Polarized-photon frequency filter in double-ferromagnetic barrier silicene junction

Journal of Magnetism and Magnetic Materials 429 (2017) 16–22

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Polarized-photon frequency filter in double-ferromagnetic barrier silicene junction Peerasak Chantngarma, Kou Yamadaa, Bumned Soodchomshomb, a b

MARK

⁎

Domain of Mechanical Science and Technology, Graduate School of Science and Technology, Gunma University, Gunma, Japan Department of Physics, Faculty of Science, Kasetsart University Bangkok 10900, Thailand

A R T I C L E I N F O

A BS T RAC T

Keywords: Silicene Spin-valleytronics Photo-sensing device

We present an analytical study of effects from circularly polarized light illumination on controlling spin-valley currents in a dual ferromagnetic-gated silicene. Two different perpendicular electric fields are applied into the ferromagnetic (FM) gates and the photo-irradiated normal (NM) area between the gates. One parallel (P) and two anti-parallel (AP) configurations of exchange fields applied along with chemical potential to the gates are used in this investigation. Interestingly, the studied junction might give rise to polarized-photon frequency filter. Spin-valley filtering can be achieved at the off-resonant frequency region with appropriate direction of electric fields and the configuration of exchange fields (AP-1 or AP-2). Under the photo irradiation, this study found that tunneling magnetoresistance (TMR) is controllable to achieve giant magnetoresistance (GMR) by adjusting electric fields or chemical potentials. Our study suggests the potential of photo-sensing devices in spin-valleytronics realm.

1. Introduction Silicene, two-dimensional silicon with out-of-plane bucklings, has recently attracted great attention after experimental discovery [1–3]. It is considered to be one of the candidates for the post-bulk-silicon era along with other artificial elemental 2D materials such as germanene (Ge) [4,5], phosphorene (P) [6], stanene (Sn) [7–9] and plumbene (Pb) [10] in electronics, spintronics, valleytronics and quantum computing applications. These novel 2D materials, especially stanene and plumbene, have large enough band gap opening and spin-orbital coupling (SOC) strong enough to maintain robust quantum spin Hall effect at high temperature. The abovementioned desirable characteristics are hard to achieve in graphene, the first elemental 2D material, due to the small ionic radius of carbon. The strong SOC also gives rise to the spinvalley coupling, which may lead to the integration of spintronics and valleytronics [11,12]. Among these 2D materials, silicene is currently considered to be the most promising candidate mainly due to the accumulation of silicon-related technology and knowhow in semiconductor industry [13,14]. Although in theory silicene has zero band gap, the buckled honeycomb lattice structure allows Dirac electron mass and its band structure to be manipulated easily by electric field [15,16]. The two-dimensional buckled honeycomb lattice structure in silicene causes tunable spin-valley coupled band structure giving rise to topological phase transition, an intriguing transport phenomenon

⁎

[17]. Silicene has rich varieties of phases such as quantum spin Hall (QSH) state, quantum anomalous Hall (QAH) state, quantum Hall effect (QHE) state, fractional quantum Hall effect (FQHE) state, band insulator (BI), and valley-polarized metal (VPM). By controlling electric field and exchange field appropriately, it is possible to materialize these phases and achieve the topological phase transition [15,18]. There have been studies on ballistic transport, and the relationship between the transmission probability and valley conductance with electric field and exchange field in silicene junctions [19,20]. More recently, double ferromagnetic-gated silicene junction was proposed to control lattice-pseudospin current along with pure spin-valley current in silicene giving a possibility for pseudospintronics [21]. Another interesting development in making silicene devices is the self-doping phenomenon caused by strain [22]. Great attention in silicene has also led to more investigation and discovery in many other aspects such as hydrogenation effect, synthesis of multilayer silicene, and photoinduced effects. Fully hydrogenated silicene is called silicane, an interesting material for FET application, while half-hydrogenated silicene is a method to introduce magnetism and generate band gap in silicene [23,24]. In addition to synthesis of monolayer silicene on Ag, ZrB2, and Ir(111) substrates, experimental groups have succeeded in synthesis of multilayer silicene using both epitaxial growth and non-epitaxial growth after theoretically proposal [25–27]. Another area that attracts attention recently is photo-induced

Corresponding author. E-mail addresses: [email protected], [email protected] (B. Soodchomshom).

http://dx.doi.org/10.1016/j.jmmm.2017.01.001 Received 14 October 2016; Received in revised form 21 December 2016; Accepted 1 January 2017 Available online 02 January 2017 0304-8853/ © 2017 Elsevier B.V. All rights reserved.

