Attenuation of Conductance in Cobalt Extended Metal Atom Chains

Article pubs.acs.org/JPCC

Attenuation of Conductance in Cobalt Extended Metal Atom Chains Vihar P. Georgiev, W. M. C. Sameera, and John E. McGrady* Department of Chemistry, Inorganic Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR, United Kingdom S Supporting Information *

ABSTRACT: Density functional theory, in conjunction with nonequilibrium Green’s functions, is used to explore the attenuation of the resistance of Cox wires along the series Co3(dpa)4(NCS)2, Co5(tpda)4(NCS)2, and Co7(teptra)4(NCS)2. At very low bias (0 < V < 25 mV) the conductance, G, decreases in the order G(Co3) > G(Co5) > G(Co7), consistent with experiment and with an anticipated inverse relationship between conductance and chain length. At higher voltages, however, the current−voltage responses of all three are striking nonlinear, and above 50 mV G(Co5) > G(Co3) > G(Co7). The very different behavior of the members of this homologous series can be traced to the different symmetries and multiplicities of their respective ground states, which in turn control the properties of the dominant transport channels.

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Å−1.29,30 Some highly conjugated and low-band gap systems, such as alkyne-bridged complexes and oligo-zinc porphyrins, have extremely low β values ( G(Co5(tpda)4(NCS)2 (the experimental values of 520 and 100 nS for Co3(dpa)4(NCS)2 and

Co(3) and Co(3)−Co(4) bonds, which are 0.07 Å shorter than those in Co3(dpa)4(NCS)2 (cf. 0.09 Å from the X-ray data). The orbitally degenerate ground state will also give rise to an additional contribution to the magnetic susceptibility that is consistent with the deviation from the spin-only value of 1.73 μB. The assignment of a 2E ground state represents a significant departure from the results of Extended Hückel theory, where a 2 A1 ground state was proposed. While this state also offers an acceptable match with the experimentally determined bond lengths, it is 0.48 eV less stable (Table 1). It should also be noted that Peng and co-workers have recently reported a study of the vibrational spectroscopy of Co5(tpda)4Cl2, including a structure optimized at the B3LYP/6-31G* level.53 The symmetry of the ground state was not discussed, but the optimized Co−Co bond lengths of 2.29 and 2.23 Å are consistent with either the 2E or 2A1 states shown in Table 1. To provide a direct comparison with Co3(dpa)4(NCS)2 and Co5(tpda)4(NCS)2, we report calculations for the analogous oligopyridylamido ligand teptra, Co7(teptra)4(NCS)2, despite the fact that this molecule has not been crystallographically characterized. Parallel computations for Co7(pzpz)4(NCS)2 (Supporting Information, Table S1), where crystallographic data are available, suggest that our chosen computational methodology reproduces ground state geometries with encouraging accuracy. In contrast to the shorter chains, the ground state of Co7(teptra)4(NCS)2 is a quartet, 4A1, the three SOMOs being the two components of the antibonding π orbital, π7, and the partially antibonding σ5 orbital (Figure 3c). The evolution of the orbital arrays in Figure 3 shows that the changes in ground state symmetry and multiplicity are a result of the progressive stabilization of the σ manifold relative to the π/δ band. Thus while the spin-β component of the central member of the σ ladder for the Co3 chain, σ2, lies above the top of the π/δ band (π3/δ3), the corresponding orbital for the Co5 chain, σ3, lies below π5/δ5. This trend continues for the Co7 chain, where σ4 lies ∼0.8 eV below the Fermi level and the next rung on the σ ladder, the partially antibonding σ5, is now stabilized such that it is almost degenerate with the top of the π/δ band (π7/δ7). The stabilization of the σ manifold relative to π/δ as the chain elongates can be understood in terms of two complementary electronic factors: (i) the two Co−NCS σ antibonding interactions are distributed over a greater number of orbitals, stabilizing the barycenter of the σ manifold, and (ii) the ratio of amide to pyridyl donors increases, destabilizing the upper levels of π/δ. The relative energies of the various states in Table 1 can be used to rationalize the absence in the longer chains of unsymmetrical structures analogous to that observed in uCo3(dpa)4Cl2.48 As was the case for Co3(dpa)4Cl2, the most stable quartet state of Co 3 (dpa) 4 (NCS) 2 , 4 E, has an unsymmetric structure with one Co−Co distance substantially longer than the other and dramatically different terminal Co−N bond lengths. In terms of the Kohn−Sham molecular orbital diagrams shown in Figure 3a, the tendency to break symmetry in the 4E excited state is related to the fact that the LUMOs of the ground state are linear combinations of weakly (δ) overlapping terminal Co−N σ* orbitals, and not the strongly Co−Co−Co antibonding σ3. Excitation to a quartet state results in population of one of these two near-degenerate orbitals, leading to symmetry breaking and localization of the electron on one center or the other. In the longer chains, in contrast, one or more vacant member of the σn manifold occupies the window between the π/δ band and the Co−N σ*

