Strain-induced ultrahigh power conversion efficiency in BP-MoSe₂ vdW heterostructure

First of all, the structural and electronic properties of the BP and MoSe₂ monolayer have been optimized. Since monolayer MoSe₂ shows large spin-splitting in both valence and conduction bands induced by SOC [56], we performed test calculations for the BP, MoSe₂ monolayer and the BP-MoSe₂ vdW heterostructure with and without SOC. As shown in figure 1(a), obviously, the SOC breaks the spin degeneracy of the valence and conduction bands in the MoSe2 monolayer. Furthermore, it is found that SOC induces spin splitting of 246.7 meV at the valence band maximum (VBM) and 14 meV at the conduct band minimum (CBM), which agrees well with the results in [56], where 188 meV at the VBM and 20 meV at the CBM. However, figure 1(b) shows that the SOC in the BP monolayer is weak enough to be ignored. This observation originates from the fact that the SOC is stronger for heavier atoms. More importantly, the positions of the VBM and CBM in the BP-MoSe2 vdW heterostructure with SOC are almost the same as those without SOC. Therefore, we believe that the SOC effects will not affect our evaluation for the photocatalytic properties of BP-MoSe2 vdW heterostructure. Therefore, the SOC is not considered in the following calculations.

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Figure 2 shows the optimized primitive cell structures of the BP and MoSe₂ monolayer (Inserts) as well as their band structures calculated at the HSE06 level. The optimal lattice constants of the BP and MoSe₂ are 3.27 Å and 3.31 Å, respectively. Thus, the corresponding lattice mismatch is evaluated to be 1.2%, which is so small that it is suitable to construct the heterostructure consisting of the BP and MoSe₂ monolayer. Moreover, the optimized bond lengths of P–P and Se–Mo are 2.26 Å and 2.54 Å, respectively. These structural parameters are in good agreement with the previous theoretical values [35, 39]. In addition, figure 2 shows that the BP monolayer is an indirect band gap semiconductor with a band gap of 2.78 eV. However, the MoSe₂ monolayer is a direct bandgap semiconductor with a bandgap of 1.94 eV, which agrees well with previous reports [35].

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Next, we construct and optimize the geometries of the BP-MoSe₂ vdW heterostructure with various stacking modes. Figure 3(a) shows the top view and side view of the optimal geometric structure of the BP-MoSe₂ vdW heterostructure, and the structural parameters are listed in table 1. It can be seen that the BP-MoSe₂ vdW heterostructure is a hexagonal system, the upper P atom is on the top of Se atom, and the bottom P atom is on the top of Mo atom. Our best geometric structure is in line with that described by Shu et al [40]. Table 1 shows that the lattice constant of the BP-MoSe₂ vdW heterostructure is 3.28 Å. The bond lengths of the P–P and Mo–Se are 2.26 and 2.54 Å, respectively, which are the same as in the individual BP and MoSe₂ monolayer, indicating that the band lengths of the P–P and Mo–Se are not affected when forming the BP-MoSe₂ vdW heterostructure. Additionally, the layer spacing of the BP-MoSe₂ vdW heterostructure is 3.17 Å, which indicates that the interlayer interactions are dominated by the vdW interactions.

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To evaluate the stability of the BP-MoSe₂ vdW heterostructure, the formation energy is calculated by the formula:

$$\begin{eqnarray}&&{E}_{\mathrm{form}}=\left({E}_{\mathrm{total}}-{E}_{\mathrm{BP}}-{E}_{\mathrm{MoSe}2}\right),\end{eqnarray} \tag{ 1 }$$

where the ${E}_{\mathrm{total}},$ ${E}_{\mathrm{BP}}$ and ${E}_{\mathrm{MoSe}2}$ are the total energy of the BP-MoSe₂ vdW heterostructure, the BP and the MoSe₂ monolayer. As a result, the formation energy of the BP-MoSe₂ vdW heterostructure is computed to be −0.21 eV, suggesting that the formation of BP-MoSe₂ vdW heterostructure is exothermic and thus the BP-MoSe₂ vdW heterostructure is energetically favorable.