Journal of Magnetism and Magnetic Materials 429 (2017) 16–22

P. Chantngarm et al.

effects, similarly to the study in graphene, where circularly polarized light is used to open a gap at the Dirac point [28]. By irradiation of circularly polarized light at fixed electric field, the topological class of silicene could be changed from quantum spin-Hall insulator (QSHI) to other phase [29,30]. Light irradiation on silicene has effects on the band structure. The photo-irradiation realizes a topological superconductor when s-wave superconductivity proximity coupling is applied [31]. It is reported that spin-valley polarization depends on the intensity of off-resonant circularly polarized light as well as electric field, and it can be inverted by reversing the direction of electric field or the circular polarization of the light [32]. Spin-valley polarizations and tunneling magnetoresistance in a ferromagnetic-normal-ferromagnetic (FNF) junction can be significantly enhanced by off-resonant circularly polarized light without electric field or magnetic field [33]. There is also a study in effective photo-induced band structure manipulation with intense terahertz irradiation beyond the off-resonant condition [34]. In this paper, we study the tight-binding model of silicene-based NM1/FM1/NM2/FM2/NM3 junction under the effects of electric fields, exchange fields, chemical potentials, and off-resonant circularly polarized light. Here, NMs stand for normal silicene and FMs stand for proximity-induced ferromagnetic silicene. Exchange field is applied to see the effects of parallel (P) and anti-parallel (AP) configurations. Then we study scattering process of the junction to obtain transmission probability. We particularly investigate the spin-valley conductance and tunneling magnetoresistance (TMR) of this junction. It is found that spin-valley filtering and TMR can be controlled by appropriate adjustment of the structure parameters and light frequency. Fig. 2. Junction types used in this investigation, where→represents h, and ←represents –h. (a) Parallel junction (P), (b) anti-parallel junction type 1 (AP-1), (c) anti-parallel junction type 2 (AP-2).

2. Model We study ballistic transport of Dirac fermions in the double-barrier silicene-based structure shown in Fig. 1. The ferromagnetic barriers have length d, and are separated from each other by distance L. There were theoretical studies showing that exchange energy of 5 meV in graphene could be induced by ferromagnetic insulators EuO due to proximity-induced exchange splitting [35,36]. However, unlike the planar structure in graphene, silicene has buckling structure consisting of A-sublattice at the top and B-sublattice at the bottom (see Fig. 1). The two out-of-plane buckled sublattices may act like two separate atom layers of one silicene sheet. Therefore, it allows us to apply two different exchange fields into a silicene monolayer. One exchange field is induced by ferromagnetic gate at the top and another exchange field is induced by ferromagnetic gate at the bottom. The exchange energies induced at A- and B-sublattices, are designated as h1A and h1B at barrier FM1, while they are designated as h2A and h2B at barrier FM2. Two controllable perpendicular electric fields, Ez1 and Ez2 , are applied to the ferromagnetic barriers FMs and NM2 region, respectively. Gate potential μ / e is applied from the top and the bottom of both ferromagnetic barriers, to induce chemical potential μ . Circularly polarized light A (t ) = A0 (sin(Ωt ), cos(Ωt )) is irradiated to the NM2

region, where Ω is frequency, A(t) is time-dependent vector potential of photon, and A0 is magnitude. The off-resonant light frequency used in this study is the frequency region where the electronic band structures are changed by virtual photon absorption processes without direct electrons excitation. In π-band tight-binding model, this can be achieved when ℏ Ω ≫ t0, where t0 is nearest hopping energy. The lowest frequency Ω to satisfy this condition can be calculated from the bandwidth 3 t0=4.8 eV=1015 Hz [29]. The perpendicular distance between A- and B-sublattices due to the buckling structure is represented with 2D=0.46 Å, where D=0.23 Å [37]. In this study, we investigate three junction types, one parallel junction (P) and two anti-parallel junctions (AP) as shown in Fig. 2. The parallel junction (P) is defined as the configuration in Fig. 2(a) where h1A=h1B=h2A=h2B=5 meV. As for anti-parallel junctions, AP-1 is the junction where -h1A=h1B=-h2A=h2B=5 meV, while AP-2 is the junction where h1A=-h1B= h2A=-h2B=5 meV as shown in Fig. 2(b) and (c), respectively. The tight-binding Hamiltonians and low-energy effective Hamiltonians are used to describe the motion of electrons in A- and B-sublattices in our analysis [38–40]. Here, the effect of Rashba interaction is negligible comparing with the other terms at low energy, so the wave equation with excited energy E can be expressed with

