Cooperative phenomenon in a rippled graphene: Chiral spin guide
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Cooperative phenomenon in a rippled graphene: Chiral spin guide
M. Pudlak, K.N. Pichugin, and R.G. Nazmitdinov , Institute of Experimental Physics, 04001 Kosice, Slovak Republic Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia (Dated: February 27, 2018)We analyze spin scattering in ballistic transport of electrons through a ripple at a normal incidenceof an electron flow. The model of a ripple consists of a curved graphene surface in the form ofan arc of a circle connected from the left-hand and right-hand sides to two flat graphene sheets.At certain conditions the curvature induced spin-orbit coupling creates a transparent window forincoming electrons with one spin polarization simultaneously with a backscattering of those withopposite polarization. This window is equally likely transparent for electrons with spin up and spindown that move in opposite directions. The spin filtering effect being small in one ripple becomesprominent with the increase of N consequently connected ripples that create a graphene sheet of thesinusoidal type. We present the analytical expressions for spin up (down) transmission probabilitiesas a function of N connected ripples. PACS numbers: 72.25.-b,71.70.Ej,73.23.Ad
I. INTRODUCTION
The extraordinary properties of graphene have at-tracted enormous experimental and theoretical attentionfor a decade (see e.g. Refs.1,2). Graphene being a zero-gap semiconductor has a band structure described by alinear dispersion relation at low energy, similar to mass-less Dirac-Weyl fermions. Such a band structure leads toexceptionally high mobility of charged carriers. A ques-tion of possible mechanisms that would allow us to throt-tle the mobility and, consequently, to control a conduc-tivity is a topical subject in graphene physics, due to itsfundamental as well as technological significance.Among various mechanisms that might affect the mo-bility, the scattering that could be induced by a ripple(see, for example, discussion in Ref.3) appears to be themost natural one, since graphene sheets are not perfectlyflat. Moreover, periodic ripples can be created and con-trolled in suspended graphene, in particular, by thermaltreatment and by placing graphene in a especially pre-pared substrate. Indeed, curvature of the surface affectsthe π orbitals that determine the electronic properties ofgraphene. It results in enhancement of spin-orbit cou-pling that could serve as a source of spin scattering. Werecall that the intrinsic (intraatomic) spin-orbit interac-tion in flat graphene is weak . It makes spin deco-herence in such a material weak as well, i.e., scatteringdue to disorder is supposed to be unimportant. In orderto get deep insight into the nature of curvature inducedscattering, it is desirable to elucidate among many ques-tions the basic one: What are the distinctive features ofcurvature induced spin-orbit coupling ? One can furtherask how to employ these features to guide an electrontransport in a graphene-based system at the theoretical,and, quite likely, practical levels.A consistent approach to introduce the curvatureinduced spin-orbit coupling (SOC) in the low energy physics of graphene have been developed by Ando andby others in the framework of effective mass and tight-binding approximations. Recent measurements in ul-tra clean carbon nanotubes (CNTs) , i.e., in an ex-treme form of curved graphene, revealed the energysplitting that can be associated with spin-orbit cou-pling. The measured shifts are compatible with theo-retical predictions , while some features regarding thecontribution of different spin-orbit terms in metallic andnon-metallic CNTs are still debatable (see, for example,discussion in ). Nowadays, nevertheless, there is aconsensus that for armchair CNTs one obtains two SOCterms: one preserves the spin symmetry (a spin projec-tion on the CNT symmetry axis), while the second onebreaks this symmetry . Thus, we have a reliableanswer to the first question, at least, for armchair CNTs.In some previous studies the role played by thesecond term was underestimated. In this paper we willattempt to answer how full curvature induced SOC, in-cluding the second term, could be used to create a po-larized spin current with a high efficiency in a rippledgraphene system.The structure of this paper is as follows. In Sec. IIwe briefly discuss the explicit expressions for the eigenspectrum and eigenfunctions of an armchair nanotubewith a full curvature induced spin-orbit coupling. Bymeans of these results we introduce a scattering modelfor one ripple and extend this model for N continuouslyconnected ripples. In Sec. III we provide a discussionof our results in terms of simple estimates. The mainconclusions are summarized in Sec. IV. II. SCATTERING PROBLEM
In order to model a scattering problem on a ripple weconsider a curved surface in the form of an arc of a circleconnected from the left-hand and right-hand sides to twoflat graphene sheets. The solution for flat graphene iswell known . The solution for a curved graphene sur-face can be expressed in terms of the results obtained forarmchair CNTs in an effective mass approximation, whenonly the interaction between nearest neighbor atoms istaken into account . A. Low energy spectrum of the armchair nanotube
Let us recapitulate the major results in the vicinityof the Fermi level E = 0 for a point K in the presence ofthe curvature induced spin-orbit interaction in an arm-chair CNT. The y axis is chosen as the symmetry and thequantizations axis. In this case the eigenvalue problemis defined asˆ H Ψ = (cid:18) f ˆ f † (cid:19) (cid:18) F KA F KB (cid:19) = E (cid:18) F KA F KB (cid:19) , (1)with the following definitions:ˆ f = γ (ˆ k x − iˆ k y ) + i δγ ′ R ˆ σ x ( ~r ) − δγpR ˆ σ y , (2)ˆ k x = − i ∂R∂θ , ˆ k y = − i ∂∂y , ˆ σ x ( ~r ) = ˆ σ x cos θ − ˆ σ z sin θ . Here, ˆ σ x,y,z are standard Pauli matrices, and the spinorsof two sub-lattices are F KA = (cid:18) F KA, ↑ F KA, ↓ (cid:19) , F KB = (cid:18) F KB, ↑ F KB, ↓ (cid:19) . (3)The following notations are used: γ = −√ V πpp a/ γ a , γ ′ = √ V σpp − V πpp ) a/ γ a , p = 1 − γ ′ / γ (seee.g. Ref.6). The quantities V σpp and V πpp are the trans-fer integrals for σ and π orbitals, respectively in a flatgraphene; a = √ d ≃ .
46 ˚A is the length of the prim-itive translation vector, where d is the distance betweenatoms in the unit cell.The intrinsic source of the SOC δ = ∆ / (3 ǫ πσ ) is de-fined as ∆ = i 3¯ h m e c h x l | ∂V∂x ˆ p y − ∂V∂y ˆ p x | y l i , (4)where V is the atomic potential and ǫ πσ = ǫ π p − ǫ σ p . Theenergy ǫ σ p is the energy of σ -orbitals, localized betweencarbon atoms. The energy ǫ π p is the energy of π -orbitals,directed perpendicular to the curved surface.By means of the unitary transformationˆ U ( θ ) = exp(i θ σ y ) ⊗ I , (5)where I is 2 × θ depen-dence in Hamiltonian (1), transformed in the intrinsic frame, and obtainsˆ H ′ = ˆ U ( θ ) ˆ H ˆ U − ( θ ) = ˆ H kin + ˆ H SOC , (6)ˆ H kin = − i γ (cid:18) ˆ τ y ⊗ I∂ y + ˆ τ x ⊗ I R ∂ θ (cid:19) , ˆ H SOC = − λ y ˆ τ y ⊗ ˆ σ x − λ x ˆ τ x ⊗ ˆ σ y . Here the operators ˆ τ x,y,z are the Pauli matrices that acton the wave functions of A- and B-sub-lattices (a pseudo-spin space), and λ x = γ (1 + 4 δp ) / (2 R ) , λ y = δγ ′ / (4 R ) (7)are the strengths of the SOC terms. In the Hamiltonian(6) the term ( ∼ λ x ) conserves, while the other one ( ∼ λ y )breaks the spin symmetry in the armchair CNT.The operator ˆ J y , being an integral of motion h ˆ H, ˆ J y i =0, is defined in the laboratory frame asˆ J y = I ⊗ (cid:18) ˆ L y + ˆ σ y (cid:19) = I ⊗ (cid:18) − i ∂ θ + ˆ σ y (cid:19) , (8)while in the intrinsic frame it isˆ J y → ˆ J ′ y = ˆ U ˆ J y ˆ U − = I ⊗ ( − i ∂ θ ) . (9)This integral allows to present the wave functions as F ′ ( θ, y ) = e i mθ e i k y y Ψ = e i mθ e i k y y ABCD . (10)These wavefunctions are also the eigenfunctions of theother integral of motion, the operator ˆ k ′ y ≡ ˆ k y . Here, m = ± / , ± / , .... is an eigenvalue of the angular mo-mentum operator ˆ J ′ y . For the components of the eigen-vector F ′ ( θ, y ) the relations | A | = | D | and | B | = | C | arefulfilled at real values of m and k y .Solving the eigenvalue problem ˆ H ′ F ′ = EF ′ , one ob-tains the eigen spectrum E = ± E m,q , E m,q = q t m + t y + λ y + λ x + 2 D m,q ,D m,q = q q λ x (cid:0) t m + λ y (cid:1) + t y λ y , q = ± , (11)where t m = γm/R , t y = γk y . B. Scattering model for one ripple
