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The role of Rashba spin-orbit coupling in valley-dependent transport of Dirac fermions
Author’s Accepted Manuscript The role of Rashba spin-orbit coupling in valleydependent transport of Dirac fermions Kobra Hasanirok, Hakimeh Mohammadpour
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PII: DOI: Reference:
S0921-4526(16)30403-3 http://dx.doi.org/10.1016/j.physb.2016.09.004 PHYSB309627
To appear in: Physica B: Physics of Condensed Matter Received date: 6 June 2016 Revised date: 21 August 2016 Accepted date: 7 September 2016 Cite this article as: Kobra Hasanirok and Hakimeh Mohammadpour, The role of Rashba spin-orbit coupling in valley-dependent transport of Dirac fermions, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.09.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The role of Rashba spin-orbit coupling in valley-dependent transport of Dirac fermions Kobra Hasanirok, Hakimeh Mohammadpour Azarbaijan Shahid Madani University, 53714-161,Tabriz, Iran
Abstract: At this work, spin- and valley-dependent electron transport through graphene and silicene layers are studied in the presence of Rashba spin- orbit coupling. We find that the transport properties of the related ferromagnetic/normal/ferromagnetic structure depend on the relevant parameters. A fully valley- and spin- polarized current is obtained. As another result, Rashba spin-orbit interaction plays important role in controlling the transmission characteristics.
PACS No.: 72.80.Vp,73.23.-b, 73.22.Pr,73.63.-b. Keywords: spin and valley-dependent transport, graphene, silicene, Rashba spin-orbit coupling, ferromagnetic tunnel junctions, magneto-resistance.
1. Introduction Graphene has drained a great deal of interest from the fundamental and applied physics community [1-3]. It is a two-dimensional allotrope of carbon with honeycomb lattice in which the valence and conduction bands meet each other at Dirac points, K and K’. This zero gap semiconductor has many unique and fantastic properties such as half-integer quantum Hall effect [4, 6, 7], Klein tunneling [8-16] and minimum conductivity [5]. Controlling the spin-orbit interaction (SOI) plays a key role for application of graphenebased structures in spintronics [20-22]. Intrinsic SOI is weak in a free-standing graphene sheet [23] so we neglect it in this paper [41]. Rashba coupling (or extrinsic SOI) arises from the structural inversion asymmetry (SIA) of the system that is caused e.g. by applying gate voltage, curvature and substrate effects [24-26]. In order to induce ferromagnetic correlation in the thin layers like graphene and silicone a metallic ferromagnetic material is used as substrate [40].
1
Beside the charge and spin degrees of freedom of electron, pseudospin (sublattice) as well as the valley degrees of freedom on honeycomb lattices can be implemented [17-19].
Recently, a honeycomb structure of silicon named Silicene was experimentally synthesized [27-30] as bulked structure. Dirac electrons in the new material have mass which is controllable by applying external agents such as electric field [31] and photo-irradiation [32]. Spin-orbit gap in silicene is large compared with graphene. This large gap couples spin and valley degrees of freedom, so we expect interesting and attracting spin and valley- coupled physics in the new two- dimensional material. In graphene covered by ferromagnetic layer [33], energy band splits in two sub-bands, so spin-dependent current can pass through the hetrojunctions of ferromagnetic grapahene [42]. In ref. [37], spin current through graphene and silicene structures has been studied and it was shown that the spin current depends strongly on gate voltage. A fully valley and spinpolarized current in normal/(anti) ferromagnetic/normal silicene
junction was also
revealed. In this paper we study spin transports of Dirac fermions through heterojunctions having Rashba spin-orbit interaction. The calculations are based on the transfer matrix method by emphasis on the valley-dependent transmission through heterojunctions of Dirac systems, containing graphene and silicene. The transmission coefficients for fermions of different spin and valley flavors of the honeycomb systems are investigated.
This research suggests transport with controllable characteristics, by adjusting the Rashba coupling strength which can remove the valley degeneracy. We mainly focus on the valley degree of freedom. According to the results, this parameter can critically affect the transport through graphene- and silicene- based ferromagnetic-normal-ferromagnetic heterostructures in the presence of Rashba coupling.
2
2. Model and calculating method At this section we present the model Hamiltonian and the corresponding eigenfunctions to be used at the transfer matrix method calculations. This section is divided in two subsections which are devoted to graphene and silicene, respectively.
