“Even” conducting superiority in molecular wires designed by porphyrin and graphene nanoribbons

Journal Pre-proof “Even” conducting superiority in molecular wires designed by porphyrin and graphene nanoribbons

Jie Li, Tao Li, Yunrui Duan, Hui Li PII:

S0264-1275(20)30020-4

DOI:

https://doi.org/10.1016/j.matdes.2020.108487

Reference:

JMADE 108487

To appear in:

Materials & Design

Received date:

18 October 2019

Revised date:

7 January 2020

Accepted date:

8 January 2020

Please cite this article as: J. Li, T. Li, Y. Duan, et al., “Even” conducting superiority in molecular wires designed by porphyrin and graphene nanoribbons, Materials & Design(2020), https://doi.org/10.1016/j.matdes.2020.108487

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© 2020 Published by Elsevier.

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“Even” Conducting Superiority in Molecular Wires Designed by Porphyrin and Graphene Nanoribbons Jie Lia,b, Tao Lia, Yunrui Duana, Hui Lia*

Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials, Ministry of

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Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI,

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53706, USA

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Email*: [email protected]

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Education, Shandong University, Jinan 250061, People’s Republic of China.

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Abstract Highly conducting molecular wires are extremely desirable in molecular-scale circuitry, but restricted by their role as tunnel barriers due to their steep conductance recession with the length. This work innovatively reveals that in equilibrium, the conductance of the newly designed porphyrin-graphene nanoribbons (GNRs) molecular wires would increase with the length,

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presenting appealing “even” conducting superiority, different with pure-GNRs molecular wires. Such anomalous transport behavior is not related with edge states of GNRs but the contribution

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of the porphyrin part in the transport. More interestingly, this “even” conducting advantage would

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be more prominent with slightly larger voltage, originated from narrower HOMO-LUMO gap of

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“even” assembled molecular wires. Besides, one of these hybrid molecular wires shows

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exceedingly strong transmission. This work suggests valuable applications of these

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new-assembled molecular wires in nano-electronics.

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Keywords: Newly designed molecular wires; Unconventional conducting behavior; Electron transport mechanisms; First principles

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I INTRODUCTION Traditionally, the conductance of some molecular wires decays exponentially with length because of their role as tunnel barriers in the electron transport, such as the molecules oligo (phenylene-ethynylenes) OPEs[1], oligo (phenylene-vinylenes) OPVs[2], oligoynes[3], alkanes[4] and siloxanes[5], which limits their potential in future molecular-scale circuitry[6]. So exploring

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new molecular wires, whose electrical properties decline slowly, has been greatly concerned in recent years[7-9].

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Graphene nanoribbons (GNRs) are promising materials because of their excellent

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mechanical[10, 11], thermal[12] and electrical response properties[13]. It is well known that

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GNRs exhibit a quantity of fascinating electronic properties, including tunable band gap[14],

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negative differential resistance (NDR)[15], rectifying behavior[16] and spin polarized electron

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transport[17]. More importantly, GNRs can potentially serve as both active device regions and electrical interconnects in nanoscale devices[18, 19]. Particularly, the construction of hybrid

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structures with GNRs and functional organic molecules has broadened the application domain of carbon-based electronics[20, 21]. Among a variety of functional organic molecules, the porphyrin molecule is more remarkable[22], due to not only the rigid geometric configuration, highly conjugated structure and chemical stability of the porphyrin-based molecular systems[23] but also their extensive application in organic electronic devices, such as switches[24], memories[25], spectroscopic markers[26], engineered nanostructures[27] and spintronic devices[28]. Based on the respective properties of the above two, their combination must be highly anticipated. Actually, such hybrid structures can be obtained experimentally. The covalent linking of molecules (including porphyrins) to graphene has already been achieved in wet-chemical

