Chirality driven topological electronic structure of DNA-like materials
CChirality induced topological nature of electrons inDNA-like materials
Yizhou Liu ∗ , Jiewen Xiao ∗ , Jahyun Koo, Binghai Yan † Department of Condensed Matter Physics,Weizmann Institute of Science, Rehovot 76100, Israel
August 21, 2020
Topological aspects of DNA and similar chiral molecules are commonly dis-cussed in geometry, while their electronic structure’s topology is less explored.Recent experiments revealed that DNA could efficiently filter spin-polarizedelectrons, called the chiral-induced spin-selectivity (CISS). However, the un-derlying correlation between chiral structure and electronic spin remains elu-sive. In this work, we reveal an orbital texture in the band structure, a topo-logical characteristic induced by the chirality. We find that the orbital textureenables the chiral molecule to polarize the quantum orbital, called the orbitalpolarization effect (OPE). The OPE induces spin polarization assisted by thespin-orbit interaction only from the metal contact and also leads to magnetism-dependent conductance and chiral separation. Beyond CISS, we predict thatOPE can also induce spin-selective phenomena even in achiral but inversion-breaking materials.
Introduction
In chemistry and biochemistry, the chirality is the geometric asymmetry of a large group ofmolecules with a nonsuperposable mirror image, either left- or right-handed. It plays a prominentrole in chemistry and biology ( ) for example in the enantioselective catalysis ( ) and drugdesign ( ). In physics, the chirality usually refers to the locking of spin and motion such asthe Weyl fermions ( ) and neutrinos ( ). Although the chirality represents seemingly irrelevantcharacters in different fields, recent experiments ( ) reveal an unexpected correlation betweenthe chiral geometry and the electronic spin. When they transmitted through DNA, electronsget highly spin-polarized, and the polarization depends on the chirality. This effect is calledchiral-induced spin selectivity (CISS)(see Refs. (
7, 8 ) for review) and is also observed in manyother chiral molecules (
9, 10, 11, 12, 13, 14, 15, 16 ) and even some chiral crystals (
17, 18, 19 ). Thehigh spin polarization is induced and manipulated in ways not previously imagined ( ).1 a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug n the last decade, the CISS effect has demonstrated appealing application potential inspintronic devices (
21, 22, 12, 23, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31 ), chiral electrocatalysis(
32, 33, 34, 35, 36 ) and enantiomer selectivity (
37, 38 ). For instance, chiral molecules were foundto adsorb on a ferromagnetic substrate with different speeds that depend on both the chirality andthe substrate magnetization, leading to efficient separation of enantiomers ( ). When contactingto magnetic leads, the chiral molecule exhibits magnetization-dependent resistivity, i.e. themagnetoresistance (MR) (
21, 12, 23, 24, 26, 27, 39, 40 ). However, the physical origin of CISS, i.e.the relation between the chiral structure and the electron spin, is still debated.The first characteristic feature of CISS experiments is their robustness at room temperature(Ref. ( ) and references therein). The second feature is that they are dynamical phenomena thatusually involve electron tunneling or electron transfer in the non-equilibrium process. Chiralmolecules like DNA exhibit no spin polarization at the ground state, and the CISS effect vanishesat equilibrium. For example, the chiral separation disappears after the substrate and moleculesreach thermodynamic equilibrium after a long enough time (
37, 38 ). Another feature that usuallyattracts little attention is that chiral molecules commonly have direct contact with a noble metal(e.g., gold) substrate, thin-film, or nanoparticles in these experiments.Although many theories have been developed to understand the CISS effect (
41, 42, 43, 44,45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69 ), theconsensus is not reached. Most models consider the chiral molecule as a spin filter and requirean effective spin-orbit coupling (SOC) in the molecule to couple the electron motion and spin.However, it is known that the experimentally measured SOC is no larger than a few meV inrelated organic systems (e.g. the curved carbon nanotubes (
70, 71 )). Thus it is challenging torationalize the robustness of CISS at room temperature (26 meV), even though several scenarioswere proposed to enhance the spin polarization. In addition, there are debates on the role ofdephasing in CISS (
42, 43, 45, 46, 48, 50, 55 ).The chiral electronic structure of Weyl semimetals (
72, 73, and references therein ) inspiresus to explore the band structure topology of DNA and similar chiral molecules. In this work, wereveal an ubiquitous topological orbital texture in the chiral lattice and propose a mechanism thatthe chiral molecule acts as an orbital polarizer or an orbital filter, rather than a spin filter, in theCISS effect. The orbital polarization effect (OPE) does not require SOC from the molecule andremains much robust against the temperature fluctuation. The orbital refers to the atomic orbitalangular momentum (OAM) of the wave function. By calculations with the Landauer-Bttikerformalism, we demonstrate that electrons get orbital-polarized after transmitting through thechiral molecule, in which the polarization depends on the chirality. (i) When electrons propagatefrom the lead through the molecule, the chirality filters the orbital and subsequently selectsthe spin, because orbital and spin are pre-locked by SOC in the heavy-metal lead. Thus, thesetransmitted electrons become spin-polarized, leading to the CISS, as illustrated in
Figure 1c .(ii) When electrons transmit from the chiral molecule into the lead, the orbital-polarizationinduces spin polarization also because of SOC in the lead. (iii) Furthermore, when it is magnetic(spin-polarized), the lead is also orbital-polarized because of SOC. If the orbital polarization ofthe lead matches that of the molecule, the electron tunneling is fast and otherwise slow, leading to2 c)(i)(ii)(iii)
Chirality (no SOC) (a) (b) L z +–0 SOC SOCOrbital / Spin PolarizerOrbital / Spin FilterSOC SOC
Figure 1:
The orbital polarization and the orbital texture. (a) The ab initio band structureof the right-handed peptide helix with orbital texture. The orbital texture refers to theparallel or anti-parallel relation between orbital polarization L z and the momentum. The insetshows the atomic structure the helix where gray, blue, red, and white spheres represent C,N, O, and H atoms. (b) The band structure of a tight-binding model of the helix. The helixhas a three-fold screw rotation (see inset), same as the helix. (c) Illustration of the orbitalpolarization effect in transport. (i) The helix (small black spheres) connects two leads that arelinear atomic chain (large black spheres). The spin-orbit coupling (SOC) exists only in leads butnot in the chiral helix. At given energy, the orbital gets polarized to + L z as the electron transmitsthrough the chiral molecule. (ii) When electrons run into the right lead where SOC exists, theorbital polarization induces spin polarization. The half circles with arrows represent the ± L z orbital. Thin arrows represent the spin. The larger orbital and spin stand for the orbital and spinpolarization, respectively, are in the right lead. (iii) The chiral molecule filters the + L z state butsuppresses the − L z state injected from the left lead. The yellow curves illustrate the scatteringtrajectory with arrows. If the spin is pre-locked to the orbital in the left lead, transmitted electronsbecome spin-polarized due to the orbital filtering. We note that the multiple-mode leads allowthe emergence of spin polarization in the two-terminal conductance despite the presence of thetime-reversal symmetry. 3R. It also induces different adsorbance / desorbance speed of molecules with opposite chirality,resulting in the chirality separation on the magnetic substrate ( ). We rationalize the CISS andrelated phenomena without requiring the presence of SOC in the molecule. We point out that theheavy metal lead plays the role of a spin-orbit translator that converts the orbital polarization intothe spin polarization. Additionally, we also clarify the debates about dephasing. Furthermore, ourwork reveals the deep connection between CISS and the magnetochiral anisotropy ( ) discussedin the condensed-matter physics community. Besides chiral molecules, we predict that OPE canalso lead to spin-selective effects in achiral but noncentronsymmetric molecules and solids. Results
Topological Orbital Texture
