Carbon Nanotubes and Nanosheets for Sustainable Solutions

A special issue of Nanomaterials (ISSN 2079-4991). This special issue belongs to the section "2D and Carbon Nanomaterials".

Deadline for manuscript submissions: closed (20 March 2024) | Viewed by 62271

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Guest Editor
Consultant, NanoWorld Innovations (NWI), 1 Jalan Mawar, Singapore 368931, Singapore
Interests: nanomaterials & nanotechnology; nanoscience for renewable energy; synthesis and applications of nanomaterials; nanoparticle- and/or nanofiber-based materials
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Special Issue Information

Dear Colleagues,

This is a broader continuation of the Special Issue in MDPI Nanomaterials “70th Year Anniversary of Carbon Nanotube Discovery - Focus on Real World Solutions”.

Carbon nanotubes (CNTs) and nanosheets (graphene) are well known for their current applications in: (1) durable battery electrodes, (2) optical/thermal nanodevices, (3) nanoscale reinforcement for strong and conductive hybrid nanocomposites, (4) ultrahigh-performance CPUs for quantum computing, (5) anti-fouling eco-friendly coatings, (6) high-efficiency recoverable and reusable catalysts and adsorbents, (7) nerve repair, (8) anti-oxidant particles combating oxidative stress, and (9) potent drug delivery.

Sustainable solutions include: energy harvesting (1) or conversion (2), ultra-strong multifunctional components (3), high-performance and secure computing and data processing (4), cost-saving green coatings (5), systematic material recovery and reuse (6), tissue regeneration (7), gradual ageing (8), and vaccines (9).

Very interestingly and inevitably in science today, many inorganic non-carbon-based nanotubes and nanosheets are also emerging as credible competitors of CNTs and graphene in this arena. These inorganic non-carbon-based nanomaterials can be tailor-synthesized and niche-applied with relative ease and are also quite durable.

The contributions sought are not limited to the above scope, and are inclusive of all forms of inorganic or carbon nanotubes and nanosheets.

Dr. Muralidharan Paramsothy
Guest Editor

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Keywords

  • nanotube
  • nanosheet
  • nanostructure
  • nanodevice
  • nanocomposite
  • biological
  • optical
  • electrical
  • catalyst
  • energy
  • antioxidant
  • drug
  • sustainable

Published Papers (12 papers)

