Spin conductances and magnetization production in chiral molecular junctions Open Access

Charge currents are routinely measured and analyzed in molecular electronics.¹ The discovery of a family of phenomena that exhibit chirality induced spin selectivity (CISS) has led to a resurge of interest in spin-related phenomena in this field.^2–7 In particular, experiments report a strong correlation between molecular chirality and a preferred spin direction in systems that exhibit a (nominally) very weak spin–orbit interaction. At present, there is no consensus concerning the explanation of many experimental CISS results.⁸ 

Motivated by the CISS phenomena, we here address charge and spin currents in chiral molecular junctions within the framework of a minimal model. As a diagnostic tool of the junction’s atomic structure, spin currents offer advantages as compared to charge currents: Spin polarized currents can, in principle, be detected in charge-transport measurements within an analyzer–polarizer setup employing, e.g., magnetized leads. However, they are expected to manifest themselves only in the non-linear regime and not in the linear charge conductance G(M). This is a consequence of Onsager’s reciprocity; its importance for the theory of the CISS effect was emphasized by Yang, van der Wal, and van Wees.^9–11 Meanwhile, the spin-conductance is less restricted by Onsager symmetries. It can be inferred, at least in principle, from measuring the pileup of magnetization, e.g., in source or drain. A brief review of the symmetry properties of transport coefficients is available in Sec. S1 of the supplementary material.

These considerations motivated us and other researchers¹² to investigate a minimal model for a chiral molecule in the presence of spin–orbit interactions. While Ref. 12 has focused on the effect of contact-enhanced spin–orbit interaction (SOI), we adopt the model proposed by Michaeli and Naaman;¹³ it exhibits SOI on the entire molecule and has the extra benefit of allowing for an analytical treatment. While Michaeli and Naaman have studied the transmission properties, our focus is on (spin) conductances. As one would have expected, the spin conductance turns out to be non-vanishing due to spin–orbit coupling. As a consequence, the transmitted and reflected currents tend to build up a non-vanishing spin-accumulation near both contacts, source and drain, already in the linear regime.¹⁴ In stationary non-equilibrium, the magnitude of induced magnetization is likely controlled by spin-relaxation processes. We predict the orientation of the magnetization in both contacts to be the same, in agreement with requirements of time-reversal invariance (TRI).¹² 

Our work is of potential impact for constructing a molecular machine. Since total angular momentum is conserved by the LS-coupling, spin-flip processes exert a mechanical torque that can drive an engine. An analog driving mechanism based on angular transfer has been investigated in Ref. 15.

Following Michaeli and Naaman,¹³ we consider electrons bound to a long tube. The left and right sides of the tube are of cylindrical shape and represent (semi-infinite) reservoirs; the central region in between takes a helical shape to mimic a chiral molecule; see Fig. 1.

In the limit of small $d/R̃$⁠, the electronic wavefunction in the central region is tightly bound to the helix and the quasi-one-dimensional nature of the model becomes manifest; here, $R̃=(2πR)2+b2>0$ denotes the distance covered when completing one helical turn. In this limit, the longitudinal and transverse motions approximately decouple and the problem simplifies. Following this idea, a systematic expansion of Eq. (1) in $d/R̃$ has been performed by Michaeli and Naaman,¹³ and keeping only leading order terms, a minimal model has been derived.

The wavefunction Ψ(s, ϱ, θ) can be conveniently expressed in a local cylindrical coordinate system: s denotes the longitudinal coordinate (distance along the helix); the motion in the plane normal to the tangential vector $ŝ$ is described by the radial coordinate ϱ, which denotes the normal distance from the centerline of the tube (Fig. 1), and θ, which denotes the corresponding angular coordinate.

For simplicity, we will assume local rotational invariance in the sense V_H(r) = V_H(ϱ). In the small $d/R̃$ limit, the longitudinal and transverse motions nearly decouple,¹³ and the wavefunction factorizes,¹⁶  $Ψ(s,ϱ,θ)≈Ψ̄m(s)ΦN,m(ϱ)eimθ$⁠. The quantum number $N∈N0$ governs the nodal structure of the wavefunction in the radial direction. Due to the stipulated rotational invariance, the $ŝ$-component of the angular momentum, −iℏ∂/∂θ, is a good quantum number; we call it ℏm and $m∈Z$⁠. Finally, owing to the presence of SOI, $Ψ̄m(s)$ represents a two-spinor.

