Thermoelectrics is an active field in nanotechnology for converting thermal energy into electricity. Researchers are trying to enhance the thermoelectric efficiency of various materials. In recent decades, owing to high-precision tools, engineering the shape of materials at nanoscales where quantum mechanical effects play a major role in determining the physical properties [1] is not just a dream.

After realizing graphene in 2004 [2], the world of two-dimensional (2D) materials attracted the attention of researchers, since they possess extraordinary properties.

Phosphorene, the two-dimensional form of black phosphorus [3, 4], is one of the materials that is of interest in the scientific community. Owing to its specific geometry, the physical properties of the phosphorene monolayer are not isotropic and are depend on the direction [3–8]. Phosphorene is proposed for a variety of applications such as transistors, batteries, water splitting, sensors, and optoelectronic applications [9–12].

The chemical stability of black phosphorus is better than that of red and white ones [1]. Phosphorene is a direct bandgap semiconductor (>1.5 eV) with high carrier mobility [8, 13–16]. The lattice thermal conductivity for phosphorene in the zigzag direction is higher than it is in the armchair direction [15, 17]. Although, phosphorene is unstable when exposed to air, to overcome this challenge, it is usually sandwiched between other materials [16]. Tailoring nanostructures can change their physical properties, and for phosphorene, despite it being an intrinsic semiconductor, its zigzag nanoribbon has a metallic behavior [18].

Quantum confinement effects become important at the nanoscale level and can hence help control the physical properties of a material, such as the Seebeck coefficient, which is essential for thermoelectric performance [19–21]. In this study, we investigate the electronic transport properties of a phosphorene nanodisk with a radius of R = 3.1 nm connected to two zigzag phosphorene nanoribbons (ZPNRs) with 12 atoms widthwise (12-ZPNR), as shown in Figure 1. We use the tight-binding and the nonequilibrium Green’s function methods for this goal. As reported in [22, 23], the five-hopping TB model is accurately reconstructed near the Fermi bands for the phosphorene monolayer. We also applied two electric fields perpendicular to the transport direction of up to 0.3 V / nm , close to the possible experimental values [24], to see how the system responds to transverse ( E x ) and perpendicular ( E z electric fields.

This article is arranged as follows: in the next section, we describe the model and method with a brief introduction to the TB and NEGF methods. In the section after that, the results and discussion are presented. In the last section, we conclude our study.

The system consists of a phosphorene disk connected to two metallic leads, as shown in Figure 1. Two zigzag phosphorene ribbon leads are labeled as the source and drain. Besides, the central region between the two leads is a nanodisk device which contains 798 atoms.

The width of the ZPNR leads includes 12 atoms. Phosphorene is a semiconductor, but cutting it into a zigzag nanoribbon turns it into a metal as mentioned in [18, 25, 26] and can be seen from the TB electronic band structure shown in Figure 1.

The electronic and transport behaviors of the system are described by the TB Hamiltonian: H = ∑ i ε i i i + ∑ < i , j > t i , j i ⟩ ⟨ j + h . c . , (1) where ε i is the on-site energy, and t i , j is the hopping parameter between atoms i and j. The TB parameters are adopted from [23, 27], which are t 1 = − 1.22 ,   t 2 = 3.665 ,   t 3 = − 0.205 ,   t 4 = − 0.105 , and t 5 = − 0.055 eV. The details of the calculation method can be found in [28]. The on-site energy, ε i , is zero in the absence of an electric field but changes when the electric field is turned on. Here, according to Figure 1, in the presence of a transverse electric field, E x , the on-site energy varies from zero to ε i = e E x x i ( x i is the distance of the atom i from the y-axis, x-coordinate of atom i) [29]. This holds for E z   as ε i = e E z z i with z i = 0 or z i = d , where d is the thickness of phosphorene.

