Electronic Structure and Carrier Mobility of Two-Dimensional Î± Arsenic

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The Journal of Physical Chemistry

Electronic Structure and Carrier Mobility of Two-dimensional α Arsenic Phosphide

Fazel Shojaei Department of Chemistry and Bioactive Material Sciences and Research Institute of Physics and Chemistry, Jeonbuk National University, Jeonju, Chonbuk 561-756, Republic of Korea and Hong Seok Kang* Department of Nano and Advanced Materials, College of Engineering, Jeonju University, Hyojadong, Wansan-ku, Chonju, Chonbuk 560-759, Republic of Korea

*Corresponding author: [email protected]

ACS Paragon Plus Environment


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ABSTRACT

Using first-principles calculations, we investigate electronic structures of α arsenic phosphide under strain. It is a two-dimensional monolayer composed of an equimolar mixture of phosphorus and arsenic, whose multilayer correspondents were synthesized very recently. According to structure optimizations and phonon calculations, the α phase branches into three distinct allotropes. Monolayers of the α1 and α3 phases are direct-gap semiconductors with band gaps that are similar to that of the α phosphorene. They exhibit anisotrpic carrier mobility. Specifically, the α3 phase exhibits the electron mobility of ~10,000 cm2V-1s-1, which is one order of magnitude larger than that for the α phosphorene. Likewise, their electronic structures display highly anisotropic behavior under strain different from that of the α phosphorene. The complex response under strain can be mostly understood in terms of the relative alignment of bonding and antibonding As-P states under a specific strain.

Keywords: first-principles calculation, dynamical stability, carrier mobility, direct/indirect gap, anisotropic behavior under strain, valleytronics


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1.

INTRODUCTION As an alternative to the gapless material graphene, phosphorene has emerged as a novel

two-dimensional (2D) semiconducting material for transistor applications. Very recently, fewlayer black phosphorene ( α phase) has been exfoliated from black phosphorus.1 This material exhibits a direct band gap of 1.51 ~ 0.59 eV, which decreases as the number of layers increases from 1 to 5.2 In addition, it possesses a high carrier mobility in the range of 103~104 cm2V-1s-1, which is more than one order of magnitude larger than that (~200 cm2V-1s-1) of another directgap 2D material, MoS2.2,3 Specifically, its monolayer exhibits anomalous elastic properties that reverse the anisotropy of the hole mobility. Moreover, biaxial strain is found to reverse the electron mobility4 or results in Dirac-shaped electronic dispersion,5 whereas uniaxial strain causes a direct-indirect-direct transition.6 Other types of dynamically stable allotropes have been proposed; among the seven possible phases, θ and the β phases (blue phosphorene) have been found to be nearly as stable as the α phase.7 Under slight compression perpendicular to its surface, bilayer black phosphorene exhibits an extraordinary room temperature electron mobility of 7 × 104 cm2V-1s-1.8

Arsenene is also beginning to attract attentions in an effort to search for other 2D materials with high mobility and excellent contact with electrode materials. Dynamically stable α arsenene is analogous to black phosphorene, whose three-dimensional (3D) crystal is orthorhombic arsenolamprite with a direct band gap of 0.3 eV.9 In contrast to the case of the α phosphorene, arsenene is an indirect-gap semiconductor with a band gap of ~0.97 eV.10 However, it transforms into a direct-gap material with anisotropic carrier mobility as a result of multilayer formation. β arsenene, which constitutes the most abundant grey arsenic phase in the 3D crystal, exhibits a semimetallic-semiconducting transition as the number of layers decreases to one.

11-13

The

indirect band gap of its monolayer can also be tuned to a direct band gap under biaxial strain.14

Herein, we investigate interesting electronic properties of 2D phases of arsenic phosphide under strain. This material differs from pure phosphorene or arsenene in that it can possess more



