Researchers have long studied the carbon atom's exceptional ability to form diverse compounds, particularly since the discovery of graphene as a 2D material [[1], [2], [3]].

As a hexagonal lattice structure of carbon atoms with exceptional physical and electronic properties, graphene has become one of the most attractive materials in nanotechnology research. A key property of this material is its zero-band gap semiconductor property, which makes it ideal for high-speed electronic applications. Graphene is known for its exceptionally high mobility of charge carriers (electron and hole carriers) with reported high values at room temperature. The dipolar behavior (Dirac cone) of graphene in its band structure leads to unique quantum properties. Additionally, its strong sp2 bonds between carbon atoms provide it with superior mechanical strength and structural flexibility. These properties have not only made graphene a prime subject in basic scientific exploration but have also fueled its revolutionary potential in technologies such as high-speed transistors, ultrasensitive sensors, composite materials, and energy storage [[4], [5], [6], [7]].

Studies of graphene structure have traditionally classified lattice defects, including pentagons and heptagons, as structural defects [8,9]. However, recent theoretical studies have revealed that these seemingly undesirable defects can be engineered as building blocks to create novel 2D carbon nanomaterials. This finding has led researchers to explore novel 2D carbon forms with unusual topologies, including:•

Novel topological structures, including pentaheptite [10], haeckelites [11], and T-graphene [12].

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Advanced allotropes, such as OPG (Oxide-Poly Graphene) [13], pentahexoctite [14], pentagraphene [15], and phagraphene [16].

We emphasize that all the carbon allotropes mentioned in this section (pentaheptite, haeckelites, T-graphene, OPG, pentahexoctite, pentagraphene, and phagraphene) are currently theoretical predictions. Their stability has been primarily confirmed through computational simulations, and their successful and unambiguous experimental synthesis remains a scientific challenge and an active area for future research.

Each of these theoretical structures possesses unique physical and electronic properties, offering potential for novel applications in nanotechnology. However, experimental challenges in synthesis have prevented the practical realization of these materials. Therefore, the search for more easily synthesized 2D carbon allotropes has emerged as a promising research direction [17].

Research confirms that a variety of graphene allotropes, including Ψ-graphene, have been identified and modeled so far. Some of these allotropes include hydrogenated forms of graphene (e.g., graphane). Meanwhile, others, such as popgraphene, have distinct crystal morphologies. The results of thermodynamic calculations show that temperature changes lead to significant fluctuations in the total energy of the system after the hydrogenation process. These findings suggest the high potential of graphene in hydrogen storage technologies [16,18,19].

Recent computational studies on various two-dimensional materials have demonstrated that atomic-scale structural engineering can significantly alter electronic and magnetic properties. For instance, research on transition metal-doped bismuthene [20] and studies on 2D SrRuO3 [21,22] illustrate how first-principles calculations can reliably predict material properties. In this context, the present investigation examines the electronic and magnetic properties of Ψ-graphene nanoribbons (ΨGNRBs) to expand our understanding of structure-property relationships in novel carbon allotropes.

A research team led by Xiaoyin Li has recently investigated the properties of Ψ-graphene. This emerging material has a molecular structure composed of 5-6-5 carbon rings and a single-atom thickness similar to graphene sheets [23]. The major limitation of graphene for use in electronic and optoelectronic devices is the lack of an energy gap in its electronic structure [24]. An effective solution for this challenge is the hydrogenation process of graphene. Notably, they showed that this process does not alter the lattice vectors of graphene. Due to the presence of strong σ bonds in the σ-graphene structure, a band gap is induced in its energy spectra [25].

Silva et al. [17] investigated the effect of edge hydrogenation on the transport properties of 1D and 2D Ψ-graphenes. They found that the positive effects of hydrogenation enhance the energy stability of the system and improve the structural stability of Ψ-graphene. In contrast, its negative impacts reduce the material's transport capabilities and limit its electronic applications [26]. The present study aims to investigate the impact of length changes and edge hydrogenation on the electronic and magnetic properties of AΨGNRBs (armchair PSI (Ψ)-graphene nanoribbons) and ZΨGNRBs (zigzag PSI (Ψ)- graphene nanoribbons). It will also compare the electronic and magnetic properties of these two types of NRBs (nanoribbons).

The investigation of Ψ-graphene nanoribbons in this work is part of the broader scientific pursuit of understanding how nanoscale structure governs macroscopic properties. This paradigm is evident across diverse material systems: in magnetic nanoparticles where doping and size control magnetic properties [20], in complex oxide systems where structural complexity creates novel electronic transport phenomena [21], and in polymer nanocomposites where interfacial engineering enables tailored optical and energy storage capabilities [22]. Similarly, in carbon nanomaterials, edge structure and termination fundamentally determine electronic behavior. While these referenced studies explore different material classes, they collectively underscore that precise structural control is the key to functional optimization in nanomaterials.