Keeping in mind a discussion that will be given here-after, we analyze the following geometry (see the con-struction profile on Fig.1). It consists of one arc of acircle that is connected from the left-hand side to a flatgraphene sheet. This (direct) arc is continuously con-nected to the inverse arc of the same radius that is con-nected to the right-hand flat graphene sheet. We put the φ θ R L RI II θ z x FIG. 1: The rippled graphene system. origin of the coordinate at the center of the direct arc ofthe circle.To give better insight into the scattering phenomenonin our model of a ripple, we study first only the directarc of the circle connected to two flat surfaces. Twoflat surfaces are: i)the region L, defined in the intervals −∞ < x < − R cos θ ; the region R, defined in the inter-vals R cos θ < x < ∞ . The region I is a part of a nan-otube of radius R , defined as − R cos θ < x < R cos θ .At θ = 0, the ripple is a half of the nanotube, while at θ = π/ φ = π − θ . To describe thescattering phenomenon one has to define wave functionsin different regions: flat (L,R) and curved (I) graphenesurfaces.Regions L and R are described by the Hamiltonianˆ H = γ (ˆ τ x k x + ˆ τ y k y ) ⊗ I (12)that does not mix spin components. For the sake of sim-plicity, we consider the electron motion at the normal in-cidence, with the electron wave vector ~k = ( k x , H Ψ = E Ψand obtains the corresponding eigenstates E = ± γ | k x | , (13)Ψ = exp (i k x x )Ψ σ ( k x ) , (14)Ψ σ ( k x ) = (cid:18) sign( γk x E )Φ σ Φ σ (cid:19) , (15)ˆ σ y Φ σ = σ Φ σ , Φ σ = (cid:18) σ (cid:19) , σ = ± . (16)Evidently, the wave functions in regions L, R, can bewritten as a superposition of all possible solutions for flatgraphene. To proceed, with the aid of eigenspinors (15),(16), we introduce an auxiliary matrix ˆ M (4 ×
4) for agiven value of energy at the normal incidenceˆ M = (cid:0) Ψ +10 ( K x ) , Ψ − ( K x ) , Ψ +10 ( − K x ) , Ψ − ( − K x ) (cid:1) . (17)Here, we define the variable K x = sign( γE ) | k x | to ensurethat the first two columns of the matrix M correspondto eigenstates that move in a positive x -direction, whilethe last two columns correspond to eigenstates that movein a negative x -direction. m -1 0 1 E n e r g y ( e V ) m m − FIG. 2: (Color online) The spectrum (19) at k y = 0 as afunction of the quantum number m . Dashed (red) and solid(blue) lines are associated with states characterized by thequantum number m s = − and m s =+1 , respectively. The val-ues of ± m s at the energy E F = 0 . R = 10˚A, δ = 0 . p = 0 . γ = . γ ′ = γ , λ x = γR (1 / δp ) = 0 . λ y = δγ ′ R = 0 . The matrix ˆ M is unitary, i.e., ˆ M − = ˆ M † . It allowsus to define a general form of the wave function Ψ L,R fora flat grapheneΨ
L,R ( x ) = ˆ M exp (cid:16) i ˆ K ( x − x L,R ) (cid:17) C L,R , (18)where ˆ K = diag( K x , K x , − K x , − K x ) is a diagonal ma-trix, x L,R are x -coordinates where flat and curved sur-faces are connected, and C L,R are corresponding vectorswith four unknown yet, normalized coefficients in eachregion. Note that we do not consider inelastic scattering.Therefore, since the electron energy is conserved, we usethe same vector ~k = ( k x ,