The band structure of the chiral lattice exhibits a topological feature that we call the orbitaltexture. We take a periodic chain of the right-handed peptide helix as an example. It isa typical secondary structure found in proteins and polypeptides, which was also studied inthe CISS experiment ( ). The helix exhibits the three-fold screw rotation around the z -axis.This symmetry induces the Dirac-like band crossings at the zone center ( Γ ) and boundary( ± Z ),as shown in the ab initio band structure in Figure 1a . It always sticks three bands togetheras a general consequence of the nonsymmorphic symmetry in the band structure topology(
76, 77 ). Thus, the nature of a chiral band structure involves multiple bands, beyond the one-banddescription. Here, SOC is ignored in the band structure since it is negligibly small.A salient feature in the band structure is the orbital polarization L z . The L z refers to theatomic OAM. Without loss of generality, the OAM operator ˆ L z in the p x,y,z basis, is knownto be, ˆ L z = − i i , where we omit the index for atomic sites. It has three eigen states p ± ≡ ( p x ± ip y ) / √ , p z with L z = ± , , respectively. Because of the inversion symmetry(IS)-breaking, the Bloch wave function is allowed to exhibit nonzero OAM at the finite momentum.The screw rotation constrains that the OAM aligns along the z direction, i.e. L z . It is worthstressing that the OAM represents the self-rotation of the wave functions around atomic centersand is not a conserved quantity. To respect the time-reversals symmetry (TRS), L z exhibitsopposite signs at the k z and − k z points, as shown in Figure 1a . Similar to the chirality of a Weylfermions (
72, 73 ), L z and k z are always parallel or anti-parallel, which depends on the moleculechirality. Different from the Weyl point that exhibits infinitely large polarization, the L z vanishesto zero at Γ and ± Z because of TRS. According to the symmetry analysis, we point out thatsuch an orbital texture is ubiquitous for a chiral lattice with and without a helix structure. Evenwhen the screw rotational symmetry vanishes in a non-helical structure, the orbital texture stillpreserves because of the IS-breaking.We adopt a right-handed helix model to represent a chiral chain. The helix includes three4ites in the primitive unit-cell and exhibits the same three-fold screw rotation as the helix.We set three p -orbitals ( p x,y,z ) on each site and consider the nearest neighboring hopping in atight-binding model (see more details in SM). As shown in Figure 1b , its band structure alsoexhibits the same band degeneracy and orbital texture. For a linear chain, in contrast, the orbitalpolarization is strictly zero at all k -points because of the presence of both inversion symmetryand TRS.The topological orbital texture originates in the anisotropic hopping of the chiral orbitalalong the chiral chain. For generality, we consider a right-hand helical chain with the n -foldscrew rotation, in which the site i has two nearest neighbors, i + 1 along + z and i − along − z . We set two bases, p ± , at each site, and set the hopping integral as the simple Slater-Kostertype ( ). Among the same orbital ( p + or p − ), the hoppings from i to i ± are the same. From p + to p − , however, the hopping from i to i + 1 is different from that from i to i − by a phasefactor e − i π/n . If the chirality reverses, the phase − π/n also changes its sign. Take a helix with n = 4 , for example. The inter-orbital hopping is different by a “–” sign between the up anddown directions. It means that p + prefers a certain direction in hopping, while p − prefers theopposite direction. Consequently, the anisotropic inter-orbital hopping induces the orbital texturein the band structure. It also indicates that the orbital texture is as robust as the hopping integral’senergy scale, i.e., the bandwidth. At a finite Fermi energy in the band structure, oppositelypropagating electrons carry opposite orbital polarization, resulting in the orbital polarizationeffect discussed in the following. Two-terminal Transport and Dephasing
In previous theoretical studies (
42, 43, 44, 48, 50, 52, 53, 55, 58, 60, 62 ), the dynamic CISS processis usually mapped to a transmission problem between two achiral leads through a chiral moleculethat exhibits effective SOC due to different mechanisms. The spin polarization of the transmittedelectrons is evaluated as evidence of the CISS. In this work, we adopt the same transmissionmodel but remove SOC from the chiral molecule. As illustrated in
Figure 2a , we add a linearpart with atomic SOC between the chiral molecule and both leads, to simulate the fact thatchiral molecules commonly contact to the noble metal substrate. Both leads and the chiralmolecule have no SOC. Achiral linear chains represent leads and the SOC part. The tight-bindingHamiltonian of the whole system (both leads and the center region) is constructed with atomic p -orbitals ( p x , p y , p z ) and the hopping parameters in the same way as that for the band structurecalculation in Figure 1c .We obtain the scattering matrix S nm from the left ( L ) to the right ( R ) lead by the scatteringtheory and obtain the conductance by the Landauer-B¨uttiker formula ( ), G L → R = e h (cid:88) n ∈ R,m ∈ L | S nm | , (1)where S nm is the transmission amplitude from m -th eigenstate in the left lead to the n -th5igenstate in the right lead. In the following discussion, we will use G for G L → R if it is not notedspecifically.In both leads, the spin ( S z = ↑↓ ) and the orbital ( L z = ± , ) are conserved quantities becauseof the disappearance of SOC and the axial rotational symmetry, respectively. Thus, one canclassify the conductance into each S z or L z channels. Given the non-polarized injection statefrom the left lead, we estimate the spin and orbital, respectively, polarized conductance oftransmitted electrons in the right lead by, G S z = G L → R ↑ − G L → R ↓ (2) G L z = G L → R + − G L → R − , (3)where G L → RS z ( L z ) represents the conductance from the L lead to the S z ( L z ) channel of the R lead. Here, G L → R L z =0 is omitted because the L z = 0 state does not contribute to thetotal polarization. Corresponding spin and orbital polarization ratios are P S z = G S z /G and P L z = G L z /G , respectively.To demonstrate the CISS, we will show that electrons go through the chiral molecule andget spin-polarized, which is caused by the orbital polarization effect, by calculating G S z and G L z . We have performed all conductance calculations by our program and verified them withthe quantum transport package Kwant ( ). Related model parameters can be found in the SM.The dephasing related to the inelastic process was frequently discussed as a necessarycondition (
42, 43, 50, 60 ) to generate CISS. The single-mode leads employed in these modelsprohibit the spin current in the presence of TRS (
81, 82, 54 ). The existence of multiple modesin our leads allows the emergence of spin current without introducing the extra dephasing. Wenote that multiple modes represent the more realistic condition of the transition metal contact,compared to the single-mode model. However, the existence of the two-terminal MR, i.e. G L → R (+ M ) (cid:54) = G L → R ( − M ) , requires the dephasing to leak electrons into virtual leads ( ).Otherwise, the charge conservation forces the reciprocity regardless of the mode number in leads.Therefore, the role of dephasing in CISS depends on whether the lead is magnetic or not, asdiscussed in the following sections. In calculations, we introduce the dephasing parameter i η as the B¨uttiker virtual probe ( ) equally to each site of the chiral molecule. As long as thedephasing exists, we find that the nonreciprocal MR manifests the CISS effect in a two-terminaldevice. Orbital Polarization Effect and CISS
The orbital texture induces the OPE as electrons go through the chiral molecule. Different fromthe band structure, the energy level of a finite-size molecule exhibits no dispersion. Nevertheless,we can still use insights from the band structure and regard the molecule wave function as thesuperposition of the right and left movers from the band structure. Since right and left moversexhibit opposite L z , the molecule displays no orbital polarization at the ground state. When anelectron tunnels through the molecule from left to right, it transmits through the right moverchannel, as illustrated in Figure 1c . 6he chiral molecule plays roles as both an orbital filter and an orbital polarizer. If weapproximate L z as conserved, then L z does not flip as traveling through the molecule. We canregard the chiral channel as an orbital filter that only allows a given L z to pass, as illustratedin Figure 1c . The orbital filter is essential at the interface between the chiral molecule and theleft lead, where the incident orbitals get filtered. However, L z is not a conserved quantity inthe molecule and will flip in the transmission. The G L z also includes substantial orbital-flipcontribution due to the transmission from p − and p z states at left lead to p + at right (see thechannel-specific conductance in SM), caused by the orbital polarizer effect. It is interesting tosee that the chiral channel can even polarize the | L z = 0 (cid:105) state. The orbital polarizer is vitalwhen electrons run from the molecule into the right lead. These emitting electrons can induceintense orbital polarization in the right lead.The chiral molecule exhibits preferred transmission for electrons with the orbital polarizationparallel or anti-parallel, which is determined by the chirality to the transmitting direction at aspecific energy. Our conductance calculations confirm the OPE, as indicated by the nonzero G L z in Figure 2b . We turn off the SOC in the whole device and observe no spin polarization in theconductance since the motion and spin separate at all. We recall that the linear chain exhibits noorbital texture. The OPE is only due to the orbital texture in the chiral region.With SOC from the contact, OPE eventually leads to the spin polarization. We turn on theatomic SOC ( λ SOC ) in the short linear chains. For simplicity, we put the SOC part attached toboth leads to make them symmetric. As shown in
