Spin-Current and Spin-Splitting in Helicoidal Molecules Due to Spin-Orbit Coupling

The use of organic materials in spintronic devices has been seriously considered after recent experimental works have shown unexpected spin-dependent electrical properties. The basis for the confection of any spintronic device is ability of selecting the appropriated spin polarization. In this direction, DNA has been pointed out as a potential candidate for spin selection due to the spin-orbit coupling originating from the electric field generated by accumulated electrical charges along the helix. Here, we demonstrate that spin-orbit coupling is the minimum ingredient necessary to promote a spatial spin separation and the generation of spin-current. We show that the up and down spin components have different velocities that give rise to a spin-current. By using a simple situation where spin-orbit coupling is present, we provide qualitative justifications to our results that clearly point to helicoidal molecules as serious candidates to integrate spintronic devices.


Model and Results
Let us start considering one electric charge, which can be controllably introduced by an AMF tip [27][28][29] or through optical exitation 46 under influence of an electrostatic potential, V, moving along a double stranded helicoidal molecule, illustrated in Fig. 1. It has been shown that the spin-orbit coupling felt by the electron is described by the Hamiltonian 19  . l a , Δϕ and θ are the arc length, the twist angle between first-neighbor sites and the helix angle, respectively. They are related by: l a cos θ = RΔϕ and l a sin θ = Δh, where R is the radius of the helix and Δh is the stacking distance between neighbor sites. H.c. is the hermitian conjugate. For DNA molecules, where the two strands have a π phase difference, we have that σ σ ϕ π = ∆ + + j ( ) j 1 2 1 . The total Hamiltonian is given by: c c c c c c Hc i n n j n j j DSM n j n j n j n j 1 1 j 2 j , , , being the usual double-strand Hamiltonian, where η j is the inter-strand hopping integral at site j, θ = + t t it cos 2 n j R s o , and t R is the intra-strand hopping overlap integral. The solution of Equation (1) is given by a four component spinor of the form: ψ = (ψ (1)↑ , ψ (1)↓ , ψ (2)↑ , ψ (2)↓ ) T , where ψ (n)σ is the wave function component on the n th strand with spin polarization σ. The time evolution is governed by the Schrödinger equation, which is given, in terms of the Wannier amplitudes, by: , thus, the time reversal symmetry is preserved. We have used eighth-order Taylor expansion in order to solve Equations (3) and (4) simultaneously with time step Δt ~ 10 −3 fs. This time step is small enough to keep the spinor normalized during the whole simulation. We follow the time evolution of an initially spin-unpolarized gaussian wave packet with width equal to l = 30 . For double strand DNA molecules, we set the parameters: ε = .
0 3eV , η = 0.3 eV, θ = 0.66 and φ ∆ ≈ π 5 31 . It is worth stressing that the effects reported here are strongly robust to chosen of the parameters (ε j Figure 2 shows the wave packet at three different time steps: t 1 = 0.4 ps, t 2 = 0.65 ps and t 3 = 0.86 ps where we have used the spin-orbit constant equal to 0.03 eV, which is one order of magnitude smaller than the hopping parameter, t R n19 . Despite the fact that the spin-orbit coupling does not produce any polarization, it can be observed that the spin up and down components move in opposite directions, leading to a spatial separation between spin polarizations. The arc length between two neighboring nucleobases is l a , which satisfies the relation l a cosθ = RΔϕ and l a sinθ = Δh, where Δϕ is the twist angle and Δh is stacking distance between two nucleobases. In order to mimic the double-strand DNA molecule, we set Δh = 0.34 nm, Δϕ = π/5, h = 3.4 nm, R = 0.7 nm, θ ≈ 0.66 rad and l a ≈ 0.56 nm. Drifting Velocity and Spin Current. In order to understand this phenomenon, we shall show that the presence of spin-orbit coupling adds an extra term to the canonical momentum. This extra term depends on the spin polarization and the canonical momentum becomes different from the kinetic momentum. Quantitative information about the separation between the spin components can be obtained by looking at the dynamics of the wave packet. One convenient quantity to be studied is the mean position of each component of the spinor, defined Figure 3(a,b) show the time evolution of the mean position of the spin up and spin down of an unpolarized initial wave packet, with l = 1 and l = 30, respectively. By observing Fig. 3(a,b), one can see clearly that each spin component has a net movement in a well defined direction. Furthermore, the mean velocity of the narrower initial wave packet, shown in Fig. 3(c), is smaller than the mean velocity of the wider initial wave packet, shown in Fig. 3(d). It is interesting to notice that the absolute value of the velocity for the up component is different than the velocity of down component when a narrow initial wave packet is considered. On the other hand, the velocity of both components has the same magnitude for the wider initial wave packet (l = 30). A qualitative justification to this behavior will be provide by exploring the simplest situation where spin-orbit coupling is present.