Figure 5. Current/voltage curves for Co 3 (dpa) 4 (NCS) 2 , Co5(tpda)4(NCS)2, and Co7(teptra)4(NCS)2. The inset shows an expanded view of the region between 0 and 50 mV. 20168

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the bias window. The Co3(dpa)4(NCS)2 device, in contrast, shows a pronounced negative differential resistance (NDR) feature between 200 and 400 mV. This NDR arises through quantum interference between the σ2β and π3β channels which are almost degenerate at this voltage as a result of the relative stabilization of the former alluded to above. The impact of quantum interference on transport has been discussed in detail by Ratner, Solomon, and co-workers,54 and also developed in Yoshizawa’s recent work on selection rules for effective transport in organic conductors.55 The matrix elements of the energy-dependent molecular Green’s function, G(E), can be expressed as a sum over the molecular orbitals, k, with eigenvalues εk:

Co5(tpda) 4(NCS)2 were measured at 25 and 30 mV, respectively). At all higher voltages we find the reverse, with the pentacobalt chain the better conductor. The origin of these very different current/voltage responses becomes apparent from a detailed analysis of the evolution of the transmission spectra at finite bias (T(E) at 150 mV are also presented in Figure 4) and the profile of the electrostatic potential drop along the principal axis at the same potential shown in Figure 6.

Gsd(E) =

∑ k

CskCdk* (Ef − εk ± iη)

where Csk and Cdk are the coefficients of the orbitals at the junction atoms on the source (s) and drain (d) side, respectively. Thus if two channels are near degenerate the denominators in the above expression are equal and their contribution to the Green’s function, and hence to the total transmission, can either be additive or subtractive depending on the phase relationship between the orbitals on the junction atoms. In the present case the σ2 and π3 channels are respectively antisymmetric and symmetric with respect to translation, and so the product of the coefficients, CskCdk*, is respectively negative and positive. The result is that the contributions of the two channels to the Green’s function offset each other, leading to the observed NDR. Our analysis of the current−voltage curves for Co3(dpa)4(NCS)2 and Co5(tpda)4(NCS)2 highlights the fact that the many striking differences between the two can ultimately be traced to the different symmetries of the ground states of the isolated molecules. The conductance, G, is striking dependent on the bias voltage, and the inequality G(Co3) > G(Co5) is true only in the very low bias regime. In light of these differences, it is not clear how we might extrapolate to longer chains such as Co7(teptra)4(NCS)2, the transport properties of which are yet to be measured. The close resemblance between the spin densities in situ ((ρCo(1):ρCo(2):ρCo(3):ρCo(4) = 0.30:0.26:0.46:0.73) and in the 4 A 1 ground state (0.40:0.25:0.47:0.75, Table 1) confirms that the singly occupied orbitals π7 and σ5 are well insulated from the electrode surface. Unlike Co5(tpda)4(NCS)2, however, the quartet multiplicity ensures that the spin-α and spin-β transmission spectra (Figure 4) are displaced to either side of the Fermi level. The zero-bias transmission is therefore very low compared to that of either of the shorter chains (74 nS vs 488 nS (Co3) and 289 nS (Co5)), the major contribution coming from a narrow peak in the spinβ transmission above the Fermi level that contains both the π7 and σ5 channels. The Co7(teptra)4(NCS)2 system therefore reflects a third type of behavior, different from either Co3(dpa)4(NCS)2 or Co5(tpda)4(NCS)2, where the spin polarization characteristic of the quartet ground state plays the dominant role.