2.1. Graphene
First we consider a graphene sheet in which the massless quasi particles near Dirac points obey the following Dirac Hamiltonian; H iv F ( x x y y ) s0
where
(1)
is Planck constant and v F 10 6 m / s is the Fermi velocity in graphene. σ (σx , σy) is
Pauli matrix in the pseudo-spin space and s0 is a unit matrix at the spin subspace. stand for two different valleys, i.e.
and
and it is difficult to discriminate between
. There is valley degeneracy in pristine graphene and
valleys experimentally.
We consider a one dimensional arrangement of ferromagnetic/normal/ferromagnetic heterojunction on the infinite sized graphene sheet in which Rashba spin-orbit coupling is applied to the normal region. At the following we refer to the heterojunction as FG/RG/FG in which F, G and R represents ferromagnetic, graphene and Rashba layers, respectively. A schematic of the model is shown in Fig. 1. The Hamiltonian of Rashba SOI with strengths R is [37] HR =
R 2
( x s y y s x ) (2)
where s is the Pauli matrix in spin subspace. Equation 2 reveals that the RSOI is not degenerate for states with different valley indices.
3
Fig.1: A ferromagnetic/normal/ferromagnetic heterojunction on the two dimensional plane with Rashba spinorbit coupling applied to the normal region
In our proposed model the charge carriers of graphene
Exchange Hamiltonian at the ferromagnetic regions is written as; H h ( h') h (' ) 0 s z
(3)
where 0 is unit matrix at pseudospin space. As shown in Fig. 1, Rashba region with length L is sandwiched between two ferromagnetic layers. The interfaces along the assigned y-direction are located at x=0 and x=L. We assume an electron at the left ferromagnetic graphene, with spin-up and energy E, which is incident to the interface at x=0 and is transmitted to the right ferromagnetic layer as up and down spins by amplitudes t and t', respectively. By solving equation ̂
, the wave function at each region are in the following spinor
forms (the corresponding results for
can be simply extracted):
i 2 e e i ( ) / 2 0 i ( ) / 2 0 i e 2 ( x 0) exp(ik x x ik y y ) e r exp( ik x x ik y y ). r exp( ik x ' x ik y y ) i ( ') / 2 . 0 e 0 i ( ) / 2 0 e 0
(4) sin 1 (k tan / k ),
4
4
(0 x L ) j 1
Wj =
f q j E
q1
W j 1 B j exp(iq j x cos j ik y y ) X j Y j 2 f q j 2
e
i j
, Xj = ( E
E 2 R E v f
2
E
1 i R
q2
, q3 q1 ,
i e 2 i ( x L) t exp(ik x x ik y y ) e 2 0 0
)
, Y j= Wj× X j,
(5 ) E 2 R E v f
, q 4 q 2 ,
0 0 t exp( i k x ik y ) x y e i / 2 i / 2 e
(6)
and ( j tan 1 (k tan / q j ) ) are the incidence angles, and wave vector amplitudes are Eh , k E h , k E h . defined as, k E h , k v f
v f
v f
v f
r and r are reflection amplitudes for up and down spins in left ferromagnetic layer and Bj
(j=1-4) are coefficients of the spin-dependent wave function in Rashba region. From the translational symmetry in y direction, y-component of the momentum is conserved:
k y k F sin k F sin i i
(7)
The scattering amplitudes are obtained by matching the wave functions at the interfaces [24] as, (0 ) (0 ) and ( L ) ( L ) .
1 t 0 ( N 44 ) 1 M 44 ( x 0) M 414 ( x L) K 44 0 . t r 0 r
(8)
N, M and K are the matrix of eigenfunctions of first, middle and last graphene layers, respectively:
5
e ik . x e i / 2 e ik . x e i / 2 e ik . x e i ( ) / 2 0 0 0 e i k . x e i / 2 e ik . x e i ( ) / 2 0 0 K ik . x ik . x N= 0 0 e e i / 2 e e i ( ) / 2 , 0 i k . x i / 2 ik . x i ( ) / 2 0 0 0 e e e e . .
0 0 0 e 0
ik . x
0 0 e i / 2 0
0 0 0 0
e ik1 . xW e ik2 . x W e ik3 . xW e ik4 . xW4 1 2 3 i k 2 . xi ik 3 . x ik 4 . x e ik1 . xm e e e i k . x i k . x i k . x i k . x M(x)= e 1 X 1 e 2 X 2 e 3 X 3 e 4 X 4 (9) e ik1 . x Y e ik2 . x Y2 e ik3 . xm Y3 e ik4 x Y4 1 .
Valley and spin conductances (GK(‘) and G↑(↓) ) are defined as,
G s =
1 2
2
d cos T
s
( )
2
Gɳ= Gɳ↑ +Gɳ↓.