From a purely quantum mechanical point of view, this phenomenon can be seem as a flux of angular momentum, without a simultaneous flux of electronic charge. Traditionally, the quantity used to quantify this angular momentum flux is the spin-current 32-34 , defined as , where V is the velocity operador given by: i  with Z being the position operator. Thus, the mean spin-current, I s = Tr (ρJ s ), where ρ is the density matrix, is given by: In fact, the spin-current is one of the most important physical quantities in spintronics and has been extensively studied from fundamental and technological perspective, such as the spin Hall effect 36,37 and spin precession 2,38 . Figure 4(a) shows the spin-current for an initial gaussian wave packet with width l = 1 for three spin-orbit coupling strength: t so = 0.01, t so = 0.02 and t so = 0.03 eV and Fig. 4(b) shows the spin-current for a initially gaussian wave packet with l = 30 for the same spin-orbit coupling constant of Fig. 4(a). It can be observed that, in both cases, there is a time persistent spin-current for any spin-orbit coupling strength. As one could anticipate by looking at Figs 3 and 4 shows that the magnitude of the spin current is smaller for the narrower initial wave packet than for the wider initial wave packet. A qualitative understanding of Fig. 4 can be obtained by analyzing , of an initially gaussian wave packet, in a double strand molecule, with width (a) l = 1 and (b) l = 30 for three spin-orbit coupling constant: t so = 0.01, t so = 0.02 and t so = 0.03 eV. (c,d) are, respectively, the mean velocity of the wave packet components for the cases studied in (a,b). the simplest case of spin orbit coupling in two limits, namely, a totally localized initial wave packet and a totally spread initial wave packet.
It is worth to emphasizing the differences between the results presented in the present manuscript and those ones reported in References [17][18][19] . The results in refs 17-19 are all based on charge transport followed by a spin polarization. This means, they show how electrons with opposite spin polarization flow differently through the helicoidal molecules when a voltage is applied. The observation of such polarized current requires, however, further ingredients besides spin-orbit coupling. For example, in reference 17 , it is necessary to break time reversal symmetry to observe the spin polarization. Loss of the electronic phase and spin memory, due to inelastic scattering, is also a mandatory ingredient in ref. 19. It was shown in ref. 18 that polarized current in double strand molecules can be obtained without electronic phase breaking or spin memory, however, the electronic coupling parameters and the spin-orbit coupling must be asymmetric in order to observe sizable spin polarized current. On the other hand, spin-orbit coupling leads to spin transfer (which is not followed by a charge transfer) without the need to include any other ingredients.

Analytical consideration for limiting cases.
It is convenient to analyze the case of a single stranded molecule with one energy level per site. Doing this: (1) all physically important features can be more easily understood in single strand molecules and (2) contrary to the polarization effects observed in the References [17][18][19] , the splitting of the spin up and down components can be observed even for single strand with one level per site. In this section, we set ħ = 1. Under these conditions, the initial wave packet is given by: The expansion coefficients ↑ a k and ↓ a k are given by: , and ψ j 0, are the value of the eigenvector ϕ σ k and the initial wave packet at the j th site. Due to the periodic nature of the potential, both eigenvectors can be written as Bloch states: where u k , v k are functions with periodicity equal to the potential.
