Blue phosphorene: Calculation of five-band k⋅p Hamiltonian based on group theory and infinitesimal basis transformations approach Narges Kafaei a , Mohammad Sabaeian a, b, * , Abdolmohammad Ghalambor-Dezfuli a, b a Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran b Center for Research on Laser and Plasma, Shahid Chamran University of Ahvaz, Ahvaz, Iran ABSTRACT In this study, we present the matrix representation of a five-band effective k⋅p Hamiltonian of monolayer blue phosphorene using the group theory and infinitesimal basis transformations approach. To the best of our knowledge, using the symmetry properties of monolayer blue phosphorene, which belongs to the D 3d point group, and considering five bands (four in the valence band (VB) and one in the conduction band (CB)), we extract the matrix representation of the effective k⋅p Hamiltonian of monolayer blue phosphorene at the Г point for the first time. The matrix elements of the Hamiltonian definitely show the anisotropy of VB and CB, and they have different effective masses at the center of the Brillouin zone. 1. Introduction Research has been conducted into the various allotropes of phos- phorus for a century [1,2]. After the discovery of graphene (a gapless honeycomb two-dimensional (2D) structure of carbon atoms with sp 2 hybridization), phosphorene was introduced as one of the very few stable 2D materials with an appropriate band gap and remarkable properties, which make it promising for applications in electronics and optoelec- tronics [3]. At present, 2D phosphorus has been obtained in black and blue monolayer forms [4,5], whereas its bulk counterparts can be found in white, violet, and red forms. Among the all bulk phosphorus allotropes, the black form possesses the most stable structure [3,4]. Monolayer black phosphorus (or black phosphorene) is a P-type semiconductor and a promising candidate for future applications in electronics and optoelectronics [6–9] due to the natural and tunable band gap of this 2D material [10–14]. N-type black phosphorene can also be synthesized under suitable doping with electron donors as well as the application of an external electric field and/or tensile strains [15]. Recently, it has been shown that the puckered structure of black phosphorus can be changed into a more symmetric buckled structure known as blue phosphorus via the translocation of phosphorus atoms. Blue phosphorus is as stable as black phosphorus [16–18], although it has a cohesive energy that is a few meV higher than that of black phosphorus [19]. Using molecular beam epitaxy, Zhang et al. successfully fabricated monolayers of blue phosphorus on Au substrates [20]. This so-called blue phosphorene can also be mechanically exfoliated from its quasi-2D pre- cursors because of the weak interlayer van der Waals interactions be- tween the successive layers [16]. Due to some differences in the structures of black and blue phos- phorus, diverse electronic properties are expected. For instance, mono- layer black phosphorus is a direct band gap semiconductor, whereas the monolayer blue phosphorus is an indirect band gap semiconductor [17]. The band gap size of black phosphorus depends on the number of layers, where it varies from 0.3 eV (bulk structure) to 1.45 eV (monolayer structure) [21–23]. These values are still much smaller than the ideal value of  2 eV, which is required to fabricate electronic, optoelectronic, and photovoltaic devices [24]. Blue phosphorene exhibits a natural band gap that exceeds 2 eV [3,24], and thus it appears to have favorable electronic properties [16,25,26]. Furthermore, it has been shown that strain can change the band gap over a relatively large interval [16,18]. Based on first principles calculations, Jiafeng et al. [24] reported that the quantum confinement in armchair nanoribbons of blue phosphorene is stronger than that in zigzag ones, thereby allowing the energy band gap to be tuned in this 2D structure. Monolayers of black phosphorene were experimentally exfoliated for the first time via mechanical cleavage on a Si substrate with a 300-nm SiO 2 capping layer [27]. Zhu et al. theoretically predicted that blue phosphorene is as thermally stable as black phosphorene [28]. Blue phosphorene will be more stable at temperatures greater than 33 K. A structural phase transition occurs from black phosphorene to blue phosphorene when the temperature exceeds 135 K [19]. Compared with MoS 2 [29], blue phosphorene has much higher carrier mobility in a * Corresponding author. Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran. E-mail address: sabaeian@scu.ac.ir (M. Sabaeian). Contents lists available at ScienceDirect Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs https://doi.org/10.1016/j.jpcs.2018.02.041 Received 26 August 2017; Received in revised form 18 February 2018; Accepted 19 February 2018 0022-3697/© 2018 Elsevier Ltd. All rights reserved. Journal of Physics and Chemistry of Solids 118 (2018) 1–5