Quantum Interference in Single-Molecule Superconducting Field Effect

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C: Physical Processes in Nanomaterials and Nanostructures

Quantum Interference in Single-Molecule Superconducting Field Effect Transistors Ciro Nappi, Francesco Romeo, Loredana Parlato, Francesco Di Capua, Alberto Aloisio, and Ettore SARNELLI J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00987 • Publication Date (Web): 04 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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The Journal of Physical Chemistry

Quantum Interference in Single-Molecule Superconducting Field Effect Transistors Ciro Nappi,∗,†,‡ Francesco Romeo,¶ Loredana Parlato,§,†,‡ Francesco Di Capua,§,‡ Alberto Aloisio,§,‡,† and Ettore Sarnelli∗,†,‡ †CNR-SPIN, Napoli, Italy ‡INFN, Sezione di Napoli, Napoli, Italy ¶Department of Physics ”E. R. Caianiello”, Università di Salerno, Fisciano (SA), Italy §Department of Physics ”E. Pancini”, Università di Napoli, Napoli, Italy E-mail: [email protected]; [email protected] Phone: +39 081 8675322; +39 081 8675263

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Abstract Single molecules can be coupled to metallic electrodes also when these latter are in the superconducting state. In such emerging hybrid molecular devices the possibility raises that the Josephson effect, i.e. the dissipation-less transport of Cooper paired electrons from one electrode to the other, appears. In this theoretical study we demonstrate that a junction formed by two superconductors linked by an annular molecule, of which benzene (phenylene group) is a prototype, can sustain indeed a supercurrent and work as a ’Single Molecule Superconducting Field Effect Transistor (SMoSFET)’. In this device, Cooper paired electrons are transmitted via the molecule in the presence of quantum interference. Through the presented model, we show that the resonant non trivial modulation of the critical current with an external gate voltage may be strongly modified by choosing a para-coupled, a meta-coupled, or an ortho-coupled molecular ring. These features are directly related to the possibility of obtaining a SMoSFET controlled by quantum interference.

Introduction Recent progress in the field of molecular electronics promises a direct integration of molecular structures into electrical on-chip circuits. 1–3 These devices are the ultimate building blocks by which standard electronic circuits can be realized. The real boosting advance would be the possibility to obtain coherent quantum transport devices in which quantum interference is under control. 4,5 There are various kinds of functions that molecules can have in single molecule devices: they can be transistors, wires, rectifiers, switches, logic gates etc. Important advancements have been made over the past decades in obtaining all such kinds of electronic molecular arrangements 6,7 in identifying important conductance properties of specific molecules 8 or in observing quantum interference effects. 9 From the practical point of view, one of the obstacles still hampering the realization of a working molecular transistor is the relatively large contact resistance associated to a 2


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single conduction channel in such type of device. The contact resistance, a concept borrowed from ordinary electronics, 10 is nature-imposed and cannot be lower than the quantum limit h/2e2 = 12.9 kΩ. Under this condition the Joule losses is an issue affecting the transistor operation efficiency. One way 11,12 to overcome this drawback is the introduction of superconducting electrodes in which Cooper-paired electrons flow without resistance through the junction, owing to the Josephson effect. 13 Indeed advances in nanostructure fabrication have made possible to couple superconductors with molecules. 12,14–19 Such devices are in practice special type of ballistic (Superconductor)-(Normal Metal)-(Superconductor) (S-NS) junctions. 20 Several nano-structures have been effectively realized, in which the part ’N’ of the S-N-S junction have been made by mesoscopic metal wires, 21 molecular wires such as carbon nanotubes, 15,22–26 DNA molecules, 16 metallofullerene, 14 a single C60 molecule, 12 an individual Fe4 single molecule magnet, 17 and one single all organic radical molecule (polychlorotriphenylmethyl (PTM)). 19 Theoretical models able to cope with these situations are now highly desirable.