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routes using graphene oxide to enable solution processing[29-31]. However, this methodology lacks specificity and thus complex multicomponent architectures may not be realized on its basis. Excitingly, He et al. [32] have demonstrated precisely covalent coupling of porphines to graphene edges in various forms by high-resolution imaging using advanced scanning-probe technology. The specific experimental implementation of the coupling process is as follows. Graphene was

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grown on an Ag (111) surface by the deposition of atomic carbon on the substrate kept at 900 K[33], which resulted in graphene islands, as shown in Figure 1a. On the partially

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graphene-covered surface, free-base porphines (2H-P) were vapour deposited at room

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temperature. After subsequent cooling to 5 K, scanning tunnelling microscopy (STM)

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measurements revealed that the porphines exclusively adsorbed as individual units on the bare Ag

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(111) surface with three different orientations[34], which also decorated graphene edges (Figure

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1b). To induce the coupling reaction, the sample was annealed to 620 K. An initial porphine coverage is usually employed in the multilayer regime. Indeed, some porphines appeared directly

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attached to graphene edges (Figure 1c), suggesting a covalent coupling. Later, Chen et al.[35] synthesized triply fused porphyrin-nanographene conjugates by the Scholl reaction from tailor-made precursors based on benzo-[m]tetraphene-substituted porphyrins. Li et al.[36] reported the construction and magnetic characterization of a fully functional hybrid molecular system composed of a single porphyrin molecule bonded to GNRs with atomically precise contacts. On-surface synthesis was used to direct the hybrid creation by combining two molecular precursors on a gold surface. High resolution imaging with a STM has shown that the porphyrin core fuses into GNRs through the formation of new carbon rings at chemically predefined positions. These ensure the stability of the hybrid and the extension of the conjugated

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character of the ribbon into the molecule.

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Figure 1[32] STM images and corresponding models that show the experimental procedure. (a) STM image (V = −0.5 V, I=0.5 nA) of graphene islands grown on Ag (111) using atomic carbon deposition. (b) After deposition at room temperature, individual 2H-P molecules decorate the bare Ag(111) terraces and the graphene edges (V = −0.1 V, I=50 pA). (c) STM image recorded after inducing the dehydrogenative coupling reactions by annealing the 2H-P on graphene/Ag(111) at 620 K (V=0.2 V, I = 90 pA). (d–f) Schemes of the experimental situations in a, b and c, respectively. The β- and meso-carbon positions are labelled in the structural model of 2H-P in e.

Actually, study of the coupled structures composed of the porphyrin and GNRs is deficiency. Besides Li’s experimental research above, related theoretical investigation is limited. In 2015, Mondal and

Sarkar

computed and

analyzed

I-V characteristics

of cobalt-centered

porphyrin-GNRs junction[37]. Nozaki et al. theoretically reported molecular switches consisting of different tautomers of metal-free porphyrin embedded between GNRs in 2016[38]. Soon after, Saraiva-Souza et al. performed theoretical study on the electronic and spin-polarized transport properties of the porphyrin molecule attached to the GNRs via different connections[39]. However, their research focused on modified porphyrin molecules coupled to general GNRs and

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the GNRs electrically acted as electrodes. How to design high-performance molecular wires has not been solved. Here, we theoretically design the hybrid molecular wires integrating the porphyrin embedded between GNRs with different lengths and investigate their electron transport properties.

II MODELS AND METHODS

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Six new-style molecular wires with different lengths are designed in this work, as shown in

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Figure 2(a1-a6). The same metal-free porphyrin molecule is embedded between 6 atoms-wide

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armchair GNRs in different sizes with hydrogen-passivated edges. There are 1-6 columns of C atoms on each side of the porphyrin molecule, respectively. Accordingly, hybrid molecular wires

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are denoted as P+Ri (i=1-6), when i=an odd number, called “odd” assembled molecular wires,

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with the same edge arrangement, but when i=an even number, called “even” assembled molecular

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wires, with another same kind of edge. The width of all molecular wires is same. Corresponding molecular devices are indicated by Device P+Ri (i=1-6), “odd” assembled molecular devices and

