Spin selective transport through helical molecular systems
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Spin selective transport through helical molecular systems
R. Gutierrez , E. D´ıaz , , R. Naaman , and G. Cuniberti , Institute for Materials Science, Dresden University of Technology, 01062 Dresden, Germany GISC, Departamento de F´ısica de Materiales, Universidad Complutense, E-28040 Madrid, Spain Department of Chemical Physics, Weizmann Institute, 76100 Rehovot, Israel Division of IT Convergence Engineering National Center forNanomaterials Technology, POSTECH, Pohang 790-784, Republic of Korea
Highly spin selective transport of electrons through a helically shaped electrostatic potential isdemonstrated in the frame of a minimal model approach. The effect is significant even for weakspin-orbit coupling. Two main factors determine the selectivity, an unconventional Rashba-like spin-orbit interaction, reflecting the helical symmetry of the system, and a weakly dispersive electronicband of the helical system. The weak electronic coupling, associated with the small dispersion,leads to a low mobility of the charges in the system and allows even weak spin-orbit interactionsto be effective. The results are expected to be generic for chiral molecular systems displaying lowspin-orbit coupling and low conductivity.
PACS numbers: 73.22.-f 73.63.-b 72.25.-b 87.14.gk 87.15.Pc
Introduction − The concept of spintronic devices oper-ating without a magnetic field has been proposed sometime ago for solid state devices in which the spin-orbitcoupling (SOC) is large [1, 2]. Recently, a new type ofmagnet-less spin selective transmission effect has beenreported [3–6]. It was found that electron transmissionthrough chiral molecules is highly spin selective at roomtemperature. These findings are surprising since carbon-based molecules have typically a small SOC that can-not support significant splitting between the spin states,splitting which is thought to be essential for any spin de-pendent property. Although it has been found both intheory [7–9] and experiments [10] that there is a cooper-ative contribution to the value of the SOC, so that thisquantity may be larger in molecules or nanotubes thanin a single carbon atom, the values calculated or exper-imentally found are still relatively small [7–11], e.g. fewmeV for carbon nanotubes [10]. Hence, even includingthis cooperative contribution, the values obtained for thespin polarization (SP) in electron transmission throughchiral molecules [6] seem to be too high and cannot berationalized by such SOC values.Recently, a theoretical model based on the first Bornapproximation in scattering theory has been proposed forexplaining the spin selectivity of chiral molecules [13]. Al-though the results are in qualitative agreement with theexperimental observations, they could not explain themusing reasonable SOC values.In what follows, a model is presented to describe elec-tron transmission through a helical electrostatic poten-tial (see Fig. 1). Although the model does not claim tofully catch the complexity of the experimentally studiedDNA-based systems [6, 14], it highlights the role of somecrucial parameters, which can determine the experimen-tally observed high SP. The key factors in the model thatallow for the high spin selectivity are: i) Lack of paritysymmetry due to the chiral symmetry of the scattering potential; ii) Narrow electronic band widths in the helicalsystem, i.e. the interaction between the units composingthe helical structure through which the electron propa-gates is relatively weak. Moreover, a physically meaning-ful estimation of the SOC can be obtained by taking intoaccount that first, like in the solid state, in the presentstudy the electric field acting on the electron needs to in-clude the effective influence of all the electrons belongingto a molecular unit [14, 15], and second, due to proximityeffects, the Coulomb interaction between the transmittedelectron and the atoms in the molecular unit can scale as1 /R for short distances R . Model and Methodology − We consider the Schr¨odingerequation for a particle moving in a helical electro-static field. Analytical results for such fields havebeen derived in Ref. 12. For the sake of simplicity,approximate expressions valid near the z-axis will beused (only x and y components will be considered, FIG. 1. A charge q in spin state σ is moving along through he-lical electric field. The parameters a , b and ∆ z are the radiusand the pitch of the helix and the spacing of the z-componentof the position vector of the charges distributed along it, re-spectively. The helical field E helix induces a magnetic field B in the rest frame of the charge and hence influences its spinstate. the z component only contributes when considering thefull three-dimensional problem, see below ): E helix = − E P i,j g i,j ( z )(cos( Qj ∆ z ) , sin( Qj ∆ z )). Here, g i,j ( z ) =(1 + [( z − ib − j ∆ z ) /a ] ) − / and Q = 2 π/b with b beingthe helix pitch and a the helix radius, see Fig. 1. Theindex m = 0 , · · · , M − M molecular units placedalong the helix. The index n = − L / , · · · , L / L being the number of helical turns) connects sites whichdiffer