Current Rectification in Mono- and Bilayer Nanographenes with

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Current Rectification in Mono- and Bilayer Nanographenes with Different Edges Aleksandar Tsekov Staykov, and Petar Boykov Tzenov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp402187a • Publication Date (Web): 15 Jun 2013 Downloaded from http://pubs.acs.org on June 18, 2013

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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

Current Rectification in Mono- and Bi-layer Nanographenes with Different Edges Aleksandar Staykov1,* and Petar Tzenov2 1)

International Institute for Carbon-neutral Energy Research (WPI-I2CNER), Kyushu

University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan 2)

Faculty of Informatics, Technical University Munich, Boltzmannstr. 3, Garching, München

D-85748, Germany

* To whom the correspondence should be addressed: [email protected]

1

ACS Paragon Plus Environment


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Abstract Graphene nanomaterials are actively used in electronics and materials science as elements of electric circuits, structural, and storage components. Their unique structure and electronic properties allow for wide variety of applications, i.e., electron and thermal conductivity, ion transport, ion storage, and electric current rectification. In this work we investigate the electric current rectifying properties of mono- and bi-layer two-terminal nanographene devices with the non-equilibrium Green’s function method combined with density functional theory. The diode-like properties are achieved by control of the nanoribbons’ edges. The sequential combination of armchair and zigzag domains leads to nanographene junctions with asymmetric current-voltage characteristics. The rectifying properties of the asymmetric armchair-zigzag carbon materials are derived from the nonequilibrium Green’s function theory. The electric current rectification is explained by the interaction of the external electric field induced between the electrodes with the localized electronic states within the junction. The model is applied on cyclophane molecules and bi-layer nanographenes for which one of the layers consists of armchair-edge nanoribbon and the second layer consists of zigzag-edge nanoribbon. Owing to the interlayer π−π stacking, the cyclophane and bi-layer nanographene junctions show higher rectification ratios compared to the monolayer junctions. The proposed devices consist of nanographenes and polycyclic aromatic hydrocarbons and the diode-like properties are obtained without heteroatom doping. The investigated carbon materials are promising candidates for current control elements in nanoelectronics.


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Introduction Graphene nanomaterials combine unique electronic and structural properties, such as electron conductivity, high charge carrier mobility, high strength, etc.1 They are actively used in nanoelectronics and nanotechnology, due to their easy fabrication and deployment, high fatigue resistance, good electric and thermal conductivity, and possibility for fine-tuning of their properties.2 Synthesis methods include top-down approaches such as adhesion3 layer separation from macroscopic graphite or catalytic formation on metal surfaces from precursor carbon-containing gases.4 Another possibility for fabrication of nanographene materials with well-defined structure is the bottom-up approach for synthesis of large polycyclic aromatic hydrocarbons (PAHs) consisting of several thousand carbon atoms proposed by Müllen and co-authors.5 That synthetic mechanism is based on the sequential addition of benzene and ethine molecules followed by dehydrogenation. The bottom-up approach allows for the fabrication of nanographene ribbons with precisely controlled stoichiometry, symmetry, edge structure, and heteroatom doping. The

two

dimensional

quasi-infinite

graphene

is

a

transparent

zero-band

semiconductor.1 However, its electronic properties strongly depend on its nanoscale structure.6,7 The band gap is strongly dependent on the width of the nanographene ribbons. Narrower graphene nanomaterials are characterized with larger band gaps. In this way, by controlling the nanoscale size of the graphene materials one can achieve control over their electronic and optical properties.8,9 Another way to achieve control over the band gap of graphene nanomaterials is to fabricate nanoribbons with well-defined edge structure.10 The two possible terminations of the nanographene edges are armchair edge, also known as phenanthrene edge (Ph), and zigzag edge, also known as acene edge (Ac). The edge and size effects were investigated with experimental and theoretical methods for both PAHs and graphene nanoribbons.5,9 The Ph edge determines higher stability and larger highest occupied molecular orbital (HOMO) – lowest unoccupied molecular orbital (LUMO) gap for PAHs compared to Ac edge PAHs with same stoichiometry. Nanographene ribbons with Ph edge are characterized with larger band gap compared to Ac edge nanoribbons with similar size and mass. Other structural characteristics, like the point group symmetry of PAHs, tend to influence the HOMO-LUMO gaps of small hydrocarbons and graphene nanoribbons while the effect diminishes with the increase in size.11 Defects in PAHs and nanographenes are used to modify their electronic properties. Heteroatoms, e.g., impurity defects, can introduce levels within the HOMO-LUMO gap, increase the charge-carrier density, or localize electrons and increase the resistance. The 3



