#### 3.1. Crystal and Electronic Structures

The new 2D monolayer materials MAs

_{2} (M = Ni, Pd, Pt), which was obtained by mechanical stripping method, that exhibited good kinetic and thermal stability [

26]. The optimal geometry structures of the monolayer MAs

_{2} (M = Ni, Pd, Pt) are obtained as showed in

Figure 1. From the top view as shown in

Figure 1a, it is clear that the monolayer MAs

_{2} has a tetragonal structure (space group

Pa3No.) and each M atom adopts a planar tetra-ligand with four As atoms. A unit-cell is constituted by two M and four As atoms, while all M atoms are always in one plane with As atoms, as shown in

Figure 1b. Three As atoms and two M atoms form a pentagonal ring network, marked with a black ellipse, as shown in

Figure 1a. The optimized lattice structure parameters are shown in

Table 1. The calculation results of the lattice constant (LC) of MAs

_{2} and the degrees of the α, β and γ angles of the pentagonal unit are consistent with the calculation results of Qian et al. [

26].

The band structure and projected density of states (PDOS) are computed by using HSE06 as plotted in

Figure 2. The presence of direct bang-gap at the S point is clearly visible, which signifies that the monolayer MAs

_{2} is the semiconductor, and the corresponding band gaps are 0.59, 0.80 and 0.34 eV for the monolayer NiAs

_{2}, PdAs

_{2} and PtAs

_{2}, respectively. The band gaps of MAs

_{2} materials is within the ideal band gap range for good thermoelectric materials (0.3–1.0 eV) [

6]. Among them, the result of PdAs

_{2} is very consistent with band gap 0.80 eV by Yuan et al. [

38] and 0.78 eV by Pan et al. [

39]. It can be seen that the monolayer MAs

_{2} possess the great band degeneracy that appears in the valence band (VB) along the X-S direction and primarily originates from the M-orbitals. This kind of band degeneracy means that is MAs

_{2} characterized by excellent thermoelectric performance. At present, the band degeneracy has been enhanced by band engineering with the aim of increasing the flatness of the density of states (DOS) to improve the power factor (PF). From the PDOS, a high density of states near the Fermi level are mainly contributed by the M-orbitals, whereas the As have only a small contribution. Besides, there are spikes near the Fermi energy level that can be observed, which effectively promotes the sharp increasing of Seebeck coefficient.

#### 3.2. Electrical Transport Properties

The electronic properties can be characterized on the basis of carrier mobility for monolayer MAs

_{2}, along the conveyor directions. We calculate them using the deformation potential (DP) theory proposed by Bardeen and Shockley [

36]. The formula of carrier mobility in 2D systems can be written as follows [

35,

40]:

where

k_{B} is the Boltzmann constant, T represents the temperature that is taken as 300 K,

m* is the effective mass for the conveyor direction,

m_{d} is the average effective mass defined by

${m}_{d}=\sqrt{{m}_{x}{m}_{y}}$,

E_{l} is the deformation potential constant, and

C^{2D} is the effective 2D elastic constants, respectively. The calculated effective mass, carrier mobility and relaxation time (τ = μm*/e) are shown in

Table 2. After calculation, it was concluded that the electrical transport properties of MAs

_{2} are isotropic, which results from its perfect lattice symmetry, m

_{d} is equal to m*. Noticeably, it shows a high hole mobility (34.27 cm

^{2}/Vs) of PdAs

_{2} at room temperature, which is much higher than that of NiAs

_{2} (~1.93 cm

^{2}/Vs) and PtAs

_{2} (~4.80 cm

^{2}/Vs). The high mobility in monolayer PdAs

_{2} is associated with the ideal band gap, which is beneficial to its electrical transport, while the mobilities of the holes are PtAs

_{2} and showed a high hole mobility (~17.07 cm

^{2}/Vs) at room temperature, which is much higher than that of NiAs

_{2} (~1.93 cm

^{2}/Vs) and PtAs

_{2} (~4.80 cm

^{2}/Vs).