0) for the left and right flatgraphene surfaces.For the curved surface we use eigenspinors of theHamiltonian (6). The general form of these eigenspinorsis defined in the intrinsic frame . Therefore, we applythe inverse transformation (5) to these eigenspinors in or-der to analyze the scattering problem in the laboratoryframe. At k y = 0 the spectrum (11) and the eigenspinorsare particularly simple E A = ± q t m + λ y + sλ x , s = ± , (19)Ψ = exp (i mθ ) Ψ sA ( t m ) , (20)Ψ sA ( t m ) = (cid:18) − s Φ sA ( t m ) σ y Φ sA ( t m ) (cid:19) , (21)Φ sA ( t m ) = exp (cid:0) − i θ ˆ σ y (cid:1) (cid:18) − i t m λ y − ( sE A − λ x ) (cid:19) . (22)Note that energies in flat graphene and in a curved sur-face are different E = E A + A ( aR ) (see details in Ref.18).This effect is caused by different hybridizations of π elec-trons in flat graphene and a graphene–based system withcurvature. In particular, A = 5 / , /
12 (eV) in the arm-chair and zig-zag nanotubes, respectively.At a fixed energy of the electron flow E ⇔ E A ,Eq.(19) yields four possible values of the quantum num-ber mm ⇒ m s = ± Rγ q ( sE A − λ x ) − λ y , s = ± . (23)Since the angular momentum is not longer the integral ofmotion, we have to consider the mixture of eigenfunctionswith all possible values of m at a given energy.As an example of the spectrum (19), a few positiveenergy branches as a function of the quantum number m are shown in Fig.2. The branches are distinguishedby the index s = ±
1. There is an anti-crossing effectbetween energy states characterized by the same m s =+1 quantum number. This anticrossing is brought about bythe interaction ( ∼ λ y ) that breaks the spin symmetry(see Sect.IIA) in the curved graphene surface. It resultsin a gap of = 2 λ y near E A = λ x indicated by the ar-row (see the inset on Fig.2). Similar gap occurs near E A = − λ x for the m -states with index s = −
1. As aconsequence of these gaps, evanescent modes arise at en-ergies λ x − λ y < | E | < λ x + λ y in our system. For thesake of illustration the positive spectrum (19) of m -statesis crossed by the horizontal line that mimics the Fermienergy. The crossing points determine quantum num-bers m that have non-quantized values when the curvedsurface (arc of circle) is connected to the flat one.With the aid of eigenspinors (21), (22), and the unitarytransformation (5), we introduce an auxiliary matrix fora given value of energy at a curved surfaceˆ M A ( θ ) = (cid:0) Ψ A ( t m ) , Ψ − A ( t m − ) , Ψ A ( − t m ) , Ψ − A ( − t m − ) (cid:1) == U ( − θ ) ˆ M A (0) . (24)As a result, in region I the wave function can be writ-ten as a superposition of all solutions for a curved sur-face in the form Ψ I ( θ ) = ˆ M A ( θ ) exp(i ˆ mθ ) C I . Here, C I is a vector of four unknown coefficients, ˆ m =diag( m , m − , − m , − m − ) is a diagonal matrix.The overlap of eigenspinors of the flat and bended re-gions can be readily calculated with the aid of Eqs.(15),(21), which results in(Ψ σ ) † (Ψ sA ) ≃ ( − sign( γk x E ) s + σ ) × (25) × (Φ σ ) † exp (cid:0) − i θ ˆ σ y (cid:1) (Φ sA ) . Evidently, the overlap is zero at σ = sign( γk x E ) s . Notethat already this result implies that some of the fourchannels between the flat and curved regions could beclosed.Matching the wave functions at the boundaries of re-gions L, I, and R, for an incoming electron flow from the left-hand side, leads us to the following equationsΨ L ( x L ) = Ψ I ( − φ/ ⇒ ˆ M (cid:18) Φ in r (cid:19) = ˆ M A ( − φ/ C I , (26)Ψ R ( x R ) = Ψ I (+ φ/ ⇒ ˆ M (cid:18) t (cid:19) = ˆ M A (+ φ/
2) exp(i ˆ mφ ) C I . (27)We recall that the angles θ and φ determine the x L,R coordinates: x L = R cos( θ + φ ), x R = R cos θ . Here, t = (cid:18) t ( L ) in ↑ t ( L ) in ↓ (cid:19) and r = (cid:18) r ( L ) in ↑ r ( L ) in ↓ (cid:19) are transmission andreflection coefficients, respectively, for incoming electroneither with a spin up | ↑i ≡ (cid:18) (cid:19) or with a spin down | ↓i ≡ (cid:18) − i (cid:19) .Solutions of Eqs.(26) (and similar equations for an in-coming electron flow from the right-hand side) yield thefollowing probabilities | t ( L ) ↑↑ | = | t ( R ) ↓↓ | = 11 + ( z − ) (28) | t ( L ) ↓↓ | = | t ( R ) ↑↑ | = 11 + ( z +1 ) (29) | r ( L ) ↑↓ | = | r ( R ) ↓↑ | = 1 −
11 + ( z − ) (30) | r ( L ) ↓↑ | = | r ( R ) ↑↓ | = 1 −