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impurity defects on the periphery (Tamm states) have different effect from the impurity defects within the PAH (Shotkey states).12,13 Beside the impurities, vacancy defects and adatoms are also observed.14,15 Both, theoretical and experimental studies have shown that the exact defect location is important for the perturbation it causes on the PAH properties. In a recent theoretical paper it was shown that rectifying properties in PAHs and cyclophanes, i.e., mono- and bi-layered nanographenes, could arise as a result of spatially asymmetric doping with electron-rich and electron-poor atoms.16 The external electric field that is induced between the electrodes interacts with the spatially asymmetric frontier orbitals leading to their localization in one of the current directions and delocalization in the opposite direction.17 As a result current rectifying properties with rectifying ratio as high as 7 could arise. With the advance of miniaturization the size of electric circuit components rapidly approaches the scale of a few to several tenths of nanometers.18 In 2012 the leading electronic manufactures reported CPUs with elements’ size of 18 nm. Any further downscaling in size will lead not only to smaller devices but also to a smooth transition between the laws of classical mechanics which govern macroscopic objects to the laws of quantum mechanics which describe the properties of molecules, atoms, and subatomic particles. Many macroscopic concepts like the band structure, Ohm’s law, conductivity, etc., would be replaced by their quantum mechanical equivalents like discrete molecular energy spectra, ballistic transport, conductance, tunneling, etc. All these concepts should be properly applied in the design of novel current controlling elements such as switches, diodes, and transistors. In the past decade, significant achievements were reported in the synthesis and fabrication of molecular photoswitches based on the diarylethene molecules.19-23 Extensive theoretical studies have contributed to the understanding and realization of reversible switching reaction.20,24-28 Most notably, the theoretical work of Datta29 and the experimental research on nanographene materials led to the design and fabrication of current controlling three-terminal devises, e.g., field effect transistors, where a gate electrode is used to shift the energy levels within a nanometer scale junction that leads to controllable conductance switching.30 Twoterminal devices, e.g., diodes, were the foundations of molecular and nanoelectronics. The concept that a single molecule could replace an element of the macroscopic electric circuits was developed in 1974 by the IBM researches Aviram and Ratner.31 They proposed a π donor-σ bridge-π acceptor molecule that could act as an electric current rectifier by allowing the electrons to tunnel in the acceptor to donor direction and block the electron flow in the opposite direction. In the recent years large number of current rectifying molecules were


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fabricated and designed theoretically32 based on Tour wires,17 donor-σ bridge-acceptors,33 donor-π bridge-acceptors,17,34 asymmetric oligophenyls,35 asymmetric coupling with the electrodes,36,37 different electrode materials,38 stacking of cyclophanes,39,40 etc. The rectifier ratio of donor-acceptor diodes could be significantly increased by adding spacer units between the electrodes and the molecule.41 It was reported that the sequential linking of several diodes increases the rectifying behavior.42 Experimental results were reported on donor-acceptor molecule consisting of a bithiophene donor and a naphthalenediimide acceptor separated by a conjugated phenylacetylene bridge and a nonconjugated end group.43 Recent studies reported that electrode contamination could significantly affect the transport properties and current rectification, respectively.44 Most of the proposed nanoscale diodes require complicated synthetic procedures of introducing acceptor and donor groups or precise doping with electron withdrawing and electron donating atoms.

Chart 1 In several recent studies technological concepts were shown for the production of nanographenes with size as small as 5 nm by cutting graphite blocks with diamante nanoedge and sequential graphene-layer separation45 or carbon nanotube unzipping.46,47 Novel concepts allowed for the precise shape and edge control of the graphene nanoribbons.48 The cutting of asymmetric nanoribbons with Ac edge structure on one side and Ph edge structure on the other (Chart 1) could lead to "easy-to-fabricate" graphene nanomaterials for use in nanoelectronics. The frontier orbitals (HOMO and LUMO) of the schematic graphene nanodevice shown in Chart 1 would be localized on the Ac part due to the smaller HOMOLUMO energy gap of Ac edge graphenes and PAHs. In the absence of applied external field only small parts of the frontier orbitals’ amplitudes would be localized on the Ph part of the 5



