Chiral Bloch states in single layer graphene with Rashba spin-orbit coupling: Spectrum and spin current density
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Chiral Bloch states in single layer graphene with Rashba spin-orbit coupling:Spectrum and spin current density
Y. Avishai and Y. B. Band Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel,New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,3663 Zhongshan Road North, Shanghai, 200062, China,and Yukawa Institute for Theoretical Physics, Kyoto, Japan Department of Physics, Department of Chemistry, Department of Electro-Optics,and The Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev,
We study the Bloch spectrum and spin physics of 2D massless Dirac electrons in single layergraphene subject to a one dimensional periodic Kronig-Penney potential and Rashba spin-orbitcoupling. The Klein paradox exposes novel features in the band dispersion and in graphene spin-tronics. In particular it is shown that: (1) The Bloch energy dispersion ε ( p ) has unusual structure:There are two Dirac points at Bloch momenta ± p = 0 and a narrow band emerges between the widevalence and conduction bands. (2) The charge current and the spin density vector vanish. (3) Yet, allthe non-diagonal elements of the spin current density tensor are finite and their magnitude increaseslinearly with the spin-orbit strength. In particular, there is a spin density current whose polarizationis perpendicular to the graphene plane. (4) The spin density currents are space-dependent, hencetheir continuity equation includes a finite spin torque density. Introduction : Following the discovery of graphene[1], novel phenomena were predicted in its electronicproperties [2, 3]. Among these, the Klein paradox[4] and chiral tunneling in single layer graphene(SLG) were reported in a seminal paper [5], andfurther findings were reported in Refs. [6, 7]. Dueto chirality near a Dirac point, electrons executeunimpeded transmission through a potential bar-rier even for energies below the barrier. This sce-nario is related to the absence of back-scattering forelectron-impurity scattering in carbon nanotubes[8]. Several extensions were reported in Refs. [9–11]. In parallel, investigation of the role of elec-tron spin in graphene led to the emergence of anew field: graphene spintronics [12–56]. The role ofKlein paradox in graphene spintronics is reported inRefs. [33, 39, 56], who studied electron transmis-sion through a barrier in the presence of Rashbaspin orbit coupling (RSOC).In this work we expose yet another facet of theKlein paradox in graphene spintronics by elucidat-ing the physics of electrons in SLG subject to aperiodic one dimensional Kronig-Penney potential(1DKPP) and uniform
RSOC. Thereby the roles ofthe Klein paradox [5] and RSOC in SLG are com-bined with the Bloch theorem, and novel aspects ofband structure and spin related observables are ex-posed. Recall that RSOC can be controlled by anexternally applied uniform electric field E = E ˆ z perpendicular to the SLG lying in the x - y plane, asin the Rashba model for the two-dimensional elec-tron gas [57]. We hope this study will motivatefurther study of graphene based spintronic devicesthat do not rely on the use of an external magneticfield or magnetic materials.Observables that are calculated include the Blochspectrum ε ( p ) ( p = crystal momentum), spin den-sity, and spin current density (related to spintorques [58]). Their properties are remarkably dif-ferent from those predicted in bulk SLG in the ab-sence of a 1DKPP, wherein the Klein paradox doesnot play a role): (1) The spin-orbit (SO) splittingof levels in the Bloch energy dispersion is ratherunusual: Recall that for