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“even” assembled molecular devices. Figure 2b displays the structure of the molecular device where the molecular wire is sandwiched between the pyramidal Au electrodes with the tip Au atom symmetrically connected to a central carbon atom on the edge of the molecular wire on each side. The tip electrode atom is bonded to the fourth C atom of each “odd” assembled molecular wire and the third one of each “even” assembled molecular wire, just as Figure 2c illustrates. For comparision, the molecular devices with pure GNRs (recorded as Device Ri, i=1-6) are also constructed, containing the same number of C columns in all, electrodes, and attached modes between electrodes and molecules to the corresponding Device P+Ri (i=1-6), as presented in Figure S1 of Supporting Information (SI). The device with the single metal-free porphyrin (called

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Device P) is built with the same gold electrode, and the tip electrode atom couples with the third

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C atom of the porphyrin molecule on each side, just as exhibited in Figure 2b.

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Figure 2 (a) Embedded structures of a metal-free porphyrin molecule and 6-atoms wide armchair graphene nanoribbons in different sizes with hydrogen-passivated edges; (a1-a6) Molecular structures with 1, 2, 3, 4, 5, 6 column(s) of C atoms on each side, respectively. (b) Schematic of studied molecular devices. The central molecule is the metal-free porphyrin. (d) Connection modes of the electrodes and molecular wires with different edges.

The simulated molecular wires and their devices are constructed in Virtual NanoLab, within a supercell with more than 10 Å (about 14 Å) of vacuum space to allow electrostatic interactions to decay for a system. All molecular wires and devices are structurally optimized before calculating electron transport properties. All calculations are performed in the Atomistix Toolkit (ATK) package within the framework of density functional theory (DFT) in combination with non-equilibrium Green’s functional (NEGF) method[40-42], using numerical LCAO basis sets

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(linear combination of atomic orbitals) and norm-conserving pseudopotentials. To achieve high-precision calculation, the Perdew-Burke-Ernzerhof (PBE)[43] formulation of the generalized gradient approximation (GGA)[44] is used as the exchange-correlation functional. Double-zeta plus polarization (DZP) basis for all atoms is adopted. The mesh cut-off for the electrostatic potentials is 75 Ha and the temperature in the Fermi function is set as 300K. All

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molecular wires and devices are optimized using the quasi-Newton method until all residual forces on each atom are smaller than 0.05eV/Å. The convergence criterion for the total energy is

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10-5 via the mixture of the Hamiltonian. The k-point sampling set is 3×3×50 for the devices and

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1×1×11 for the molecular wires. In our calculations, the average Fermi level, which is the average

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chemical potential of left and right electrodes, is set as zero.

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The conductance G can be expressed in terms of the transmission function T  E , V  within the Landauer-Büttiker formalism[45-47],

 f  E   dET  E ,V     E  

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G  G0 



(1)

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

where G 0=2e2/h is the quantum unit of conductance, h is the Planch’s constant, e is the electron charge, V is the bias voltage applied on the system, f (E) is the Fermi distribution function and T(E, V) is the quantum mechanical transmission probability of electrons, it can be given as[45]

T  E,V   tr  L  E ,V  G R  E ,V   R  E ,V  G A  E ,V 

(2)

Where GR and GA are the retarded and advanced Green functions of the conductor part respectively and ΓL and ΓR are the coupling functions to the left and right electrodes respectively.

