PAPER • OPEN ACCESS Spin dynamics in helical molecules with nonlinear interactions To cite this article: E Díaz et al 2018 New J. Phys. 20 043055   View the article online for updates and enhancements. Related content A new approach towards spintronics–spintronics with no magnets Karen Michaeli, Vaibhav Varade, Ron Naaman et al. - Lectures on Yangian symmetry Florian Loebbert - A Green's function approach to giant- dipole systems Thomas Stielow, Stefan Scheel and Markus Kurz - This content was downloaded from IP address 147.96.14.16 on 04/06/2018 at 19:25 https://doi.org/10.1088/1367-2630/aabb91 http://iopscience.iop.org/article/10.1088/1361-648X/aa54a4 http://iopscience.iop.org/article/10.1088/1361-648X/aa54a4 http://iopscience.iop.org/article/10.1088/1751-8113/49/32/323002 http://iopscience.iop.org/article/10.1088/1361-6455/aa9763 http://iopscience.iop.org/article/10.1088/1361-6455/aa9763 New J. Phys. 20 (2018) 043055 https://doi.org/10.1088/1367-2630/aabb91 PAPER Spin dynamics in helical molecules with nonlinear interactions EDíaz1, PAlbares2, PGEstévez2, JMCerveró2, CGaul1,3, EDiez2 and FDomínguez-Adame1 1 GISC,Departamento de Física deMateriales, UniversidadComplutense, E-28040Madrid, Spain 2 NANOLAB,Departamento de Física Fundamental, Universidad de Salamanca, E-37008 Salamanca, Spain 3 Cognitec SystemsGmbH,Großenhainer Str. 101, D-01127Dresden, Germany E-mail: elenadg@ucm.es Keywords: spin dynamics, chiralmolecules, solitons Abstract It is widely admitted that the helical conformation of certain chiralmoleculesmay induce a sizable spin selectivity observed in experiments. Spin selectivity arises as a result of the interplay between a helicity-induced spin–orbit coupling (SOC) and electric dipole fields in themolecule. From the theoretical point of view, different phenomenamight affect the spin dynamics in helicalmolecules, such as quantumdephasing, dissipation and the role ofmetallic contacts.With a few exceptions, previous studies usually neglect the local deformation of themolecule about the carrier, but this assumption seems unrealistic to describe charge transport inmolecular systems.We introduce an effectivemodel describing the electron spin dynamics in a deformable helicalmolecule withweak SOC.Wefind that the electron–lattice interaction allows the formation of stable solitons such as bright solitonswithwell defined spin projection onto themolecule axis.We present a thorough study of these bright solitons and analyze their possible impact on the spin dynamics in deformable helical molecules. 1. Introduction Manipulation and control of the electron spin degree of freedom in nanoscalematerials lies at the very core of spintronics. Among the large variety ofmaterials with technological interest in this field, organic systems are gaining significance as active components in spintronics nanodevices. Although large spin–orbit coupling(SOC) is uncommon in carbon-basedmaterials, recent experiments on electron transport have brought with them a considerable effort to uncover the origin of the observed high spin selectivity inDNA [1, 2] and bacteriorhodopsin on non-magneticmetallic substrates [3]. As aworking hypothesis, it has been suggested that spin selectivitymay be related to the specific geometric structure of the involvedmolecular systems, namely their helical conformation [2]. A number of theoreticalmodels have been put forward to explain the observed spin selectivity in helicalmolecules. Usually they rely on large SOC [4–12], the need for dephasingwhen SOC is weak [13, 14], the leakage of electrons from themolecule to the environment [15], the role of the bonding of the molecule to themetallic leads that enhance the effect [16, 17], or the interplay between a helicity-induced SOC and a strong dipole electric field, which is characteristic of thesemolecules [18] (see [19, 20] for a recent review). Theoreticalmodels