Based on the Boltzmann transport equation and rigid band approximation, the electricity transport properties under the relaxation time approximation are calculated. After calculation, it was found that the electrical transport properties of MAs

_{2} are isotropic, which results from their perfect lattice symmetry. As shown in

Figure 3, we can find that the Seebeck coefficient (S), electrical conductively (σ/τ), electron thermal conductively (κ

_{e}/τ) and power factor (S

^{2}σ

/τ) as the function of chemical potential (μ) at 300, 500 and 700 K are obtained, whereas the positive and negative μ correspond to n-type and p-type of monolayer MAs

_{2}. The electricity transport properties can be obtained by

where

α and

β are cartesian indices,

V is the volume of the primitive cell and

$\sum _{\alpha \beta}\left(\epsilon \right)$ is the transport distribution function. The S is inversely proportional to temperature, which is proven in

Figure 3a–c. We can clearly find that the absolute value of S of monolayer MAs

_{2} decreases with the increase in temperature. The p-type and n-type doping of monolayer PdAs

_{2} surprisingly possess a very large absolute value of S up to 440 and 460 μV/K at 300 K, which is significant to improve power factor. Meanwhile, the absolute value of p-type S of monolayer NiAs

_{2} and PtAs

_{2} are also observed about 140 and 135 μV/K, respectively. The calculated large S of monolayer MAs

_{2} can be benefited from the PDOS.

The electrical conductively (σ) is one of the important parameters for analyzing thermoelectric properties. As

Figure 3d–f presents, the pretty high σ can be observed, which is very beneficial for optimizing PF and thus improving thermoelectric performance. In addition, we can find that the change in σ is independent of the change in temperature, which is different from the trend of

S. Then, further decomposition suggests that the σ of monolayer PdAs

_{2} is less than that of monolayer NiAs

_{2} and PtAs

_{2}, while the σ of n-type doping is always superior to that of p-type doping.

The electronic thermal conductivity (κ

_{e}) can be calculated by the Wiedemann-Franz law:

where L = π

^{2}κ

_{B}^{2}/3e

^{2} is the Lorenz number. From

Figure 3g–i, we can clearly find that the function curve of

κ_{e} is similar to that of the

σ, which is contributed by the proportional relationship between them.

According to Equation (1), the PF can be evaluated and illustrated in

Figure 3j–l, which is obtained by combining

S with

σ. The maximum value of PF is n-type monolayer PdAs

_{2} up to 3.9 × 10

^{11} W/K

^{2}ms, which is much higher than n-type monolayer NiAs

_{2} and p-type monolayer PtAs

_{2} corresponding to 2.3 × 10

^{11} W/K

^{2}ms and 1.7 × 10

^{11} W/K

^{2}ms, respectively. This phenomenon is mainly caused by the dominant advantage of

S. The calculated results suggest that monolayer MAs

_{2} possess the great merits to be a promising thermoelectric material.

#### 3.3. Thermal Transport Properties

In order to accurately analyze the effect of phonon transport properties on TE performance, we computed the phonon spectrum of monolayer MAs

_{2} and corresponding phonon DOS (PhDOS) as plotted in

Figure 4. The lack of virtual frequency in the phonon spectrum indicates that monolayer MAs

_{2} is dynamically stable, which is consistent with the previous theoretical date. There are two M and four As atoms, corresponding to eighteen curves which include three phonon–phonon and fifteen optical–phonon curves. Additionally, the phonon spectrum consisting of two parts that correspond to the two parts of PhDOS can be clearly observed. Among them, the phenomenon is that the low-frequency phonon–phonon curves are mainly controlled by the vibration of M atoms, while the vibrations of M and As atoms jointly contribute to the optical-phonon curves.

The lattice thermal conductivity (

κ_{l}) is one of important factors for evaluating TE properties, which can be proved from Equation (2). Based on the Boltzmann transport theory with implementing in ShengBTE mode, the

κ_{l} of monolayer NiAs

_{2}, PdAs

_{2} and PtAs

_{2} can be computed by

where

C_{λ},

v_{λ} and

τ_{λ} are the mode heat capacity, phonon group velocity and relaxation time, respectively. The

κ_{l} as a function of temperature is presented in

Figure 5a. We can find that the

κ_{l} of monolayer MAs

_{2} gradually reduce with the increase in temperature following the inverse relation, which is mainly caused by increasing phonon scattering with the elevating temperature. At 300 K, The

κ_{l} of monolayer NiAs

_{2}, PdAs

_{2} and PtAs

_{2} are 5.9, 2.9 and 3.6 W/mK, respectively.

The notion that nanostructures can effectively reduce thermal conductivity and thus improve thermoelectric performance has been proved, because nanostructures can hinder phonon transport and reduce the lattice thermal conductivity while having little impact on the electronic thermal conductivity, which greatly reduces the interaction between transport parameters. Consequently, the influence of size effect on

κ_{l} is considered and calculated. As

Figure 5b demonstrates, the phonon mean free path (MFP) as a function of accumulated

κ_{l} of monolayer MAs

_{2} at 300 K exhibits that the value is optical within the range of 1 nm, due to the accumulated

κ_{l} having no change with transforming size. Surprisingly, a positive phenomenon we can observe is that the slope curves of monolayer PdAs

_{2} is very small, which can actively promote the application in TE materials.

