Two-dimensional (2D) materials have interesting unique properties, and due to their applications in various fields, they are currently the subject of many researches. Graphene, which is composed of carbon atoms arranged in a hexagonal configuration and it is the most well-known example of the two-dimensional materials. Its lack of a band gap stems from the touching of the valence and conduction bands at the Dirac point [1]. As a result, Graphene has massless fermions at the Dirac point and some studies have shown that controlling the Dirac point voltage is necessary to realize the various practical applications of Graphene [2]. It should be emphasized that near the Dirac point, the band structure of Graphene exhibits linearity with respect to momentum [3].

Beyond the Graphene, the 2D material structures with a Graphene-like structure but containing one or two non-carbon atoms have attracted much research. The 2D materials with single element like Silicene and Germanene have buckled structures and similar to Graphene, they exhibit a lack of a band gap at the Dirac point [4,5] and become semiconductor with a nonzero band gap, when they are influenced with a bias voltage [6]. The graphene analog materials containing two different atom types like GeC, SiC, GaN and BN, are energetically favorable with flat hexagonal lattices and they are semiconductors with non-zero band gap [[7], [8], [9], [10]].

In addition to Graphene, in recent years, many studies have investigated the properties of 2D hexagonal boron nitride (h-BN) due to its highly stable structure, mechanical properties and good thermal conductivity [11,12]. The high thermal stability in the BN structure allows it to be used for thermoelectric applications such as high thermal conductivity [13]. Two-dimensional hexagonal boron nitride (h-BN) is made up of Boron (B) and Nitrogen (N) atoms that, like Graphene, are configured in a hexagonal lattice and it has a wide band gap of approximately 4 eV [14]. It should be noted that h-BN systems can also form boron nitride nanotubes (BNNTs) and boron nitride nanoribbons (BNNRs).

Hexagonal boron nitride possesses an atomically flat surface free of dangling bonds and charge traps, enabling its use as a substrate for mono- and bilayer graphene [15]. Due to having the same lattice constant and hexagonal structure similar to graphene and based on the insulating properties, the h-BN can be used as a charge leakage barrier layer for application in electronic devices and also it can be employed to increase the stability and quality of electrical properties of Graphene in future graphene electronics [16]. The characteristics of hexagonal boron nitride have been explored in several studies and it shows interesting optical applications for optical storage and nanosurgery due to its direct band gap in the ultraviolet region [17,18]. It has no optical absorption peak in the visible region while it shows a strong absorption peak in the deep ultraviolet range [19]. Also, the application of this structure has been reported for use as a the deep ultraviolet light emitter and it can be applied for development of deep ultraviolet photoconductive detectors [20,21]. Another application that has been investigated for h-BN is gas sensing properties, which have recently been studied for different gas molecules such as NO2, NO, NH3 and CO [22,23].

The wide band gap in the monolayer BN nanostructure can be decreased and controlled in different ways such as applying an external electric field [24,25], strain [[26], [27], [28], [29]] and applying impurities [[30], [31], [32], [33], [34]]. This work focuses on analyzing the influence of carbon dopants on the thermoelectric performance of single-layer h-BN. Compared with the boron and nitrogen atoms, the Carbon atom has approximately the same atomic radius, making it a viable option for use as an impurity in the BN structure.

Synthesizing doped 2D materials is more challenging than pure 2D materials. Doping can be achieved through several methods like mechanical exfoliation, surface functionalization, vapor deposition, plasma processing, absorption, electrochemical techniques, and thermal evaporation. Each of these methods has its own strengths and limitations [35]. Mechanical exfoliation can produce high-quality, ultrathin layers via weak van der Waals forces [36,37], but lacks control over thickness [35]. Surface functionalization allows various compositions of 2D functional units on few-layered 2D materials but needs improved stability [35]. Vapor deposition techniques like PVD and CVD are widely used to synthesize large-area, high-quality doped 2D materials [38], with precise control over thickness and dopant concentration during growth [39]. The technique of electrochemical intercalation has been employed to introduce dopants into 2D materials, and generate diverse superlattices with customized molecular structures, interlayer distances, as well as electronic and optical properties [40].

2D compounds consisting of B, N and C atoms are stable and have semiconducting properties [41] and the electronic applications of BCN materials, can be increased by controlling the carbon impurity concentration [42]. The C doping reduces the band gap of the BN monolayer and changes its adsorption abilities [43,44]. Also, the poor performance for h-BN-based photodetectors can be increased and improved by C doping [45]. In addition to the Carbon atom, the BN doped with different dopants such as Si [32], O [46] and transition metals (Co, Cu, Ni, Zr, and Bi) [33] has been investigated. Introducing impurities modifies the electronic properties of pure h-BN, such as decreasing the band gap. As an example, replacing B and N sites with Si dopants lowers the energy gap of the h-BN sheet to 1.24 eV and 0.84 eV, respectively [32].

The thermal characteristics of the h-BN have attracted much research attention and been explored in several works. The monolayer h-BN has thermal conductivity up to 600 W/mK which is less than that of the thermal conductivity of Graphene which is about 3500–5300 W/mK [47,48]. The thermal conductivity of BN structures depends on the number of layers; increasing the layers causes a reduction in thermal conductivity [49]. The multilayer BN with five layers has thermal conductivity about 250 W/mK which is nearly comparable to the thermal conductivity of bulk BN [50].

By creating multilayers consisting of graphene and BN, a higher thermal conductivity can be achieved which has been shown for the bilayer Graphene/BN structure [51]. In addition to the above mentioned structures, the thermal conductivity of BCN structures shows less temperature dependence and decreases with compressive strain and increases/decreases with tensile strain, respectively [52]. Additionally, the BC2N is more stable than BCN and has higher thermal conductivity than BCN with dependence on temperature, size and strain [53]. Note that, two-dimensional carbon and BN allotropes, such as penta-graphene, BN-doped graphyne, and BN-like graphyne, have promising thermoelectric properties [54,55]. Their thermal conductivity increases with temperature, and they exhibit a large Seebeck coefficient.

While BN nanostructures have been extensively investigated for their electronic properties, less attention has been given to their thermal characteristics. In particular, the thermal performance of C-doped h-BN under a magnetic field, has not been explored. The present work investigates the effects of impurity type on the thermoelectric features of carbon doped boron nitride, using the Kubo-Greenwood formula and Green function approach.