In representation (6), flipping the spin boosts the momentum along the tube axis by a reciprocal lattice vector $2π/R̃$⁠. Alternatively to s, one can label the helical motion with an angle $δ=λ2πs/R̃$ that takes unique values on the entire real axis; moving up one pitch implies a change of δ ↦ δ ± 2π. From this perspective, the phase-factors in (6) boost the angular δ-dependency of the spinor $Ψ(δλR̃2π,ϱ,θ)$ by an extra factor e^iδ, and in this sense, angular momentum is conserved.

While the phase factor in (6) accounts for the conservation of the total angular momentum and therefore is crucial, it also obstructs an easy analytical solution of the model because it is not translationally invariant. The Hamiltonian considerably simplifies after a gauge transformation, $Ψ̄m(s)↦eiλσ̂zπs/R̃Ψ̄m(s)$⁠, and accordingly for operators.¹⁸ 

The $γ̃$-term corresponds to a momentum shift and thus represents a “synthetic” vector potential. It can formally be removed from $ĤN,m′$ by dressing the wavefunctions with a gauge-factor $eiγ̃ms′$¹⁹ and therefore leaves the spectrum invariant. If $γ̃m$ has a natural interpretation as a vector potential, the term proportional to b/R in the second line of (9) is the corresponding “synthetic Zeeman term.” Notice that the vector potential, $γ̃m$⁠, and the Zeeman term change sign under m ↦ −m; they do not break TRI because the full Hamiltonian sums over all m.

Figure 2 shows the dispersion law; the horizontal shift of the two parabolas, red and blue, reflects the non-Abelian gauge transformation. The spin–orbit term is effective only at the crossing of parabolas, where it opens a spin–orbit gap, $2|m|κ̃$⁠, as is easily confirmed by degenerate perturbation theory at the crossing point k = 0. Chirality-induced spin-selective phenomena are expected to be strong in this region of energies.

In the gap region, there remain four ungapped bands—a factor of two for spin and angular momenta each—where the spin projection and the sign of the velocity, dE_m,α(k)/dk, are locked. In all four bands, the projection of spin onto the direction of velocity equals −λℏ/2 (“spin-momentum locking”). Such states are termed helical, in full analogy to the helicity concept for edge states in the quantum spin Hall effect.²⁰ The band structure arising from (10) was also discussed in the context of Rashba quantum wires^21,22 and chiral carbon nanotubes,²³ although the physical origin of the terms of the Hamiltonian was different from our situation. In particular, the gap of the minimal model (7) opens due to the SOI term, $−λκ̃mσ̂y$⁠, while in Rashba wires and nanotubes such a term originated from a transverse magnetic field.

To facilitate transport studies, we attach two straight tubes, Fig. 1, which serve as reservoirs. Formally, the reservoirs are included by extending model (9) so that $γ̃,κ̃=0$ if s′ < 0 or s′ > L. Importantly, the non-Abelian gauge transformation restores translational invariance also after attaching leads provided that it is performed in the reservoirs, too. Similarly, the synthetic vector potential, $γ̃(s′)m$⁠, can still be removed by applying a (non-local) gauge factor $eimΓ̃(s′)$⁠, with $γ̃(s′)=∂s′Γ̃(s′)$⁠.

Due to local charge conservation, the charge current is well defined and, in particular, independent of where it is measured along the current flow. In contrast, spin is not locally conserved in the presence of spin–orbit coupling. It is only in the leads of the extended model where (longitudinal) spin-currents are well-defined observables. Notice that due to the loss of spin-conservation in the central region, 0 < s < L, spin-currents in the left- and right-reservoirs, $Ii(L)$ and $Ii(R)(V)$⁠, may differ in a quasi-stationary non-equilibrium situation.

We adopt here the following sign convention for the spin currents: $Ii(L)$ measures spin entering the junction from the left $(L)$ contact and $Ii(R)(V)$ measures spin exiting the junction into the right $(R)$ contact. This is fully analogous to the definition of the charge current.

We address the transport problem by calculating the scattering matrix using conventional wavefunction matching. In this process, angular momentum m matches at the two interfaces: it is conserved in the scattering process; for further details, see the supplementary material.

