The reconstruction of the symmetry between sublattices: a strategy to improve the transport properties of edge-defective graphene nanoribbon transistors

Shizhuo Ye, Hao Wang, Minzheng Qiu, Yi Zeng, Qijun Huang, Jin He and Sheng Chang*
School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, People's Republic of China. E-mail:

Received 28th March 2020 , Accepted 8th June 2020

First published on 9th June 2020

A numerical study that combines device simulation and first-principle calculations is performed, aiming to alleviate the performance degradation of graphene nanoribbon field-effect devices with edge defects. We believe that investigating the symmetry between the sublattices of graphene is a novel approach to understand this key problem. The results show that the edge defects that break the symmetry between the sublattices of graphene cause more severe degradation of the device performance because they induce highly localized electronic states, which dramatically affect the transport of carriers. We propose a strategy to alleviate the localization of electronic states by rebuilding the symmetry between the sublattices. This strategy can be realized by introducing foreign radicals to modify the defective edge. A stability analysis is performed to find the most stable modified structures. The final effect of our strategy on the corresponding devices demonstrates that it can effectively address specific edge defects and remarkably improve the ON-state current and subthreshold swing.


Armchair graphene nanoribbons (AGNRs) are considered a competitive channel material in the post-silicon era due to their width-dependent band gap and good gate control of their electrostatics.1,2 Previous numerical simulations have demonstrated that AGNR-based field-effect devices can achieve high on-current densities and very small subthreshold swings (SSs).3–5 However, the ON-state current (ION), mobility, and SS are affected by edge defects.6,7 This accounts for the experimentally observed performance degradation of the AGNR devices compared to the theoretical prediction.8 Although recent advances in fabrication technology have partly alleviated this problem, a complete solution for achieving atomically smooth edges in AGNRs still requires more effort.9–12 Edge defects have become a major obstacle for the future application of AGNR-based logical devices.13 In addition to waiting for breakthroughs in fabrication techniques, we should actively seek more strategies to address this problem. To date, this aspect has received scant attention in the research literature.

The study of the physical mechanism of edge defects is helpful to understand the relationship between edge defects and the degradation of device performance. Previous research suggested that localized electronic states, induced by edge defects, provide a key insight into the underlying physics of the device. Based on this, one can successfully explain the characteristics of AGNR transistors with different device parameters and densities of edge defects.6,14 In addition, considering the unique structure of AGNRs and the resulting quantum effects, it is important to investigate the relationship between the specific edge defect configuration and the degree of localization of the corresponding localized electronic states, which may provide potential solutions to the above problems.

In this study, by performing numerical calculations, we investigate the transport properties and electronic structures of AGNRs with edge defects. The devices with specific edge defects show more severe performance degradation than the other devices in the study, which implies that the electronic states induced by these edge defects are highly localized. This phenomenon is a result of the broken symmetry between the sublattices of graphene. Based on this, we propose a strategy to mitigate the localization of the electronic states by reconstructing the symmetry between the sublattices. This strategy can be realized by modifying the defective edges with foreign radicals. Simulations of the corresponding devices show that our strategy can restore the transmission channels reflected by the highly-localized edge states and significantly improve the ION and SS properties in most cases.

Simulation model and method

In this paper, the calculations are in two parts. The first part is for simulating the transport properties of the devices and the second part is for computing the properties of the defective AGNR segment. As shown in Fig. 1a, the atom-resolved device model adopted in the device simulations is a typical double-gate metal-oxide-semiconductor field-effect transistor (MOSFET). The thickness and relative dielectric constant of the oxide layers are 1 nm and 3.9, respectively. The source and drain extensions are n-type doped with a concentration of 1.0 × 10−2 electrons per atom by the atomic compensation charge method and their lengths are 5 nm. The length of the channel is 10 nm. Due to the short length of the channel, only one edge defect is considered for each device. Additionally, the edge defect is placed in the center of the device. The influence of the location of the edge defects has been investigated as shown in Fig. S1 of ESI. The location of the edge defects does not influence the central idea of this work. The AGNRs were prefixed with their widths (e.g. 9AGNR in Fig. 1a). The AGNRs of the 3m and 3m + 1 families both manifest band gaps suitable for transistors and the smaller nanoribbon widths are in line with future development trends. Therefore, we chose 9AGNR and 10AGNR as the representatives for device simulations.15
image file: d0cp01684e-f1.tif
Fig. 1 (a) Atom-resolved device model. SR, LE, and RE represent the scattering region, left electrode, and right electrode, respectively. (b) Unit cells of defective AGNR segments.