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diverse phases depending on the topological arrangement of the two types of atoms. Very recently, α AsxP1-x was indeed synthesized adopting alloying strategy.15Its composition is highly tunable, and the band gaps of its multilayers with thickness greater than 30 nm decrease with increasing As content. In addition, the material was mechanically exfoliated into bilayers, whose electronic structures were not reported. Therefore, it is highly desirable to investigate the electronic structure of its monolayer. In this work, we focus on AsP, which is formed from a 1:1 stoichiometric mixture of P and As. Note that there are many different approaches for forming equimolar α AsP depending upon the topology of two types of atoms. Specifically, there are three uniform allotropes shown in Figure 1: α1 and α 2 indicate that there are As-As bonds and they are directed almost normal (out-of-plane) to or almost parallel (in-plane) to the layer, respectively; α 3 indicates that there are no As-As bonds but As-P bonds. As a reference, it is worth referring to the data for the atomic radii (0.70, 1.10, and 1.21 Å, respectively) and the electronegativities (3.0, 2.1, and 2.0, respectively) of three isoelectronic elements around P, i.e., N, P, and, As.16 These data indicate that the P-As pair is quasi-homonuclear, whereas the N-P pair is clearly heteronuclear. We will demonstrate that their homonuclear nature analogous to the

α phosphorene is manifested in the direct band gaps at zero strain, which are clearly different from the case of the α arsenene. Furthermore, we will also demonstrate that their heteronuclear characteristics are magnified under strain.

2.

COMPUTATIONAL METHODS

Geometry optimizations were performed using the Vienna ab-initio simulation package (VASP).17-18 The electron-ion interactions were described using the projector-augmented wave (PAW) method, which is primarily a frozen-core all-electron calculation.19 Attractive van der Waals interactions were included through Grimme’s correction in the PBE-D3 method.20 For structure optimization, the atoms are relaxed in the direction of the Hellmann-Feynman force using the conjugate gradient method with a high cut-off energy of 400 eV until a stringent convergence criterion (= 0.01 eV/Å) is satisfied.


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The lattice constants were also optimized using this method. When necessary, accurate electronic structure calculations were performed using the HSE06 method using the lattice constants. This combination is known to be the most reliable for pure phosphorene.2 We maintained a sufficiently large vacuum space (= 17 Å) along the direction normal to the sheet plane to guarantee no appreciable interactions between two adjacent layers. k-point sampling was performed using 25 × 25 × 1 points for structural optimization and electronic structure calculations. Phonon dispersion calculations were performed using the supercell finitedisplacement method implemented in the PHONOPY package21, with VASP used as the forceconstant calculator.22 Force evaluations were performed on 3×3×1 supercells using reduced kpoint sampling meshes of 11×11×1 points.

In the 2D system, the carrier mobility is calculated from the electron-phonon coupling in the longitudinal acoustic phonon limit23

µ2 D =

eh 3C2 D i kTm * md ( E1 ) 2

(1)

where C 2 D is the elastic modulus of the longitudinal strain in the transport direction, m * is the effective mass of the carrier in the direction, md is the average effective mass in the two directions given by md = mx * my * and E1i mimics the deformation energy constant of the carrier due to phonons for the i-th edge band along the transport direction by the relation: 1 E i = ∆ E i /( l / l 0 ) , where ∆ Ei and l / l 0 represent the energy change of the i-th band and dilation,

respectively. More details on the calculation procedure can be found in the Supporting Information.



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3.

RESULTS AND DISCUSSION

Table 1 shows that the α1 is the most stable among the three allotropes of the α phase, in which all zigzag bonds are composed of alternating types of atoms. Figure 2(a) shows that the allotrope exhibits no imaginary frequency in the phonon dispersion curve, which is also indicated in Table 1. For the α1 , we find that the LY constant along the zigzag direction is 6% longer than that for pure phosphorene, whereas there is only a marginal elongation in the LX constant along the armchair direction. For comparison, we recall that our lattice parameters for the pure phosphorene are in very good agreement with those obtained from the optB86b-vdW calculation, 2

thereby confirming the reliability of our PBE-D3 calculation for the structure optimization. As

shown in Table 1, the elongation can be attributed to the As-P bond length of 2.38 Å which is ~ 0.16 Å longer than the in-plane P-P bond length in α P. We note that the P-As-P bond angle As (ψ zz ) of 95.0º along the zigzag direction is almost the same as that (= 96.0º) in the pure

phosphorene, as shown in Table 1. However, there are two puckering angles along the armchair direction: the P-As-As (ψ ac As ) angle of 99.9º is appreciably smaller than that (= 103.9º) in pure phosphorene, thereby almost compensating for the effect of the longer P-As bond length in the LX parameter. Meanwhile, the As-P-P (ψ ac P ) angle of 103.2º is only slightly smaller.