Moreover, the electric fields are only applied to the nanodisk, i.e., the device section. Therefore, the electrodes do not feel any electric field. The calculation method is available in [29]. To study the transport properties, the TB Hamiltonians are implemented in the NEGF formalism. The retarded Green’s function can be evaluated as follows [30]: G E = E + i η I − H C − Σ S C E − Σ D C E − 1 , (2) where E is the electron energy, I is the identity matrix, η is an arbitrarily small positive number, H C is the central region (or device) Hamiltonian, and Σ S C D C is the self-energy for the source (drain) lead. Details about this formalism can be found in [16, 31].

The spectral density operator is given by Γ S D E = i Σ S C D C E − Σ S C D C E † , (3)

The transmission probability for the electron can be obtained as follows: T E = Trace Γ S E G E Γ D E G E † . (4)

Besides, the local density of state (LDOS) for a given atom (indicated by index i) can be derived by evaluating LDOS E i = − 1 π Im G E i , (5) Now, one can evaluate the electronic conductance, g; the Seebeck coefficient, S; and the electronic thermal conductance, κ as given in [32, 33]: S μ , T = 1 e T L 1 μ , T L 0 μ , T , (6)

The elementary charge is indicated by e in Eq. 6, and L n is given by L n μ , T = − 2 h ∫ − ∞ ∞ T e E E − μ n k B T exp ⁡   ( E − μ k B T ) exp ⁡   ( E − μ k B T ) + 1 2 d E , (7) with h as the Plank constant and k B as the Boltzmann constant. The temperature is T = 300 K in the calculations.

The electronic transport properties and the Seebeck coefficient are studied in a phosphorene nanodisk with a radius of R = 3.1 nm with the help of the TB method and NEGF formalism. As mentioned earlier, phosphorene is not an isotropic structure.

The electronic conductance in the zigzag direction is about an order of magnitude lower than the armchair direction, as reported in [6]. In contrast to electronic conductance, the phononic conductance in the zigzag direction is about 40% higher than that in the armchair direction. As one can see in Eqs. 6, 7, the Seebeck coefficient is calculated based on the transmission coefficient. Hence, to have a larger S, a zero transmission coefficient with respect to the Fermi energy is of high importance (the wider energy gap, the higher S) or a transmission coefficient with sharp peaks followed by low values. Therefore, we first investigate the transmission coefficient. In the absence of any electric field, the ZPNR is a metal, as evidenced by its electronic band structure (Figure 1) and the transmission spectrum (thick red line in the panels of Figure 2). In the case of the nanodisk, the structure has an energy gap of 3.88 eV, as shown in Figure 2A (magenta line). With the increase of the transverse electric field strength (blue line for E x = 0.15 V/nm and green line for 0.3 V/nm), the electronic transmission coefficient of the system decreases. The effect of E z is shown in Figure 2B. This field has a weaker effect on the transmission spectra but is still in the form of a reduced transmission coefficient.

We note that colors are preserved in all figures respect to the structures and studied electric fields. The behavior of the transmission coefficient is important in determining the Seebeck coefficient. On the other hand, based on the formalism used in this work, at a fixed temperature, the overall transport coefficient is important in determining the Seebeck coefficient, as the integral in Eq. 7 says.

It is important to know what states are present in the device, so that the energy levels are plotted against the state index by solving the eigenvalue problem for each state in the device [34], as shown in Figure 3A. The bulk energy gap in the device, based on these energy levels, is ∼1.66 eV, close to the phosphorene bandgap, indicating that from the electronic energy gap point of view, the behavior of this nanodisk is almost close to that of a single-layer phosphorene.