As example, we focus on N = 1 so that m = ±1. We further continue to ignore the effect of the synthetic fields. In passing, we briefly mention that their effect is to assign a preferred spin direction, up or down, to a given angular momentum m. Hence, they will result in a circulating (transverse) spin current. In the following, our focus is on the longitudinal currents.

The energy dependence of the resulting conductances for charge and spin in this model is displayed in Fig. 3. We offer a few comments as follows:

• With E_F well outside the spin–orbit gap, the effect of spin–orbit coupling is small and translational invariance is hardly broken. In this case, backscattering is weak and the (charge) conductance reaches a maximum of four conductance quanta reflecting two spin and two orbital (m = ±1) channels.

• In the off-gap regime, mesoscopic oscillations are visible. The oscillation frequency (in E_F) is seen to decrease with the inverse length, L⁻¹, and therefore, we assign the oscillations to Fabry–Pérot interference.

• With E_F inside the spin–orbit gap, backscattering inside the wire is suppressed and the Fabry–Pérot oscillations quickly die out.

• In the in-gap regime, electrons can tunnel via two evanescent modes that result from the two “gapped” bands. Accordingly, the charge conductance in the middle of the gap approaches two conductance quanta from above.

• The absolute value of the spin conductances in either one of the leads approaches $2e4π$⁠. This can be understood from Fig. 2: in the spin–orbit gap, the right (or left) moving modes have identical spin regardless of m, i.e., there are two channels distinguished only by the orbital angular momentum.

• Importantly, the spin conductances are non-zero for energies even far from the spin–orbit gap, where the oscillations peak at $≈0.2e4π$⁠. It can be shown that $Gx(L,R)(EF)=0$ for any E_F since the Bloch functions have zero average $σ̂x$⁠. Moreover, $Gy(L,R)=0$ because the expectation values of $σ̂y$ change sign along with the sign change of m (see Fig. 2), while the lead Hamiltonian is m-independent. In other words, the y components of Bloch waves exactly cancel upon the summation over m.

• Remarkably, the $L$ and $R$ spin conductances differ only up to a sign. By combining TRI and left–right reflection, it is possible to derive relations between scattering matrix elements that lead to the exact identity $Gz(R)=−Gz(L)$ at any energy (see Secs. S4 and S5 of the supplementary material for the details of this symmetry analysis). In realistic molecular junctions, a symmetry of couplings to the left and right leads cannot be expected. We show in Sec. II F that in this more general situation, the magnitude of $Gz(R)$ and $Gz(L)$ no longer is the same, while the sign is still opposite due to TRI.

We begin the discussion by addressing a common misconception of the minimal model: Fig. 2 is frequently interpreted as predicting a nonzero spin current even in equilibrium. Indeed, Fig. 2 seems to suggest that for a Fermi level situated in the spin–orbit gap, there is an excess of spin-up right movers [velocity dE_m,α(k)/dk > 0] over spin-down right movers. Taken at face value, this observation would imply that due to the wire connecting $L$ and $R$ reservoirs, both reservoirs would become magnetized at equilibrium. Clearly, such a transport of magnetization is violating the second law of thermodynamics. The vanishing of equilibrium spin currents is a rigorous consequence of unitarity; see Sec. S5 of the supplementary material. The fact that spin transport cannot be derived from the band structure alone is known in the field of two-dimensional materials; see Sec. 4.1 of Ref. 25 for a review and a discussion in the context of CISS in Ref. 9. The paradox will be resolved on a more intuitive level in the subsequent discussion.

We consider λ = −1 and the limit L ≫ d, R and Fermi energy, E_F, inside the spin–orbit gap. In this limit, at observation points deep inside the one-dimensional wire all up-spin fermions flow in one direction, while all down-spin fermions flow into the other direction; see Fig. 2. We infer that $T↓↓,m(LR),T↑↑,m(RL)≃1$⁠; there is no spin-flip inside the wire. Meanwhile, deep inside either reservoir, the spin–orbit gap vanishes. Therefore, spin-up and spin-down currents flow alike in either direction. It is easy to see that both limits match if $R↑↓,m(LL),R↓↑,m(RR)≃1$⁠.

The situation is illustrated in Fig. 4(a). The central region of the junction shows counter-propagating electrons with opposite spin, as suggested in Fig. 2. These modes are properly interpreted as carrying a conductance quantum for charge, because charge is conserved inside the wire; hence, in Fig. 3, the conductance is seen to be G ≈ e²/h per angular-momentum channel inside the gap. These same modes are not properly interpreted in terms of spin-conductances, however, because spin is not conserved inside the wire.

