Electric field effect of GaAs monolayer from first principles

Using first-principle calculations, we investigate two-dimensional (2D) honeycomb monolayer structures composed of group III-V binary elements. It is found that such compound like GaAs should have a buckled structure which is more stable than graphene-like flat structure. This results a polar system with out-of-plane dipoles arising from the non-planar structure. Here, we optimized GaAs monolayer structure, then calculated the electronic band structure and the change of buckling height under external electric field within density functional theory using generalized gradient approximation method. We found that the band gap would change proportionally with the electric field magnitude. When the spin-orbit coupling (SOC) is considered, we revealed fine spin-splitting at different points in the reciprocal space. Furthermore, the valence and conduction bands spin-splitting energies due to SOC at the K point of buckled GaAs monolayers are found to be weakly dependent on the electric field strength. Finally elec...


INTRODUCTION
][7][8][9][10] In particular, graphene is a 2D honeycomb structure of carbon with unique symmetry, which gives rise to the energy band structure of the Dirac points at the Fermi level. 1,11,12The discovery of graphene also allowed us to observe the quantum Hall effect using the graphene layered devices. 13,14Silicene is predicted to be a stable 2D allotrope of silicon with a buckled hexagonal structure closely related to the structure of graphene. 15he properties of silicene have been studied intensively very recently, whose hexagonal honeycomb structure also gives rise to the linear dispersion relation in band structure.On the other hand, silicene may be implemented with the silicon technology and the buckled geometry may offer some excellent properties beyond the graphene.For instance, an external electric field opens band gap in silicene but not in graphene. 16The monolayer geometry of group III-V binary compounds 17 are structurally similar to the group IV silicene.The buckled monolayer structure with trigonal symmetry is more stable with respect to the planar geometry.This is due to the destabilization of the π bonds in the sp 2 hybridization because of the different chemistry between two neighboring atoms.When spin-orbit coupling is considered, both graphene and silicene can have finite band gaps as Kane-Mele 2D topological insulator. 18For buckled germanium monolayer, spin-orbit coupling may even open a much larger band gap. 19sing first-principles calculations, S ¸ahin et al. investigated 2D honeycomb structures of group-IV elements and of group III-V elements. 205][26][27] Other than that, it was found that many other 2D III-V compounds were buckled honeycomb, and such AB monolayers have trigonal symmetry (e.g., group IV binary monolayers SiGe, SiSn, GeSn and group III-V binary monolayers AlSb, GaP, GaAs, InP, InAs, InSb), see the Fig. 1.These group III-V binary AB monolayers are polar semiconductors with the presence of the electronegativity difference and dipole moment at each A-B bond.All the structures have the spin-orbit coupling (SOC) induced spin-splitting in valence and conduction bands at the high-symmetry K point.Although the systems are nonmagnetic, electrons around K valley experience strong Zeeman-like magnetic field.The spin polarization may be switched by changing the spatial buckling orientation, or by changing the time-reversed K and -K points.The spin-valley coupled physics properties in two-dimensional honeycomb binary compounds provide us possibility for exploring new quantum materials.In this work, we investigate the electric field effects on the electronic properties of buckled 2D GaAs monolayer.

METHODS
We have performed first-principle calculations for structure relaxation, total energy and electronic structures.All calculations were done by using density functional theory (DFT) at the PBE level 28 unless otherwise stated, in which the exchange-correlation potential is approximated by generalized gradient approximation (GGA).A set of (7×7×1) k-point sampling is used for Brillouin zone in the reciprocal space, and the k-point mesh is generated by Monkhorst-Pack scheme. 29The energy cutoff is 500 eV, and the convergence criterion of self-consistent calculations is 10 -8 eV for total-energy values.Fermi level smearing is taken as 0.05 eV for geometry optimization and the same value for energy band calculations.All atomic positions and unit cell were optimized until the atomic forces were less than 0.005 eV/Å.The supercell contains a minimal vacuum spacing of 20 Å to prevent interactions between the adjacent periodic layers.To consider the buckling height and band gap values obtained by PBE, we also carried out calculations based on LDA 30 and PBEsol 31 exchange correlation potential.Moreover, we study the important SOC effect by considering the relativistic correction in the pseudopotential DFT calculations, as implemented in the Vienna Ab Initio Simulation Package (VASP).
We calculate the second harmonic generation (SHG) to study its dependence on the buckling structure change.The SHG spectra of the second order susceptibility tensor are obtained by using the "exciting" software, 32 which is a full-potential all-electron DFT package to calculate the many physical properties of condensed matters.