(10)
Then one can obtain the spin polarization in the following way, Gsɳ =Gɳ↑ -Gɳ↓ . (11)
Ts (T and T') are spin-dependent transmission probabilities where T= T ( ) t *t and T'= T ( ) t '* t ' with t and t' being the transmission amplitudes as spin up and spin down, respectively . The other parameter that we study here is the “mangetoresistance”. This term stands usually for the effect of external magnetic fields or intrinsic magnetization on the resistivity. However it is also referred to the contribution to the conductance of other spin-dependent interactions such as magnetic impurities and spin-orbit coupling [36]. In this paper, by employing the second definition, we have used "magnetoresistance" as the effect of SOI on the conductance of system, so it is defined as; MR( )
G( 0) G( ) G ( )
6
(12)
2.2 . Silicene At this section we consider a ferromagnetic/Rashba /ferromagnetic heterojunction on silicene, again in the arrangement of Fig. 1 (FS/RS/FS) where S stands for silicene. The Hamiltonian of silicene is described by [37] H iv F ( x x y y ) s0 (s so z hs s z ) z h z s0
(13)
Large ionic radius of silicone atoms distorts the honeycomb lattice and forms a buckled structure. We assume a silicene sheet in the x-y plane, and apply an electric field Ez perpendicular to the sheet. This field at the buckled honeycomb lattice leads to separation of sublattices, A and B and generates a staggered potential between silicon atoms at A and B sublattice sites. We assign ∆z to the potential difference between the two sublattices which is tunable by Ez. The large value of spin-orbit coupling, of about so =3.9 meV [39], makes a cross correlation between spin and valley degrees of freedom which is an outstanding departure from the corresponding graphene structure. The wave functions at the FS regions are obtained from Hamiltonian (13) but the results are not presented here for brevity. By using the T-Matrix approach and the same method explained at graphene section, we can calculate the transmission coefficient, spin coductance and spin polarization at Dirac points K and K’ in heterostructures of silicene.
3. Results and discussion The calculated transport properties in FG/RG/FG (graphene) structure are exhibited at the first part of this section and at the rest, the results for FS/RS/FS (silicene) are presented. We assume in both cases that the incoming electrons from the left ferromagnetic layer are polarized in up spin state with energy
with incidence angle
to the first intersection at
. We choose the units of the energies and lengths as the Fermi energy, [38] and Fermi wave vector,
, respectively. It should be mentioned that as denoted at the
bottom of the axes, the incidence angle is written in units of pi/3 at the depicted results.
Let us first study graphene structure and present the spin-depend transmission probabilities. In Fig. 2, the transmission probability without spin inversion, T, is plot as a function of the incidence angle, , for different Dirac points K and K’. Rashba coupling is not 7
invariant by
→-
and will give rise to asymmetry of T with respect to the angle, i.e.
T( )≠T(- ), which increases by
. As expected, T decreases by increasing incidence angle
however meanwhile the asymmetry acts in disturbing this behavior at about |
|= in
which T has the Maximum asymmetry. In the other hand, RSOI leads to a valley-dependent transport. As shown in the figure, TK’ has less values than TK except for
≈
in which the asymmetry of TK’ is the largest.
Fig. 2: Transmission coefficient T as a function of the incident angle =0.4,h'=h=-0.01, L=2, E=1).
for different Dirac points, K and K’ (λ
Valleytronic opens a rich parameter space for controlling the transport and may lead to development of the new quantum devices. The transmission Probabilities of the incoming spin-up electron with and without spin inversion, T’ and T, are plot versus
for Dirac point K′ in Fig. 3. The asymmetry induced By
Rashba coupling can affect the probability T' at all the angle range, however T is asymmetric just at about |
|= .
Fig. 3: Transmission coefficient T and T’ versus
for Dirac point K’ (λ =0.4,h'=h=-0.01, L=2, E=1). 8
It is deduced from the above diagrams that the transmission of electrons depends on both of
and Dirac point, so that the spin-current of the system can be effectively controlled.
In fig. 4 the spin conductances G↑ and G↓ are plot in terms of Rashba coupling strength, λ, for ferromagnetic exchange interactions h'=h=-0.01, at Rashba region length, and energy (we remind that the length and the energy are normalized to the Fermi wavelength and Fermi energy, respectively). These spin-depend parameters reveal different behaviors for carriers at different Dirac points.
(a) (b)
Fig. 4: Spin conductance G↑ and G↓ versus Rashba coupling strength λ for carriers at (a) K (b) K’ (h'=h=0.01, L=2, E=2).