Figure 2 , the spin polarization ( P S z ) increasesas λ SOC increases. At given λ SOC , one can find that G S z is roughly proportional to G L z . Here,the SOC converts the orbital polarization to the spin polarization, since SOC locks the spin andorbital together. For example, p + ( p − ) is locked to the ↑ ( ↓ ) spin in | j m = (cid:105) ( | j m = − (cid:105) ) state( j m is the z projection of the total angular momentum). Although | j m = ± (cid:105) are degenerate inthe SOC regime, the chiral molecule filters the p + ( p − ) state and consequently selects the ↑ ( ↓ )spin around the energy –1.8 (+1.8) eV in Figure 2d . If reversing the chirality of the molecule,the orbital texture gets inverted, and thus, the spin polarization can be flipped.As increasing the atomic number ( N ) in the chiral chain, the calculated P S z first increasesand soon gets nearly saturated after N = 7 . In reality, the critical length depends on thematerial details. The region of increasing P S z can interpret the length-enhanced spin polarizationin the experiment (
6, 12, 84 ). We note that N = 1 , are actually achiral segments and thuscorresponding P S z is zero. The OPE is a robust effect compared to the temperature because theorbital texture is in the order of magnitude of the bandwidth (e.g. ∼ . eV for the helix inFigure 1a) and thus is much larger than the room temperature. Given than SOC in the heavymetal lead lies in a similar magnitude, the resultant spin polarization in CISS becomes a strongphenomenon in the ambient condition.The existence of spin polarization does not require extra dephasing in our calculations. Thisis because we have three modes in each lead. We find that the spin conductance G S z is insensitiveto the moderate dephasing with strong SOC and turns to decrease when the large dephasingviolates the coherence (see Figure 2c and SM). If single-mode leads are employed, we verify G S z = 0 without dephasing (see SM). 7igure 2: Orbital polarization and spin polarization in the conductance. (a) The transportmodel includes two linear leads and the chiral molecule with two helical units long. The SOCis only added to the intermediate site between the chiral molecule and the lead. (b) The orbitalpolarization G L z exists while the spin conductance G S z = 0 for λ SOC = 0 . G is the totalconductance. (c)(e)&(f) G S z and the spin polarization rate P S z increases as turning on SOC. The G S z (peak value around 1.8 eV) dependence on the dephasing term η is shown in the inset of (c).(d) Peaks of P S z [noted in (c)] increase quickly as increasing the length of the chiral moleculeand get almost saturated after the number of atoms is 7 [the same length as the model shown in(a)]. No dephasing is included in calculations except the inset of (c).8s discussed above, the required SOC comes from heavy metal leads such as gold (sometimesby contacts with the In-doped SnO ( ), GaN, and CdSe ( ) in experiments). The λ SOC dependence of CISS can be examined in transport experiments to verify our OPE theory. However,photoemission experiments of CISS deserve subtle treatment since they do not involve two-terminal devices. In intense light irradiation, the substrate ejects electrons into the vacuumthrough a layer of chiral molecules, and then the magnetization of photoelectrons is measuredby a Mott polarimeter (
11, 14 ). We note that the Mott detector is sensitive to both the orbitalmoment and spin moment. Since the orbital moment does not rely on any SOC, the detectedtotal magnetization of photoelectrons may be less sensitive to the substrate SOC, compared tothe transport measurement.
Magnetoresistance and Magnetic Chiral Separation
The orbital polarization can also rationalize these experiments on the MR (
21, 22, 12, 23, 13, 14,24, 26, 27 ) and chiral separation by the magnetic substrate. (
37, 38 ). In MR experiments, agold nanoparticle was included between one lead and the chiral molecule and the other lead ismagnetic such as nickel. Switching the lead magnetization induces the change of resistance. Forthe chirality selection, the substrate is a ferromagnetic Co film covered by a thin layer (severalnanometers) of gold. Molecules with opposite chirality get adsorbed to the substrate at a differentspeed, leading to the separation of chiral enantiomers. In the transient state when the moleculegets adsorbed on a metal surface, a small amount of charge transfer occurs between them ( ).The speed of the charge transfer, which is a quantity similar to the conductance, characterizesthe speed of the adsobance. Therefore, we can gain useful insights both for the adsorbance andMR from the conductance calculations.As injecting spin-polarized electrons from the substrate, the gold regime becomes spin-polarized and also orbital polarized because of SOC. Then the orbital direction is locked withthe magnetization direction. If this orbital matches the following OPE in the chiral molecule,the total conductance is large and otherwise small. As a consequence, different chirality andmagnetization can lead to different MR and speed of adsorbance as well.We employ the same two-terminal model and add an exchange field ( M ) along the z -directionto the spin components in both leads. The intermediate SOC regime mimics the noble metalpart. As shown in Figure 3b , the conductance changes as switching the sign of magnetization.The change of conductance ∆ G is proportional to the magnitude of M . One can understandthe role of the OPE by observing G S z and G L z , as shown in Figure 3d . When flipping themagnetization, it is not surprising G S z changes its sign at a certain energy. Subsequently, themagnitude of G L z varies because of the SOC. In Figure 3d, the increase (decrease) of | G L z | around 1.8 (–1.8) eV rationalizes the increase (decrease) of G at the same energy regime inFigure 3b, as a consequence of the OPE. Here, we include a finite dephasing parameter.As shown in Figure 3c , ∆ G = 0 if η = 0 . As increasing η , ∆ G first increases quicklyand then reduces if η is too large (see more in SM). In the coherent two-terminal measurement( η = 0 ) ( ), the reciprocity theorem requires G L → R ( M ) = G L → R ( − M ) . However, the virtual9igure 3: Magnetoresistance with magnetic leads. (a) The device model with two magnetized( ± M ) leads. (b) The total conductance varies when flipping the lead magnetization. (c) Thechange of the conductance ∆ G = G (+ M ) − G ( − M ) increases as increasing the magnitude of M and the dephasing term η . The inset of the lower panel shows the dependence of ∆ G (theright peak at about 1.8 eV on η in a larger scale. (d) As flipping M from – to +, G S z changessign in the general energy window, leading to changes of G L z . The increase (decrease) of G L z accounts for the increase (decrease) of G in (b), as an inverse effect of the orbital polarization.We set λ SOC = 0 . in all these calculations. 10ead ( i η ) releases this constrain, resulting in nonzero MR. The nonreciprocal MR is well consistwith the fact that the differential conductance ( dI/dV ) changes as reversing M in experiment.It should be noted that the dephasing is actually zero if no current flows in the device. Ifthe dephasing is too large, electrons are completely incoherent and feel no OPE in the transport.This explains the decreasing of MR for large dephasing. In addition, we verify that calculatedconductance satisfies the global Onsager’s reciprocal relation, G L → R ( M ) = G R → L ( − M ) (seeSM).The unidirectional conductance can also rationalize the chiral separation. We note thatthe adsorbance and desorbance correspond to opposite charge transfer directions between thesubstrate and the chiral molecule. Therefore, a chiral molecule releases slower (faster) from thesurface if it adsorbs faster (slower). Both adsorbance and desorbance guarantees that a substratewith certain magnetization attracts one chirality faster than the opposite chirality. Discussion
