Control of electronic transport in nanohole defective zigzag graphene nanoribbon by means of side alkene chain

Abstract

Using the nonequilibrium Green function formalism combined with density functional theory, we studied the electronic transport properties of nanohole defective zigzag graphene nanoribbon (ZGNR) junctions. A side alkene chain is connected to the edge of the defective ZGNR in the scattering region. We find that the transport properties of the defective ZGNR junction are strongly dependent on the parity of the number of carbon atoms in the side alkene chain. The side chain can switch on (even) and off (odd) the transport channel of our proposed junction. It is found that the transmissions for the side chains with an even number of carbon atoms are around 2G₀, but they are around 1G₀ for the side chains with an odd number of carbon atoms. The origin of this peculiar behavior is analyzed as due to the electronic states at the edge of the defective ZGNR which are modulated by the side-chain length. Our theoretical study shows that it is feasible to control the conduction of ZGNR by changing the side-chain length via external modulations such as chemical methods, which may stimulate experimental investigations in the future.

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Introduction

Electron transport through a graphene nanoribbon (GNR) junction has recently received considerable experimental^1–3 and theoretical^4–7 attention. Different from bulk graphene, GNRs have energy gaps, which make them suitable for the realization of electronic devices.⁸ Until now, many electronic devices have been realized in the GNR nanostructures, such as field-effect transistors (FET),^9–11 switches,^12,13 gas molecular sensors,^14,15 spintronic devices^16,17 and rectifiers.^18,19 According to the geometry of the edge, GNRs can be divided into two kinds: armchair GNR (AGNR) and zigzag GNR (ZGNR). In the case that the spin effect is disregarded, previous studies show that ZGNRs are expected to be metallic regardless of their width whereas AGNRs can be metallic or semiconducting depending on their width.²⁰ Among them, ZGNRs have attracted much more attention due to the unique edge states and edge magnetism. Owing to the potential applications of ZGNRs in nanosized spintronic devices, a number of researches have been done, and the zigzag-edge states are shown to be modified by doping atoms, side groups, and defects at the edges.^21,22

Generally, defects and side groups in GNRs can affect the magnetic and electrical properties in unexpected ways, which are commonly used to modulate artificially the electronic structures of GNRs. Punching nanoholes on graphene could increase the band gap of graphene and make it change from the semimetal to semiconductor.^23–25 Moreover, theoretical studies showed that the molecular electronic transport properties of a ZGNR are strongly dependent on the edge states,²⁶ introducing side groups is an effective way for modulating edge structure ZGNRs, providing us a method to control the transport properties of ZGNR devices. Side chains modulation can bring many novel properties such as the Fano effect^27,28 and the standing wave effect.^29,30 Moreover, the electronic transport properties can also be controlled through chemical conformational modification of side chains to aromatic molecules.^31–33

But up to now, as we know, there is no literature to report the control of electronic transport properties by means of side-chain length in defective graphene nanoribbons. In fact, the GNRs are important nanostructures for developing nano-devices in the future, and the nanohole defects and side chain are effective methods for modulating the electronic structures of ZGNRs. Thus, it is very necessary to study the electronic transport behaviors in nanohole-defected ZGNRs with side chains. In this paper, we design a nanohole in a ZGNR and study the spin-dependent transport properties of the defective ZGNR device with different lengths of side alkene chain. The goal of this work is to improve the existing understanding of a mechanism to control the current in GNRs-based molecular electronic devices.

Models and methods

We study the spin-dependent transport properties of a nanohole defective ZGNR with a side alkene chain. As shown in Fig. 1, six carbon atoms are removed from the scattering region in the zigzag graphene nanoribbon, and the scattering region with a nanohole in the center forms an 18-membered carbon ring, which follows the molecular structure of an annulene. Previous studies have shown that the geometric structure of annulene is stable and it has good electrical conductivity,³⁴ so it is interesting for us to study the transport properties. A side alkene chain (CH)_nH is linked to the ZGNR, where n is an integer number (n = 0, 1, 2…). In our numerical simulation, six systems with n = 0, 1…5 are considered, for example, when n = 2, the side chain is C₂H₃ (M2, n = 2). In this work, all calculations are performed by the Atomistix ToolKit (ATK) package,^35–37 in which the nonequilibrium Green’s function formalism (NEGF) and density functional theory (DFT) are implemented. A cutoff energy of 150 Ry and a Monkhorst–Pack k-mesh of 1 × 1 × 100 with a convergence criterion of 10⁻⁶ eV are chosen to achieve the balance between calculation efficiency and accuracy. We have checked total energy with different cutoff energies for our systems, and we found that 150 Ry is a suitable value for our numerical calculation. That is why 150 Ry has been employed in many previous studies.^38,39 Moreover, a double zeta plus polarized (DZP) basis set is adopted for electron wave function. The NEGF-DFT self-consistency is controlled by a numerical tolerance of 4 × 10⁻⁵ eV. And also, the double contour integral has been used to integrate the Green’s function in the real axis, the convergence is 10⁻⁵ eV, and the point density 0.002 eV has been used in the integral. The vacuum layers between two ribbons along the x and y directions (defined in Fig. 1) are 15 Å to avoid the artificial Coulomb interactions between the contents in two neighboring cells. To obtain accurate results, all the systems after functionalization with a side chain have had geometry optimization performed until the force on each atom is less than 0.05 eV Å⁻¹. The spin-polarized current through the system is calculated using the Landauer–Büttiker formula,⁴⁰where e is the electron charge, h is Planck’s constant, and T_σ is the transmission of an electron with spin σ. f_L(R)(E, V_b) is the Fermi–Dirac distribution function of the left (right) electrode, and the difference in the chemical potentials between the left and right electrodes is µ_L(V_b) − µ_R(V_b) = eV_b, where V_b denotes the external bias voltage. T_σ can be obtained from the equation,where G^r (G^a) is the retarded (advanced) Green’s function matrix, and is the retarded self-energy matrix for the left (right) electrode.