Table 1 shows that the LY constant of the α2 is larger than that in the pure phosphorene but smaller than that in the α1. Recall that zigzag bonds along the axis are exclusively composed of As-As or P-P bonds in α2. The As-As bond length of 2.45 Å is 0.07 Å longer than the P-As bond length in the α 1 . However, the elongation is compensated by the As-As-As bond angle (ψ zz As ) of 89.80º, which is even smaller than the ψ zz As angle in the α 1 . Moreover, the P-P bond length is elongated to 2.27 Å to achieve the common lattice constant along the direction. Likewise, the PP-P bond angle (ψ zz P ) of 99.4º becomes appreciably larger than the As-As-As angle (ψ zz As ) of 89.8º. This types of disproportion in the bond lengths and bond angles might result in the dynamical instability of the phase. However, the phonon dispersion relation in Figure 2(b) shows


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only very small negative frequencies for a long wavelength acoustic mode along the Γ → Y direction. Therefore, the dynamical stability of the α 2 phase cannot be ruled out.

Regarding the α 3 , we find that this phase is as energetically stable as the α 2 . However, it is also dynamically stable, as shown in the phonon dispersion curve in Figure 2(c). The lattice constants of this phase are approximately the same as those of the α1 . As in the case of the α1 AsP, there are two types of As-P bonds with slightly different bond lengths of 2.37 and 2.39 Å : one inAs

plane bond and the other out-of-plane bond, respectively. The P-As-P bond angle (ψ zz ) of 95.0o along the zigzag direction is similar to that (= 95.0º) in the α1 . In contrast to the case of the

α1 , there are no P-P or As-As bonds along the armchair direction. Consequently, there are two puckering angles: the P-As-P (ψ ac As ) and As-P-As angles (ψ ac P ) of 99.7º and 103.9º along the armchair direction are also very simillar to the corresponding angles in the α1 . Regarding the outof-plane bonds, the heteronuclear As-P bonds in the α 3 phase are less stable than the sum of corresponding P-P and As-As bonds in the α1 phase.

Figure 3(b) shows that the α1 phase is a direct-gap semiconductor at the Γ point with an HSE06 band gap of 1.63 eV. This gap is very close to that (= 1.65 eV) of the α phosphorene. For the latter, it is worth recalling the valence band maximum (VBM) is slightly off the Γ point, not exhibiting a perfect direct gap.24 In the later paragraph, we will note that the PBE band gap at the k-point is 0.88 eV, which is also closer to the PBE gap of 0.84 eV for the α P at the same kpoint. As a reference, indirect and direct PBE gaps for the α As are 0.62 and 0.72 eV at X ' → Γ and X ' → X ' transitions, respectively, where X ' denotes a k-point between Γ and X

points. This result is different from the corresponding report for the multilayer, in which the observed band gap is rather closer to that of the α arsenene.15 Similar to the case of the α phosphorene, Figures S1(a) and (b) show that the VBM and the conduction band minimum (CBM) represent out-of-plane pz orbitals. However, there is a second VBM at the X ' point. We



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find that Ev ( X ' ) − Ev (Γ) = −0.08eV , where Ev ( X ' ) and Ev (Γ) denote the energy eigenvalues of the valence band at appropriate k-points. We can conjecture that the two VBMs are almost degenerate because a PBE calculation in a later paragraph will show that their relative stabilities are reversed. In contrast, the corresponding difference of 0.64 eV in the α phosphorene is considerably larger. This observation indicates that the α1 AsP will be useful in valleytronics,25 whereas the unstrained α phosphorene will not. As shown in Figure S2 (a), the second VBM at the X ' point essentially represents in-plane σ bonds through px orbitals. Recall that the VBM is located at the X ' point for the α arsenene.10 This observation can be understood if we note that the in-plane P-As bonding σ states of the α1 AsP are less stable than the corresponding P-P states of the α phosphorene because the atomic px(As) state is located higher than the atomic px(P) state. In short, the near degeneracy of the valence band at the two k-points originates from the heteronuclear As-P bonds.