usually assume rigid lattices and neglect the local deformation of themolecule about the carrier. However, this assumption seems unrealistic to describe charge transport inmolecular systems likeDNA [21]. It is worthmentioning that theoreticalmodels have taken into account the effects ofmolecular deformation only very recently [22, 23]. Depending on the various energy scales involved (electron bandwidth, zero-point energy ofmolecular vibrations, thermal energy), lattice deformation can play a significant role on transport properties. This is particularly relevant when charge carriers interact with intramolecularmodes that occur at high frequency due to the stretching of stiff covalent bonds. Coupling to thosemodesmay strongly alter charge transport [24] and even lead to self-trapping of carriers, provided that the relaxation energy (the energy gained upon the deformation of the lattice about the carrier) exceeds the bandwidth [25]. Self-trapping has been commonly OPEN ACCESS RECEIVED 31October 2017 REVISED 19March 2018 ACCEPTED FOR PUBLICATION 4April 2018 PUBLISHED 27April 2018 Original content from this workmay be used under the terms of the Creative CommonsAttribution 3.0 licence. Any further distribution of this workmustmaintain attribution to the author(s) and the title of thework, journal citation andDOI. © 2018TheAuthor(s). Published by IOPPublishing Ltd on behalf ofDeutsche PhysikalischeGesellschaft https://doi.org/10.1088/1367-2630/aabb91 https://orcid.org/0000-0001-7964-4148 https://orcid.org/0000-0001-7964-4148 mailto:elenadg@ucm.es http://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/aabb91&domain=pdf&date_stamp=2018-04-27 http://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/aabb91&domain=pdf&date_stamp=2018-04-27 http://creativecommons.org/licenses/by/3.0 http://creativecommons.org/licenses/by/3.0 http://creativecommons.org/licenses/by/3.0 formulatedwithin the framework of the small polaron theory based on a localHolstein-type coupling [26] between the carrier and the intramolecularmode. Thismodel was later extended by Peyrard andBishop to study the ac response of aDNAmolecule, where the charge in theπ-stack interacts with the base-pair opening dynamics of the double strand [27–30]. In a recent paper, Behnia et al.have studied the impact of the base-pair opening on the spin selectivity inDNAby using a generalization of the Peyrard–Bishop–Holstein [22]. Davydovʼs soliton theory of charge and energy transfer inα-helix and acetanilide provides another paradigmatic example on how the interaction of carriers and vibrational degrees of freedom can induce self- trapping phenomena [31]. Starting from a Frölich-likeHamiltonian [32] and assuming the adiabatic approximation, Davydov put forward a soliton theory of long-range energy transfer of excitations interacting with intramolecular vibrationalmodes in a quasi-one-dimensional lattice. In the adiabatic approximation, the continuous limit of theDavydovʼs equations reduce to the nonlinear Schrödinger (NLS) equation for the elementary excitations. Inspired by the success of theDavydovʼs approach, in this workwe introduce an effective self-focusing nonlinearmodel describing the dynamics of a single charge carrier in the electrostatic potential created by a helical arrangement of dipoles. The proposal generalizes the linearmodel formerly introduced byGutiérrez et al. considering spin-selective transport of electrons through a helically shaped electrostatic potential [6]. This model has been recently revisited and extended to study the coherent spin dynamics in helicalmolecules [11]. The strong interactionwith the lattice vibrationswill be addressed by adding a nonlinear term to the Schrödinger equationwithin the adiabatic approximation [33]. The resulting equation turns out to be integrable, thus allowing us to obtain a family of bright solitons describing the coherent spin dynamics in deformable helical molecules. 