Interplay between friction and spin-orbit coupling as a source of spin polarization
Artem G. Volosniev, Hen Alpern, Yossi Paltiel, Oded Millo, Mikhail Lemeshko, Areg Ghazaryan
IInterplay between friction and spin-orbit coupling as a source of spin polarization
Artem G. Volosniev, Hen Alpern,
2, 3
Yossi Paltiel, Oded Millo, Mikhail Lemeshko, and Areg Ghazaryan IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria Applied Physics Department and the Center for Nanoscience and Nanotechnology,The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Racah Institute of Physics and the Center for Nanoscience and Nanotechnology,The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
We study an effective one-dimensional quantum model that includes friction and spin-orbit cou-pling (SOC), and show that the model exhibits spin polarization when both terms are finite. Mostimportant, strong spin polarization can be observed even for moderate SOC, provided that frictionis strong. Our findings might help to explain the pronounced effect of chirality on spin distributionand transport in chiral molecules. In particular, our model implies static magnetic properties of achiral molecule, which lead to Shiba-like states when a molecule is placed on a superconductor, inaccordance with recent experimental data.
One open question in quantum dissipation is how fric-tion manifests itself in systems with spin-orbit coupling,such as an electron in an insulating medium. Dissipa-tive systems do not obey time-reversal symmetry, andhence, they are beyond the applicability region of the cel-ebrated Kramers degeneracy theorem. This means thatfriction may lead to the essential change in the behaviorof systems with spin. Our work is set to study the SOC-friction interplay in a basic one-dimensional setting. Weillustrate our general findings by discussing the spin sen-sitivity of chiral molecules – a phenomenon observed inmany experiments, which still requires a satisfying theo-retical explanation. A number of theoretical models havebeen developed [1–17] to understand this sensitivity inconnection to the chiral induced spin selectivity (CISS)effect [18–20]. However, certain key questions remainunanswered. In particular, the CISS effect is too pro-nounced to be accounted for by weak SOC of light atomsthat constitute organic molecules. In our work, SOC andfriction act in unison, so that weak SOC can effectivelybe enhanced by dissipation, and lead to an efficient spinpolarizer.In the absence of dissipation, the standard route tostudy the motion of a particle in a medium is to mapthe problem onto some effective single-particle Hamilto-nian. This approach is at the heart of our understandingof electrons in materials [21]. There is no simple strat-egy of adding dissipation to such effective models, sincea naive quantization of classical dissipative terms is in-compatible with the Heisenberg uncertainty relation, ascan be demonstrated using a simple harmonic-oscillatormodel [22]. It is possible to include a quantum analogueof the classical frictional force in time evolution of theone-body density matrix [23]. The corresponding termcan either be derived using solvable models (e.g., theCaldeira-Leggett model [24]) or matched to the classi-cal Stokes’ drag, − γp . A somewhat simpler approach isto work with dissipative Schr¨odinger equations [25–28],which are constructed from the classical limit and satisfycertain conditions, e.g., the Heisenberg uncertainty prin- e e No friction or spin-orbit coupling Friction and spin-orbit coupling e ee bac d ee ee eeee e FIG. 1. (a) and (b) An illustration of the effective model in-troduced in Eq. 1. (a) Without friction ( γ = 0) or spin-orbitcoupling ( α = 0), the spatial distribution of spins in a quan-tum wire is mirror symmetric. (b) If both terms are included,spin-up and spin-down particles move in opposite directions.(c) The average position of spins in the Albrecht’s model ofdissipation, as a function of the only parameter in the model, γα . Here, we use dimensionless quantities defined in the text.(d) For chiral molecules on a substrate, this separation mightimply magnetic properties. ciple, and the Ehrenfest theorem. This way is purely phe-nomenological and leads to a number of possible modifiedSchr¨odinger equations. However, they are compatiblewith each other and with master-equation approaches,at least for simple systems [27, 29–31].In this work, we employ a modified Schr¨odinger equa-tion to study the interplay between friction (that mayemerge from electron-bath interactions) and SOC. Weassume that the timescale of the spin coherence is verylarge, i.e., there are no spin changing interactions in thesystem (no spin-flips). This assumption allows us to useeigenstates of modified Schr¨odinger equations to under-stand the spin distribution of an electron confined in a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n a well. Our main finding is that the spin-momentumlocking leads to a position-dependent spin distribution:Spin-up particles are spatially separated from spin-downparticles. We speculate that the predicted effect hasbeen observed with chiral molecules adsorbed on vari-ous substrates. For example, it has been shown thatchiral molecules adsorbed onto s-wave superconductorsinduce triplet superconductivity [32–34], or Shiba-likestates [34]. Chiral molecules on ferromagnetic surfacesinduce magnetic ordering [35, 36]. Theoretical model.
