Magnetoconductance from spin-charge inter-conversion in two-terminal molecular nanojunctions with strong spin-orbit coupling electrodes.

Spin-charge inter-conversion mediated by spin-orbit coupling can lead to finite magnetoconductance in two-terminal molecular nanojunctions under non-equilibrium conditions. Here, we demonstrate how such a finite magnetoconductance can emerge in model two-terminal molecular nanojunctions by means of density functional theory based transport calculations with spin-orbit coupling in a first-order perturbation approximation. The junctions are built from the two chiral partners of an idealized helical molecule and tungsten or gold electrodes with two layers of magnetic nickel at the interface of the drain electrodes. Using Au source electrodes and a low applied bias of 0.1 V, we find percentage relative magnetoconductance values in excess of the lower bound reported in recent low-temperature, low-bias, experiments. The left-handed molecule is seen to exhibit greater magnetoconductance than its right-handed chiral partner for both Au and W source electrodes, thus demonstrating that our calculations can also exhibit enantioselectivity.


Introduction
Magnetoconductance experiments involve measuring changes in electrical conductance under an applied magnetic field.In the realm of the chirality-induced spin selectivity (CISS) phenomenon, these measurements offer valuable perspectives on the spin-specific transport characteristics of chiral systems [1][2][3][4][5][6][7][8].By examining the variation in conductance with changes in the magnetic field in suitably designed experiments [6,8], information can be gathered about the spin selectivity in chiral systems that are ultimately induced by spatial symmetry-breaking [9,10].
The CISS effect holds great promise for the development of molecular quantum computing devices [11], but its theoretical underpinnings are still a subject of intense research and debate [12][13][14][15][16][17][18].One prominent theory suggests that the CISS effect arises from spin-orbit coupling (SOC).This is essentially a manifestation of the Spin Hall [19][20][21] or Edelstein effects [22] in a two-terminal junction setting involving a strong SOC electrode material.

Methods
We employ the latest SOC-corrected version of the NEGF quantum transport code ANT.Gaussian [30], built on top of Gaussian09 [31], in open-boundary transport calculations of two-terminal molecular nanojunctions under finite bias in the linear-response regime [28].The structures used in our calculations are shown in Fig. 1.They are oriented along the +z-axis and consist of a simple helical molecule [9], of both chiral handedness, sandwiched between either gold or tungsten electrodes, consisting of two layers stacked along the (001) crystallographic axis of bulk face-centred cubic (FCC) Au or body-centred cubic (BCC) W. In addition, two layers of magnetic FCC Ni(001) have been placed at the molecule-metal interface of the drain electrodes, with the initial magnetization of the Ni atoms in these layers pointing along the +z-axis.In this way, Ni serves as the ferromagnetic (FM) detector component in our system [32,33].We employ the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional in our DFT computations.[34].DFT+PBE is known to underestimate the band gap; however, in this case, because we use an idealized molecule, underestimation of the bandgap by DFT is irrelevant.To drive our systems out of equilibrium, we apply a low bias of 0.1 V across the junctions.SOC is only applied to the first metallic layer on either side of the molecules and imposes a global quantization axis along the +z direction in all our calculations [9].To gauge the influence of SOC on our results, we vary its strength in the Au or W layers in the source electrodes by uniformly scaling the multiplicative SOC factors [35] of their P and D Contracted Gaussian-Type Orbital (CGTO) shells from zero up to five times relative to the values that give a good fit to SOC-corrected bands [36].The SOC factors of the first layer of Ni in the drain electrodes are not varied in this way except when SOC is set to 0 in the whole system.Instead, when finite, the multiplicative SOC factors are kept at values that give a good fit to the SOC bands of bulk FCC Ni.Those interested can find comprehensive information about our implementation of SOC in ANT.Gaussian in our previous publications [9,10,35,37].

Results and discussion
Figure 2 shows the percentage relative magnetoconductance (%M C) as a function of energy about the Fermi level (zero on the horizontal axis) the four structures from Fig. 1 (included as insets here).These results have been obtained at a finite bias of 0.1V and for SOC strengths ranging from zero up to five times the SOC-band-fitting values of the Au or W source electrodes.The percentage M C has been calculated as where T (+M ) and T (−M ) are the transmission functions that have been obtained for the mutually reversed magnetizations of each system, calculated following the approach described in Ref. [10].(The magnetization in the FM Ni layers point along the −z-axis in the reversed case.)The zero-bias results are also included as an inset in each subfigure and, as dictated by the Onsager relation T (+M ) = T (−M ) in force at equilibrium [32,33], exhibit vanishing %M C compared to their finite bias counterparts in the main panels.
In the case of the Au source electrode in Figs. 2 a) and b), the %M C of the system with the left-handed helix in a) achieves values in excess of the 9% lower bound reported in Ref. [6].This trend remains consistent across various energy levels around the Fermi level.The experimental value in Ref. [6] is considered a lower bound because there are other contributions to transport across the junctions beside scattering through the junction molecules.By contrast, the righthanded enantiomer in b) exhibits lower %M C values, and the experimental lower bound is exceeded only at energies between 2 and 3 eV on either side of the Fermi level.When SOC is absent from the system (red line in the plots), the %M C is necessarily vanishing.
Turning now to the systems with the W source electrode in Figs. 2 c) and d): much lower %M C values compared to a) and b) are observed.In both cases we see that values approaching the experimental lower bound only occur at energies of 2-3 eV above the Fermi level.Once again, the left-handed molecule tends to give rise to higher %M C across the entire range of energies shown.However, both systems with W source electrode (Fig. 2 c,d) exhibit much lower %M C than the two systems (Fig. 2 a,b), with Au source layers.The SOC-corrected D-shells of the Au source electrode, which occur below the Fermi level in bulk Au, thus appear to better confer SOC to the molecules than the SOC-corrected P -shells of W, which instead occur above the Fermi level in bulk W.

Conclusion
We have demonstrated that percentage relative magnetoconductance values observed in recent low-temperature, low-bias, two-terminal experiments employing Au as electrode material, can be reproduced in SOC-corrected linear-response DFT transport calculations using Au as source electrode.Interestingly, the left-handed partner of the model helical molecule in this work exhibits higher %M C than its right-handed counterpart irrespective of which source electrode material, Au or W, is used.Thus, the chirality of the same molecule affects the extent to which it can filter spins, reflecting the enantioselectivity of the CISS effect.We speculate that using W as source electrode leads to much lower %M C on account of SOC being conferred to the molecule via valence P shells of W instead of the valence D shells in Au.

Figure 1 .
Figure 1.(Colour online) a) Left-handed helical molecule sandwiched between an Au source electrode and a Ni-Au drain electrode.b) Right-handed helical molecule between the same set of electrodes as in a).c) Left-handed helical molecule between a W source and Ni-W drain electrode.d) Same electrodes as in c) but with left-handed helical molecule in the constriction.These structures have been rendered using OVITO [29].

Figure 2 .
Figure 2. (Colour online) %M C versus energy via Eq.(1) under bias = 0.1 V for a), the left-handed helix between Au source electrode and Ni-Au drain electrode; b), the right-handed helix with the same electrodes as in a); c), the left-handed helix between W source electrode and Ni-W drain electrode; and d), the right-handed helix with the electrodes in c).The inset plots show %M C under zero bias in each case.