Silicene has attracted remarkable attention in the semiconductor research community due to its silicon (Si) nature. It is predicted as one of the most promising candidates for the next generation nanoelectronic devices. In this paper, an efficient non-iterative technique is employed to create the SPICE models for p-type and n-type uniformly doped silicene field-effect transistors (FETs). The current-voltage characteristics show that the proposed silicene FET models exhibit high on-to-off current ratio under ballistic transport. In order to obtain practical digital logic timing diagrams, a parasitic load capacitance, which is dependent on the interconnect length, is attached at the output terminal of the logic circuits. Furthermore, the key circuit performance metrics, including the propagation delay, average power, power-delay product and energy-delay product of the proposed silicene-based logic gates are extracted and benchmarked with published results. The effects of the interconnect length to the propagation delay and average power are also investigated. The results of this work further envisage the uniformly doped silicene as a promising candidate for future nanoelectronic applications.
Citation: Chuan MW, Wong KL, Riyadi MA, Hamzah A, Rusli S, Alias NE, et al. (2021) Semi-analytical modelling and evaluation of uniformly doped silicene nanotransistors for digital logic gates. PLoS ONE 16(6): e0253289. https://doi.org/10.1371/journal.pone.0253289
Editor: Talib Al-Ameri, University of Glasgow, UNITED KINGDOM
Received: April 16, 2021; Accepted: June 1, 2021; Published: June 14, 2021
Copyright: © 2021 Chuan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting Information files.
Funding: 1.) Munawar Agus Riyadi - World Class Research Universitas Diponegoro (WCRU) 2021 Grant no. 118-16/UN7.6.1/PP/2021; The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. 2.) Michael Tan Loong Peng - UTM Fundamental Research (UTMFR) Vote no.: Q.J130000.2551.21H51; The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Digital logic gates are the foundation of modern computation and information processing in various systems such as nanostructure computers [1,2], photonic technology [3,4] and biomedical engineering [5,6]. The development of these systems is governed by Moore’s Law [7,8] for more than four decades. However, the present digital logic gates, primarily based on bulk silicon (Si) field-effect transistors (FETs), are reaching the fundamental device limitations [9,10]. Therefore, the quest for next-generation FETs to leverage nanotechnologies, for more than Moore’s applications, has become one of the mainstream research topics.
The development of two-dimensional (2D) materials has been motivated by the success of monolayer honeycomb carbon (C)—graphene . While the discovery of graphene has more than 15 years of history, the honeycomb Si-based monolayer—silicene has only attracted research interest in the recent years as shown by the trend of publication numbers [12,13] despite its potential compatibility with the present Si-dominant fabrication processes. In 2015, Tao et al.  demonstrated the first silicene-based transistor using synthesis-transfer fabrication technique. Interestingly, silicene is envisaged as an alternative material for transistor scaling in the International Roadmap for Devices and Systems (IRDS) . Although silicene sheets have successfully been formed on various substrates in their buckled [16–18] and planar  forms, the fabrication of stable free-standing silicene still remains a major challenge. At this early stage of development, computational modelling and simulation are very useful in providing fundamental insights of silicene-based devices and circuits.
Silicene, as a counterpart of graphene [20,21], exhibits the Dirac cone properties and in addition, an extremely small energy bandgap of 1.55 meV . Similar to graphene, bandgap engineering techniques are required in order to build silicene-based transistors, where the transistors for digital switching applications typically require an energy bandgap of at least 0.4 eV  to suppress the unwanted subthreshold conduction. Silicene sheets can be sliced into semiconducting silicene nanoribbons (SiNRs) which have shown promising transistor performances , but their energy bandgap values and electronic properties are heavily dependent on the nanoribbons widths . Although altering nanoribbon width is proven to be an viable bandgap engineering option, the fabrication technique to produce nanoribbon with perfect edge control is yet to be discovered, even for the well-known and matured graphene .