11 + ( z +1 ) . (31)Here, we have also introduced the variable z s z s = λ y sin( m s φ ) t m s = λ y Rγ × sin( m s φ ) m s , s = ± , (32)related to the characteristics of the curved surface (seeSec.IIA).Evidently, there is no backscattering for incoming elec-trons, if λ y = 0 [see Eqs.(30)-(32)]. However, at λ y = 0backscattering with a spin inversion takes place. Thereflection probabilities without the spin inversion are | r ( L ) ↑↑ | = | r ( L ) ↓↓ | = 0. The same is true for the trans-mission probabilities with a spin inversion, i.e., | t ( L ) ↑↓ | = | t ( L ) ↓↑ | = 0. Thus, backscattering with a spin inversionis nonzero in the ripple due to the curvature inducedSOC produced by the λ y -term. In addition, incomingelectrons with different spin orientations choose differentchannels (different m s , s = ± | t ( L ) ↑↑ | = 1takes place at the condition m − φ c = πn , n = 1 , , . . . , (33)[see Eqs.(28),(32)]. Evidently, this probability becomesdominant at the minimum transmission | t ( L ) ↓↓ | . Thelowest minimum of the transmission | t ( L ) ↓↓ | occurs at thecondition E A = λ x , when m +1 becomes imaginary [seeEq.(23)]. In other words, the propagating mode trans-forms to the evanescent mode for the channel m +1 . Tak-ing into account the condition E A = λ x in Eq.(23), oneobtains the critical angle of the curved surface (in formof the arc) for a maximum of spin up filter efficiency φ c = πnm − = πnγR q λ x − λ y , (34)where the SOC strengths λ x,y are defined by Eq.(7).For parameters listed in the caption of Fig.2 we have | φ c | ≈ . π ( n = 1). For the same critical angle φ c and E A = − λ x we obtain a maximum for the spin down filterefficiency, when m − becomes imaginary.Thus, there are different channels for the spin up andspin down electron (hole) flows. Note that the deviationfrom the energy value E A = ± λ x could produce the equaltransmission for spin up and spin down electrons (seeFig.3.a). Therefore, it is important to choose the energy | E A | to be in the close vicinity of the energy value ≃ λ x . For the considered parameters the filter efficiencyis, however, small. So far this result has met with onlylimited success. C. Scattering model for N ripples To increase the efficiency we suggest connecting thebent parts sequentially, as shown in Fig. 1. In particular,the construction with the direct+inverse arcs (with thesame angle φ ) transforms Eqs.(26)(27), to the formsˆ M (cid:18) Φ in r (cid:19) = ˆ M A ( − φ/ C I , ˆ M (cid:18) t (cid:19) = ˆ M A ( π − φ/
2) exp( − i ˆ mφ ) × (35)ˆ M − A ( φ/ − π ) ˆ M A (+ φ/
2) exp(+i ˆ mφ ) C I . (36)Since in the inverse arc the phases, accumulated fromthe point of connection with the direct arc to the pointof connection with a straight line ( flat graphene), havea sign opposite that of the first one, we use exp( ± i ˆ mφ ).Matching the wave functions at the boundaries of re-gions L, I, II, and R, for electron coming from the left-hand (L) and right-hand (R) sides of the construction,leads us to the following nonzero probabilities | t ( L ) ↑↑ | = (cid:20)
11 + 2( z − ) (cid:21) = | t ( R ) ↓↓ | , (37) | t ( L ) ↓↓ | = (cid:20)
11 + 2( z +1 ) (cid:21) = | t ( R ) ↑↑ | , (38) | r ( L ) ↑↓ | = 1 − | t ( L ) ↑↑ | = | r ( R ) ↓↑ | , (39) | r ( L ) ↓↑ | = 1 − | t ( L ) ↓↓ | = | r ( R ) ↑↓ | . (40) Thus, the transmissions through one and two (di-rect+inverse) arcs are accompanied by the inversebackscattering. The considered cases imply that thelarger the number of arcs is, the stronger the inversebackscattering is for one of the spin components.Following the recipe described in Ref.19 (interferingFeynman paths), and combining S-matrices for N con-nected arcs, we obtain | t ( L ) ↑↑ | = C (+) − N + C ( − ) − N = | t ( R ) ↓↓ | , (41) | t ( L ) ↓↓ | = C (+)+1 N + C ( − )+1 N = | t ( R ) ↑↑ | . (42)Here, the variable C ( ± ) s is defined as C ( ± ) s = p z s ) ± z s , s = ± . (43)Evidently, at z s = 0 the transmission probability is 1for any number of arcs, while z s = 0 leads to the decreaseof the transmission probability with the increase of thenumber of arcs. The suppression is, however, different forvarious transmission probabilities due to their differentdependence on the quantum number m s .As shown above, conditions (33), (34), determine thedominance, in particular, of the transmission probabilityof spin up incoming electrons at E A >