2. Electron spin dynamics in a rigid helicalmolecule Following [11], we start out by revisiting and amending themodel introduced in [6]. Twomain factors determine the high spin selectivity found: an unconventional Rashba-like SOC, reflecting the helical symmetry ofmolecules, and aweakly dispersive electronic band.α-helix in the secondary structure of proteins and other macromolecules present a net dipolemoment along the helix axis due to the helical arrangement of peptide dipoles [34]. In anα-helix, the peptide dipoles are aligned in such away that about 97%of the peptide dipole moments point in the direction of the helix axis [35] (see figure 1(a)). Therefore,α-helices can be regarded as macrodipoles, having a total dipolemoment of the order of 3.5NDebye (N is the number of residues), pointing from the negative C-terminus to the positiveN-terminus of the helix (see [36] and references therein). The magnitude of the electric dipole of themolecule is not afixed value but depends on themolecular conformation and on its local environment. For instance, hydrogen bonds can yield a significant enhancement of the dipole moments, as pointed byHol [35]. Figure 1. (a)Dipolemoment of a peptide unit in anα-helix, according toHol [35]. (b)Schematic view of the arrangement of the peptide dipoles = d edn z , aligned along themolecule axis. The dipoles shownhere span a length along theZ axis equal to themolecule pitch b. 2 New J. Phys. 20 (2018) 043055 EDíaz et al We thus consider the electronmotion through a very long helical arrangement of peptide dipoles directed along theZ axis, as shown schematically infigure 1(b). The dipoles are located at r= D + r en z an z n, a being the radius of themolecule, and their dipolemoments are = d edn z . Here, we have used cylindrical coordinates with r j j= ( )cos , sin , 0n n n , andjn=2π n/Nd+π, beingNd the number of dipoles per turn. The orientation of the individual dipoles does not affectmuch the results, provided they are arranged helically and parallel to each other [11]. The electric field generated by the dipoles aligned along themolecule axis reads  åp = - - - - -=-¥ ¥ ⎡ ⎣⎢ ⎤ ⎦⎥( ) ( ) · ∣ ∣ ( ) ∣ ∣ ( )E r r r d r r r r d r r 1 4 3 . 1 n n n n n n n0 5 3 Wewill constrain the electronmotion along theZ axis the position vector the is = r ez z . Thus, we get the following components of the electricfield(1) in theXY plane   å å p j p j =- - D + - D =- - D + - D =-¥ ¥ =-¥ ¥ ( ) ( ) [ ( ) ] ( ) ( ) [ ( ) ] ( ) E z ad z n z a z n z E z ad z n z a z n z 3 4 cos , 3 4 sin . 2 x n n y n n 0 2 2 5 2 0 2 2 5 2 The transverse components of the electric field are needed to evaluate  = - -( ) ( ) ( )z E z E zi x y (see [12] for further details). From equation (2)we obtain   åp = - - D + - D p =-¥ ¥ - ( ) ( ) [ ( ) ] ( )z ad z n z a z n z i 3 4 e . 3 n n N 0 i2 2 2 5 2 d For further estimations, notice that typical values areNd∼10 dipoles per turn inDNAorNd∼5 inα-helix. In the limit N 1d , equation (3) can be approximated by the formal substitution òå  D( )b z udn , u=(z −nΔ z)/b being the dummy variable òå - D + - D D + p p =-¥ ¥ - - D -¥ ¥ ( ) [ ( ) ] ( ) ( ) ( )z n z a z n z b z u u u a b e 1 d e . 4 n z n z b ui2 2 2 5 2 3 i2 2 2 2 5 2 After performing the integrationwe obtain themagnitude ( )z needed to calculate the SOC    p p= D ºp p- -( ) ( ) ( )z d zb K a b 2 2 e e , 5z b z b 0 2 1 i2 0 i2 whereK1 is themodified Bessel function of the second kind. The SOCHamiltonian stems from the classical formula s ´· ( )p E , symmetrized such that the Hamiltonian isHermitian. Heres is a vector whose components are the Paulimatricesσx,σy, andσz. For =  p epz z the SOCHamiltonian simplifies to * *     l = +   ⎡ ⎣⎢ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎤ ⎦⎥ ( ) ( ) ( ) ( ) ( )H p z z z z p 2 0 0 0 0 , 6z zSO where l = ( )mce 2 2. The electronHamiltonian = + p m H2z 2 SO can be cast in the form = E Hb where the dimensionlessHamiltonian H reads pg= -¶ -x   ( )H M a2 . 