RESULTS AND DISCUSSION
We begin our analysis by confirming the 2D buckled monolayer structures of trigonal symmetry, namely group IV binary monolayers SiGe, SiSn, GeSn and group III-V binary monolayers AlSb, GaP, GaAs, InP, InAs, InSb.The calculated buckling angle θ (deg), height h (Å), band gaps (eV) and spin-splitting energies of valence band (conduction band) ∆E VB (∆E CB ) (meV) at the K point are in good agreement with Ref. 17, 20.We note that group-IV AB compounds generally have band gaps smaller than those of III-V groups.Group III-V compound semiconductors are important for making the light emitting diode, and for the solar cell materials.Among them the GaAs is the representative, thus in the following we focus our attention on the physical properties of GaAs and the electric field effects.In Fig. 2, we illustrate the electronic band structures of GaAs monolayer with and without taking SOC 33 into account.The first Brillouin zone of the hexagonal primitive cell is used in our electronic structure calculation.From both results with and without SOC, we obtain a semiconductor with a band gap of 1.09 eV.The valence band maximum (VBM) is located at the high-symmetry K point of the Brillouin zone, whereas the conduction band minimum (CBM) is located at the highsymmetry Γ point.The VBM at K point is slightly higher than Γ point by 0.01 eV, and our PBEsol functional results show both VBM and CBM are located at Γ point.In addition, we find that both the valence and conduction bands are splitted by SOC at K point, but the states are degenerate at the high-symmetry M point.The spin splitting at M point is so-called Rashba-like 34 along horizontal axis direction, and at K point it is Zeeman-like along the vertical energy scale.Furthermore, as shown in the insets of Fig. 2, the spin splitting value ∆E VB at the K point in GaAs is found to be 13.6 meV, and the spin splitting value ∆E CB at the K point is 34.2 meV.
The electric field tunes the carrier concentration in a semiconductor device, so that electron transport through it can be controlled.Now we discuss the change of physical properties by external electric field effect 35 on our 2D system.The buckling height can be modified by the external electric field along c axis, as shown in the Fig. 3.Although we get different values of buckling from the GGA, LDA and PBEsol calculations, the electric field induced buckling variations have the same trend in all cases.When the monolayer structure is under positive electric field, the heights increase linearly by about 0.02 Å when the field strength is 0.6 eV/Å.On the other hand, when we switch the electric field along the opposite c axis, the buckling height decreases linearly, with the same range of change about 0.02 Å.In the monolayer structure of GaAs, there is an internal equivalent field E int which is alone the c axis direction.The positive external electric field assists to increase the E int within the structure, so the height of GaAs monolayer could be increased.When the reversed external electric field is applied, it effectively decreases the E int within the GaAs structure, so the height would be decreased.This field-induced structural change has impact on the Ga-As bond length FIG. 2. Band structures of GaAs monolayer without (blue) and with (red) spin-orbit coupling (SOC).The calculated band gap is 1.09 eV, from the valance band maximum (VBM) at K point to the conduction band minimum (CBM) at Γ point.The insets show the valence and conduction bands splitting zoom-in at K points.and angle change, and consequently we expect the band gap should also have a response to the electric field.
The lattice parameter a=b=4.0557Åunder zero electric field, and the positive electric field of 0.56 eV/Å may change the lattice parameter to a=b=4.0536Å; whereas lattice parameters (a=b) are increased from 4.0557 Å to 4.0609 Å, under the negative electric field of 0.52 eV/Å.This can be understood that the change by the positive and negative external electric field is effectively applying a stress along ab plane, such that the bond with a mixed covalent bond to ionic character shows a reverse piezoelectric-like phenomenon in GaAs monolayer.