Due to this shortcoming, we propose to employ the n-type and p-type uniformly doped silicene as the semiconducting channel of the silicene-based FETs, by using phosphorus (P) and aluminium (Al) as the dopant atoms, respectively. This uniform doping technique has been proven previously to be an effective way to obtain semiconducting silicene nanosheets, that are suitable for digital switching applications . The silicene sheets that are uniformly doped using P and Al will be denoted as PSi3 and AlSi3, respectively in the rest of this paper. Unlike the selective doping technique [28,29], where the electronic properties of doped silicene vary with dopant sites, the uniform doping technique is independent on the dopant sites . After obtaining semiconducting silicene nanosheets, these n-type and p-type silicene-based FETs can be employed to build various complementary metal–oxide–semiconductor (CMOS) circuits such as inverter, NAND, and NOR gates.
Despite rigorous efforts to model and simulate silicene-based devices [12,31], there is still minimal work focusing on the circuit-level performance analysis featuring silicene-based transistor. In this paper, the circuit-level performance of n-type and p-type uniformly doped silicene FETs, as shown in Fig 1, are assessed by developing a SPICE-compatible model . This circuit-level model is developed by extending our previous works on uniformly doped silicene model at the material-level  and device-level . Section 2 describes the detailed modelling procedures employed in this work. In Section 3, the simulation results of the silicene-based logic gates are shown by plotting the timing diagrams. Subsequently, the circuit performance of the logics gates are analysed based on their propagation delay, average power, power-delay product (PDP) and energy-delay product (EDP). Finally, the conclusion of this work is drawn and potential future work is recommended in Section 4.
2. Modelling procedures
This section describes the step-by-step modelling procedures for the proposed silicene-based nanotransistors from material-level up to circuit-level, where the overall flow chart is depicted in Fig 2.
2.1 Uniformly doped silicene transistors
At the material-level, the electronic properties of p-type (AlSi3) and n-type (PSi3) nanosheets as shown in Fig 1, are modelled using nearest neighbour tight-binding (NNTB) model by fitting the published DFT band structures in Ref . The nanosheets are assumed to be in their perfect planar honeycomb lattice. This assumption is applicable because the successful fabrication of planar silicene has recently been reported . Subsequently, the transverse effective masses (in the zigzag direction) of these materials are obtained by using parabolic band approximation. The detailed procedures to obtain the effective masses can be found in Ref . Although previous work  shows only the modelling procedures for the p-type AlSi3 nanosheet, the same technique is repeated to compute the electronic properties of n-type PSi3 nanosheet, in order to obtain both type of transistors for CMOS applications. Table 1 summarises the important electronic properties of the AlSi3 and PSi3 nanosheets, where m0 is the constant for electron rest mass. The results show that both uniformly doped silicene nanosheets have achieve energy bandgap values of 0.4 eV≤Eg≤3.0 eV, making them suitable for nanoelectronic digital switching applications .
With the obtained electronic properties, the work then proceeds with device-level modelling by employing the top-of-the-barrier (TOB) ballistic nanotransistor model , which has been widely used to predict the performance limits of various low-dimensional materials . In this work, a double-gated FET structure with Lg = 10 nm is employed, where the gate oxide layers are SiO2 (εr = 3.9) with thickness of tOX = 1.5 nm. All simulations are done at the room temperature of T = 300 K. For simplification, the W/Lg aspect ratio is set as unity for both type of silicene FETs. The schematic diagrams of the FETs are illustrated in Fig 1. In n-type 2D FETs, the current transport primarily depends on the electron mobility via the conduction band, and vice versa for p-type 2D FETs (where the structures of 2D FETs are similar to those of junctionless FETs ). Therefore, electron effective mass is used for n-type (PSi3) FET while hole effective mass is used for p-type (AlSi3) FET.
The self-consistent potentials USCF in the TOB model are calculated iteratively in MATLAB until the solutions for the charge carriers at the TOB converge. Therefore, further modifications are required to obtain a non-iterative model, in terms of the drain-source voltage VDS and gate-source voltage VGS, to allow cross-platform simulation and reduce computational cost . In order to create a model compatible with the Simulation Program with Integrated Circuit Emphasis (SPICE), the self-consistent potential USCF in the TOB model is computed through fifth order polynomial equation within the non-linear regression model , expressed as (1) where Pij is the coefficient for each respective |VGS|i|VDS|j term. The fifth order binomial equation as shown in Eq (1) can be expanded via Pascal’s triangle rule. The coefficients are extracted and optimised using MATLAB curve fitting tool, where Fig 3 shows the results of the non-linear regression model. The full equation and coefficients for p-type and n-type uniformly doped silicene is attached in the supplementary data file (S1 File). We employed the fifth order binomial equation because the lower order binomial equations are unable to produce decent fit for the USCF and the fifth order is the highest available within the curve fitting tool.
