Spin-dependent transport properties in strained silicene with extrinsic Rashba spin–orbit interaction

Accepted Manuscript Research articles Spin-dependent transport properties in strained silicene with extrinsic Rashba spin–orbit interaction Farhad Sattari, Soghra Mirershadi PII: DOI: Reference:

S0304-8853(17)31863-2 http://dx.doi.org/10.1016/j.jmmm.2017.08.067 MAGMA 63101

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

16 June 2017 26 July 2017 22 August 2017

Please cite this article as: F. Sattari, S. Mirershadi, Spin-dependent transport properties in strained silicene with extrinsic Rashba spin–orbit interaction, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/ 10.1016/j.jmmm.2017.08.067

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Spin-dependent transport properties in strained silicene with extrinsic Rashba spin–orbit interaction Farhad Sattari a,*, Soghra Mirershadi b a

Department of Physics, Faculty of Sciences, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran b Department of Engineering Sciences, Faculty of Advanced Technologies, University of Mohaghegh Ardabili, Namin, Iran Abstract

We theoretically investigate the spin-dependent transport properties in a silicene junction, with extrinsic Rashba spin–orbit interaction (RSOI), in the presence of strain. Due to the RSOI coupling, spin-inversion can be achieved. The spin resolved conductance and spininversion effect can be efficiently tuned by RSOI and strain strength. In addition, for particular values of RSOI strength, electrons with perfect spin-inversion transmit through the junction. It is found that for the armchair direction strain, unlike the zigzag direction the spin polarization can be observed and it increases with increasing the RSOI strength. The magnitude and sign of spin polarization can be manipulated by strain. The spin polarization reaches a maximum value at 2% strain.

Keywords: Silicene junction; Extrinsic Rashba spin–orbit interaction; Strain; Spin-inversion; Spin polarization.

*E-mail

address: [email protected] (F. Sattari)

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1. Introduction Silicene, a monolayer of silicon atoms packed into a honeycomb lattice much like graphene, has been synthesized through epitaxial growth on the Ag(110) [1,2] and then on variety of surfaces [3-6]. In low energy regime and in the absence of spin–orbit coupling (SOC), charge carriers in silicene are described by the massless Dirac-like equation near the K and K′ points. Moreover, silicene has a large intrinsic spin–orbit coupling which is about 1000 times larger than graphene [7,8], due to the buckling nature of silicene lattice that originates from the

sp 2 / sp 3 mixed hybridization [9]. Intrinsic spin–orbit interactions leads to a gap of 1.55 meV in the band structure. In addition, there are two kinds of Rashba spin–orbit interactions in silicene; intrinsic and extrinsic Rashba spin–orbit interaction. Intrinsic relates to the lowbuckled geometry of silicene. Since this term has a negligible effect on the energy dispersion relation, it can be neglected for most purposes [10,11]. The extrinsic Rashba spin–orbit coupling arises from an external electric field perpendicular to the silicene sheet, or interaction with a substrate [12]. In contrast to graphene, silicene is a good candidate for spintronics applications since the spin–orbit coupling, spin relaxation time and spin coherence length is long compared to graphene. Also, many features of the silicene can be changed by strain. For instance, a semimetal-metal transition occurs with the usage of external strain larger than 7.5% [13]. When the strain is applied in the armchair direction an indirect–direct gap transition exists at a strain of 5% [14]. In recent years, lots of attention is attracted to electronics and spintronics properties of silicene [15-31]. The valley and spin resolved thermoelectric transport in a normal/ferromagnetic/normal silicene has been investigated in Ref. 15. Niu and Dong [15] showed that the valley current can be tuned by ferromagnetic exchange field. Wang et al. studied the magnetoresistance in a silicene-based asymmetrical magnetic tunnel junction and found out that the magnetoresistance of the system can be effectively modified by external electric field [16]. Yamakage et al. predicted

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the quantized conductance in p–n and n–p–n silicene junctions is almost to be 0, 1, and 2 [26]. However, the effect of the strain on the spin-dependent transport and the spin polarization properties in silicene with extrinsic Rashba spin–orbit interaction, has not yet been considered. In this paper, the spin transport in a silicene junction with extrinsic Rashba spin–orbit interaction (RSOI), in the presence of zigzag and armchair direction strain has been studied. We apply strain up to 20% and calculate the spin-dependent conductance, with and without spin flip. Spin-inversion can be efficiently controlled by extrinsic RSOI and strain strength. When the armchair direction strain is applied, the spin polarization current can be observed, whereas for the zigzag direction strain is zero. It is also observed that the spin polarization reaches to a maximum value when the strain is beyond 2%.