Hˆ ησ Ψησ = EΨησ

(1)

when k(k′) valley is represented by η = 1(−1), and spin ↑(↓) is ⎛ φA, ησ ⎞ represented by σ = 1(−1), respectively [39–41]. Here, ψησ = ⎜ φ ⎟ is ⎝ B, ησ ⎠ spin-valley-dependent "lattice-pseudospinnor field", where φA, ησ and φB, ησ are wave functions at A- and B-sublattices, respectively. In ferromagnetic regions FM1(2), the Hamiltonian is defined as Fig. 1. Cross-sectional schematic model of double-barrier silicene-based NM1/FM1/ NM2/FM2/NM3 structure.

Hˆ ησ = HˆF1(2) = vF ( pˆx τ x − ηpˆy τ y ) − Δησ1(2) τ z − μσ1(2) , 17

(2)

Journal of Magnetism and Magnetic Materials 429 (2017) 16–22

P. Chantngarm et al. ∂

∂

where pˆx = −iℏ ∂x , pˆy = −iℏ ∂y and τ x , τ y , τ z are elements of Pauli spinoperators acting on lattice-pseudospin states, and vF indicates Fermi velocity near Dirac point. Δησ1(2) = ησΔSO − ΔE1 + σΔM1(2) is spin-valleydependent energy gap, where effective spin-orbit interaction ΔSO = 3.9meV [42], electric field ΔE1 = eDEz1, and exchange field(h − h ) (h − h ) induced gap ΔM1 = 1A 2 1B and ΔM 2 = 2A 2 2B [40]. The spin-dependent chemical potential is characterized with μσ1(2) = μ + σuM1(2), where (h

(h

+h )

+h

kx = mx =

)

k// = =

(3)

(4)

2 E2 − Δησ

, and Nησ =

E + μσ 2 + Δησ 2

. (5)

The coefficients rησ , a ησ , bησ , gησ , fησ , pησ , qησ , tησ are obtained by using the boundary conditions where

ψNM1 (0) = ψFM1 (0), ψFM1 (d ) = ψNM 2 (d ), ψNM 2 (d + L ) = ψFM 2 (d + L ), and ψFM 2 (2d + L ) = ψNM 3 (2d + L ).

2 Δησ 2

2 (E + μσ1)2 − Δησ 1 sin α1

ℏvF sin α2

ℏvF

=

2 E2 − Δησ sin β

,

ℏvF (8)

e2 N (E )|tησ |2 = G 0 h

2 E2 − ΔSO

|E |

π /2

∫−π /2 81 dθ cos(θ ) Tησ .

(9) W |E | π ℏvF

(Gk ↑ + Gk′↑) − (Gk ↓ + Gk′↓) GT (Gk ↑ + Gk ↓) − (Gk′↑ + Gk′↓) GT GP − GAP GP

× 100, × 100,

× 100,

(10)

Throughout this numerical study, we set parameters of the base case to be L=25 nm, d=25 nm, h=5 meV, and E=4 meV. Firstly, spinvalley currents are investigated. Fig. 3 depicts spin-valley conductance as a function of the frequency of irradiated circularly polarized light when there is no electric field, EZ1 and EZ2 . The conductance shows pure spin polarization in parallel junction (see Fig. 3(a)), which coincides with the results from previous study [21] where there is no photo irradiation. Interestingly, the conductance shows complete spinvalley polarization under varying photon frequency only in anti-parallel junctions ie., Gk ↑ ≠ Gk ↓ ≠ Gk ′↑ ≠ Gk ′↓ (see Figs. 3(b) and 3(c)). It is noticeable here that the magnitude of conductance is lower than 0.03, which is a very low number. The cause of this low magnitude is the result of low energy E approaching Δso (see Eq. (9) to get

,

2 (E + μσ 2 )2 − Δησ 2

−

=

5. Result and discussion

where

E + Δησ

(7)

where GT = Gk ↑ + Gk ↓ + Gk ′↑ + Gk ′↓ is total conductance, while GP and GAP are total conductance in parallel and anti-parallel junctions, respectively.