III RESULTS AND DISCUSSION To explore the electron transport of new assembled molecular devices, the quantum

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conductance is calculated, which quantifies the charge transport properties, shown on a logarithmic scale against the number of C columns on each side (denoted as Number) of molecular wires in Figure 3. For comparison, the conductance of pure GNRs-based devices is also shown. The conductance is calculated at the bias energy region of [-3eV, 3eV]. From Figure 3a, the steep decay of conductance with increasing length is clearly seen in pure GNRs-based

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devices. Traditionally, the conductance of single molecular wires decays exponentially by length, leading to their role as tunnel barriers[5, 6]. For example, the study of Li et al. has showed the

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length-dependent conductance decay of alkanes, silanes and siloxanes molecular wires, presented

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in Figure S2 of SI[5]. In the work of Kuang et al., the molecular wires of polyporphyrin

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oligomers also exhibit exponentially decay laws[48]. Essentially, the conductance change with

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length of pure GNRs follows the general law of traditional molecular wires. Differently, the

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conductance of devices with hybrid molecular wires does not show the overall downward trend but unusual mutations in sharp contrast to traditional molecular-wire junctions, suggesting

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promising application in molecular-scale circuitry. In detail, the descendent tendency of the conductance appears among “odd” and “even” assembled molecular devices respectively, which can be seen from Figure 3 (b-c). Of course, the pure GNRs-based devices definitely present the declining trend depending on the odd and even numbers respectively because they possess the overall descending trend. But for new-type molecular devices, such decrease relying on the odd numbers is independent of that on the even numbers. The declining speed of the conductance slows sharply for “even” assembled molecular devices while it increases gradually for “odd” ones, which is well reflected at two abrupt changing points (when Number=2, 6), that unexpectedly the conductance of Device P+R2 is

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larger than Device P+R1 and the conductance of Device P+R6 is stronger than Device P+R5. It all goes to show the difference between “odd” and “even” assembled molecular devices, further highlighting the advantage of “even” ones. In this way, Device P+R2 possesses the strongest conductance among all devices. Besides, the conductance of Device P, which has the shortest

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length, is abnormally weaker than that of Device P+Ri (i=1-3).

Figure 3 (a) Conductance of devices based on assembled molecular wires (Device P+R) and pure GNRs-based molecular devices (Device R) as a function of Number on a log scale; the purple triangle indicates the conductance of Device P. (b-c) Conductance and corresponding rate of decline for “odd” and “even” assembled molecular devices, respectively.

To clarify why the conductance of devices with hybrid molecular wires displays such an unconventional changing trend and “odd-even” difference, we firstly plot the transmission versus energy curves, which reflect interiorly the conductance[49, 50]. Figure 4 shows zero-bias quantum mechanical transmission functions of all molecular devices. The transmission at each energy point is calculated by summing up the transmission eigenvalues of every electron transport channel, which are obtained by diagonalizing the transmission matrix[40]. The

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transmission spectrum is a direct reflection of the electronic structures and transport properties, and the large transmission coefficient near EF means the strong transport capacity[51]. Most strikingly, the resonant transmission peaks of Device P+R2 present the maximum amount, pretty high peak values and the nearest distance to EF, meaning the strongest conductance. For Device P+R1, the number of the peak declines to 2 and the height of peaks becomes lower,

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showing its slightly weaker conductance. As the molecular length increases, the transmission near EF gradually evolves from multiple peaks to single lower peak, and the peak shifts away from EF.

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During this changing process, a split peak appears in Device P+R3 as a transition state.

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Noticeably, the peak of Device P+R5 is too weak to be seen in the same transmission range.

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Whereas, the peak of Device P+R6 comes into view again but weaker than that of Device P+R4.

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All these are in line with the conductance change of new-style molecular devices with the number.

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By contrast, there is only one single transmission peak in every device with pure GNRs and the peak value becomes small with the increasing number, resulting in the conventionally decreasing

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conductance. As for Device P, the resonant peak is not lower than Device P+Ri (i=1-3) but far away from EF, explaining the weaker conductance of Device P to some degree. In a more quantitative way, the transmission coefficient at EF as a function of Number is presented on a log scale in Figure 4f and g. For a more direct view, the conductance changing with Number is also provided accordingly. It is noticed that Device P+R2 possesses the largest transmission value at EF (about 0.83, a highly conducting value), corresponding to the highest point on the conductance curve. Clearly, the changing trend of the transmission at EF fits well with the variation of the quantum conductance, for both the assembled molecular devices and pure GNRs-based devices.

