Giant anisotropic photogalvanic effect in a flexible AsSb monolayer with ultrahigh carrier mobility†
Received
1st August 2017
, Accepted 16th September 2017
First published on 18th September 2017
Searching for novel twodimensional (2D) materials with desirable properties is of great significance for the design of next generation nanodevices. In this work, we address the electronic and optoelectronic properties of monolayer AsSb on the basis of density functional theory (DFT) combined with quantum transport simulations. We find three stable phases of monolayer AsSb, that is, the α, γ and ε phases, and all of them show direct bandgaps, which are beneficial in increasing the transition probability of photongenerated electrons and improving the efficiency of photoelectric conversion. In addition, these systems could attain meaningful straininduced band engineering and a phase transition from semiconductor to metal occurs. It is highly interesting that the monolayer AsSb has an ultrahigh carrier mobility (∼10^{4} cm^{2} V^{−1} s^{−1}), which is evidently larger than most of the reported 2D materials. In light of the nonequilibrium Green's function formalism, a linear photogalvanic effect (PGE) is observed along both the zigzag and armchair directions, suggesting that monolayer AsSb exhibits excellent photoresponse in a broad spectrum ranging from ultraviolet to infrared light, which is favorable for serving as a potential outstanding photovoltaic material. Our results highlight that these monolayer AsSb are promising candidates for future applications in electronics and optoelectronics.
Introduction
The emergence of atomic twodimensional (2D) materials, such as graphene and transitionmetal dichalcogenides (TMDs), has attracted growing interest due to their exotic properties, which are desirable for electronic and optoelectronic devices.^{1–3} However, it is difficult for graphenebased field effect transistors (FETs) to acquire a high on/off current ratio due to their gapless features, while for TMDs materials, the low carrier mobility (∼200 cm^{2} V^{−1} s^{−1}) limits their applications in fastdeveloping electronic and optoelectronic devices.^{4–7} Therefore, searching for novel 2D materials besides graphene and TMDs with desirable properties has become an important scientific topic. Recently, GroupVA monolayer materials, such as black phosphorene (BP), Sb, Bi as well as black ArsenicPhosphorus (bAsP), have been successfully fabricated and their advanced electronic structures and high carrier mobility have been demonstrated as well.^{8–16} These novel monolayer materials infuse new blood into the family of 2D materials.
Inspired by the experiments, a new member of the family of groupVA compounds, namely the AsSb monolayer has been reported recently.^{17–20} Nie and Zeng et al. find that monolayer AsSb exhibits semiconductortopological insulator transitions under tensile strain. Zhang et al. consider that fluorinated monolayer AsSb breaks its inversion symmetry, resulting in the Rashba effect. Sun et al. report that typical point defects in 2D AsSb can significantly reduce its band gap and induce magnetic moments. Although these reports show intriguing results, investigations on the optoelectronic properties of monolayer AsSb are still scarce. It is reported that the AsSb monolayer has five stable and distinctive phases, three of which (i.e. α, γ, εAsSb) show direct band gaps. To explore the potential applications of these proposed 2D materials, in this work, we systematically investigate the electronic and optoelectronic properties of α, γ, and εAsSb monolayers based on density functional theory simulations. The thermal stability is firstly evaluated using Born–Oppenheimer molecular dynamics (BOMD) simulations.^{21} Based on these stable structures, we next study the strain effect and discover that strain can be an effective method to modulate the electronic properties of three kinds of AsSb (α, γ, ε) monolayers. Interestingly, phase transition from semiconductor to metal occurs when strain is applied. Our results also indicate that these monolayer materials exhibit excellent flexible mechanical properties and ultrahigh carrier mobility, with a value of up to 5.4 × 10^{4} cm^{2} V^{−1} s^{−1}, which is higher than most of the known 2D materials. In addition, we find that the PGE occurs in materials lacking a center of symmetry when irradiated by polarized light.^{22–24} Due to the lack of space inversion symmetry, the PGE is also observed in AsSb monolayers, in which the photocurrent can be detected under light illumination without external bias. Upon illumination, the proposed AsSb monolayers exhibit high photoresponsivity over a broad range of solar energy, suggesting great potential in optoelectronic and photovoltaic applications.
Calculation methods
Atomic structure, energetic, and electronic properties are analyzed based on firstprinciples DFT calculations by using the PAW potential^{25} which is implemented in the planewave basis Vienna ab initio Simulation Package (VASP). Perdew Burke Ernzerhof (PBE)^{26} within the generalized gradient approximations (GGA)^{27} is used to describe the electron exchange and correlation potential. For all calculations, we choose 400 eV as the kinetic energy cutoff. For the structure optimization, we employ a 9 × 11 × 1 grid for the Brillouinzone sampling to relax the primitive cells. The calculated lattice parameters are in agreement with previous study.^{17} To avoid the interaction between repeat images, a vacuum space of about 20 Å perpendicular to the plane is used. The convergent value for energy is 10^{−4} between two neighboring steps and as the values of the Hellmann–Feynman forces experienced by each atom are less than or equal to 0.02 eV Å^{−1}, structure relaxation will be completed. The Gaussian smearing is chosen to address the problem in our calculation process. Considering PBE would underestimate the band gap, hence, the electronic properties of these three types of AsSb (α, γ, ε) are calculated based on the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional,^{28} which is composed of 75% PBE exchange and 25% Fock exchange. Furthermore, to demonstrate the thermodynamic stability of these 2D materials, ab initio molecular dynamics (AIMD) simulations are performed in the microcanonical ensemble. The systems are heated to 300 K for 3 ps with a 1 fs time step.
The photocurrent is calculated based on DFT methods within Keldysh nonequilibrium Green's functions (NEGFDFT).^{29,30} The total Hamiltonian are defined as:

 (1) 
where
Ĥ_{0} is the selfconsistent Hamiltonian of the system,
A and
are the electromagnetic vector potential and momentum of electrons respectively.
^{31,32} The photocurrent is then determined as:

 (2) 
where
Γ represents the coupling of the device scattering region to the electrodes,
G^{>(ph)},
G^{<(ph)} are the greater and smaller Green's function including electron–photon interaction.
^{33,34}
Results and discussion
In this work, we firstly investigate the structural characteristics of these monolayers. The α, γ, εAsSb monolayers have rectangular primitive cells, among which αAsSb and γAsSb contain four atoms per unit cell, while εAsSb has eight atoms in its primitive cell. As shown in Fig. 1, all these structures have a puckered surface, and each atom bonds with the other three neighboring atoms in a buckled configuration. The bond lengths are in the range of 2.6–2.8 Å, which are slightly shorter than that of the Sb monolayer (2.89 Å), while longer than that of the As monolayer (2.57 Å). The calculated structural parameters are summarized in Table S1 (see the ESI†), which are in good agreement with previous study.^{17}

 Fig. 1 The structure of monolayer AsSb allotropes, which are plotted in both the top and side views. (a), (b) and (c) are the top views and the corresponding side views. (d) is the Brillouin zone of monolayer AsSb. a and b are the lattice constants, d_{1}, d_{2}, d_{3} and θ_{1}, θ_{2} are the bond lengths and bond angle respectively.  
To examine the thermal stability of the proposed AsSb monolayers, ab initio molecular dynamics (AIMD) calculations are carried out in this work. As shown in Fig. S1 (see the ESI†), structure reconstruction or broken bonds are not observed after heating the system at 300 K for 3 ps, implying these structures may be stable at the room temperature. The formation energy (E_{f}) of these 2D structures is also studied, which is defined by the following equation:

E_{f} = (E_{AsSb} − nE_{As} − nE_{Sb})/2n,  (3) 
where
E_{AsSb} is the total energy of AsSb monolayer structures,
E_{As} and
E_{Sb} are the energies of As and Sb atoms,
n is the number of atoms. The calculated formation energies are 0.07 eV per atom, 0.09 eV per atom, and 0.14 eV per atom for the α, γ, and εAsSb monolayers, respectively, which are close to the experimental value of the Sb monolayer (∼0.09 eV per atom).
It is well known that the Young's modulus (Y) is an effective characterization of the flexibility or rigidity of materials, which depends on the geometry of the considered structures. Different from the bulk Young's modulus, the Young's modulus in the 2D monolayer (inplane stiffness) along different axes can be determined by the elastic constants C_{ij}:

 (4) 
The Young's modulus presents direct information on the system behavior under uniaxial strain. From the results listed in
Table 1, we notice that the 2D Young's modulus for αAsSb and εAsSb exhibit obvious anisotropy, while for γAsSb, the values are almost the same along the
xaxis and
yaxis. The calculated Young's modulus of α, ε, and γAsSb is as low as 24.30, 45.32 and 37.45 N m
^{−1} respectively, which is comparable or smaller than those of SnS, GeS, MoS
_{2} and hBN.
^{35–37} Such a small Young's modulus indicate that the AsSb monolayers have flexible structures, which can be ascribed to their puckered natures. On the other hand, layer modulus (
γ) is another important parameter that measures the material's resilience to hydrostatic stretching, which can be obtained by the equation:

 (5) 
As shown in
Table 1, all AsSb monolayers have a smaller layer modulus (
γ) as compared with graphene and GeC.
^{38} These results illustrate that the AsSb monolayers possess relatively good expandability, enabling them to be applied in flexible microelectronic devices. Interestingly, we find that the γAsSb is an auxetic material with a negative Poisson's ratio (
ν ∼ −0.02), which means this kind of material expands laterally (gets fatter) when stretched. This partially stems from its puckered triangle structure, and the result is also confirmed by previous study.
^{39} In addition, we also estimate the outofplane deformation based on the following equation:

h/l ≈ (ρgl/Y_{2D})^{1/3},  (6) 
where
h,
l,
g and
ρ represent the height of deformation, length of side, and the acceleration of the gravity and density of the AsSb monolayers respectively. For a square AsSb nanosheet with
l = 20 μm, the gravity induced bending
h/
l are about 1.42 × 10
^{−4}, 1.1 × 10
^{−4}, 1.0 × 10
^{−4} for the α, γ, and εAsSb monolayers, which are comparable with that of graphene.
^{40} This manifests that the AsSb materials exhibit small fluctuations under their own gravity.
Table 1 Calculated coefficients of inplane elastic stiffness C_{11}, C_{22} and C_{12}. The Young's modulus and Layer modulus are also listed here
Phase 
C
_{11} (N m^{−1}) 
C
_{22} (N m^{−1}) 
C
_{12} (N m^{−1}) 
Young's modulus (N m^{−1}) 
Layer modulus (N m^{−1}) 
Y
_{
x
}

Y
_{
y
}

γ

αAsSb 
50.39 
15.14 
19.18 
24.30 
7.36 
25.98 
γAsSb 
38.61 
45.34 
−0.90 
45.32 
38.60 
20.54 
εAsSb 
39.17 
19.45 
5.78 
37.45 
18.60 
17.55 
We then investigate the electronic properties of these 2D materials. In Fig. 2, the band structures of the α, γ, and εAsSb monolayers are plotted. Clearly, all three types of AsSb monolayers exhibit direct band gaps, which are favorable for electron excitation from the valence band to the conduction band. Note that the band edge positions locate at the Γpoint in the γAsSb monolayer, while they change to certain points between the Γ and Xpoints for the other type of AsSb monolayers. The calculated band gap of α, γ, and εAsSb are 0.31, 0.82 and 1.28 eV respectively at the PBE level. To accurately account for the band edge positions, we also calculate the band structures based on the HSE method, and the results are 0.45, 1.36 and 1.81 eV for α, γ, and εAsSb, respectively. Compared with the PBE results, we find that the band structures show almost the same parabolic relations except for the discrepant band gap. Therefore, in the following content, the results are mainly discussed at the PBE level.

 Fig. 2 Band structures of (a) αAsSb, (b) γAsSb and (c) εAsSb monolayer. The black and red solid lines represent the band structures based on the PBE and HSE level respectively.  
In Fig. 3, we show the partial density of states (PDOS). For all studied cases, the valence band maximum (VBM) and conduction band minimum (CBM) are mainly attributed to the pstates of the As and Sb atoms. Meanwhile, we also present the partial charge densities respectively corresponding to VBM and CBM of AsSb monolayer materials. Notably, the distribution of electrons and holes is along the y and x axes respectively in the γAsSb monolayer.