⎡ ⎛ 1 ⎞ ⎤ ikx x ⎥ e ik//y , ψNM 3 = ⎢tησ ⎜ −iηθ ⎟ e A e ⎠ ⎣ ⎝ ησ ⎦

Mησ =

(E + μσ 2

)2

and TMR (%) =

⎤ ⎡ ⎛ ⎞ ⎛ ⎞ 1 1 inx x + q −inx x ⎥ e ik//y , ψFM 2 = ⎢ pησ ⎜ −iηα ⎟ e ησ ⎜− N e iηα2 ⎟ e ησ ⎠ ⎝ ⎦ ⎣ ⎝ Nησ e 2 ⎠

E + μσ1 + Δησ1

ℏvF

VP (%) =

⎡ ⎛ ⎤ ⎞ ⎛ ⎞ 1 1 imx x + f −imx x ⎥ e ik//y , = ⎢gησ ⎜ −iηβ ⎟ e ησ ⎜− M e iηβ ⎟ e ησ ⎝ ⎠ ⎣ ⎝ Mησ e ⎠ ⎦

2 (E + μσ1)2 − Δησ 1

2 E2 − ΔSO sin θ

SP (%) =

⎤ ⎡ ⎛ ⎞ ⎛ ⎞ 1 1 −ilx x ⎥ e ik//y , ilx x + b ψFM1 = ⎢a ησ ⎜ ησ ⎜ −iηα1⎟ e iηα1⎟ e B e − B e ησ ησ ⎠ ⎝ ⎠ ⎦ ⎣ ⎝

, Bησ =

.

is unit conductance, where N0 (E ) = is Here, G0 = density of state at transport channel in silicene excluding the spin-orbit interaction effect, h is Planck's constant, and W indicates the width of W 2 in Eq. (9) represents density of silicene sheet. N (E ) = π ℏv E2 − ΔSO F state at the transport channel of normal silicene junction. The spin polarization (SP), valley polarization (VP) and tunneling magnetoresistance (TMR) are defined respectively as

⎡⎛ 1 ⎞ ⎤ ⎛ ⎞ 1 ikx x + r −ikx x ⎥ e ik//y , ψNM1 = ⎢ ⎜ ησ ⎜ −iηθ ⎟ e iηθ ⎟ e A e A e − ησ ⎠ ⎝ ⎠ ⎣ ⎝ ησ ⎦

2 E2 − ΔSO

ℏvF

4e2 N (E ) h 0

In this section, we describe the motion of electrons where the spinvalley currents flow along the x-axis. Using the Hamiltonians in Eqs. (1) to (4), the wave functions in the NM1, FM1, NM2, FM2, and NM3 regions can be defined respectively as

E + ησΔSO

and nx =

,

We describe the spin-valley conductance and tunneling magnetoresistance in this section. Spin-valley conductance in ballistic regime at zero temperature can be obtained with integration on all incident angles using the standard Landauer's formalism [43] as shown below

3. Scattering process

Aησ =

ℏvF

ℏvF

2 cos α (E + μσ 2 )2 − Δησ 2 2

4. Transport formulae

Gησ =

ψNM 2

2 cos β E2 − Δησ

where θ is the incident angle at the NM1/FM1 junction, α1 is the incident angle at FM1/NM2, β is the incident angle at NM2/FM2, and α2 is the incident angle at FM2/NM3. The transmission probability amplitude Tησ is calculated via the formula Tησ = Jt / Jin = |tησ |2 , where Jt and Jin are current densities of transmitted electrons and injected electrons, respectively.

where Δησ = ησΔSO − ΔE 2 + ηλΩ , ΔE 2 = eDEz2 , and λΩ = (eΛVF )2 /ℏΩ . We have Λ = eAa /ℏ being a dimensionless number used to characterize the light intensity with "e" represents elementary charge and "a" represents the lattice constant of silicene [28,33]. The value of Λ is typically less than 1 for the intensity from light sources in the frequency range of our interest [29]. In the normal regions NM1(3), where there is neither photo irradiation or electric field, the spin- and valley-dependent energy gap is Δησ 3 = ησΔSO . Hence, the Hamiltonian in these regions is defined as

Hˆ ησ = HˆNM1(3) = vF ( pˆx τ x − ηpˆy τ y ) − ησΔso τ z.