The difference between curves clearly demonstrates the importance of Rashba coupling. By increasing the Rashba strength, electrons can’t stay at the up spin state. So the decrement rate of probability, T, results in decreasing (increasing) of the spin-conductances G↑ ( G↓) (Fig. 4a), while we have opposite behavior at K' (Fig. 4b). Magnetoresistance (MR) and valley-resolved conductances GK and GK’ can be effectively control by Rashba coupling, as expected.
9
Fig. 5: Valley resolved conductances GK and GK’ versus λ for h'=h=0.01, L=2, E=2.
Fig. 6: Valley-dependent Magneto Resistances, MRK and MRK' , versus λ for h'=h=0.01, L=2, E=2.
At this part of the results, we present the spin-dependent transport properties for silicene heterojunction, FS/RS/FS.
In the following, we assume that the Rashba layer is also
ferromagnetic, i.e. spin up energy sub-band is shifted vertically (along the energy axis) with respect to spin down energy sub-band. In fig.7 both spin-conserved and spin-flipped transport amplitudes are depicted versus angle,
, for (a) K and (b) K’. Transmission
probabilities, T and T', show behaviors different from the graphene case.
10
(a)
(b)
Fig. 7: MRK and MRK' as a function of
for ∆z =2, ∆=0.2, E=4, hR =-0.06,h=-0.01;h'=-0.04, λ =2, L=3.
As shown in Fig. b. the probability that electron keeps it’ s spin in transmission through the Rashba layer is very weak. In Fig. 8, is plot as a function of length of RSOI layer, , in valley. Rashba SOI coupling acts like an effective magnetic field, so the spins of the moving electrons along axis rotate in plane.
Fig. 8: Transmission coefficient, T as a function of λ for ∆z =2, ∆=0.6,
= , E=2, hR =-0.1,h=0.1;h'=-0.8
The transmission coefficients T and T’ are plot as functions of the Rashba strength λ for valley at Fig. 9.
11
Fig. 9: Transmission coefficients T and T' versus λ, for ∆z =2, ∆=0.5,
= , E=2, hR =-0.1,h=0.1;h'=-0.8, L=2.
By inspection of diagrams of Fig. 9 we clearly observe that by increasing the Rashba coefficient, T increases while T' decreases as mentioned for K' valley of graphene. So, the transmission properties are valley-dependent and may be totally changed for ɳ=+1 and ɳ=1. Beside the critical role played by Rashba constant, the incidence angle and the length of the Rashba layer can effectively control transport characteristics. The transmission coefficients with and without spin-inversion, T' and T, are plot versus λ for various values of and different Rashba region length, in Figs., 10 and 11, respectively.
Fig. 10: Transmission coefficient T' as a function of λ for ∆z =2, ∆=0.2, L=2,E=2, hR =-0.1,h=0.1;h'=-0.3.
Fig. 11: Transmission coefficient T as a function of the Rashba strength λ for ∆z =2, ∆=0.2, 0.1,h=0.1;h'=-0.2.
12
= , E=2, hR =-
These diagrams demonstrate that transmission coefficients.
λ,
and Dirac points are deeply involved in
In order to investigate the silicene structure parameters, the transmission, T, is plot as a function of the strength of the Rashba coupling, , and angle for different Dirac points in fig. 12.
(a)
(b)
Fig. 12: Transmission coefficient T as a function of λ for (a) ɳ=+1,∆z =1.5, ∆=0.2,E=2, hR =-0.1,h=-0.02;h'=-0.06, L=1.8, and for (b) ɳ=-1,∆z =2, ∆=0.3,E=2, hR =-0.06,h=-0.02;h'=-0.08, L=3.
As is shown at fig. 12a (ɳ=+1), incidence angle, , controls the transmission coefficient, T, in such a way that T decreases by increasing the magnitude of the angle, | |, meanwhile λ, plays a minor role. But fig. 12b (ɳ=-1) reveals that T depends strongly on both of and λ.
Finally we are going to consider how Rashba coupling can influence the spin conductances. The spin conductances, G↑ and G↓, are plot as a function of the Rashba SOI constant at Fig. 13. Comparing the main figure with the inset one, reveals that the increment for G↑ by λ is more pronounced than that for G↓.
13
Fig. 13: Spin conductances G↑ and G↓ as versus λ for ɳ=-1, ∆z =2, ∆=0.2, E=4, hR =-0.06, h=-0.01, h'=-0.04, L=3.
4. Conclusion In this work, by using the transfer matrix method, we studied spin- and valley-dependent transport of the chiral particles through the graphene-based as well as silicene-based ferromagnetic three-layers heterojunction in the presence of Rashba spin-orbit coupling. Results manifests that the transport properties of the systems depend critically on the valleys, Rashba strength and structural parameters such as length of the Rashba SOI region and incidence angle.