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Figure 4 (a1-a6, b) Equilibrium transmission spectra of Device P+Ri (i=1-6) and Device P in the energy region [-0.5eV, 0.5eV] respectively, showing the same transmission range. (c) Equilibrium transmission spectra of Device P+R5 in the same energy interval with a narrower range of the vertical coordinate. (d) Zero-bias transmission curves of pure GNRs-based devices in the energy range [-0.5eV, 0.5eV]. (e) Transmission functions of Device R4-R6, displaying a smaller transmission range. (f) Conductance and transmission at EF of new-assembled molecular devices and Device P depending on Number on a log scale. (g) Conductance and transmission at EF of pure GNRs-based devices depending on Number on a log scale.

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Fundamentally, these resonant transmission peaks are associated with the resonant electron transport through the discrete energy levels of the molecule[52], induced by the electron transport through different frontier molecular orbitals, which couples to electrodes[53, 54]. For Device P+R1, the resonant transmission is induced by the highest occupied molecular orbital (HOMO), same as Device R1. The dominating orbital of Device P+R2 is still the HOMO, while the conductance of Device R2 is dominated by the lowest unoccupied molecular orbital (LUMO). Nevertheless, as Number rises (from 3 to 6), the transmission of Device P+Ri and Device Ri is always derived from the LUMO. So the new assembled molecular devices are basically consistent with the pure GNRs-based molecular devices in the dominating state of the

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conductance, except Device P+R2 with unusual high conductance. The HOMO and LUMO states of all molecular wires are presented in Figure S3 and S4 of SI. From the above analysis of the transmission near EF, Device P+R2 possesses the strongest transmission capacity, contributing to the largest conductance. Such unconventional conductance is essentially originated from the uninheritance of the dominant orbital from the GNRs. The transmission condition has been considered to preliminarily analyze the unusual changing

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tendency of the conductance and “odd-even” difference for new-type molecular devices. For

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deeper analysis, Figure 5 (a1-a6) and (b1-b6) show the edge situations during the transport process for all devices, considering the structural difference in the edge (from the GNRs) between

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“odd” and “even” molecular wires. It is observed that the wave function on the joint gold atom

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has angular nodes, which are characteristics of Au-dxz, and the presence of the Au-px can be seen because two of four nodes are bigger. This phenomenon is more noticeable in the devices with

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longer length. As for the shorter molecular devices (Device P+Ri, i=1-2 and Device R1), these four nodes have little difference. This indicates the importance of the d-states in hybridizing with

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the dominant orbitals of molecular wires. Prior research shows that the coupling of molecular dominant transport orbitals to metal electrodes, closely related to the electron transport, is

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associated with the DOS at EF of d orbitals in electrode atoms[55]. Thus, Figure S5 of SI shows the DOS at EF of the apex orbitals involved in coupling. Obviously, no noPorphyrin-GNRs

molecular wires are innovatively designed. The

new-assembled

molecular

wires

show

promising

potential

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molecular-scale circuitry because of their unusual conductance change with length, different from common molecular wires acting as tunnel barriers. It is demonstrated theoretically that molecular wires present attractive “odd-even” conducting difference and “even” superiority which would be more prominent with slightly larger voltage.

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The inner physical mechanisms of the conducting behavior are elucidated.

difference exists in the spectral density of the d-states for new-style molecular devices except Device P+R2 with significantly large DOS, which contributes to its most robust transport. So, the extraordinarily high conductance of Device P+R2 is derived from the change of the dominating

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orbital inheriting from the GNRs, which results in the enhanced coupling between the leading transport orbital and electrodes. Likewise, the DOS keeps almost same among the devices with

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pure GNRs except Device R1 showing slightly large value, which is the important internal factor

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affecting its electron transport. Anyway, the DOS at EF of d orbitals in electrode atoms does not

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take the lead in “odd-even” conducting difference of devices with assembled molecular wires.