λ = 0, there are two de-generate levels in the valence and the conductionband and the gap is closed at a single Dirac pointat Bloch momentum p = 0 [see Fig. 1(a) below].For λ > ± p = 0 [see Fig. 1(c) below]. (2)Although the charge current and the spin densityvector vanish, the non-diagonal elements of the spin current density tensor J ij are finite (here i = x, y, z is the polarization direction and j = x, y is the prop-agation direction). Thus, unlike in bulk SLG [47], J zx = 0 and J zy = 0 (current is polarized per-pendicular to the SLG plane). (3) J ij ( x ) is space-dependent so that there is a finite spin torque [58].(4) The response of the spin current densities tothe RSOC strength λ is substantial even for small λ (the magnitude of λ due to a strong perpendic-ular electric field in SLG as reported in Ref. [44]is a fraction of meV). These predictions regardinggraphene spintronics are experimentally verifiable. Formalism : Consider a system of massless 2DDirac electrons in SLG lying in the x - y plane sub-ject to a uniform electric field E = E ˆ z and a 1Dperiodic Kronig-Penney potential, u ( x ) = u ∞ X m = −∞ Θ( x − mℓ )Θ( mℓ + d − x ) . (1)The (Fermi) energy ε and the potential height u satisfy the inequality u > ε > K ′ . Since thetransverse wave number k y is conserved, the wavefunction can be factored: Ψ( x, y ) = e ik y y ψ ( x ). Re-call that, in addition to the isospin τ encoding thetwo-lattice structure of SLG, there is now a realspin , σ . Hence, the wave function ψ ( x ) is a fourcomponent spinor in σ ⊗ τ (spin ⊗ isospin) space. Ithas dimensions of 1 / √ A where A is some relevantarea. Hereafter we take A = ( d + ℓ ) × ,and omit this factor when no confusion arises. TheHamiltonian is, h ( − i∂ x , k y , λ ) = γ { [ − i∂ x + λ (ˆ z × σ ) x ] τ x +[ k y + λ (ˆ z × σ ) y ] τ y } + u ( x ) ≡ h ( − i∂ x , k y , λ ) + u ( x ) . (2)which is a 4 × γ = ~ v F = 659 .
107 meV · nm is the kineticenergy parameter, and λ is the RSOC strength pa-rameter [59] (it is also the inverse SO length pa-rameter λ = 1 /ℓ so ∝ E ). The products, σ x τ x , σ y τ y , implicitly incorporate a Kronecker product. ψ ( x ) is a combination of four component plane-wave spinors, e ± ik x x v ( ± k x ) (between barriers),and e ± iq x x w ( ± q x ) (in the barriers). The constantvectors v ( ± k x ) and w ( ± q x ) satisfy the algebraic lin-ear equations, h ( ± k x , k y , λ ) v ( ± k x ) = εv ( ± k x ) ,h ( ± q x , k y , λ ) w ( ± k x ) = εw ( ± k x ) . (3)The vectors v ( ± k x ) and w ( ± k x ) cannot be chosenas spin eigenfunctions because spin is not conserved .Moreover, Eqs. (3) are not eigenvalue equations. In-deed, assuming fixed transverse wave number k y ,potential parameters u , d, ℓ and RSOC strength λ , the wave numbers k x and q x must depend onthe (yet unknown) energy ε . For ε > s = ± , there are two wave numbers that solvethese implicit equations: sk xn ( ε ) for x / ∈ [0 , d ], and sq xn ( ε ) for x ∈ [0 , d ] ( n = 1 , sk xn ( ε ) and sq xn ( ε ) as well as for v ns ≡ v [ sk xn ( ε )]and w ns ≡ w [ sk xn ( ε )] (where n = 1 , s = ± ).The solution of Eqs. (3) is given by, k xn = [ ε + ( − n +1 λ ] − λ − k y ,q xn = [ ε + ( − n +1 λ − u ] − λ − k y , (4)together with the vectors v ns and w ns (their ana-lytic expressions will not be explicitly given here).They are normalized as h v ns | v n ′ s i = h w ns | w n ′ s i = δ nn ′ , but h v n + | v n − i 6 = 0 and h w n + | w n − i 6 = 0.The general form of the wave functions betweenand within the barriers is then: ψ ( x ) = X n,s = ± ( a ns e isk xn x v ns , ( u ( x ) = 0) ,b ns e isq xn x w ns , ( u ( x ) = u ) . (5)The constants a ns ( ε ) and b ns ( ε ) with n = 1 , s = ± , are determined by matching the