0. Indeed, a set N ≫ m s = − channel. However, this set suppresses thespin down transmission probability for the m s =+1 chan-nel that is proportional to x < ⇒ x N → E A > spin-up electrons from the left side of our system, one obtainsthe same magnitude for the transmission probability for spin-down electrons from the right side. III. DISCUSSIONA. N-factor
To obtain a simple picture of the physics behind theenhancement of the spin filtering effect, let us considerthe transmission at the energy E A ≃ λ x , when m +1 be-comes imaginary [see Eq.(23)] and the propagating modetransforms to the evanescent mode for the channel m +1 .In light of Eqs.(23),(7), one obtains m +1 = i Rγ λ y = ix , x = δγ ′ γ ≈ .
007 (44) E A ( eV ) -1 0 1 T r a n s m i ss i o n E A ( eV ) -0.5 0 0.5 T r a n s m i ss i o n E A ( eV ) -0.5 0 0.5 T r a n s m i ss i o n E A ( eV ) -0.5 0 0.5 T r a n s m i ss i o n FIG. 3: (Color online) Dependence of transmission probabili-ties | t ( L ) ↑↑ | (blue, dashed lines) and | t ( L ) ↓↓ | (red, solid lines)on the energy E A at k y = 0 for 1 (a), 20 (b), 100 (c) and 200(d) sequentially connected ripples ( π -arcs). The parametersare the same as in Fig.2. As a result, the variable z + (Eq.(32)) transforms in theform z + ≃ xφ ≪ . (45)Taking into account Eqs.(43-45), one can readily estimatethat at N ≫ C (+)+1 N + C ( − )+1 N ≃ N xφ ) ⇒ (46) ⇒ | t ( L ) ↓↓ | ≈ (cid:20)
22 + (
N xφ ) (cid:21) (47)With our choice of parameters and φ ≃ π , this resultyields | t ( L ) ↓↓ | ≪ ⇐⇒ N ≫ xπ ≈ . (48)The illustration of this phenomenon is displayed in Fig.3for the transmission probabilities through 1, 20, 100 and200 sequentially connected ripples ( π -arcs). Here, weconsider the transmission as a function of the curved sur-face energy E A of the incoming electrons (holes). A smalldifference between spin up and spin down transmissionprobabilities for one ripple (Fig.3a) at E A > ≃ N = 200ripples (Fig.3d). The opposite picture takes place for thespin down transmission probabilities at E A <
0. To re-alize such a situation one might use the SiO substrateas a gate of the curved surface, which helps control theconcentration of charge carriers in graphene. As a result,one can change the charge carrier type from electron tohole . B. Spin filtering and ripple parameters R and φ In light of the above analysis, without loss of gener-ality, we can consider m s φ < λ x ≫ λ y , this requirement leadsto the following inequality λ x − γRφ < | E A | < λ x + γRφ . (49)To remain at the maximum, for example, the transmis-sion probability | t ( L ) ↑↑ | = 1, it is necessary to fulfill con-dition (33). As a result, in light of Eq.(23) and the con-dition λ x ≫ λ y , taking into account Eq.(7), one obtains R ≃ γ | E A | ( πφ − β ) , β = 1 + 4 δp . (50)Combining this equation with Eq.(49), we have π − β < φ < π + 12 β . (51)Thus, Eqs.(50),(51) determine the region of feasibilityof the parameters R and φ , where the spin filtering ef-fects could exist at fixed system (graphene) parameterssuch as γ , δ , and the electron energy E A . From this ob-servation, two arguments follow in favor of our findings.First, even at φ = φ c (see Eq.