Figure 1: a) A tight binding representation of the system under study in this paper: a (superconductor)-(six atom ring molecule)-(superconductor) structure. b) An auxiliary system of a four atom chain molecule between superconductor electrodes. The structure a) (a Mach-Zehnder interferometer) may be ideally thought as derived from the structure b) by adding a parallel pathway of two extra atoms. c) Schematic representation (case of the chain molecule) of the Andreev bound states (0 < E < ∆) through which Cooper pairs are transferred from left to right electrode or from the right to the left. In this paper we report on the transport modelization of supercurrent through a ring 3



molecule sandwiched between superconducting electrodes and subject to an electrostatic gating operation (see Figure 1 a and b). Andreev bound states (infinite loops of Andreev reflections e → h → e . . . , see Figure 1c) 27–30 can form in such an ’atomistic’ junction structures and permit a controlled Josephson current transport. In this device, which is a basic configuration for a single molecule superconducting field effect transistor, the extra degrees of freedom introduced by the structural conformation of the molecule can be utilized along with the dissipationless transport properties of the superconductor. For simplicity we have focused on an idealized model for a benzene molecule (Benzene1,4-dithiol), or a 1,4-phenilene group, between simple superconducting electrodes. Figure 1, a and b, represent the tight binding model (H¨ uckel type) structures under study: a six atom ring molecule (a), and a simple chain molecule (b), between two electrodes. We assume that the molecule is contacted by the electrodes in two of the six atomic sites forming a natural Mach-Zender interferometer. Figure 1a can be viewed as a simple model of π electrons in a phenyl ring, with one electron per orbital, positioned between electrodes. Figure 1b can represent a simple model for a monovalent atomic wire, between electrodes, that we use as a comparative system in which electron/hole propagation occurs along a single pathway. Similar simple structures have been nicely studied by Sparks et al., 31 and Lambert 32 in a context of a single molecule electronic system with no reference to superconductivity, although an interesting reference to Andreev reflections was already present in an earlier study by Kormányos et al. 33 who calculated the differential conductance of a normal conductormolecule-superconductor junction. The two electrodes are supposed here to be both in the superconducting state (superconducting energy gap ∆(T ) 6= 0), while the molecular atomic sites are characterized by a normal state (superconducting energy gap ∆(T ) = 0). The superconducting state of the electrodes, usually a very thin layer of gold in the experiments, is typically obtained by proximity effect with a superconductor like aluminum, niobium, molybdenum-rhenium, etc. The orbital energy of each atom is denoted by and the atoms themselves are coupled to

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each other by a nearest-neighbour matrix element −γ. Both and γ acquire appropriate site indexes (cf Figure 1). The quantities αL and βR play a relevant role: they measure the degree of coupling of the molecule with the gold electrodes, that is, the degree of hybridization of the molecular orbitals with those of the electrodes. Coupling different sites to the two electrodes, sites 0 and 1, 0 and 2, and 0 and 3 results in a ortho, meta, and para configuration, respectively. Note also that the electrodes are considered infinitely long (open quantum systems) and connected to macroscopic reservoirs (not shown).

Theoretical Methods Instead of using the conventional perturbative tunnel-Hamiltonian approach, we treat the Cooper pair transfer through the molecule-superconducting electrodes system, by means of a scattering formalism. The scattering theory of transport (Landauer-Buttiker formalism) has been shown to provide a general framework for the description of quantum transport, Josephson effect included. 34–39 Starting point for the current calculation is the tight-binding Bogoliubov-de Gennes 40 equations for the electron-like (fµ ) and hole-like (gµ ) components of the quasi-particle wavefunction on the site µ, which can be written as

µ fµ −

X

γµ,ν fν + ∆µ gµ = Efµ

(1)

ν

−µ gµ +

X

∗ γµ,ν gν + ∆∗µ fµ = Egµ

ν

In eqs 1 the summation is over all nearest-neighbours ν of site µ of the molecule and the electrodes. The quantity ∆µ is the superconducting energy gap of the electrodes. We assume (see Figure 1), ∆µ = |∆| for µ = −1, −2, −3... (left electrode) and ∆µ = |∆|eiϕ for µ = 1, 2, 3...(right electrode). Moreover, on the six molecular sites, ∆µ = 0. The quantity ϕ is the gauge invariant phase difference between the two superconducting electrodes and E is the energy of the scattering state.