Unidirectional Conductance and Electric Magnetochiral Anisotropy
In the CISS-induced MR, as discussed above, the lead magnetization and the chirality togetherpick up one direction, along which the current flow is favored in the device. The conductance( G ) and resistance ( R ) can be describe to the leading order as, G ( M , I ) = G + G χ M · I (4) R ( M , I ) = R − R χ M · I (5)where R = 1 /G , R χ = G χ / ( G ) , M stands for the magnetization in the lead, I for thecurrent, G χ for the chirality ( χ ) determined conductance ( G χ = − G − χ ). G is the ordinaryconductivity while G (2) characterizes the unidirectional contribution. We point out that the linearrelation of G on I comes from the approximate linear dependence on dephasing in the small η region. So the specific form of Eqs. 4 & 5 can be modified by the dephasing dependence. Forthe I-V relation, we obtain I = G V / (1 − G χ M V ) ≈ G V + G G χ M V + O ( V ) (6)where the sign of M can be ± . Equation 6 agrees with the nonlinear I-V curves in experiments(
21, 12, 23, 24, 26, 27, 84 ). It indicates that the CISS-induced MR can be probed by two-terminalexperiments, to be accurate, in the nonlinear regime, which was debated recently (
60, 86, 62, 87 ).However, one can gain significant insights into MR by the linear-response scenario by assuming adephasing term, reminding that the dephasing commonly exists due to the energy dissipation andinelastic effects at finite current. The anisotropic conductance can lead to the current rectification.Suppose applying an ac electric field E ( t ) = E cos ωt between two leads, the V term in Eq. 6leads to a dc current density averaged in the driving period. The dc current (or open-circle11oltage) may be considered for photon-detection or energy-harvesting from the long-wavelengthlight without requiring specific polarization in light.The nonreciprocal conductance described by Eqs. 4 & 5 reminisces the electrical magnetochi-ral anisotropy (EMChA) discussed in literature (
74, 88 ), which was observed in chiral conductors(e.g., bismuth helix ( )) in the presence of a magnetic field. Corresponding resistance change( ∆ R ) of a chiral conductor (chirality χ = ± ) subject to a longitudinal magnetic field B isexpressed as ∆ R = R χ B · I . This effect was heuristically derived by generalizing the Onsager’sreciprocal theorem into the nonlinear regime ( ).Rikken et al once speculated the underlying connection between EMChA and CISS (
20, 89 ).Our work reveals the unambiguous link between EMChA and the CISS-induced nonreciprocalMR (rather than the spin current). Both EMChA and the CISS-induced MR satisfy the globalOnsager’s reciprocity. The EMChA refers to the conductor regime where both the TRS- andIS-breaking occur. In comparison, the CISS-induced MR is induced by the IS-breaking in theconductor but the TRS-breaking in leads. By generalizing this symmetry condition in a two-terminal device, we can distribute the IS- and TRS-breaking to any of three regimes, includingtwo leads and the conductor, to induce the nonreciprocity. We note that the magnetic leadscan also be antiferromagnetic since some noncollinear antiferromagnets can also generate spin-polarized current ( ). Also, the OPE provides one possible microscopic scenario for EMChAby revealing the role of the quantum orbital and SOC. Beyond the Chiral Structure
We point out that the OPE can also generate nontrivial spin-transport phenomena in non-helicaland even non-chiral systems. The OPE is caused by orbital texture, which only requires theIS-breaking if TRS exists. The chiral structure represents a strong case of the inversion-breaking.F thee, the induced orbital polarization may differ from the current direction, depending on theway of inversion-breaking.Take an achiral chain for example, see
Figure 4a . It is periodic along the z axis and hasmirror reflection for x . The mirror symmetry forces orbitals L z , L y to vanish but allows theexistence of L x . In the band structure, the orbital texture refers to the locking between L x and k z (see Figure 4b). If leads are nonmagnetic, OPE induces nonzero G S x rather than G S z in thepresence of SOC from the contact. If leads exhibit magnetization along the x direction, OPEinduces the MR as reversing the magnetization. This model shows that the spin polarization doesnot necessarily align with the current flow. For a general noncentrosymmetric material (chiral orachiral), the orbital polarization depends on the specific symmetry. Therefore, we can engineerthe geometric atomic structure to tailor the direction and magnitude of the spin-polarization.The OPE-induced nonreciprocal MR in noncentrosymmetric systems coincides with the factthat EMChA was recently been generalized to ordinary IS-breaking materials for example, SiFET ( ) and the polar semiconductor BiTeBr ( ) and also predicted for noncentrosymmetricWeyl semimetals ( ). 12igure 4: The orbital and spin polarization in an achiral system. (a) The device structure. Thecenter region represents a molecule that has reflection symmetry but breaks inversion symmetry.(b) Corresponding band structure with L x orbital polarization. (c) The conductance is bothorbital and spin-polarized when leads are nonmagnetic. The spin-polarization is along the x direction rather than the z axis. (d) It also exhibits magnetoresistance (the dephasing η = 0 . )as switching on the magnetization from the + x to − x directions in leads.13 ummary In summary, our theory brings the missing block, the orbital degree of freedom, to understandthe consequence of chiral atomic structures. The orbital polarization effect circumvents theweak SOC in organic molecules and explains the robustness of the CISS-induced phenomena,by the intense orbital texture in the molecule and the large SOC in the lead. The orbital textureprovides an insightful quantity in the band structure to estimate the CISS effect for real materials.Additionally, our work resolves the debate on the dephasing. We found that the dephasing isunnecessary to generate the spin current when employing multi-channel leads while it is requiredto induce the MR and magnetic chiral selection. The nonreciprocal conductance can lead tocurrent rectification and may be applied for photodetection or energy harvesting. From the OPE,we can deduce the EMChA independently, which refers to the nonreciprocal MR observed in thesolid-state materials. Beyond helical molecules and even beyond the chiral structure, the OPEpaves a way to manipulate the spin polarization by engineering the atomic structure in generalnoncentrosymmetric materials. Since the chirality is a common feature of many chemical andmost biochemical systems, possibly the extent of OPE may be larger than one can imagine fromthe CISS, which calls for further investigations. Our work may provide a topological perspectiveto understand the fundamental role of chirality in the biological ( ) and chemical (
38, 95 )systems.
Methods
We calculate the band structure of helix by the density-functional theory within the general-ized gradient approximation ( ) using the the Vienna Ab initio Simulation Package (VASP) ( ).The orbital moment L z is extracted from the phase-dependent atomic-orbital projection of theBloch wave function. Information for the transport calculations can be found in the SM. Supplementary Materials
Supplementary material for this article is available at http://XXX.Section S1. Tight-binding model of the helical chain.Section S2 and Figure S1. Anisotropic hopping along the chiral chainSection S3 and Figure S2. Model parameters for the transport calculations.Figure S3. Band structures of the lead, the SOC region and the chiral molecule.Figure S4. Verification of the global Onsager’s reciprocal relation.Figure S5. Orbital channel-specific conductance of the chiral chain.Figure S6. Orbital channel-specific conductance of the achiral chain.Figure S7. Influence of dephasing parameter η on spin conductance.Figure S8. The orbital conserved lead and non-conserved lead in the achiral chain device.14 eferences and Notes Acknowledgements:
We thank gratefully the advice and help from Ady Stern and Yuval Oreg.We also acknowledge inspiring discussions with Ron Naaman, Yossi Paltiel, Zhong Wang,ChiYung Yam, Lukas Muechler, Tobias Holder, Raquel Queiroz, Karen Michaelia, Leeor Kronik,Shuichi Murakami, Eugene J. Mele and Claudia Felser. B.Y. honors the memory of ShouchengZhang who inspired him to study the chirality in both physics and biology. B.Y. acknowledgesthe financial support by the Willner Family Leadership Institute for the Weizmann Institute ofScience, the Benoziyo Endowment Fund for the Advancement of Science, Ruth and HermanAlbert Scholars Program for New Scientists, the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (Grant No. 815869).