Here, we briefly describe the band structure of the α 2 AsP. Figure 3(c) indicates that the

α 2 AsP is also a direct-gap semiconductor with the band gap (= 1.63 eV) at the Γ point, which is the same as in the α1 AsP. Now, we focus on the band structure of the α 3 AsP. Figure 3(d) shows that its band structure appears very similar to that of the α 1 phase. The band gap is direct and is located at the Γ point. In addition, the HSE06 gap of 1.60 eV is also close to that of the α1 phase. This result is not surprising considering the structural similarity of the two phases. As previously mentioned, the only difference is that the out-of-plane bonds are homonuclear and heteronuclear in the α1 AsP and the α 3 AsP, respectively. This difference is reflected in the relative energy of the second VBM at the X ' point with respect to the VBM, which is considerably more (~ 0.22 eV) pronounced than in the α1 phase.

Table 2 shows that the effective masses of carriers for the α1 and the α 3 AsP are similar to those for the α P, except for the hole mass which is somewhat lighter along the zigzag direction.


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However, the carrier mobility is more sensitive to the deformation energy, decreasing the hole mobility along that direction. Surprisingly, the electron mobility for the α 3 AsP along the armchair direction is one order of magnitude larger than that for the α P, because corresponding deformation energy is considerably smaller (< 1/4) than that for the latter. In turn, the smaller deformation energy can be ascribed to the flexibility of the structure along the armchair direction under the strain, which brings about minimal strain in the in-plane As-P bonds responsible for the energy of the CBM. Other kinds of carrier mobilities are similar to those for the α P, rendering the allotropes as useful as the α P for nanoelectronics. For the α1 AsP, the hole mobilities shown in the table will correspond to the upper bound. This is because the X ' point, which is almost degenerate to the Γ point in the VB, provides the other quantum state for the hole transport. Our calculated hole mobility is also anisotropic, and our data for the two directions can be favorably compared with the measured value of ~110 cm2V-1s-1 for multilayers of As0.83 P0.17.15

Now, we focus on changes in the electronic structure under strain for various AsP phases. As Figure S4 shows, Poisson’s contraction is taken into account along the direction normal to the applied strain. In contrast to the case at zero strain, most of the electronic structure calculations are performed using the PBE functional. As shown in Figure S3, the α AsP phases are softer than the α phosphorene because P-As bonds are weaker than P-P bonds. Uniaxial strain ( σ ac ) along the softer armchair direction can be achieved quite easily up to 10%. The α phosphorene was shown to be capable of withstanding tensile stress and strain along the same direction up to 10 N/m and 30%, respectively.6 The strain energy in our system is less than 0.02 eV per atom at 10% strain, and the tensile stress is ~1 N/m at 10% strain. Although stretching along the zigzag direction is more difficult, the strain energy of 0.04 eV per atom is still comparable to the thermal energy at 10% strain. However, biaxial strain is considerably more difficult to achieve. In fact, the strain energy at 6% of isotropic ( σ ac : σ zz = 1 : 1 ) biaxial strain is equivalent to that at 10% uniaxial strain along the zigzag direction.



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In the HSE06 band structure for the α1 AsP, it was observed that the energy levels of the VBM and the second VBM are similar. Therefore, it is not surprising that their levels are reversed depending on the type of exchange-correlation functions employed. When the PBE functional is adopted, the reversal actually occurs even at 0% strain. Keeping this fact in mind, we find that

Ev ( X ' ) − Ev (Γ) = 0.02eV . Therefore, the material can still be considered to be a direct-gap semiconductor with the band gap of 0.88 eV at the PBE level. Now, we focus on the change in the electronic structure under strain. Figure 4 shows that the PBE band gap increases monotonically with uniaxial strain along the armchair direction, reaching 1.14 eV at 10%. Our separate analysis shows that the PBE band gap remains direct over the entire range of strain considered. For comparison, the α phosphorene was shown to exhibit a quite similar behavior up to 8%.6 However, our HSE06 calculation demonstrates that the CB at the Γ point is almost degenerate to that at the Y point at 7% of σ ac , where the band gap of 1.84 eV is larger than that (= 1.63 eV) at zero strain. Consequently, a direct-indirect transition is observed at a larger value of σ ac in the HSE06 calculation. Figure S5 shows that this is indeed the case at 10%, where the indirect gap is 1.80 eV. Figures S2 (c) and (d) show that the CB at the Y point represents the py(P) orbitals, which is stabilized by larger bonding interactions between P atoms not directly bonded along the zigzag direction due to the Poisson’s contraction at the value of σ ac . In short, the HSE06 energy eigenvalues of the CB at Γ and Y cross each other at ~7% of σ ac .