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Table 1: Energy Levels of the isolated six atom ring molecule. Parameters values used are = 0, γ = 2eV Level 1 2 3 4

Energy (eV) Degeneracy 4 single 2 double LUMO -2 double HOMO -4 single

The structure of the molecule defines the actual discrete energy levels of the scattering region between electrodes. Table 1 summarize the energy-level spectrum of the ring molecule calculated in the tight binding approximation, eqs 1 (see Supporting Information). Note that to illustrate the general possibilities of the formalism, we have not attempted any optimization of the parameters (, γ ) to stay as close as possible to the case of real benzene molecule. The potential ∆/e is responsible for the scattering of electrons into holes or vice-versa (Andreev scattering) at the boundary between molecule and electrodes. In obtaining the Josephson current, we solve the Andreev scattering problem 41 for an electron incoming from the left (process 1: transfer of a Cooper pair from left to right) and for a hole incoming (again) from the left (process 2: transfer of a Cooper pair from right to left) with energy E > ∆ or (E < −∆). This provides, through eq 1, the four probability amplitudes a(E), r(E), b(E), t(E) for electron scattering and a∗ (E), r∗ (E), b∗ (E), t∗ (E) for hole scattering, such that the four coefficients A = |a|2 , R = |r|2 , B = |b|2 , T = |t|2 , (A + B + R + T = 1), corresponding respectively to Andreev reflection, normal reflection, anomalous transmission and normal transmission, can be determined. Knowing the Andreev scattering probabilities amplitudes, a(E) and a∗ (E), determines the entire Cooper pair current or the dc Josephson current (see Supporting Information). Notice that eqs 1, written for the ring junction, determines a typical quantum which-way problem, which is reflected in the form assumed by a(E) and a∗ (E). Any energy change δ in the atomic orbital energy of the molecule results in a uniform shift δ of all the energy levels of the molecule with respect to the Fermi level of the superconducting electrodes. Such energy shift is fundamental in controlling the device and may 6


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be generated by the application of an external electrostatic potential Vg to the molecular region (it can be thought as an electric field normal to the sheet plane in Figure 1 such that → ± |e|Vg . Actually, applying an external potential Vg to the molecule has various effects both electrical and mechanical. A complete treatment of the electrical effect requires a self-consistent solution of the Schrödinger-Poisson equations for the potential. A useful qualitative description of the result is indeed that a gate bias rigidly shifts the energy levels of the molecule and hence the transmission: T (E, Vg ) ∼ T (E − |e|Vg ), as we assume in this paper. 42 The Cooper pair current is evaluated through the Furusaki-Tsukada formula 37,38 which is directly linked with the Green function method. 43,44 The final result depends only on the scattering amplitudes a(E) and a∗ (E) for the two processes 1, 2, respectively (see Supporting Information). Among the merits of this procedure is the automatic inclusion in the supercurrent evaluation of both the Andreev bound state (Figure 1c) and the continuous Andreev spectrum contributions. Through the model, the superconducting current-phase relation of the junction can be calculated as the various parameters characterizing the molecular arrangement and the environment change (, γ, Vg , temperature T, etc.). Most importantly, the critical current of the junction Ic = max(I)ϕ∈[0,2π] , i.e. the maximum supercurrent the device can sustain before developing a finite voltage between electrodes, can be easily obtained. The above model assumes non-interacting quasiparticles. Generally speaking, electron correlations tends to suppress Andreev reflections so that the results discussed hereafter apply properly to a molecule arrangement with a small charging Coulomb energy U . The charging energy truly dominates only when the resonance width Γ is very small (case of weak coupling with the electrodes, Γ U, ∆). 45 In this regime the charge tunneling rate towards the electrodes, Γ/¯ h, is very small too, and the Josephson effect disappears due to Coulomb blockade phenomena. 46,47 For wide resonances the Coulomb repulsion affects only moderately the Andreev processes and one can see that the Josephson current is barely

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modified. 46,48

Discussion In Figures 2,3,4 we provide specific examples of current calculation characterizing the superconducting state of the devices shown in Figure 1a and b.