Competing Interests:
The authors declare that they have no competing interests.
Data and materials availability:
All data needed to evaluate the conclusions in the paper arepresent in the paper and/or the Supplementary Materials. Additional data related to this papermay be requested from the authors. ∗ Both authors contributed equally to this work. † Email of correspondence: [email protected]
References
1. J. S. Siegel, Homochiral imperative of molecular evolution.
Chirality , 24–27 (1998).2. L. Ma, C. Abney, W. Lin, Enantioselective catalysis with homochiral metal–organic frame-works. Chemical Society Reviews , 1248–1256 (2009).3. E. Francotte, W. Lindner, Chirality in drug research , vol. 33 (Wiley-VCH Weinheim,Weinheim, Germany, 2006).4. H. Weyl, Gravitation and the electron.
Proceedings of the National Academy of Sciences ofthe United States of America , 323 (1929).5. M. Goldhaber, L. Grodzins, A. W. Sunyar, Helicity of neutrinos. Phys. Rev. , 1015–1017(1958).6. B. Gohler, V. Hamelbeck, T. Z. Markus, M. Kettner, G. F. Hanne, Z. Vager, R. Naaman,H. Zacharias, Spin Selectivity in Electron Transmission Through Self-Assembled Monolay-ers of Double-Stranded DNA.
Science , 894 – 897 (2011).7. R. Naaman, D. H. Waldeck, Chiral-Induced Spin Selectivity Effect.
The Journal of PhysicalChemistry Letters , 2178–2187 (2012). 15. R. Naaman, Y. Paltiel, D. H. Waldeck, Chiral molecules and the electron spin. NatureReviews Chemistry , 250–260 (2019).9. K. Ray, S. P. Ananthavel, D. H. Waldeck, R. Naaman, Asymmetric Scattering of PolarizedElectrons by Organized Organic Films of Chiral Molecules. Science , 814–816 (1999).10. I. Carmeli, V. Skakalova, R. Naaman, Z. Vager, Magnetization of Chiral Monolayers ofPolypeptide: A Possible Source of Magnetism in Some Biological Membranes We are grate-ful to Prof. M. Fridkin and his group for helping us in the synthesis of the polyalanine. Partialsupport from the USIsrael Binational Science Foundation is acknowledged.
AngewandteChemie International Edition , 761 (2002).11. D. Mishra, T. Z. Markus, R. Naaman, M. Kettner, B. Ghler, H. Zacharias, N. Friedman,M. Sheves, C. Fontanesi, Spin-dependent electron transmission through bacteriorhodopsinembedded in purple membrane. Proceedings of the National Academy of Sciences ,14872–14876 (2013).12. M. Kettner, B. Goehler, H. Zacharias, D. Mishra, V. Kiran, R. Naaman, C. Fontanesi, D. H.Waldeck, S. Sek, J. Pawaowski, J. Juhaniewicz, Spin Filtering in Electron Transport ThroughChiral Oligopeptides.
The Journal of Physical Chemistry C , 14542–14547 (2015).13. T. J. Zwang, S. Hrlimann, M. G. Hill, J. K. Barton, Helix-Dependent Spin Filtering throughthe DNA Duplex.
Journal of the American Chemical Society , 15551–15554 (2016).14. M. Kettner, V. V. Maslyuk, D. Nrenberg, J. Seibel, R. Gutierrez, G. Cuniberti, K.-H. Ernst,H. Zacharias, Chirality-Dependent Electron Spin Filtering by Molecular Monolayers ofHelicenes.
The Journal of Physical Chemistry Letters , 2025–2030 (2018).15. M. Eckshtain-Levi, E. Capua, S. Refaely-Abramson, S. Sarkar, Y. Gavrilov, S. P. Mathew,Y. Paltiel, Y. Levy, L. Kronik, R. Naaman, Cold denaturation induces inversion of dipoleand spin transfer in chiral peptide monolayers. Nature communications , 10744 (2016).16. C. Kulkarni, A. K. Mondal, T. K. Das, G. Grinbom, F. Tassinari, M. F. J. Mabesoone,E. W. Meijer, R. Naaman, Highly Efficient and Tunable Filtering of Electrons’ Spin bySupramolecular Chirality of Nanofiber-Based Materials. Advanced Materials , 1904965(2020).17. F. Tassinari, J. Steidel, S. Paltiel, C. Fontanesi, M. Lahav, Y. Paltiel, R. Naaman, Enan-tioseparation by crystallization using magnetic substrates. Chemical Science , 5246–5250(2019).18. H. Lu, J. Wang, C. Xiao, X. Pan, X. Chen, R. Brunecky, J. J. Berry, K. Zhu, M. C. Beard, Z. V.Vardeny, Spin-dependent charge transport through 2D chiral hybrid lead-iodide perovskites. Science Advances , eaay0571 (2019). 169. A. Inui, R. Aoki, Y. Nishiue, K. Shiota, Y. Kousaka, H. Shishido, D. Hirobe, M. Suda, J.-i.Ohe, J.-i. Kishine, H. M. Yamamoto, Y. Togawa, Chirality-induced spin-polarized state of achiral crystal crnb s . Phys. Rev. Lett. , 166602 (2020).20. G. L. J. A. Rikken, A New Twist on Spintronics.
Science , 864–865 (2011).21. Z. Xie, Z. Xie, T. Z. Markus, S. R. Cohen, Z. Vager, R. Gutierrez, R. Naaman, Spin specificelectron conduction through DNA oligomers.
Nano letters , 4652–5 (2011).22. S. P. Mathew, P. C. Mondal, H. Moshe, Y. Mastai, R. Naaman, Non-magnetic or-ganic/inorganic spin injector at room temperature. Applied Physics Letters , 242408(2014).23. R. Naaman, D. H. Waldeck, Spintronics and Chirality: Spin Selectivity in Electron TransportThrough Chiral Molecules.
Annual Review of Physical Chemistry , 263 – 281 (2015).24. V. Kiran, S. P. Mathew, S. R. Cohen, I. H. Delgado, J. Lacour, R. Naaman, Helicenes-ANew Class of Organic Spin Filter. Advanced Materials , 1957–1962 (2016).25. O. B. Dor, S. Yochelis, A. Radko, K. Vankayala, E. Capua, A. Capua, S.-H. Yang, L. T.Baczewski, S. S. P. Parkin, R. Naaman, Y. Paltiel, Magnetization switching in ferromagnetsby adsorbed chiral molecules without current or external magnetic field. Nature Communi-cations , 14567 (2017).26. H. Al-Bustami, G. Koplovitz, D. Primc, S. Yochelis, E. Capua, D. Porath, R. Naaman,Y. Paltiel, Single nanoparticle magnetic spin memristor. Small , 1801249 (2018).27. V. Varade, T. Markus, K. Vankayala, N. Friedman, M. Sheves, D. H. Waldeck, R. Naaman,Bacteriorhodopsin based non-magnetic spin filters for biomolecular spintronics. PhysicalChemistry Chemical Physics , 1091–1097 (2018).28. E. Z. B. Smolinsky, A. Neubauer, A. Kumar, S. Yochelis, E. Capua, R. Carmieli, Y. Paltiel,R. Naaman, K. Michaeli, Electric Field-Controlled Magnetization in GaAs/AlGaAs Het-erostructuresChiral Organic Molecules Hybrids. The Journal of Physical Chemistry Letters , 1139–1145 (2019).29. L. Jia, C. Wang, Y. Zhang, L. Yang, Y. Yan, Efficient Spin Selectivity in Self-AssembledSuperhelical Conducting Polymer Microfibers. ACS Nano (2020).30. S.-H. Yang, Spintronics on chiral objects.