Figure 6. Profile of the potential drop along the principal axis for Co3(dpa)4(NCS)2 and Co5(tpda)4(NCS)2 at 150 mV. Dashed lines indicate a hypothetical linear ramp potential.

In Co3(dpa)4(NCS)2, the dominant σ2 channel is delocalized over the two terminal Co−NCS units and so, in the presence of a linear ramp potential (dashed line in Figure 6), this channel would experience an average potential of 0 V. However, the real electrostatic potential profile shown in Figure 6 shows that the majority of the 150 mV drop is concentrated over the Co−N bond on the source side of the molecule, and the majority of the molecular region experiences a potential close to that of the drain rather than to the average. The channel therefore tracks the drain potential as the bias window opens, such that it never fully enters the window. The lower-lying π3 level, in contrast, has maximum amplitude on the central Co atom, and so is stabilized to a lesser extent at finite bias. The peaks due to the spin-β σ2 and π3 channels therefore merge as the bias increases. In the Co5 chain the dominant π5 channel is also localized on the central portion of the chain and as a result the voltage drops more symmetrically across the molecular region. The π5 resonance therefore does not track the drain potential to the same extent, and instead enters entirely within the bias window at 150 mV, leading to a typical resonant transfer profile, the current rising to a plateau at ∼1200 nA. At even higher potentials (200 mV < V < 500 mV) the current−voltage responses of the two are again very different. In the longer Co5(tpda)4(NCS)2 chain the plateau region persists to ∼300 mV, after which a further increase coincides with the entry of the spin-α component of the σ4 channel into

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CONCLUSIONS In summary, our calculations suggest that the electron transport properties of the cobalt-based EMACs evolve in a very subtle way with increasing chain length. The presence of multiple levels (σ, π, δ) in the frontier region means that the symmetry and multiplicity of the ground state changes, and this has a 20169

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tions” key. All structures were optimized by using the gradient algorithm of Versluis and Ziegler.63 Transport calculations were performed with the ATK201164−67 package, using the LDA functional with self-interaction corrections due to Perdew and Zunger.68 The methodology combines a density functional theory treatment of the electronic structure with the Keldysh nonequilibrium Green’s function approach to simulating coherent transport.69−71 All atoms were treated with the double-ζ basis set, extended with a single polarization function. Core electrons were described by norm-conserving pseudopotentials.72 The electronic structure of the two-probe systems at equilibrium was converged by using a 100 Ry mesh cutoff, finite temperature of 300 K at the electrodes, and the real space density constraint at the electrodes. Sampling of the Brillouin zone was performed with use of a Monkhorst−Pack grid73 with 300 k-points along the transport direction. In the calculation of the transmission spectra and currents, a 5 × 5 grid was used to sample the Brillouin zone, and the bias window was sampled at 0.005-eV intervals. The computed current is particularly sensitive to the sampling of the bias window in cases such as Co3(dpa)4(NCS)2, where only the tail of a resonance lies in the window. The initial spin density for the two-probe calculations was polarized to be consistent with the net spin densities of the isolated molecules in their gas phase ground states. The scattering region contained the EMAC sandwiched between the 6 × 6 layer of the Au (111) surface of the source and the drain, respectively. The inclusion of the entire oligo-polypyridyl ligands in our model represents a significant development from the protocol described in our previous work in ref 9, where a truncated model was used. The reason for this change is simply that truncation of the tpda or teptra ligands is not possible without leaving unsaturated valences on one or more carbon, which would in turn introduce spurious levels around Ef. To maintain strict comparability, we have recalculated the transport properties of the Co3(dpa)4(NCS)2 chain using the full ligand. The greater extent of the molecular scattering region in the x and y directions (i.e., orthogonal to the transport axis) means that a 6 × 6 array of gold atoms (rather than the 4 × 4 unit cell used previously) was required to avoid Coulomb interactions between molecules in neighboring unit cells. The dramatic expansion of both the central region and the electrodes necessitates some compromises in other aspects of the calculation, most notably a decrease of the mesh cutoff from 350 to 100 Ry. The sulfur atoms of the two NCS− ligands are located in a hollow site on the Au (111) surface with Au−S distance 2.52 Å. The precise details of the contact geometry remain a significant issue in all transport calculationsthe “hollow-site” geometry with a gold−sulfur distance of 2.52 Å (corresponding to a distance of ∼1.9 Å between the sulfur and the surface) adopted here has been established as the global minimum for many examples of sulfur coordination to Au (111)74 and is used in the majority of comparative studies.75,76 A full set of Cartesian coordinates for the two-probe system is provided in the Supporting Information.