2.2 Device and circuit simulation
Following that, the current-voltage (I-V) characteristics of n-type and p-type uniformly doped silicene FETs can be obtained in terms of VDS and VGS, by using Landauer-Büttiker ballistic transport equation  with Fermi-Dirac integral solution , given as (2) with the normalised source and drain energies of (3) and (4) where g is the degeneracy factor (set as 2 to include up and down spins); ℏ is the Planck’s constant; q is the constant for electric charge; and kB is the Boltzmann constant.
The magnitude of the supply voltage |VDD| proposed in this work is 0.60 V and the Fermi energy level EF is adjusted such that the off-current Ioff = 100 nA/μm, for low-standby power (LSTP) CMOS applications . In this work, the original gate control (αD = 0.880) and drain control (αD = 0.035) parameters from Ref  are employed and the source terminal is always tied to the ground (VS = 0 V). Eqs (1) to (4) are used to create the SPICE library files for n-type and p-type uniformly doped silicene FET. With these operating conditions, the n-type and p-type silicene FETs have achieved on-to-off current (Ion/Ioff) ratio of Ion/Ioff = 2.8×105 and Ion/Ioff = 2.6×105, respectively at room temperature T = 300 K. The n-type silicene FET has higher Ion/Ioff ratio because the electron effective mass of PSi3 nanosheet is lighter than the hole effective mass of AlSi3. The Ion/Ioff ratio of the proposed device is higher than Si FinFET  by two orders. In addition, the Ion/Ioff ratios of the n-type and p-type uniformly doped silicene FETs are improved by 35.7% and 19.2%, respectively, when compared to n-type and p-type Si nanowire FETs (where the Ion/Ioff ratios of n-type and p-type Si nanowire FETs were found to be 1.8×105 and 2.1×105, respectively in Ref ). In addition, Table 2 compares the Ion/Ioff ratios of the proposed model with published 2D material-based FET models. It is shown that the proposed FET models outperform Phosphorene and graphene nanoribbon (GNR) FET models. Although 27-ASiNR FET outperforms the proposed FETs, it still remains a huge challenge to precisely control the size of 2D nanoribbons with specific widths, even for graphene which was discovered in the laboratory more than 15 years ago [26,45].
The simulated I-V characteristics for the original iterative TOB model and non-iterative SPICE model are plotted on the same graph in Fig 4. The results show that the fifth order binomial equation for USCF is capable of reproducing the iterative TOB model in the HSPICE circuit simulator with minimal error. With the p-type and n-type FET SPICE models ready, the work proceeds to build and simulate digital logic circuits using HSPICE simulator. In order to make the circuit simulation more practical, copper (Cu) interconnect capacitance is incorporated as the load capacitance for all circuits. The Cu interconnect capacitance is identified as Cint = 121.3 aF/μm by using the ITRS projected interconnect capacitance value for transistor with 10 nm gate length . The length Lint of the Cu interconnect is varied from 10 nm to 50 μm to investigate its effects on the logic circuit performance.
3. Results and discussions
In this section, the simulation results of digital logic gates, including inverter, 2-input NAND and 2-input NOR gates are shown. Their circuit performances are also evaluated by extracting the propagation delay (tp), average power (Pavg), PDP and EDP. The propagation delay of the proposed model is also benchmarked with selected published results.
3.1 Timing diagrams
The silicene-based logic circuits simulated in HSPICE are then plotted using Avanwaves. The high voltage (representing ‘1’ digital signal) of the input pulses are set to the supply voltage of 0.60 V; and low voltage (representing ‘0’ digital signal) of the input pulses are set to the ground voltage of 0 V. A rise and fall time of tr = tf = 0.1 ps are used for the input waveforms in order to obtain sharp rising and falling edges. Figs 5–7 clearly show that the silicene-based logic circuits are able to function correctly according to the intended Boolean logics for inverter, 2-input NAND and 2-input NOR gates , respectively.