in their z-coordinate by b [16]. We note that theconsidered helical potential is assumed to be related tothe charge distribution along the stack of molecular unitsbuilding the helical structure; hence the factor E is pro-portional to the local charge density.For a charge moving with momentum p through thehelix, the field E helix induces a magnetic field in thecharge’s rest frame, from which a SOC arises: H SO = λσσσ ( p × E helix ). The SOC strength is λ = e ¯ h/ m c and σσσ is a vector whose components are the Pauli matrices σ x , σ y , σ z . The general problem is three-dimensional;however, in order to get first insights into the behaviorof the SP, we will assume p x = p y = 0 , p z = 0, so thatthe Schr¨odinger equation takes the form [17]: h − ¯ h m ∂ z + U ( z ) + α (cid:18) z ) − Ψ ∗ ( z ) 0 (cid:19) ∂ z − α (cid:18) f ( z ) f ∗ ( z ) 0 (cid:19) i χ ( z ) = Eχ ( z ) . (1)Here, χ ( z ) = ( χ ↑ ( z ) , χ ↓ ( z )) T is a spinor, Ψ( z ) = E x − i E y = P i,j e − i Qj ∆ z g i,j ( z ), f ( z ) = ∂ z Ψ( z ), and U ( z ) thehelical electrostatic potential. The terms ∼ f ( z ) , f ∗ ( z )are introduced to make the Hamiltonian hermitian inthe continuum representation. The SOC parameter α =¯ hλE (with dimensions of energy × length) depends onthe effective charge density through E . The problemposed by Eq. 1 can be written as an effective two-channelnearest-neighbor tight-binding model [17]: H = X σ = ↑ , ↓ N X n =1 U n c † n,σ c n,σ + V X σ = ↑ , ↓ N − X n =1 ( c † n,σ c n +1 ,σ + h . c . )+ N X n,m =1 ( c + n, ↑ W n,m c m, ↓ + c + m, ↓ W × m,n c n, ↑ ) + H leads . (2)The operators { c n,σ , c + n,σ } n =1 ,...,N,σ = ↑ , ↓ create or destroy,respectively, an excitation at the tight-binding site n withspin index σ . The only non-zero elements of the inter-channel coupling matrix W are given by [17]: W n,n = − αf ( n ∆ z ), W n,n +1 = α Ψ( n ∆ z ) / z , and W n +1 ,n = − α Ψ(( n + 1)∆ z ) / z . Further, the matrix W × n,m satis-fies W × n,m = − ( W n,m ) ∗ for n = m , and W × n,n = ( W n,n ) ∗ .The hopping V should in general be estimated on thebasis of a first-principle calculation of the electronic cou-pling for a given system. However, we will consider itas a free parameter, whose order of magnitude for helical organic systems is expected to lie in the range of few tensof meV (e.g. for DNA, electronic structure calculationsyield values of the order of 20 −
40 meV [18]). Finally,the operator H leads includes the semi-infinite chains tothe left (L) and right (R) of the SO active region [17].A schematic representation of this two-channel model isshown on the top panel of Fig. 2. Transport properties − We focus on the spin-dependenttransmission probability, T ( E ), of the model Hamilto-nian given by Eq. 2, as a function of the electron’s in-jection energy E . The problem can be considered as ascattering problem where a finite-size region (with non-vanishing SOC) is coupled to two independent L (left)-and two independent R (right)-electrodes, each electrodestanding for a spin channel and being represented by asemi-infinite chain, see Fig. 2. T ( E ) encodes the influ-ence of multiple scattering events in the SOC region; us-ing Landauer’s theory [20] we obtain [17]: T ( E ) = Γ R ↑ (Γ L ↑ | G ↑ ,N ↑ | + Γ L ↓ | G ↓ ,N ↑ | )+ Γ R ↓ (Γ L ↑ | G ↑ ,N ↓ | + Γ L ↓ | G ↓ ,N ↓ | )= t up ( E ) + t down ( E ) . (3)In Eq. 3, G nσ,mν ( E ) with σ, ν = ↑ , ↓ are matrix elementsof the retarded Green’s function of the SOC region in-cluding the influence of the L - and R -electrodes. Theindividual contributions in Eq. 3 can be related to dif-ferent transport processes without (e.g. Γ L ↑ Γ R ↑ | G ↑ ,N ↑ | )or with (e.g. Γ L ↑ Γ R ↓ | G ↑ ,N ↓ | ) spin-flip scattering, seeFig. 2. Notice that t up ( E ) and t down ( E ) − the trans-missions for the up and down channels respectively, asdefined by Eq. 3 − , contain contributions arising bothfrom direct transmission without spin-flip as well asspin-flip. An energy-resolved SP for different initialspinor states can be defined as: P ( E ) = ( t up ( E ) − t down ( E )) /T ( E ). The energy-average SP h P ( E ) i E = P ( h t up ( E ) i , h t down ( E ) i , h T ( E ) i ) will also be used. Wefocus only on electron-like contributions ( E <
0) andon energies | E | ≥ k B T ≈
23 meV, so that h . . . i E = R − k B T − V dE ( . . . ). Results − A crucial parameter in the model is the SOCcoupling α . Realistic values are obviously very diffi-cult to obtain [21, 22], since α is not simply the atomicSOC, but contains the influence of the charge distri-bution in the system via the field factor E . For thesake of reference, a rough value of E for DNA maybe estimated along the following lines. A single DNAbase is considered as composed of discrete point-likecharge centers A , representing the atoms. We associatewith each center A at position R A a Gaussian-shapedcharge distribution of width w ∼ . − . ρ for C, N, and O atoms (considered as spheres witha radius of the order of the corresponding covalent ra-dius). The local field of this charge distribution, E = − (1 / πǫ )( ∂/∂r ) R d r ′ ρ ( r ′ − R A ) | r − r ′ | − , can be com- -100-50050100 P ( E )[ % ] P P -60 -40 -20 0 20 40 60 Energy[meV] -50050100 P FIG. 2.