In order to analyze the influence of lattice thermal conductivity on TE performance in detail, we calculated the phonon group velocity (

v), relaxation time (

τ), Grüneisen parameters (γ) and the three-phonon scattering phase space (P

_{3}) as presented in

Figure 6.

An important factor (

v) affecting the evaluation of thermal transport ability is determined by employing the phonon dispersion, which can be calculated by

where ω

_{λ,q} is the phonon frequency. In

Figure 6a, we can clearly find that the

v of monolayer MAs

_{2} of low-frequency acoustic breaches are much higher than that of high-frequency optical breaches, which indicates that acoustic breaches make a contribution to the

κ_{l}. The value of

v in the low-frequency region at 300 K can be obtained of 7.6, 5.2 and 6 Km/s for NiAs

_{2}, PdAs

_{2} and PtAs

_{2}, respectively. The order of magnitude is NiAs

_{2} > PtAs

_{2} > PdAs

_{2}, which suggests that the magnitude relationship of the

κ_{l} in

Figure 5a is consistent. As

Figure 6b shows, another key parameter phonon relaxation time (

τ) is evaluated according to Equation (5). We can find that the

τ of monolayer PtAs

_{2} is smaller than that of monolayer NiAs

_{2} and PdAs

_{2}, which is useful for receiving desired

κ_{l}.

Usually, the anharmonic interactions are used to determine the intensity of interactions and described by

γ; thus, the greater anharmonic interaction can promote the generation of a much stronger phonon-phonon interaction as well as a smaller lattice thermal conductivity.

Figure 6c displays the Grüneisen parameters (

γ) of monolayer MAs

_{2} with respect to frequency at 300 K, which can be calculated by

where

V is the volume. We can find that monolayer MAs

_{2} possess a very high value for

γ at a low frequency, corresponding to 10, 32 and 13 of NiAs

_{2}, PdAs

_{2} and PtAs

_{2}, respectively. Obviously, the value of

γ of monolayer PdAs

_{2} is much higher than that of NiAs

_{2} and PtAs

_{2} sheet, indicating that monolayer PdAs

_{2} has a large anharmonic interaction, which causes the smallest

κ_{l} in three arsenic compounds.

The three-phonon scattering phase space (P

_{3}) is used to describe the

τ, and a larger value of P

_{3} shows that more space is adopted to the three-phonon scattering, while a shorter

τ can be reaped. As shown as

Figure 6d, the P

_{3} of monolayer MAs

_{2} as function of phonon frequency is obtained. We can clearly find that monolayer MAs

_{2} possess a large scattering phase space at a low frequency, indicating that they can promote little

τ for acoustic phonon breaches.

#### 3.4. Thermoelectric Figure of Merit (ZT)

The large power factor (PF) and very low thermal conductivity of monolayer MAs

_{2} are obtained through the calculation of electronic and phonon transport properties, i.e., a high thermoelectric figure of merit (ZT) is generated. By combining the phonon and electron transport coefficients, we calculate the ZT of monolayer MAs

_{2}.The electronic scattering time τ is obtained by the DP theory, as shown in

Table 2. The ZT values of monolayer NiAs

_{2}, PdAs

_{2} and PtAs

_{2} as functions of chemical potential at 300, 500 and 700 K are plotted in

Figure 7 corresponding to (a), (b) and (c), respectively. We can clearly note that the p-type doping ZT value of NiAs

_{2} and PtAs

_{2} sheet are greater than n-type doping, while the ZT value of sing-layer PdAs

_{2} is contrary to them and belongs to n-type doping, which is consistent with the type of PF. The maximum ZT value of monolayer NiAs

_{2} (p-type), PdAs

_{2} (n-type) and PtAs

_{2} (p-type) are 0.58, 2.1 and 0.64 at 700 K, respectively. The predicted

ZT value of PdAs

_{2} is larger than those of the commercial TE material p-type penta-PdX

_{2} (X = S, Se) [

41] and some other arsenic compound [

42].

Besides, the ZT value of PdAs_{2} was three to four times higher than that of the other two arsenic compounds, mainly due to the combination of larger Seebeck coefficient and lower lattice thermal conductivity. The monolayer PdAs_{2} can be expected for application in thermoelectric material.