Under uniaxial strain ( σ zz ) along the zigzag direction, the PBE band gap exhibits almost opposite behavior under strain, i.e., it increases and then decreases with strain after 1%. In greater detail, the α 1 phase undergoes a direct-indirect-direct transition at 1 and 4% strain within the PBE. Apparently, similar double transitions were also observed for the α phosphorene, but these transitions occurred at larger values of strains, i.e., at 8 and 11.3%, respectively.6 In our case, the Ev ( X ' ) is destabilized under strain because the k-state represents in-plane bonds. Therefore, the PBE gap becomes indirect at 1% strain. Consistent with the PBE result, Figure S6(a) shows that the HSE06 band gap also corresponds to the X ' → Γ transition at 2% strain: the direct Γ → Γ transition energy of 1.66 eV is slightly larger than that (= 1.63 eV) for the X ' → Γ


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transition because Ev ( X ' ) − Ev (Γ) = 0.03eV . In short, the HSE06 energy eigenvalues of the VB at Γ and X ' cross each other at ~2% of σ zz , causing the direct-indirect transition. Apparently, a similar transition was observed at 8% strain for α phosphorene, where the PBE Ev ( X ' ) is still located below Ev (Γ) and the PBE band gap instead corresponds to the Γ → X ' transition.

As the strain increases further, we expect that Ec ( X ' ) of the α 1 phase to be stabilized. This is because this strain increases the in-plane bond lengths and therefore stabilizes the k-state that corresponds to an antibonding interaction between the bonds, as shown in Figure S2(b). Moreover, the Ec (Γ) is destabilized because the in-plane bonding interaction at the k-state is weakened. These effects are reminiscent of the HSE06 band structure at ~7% shown in Figure S6(b), where the band gap of 0.71 eV is now direct at the X ' point. This indirect-direct transition is also apparently similar to the case of that at 11.3% for the α phosphorene within the PBE. However, there is a clear difference in the transition mechanism of the two cases. Although we will not go into detail, we find that the difference originates from the difference of the As-P antibonding interaction from the P-P antibonding interaction. In summary, the nature of the double transitions in the α 1 phase within the PBE is different from that of the α phosphorene, which evidently originates from the difference between As-P bonds and P-P bonds. This is also the reason why the transitions occur at quite different values of strain in the two systems. In short, the HSE06 energy eigenvalues of the CB at Γ and X ' cross each other at 10,000 cm2V-1s-1) along the armchair direction is one order of magnitude larger than that for the α P. The predicted high electron mobility can be partly deteriorated by defects, electron-electron scattering, and semiconductor-substrate interaction. The spin-orbit effect can be also important for As atoms. However, our preliminary calculation shows that the effect brings about the change in the PBE band gap of the α 1 phase only by 1%.



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Likewise, the electronic structures of the two α AsP also exhibit highly anisotropic behavior under strain. Under uniaxial strain along the armchair direction, their band gaps remain direct and increase monotonically up to ~7%. On the one hand, the allotropes experience more complicated electronic changes under other types of strain. Specifically, the two allotropes will also be useful for n-type valleytronics at a proper value of isotropic biaxial stress, where degeneracy of two or three k-states occurs in the conduction band. The complexity can be understood in terms of the relative alignment of four states under specific strain, i.e., in-plane and out-of-plane bonding and antibonding states of As-P bonds, which is certainly different from that of P-P bonds in the α P. We hope that the present work will facilitate the experimental fabrication of more diverse phosphorene analogues for applications in nanoelectronics.

ACKNOWLEDGMENTS This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant 2012-R1A1A-2039084). We would also like to thank Jeonju University for partial financial support.