Magnitudes of the hy-

bridization molecule-electrode parameters, αL and βR are chosen equal for simplicity, i.e. αL = βL = δ. For the same reason we will set α1 = α2 = β1 = β2 = γ1 = γ2 = γ, as well as 1 = 2 = 0L = 0R = in the molecule. By changing the values of the parameter δ, the effect of the coupling strength of the molecule with electrodes can be studied. Figure 2 a shows a low temperature (T = 0.01Tc ) plot of the critical current versus δ. The simulation shows that there is an optimal coupling δmax in correspondence to which the critical current sharply maximizes (the Josephson coupling maximizes) in both junction arrangements to about 97% of e∆(T )/¯ h (the full value Ic,max = e∆(T )/¯ h is reached only asymptotically as T /Tc → 0). One finds δmax = 2.0 eV and δmax = 2.8 eV as optimal coupling in the chain and ring junction (para-linked ring), respectively. Since γR = γL = γ = 2 eV, the result δmax = 2 eV for the chain junction simply demonstrates that the optimal molecule-electrode coupling corresponds to the uniform hopping element distribution γR = γL = γ = δ = 2 eV. Less trivial is the result for the ring junction, which is directly determined by the presence of a second pathway. It is important to observe that, in the present model, changing the parameter δ means modifying the scattering amplitude probability of the Andreev reflection at the two (normal molecule)/(superconducting electrode) ’interface’. Modified hopping conditions at the two molecule/leads boundaries makes more or less efficient the mechanism of Cooper pair transport through the Andreev states. This altered reflection mechanism is at the base of the curves shown in Figure 2. A further key observation is that the Andreev scattering probability amplitude depends crucially on the values the wavefunctions assume at the edge

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molecule points, the nodal points L and R in the tight binding configuration. These values are ultimately sensitive to whether or not a single molecular propagation pathway is added or subtracted from the molecule structure. From the mathematical point of view they encode information about quantum interference effect in the structure. Also we notice that

Figure 2: Panel a): Josephson critical current of the junction as a function of δ, the coupling strength of the molecule with electrodes: curve 1, chain junction, curve 2 ring junction in para-coupling with the electrodes. The critical current (or the Josephson coupling energy) maximizes for δmax = 2.0 eV and δmax = 2.8 eV in 1) and 2), respectively. Panel b): Comparison of the curve 2 of Panel a) (critical current vs δ in the ring junction, blue line) with the normal state transmission of the ring junction versus δ (panel b, curve 1, green line). The normal transmission probability is evaluated at the Fermi energy, i.e. T = T (EF ). in both cases depicted in Figure 2 a, the critical current resonance curve at low temperature has a cusp-like character. In the case of a quantum dot with a single level, Beenakker and van Houten 46 have shown that the presence of cusp shaped resonances characterizes the superconducting case, in contrast with the normal state case in which resonances show Lorentzian behavior. This shape difference can be fully appreciated in the present work in Figure 2b where the dependence of the normal transmission on δ is compared with the plot of the critical current vs δ in the ring junction (see also Figure 4). Analogous different shape result is obtained for the chain molecule junction case (not shown). It is remarkable that, by using a different mathematical approach for the Josephson current evaluation, and treating

