The electron–phonon scattering and charge transport of the two-dimensional (2D) polar h-BX(X = P, As, Sb) monolayers

In this paper, the electron–phonon scattering and phonon-limited transport properties of the two-dimensional polar h-BX(X = P, As, Sb) have been studied through first-principles calculations in combination with Boltzmann transport theory. The electron–phonon scattering in these three systems is systematically assessed. Remarkably, intravalley scattering and intervalley scattering are separately investigated, of which the contribution to total scattering is found to be relatively comparable. The carrier mobility is determined over a broad range of carrier concentrations. The results indicate that h-BX (BP, BAs, BSb) simultaneously possess ultrahigh electron mobilities (4097 cm2 V−1 s−1, 4141 cm2 V−1 s−1, 12 215 cm2 V−1 s−1) and hole mobilities (7563 cm2 V−1 s−1, 7606 cm2 V−1 s−1, 22 282 cm2 V−1 s−1) at room temperature as compared to the most known two-dimensional (2D) materials. Additionally, it is discovered that compressive strain can induce a further increase in carrier mobility. The exceptional charge transport properties exhibited by these 2D semiconductors are attributed to the small effective masses in combination with the significant suppression of scattering due to high optical longitudinal optical- and transverse optical-phonon frequencies. This is the first time that we have provided a systematic interpretation of the reason for the exceptional charge transport properties exhibited by the 2D h-BX(X = P, As, Sb) semiconductors. Our finding can provide a theoretical perspective regarding the search for 2D materials with the high carrier mobility.


Introduction
With the rapid development of the semiconductor industry and the sustained scaling of transistors according to Moore's law [1], the search for high-performance semiconductor materials has become a pressing task for scientific researchers.In recent years, two-dimensional (2D) materials have attracted considerable interest owing to their outstanding photoelectric properties and ultrathin features [2][3][4].Recently, Wu et al have successfully prepared an ultra-small MoS 2 transistor with a vertical structure, achieving an effective gate length of 0.34 nm for the first time [5].This work has pushed Moore's Law to further develop to the sub-nanometer level while providing an essential path for applying two-dimensional electronic materials in future integrated circuits.
Although 2D semiconductors have many advantages and unique characteristics, they usually exhibit a low carrier mobility [6].It is well known that carrier mobility is used to characterize how fast carriers (electrons/holes) can move under the influence of an electric field.Generally, a high carrier mobility is more desirable in practical applications because it determines the switching frequency in transistors, the photoconductive gain in photodetectors, as well as the transport properties in solar cells and light-emitting devices [4,[7][8][9][10].To our knowledge, the WS 2 and MoS 2 monolayers have a room-temperature carrier mobility of ∼200 cm 2 V −1 s −1 [11].The electron mobility of the InSe monolayer is predicted to be 120 cm 2 V −1 s −1 [12].These carrier mobilities are much lower than those of conventional Si and GaAs [13].The limitation hinders their potential applications in the field of optoelectronics.Among the known 2D materials, graphene is an exception, which is confirmed to have the ultrahigh carrier mobility of ∼1.35 × 10 5 cm 2 V −1 s −1 due to the unique Dirac cone band structure at K point, but the zero bandgap limits its applications [14].
Recently, the 2D h-BX(X = P, As, Sb) semiconductors have been recognized as promising candidates for (opto)electronic and thermoelectric applications [15].The semiconductors possess the like-Dirac cone band structure and are therefore expected to possess a high carrier mobility.In recent studies, their high room-temperature carrier mobilities have been confirmed by the deformation potential theory [16] and the Boltzmann transport theory [17,18].However, the systematic interpretation of the underlying reason for the high carrier mobility has not been reported.As we know, the electron-phonon scattering plays a crucial role in determining charge transport properties of materials.Elucidating the electron-phonon scattering mechanism can deepen the understanding on charge transport and thermoelectric performance of the h-BX systems.In particular, more complicated intravalley scattering and intervalley scattering are involved due to the multiple-valley band structure around the Fermi level.In addition, because the 2D h-BX systems are a polar material, the well-known Fröhlich interaction between electrons and longitudinal optical (LO) phonon modes is of great significance in the scattering process [19].On the other hand, it should be noted that the carrier mobility in materials is usually not a fixed value, which is greatly influenced by carrier concentrations.Different materials may exhibit diverse behaviors of the carrier mobility in response to the change of carrier concentrations.For instance, the electron mobility in graphene can be enhanced from 1.45 × 10 5 cm 2 V −1 s −1 to 5.5 × 10 6 cm 2 V −1 s −1 by altering the carrier concentration [14].For h-BeO [20], the mobility almost remains unaffected despite experiencing an increase in carrier concentration from 1 × 10 10 cm −2 to 1 × 10 13 cm −2 .Particularly, once the carrier concentration increases to a certain extent, the carrier mobility always sharply decreases, and the device performance will be severely influenced.Therefore, understanding how carrier concentrations impact the carrier mobility is of great importance in designing and developing high-efficient devices based on the h-BX semiconductors.
For the above reasons, we systematically investigate the electron-phonon scattering and charge transport of the h-BX(X = P, As, Sb) monolayers by using first-principles calculations followed by the Boltzmann transport theory.Their stability is confirmed via the ab initio molecular dynamics (AIMD) simulation.The electron-phonon scattering, which are divided into intravalley scattering and intervalley scattering, are comprehensively investigated.The carrier mobility is evaluated over a broad range of carrier concentrations ranging from 1 × 10 10 cm −2 to 1 × 10 14 cm −2 .The results demonstrate that h-BX exhibits remarkably high electron and hole mobilities simultaneously.Furthermore, a systematic interpretation is provided to shed light on the underlying reason for the exceptional charge transport properties.Additionally, the response of charge transport to biaxial strain is investigated, revealing the mechanism behind the variation in carrier mobility under biaxial strain.

