagnetic barrier in the NFN silicene junction induces exchange splitting in the charge conductance and the charge conductance is a periodic function of the barrier potential and exchange field.23–25 Also the valley and spin transport in ferromagnetic/ferromagnetic barrier/ferromagnetic26 and ferromagnetic/nonmagnetic/ ferromagnetic27 silicene junctions strongly depends on the local application of a vertical electric field in the middle regions and effective magnetization configuration of the ferromagnetic layers The spin-valve effect which is the resistance of a device against switching the relative orientation of the magnetizations is in the heart of spintronics.28 Spintronics aims to study controlling and manipulating the spin degree of freedom in solid state systems.28–32 Investigations of spin transport, spin dynamics and spin relaxation are fundamental studies of spintronics Hence, one of the typical questions posed in spintronics is: what is an effective way to polarize the spin current in solid state systems? Motivated by the development of spintronics devices with the novel 2D-materials, we study the spin and valley polarized current through the silicene-based ferromagnetic/nonmagnetic/ferromagnetic (FNF) spin-valve junction where an electrostatic gate potential is attached to the nonmagnetic segment Actually, due to the importance of nonmagnetic tunneling junction for making relativistic devices26,33,34 and novel physics in ferromagnetic silicene junction, here, we study the valley and spin transport in a silicene-based FNF junction in the presence of the on-site potential difference ∆z and the electrostatic gate potential U Note that our proposed FNF silicene junction is different from Refs 26 and 27 in those there is no electrostatic gate potential in the middle regions In fact, we propose a nonmagnetic barrier spacer layer between two ferromagnetic silicene layers The magnetism in the left and right silicene regions could be induced by the magnetic proximity effect with a magnetic insulator EuO, which is proposed and realized for graphene.21,35,36 The magnetization direction in two ferromagnetic regions can be up and down, so there are two effective magnetization configurations We show that the electrostatic gate potential U is a useful probe to control the band structure, quasi-bound states in the nonmagnetic barrier as well as the transport properties of the FNF silicene junction In particular, by introducing the electrostatic gate potential, the spin and valley conductances of the junction show an oscillatory behavior The amplitude and frequency of such oscillations can be controlled by U Interestingly, we obtained that by increasing U, the second characteristic of the Klein tunneling is satisfied because of the chiral nature of the quasiparticles which can penetrate through the barrier In addition, it is shown that at special height of U, the junction exhibits a gap in the spin and valley conductances and the amplitude of this gap can be controlled by the on-site potential difference, ∆z The rest of this paper is organized as follow: In Sec II, we explain our formalism and analytical calculations of the spin and valley dependent transmittance and conductance through the junction The results are given in Sec III, where we in particular treat the role of the electrostatic gate potential U on the transport properties of the junction Finally, we end the paper with conclusion in Sec IV II THEORETICAL CONSIDERATIONS For the model of calculation, we consider a wide planar two-dimensional ferromagnetic/nonmagnetic/ferromagnetic (FNF) silicene junction in the xy plane The proposed experimental setup of our model is shown in figure The two ferromagnetic silicene regions which can be produced by the magnetic proximity effect with a magnetic insulator substrate have been separated by a nonmagnetic 025307-3 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG Schematic diagram of ferromagnetic/nonmagnetic/ferromagnetic silicene junction There are on-site potential difference ∆z and electrostatic gate potential U in the nonmagnetic region The magnetization direction is parallel in both ferromagnetic regions region and the interfaces between the left and right ferromagnetic silicene are located at x=0 and x=L where L is the length of the nonmagnetic region The magnetization configuration of both ferromagnetic regions can be easily reversed from up and down alignment Here, we only consider parallel configuration for the magnetization direction The effective low-energy Hamiltonian including ferromagnetic and nonmagnetic silicene around Dirac points can be read as8,21 H = ~vF (k x τx − ηk y τy ) − (∆z − ησ∆SO)τz − σh + U (1) Here, vF = 5.5 × 105m/s is the Fermi velocity and σ = +1(↑) or −1(↓) denotes the spin indices in the left and right ferromagnetic silicene regions η(η ′) = +1(−1) corresponds to the K (K ′) valley τx , τy and τz are the pauli metrices in sublattice pseudospin space ∆SO = 3.9meV is the spin-orbit coupling term and ∆z is the on-site potential difference between A and B sublattices which can be tuned by an external electric field U is the electrostatic gate potential where it is attached to the nonmagnetic region like that in graphene.35 This electrostatic gate potential can be used to tune the Fermi level in the nonmagnetic region h which is the exchange field in the left and right ferromagnetic regions has the same value and orientation The eigenvalues of the Hamiltonian in Eq (1) in the left and right ferromagnetic regions and in the nonmagnetic region can be determined as  (2) E = ± (~vF )2(k 2x + k 2y ) + ∆2F − σh  E = ± (~vF )2(k x′ + k y′ 2) + ∆2N + U (3) where ∆ F = ση∆so and ∆ N = ∆z − ση∆so The components of the wave vector along the x-axes in the ferromagnetic regions and nonmagnetic region are given by  kx = (E + σh)2 − ∆2F − (~vF k y )2 (4) ~vF and k x′ = ~vF  (E − U)2 − ∆2N − (~vF k y′ )2 Also the wave vectors in the ferromagnetic (k F ) and nonmagnetic regions (k N ) can be read as  k F = (k x )2 + (k y )2 (5) (6) and kN =  (k x′ )2 + (k y′ )2 (7) We should also mention that in the left and right ferromagnetic regions, an external electric field is not applied Here, x-axis is perpendicular to the interfaces and due to the translational invariance in the y direction, k y , the momentum parallel to the y axis is conserved (k y = k y′ ) By solving equation (1) in each region, the wavefunctions can be obtained analytically For an electron with 025307-4 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) an energy E, incident on the junction from the left ferromagnetic region, the wavefunctions for the valley η and spin σ in three regions can be read as  ~v f (k x + iηk y ) ~v f (−k x + iηk y ) +/ + σh + E + ∆ F * σh + E + ∆ F −i k x x cos ϕ * E + σh + ∆ + r ψ(x ≤ 0) = e e F / ησ E + σh + ∆ F // 2(E + σh) 2(E + σh) 1 , , ~v f (k ′x + iηk ′y ) ~v f (−k ′x + iηk ′y ) * +/ +/ ′ −i k ′ x cos θ * ψ(0 ≤ x ≤ L) = Aη σ e i k x x cos θ E + ∆N E + ∆N / + Bη σ e x / 1 , ,  ~v f (k x + iηk y ) + σh + E + ∆ F * E + σh + ∆ F // ψ(x ≥ L) = t η σ e i k x (x−L) cos ϕ 2(E + σh) ,  i k x x cos ϕ Here, ϕ and θ are the incident and refraction angles in the ferromagnetic and nonmagnetic regions, respectively The amplitude of spin and valley-dependent transmission coefficient (t ησ ) can be calculated by using the wavefunctions continuity at the boundaries x=0 and x=L The spin and valley-dependent charge conductance formula of the junction (Gησ ) at zero temperature can be described by using the standard Landauer-buttiker formalism21,37:  π Gησ = G0(EF ) |t ησ |2 cos ϕdϕ (8) −π 2 EF where EF is the Fermi energy of the system, G0(EF ) = 4eh W π~v F is the conductance unit and W is the width of silicene sheet in the y direction The valley-resolved conductance of the junction can be read as Gη = 1/2(Gη ↑ + Gη ↓), η = K, K ′ (9) The spin-resolved conductance of the junction one as Gσ = 1/2(G K σ + G K ′σ ), σ =↑,↓ (10) III RESULTS AND DISCUSSION In the following calculations, we consider different values for the exchange field h, the on-site potential difference ∆z , the length of nonmagnetic region K F L and the electrostatic gate potential U in each figure We only take the spin-orbit gap ∆so/E = 0.5 and K F is defined as K F = E/~v f We have normalized all the parameters with the Fermi energy E, but the valley and spin-resolved conductances are normalized with G0 In figures 2–7 we have studied the effect of electrostatic gate potential U on the transmittance and the valley and spin-resolved conductance of the junction In these figures the black, red and blue curves correspond to U=0, U=5E and U=10E, respectively A Valley and spin dependent transmittance Figure shows the valley and spin dependent transmission probability (T η σ = |t ησ |2) as a function of incident angle φ for different values of the electrostatic gate potential U We see that for the normal incidence (φ = 0), the junction is not totally transparent in the case of U=0 (figure 2(a)) In this case, the junction is totally transparent at different angles (see the black curve in figures 2(a) and 2(d)) Note also that there is no transmission for the quasiparticles for valley K with spin up (T K ↑) and valley K ′ with spin down (T K ′↓) configurations (see the black curve in figures 2(b) and 2(c)).27 By introducing the electrostatic gate potential (U=5E), one can see that T η σ changes significantly