The propagation pattern of current channels in the leads is also displayed in Fig. 4(a); the matching condition at the interface shown there follows from bulk limits. Since spin is conserved inside the reservoirs, the channel pattern may be interpreted in terms of spin currents. In accordance with second law of thermodynamics (Sec. II E 1), there are no net spin currents flowing into the reservoirs in equilibrium, because the propagating spin-current is exactly compensated by the (reflected) counter-propagating current.

Upon applying a finite bias, states with energy inside the bias window are incoming only from one reservoir. In this case, a spin current of one spin-conductance quantum (per angular momentum channel) survives inside the reservoirs. Also with respect to spin-conductances, Figs. 4(a) and 3 provide a consistent picture.

The qualitative discussion given here is fully backed up by an explicit calculation of all scattering probabilities; for explicit results, we refer the reader to Fig. S2 of the supplementary material.

An implication of spin-flip scattering is that both reservoirs accumulate spin in the presence of a current flow. To see how this happens, we once again consult Fig. 4(a). For a charge current flowing from $L$ to $R$⁠, the drain acquires a spin-up magnetization (red). Simultaneously, the incoming flow of spin-down particles (blue) is reflected and spin flipped so that the $L$-lead acquires a spin-up magnetization (red), too. The observation represents a one-dimensional analog of the Rashba–Edelstein effect.²⁶ 

For further illustration, we confront the spin-flip scattering in the helical junction with the more familiar case of a magnetized junction that operates as a spin-filter, Fig. 4(b). Also in this archetypal situation, the reservoirs accumulate spin; however, they do so in opposite directions (see the schematics of scattering in Fig. 4). The key difference to the previous case is that spin is conserved everywhere so that the (minority) spins removed from the source accumulate in the drain.

Figure 5 displays the conductances for increasing barrier strength. The charge conductance exhibits a gradual crossover from the transparent, weak-barrier limit, c ≪ 1, to the strong-barrier limit, c ≫ 1, in which the transmission grows linearly with E_F. Concomitantly, the spin-conductances evolve in a strikingly asymmetric fashion. For qualitative insights into the strong barrier limit, we consult again Fig. 4(a). For a current flowing from $L$ to $R$ and E_F inside the spin–orbit gap, the barrier suppresses the transmitted current and spin-flip processes alike; hence, $Gz(L)$ is strongly suppressed inside the source. Conversely, the spin-current flowing in the drain equals the transmitted current (in units of the conductance quanta). Our argument implies that the spin conductances continue to exhibit opposite signs in the presence of asymmetries. Therefore, we propose that the property of source and drain to magnetize into the same direction upon a current flowing is a general result robust with respect to generic deformations of the minimal model (14).

We supplement the transport results with expressions for the charge and spin density in the $L$-lead (s < 0) in linear response. The latter quantities will be given in units of e and ℏ/2, respectively. Our formulas follow from scattering theory and are thus independent of the microscopic details of the Hamiltonian. The details of the derivation are moved to Sec. S6 of the supplementary material for the sake of brevity.

Notice that unlike spin-conserving processes, spin-flip processes, such as incorporated by $r↓↑,m(LL)$⁠, do not contribute to the oscillating term in (17) because the superposition of two probability currents with opposite spins has no cross (interference) term. Spin-flip terms do enter the conductance, G, of course.

• Spin currents and spin accumulation are a central topic in the field of Spintronics.^29,30 In particular, the possibility of spin accumulation near interfaces between materials with and without spin–orbit interaction is well understood;³¹ the results we report here on the minimal level confirm the validity of the general picture down to the molecular scale. With an eye on experiments, we mention that the amount of accumulated spin should depend on the spin-relaxation time. Our result for the spin accumulation applies to local measurements a distance no longer than the spin relaxation length away from the contact.

  In Sec. II, we have mentioned that the bands for energies in the spin–orbit gap are analogous to edge states associated with a quantum spin Hall device. As a consequence, the minimal model exhibits spin-momentum locking with the consequence that backscattering off defects is suppressed. In other words, we expect that our transport results are robust against weak (non-magnetic) disorder.