72 Herewe have defined =E mb2b 2 2, ξ=z/b,∂ξ=∂/∂ξ and the dimensionless spin–orbit parameter  g l p= ( )bE2 b0 . Thematrix operator M is given by p p = ¶ - ¶ + px px x x - ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ( )M b0 e e 0 i 0 0 i . 7 i2 i2 The dimensionlessHamiltonian(7) is readily diagonalized since it commutes with the helical operator ps= + q pz z . The corresponding normalized eigenfunctions and eigenenergies are c x b b e p a= = + - p x p x  -  + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟( ) ( ) ( ) ( ) ( ) ( ) s s q sq e e , , 8qs q q qs i i 2 2 3 New J. Phys. 20 (2018) 043055 EDíaz et al where a p g= +2 1 2 , = s 1and b f f b f f = + + - = - - +   ( ) [( ) ( ) ] ( ) [( ) ( ) ] ( ) s s s s s s 1 2 1 cos 1 sin , 1 2 1 cos 1 sin , 9 satisfying b b+ = ( ) ( )s s 12 2 , and f g g = + + ( )tan 1 1 . 10 2 Notice that the helical conformation of the electric dipoles gives rise to an the additional effectivemomentum p g= +q s 1s 2 even in the absence of SOC. Previous studies report on this fact as having an impact on the linear optical response aswell [37]. Once the eigenvectors of theHamiltonian(7a) have been obtained, we focus on the dynamics of an electron wave packet of the form òåc cx p x= e -¥ ¥ -( ) ( ) ( )t q C a, d 2 e , 11 s qs qs ti qs where time is expressed in units of ÿ/Eb and ò c cx x x= -¥ ¥ ( ) · ( ) ( )†C bd , 0 . 11qs qs Ourmagnitude of interest will be the time-dependent spin projection onto themolecule axis, also referred as helicity, which is calculated as follows * ò òc cx x s x p f f = = - + e e -¥ ¥ -¥ ¥ + - + - -+ - ( ) ( ) ( ) [(∣ ∣ ∣ ∣ ) ( ) ( ) ( )] ( ) † ( ) t t t q C C C C SP d , , d 2 cos 2 2 sin 2 Re e . 12 z q q q q t , 1 2 , 1 2 , 1 , 1 i q q, 1 , 1 Consider an initial wave packet with an arbitrary state of helicityc x x q q= + j  ( ) ( )[ ( ) ( ) ]u uf, 0 cos e sini , where f (ξ) is an arbitrary bell-shaped function of dimensionless widthW. Here su withσ= , denotes an eigenvector ofσz and the spin projection is defined by the angle θ. For the sake of concreteness we setj= 0 and we refer to as fully polarized or fully unpolarized state when the spin is parallel to (θ=0,π/2) or out of (θ=π/4, 3π/4) themolecular axis. After a straightforward calculation one can obtain a closed expression for SP(t) that has a transient contributionwhich vanishes at large times p g+ ( )t W 4 1 2 . A transient time of ~t fs40 is roughly estimated for a highly localized initial statewithW∼1 passing through aDNAmolecule with a SOC parameter of the order of γ∼0.1. Thus, after a quick transient state, the spin projection reaches the asymptotic value given as =  ¥¥ ( )tSP SP , where òg q g q x x px= + -¥ -¥ ¥⎡ ⎣⎢ ⎤ ⎦⎥( ) ( ) ∣ ( )∣ ( ) ( )fSP 1 1 cos 2 sin 2 d cos 2 . 13 2 2 Notice that if we consider an initial fully polarized state with θ=0 (θ=π/2)namelywith spin parallel (antiparallel) to themolecule axis, the larger the SOCparameter, the smaller the asymptotic spin projection, as expected. In experiments, however, an initially unpolarized current becomes spin polarized after being transmitted through the helicalmolecule. Therefore, our case of interest is an initial fully unpolarizedwave packet with spin projection out of themolecule axis, i.e. along theX axis such asc x x= - + ( ) ( )( )u uf, 0 2 . Let us have a deeper look at this particular case, for which the coefficients(11b) read ò ò ò ò f x x f x x f x x f x x = - = + p x p x p x p x + -¥ ¥ - - -¥ ¥ - + - -¥ ¥ - - -¥ ¥ - + ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) C f f C f f 1 2 cos d e sin d e , 1 2 sin d e cos d e , 14 q q q q q q , 1 i i , 1 i i which leads to the following expression for SP(t) * * ò ò ò f x x px x x x a f pa x x x a =- - + + + pa px -¥ ¥ - -¥ ¥ - -¥ ¥ ⎧⎨⎩ ⎡ ⎣⎢ ⎤ ⎦⎥ ⎫⎬⎭ ⎡ ⎣⎢ ⎤ ⎦⎥ ( ) ( ) ∣ ( )∣ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f f f t t f f t SP sin 4 2 d cos 2 Re e d e sin 2 sin Im d . 