Computational details
The electron and phonon band structures are computed within the density-functional theory (DFT) and the density-functional perturbation theory (DFPT) framework by employing the QUANTUM ESPRESSO code [21].AIMD calculations are carried out via the Vienna ab initio Simulation Package (VASP) [22].In DFT calculations, we use the fully relativistic norm-conserving pseudopotential optimized with the SG15 norm-conserving Vanderbilt pseudopotentials.The Perdew-Burke-Ernzerhof (PBE) exchange-correlation function is used to consider the exchange and correlation energies [23].The kinetic energy cutoffs for wavefunctions and charge density are set to be 80 and 320 Ry, respectively.For the structural optimization, we employ a 36 × 36 × 1 k-point grid and set a convergence criterion of 10 −4 Ry Bohr −1 for force and 10 −6 Ry Bohr −1 for energy.The spurious interaction between layers of the vacuum slab model is mitigated via the 2D Coulomb cutoff approach [24].We integrate over the Brillouin zone (BZ) using an 18 × 18 × 1 k-point grid to obtain the electronic properties.To address the underestimated bandgaps at the PBE level, we compute band structures by using the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) [25] in VASP.In DFPT calculation, an 18 × 18 × 1 q-point mesh is used to determine the phonon dispersion and electron-phonon matrix elements with a convergence threshold of 10 −20 Ry.The Wannier90 code [26,27] is employed to obtain the localized Wannier function, where the valence p orbitals of B and X (X = P, As, Sb) atoms are used as an initial guess (see figure S1 in the supplementary materials).All the charge transport calculations are performed by the PERTURBO package [28].The determined electronic energies and phonon frequencies are interpolated to 800 × 800 × 1 k and q meshes, respectively (figure S2 of the supplementary materials shows the convergence test on k and q grids).Meanwhile, the electron-phonon matrix is achieved, which is defined as: where Ψ mk+q and Ψ nk are the final and initial states, respectively.When dealing with 2D polar materials and their distinct phonon dispersion featuring the widely recognized LO-TO splitting at q → 0, it becomes crucial to incorporate the Fröhlich interaction [29].As a result, g can be represented as: where g L mn,v (k, q) is the long-range contribution, which can be evaluated directly using an analytical formula involving the Born effective charges and the dielectric tensor, and g S mn,v (k, q) is the short-range contribution, which can be interpolated via the Fourier transformation.
The scattering rate (1/τ nk ) of electrons upon the electron-phonon interaction can be directly calculated from the imaginary part of the electron-phonon (e-ph) self-energy, which is expressed as: ˆdq where ω vq represents the phonon frequency in mode v at wave vector q, f mk and f nk are the Fermi-Dirac distribution, n is the Bose-Einstein distribution functions and g mn,v is the electron-phonon coupling matrix elements.
The phonon-limited intrinsic carrier mobility is computed via the ab initio Boltzmann transport equation within the self-energy relaxation time iterative method.It is known that the carrier mobility is a second-order tensor: Because the h-BX systems possess a 2D hexagonal structure with the D 3h symmetry, the diagonal terms µ xx and µ yy (µ zz = 0) are usually quite similar, while the non-diagonal term µ xy (µ xz = µ yz = 0) is extremely small and negligible.Therefore, we only give the diagonal term µ xx in current work.The component µ αβ can be computed by [30]: ˆdk where α and β run over the three Cartesian directions, Ω(Ω BZ ) is the area of the unit cell (Brillouin zone), v nk is the electronic band velocity (n is the band index and k is the wave vector), f nk is the Fermi-Dirac distribution, ∂E β f nk is the derivative of f nk with respect to the static electric field along the β direction E β and e is the electron charge.

