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: />View Table of Contents: />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
1

Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz,
Ahvaz, Iran
2
Department of Physics, Faculty of Science, University of Lorestan, Lorestan, Iran

(Received 26 October 2015; accepted 3 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.
[ />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:

2158-3226/2016/6(2)/025307/13

6, 025307-1

© Author(s) 2016


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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 1. 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

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FIG. 1. 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′ 2 + 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

1
kx =
(E + σh)2 − ∆2F − (~vF k y )2
(4)
~vF
and
k x′ =

1
~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


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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
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
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) .
1
,


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
2
Gησ = G0(EF )

|t ησ |2 cos ϕdϕ
(8)
−π
2

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 2 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


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FIG. 2. 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 3 shows the valley and spin-resolved conductance Gησ as a function of K F L, for several
values of U/E for h/E = 0 and ∆z /E = 0. As we can see, in the absence of the exchange field h

FIG. 3. Valley and spin-resolved conductance Gησ as a function of K F L, for several values of U/E for h/E = 0 and
∆ z /E = 0. In this figure, black, red and blue curves correspond to U=0, U=5E and U=10E, respectively.


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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 4 displays Gησ as a function of K F L for several values of U/E for ∆z /E = 0 and
h/E = 1.6. Actually, in this figure we take the nonzero value for the exchange field h/E as
compared to figure 3. 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 = 0. 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. 4. 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. (a) spin up and (b) spin down configurations.


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FIG. 5. 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 5 as
compared to figures 3 and 4. In figures 3 and 4 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 4. 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 4 and 5. It is easy to see in figures 6(a)–6(d) that for U = 0 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


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FIG. 6. 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 4 and 5).
In figure 7 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. 7. 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 ′↓.


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FIG. 8. The energy dispersion relation near K and K ′ valley in the nonmagnetic region. The upper panels correspond to
∆ z /E = 0 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 5
and 6 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 8. 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 9. 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
1 − (∆z /E − ση∆so/E) < U/E < 1 + (∆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


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FIG. 9. Valley and spin-resolved conductance Gησ as a function of U /E, for several values of ∆ z for K F L = 8 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 9. 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.

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FIG. 10. Valley and spin-resolved conductance Gησ as a function of U /E, for several values of ∆ z for K F L = 8 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 <| (1 − U/E) + ησ∆SO |, the spin and valley resolved conductance Gησ can be resolved
because in this area the wave vector in the middle region, k N , is real. However, beyond this criteria,
k N will be imaginary and Gησ decreases to zero. For example in the case of U/E=5 (figure 11(b)),
we can see G K ↑ and G K ′↓ curves for ∆z /E < 3.5, but we can see G K ↓ and G K ′↑ curves for
∆z /E < 4.5. It can be seen that U/E has a strong effect on the amplitude of Gησ oscillations. By
increasing U, we see from figures 11(b) and 11(c) that Gησ goes to zero for the larger values of
the on-site potential difference ∆z /E. In addition, by introducing U the Gησ shows an oscillatory
behaviour and the amplitude and frequency of the valley and spin-resolved conductance oscillations
enhance by U. As an important result, we found that by changing U/E one can control the critical
values of ∆z /E and also the amplitude and frequency of the Gησ oscillations. It is worth mentioning


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Y. Hajati and Z. Rashidian

AIP Advances 6, 025307 (2016)

FIG. 11. Valley and spin-resolved conductance Gησ as a function of ∆ z , for several values of U for K F L = 8 and h/E = 1.6.
(a) U=0, (b) U=5E and (c) U=10E. The red, blue, green and gray curves correspond to G K ↑, G K ′↑, G K ↓ and G K ′↓,
respectively.

that the gate voltage engineering by U-modulation, offering new ideas to design nano-electronics
and spintronics silicene-based devices.
We should emphasize that many of the physical properties of the FNF silicene junction presented here have no analogous features in the graphene junctions. More precisely, as the spin-orbit
interaction in graphene is very weak, we cannot see the formation of the bandgap in the similar
ferromagnetic/normal graphene junctions.35,42 Due to the controllable valleytronics applications of
our silicene junction, we expect that our results would be applicable to other 2D-materials with
the spin and valley coupled systems such as molybdenum disulfide and group VI-transition metal
dichalcogenides.

IV. CONCLUSION

To sum up, we have investigated the valley and spin transport through a silicene-based ferromagnetic/nonmagnetic/ferromagnetic (FNF) junction with an attached electrostatic gate potential U


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AIP Advances 6, 025307 (2016)

in the nonmagnetic region. The addition of this U potential leads to high controllability of the spin
and valley current in the concerned silicene junction. More precisely, in presence of U, both the
valley and spin conductances of the junction show an oscillatory behavior. Furthermore, we have
shown that by increasing U the following two fundamental phenomena happen: (i) the second characteristic of the Klein tunneling is satisfied and (ii) the amplitude and frequency of the oscillations
enhance. Also, it is found that the valleytronics application of this junction can be tuned by the

values of U/E. This property shows the potential applications of the concerned silicene junction for
improving the performance of nano-electronics devices. It is expected that the theoretical results
obtained in our model may be confirmed by future experiments.
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