Combining these properties, the time evolution of the spin components can be written as:  where we have explored the time reversal symmetry by doing . At this point, the knowledge of the eigenstates and eigenvalues of Equation (2) is necessary. Using the Ansatz 39,40 : where k 0 is determined by the periodicity of the spin-orbit coupling potential. By applying Equation (11) for H in Eq. (2), it is possible to find the eigenvalues of H. However, the expression for the eigenvalues is rather complicated for a general k 0 and no useful analytical expressions can be obtained. On the other hand, the simple case where k 0 = 0 captures many important features. In this case, the eigenenergies are: here, we have defined t i = 2t so cosθ and λ = 2t so sinθ. The eigenvector associated with E ↑ and E ↓ are, respectively: We shall apply these results for two interesting limiting cases: (i) a totally localized initial wave packet and (ii) a completely delocalized initial wave packet.
Localized initial wave packet. The first case that we analyze is the case where the two components of the initial wave packet are localized on a single site, j 0 , on the molecule: , 0 . Without loss of generality, let us consider that the initial wave packet is localized at j 0 = 0. In this limit, the coefficients a k 's in equations 9 and 10 are: One should notice that Equations (16) and (17) do not depend on k. This means that the Equations (9) and (10) have contributions associated with all k's within the first Brillouin zone and can be written as: The spin current, thus, is given by: can be understood as follow: The narrow initial wave packet is written as a sum of many terms associated with the wave vector k (in this limit case, the sum contains an infinity numbers of terms). Thus, the spin components of the initial wave packet have terms that can be identified as plane wave traveling in opposite directions (see Equations (18) and (19)) that lead to a vanishing spin-current.
Delocalized initial wave packet. The other limit we must study is the one where the initial wave packet is totally delocalized over the whole system: where N + 1 is the number of sites. Since  N 1, we have made: 2(N + 1) ≈ 2N. The coefficients in Equations (9) and (10) are: Therefore, the spinor components can be written as: and, thus, the spin-current is: It is interesting to notice that only the state associated with k = 0 is present. However, contrary to what one could have expected, the spin current is nonzero, meaning that there is a relative velocity between |↑ 〉 and |↓ 〉 states even for k = 0. When spin-orbit coupling is taken into account, the canonical momentum is not equal to the kinetic momentum, which is similar to the case of an electron moving in a magnetic field. One can see this clearly by looking at the phase velocity of the electron in both energy bands: One can see from Equations (26) and (27) that, even for k = 0, the velocity is nonzero. Furthermore, the additional velocity introduced by the spin-orbit coupling is opposite for different spin polarization.
At this point, we can qualitatively justify the behavior shown in Fig. 3. In particular, Fig. 3(c) shows that the packets with opposite polarization move in opposite directions; however, the magnitude of their velocities is different. Consider an extremely narrow initial wave packet with nonzero width. One can construct such wave packet by summing over every wave vector k (as done in Equations (18) and (19), except over a specific wave vector k 0 : 2 (1 ) 2 and Ω =  One can identify the four first terms in equation 30 as the velocity of up component of the spinor, v ↑ , and last four terms can be identified as the velocity of the down component, v ↓ . Clearly, either v ↑ or v ↓ can be complex number. The physical meaning of the imaginary part of v ↑ and v ↓ is the population changing in each polarization and the real part gives variation of the position of the center of mass of the packet. In this sense, we are interested only in real parts of v ↑ and v ↓ , which are given by: Indeed, the magnitude of the two velocities is different. From another perspective, it is possible to understand this fact by considering that, in order to get an unpolarized initial wave packet a little wider than a delta function, it is necessary to remove at least one state from each band. As shown above, the phase velocities are different in each band. Thus, the velocity of the up component must be different from the velocity of down component. This behavior can be clearly seen in Fig. 3(c). On the other hand, the velocities of both polarization are the same for an initial wide wave packet [ Fig. 3(d)]. As was already pointed out in Equations (23) and (24), only the wave vector k = 0 is present. Thus, the velocities of the spinor components are: Thus, the velocities of spinor components are time independent and are opposite but equal in magnitude.

Spin Spatial Separation.