Geometry and electronic structures of h-BX
Figure 1(a) exhibits the geometry structures of the 2D hexagonal boron compounds h-BX(X = P, As, Sb).These h-BX systems possess a planar geometry with the P-6m2 space group.Relaxed lattice constants for the BP, BAs, and BSb monolayers are 3.207 Å, 3.382 Å, and 3.727 Å, respectively, in good agreement with reported values.To ensure the stability of the h-BX systems, we carried out the AIMD simulations.The simulations are implemented at 300 K and 700 K, utilizing the Nosé-Hoover thermostat with a time step of 3 fs.Figures 1(b)-(d) depict the time evolution of MD trajectories at both temperatures.The simulations have been carried out for sufficient time to achieve a steady state.The structures of the h-BX monolayers remain unchanged throughout the process, demonstrating their thermal stability at both temperatures.Figure 2 presents the band structures of h-BX at the PBE (black line) and HSE06 (red line) levels of theory.It is observed that all three materials exhibit a direct band gap, with the valence band maximum (VBM) and the conduction band minimum (CBM) located at the high-symmetry K point in the BZ.At the  PBE level, the determined bandgaps are 0.902 eV, 0.755 eV, and 0.321 eV for BP, BAs, and BSb, respectively.However, it is known that the PBE calculations tend to underestimate the band gaps.In order to address this issue, HSE06 calculations are performed and the resulting band gaps are 1.480 eV, 1.183 eV, and 0.608 eV for BP, BAs, and BSb, respectively.These values are in good agreement with previous studies [31].In this work, we mainly focus on the charge transport properties of h-BX.In terms of the Drude mode [32], the charge transport characteristics strongly depend on the effective mass and scattering rate simultaneously.Through the band-structure analysis of the h-BX systems, the effective masses around CBM and VBM are achieved and listed in table 1, which align well with the results of the previous study [33].The relatively small effective masses suggest the possibility of high carrier mobilities [17,34].On the other hand, due to the existence of six K valleys in the band structures of h-BX near the Fermi level, the carriers (electron/hole) will only occupy these K valleys over a wide range of carrier concentrations.As a result, both the intravalley scattering (within the same K or K ′ valley) and the intervalley scattering (between the K and K ′ valleys) for carriers are possible to occur.To gain a comprehensive understanding on the scattering mechanism, we have to take the Fermi Table 1.Calculated electron effective masses of the h-BX(X = P, As, Sb) systems.

Electron (CBM)/me
Hole (VBM)/me  When n 2D is lower than the critical value, the corresponding Fermi level will be situated between the VBM and CBM.However, as n 2D increases beyond the critical value, the Fermi level will move inside the valence band for holes or the conduction band for electrons, which may significantly impact the scattering.