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a multi-level case, we obtain here the same result of ref. 46 Figure 3 shows the calculated low temperature current-phase relations of the ring (blue line) and the wire (magenta line) molecular junction. The parameters have been chosen such that the Josephson current is optimized for the ring junction, i.e. γL = γR = γ = 2 eV, and δ = 2.8 eV, as discussed above. The steplike current-phase relation obtained under this condition for the ring junction (Figure 3, blue line) reproduces the current-phase relation of a zero temperature (T = 0) ballistic SNS junction (transmission T (EF ) ∼ 1) in which the size L/ξ0 of the ’normal part’ portion of the junction tends to zero (Kulik Omel’yanchuck clean limit). 29,30,49,50 Here ξ0 is the coherent length of the two superconductors. The simulation in Figure 3 shows the effect of quantum interference on the Josephson current. In fact, we can ideally start from a two pathway, well tuned (δ = 2.8 eV) ring molecular structure displaying a critical current of 0.97e∆/¯ h (Figure 3, blue line), eliminate one of the pathways (made by two atoms), obtaining this way a chain junction, and observe: i) a reshaping of the current-phase relation, which moves from a steplike curve to a sinusoid; ii) an associated fall in the junction critical current: the critical current, falls off from 0.97 of the ring junction to 0.40 e∆/¯ h of the chain junction (Figure 3, magenta line; see also Figure2a). In the absence of quantum interference, e.g. in a ’macroscopic’ device, the same ideal experiment would have lead to no current-phase curve reshaping and to a simple halving of the critical current, that would have passed from 0.97 to 0.48 e∆/¯ h. The 20% of extra suppression and a modification of the current-phase relation here are purely quantum interference effects. We turn now to the influence of a voltage gate. The critical current of the junction can be possibly controlled by applying an external gate potential Vg to the molecular region by means of a third electrode (not shown in Figure 1). Figure 4a shows the critical current of the para-linked ring junction as a function of the gate voltage Vg for moderate coupling (δ = 1 eV) of the molecule with the electrodes, and far from the optimal coupling condition. Under this condition the resonances are well separated but still characterized by a non vanishing

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Figure 3: The Josephson current as a function of the phase difference ϕ between superconducting electrodes in an annulene molecular structure of six atoms (blue line) and in a simple atomic chain of four atoms (magenta line) linked to superconducting electrodes. The value of the coupling parameter δ has been set to δ = 2.8 eV in both structure, i.e. the optimal value of the molecule/electrodes coupling for the ring junction. Indicated (light green vertical segments) are the values of the normalized critical current Ic in the two cases: Ic = 0.97 and Ic = 0.40, respectively. width Γ. As Vg is made to vary around zero, the energy levels of the molecule are upshifted or down-shifted uniformly with respect to the Fermi energy EF of the electrodes. The figure illustrates the characteristic resonant phenomenon at the heart of a SMoSFET: when a molecule energy level (cf Table 1 ) aligns with the Fermi energy of the electrodes, a supercurrent can flow (on-state) as a result of resonant Cooper pair transfer between the electrodes. Away from the resonance, when the Fermi level of the electrodes is between two molecule levels, Ic is suppressed (off-state). Figures 4 b, c show the important changes in the critical current vs Vg pattern of a ring molecule resulting from a different molecule-electrode contact scheme. Figure 4b shows the meta coupling case and Figure 4c the ortho-linked case (these may be realized in practice by using different molecules: 1,3-Benzenedithiol C6 H6 S2 and Benzene-1,2-dithiol C6 H4 (SH)2 , respectively). The interference condition, this time, is altered by the different lengths of two molecular pathways. When compared with the pattern of the critical current of Figure 4a one sees the emerging of anti-resonances (green arrows in the figures), in both the meta and 11



ortho case, i.e. there are gate potential values in correspondence to which the critical current vanishes. There are three of these points in the meta case and four in the ortho case. Two of these anti-resonances appear where in the para case a peak was present, i.e. at Vg ≈ ±2 eV. In the meta case, Vg = 0 (the center of the HOMO-LUMO gap) is an anti-resonance point, meaning that the meta-coupling corresponds to destructive interference for the Josephson current. This result has to be compared with ortho and para couplings where at Vg = 0 the same finite critical current value is obtained (constructive interference). In the recent