15 t2 i i2 2 Notice that the last two integral involves the overlap of f (ξ) and f*(ξ+αt) and consequently vanishwhen at W . Therefore, at times larger thanW/α the spin projection reaches a steady value, corresponding to the 4 New J. Phys. 20 (2018) 043055 EDíaz et al time-independent term in equation (15). In this case, the asymptotic spin projection has a non-monotonous dependence on themagnitude of the SOCparameter and the time-independent integral approaches unity for a narrowwave packet. The asymptotic spin projection along themolecule axis becomes f g g = = + ¥∣ ∣ ( ) ( )SP sin 4 2 1 , 16 2 in agreementwith Eq. (13). Therefore, the SOC rotates the electron spin and, after a quick transient, the spin projection along themolecule axis becomes nonzero. 3. Electron spin dynamics in a deformable helicalmolecule In order to describe a deformable helicalmolecule where electron dynamics is affected by the lattice vibrations, wewill assume that dipoles vibrate along themolecule axis independently of each other. The potential energy of an electron at position = r er z in the electric field created by the nth dipole is p = - - + - ( ) [ ( ) ] ( )V z ed z z a z z4 , 17n n n0 2 2 3 2 where zn=nΔ z+un. After a Taylor expansion in the small displacement un from equilibrium and keeping up to thefirst order term,we get d» +( ) ( ) ( )V z V z V zn n n 0 . The potential energy ( )V zn 0 corresponds to the electron interactionwith the dipole at equilibrium, as discussed in the previous section. The first order correction is given as  d p = - D⎜ ⎟⎛ ⎝ ⎞ ⎠( ) ( )V z ed a f z n z a u 4 , 18n n 0 3 with = - +( ) ( ) ( )f x x x1 2 12 2 5 2. This function is sharply peaked at x=0 and it can be replaced by aDirac δ-functionwith the proper coupling constant (details will be given elsewhere). Therefore, the first order correction to the potential energy of the electron in the electric field created by the dipoles is å åd d d= = - D =-¥ ¥ =-¥ ¥ ( ) ( ) ( ) ( )V z V z V u z n z , 19 n n n n1 V1 being a positive constant. Following a lengthy but straightforward procedure introduced by Bang et al. [38], in the adiabatic limit it is found that c c= - D D( ) · ( )†u C n z t n z t, ,n withC a positive constant that depends on themass of the dipole units and their frequency.We nowproceed to the continuum limit and replace the summation in equation (19) by an integration ò c cd q q d q= - - -¥ ¥ ( ) ( ) · ( ) ( ) ( )†V z V C t t z, , , 201 andfinally we get c cd = -( ) ( ) · ( )†V z V C z t z t, ,1 . This energy is to be added to theHamiltonian H discussed in the previous section. In this scenario, the dimensionless NLS describing the dynamics of the spinor statec x( )t, including the first order correction of the interaction due to the vibrating dipoles reads c c c c cx x x x x¶ = -( ) ( ) [ ( ) · ( )] ( ) ( )†t H t g t t ti , , 4 , , , , 21t where H is given in equation (7a) and g>0 is a dimensionless constant. The integrability of this equation has been recently analyzed by using the Painlevé test [39]which requires that the components ofc admit the Laurent expansion åc x x f x x f x = = ¥ - - ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟( ) ( ) ( ) ( ) ( ) ( )t a t t b t t , , , , , , 22 j j j j j 0 1 1 around themovable singularitymanifoldf(x, t)=0. Bymeans of this test, it can also be proved that equation (21) is the only integrable case of amodel very recently put forward byKartashov andKonotov to study the dynamics of Bose–Einstein condensates with helical SOC [40]. Furthermore, it reduces to theManakov nonlinear systemwhen the SOC vanishes [41, 42]. Thewell known singularmanifoldmethod [43] arises from the truncation of the Painleve series(22) and yields the following three component Lax pair for equation (21) 5 New J. Phys. 20 (2018) 043055 EDíaz et al m m p pg pg m p m m m m m m m p p g g g g g Y Y Y Y Y ¶ = - F F -F + - -F - - ¶ = F F + F F + - ¶ F - F - ¶ F - F - ¶ F + F -F F - -F F - ¶ F + F -F F -F F - + - + - F + F - F - F F - F F + F x x x x x * * * * * * * * * * * * * * * * ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ ( ) ( ) i i 2 i 2 i 2 i 2 2 i 2 2 i 1 0 0 0 0 , 23 t 1 2 1 2 1 1 2 2 2 1 1 2 2 1 1 1 1 2 1 2 2 2 2 1 2 2 2 2 2 1 1 2 2 1 1 2 whereμ is the spectral parameter,Y is a three component eigenvector and, to avoid exponentials, we have defined the transformation c x x x = F F p x p p x p - + - ⎛ ⎝⎜ ⎞ ⎠⎟( ) ( ) ( ) ( ) ( ) ( ) t g t t , 1 2 e , e , . 24 t t i 1 i 2 The singularmanifoldmethod has been applied in [39] to this Lax pair in order to deriveDarboux transformations and an iterative procedure that yields solitonic solutions for equation (21). For self-focusing nonlinear interaction (g>0), it can be demonstrated that the following bright solitons (similar to the case ofDavydovʼs soliton) are a solution to equation (21) c x x b b = + = j x px px -  -  ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟( ) [ ( )] ( ) ( ) ( )( )t g g ct s s s, 2 sech e e e , 1, 25s ti , i i s whereβ↑(s) andβ↓(s) are given by(9). Here c is a free parameter representing the velocity of the soliton. The phase is defined asj x p f x p g= - + - --( ) [ ( )] ( )t c s c g t, 2 cos 2 4s 1 2 2 2 2 . The existence of two different bright solitons due to the arbitrary choice of the constant = s 1 is known as theKramer doublet and it is directly related to the preservation of the time-reversal symmetry in themodel. Notice that these solitons have a well defined helicity that depends on the SOCdue to equation (10).Most importantly, this helicity is preserved along its propagation and it is found to be b b g = - = +  ∣ ∣ ∣ ( ) ( )∣ ( )s sSP 1 1 . 26sol 2 2 2 4. Connectionwith experiments Having presented the salient features of solitons in deformable helicalmolecules, we now turn to discuss their relevance in experiments. In recent experiments on electron transport in organic helicalmolecules, it has been clearly demonstrated that an initially unpolarized current turns out to get highly polarizedwhen passing through themolecule [1–3, 44–49]. Since the intrinsic SOC effects are rather weak in thesemolecules, theoreticalmodels proposed to describe the experiments rely on SOC related to the peculiarities of the helical geometry. It is worth mentioning that all thesemodels strongly depend on a phenomenological SOCwhichwas roughly estimated to beα=4–12 meV nm [6]. It seems that this coupling, even in the best scenario, is not large enough to support the high degree of spin polarization observed in the experiments. All these approaches, however, neglected lattice deformations that strongly affect the electron dynamics in organicmolecules. In order to show the relevance of the nonlinear interaction between the lattice and the spin degree of freedom, figure 2 compares the spin projection achievedwith the linear, ¥SP , and the nonlinearmodel, SPsol, as given by equations (16) and(26) respectively. To understand the experimental situationwhen a spin unpolarized current is injected into themolecule, wewill consider an initial localizedwave packet with a spin projection such that θ=3π/4 (fully unpolarized in the sense discussed above). Thewave packet evolves in time and, after a short transient time, it reaches a steady spin proyection given by equation(16), to be comparedwith that obtained for the stable soliton(26). Figure 2 clearly shows an outstanding result, namely, the nonlinearity strongly enhances the resulting spin projection of a coherent electron passing through a deformable helical molecule. Thus, lattice vibrations lead to a larger effective SOC in themolecule.We define the enhancement factor as h = - +¥ ¥(∣ ∣ ∣ ∣) (∣ ∣ ∣ ∣)SP SP SP SPsol sol , as depicted in the inset offigure 2. This parameter assesses the effects of the lattice on the spin rotation capability of the deformable helicalmolecule. According to the curve shown in the inset offigure 2, the self-focusing nonlinear interaction has amajor effect when the SOC isweak. And this is precisely the case of interest in experiments, as discussed below. 