 Fig. 3 Orbitaldecomposed DOS of monolayer (a) αAsSb, (b) γAsSb and (c) εAsSb. The corresponding partial charge density at VBM (left panel) and CBM (right panel) are presented in (d), (e) and (f) respectively. All the Fermi levels are pinned in 0 eV and the isosurface of the partial charge density is at 0.004 e Å^{−3}.  
As semiconductor materials applied in modern electronic devices, it is of great significance for them to maintain a tunable band gap, and particularly, a suitable strain could give rise to a phase transition from semiconductor to metal. In addition, we also notice that the strain has sparked novel physics properties in arsenene and antimonene, such as the quantum spin hall effect and the transition from indirect to direct band gap.^{41–43} Hence, we are curious about the effect of strain imposed on AsSb monolayer materials. In order to clarify our doubts, we place a −10% to 10% biaxial strain on the system. Here, we define ε = a − a_{0}/a_{0} as the strain level, where a_{0} and a are the lattice constants of the pristine supercell and the strained supercell respectively. As plotted in Fig. 4, for αAsSb, we can find that the band gap monotonically increases with the increasing tensile strain, while the band gap monotonically decreases with the increasing compressive strain. Expressly, when the strain reaches −4%, a phase transition from semiconductors to metal occurs. Different from αAsSb, the εAsSb has a maximum band gap when the system is in the unstrained status. The band gap of γAsSb system shares a similar trend with εAsSb. Encouragingly, a phase transition from semiconductors to metal appears under −8% compressive strain in both γAsSb and εAsSb systems.

 Fig. 4 Band gap of monolayer AsSb as a function of strain. The black, red and purple solid lines signify αAsSb, γAsSb and εAsSb respectively.  
In addition to the appropriate band structures, carrier mobility is another critical factor that determines the performance of the devices. Thus, we firstly calculated the carrier effective masses of the αAsSb, εAsSb and γAsSb systems, which can qualitatively describe the properties of transport of electrons–holes along a certain specific direction. The effective masses of carriers are investigated by second polynomial fitting to the VBM and CBM of the AsSb system on the basis of eqn (7):

 (7) 
where
k is the wave vector, and
E_{k} is the energy corresponding to the wave vector
k. Generally speaking, the effective mass of electrons and holes are positively correlated to the slope of the band edge. Therefore, we select different directions as the transmission path of migratory carriers to explore their effective masses in AsSb monolayers. For αAsSb, we choose two perpendicular directions (
xaxis,
yaxis) to analysis the effective mass of electrons and holes. We find that both electrons and holes along the
xaxis (
yaxis) have small effective masses, and the effective masses of electrons and holes have little disparity. However for γ and εAsSb, we only calculate the effective mass along the
xaxis. This is because the band edge along the
xaxis shows a steep slope in comparison to the band edge along the
yaxis. The predicated
m_{e}* and
m_{h}* values are considerably smaller than those of other 2D materials, such as the MoS
_{2} monolayer (
m_{e} = 0.48
m_{0}) and InSe (
m_{e} = 0.18
m_{0}).
^{44} Commonly, a small effective mass corresponds to high carrier mobility. To further examine the mobility properties of the three types of monolayer AsSb, we provide a theoretical prediction for the carrier mobility (
μ). As we know, in semiconductors, the coherent wavelength of thermally activated carriers at room temperature is close to the acoustic phonon wavelength.
^{45} And the carrier mobility in monolayer materials is obtained by electronacoustic phonon coupling based on deformation potential (DP) theory at room temperature. And DP theory has been diffusely applied to calculate the carrier mobility in 2D materials.
^{46,47} The carrier mobility equation is determined as follows:

 (8) 
Here, e is the electron charge, C = [∂^{2}E/∂ε^{2}]/S_{0} is the elastic constant of the 2D system, wherein E and S_{0} represent total energy and surface area of the equilibrium system, k_{B}, T, and m* are the Boltzmann constant, temperature and carrier effective mass, respectively. E_{d} is the DP constant which describes the shift of band edges induced by the strain. For a certain system, the C and DP constant are determined by the band edge positions and the total energy of the system under strain. As shown in Fig. S3(a)–(c) (see the ESI†), we calculate the band edge positions of the monolayer AsSb as a function of uniaxial strain along transport directions corresponding to the calculated carrier effective mass. For the αAsSb and εAsSb monolayers, both CBM and VBM decrease monotonously with the increase of uniaxial strain. For the γAsSb monolayer, however, the VBM and CBM show opposite tendencies. The total energies of the monolayer AsSb as a function of strain are plotted in Fig. S3(d)–(f) (see the ESI†), which show quadratic behaviors. In addition, strong anisotropic characteristics are observed in the αAsSb monolayer. The calculated elastic constant C along the xaxis is only 16.35 N m^{−1}, which is about 3.4 times smaller as compared with the value along the yaxis. This indicates that when the same strain is imposed, the required energy along the xaxis is smaller than the value along the yaxis. This also manifests that the structure along the xaxis ise liable to be deformation compared with that along the yaxis. The predicted mobility of electrons and holes at 300 K is presented in Table 2. For αAsSb, the electron mobility along the xaxis direction is up to 4.5 × 10^{4} cm^{2} V^{−1} s^{−1}, which is higher than most of the known 2D materials, and even approaches the record of graphene (∼×10^{5} cm^{2} V^{−1} s^{−1}).^{48,49} However the hole mobility along the xaxis is about 3.4 × 10^{3} cm^{2} V^{−1} s^{−1}, which can be attributed to the much larger DP constant. The huge difference of mobility between electrons and holes facilitates the separation of electron–hole pairs. This high carrier mobility is also found in γAsSb and εAsSb monolayers. The high carrier mobility makes AsSb monolayers extremely appealing for applications in FETs and optoelectronic technology.
Table 2 Calculated effective masses of electrons and holes, elastic modulus (C), deformation potentials (E_{d}), and carrier mobility (μ)
Phase 
Carrier type 
m*/m_{0} 
C (N m^{−1}) 
E
_{d} (eV) 
μ (cm^{2} V^{−1} s^{−1)} 
αAsSb 
Electrons (x) 
0.040 
16.35 
1.79 
4.5 × 10
^{
4
}

Hole (x) 
0.039 
16.35 
6.67 
0.3 × 10^{4} 
Electrons (y) 
0.051 
56.04 
8.90 
0.4 × 10^{4} 
Hole (y) 
0.062 
56.04 
4.00 
1.3 × 10
^{
4
}


γAsSb 
Electrons (x) 
0.038 
40.67 
2.72 
5.4 × 10
^{
4
}

Hole (x) 
0.080 
40.67 
3.88 
0.6 × 10^{4} 

εAsSb 
Electrons (x) 
0.140 
39.43 
6.52 
0.7 × 10^{4} 
Hole (x) 
0.470 
39.43 
0.27 
3.4 × 10
^{
4
}

We now turn on the light and investigate the photoinduced current in AsSb monolayers. Due to the lack of space inversion symmetry, the photogalvanic effect is observed in AsSb monolayers, in which the photocurrent can be detected under light illumination without external bias. In this work, a twoprobe model is employed to evaluate the photoresponsive properties of the AsSb monolayers. This model consists of a central scattering region sandwiched by two semiinfinite electrodes. Here, we redefine the z axis as the transport direction, and the device constructed by the αAsSb monolayer along the armchair direction is shown in Fig. 5 as an example. When light irradiates the surface of monolayer AsSb materials, electrons–hole pairs are created and traverse the device from the left lead to the right lead, leading to the formation of photoinduced current. One critical figureofmerit to determine the performance of the AsSb monolayers is the photoresponse function (R_{ph}), which is calculated as:

 (9) 
where
e is the electron charge.
F_{ph} is the photon flux defined as the number of photons per unit time per unit area.