wavefunctions on the walls of the barrier and employingBloch condition to which we now turn.Consider the unit cell [0 , R ] consisting of the bar-rier region [0 , d ] and the spacing [ d, d + ℓ = R ],corresponding to the case m = 0 in Eq. (1). Thematching equations at the left wall of the barrier x = 0 implies ψ (0 − ) = ψ (0 + ). It is written interms of { a ns } , { b ns } using the following notation: a = ( a + , a + , a − , a − ) T , b = ( b + , b + , b − , b − ) T . (6) a and b are the 4 × V = ( v + , v + , v − , v − ) ,W = ( w + , w + , w − , w − ) , (7)are 4 × × x = 0 and the transfer matrix carrying ψ (0 − ) to ψ (0 + ) are then given by, V a = W b , ⇒ T − → + = W − V, (8)so that T − → + a = b . Similarly, the transfer ma-trix carrying ψ ( d − ) → ψ ( d + ) across the right wallof the barrier is T d − → d + = V − W . To completethe construction of the transfer matrix T that car-ries the wave function across a unit cell from x = 0 − to x = R − = ℓ + d − recall that the propagation of ψ ( x ) from 0 + → d − and from d + → R − is respec-tively controlled by the 4 × q = diag[ e iq x d , e iq x d , e − iq x d , e − iq x d ] , Φ k = diag[ e ik x ℓ , e ik x ℓ , e − ik x ℓ , e − ik x ℓ ] , (9) which leads eventually to the expression, T =Φ k T d − → d + Φ q T − → + . T is a symplectic 4 × T ] = 1 and T † Σ z T = Σ z , whereΣ z = × ⊗ τ z . The Bloch theorem (for fixed λ, k y , u , d, ℓ ) requires that ψ ( x + R ) = e ipR ψ ( x )where p is the crystal wave number. This impliesthe eigenvalue equation T ( ε ) a ( ε ) = e ipR a ( ε ) . (10)Equation (10) defines a relation between the foureigenvalues { λ j ( ε ) } ( j = 1 , , ,
4) of T ( ε ) and theBloch wave number p , that is, Im[ λ j ( ε )] = sin pR .Thereby we get the dispersion curves ε j ( p ) =[ λ Ij ] − (cid:0) sin pR (cid:1) . The eigenvalues of T satisfy theequalities λ = 1 /λ , λ = 1 /λ so that if λ j ( ε ) isreal the energy ε j ( p ) is in the gap. Otherwise, theeigenvalues consist of two pairs of conjugate com-plex numbers lying on the unit circle, re-numberedas λ = 1 /λ ∗ , λ = 1 /λ ∗ . Consequently, there are two symmetric dispersion curves ε ( p ) = ε ( − p )and ε ( p ) = ε ( − p ) corresponding to the two SOsplit levels. As we shall see below, for fixed k y = 0and RSOC strength λ →
0, the two curves coin-cide, forming valence and conduction bands thatdisplay a Dirac point at p = 0, with linear disper-sion ε j ( p ) = ε j (0) + ( − j a | p | for small p (where a > λ > Choice of parameters : Our objectives are twofold:(1) to elucidate the Bloch dispersion and its de-pendence on the RSOC strength λ (tunable by theelectric field). (2) To calculate wave functions andspin-related observables, to asses their space depen-dence and their response to variation of λ . As wehope to enrich our understanding of graphene spin-tronics, it is important to choose potential param-eters u , d, ℓ and RSOC strength λ in accordancewith experimental capability.Below, the lengths x, y, d, ... are given in nm,and energies ε, u λ as well as the wave numbers k x , k y , q x (introduced above) are given in (nm) − ,[1 (nm) − corresponds to 659.107 meV]. The sizeof λ is dictated by experiments on Rashba spin-splitting in SLG. In Ref. [44], it is shown that λ is in the order of fraction of 1 meV. Here we let0 < λ ≤ − (0 < λ ≤ . k xn and q xn should be real [see Eq. (4)]. For k y = 0, this implies ε > λ (for real k xn ) and ( u − ε ) + 2 λ ( u − ε ) > q xn . Finally, for simplicity, we consider forwardscattering, k y = 0. The case k y = 0 will be exploredin a future communication. (Note that it is exper-imentally difficult to tune k y for fixed Fermi en-ergy ε ). In summary: (1) The fixed parameters are: k y = 0 , u = 98 .