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nanographene. When bias is applied between the electrodes, this localization would be further enhanced in one of the bias directions. In the opposite applied bias direction some of the orbitals would be delocalized through the nanographene junction. The amplitudes of the orbitals close to the Fermi energy at the electrode-nano device termination are crucial for the conductance. Large amplitudes on both sides would determine good electron transport through the junction, whereas localized amplitudes on one side of the device would determine poor conductance.16 Thus, the edge engineering of nanographenes and the interaction of the external electric field with the energy levels would allow for the design an "easy-to-fabricate" carbon-only device with electric current rectifying properties. Theoretical background The electron transport calculations were performed using the non-equilibrium Green's function (NEGF) method.29 For a macroscopic system the conductance (g) is related to the conductor’s intersection (A) and length (L) through the Ohm’s law, as it is shown in eq 1.

g =σ

A L

(1)

In eq 1 σ is denoted the conductivity, which is a material specific property related to the free path of the electrons, e.g., the distance that the electrons travel before they collide with ions from the crystal lattice. However, the devices used in nanoelectronics are often shorter than the electron’s free path. Within such nanoscale device the wave function of an electron residing in the source electrode possesses small amplitude on the drain electrode, which determines a finite probability for tunneling through the conductor. That probability will depend on the electronic structure of the conductor placed between the electrodes, i.e., its energy levels and spatial distribution of the wave function. The conductance through such device does not obey Ohm's law but is described with Landauer’s formula shown in eq 2.

g=

2e 2 T h

(2)

In eq 2 with T is denoted the transmission probability for the electron to propagate through the device, e is denoted the electron charge and, h is denoted the Planck constant. The transmission probability can be calculated using the Green’s function of the semi-infinite


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electrode-conductor-electrode system. The transmission probability is a function of the applied bias between the electrodes (V) and the energy of the electrons within the conductor (E). In eq 3 GA and GR are denoted the advanced and the retarded Green’s functions, respectively. ΓL and ΓR are denoted the broadening functions of the electrodes. (3)

T (E,V ) = trace[Γ L G R Γ RG A ]

The Green's function of the electrode-conductor-electrode system is given with the following equation:

" EI − H L L $ G = $ −VCL $$ 0 #

−VLC

0

EI C − H C

−VCR

−VRC

EI R − H R

−1

% ' ' '' &

(4)

In eq 4 with HC is denoted the Hamiltonian matrix of the conductor, with HL and HR are denoted the Hamiltonian matrices of the left and the right electrodes. VCL and VLC denote the interaction matrices between the left electrode and the channel. VCR and VRC denote the interaction matrices between the right electrode and the channel. IL, IC, and IR denote the unity matrices. The Hamiltonian matrices can be constructed using different level of theories depending on the complexity of the investigated system and the desired accuracy, i.e., Hückel theory (HMO), Hartree-Fock theory (HF), and density functional theory (DFT). Major computational problem of eq 5 is the different dimensions of HC, HL, and HR. While HC is a finite square matrix whose dimension depends on the number of atoms and used basis set, HL, and HR are infinite, due to the semi-infinite left and right electrodes. A series of linear algebra operations transforms eq 4 to eq 5 where all matrices have finite dimensions.

GC (E) = [EI C − H C − ΣL − Σ R ]−1

(5)

In eq 5 ΣL and ΣR are the self-energies of the electrodes, which are given with the following expressions: (6)

ΣL = (−VCL )(EI L − H L )(−VLC ) 7



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(7)

Σ R = (−VCR )(EI R − H R )(−VRC )

The matrices VCL, VLC, VCR, and VRC depend on the overlapping of HC with HL and HR, respectively. VCL, VLC, VCR, and VRC represent the escape rates of an electron from a conductance channel, e.g., the probability of an electron to enter or leave the junction. The broadening functions ΓL and ΓR are derived from the self-energies of the electrodes and are given with the following expressions: ΓL = i[ΣL – ΣL†]

(8)

ΓR = i[ΣR – ΣR†]

(9)

An important consequence of eqs 6-9 is that if one of the matrices VCL, VLC, VCR, and VRC is a zeros matrix, e.g., there is no overlap between the electrode and a conductance channel, then one of the self-energy matrices will be a zeros matrix and one of the broadening functions will be zeros matrix, which will lead to zero transmission probability in eq 3. The steady state current through the device is given with the following equation:

2e +∞ I= ∫ dET (E)[ f (E − µ L ) − f (E − µ R )] h −∞

(10)

where e is denoted the electron charge, f is denoted the Fermi function of the electrodes, µL and µR are denoted the chemical potential at the left and the right electrodes, respectively, and T(E) is denoted the transmission probability as a function of electron energy. When the current is calculated for different applied biases the current/bias (I/V) curve of the device is obtained. Electric current rectifiers show different values for the current for the same bias applied in opposite directions. The external electric field induced between the electrodes plays crucial role for the rectifying properties. Eqs 4-10 are independent from the current direction and would not yield current rectification. However, the induced electric field will affect the molecular orbitals of the device. The external electric field, F, is added as a perturbation matrix,