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Research

10 pages, 2732 KiB  
Communication
Directed Evolution of Near-Infrared Serotonin Nanosensors with Machine Learning-Based Screening
by Seonghyeon An, Yeongjoo Suh, Payam Kelich, Dakyeon Lee, Lela Vukovic and Sanghwa Jeong
Nanomaterials 2024, 14(3), 247; https://doi.org/10.3390/nano14030247 - 23 Jan 2024
Viewed by 907
Abstract
In this study, we employed a novel approach to improve the serotonin-responsive ssDNA-wrapped single-walled carbon nanotube (ssDNA-SWCNT) nanosensors, combining directed evolution and machine learning-based prediction. Our iterative optimization process is aimed at the sensitivity and selectivity of ssDNA-SWCNT nanosensors. In the three rounds [...] Read more.
In this study, we employed a novel approach to improve the serotonin-responsive ssDNA-wrapped single-walled carbon nanotube (ssDNA-SWCNT) nanosensors, combining directed evolution and machine learning-based prediction. Our iterative optimization process is aimed at the sensitivity and selectivity of ssDNA-SWCNT nanosensors. In the three rounds for higher serotonin sensitivity, we substantially improved sensitivity, achieving a remarkable 2.5-fold enhancement in fluorescence response compared to the original sequence. Following this, we directed our efforts towards selectivity for serotonin over dopamine in the two rounds. Despite the structural similarity between these neurotransmitters, we achieved a 1.6-fold increase in selectivity. This innovative methodology, offering high-throughput screening of mutated sequences, marks a significant advancement in biosensor development. The top-performing nanosensors, N2-1 (sensitivity) and L1-14 (selectivity) present promising reference sequences for future studies involving serotonin detection. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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11 pages, 8711 KiB  
Article
Spectromicroscopy Study of Induced Defects in Ion-Bombarded Highly Aligned Carbon Nanotubes
by Sammar Tayyab, Alice Apponi, Maria Grazia Betti, Elena Blundo, Gianluca Cavoto, Riccardo Frisenda, Nuria Jiménez-Arévalo, Carlo Mariani, Francesco Pandolfi, Antonio Polimeni, Ilaria Rago, Alessandro Ruocco, Marco Sbroscia and Ravi Prakash Yadav
Nanomaterials 2024, 14(1), 77; https://doi.org/10.3390/nano14010077 - 27 Dec 2023
Viewed by 721
Abstract
Highly aligned multi-wall carbon nanotubes were investigated with scanning electron microscopy (SEM), Raman spectroscopy and X-ray photoelectron spectroscopy (XPS) before and after bombardment performed using noble gas ions of different masses (argon, neon and helium), in an ultra-high-vacuum (UHV) environment. Ion irradiation leads [...] Read more.
Highly aligned multi-wall carbon nanotubes were investigated with scanning electron microscopy (SEM), Raman spectroscopy and X-ray photoelectron spectroscopy (XPS) before and after bombardment performed using noble gas ions of different masses (argon, neon and helium), in an ultra-high-vacuum (UHV) environment. Ion irradiation leads to change in morphology, deformation of the carbon (C) honeycomb lattice and different structural defects in multi-wall carbon nanotubes. One of the major effects is the production of bond distortions, as determined by micro-Raman and micro-X-ray photoelectron spectroscopy. We observe an increase in sp3 distorted bonds at higher binding energy with respect to the expected sp2 associated signal of the carbon 1s core level, and increase in dangling bonds. Furthermore, the surface damage as determined by the X-ray photoelectron spectroscopy carbon 1s core level is equivalent upon bombarding with ions of different masses, while the impact and density of defects in the lattice of the MWCNTs as determined by micro-Raman are dependent on the bombarding ion mass; heavier for helium ions, lighter for argon ions. These results on the controlled increase in sp3 distorted bonds, as created on the multi-wall carbon nanotubes, open new functionalization prospects to improve and increase atomic hydrogen uptake on ion-bombarded multi-wall carbon nanotubes. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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12 pages, 6152 KiB  
Article
Growth and Characterization of Carbon Nanofibers Grown on Vertically Aligned InAs Nanowires via Chemical Vapour Deposition
by Muhammad Arshad, Lucia Sorba, Petra Rudolf and Cinzia Cepek
Nanomaterials 2023, 13(24), 3083; https://doi.org/10.3390/nano13243083 - 05 Dec 2023
Viewed by 884
Abstract
The integration of carbon nanostructures with semiconductor nanowires holds significant potential for energy-efficient integrated circuits. However, achieving precise control over the positioning and stability of these interconnections poses a major challenge. This study presents a method for the controlled growth of carbon nanofibers [...] Read more.