Phonon dispersions of h-BX
Figures 3(a)-(c) show the phonon dispersions of the h-BX monolayers along the high symmetry path of Γ → M → K → Γ.For these systems, there are a total of six phonon modes, including longitudinal acoustic (LA) mode, transverse acoustic (TA) mode, out-of-plane acoustic (ZA) modes, longitudinal optical (LO) mode, transverse optical (TO) modes, and out-of-plane (ZO) modes.Since their phonon frequencies are all positive, the dynamic stability is confirmed.Because the systems are typically polar materials, there exists an obvious LO-TO splitting around Γ point, but the splitting vanishes at the center of the BZ.This is because the LO-TO splitting of 2D materials is independent of q in the long-wavelength limit, which is quite different from that of a 3D material.From figure 3, it is shown that the optical LO and TO modes possess very large phonon frequencies throughout the entire BZ.Especially at Γ point, the computed phonon frequencies (ω LO = ω TO ) of BP, BAs, and BSb are 120 meV, 100 meV, and 89 meV, respectively.The acoustic modes also exhibit a nonzero phonon frequency, except at the Γ (gamma) point, where all acoustic phonon frequencies are zero.However, the phonon frequencies of acoustic modes are much smaller than those of the LO/TO modes.Generally, there exists the strong interaction between electrons and phonons and therefore it plays a crucial role in the scattering process for carriers (electron/hole).In terms of momentum conservation, intravalley scattering, and intervalley scattering separately involve the phonons associated with Γ and K points in the BZ.Along with the phonon dispersions, the room-temperature phonon occupancy of the phonon modes is evaluated (see figures 3(d)-(f)).It is found that the LO-and TO-phonon occupancy is relatively low throughout the BZ, which usually occurs when these optical modes possess high vibrational frequencies.The low occupancy suggests a substantial suppression of scattering induced by these optical phonons.

Electron-phonon matrix element and scattering rate
In figure S4, we present the calculated absolute values of e-ph matrix elements, denoted as g mnν (k, q), between two electron states in the lowest conduction band and in the highest valence band (Initial electronic state k = K) with a phonon branch ν.It is worth noting that the LO e-ph matrix elements have incorporated the long-range e-ph interaction stemming from the polarization field.We consider the intraband electron interaction with all phonon modes, and the resulting e-ph matrix elements are plotted as a function of phonon wavevector q along the high symmetry path (Γ → M → K → Γ) of the BZ.Around the Γ point, the e-ph matrix elements for the LO mode are noticeably larger than those of the other phonon modes.The electron-LO phonon interaction shows a strong 1/q dependence at small q mainly due to the contribution of the long-range term g L mnv (k, q).However, the values are not always the largest over the entire BZ.The values of e-ph matrix elements involving LO, TO, LA, and TA modes are comparable around the K point.For the remaining ZA and ZO modes, the matrix elements are significantly smaller across the entire BZ and can thus be disregarded.
Using the calculated e-ph matrix elements, we further assess the electron-phonon scattering rate of the h-BX systems at room temperature and a carrier concentration of 1 × 10 12 cm −2 .Because the e-ph matrix is calculated at T = 0, the temperature dependence of the scattering rate is incorporated via the electron and phonon occupation functions.Figure 4 shows the total scattering rates versus carrier energy in the conduction band and the valence band of h-BX, where we have shifted the CBM and VBM to 0 eV, respectively.The total scattering rates for BP and BAs exhibit a nearly identical magnitude, whereas a smaller value is observed for BSb.To investigate the contributions of different phonon modes to the total scattering rate, the mode-resolved scattering rate of h-BX is also calculated. Figure S5 illustrates that the total scattering rate is primarily dominated by four prominent phonon modes (TA, LA, TO, and LO), while the contributions from the ZA and ZO modes are negligible.The low-energy electrons are strongly scattered by the LA and TA modes, while the scattering dominance shifts towards the TO and LO modes as the energy increases.Due to the presence of multiple K valleys in the band structures of h-BX near the Fermi level, both the intravalley and intervalley transitions are possible.To determine which one is more significant to total scattering, it is necessary to separate intravalley and intervalley contributions.By examining the magnitude of the phonon wave vector (q), we can distinguish the scattering rates into distinct intravalley and intervalley components.The individual contributions of intravalley and intervalley transitions for h-BX are plotted in figure 5. Based on the obtained scattering rates, it can be confirmed that both intravalley scattering and intervalley scattering play a significant role, and therefore, neither can be ignored.Especially for BSb, the intravalley and intervalley transitions are almost equally important.Furthermore, it is seen that the intravalley scattering rate follows a one-step change trend, while the intervalley scattering rate exhibits a two-step change pattern.The step-change behaviors can be attributed to the non-vanishing phonon frequencies for specific phonon modes around Γ and K points in the BZ.It is obvious that the situation is relatively complicated and cannot be fully comprehended solely based on the observation.To gain a comprehensive understanding of the electron-phonon scattering of h-BX, we calculate the mode-resolved scattering rates (Taking the conduction-band electrons as an example), as depicted in figure 6.The results demonstrate the main contribution of two acoustic modes (TA and LA) and two optical modes (TO and LO) to the overall scattering process.For intravalley scattering, the scattering rates due to TA and LA modes increase with carrier energy without any specific threshold frequency.However, there is an observable threshold frequency for the optical LO and TO modes, which is associated with the phonon frequencies (ω LO = ω TO ) at the Γ point.When the energy is lower than E CBM + hω LO for electrons, only the LO-and TO-phonon absorption is allowed, and the corresponding scattering rates are almost zero.This is due to the remarkably low LO-and TO-phonon occupation numbers.On the contrary, once the energies are higher than E CBM + hω LO , the LOand TO-phonon emission begins to dominate the scattering channel, thus leading to a significant rise in the scattering rate.In particular, the LO scattering rate is significantly larger than that arising from the TO phonon due to the Fröhlich interaction.Regarding the intervalley scattering, on the other hand, all these four phonon modes (TA, LA, TO, and LO) exhibit distinct threshold frequencies associated with the phonon frequencies at K point.It is shown that the LO and TO threshold frequencies are much larger than those of LA and TA.Also, one can observe that the scattering arising from the LO-and TO-phonon absorption is extremely weak due to their low phonon occupation.At energies above E CBM + hω LO (hω TO ), the LO(TO) mode starts to make a significant contribution to the total scattering.