6 New J. Phys. 20 (2018) 043055 EDíaz et al The gray area offigure 2 highlights the regionwhere the dimensionless SOCof ourmodel takes values consistent with previous estimations g a p= ( )bE2 0.2b inDNA.Our approach explains the physical scenario in experiments as follows. The initial unpolarized electron is injected in a deformable helicalmolecule whose vibrations interact continuously with the electronic dynamics inside the system. Such nonlinear interaction transforms the initial arbitrary state till it reaches themost stable configuration, namely, a solitonic solution.Once the soliton state is conformed, it can propagate with no dispersion along themolecule with awell defined high degree of helicity. The formation of solitons after a short transient time could be an important effect for the resulting highly polarized currents found in experiments. 5. Conclusions In conclusion, we have presented a nonlinearmodel to study the spin dynamics of electrons in a deformable helicalmolecule. Such dynamics is subjected to: (i) the electricfield created by the helical arrangement of molecular dipoles and (ii) the interaction between the electron and the lattice vibrations. On the one hand, the dipole electricfield induces a Rashba-like SOC for electronsmoving along the helical axis. On the other hand, the electron–lattice interaction allows the formation of stable solitons. Once themodel was presented, wewere able to prove that the system supports the formation of stable bright solitons. Remarkably, the helicity of such solitons is preserved during the propagation across the helical molecule.We also calculated the spin projection onto themolecule axis as afigure ofmerit to assess the spin dynamics. For completeness, we also compared this result with those obtained for the linearmodel corresponding to the rigidmolecule. In particular, we focused on themost relevant situation for experiments, namely the partial spin projection achieved by an initial unpolarized electronwhen passing through the helical molecule. In such scenario, the spin rotation capability of a deformable helicalmolecule is largely increased compared to the rigid one, even in the case of weak SOC. Acknowledgments The authors thankRGutiérrez andVMujica for helpful discussions. This research has been supported by MINECO (GrantsMAT2013-46308 andMAT2016-75955) and Junta deCastilla y León (Grant SA045U16) and Junta deCastilla y León (Grant SA045U16) and P. Albares Ph.D. Fellowship. ORCID iDs EDiez https://orcid.org/0000-0001-7964-4148 References [1] Göhler B,HamelbeckV,Markus TZ, KettnerM,HanneGF, Vager Z,NaamanR andZachariasH 2011 Science 331 894 [2] Xie Z,Markus TZ,Cohen SR,Vager Z, Gutiérrez R andNaamanR 2011Nano Lett. 11 4652 [3] MishraD,Markus TZ,NaamanR, KettnerM,Gohler B, ZachariasH, FriedmanN, ShevesM and Fontanesi C 2013 Proc. Natl Acad. Sci. USA 110 14872 Figure 2.Asymptotic spin projection as a function of the dimensionless SOCparameter γ for a rigid helicalmolecule (dashed line) and a nonlinear deformable helicalmolecule (solid line). The inset shows the enhancement factor h = - +¥ ¥(∣ ∣ ∣ ∣) (∣ ∣ ∣ ∣)SP SP SP SPsol sol . The gray area highlights the regionwith realistic values of γ according to to previous estimations [6]. 7 New J. 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Introduction 2. Electron spin dynamics in a rigid helical molecule 3. Electron spin dynamics in a deformable helical molecule 4. Connection with experiments 5. Conclusions Acknowledgments References