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www.elsevier.com/locate/physb
PII: DOI: Reference:
S0921-4526(16)30403-3 http://dx.doi.org/10.1016/j.physb.2016.09.004 PHYSB309627
To appear in: Physica B: Physics of Condensed Matter Received date: 6 June 2016 Revised date: 21 August 2016 Accepted date: 7 September 2016 Cite this article as: Kobra Hasanirok and Hakimeh Mohammadpour, The role of Rashba spin-orbit coupling in valley-dependent transport of Dirac fermions, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.09.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The role of Rashba spin-orbit coupling in valley-dependent transport of Dirac fermions Kobra Hasanirok, Hakimeh Mohammadpour Azarbaijan Shahid Madani University, 53714-161,Tabriz, Iran
Abstract: At this work, spin- and valley-dependent electron transport through graphene and silicene layers are studied in the presence of Rashba spin- orbit coupling. We find that the transport properties of the related ferromagnetic/normal/ferromagnetic structure depend on the relevant parameters. A fully valley- and spin- polarized current is obtained. As another result, Rashba spin-orbit interaction plays important role in controlling the transmission characteristics.
PACS No.: 72.80.Vp,73.23.-b, 73.22.Pr,73.63.-b. Keywords: spin and valley-dependent transport, graphene, silicene, Rashba spin-orbit coupling, ferromagnetic tunnel junctions, magneto-resistance.
1. Introduction Graphene has drained a great deal of interest from the fundamental and applied physics community [1-3]. It is a two-dimensional allotrope of carbon with honeycomb lattice in which the valence and conduction bands meet each other at Dirac points, K and K’. This zero gap semiconductor has many unique and fantastic properties such as half-integer quantum Hall effect [4, 6, 7], Klein tunneling [8-16] and minimum conductivity [5]. Controlling the spin-orbit interaction (SOI) plays a key role for application of graphenebased structures in spintronics [20-22]. Intrinsic SOI is weak in a free-standing graphene sheet [23] so we neglect it in this paper [41]. Rashba coupling (or extrinsic SOI) arises from the structural inversion asymmetry (SIA) of the system that is caused e.g. by applying gate voltage, curvature and substrate effects [24-26]. In order to induce ferromagnetic correlation in the thin layers like graphene and silicone a metallic ferromagnetic material is used as substrate [40].
1
Beside the charge and spin degrees of freedom of electron, pseudospin (sublattice) as well as the valley degrees of freedom on honeycomb lattices can be implemented [17-19].
Recently, a honeycomb structure of silicon named Silicene was experimentally synthesized [27-30] as bulked structure. Dirac electrons in the new material have mass which is controllable by applying external agents such as electric field [31] and photo-irradiation [32]. Spin-orbit gap in silicene is large compared with graphene. This large gap couples spin and valley degrees of freedom, so we expect interesting and attracting spin and valley- coupled physics in the new two- dimensional material. In graphene covered by ferromagnetic layer [33], energy band splits in two sub-bands, so spin-dependent current can pass through the hetrojunctions of ferromagnetic grapahene [42]. In ref. [37], spin current through graphene and silicene structures has been studied and it was shown that the spin current depends strongly on gate voltage. A fully valley and spinpolarized current in normal/(anti) ferromagnetic/normal silicene
junction was also
revealed. In this paper we study spin transports of Dirac fermions through heterojunctions having Rashba spin-orbit interaction. The calculations are based on the transfer matrix method by emphasis on the valley-dependent transmission through heterojunctions of Dirac systems, containing graphene and silicene. The transmission coefficients for fermions of different spin and valley flavors of the honeycomb systems are investigated.
This research suggests transport with controllable characteristics, by adjusting the Rashba coupling strength which can remove the valley degeneracy. We mainly focus on the valley degree of freedom. According to the results, this parameter can critically affect the transport through graphene- and silicene- based ferromagnetic-normal-ferromagnetic heterostructures in the presence of Rashba coupling.
2
2. Model and calculating method At this section we present the model Hamiltonian and the corresponding eigenfunctions to be used at the transfer matrix method calculations. This section is divided in two subsections which are devoted to graphene and silicene, respectively.
2.1. Graphene
First we consider a graphene sheet in which the massless quasi particles near Dirac points obey the following Dirac Hamiltonian; H iv F ( x x y y ) s0
where
(1)
is Planck constant and v F 10 6 m / s is the Fermi velocity in graphene. σ (σx , σy) is
Pauli matrix in the pseudo-spin space and s0 is a unit matrix at the spin subspace. stand for two different valleys, i.e.
and
and it is difficult to discriminate between
. There is valley degeneracy in pristine graphene and
valleys experimentally.