ℏvF

2 cos α (E + μσ1)2 − Δησ 1 1

, lx =

Incident angles can be obtained through the conservation of components in y-direction as given by

uM1 = 1A 2 1B and uM 2 = 2A 2 2B since the chemical potential in the barrier is spin-dependent relating to the exchange field. The Hamiltonian in normal regions NM2 with photo irradiation and electric field is defined as Hˆ ησ = HˆNM 2 = vF ( pˆx τ x − ηpˆy τ y ) − Δησ τ z,

2 cos θ E2 − ΔSO

(6)

E2 − Δ 2

SO G ∝ 2G0 × transmission). This is because in the carrier |E | transport near the bottom of the energy band E ≅ ΔSO , we get fewer 2 / π ℏvF ≈ density of states in transport channel ie.,N (E ) = W E2 − ΔSO small. When E is lower than ΔSO , the junction becomes an insulator and

Here, rησ and tησ represent reflection and transmission coefficients, respectively. The wave vectors in the x-direction in each region are given by 18

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P. Chantngarm et al.

Fig. 3. Conductance as the function of light frequency without electric field (Ez1=Ez2 =0 meV) in P and AP junctions when L=25 nm, d=25 nm, E=4 meV, h=5 meV, and µ=2.5 meV. (a) In P junction, (b) in AP-1 junction, and (c) in AP-2 junction.

of TMR goes beyond −2,200% which is a significant increase from −73% when the electric field is 4 meV. This indicates giant magnetoresistance (GMR) in which the total resistance is low in P junction but high in AP junction. Fig. 5(b) also shows similar correlation with gateinduced chemical potential, where higher chemical potential results in lower minimum point of TMR at lower frequency. Similarly to Fig. 5(a), GMR also shows up here. The minimum of TMR increases significantly from −19% when µ=5.0 meV to approximately −570% when µ=7.5 meV. The results shown in Fig. 5 indicate that external electric field and chemical potential may have a significant impact on electron scattering, when their intensities reach a certain level. Spin polarization (SP) in P junctions and valley polarization (VP) in AP-1 junctions as the function of light frequency with varying chemical potential µ when there is no electric field are shown in Fig. 6(a) and (b), respectively. This control of SP and VP with chemical potential is mainly due to quantum interference of electrons in ferromagnetic gates. Another interesting result is that both Fig. 6(a) and (b) depict almost exactly the same characteristics of photo-frequency-dependency for both SP and VP in µ=0 meV, and µ=2.5 meV. However, when the chemical potential µ increase to 5.0 meV and 7.5 meV, SP shows less dependency on light frequency, while VP keep almost the same dependency on light frequency. Finally, the effects of the distance L between ferromagnetic gates on the spin-valley currents in AP junction are examined, and possible photon frequency filtering is proposed. Fig. 7(a) shows k↑ current under the same condition as in Fig. 4(a) except that the value of distance L is varied. The results implies that the filtering effect on k↑ comes from resonant, where the shorter L allows resonant to occur at lower photon frequency and vice versa. Fig. 7(b) confirms our under-

the conductance may turn to zero. The junction may be a good conductor when E > > ΔSO and E > > μ, h, eDEz leading to G → 2G 0 . For the case of E ≅ ΔSO , the effect of spin-valley dependent energy gap in the barriers is strong, giving rise to resonant conductance peaks as a spin-valley filter [40]. We therefore focus on this regime to investigate spin-valley filtering effect induced by frequency of polarized photon. When electric field EZ1 is applied to ferromagnetic regions, FM1 and FM2, with another electric field EZ2 applied to normal region NM2 along with circularly polarized light, this double-barrier silicene-based structure might behave as a spin-valley filter. Fig. 4 shows the conductance in AP-1 and AP-2, which indicates that by properly manipulating the direction of electric fields EZ1 and EZ2 as well as the direction of exchange fields at the barriers, we can select the spin-valley polarization to be filtered. When EZ1=EZ2 =4 meV, k↑ and k↓ show sharp peaks near 2,000 THz in AP-1 and AP-2 configurations as indicated in Fig. 4(a) and (b), respectively. Similarly, when EZ1=EZ2 =−4 meV, k′↓ and k′↑ show peaks near 2000 THz in AP-1 and AP-2 as seen in Fig. 4(c) and (d), respectively. Particularly, the comparison of Fig. 3(b) and Fig. 4(a) shows that the k′↓ component can be suppressed by applying positive electric field. This is considered to be the results of filtering effects from the ferromagnetic barriers with electric field. Tunneling magnetoresistance (TMR), an interesting phenomenon widely used in sensing applications, is also controllable by electric fields and chemical potential under the influence of photo irradiation as depicted in Fig. 5. Here, AP-1 configuration is used to represent GAP . Fig. 5(a) shows the dependency of TMR on electric fields EZ =EZ1=EZ2 and light frequency. It shows that the minimum of TMR becomes lower and shifts toward lower light frequency, when the electric fields increase. Particularly when the electric field is 6 meV, the minimum 19