Our work rests on the two observa-tions: (i) A spin-orbit interaction couples particle’s spinto its momentum; (ii) the classical frictional force is de-termined by the direction of the momentum. The obser-vations suggest that dissipation could potentially coupleto spin, hence, lead to spin currents, and steady statesthat do not respect the Kramers degeneracy theorem.To investigate these possibilities, we study a quantumwire, i.e., a one-dimensional (1D) confined system, withspin-orbit coupling, αpσ z , where α is the SOC ampli-tude, p is the momentum operator, and σ z is the Paulimatrix. Without loss of generality, we have SOC in the ˆ z -direction. The 1D SOC interaction does not couple spin-up and spin-down particles; we define the correspondingquantum number as s = ±
1. To include friction for agiven s , we use a phenomenological potential: γW , where γ > W ) determines the strength (spatial distribution)of dissipation [27]. W depends on the state of the system,so that the resulting time evolution is non-linear. Motionof a particle of mass m , and spin s obeys the followingequation i (cid:126) ∂ Ψ s ∂t = H s Ψ s ; H s = p m + αps + V ( x ) + γW, (1)where we assume a box potential V ( x ), i.e., V ( x ) = 0 for x ∈ ( − a, a ), and V → ∞ otherwise. A clear advantage ofEq. (1) over other models of dissipation is its simplicity.It will allow us to analyze the problem analytically.If γ = 0, then the eigenstates of Eq. (1) are Ψ s = e − i ( E t + mαsx ) / (cid:126) sin( πn ( x + a ) / (2 a )), where n is integerand E = (cid:126) π n / ma + mα / | Ψ s | is illustrated inFig. 1 (a) for n = 1. For each value of s , there is a spincurrent, which, however, does not lead to any observableflux when we average over s (cf. Ref. [37, 38]). If γ (cid:54) = 0and α = 0, s does not enter Eq. (1), and there can be nointeresting effects associated with spin.Let us now consider a system with non-vanishing SOCand friction. We focus on the Albrecht’s potential [28], W = (cid:104) p (cid:105) s ( x − (cid:104) x (cid:105) s ) , (2)where (cid:104) O (cid:105) s = (cid:82) Ψ ∗ s O Ψ s d x . Time evolution of quantumaverages that obey Eq. (1) is connected to classical timeevolution: (cid:28) dH s dt (cid:29) s = − γ (cid:104) p (cid:105) s d (cid:104) x (cid:105) s dt , d (cid:104) x (cid:105) s dt = (cid:104) p (cid:105) s m + αs, (3) which explicitly demonstrates that the potential W leadsto dissipation, and breaking of the time-reversal sym-metry. Solutions of Eqs. (3) are time-independent if (cid:104) p (cid:105) s = − αms [39]. The system is dissipative, there-fore, only the ‘lowest-energy’ steady-state of H s can beinterpreted as the fixed point of time evolution, see,e.g., Ref. [28]. Below, we explore the physical natureof this state. In what follows, we utilize dimensionlessunits: ˜ x = x/a , ˜ t = (cid:126) t/ (2 ma ), ˜ α = 2 mαa/ (cid:126) and˜ γ = 2 mγa / (cid:126) . For simplicity, we shall omit tilde.To find the steady state of the Albrecht’s model withSOC, we apply the gauge transformation: Ψ s ( x, t ) = e − iαsx − iEt f s ( x ), which transfers the SOC term intorenormalization of dissipation. We use the steady-statevalue of (cid:104) p (cid:105) s to derive the equation for f s − ∂ f s ∂x + V ( x ) f s + βxf s = E (cid:48) f s , (4)where E (cid:48) = E − α / β (cid:104) x (cid:105) s . Note that the definingparameter here is β = − γαs/
2, which means that theeffect of the linear potential in Eq. (4) can be large evenfor weak SOC. In this sense, SOC and friction are inter-changeable in our work: the steady state obtained withweak friction and strong SOC is identical to that withstrong friction and weak SOC. This happens because thefrictional force is determined by γ (cid:104) p (cid:105) s , where the valueof (cid:104) p (cid:105) s is dictated by the SOC strength, α .Equation (4) is the Airy (Stokes) equation. The corre-sponding solution reads as f s = κ Ai (cid:104) (cid:112) β (cid:0) x − x s (cid:1)(cid:105) + κ Bi (cid:104) (cid:112) β (cid:0) x − x s (cid:1)(cid:105) , (5)where x s = E (cid:48) /β , Ai( x ) and Bi( x ) are Airy functions ofthe first kind and the second kind, respectively. The pa-rameters κ , κ and E (cid:48) are determined from the boundaryconditions due to V and