Carrier mobility
By solving the iterative Boltzmann transport equation, we determine the room-temperature carrier mobilities of h-BX over a wide range of carrier concentrations (n 2D ) from 1 × 10 10 cm −2 to 1 × 10 14 cm −2 based on the calculated scattering rates.The obtained results are presented in figure 7. When  n 2D = 1 × 10 10 cm −2 , the BP, BAs, and BSb monolayers simultaneously possess ultrahigh electron and hole mobilities at room temperature.The calculated electron mobilities are 4097 cm 2 V −1 s −1 , 4141 cm 2 V −1 s −1 , and 12 215 cm 2 V −1 s −1 , respectively.At the same time, the hole mobilities are evaluated to be 7563 cm 2 V −1 s −1 , 7606 cm 2 V −1 s −1 , and 22 282 cm 2 V −1 s −1 , respectively.It is found that the carrier mobilities are significantly greater than those of the common 2D semiconductors such as MoS 2 [11], InSe [12], and black phosphorus [35] in the monolayer form.The ultrahigh carrier mobilities in combination with the suitable bandgaps makes h-BX monolayers a very promising material in electronic and thermoelectric applications.However, it should be noted that the values of the carrier mobility do not always remain unchanged with an increasing carrier concentration.Once the carrier concentration (n 2D ) increases beyond 1 × 10 12 cm −2 , one can observe that a rapid decline occurs in the mobility.Especially when n 2D increases to 1 × 10 14 cm −2 , the mobilities have been dramatically reduced to merely ∼500 cm 2 V− 1 s −1 .Hence, for the three semiconductors, the carrier concentration (n 2D ) lower than 1 × 10 12 cm −2 is recommended in practical applications.
To better understand the reason behind these behaviors, a closer look into the scattering mechanisms would be helpful because electron-phonon scattering plays a key role in determining carrier mobility.We thus expect that the changing trend of the carrier mobility with n 2D can be comprehended with the help of the scattering rate.For the sake of simplicity, we only focus on the scattering rate of conduction-band electrons.Figure 8 shows the scattering rate versus the carrier energy with respect to total six different carrier concentrations (1 × 10 10 cm −2 , 1 × 10 11 cm −2 , 1 × 10 12 cm −2 , 1 × 10 13 cm −2 , 5 × 10 13 cm −2 , 1 × 10 14 cm −2 ).As depicted in figure 8, the scattering rates for n 2D = 1 × 10 10 cm −2 and 1 × 10 11 cm −2 follow quite similar trends, providing an explanation of the almost unchanged carrier mobility within this range of n 2D .Near the band edge, the exceptionally weak scattering induced by the LO and TO phonons accounts for the remarkably high carrier mobilities in the h-BX systems.Among them, BSb exhibits the highest carrier mobility as a result of the lowest scattering rate.
We note that only electrons or holes within a relatively narrow energy range around the Fermi level (within ∼0.2 eV of the Fermi level at room temperature) are involved in scattering and contribute to carrier mobility.When the Fermi level approaches the CBM, the scattering rate near the CBM starts to rise.This is because there are more carriers around the Fermi level participating in the scattering.Especially when the Fermi level significantly exceeds the CBM, the scattering for these carriers will be greatly enhanced, thus leading to a sharp reduction in carrier mobility.Moreover, for large carrier concentrations such as n 2D = 5 × 10 13 cm −2 and 1 × 10 14 cm −2 , there is always a dip in the scattering rate around the Fermi level.By analyzing mode-resolved scattering rates (in figure 9), we find that the dip is mainly caused by the electron-LO phonon scattering and the electron-TO phonon scattering, specifically referring to the LO/TO phonon emission and absorption.It is widely recognized that the scattering rate heavily relies on both the electron occupation function and the phonon energy.It can be expressed in terms of the energy difference between the initial state and the Fermi level (|ε k − ε F |).When the initial states are exactly at the Fermi level, the scattering rate reaches its minimum, thus resulting in the observed dip.The fundamental reason is the pronounced suppression of electron scattering induced by the LO and TO phonons below their phonon energies.When the electron energy is below the Fermi level, the absorption process dominates, while the emission process becomes dominant as the energy exceeds the Fermi level.