, to the

Hamiltonian matrix, HC, which yields to effective Hamiltonian matrices in both current directions perturbation matrix

and

, respectively. Depending on the level of theory used, the can be a linear potential along the main diagonal (HMO) or can


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contain off-diagonal components leading to polarization of the orbitals (HF and DFT).29 For an asymmetric device with localized orbitals the field might lead to further localization of the orbitals in one direction and delocalization in the opposite direction. Owing to the different overlap of the effective Hamiltonian matrix of the central region with the electrodes’ Hamiltonians for the different current directions rectifying properties could arise. Strongly localized orbitals will yield in values of the elements of some of the matrices VCL, VLC, VCR, and VRC close to zero, which will determine poor conductance. Well-delocalized orbitals will yield in almost uniform, different from zeros, values of the elements of matrices VCL, VLC, VCR, and VRC, which will determine good conductance. Methods of calculation The electron transport calculations in this study were performed with the NEGF-DFT method implemented in ATK 12.8.49-51 That implementation includes full self-consistent treatment of the electrode-molecule-electrode system. The effect of the external electric field in the central region is implemented as a linear potential and polarization of the orbitals. The calculations were performed with the local density approximation (LDA) and Perdew-Zunger (PZ) functional with double ζ basis set (DZ) for C, H, S and single ζ basis set (SZ) for Au. The results were verified with double ζ basis set with polarization (DZP) for the smaller models. Two types of electrodes were considered in this study: gold electrodes were used for the study of electron transport through small PAH or cyclophane junctions, and graphene electrodes were used to study the electron transport through larger graphene nanoribbons and bi-layer nanographenes. The gold electrodes were modeled by 3x3 elementary unit cells. Three layers of each electrode with Au (111) surface were included in the central region. The graphene nanoribbon electrodes were modeled with 17-atom layers wide and 4 atom layers long Ph edge nanoribbon. The electrode interface is characterized with Ac edge. The nanographene electrodes’ edges and interfaces are considered to be hydrogen passivated. Two layers of each graphene nanoribbon electrode were included in the central region. The geometry and electronic properties of the investigated PAHs and cyclophanes were studied with the second order Møller–Plesset perturbation theory (MP2) implemented in the TURBOMOLE program.52 The def2-SV(P) basis set was used in the calculation. DFT could not be applied for all the structures because it was shown that DFT underestimates the distortion due to the antibonding interaction between full face-to-face π-π stacked aromatic molecules.53,54 The calculations were performed using the RI-MP2 approximation, which

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significantly reduces the computational demands.55 Results and discussion In this study we investigate the electron transport properties of nanographenes with combined Ac-type and Ph-type edge structures. The investigated species are shown in Figure 1. The larger species could be fabricated by precise cutting of graphene nanolayers. Smaller species represent PAH and cyclophane and could be synthesized in bottom-up approach. In Figure 1, 1 is denoted a linear Ac-Ph oligobenzene, 2 is denoted anthracene-phenanthrene cyclophane, 3 is denoted nanographene wafer with Ac and Ph edges, and 4 is denoted a π-π stacked Ac-Ph bi-layer nanographene. The edges of all investigated materials are considered to be hydrogen passivated. That passivation is necessary to ensure the stability of the Ac edge, which could exhibit structural deformations leading to the more stable Ph edge.

Figure 1. Investigated carbon nanomaterials with Ac and Ph edges. 1: Ac-Ph oligobenzene structure; 2: anthracene-phenanthrene cyclophane; 3: Ac-Ph nanographene wafer with Ph edge electrodes; 4: π-π stacked Ac-Ph bi-layer nanographene wafer with Ph edge electrodes. For sake of simplicity, the double bonds in 3 and 4 are omitted. Structure 1 consists of naphthacene and chrysene subunits. The energy spectrum of