Schematic circuit diagram of silicene-based inverter (Lint = 1 μm) with its input (blue) and corresponding output waveforms (red).
Schematic circuit diagram of silicene-based 2-input NAND gate (Lint = 1 μm) with its input (blue) and corresponding output waveforms (red).
3.2 Performance analysis of digital logic circuits
The propagation delay (tp) and average power (Pavg) for the simulated logic gates are extracted and plotted against the length Lint of Cu interconnect, as depicted in Fig 8. It is clearly shown that the tp for all three logic gates increases as the Lint increases. Nevertheless, the 2-input NAND gate has the highest tp for all interconnect lengths. On the other hand, the Pavg for all three logic gates remain almost constant until Lint = 1 μm, regardless of the type of logic gates. Thus, it is crucial to optimise the Lint in digital system design in order to achieve minimal propagation delay and suppress the power consumption. Similar circuit degradation due to long Lint was also previously reported for GNR FETs with interconnect analysis .
INV, NAND2 and NOR2 represent inverter, 2-input NAND and 2-input NOR gates, respectively.
Subsequently, the figure of merits for digital logic circuits are calculated using the extracted values in Fig 8 and the equations of the PDP and EDP , given as (5) and (6) where the average power Pavg and propagation delay tp. Fig 9 shows the PDP and EDP of proposed silicene-based digital logic circuits when Lint is varied from 10 nm to 50 μm. The 3D plot in Fig 9(b) show that, at all values of Lint, 2-input NAND gate has the highest EDP due to its high propagation delay tp compared to inverter and 2-input NOR gates.
(a) PDP and (b) EDP of silicene-based digital logic circuits. INV, NAND2 and NOR2 represent inverter, 2-input NAND and 2-input NOR gates, respectively.
As this study aims to assess the circuit-level performance of the proposed uniformly doped silicene FETs for digital logic gates, the results are benchmarked with published works that are based on low-dimensional materials, including GNR FET and 7 nm FinFET from Ref ; as well as 10 nm carbon nanotube (CNT) FET and 10 nm FinFET from Ref . Due to the unavailability of complete data, we have only compared the propagation delay tp among the models for inverter and 2-input NAND gates as shown in Fig 10. The bar graph clearly shows that the proposed silicene-based inverter gate outperforms all the published models in terms of the propagation delay. However, the propagation delay of proposed silicene-based 2-input NAND gate is higher than that of the graphene-based logic circuit . Despite this slight disadvantage, silicene-based circuits are still a prospective choice for the future nanoelectronic applications due to its potential compatibility with Si CMOS technology .
In this paper, we have investigated the circuit-level performance of digital logic gates built using the p-type and n-type uniformly doped silicene FETs. By fitting the self-consistent potential at the TOB using fifth order binomial equations, a non-iterative SPICE model for the proposed FETs are created, where the model is then utilised to perform circuit-level simulations. Following that, the timing diagrams for the proposed silicene-based logic gates are computed and verified. In order to gain more insights from the digital logic output waveforms, the figure of merits for inverter, 2-input NAND, and 2-input NOR gates are extracted and compared to recent published results. Based on the benchmark of the results, the proposed silicene-based inverter has achieved the lowest propagation delay. Although the propagation delay of the proposed silicene-based 2-input NAND gate is outperformed by GNR-based gate, it is still optimistic that silicene-based CMOS logic circuits are promising substitutes for future nanoelectronic devices because graphene-based systems might require an entirely redesigned fabrication technique and equipment for mass production in the semiconductor industry. In future work, it may be useful to extend the present study on the basic logic gates to explore more complex silicene-based digital circuits and systems.
The authors acknowledge the Research Management Centre (RMC) and School of Graduate Studies (SPS) of Universiti Teknologi Malaysia (UTM) for providing excellent and stimulating research environment.
