Weak Electron-Phonon Coupling Is Insufficient to Generate Significant CISS in Two-Terminal Transport

TL;DR

The paper investigates whether vibrational (electron-phonon) effects can generate sizable chiral-induced spin selectivity (CISS) in two-terminal transport through a helical molecule. It deploys a fully self-consistent nonequilibrium Green's function framework within the self-consistent Born approximation ($SCBA$) for a spin-orbit coupled, helical tight-binding model, comparing global, local, and diagonal e-ph couplings. Across energy-dependent and energy-integrated analyses, the study finds negligible spin polarization in the weak-coupling regime, with electron-phonon interactions primarily renormalizing the spectrum and preserving quasi-ballistic transport; diagonal approximations alter size-dependence but do not yield large CISS. The results suggest that additional ingredients—such as multi-orbital structure, stronger coupling, or electron-electron interactions—may be required to realize appreciable CISS in similar two-terminal setups, guiding future theoretical directions.

Abstract

A central open question in chiral-induced spin selectivity (CISS) is whether weak electron-phonon coupling in a helical molecular junction can generate a sizable spin polarization in two-terminal transport without invoking additional strong symmetry-breaking ingredients. We address this question by implementing a self-consistent nonequilibrium Green's function (NEGF) calculation for a helical tight-binding model with spin-orbit coupling and electron-phonon interactions. The electron-phonon self-energies are evaluated self-consistently, and the transport signal is extracted using the standard magnetization-reversal protocol with a spin-polarized analyzer lead. We benchmark a fully self-consistent NEGF within the self-consistent Born approximation (SCBA) treatment for both global and local electron-phonon couplings against commonly used approximations, including diagonal self-energy schemes. We quantify how the resulting transport regime and spin polarization depend on phonon frequency, coupling strength, bias, temperature, and system size. In contrast to large polarizations and anomalous size trends reported under approximate treatments, the fully self-consistent calculation yields negligible spin polarization, additionally the electron-phonon coupling mainly renormalizes the spectrum, and transport remains quasi-ballistic across the explored parameter range.

Figures (5)

• Figure 1: Schematic of the theoretical setup. A helical molecule is kept between two leads with chemical potentials, $\mu_L = V/2$ and $\mu_R = -V/2$. The left lead is magnetised one way and the particle current is measured. The magnetization is then reversed and the current is measured again. The difference between the two currents quantifies the CISS-induced spin polarization. Specific parameters for this figure are $n_l=10, m_l=5$, radius of the helix $a=1$, and length of the molecule $c=8$.
• Figure 2: Energy-dependent Figures with Global e-ph interaction:(a) Variation of density of states for different e-ph interaction strength $\chi$, such that $t_1=\chi t_0, \lambda_1=\chi \lambda_0$. (b) Difference in density of electronic states for opposite lead magnetization, $\rho_\uparrow-\rho_\downarrow$, for different e-ph interaction strengths. (c) Variation of particle current coming out of the left bath, for opposite lead magnetization and different phonon frequencies. (d) Energy-dependent Spin polarization for different phonon frequencies. (e) Energy-dependent Phonon-bath current and numerical current noise denoted by the symbol 'n' for different phonon frequencies and left lead polarisation, p=0.5. The numerical noise is defined in the subsection \ref{['numerical_details']}. (f) Energy-dependent current for different system sizes. The default system size considered above is $n_l=4, m_l=2$. The e-ph coupling is similar to the study Fransson_PhysRevB.102.235416, with $\varepsilon_1=0$, and the default values of the parameters used in the model and the code in units of 0.1 ev are $t_0=0.4,\varepsilon_0=-6t_0, t_1=0.1t_0, \lambda_0=t_0/40,$$\lambda_1=0.1\lambda_0$, $\Gamma_0=t_0/4, \omega_0=0.01t_0, T=0.25$$(300 K)$. Also, radius and length of helix are equal, $a=c=1$. The chemical potentials of the two leads are given as, $\mu_L=V/2, \mu_R=-V/2$, and $V=15t_0$.
• Figure 3: Energy-dependent figures with local e-ph interaction:(a) Variation of Energy-dependent particle current coming out of the left bath, for opposite lead magnetization and different phonon frequencies, (b) Variation of phonon-bath current with energy for different phonon frequencies (c) Energy-dependent left lead particle current for different system sizes, and (d) Variation of polarization with energy for different system sizes. Currents characteristics under the diagonal approximation described in subsection \ref{['diagonal_approximation']}, (e) Energy-dependent left lead particle current for different system sizes with diagonal approximation, the horizontal dashed lines show the energy integrated particle current for their respective system size, (f) Energy-dependent polarization for the case with diagonal approximation. As earlier, the default system size considered above is $n_l=4, m_l=2$. The default values of the parameters used in the model and the code in units of 0.1 eV are $t_0=0.4,\varepsilon_0=-6t_0, \epsilon_1=0.1t_0, \lambda_0=t_0/40,$$\lambda_1=0.1\lambda_0$, $\Gamma_0=t_0/4, \omega_0=0.1t_0, T=0.25$ ($300$ K). Also, $t_1=0.1t_0$ for the mixed coupling with diagonal approximation figs, while it is '0' for the purely local case. The chemical potentials of the two leads are given as $\mu_L=V/2, \mu_R=-V/2$, and the voltage is $V=15t_0$.
• Figure 4: Energy integrated figures:(a) Variation of particle current for global (G), local (L), and mixed-diagonal (MD) couplings with voltage. For the global case, two different phonon frequencies are considered, '$\omega_0=\omega_L=0.01t_0, \omega_0=\omega_H=0.1t_0$', for the other two cases the phonon frequency is $\omega_0=0.1t_0$. Variation of particle current with (b) Temperature, and (c) e-ph coupling strength $\chi$. (d) Difference between left and right bath current with normalised Voltage, Temperature, and $\chi$ for p=0.5. The x-axis for the different parameters has been normalised according to their respective ranges, as shown in earlier graphs. polarization plots with normalised Voltage, Temperature, and e-ph interaction strength for (e) global coupling, (f) Local coupling, and mixed coupling with Diagonal Approximation. The default parameters for Global and Local figures are the same as stated in earlier figures. For Mixed-Diagonal, we use the system size $m_l=5, n_l=4$.
• Figure 5: (a) Energy-dependent density of states obtained using the approximations introduced in Appendix \ref{['fransson_approx']}, (b) Current-voltage characteristics for opposite left-lead magnetizations. (c) Spin polarization as a function of bias voltage.