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each separate subunit was calculated. The HOMO-LUMO gap of naphthacene is 2.78 eV and the HOMO-LUMO gap of chrysene is 4.42 eV. The energy gap is significantly narrower for the naphthacene, which is expected to lead to localization of the frontier orbitals of 1 at the naphthacene subunit. Such localized orbitals are promising for rectifying properties. The frontier orbitals of 1 are shown in Figure 2 A. The expected localization of the frontier orbitals is observed with only minimal orbital amplitudes on the chrysene subunit. Structure 2 is a cyclophane molecule, for which one of the aromatic systems is anthracene and the other is phenanthrene. The anthracene is covalently linked to the phenanthrene at 1-st and 4-th positions. The phenanthrene is linked covalently to the anthracene at 1-st and 4-th position. That linking results in the full face-to-face overlapping of one benzene ring from the anthracene with one benzene ring from the phenanthrene. Such full face-to-face overlapping of aromatic molecules leads to antibonding π-π interaction. That antibonding interaction is avoided by symmetry-breaking distortions, i.e., slipping or rotation.

Figure 2. Optimized structures and spatial distribution of frontier orbitals. A: Geometry and frontier orbitals of 1; B: Geometry and frontier orbitals of 2. The relaxed geometry of 2 is shown in Figure 2 B. The interlayer distance is calculated at 3.3 Å, which is comparable to the interlayer distance of multilayer graphene materials, 3.35 Å.56 Due to steric hindrance the planar structure of the phenanthrene is 11



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slightly distorted. The antibonding π-π stacking interaction is avoided with small angle rotations. The estimated angle is 6° which is comparable with the experimentally estimated distortion of cyclophanes of 5°.57 The HOMO-LUMO gap of anthracene is 3.60 eV and the HOMO-LUMO gap of phenanthrene is 4.74 eV. The energy gap is significantly narrower for anthracene, which is expected to lead to localization of the frontier orbitals of 2 at the anthracene layer. The frontier orbitals of 2 are shown in Figure 2 B with large amplitudes on the anthracene layer and, only for HOMO, small amplitude on the phenanthrene layer. The electron transport properties of structures 1 and 2 were calculated with the formalism described in eqs 3-10 and implemented in ATK 12.8. The anchoring to the electrodes is achieved through thiol groups. Several studies have reported that on the gold electrode surface the sulfur-hydrogen bond is dissociated and the sulfur atom is bound covalently with the gold atoms at a three-fold hollow position.58,59 Beside the hollow site adsorption of the sulfur atom, bridge site over two gold atoms, and on top adsorption over a single gold atom are also possible. The adsorption geometry was investigated based on the orbital interaction and on the Au(111) surface the site preference was found to be hollow > bridge > on top.60 STM break junction experiments reveal that in the realistic experimental conditions all three adsorption sites might exist and the adsorption site affects the electron transport properties.61 The fluctuation of the transport properties could be avoided when the thiol anchoring groups were replaced by amine anchoring groups.62 The distance between the gold atoms and the sulfur atom is set at 2.4 Å.60 The connecting sites at the carbon nanostructures were selected according to a recently proposed orbital symmetry rule for electron transport. Symmetry allowed sites were selected for which the orbital symmetry rule predicts large transmission probabilities.53,63-66 Two possible applied bias directions are defined: positive bias that corresponds to positive electrode-Ac subunit-Ph subunit-negative electrode and negative bias that corresponds to negative electrode-Ac subunit-Ph subunitpositive electrode. The electrode-molecule-electrode junction of 1 is shown in Figure 3 A. The electron transport calculations are performed for applied biases in the range 2.0 V to -2.0 V with step of 0.4 V. The results for the transmission probability of 1 as a function of the electron energy and applied bias are shown in Figure 3 B. In the zero bias spectrum the closest peaks to the Fermi level correspond to HOMO and LUMO of the insolated molecule 1, e.g., HOMO and LUMO of 1 without the electrodes. With the increase of the applied bias in positive direction to 1.6 V the peak above the Fermi level (LUMO) enters in the bias window. That would lead


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to resonance tunneling and increased current through the junction. With the increase of the bias in negative direction to -2.0 V the peak below the Fermi level (HOMO) enters in the bias window that would lead to resonance tunneling and increased current through the junction. However, for applied bias of -1.6 V HOMO is still outside of the bias window and its intensity is significantly reduced, which might lead to rectifying properties. The I/V curve of 1 is shown in Figure 3 C. In the low-bias regime between -1 V and 1 V symmetric I/V curve is obtained. For applied bias of 1.6 V the obtained value of the current is 1534 nA while for applied bias in opposite direction, -1.6 V, the obtained value for the current is -605 nA. The calculated rectifying ratio is 2.5.

Figure 3. Electron transport properties of 1. A: Geometry of the junction; B: Transmission spectra; C: I/V curve. The electron energies are aligned to the Fermi level (EF = 0.0 eV).

