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Switchable Negative Differential Resistance Induced by Quantum Interference Effects in Porphyrin-based Molecular Junctions Daijiro Nozaki,*,†,§ Lokamani,† Alejandro Santana-Bonilla,†,‡ Arezoo Dianat,† Rafael Gutierrez,† and Gianaurelio Cuniberti†,§,∥ †

Institute for Materials Science, TU Dresden, 01062 Dresden, Germany Dresden Center for Computational Materials Science, TU Dresden, 01062 Dresden, Germany ‡ Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany ∥ Center for Advancing Electronics Dresden, TU Dresden, 01062 Dresden, Germany §

S Supporting Information *

ABSTRACT: Charge transport signatures of a carbon-based molecular switch consisting of different tautomers of metal-free porphyrin embedded between graphene nanoribbons is studied by combining electronic structure and nonequilibrium transport. Different lowenergy and low-bias features are revealed, including negative differential resistance (NDR) and antiresonances, both mediated by subtle quantum interference effects. Moreover, the molecular junctions can display moderate rectifying or nonlinear behavior depending on the position of the hydrogen atoms within the porphyrin core. We rationalize the mechanism leading to NDR and antiresonances by providing a detailed analysis of transmission pathways and frontier molecular orbital distribution.

G

raphene-nanoribbons (GNRs),1 quasi-one-dimensional graphene stripes, are promising materials due to their superior mechanical,2,3 thermal,4 and electrical response properties.5 A number of fascinating electronic properties including tunability of the band gap,6 negative differential resistance (NDR),7−14 electrical rectification behavior13−18 as well as spin polarized electron transport19 have been demonstrated. More importantly, GNRs can potentially act as both active device regions and electrical interconnects, in nanoscale devices.20−25 Especially, building hybrid structures combining GNR electrodes and functional organic molecular systems such as molecular switches26 opens a variety of opportunities to build carbon-based electronics.27 To date, several kinds of molecules such as diarylethene-derivative28−31 or azobenzene-derivatives32,33 have been synthesized and demonstrated to work as nanoscale switches. In general, most of the molecular switches demonstrated to date involve either conformational changes or local chemical modifications (bond opening and closing). More subtle switching processes have, however, also been demonstrated,34,35 the case of currentinduced tautomerization in single naphthalocyanine molecules being meanwhile paradigmatic.34 Molecular switching has been theoretically studied in this latter molecule, and strong NDR effects revealed.36 Another molecular system where hydrogen tautomerization effects are important is porphyrin.37,38 Porphyrin-based molecular systems have found broad applications as building blocks in switches,39 engineered molecular nanostructures,40,41 and molecular spintronics devices.42 Here, we theoretically investigate the charge transport signatures of a molecular-scale device integrating GNR as © 2015 American Chemical Society

mesoscopic electrodes and metal-free porphyrin molecules as active switching elements. The strong covalent bonds between GNR and the porphyrin molecule suppress structural fluctuations of the molecular core as well as the problems often caused by the dependence of the conductance to the contacting position when using metallic electrodes. The possibility of tuning the conductivity of GNRs by varying their width and edge shape represents an additional advantage of using them as pure carbon electrodes. We show that molecular junctions differing only in the arrangement of the hydrogen atoms in the porphyrin core (tautomers) can display a variety of subtle charge transport features such as NDR and quantum interference (QI) effects. We rationalize the mechanism of the NDR by analyzing QI-induced antiresonances in the transmission spectra. Clear quantitative differences in the electrical current of the different tautomers suggest the relevance of the studied carbon-based devices as efficient molecular switches. Interference effects in molecular junctions with graphene nanoribbon electrodes have been theoretically investigated in relation to spintronics applications,43 but QIbased effects have also been addressed in different contexts, see, e.g., refs 44−47. We remark that NDR has recently been demonstrated experimentally in anthraquinone derivatives;48 its origin, however, seems related to mechanically or bias-induced Received: July 24, 2015 Accepted: September 16, 2015 Published: September 16, 2015 3950