To obtain the transport properties of the devices, the device system is simulated based on the non-equilibrium Green's function (NEGF) method, with a self-consistent-charge density-functional tight-binding (DFTB) Hamiltonian built from the CP2K parameters.16–18 This method has been used in previous work to study the transport properties of graphene systems.19–22 The DFTB Hamiltonian has significantly improved computing efficiency by introducing a parameterized matrix. Meanwhile, due to the introduction of the self-consistent-charge process, it shows good agreement with experimental data and results from the more accurate density functional theory (DFT). Besides, the atomic structure used in the device is optimized by the so-called “Bulk Rigid Relaxation” method with a maximum force tolerance of 0.02 eV Å−1.23 1 × 1 × 200 grids are adopted to sample the k-space in the device simulation.

QuantumATK (Ver.P-2019.03) has been used to solve the NEGF equations, of which the key quantity is the retarded Green's matrix:23

image file: d0cp01684e-t1.tif
where E is the energy, i is the imaginary unit, and ε is an infinitesimally positive number. S and H are the overlap and Hamiltonian matrices of the entire system, respectively. With the retarded Green's matrix, the transmission coefficient may also be obtained using:
T(E) = G(E)ΓL(E)G(E)ΓR(E),
where ΓL and ΓR are the broadening functions of the left and right electrodes, respectively.

The current is calculated using the Landauer–Büttiker formula:

image file: d0cp01684e-t2.tif
where e is the charge of the electron, h is Planck's constant, and f is the Fermi–Dirac distribution. μL and μR are the chemical potentials of the left and right electrodes, respectively.

In Fig. 1b, the unit cells of defective AGNR segments are depicted. To imitate the cases of missing carbon atoms at the edges, we consider the cases with two limitations. First, one to three carbon atoms are lost. Second, the missing carbon atoms are adjacent to each other. Without these limitations, the cases will be too complex to study. The A–G defects are created by an enumeration method. The A defect corresponds to the case of one missing carbon atom. The B–D defects correspond to the case of two missing carbon atoms and the E–H defects correspond to the case of three missing carbon atoms.

The geometry optimization, total energy, and electronic structure of these unit cells are calculated using the DFT method. In DFT calculations, local density approximation is used to treat the exchange–correlation functional. The double-ζ-plus-polarization basis-sets and an energy-mesh cutoff of 150 Ry are employed.24,25 Additionally, a vacuum thickness of 1.5 nm is set in the out-of-plane and vertical direction to decouple the periodic images. 1 × 1 × 10 k-point grids are used in the DFT calculations. In addition, the geometry optimization of the unit cell is performed with a maximum force tolerance of 0.02 eV Å−1.

Simulation results and discussion

The transfer characteristics of 9AGNR and 10AGNR MOSFETs with and without the edge defects are shown in Fig. 2. These defective MOSFETs are suffixed according to their defect types, such as 9AGNR-A. The gate voltage (VGS) ranges from 0 to 1.0 V and the drain-to-source voltage (VDS) is 0.1 V. The device OFF-state is aligned with VGS = 0 V and the ON-state current of MOSFET, based on 9AGNR and 10AGNR, is defined as the current at VGS = 1.0 V. Defects always induce new behavior in low-dimensional materials.26,27 From the figure, we see that the edge effects lead to degraded performances. The average ION of the defective 9AGNR MOSFETs is 31% of the ION of the perfect 9AGNR MOSFET and this value is 22% for the 10AGNR case. The SS of the perfect 9AGNR (10AGNR) MOSFET is 70 (64) mV dec−1, whereas the average SS of the defective 9AGNR (10AGNR) MOSFETs is 90 (76) mV dec−1.
image file: d0cp01684e-f2.tif
Fig. 2 Transfer characteristic curves of (a and b) the perfect and defective 9AGNR MOSFETs and (c and d) the perfect and defective 10AGNR MOSFETs. The left and right panels are on the linear and logarithmic scale, respectively.

The edge defects may induce highly-localized electronic states and thereby suppress the conductance.6 This partly explains the transport behavior of the defective AGNRs in Fig. 2. Moreover, it is worth noting that the performance degradation caused by the C defect is not as severe as the other edge defects. Moreover, 9AGNR-B and 10AGNR-D also show strong transfer characteristics. This means that the degree of localization induced by the edge defects in these cases may not be as high as that of the other defects.