Fig. 1 Schematic illustration of the two-probe ZGNR system: LE, RE, and CR denote the left electrode, the right electrode, and the central scattering region, respectively. A side chain (CH)_nH is connected to the central edge of the defective ZGNR, and the edge carbon atoms are saturated with hydrogen atoms.

Results and discussions

Fig. 2 presents the transmission spectra of nanohole defective ZGNRs for side chains with different numbers of carbon atoms (n = 0, 1…5) under zero bias. Generally, there are two different spin configurations for the ZGNR, parallel (P) and antiparallel (AP) spin configurations. The spin-up and spin-down electron in the transmission spectra of ZGNR are split when the electrodes are in P spin configuration, while they are degenerate when the electrodes are in AP spin configuration.^41,42 Since the behavior of the spin splitting on ZGNR may have more interesting potential applications in spintronics devices, the P spin configuration of the ZGNR has been chosen in our calculations. It is noted that the transmission coefficients at the Fermi level (FL) still have high values even if ZGNR is punched. We can find that the transmission peaks near FL for the side chains with an even number of carbon atoms are around 2G₀ (G₀ = e²/h is the quantum conductance), however, they are around 1G₀ for the side chains with an odd number of carbon atoms, which means the conduction of nanohole defective ZGNRs can be modulated by the parity of their side alkene chains. For the structure under consideration, the atoms on the two edges provide the conduction channels in the scattering region, so the side chains connected to the edge atoms have evident effects on the transport properties.

Fig. 2 The spin-dependent transmission spectra T(E, V_b) for all systems under the bias of zero. The Fermi level is set to zero. SU (SD) is the transmission spectrum for spin-up (spin-down) electrons.

To understand the different effects of side chains with an odd and even number of carbon atoms, we present the electron transmission pathways at the Fermi level (0.0 eV) under zero bias for the spin up and spin down states of M0, M1, M2, M3, M4 and M5 systems. As shown in Fig. 3(a–l), the volume of each arrow indicates the magnitude of the local transmission between each pair of atoms, and the arrow and the color designate the direction of the electron flow. The transmission pathways T_ij can show us the local bond contributions to the transmission coefficient, for example, the total transmission coefficient between two parts A and B can be expressed as . We can clearly find that the transmission pathways of our proposed systems are strongly dependent on the parity of the number of carbon atoms for the side chain. For the case of a side chain with an even number of carbon atoms, such as n = 0, 2, 4, as shown in Fig. 3(a, b, e, f, i and j), there are two transmission pathways at the two edges of the nanoribbon, and both the spin-up and spin-down electrons can move from the left electrode to the right electrode, and they both have two conduction channels. Nevertheless, for the odd case, such as n = 1, 3, 5, as shown in Fig. 3(c, d, g, h, k and l), we can find that electrons can’t pass through the edge connected to the side chain, and there is only one conduction channel on the other edge for each system.

Fig. 3 The electron transmission pathways at the Fermi level (0.0 eV) under zero bias. (a–l) Refer to the spin up and spin down states of M0, M1, M2, M3, M4 and M5 systems, respectively.