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In order to understand the mechanism of the current rectification and interaction with the external electric field we perform analysis of the molecular projected self consistent Hamiltonian (MPSH) states of 1. The MPSH states are the eigenstates of HC (eq 4). The matrix HC does not include directly the self energies of the electrodes (eq 6 and eq 7), however, in the course of the SCF procedure the matrix HC is modified by the electrode environment. It is worth noting that the energy levels of the MPSH states do not coincide precisely with the peaks in the transmission spectrum because the self energies of the electrodes are included in the transmission calculations (eqs 3, 5, 8, 9).

Figure 4. Molecular projected self consistent Hamiltonian states of 1 and external potential for applied biases of 1.6 V and -1.6 V. The electron energies are aligned to the Fermi level (EF = 0.0 eV). In Figure 4 are plotted the MPSH states of 1 corresponding to HOMO and LUMO for applied biases of 1.6 V and -1.6 V. In the background of the MPSH states is plotted the external potential, i.e., the external electric field between the electrodes. Blue color on the external potential plot corresponds to the electrode with chemical potential below the Fermi


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level, e.g., positive electrode, and red color corresponds to the electrode with chemical potential above the Fermi level, e.g., negative electrode. HOMO tends to localize in the vicinity of the negative electrode while the LUMO localizes in the vicinity of the positive electrode.17 For the insulated molecule 1, e.g., 1 without the electrodes, both orbitals are localized on the naphthacene subunit. When positive bias is applied, e.g., the negative electrode is bound to the chrysene subunit, HOMO delocalizes over the entire molecule, while LUMO remains localized on the naphthacene subunit. When negative bias is applied, e.g., the negative electrode is bound to the naphthacene subunit, HOMO is localized on the naphthacene subunit, while LUMO delocalizes through the junction. From the analysis of the transmission spectra and the MPSH states can be concluded that the current rectification for 1.6 V / -1.6 V applied biases occurs due to resonance tunneling through LUMO for 1.6 V bias which is not observed in the case of -1.6 V bias. The second reason is the contribution to the current from the delocalized HOMO for 1.6 V bias which is not observed in the case of -1.6 V bias due to localization of the orbital. The π-electron system of 2 differs significantly from that of 1. While the πconjugation of 1 consists of in-plane π-type overlapped carbon 2pz atomic orbitals (AOs), 2 consists of two separate planes with π-type overlapped carbon 2pz AOs. The both plains are linked by σ-type overlapped carbon 2pz AOs on each adjutant PAH. Analysis of the overlap integrals shows that in cyclophanes the intraplanar π-type overlap integrals are one order of magnitude larger than the interplanar σ-type overlap integrals.16,53 That leads to a conjugated π-electron system that spreads throughout the nanostructure, which is important for the ballistic electron transport. Owing to the weak interplanar σ-type overlap of the carbon 2pz AOs, the frontier orbitals would be better localized on the anthracene part and significantly stronger external electric fields would be necessary to delocalize them throughout the nanostructure. In the reverse bias direction even small field would be able to localize the frontier orbitals. That should lead to improved current rectifying properties and higher rectifying ratio in the high bias regime. Such increased rectification was already estimated for boron, nitrogen doped cyclophanes compared to the boron, nitrogen doped nanographenes.16

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Figure 5. Electron transport properties of 2. A: Geometry of the junction; B: Transmission spectra; C: I/V curve. The electron energies are aligned to the Fermi level (EF = 0.0 eV). The junction geometry of 2 is shown in Figure 5 A. The electron transport calculations are performed for applied biases in the range 2.0 V to -2.0 V with step of 0.4 V. The results for the transmission probability of 2 as a function of electron energy and applied bias are shown in Figure 5 B. In the zero bias spectrum the closest peaks to the Fermi energy correspond to HOMO and LUMO. Due to the fragmentation in the π-electron system the intensities of those peaks are very low. Careful analysis of the transmission spectra shows that with increase of the applied bias in positive direction the intensity of the peak below the Fermi level (HOMO) increases while with increase of the applied bias in negative direction its intensity is close to zero. The peak above the Fermi level (LUMO) increases its intensity as the applied bias in the negative direction increases. For applied bias of 2.0 V LUMO is


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within the bias window while HOMO is close to the bias window and contributes to the transmission probability. For applied bias of -2.0 V HOMO is within the bias window but its intensity is close to zero while LUMO is outside the bias window with energy of 1.6 eV. That leads to strongly asymmetric I/V curve shown in Figure 5 C and calculated rectifying ration of 8. In the low-bias regime between -1 V and 1 V symmetric I/V curve is obtained with very low calculated values for the current. For the positive bias direction the calculated values for the current increase significantly for 2.0 V applied bias. For the negative bias direction and applied bias of -2.0 V the current values remain similar to those at the low-bias regime. MPSH state analysis was performed to elucidate the rectification mechanism.