normalization condition; (cid:104) x (cid:105) s and E that enter E (cid:48) are determined self-consistently.The Hamiltonian (1) commutes with U P , where U = e iσ x π/ is the spin rotation and P is the inversion opera-tor. Therefore, the eigenstates for s = ± (cid:104) x (cid:105) +1 = −(cid:104) x (cid:105) − [40]. The sign of (cid:104) x (cid:105) s is deter-mined by s , and the sign of the SOC amplitude, α . Fig-ure 1 (c) shows the mean value of the position operator, (cid:104) x (cid:105) s , as a function of γα . Opposite spin states are sepa-rated in space in different regions of the system, and thisseparation increases linearly for small values of γα . Theproperty (cid:104) x (cid:105) +1 = −(cid:104) x (cid:105) − is also clearly observed. Notethat (cid:104) x (cid:105) is measured in units of a , which means that theabsolute strength of the effect is larger for longer systems,at least for systems where the spin coherence length islarger than a .If the system is initialized in some state of the Hamil-tonian H s with γ = 0, see Fig. 1 (a), then at later timesthe frictional force drives particles with different spinsinto opposite directions. Figure 1 (b) illustrates the so-lution to the static problem of Eq. (5), which we expect todefine a fixed point of time evolution. The amplitude ofthe frictional force is given by γα , and therefore, a strongdissipative system can act as an efficient spin filter evenfor weak SOC. This is not an artifact of the specific (Al-brecht’s) form of W . We have observed similar physicsfor other standard (e.g., S¨ussmann’s and Kostin’s) formsof W [41], and for traditional master-equation approachesto friction. A detailed analysis of different models withSOC and friction will be published elsewhere.In this work, we do not attempt to derive from thefirst principles the effective model introduced in Eq. (1).However, we must outline a few necessary conditions forthe validity of that model. First of all, it should not bepossible to gauge out the SOC term from the microscopicHamiltonian that underlies Eq. (1). In one spatial dimen-sion, this implies that α must depend on position, alter-natively, beyond-1D effects must be included, see, e.g.,Ref. [42]. The microscopic Hamiltonian may in princi-ple allow for spin-flip processes in two-body scattering,however, their rate should be smaller in comparison tospin-preserving collisions, i.e., the spin coherence lengthmust be much larger than the mean-free path of parti-cle. This requirement is needed as our effective modelpreserves spin. Spin separation in chiral molecules.
We now considera chiral molecule – a system where the discussed phe-nomena could potentially be observed. Note that spinand charge reorganization have already been observed ina layer of chiral molecules by using a modified Hall de-vice [43].We start with the 1D tight-binding model of a chiralmolecule with M sites [8] H mol = (cid:15) M (cid:88) m =1 c † m c m − J M (cid:88) m =1 (cid:104) c † m +1 V m c m + H . c . (cid:105) , (6)where c † m = (cid:16) c † m, ↑ , c † m, ↓ (cid:17) is the creation operator at thesite m , V m = e i K m · σ is a unitary matrix which definesSOC, and J is a hopping amplitude. The parameter K m reads as K m = λl (cid:20) hN ( − S m ˆ x + C m ˆ y ) − R sin (0 . ϕ ) ˆ z (cid:21) , (7)where l = (cid:113) ( h/N ) + [2 R sin (0 . ϕ )] is the distancebetween nearest neighbors in a helical molecule of ra-dius R and pitch h ; S m = sin [( m + 0 .
5) ∆ ϕ ], C m =cos [( m + 0 .
5) ∆ ϕ ], where ∆ ϕ = ± π/N is the twist an-gle between nearest neighbors (the notation + ( − ) cor-responds to a right (left) helix, respectively), N is thenumber of sites in each turn, and λ is a dimensionlessquantity that parametrizes SOC on a lattice. It is clearthat V m is periodic after each turn of the helix V N = V .To connect H mol to H s with γ = 0, we perform thegauge transformation c † m = a † m V m (cid:48) /N V N V N − . . . V m (cid:48) , (a) (b) α H R ( l ) N = 15 N = 20 N = 30 N = 40 α H h ( l ) N = 15 N = 20 N = 30 N = 40 FIG. 2. The SOC amplitude α H for a helical molecule asa function of the radius R (a) and pitch h (b) of the helix.Different curves correspond to different numbers of sites ineach turn, N . The SOC parameter λ = 0 .