2. Model and theory We consider a silicene barrier with RSOI and uniaxial strain as shown in Fig. 1. The barrier region with the extrinsic RSOI strength λ R 2 = const is separated by a normal silicene in which there is no extrinsic RSOI interaction. The effect of strain on the Hamiltonian is given by the two-dimensional reductions of the strain tensor, which can be written as follows [32]

 cos 2 α − µ sin 2 α є= ε   (1 + µ ) cos α sin α

(1 + µ ) cos α sin α  , sin 2 α − µ cos 2 α 

(1)

where, α denotes the direction of applied strain with respect to the y axis. α = 0 and

α = π / 2 refer to strains along the zigzag and armchair directions, respectively. Poisson’s ratio µ for the silicene is 0.3 [33,34]. ε is the strain modulus inside the barrier and its value

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is zero otherwise. Using the single electron picture; the Hamiltonian of the charge carriers in our model is expressed by the following equation

(2)

Hˆ = Hˆ 0 + Hˆ 1 + Hˆ IRSO + Hˆ ERSO + V ( x ) Iˆ,

in which, Hˆ 0 =
v F U † (α )[σ x (1 − λ x ε ) q x + σ y (1 − λ y ε ) q y ]U (α ), Hˆ 1 = − (ηλ SO − ∆ z )τ z , Hˆ IRSO = aτ z λ R1 ( k y σ x − k xσ y ),

(3)

Hˆ ERSO = λ R 2 (τ xσ y − τ yσ x ),

U , V (x) =  0  0,

0< x
x≤L

where σˆ = (σ x , σ y ) and τˆ = (τ x ,τ y ,τ z ) are the real spin Pauli matrices and sublattice pseudospin

respectively,

v F ≈ 5.5 × 10 5 m / s

is

the

Fermi

velocity

in

silicene,

a ∂t ) ≈ 1.6 denotes the logarithmic derivative of the nearest2t ∂a

λ x = 2(κ 0 − 1 / 2) with κ 0 = ( )(

neighbor hopping t with respect to the lattice parameter (a=0.386 nm) at ε = 0 , λ y = −0.31 , and Iˆ is the 2×2 unit matrix. q x and q y are components of the quasi-particle wave vectors along the x and y direction, respectively. η = ±1 denotes the spin-up and spin-down, respectively. The first term in Hˆ 1 represents the effective spin–orbit coupling with λ SO = 3.9 meV [11,35]. The third term in Eq. (2) represents intrinsic Rashba spin–orbit coupling with

λ R1 = 0.7 meV. However, this term has a negligible effect on the energy dispersion relation [36] and we neglect it in our calculations. λ R 2 = λ R in Hˆ ERSO is extrinsic Rashba spin–orbit coupling, which is induced by external electric field. ∆ Z is the on-site potential difference between the A and B sublattices, which is tunable by an electric field applied perpendicular to the plane. In the normal region, we set ∆ z = V ( x) = 0. So, the strain segment is in contact 4

with the gate electrode. The barrier height U 0 is controlled by the gate voltage. U (α ) is the unitary matrix [37] 1 0  . U (α ) =  −iα  0 e 

(4)

Consider an electron with the spin of s, incident on the left side with energy E and incident angle of ϕ with respect to the k x axis in the reciprocal space. The general solutions of the Eq. (2) in the three regions can be represented as + − Ψ = aψ Ns + bψ Ns + cψ N+s ′ + dψ N−s ′ ,  + − + −  Ψ′ = a′ψ s + b′ψ s + c′ψ s ′ + d ′ψ s ′ ,

0< x< L x≤0

(5) or

x≥L

where s(s ′) = +1(−1) is the eigenvalue of the spin projection. a, b, c, d and a', b', c', d' represent the ampliudes of quasi-particles in the normal and barrier regions, respectively. ± Also ψ Ns andψ s± are the wave functions moving along the ± x , in the normal and barrier

i(±k x x +k y y )