Moreover, the mechanical stability is also checked through determining whether the elastic constants satisfies the Born–Huang criteria (${C}_{11}{C}_{22}-{C}_{12}^{2}\gt 0\,and\,{C}_{66}\gt 0$). By using the strain-stress method, the elastic constants of the BP-MoSe₂ vdW heterostructure can be computed: C₁₁ = 196.3 N m⁻¹, C₂₂ = 187.8 N m⁻¹, C₁₂ = 29.4 N m⁻¹, and C₆₆ = 77.6 N m⁻¹. Apparently, the calculated elastic constants meet the Born–Huang criteria, implying the mechanical stability of the BP-MoSe₂ vdW heterostructure.

Figure 3(b) shows the weighted energy band structure of the BP-MoSe₂ vdW heterostructure calculated with the HSE06 functional, from which we can see that the BP-MoSe₂ vdW heterostructure is an indirect gap semiconductor with a bandgap of 1.73 eV, which is lower than those of isolated monolayers. In addition, it is evident that the VBM and CBM are contributed by the MoSe₂ and BP monolayer, respectively, indicating that the BP-MoSe₂ vdW heterostructure forms a type-II band structure. This is also reflected in the band decomposed charge density of the BP-MoSe₂ vdW heterostructure, as shown in figure 3(c), from which we can see that the VBM and CBM are localized on the MoSe₂ and BP monolayer, respectively. Therefore, the BP-MoSe₂ vdW heterostructure is a typical type-II semiconductor, and then the separation of electron–hole pairs in space can be realized, overcoming the electron–hole recombination issue. Moreover, the conduction band offset (CBO) and valence band offset (VBO) are 0.31 and 1.09 eV, respectively. As shown in figure 3(d), the photogenerated electrons in the CBM of MoSe₂ could transfer to the CBM of BP by the chemical potential difference of CBO, while the photogenerated holes in the VBM of BP can migrate to the VBM of MoSe₂ via the chemical potential difference of VBO. The results further ensure the effective spatial separation of electron–hole pairs.

Since the BP-MoSe₂ vdW heterostructures has not been experimentally synthesized, an AIMD simulations under 300 K was carried out to evaluate its thermodynamic stability. As shown in figure 4, the temperature and total energy fluctuations of the BP-MoSe₂ vdW heterostructure at room temperature is in a very small range. Furthermore, it is found that the distortion of the system is negligible and no bond breaks after heating 10 ps at 300 K. These results indicate that the BP-MoSe₂ vdW heterostructure has robust thermodynamic stability at room temperature.

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Applying external strain has been known as an effective strategy to control electronic, optical, and transport properties of materials for semiconductors [35, 38, 39, 57, 58]. To examine the strain effects on the electronic and optical properties of the BP-MoSe₂ vdW heterostructure, we calculate the electronic band structures, band edge positions, PCE and optical absorption of the BP-MoSe₂ vdW heterostructure under biaxial strains up to ±3% with the HSE06 functional. The biaxial strain is defined as (l–l₀)/l₀, where the l and l₀ are the lattice constants of the strained and initial structures, respectively. Figure 5 shows the weighted band structures of the BP-MoSe₂ vdW heterostructure under biaxial strains from −3% to 3%. The results indicate that the bandgap of the BP-MoSe₂ vdW heterostructure monotonously decreases with increasing of the tensile strains from 1% to 3%, and a type-II to type-I band structure transition occurs under 3% tensile strain due to the decrease of CBM of the MoSe₂. In addition, we notice that the VBM of the BP-MoSe₂ vdW heterostructure (i.e. the MoSe₂) shifts from the K point to the Γ point when the tensile strains of 1%–3% are applied. On the other hand, the bandgap of the BP-MoSe₂ vdW heterostructure increases under compressive strain of −1%, whereas it decreases as the compressive strain increases from −2% to −3%. More important, it is notable that the BP-MoSe₂ vdW heterostructure is changed to a direct bandgap semiconductor due to the shifts of CBM of the BP from Γ and M to point K. The direct bandgaps are 1.72 eV and 1.63 eV corresponding to the compressive strains of −2% and −3%, respectively.

