First principles design of divacancy defected graphene nanoribbon based rectifying and negative differential resistance device

We have elaborately studied the electronic structure of 555-777 divacancy (DV) defected armchair edged graphene nanoribbon (AGNR) and transport properties of AGNR based two-terminal device constructed with one defected electrode and one N doped electrode, by using density functional theory and non-equilibrium Green's function based approach. The introduction of 555-777 DV defect into AGNRs, results in a shifting of the {\pi} and {\pi}* bands towards the higher energy value which indicates a shifting of the Fermi level towards the lower energy. Formation of a potential barrier, very similar to that of conventional p-n junction, has been observed across the junction of defected and N doped AGNR. The prominent asymmetric feature of the current in the positive and negative bias indicates the diode like property of the device with high rectifying efficiency within wide range of bias voltages. The device also shows robust negative differential resistance (NDR) with very high peak-to-valley ratio. The analysis of the shifting of the energy states of the electrodes and the modification of the transmission function with applied bias provides an insight into the nonlinearity and asymmetry observed in the I-V characteristics. Variation of the transport properties on the width of the ribbon has also been discussed.


Introduction
The remarkable technological progress and reduction in the size of electronic devices over the last two decades have driven the scientific community and industries to find the significance of nanomaterials in the electronic devices such as rectifiers, 1,2 switching devices, 3,4 field effect devices, 5,6 spin-filters, [7][8][9] negative differential resistance (NDR) based devices, [10][11][12] etc. Particularly at present rapid developments of NDR and rectifying devices are going on, which include nanoribbon, 12 nanowire, 13 quantum dot, 14 nanotube, 1, 11 molecular junction 15 etc. Also the search of suitable nanostructures having tunable topology with versatile electronic properties is going on to look beyond Si based devices. In the recent past, graphene has attracted a lot of attention, due to its unique physical properties, high mobility, low power consumption and above all the ease to synthesize. [16][17][18][19][20] However the zero band gap semi-metallic nature of graphene is not suitable for device application. Tailoring the width of graphene sheet less than 10 nm results in one-dimensional (1D) graphene nanoribbon (GNR) that exhibits finite band-gap. 21 Depending on the edge geometry there are two primary types of GNR, viz. (i) Armchair edged graphene nanoribbon (AGNR) and (ii) Zigzag edged graphene nanoribbon (ZGNR). Both these GNR with each edge atoms passivated by single hydrogen atom, exhibit energy gaps that decrease with increasing GNR width. 22 Moreover, the electronic properties of GNR are very sensitive to many other factors, such as application of electric field, 23 modification of edges, [24][25][26] doping, 6,12,[27][28][29][30] introduction of topological defects, [31][32][33][34] chemical functionalization 35,36 etc. Such wide range of functionalities 37 of GNR has established it as a potential candidate for the post-Si device applications.
Doping is one of the most fundamental and frequently used ways to tailor the electronic property of GNR. There are mainly two different processes of doping (i) doping with foreign elements and (ii) self doping by introduction of defects. In the context of foreign element doping, the introduction of Boron (B) and Nitrogen (N) resulting in hole and electron doped GNR, has been reported in the literature. 6,28,38,39 Several B and N doped GNR based rectifiers and NDR devices have been reported and underlying mechanisms have been proposed. 12,[40][41][42][43][44] The modification of transmission function due to the shifting of electrode energy levels with applied bias was proposed to be the main reason behind the two above phenomena as reported by Zhang et al. 42 Pramanik et al. 40 explained the origin of rectification and NDR of a B and N doped AGNR based device on the basis of relative shifting of different energy levels of the total system with applied source-to-drain voltage. Deng et al. 41 explained the observed rectifying behavior of a zigzag-edged trigonal GNR device in terms of the asymmetric distribution of the electrostatic potential across the devices and the spatial distribution of electronic states at different applied voltages.
Width dependent rectifying character was observed by Zheng et al. 44 in a Z-shaped GNR device which was explained by the analysis of spatial distribution of molecular energy levels. On the other hand selfdoping 18,45 via defects interaction plays an important role in the modification of electronic structure of GNR due to disorder and localization. Topsakal et al. 46 have shown that vacancy defect in AGNR leads to a modification of band-gap that depends on the position of the defects with respect to the edges.
Suppression of conductance was observed in vacancy defected GNR originating from the localization of electronic states that eventually weakened the coupling between the electrode and the device. 47 Zhao et al. 48 observed improvement of the transport property of AGNR with the 5-8-5 (pentagon-octagonpentagon) double vacancy defect, while ZGNR with 5-8-5 defect was reported to be unfavorable for electronic transport.
Recently the technological progress in highly focused and energetic electron and ion beam irradiation technique has made possible the controlled and selective generation of defects and monitoring structural reconstruction such as Stone-Wales defect vacancy defects, disorder etc in carbon based nanostructures. 32,49 Specifically divacancy (DV) defects with removal of two carbon atoms followed by a structural reconstruction is one of the most abundant defects in carbon based materials and thermodynamically more favorable than single vacancy defect. 45,[50][51] Among the various possible configurations of DV defect such as 5-8-5 (two pentagons and one octagon), 555-777(three pentagons and three heptagons four pentagons), 5555-6-7777 (one hexagon and four heptagons), the 555-777 DV defect configuration is found to be the most stable one in GNR as reported via ab-initio simulations. 32,50,52 In the present work, we investigated the modification of electronic structure of AGNR due to the introduction of DV 555-777 defect using state-of-the art density functional approach. We observed that there is a shifting of Fermi level towards the lower energy value, which is a signature of p-type doping.
This result has motivated us to design and calculate the transport properties of AGNR based two-terminal device with one DV defected electrode and one N doped electrode. An asymmetric distribution of the electrostatic potential similar to conventional p-n junction device was observed across the scattering region. Our theoretically modeled device exhibits diode like property with high rectifying efficiency and also NDR phenomena with large peak-to-valley ratio has been observed.