Applied Physics Letters , 120502 (2020).31. T. Liu, X. Wang, H. Wang, G. Shi, F. Gao, H. Feng, H. Deng, L. Hu, E. Lochner,P. Schlottmann, S. v. Molnr, Y. Li, J. Zhao, P. Xiong, Spin selectivity through chiralpolyalanine monolayers on semiconductors. arXiv 2001.00097 (2019).172. W. Mtangi, V. Kiran, C. Fontanesi, R. Naaman, Role of the electron spin polarization inwater splitting.
The journal of physical chemistry letters , 4916–4922 (2015).33. W. Mtangi, F. Tassinari, K. Vankayala, A. Vargas Jentzsch, B. Adelizzi, A. R. Palmans,C. Fontanesi, E. Meijer, R. Naaman, Control of electrons spin eliminates hydrogen peroxideformation during water splitting. Journal of the American Chemical Society , 2794–2798(2017).34. P. C. Mondal, W. Mtangi, C. Fontanesi, Chiro-Spintronics: Spin-Dependent Electrochemistryand Water Splitting Using Chiral Molecular Films.
Small Methods , 1700313 (2018).35. F. Tassinari, K. Banerjee-Ghosh, F. Parenti, V. Kiran, A. Mucci, R. Naaman, Enhancedhydrogen production with chiral conductive polymer-based electrodes. The Journal ofPhysical Chemistry C , 15777–15783 (2017).36. K. B. Ghosh, W. Zhang, F. Tassinari, Y. Mastai, O. Lidor-Shalev, R. Naaman, P. Mllers,D. Nrenberg, H. Zacharias, J. Wei, E. Wierzbinski, D. H. Waldeck, Controlling ChemicalSelectivity in Electrocatalysis with Chiral CuO-Coated Electrodes.
The Journal of PhysicalChemistry C , 3024–3031 (2019).37. K. Banerjee-Ghosh, O. B. Dor, F. Tassinari, E. Capua, S. Yochelis, A. Capua, S.-H. Yang,S. S. P. Parkin, S. Sarkar, L. Kronik, L. T. Baczewski, R. Naaman, Y. Paltiel, Separation ofenantiomers by their enantiospecific interaction with achiral magnetic substrates.
Science , 1331–1334 (2018).38. T. S. Metzger, S. Mishra, B. P. Bloom, N. Goren, A. Neubauer, G. Shmul, J. Wei, S. Yochelis,F. Tassinari, C. Fontanesi, D. H. Waldeck, Y. Paltiel, R. Naaman, The Electron Spin as aChiral Reagent.
Angewandte Chemie International Edition , 1653–1658 (2020).39. D. M. Stemer, J. M. Abendroth, K. M. Cheung, M. Ye, M. S. E. Hadri, E. E. Fullerton,P. S. Weiss, Differential Charging in Photoemission from Mercurated DNA Monolayers onFerromagnetic Films. Nano Letters , 1218–1225 (2020).40. S. Ghosh, S. Mishra, E. Avigad, B. P. Bloom, L. T. Baczewski, S. Yochelis, Y. Paltiel,R. Naaman, D. H. Waldeck, Effect of Chiral Molecules on the Electrons’ Spin Wavefunctionat Interfaces. The Journal of Physical Chemistry Letters , 1550–1557 (2020).41. S. Yeganeh, M. A. Ratner, E. Medina, V. Mujica, Chiral electron transport: Scatteringthrough helical potentials. The Journal of Chemical Physics , 014707 (2009).42. A.-M. Guo, Q.-f. Sun, Spin-Selective Transport of Electrons in DNA Double Helix.
PhysicalReview Letters , 218102 (2012).43. A.-M. Guo, Q.-f. Sun, Sequence-dependent spin-selective tunneling along double-strandedDNA.
Physical Review B , 115441 (2012).184. R. Gutierrez, E. D´ıaz, R. Naaman, G. Cuniberti, Spin-selective transport through helicalmolecular systems. Phys. Rev. B , 081404 (2012).45. R. Gutierrez, E. Diaz, C. Gaul, T. Brumme, F. Dominguez-Adame, G. Cuniberti, ModelingSpin Transport in Helical Fields: Derivation of an Effective Low-Dimensional Hamiltonian. The Journal of Physical Chemistry C , 22276–22284 (2013).46. J. Gersten, K. Kaasbjerg, A. Nitzan, Induced spin filtering in electron transmission throughchiral molecular layers adsorbed on metals with strong spin-orbit coupling.
The Journal ofChemical Physics , 114111 (2013).47. A.-M. Guo, E. D´ıaz, C. Gaul, R. Gutierrez, F. Dom´ınguez-Adame, G. Cuniberti, Q.-f. Sun,Contact effects in spin transport along double-helical molecules.
Phys. Rev. B , 205434(2014).48. A.-M. Guo, Q.-F. Sun, Spin-dependent electron transport in protein-like single-helicalmolecules. Proceedings of the National Academy of Sciences , 11658–11662 (2014).49. E. Medina, L. A. Gonzlez-Arraga, D. Finkelstein-Shapiro, B. Berche, V. Mujica, Continuummodel for chiral induced spin selectivity in helical molecules.
The Journal of ChemicalPhysics , 194308 (2015).50. S. Matityahu, Y. Utsumi, A. Aharony, O. Entin-Wohlman, C. A. Balseiro, Spin-dependenttransport through a chiral molecule in the presence of spin-orbit interaction and nonunitaryeffects.
Physical Review B , 075407 (2016).51. S. Varela, V. Mujica, E. Medina, Effective spin-orbit couplings in an analytical tight-bindingmodel of DNA: Spin filtering and chiral spin transport. Physical Review B , 155436(2016).52. T.-R. Pan, A.-M. Guo, Q.-F. Sun, Spin-polarized electron transport through helicene molecu-lar junctions. Physical Review B , 235448 (2016).53. K. Michaeli, V. Varade, R. Naaman, D. H. Waldeck, A new approach towards spintronics–spintronics with no magnets. Journal of Physics: Condensed Matter , 103002 (2017).54. S. Matityahu, A. Aharony, O. Entin-Wohlman, C. A. Balseiro, Spin filtering in all-electricalthree-terminal interferometers. Physical Review B , 085411 (2017).55. V. V. Maslyuk, R. Gutierrez, A. Dianat, V. Mujica, G. Cuniberti, Enhanced Magnetoresis-tance in Chiral Molecular Junctions. The Journal of Physical Chemistry Letters , 5453–5459(2018).56. E. D´ıaz, A. Contreras, J. Hern´andez, F. Dom´ınguez-Adame, Effective nonlinear model forelectron transport in deformable helical molecules. Phys. Rev. E , 052221 (2018).197. E. D´ıaz, F. Dom´ınguez-Adame, R. Gutierrez, G. Cuniberti, V. Mujica, Thermal decoherenceand disorder effects on chiral-induced spin selectivity. The journal of physical chemistryletters , 5753–5758 (2018).58. K. Michaeli, R. Naaman, Origin of Spin-Dependent Tunneling Through Chiral Molecules. The Journal of Physical Chemistry C , 17043–17048 (2019).59. D. Nuerenberg, H. Zacharias, Evaluation of spin-flip scattering in chirality-induced spinselectivity using the Riccati equation.