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From aforementioned, we know that the BP-MoSe₂ vdW heterostructure under tensile strains of 1% to 2% and compressive strains of −1% to −3% still maintains type-II band structures and the band gaps exceed the water redox potential of 1.23 eV. It is known that as a photocatalyst for water splitting, in addition to the type-II band structure, the band edge position is also important. So we further examined the band edge positions of the BP-MoSe₂ vdW heterostructure under strains from −3% to 2% and compared to the redox potential of the water splitting. Generally, the redox potential of the water splitting reaction depends on the pH [59–63], which is calculated using (−4.44 + pH × 0.059) eV for the reduction potential of H⁺ → H₂ and (−5.67 + pH × 0.059) eV for the oxidation potential of H₂O → O₂. Figure 6 shows the band edge positions of the BP-MoSe₂ vdW heterostructure under strains from −3% to 2% referencing to the vacuum level with respect to the redox potential of water splitting at pH = 0. Results show that the CBM and VBM of the BP-MoSe₂ vdW heterostructure under strains from −3% to 2% at pH = 0 all straddle the redox potential of water, indicating that the BP-MoSe₂ vdW heterostructure under strains of −3% to 2% is a promising candidate for photocatalytic water splitting without an external bias voltage in acidic environments (pH = 0).

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The PCE is another important index to evaluate photocatalyst, and thus the PCE of the BP-MoSe₂ vdW heterostructure under strains from −3% to 2% has been evaluated using the formula [64]:

$$\begin{eqnarray}&&\eta =\frac{0.65{J}_{\mathrm{sc}}{V}_{\mathrm{oc}}}{{P}_{\mathrm{solar}}}=\frac{0.65({E}_{g}^{d}-{\rm{\Delta }}{E}_{c}-0.3)\displaystyle {\int }_{{E}_{g}^{d}}^{\infty }\frac{P\hslash \omega }{\hslash \omega }d(\hslash \omega )}{\displaystyle {\int }_{0}^{\infty }P(\hslash \omega )d(\hslash \omega )},\end{eqnarray} \tag{ 2 }$$

Where ${E}_{g}^{d}$ and ${\rm{\Delta }}{E}_{c}$ are the bandgap of the donor and the CBO, respectively. P($\hslash$ ω) is the AM 1.5 solar flux at the photon energy $\hslash$ ω, and $\displaystyle {\int }_{0}^{\infty }P(\hslash \omega )d(\hslash \omega )$ is the total incident solar power per unit area. The calculated CBM, VBM, bandgaps, bandgaps of the donor, CBO and PCE under strains from −3% to 2% are listed in table 2, where we can see that the PCE of the BP-MoSe₂ vdW heterostructure without any strain is 12.7%. Evidently, the PCE is 12.0%, 12.7% and 12.8% under compressive strains of −1%, −2% and −3%, which is not sensitive to the compressive strain. However, the PCE is 16.3% and 20.2% under tensile strains of 1% and 2%, which increases rapidly under the tensile strain. Specially, the PCE is remarkably enhanced to 20.2% under biaxial tensile strain of 2%, which is higher than that of 19.9% in the MoSSe/BlueP vdW heterostructure under a 2% tensile strain [41]. We know from formula (2) that the PCE is inversely proportional to CBO and directly proportional to bandgap of the donor. That is, the value of PCE depends on the competition of CBO and bandgap of the donor. As a result, the PCE can achieve 20.2% under strain of 2% due to the very small CBO value of 0.07 eV. On the other hand, the carrier concentration depends on the temperature and bandgap. The high temperature and narrow bandgap can facilitate the electron excitation, increasing the carrier concentration, and vice versa. Therefore, the BP-MoSe₂ vdW heterostructure under strain of 2% has the highest carrier concentration due to its minimum bandgap.

Table 2. Calculated CBM, VBM, bandgaps (${E}_{g}^{\mathrm{HSE}}$), bandgaps of the donor (${E}_{g}^{d}$), CBO (ΔE_c ) and PCE under strains from −3% to 2%.

Strain (%)	CBM (eV)	VBM (eV)	${E}_{g}^{\mathrm{HSE}}$ (eV)	${E}_{g}^{d}$ (eV）	ΔEc (eV)	PCE (%)
−3	1.43	−0.20	1.63	1.99	0.36	12.8
−2	1.49	−0.24	1.72	2.04	0.32	12.7
−1	1.51	−0.23	1.75	2.08	0.34	12.0
0	1.50	−0.23	1.73	2.04	0.31	12.7
1	1.41	−0.23	1.64	1.83	0.20	16.3
2	1.33	−0.22	1.56	1.63	0.07	20.2

To gain deep insight into the excellent PCE under tensile strains, we further examined the project density of states of the BP-MoSe₂ vdW heterostructure under strains of 0%, 1% and 2%. As shown in figure 7, the lowest unoccupied molecular orbital (LUMO) of Mo atom, represents by the light purple line, rapidly decreases as the tensile strain increases from 0% to 1% to 2%. However, the LUMO of Se atom (green line) decreases very slowly with the increase of the tensile strain. Therefore, the CBM of MoSe₂ is mainly contributed by 4d of Mo atom. Moreover, the LUMO of Mo atom approaches that of P atom under a 2% tensile strain, which results in the small value of CBO, causing the high PCE value.