 Fig. 5 The schematic diagrams of twoprobe devices constructed by the αAsSb monolayer. (a) and (b) are the top and side views respectively. zAxis is redefined as the transport direction.  
In this work, linearly polarized light which propagates perpendicular to the AsSb monolayer is used to irradiate the entire scattering region of the device. For the linearly polarized light vector, the polarization angle θ is defined as the included angle with respect to the yaxis (_{1}) direction (see Fig. 5a). The photoresponses along the zigzag (y) and armchair (z) directions are determined at different θ for photon energies ranging from 0 to 3.8 eV, which are shown in Fig. 6. The results indicate that the three types of AsSb monolayers show a high photoresponse in the broad range of the spectrum from ultraviolet to infrared light. Surprisingly, the photoresponse coefficients can reach 0.6, 0.7 and 3 a_{0}^{2} per photon (a_{0} stands for Bohr radius) for α, γ, and εAsSb respectively, which are up to two orders of magnitude larger as compared with that of Sdoped black phosphorus.^{23}

 Fig. 6 The photoresponse coefficients of (a) αAsSb, (b) γAsSb and (c) εAsSb for the armchair direction, (d), (e) and (f) are for the corresponding zigzag directions. The abscissa stands for the energy of photon, and the ordinate stands for the polarizing angle. The color of the inserted bar graph stands for the values of the photoresponse coefficient.  
It is important to note that AsSb monolayers display strong anisotropy along different directions, in particular for εAsSb. As shown in Fig. 6(c), the photoresponse along the armchair direction is up to about 3 while it reduces to 0.6 along the zigzag direction. Similar results are also found for α and γAsSb. The strong anisotropy can be analyzed in terms of the band structures and charge transport direction. For αAsSb, the VBM and CBM locate within Γ–X points (see Fig. 2(a)), i.e. along the armchair direction, leading to a higher probability of photoinduced electron–hole separation than that along the zigzag direction. As a result, the photoinduced current along the armchair direction shows a considerably large value. For the γAsSb monolayer, the VBM and CBM locate at the Γpoint. However, as shown in Fig. 3(e), electrons are distributed along the zigzag direction. Thus, the excited electrons are liable to transport along this direction. As for the εAsSb monolayer, VBM locates within the range of Γ–X, i.e. the armchair direction. In addition, the excited electrons tend to transport along the armchair direction (see Fig. 3(f)). As a result, a higher photocurrent is observed along the armchair direction in the εAsSb monolayer.
Based on the phenomenological theory, we know that the values of photocurrents are related to the incident angle (β) of light. In order to clarify the effect of incident angle on the photocurrent, we select the maximum photoresponse to analyze its variation with the incident angle. As plotted in Fig. 7, for both the armchair and zigzag directions, the photoresponse monotonously decreases with the increase of β, in other words, when light vertically incidents on the surface of AsSb monolayers, the photoresponse of the system reaches its maximum value.

 Fig. 7 The photoresponse of PGE for (a) the zigzag and (b) armchair direction with increasing incidence angle. The x–y plane is the incident plane.  
Conclusion
In summary, we systematically study the geometric, mechanical and electronic properties of the AsSb monolayer based on firstprinciples calculations and elastic theory. Our results indicate that all these monolayer materials possess excellent flexible structures and exhibit anisotropic characteristics based on the analysis of their Young's modulus. Furthermore, the α, γ and εAsSb monolayers have a direct band gap which could be effectively tuned by a suitable strain, and a transition from semiconductor to metal appears when compressive strain is applied. The most overriding matter is that all the proposed monolayer materials have an ultrahigh carrier mobility (∼ 10^{4} cm^{2} V^{−1} s^{−1}), which is larger than that of the widely studied 2D materials, even approaching the value of graphene. The performance of the photoresponse of these AsSb monolayers is also evaluated based on the NEGFDFT calculations. Under illumination, an anisotropic photogalvanic effect (PGE) is observed along both the zigzag and armchair directions, and the AsSb based systems exhibit excellent photoresponse properties in a broad range of the spectrum. These outstanding mechanical and optoelectronic properties endow the monolayer AsSb with great promise to be potential materials applied in designing various optoelectronic and microelectronic devices.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This work is supported by the National Basic Research Program of China (973 program, 2013CB632401), the National Natural Science foundation of China (11604213, 11374190, and 21333006), 111 Project (B13029), and the Taishan Scholar Program of Shandong Province.
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Footnote 
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp05201d 

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