Figure 4: a, b, c: Josephson critical current as a function of the gate voltage Vg in a six atoms ring molecule between superconducting electrodes. The curves in a, b, and c correspond to para, meta, and ortho type coupling, respectively. The temperature is T = 0.01Tc . The parameters describing the molecule are those reported in Table 1. For the electrodes we choose, R =L =0 with an hopping parameter γR =γL =2.2 eV, such that the bandwidth of the electrodes extends from -4.4 eV to 4.4 eV. The quantity δ, i.e. the coupling of the molecule with the electrodes, has been set to δ=1eV in each panel of the Figure. Green arrows indicate anti-resonance points. Panels d, e, f: Normal transmission (∆ = 0) vs energy corresponding to panels a, b, c, respectively. past, this kind of quantum interference effect, of ’conformational’ type, has been highlighted, either in benzene or in multi-pathway simple molecules, with reference to the presence of resonances in the conductance of junctions realized with normal metal electrodes (normal 12


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conductance). 4,31,32 By direct comparison of Panels a, b, c with Panels d, e, f in Figure 4, one can see that the resonances (anti-resonances) appearing in the critical current are strictly correlated with the analogous resonances or anti-resonances in the normal conductance of the junction. It is well known that the critical current of a conventional Josephson junction (e.g. superconductor-oxide-superconductor junction), which is ohmic in the normal state, is generally proportional to its normal-state conductance. 20 This statement has limited meaning in the case of a molecular junction. We note, in particular (Figure 4, Panels a and d), that the critical current modulation as a function of the voltage gate Vg at low temperature, reveals the same cusp-like character (Panel a) discussed above for the curves of Figure 2 and absent in the normal state conductance (Panel d). Also, the energy positions of the resonances do not perfectly match in Panels a and d. As far as anti-resonances are concerned, a comparison between Panels b, c and e, f of Figure 4 confirms the existing strict correlation between the superconducting case and the normal state case (for the anti-resonances in fact the energy positions coincide in the two cases, normal and superconducting), but also shows an even stronger deviation from a simple proportional correlation law between normal state transmission and critical current. To complete the above evidences, Figure 5 shows that by a different choice of the parameters, the critical current and normal transmission curves (the blue and magenta line on the same plot) can also deviate markedly each other as far as the shape is concerned. With respect to Figure 4, in Figure 5, δ has been incremented to 2 eV, and γR , γL have been slightly decreased (2 eV instead of 2.2 eV).

Conclusions In summary, we have presented a comprehensive formalism allowing the study of a SMoSFET based on a simple annulene molecule trapped between superconducting electrodes. Through the model, the effects of the quantum interference can be explored in detail for the operation of the device as a superconducting single-molecule transistor. We have demonstrated the

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Figure 5: a, b, c, blue line, Josephson critical current as function of the gate voltage Vg in a six atoms ring molecule between superconducting electrodes. The curves in a, b and c correspond to para, meta and ortho type coupling, respectively. On the same plot, the magenta line curves, are the normal transmissions T (E) of the same molecular arrangements with the electrodes in the normal state (∆ = 0). We choose here γR =γL =2 eV and δ = 2 eV. The remaining parameters are identical to those used in Figure 4. existence of an optimal molecule-electrode coupling value which maximizes the Josephson current carried by the junction. Successive discrete energy states may be tuned on- and off-resonance with the Fermi energy in the superconducting electrodes by means of a gate voltage, that is the heart of the resonant mechanism of a SMoSFET. In the ring junction the Josephson critical current versus gate voltage pattern changes dramatically on passing from a para-linked configuration to a meta- or ortho-linked configuration, due to quantum interference on two pathways transport channel. The model presented in this paper can be easily generalized to more complex annulenes or chains of annulenes or cyclic compound molecules. It provides a simple, effective tool in designing and figuring out various field effect mechanisms in superconducting molecular transistors.

Acknowledgement The INFN-CNR national project (PREMIALE 2012) EOS Organic Electronics for Innovative research instrumentation is gratefully acknowledged

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Supporting Information Available Details of the model and calculation procedure. This material is available free of charge via the Internet at http://pubs.acs.org/.