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The ability to capture Sunlight is another important requisite for an excellent photocatalyst. To examine the light absorption performance of the BP-MoSe₂ vdW heterostructure under light, the optical absorption coefficients have been calculated based on the HSE06 functional. Since the tensile strains can enhance the PCE of the BP-MoSe₂ vdW heterostructure, the optical absorption coefficients of the BP-MoSe₂ vdW heterostructure with 0%, 1% and 2% tensile strains have been considered by the formula [65]:

$$\begin{eqnarray}&&\alpha \left(\omega \right)=\displaystyle \frac{\sqrt{2}\omega }{c}{\left\{{\left[{\varepsilon }_{1}^{2}\left(\omega \right)+{\varepsilon }_{2}^{2}\left(\omega \right)\right]}^{1/2}-{\varepsilon }_{1}\left(\omega \right)\right\}}^{1/2},\end{eqnarray} \tag{ 3 }$$

where $\omega$ and c represent the angular frequency and the speed of light in vacuum, respectively. ${\varepsilon }_{1}\left(\omega \right)$ and ${\varepsilon }_{2}\left(\omega \right)$ are the real and imaginary parts of the dielectric function, respectively. As shown in figure 8, the BP-MoSe₂ vdW heterostructure without any strains has two notable absorption peaks, the corresponding absorption intensity is ∼4.4 × 10⁵ cm⁻¹ at the wavelength of 415 nm and ∼5.0 × 10⁵ cm⁻¹ at the wavelength of 560 nm. These intensities are even higher than those of other high-efficiency photocatalyst systems [11, 37,41, 63]. Therefore, the BP-MoSe₂ vdW heterostructure is efficient light harvesting photocatalysts. In addition, it is found that the first absorption peak has a redshift under the tensile strains of 1% and 2%. Moreover, it should be noted that the tensile strains could broaden the optical absorption of the BP-MoSe₂ vdW heterostructure to the near-ultraviolet light spectrum. Specifically, the absorption intensity is ∼5.5 × 10⁵ cm⁻¹ at the wavelength of 358 nm under a 1% strain and ∼4.3 × 10⁵ cm⁻¹ at the wavelength of 379 nm under a 2% strain. Therefore, the tensile strains greatly improve the utilization of Sunlight.

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We know from the above analysis that the BP layer is an electron accepter, while the MoSe₂ layer is an electron donor in the BP-MoSe₂ vdW heterostructure. Once the BP-MoSe₂ vdW heterostructure is radiated by sunlight as a photocatalysts, the BP layer will accumulate a large number of electrons, while the MoSe₂ layer will accumulate a large number of holes. Therefore, the reduction reaction of $2{{\rm{H}}}^{+}+2{e}^{-}\to {{\rm{H}}}_{2}$ (HER) will occur on the BP layer due to aggregation of electrons, and the oxidation reaction of $2{{\rm{H}}}_{2}{\rm{O}}+4{{\rm{h}}}^{+}\to {{\rm{O}}}_{2}+4{{\rm{H}}}^{+}$ (OER) will occur on the MoSe₂ layer due to accumulation of holes. Therefore, the HER performance of BP layer and OER catalytic activity of MoSe₂ layer are investigated in the following calculations.