- 1. Hills G, Lau C, Wright A, Fuller S, Bishop MD, Srimani T, et al. Modern microprocessor built from complementary carbon nanotube transistors. Nature. 2019;572(7771):595–602. pmid:31462796
- 2. Shulaker MM, Hills G, Patil N, Wei H, Chen H-Y, Wong H-SP, et al. Carbon nanotube computer. Nature. 2013;501(7468):526–30. pmid:24067711
- 3. Goudarzi K, Mir A, Chaharmahali I, Goudarzi D. All-optical XOR and OR logic gates based on line and point defects in 2-D photonic crystal. Optics & Laser Technology. 2016;78:139–42. https://doi.org/10.1016/j.optlastec.2015.10.013.
- 4. Pirzadi M, Mir A, Bodaghi D. Realization of ultra-accurate and compact all-optical photonic crystal OR logic gate. IEEE Photonics Technology Letters. 2016;28(21):2387–90. https://doi.org/10.1109/LPT.2016.2596580.
- 5. Siuti P, Yazbek J, Lu TK. Synthetic circuits integrating logic and memory in living cells. Nat Biotechnol. 2013;31(5):448–52. pmid:23396014
- 6. Hasty J, McMillen D, Collins JJ. Engineered gene circuits. Nature. 2002;420(6912):224–30. pmid:12432407
- 7. Waldrop MM. More than Moore. Nature. 2016;530(7589):144–8. https://doi.org/10.1038/530144a.
- 8. Zhang G, Allaire D, Shankar V, McAdams DA. A case against the trickle-down effect in technology ecosystems. PLoS One. 2019;14(6):e0218370. https://doi.org/10.1371/journal.pone.0218370.
- 9. Chien AA, Karamcheti V. Moore’s law: The first ending and a new beginning. Computer. 2013;46(12):48–53. https://doi.org/10.1109/MC.2013.431.
- 10. Peng L-M, Zhang Z, Wang S. Carbon nanotube electronics: recent advances. Mater Today. 2014;17(9):433–42. https://doi.org/10.1016/j.mattod.2014.07.008.
- 11. Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric field effect in atomically thin carbon films. Science. 2004;306(5696):666–9. https://doi.org/10.1126/science.1102896.
- 12. Zhao J, Liu H, Yu Z, Quhe R, Zhou S, Wang Y, et al. Rise of silicene: A competitive 2D material. Prog Mater Sci. 2016;83:24–151. https://doi.org/10.1016/j.pmatsci.2016.04.001.
- 13. Chuan MW, Wong KL, Hamzah A, Rusli S, Alias NE, Lim CS, et al. 2D Honeycomb Silicon: A Review on Theoretical Advances for Silicene Field-Effect Transistors. Current Nanoscience. 2020;16(4):595–607. https://doi.org/10.2174/1573413715666190709120019.
- 14. Tao L, Cinquanta E, Chiappe D, Grazianetti C, Fanciulli M, Dubey M, et al. Silicene field-effect transistors operating at room temperature. Nature nanotechnology. 2015;10(3):227. https://doi.org/10.1038/nnano.2014.325.
- 15. IEEE. International Roadmap for Devices and Systems (IRDS) [Internet]. 2018. Available from: https://irds.ieee.org/images/files/pdf/2017/2017IRDS_ERM.pdf.
- 16. Huang L, Zhang Y-F, Zhang Y-Y, Xu W, Que Y, Li E, et al. Sequence of silicon monolayer structures grown on a Ru surface: From a herringbone structure to silicene. Nano Lett. 2017;17(2):1161–6. https://doi.org/10.1021/acs.nanolett.6b04804.
- 17. Stępniak-Dybala A, Krawiec M. Formation of Silicene on Ultra-Thin Pb (111) Films. The Journal of Physical Chemistry C. 2019. https://doi.org/10.1021/acs.jpcc.9b04343.
- 18. Hsu H-C, Lu Y-H, Su T-L, Lin W-C, Fu T-Y. Single crystalline silicene consist of various superstructures using a flexible ultrathin Ag (111) template on Si (111). Semicond Sci Technol. 2018;33(7):075004. https://doi.org/10.1088/1361-6641/aaad88.