5. The parameters R and h are measured in the units of l . with m (cid:48) = m mod N and V = V N V N − . . . V , whichbrings the Hamiltonian (6) into a translationally invari-ant form. We impose periodic boundary conditions, andwrite the Fourier transform of the Hamiltonian as: H ( k ) = (cid:15) − J cos (cid:18) k − θ ˆ n · σ N (cid:19) , (8)where k = 2 πr/M [ r = 0 , . . . , M −
1] is the wave vectorin units l − . The operator V now reads as V = e iθ ˆ n · σ ,where ˆ n determines the quantization direction for a spinof particle in a chiral molecule. It can be expressedthrough the parameters in Eq. (6) [44]. Taking intoaccount that SOC is expected to be weak for organicmolecules ( θ/N (cid:28) k → m = (cid:126) / (2 Jl ) and α = θ (cid:126) / ( N ml ) [45]. Ifwe now include dissipation then the edge of a moleculebecomes spin polarized in the n -direction, see Fig. 1 (d).Therefore, according to our effective model, the chargedistribution of donor electrons in a chiral molecule is ac-companied by spin polarization in the n -direction deter-mined by the sign of SOC. For given SOC, transform froma right-handed to a left-handed enantiomer correspondsto ( n x , n y , n z ) → ( − n x , n y , − n z ) [44]. Therefore, as longas n y (cid:54) = 0 spin polarizations in right- and left-handedenantiomers are not connected under a point reflection.To demonstrate how the geometrical structure of themolecule manifests itself in the SOC amplitude, and,hence, in spin polarization of the molecule, we show inFig. 2 the dependence of the dimensionless SOC ampli-tude α H = 2 θ/N on R and h . The figure shows thatby increasing (decreasing) the radius (pitch), one can re-duce the strength of SOC and spin polarization. When h is small, geometrical chirality is virtually absent. De-spite that, spin separation can still occur in the directiongiven by n = ˆ z , provided that one can identify edges of amolecules. For R = 0 the system is chiral with respect tospin rotation as is evident from Eq. (7), and the moleculebecomes a quantum wire.In the presence of dissipation, the opposite regions ofa molecule have opposite magnetizations, and a chiralmolecule is effectively acting as a magnet. The spin sen-sitivity of chiral molecules in our model is a static effectand does not require a current of electrons, in contrastto previous theoretical descriptions of the CISS effect.Below, we discuss Shiba-like states [46] caused by a chi-ral molecule placed on a superconductor, which originatefrom the predicted here spin polarization of a molecule.Shiba-like states have already been observed in a lab-oratory [34], thus, our model provides a possible theo-retical explanation of the existing experimental data. Itis worthwhile noting that for short chiral molecules likehelicenes or amino acids, Shiba-like states were not ob-served, which is consistent with our prediction that thestrength of spin polarization is controlled by the lengthof a system. Chiral molecules on a superconductor.
We are inter-ested here in properties of a two-dimensional (2D) su-perconductor coupled to a chiral molecule. To make thediscussion in this section as simple as possible, we intro-duce the two-site model of a molecule H = (cid:15) (cid:88) m =1 c † m c m − J (cid:104) c † vc + c † v † c (cid:105) + γασ z (cid:16) c † c − c † c (cid:17) , (9)with v = e iασ z . The first two terms on the right-hand-side of Eq. (9) are taken from Eq. (6). The last termis taken from the lattice representation of Eq. (4). Notethat the term v is not important as we can choose a gaugetransformation to eliminate it. Non-trivial effects of SOCare contained in the last term of Eq. (9), where the signof α is determined by chirality.To describe the superconductor, we consider the 2Dtight-binding model (see, e.g., Ref. [47]) H sup = N s (cid:88) i,j =1 [ (cid:15) S τ z + ∆ τ x ] σ b † i,j b i,j − J S N s (cid:88) (cid:104) i,j,i (cid:48) ,j (cid:48) (cid:105) =1 (cid:104) τ z σ b † i,j b i (cid:48) ,j (cid:48) + H . c . (cid:105) , (10)where b † i,j = (cid:16) b † i,j, ↑ b † i,j, ↓ − b i,j, ↓ b i,j, ↑ (cid:17) is the creation op-erator at the ( i, j )th site of the superconductor, b † i,j, ↑ isthe bare operator, τ acts in a particle-hole sector, (cid:15) S isthe on-site energy, J S is the hoping amplitude, and ∆is the s − wave superconducting gap. Finally, the centralsite of the superconductor is coupled to the molecule via H int = − J I (cid:20) b † NS +12 , NS +12 c + c † b NS +12 , NS +12 (cid:21) , (11)where J I is the hopping amplitude between the super-conductor and the molecule, and, for convenience, weassumed that N S is odd. -1-0.500.51 1 2 3 4 5 6 7 (a) (b) E n e r g y / ∆ J I /J j i . . . . . . . . . A m p m FIG. 3. (a) Energy levels inside the superconducting gap forthe two-site molecule (9) on top of a superconductor as func-tion of the hopping amplitude, J I . (b) The probability am-plitude of the lowest Shiba-like states in the superconductor(main) and in the molecule (inset) when J I /J = 2 .