The effect of strain on the transport property
Strain engineering is a powerful tool to design and optimize semiconductors for specific electronic properties [36].Through adjusting the lattice constant, the strain can be introduced into the materials, which can significantly influence the physical properties of materials.Specifically, the biaxial strain has been shown to have a significant impact on the properties of the 2D materials [37,38].Given this background, we have considered the influence of biaxial strain on the carrier mobility of the h-BX systems.The strain capacity (ε) is defined as the percentage change in lattice constant from its equilibrium value a 0, and it can also be used to tailor the carrier mobility of the material.For the h-BX systems, a total of four strains (ε = −2%, −1%, 1%, and 2%) have been taken into account.Under these strains, carrier mobility has been evaluated as a function of the carrier concentration.
Figure 10 demonstrates that the biaxial strain has a significant effect on the carrier mobility of h-BX.When the systems are subjected to tensile strain, the mobility of electrons and holes starts to decrease.On the other hand, applying compressive strain to the systems can give rise to a significant increase in carrier mobility.For instance, in the case of BSb with a carrier concentration of 1 × 10 12 cm −2 , when a 2% compressive strain is applied and then switches to a 2% tensile strain, the hole mobility remarkably decreases from 8463 cm 2 V −1 s −1 to 2625 cm 2 V −1 s −1 .Similarly, the electron mobility is also quickly varied, ranging from 4884 cm 2 V −1 s −1 to 2063 cm 2 V −1 s −1 .It is noticeable that the carrier mobility of h-BX shows a significant dependence on the applied strain.However, it should be noted that the notable variation in carrier mobilities with the strain is sustained effectively only when n 2D falls within the range of 1 × 10 10 cm −2 -1 × 10 12 cm −2 .This finding unveils the promising prospects of employing these semiconductors as stress or strain sensors.
As mentioned previously, electron-phonon scattering is the key factor in determining the carrier mobility for intrinsic semiconductors.To enhance the understanding on the correlation between strain and carrier mobility, we also compute the corresponding scattering rates (taking the conduction-band electrons as an example).Figure 11 depicts the corresponding intravalley, intervalley, and total scattering rates for the h-BX systems under strains of −2%, −1%, 1%, and 2%.Notably, compressive strain results in a remarkable decrease in the total scattering rate, while tensile strain leads to an increase in the total scattering rate.The behavior is in good agreement with the changing trend of the carrier mobility.We also observe that the change of the total scattering rate mainly stems from the change of the intravalley scattering in both BP and BAs systems.The case of BSb is quite different, in which the changes of the intravalley scattering and the intervalley scattering are almost equally important.On the other hand, considering the inherent polarity of h-BX, we partitioned the total scattering rate into two constituents: the polar component and the non-polar   component, as illustrated in figure 12.It has been observed that strain induces a significant alteration in the non-polar component, thus affecting the carrier mobility.In contrast, the changed polar scattering rate is still significantly suppressed because of the high LO/TO-phonon frequencies and thus has a minor impact on the carrier mobility.In summary, once a compressive strain is applied, the total scattering rate is notably decreased, leading to a significant increase in carrier mobility.On the contrary, a tensile strain increases the total scattering rate and results in an obvious reduction of carrier mobility.It is important to note that the influence of strain on the scattering rate has a complicated physical mechanism.The possible reason for the change in scattering rate under the strain is the variations of phonon occupancy and DOS.Under tensile strain, we find that the phonon occupancy of the h-BX systems give rise to obvious enhancement due to the decrease of the phonon frequencies for the phonon modes, which is called the phonon softening [39].The enhanced phonon occupancy can increase the scattering rate in the low-energy region related to the phonon-absorption process.On the other hand, tensile stain increases the DOS around the VBM and CBM.The increasing DOS can increase the number of the final states in the scattering process, thus enhancing the total scattering rate for the three systems.Also, the opposite change trend is obtained under the compressive strain.

