Nonmagnetic single-molecule spin-filter based on quantum interference

Key spin transport phenomena, including magnetoresistance and spin transfer torque, cannot be activated without spin-polarized currents, in which one electron spin is dominant. At the nanoscale, the relevant length-scale for modern spintronics, spin current generation is rather limited due to unwanted contributions from poorly spin-polarized frontier states in ferromagnetic electrodes, or too short length-scales for efficient spin splitting by spin-orbit interaction and magnetic fields. Here, we show that spin-polarized currents can be generated in silver-vanadocene-silver single molecule junctions without magnetic components or magnetic fields. In some cases, the measured spin currents approach the limit of ideal ballistic spin transport. Comparison between conductance and shot-noise measurements to detailed calculations reveals a mechanism based on spin-dependent quantum interference that yields very efficient spin filtering. Our findings pave the way for nanoscale spintronics based on quantum interference, with the advantages of low sensitivity to decoherence effects and the freedom to use non-magnetic materials.


Electronic measurement circuit
Supplementary Fig. 1 shows the electronic setup connected to the sample. The circuit can be switched between a Conductance Mode, which is used to measure d.c. conductance as well as a.c. (differential) conductance spectra, and a Noise Mode, in which thermal noise and shot noise are measured. In the latter mode of measurement, the relatively noisy instruments used in the conductance mode are disconnected from the sample because of the high sensitivity of the noise measurements. Figure 1. Electronic circuit of the measurement setup. Schematic presentation of the electronic circuit for conductance and noise measurements. Two switchable measurement circuits are observed: a conductance circuit (orange) and a noise circuit (blue). The section cooled to 4.2 K is marked in light gray.

Shot noise measurements
The overall zero-frequency current noise (both thermal and shot noises) generated in a quantum coherent conductor can be expressed in the framework of Landauer formalism as 1,2 where = /2 B provides the ratio between an applied voltage V and temperature T. At near-equilibrium conditions (x≪1), equation (1) reduces to the thermal noise expression 1 = 4 B . At high applied voltage (x≫1), the current noise depends linearly on the current, = 2 . Supplementary Fig. 2 presents an example of shot noise analysis in an Ag/vanadocene molecular junction. Before and after each set of noise measurements as a function of bias voltage, differential conductance spectra (dI/dV versus V; Supplementary   Fig. 2a), are recorded in order to confirm that the junction has remained stable during the noise measurements.
The zero-bias conductance of the junction is determined from the average differential conductance in a window of |V| ≤ 5 mV. Variations in the conductance at this range are taken into consideration as an experimental error.
Supplementary Fig. 2b presents a set of noise spectra for different applied current. Note that the bias current produces bias voltage across the junction ( = / ). Each spectrum is obtained from the Fourier transform of voltage fluctuations produced by the junction, and averaged for 3,000 consecutive measurements. The voltage noise produced by the amplifier was measured separately and subtracted from the recorded spectra. The measured voltage noise is presented as a current noise using the relation = • 2 . Next, the spectra were corrected to account for low-pass RC filtering due to the resistance of the sample and cabling, as well as the amplifier input capacitance and the capacitance of the cabling (total capacitance of ∼40 pF). The noise power is averaged in a frequency window, in which 1/f noise contributions are negligible (marked by red lined in Supplementary Fig. 2b). Supplementary Fig. 2c shows the obtained average current noise power as a function of the generated bias voltage across the junction. Following Ref. 2, Eq. (1) can be expressed as Here, = ℎ( ) and = [ ( ) − (0)]/ (0). The Fano factor is obtained by calculating the reduced parameters and and obtaining a linear fit of ( ) according to Eq. (2), as shown in Supplementary Fig. 2d.
Occasionally, we observed zero-bias anomalies in the differential conductance curves that may originate from different possible mechanisms, including Kondo physics. The data obtained in these specific cases will be analyzed and reported elsewhere, since the physics involved is beyond the Landauer formalism considered here. Figure 2: Analysis of shot noise measured in an Ag/vanadocene junction. a, Differential conductance versus voltage curves measured for an Ag/vanadocene junction before (black) and after (dashed red) noise measurements. b, A series of current noise spectra for different bias voltage across the junction. Red lines indicate the frequency window that is selected to obtain the average noise power. c, Average current noise power as function of bias voltage calculated using the presented noise in (b). d, Reduced parameter as a function of (blue) calculated using the measured noise in (c), and a linear fit (red), giving F = 0.15 ± 0.01 according to Supplementary Eq. (2).