The integration of carbon nanostructures with semiconductor nanowires holds significant potential for energy-efficient integrated circuits. However, achieving precise control over the positioning and stability of these interconnections poses a major challenge. This study presents a method for the controlled growth of carbon nanofibers (CNFs) on vertically aligned indium arsenide (InAs) nanowires. The CNF/InAs hybrid structures, synthesized using chemical vapor deposition (CVD), were successfully produced without compromising the morphology of the pristine nanowires. Under optimized conditions, preferential growth of the carbon nanofibers in the direction perpendicular to the InAs nanowires was observed. Moreover, when the CVD process employed iron as a catalyst, an increased growth rate was achieved. With and without the presence of iron, carbon nanofibers nucleate preferentially on the top of the InAs nanowires, indicating a tip growth mechanism presumably catalysed by a gold-indium alloy that selectively forms in that region. These results represent a compelling example of controlled interconnections between adjacent InAs nanowires formed by carbon fibers. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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16 pages, 5521 KiB  
Article
Polypyrrole Solid-State Supercapacitors Drawn on Paper
by Antonella Arena, Graziella Scandurra, Caterina Branca, Mariangela Ruggeri, Mauro Federico, Valentino Romano, Giovanna D’Angelo and Carmine Ciofi
Nanomaterials 2023, 13(23), 3040; https://doi.org/10.3390/nano13233040 - 28 Nov 2023
Cited by 1 | Viewed by 683
Abstract
Solid-state supercapacitors with areal capacitance in the order of 100 mF⋅cm−2 are developed on paper substrates, using eco-friendly, low-cost materials and a simple technology. The electrochemically active material used as the electrode is prepared from a stable water-based ink, obtained by doping [...] Read more.
Solid-state supercapacitors with areal capacitance in the order of 100 mF⋅cm−2 are developed on paper substrates, using eco-friendly, low-cost materials and a simple technology. The electrochemically active material used as the electrode is prepared from a stable water-based ink, obtained by doping commercial polypyrrole (PPY) powder with dodecylbenzene sulfonic acid (DBSA), and characterized by optical and electrical measurements, Raman investigation and Atomic Force Microscopy. The PPY:DBSA ink can be directly applied on paper by means of rechargeable water pens, obtaining, after drying, electrically conducting solid state tracks. The PPY:DBSA layers are then interfaced to one another through a polymer gel based on potassium hydroxide and chitosan, acting both as the ion-conducting medium and as the separator. The areal capacitance of the devices developed by following such a simple rule can be improved when the PPY:DBSA ink is applied in combination with other nanostructured carbon material. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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12 pages, 2305 KiB  
Article
Transition of Carbon Nanotube Sheets from Hydrophobicity to Hydrophilicity by Facile Electrochemical Wetting
by Myoungeun Oh, Hyunji Seo, Jimin Choi, Jun Ho Noh, Juwan Kim, Joonhyeon Jeon and Changsoon Choi
Nanomaterials 2023, 13(21), 2834; https://doi.org/10.3390/nano13212834 - 26 Oct 2023
Viewed by 913
Abstract
The present study delves into the transformative effects of electrochemical oxidation on the hydrophobic-to-hydrophilic transition of carbon nanotube (CNT) sheets. The paper elucidates the inherent advantages of CNT sheets, such as high electrical conductivity and mechanical strength, and contrasts them with the limitations [...] Read more.
The present study delves into the transformative effects of electrochemical oxidation on the hydrophobic-to-hydrophilic transition of carbon nanotube (CNT) sheets. The paper elucidates the inherent advantages of CNT sheets, such as high electrical conductivity and mechanical strength, and contrasts them with the limitations posed by their hydrophobic nature. A comprehensive investigation is conducted to demonstrate the efficacy of electrochemical oxidation treatment in modifying the surface properties of CNT sheets, thereby making them hydrophilic. The study reveals that the treatment not only is cost-effective and time-efficient compared to traditional plasma treatment methods but also results in a significant decrease in water contact angle. Mechanistic insights into the hydrophilic transition are provided, emphasizing the role of oxygen-containing functional groups introduced during the electrochemical oxidation process. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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11 pages, 2757 KiB  
Article
Electrochemically Oxidized Carbon Nanotube Sheets for High-Performance and Flexible-Film Supercapacitors
by Jun Ho Noh, Jimin Choi, Hyunji Seo, Juwan Kim and Changsoon Choi