Figure 6. Molecular projected self consistent Hamiltonian states of 2 and external potential for applied biases of 2.0 V and -2.0 V. The electron energies are aligned to the Fermi level (EF = 0.0 eV). In Figure 6 are plotted the MPSH states of 2 corresponding to HOMO and LUMO for applied biases of 2.0 V and -2.0 V. In the background of the MPSH states is plotted the external potential, i.e., the external electric field between the electrodes. Blue color on the external potential plot corresponds to the electrode with chemical potential below the Fermi level, e.g., positive electrode, and red color corresponds to the electrode with chemical 17



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potential above the Fermi level, e.g., negative electrode. For applied bias of 2.0 V the main part of the HOMO amplitude can be found at the phenanthrene fragment, however, significant amplitude is also delocalized over the anthracene fragment. Such spatial delocalization leads to larger peak in the transmission spectrum. LUMO is fully localized at the anthracene fragment which leads to small intensity of its peak in the transmission spectrum. Both HOMO and LUMO contribute to the calculated value of the current of 217 nA. In case of applied bias of -2.0 V HOMO is fully localized on the anthracene fragment, with only small amplitude in the overlap region between the anthracene and phenanthrene fragments. Such strong localization leads to close to zero intensity of its peak in the transmission spectrum. LUMO is mainly localized on the phenanthrene fragment, however, small amplitude can be found on the anthracene fragment. This partial delocalization leads to peak with larger intensity in the transmission spectrum, however, due to its high energy (1.45 eV) it does not contribute to the calculated value of the current. The only contribution to the computed value of the current for applied bias of -2.0 V comes from HOMO. However, due to its small peak in the transmission spectrum the value of the current for -2.0 V applied bias is -27 nA. Analysis on the transmission spectra shown in Figure 5 B shows that the larger current for 2.0 V applied bias is calculated due to the effect of HOMO that is just outside the bias window, however, the right shoulder of its peak affects the transmission coefficients within the bias window and gives rise to larger current. For negative applied bias LUMO is far above the bias window and does not affect the transmission coefficients within the bias window resulting in small values for the current. With these results we demonstrate that it is possible to design a high-rectifying ratio nanoscale rectifier without the necessity of precise donor and acceptor doping. The necessary conditions are: π-conjugated system throughout the nanostructure; spatially localized frontier orbitals in zero bias case; effective fragmentation of the π-electron system. The external electric field induced between the electrodes could alter the spatial distribution and shift the energy levels of the orbitals within the scattering region, which was found to be the reason for the diode-like properties. The results for the electron transport through 1 and 2 have demonstrated that the edge type of small PAHs and cyclophanes could determine the direction of the electric current flow and could lead to the design of molecular rectifiers. It is important to apply the edge-controlled molecular rectifier concept to the mono- and bi-


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layered nanographenes 3 and 4 shown in Figure 1.

Figure 7. Electron transport properties of 3. A: Geometry of the junction; B: Transmission spectra; C: I/V curve. The electron energies are aligned to the Fermi level (EF = 0.0 eV). The graphene wafer 3 is a single layer nanomaterial. The electrodes are modeled by wider nanoribbons with Ph edge type characterized with narrower band gap and better conductivity compared to the nanoribbons within the scattering region. The scattering region is modeled by a narrower graphene nanoribbon with sequential Ac and Ph edges. It is worth noting that monolayer graphene nanomaterials are characterized with high charge-carrier 19



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mobility and good π-electron delocalization. The results of the performed electron transport calculations are shown in Figure 7. In Figure 7 A is shown the computational model. The electron transport calculations were performed for applied biases in the range of 2.0 V to -2.0 V with step of 0.5 V. In Figure 7 B are plotted the transmission spectra as a function of the applied bias and electron energy. In Figure 7 C is plotted the I/V curve of the system 3. The I/V curve is nearly symmetric showing very weak rectifying properties. The analysis of the transmission spectra shows that the applied external electric field is not sufficient to result in significant localization of the electron levels in one direction of the current and delocalization in the opposite direction. The results are consistent with the recent study of boron-nitrogen doped polycyclic aromatic hydrocarbons,16 which shows small rectifying properties for monolayer nanographenes for high applied biases. Beside the high charge-carrier mobility, another reason for the limited rectifying properties is the magnitude of the induced external electric field. The field (F) is given with eq 11: F = V/L