85 meV, d = 200 nm, ℓ = 260 nm.(2) In the calculations of the spectrum the Blochenergy is varied in the interval [2 λ, u / − = 13 . λ = 0 . − = 0 . λ = 0), and λ = 0 . − = 1 . Results : In the series of figures below we show ourresults for the Bloch spectrum, the charge density,and the non-diagonal elements of the spin currentdensity. It is argued that the charge current densityand the spin density vector vanish. Expressions forall these quantities are given below.First, we discuss the Bloch spectrum. In Fig. 1(a)the dispersion curves ε ( p ) are shown for very small(actually vanishing) RSOC strength λ = 0 . − =0.10054 meV. It consists of two (virtually)degenerate levels in the valence and the conduc-tion bands with a single Dirac point at p = 0.Strictly speaking, the periodic potential is 1D, sowe should refer this linear dispersion as a Dirac tri-angle and not a Dirac cone. As we increase λ to0.0016(nm) − = 1 . p ( ε ) R as function of ε (restricted to positive p for simplicity). In Fig. 1(c) it is shown for λ → p levels coincide and form a Diracpoint with linear dispersion. For λ > p level “pulls the Dirac point up”, and the two blue p levels repel each other. As a result, (taking intoaccount the symmetric pattern for p <
0) it impliesthat RSOC causes level repulsion in both energy(except at the Dorac points) and momentum. Thesingle Dirac point at p = 0 is now split into a coupleof Dirac points ± p = 0. But the dispersion at thesetwo Dirac points remains linear, unlike in the pat-tern encountered in bulk SLG [41]. From the pointof view of band structure, the central rhombus inFig. 1(b) specifies a narrow “semi-metallic band”between the valence and conduction bands.Now we consider Bloch wave functions andderivation of local observables. Calculations arecarried out at a given energy ε =0.025 nm − thatpasses through the two Dirac points at pR = ± four wave functions { ψ p i ( x ) } , i =1 , , ,
4, corresponding to the four points { p i } atwhich the constant energy line crosses the four dis-persion curves. The expressions of the wave func-tions are given in Eq. (5), wherein the coefficients { a ns } are the component of the vector a [definedin Eq. (6)] that is an eigenvector of T with eigen-value e ip i R . Similarly, the coefficients { b ns } are thecomponent of the vector b defined after Eq. (8).An operator ˆ O , is representable as 4 × σ ⊗ τ (spin ⊗ isospin) space. Localobservables are obtained by O ( x ) = 14 X i =1 ψ † p i ( x ) ˆ Oψ p i ( x ) . (11)(this is not an expectation value: observables maydepend on x ). Below we will consider operators ofcharge density, charge current (or velocity), spindensity and spin current density, and check thespace dependence of the corresponding