Nanomaterials 2023, 13(20), 2814; https://doi.org/10.3390/nano13202814 - 23 Oct 2023
Cited by 1 | Viewed by 952
Abstract
The development of flexible, high-performance supercapacitors has been a focal point in energy storage research. While carbon nanotube (CNT) sheets offer promising mechanical and electrical properties, their low electrical double-layer capacitance significantly limits their practicability. Herein, we introduce a novel approach to address [...] Read more.
The development of flexible, high-performance supercapacitors has been a focal point in energy storage research. While carbon nanotube (CNT) sheets offer promising mechanical and electrical properties, their low electrical double-layer capacitance significantly limits their practicability. Herein, we introduce a novel approach to address this challenge via the electrochemical oxidation treatment of CNT sheets stacked on a polyethylene terephthalate substrate. This introduces oxygen-containing functional groups onto the CNT surface, thereby dramatically enhancing the pseudocapacitive effect and improving ion adsorption. Consequently, using the material in a two-electrode system increased the capacitance by 54 times compared to pristine CNT. The results of electrochemical performance characterization, including cyclic voltammograms, galvanostatic charge/discharge curves, and capacitance retention testing data, confirm the efficacy of the electrochemical oxidation approach. Furthermore, the mechanical flexibility of the electrochemically wetted CNT sheets was validated through resistance and discharge retention testing under repetitive bending (98% capacitance retention after 1000 bending cycles). The results demonstrate that electrochemically wetted CNT sheets retain their intrinsic mechanical and electrical properties while significantly enhancing the electrochemical performance (0.59 mF/cm2 or 97.8 F/g). This work represents a significant advancement in the development of flexible, high-performance supercapacitors with potential applicability to wearable electronics, flexible displays, and next-generation energy storage solutions. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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13 pages, 4577 KiB  
Article
Effect of Atmospheric Temperature on Epoxy Coating Reinforced with Carbon Nanotubes for De-Icing on Road Systems
by Seung-Jun Lee, Yu-Jin Jung, Chunhee Cho and Sung-Hwan Jang
Nanomaterials 2023, 13(15), 2248; https://doi.org/10.3390/nano13152248 - 03 Aug 2023
Viewed by 1024
Abstract
Traffic accidents caused by road icing are a serious global problem, and conventional de-icing methods like spraying chemicals have several limitations, including excessive manpower management, road damage, and environmental pollution. In this study, the carbon nanotubes reinforced de-icing coating for the road system [...] Read more.
Traffic accidents caused by road icing are a serious global problem, and conventional de-icing methods like spraying chemicals have several limitations, including excessive manpower management, road damage, and environmental pollution. In this study, the carbon nanotubes reinforced de-icing coating for the road system with a self-heating function was developed as part of the development of a new system to prevent accidents caused by road icing. The electrical characteristics of the fabricated coating were analyzed, and the carbon nanotube coating heating performance experiment was conducted to measure the temperature increments by applying a voltage to the coating at a sub-zero temperature using an environmental chamber. In addition, the coating was installed on the road pavement and the applicability was investigated through a heating test in winter. As a result of the experiment, the coating made with the higher carbon nanotube concentration presented higher heating owing to its higher electrical conductivity. In addition, the coating showed sufficient heating performance, although the maximum temperature by Joule heating decreased for the entire coating at sub-zero temperatures. Finally, field tests demonstrated the potential of electrically conductive coatings for de-icing applications. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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15 pages, 4024 KiB  
Article
Microstructure and Superconducting Properties of Bi-2223 Synthesized via Co-Precipitation Method: Effects of Graphene Nanoparticle Addition
by Siti Nabilah Abdullah, Mohd Mustafa Awang Kechik, Aliah Nursyahirah Kamarudin, Zainal Abidin Talib, Hussein Baqiah, Chen Soo Kien, Lim Kean Pah, Muhammad Khalis Abdul Karim, Muhammad Kashfi Shabdin, Abdul Halim Shaari, Azhan Hashim, Nurbaisyatul Ermiza Suhaimi and Muralidhar Miryala
Nanomaterials 2023, 13(15), 2197; https://doi.org/10.3390/nano13152197 - 28 Jul 2023
Cited by 2 | Viewed by 43245
Abstract