Supporting Information Available: A full list of Ref. 15, detailed procedures for calculating the effective mass, the elastic modulus, and the deformation energy as well as Figures S1-S6. This information is available free of charge via the Internet at http://pubs.acs.org


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26.

Han, X.; Stewart, H. M.; Shelvin, S. A.; Catlow, R. A.; Guo, Z. X. Strain and Orientation Modulated Band Gaps and Effective Masses of Phosphorene Nanoribbons. Nano Lett. 2014, 14, 4607.



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Table 1 Various properties for three allostropes of the α AsP in comparison with those of the α P. All lengths are in units of Å. The superscript in the bond angle designates the central atom. LX, LYa

lAs-P

lAs-As

ψacP

ψacAs

ψzzP

ψzzAs

-

-

103.9

-

96.0

-

2.38

2.53

103.2

99.9

95.0

95.0

2.39

2.45

-

103.6

99.4

89.8

2.37,2.39b

-

103.9

99.7

95.0

95.0

lP-P

4.59,3.30 2.22,2.26b αP α1 4.62,3.50 2.22 AsP α2 4.79,3.46 2.27 AsP α3 4.65,3.50 AsP a Two lattice parameters.

∆Εc 0

Dynamic stabilityd O O

0.02

∆

0.02

O

b

Two values for lP-P in the α P and those for lAs-P in the α3 ΑsP correspond to in-plane and out- of- plane P-P and As-P bond lengths, respectively. c

Relative stability of different allotropes with respect to the α1 ΑsP in units of eV/atom.

d

O sign indicates that there is no negative frequency in the phonon dispersion curve, whereas ∆ symbol implies that there are small negative frequencies in a longitudinal acoustic mode.

Table 2 The effective mass ( me *, mh * ) with respect to a free-electron mass ( m0 ), and the mobility ( µe , µh ) in (103 cm2 V-1s-1) of an electron and a hole along two directions for the α1 and the α3 AsP in comparison with those for the α phosphorene obtained from the HSE06 calculation. Data for the elastic modulii (C2D) are also shown.

αP α1 AsP

α3 AsP a

µe

mh * / m0

Hole i E1 (eV)

5.80(7.11)a

0.13(0.08)a

6.40 (6.35)a

0.27(0.15)a

0.18(0.17)

1.4(2.72)

0.16(0.15)

3.13 (2.50)

1.11 0.24 1.10 0.22

5.10 1.15 5.16 0.45

2.873.40(1.10) 0.11-0.12 1.68-2.38 0.11 10.4625.84

3.636.56(1026)a 0.32(0.60)

1.59 0.20 1.69 0.19

1.45 3.70 1.34 3.29

0.75-1.03 0.20-0.22 0.86-1.19 0.29

C2D (J/m2)

me * / m0

Zigzag

106.27(101.60)a

1.12(1.12)a

Armchair

23.92(28.94)

Zigzag Armchair Zigzag Armchair

77.76 15.27 76.36 16.12

Electron i E1 (eV)

Values inside parenthese denote those reported in Ref. 2.


µh

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Figure 1 Two different views for the structures of three allotropes of the α AsP.



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Figure 2 Phonon dispersion curves for three allotropes of the α AsP, i.e., α 1 (a), α 2 (b), and α 3 (c). In addition, the first Brillouin zone and points of special symmetry are also shown in (d).


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Figure 3 The HSE06 electronic band structures for three allotropes of the α AsP, i.e., α 1 (b), α 2 (c), and α 3 (d), in comparison with that of the α P (a).



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Figure 4 The PBE band gap versus isotropic biaxial ( σ ac : σ zz = 1 : 1 ), anisotropic biaxial (

σ ac : σ zz = 3 :1 or σ ac : σ zz = 4 : 1 ) , or uniaxial strain ( σ zz or σ ac ) for the α1 AsP.


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Figure 5 The HSE06 (a) and the PBE (b) band structures of the α1 AsP at the anisotropic biaxial strain of σ ac = 8% for series B.



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Figure 6 The PBE band gap versus isotropic biaxial, uniaxial and anisotropic biaxial strains for the α 3 AsP.


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Table of Contents (TOC) Graphics


Electronic Structure and Carrier Mobility of Two-Dimensional Î± Arsenic

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