observables.For the charge density, the relevant opera-tor is ˆ I × and the density is then ρ ( x ) = P i =1 ψ † p i ( x ) ψ p i ( x ). ρ ( x ) is shown in Figs. 2(a)for λ = 0 . − and 2(b) for λ = 0 . − . Note the concentration of oscillationsaround 1. The reason is that Bloch waves propagatein the longitudinal direction (recall that k y = 0).In the absence of RSOC the Klein paradox impliesthat transmission through a barrier is unimpeded.As shown in Ref. [60], in the presence of RSOC thetransmission is still high but not perfect. Increas-ing λ implies larger oscillation amplitudes. Thisis manifested here by noting that the amplitudeof oscillations of the density at higher λ as shownin Fig. 2(b), is larger than those for λ → < x < d (compared with the spacing region − ℓ < x <
0) reflects the inequality of wave numbers q xn > k kn , see Eq. (4).Next we consider the velocity operator (which is - - sin pR ε ( n m ) - (a) - - sin pR ε ( n m ) - (b) ε ( nm ) - s i n p R (c) ε ( nm ) - s i n p R (d) FIG. 1. (a) Bloch spectrum at λ = 0 . − =0 . λ = 0 . − =1 . two Dirac points . (c) and (d) Compare theinverse function sin p ( ε ) R : (c) is for λ = 0 . − =0 . λ = 0 . − = 1 . λ = 0) while maintaining the Dirac points. also the charge current),ˆ V = I × ⊗ τ . (12)As expected, we find that V x = 0, due to left-rightsymmetry. Also, V y = 0 because we have chosen k y = 0. However, the velocity operator will con-tribute to the spin current density (see below).The spin density operators ˆ S (from which thespin density observables S ( x ) are derived viaEq. (11)), are given by, ˆ S = ( ˆ S x , ˆ S y , ˆ S z ) = ~ σ ⊗ I . The unit of the observable S ( x ) is S = ~ /A . Butin the present case, it is found that S = 0. For S x ,it is expected that there is no spin density alongthe direction of motion. For S z , it is expected thatthere is no spin density along the direction of mo-tion outside the SLG plan. For S y , there is cancel- - (cid:1)(cid:2)(cid:2) - (cid:3)(cid:2)(cid:2) (cid:2) (cid:3)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2)(cid:2)(cid:4)(cid:5)(cid:5)(cid:5)(cid:6)(cid:2)(cid:4)(cid:5)(cid:5)(cid:5)(cid:7)(cid:2)(cid:4)(cid:5)(cid:5)(cid:5)(cid:5)(cid:3)(cid:4)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:4)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)(cid:4)(cid:2)(cid:2)(cid:2)(cid:1)(cid:3)(cid:4)(cid:2)(cid:2)(cid:2)(cid:8) (cid:1) ψ (cid:1) (cid:1) (a) - (cid:1)(cid:2)(cid:2) - (cid:3)(cid:2)(cid:2) (cid:2) (cid:3)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2) (cid:2)(cid:4)(cid:5)(cid:6)(cid:2)(cid:4)(cid:5)(cid:7)(cid:3)(cid:4)(cid:2)(cid:2)(cid:3)(cid:4)(cid:2)(cid:1)(cid:3)(cid:4)(cid:2)(cid:8) (cid:1) ψ ( (cid:1) ) (cid:1) (b) FIG. 2.