In the next step, we take a look into the local DOS. The normalized LDOS is obtained respect to the highest LDOS in the particular system. Figure 3B presents the normalized LDOS of the PDisk with no bias. The LDOS shows that atoms in the upper and lower layers of the puckered structure of phosphorene have different local electron densities, and according to the central point of the device, there is a mirror symmetric behavior from the LDOS point of view in all directions. Here, we arrange the structure according to the origin of the coordination system (0, 0, 0). The high LDOS states are located where the electrodes are connected to the device. In the disk region, the LDOS distribution is not too different, a clue of lower localization and thus of better conductivity. Higher localized states can enhance the Seebeck coefficient by reducing the contribution of electrons in conductivity. In this case, the high LDOS states in the path of entry and exit of the electrons in the system suppresses the participation of electrons in their related properties. Figure 3C shows the effect of an electric field of E x = 0.15 V/nm on the LDOS distribution. This field breaks the symmetric LDOS distribution. Note that for this electric field, if one assumes the puckered structure of the phosphorene as two adjacent layers, the LDOS is not equally distributed in these layers.

The path for moving an electron between the two leads is not straight, since high LDOS atoms are surrounded by relatively low LDOS ones, making it an uneven path. It should be noted that atoms at the lowest x-coordination do not experience electric potential, but the ones at the highest x-coordination experience the highest electric potential, so the atoms between these hypothetical lines feel a relative electric potential with respect to their position.

For E x = 0.3 V/nm, the LDOS distribution changes, according to Figure 3D. This field suppresses the transmission probability by localizing the states more than the weaker E x field, as evidenced in Figure 2A by the green area. High LDOS values are concentrated on the +x and −x coordination of the PDisk, meaning the electronic contribution is not favorable in these regions in the transmission probability.

The LDOS in the presence of perpendicular electric fields of 0.15 and 0.3 V/nm are presented, respectively, in the panels of Figure 3E, F. For the electric fields, the LDOS distribution is mirror symmetric with respect to the center, similar to the system without applying an electric field.

The Seebeck coefficient is plotted in Figure 4 for the leads and the transport system and in the presence of the different applied electric fields. As we mentioned earlier, more suppression of the overall transmission coefficient, especially close to the Fermi level of the system can lead to a higher Seebeck coefficient.

The Seebeck coefficient of the leads is small, which is a consequence of metallicity (thick red line in both and panels in Figures 4A, B). For an applied transverse electric field (blue line in Figure 4A), one can see that a moderate electric field, E x = 0.15 V/nm, suppresses S in comparison to the unbiased system, but a strong field, E x = 0.3 V/nm, makes the absolute value of S slightly higher than the unbiased one.

This behavior is also evident in the case of a perpendicular electric field (Figure 4B), although the Seebeck coefficient does not reach to the case in the absence of an electric field. The sign change of the Seebeck coefficient is attributed to the change of the carrier type, meaning that the high S for E x = 0.3 V/nm is caused by holes. These results suggest that the effect of a perpendicular electric field on S is weaker than of the transverse field.

From a thermoelectric point of view, the square of the Seebeck coefficient is important when calculating the thermoelectric efficiency [35]. The maximum absolute value of S in the absence of an electric field is ∼2.03 mV/K at a chemical potential μ of − 0.8 eV. The highest S, 2.09 mV/K, is achieved for E x = 0.3 V/nm at μ = 0.58 eV. The effect of E z on S is weaker and does not reach the no applied electric field. The results are listed in Table 1.

In this work, we have studied the transmission coefficient and the Seebeck coefficient of a phosphorene nanodisk with a radius of 3.1 nm within the framework of the five-hopping TB model and with the help of NEGF formalism. According to our numerical results, engineering phosphorene to a nanodisk at the nanoscale increases the Seebeck coefficient to 2.03 mV/K and induces an energy gap of 3.88 eV. Our numerical results have shown that with an increase of the magnitude of electric field, the transmission coefficient decreases and the Seebeck coefficient changes. Moreover, an electric field perpendicular to the disk surface has a smaller impact on the Seebeck coefficient than a transverse electric field. A transverse electric field with the strength of 0.3 V/nm enhances the Seebeck coefficient to 2.09 mV/K. These results can help in the design of phosphorene-based materials in thermoelectrics.