In this work, the changes of Gibbs free energy (ΔG) for the HER and OER processes are calculated using formula as [11]:

$$\begin{eqnarray}&&{\rm{\Delta }}G={\rm{\Delta }}E+{\rm{\Delta }}{E}_{\mathrm{ZPE}}-T{\rm{\Delta }}S\,,\end{eqnarray} \tag{ 4 }$$

where ${\rm{\Delta }}E$ is the adsorption energy, ${\rm{\Delta }}{E}_{\mathrm{ZPE}}$ is the difference of zero point vibration energy, $T$ and ${\rm{\Delta }}S$ are the absolute temperature and entropy change, respectively. Here, for the zero point energy contribution, only the vibrational frequencies of absorbed species are calculated with the BP-MoSe₂ heterostructure fixed. Moreover, only vibration entropy contributions are considered for adsorbates, and we take the temperature as 298.15 K. Firstly, the HER process in the acidic environment has been considered. There are two elementary reaction steps for HER: ${{\rm{H}}}^{+}+{e}^{-}\to {{\rm{H}}}^{* }$ and ${{\rm{H}}}^{* }\to \tfrac{1}{2}{{\rm{H}}}_{2},$ where * represents the active site of the substrate, and ${{\rm{H}}}^{* }$ represents the hydrogen is attached to the substrate. The first step is to describe the adsorption process of hydrogen atom on the substrate, and the second step is to describe the desorption process of hydrogen molecule from the substrate. Here, the standard hydrogen electrode (SHE) approximation is considered, i.e. the ${{\rm{H}}}^{+}$ ion is converted to ${{\rm{H}}}_{2}$ molecule without Gibbs free energy change in the reaction of ${{\rm{H}}}_{(\mathrm{aq})}^{+}+{{\rm{e}}}^{-}\rightleftharpoons \frac{1}{2}{{\rm{H}}}_{2({\rm{g}})}$ (ΔG = 0).

Figure 9(a) shows the Gibbs free energy profiles with different hydrogen coverages on BP layer in HER process. Evidently, the first step is the limiting step of the HER for all hydrogen coverages from 1H to 4H, and the positive ΔG_H* indicates that this process is endothermic and cannot proceed without light illumination or external potential. In addition, the ΔG_H* of the third hydrogen is minimum (0.64 eV), whereas the ΔG_H* of the fourth hydrogen is maximum (1.34 eV), indicating that the BP-MoSe₂ vdW heterostructure is potentially catalyst for the HER at an appropriate hydrogen coverage.

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Then, we turn our attention to the OER process also in the acidic environment. There are four elementary reactions steps in OER process:

$$\begin{eqnarray}&&{{\rm{H}}}_{2}{\rm{O}}({\rm{l}})+* \,={{\rm{OH}}}^{* }+{{\rm{H}}}^{+}+{{\rm{e}}}^{-}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{\rm{OH}}}^{* }={{\rm{O}}}^{* }+{{\rm{H}}}^{+}+{{\rm{e}}}^{-}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{\rm{O}}}^{* }+{{\rm{H}}}_{2}{\rm{O}}\left({\rm{l}}\right)={{\rm{OOH}}}^{* }+{{\rm{H}}}^{+}+{{\rm{e}}}^{-}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{\rm{OOH}}}^{* }=* +{{\rm{O}}}_{2}({\rm{g}})+{{\rm{H}}}^{+}+{{\rm{e}}}^{-}\end{eqnarray} \tag{ 8 }$$

and the corresponding ΔG is calculated using

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{1}={G}_{{\mathrm{OH}}^{* }}+\frac{1}{2}{G}_{{{\rm{H}}}_{2}}-{G}_{* }-{G}_{{{\rm{H}}}_{2}{\rm{O}}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{2}={G}_{{{\rm{O}}}^{* }}+\frac{1}{2}{G}_{{{\rm{H}}}_{2}}-{G}_{{\mathrm{OH}}^{* }}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{3}={G}_{{\mathrm{OOH}}^{* }}+\frac{1}{2}{G}_{{{\rm{H}}}_{2}}-{G}_{{{\rm{O}}}^{* }}-{G}_{{{\rm{H}}}_{2}{\rm{O}}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{4}=4.92-({\rm{\Delta }}{G}_{1}+{\rm{\Delta }}{G}_{2}+{\rm{\Delta }}{G}_{3}).\end{eqnarray} \tag{ 12 }$$

Therefore, the species of ${{\rm{OH}}}^{* },$ ${{\rm{O}}}^{* }$ and ${{\rm{OOH}}}^{* }$ are the intermediates of the elementary reactions.

Figure 9(b) shows the overall Gibbs free energy profiles in the OER process on the MoSe₂ layer of the BP-MoSe₂ vdW heterostructure. The inserts are the optimal adsorption configurations of OH, O and OOH adsorbed on the MoSe₂ layer. Obviously, ΔG of the first three steps is positive while the last step is negative. The third step is rate-limiting with ${\rm{\Delta }}{G}_{3}$ of 2.57 eV. At the standard potential of water oxidation (U = 1.23 V), the ${\rm{\Delta }}{G}_{3}$ of the rate-limiting step will decrease to 1.34 eV, which is comparable to that of the C₂N/GaTe type-II heterojunctions [11].