Physical Chemistry Chemical Physics , 3761–3770(2019).60. X. Yang, C. H. v. d. Wal, B. J. v. Wees, Spin-dependent electron transmission model forchiral molecules in mesoscopic devices. Physical Review B , 024418 (2019).61. S. Dalum, P. Hedegrd, Theory of Chiral Induced Spin Selectivity. Nano Letters , 5253–5259 (2019).62. X. Yang, C. H. v. d. Wal, B. J. v. Wees, Detecting chirality in two-terminal electronic devices. arXiv 1912.09085 (2019).63. X. Li, J. Nan, X. Pan, Chiral Induced Spin Selectivity as a Spontaneous Intertwined Order. arXiv 2005.03656 (2020).64. J. Fransson, Vibrational Origin of Exchange Splitting and Chirality Induced Spin Selectivity. arXiv 2005.06753 (2020).65. A. Ghazaryan, Y. Paltiel, M. Lemeshko, Analytic Model of Chiral-Induced Spin Selectivity. The Journal of Physical Chemistry C (2020).66. S. Varela, I. Zambrano, B. Berche, V. Mujica, E. Medina, Spin-orbit interaction and spinselectivity for tunneling electron transfer in DNA. arXiv 2003.00582 (2020).67. J. D. Torres, R. Hidalgo, S. Varela, E. Medina, Mechanically modulated spin orbit couplingsin oligopeptides. arXiv 2005.08474 (2020).68. A. Shitade, E. Minamitani, Geometric spin-orbit coupling and chirality-induced spin selec-tivity. arXiv preprint arXiv:2002.05371 (2020).69. Y. Utsumi, O. Entin-Wohlman, A. Aharony, Spin selectivity through time-reversal symmetrichelical junctions. arXiv 2005.04041 (2020).70. F. Kuemmeth, S. Ilani, D. C. Ralph, P. L. McEuen, Coupling of spin and orbital motion ofelectrons in carbon nanotubes.
Nature , 448–452 (2008).71. G. A. Steele, F. Pei, E. A. Laird, J. M. Jol, H. B. Meerwaldt, L. P. Kouwenhoven, Largespin-orbit coupling in carbon nanotubes.
Nature Communications , 1573 (2013).202. B. Yan, C. Felser, Topological Materials: Weyl Semimetals. Annu. Rev. Cond. Mat. Phys. ,337 – 354 (2017).73. N. P. Armitage, E. J. Mele, A. Vishwanath, Weyl and Dirac semimetals in three-dimensionalsolids. Rev. Mod. Phys. , 015001 (2018).74. G. L. J. A. Rikken, J. Flling, P. Wyder, Electrical Magnetochiral Anisotropy. PhysicalReview Letters , 236602 (2001).75. F. Tassinari, D. R. Jayarathna, N. Kantor-Uriel, K. L. Davis, V. Varade, C. Achim, R. Naaman,Chirality Dependent Charge Transfer Rate in Oligopeptides. Advanced Materials ,1706423 (2018).76. S. A. Parameswaran, A. M. Turner, D. P. Arovas, A. Vishwanath, Topological order andabsence of band insulators at integer filling in non-symmorphic crystals. Nature Physics ,299 – 303 (2013).77. S. M. Young, C. L. Kane, Dirac semimetals in two dimensions. Phys. Rev. Lett. , 126803(2015).78. J. C. Slater, G. F. Koster, Simplified lcao method for the periodic potential problem.
Phys.Rev. , 1498–1524 (1954).79. M. B ¨uttiker, Four-terminal phase-coherent conductance. Phys. Rev. Lett. , 1761–1764(1986).80. C. W. Groth, M. Wimmer, A. R. Akhmerov, X. Waintal, Kwant: a software package forquantum transport. New Journal of Physics , 063065 (2014).81. A. A. Kiselev, K. W. Kim, Prohibition of equilibrium spin currents in multiterminal ballisticdevices. Physical Review B , 153315 (2005).82. J. H. Bardarson, A proof of the Kramers degeneracy of transmission eigenvalues fromantisymmetry of the scattering matrix. Journal of Physics A: Mathematical and Theoretical , 405203 (2008).83. M. Bttiker, Role of quantum coherence in series resistors. Physical Review B , 3020–3026(1986).84. S. Mishra, A. K. Mondal, S. Pal, T. K. Das, E. Z. B. Smolinsky, G. Siligardi, R. Naaman,Length-Dependent Electron Spin Polarization in Oligopeptides and DNA. The Journal ofPhysical Chemistry C , 10776–10782 (2020).85. M. B¨uttiker, Symmetry of electrical conduction.
IBM Journal of Research and Development , 317–334 (1988). 216. R. Naaman, D. H. Waldeck, Comment on “Spin-dependent electron transmission model forchiral molecules in mesoscopic devices”. Physical Review B , 026403 (2020).87. X. Yang, C. H. van der Wal, B. J. van Wees, Reply to “comment on ‘spin-dependent electrontransmission model for chiral molecules in mesoscopic devices”’.
Phys. Rev. B , 026404(2020).88. Y. Tokura, N. Nagaosa, Nonreciprocal responses from non-centrosymmetric quantum mate-rials.
Nature Communications , 3740 (2018).89. G. L. J. A. Rikken, N. Avarvari, Strong electrical magnetochiral anisotropy in tellurium. Phys. Rev. B , 245153 (2019).90. J. Zelezny, Y. Zhang, C. Felser, B. Yan, Spin-Polarized Current in Noncollinear Antiferro-magnets. Physical review letters , 187204 (2017).91. G. L. J. A. Rikken, P. Wyder, Magnetoelectric Anisotropy in Diffusive Transport.
PhysicalReview Letters , 016601 (2005).92. T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu, Y. Kaneko, Y. Tokura,N. Nagaosa, Y. Iwasa, Bulk rectification effect in a polar semiconductor. Nature Physics ,578–583 (2017).93. T. Morimoto, N. Nagaosa, Chiral Anomaly and Giant Magnetochiral Anisotropy in Noncen-trosymmetric Weyl Semimetals. Physical Review Letters , 146603 (2016).94. A. Kumar, E. Capua, M. K. Kesharwani, J. M. L. Martin, E. Sitbon, D. H. Waldeck,R. Naaman, Chirality-induced spin polarization places symmetry constraints on biomolecularinteractions.
Proceedings of the National Academy of Sciences , 2474–2478 (2017).95. M. Innocenti, M. Passaponti, W. Giurlani, A. Giacomino, L. Pasquali, R. Giovanardi,C. Fontanesi, Spin dependent electrochemistry: Focus on chiral vs achiral charge trans-mission through 2D SAMs adsorbed on gold.
Journal of Electroanalytical Chemistry ,113705 (2020).96. J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple.