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On another hand, the electronic states are more widely distributed on the edges of these three

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shorter molecular wires. In detail, electronic states can be found on the 1st, 3rd, 4th, and 6th marginal carbon atoms of the molecule P+R1, and on the 1st, 3rd, 4th, 5th and 6th atoms of the edge

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for the molecule P+R2. The molecule R1 shows electronic states on the 1st, 2nd, 3rd, 4th, and 6th carbon atoms of the edge. These just contribute to their relatively higher conductance in some way. Whereas, as the length increases, the electronic states spread on the 2nd, 4th, and 6th atoms of the edge for the molecules P+R3, P+R5, R3 and R5, and on the 1st, 3rd, and 5th atoms of the edge for the molecules P+R4, P+R6, R2, R4 and R6. Therefore, the distributed positions of the electronic states indeed make a difference between the “odd” and “even” molecular wires, but such discrepancy is same for the new-assembled molecular wires and the pure GNRs. It should be noted that the “odd” molecular wires are not same as the “even” ones in the position of the linked C atom to the apex Au atom.

Journal Pre-proof Comprehensively, the edge state makes no difference in essence between “odd” and “even” molecular wires. The electronic state must emerge on the directly connected C atom from which it appears alternately along two sides, except that the three shorter molecules with higher conductance exhibit some continuous distribution. In summary, during the transport process the hybrid molecular wires are consistent with the pure GNRs in edge states. Such anomalous

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“odd-even” conductance difference displaying in the new-type molecular devices might not be related with the distribution of electronic states on the GNRs edges, so it should be closely

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associated with the part of the porphyrin.

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In fact, from Figure S6 (the electronic states of molecular wire P+Ri, i=1-6) of SI, there is no

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big difference in the edge states among all assembled molecular wires but the part of the

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of the part of the porphyrin.

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porphyrin varies much in the distribution of electronic states, which also stresses the importance

Whereas, it’s different for pure GNRs. The decay of the conductance with the length is

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associated with the change of the edge states, which can be well reflected in the local density of states (shown in Figure S7 of SI)[56]. When the edge states weaken, the conductance decreases. Or more precisely, the electronic states of the entire GNRs show gradual localization with the increasing length, resulting in the decline of the conductance. What’s more, it is the edge structure that makes the conductance decay heavily. Koch et al.[57] have compared the calculated conductance for different ribbon structures. Results show that in comparison to the armchair edge structure at the sides of the GNRs, zigzag edge structure strongly improves the conductance, and there is no conductance decay in such edge structure. Thus, the conductance decays heavily only with the armchair edge structure, which is just the edge structure in our work.

Journal Pre-proof To further elucidate the underlying physical mechanism behind the “odd-even” conducting properties, the projected density of states (PDOS) on the part of porphyrin (P) at EF is calculated. Figure 5c presents the value of the PDOS on P as a function of Number and Figure 5d exhibits the percentage of PDOS on P in the total DDOS depending on Number, representing the contribution rate of PDOS on P in the total DDOS. Obviously, Device P+R2 shows the largest value and the value of Device P+R6 is higher than that of Device P+R5. The value of PDOS on P

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shows the same “odd-even” difference to the conductance. For clarity, Figure 5c-d describes

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simultaneously the trend of the conductance changing with Number. Like the value of PDOS on P,

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the contribution rate of PDOS on P coincides well with the change of the conductance with

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Number. Thus, the value of PDOS on P and its contribution rate well account for the

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extraordinary “odd-even” conducting difference in the devices with bybrid molecular wires. The

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porphyrin is not just a simple linker and it has been fused with GNRs together. It plays a key role in the electron transport, which is well reflected not only in the same changing tendency of the

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DOS to the conductance with the length but also in the electronic distribution of dominant orbitals. From Figure S3 (HOMO and LUMO states), all the dominating orbitals show strong electronic states in the part of the porphyrin.