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P. Chantngarm et al.

Fig. 4. Conductance as the function of light frequency with electric field in AP junctions when L=25 nm, d=25 nm, E=4 meV, h=5 meV, and µ=2.5 meV. When EZ1=EZ 2 =4 meV (a) in AP-1 junction, (b) in AP-2 junction, and when EZ1=EZ 2 =−4 meV (c) in AP-1 junction (d) in AP-2 junction.

quency filter. The filtered frequency that yields maximum conductance may be controllable by varying the thickness of the barriers.

standing that the distance L affects only filtered spin-valley current, k↑ in this case, since other spin-valley currents are almost unchanged comparing to Fig. 4(a) with different L. Since total conductance is an important parameter from practical point of view, Fig. 7(c) provides an evidence to support our argument that the light frequency at which the peak of the filtered spin-valley current k↑ appears is determined by the resonant. This is affected by the distance L between the two FM gates, while L does not have impact on other spin-valley currents. Since maximum total conductance resonance is found at certain frequency of polarized photon, this shows the presence of polarized-photon fre-

6. Summary and conclusion We have investigated spin-valley transport in a double ferromagnetic silicene-based junction. We found the possibility of doublebarrier silicene-based structure to completely control spin-valley current with electric fields and direction of exchange fields at gates under irradiated circularly polarized light. When there is no electric

Fig. 5. Tunneling magnetoresistance as the function of light frequency with electric field in AP-1 junctions when L=25 nm, d=25 nm, E=4 meV, h=5 meV. When EZ =EZ1=EZ 2 (a) µ=2.5 meV and EZ is varied from 0 to 6 meV, (b) EZ =0 meV and µ is varied from 0 to 7.5 meV.

20

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P. Chantngarm et al.

Fig. 6. (a) Spin polarization in P junctions, and (b) valley polarization in AP junctions (AP-1) as the function of light frequency without electric field (Ez1=Ez2 =0 meV) when L=25 nm, d=25 nm, E=4 meV, h=5 meV, and µ is varied from 0 to 7.5 meV.

field and the configuration of exchange energies direction is parallel junction, the conductance shows pure spin polarization similar to the case of no photo irradiation. Interestingly, when electric fields are applied, the structure shows spin-valley filtering characteristics where we can select specific spin-valley polarization to be filtered by properly manipulating the direction of electric fields and exchange fields. Furthermore, we found that by adjusting the distance between two FM gates, the light frequency required to trigger the peak of the filter spin-valley current can be controlled. This structure also shows a possibility to control tunneling magnetoresistance (TMR) with electric fields and gate-induced chemical potentials under the photo-irradiation

regime. This study also found that TMR magnitude changes significantly by varying frequency of polarized-photon. Particularly, when the value of electric field and gate-induced chemical potential reach a certain level, GMR of as large as 2,200% and 570% appear, respectively. In addition, we show the control of SP and VP by adjusting gateinduced chemical potentials µ. In summary, our study demonstrates a possibility to control spin-valley filtering using ferromagnetic silicene junction under polarized-photo irradiation, and suggests the potential of photo-sensing devices in spin-valleytronics realm.

Fig. 7. Effects of the distance L between FM gates on the spin-valley current in AP junction. (a) Conductance of k↑ current in with varied L values, (b) conductance of each spin-valley currents when L=50 nm, (c) total conductance of all combined spin-valley currents with varied L values. Photon frequency can be filtered.

21

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P. Chantngarm et al. [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

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