As we know, the charge can move in the ZGNR system because of the channels provided by the molecular orbital. The main channels for charge transport are determined by the frontier molecular orbitals near the Fermi energy, which are called the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital). To further understand the transmission spectra presented in Fig. 2, we plot the frontier molecular orbitals of our proposed structure in Fig. 4. Since the spin-dependent transport properties of our proposed systems are strongly dependent on the parity of the number of carbon atoms in the side chain, we only present the numerical results for three ZGNR devices with n = 0, 4, 5 due to the space limitation. Considering the fact that the HOMO is closer to the FL than the LUMO and mainly contributes to the electronic transport for each device, we only show the spatial distribution of the contour map for the HOMO. And also the three devices with the side chains of n = 0, 4, 5 have been selected here. It is noted in Fig. 4(a, c and e) that the spin-down HOMO states for side chains with different numbers of carbon atoms M0, M4, M5 are all localized. For the spin-up HOMO states, they are all delocalized, however, it is easy for us to note that there are obvious differences between structures with an even number chain and odd number chain. The HOMO states for side chains with an even number of carbon atoms are delocalized on the two edges of the ZGNR, but the HOMO states for side chains with an odd number of carbon atoms are only delocalized on one edge. Since the delocalized states contribute to the transmission of the carriers we have the differences in transmission for the structures with an even number side chain and odd number side chain.

Fig. 4 The frontier molecular orbitals of HOMO eigenstates at 0.0 eV under zero bias. (a–f) The spin up and spin down states of M0, M4 and M5 systems, respectively. The isovalue is 0.05 a.u. Red and blue are used to indicate the positive and negative signs of the wavefunctions, respectively.

Moreover, as shown in Fig. 5, we also study the local density of states (LDOS) at the Fermi level under zero bias for M0, M4, and M5. The previous studies have shown that the transport properties depend on the edge states and symmetry of ZGNRs.^37,38 It is noted in Fig. 5(a and b) that transport properties of the defective ZGNR are determined by the two edge states. For the chain with an even number of carbon atoms M4, as shown in Fig. 5(c and d), the LDOS are similar to M0, we have the LDOS localized over the two edges of the defective ZGNR. However, it is different for the structures connected to a chain with an odd number of carbon atoms, and we can see from Fig. 5(e and f) that the LDOS are zero around the site of the upper edge that links to the side chains. As we know, the delocalized edge states play a significant role in the electronic transport properties in defective ZGNRs, but the localized states give very little contribution to electronic transport, which corresponds to the suppressed transmission coefficient in Fig. 2(b, d, and e). The defective ZGNRs linked to even side chains have two transport channels, and we can see from Fig. 5(c and d) that both the channels of the upside edge and downside edge are opened. But for the defective ZGNRs linked to odd chains, the channel of the upside edge is closed. The analysis agrees well with the results presented in Fig. 2.

Fig. 5 The spin-dependent LDOS at the Fermi level under zero bias. (a–f) The distributions of the LDOS plot at the isovalue of 0.1e for the spin up and spin down states of M0, M4 and M5 systems, respectively.

In order to understand the physical mechanism of the coupling between the side alkene chain and the zigzag edge states, as shown in Fig. 6, we also present the spin dependent transmission spectra under zero bias for the side alkene chain doped non-defective ZGNR systems. Three perfect ZGNRs with side chains with the number of carbons n = 0, 4, 5 have been chosen, which are named as Z0, Z4 and Z5 for short. Comparing with the results of the defective ZGNRs systems M0, M4 and M5 in Fig. 2(a, e and f), it is clearly seen that the transmission spectra in Fig. 6 are strongly correlated to the non-defective and defective ZGNRs, especially at the location of their peaks. The difference is that there is an initial conductance platform about 1.0G₀ for the non-defective systems, while that is about zero for the defective systems. Thus, in the defective systems, we can see that the transmission channels are almost contributed from the transmission peeks near the Fermi level, and transport behaviors of the defective ZGNR devices would be almost modulated by the doped side chain. Moreover, we can find the zero values of transmission coefficients at some higher energy points in Fig. 6(b and c), which is caused by the resonant backscattering states formed in the side chains.^6,21

Fig. 6 The spin-dependent transmission spectra of the non-defective 4ZGNR under zero bias. (a–c) Z0, Z4 and Z5 refer to the systems with different number of carbons in side chains as n = 0, 4, 5, respectively. The insets indicate the corresponding spin-dependent LDOS at the Fermi level under zero bias, and the isovalue is 0.1e.

Furthermore, we also plot the LDOS at the Fermi level as insets in Fig. 6. We can clearly see that the distributions of the LDOS are delocalized in the whole scattering region for each system. Only very little edge states around the side chain are suppressed in Z5 rather than Z0 and Z4, so it is impossible to observe the dependence of transport properties on the parity of the number of carbon atoms in the side alkene chain in side alkene chain doped non-defective ZGNR systems. Thus, the side chains could play a more prominent role in the transport properties of the nanohole defective ZGNRs than that of the perfect ones.