We can understand this phenomenon with a simple model of the symmetry of the π electrons between the sublattices in graphene. As is known, the graphene hexagon network contains two unequal sublattices, denoted by α and β in this paper. According to the resonating-valence-bond theory for unpaired π-electrons in benzenoid carbon species, the unpaired π-electrons can be qualitatively described by “resonant-theoretic zeroth-order free valences” (referred to as free valences below), which are defined as the deficit from 1 of 1/3 of the π-pairing degree of a π-center.28 Resonant-theoretic free valences on typical AGNR edges are displayed in Fig. 3a. The central carbon atom in the first AGNR edge is sp2-hybridized. Because it has two neighboring sp2-carbons, its π-pairing degree is 2. Thus it shows free valences of 1/3 of an electron. The central carbon atom in the second AGNR edge is also sp2-hybridized but it only has one neighboring sp2-carbon and thus it shows free valences of 2/3 of an electron. The sp3-hybridized carbon atoms in the three bonding patterns on the right are not π-centers. Their neighboring sp2-carbons still show free valences of 1/3 of an electron.

image file: d0cp01684e-f3.tif
Fig. 3 (a) Resonant-theoretic free valences on typical AGNR edges. Green and orange circles represent free valences on the α and β lattices, respectively. Free valence distributions of (b–e) the B–E defects accompanied by band structures and Bloch states (represented as red isosurfaces) near the Fermi level.

By applying these free valence distributions to treat the edge defects in Fig. 1b, the broken symmetry between the sublattices induced by specific edge defects is revealed. For example, the free valence distributions near the B–E defects along with the band structures and Bloch states near the Fermi level are shown in Fig. 3b–e. First, the free valences of the C and E defects are compared. The C defect leaves an equal number of free valences on the α and β sublattices, whereas the E defect induces more free valences on the α sublattice. Thus, the C defect preserves the symmetry of the π electrons between the sublattices but the E defect destroys it. This is also reflected in the Bloch states near the Fermi level. The Bloch states of the E defect primarily appear on the neighboring sites of edge defects and are only distributed on the α sublattice. However, those of the C defect are not localized near the edge defects and are equally distributed between the sublattices.

Now, the free valences of the B and D defects will be discussed. As far as the entire unit cell is concerned, both the B and D defects show equal free valences on the α and β sublattices. Therefore, the electronic states corresponding to the B and D defects are not as localized as they are for the E defect. This explains why 9AGNR-B and 10AGNR-D also exhibit good transfer characteristics. However, the electronic states of the B and D defects are more localized compared to the C defect from the Bloch states. In the B defect, the longest distance between the free valences of the opposite-sublattice sites is approximately four times the C–C bond length. This large distance causes two local asymmetric distributions of the free valences. In the D defect, the largest free valences are of 2/3 of an electron and the rest are of 1/3 of an electron but all of the free valences in the C defect are of 1/3 of an electron. The unequal number of free valences breaks the local symmetry between the sublattices in the D defect and reinforces the localization of the electronic states. In summary, the B and D defects are considered as defects that break the symmetry between the sublattices in terms of local atomic structure.

Broken symmetry between the sublattices accounts for the highly-localized electronic states.29 Therefore, to address the degradation of performance, we propose a strategy to alleviate the localization of the electronic states by reconstructing the symmetry of the π electrons between the sublattices. The above strategy can be realized by modifying the defective edge with foreign radicals. To be more specific, the highly-localized electronic states are attracted to the radicals and thus carbon atoms with highly-localized electronic states easily adsorb the free radicals.30 After the carbon atoms adsorb the foreign radicals, their hybridization states will change. Therefore, the distribution of π electrons near the edge defect will be modified.

For simplicity, we choose the hydrogen atom as the radical in this work. The formation energy and free valence distributions of the modified structures are studied to verify our viewpoint. The formation energy is defined as:

image file: d0cp01684e-t3.tif
where Em, Ed, and EH2 are the total energies of the final modified AGNR segment, defective AGNR segment, and isolated H2 molecule, respectively. NH2 is the number of hydrogen atoms used to modify the defective AGNR segment. An investigation of hundreds of structures is required if the enumeration method is used to find the modified final structure. To simplify the problem, the modification is divided into several steps. In each modification step, one hydrogen atom is used to modify the initial structure. Thus, the formation energy of the i-th modification step can be obtained as:
image file: d0cp01684e-t4.tif
where Et and Eb are the total energies of the tentatively modified and initial structures in the i-th modification step. In the first modification step, the defective AGNR segment is the initial structure. As the carbon atoms with highly-localized electronic states are more attracted to the free radicals, only the carbon atoms close to the edge defects are considered to be modified.30 A tentatively modified structure is created by bonding a foreign hydrogen atom with one of the carbon atoms close to the edge defects. Then, this initial structure is optimized and the total energy of the optimized structure is calculated. In this way, a series of optimized structures and related Efi is obtained. In order to make the data intuitive, we project the Efi values onto the site of the corresponding carbon atom and represent them as a two-dimensional map using the interpolation method. The optimized structure with the lowest Efi is the most stable modified structure for this modification step. Additionally, the most stable modified structure serves as the new initial structure (similar to the defective AGNR segment in the first step) for the next modification step unless the symmetry between the sublattices is rebuilt.