Physical Review Letters , 3865 (1996).97. G. Kresse, J. Furthm¨uller, Efficient iterative schemes for ab initio total-energy calculationsusing a plane-wave basis set. Physical Review B , 11169 (1996).22 upplementary Materials for “Chirality Induced Quantum Orbital Polarization Effectin Chiral Molecules” Yizhou Liu ∗ , Jiewen Xiao ∗ , Jahyun Koo, Binghai Yan † Department of Condensed Matter Physics,Weizmann Institute of Science, Rehovot 76100, Israel
S1. TIGHT-BINDING MODEL
For the helix model with the three-fold skew rotation, there are three atomic sites ( R cos( i − π/ , R sin( i − π/ , z ( i − i = 1 , ,
3) within the unit cell, where i is the site number. We set the orbital bases p x , p y , p z oneach site. The onsite energy of p orbitals are set to zero while the nearest neighboring hopping is modelled by theSlater-Koster hopping. Therefore, the spinless Hamiltonian H ( k z ) can be expressed in the p x,y,z orbitals of threeatomic sites, H ( k z ) = T exp ( − ik z · a/ T exp (+ ik z · a/ T exp (+ ik z · a/
3) 0 T exp ( − ik z · a/ T exp ( − ik z · a/ T exp (+ ik z · a/
3) 0 , (S1)where 1 , , k z and a are the wave vector and lattice constant in the z direction, respectively. T ij is the nearest-neighbor hopping matrix from site i to site j and can be written as T ij ( φ ij , θ ij ) = t ij,xx t ij,xy t ij,xz t ij,yx t ij,yy t ij,yz t ij,zx t ij,zy t ij,zz (S2) t ij,xx = t π sin φ ij + cos φ ij ( t σ sin θ ij + t π cos θ ij ) t ij,yy = t π cos φ ij + sin φ ij ( t σ sin θ ij + t π cos θ ij ) t ij,zz = t σ cos θ ij + t π sin θ ij t ij,xy = t ij,yx = sin φ ij cos φ ij ( t σ sin θ ij − t π cos θ ij ) t ij,xz = t ij,xz = cos φ ij sin θ ij cos θ ij ( t σ − t π ) t ij,yz = t ij,zy = sin φ ij sin θ ij cos θ ij ( t σ − t π ) (S3)Here, φ ij and θ ij are the spherical coordinates of site j relative to site i . For the helix molecule, θ ij is set to ± π/ z = 2 √ R ), and φ ij can adopt ± π/ , ± π/ , ± π/ t σ and t π are 1.5 eVand -0.5 eV respectively. With the above parameter, band structure is calculated and shown in Figure 1(b). S2. ANISOTROPIC HOPPING ALONG THE CHIRAL CHAIN
In the last section, we derive the hopping matrix from site i to site j under p x , p y , p z bases. To understand thechiral selection from the anisotropic hopping, we write T ij under the p + , p , p − orbital basis, T ij ( φ ij , θ ij ) = t ij, ++ t ij, +0 t ij, + − t ij, t ij, t ij, − t ij, − + t ij, − t ij, −− (S4) t ij, ++ = ( t π (1 + cos θ ij ) + t σ sin θ ij ) / t ij, = t σ cos θ ij + t π sin θ ij t ij, −− = ( t π (1 + cos θ ij ) + t σ sin θ ij ) / t ij, +0 = t ij, − = (( t σ − t π ) sin θ ij cos θ ij exp( − iφ ij )) / √ t ij , + = t ij, − = (( t σ − t π ) sin θ ij cos θ ij exp(+ iφ ij )) / √ t ij, + − = (( t σ − t π ) sin θ ij exp( − i φ ij )) / t ij, − + = (( t σ − t π ) sin θ ij exp(+ i φ ij )) / i toadjacent site i − i + 1, and such process can be denoted as ’d’ (down) and ’u’ (up) respectively. Therefore,the hopping matrix is written as T d = T i ( i − ( π − φ/ , − θ ) and T u = T i ( i +1) ( φ/ , θ ), and their relation can be furtherderived: T u = t u ++ t u +0 t u + − t u t u t u − t u − + t u − t u −− = t d ++ · e − i π · ∆ L z ++ /n t d +0 · e − i π · ∆ L z +0 /n t d + − · e − i π · ∆ L z + − /n t d · e − i π · ∆ L z /n t d · e − i π · ∆ L z /n t d − · e − i π · ∆ L z − /n t d − + · e − i π · ∆ L z − + /n t d − · e − i π · ∆ L z − /n t d −− · e − i π · ∆ L z −− /n (S6)Therefore, suppose there are two orbitals p and q , the phase factor that connects the hopping term t upq and t dpq canbe expressed as e − i π · ∆ L zpq /n , where ∆ L zpq is the z-component angular momentum ( L z ) differences between orbital p and orbital q . For the intra-orbital and inter-orbital hopping, we further have: t u ++ = t u −− = t d ++ = t d −− (S7) t u = t d (S8) t u + − = t d + − e − i π/n (S9) t u +0 = t u − = t d +0 e − i π/n = t d − e − i π/n (S10)To validate the above T u and T d relation expressed by the differences of angular momentum, we further consider the d -orbital hopping in the chiral chain, with L z = 2 , , , − , −
2. Similarly, the hopping matrix from site i to site j under the d i , d i , d i , d − i , d − i orbital basis can be written as: T ij ( φ ij , θ ij ) = t ij − − t ij − − t ij − t ij − t ij − t ij − − t ij − − t ij − t ij − t ij − t ij − t ij − t ij t ij t ij t ij − t ij − t ij t ij t ij t ij − t ij − t ij t ij t ij (S11) t ij = t ij − − = (3 t σ − t π + t δ ) sin θ/ t π sin θ + t δ cos θt ij = t ij − − = (3 t σ − t π + t δ ) sin θ cos θ/ t π (1 + cos θ ) + t δ sin θt ij = t σ (sin θ − cos θ ) / t π sin θ cos θ + 3 t δ sin θ/ FIG. S1. Anisotropic hopping in the chiral chain. For the right hand helical chain, the hopping process from site i to i − i to site i + 1 can be denoted as ’d’ (down) and ’u’ (up), respectively. t ij = t † ij = − t ij − − = − t † ij − − = (4( t δ − t π ) sin 2 θ − (3 t σ − t π + t δ ) sin θ sin 2 θ ) e − iφ t ij = t † ij = t ij − = t † ij − = ( √ θ/ t σ (cos θ − sin θ ) − t π cos θ + t δ (1 + cos θ )) e − i φ t ij − = t † ij − = − t ij − = − t † ij − = (sin 2 θ sin θ/ t σ − t π + t δ ) e − i φ t ij − = t † ij − = (sin θ/ t σ − t π + t δ ) e − i φ t ij = t † ij = − t ij − = − t † ij − = ( p (3) sin θ cos θ/ t σ (2 cos θ − sin θ ) − t π cos 2 θ + t δ sin θ ) e − iφ t ij − = t † ij − = (sin θ/ t δ − t π ) − (3 t σ − t π + t δ ) cos θ ) e − i φ (S13)With T d = T i ( i − ( π − φ/ , − θ ) and T u = T i ( i +1) ( φ/ , θ ), the general relation between the matrix element t upq in T u and t dpq in T d can be written as (where p and q are d orbitals, and ∆ L zpq is the L z difference between them): t upq = t dpq exp ( − i π · ∆ L zpq /n ) (S14) S3. MODEL PARAMETERS FOR THE TRANSPORT CALCULATIONS
Hopping parameters for the two terminal device are presented in Figure S2, and their values are specified in theattached codes. To calculate the orbital channel-specific conductance of the achiral chain, we set the isotropic hopping t in leads. We also test the spin channel-specific conductance for both the isotropic hopping ( t = t σ = t π = 1 . L x conserved) and the general anisotropic hopping ( t σ = 1 . t π = 1 . L x non-conserved) inleads, and results in Figure S6 display the similar feature. FIG. S2. Hopping parameters for the two-terminal device: (a) chiral molecule, and (b) achiral molecule.FIG. S3. Band structures of the lead, the SOC region and the chiral molecule. The chiral molecule is represented by theright-hand helix model discussed above and related hopping parameters and SOC strength are detailed in the section S3.FIG. S4. Verification of the global Onsager’s reciprocal relation. When dephasing term η is set to zero, ∆ G = G L → R ( M ) − G R → L ( − M ) = 0. FIG. S5. Orbital channel-specific conductance of the chiral chain. Nine kinds of conductance from L + , L , L − orbital channelin the left lead to the L + , L , L − orbital channel in the right lead are presented (SOC strength is set to zero.) FIG. S6. Orbital channel-specific conductance for the achiral chain. Nine kinds of conductance from L + , L , L − orbital channelin the left lead to the L + , L , L − orbital channel in the right lead are presented (SOC strength is set to zero.) FIG. S7. Influence of dephasing parameter η on spin conductance G S z of (a)-(b) multiple-mode leads and (c)-(d) single-modeleads, respectively. (a), (c) G S z as function of energy for various η . (b), (d) G S z as function of η and λ SOC at fixed energy(indicated by the arrows in (a) and (c), respectively). The single-mode leads are created by setting t leads π = 0 so that only p z modes (with two spins) contribute to spin and charge transport. For single-mode leads a finite dephasing is needed to generatenonzero G S z whereas for multiple-mode leads it is not necessary.FIG. S8. Spin conductance G Sx for the L xx