Conclusions
To summarize, we performed first-principles calculations along with Boltzmann transport theory to gain an insight into the electron-phonon scattering and charge transport of the 2D polar h-BX (X = P, As, Sb) semiconductors.Our findings reveal that both intravalley scattering and intervalley scattering significantly contribute to the overall scattering process.Remarkably, these 2D semiconductors are found to possess both high hole and electron mobilities at room temperature.Furthermore, it is discovered that carrier mobility shows a pronounced dependence on the biaxial strain.Compressive strain induces a substantial increase in carrier mobility, while tensile strain leads to a noticeable decrease.Through our analysis, it is determined that the prominent charge transport properties of the 2D h-BX (X = P, As, Sb) semiconductors are attributed to the relatively small effective masses in combination with the substantial suppression of scattering due to the

Figure 1 .
Figure 1.(a) Top view and side view of monolayer h-BX.MD simulations of BP (b), BAs (c), and BSb (d) at 300 K and 700 K.

Figure 2 .
Figure 2. The band structures of BP (a), BAs (b), and BSb (c).The black line represents the band calculated using PBE functional and the red line represents the band after considering the HSE correction.

Figure 4 .
Figure 4.The total scattering rates of h-BX at 300 K and a carrier concentration of 1 × 10 12 cm −2 .

Figure 5 .
Figure 5.The total (blue), intravalley (black), and intervalley (red) scattering rates versus the carrier energy (both CBM and VBM are set to zero).

Figure 7 .
Figure 7.The carrier mobility of h-BX at 300 K as a function of carrier concentrations.

Figure 8 .
Figure 8.The carrier scattering rate of BP (a), BAs (b), and BSb (c) at 300 k as a function of the carrier concentration.

Figure 10 .
Figure 10.The hole mobilities (a)-(c) and electron mobilities (e), (f) as the function of the carrier concentration under different biaxial strains.

Figure 11 .
Figure 11.The electron scattering rate of total (blue), intravalley (black), and intervalley (red) above the CBM at 300 K with five strains.

Figure 12 .
Figure 12.The rate of the polar part and nonpolar par above the CBM at 300 K under different biaxial strains.