Figure 7. Co5(tpda)4(NCS)2 in a two-probe configuration between (111) faces of two semi-infinite gold electrodes.

direct impact on the response to applied voltage. Only at very low voltages does the experimentally measured trend in conductance, G(Co3) > G(Co5), emerge. Above 25 mV marked nonlinearity is observed in the current voltage response of both Co3(dpa)4(NCS)2 and Co5(tpda)4(NCS)2. In the former the dominant σ2 channel is delocalized over both sides of the molecule, and, as a result, the channel tracks the drain potential, never fully entering the bias window even at 500 mV. The SOMO of Co5(tpda)4(NCS)2, in contrast, has π symmetry with maximum amplitude in the center of the molecule and the potential drop across the channel is more linear. The channel therefore does enter the bias window, giving a sharp rise in current such that I(Co5) > I(Co3). At higher voltages (>300 mV) the current increases for Co5(tpda)4(NCS)2 but a negative differential resistance feature emerges in Co3(dpa)4(NCS)2 as a result of interference between near-degenerate σ and π channels. The behavior of the Co7(teptra)4(NCS)2 chain is different again: the ground state is a quartet and the large spin polarization results in a relatively featureless transmission spectrum around the Fermi level. The extent to which these computed properties at finite bias can be related to the experimental observations depends on a number of issues, not least the assumption that the measurements reflect coherent scattering and not thermally activated hopping. Moreover, we have assumed a bias-independent molecular geometry identical with that in the gas phase and a single idealized contact structure with the molecule aligned along the transport direction in a 3-fold hollow site. The considerable spread within the experimental data5 clearly suggests that the presence of a range of contact geometries. The presence of many lowlying electronic states also suggests that the structure of the molecule itself may indeed be bias dependent. Nevertheless, it is clear that an intimate understanding of the transport pathways is essential if the properties of the shorter chains are to be extrapolated to their longer counterparts.

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METHODS All gas-phase electronic structure calculations were performed with use of the Amsterdam Density Functional package ADF2012.56−58 A double-ζ Slater-type basis set, extended with a single polarization function, was used to describe the main group atoms, while cobalt was modeled with a triple-ζ basis set. Electrons in orbitals up to and including 1s {C, N}, 2p{Co, Cl} were considered part of the core and treated in accordance with the frozen core approximation. The local density approximation was employed for the optimizations,59 along with the local exchange-correlation potential of Vosko, Wilk, and Nusair60 and gradient corrections to exchange and correlation proposed by Becke and Perdew (BP86).61,62 Different configurations were defined by using the “occupa-

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ASSOCIATED CONTENT

S Supporting Information *

(i) Electronic configurations corresponding to various states, (ii) relative energies and optimized structural parameters of the Co7(teptra)4(NCS)2 and Co7(pzpz)4(NCS)2, and (iii) Cartesian coordinates and total energies of all stationary points. This material is available free of charge via the Internet at http:// pubs.acs.org. 20170

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the EPSRC for financial support (EP/F019327/1, EP/G002789/2) and the Oxford Supercomputer Center (OSC) for computational resources.

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