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Figure 5 (a1-a6) Edge states of Device P+Ri (i=1-6), respectively. (b1-b6) Edge states of Device Ri (i=1-6), respectively. Isosurface plots in bottom left corners show the states in xz plane and the right plots present the states in yz plane. (c) PDOS on P and the overall trend of the conductance of new-style molecular devices at EF as a function of Number on a log scale. (d) Percentage of PDOS on P in total DOS and the overall trend of the conductance of new-style molecular devices at EF depending on Number on a log scale.

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In view of the distinctive equilibrium-conducting behavior of hybrid molecular devices, the

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conductance at low voltage is calculated, shown in Figure 6a. It shows that Device P+R2 always

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possesses the largest conductance at low biases, followed by Device P+R1. To be more clearly, the comparison of the conductance is also displayed on a log scale in Figure 6b. Distinctly, the tendency of the conductance varying with the voltage can be classified into two cases with V0 as a bound, which is qualitatively summarized in Figure 6c. The upper half shows the trend of the conductance changing with Number with the bias less than V0 while the bottom half reveals the variation of the conductance depending on the molecular length when the voltage is larger than V0. When the bias is not very large (not above V0), the changing tendency of the conductance is identical with that in equilibrium condition, also showing noticeable odd-even difference. More intriguingly, such phenomenon is more prominent with the increasing voltage. When the bias is

Journal Pre-proof beyond V0, the conductance of each “even” assembled molecular device is correspondingly larger than that of each “odd” assembled molecular device with slightly smaller length. The conductance presents an overall state of fluctuation changing with Number. Such fold line

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highlights the superiority of “even” assembled molecular devices.

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Figure 6 (a) Quantum conductance of new assembled molecular devices under low biases. (b) Presentation of plots in (a) on a log scale. (c) The rough trend of the conductance (G) changing with Number with the bias below V0 (exhibited in the upper part) and above V0 (displayed in the lower part). (a) and (b) have the same legend, shown on the top of the figure.

What is the origin of such unusual difference in conductance under low bias? It has been

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analyzed that the contribution of the porphyrin part (P) in transport determines the “odd-even”

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conducting difference at 0V. Based on this, we have also calculated the PDOS of P under non-equilibrium conditions. One representative voltage has been selected from each of the two bias intervals (i.e. 0.1V from [0V, V0] and 0.25V from [V0, 0.3V]). Figure 7 (a) and (b) illustrate the P-PDOS of the new assembled molecular devices near EF at the bias of 0.1V and 0.25V, respectively. The energy range in the bias window ([-V/2, +V/2]) is enough, i.e. [-0.05eV, 0.05eV] for 0.1V and [0.125eV, 0.125eV] for 0.25V. It can be clearly seen that the quantitative comparison of P-PDOS near EF at 0.1V keeps consistent with that at EF under equilibrium condition, contributing to the attractive conducting behavior. When the voltage is above V0, the P-PDOS does not change a lot for the “odd” assembled molecular devices, comparing with the

Journal Pre-proof P-PDOS at the bias below V0. But, for the “even” assembled molecular devices, the shape of the P-PDOS curve becomes completely different and the value gets large close to EF that mostly exceeds that of the other devices, resulting in more prominent “even” conducting preponderance. To shed light onto the significant changes of the DOS at higher voltage, the electronic structures of the new-assembled molecular wires are investigated during the process of electron

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transport, which are called the molecular projected self-consistent Hamiltonian (MPSH), obtained by projecting the self-consistent Hamiltonian onto the Hilbert space spanned by the basis

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functions of the molecule. The frontier orbitals are of huge importance (the energies can be

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extracted from MPSH spectrum, i.e. Molecular Energy Spectrum), especially the HOMO and the

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LUMO, due to the energies of these orbitals being the closest of any orbitals of different energy

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levels. Therefore, the energy difference between the HOMO and LUMO or HOMO-LUMO gap

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is generally the lowest energy electronic excitation that is possible in a molecule, which plays a decisive role in the electron transport. Figure 7c shows the HOMO-LUMO gap of new-type