for both valleys and spin configurations and also for all the incident angles of the electrons It is known that Klein tunneling for chiral relativistic particles has two remarkable characteristics: First, the transmission is not always suppressed by a barrier and second, for normal incidence, the barrier is perfectly transparent.14,38,39 As an interesting result, we found that by increasing the height of the gate potential U (form 5E to 10E), the junction will be totally transparent at the normal incidence and the second characteristic of Klein tunneling is satisfied for the normal incidence.14 025307-5 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG Valley and spin dependent transmission probability T η σ as a function of incident angle φ for different values of electrostatic gate potential U, U=0 (black curve), U=5E (red curve) and U=10E (blue curve) (a) T K ↑, (b) T K ′↑, (c) T K ↓ and (d) T K ′↓ In this figure, h/E = 1.6, ∆ z /E = 1.4 and K F L = 20 Klein tunneling is due to the relativistic nature of the massless qausiparticles which can be also seen in graphene.39 In the case of silicene, as the quasiparticles are massive and the mass of them can be controlled by the on-site potential difference ∆z ,10,11,40 hence we can not see the perfect junction for the normal incidence39 (see the black curves in figure 2) Remarkably, with increasing U especially at U/E=10, the junction will be totally transparent for the normal incidence for both valleys and spin configurations Hence, it is worth mentioning that the gate potential U is a useful probe to control the transmittance of the quasiparticles through the junction Moreover, the tunneling through the junction exhibits a strong valley dependent feature B Valley and spin-resolve conductance Figure shows the valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = and ∆z /E = As we can see, in the absence of the exchange field h FIG Valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = and ∆ z /E = In this figure, black, red and blue curves correspond to U=0, U=5E and U=10E, respectively 025307-6 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) and on-site potential difference ∆z , the conductance for both spin configurations and both valleys is the same, G K ↑=G K ′↑=G K ↓=G K ′↓ For U=0 case, the valley and spin-resolved conductance Gησ does not depend on K F L By introducing the gate potential (U=5E), the valley and spin-resolved conductance Gησ changes significantly and it shows an oscillatory behaviour with K F L Note that the period of the Gησ oscillations decreases by U The reason for the oscillatory behaviour is the chiral nature of the Dirac quasiparticles in silicene which leads to penetration of the quasiparticles through the barrier (Klein tunneling) Moreover, due to the existence of the quasi-bound state inside the gate potential barrier, the tunneling conductance oscillations increase by U Note also that the electrostatic gate potential can change the position of the Fermi energy in silicene which can change the band structure of silicene.23,24 So, one can conclude that the electrostatic gate potential is a useful probe to change the band structure and the Fermi energy of silicene It can also control the period of the valley and spin-resolved conductance oscillations Figure displays Gησ as a function of K F L for several values of U/E for ∆z /E = and h/E = 1.6 Actually, in this figure we take the nonzero value for the exchange field h/E as compared to figure In the absence of the on-site potential difference (∆z /E = 0), Gσ=↑ and Gσ=↓ curves are identical for both valleys as seen in figures 4(a) and 4(b), respectively This is why we hold the symmetry between the sublattices by taking ∆z /E = By increasing h in this figure, we can see that both curves for U=0 depend on K F L; G ↑ decays with K F L and G ↓ oscillates with K F L The reason for the damping behaviour is that the Fermi energy lies inside the band gap near K and K ′ valleys for spin up configuration and the reason for the oscillatory behaviour is that the Fermi energy crosses the band gap near K and K ′ valleys for spin down configuration.21,34 With increasing U to 5E and 10E, we observe that Gησ in figures 4(a) and 4(b) starts to oscillate with K F L but in the FIG Valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = 1.6 and ∆ z /E = (a) spin up and (b) spin down configurations 025307-7 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG Valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = 1.6 and ∆ z /E = 0.8 (a) G K ↑, (b) G K ′↑, (c) G K ↓ and (d) G K ′↓ case of Gσ=↑, the amplitude of the oscillations decays with K F L (figure 4(a)), while in the case of Gσ=↓ we see the