Top panel : Schematic representation of the tight-binding model, see Eq. 2. The two channels interact via theSOC (framed region). To the left and right of the spin scatter-ing region, both channels are independent and are modeledby semi-infinite chains.
Bottom panel : Energy dependenceof the SP P ( E ) for L =3 helical turns, and for injected elec-trons polarized with their spin pointing up ( P ), down ( P ),or unpolarized ( P ). A spin-filter effect takes place only forenergies near the band edges, where all SPs have the samesign. Notice also that near the band edges the SP has oppo-site signs for electrons ( E <
0) and holes (
E > P ( E ) is not exactly antisymmetric. Parameters: α = 5 meVnm, V = 30 meV, U = 3 meV. puted analytically [17] and it scales for R = | r − R A | ≪ w like E ≈ ( N ρ / πǫ )( w/ √ π ) R − ( E has been mul-tiplied by a factor N ∼
10, the number of atoms in abase, to approximately account for other charge centers.For
R/w ∼ . − .
4, values of α = ¯ hλE ≈ . − . α ∼ − ρ and N as well as proximity effects (short-distance scaling of E ) in the estimation of α .Fig. 2 presents the energy dependent SP for differentincoming spin states when the spin is pointing up (10),down (01) or the electrons are unpolarized (11). Thecoupling α was assumed to be 5 meV nm. Althoughthis value is larger than the previously estimated one, itserves to illustrate the behavior of the model in a clearway. In the case of (10) and (01) states, the interestingenergy windows are those where both SPs have the samesign, which indicates that the outgoing state will alwayshave the same SP independently of the initial condition.This behavior occurs mainly for energies near the bandedges. A similar situation is found for the (11) state, seeFig. 2. Near the band center, P ( E ) and P ( E ) have -60 -40 -20 0 20 40 6000.20.40.60.81 T r a n s m i ss i on T=t up +t down -60 -40 -20 0 20 40 60 Energy[meV] t up -60 -40 -20 0 20 40 60 t down State (10) State (01)State (11)
FIG. 3. Different components of the transmission t up ( E ), t down ( E ) and T ( E ) as defined in Eq. 3, and for thesame parameters of Fig. 2. Focusing on electron-like contri-butions, it is only near the lower band edge ( E ≤ −
22 meV)where a positive SP for all incoming states (10), (01), and(11) is obtained, see also Fig. 2. opposite signs and hence the SP depends on the incomingspin state. The average SPs, as defined above, amountto approximately h P i E = h P i E = h P i E ≈ L = 0 [17], and hence, the total transmission cannotexceed one. For (11) both channels are open and themaximum transmission is 2.In the top panel of Fig. 3, for (10) and (01), we findsome degree of spin-dependent back-scattering, whichis reflected in the different total transmissions T ( E )for each polarization. In what follows, for the sakeof reference, only the behavior in the energy window[ − V, − k B T ] , k B T ∼
23 meV is discussed. For the (10)state, transmission without spin flip is dominant in thisenergy region, and this leads to the positive SP. How-ever, for (01), spin-flip processes become dominant in thesame energy region, and hence the outgoing up-channelacquires a larger weight. As a result the SP for (01)is also positive. This behavior is closely related to thechiral symmetry, which basically manifests in the spe-cial structure of the
W, W × matrices. For the (11) state,bottom panel of Fig. 3, the outgoing up-channel clearlydominates the transmission in the considered energy win-dow , thus indicating that for unpolarized electrons back-scattering and spin-flip of the down-component will ul-timately lead to a positive SP. A similar analysis canbe performed for the hole-like energy region E >
0. Ingeneral terms, SP may occur either by spin-flip (with nonet change of the total transmission) or by spin selec-tive back-scattering. The results of Fig. 3 suggest that