Determination of conductance spin polarization by shot noise measurements
Shot noise measurements were used in a variety of nanoscale and mesoscopic systems to gain information on spin transport effects [2][3][4][5][6][7] . In a quantum coherent conductor with an unknown number of transmission channels, the combination of conductance and shot noise measurements provides a lower bound for the degree of conductance spin polarization (CSP) 6 . In the following, we wish to elaborate on the meaning of the ( , ) plot presented in Fig. 2  For zero spin-polarization, the conductance is carried by one or more pairs of spin-polarized transmission channels for opposite spins with equal transmissions ( 1,↑ = 1,↓ , 2,↑ = 2,↓ , … ). For example, the conductance of a single-atom junction of silver is carried by two equal and opposite spin-polarized transmission channels that are associated with the single s valence orbital of silver. For zero CSP, the ( , ) data points can have any value in the white region of Fig. 2 in the main text. However, the dark and light gray regions are forbidden. For finite spinpolarization, the constraint ,↑ = ,↓ does not hold and the forbidden region (presented now in dark gray) is "squeezed" by a factor of 2 along the conductance axis. As a result, any ( , ) combination that is located in the light gray region indicates a finite spin-polarization.
A lower bound for the CSP can be given by the following analysis. Assuming only two spin-polarized channels of opposite spin type, the Fano factor can be rewritten in terms of 2ch ( is defined in the main text, and 2 ℎ is for two channels) as follows: = 1 − (1 + ( 2ch ) 2 )/2 0 s , and hence 2ch = [2 0 s (1 − )/ − 1 ⁄ ] 1/2 . In contrast, for two spin-polarized channels of the same spin type, the CSP is 100%. Since shot noise does not provide information about the spin type, for two spin channels of an unknown spin type the CSP is either 2ch or 100%. According to the derivation that appears in the Supplementary

Conductance measurements of Ag/ferrocene and Ag/vanadocene molecular junctions
In Supplementary Fig. 5a,b, left panels, plateaus and tilted plateaus below the peak of 2 0 s and above 0.1 0 s can be observed. These conductance values are repeatable despite the fact that 2-5·10 -2 0 s is the most probable conductance of Ag/ferrocene junctions as seen in Supplementary Fig. 5c,d, middle panels. Based on comparison of the conductance histograms in Supplementary Fig. 5a,b, middle panels, it is clear that the contribution of these plateaus to the histogram is higher during the formation process (Supplementary Fig.   5b; i.e., they are more abundant during this process). In view of these observations, the shot noise measurements that are performed on Ag/ferrocene junctions and presented in Fig. 2b were taken on junctions with different fixed inter electrode distance, following a formation process. During this procedure, the opposite electrodes are brought together until a contact is formed. Repeating this procedure provides shot noise measurements of different junction configurations characterized by different conductance.

Numerical analysis of transmission channel distributions
We briefly describe the numerical procedure used to determine the channel distribution in Fig where ∆ and ∆ are the experimental errors in and , respectively. We define { , , } =1 as the j th set out of k transmission sets that match the experimental values. The transmission coefficient τ i,σ can now be determined to be in the range between , , min = min{ i,σ } =1 − ∆ and , , max = {τ i,σ } =1 + ∆ . Here, ∆ are added to certify that all possible solutions for , are included in this range. In this analysis, we assume a priori a given number of channels. However, we can repeat the procedure for N+1, N+2,… to obtain convergence of the channel distribution. This procedure helps us avoid a deficient estimation of N.

Supplementary Figure 7:
Experimental analysis of spin-polarized transport during elongation of Ag\vanadocene junctions, as in Fig. 3 in the main text. a, Fano factor versus conductance during the elongation of three different Ag\vanadocene junctions. Following each measurement, the junction is stretched by up to 0.25 Å. The uncertainty, corresponding to systematic errors in our measurements, is comparable to the diameter of the semitransparent symbols, as shown in Supplementary Fig. 3 and 4. b, Total conductance (black dots, error range is given by dots' diameter) and transmission probabilities of the largest four spin-polarized transmission channels (colors). The large uncertainty in the channel transmissions stems from the numerical analysis (Supplementary Note 5). c, lower bound for CSP ( is the minimal ), as determined by the experimentally obtained Fano factor and conductance, with ± 3% experimental uncertainty. The different panels exemplify reduction (I), insensitivity (II), and non-monotonic (III) CSP response to junction stretching.

I.
II.

Experimental analysis of spin-polarized transport during elongation of Ag/vanadocene junctions
The goal of Fig. 3 in the main text is to show that in some cases the conductance can approach the limit of ideal spin polarized ballistic conductance. The limit is given by a single fully open spin polarized transmission channel, namely G=e 2 /h, F=0. As a result, we focus in Fig. 3, main text on stretching sequences, approaching this limit.
Examining the entire measured data of transport vs. stretching, including cases that do not approach the spintransport ballistic limit, we find that the CSP can evolve in different ways in response to junction elongation. This is actually expected considering the non-monotonic evolution of spin polarization as a function of inter-electrode distance as revealed by the different DFT and transport calculations presented in Supplementary Fig. 11 and 12.
Over all, we examined 43 shot noise vs. junction elongation traces (see some example in Supplementary Fig. 7) and found 11 traces with increased CSP, 8 traces with decreased CSP, 16 traces with a non-monotonic behavior, and 8 traces with CSP rather insensitive to stretching. Since not always a single trace can capture the entire nonmonotonic behavior, these results are in line with the outcome of our calculations.