(34)) one of the spin com-ponent in the incoming electron (hole) flow is suppressedfor a large enough number of ripples at some particularenergy region. Second, we assume that all ripples areidentical. Practically, the graphene surface is randomlycurved, and it is a real challenge to create identical, con-sequently connected ripples. However, it is our belief thatmodern technology will allow us to realize this situationsoon or later. Whatever the case, the spin filtering effectshould survive if small variations of radii and angles ofconsequently connected ripples are subject to conditions(50),(51), at a fixed value E A of the electron energy flow. C. Effect of a finite k y -momentum In our model a single ripple is modeled as part of ananotube that is infinite in the y direction. Evidently,realistic ripples are limited in space in both the x − and y − directions. In particular, graphene nanoribbons areconsidered prominent candidates to control the electronicproperties of graphene based devices. This issue requires,however, a dedicated study, and is the subject of a forth-coming paper.In order to have some idea of what should be expectedin graphene nanoribbons, we analyze the case with a fi-nite k y = 0. Nonzero k y could mimic the case of a ripplelimited in the y -direction. Indeed, a finite width in the y -direction introduces the quantization of the k y momen-tum on the curved surface. As a result, the eigenspinorsat the curved surface would depend on the mixture ± k y values for s = ±
1, i.e., altogether four momentum k y (see details in Ref.17). In this case analytical expres-sions are too cumbersome, even in a simple case of oneconserved momentum k y on the curved surface. There-fore, we proceed with a numerical analysis that providesa vivid presentation of a simple case with a single valueof the k y momentum on the curved surface.Let us suppose that the incoming electron flow pos-sesses a momentum ~k = ( k x , k y ) in regions L (R). Evi-dently, in this case E = ± γ q k x + k y . For simplicity,we consider E >
0, and obtain for the momentum onthe curved surface k y = t y /γ = E /γ × sin( α ) . (52)The results of the calculations exhibit a degradation ofthe spin filter ability of our system. At a fixed value ofthe energy E = λ x the transmission probability | t ( L ) ↑↑ | decreases drastically at | α | ≥ π/ | α | < π/ γ and E . α - π /2 - π /4 0 π /4 π /2 T r a n s m i ss i o n E A ( eV ) -0.5 0 0.5 T r a n s m i ss i o n FIG. 4: (Color online) Transmission probabilities | t ( L ) ↑↑ | (blue, dash lines) and | t ( L ) ↓↓ | (red, solid lines): (a) as a func-tion of the incidence angle α at E = E A = λ x = 0 . E A at k y = 0 . − . Thecalculations are done for 200 sequentially connected ripples( π -arcs) The other parameters are the same as in Fig.2. At a fixed value of the momentum k y the effectivityof spin filtering is reduced by ∼
10% [see Fig.4(b)]. Atthe same time, our systems manifests a zero transmis-sion for all spin orientation for charge carriers at energies − . eV < E A < . eV due to our choice of the value k y . D. The graphene purity
We restricted our consideration to a ballistic regime.This approximation is well justified due to the followingfactors. The remarkable strength of the carbon honey-comb lattice makes it quite difficult to introduce any de-fects into the lattice itself. Charge impurities that couldlimit electron mobility in graphene are still an open prob-lem from both experimental and theoretical points ofview (see, for example, discussion in Ref.1). It is also wellknown that the difference in conductivity in graphene be-tween T ≈ ∼ R =1nm. As a result, our system length is πR × ≈ ℓ ≈ µ m (see,e.g., Ref.2), it seems our consideration is on a reasonablebasis.Thus, in our model the basic mechanism that is respon-sible for spin filtering effects is an attenuation of one ofthe transmitting modes. It transforms to the evanescentmode in the energy gap created by the SOC in the curvedsurface. The multiplicative action of a large enough num-ber of ripples suppresses this transmitting mode at cer-tain conditions that provide a high efficiency for the otherone. IV. SUMMARY