(11)

where V denotes the applied bias and L denotes the left electrode - right electrode distance. Due to the longer electrode-electrode distance in system 3 compared to system 1, rectifying properties would be observed for significantly higher applied biases. Graphene wafer 4, shown in Figure 1, is a bi-layer nanomaterial. The electrodes are modeled by wider nanoribbons with Ph edge characterized with narrow band gap and high conductivity. The both layers are separated by 3.35 Å distance which is typical for graphite and few layers nanographene materials. The central region is modeled by narrow graphene nanoribbons characterized with wider band gap. The first layer possesses Ac edge while the second layer possesses Ph edge. The both layers overlap to assure π−π stacking interaction between the Ac nanoribbon and the Ph nanoribbon.


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Figure 8. Electron transport properties of 4. A: Geometry of the junction; B: Transmission spectra; C: I/V curve. The electron energies are aligned to the Fermi level (EF = 0.0 eV). The geometry of the nanographene junction is shown in Figure 8 A. The electron transport calculations are performed for applied biases in the range of 2.0 V to -2.0 V with step of 0.5 V. The transmission spectra as a function of the applied bias and electron energy are plotted in Figure 8 B. The transmission spectra differ significantly from those of monolayer nanographenes plotted in Figure 7 B and show that the external electric field induced between the electrodes is sufficient to localize the molecular orbitals in one direction and delocalize them in the opposite direction. In Figure 8 C is plotted the I/V curve which is asymmetric with rectifying ratio of 17 for the low bias regime between -1.0 V and 1.0 V. In 21



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the high bias regime between 2.0 V and -2.0 V rectifying ratio of 7 was calculated. The calculated current for 1.0 V applied bias is 10 nA and the calculated current for -1.0 V applied bias is -167 nA. The calculated current for 2.0 V applied bias is 479 nA and the calculated current for -2.0 V applied bias is 3326 nA. The obtained results from the electron transport calculations are in agreement with the previously obtained results for electron transport through bi-layer nanographenes with boron, nitrogen doping.16 The separation of the π-electron systems with weak σ−type interlayer overlap of the 2pz AOs is crucial for the electric current rectifying properties. The other necessary condition is the spatial localization of the frontier orbitals close to one of the electrodes. The proposed rectifying model relies only on the different electronic properties of nanographene wafers with different edges. Conclusions In this study we have investigated the electric current rectifying properties of polycyclic aromatic hydrocarbons, cyclophanes, mono- and bi-layer nanographenes with different edge structure. Our calculations show that the current rectification occurs as a result of interaction of the external electric field induced between the electrodes and the electronic states within the junction. The external field applied in one of the directions localizes the orbitals of asymmetric molecules and nanographenes, while when applied in the opposite direction it delocalizes them throughout the entire system. From the non-equilibrium Green's function theory follows that localized orbitals with small amplitudes at one of the electrodes would not contribute significantly to the transmission probability. Well delocalized orbitals with large amplitudes at both electrodes lead to large peaks in the transmission spectrum. Thus, asymmetric junctions can show significant rectifying properties. The concept was applied to mono- and bi-layered nanographene ribbons and their finite models polycyclic aromatic hydrocarbons and cyclophanes. Two types of electrodes were considered: gold electrodes through sulfur covalent bonds and wider graphene nanoribbons. The asymmetry within the junctions was achieved by using the well-known rule for band-gap width of nanographenes and PAHs with Ac and Ph edges. The calculations show that while rectifying properties were observed for both mono- and bi-layered carbon materials, the rectifying ratios of the bi-layered nanographenes and cyclophanes were significantly larger compared to the rectifying ratios of monolayer nanographenes and PAHs. The reason for that is the effective barrier introduced by the interlayer π−π stacking which helps for the improved localization of the electronic states.


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Acknowledgements This work was supported by World Premier International Research Center Initiative (WPI), Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), Japan.

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Table of content (TOC) graphics

Captions: Polycyclic aromatic hydrocarbons and graphene nanoribbons with spatially asymmetric armchair and zigzag edges show diode-like properties. The electric current rectification arises as a result of electronic states’ perturbation by the electric field induced between the electrodes. The asymmetric electronic properties are determined by the edge topology.

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Current Rectification in Mono- and Bilayer Nanographenes with

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