We consider a one dimensional arrangement of ferromagnetic/normal/ferromagnetic heterojunction on the infinite sized graphene sheet in which Rashba spin-orbit coupling is applied to the normal region. At the following we refer to the heterojunction as FG/RG/FG in which F, G and R represents ferromagnetic, graphene and Rashba layers, respectively. A schematic of the model is shown in Fig. 1. The Hamiltonian of Rashba SOI with strengths R is [37] HR =
R 2
( x s y y s x ) (2)
where s is the Pauli matrix in spin subspace. Equation 2 reveals that the RSOI is not degenerate for states with different valley indices.
3
Fig.1: A ferromagnetic/normal/ferromagnetic heterojunction on the two dimensional plane with Rashba spinorbit coupling applied to the normal region
In our proposed model the charge carriers of graphene
Exchange Hamiltonian at the ferromagnetic regions is written as; H h ( h') h (' ) 0 s z
(3)
where 0 is unit matrix at pseudospin space. As shown in Fig. 1, Rashba region with length L is sandwiched between two ferromagnetic layers. The interfaces along the assigned y-direction are located at x=0 and x=L. We assume an electron at the left ferromagnetic graphene, with spin-up and energy E, which is incident to the interface at x=0 and is transmitted to the right ferromagnetic layer as up and down spins by amplitudes t and t', respectively. By solving equation ̂
, the wave function at each region are in the following spinor
forms (the corresponding results for
can be simply extracted):
i 2 e e i ( ) / 2 0 i ( ) / 2 0 i e 2 ( x 0) exp(ik x x ik y y ) e r exp( ik x x ik y y ). r exp( ik x ' x ik y y ) i ( ') / 2 . 0 e 0 i ( ) / 2 0 e 0
(4) sin 1 (k tan / k ),
4
4
(0 x L ) j 1
Wj =
f q j E
q1
W j 1 B j exp(iq j x cos j ik y y ) X j Y j 2 f q j 2
e
i j
, Xj = ( E
E 2 R E v f
2
E
1 i R
q2
, q3 q1 ,
i e 2 i ( x L) t exp(ik x x ik y y ) e 2 0 0
)
, Y j= Wj× X j,
(5 ) E 2 R E v f
, q 4 q 2 ,
0 0 t exp( i k x ik y ) x y e i / 2 i / 2 e
(6)
and ( j tan 1 (k tan / q j ) ) are the incidence angles, and wave vector amplitudes are Eh , k E h , k E h . defined as, k E h , k v f
v f
v f
v f
r and r are reflection amplitudes for up and down spins in left ferromagnetic layer and Bj
(j=1-4) are coefficients of the spin-dependent wave function in Rashba region. From the translational symmetry in y direction, y-component of the momentum is conserved:
k y k F sin k F sin i i
(7)
The scattering amplitudes are obtained by matching the wave functions at the interfaces [24] as, (0 ) (0 ) and ( L ) ( L ) .
1 t 0 ( N 44 ) 1 M 44 ( x 0) M 414 ( x L) K 44 0 . t r 0 r
(8)
N, M and K are the matrix of eigenfunctions of first, middle and last graphene layers, respectively:
5
e ik . x e i / 2 e ik . x e i / 2 e ik . x e i ( ) / 2 0 0 0 e i k . x e i / 2 e ik . x e i ( ) / 2 0 0 K ik . x ik . x N= 0 0 e e i / 2 e e i ( ) / 2 , 0 i k . x i / 2 ik . x i ( ) / 2 0 0 0 e e e e . .
0 0 0 e 0
ik . x
0 0 e i / 2 0
0 0 0 0
e ik1 . xW e ik2 . x W e ik3 . xW e ik4 . xW4 1 2 3 i k 2 . xi ik 3 . x ik 4 . x e ik1 . xm e e e i k . x i k . x i k . x i k . x M(x)= e 1 X 1 e 2 X 2 e 3 X 3 e 4 X 4 (9) e ik1 . x Y e ik2 . x Y2 e ik3 . xm Y3 e ik4 x Y4 1 .
Valley and spin conductances (GK(‘) and G↑(↓) ) are defined as,
G s =
1 2
2
d cos T
s
( )
2
Gɳ= Gɳ↑ +Gɳ↓.