We have analysed the transmission and reflection ofballistic electron flow through a ripple in an effective massapproximation, when only the interaction between near-est neighbor atoms is taken into account. In our consid-eration a ripple consists of the curved surface in the formof an arc of a circle connected from the left-hand andthe right-hand sides to two semi-infinite flat graphenesheets. Considering the curved surface as a part of thearmchair nanotube, we have shown that the curvatureinduced spin-orbit coupling yields a backscattering [seeEqs.(30,31)] with spin inversion. This spin inversion iscaused by the spin-orbit term that breaks spin symmetry(a spin projection on the symmetry axis) in the effectiveHamiltonian of the armchair CNT.In the energy gap created by the curvature inducedspin-orbit coupling there is a preference for one spin ori-entation, depending on the direction of the electron flowat normal incidence. The width of the energy gap de-pends in inverse proportion on the radius of the ripple.At this energy range the ripple acts as a semipermeablemembrane which is more transparent for the incomingelectrons with spin up from the left-hand side and withspin down from the right-hand side, and vise versa forthe holes. In other words, there is a precursor of chiral transmission of spin components of the incoming electron(hole) flow at a fixed energy. For one ripple system thiseffect is, however, small. In order to enforce this effect,we extended our consideration to a curved surface of thesinusoidal wave type with N arcs. This step is of crucialimportance to suppress one of the spin components andto support the spin inversion symmetry for the transmis-sion probability. The larger the number of consistentlyconnected ripples (arcs) is, the stronger the dominanceof a specific spin component is in comparison with theother in the transmission from the same direction. Thereis a cooperative effect of chiral spin transmissions pro- duced by a large number of ripples. To trace the N -dependence we have derived a formula for a compositetransmission probability for well-polarized spin compo-nents: i) Eq.(41) for spin up electrons; ii) Eq.(42) forspin down electrons. Based on these results, we predict astrong spin filtering effect for a sufficiently large numberof arcs in the rippled graphene system. In contrast to theusual waveguide that guides optical or sound waves of achosen frequency in a well–defined direction, our systemguides spin electron (hole) waves with a well–defined po-larization in one or another direction at a certain energy.It seems, therefore, natural to name this system chiralspinguide .We have considered only a curved surface that owes itsorigin to an armchair nanotube. Evidently, our modelcan be extended to other types of origins. However, thecorresponding analysis requires a separate study. We alsoneglected the effective magnetic field that arises from thedependence of the hopping parameter γ on the curva-ture (see discussion in ). This effect influences the localdensity of states . It can cause the localization of theelectrons on the boundary between flat graphene and thecurved surface, similar to the boundary state for sometypes of carbon nanoparticles . As a result, it mightaffect the efficiency of the spinguide . Last, but not least,many body effects such as electron-electron interactionshould be incorporated and analyzed as well. It is espe-cially noteworthy that electron-electron interaction, de-signed in the form of a specific potential barrier on thegraphene sheet , leads to separation of spin-polarizedstates. In fact, this result is in close agreement with ourfinding, obtained for one ripple. As mentioned above,the curvature induced SOC simulates a penetrable bar-rier preferable for transmission of only one of two spincomponents, depending on the direction and energy ofthe incoming electron (hole) flow. It would be interest-ing to study the interplay between the SOC and electron-electron interaction on the electron transport in our sys-tem. Evidently, this consideration would allow us tostudy in more detail the effect of impurities on the elec-tron mobility in our system.In conclusion, the transparency and the mathemati-cal rigor of our results provide good grounds to believethat spin filtering effects found in this paper, giving riseto a chiral spinguide phenomenon, will be observable inexperiment. Acknowledgments
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