Structural analysis based on DFT and transport calculations
The connection of the vanadocene molecule to the electrodes is based on chemical binding (see further analysis of the total binding energy in Supplementary Note 9). Therefore, the number of stable junction configurations is limited to molecular orientations that allow such binding, and can be classified into several archetypical configurations. The classification describes the different symmetries possible for the chemical binding between the molecule and the electrodes.
Supplementary Figure 8 presents five different architypes of molecular junction configurations, with different chemical binding sites. For convenience, we also present the perpendicular (in-axis) and parallel (off-axis) molecular orientation ( Supplementary Fig. 8a,b) considered in the main text, and we add the following configurations: parallel in-axis ( Supplementary Fig. 8c), tilted ( Supplementary Fig. 8d), and perpendicular off-axis ( Supplementary Fig. 8e). Interestingly, when the structures are allowed to energetically relax, both the parallel in-axis molecular configuration (Supplementary Fig. 8c) and the tilted configuration ( Supplementary Fig. 8d) relax to parallel off-axis orientation ( Supplementary Fig. 8b), indicating that these two configurations are not energetically stable. When the perpendicular off-axis configuration ( Supplementary Fig. 8e) is allowed to relax, the molecular junction breaks and the molecule is adsorbed on the facet of one of the electrodes, pointing that also this configuration is instable. These observations reveal that out of the different binding possibilities between the molecule and the electrodes considered here, only the parallel off-axis and perpendicular in-axis configurations ( Supplementary Fig. 8a,b) are energetically stable. We further note that the calculated conductance and conductance spin polarization (CSP) for the configurations presented in Supplementary Fig. 8ce deviate considerably from the experimental observations of interest, where the conductance is around 1 0 s and the CSP is high. In contrast, only the perpendicular (in-axis) configuration reveals conductance and CSP that match well the mentioned experimental observations. In view of the stability of the different configurations and their transport properties, we conclude that the perpendicular in-axis and parallel off-axis configurations (denoted in the main text as perpendicular and parallel configurations for convenience) can be assigned to the experimentally found most probable high and low conductance cases, respectively. Furthermore, based on transport properties, only the perpendicular in-axis configuration can be associated with the measured high spin filtering.
Focusing on the perpendicular in-axis molecular orientation that is associated with high spin filtering, we now examine the influence of the electrode structure on the conductance (i.e., the transmission at the Fermi energy in 0 units) and conductance spin polarization. As seen in Supplementary Fig. 9, we find that it is enough to have one flat electrode or even one diatomic tip to lose the spin filtering effect and have a significantly lower conductance than in the experimentally observed cases of interest. This is simply due to the lack of direct binding between the vanadium atom and a protruding silver atom on each of the two electrodes that is required in order to have significant electronic transport via this pathway.
Based on the above comparison between the calculations described here and the experimental results, we conclude that the ensemble of molecular junctions that show high conductance and high spin filtering in the experiments should have the following typical structural properties: (i) perpendicular in-axis molecular orientation; and (ii) direct binding between the vanadium atom and a protruding atom on each silver electrode.
The abundance of high spin filtering (for example, CSP<70%) found in the experiments together with high conductance suggests that these structural properties of the Ag/vanadocene junction are not very rare. As discussed in the main text, the significant spin filtering found for the calculated perpendicular molecular junction configuration is an outcome of spin dependent quantum interference. We note that the examination of additional archetypical structures of molecular junctions does not reveal an alternative explanation for the high spin filtering found in some of the measured molecular junctions.