ψ N± ↑ = ((±kx − iky ), (E + ηλSO) /
vF ,0, 0)e ψ

± N↓

= (0, 0, ( E −ηλSO ) /
vF , (±kx − ik y ))e

× A,

(i ± k x x + k y y )

(6)

× A,

2

A = 1/ 2([ k x + k y2 + ((E −ηλSO) /
vF )2 ) , kx = (E + ηλSO)(E −ηλSO ) /(
vF )2 − k y2 ,

ψ s±′(s ) = {[(±ks′(s) )(1 − λxε ) − (1 − λyε )iqy ],[ E −ηλSO + ∆Z − U0 ] /
vF , − is′(s)[E −ηλSO + ∆Z − U0 ] /
vF , − is′(s)(±ks′( s ) )[(1 − λxε ) + (1 − λyε )iqy ]}× e

i ( ± k s ′ ( s ) x + k y y)

(7)

Ds′( s) ,

2

Ds′( s ) = 1/ 2(( ks′( s ) × (1 − λxε )2 ) + (q2y × (1 − λxε )2 ) + ((E −ηλSO + ∆ Z − U0 ) /
vF )2 ) ,

with k s′( s ) = ( E − ηλ SO + ∆ z − U 0 )( E + ηλ SO − ∆ z − U 0 − s ′( s )λ R ) /
2 v F2 − (1 − λ y ε ) 2 q y2 /(1 − λ x ε ). (8) Where E is the energy of incident electron. k s′( s ) and k x = E cos ϕ /
vF are the wave vectors along

the

x

direction

in

the

barrier 5

and

well

regions

respectively,

while

q y = k y − [κ 0 × ε (1 + µ ) sin( 3α )] / a and k y = E sin ϕ /
vF is the wave vectors along the y

axis. After the transmission probability T s′s is derived from the wave function continuity, spin-dependent conductance at zero temperature is given by: π /2

Gs′s = G0

∫T

s′s

(ϕ ) cos(ϕ )dϕ ,

(9)

−π / 2

with G0 = e 2 k F L y / π
, where k F = E 2 − ηλ2SO and L y denotes the width of the barrier along the y direction. Also, T s ′s is the probability for an incident quasi-particle with the spin s =↑, ↓ to be transmitted with the spin s′ =↑, ↓ . Then, the spin polarization of conductance

can be defined by [38,39]

P=

G↑↑ + G↑↓ − G↓↑ − G↓↓ G↑↑ + G↑↓ + G↓↑ + G↓↓

(10)

.

3. Numerical result and discussion We perform numerical calculations under U 0 = 100 meV, E = 40 meV, ∆ z = 5 meV,

λ SO = 3.9 meV, λ R = 20 meV, and barrier width L= 100 nm, unless otherwise specified. The spin-dependent conductance for an incoming electron with the spin ↑ as a function of extrinsic RSOI strength is illustrated in Fig. 2(a). G↑↑ / G 0 is the conductance for collected the electrons with the same polarization, while the G↓↑ / G0 is for the outgoing electrons with opposite spins. As shown in Fig. 2(a) the spin-dependent conductance of the system has an oscillatory behavior with respect to λ R . Oscillation of conductance with respect to the extrinsic RSOI strength originates from the interference of two RSO spin-split electronic waves in one subband [40]. It can be seen clearly that, when one of G↑↑ / G 0 or G↓↑ / G0 has a maximum value at a certain λ R , the other has a minimum at the same λR . Also the maximum 6

(minimum) value of G↑↑ / G 0 or G↓↑ / G0 is very close to 1(0). In other words, by changing the λR , one can find an appropriate value of extrinsic Rashba constant at which a perfect spin-inversion can occur for silicene junction. In order to obtain proper values of extrinsic RSOI strength λ R for spin-inverter with high efficiency, the transmission probability with and without spin flip versus λ R and incident angle are plotted in Figs. 2(b) and 2(c). It is observed that, due to the interface of normal and barrier regions some resonant peaks (valleys) appear in T↑↑ and T↓↑ at some λ R . One can see that in a certain range of λ R the transmission of electron with the same spin as the incoming spin is impossible and electron can be transmitted through the system only with spin-inversion. In this case, perfect spininversion can occur for all incident angles. We assumed that the strain strength is set to zero, in Fig. 2. In Fig. 3(a), we show the spin-dependent conductance G↑↑ / G 0 (G↓↑ / G0 ) as a function of strain strength, when the zigzag direction strain (α = 0) is applied to silicene barrier. Similar to Fig. 2(a), the spin-dependent conductance has an oscillatory behavior with respect to the strain strength. The conductance with spin-inversion has maxima values at ε = 20 % and