DOI: 10.1021/acs.jpclett.5b01595 J. Phys. Chem. Lett. 2015, 6, 3950−3955

Letter

The Journal of Physical Chemistry Letters

conductance G of the system is proportional to the transmission function evaluated at the Fermi energy: G = (2e2/h) T(E ≡ EF). At finite bias, the electrical current will be computed as I(V) = (2e/h)∫ ∞ −∞T(E,V)[f L(E) − f R(E)] dE, including the bias dependence of the transmission T(E,V) within a self-consistent procedure, and f L(R)(E) are the left (right) electrode Fermi functions. For a stable operation of the molecular switches, each configuration in Figure 1a−d should remain in a metastable state at room temperature, if no external perturbation is acting on the junction. To assess the mechanical stability of the four configurations, we computed the energy barriers between the different states. Figure 1e summarizes the energetics of the four states and the corresponding energy barriers along the reaction coordinates from one state to the other. The transition between tautomers was modeled by linear translations of the corresponding hydrogen atoms, and then the resulting structure was subsequently optimized while fixing the spatial position of the corresponding hydrogen atoms. For each junction, the relative total energy with respect to the energetically most stable configuration (junction 1) is shown. It is clear that the asymmetric structures (junctions 2 and 4) have higher total energies than symmetric ones (junctions 1 and 3) because of steric repulsion of the hydrogen atoms. The energetic barriers for transitions where two hydrogen atoms migrate are approximately double as for the case with a single hydrogen displacement. The energy barriers are in all cases much higher than the energy scale of thermal fluctuations, so that we can safely conclude that the four states will be mechanically stable at ambient conditions as far as no external perturbations act on the system. In Figure 2a−d, the corresponding transmission spectra for the four junctions in Figure 1a−d are displayed. Although changing the position of the hydrogen atoms in the porphyrin core represents a rather subtle modification of the atomic structure of the junction, we see clear differences in the lowenergy (energy window [−1,1] eV) behavior of the transmission function and hence of the linear conductance of the systems. In all junctions, antiresonances resulting from quantum interference effects are found (see below for further discussion). Especially for junction 1, the transmission around the Fermi energy is strongly suppressed due to the two almost symmetrically positioned antiresonances, so that a high on/off ratio in current at low bias might be expected. The spectral features in the transmission function are in general quite broadened when entering higher energy regions as a result of the strong coupling between the porphyrin molecule and the GNR electrodes. To address in more detail the electrical response of the molecular junctions, we have self-consistently computed the bias-dependent transmission spectra for these systems, and the results are shown as surface plots in Figure 2e−h. The corresponding I−V characteristics obtained by integrating the voltage-dependent transmission spectra over energy are summarized in Figure 2i and the corresponding rectification ratios and on/off ratios in Figure 2j,k, respectively. We defined the rectification ration as max[|I(V)|/|I(−V)|,|I(−V)|/I(V)], while the on−off ratio of junction number j (j = 1,2) was defined by the expression |Ij(V)|/max|Ik≠j(V)|, with Ij(V) being the current value for junction j at a given bias voltage V, and max|Ik≠j(V)| being the largest current at voltage V in the set {I2(V),I3(V),I4(V)} for j = 1 and {I1(V),I3(V),I4(V)} for j = 2. The very large currents in the μA region result from the

induced degeneracy breaking, leading to current reduction beyond a certain threshold voltage. Figure 1a−d shows the different structures of the molecular junctions considered in this work. The porphyrin molecules are

Figure 1. (a−d) Molecular devices based on metal-free porphyrin covalently bonded to mesoscopic zigzag graphene nanoribbons with hydrogen-passivated edges. The four devices differ in the arrangement of the hydrogen atoms in the porphyrin core (tautomers). (e) Calculated potential energy barriers between different configurations. In each configuration, the total energy is shown with respect to the lowest one (junction 1).