- 19. Stępniak-Dybala A, Dyniec P, Kopciuszyski M, Zdyb R, Jałochowski M, Krawiec M. Planar silicene: a new silicon allotrope epitaxially grown by segregation. Adv Funct Mater. 2019;29(50):1906053. https://doi.org/10.1002/adfm.201906053.
- 20. Borowik P, Thobel J-L, Adamowicz L. Monte Carlo study of electron transport in monolayer silicene. Semicond Sci Technol. 2016;31(11):115004. https://doi.org/10.1088/0268-1242/31/11/115004.
- 21. Safina I, Bourdo SE, Algazali KM, Kannarpady G, Watanabe F, Vang KB, et al. Graphene-based 2D constructs for enhanced fibroblast support. PLoS One. 2020;15(5):e0232670. https://doi.org/10.1371/journal.pone.0232670.
- 22. Liu C-C, Feng W, Yao Y. Quantum spin Hall effect in silicene and two-dimensional germanium. Phys Rev Lett. 2011;107(7):076802. https://doi.org/10.1103/PhysRevLett.107.076802.
- 23. Ni Z, Zhong H, Jiang X, Quhe R, Luo G, Wang Y, et al. Tunable band gap and doping type in silicene by surface adsorption: towards tunneling transistors. Nanoscale. 2014;6(13):7609–18. https://doi.org/10.1039/C4NR00028E.
- 24. Poljak M. Impact of Width Scaling and Parasitic Series Resistance on the Performance of Silicene Nanoribbon MOSFETs. IEEE Trans Electron Devices. 2020. https://doi.org/10.1109/TED.2020.3017465.
- 25. Ding Y, Ni J. Electronic structures of silicon nanoribbons. Appl Phys Lett. 2009;95(8):083115. https://doi.org/10.1063/1.3211968.
- 26. Celis A, Nair MN, Taleb-Ibrahimi A, Conrad E, Berger C, De Heer W, et al. Graphene nanoribbons: fabrication, properties and devices. J Phys D: Appl Phys. 2016;49(14):143001. https://doi.org/10.1088/0022-3727/49/14/143001.
- 27. Ding Y, Wang Y. Density functional theory study of the silicene-like SiX and XSi3 (X = B, C, N, Al, P) honeycomb lattices: The various buckled structures and versatile electronic properties. The Journal of Physical Chemistry C. 2013;117(35):18266–78. https://doi.org/10.1021/jp407666m.
- 28. Zhang J-M, Song W-T, Xu K-W, Ji V. The study of the P doped silicene nanoribbons with first-principles. Computational Materials Science. 2014;95:429–34. https://doi.org/10.1016/j.commatsci.2014.08.019.
- 29. Lopez-Bezanilla A. Substitutional doping widens silicene gap. The Journal of Physical Chemistry C. 2014;118(32):18788–92. https://doi.org/10.1021/jp5060809.
- 30. Chuan MW, Wong KL, Hamzah A, Rusli S, Alias NE, Lim CS, et al. Two-dimensional modelling of uniformly doped silicene with aluminium and its electronic properties. Advances in nano research. 2020;9(2):105–12. http://doi.org/10.12989/anr.2020.9.2.105.
- 31. Kharadi MA, Malik GFA, KHANDAY FA, Shah K, Mittal S, Kaushik BK. Silicene: From Material to Device Applications. ECS Journal of Solid State Science and Technology. 2020. https://doi.org/10.1149/2162-8777/abd09a.
- 32. Kazmierski TJ, Zhou D, Al-Hashimi BM, Ashburn P. Numerically efficient modeling of CNT transistors with ballistic and nonballistic effects for circuit simulation. IEEE Transactions on Nanotechnology. 2009;9(1):99–107. https://doi.org/10.1109/TNANO.2009.2017019.
- 33. Chuan MW, Wong KL, Hamzah A, Rusli S, Alias NE, Lim CS, et al. Electronic properties and carrier transport properties of low-dimensional aluminium doped silicene nanostructure. Physica E: Low-dimensional Systems and Nanostructures. 2020;116:113731. https://doi.org/10.1016/j.physe.2019.113731.