FIG. 4. 2D plot of the energy average SP h P ( E ) i E as a func-tion of both the hopping parameter V and the SOC α . Onlyfor small V a relative large SP is found. With increasing elec-tronic coupling, larger SOC strengths are required to get asizeable SP. both processes are playing a role; their relative contribu-tion to the SP turns out however to sensitively dependon the specific energy window considered. The selectiv-ity found in this model relates to two special features ofthe chiral system: (i) the symmetry of the field whichtranslates into an unconventional SOC, and (ii) the nar-row electronic band width in chiral organic systems. Theterm band width serves only as a keyword for the aver-aged value of the coupling matrix elements, V , betweenneighboring molecular states mediating charge motion.As shown in Fig. 4, the size of the hopping parameterstrongly affects the energy average SP, ultimately lead-ing to h P ( E ) i E → V . For small hopping,however, the SP can achieve very large values by onlya moderate increase of the SOC α . The interplay be-tween α and V seems related to the relatively long time(roughly proportional to ¯ hV − ) the electron will spendin the conducting channel in a real system, allowing forthe SOC to become more effective. Conclusions − The present study based on a genericmodel sheds new light on a chirality-induced spin selec-tivity (CISS) effect. It suggests that beyond the symme-try itself, CISS depends on the organic molecules beingpoor conductors. Weak electronic coupling along the he-lical structure is expected to lead to low mobility of theelectrons through the system and allows enough time forthe SOC, although being weak, to influence spin trans-port. The effect depends on the electron momentumand once the electrons have kinetic energy above k B T ,the SP increases and becomes weakly energy dependent.One open issue for further inquiry is the influence of theelectrode-molecule interface. If the electrodes are mag-netic, spin-dependent tunnel barriers emerge, which may influence the SP. The present study indicates that CISSmay be a very general phenomenon, existing in chiral sys-tems having low SOC and low conductivity, and hencemay play a role in charge transport through biosystems.The effect could also be of great interest to control thespin injection efficiency in the context of semiconductor-based spintronics by interfacing chiral molecules withsemiconductor materials.RG and ED thank H. Pastawski, R. Bustos-Marun,T. Brumme, and S. Avdoshenko for fruitful discussions.This work was partially funded by the DFG under CU44/20-1, MAT2010-17180 and by the South Korea Min-istry of Education, Science, and Technology Program“World Class University” (No. R31-2008-000-10100-0).Computational resources were provided by the ZIH atTU-Dresden. ED thanks MEC and RN thanks theGerman-Israel Science Foundation and the Israel ScienceFoundation for financial support. [1] B. Datta, and S. Das, Appl. Phys. Lett. , 665 (1990).[2] J. P. Lu e t al., Phys. Rev. Lett. , 1282 (1998).[3] K. Ray, S. P. Ananthavel, D. H. Waldeck, and R. Naa-man, Science , 814 (1999).[4] R. Naaman, and Z. Vager, MRS Bul. , 429 (2010).[5] S. G. Ray, S. S. Daube, G. Leitus, Z. Vager, and R.Naaman, Phys. Rev. Lett. , 036101 (2006).[6] B. Goehler et al. , Science , 894 (2011).[7] T. Ando, J. Phys. Soc. Jpn. , 1757 (2000).[8] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.Rev. B , 155426 (2006).[9] A. De Martino, R. Egger, K. Hallberg, and C. A. Bal-seiro, Phys. Rev. Lett. , 206402 (2002).[10] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,Nature , 448 (2008).[11] E. I. Rashba, Sov. Phys. Solid State , 1109 (1960)[Fiz.Tverd.Tela (Leningrad) , 1224 (1960)].[12] D. Hochberg, G. Edwards, and Th. W. Kephart, Phys.Rev. E , 3765 (1997).[13] S. Yeganeh, M. A. Ratner, E. Medina, and V. Mujica, J.Chem. Phys. , 014707 (2009).[14] Z. Xie, et al. , Nano Letters , 4652 (2011).[15] G. Bihlmayer, S. Bl¨ugel, and E. V. Chulkov, Phys. Rev.B , 195414 (2007).[16] We chose for the sake of reference b ∼ . a ∼ . z = 0 .
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