embedded between GNR electrodes. In each junction, the hydrogen atoms are arranged in different positions. In order to determine the stable configurations of the porphyrin cores between GNR contacts, conjugate-gradient geometry optimization of the extended molecule (porphyrin molecule plus one unit cell of the GNR on each side) was performed using the density-functional tight-binding method (DFTB) with periodic boundary conditions. This DFT-parametrized semiempirical approach has been shown to be computationally highly efficient.49 We use the parametrization of ref 50 for the carbon, nitrogen, and hydrogen atoms. The structurally optimized unit cells of the extended molecules were then covalently attached to semi-infinite GNRs, and the extended molecules were relaxed again to take into account local structural modifications resulting from the covalent bonding between the molecule and the electrodes. The computation of the quantum mechanical transmission functions for the different junctions was carried out using nonequilibrium Green’s functions as implemented in the gDFTB code.51 The (in general bias-dependent) transmission function T(E) is given by the standard expression: T (E) = Tr[ΓL(E)Gr (E)ΓR (E)Ga(E)]

(1)

where ΓL/R correspond to the spectral densities of the left (right) contacts and are defined as ΓL/R ≡ i[ΣL/R − Σ†L/R] . The self-energies ΣL/R encode both the electronic structure of the semi-infinite electrodes as well as the electronic properties of the electrode-molecule interface. In the linear regime, the 3951

DOI: 10.1021/acs.jpclett.5b01595 J. Phys. Chem. Lett. 2015, 6, 3950−3955

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The Journal of Physical Chemistry Letters

Figure 2. Transport signatures of the molecular devices shown in Figure 1. (a−d) Zero-bias quantum mechanical transmission functions. (e−h) Surface plots of the voltage-dependent transmission function as a function of the incoming electron energy E energy and applied bias voltage V. The transmission is plotted in logarithmic scale. Antiresonances are observed as dark spots. The bias windows are shown as red lines. (i) Current−voltage characteristics of the junctions. Junctions 1 and 2 display nonlinear features in the low-bias regime, while junctions 2 and 4 show rectifying behavior because of asymmetries in the molecular structures (arrangement of the H atoms). The inset shows NDR behavior in junction 2 at low bias. (j) Rectification ratio and (k) on/off ratio for junctions 1 and 2. Both quantities are defined in the main text.

covalent coupling between the porphyrin molecule and the electrodes. In junction 1, the current at low voltages (|V| ≤ 0.3 V) is strongly suppressed due to the presence of the antiresonance around the Fermi energy; with increasing applied voltage, transmission resonances at higher energies start entering the bias window (see Figure 2e) and the current increases resulting in weakly nonlinear I−V characteristics. Junctions 2 and 4 display a moderate rectifying behavior (see Figure 2j), which reflects the structural asymmetry of the junctions related to the position of the H atoms in the porphyrin cores (see Figure 1). This is also seen in the surface plots in Figure 2f,h, which clearly show the asymmetries of the transmission function when opening the bias window. Moreover, Figure 2k shows that a peak in the on−off ratio ∼6 can be achieved around 0.6 V for junction 2. Interestingly, Figure 2i also shows strong NDR behavior in the junction 2 at low applied bias. This NDR effect originates as a result of a delicate variation of the transmission around the Fermi energy when varying the applied voltage. This behavior is illustrated in Figure 3 for selected bias voltages. The antiresonances in the transmission function are already present at zero bias, sinceas discussed later onthey result from quantum interference effects involving selected frontier orbitals. However, once a finite bias is applied, the transmission spectrum is modified as a result of the charge rearrangement and the electrostatic potential redistribution under the action of the bias-induced electric field. This manifests in an energetic shift of the spectral features of the transmission at different voltages as well as in a modification of their spectral weights. As a result, there is a certain bias where the antiresonances enter the integration region, reducing the current, and this is what leads to the observed NDR effect at low bias.