(10)
Then one can obtain the spin polarization in the following way, Gsɳ =Gɳ↑ -Gɳ↓ . (11)
Ts (T and T') are spin-dependent transmission probabilities where T= T ( ) t *t and T'= T ( ) t '* t ' with t and t' being the transmission amplitudes as spin up and spin down, respectively . The other parameter that we study here is the “mangetoresistance”. This term stands usually for the effect of external magnetic fields or intrinsic magnetization on the resistivity. However it is also referred to the contribution to the conductance of other spin-dependent interactions such as magnetic impurities and spin-orbit coupling [36]. In this paper, by employing the second definition, we have used "magnetoresistance" as the effect of SOI on the conductance of system, so it is defined as; MR( )
G( 0) G( ) G ( )
6
(12)
2.2 . Silicene At this section we consider a ferromagnetic/Rashba /ferromagnetic heterojunction on silicene, again in the arrangement of Fig. 1 (FS/RS/FS) where S stands for silicene. The Hamiltonian of silicene is described by [37] H iv F ( x x y y ) s0 (s so z hs s z ) z h z s0
(13)
Large ionic radius of silicone atoms distorts the honeycomb lattice and forms a buckled structure. We assume a silicene sheet in the x-y plane, and apply an electric field Ez perpendicular to the sheet. This field at the buckled honeycomb lattice leads to separation of sublattices, A and B and generates a staggered potential between silicon atoms at A and B sublattice sites. We assign ∆z to the potential difference between the two sublattices which is tunable by Ez. The large value of spin-orbit coupling, of about so =3.9 meV [39], makes a cross correlation between spin and valley degrees of freedom which is an outstanding departure from the corresponding graphene structure. The wave functions at the FS regions are obtained from Hamiltonian (13) but the results are not presented here for brevity. By using the T-Matrix approach and the same method explained at graphene section, we can calculate the transmission coefficient, spin coductance and spin polarization at Dirac points K and K’ in heterostructures of silicene.
3. Results and discussion The calculated transport properties in FG/RG/FG (graphene) structure are exhibited at the first part of this section and at the rest, the results for FS/RS/FS (silicene) are presented. We assume in both cases that the incoming electrons from the left ferromagnetic layer are polarized in up spin state with energy
with incidence angle
to the first intersection at
. We choose the units of the energies and lengths as the Fermi energy, [38] and Fermi wave vector,
, respectively. It should be mentioned that as denoted at the
bottom of the axes, the incidence angle is written in units of pi/3 at the depicted results.
Let us first study graphene structure and present the spin-depend transmission probabilities. In Fig. 2, the transmission probability without spin inversion, T, is plot as a function of the incidence angle, , for different Dirac points K and K’. Rashba coupling is not 7
invariant by
→-
and will give rise to asymmetry of T with respect to the angle, i.e.
T( )≠T(- ), which increases by
. As expected, T decreases by increasing incidence angle
however meanwhile the asymmetry acts in disturbing this behavior at about |
|= in
which T has the Maximum asymmetry. In the other hand, RSOI leads to a valley-dependent transport. As shown in the figure, TK’ has less values than TK except for
≈
in which the asymmetry of TK’ is the largest.
Fig. 2: Transmission coefficient T as a function of the incident angle =0.4,h'=h=-0.01, L=2, E=1).
for different Dirac points, K and K’ (λ
Valleytronic opens a rich parameter space for controlling the transport and may lead to development of the new quantum devices. The transmission Probabilities of the incoming spin-up electron with and without spin inversion, T’ and T, are plot versus
for Dirac point K′ in Fig. 3. The asymmetry induced By
Rashba coupling can affect the probability T' at all the angle range, however T is asymmetric just at about |
|= .
Fig. 3: Transmission coefficient T and T’ versus
for Dirac point K’ (λ =0.4,h'=h=-0.01, L=2, E=1). 8
It is deduced from the above diagrams that the transmission of electrons depends on both of
and Dirac point, so that the spin-current of the system can be effectively controlled.
In fig. 4 the spin conductances G↑ and G↓ are plot in terms of Rashba coupling strength, λ, for ferromagnetic exchange interactions h'=h=-0.01, at Rashba region length, and energy (we remind that the length and the energy are normalized to the Fermi wavelength and Fermi energy, respectively). These spin-depend parameters reveal different behaviors for carriers at different Dirac points.
(a) (b)
Fig. 4: Spin conductance G↑ and G↓ versus Rashba coupling strength λ for carriers at (a) K (b) K’ (h'=h=0.01, L=2, E=2).
The difference between curves clearly demonstrates the importance of Rashba coupling. By increasing the Rashba strength, electrons can’t stay at the up spin state. So the decrement rate of probability, T, results in decreasing (increasing) of the spin-conductances G↑ ( G↓) (Fig. 4a), while we have opposite behavior at K' (Fig. 4b). Magnetoresistance (MR) and valley-resolved conductances GK and GK’ can be effectively control by Rashba coupling, as expected.