- 34. Chuan MW, Wong KL, Hamzah A, Rusli S, Alias NE, Lim CS, et al. Device modelling and performance analysis of two-dimensional AlSi3 ballistic nanotransistor. Advances in nano research. 2021;10(1):91–9. http://doi.org/10.12989/anr.2021.10.1.091.
- 35. Lim WH, Hamzah A, Ahmadi MT, Ismail R. Band gap engineering of BC2N for nanoelectronic applications. Superlattices Microstruct. 2017;112:328–38. https://doi.org/10.1016/j.spmi.2017.09.040.
- 36. Rahman A, Guo J, Datta S, Lundstrom MS. Theory of ballistic nanotransistors. IEEE Trans Electron Devices. 2003;50(9):1853–64. https://doi.org/10.1109/TED.2003.815366.
- 37. Chuan MW, Wong KL, Hamzah A, Rusli S, Alias NE, Lim CS, et al. A review of the top of the barrier nanotransistor models for semiconductor nanomaterials. Superlattices Microstruct. 2020;140:106429. https://doi.org/10.1016/j.spmi.2020.106429.
- 38. Schwierz F, Pezoldt J, Granzner R. Two-dimensional materials and their prospects in transistor electronics. Nanoscale. 2015;7(18):8261–83. https://doi.org/10.1039/C5NR01052G.
- 39. Kazmierski TJ, Zhou D, Al-Hashimi BM, editors. A fast, numerical circuit-level model of carbon nanotube transistor. 2007 IEEE International Symposium on Nanoscale Architectures; 2007: IEEE. https://doi.org/10.1109/NANOARCH.2007.4400855.
- 40. Chin HC, Lim CS, Wong WS, Danapalasingam KA, Arora VK, Tan MLP. Enhanced device and circuit-level performance benchmarking of graphene nanoribbon field-effect transistor against a nano-MOSFET with interconnects. Journal of Nanomaterials. 2014;2014. https://doi.org/10.1155/2014/879813.
- 41. Kim R, Lundstrom M. Notes on Fermi-Dirac Integrals (3rd Edition). 2008. https://arxiv.org/abs/0811.0116.
- 42. Lam K-T, Dong Z, Guo J. Performance limits projection of black phosphorous field-effect transistors. IEEE Electron Device Letters. 2014;35(9):963–5. https://doi.org/10.1109/LED.2014.2333368.
- 43. Yu B, Chang L, Ahmed S, Wang H, Bell S, Yang C-Y, et al., editors. FinFET scaling to 10 nm gate length. Digest International Electron Devices Meeting; 2002: IEEE. https://doi.org/10.1109/IEDM.2002.1175825.
- 44. Chen X, Tan CM. Modeling and analysis of gate-all-around silicon nanowire FET. Microelectronics Reliability. 2014;54(6–7):1103–8. https://doi.org/10.1016/j.microrel.2013.12.009.
- 45. Ma L, Wang J, Ding F. Recent progress and challenges in graphene nanoribbon synthesis. Chemphyschem. 2013;14(1):47–54. https://doi.org/10.1002/cphc.201200253.
- 46. Eshkalak MA, Faez R, Haji-Nasiri S. A novel graphene nanoribbon field effect transistor with two different gate insulators. Physica E: Low-dimensional Systems and Nanostructures. 2015;66:133–9. https://doi.org/10.1016/j.physe.2014.10.021.
- 47. Tan MLP, Lentaris G, Amaratunga GA. Device and circuit-level performance of carbon nanotube field-effect transistor with benchmarking against a nano-MOSFET. Nanoscale research letters. 2012;7(1):467. https://doi.org/10.1186/1556-276X-7-467.
- 48. Jiang Y, Laurenciu NC, Cotofana S, editors. Complementary arranged graphene nanoribbon-based Boolean gates. 2018 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH); 2018: IEEE. https://doi.org/10.1145/3232195.3232199.
- 49. Moaiyeri MH, Razi F. Performance analysis and enhancement of 10-nm GAA CNTFET-based circuits in the presence of CNT-metal contact resistance. Journal of Computational Electronics. 2017;16(2):240–52. https://doi.org/10.1007/s10825-017-0980-0.