Density ρ ( x ) in the unit cell − l < x < d for (a) λ = 0 . − = 0 . λ = 0 . − = 1 . lation between the four contributions in Eq. (11).Now let us focus on spin-current density. Thecorresponding operator is ˆ J (a tensor) from whichthe observed components of the spin current den-sity observables J ij ( x ) are derived via Eq. (11)), isdefined as ˆ J = 12 { ˆ S , ˆ V } , (13)where ˆ S is the spin density operator defined above,and ˆ V = I ⊗ τ is the velocity operator defined inEq. (12). In Eq. (13), i = 1 , , x, y, z speci-fies the polarization direction, and j = 1 , x, y specifies the axis along which electrons propagate.The unit of spin current density observables J ij is J = S v F = γ/A meV/nm.Our calculations show that the non-zero spin cur-rent density observables are the non-diagonal ele-ments of the spin current density observable, ex-plicitly, J xy ( x ) = J yx ( x ) , J zx ( x ) and J zy ( x ). Theyare shown in Figs. 3, 4 and 5 respectively for λ =0.00016(nm) − =0.10054 meV in panel (a) and λ = 0 . − = 1.0054 meV in (b). Notethat (1) Despite the fact that V = S = 0, the spincurrent density does not vanish. (2) Increasing λ bya factor 10 increases the amplitudes of the spin cur-rent density by a factor of about 15 for J xy and J zx and about 50 for J zy . (3) The spin current densitieshave a rich space dependence implying a non-zerotorque, see below.The spin current density was calculated in bulkSLG in Ref. [47]. The authors found that (1) J xx = J yy = J zx = J zy = 0, (2) J xy = − J yx , and(3) the spin currents are not space dependent (seeEq. (5) in Ref. [47]). In our calculations it is shownthat in the presence of a 1D potential (where thereis no rotational symmetry around the z -axis), thesymmetry relation (valid in bulk SLG [47]) is re-versed, J xy = + J yx . Moreover, although the valueof λ used in our calculations is about two ordersof magnitude smaller than that used in Ref. [47],the size of the spin current densities in both sys-tems are the same order of magnitude. Another no-ticeable difference from SLG is that in the presentsystem, spin current densities are space dependentand the divergence of the spin current density doesnot vanish. The continuity equation for the spin - (cid:1)(cid:2)(cid:2) - (cid:3)(cid:2)(cid:2) (cid:2) (cid:3)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2) - (cid:2)(cid:4)(cid:2)(cid:1) - (cid:2)(cid:4)(cid:2)(cid:3)(cid:2)(cid:4)(cid:2)(cid:2)(cid:2)(cid:4)(cid:2)(cid:3)(cid:2)(cid:4)(cid:2)(cid:1) (cid:1) < (cid:1) (cid:1)(cid:2) > (a) - (cid:1)(cid:2)(cid:2) - (cid:3)(cid:2)(cid:2) (cid:2) (cid:3)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2) - (cid:2)(cid:4)(cid:5) - (cid:2)(cid:4)(cid:1) - (cid:2)(cid:4)(cid:3)(cid:2)(cid:4)(cid:2)(cid:2)(cid:4)(cid:3)(cid:2)(cid:4)(cid:1)(cid:2)(cid:4)(cid:5) (cid:1) < (cid:1) (cid:1)(cid:2) > (b) FIG. 3.
Spin current density J xy ( x ) in the unit cell − l
Spin current density J zx ( x ) in the unit cell − l
0. When,in addition, a uniform perpendicular electric field E = E ˆ z is applied, the role of electron spin entersdue to RSOC. This system was studied in relationto transmission [33, 39, 56] and spin current densi-ties [56] with the quest to reveal interesting facetsof graphene spintronics within a time-reversal in-variant formalism. Its study is appealing due tothe fact that u and the RSOC strength λ can beexperimentally controlled, making it verifiable. - (cid:1)(cid:2)(cid:2) - (cid:3)(cid:2)(cid:2) (cid:2) (cid:3)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2) - (cid:2)(cid:4)(cid:2)(cid:2)(cid:5) - (cid:2)(cid:4)(cid:2)(cid:2)(cid:1)(cid:2)(cid:4)(cid:2)(cid:2)(cid:2)(cid:2)(cid:4)(cid:2)(cid:2)(cid:1)(cid:2)(cid:4)(cid:2)(cid:2)(cid:5) (cid:1) < (cid:1) (cid:1) (cid:2) > (a) - (cid:1)(cid:2)(cid:2) - (cid:3)(cid:2)(cid:2) (cid:2) (cid:3)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2) - (cid:2)(cid:4)(cid:1) - (cid:2)(cid:4)(cid:3)(cid:2)(cid:4)(cid:2)(cid:2)(cid:4)(cid:3)(cid:2)(cid:4)(cid:1) (cid:1) < (cid:1) (cid:1) (cid:2) > (b) FIG. 5.
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