Geometric Spin-Orbit Coupling and Chirality-Induced Spin Selectivity

We report a new type of spin-orbit coupling (SOC) called geometric SOC. Starting from the relativistic theory in curved space, we derive an effective nonrelativistic Hamiltonian in a generic curve embedded into flat three dimensions. The geometric SOC is $O(m^{-1})$, in which $m$ is the electron mass, and hence much larger than the conventional SOC of $O(m^{-2})$. The energy scale is estimated to be a hundred meV for a nanoscale helix. We calculate the current-induced spin polarization in a coupled-helix model as a representative of the chirality-induced spin selectivity. We find that it depends on the chirality of the helix and is of the order of $0.01 \hbar$ per ${\rm nm}$ when a charge current of $1~{\rm \mu A}$ is applied.

Introduction-Spin-orbit coupling (SOC) is a relativistic interaction between the spin and the orbital motion of an electron. It gives rise to many intriguing phenomena such as the spin Hall [1] and the Edelstein effects [2][3][4][5]. The spin Hall effect is a phenomenon in which the spin current flows perpendicular to an applied electric field, leading to the spin accumulation at the boundaries. Its theoretical rediscovery [6,7] motivated the recent development in topological insulators [8,9]. On the other hand, the Edelstein effect is a phenomenon in which spin polarization is induced by the electric field only when the inversion symmetry is broken. These phenomena are governed by the energy scale of the SOC, which is proportional to Z 4 in atomic limit, Z being the atomic number. Thus, heavy elements are more likely to demonstrate nontrivial effects caused by the SOC. In fact, the large spin Hall effect was observed in heavy metals including Pt [10][11][12][13] and Au [11,12,14], and the Edelstein effect was observed in a Bi/Ag interface [15] and topological insulator surfaces [16,17].
In contrast, a spin filtering effect that resembles the Edelstein effect was reported in chiral molecules composed of light elements [18][19][20][21][22]. The spin polarization of photoelectrons transmitted through the molecules depends on the molecular chirality. Recently, the effect of chirality on the magnetoresistance was reported [23,24]. These phenomena are called chirality-induced spin selectivity (CISS) [25,26]. Surprisingly, the energy scale of the SOC relevant to the CISS was experimentally estimated as hundreds of meV [22], which is unexpected in light elements. Previous theoretical explanations of the CISS relied on the existence of a large SOC [27][28][29][30][31][32][33][34][35][36]. However, the origin of the SOC remains unclear.
The CISS strongly indicates the existence of a large unknown SOC in chiral molecules. The conventional SOC is derived from the Dirac Lagrangian density in electromagnetic field in flat spacetime. Furthermore, a novel coupling between the spin and mechanical rotation was derived from relativistic quantum mechanics in curved spacetime [37,38]. Since the chiral molecules are modeled as a one-dimensional (1d) curve embedded in 3d flat space, we can assume that the large SOC in chiral molecules originates from the relativistic effect in the 1d curve.
In this Letter, we derive an effective nonrelativistic Hamiltonian in a generic curve from the Dirac Lagrangian density in curved space. We use the Frenet-Serret (FS) frame to describe the curve embedded in 3d flat space [32,[39][40][41], apply the thin-layer quantization to derive an effective Lagrangian density in the curve [39,42], and then perform the Foldy-Wouthuysen (FW) transformation to take the nonrelativistic limit [43,44]. We find what we call geometric SOC of O(m −1 ), in contrast to the conventional one of O(m −2 ), where m is the electron mass. The energy scale is estimated to be a hundred meV. We also calculate the current-induced spin polarization in a coupled-helix model and find that it is of the order of 0.01 per nm when a charge current of 1 µA is applied.