9
Fig. 5: Valley resolved conductances GK and GK’ versus λ for h'=h=0.01, L=2, E=2.
Fig. 6: Valley-dependent Magneto Resistances, MRK and MRK' , versus λ for h'=h=0.01, L=2, E=2.
At this part of the results, we present the spin-dependent transport properties for silicene heterojunction, FS/RS/FS.
In the following, we assume that the Rashba layer is also
ferromagnetic, i.e. spin up energy sub-band is shifted vertically (along the energy axis) with respect to spin down energy sub-band. In fig.7 both spin-conserved and spin-flipped transport amplitudes are depicted versus angle,
, for (a) K and (b) K’. Transmission
probabilities, T and T', show behaviors different from the graphene case.
10
(a)
(b)
Fig. 7: MRK and MRK' as a function of
for ∆z =2, ∆=0.2, E=4, hR =-0.06,h=-0.01;h'=-0.04, λ =2, L=3.
As shown in Fig. b. the probability that electron keeps it’ s spin in transmission through the Rashba layer is very weak. In Fig. 8, is plot as a function of length of RSOI layer, , in valley. Rashba SOI coupling acts like an effective magnetic field, so the spins of the moving electrons along axis rotate in plane.
Fig. 8: Transmission coefficient, T as a function of λ for ∆z =2, ∆=0.6,
= , E=2, hR =-0.1,h=0.1;h'=-0.8
The transmission coefficients T and T’ are plot as functions of the Rashba strength λ for valley at Fig. 9.
11
Fig. 9: Transmission coefficients T and T' versus λ, for ∆z =2, ∆=0.5,
= , E=2, hR =-0.1,h=0.1;h'=-0.8, L=2.
By inspection of diagrams of Fig. 9 we clearly observe that by increasing the Rashba coefficient, T increases while T' decreases as mentioned for K' valley of graphene. So, the transmission properties are valley-dependent and may be totally changed for ɳ=+1 and ɳ=1. Beside the critical role played by Rashba constant, the incidence angle and the length of the Rashba layer can effectively control transport characteristics. The transmission coefficients with and without spin-inversion, T' and T, are plot versus λ for various values of and different Rashba region length, in Figs., 10 and 11, respectively.
Fig. 10: Transmission coefficient T' as a function of λ for ∆z =2, ∆=0.2, L=2,E=2, hR =-0.1,h=0.1;h'=-0.3.
Fig. 11: Transmission coefficient T as a function of the Rashba strength λ for ∆z =2, ∆=0.2, 0.1,h=0.1;h'=-0.2.
12
= , E=2, hR =-
These diagrams demonstrate that transmission coefficients.
λ,
and Dirac points are deeply involved in
In order to investigate the silicene structure parameters, the transmission, T, is plot as a function of the strength of the Rashba coupling, , and angle for different Dirac points in fig. 12.
(a)
(b)
Fig. 12: Transmission coefficient T as a function of λ for (a) ɳ=+1,∆z =1.5, ∆=0.2,E=2, hR =-0.1,h=-0.02;h'=-0.06, L=1.8, and for (b) ɳ=-1,∆z =2, ∆=0.3,E=2, hR =-0.06,h=-0.02;h'=-0.08, L=3.
As is shown at fig. 12a (ɳ=+1), incidence angle, , controls the transmission coefficient, T, in such a way that T decreases by increasing the magnitude of the angle, | |, meanwhile λ, plays a minor role. But fig. 12b (ɳ=-1) reveals that T depends strongly on both of and λ.
Finally we are going to consider how Rashba coupling can influence the spin conductances. The spin conductances, G↑ and G↓, are plot as a function of the Rashba SOI constant at Fig. 13. Comparing the main figure with the inset one, reveals that the increment for G↑ by λ is more pronounced than that for G↓.
13
Fig. 13: Spin conductances G↑ and G↓ as versus λ for ɳ=-1, ∆z =2, ∆=0.2, E=4, hR =-0.06, h=-0.01, h'=-0.04, L=3.
4. Conclusion In this work, by using the transfer matrix method, we studied spin- and valley-dependent transport of the chiral particles through the graphene-based as well as silicene-based ferromagnetic three-layers heterojunction in the presence of Rashba spin-orbit coupling. Results manifests that the transport properties of the systems depend critically on the valleys, Rashba strength and structural parameters such as length of the Rashba SOI region and incidence angle.
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