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molecular devices as a function of Number at 0.25V. Obviously, the gap of each “even” assembled molecular wire is narrower than that of each “odd” assembled molecular wire, bringing about an enhanced contribution of the porphyrin part to the transport, which essentially supports the advantages of the “even” assembled molecular devices in the electron transport. Actually, with the increasing length, the tendency of electrical conductance for each “even” assembled molecular device to decay is compensated by a decrease in the HOMO-LUMO gap. Furthermore, the longer the length gets, the more the gap compensates, which is reflected by the smaller gap of the “even” assembled molecular wires with larger Number.

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Figure 7 (a-b) PDOS on P for the devices with hybrid molecular wires near EF at the bias of 0.1V and 0.25V, respectively. (c) The HOMO-LUMO gap (Δgap) of new-type molecular devices (Device P+R) depending on Number at 0.25V. (a) and (b) have the same legend, shown on the top of the figure.

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In addition, zigzag edges of GNRs (like the zigzag termini of AGNRs and the extended zigzag

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edge of ZGNRs) hold spin-polarized electronic states and show promise for spintronic

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devices[56], which has gained much attention in recent years. We have also been passionate about it and plan to focus on the spin transport properties next. Except the application as common

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electronics in molecular-scale circuitry, the new-assembled molecular wires are also promising

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candidates for thermoelectric devices[58, 59]. What we have revealed about the change of the

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conductance with the increasing length is necessary for studying thermal properties, because the charge conductance, thermal conductance and Seebeck coefficient are very sensitive to the structure and size of the molecular junction[58]. More importantly, the unconventional “odd-even” conducting behavior and the inner physical mechanisms related to frontier orbitals are no doubt valuable to design high-performance thermoelectric devices[58, 59].

IV Conclusion In summary, we have theoretically investigated the electron transport of the devices with new-type molecular wires assembled by the porphyrin and graphene nanoribbons (GNRs). Results show that the conductance of these devices presents unusual changing tendency with the length, comparing with conventional molecular wires-based devices (including pure-GNRs based

Journal Pre-proof ones) at equilibrium condition, characterized as “odd-even” conducting behavior. Further analysis indicates that such attractive performance is determined by the achievement of the porphyrin part in transport. Moreover, with higher voltage, the conductance of each “even” assembled molecular junction is stronger than the “odd” one with the slightly shorter length, essentially due to smaller HOMO-LUMO gap. Besides, one molecular device shows extraordinarily high conductance,

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derived from the change of the dominating orbital inheriting from the GNRs, which causes the enhanced coupling between the leading transport orbital and electrodes. This work uncovers the

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potential of these molecular wires as interconnects in future molecular-scale circuitry.

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Acknowledgements

The authors would like to acknowledge the support from the National Natural Science

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Foundation of China (Grant No. 51671114 and U1806219). This work is also supported by the

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Special Funding in the Project of the Taishan Scholar Construction Engineering. The financial

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support by the China Scholarship Council (grant number 201806220169) for Jie Li to work as a visiting graduate student at UW-Madison is also acknowledged.

References

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Credit author statement Jie Li: Conceptualization, Formal analysis, Investigation, Writing - Original Draft, Visualization, Writing - Review & Editing, Project administration Tao Li: Software, Data Curation

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Yunrui Duan: Methodology, Resources

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Hui Li: Validation, Supervision, Funding acquisition

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Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Graphical abstract

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Highlights

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 Porphyrin-GNRs molecular wires are innovatively designed.

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 The new-assembled molecular wires show promising potential in

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molecular-scale circuitry because of their unusual conductance change

barriers.

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with length, different from common molecular wires acting as tunnel

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 It is demonstrated theoretically that molecular wires present attractive

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“odd-even” conducting difference and “even” superiority which would be more prominent with slightly larger voltage.  The inner physical mechanisms of the conducting behavior are elucidated.