Derivation of the geometric SOC-We begin with the Dirac Lagrangian density in curved spacetime, e a µ is a vielbein, which is related to a metric as g µν = η ab e a µ e b ν , whereas e µ a and e ≡ det e a µ are the inverse and the determinant of the vielbein, respectively.ω abµ is the torsion-free spin connection calculated as with t aµν ≡ ∂ µ e aν − ∂ ν e aµ . γ a is the γ matrix that satisfies {γ a , γ b } = 2η ab , and Σ ab ≡ [γ a , γ b ]/2i is proportional to the spin. We take the Minkowski metric as η ab = [−1, +1, +1, +1]. In this convention, the γ matrices are expressed as γ0 = iβ, γî = iβαî with use of the Dirac matrices that satisfy {αî, α} = 2ηî, {αî, β} = 0, β 2 = 1. ψ is a four-component spinor, andψ ≡ ψ † β is the Dirac conjugate. First, we define a coordinate system. We introduce the FS frame to describe a generic curve r(s) parametrized by its arc length s. The tangential, normal, and binormal vectors are defined as T ≡ r , N ≡ T /κ, B ≡ T × N and arXiv:2002.05371v1 [cond-mat.mes-hall] 13 Feb 2020 satisfy the FS formula, Here, κ and τ are the curvature and the torsion, respectively. Right-(χ = +1) and left-handed (χ = −1) helices are described by r = [R cos s/L, R sin s/L, χP s/L], where R and 2πP are the radius and the pitch, respectively, and L ≡ √ R 2 + P 2 . The FS vectors are expressed as The curvature and the torsion are κ = R/L 2 and τ = χP/L 2 , respectively. We now shift to a rotated FS frame { t, n, b} following Refs. [32,39,40]. These vectors are defined as in which θ is related to the torsion as From Eq. (3), we obtain The original and the rotated FS frames in a right-handed helix are depicted in Fig. 1. We define a coordinate system for this frame as x(s, q 2 , q 3 ) = r(s) + n(s)q 2 + b(s)q 3 . q 2 = q 3 = 0 describes the curve. We obtain d x = teds + ndq 2 + bdq 3 with e = 1 − κq 2 cos θ − κq 3 sin θ being the determinant of a vielbein chosen below. The rotated FS frame is better than the original one because its metric is diagonal. We choose a vielbein as e0 0 = 1, eî s = tîe, eî q2 = nî, eî q3 = bî and e 0 0 = 1, e ŝ ı = tî/e, e q2 ı = nî, e q3 ı = bî. With this choice, t aµν defined above and the torsion-free spin connection Eq. (2) vanish.
Applying the thin-layer quantization, we derive an effective Lagrangian density in the curve [39,42]. We introduce a strong confinement potential in the normal and binormal directions. This enables us to assume a separable wave function and integrate Eq. (9) with respect q 2 , q 3 . We obtain an effective Lagrangian density for the tangential part ψ Carrying out the FW transformation to take the nonrelativistic limit [43,44], we find that under a unitary transformation ψ t , the Lagrangian density can be approximated as Retaining the upper block in the diagonal terms and dropping the rest mass energy, we obtain the effective nonrelativistic Hamiltonian for a twocomponent spinor χ The third term includes the momentum p s ≡ −i ∂ s and the spin σ · B, in which B is the binormal vector in the original FS frame. We term this as the geometric SOC. A similar but non-Hermitian result was reported in Ref. [40]. We believe that our result is physically correct, since we started with the Hermitian Lagrangian density and performed the unitary transformation. We emphasize that the geometric SOC is completely different from the conventional SOC because the former is O(m −1 ), while the latter is O(m −2 ). We may carry out the FW transformation before the thin-layer quantization, but this leads to a different result, The second term is the quantum geometric potential owing to the curvature [39,42]. The geometric SOC does not appear. Similar noncommutativity has already been pointed out in the context of a curved surface [45,46]. The relevance of the geometric SOC becomes evident when we consider a 1d ring. In this case, s = Rφ, κ = R −1 , τ = 0, B = z, and Eq. (12) is reduced to Since 3 ≡ −i∂ φ is the orbital angular momentum, 3 + σ3/2 is the total one. In the relativistic theory, only the total angular momentum is conserved. When the subspace is curved, the same occurs even in the nonrelativistic limit.
Edelstein effect caused by the geometric SOC-The geometric SOC is of the same form as the SOC that has been assumed in the theoretical literature [28,29,31,33,35,36]. In the case of DNA, the radius and the pitch are R = 1 nm and 2πP = 3.2 nm, respectively [25], leading to the curvature κ = 0.8 nm −1 . The energy scale which is estimated to be v F κ/2 = 160 meV, using a typical Fermi velocity v F = 6 × 10 5 m/s [26], is of the same order of magnitude as the experimental result [22]. In addition, the obtained geometric SOC is consistent with the experimental fact that the CISS was not observed in a singlestranded DNA but in a double-stranded one [21]. As already pointed out [29], any SOC of the first order with respect to the momentum can be eliminated in a purely 1d curve by a unitary transformation. This is also true for the geometric SOC. In fact, we obtainH (1) t = p 2 s /2m by the following unitary transformation, t (s). (15) Here, P is the path-ordered product along s. Therefore, the sublattice degree of freedom is essential for spinrelated phenomena.