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van der Zant, S. J. Superconducting Molybdenum-Rhenium Electrodes for SingleMolecule Transport Studies. Appl. Phys. Lett. 2015, 106, 222602. (18) Katzir, E.; Sukenik, N.; Kalcheim, Y.; Alpern, H.; Yochelis, S.; Berlin, Y. A.; Ratner, M. A.; Millo, O.; Paltiel, Y. Probing Molecular-Transport Properties Using the Superconducting Proximity Effect. Small Methods 2017, 1, 1600034. (19) Island, J. O.; Gaudenzi, R.; de Bruijckere, J.; Burzur`ı, E.; Franco, C.; Mas-Torrent, M.; Rovira, C.; Veciana, J.; Klapwijk, T. M.; Aguado, R. et al. Proximity-Induced Shiba States in a Molecular Junction. Phys. Rev. Lett. 2017, 75, 117001. (20) Barone, A.; Paternò, G. Physics and Applications of the Josephson Effect; John Wiley and Sons: New York, 1982. (21) Dubos, P.; Courtois, H.; Pannetier, B.; Wilhelm, F. K.; Zaikin, A. D.; Schön, G. Josephson Critical Current in a Long Mesoscopic S-N-S Junction. Phys. Rev. B 2001, 63, 064502. (22) Yang, Y.; Fedorov, G.; Shafranjuk, S. E.; Klapwijk, T. M.; Cooper, B. K.; Lewis, R. M.; Lobb, C. J.; Barbara, P. Electronic Transport and Possible Superconductivity at Van Hove Singularities in Carbon Nanotubes. Nano Lett. 2015, 15, 7859–7866. (23) Pillet, J.-D.; Quay, C. H. L.; Morfin, P.; Bene, C.; Levi Yeyati, A.; Joyez, P. Andreev Bound States in Supercurrent-Carrying Carbon Nanotubes Revealed. Nature Phys. 2010, 6, 965–969. (24) Jarillo-Herrero, P.; van Dam, J. A.; Kouwenhoven, L. P. Quantum Supercurrent Transistors in Carbon Nanotubes. Nature 2006, 439, 953–956. (25) Buitelaar, M. R.; Belzig, W.; Nussbaumer, T.; Babic, B.; Bruder, C.; Schönenberger, C. Multiple Andreev Reflections in a Carbon Nanotube Quantum Dot. Phys. Rev. Lett. 2003, 91, 057005. 17



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Graphical TOC Entry

Schematic of a Single-Molecule Superconducting Field Effect Transistor (SMoSFET).

21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

a) D -3

-2

eL

branch 1 g1

1

a1 -1

e1

aL

gL

a2

b) D -2

-3

eL

gL

e1 b 1

De bR

1

L aL

Superconductor

2

1

e

Molecule

D e ij 2

R

g e0L

Cooper pair

2

g2 branch 2

e

c)

ij

gR e R

D=0 -1

Page 22 of 27

e

e0R

1

bR

e

electron

h

hole

3

2

e0R R e b2

e2 1

2

D=0

e0L

L


2

h

3

gR

eR

e ACS Paragon Plus Environment

h

Page 23 of 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


a)

b)

2)

1)

1)

T/Tc=0.01, Vg=0

T/Tc=0.01, Vg=0 eL= eR = e= 0 eV gL= gR= g = 2 eV


2)

eL= eR = e= 0 eV gL= gR= g = 2 eV

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Ic Ic

Page 24 of 27

eL=eR=e=0 gL=gR=g=2 eV T/Tc=0.01 D=0.001eV

d=2.8


Page 25 of 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38


a)

b)

d)

c)

e)


f)

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

a)

b)

Page 26 of 27

c)


Page 27 ofThe 27 Journal of Physical Chemistry

Source

Drain

1 Superconductor Superconductor 2 3 Oxide 4 Gate ACS Paragon Plus Environment 5 6 Vg 7

Quantum Interference in Single-Molecule Superconducting Field Effect

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