Figure 3. Origin of the negative differential resistance effect in the molecular junction 2 at low bias: the voltage-dependent transmission spectrum is shown at four different applied voltages. The NDR emerges as the antiresonances enter the integration window when computing the total current.

What is the origin of the low-energy antiresonances in the transmission spectrum? To shed light onto it, we consider a simplified model of the junction. First, we assume the wideband limit (WBL) for the electrode self-energies ΣL/R(E), since within the narrow spectral range around the Fermi energy we are going to discuss, the surface density of states of the electrodes can approximately be assumed to have a nearly constant value. We thus assume ΣL/R(E) ≡ (−iγ/2) with γ being an energy-independent constant and set γ = 10 meV. We stress that the WBL is obviously not valid for the whole spectral range, but only around the energy window enclosing the antiresonance. Although the real broadening of the molecular states is much larger due to the strong coupling with the carbon nanoribbonon electrodes, we assume a smaller coupling γ in order to clearly correlate the position of the transmission resonances in our minimal model with the energetic position of the frontier molecular orbitals of the scattering region. Second, we only consider a single pz orbital per site, since our 3952

DOI: 10.1021/acs.jpclett.5b01595 J. Phys. Chem. Lett. 2015, 6, 3950−3955

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The Journal of Physical Chemistry Letters calculations show that the frontier orbitals, i.e., those energetically closest to the Fermi energy, are mostly derived from atomic pz states. Based on these simplifications we can now write the transmission function connecting an atomic site iL (or a pz atomic orbital on that site) at the interface with the left electrode with a site jR at the right interface as Ti L, jR (E) = γ 2 |GirL, jR (E)|2

(2)

The condition for an antiresoannce, TiL,jR(E) = 0, can now be restated as GriL,jR(E) = 0. In the next step, we use the Lehmann representation of the retarded Green’s function to get its spectral decomposition in terms of the molecular orbitals Ψm of the scattering region:52,53 N

GirL, jR (E)

=

∑ m=1

⟨pizL |Ψm⟩⟨Ψm|pjzR ⟩ E + iγ − εm

(3)

where pziL (pzjR) are atomic pz orbitals on the corresponding sites. Since we have in general multiple contacts to each electrode, the total Green’s function is then built by summing over all contact sites: G L/R (E) =

∑ GirL,jR (E) i L, j R

(4)

Therefore, the transmission within the WBL simplifies to by TWBL(E) = γ2|GLR(E)|2. The scattering region in each device in Figure 1 includes three carbon atoms at each interface, each atom carrying a pz orbital. Hence, GLR(E) in eq 4 contains a sum over nine pairs of sites. The following discussion focuses on junction 2. Although this junction is not the minimal energy configuration at zero temperature, we chose it since (i) it displayed the NDR effect and (ii) we are also interested in the interconvertion between different tautomers under specific external perturbations (see the discussion at the end of the article), so that all junctions are of potential interest regardless their energetic order at zero temperature. Figure 4b,c shows the transmission spectra of junction 2: panel b displays a zoom of the low-energy part of the full transmission spectrum already shown in Figure 2b, while panel c shows the transmission using our simplified model of the molecular junction. The vertical bars in each panel indicate the position of the molecular orbital energies computed for the scattering region, i.e., without inclusion of the electrodes, in a single point calculation. In order to take into account the shift of the eigenvalues accompanying the charge transfer between leads and the scattering region, the original charge distribution was kept fixed when performing the single point calculation of the isolated scattering region. Also shown for reference is the spatial distribution of the HOMO−1. HOMO, and LUMO states of the isolated scattering region. The blue line in Figure 4c shows the transmission function including all molecular orbitals of the scattering region when carrying out the summation in eq 3. We clearly see that the transmission resonances nicely correlate with the energetic position of the molecular orbitals. Due to the WBL approximation, these resonances can be resolved in energy; this is not the case in the original transmission with graphene electrodes as a result of the strong molecule-electrode coupling (compare with Figure 4b). The red line in Figure 4c corresponds to the transmission when only including a subset Ψm of frontier orbitals in the spectral decomposition eq 3, with