Thus, we consider a model in which H (1) t (s) and H (1) t (s + πL) are coupled via a constant coupling Λ [29]. This model describes two coupled helices as in the doublestranded DNA [29] or a single helix with the long-range transfer integral as in α-helical protein [35]. According to Eq. (4c), the1,2 components of B change their signs by s → s + πL, while the3 component does not. Thus, the Hamiltonian of the coupled-helix model is expressed as Here, we have added the Zeeman coupling Bσ3 only for calculating the expectation value of σ3 for each band and p s . The s dependence in B ⊥ can be eliminated by a unitary transformation χ = e −i(s/L)σ3/2χ , which sends p s → p s − σ3/2L, σ ⊥ · B ⊥ → −(χP/L)σ2. For convenience, we introduce the dimensionless parameters p s = s /L, R = Lr, P = Lp, Λ = E 0 λ, B = E 0 b with r 2 + p 2 = 1, E 0 ≡ 2 /2mL 2 . The dimensionless transformed Hamiltonian now takes the form h = 2 s + p 2 /4 + λρ 1 + bσ3 − (p 2 σ3 + χrpρ 3 σ2) s , (17) and its eigenvalues are for each ρ = ±1, σ = ±1, , from which we obtain the group velocity v z ρσ = (χP/ )E 0 ∂ ρσ and the spin σ3 ρσ = ∂ b ρσ . All the quantities correspond to b = 0. The band structure for R = 1 nm, 2πP = 3.2 nm, and λ = 0.2 is shown in Fig. 2(a).
When we apply an electric field E z in the z direction, the charge current j z and the spin s z are induced in the same direction. We calculate the electric conductivity σ zz and the Edelstein coefficient α z z , which characterize j z = σ zz E z , s z = α z z E z , respectively. In the helix, since z = χP s/L, the electric field reduces to E z (χP/L) in terms of the arc length s. Within the relaxation time approximation at zero temperature, we obtain in which τ re is the relaxation time, q is the electron charge, and E 0 µ is the chemical potential. α z z changes its sign when the chirality changes. In Fig. 2(b), we show the µ dependence of the dimensionless Edelstein coefficient χ zz = (χqp/mL)α z z /σ zz = (qτ /m)α z z /σ zz , which characterizes s z = (m/qτ )χ zz j z . Note that χ zz is independent of χ, τ re . When the chemical potential lies only in the two lower bands, whose spin quantum number is σ = +1, χ zz changes from negative values to positive ones, because the integrand in Eq. (19b) is approximated as −p 2 < 0 for small | | and 2σp| | > 0 for large | |. The two upper bands characterized by σ = −1 have a negative contribution to χ zz . Thus, χ zz attains its the maximum value at the edge of the two upper bands. We also show the λ dependence of χ zz for µ = 0.2, 0.4, 0.6 in Fig. 2(c). χ zz vanishes for λ = 0 since the geometric SOC can be eliminated by a unitary transformation, as mentioned above. Each peak position corresponds to the parameter for which the band edge is equal to the chemical potential. We find that χ zz is of the order of 0.1 in a wide range of the parameters λ, µ. Therefore, s z 0.01 can be observed per nm when a charge current of 1 µA is applied.
To summarize, we have derived the geometric SOC of O(m −1 ) starting from the Dirac Lagrangian density in curved space, then applying the thin-layer quanti-zation [39,42] and finally, taking the nonrelativistic limit [43,44]. If the order is reversed, the geometric SOC does not appear. The estimated energy scale is a hundred meV for a nanoscale helix, much larger than the conventional SOC expected in light elements. We have also calculated the Edelstein coefficient in the coupledhelix model, which describes two coupled helices or a single helix with the long-range transfer integral. The current-induced spin polarization depends on the chirality and is of the order of 0.01 per nm when a charge current of 1 µA is applied. Although we have not considered the detailed compositions or the structures of chiral molecules, we believe that the emergence of the geometric SOC is general and provides a theoretical foundation for the CISS.