Model Structure
There exists three distinct groups of P-AGNR (where P is the number of dimmer lines across the ribbon width) viz. P=3n-1, 3n, 3n+1, with n integer, and we have considered mainly 8-AGNR, 9-AGNR and 10-AGNR in this study. For electronic structure calculation we took (1×1×4) supercell of the AGNRs as illustrated for pure 9-AGNR in Fig. 1a. 555-777 DV defect was introduced in the (1×1×4) supercell of the AGNRs, by removing two carbon atoms and rotating the bonds as required and the resulting structures were geometrically optimized. Fig. 1b shows the optimized structure of 555-777 DV defected 9-AGNR.
All the edge carbon atoms of both pure and the defected AGNRs have been passivated by Hydrogen atoms. Spurious interactions between the periodic images are minimized in our theoretical models by considering a vacuum of greater than 15 Å along X and Y direction.
The model structure of the two terminal devices considered for transport calculation is shown in

Computational details
Geometry relaxation and the electronic structure calculations were performed using density functional theory (DFT) based code Vienna Ab Initio Simulation Package (VASP). 53 The exchange-correlation part was approximated by generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE). 54 Projector augmented wave (PAW) 55  Electronic transport properties of the two-terminal devices were computed using NEGF-DFT (Non-equilibrium Green's function method combined with DFT) technique as implemented in TranSIESTA code. 57 The optimized geometries of the two terminal devices, obtained from VASP code, were used in the transport calculations. As the size of the two terminal devices are very large, single-zeta (SZ) basis set has been used and the real space grid cutoff was set to 150 Ry in our transport calculations, in order to overcome the computational burden. In fact, SZ basis set yields reasonably good results for carbon based systems and has been used in several previously published reports. [58][59][60] For further justification of using SZ basis set, we performed some test calculations on defected 8-AGNR and N doped 8-AGNR with SZ basis (150 Ry mesh cutoff) as well as with double-zeta polarized (DZP) basis set (350 Ry mesh cutoff). The resulting band structures of SZ and DZP calculations were found to be nearly identical (see Fig. A1 and Fig. A2 of Appendix). Norm conserving Toullier-Martins pseudopotential 61 and PBE exchange-correlation functional was used during the calculation of transport properties. Self consistent calculation was carried out to obtain the current-voltage characteristics for the two-terminal devices with bias voltage ranging from -1.5 V to 1.5 V in steps of 0.1 V. The current through the scattering region at a finite bias (V b ) is calculated by integrating the transmission function at that bias within the bias energy window -eV b /2 to +eV b /2 using the Landauer-Buttikeformula 62 : Where ( ) is the Fermi-Dirac distribution function for the left and right electrode and µ ( ) is the electrochemical potential of the left (right) electrode such that µ ( ) = ± 2 , with being the equilibrium Fermi energy of the system which was set to zero.