undamped oscillatory behaviour with K F L (figure 4(b)) Interestingly, one can see that with increasing the gate potential from U=0 to 10E, the frequency of the charge conductance oscillations increases for both spin states, up and down Thus, one can conclude that the electrostatic gate potential has a strong effect on the amplitude and frequency of the spin and valley-resolved conductance oscillations This finding reveals the application of the electrostatic gate for controlling the current in nano-electronics and spintronics silicene-based devices To demonstrate the effect of the on-site potential difference and the electrostatic gate potential U on the transport properties of the junction, we apply ∆z /E across the junction in figure as compared to figures and In figures and as ∆z /E = 0, ∆ N and ∆ F are equal (see equation (2) and (3)), so both valleys have the same contribution in the charge conductance Here, in the case of U=0, by introducing ∆z /E the conductance for each valley is not the same owing to symmetry breaking between the two sublattices So, the contribution of each valley to the charge conductance should be computed separately in silicene where it is in contrast to the valley degeneracy in graphene.41 We note that for G K ↓ and G K ′↑ curves, the Fermi energy is located inside the band gap, so these curves show the decaying behaviour (see figures 2(b) and 2(c)) But for G K ↑ and G K ′↓ curves, the Fermi level crosses the band gap and hence the charge conductances oscillate with K F L (see figures 2(a) and 2(d)).21,34 By increasing the height of U to 5E and 10E, we observe the same behaviour as compared to figure The amplitude of the valley and spin-resolved conductance oscillations for spin up configuration (for both K and K ′ valleys) decays with K F L (see figures 5(a) and 5(b))), but in the case of spin down configuration, the amplitude of the valley and spin-resolved conductance oscillations oscillates with K F L (see figures 5(c) and 5(d))) In figure 6, we further examine the effect of the on-site potential difference and the electrostatic gate potential U on the transport properties of the concerned silicene junction by taking ∆z = 3.4E as compared to figures and It is easy to see in figures 6(a)–6(d) that for U = case, the conductance for each valley and spin configuration decays with K F L because the Fermi level lies inside the band gap for ∆z /E = 3.4 as compared to ∆z /E = 0.8 case (in figure 5) Again, it is easily seen that by enhancing the height of U to 5E and 10E, the amplitude of the conductance oscillations 025307-8 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG Valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = 1.6 and ∆ z /E = 3.4 (a) G K ↑, (b) G K ′↑, (c) G K ↓ and (d) G K ′↓ for spin up and spin down configurations decays and oscillates with K F L, respectively (similar to those seen in figures and 5) In figure we have increased the exchange field to h/E = 4.5 (as compared to h/E = 1.6 in figure 6) and studied its effect on the transport properties of the junction It is easily seen that for each valley and spin configuration, the amplitude of the conductance oscillations decays with K F L FIG Valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = 4.5 and ∆ z /E = 3.4 (a) G K ↑, (b) G K ′↑, (c) G K ↓ and (d) G K ′↓ 025307-9 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG The energy dispersion relation near K and K ′ valley in the nonmagnetic region The upper panels correspond to ∆ z /E = and the lower panels correspond to ∆ z /E = 0.8 Figures a and d correspond to U=0, figures b and e correspond to U=1E and figures c and f correspond to U=2.5E Red and blue lines corresponds to K and K ′ valley, respectively and the amplitude of the conductance oscillations for each U is lower than those seen in figures and with smaller h Increasing the exchange field induces a large spin splitting for both K and K ′ valleys (equation (2)) leading to a larger shift in the energy difference between the spin states up and down as compared to the smaller h To explore the effect of electrostatic gate potential U on the transport properties of the junction, we have plotted the band structure near K and K ′ point in the nonmagnetic region in figure The horizontal lines denote the Fermi energy E Figures 8(a), 8(b) and 8(c) correspond to U=0, 1E and 2.5E, respectively In these figures the on-site potential difference ∆z /E is zero In this case, due to the absence of the ∆z /E, the band structure for K and K ′ valleys is the same We see that by increasing the electrostatic gate potential for U=1E, the Fermi energy lies inside the energy gap (see figure 8(b)) and by enhancing the electrostatic