The effects of graphene addition on the phase formation and superconducting properties of (Bi1.6Pb0.4)Sr2Ca2Cu3O10 (Bi-2223) ceramics synthesized using the co-precipitation method were systematically investigated. Series samples of Bi-2223 were added with different [...] Read more.
The effects of graphene addition on the phase formation and superconducting properties of (Bi1.6Pb0.4)Sr2Ca2Cu3O10 (Bi-2223) ceramics synthesized using the co-precipitation method were systematically investigated. Series samples of Bi-2223 were added with different weight percentages (x = 0.0, 0.3, 0.5 and 1.0 wt.%) of graphene nanoparticles. The samples’ phase formations and crystal structures were characterized via X-ray diffraction (XRD), while the superconducting critical temperatures, Tc, were investigated using alternating current susceptibility (ACS). The XRD showed that a high-Tc phase, Bi-2223, and a small low-Tc phase, Bi-2212, dominated the samples. The volume fraction of the Bi-2223 phase increased for the sample with x = 0.3 wt.% and 0.5 wt.% of graphene and slightly reduced at x = 1.0 wt.%. The ACS showed that the onset critical temperature, Tc-onset, phase lock-in temperature, Tcj, and coupling peak temperature, TP, decreased when graphene was added to the samples. The susceptibility–temperature (χ′-T) and (χ″-T) curves of each sample, where χ′ and χ″ are the real and imaginary parts of the susceptibility, respectively, were obtained. The critical temperature of the pure sample was also measured. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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30 pages, 6277 KiB  
Article
Semi-Empirical Pseudopotential Method for Graphene and Graphene Nanoribbons
by Raj Kumar Paudel, Chung-Yuan Ren and Yia-Chung Chang
Nanomaterials 2023, 13(14), 2066; https://doi.org/10.3390/nano13142066 - 13 Jul 2023
Viewed by 1603
Abstract
We implemented a semi-empirical pseudopotential (SEP) method for calculating the band structures of graphene and graphene nanoribbons. The basis functions adopted are two-dimensional plane waves multiplied by several B-spline functions along the perpendicular direction. The SEP includes both local and non-local terms, which [...] Read more.
We implemented a semi-empirical pseudopotential (SEP) method for calculating the band structures of graphene and graphene nanoribbons. The basis functions adopted are two-dimensional plane waves multiplied by several B-spline functions along the perpendicular direction. The SEP includes both local and non-local terms, which were parametrized to fit relevant quantities obtained from the first-principles calculations based on the density-functional theory (DFT). With only a handful of parameters, we were able to reproduce the full band structure of graphene obtained by DFT with a negligible difference. Our method is simple to use and much more efficient than the DFT calculation. We then applied this SEP method to calculate the band structures of graphene nanoribbons. By adding a simple correction term to the local pseudopotentials on the edges of the nanoribbon (which mimics the effect caused by edge creation), we again obtained band structures of the armchair nanoribbon fairly close to the results obtained by DFT. Our approach allows the simulation of optical and transport properties of realistic nanodevices made of graphene nanoribbons with very little computation effort. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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21 pages, 10858 KiB  
Article
A Comparison of Shell Theories for Vibration Analysis of Single-Walled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model
by Matteo Strozzi, Isaac E. Elishakoff, Michele Bochicchio, Marco Cocconcelli, Riccardo Rubini and Enrico Radi
Nanomaterials 2023, 13(8), 1390; https://doi.org/10.3390/nano13081390 - 17 Apr 2023
Cited by 6 | Viewed by 1186
Abstract
In the present paper, a comparison is conducted between three classical shell theories as applied to the linear vibrations of single-walled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders, and Flügge shell theories. The actual discrete [...] Read more.
In the present paper, a comparison is conducted between three classical shell theories as applied to the linear vibrations of single-walled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders, and Flügge shell theories. The actual discrete SWCNT is modelled by means of a continuous homogeneous cylindrical shell considering equivalent thickness and surface density. In order to take into account the intrinsic chirality of carbon nanotubes (CNTs), a molecular based anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and a complex method is applied to solve the equations of motion and to obtain the natural frequencies. Comparisons with the results of molecular dynamics simulations available in literature are performed to check the accuracy of the three different shell theories, where the Flügge shell theory is found to be the most accurate. Then, a parametric analysis evaluating the effect of diameter, aspect ratio, and number of waves along the longitudinal and circumferential directions on the natural frequencies of SWCNTs is performed in the framework of the three different shell theories. Assuming the results of the Flügge shell theory as reference, it is obtained that the Donnell shell theory is not accurate for relatively low longitudinal and circumferential wavenumbers, for relatively low diameters, and for relatively high aspect ratios. On the other hand, it is found that the Sanders shell theory is very accurate for all the considered geometries and wavenumbers, and therefore, it can be correctly adopted instead of the more complex Flügge shell theory for the vibration modelling of SWCNTs. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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16 pages, 2068 KiB  
Article
Seed Priming with Single-Walled Carbon Nanotubes Grafted with Pluronic P85 Preserves the Functional and Structural Characteristics of Pea Plants
by Sashka Krumova, Asya Petrova, Nia Petrova, Svetozar Stoichev, Daniel Ilkov, Tsonko Tsonev, Petar Petrov, Dimitrina Koleva and Violeta Velikova
Nanomaterials 2023, 13(8), 1332; https://doi.org/10.3390/nano13081332 - 11 Apr 2023
Cited by 5 | Viewed by 2031
Abstract
The engineering of carbon nanotubes in the last decades resulted in a variety of applications in electronics, electrochemistry, and biomedicine. A number of reports also evidenced their valuable application in agriculture as plant growth regulators and nanocarriers. In this work, we explored the [...] Read more.
The engineering of carbon nanotubes in the last decades resulted in a variety of applications in electronics, electrochemistry, and biomedicine. A number of reports also evidenced their valuable application in agriculture as plant growth regulators and nanocarriers. In this work, we explored the effect of seed priming with single-walled carbon nanotubes grafted with Pluronic P85 polymer (denoted P85-SWCNT) on Pisum sativum (var. RAN-1) seed germination, early stages of plant development, leaf anatomy, and photosynthetic efficiency. We evaluated the observed effects in relation to hydro- (control) and P85-primed seeds. Our data clearly revealed that seed priming with P85-SWCNT is safe for the plant since it does not impair the seed germination, plant development, leaf anatomy, biomass, and photosynthetic activity, and even increases the amount of photochemically active photosystem II centers in a concentration-dependent manner. Only 300 mg/L concentration exerts an adverse effect on those parameters. The P85 polymer, however, was found to exhibit a number of negative effects on plant growth (i.e., root length, leaf anatomy, biomass accumulation and photoprotection capability), most probably related to the unfavorable interaction of P85 unimers with plant membranes. Our findings substantiate the future exploration and exploitation of P85-SWCNT as nanocarriers of specific substances promoting not only plant growth at optimal conditions but also better plant performance under a variety of environmental stresses. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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10 pages, 387 KiB  
Article
Lateral Controlled Doping and Defect Engineering of Graphene by Ultra-Low-Energy Ion Implantation
by Felix Junge, Manuel Auge, Zviadi Zarkua and Hans Hofsäss
Nanomaterials 2023, 13(4), 658; https://doi.org/10.3390/nano13040658 - 08 Feb 2023
Cited by 3 | Viewed by 1704
Abstract
In this paper, the effectiveness of ultra-low-energy ion implantation as a means of defect engineering in graphene was explored through the measurement of Scanning Kelvin Probe Microscopy (SKPM) and Raman spectroscopy, with boron (B) and helium (He) ions being implanted into monolayer graphene [...] Read more.
In this paper, the effectiveness of ultra-low-energy ion implantation as a means of defect engineering in graphene was explored through the measurement of Scanning Kelvin Probe Microscopy (SKPM) and Raman spectroscopy, with boron (B) and helium (He) ions being implanted into monolayer graphene samples. We used electrostatic masks to create a doped and non-doped region in one single implantation step. For verification we measured the surface potential profile along the sample and proved the feasibility of lateral controllable doping. In another experiment, a voltage gradient was applied across the graphene layer in order to implant helium at different energies and thus perform an ion-energy-dependent investigation of the implantation damage of the graphene. For this purpose Raman measurements were performed, which show the different damage due to the various ion energies. Finally, ion implantation simulations were conducted to evaluate damage formation. Full article
(This article belongs to the Special Issue Carbon Nanotubes and Nanosheets for Sustainable Solutions)
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