The carrier mobility has been investigated to further evaluate the photocatalytic performance of the BP-MoSe₂ vdW heterostructure. Generally, high carrier mobility could potentially improve the charge utilization for redox reactions before their recombination [10]. On the basis of the Takagi model within the deformation potential approximation [66–68], the acoustic phonon-limited carrier mobility μ can be calculated as

$$\begin{eqnarray}&&\mu =\frac{e{\hslash }^{3}{C}_{2D}}{{k}_{{\rm{B}}}Tm{m}_{d}{\left({E}_{1,i}\right)}^{2}},\end{eqnarray} \tag{ 13 }$$

where e, $\hslash ,$ and ${k}_{{\rm{B}}}$ are the electron charge, Planck constant and Boltzmann constant, respectively. T is the absolute temperature, here a room temperature of 300 K is used in the calculations. m is the effective mass of carriers along the transport direction, which is defined as $\tfrac{1}{m}=\tfrac{1}{\hslash }\tfrac{{\partial }^{2}E(k)}{\partial {k}^{2}}.$ m_d is the average effective mass, which equals $\sqrt{m{m}^{* }}$ (${m}^{* }$ is the effective mass perpendicular to the transport direction). ${E}_{1,i}$ is the deformation potential constant of the VBM for a hole or the CBM for an electron along the transport directions, which is defined as ${E}_{1,i}=\tfrac{{\rm{\Delta }}V}{{\rm{\Delta }}L/L}$ ($\mathrm{\Delta V}$ is the band edge shift under lattice deformation, ${\rm{\Delta }}l$ is the lattice deformation). ${C}_{2{\rm{D}}}$ is the elastic modulus of the 2D heterostructure, which is computed by ${C}_{2{\rm{D}}}={\rm{\Delta }}E/\left[{\left(\tfrac{{\rm{\Delta }}l}{l}\right)}^{2}{S}_{0}\right]$ (${\rm{\Delta }}E$ is the energy change of the system under ${\rm{\Delta }}l,$ and ${S}_{0}$ is the area of the BP-MoSe₂ vdW heterostructure). It has been demonstrated this model is not only computational efficient but also physically reasonable [69, 70], and has been successfully used in predicting the carrier mobility of 2D semiconductors such as GeAsSe and SnSbTe [63].

Table 3 shows that the carrier mobility displays a fairly pronounced anisotropy, and thus is essential to control the carrier transport directionally to enhance device performance. For example, the mobility of electrons along the x and y directions are 17.01 and 3613.68 ${{\rm{cm}}}^{2}\cdot {{\rm{V}}}^{-1}\cdot {{\rm{s}}}^{-1},$ respectively, and the mobility of holes along the x and y directions are 523.45 and 235.46 ${{\rm{cm}}}^{2}\cdot {{\rm{V}}}^{-1}\cdot {{\rm{s}}}^{-1},$ respectively. These mobilities of carriers are comparable with the theoretically proposed β-AsP monolayer [71] and C₂N-based type-II heterojunctions [11]. In addition, the deformation potential constant reflects the strength of electron-phonon coupling, and is determined by the band edge shift under lattice deformation. Further observation finds that the low mobility of electrons along x direction is attributed to its large deformation potential constant (12.86 eV), and thus significantly reduces the electron mobility.

Table 3. The effective mass of carriers (in m₀), deformation potential constants (in eV), elastic modulus (in ${\rm{J}}\cdot {{\rm{m}}}^{-2}$), and mobility of carriers (in ${{\rm{cm}}}^{2}\cdot {{\rm{V}}}^{-1}\cdot {{\rm{s}}}^{-1}$) along x and y directions.

Material	Carrier	${m}_{x}$	${m}_{y}$	${E}_{x}$	${E}_{y}$	${C}_{x}$	${C}_{y}$	${\mu }_{x}$	${\mu }_{y}$
BP-MoSe2	e	1.42	0.78	12.86	1.17	196.47	189.85	17.01	3613.68
	h	0.57	0.83	4.52	5.49	196.47	189.85	523.45	235.46