gate potential for U=2.5E the Fermi energy crosses the bands near K and K ′ valleys (see Fig 8(c)) Figures 8(d), 8(e) and 8(f) correspond to U=0, 1E and 2.5E, respectively In these figures, we have increased the strength of the on-site potential difference to ∆z /E = 0.8 to see how the band structure changes upon varying U near K and K ′ valleys In the case of U=0 in figure 8(d) the Fermi energy crosses the band structure near K and K ′ valleys This means that both valleys contribute to the current It should be noted that by increasing U, for special value (for example U=1E), the Fermi energy lies inside the energy gap (see figure 8(e)), but for higher values of U (U=2.5E), the Fermi energy crosses the band structure near K and K ′ valleys (see figure 8(f)) From the band structure in the nonmagnetic region, we realize that for special values of the electrostatic gate potential, the Fermi energy lies inside the energy gap Thus, only the evanescent modes contribute to the current With increasing U, the Fermi energy crosses both valleys Hence, the current is carried by both valleys leading to an oscillatory behaviour of the conductance To understand more systematically the effect of the electrostatic gate potential on the transport properties of the concerned junction, we have plotted Gησ as a function of U for different values of ∆z /E for h = 1.6E, as depicted in figure In figure 9(a), we show that in the absence of ∆z /E, G K ↑ = G K ′↓ and G K ↓ = G K ′↑ Interestingly, we observe that at some values of the gate potential U, the junction shows a gap in the valley and spin conductance Gησ curves and there is no conductance at these U Creation of this gap is due to the fact that for these values of U, the component of the wave vector in the middle region, k x′ , will be imaginary and the evanescent modes contribute to the current, so there is no conductance (see also the band structure in figures 8(b) and 8(e)) These results are quite different from those for FNF graphene junction35,42 and two-dimensional electron gas By increasing ∆z /E we see that the amplitudes of the gap for G K ↑ and G K ′↓ are different from G K ′↑ and G K ↓ (figures 9(b) and 9(c)) Actually the condition for the formation of the gap can be obtained by − (∆z /E − ση∆so/E) < U/E < + (∆z /E − ση∆so/E) (11) ′ for these values of U/E, the wave vector in the middle region (k N ) is real For example in the ′ case of ∆z = 3.4E, the gap for G K ↑ and G K ↓ is in the range of −1.9 < U/E < 3.9 and for G K ↓ and G K ′↑ the gap is in the range of −2.9 < U/E < 4.9 (figure 9(c)) By increasing the amplitude 025307-10 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG Valley and spin-resolved conductance Gησ as a function of U /E, for several values of ∆ z for K F L = and h/E = 1.6 (a) ∆ z /E = 0, (b) ∆ z /E = 0.8 and (c) ∆ z /E = 3.4 The red, blue, green and gray curves correspond to G K ↑, G K ′↑, G K ↓ and G K ′↓, respectively of the electrostatic gate potential beyond the gap, Gησ shows an oscillatory behaviour (see the band structure in figures 8(c) and 8(e)) These oscillations are due to the chiral nature of the quasiparticles in silicene.23,24 Hence, as an important result we found that by changing ∆z /E, one can control the amplitude of the gap that could be a useful probe for the applications in silicene-based nano-electronic devices Also note that the amplitude of the valley and spin-conductance oscillations decreases as the exchange field h increases In figure 10 we have increased the exchange field to h = 4.5E (as compared to h = 1.6E in figure 9) and studied its effect on the valley and spin-resolved conductance Gησ as a function of U/E As can be seen from figures 10(a)–10(c), the amplitudes of Gησ oscillations are strongly suppressed, as compared to figure This behaviour arises from the fact that by increasing the exchange field h/E from 1.6 to 4.5, the component of the wave vector along the x-axes in the ferromagnetic regions (k x ) increases, see equation (4) Note that increasing h does not have any effect on the amplitude of the gap 025307-11 Y Hajati and Z Rashidian AIP Advances 6, 025307 (2016) FIG 10 Valley and spin-resolved conductance Gησ as a function of U /E, for several values of ∆ z for K F L = and h/E = 4.5 (a) ∆ z /E = 0, (b) ∆ z /E = 0.8 and (c) ∆ z /E = 3.4 The red, blue, green and gray curves correspond to G K ↑, G K ′↑, G K ↓ and G K ′↓, respectively Finally, let us investigate how Gησ depends on ∆z /E for different values of the electrostatic gate potential U in figure 11 To check the validity of our calculations, we emphasize that the results for U=0 in figure 11(a) coincide with Ref 27 From figures 11(a)–11(c), it is clear that for ∆z /E