0. Otherparameters are J S /J = 2 . /J = 0 . J ], (cid:15) = (cid:15) S = 0 . αγ = π/
2. The superconductor is comprised of 31 ×
31 lat-tice sites, i.e., i ∈ [1 ,
31] and j ∈ [1 , We solve H + H sup + H int using the numerically ex-act diagonalization method. Our results for the in-gapShiba-like states are presented in Fig. 3. Their presenceis largely determined by the hopping amplitude J I evenwhen the molecule acts as a magnet, i.e., for finite valuesof αγ ; J I controls the penetration of spin-polarized elec-trons into the superconductor. By changing the value of J I , the energy of the in-gap states can be modified andthe crossing at zero energy can be observed. The behav-ior of energies in Fig. 3 (a) can be modified by changingthe parameter αγ , however, as long as αγ (cid:54) = 0, only quan-titative changes occur. Figure 3 (b) shows the probabilityamplitudes of the state at J I /J = 2 .
0, which demonstratethat the state is mainly located inside the superconduc-tor. All other essential features of Shiba states can beobtained by considering our model of a chiral molecule.A detailed matching to the experimental data requires amore elaborate study and will be considered elsewhere.
Discussion and conclusions:
We have shown that a 1Dconfined system in the presence of SOC and dissipationcan act as a spin-polarizer and spatially separate particleswith opposite spins. In the presented model, SOC andfriction act in unison, so that pronounced spin polariza-tion can be observed even if SOC is weak. We speculatethat our findings may be related to the long-standingpuzzle of the CISS effect, in which chiral molecules actas efficient spin filters although the SOC amplitude isexpected to be small. The present model leads to spinpolarization even in equilibrium, i.e., without currents,which tells it from the previously considered models ofCISS.The friction term in our model is a phenomenologicaldescription of the electron-bath interactions. An impor-tant future milestone will be a derivation (from some mi-croscopic Hamiltonian) of the proposed effective modelthat will establish a rigorous connection between frictionand electron-bath interactions, as well as between molec-ular chirality and the direction of SOC.Our model suggests a metastable spin accumulation onthe opposite ends of chiral molecules. This accumulationshould lead to a small localized magnetic moment, whichcan be probed experimentally. In solid-state systems,the magnetic moment may be measured using anomalousHall measurements, as well as by tunneling spectroscopyof a Kondo resonance. We note that the spin accumu-lation at the edges of a chiral molecule could be highlyrelevant for molecule-molecule interactions in biologicalsystems. Those interactions can be probed similarly tointeractions between spin-polarized chiral molecules andmagnetic surfaces [48, 49]. Lastly, our results imply astrong dependence of spin polarization on temperature.Spin coherence at low temperatures along with increaseddissipation at high temperatures suggest the existence ofsome optimal temperature, which could be relevant forbiological processes.
Acknowledgements
We thank Rafael Barfknecht foruseful discussions. This work has received funding fromthe European Union’s Horizon 2020 research and inno-vation programme under the Marie Sk(cid:32)lodowska-CurieGrant Agreement No. 754411 (A. G. and A. G. V.).M. L. acknowledges support by the European ResearchCouncil (ERC) Starting Grant No. 801770 (ANGU-LON). Y.P. and O.M. acknowledge funding from theNidersachsen Ministry of Science and Culture, and fromthe Academia Sinica Research Program. O.M. thankssupport through the Harry de Jur Chair in Applied Sci-ence. [1] S. Yeganeh, M. A. Ratner, E. Medina, and V. Mujica,J. Chem. Phys. , 014707 (2009).[2] E. Medina, F. L´opez, M. A. Ratner, and V. Mujica, EPL(Europhysics Letters) , 17006 (2012).[3] S. Varela, E. Medina, F. Lopez, and V. Mujica, J. Phys.:Condens. Matter , 015008 (2014).[4] A.-M. Guo and Q.-f. Sun, Phys. Rev. Lett. , 218102(2012).[5] R. Gutierrez, E. D´ıaz, R. Naaman, and G. Cuniberti,Phys. Rev. B , 081404 (2012).[6] R. Gutierrez, E. D´ıaz, C. 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