Defect induced modification in the electronic structure
First we discuss the modification of electronic structure of three different types of AGNR (viz. 8, between the highest π and lowest π* band of pure 9-AGNR decreases by an amount of energy of 0.12 eV due to defect. For 9-AGNR the defect state (the green band in Fig. 2d) near E f is very localized having band width of only 0.06 eV and this localized band yields a peak in the DOS of defected 9-AGNR at -0.06 eV. However the defect induced change in the Γ point energy gap, between the π and π* band of 10-AGNR is very small, as is clearly evident from Fig. 2e and Fig. 2f. The band structure plot of defected 10-AGNR also shows that two localized bands crosses the Fermi level because of which there is a peak at E f in the DOS of defected 10-AGNR (Fig. 2f). Calculations performed with AGNR of larger width (11-AGNR, 12-AGNR, and 13-AGNR) also gave similar trends. So the main point to be emphasized here is that the introduction of 555-777 DV in three different classes of AGNR namely 3n-1, 3n, 3n+1 results in a shifting of the Fermi-level towards the lower energy value along with a upward shifting of the highest π and lowest π* bands with a slight change in their shapes.
Now before going to discuss the transport properties of the device shown in Fig. 1c it is worth mentioning the electronic properties of N doped AGNR in brief. As N doping is preferred at the edge 63

Potential distribution across the scattering region
In order to gain an insight into the internal polarization of the scattering region, we have analyzed the equilibrium electrostatic potential across the junction of defected and N doped electrode. The zero bias potential profile for the 9-AGNR junction is shown in Fig. 3. An asymmetric distribution of the potential is clearly observed from the 2-D potential plot (Fig. 3a). The color codes indicate that the N doped region is at a lower potential as compared with the defected region. In Fig. 3b we have plotted the potential, averaged along the X and Y direction, showing the oscillating behavior as indicated by the blue line. The averaged trend of the oscillating potential, denoted by the red line clearly shows the formation of a barrier across the junction, analogous to the case of conventional p-n junction. Therefore a natural rectifying character of the two-terminal device (shown in Fig. 1c) is expected. Equilibrium potential across the scattering region for the 8-AGNR and 10-AGNR devices shows similar trend (See Fig. A4 and A5 of Appendix).

Current-voltage characteristics
Now we discuss the transport properties of the AGNR based two-terminal devices. Here the point to be mentioned is that rectifiers with length, scaled down to a few nanometer, can show negative rectification as reported in several theoretical studies. 40,44,64 The rectifying efficiency of all the three AGNRs based devices are quite high, making them potential candidate for nano-electronic device applications.

Interpretation via transmission function
In order to elucidate the nonlinearity and asymmetry observed in the I-V characteristics, we have plotted (in Fig. 5 compared with the high current in positive bias (Fig. 4b) explains the high rectifying efficiency of the device. However, at -0.9 V there is a noticeable peak in the current (Fig. 4b) originating from the synchronized states inside the bias window (as indicated by the shaded region in Fig. 5f). Thus the 'matching' and 'mismatching' of the electrode energy states within the bias window qualitatively dictates the development and suppression of the transmission function. This bias dependent transmission function (within the bias window) provides a clear picture of the nonlinear and asymmetric behavior observed in the I-V characteristics. Similar conclusion can be derived for the 8-AGNR and 10-AGNR based devices, whose bias dependent transmission functions are shown in Fig. A6 and A7 of Appendix.

Conclusions
In summary, we have studied the 555-777 DV defect induced modification in the electronic structure of AGNRs and the transport properties of AGNR based two-terminal devices. It has been observed that the DV defect create electron-hole asymmetry which moves the highest π and lowest π*

APPENDIX
In order to ensure continuity of the essential physics in this paper, we have included these figures in the APPENDIX.