ε = −20% . In this case, as shown in Figs. 3(b) and 3(c) electron can be transmitted through the silicene barrier only with spin flip from up to down. Thus, according to Figs. (2) and (3) conductance with complete spin-inversion can be obtained via adjusting λ R or ε . Therefore, the silicene barrier can invert the spin of electrons and can be used as a perfect spin-inverter or spin NOT gate. Figs. 4(a) and 4(b) show the spin polarization as a function of extrinsic RSOI strength and the barrier height, respectively with different values of strain strength, when the armchair direction strain (α = π / 2) is applied to silicene barrier. As it is obvious from Fig. 4, unlike the zigzag direction strain spin polarization rises by increasing the strain strength and reaches maximum value at ε = −2% or ε = 2% , and then decreases with further

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increase of strain strength. Furthermore, sign of the spin polarization can switch from positive to negative when the strain strength changes from positive to negative values. In other words, the spin-polarized current can be switched mechanically, without switching the external magnetic fields. To further understand this behavior, we calculated the spin polarization as a function of strain strength with several values of extrinsic RSOI strength. The result is plotted in Fig. 5. We observe that, the spin polarization is an odd function of the strain strength, P(ε ) = − P (−ε ) , with its amplitude and sign varying with ε . It is zero when only extrinsic

RSOI [ ε = 0 ] or zigzag direction strain (α = 0) is applied to silicene barrier. When both extrinsic RSOI and strain strength are finite P is also finite and it reaches its highest value at

ε = −2% or ε = 2%. Moreover, for ε ≠ −2% or ε ≠ 2% the spin polarization increases as a function of extrinsic RSOI strength. Thus, silicene generate a highly spin-polarized current when used as a tunnel barrier, even for an unpolarized injection.

4. Conclusion In conclusion, we have investigated the spin transport in a normal silicene/strained silicene/normal silicene junction. The extrinsic Rashba spin–orbit interaction is assumed to be induced into the strained region via local external electric field. Remarkably, it was found that the spin-dependent conductance, with and without spin flip can be controlled by extrinsic RSOI and strain strength. For appropriate values of extrinsic RSOI and strain strength the silicene junction can act as a perfect electron spin-inverter. We also found that the magnitude and sign of spin polarization can be tuned by local application of a gate voltage and strain. Also we have attained the condition for observing a nearly pure spin-polarized current. Thus, making the strained silicene junction with a region of finite extrinsic RSOI sandwiched between two normal regions is ideal for future spintronics applications.

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Figure captions:

Fig. 1. (a) Schematic of the setup of a normal silicene/strained silicene/normal silicene junction with extrinsic Rashba spin–orbit interaction and barrier potential in the strain region. (b) Honeycomb structure of silicene, with two sublattices A and B. Fig. 2. (a) The spin-dependent conductance without spin flip G↑↑ / G0 (blue solid line), and with spin flip G↓↑ / G0 (red dashed line) as a function of extrinsic RSOI strength for ε = 0. The spin transmission probabilities (a) T↑↑ and (b) T↓↑ as a function of RSOI and incident angle for ε = 0.

Fig. 3. (a) The spin-dependent conductance G↑↑ / G0 (blue solid line), and G↓↑ / G0 (red dashed line) as a function of strain strength. The spin transmission probabilities (a) T↑↑ and (b) T↓↑ as a function of strain strength and incident angle.

Fig. 4. The spin polarization as a function of (a) extrinsic RSOI strength and (b) barrier height with several values of the strain strength for the armchair direction strain.

Fig. 5. The spin polarization as a function of strain strength for the armchair direction strain with several values of extrinsic RSOI strength.

11

Fig. 1.

12

Fig. 2.

13

Fig. 3.

14

Fig. 4.

15

Fig. 5.

16

► We investigate the spin-dependent transport properties in a strained silicene junction. ► Under appropriate conditions, electrons can be transmitted only with spin-inversion. ► The silicene junction can be used as a perfect spin-inverter or spin NOT gate. ► The magnitude and sign of spin-polarized current can be switched mechanically. ► The spin polarization reaches a maximum value at 2% strain.

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