• Recently, a related study on parallel spin-accumulation and spin transport has been published.¹² The tight-binding chain investigated by these authors is essentially equivalent to our toy model with a distinctive difference: In Ref. 12, the SOI is confined to the contact bond only, while in our toy model, the SOI is a property of the molecule and non-vanishing along the entire helical structure.

  The spin conductances reported in Ref. 12 show the same symmetry, $Gz(L)=−GzR$⁠, as in our work. However, the energy dependences of the spin conductances exhibit pronounced differences: $Gz(L/R)(E)$ in Ref. 12 does not exhibit a fixed sign; in our case it does so that the sign of the spin accumulation is independent of the Fermi energy. This difference can be traced back to the different ways how SOI is implemented; it may, at least in principle, be used to discriminate one situation from the other, experimentally.

• It is well-known in spintronics that spin conductances are non-zero in junctions with multiple conduction channels only (see Sec. S1, Principle 1, of the supplementary material). In our model, the absence of spin polarization for a single channel is immediately obvious by setting m = 0 in Eq. (9). Molecular junctions generically exhibit multiple channels and are thus prepared for hosting spin currents (e.g., see Ref. 32 for a channel analysis of helical peptides). Frequently employed molecular linker units (e.g., thiol groups linked to Au) effectively suppress all conduction channels but one. Moreover, the generation of significant spin currents benefits from two conduction channels of similar transmission probabilities, which in turn requires quasi-degenerate molecular orbitals. To favor such conditions, linker-free benzene-type structures are natural candidates. They can be functionalized with heavier elements (see, e.g., Ref. 33) to boost the spin–orbit coupling and promote chirality.

We have investigated charge and spin transport in a minimal model of a helical molecule with spin–orbit coupling attached to two spin-conserving leads. The minimal model was first devised by Michaeli and Naaman and allows for a full analytic treatment of transport properties. While the earlier authors have focused on spin polarizations, we calculated the full conductance matrix, including charge and spin conductances.

The band structure of the minimal model hosts four helical bands that exhibit spin-momentum locking. We first clarify the connection between transport properties and band structure. In particular, first glances could suggest the existence of an equilibrium spin current, which sometimes is used as an argument against the validity of the model. We explain the origin of the misunderstanding, which results from neglecting the contact scattering that always exists in a transport geometry. In actuality, there is no equilibrium spin transport in this model.

Our explicit calculation of charge and spin conductances in the Landauer formalism shows that the spin conductance reaches a maximum (e/2π) for energies inside the gap, corresponding to two fully polarized conduction channels. Outside the gap region, the spin conductances remain sizable, too. Furthermore, we find that at small biases and for incoming currents being unpolarized, there are spin currents with opposite signs in each lead.

Upon a current flowing, spin accumulates in the vicinity of each contact as a consequence of the spin currents; the magnetizations at left and right contacts are the same and reverse with voltage and helicity. The magnitude of the accumulated spin polarization is directly proportional to the spin conductance of the respective lead; therefore, we reveal a new route toward the measurement of $Gz(L/R)$⁠, as has also been proposed for specific device geometries by Yang and van Wees.²⁸ 

The minimal model therefore directly reveals important features of the analytic structure of spin transport in helical molecules. It can serve as a guidance for the interpretation of ab initio calculations of transport coefficients in chiral molecular systems, e.g., based on nonequilibrium Green’s functions. Furthermore, it lends itself to straightforward generalizations, e.g., the analytic calculation of current-induced mechanical torques in a quantum model.

The supplementary material contains Secs. S1–S6 with methodological details.

Financial support for this project was provided by the Czech Science Foundation (Project No. 22-22419S), the Netherlands Organisation for Scientific Research (NWO), Grant No. 680.92.18.01. FE acknowledges the support from the German Research Foundation (DFG) through the Collaborative Research Center, SFB 1277 (Project No. A03), through GRK 2905, Project-ID. 502572516, and through the State Major Instrumentation Program, INST 89/560-1; project No. 464531296. The authors thank K. Michaeli, O. Tal, K. Richter, J. Fabian, J. Schliemann, D. Weiss, W. Wulfhekel, D. Kochan, and B. Yan for their discussions. We thank I. Dimitriev for correcting our expression of the current operator.

The authors have no conflicts to disclose.

Richard Korytár: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jan M. van Ruitenbeek: Conceptualization (supporting); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Ferdinand Evers: Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.