In addition, for a further insight into the spin-dependent transport properties of the nanohole defective ZGNRs devices, we present the current–voltage (I–V) curves of all systems in Fig. 7. The I–V curves for the systems with even side chains and odd side chains are shown in Fig. 7(a and b), respectively. We know from Fig. 6 that the currents of all systems in the area of low bias increase quickly with the increase of the bias, which shows the conductive properties of metals. And the current of systems with even chains is bigger than those with odd chains, and which can be explained by the transmission shown in Fig. 2. This is because the transmissions for the side chains with an even number of carbon atoms are larger than the transmissions for the side chains with an odd number of carbon atoms. With the increase in bias, comparing Fig. 7(a) with Fig. 7(b), it is found that the currents of odd chains and even chains present completely different changing laws. In the region of bias 0.2 to 1.0 V, the currents for side chains with an even number of carbon atoms almost remain the same. The currents for side chains with an odd number of carbon atoms are completely different, and it is noted that the relation between currents and the gate voltages agrees well with Ohm’s law. Furthermore, we also can find the currents of spin-up and spin-down are split with the increase in the bias for each system, which means that the systems we proposed can appear with spin undegenerated transport behaviors. And the nanohole defective ZGNR devices would have potential applications in the field of spintronics. More interestingly, as shown in the I–V curves of M0, when V_b > 0.4 V for the spin-up state, and V_b > 0.5 V for the spin-down state, the spin-up and spin-down currents are decreased by the increasing of the bias, and the obvious negative differential resistance (NDR) behaviors can also be observed for both the spin-up and spin-down states of M0. From the above results, we can know that different transport behaviors of the nanohole defective ZGNRs originate from different electronic states at the edge of the defective ZGNR which can be modulated by the side alkene chains.

Fig. 7 The current as function of applied bias voltage for the systems with different side alkene chains: (a) all even and (b) all odd.

As the current is determined by the values of T(E, V_b) in the bias window (eqn (1)), to further understand the spin-dependent transport behaviors, taking M0, M4 and M5 as examples, the spin-dependent transmission spectra as a function of the electron energy and bias are plotted in Fig. 8. At low bias, the large transmission peaks can be observed around the Fermi level and is broadened with the increase of bias in each panel, which leads to a quick increase in the current, and the Ohmic I–V curves can be found for each system under the lower bias. With the increase of bias, it is noted that the values of transmission are different between the spin-up and spin-down electrons for each system, so the spin undegenerated transport behavior can be observed in the I–V curves in Fig. 7. Especially, it is clearly seen that the transmission spectra of spin-up in Fig. 8(d) are significantly larger than that in Fig. 8(c) in a wide bias region, which results in the obvious spin-filter effects for M4. For M4 and M5, we find that the transmission in the bias windows increased steadily, which results in a continuous growth of current in Fig. 7. However, for M0, we can clearly find that both the transmissions of spin-up and spin-down states in the bias widow are reduced with the increase of bias, which results in the NDR behaviors in the I–V curves in Fig. 7.

Fig. 8 Calculated transmission spectra as a function of electron energy E and bias for spin-down and spin-up states of M0, M4 and M5. The region between the white solid lines refers to the bias window.

Conclusion

In summary, using non-equilibrium Green’s functions combined with density functional theory, we have investigated the electronic transport properties of nanohole defective ZGNRs linked to side alkene chains. Different numbers (odd or even) of carbon atoms for the side chains linked to defective ZGNR junctions were considered. The calculated results show that the side alkene chains can significantly change the electronic transport behaviors of the defective ZGNRs. We find that the defective ZGNRs linked to side chains with an even number of carbon atoms are semiconductor, while metallic character can be observed for those with an odd number of carbon atoms. Our theoretical analysis shows that the substituted-odd side chains can change the distribution of the electronic states at the edges of the nanohole defective ZGNRs and results in the peculiar transport characteristics. We find that the odd side chains generally break down the edge states along the same edge, which carries less current in the junction in the area. The transmission for a side chain with an odd number of carbon atoms is half of the transmission for a side chain with an even number of carbon atoms around the Fermi level. As a consequence electron transport through the molecule can be controlled either by chemically modifying the side group, or by changing the conformation of the side group. Moreover, the interesting NDR behaviors also can been observed on the proposed nanohole defective ZGNR junction. These results suggest that the edge modified ways make the graphene-based nanomaterials present more abundant electronic transport phenomena and can be useful for the design of future nanoelectronic devices.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos 21103232, 61306149, and 11334014), the Natural Science Foundation of Hunan Province (no. 14JJ3026), Postdoctor Foundation of China (nos 2013M542130 and 2014M552145), the Hunan Postdoctoral Scientific Program (2013RS4048), the Fundamental Research Funds for the Central Universities of Central South University (2013zzts158) and Hunan Key Laboratory for Super-microstructure and Ultrafast Process.

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