Another very desirable property of materials that are candidates to integrate spintronics devices is the ability to select a given spin polarization without the need of an external magnetic field. These components are known as spin-filters and play a central role on logical operations in spintronics. In this sense, a spatial separation between two polarizations fulfills this need. In order to quantify the degree of separation of the spin components in the wave packet, we look at the the projection of the up component on the down component of the wave packet: In the limit where the two components are in the same region, ξ(t) is one. On the other hand, if the two components are completely separated, ξ(t) is zero.
Once more, we turn our attention to a much simpler system: the single strand molecule, with a single orbital per base. Figure 5(a) shows the time evolution of the ζ function of an initially gaussian wave packet with width equal to 1 for three values of spin-orbit coupling: t so = 0.01, 0.02 and 0.03 eV. One can observe that, even for the largest spin-orbit coupling constant, there is not a substantial separation between up and down components. The dominant effect here is the spreading of both components. This spread can be directly seen by looking at the electronic wave packet in a system with spin-orbit constant equal to 0.03 eV at t = 0.49 ps, as shown in Fig. 5(c). It can be noticed that both components occupy nearly the same region. Qualitatively, this behavior can be understood by looking at Equations (31) and (32). Albeit the velocities of the spinor components are different, the dominant terms are the one which involve the first order Bessel function. Thus, the mean position of each component performs an oscillatory movement around the initial position and no separation can be observed. On the other hand, for a wider initial wave packet, there is a clear spatial separation with the increase of the spin-orbit coupling, as Fig. 5(b) shows. Again, this separation between the components can be seen by looking at the wave packet. Figure 5(d) shows the components of the wave packet at t = 0.49 ps in a system with spin-orbit coupling equal to Scientific RepoRts | 6:23452 | DOI: 10.1038/srep23452 0.03 eV. Clearly spin up and spin down wave packet have different spatial functions. Likewise done previously, this action can be understood in the lights of Equation (33) and (34). Since both components have opposite constant velocities, the mean position of each components moves away from each other, leading to a decreasing of the ξ function.

Conclusions
In conclusion, we have considered single and double strand helicoidal molecules in a presence of Rashba spin-orbit coupling. We have demonstrated that, within the model proposed by Guo and Sun 19 , due to the spin-orbit coupling, a spatial separation between spin up and down of an initially unpolarized wave packet emerges. We also have shown that the spin-orbit coupling gives rise to a spin-current. An important feature induced by the spin-orbit coupling it the fact that it makes the canonical momentum different from the kinetic momentum. Furthermore, each component of the spinor has an extra and opposite term added to the velocity. With the use of the wave packet to describe the electron, it was possible to study different scenarios and to establish what are the experimental conditions to get an efficient spin separation and high spin current. In order to get an efficient spatial separation and higher spin-current, one should prepare, as the initial state, a well defined wave packet in the momentum space. It is worth to call the attention that the results shown here do not depend on a break of phase, however, its quantitative importance needs a further investigation. Furthermore, different from electrical properties, which are based on an unbalance of between up and down polarization, the properties shown here are also present in single strand molecules with only one level per site. Thus, single strand molecules with one site per site (single strand DNA, for instance) can also be considered for technological applications. It is important to keep in mind that, formally there is no need of a net polarization in any system presented here. There is only a spacial separation between spin up and down. Thus, our results do not violate the statement done by Guo and Sun that forbids spin polarization in single strand molecules 19 . From the experimental point of view, the detection of spin current is one of the most important and defiant issue in spintronics. Most of the progress in this field uses spin Hall effect to detect spin current. However, despite the spin accumulation at the edges of the sample, it does not produce a measurable electrical signal 41,42 . However, it has been proposed that spin current can be electrically probed by using the Inverse spin Hall effect, which converts spin current into charge current, in ferromagnetic materials 43 . Very recently, a new experimental technique has been reported in the literature allowing to measure spin-current in organic semiconductor 44 . We believe that something in this direction could be used in order to measure pure spin current in helicoidal molecules. We hope that, due to the great potential of organic chiral molecules, our work can stimulate experimental progresses in this direction.

Method
Let us derive the expression of the spin current for an initial completely localized wave packet [equation (20)] Each term of Equation (5) can be calculated by using the expressions (18) and (19). We start calculating the first term of the sum in Equation (5):