Figure 4. Emergence of antiresonances in the transmission spectrum as a result of the mutual cancelation of contributions of subsets of molecular orbitals to the transmission amplitude. (a) Molecular structure of junction 2. The device is divided into the central scattering region and the left and right lead as contacts. (b) Transmission spectrum calculated using the gDFTB code and corresponding position of the molecular eigenvalues (vertical lines) of the isolated scattering region. The corresponding density distribution of selected molecular orbitals is shown above the plot. (c) Transmission spectra within the wide-band limit (WBL) considering contributions to eq 3 and eq 4 arising from three frontier molecular orbitals (HOMO−1, HOMO, and LUMO) only (red solid line) and with full inclusion of all eigenstates (solid blue line). The broadening of the resonances in the WBL approximation is set to 10 meV.

m = HOMO−1, HOMO, LUMO. We see that this subset is enough to recover the behavior of the transmission for |E| < 0.3 eV and the emergence of the antiresonances, thus suggesting that their coherent addition in eq 3 leads to mutual cancelation of their contributions below and above the Fermi energy. Adding other orbitals may lead to a slight shift of the energetic position of the antiresonances, but in general such terms are considerably less effective in mediating quantum interference due to the larger denominators in eq 3. The presented analysis thus allows one to efficiently identifying the main eigenstates involved in quantum interference effects (a similar analysis could be carried out to discuss the emergence of, e.g., Fano resonances). We have studied a set of molecular devices based on a porphyrin core covalently bonded to graphene nanoribbons. The differences in the devices lie in the arrangement of the hydrogen atoms in the molecular core (tautomers). We have shown that a rich physical behavior of the low-energy, lowvoltage properties of the junctions results from this rather subtle modification of the atomic configuration of the junction (hydrogen switching). This includes quantum interference 3953

DOI: 10.1021/acs.jpclett.5b01595 J. Phys. Chem. Lett. 2015, 6, 3950−3955

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effects, negative differential resistance, and rectifying behavior. In closing, we mention that it is possible to fabricate GNR stripes by organic synthethic approaches54 or e-beam lithography techniques.55 Using micromechanical1 or liquidphase exfoliation56−59 methods, it is possible to place GNRs on arbitrary insulating substrates.60 By depositing metallic contacts, GNR-based electrodes can be contacted to power sources electrically. Concerning the coupling of porphyrin cores to GNRs, the hints for developing such synthetic routes are given in refs 61 and 62. Changing the positions of the hydrogens in the porphyrin core is also another challenging issue in the studied devices. Quantum molecular dynamics simulations of the molecular junctions and their energetics have shown that switching of hydrogen atoms between different configurations is feasible under the action of, e.g., a heat pulse, which provides the necessary energy to overcome the transition barriers between the different metastable conformations. Details of these simulations are provided in the Supporting Information.63



ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b01595. Results of molecular dynamics simulations of the porphyrin cores with and without external perturbation demonstrating switching events from one configuration to others (PDF) Actual MD trajectory for Figure S3a (MPG)



AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS D.N. and R.G. thank Göran Wendin for useful discussions. This work has been funded by the European Union within the project Synaptic Molecular Networks for Bioinspired Information Processing (SYMONE, Project No. 318597). This work has also been partly supported by the German Research Foundation (DFG) within the Cluster of Excellence “Center for Advancing Electronics Dresden”. We acknowledge the Center for Information Services and High Performance Computing (ZIH) at the Dresden University of Technology for computational resources.



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DOI: 10.1021/acs.jpclett.5b01595 J. Phys. Chem. Lett. 2015, 6, 3950−3955

Letter

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DOI: 10.1021/acs.jpclett.5b01595 J. Phys. Chem. Lett. 2015, 6, 3950−3955