Total energy and spin dependent electronic transport of Ag/vanadocene molecular junctions
To verify that the outcome of our calculations is robust and independent of the calculation technique and methodology, we present here additional calculations done using a different methodology with respect to the calculations presented in the main text and in the rest of the Supplementary Information. Specifically, we used the SIESTA 9 and VASP 10 codes for DFT calculations, while the spin-dependent electron transport calculations were performed using a TRANSIESTA code, which is based on non-equilibrium Green's-function (NEGF) formalism with a self-consistent DFT (see Methods).
The relative stability of the parallel and perpendicular Ag/vanadocene junction configurations are presented in Supplementary Fig. 10. While the perpendicular configuration is more stable in general, the parallel configuration is more stable at a larger inter-electrode separation. Note that the most stable perpendicular orientations take place at D=6-6.8 Å, in some deviation from the calculations used in the main text and in Supplementary Fig. 12, in which the minimum energy is obtained at D= 5.8 Å, likely due to different calculation details.
The calculated spin-polarized transmission of a perpendicular Ag/vanadocene junction at different electrode separations is presented in Supplementary Fig. 11. The suppression of spin-up transmission is clearly seen for a wide energy range around the Fermi energy in all cases. In contrast, the spin-down transmission at the Fermi energy is non-monotonic with respect to electrode separation, being maximal (very close to 1) for an intermediate inter-electrode separation of 6.0 Å. This gives rise to essentially perfect conductance spinpolarization, and ideal ballistic spin-polarized transport for this junction geometry.

Spin-resolved transmissions of perpendicular Ag/vanadocene junctions for different electrode separations
We plot in Supplementary Fig. 12  The calculations presented in Supplementary Fig. 10 and 11, as well as in Supplementary Fig. 12

Interference in terms of vanadium-and carbon-originated molecular states
The relevant interference takes place between the main transport contribution via the vanadium 2 level and the transport contributions from many states of the carbon rings located in a wide range of different energies. The latter states are located far from the Fermi energy at different energies, and can be clearly recognized in Supplementary Fig. 13a, at E > 3eV and E < -2eV. Our analysis (carried by switching off, one by one, different carbon states) has shown that one cannot single out a specific carbon state that dominates the interference. All these states are of importance, contributing to the efficient suppression of transmission at the Fermi energy for spin up, and to the finite phase shift in the even-symmetry scattering channel, as simulated in Supplementary  Fig. 16b,c using a simple toy model. where the positions of the (renormalized) vanadium 2 -originated level (relevant for transport) are indicated by colored arrows for different coupling strengths. We see that the energy level's position (arrows) is shifted, following very well the position of the corresponding transmission peak. Thus, the shift in the transmission peak reflects change in the coupling strength of the vanadium 2 orbital to the electrodes for the tree different cases: i) coupling by only vanadium 2 -orbital to the electrodes (green); ii) coupling to the electrodes by the 2 -orbital and by the π orbitals of the carbon rings (red); and iii) direct electrode-electrode coupling is added (dashed black). The shift is due to the different coupling matrices : for cases (i) and (ii), simply because of different hopping matrices V, while the electrode's Green function G is given by two independent contributions from left and right electrodes, G=G L +G R . In case (iii), the last statement is not true due to direct coupling between electrodes, so that G=G L +G R +G LR with a nonzero crossing term G LR . To conclude, the shift is related to changes in the hybridization, since in each case (i-iii) different orbital combinations are taken into account, while the asymmetric shape of the transmission peaks is an outcome of the interference discussed in the main text.

Supplementary
Supplementary Figure 14: The role of hybridization and electrode-molecular levels coupling on the position of the transmission peaks.

Toy model for understanding spin-dependent interference
To illustrate the mechanism of spin filtering by quantum interference, we set up a simple tight-binding model shown in Supplementary Fig. 15. It consists of three levels: two levels simulate the energy levels of the carbon rings (at 6 eV) and one spin-split level simulates the spin-polarized d-level ( 2 ) of vanadium, placed symmetrically (with respect to the Fermi level) at -1 eV and 1 eV for spin-up and down, respectively. The levels are coupled to two semi-infinite 1D chains, representing the silver electrodes.
When the two pathways are treated separately, no spin filtering at the Fermi level is observed. Specifically, the transmission through the spin degenerate carbon levels (pink lines) is obviously non-polarized, while the transmission through the spin-polarized d-level (green lines) gives two spin-split Lorentzian structures with equal transmission at the Fermi energy. In contrast, if interference between the two electronic pathways is taken into account (black lines), a full reflection of spin-up electrons and enhanced spin-down transmission are found at the Fermi energy. Note that in this symmetric case (regarding the d-level spin up/down position) the perfect spin filtering at the Fermi energy is purely due to quantum interference since the d-level density of states are not spin-polarized.
The actual asymmetric shape of the transmission function due to the above-discussed interference could be readily rationalized in terms of the so-called scattering phase shifts. In Supplementary Fig. 16, we analyze the spin-up case (as spin-down transmission has the same shape). For our single band case, the transmission probability can be expressed 12  Consequentially, the transmission curve would change its shape (I), and drop to zero at the left side of the PDOS peak, remaining finite at its right side.
We can conclude that the nature of interference at zero energy (at the Fermi energy) depends not only on the dlevel position (with respect to the Fermi energy) but also on its symmetry. Note also that the PDOS on the molecule, on the contrary, is very similar in both cases, (d), (g).