The formation energy maps and the final modified structures corresponding to the edge defects are shown in Fig. 4. The lowest formation energy to modify the C defect is at least 1.5 eV larger than that of the other defects and the C defect does not break the symmetry between the sublattices. Therefore, the C defect is not considered in our strategy. From the figure, the A, E, F, and G defects require one hydrogen atom, the B and D defects require two hydrogen atoms, and the H defect requires three hydrogen atoms to rebuild the symmetry between the sublattices. For the D and E defects, there are two final modified structures that have very close formation energies. It is noted that the formation energy becomes larger as the hydrogen atoms move away from the defects. This means that the location of the foreign hydrogen atom is limited to near the edge defects. In other words, our strategy will not affect the intact portion of the AGNRs, which is favorable for practical applications. Moreover, most of the modified structures, except for modified 10AGNR-E, show lower edge formation-energies than 10AGNR, as shown in the ESI. Thus, the stability of the AGNR segments is not destroyed by our modification strategy. In regards to the free valences distribution of the modified structures, they all show equal free valences between the sublattices. As the final modified structures are the most stable structures after modification according to the formation energy analysis and they show a balance between the sublattices, our strategy is feasible.

image file: d0cp01684e-f4.tif
Fig. 4 Formation energy maps and the final modified structures corresponding to the (a) A defect, (b) B defect, (c) D defect, (d) E defect, (e) F defect, (f) G defect, and (g) H defect.

Our strategy shows similar improvements for defective 9AGNR and 10AGNR devices, except for the cases of the B and G defects. The data for all of the modified 10AGNR devices and modified 9AGNR-G are shown as representatives in Fig. 5 (all of the data for the modified 9AGNR devices are shown in Fig. S2 in ESI). As shown in the figure, a part of the modified devices can almost reproduce the transfer characteristic curve of the perfect 10AGNR devices, such as modified 10AGNR-F. Another part of them cannot completely restore the device performances but compared to the defective systems, there are performance improvements, for example, in modified 10AGNR-A. In Fig. 5a, the ION of modified 10AGNR-A is obviously improved compared to 10AGNR-A. From the transmission eigenstates, the A defect almost completely reflects the incident states, whereas the transmission channel of modified 10AGNR-A is well restored as the symmetry between the sublattices is rebuilt. This intuitively confirms the effectiveness of our strategy from a physical standpoint. Moreover, our strategy does not show significant improvements for modified 10AGNR-B and modified 10AGNR-G. However, as shown in Fig. 5j, modified 9AGNR-G shows better performances than those of 9AGNR-G. This is because the B and G defects also induce a device density of states gap (DDOS) in addition to the highly-localized electronic states, resulting in a suppression of transportation. The DDOS data are shown in Fig. S3 in ESI. In fact, our strategy still alleviates the localization of electronic states in the modified 10AGNR-B device, as seen in the transmission eigenstates in Fig. 5b. The DDOS problem is beyond the scope of our modification strategy, which aims to alleviate highly-localized electronic states to improve device performances. The transport properties of the perfect, defective, and perfect 12AGNR and 13AGNR devices are shown in Fig. S4 in ESI. Our strategy shows similar results from the 12AGNR (13AGNR) devices to the 9AGNR (10AGNR) devices. This implies that our strategy may be still useful for larger defective AGNR structures. In summary, most of the modified devices show better performances than the defective devices. The average ION of the modified 9AGNR MOSFETs is 73% of the ION of the perfect 9AGNR MOSFET and this value is 64% for the 10AGNR group. The average SS of the modified 9AGNR (10AGNR) MOSFTEs is 72 (64) mV dec−1, which is almost the same as the perfect case. The performance improvement observed in most of the defective AGNR devices indicates that our strategy effectively addresses highly-localized electronic states induced by the edge defects.

image file: d0cp01684e-f5.tif
Fig. 5 The transfer characteristic curves and the transmission eigenstates at VGS = 1.0 V of representative defective and modified devices.


The performance degradation induced by highly-localized edge states on AGNR MOSFETs is studied through the combination of device simulation and first-principles calculations. Our understanding is that specific edge defects break the symmetry between the sublattices of graphene, inducing highly localized electronic states, which severely affect transportation. A simple but intuitive π electron model is illustrated to uncover this relationship. Based on this, we propose a strategy to alleviate the localization of electronic states by reconstructing the symmetry between the sublattices of graphene. This strategy can be realized by modifying the defective edge with foreign radicals. The final effect of this strategy is demonstrated by improved device performances and physical pictures. Our strategy can effectively deal with the highly-localized electronic states induced by edge defects and shows obvious performance improvements for most of the defective AGNR devices.

Conflicts of interest

There are no conflicts to declare.


This work was supported by the National Natural Science Foundation of China under Grant 61874079, Grant 81971702 and Grant 61774113, Wuhan Research Program of Application Foundation under Grant 2018010401011289, and the Luojia Young Scholars Program. The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

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Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp01684e

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