In Fig. 4, we present the electric field induced change of band gap, 36 with and without SOC.When SOC is not considered, the band gap amplitudes are increased proportionally to the field strength, with the direction of electric field along the positive c axis.Similar to the height variation trend, the band gap is linearly enhanced by 0.03 eV when the field strength is 0.6 eV/Å.For the electric field of opposite direction, the band gap values are suppressed by about 0.06 eV under electric field of -0.6 eV/Å.Therefore the buckling height value is correlated with the band gaps under the externally applied electric field.Such field effect is dramatically affected by the SOC inclusion in the calculation.Fig. 4 show that the positive external electric field very slightly increases the band gap, and then decreases the band gap.Meanwhile we have found that the indirect band gap between K and Γ points can be changed to a direct band gap at Γ point.On the other hand, the negative external electric field decreases band gap with SOC more dramatically, and further negative electric field is found to close the band gap.This effect is confirmed by using different functionals, which give us a critical electric field for gap closing at about 0.5eV/Å.In the inset of Fig. 4 we present two band structures to show how the conduction bands are splitted and shifted down below the Fermi level near Γ point.
We calculate the change of bands spin-splitting due to SOC, now in the presence of external electric field.For that, the spin-splitting energy values ∆E VB (meV) and ∆E CB (meV) of valence band and conduction band at the K point are shown in Fig. 5.Under the positive external electric field of 0.5 eV/Å, we can see that the ∆E VB increases from 13.6 meV to 14.0 meV; for the negative field of 0.5 eV/Å, the ∆E VB decreases from 13.6 meV to 13.1 meV.Similar behaviors are found for the conduction band spin-splitting (∆E CB ) at K point.The positive/negative electric field increases/decreases ∆E VB from 34.2 meV to 34.4/33.8meV.
The external electric field might not be able to switch the displaced positions of Ga and As atoms in the buckled GaAs monolayer, if the energy barrier is too large for ferroelectric-type structural inversion.Nonetheless we have made another monolayer configuration of opposite polarization by manual operation.The height of this equally stable degenerate structure is also equal to that in Fig. 1, but now the Ga atomic layer is above As layer with a same height of 0.6 Å.Then we calculated  Finally we calculate the second harmonic generation spectra of GaAs two-dimensional structures with different buckling strength, as shown in Fig. 7.For the flat monolayer h=0Å, the calculated second-order nonlinear susceptibility is equal to 0 nm/V because of the symmetry of this structure.The effective bulk-like second-order susceptibility of the GaAs monolayers, can be defined as d eff = χ (2) /2T, where T is the thickness of a GaAs monolayer.When we use DFT to calculate the SHG spectra, the input parameters are carefully checked including the lattice constant, coordinate of each atom, K-mesh and so on.In this paper we calculate the SHG spectra of GaAs monolayer of different heights (0, 0.12, 0.24, 0.36, 0.48 and 0.6 Å).For that, we take the initial GaAs structure in Fig. 1 to find the best K-mesh for all calculations.From a tradeoff between the computational expense and precision, we select the 32×32×1 K-mesh for the following calculations.The calculated second-order FIG. 7. The SHG spectra of GaAs two-dimensional structures with different buckling heights.The values of buckling height are 0, 0.12, 0.24, 0.36, 0.48 and 0.6 Å. nonlinear optical susceptibility χ (2) values are generally increased by the buckling height after the symmetry is broken.In particular, the low energy excitations are significantly enhanced with the prominent peaks appearing near the visible light region.The predicted SHG spectra of monolayers with different thickness can be employed to probe the relationship between buckling structural change and optical properties of the 2D materials in general.

CONCLUSION
In summary, we investigate the physical properties of buckled GaAs monolayers, taking into account the roles of spin-orbit coupling and external electric field.Even the external electric field might not be able to induce polar structural inversion, it can modify delicately the buckling amplitude and electronic structure.When SOC is absent, the buckling height and band gap would linearly increase or decrease by positive or negative external electric field.When we consider the SOCinduced band spin-splitting, the band gap and band splitting values have different dependence on the external electric field.Furthermore, we have also present the interesting spin texture and second harmonic generation spectra from the first principles, and predict some electric field effect to be investigated in future experiments.

aFIG. 1 .
FIG. 1. Monolayer structure composed of binary elements GaAs.(a) Top view of a buckled GaAs monolayer along the c-axis.(b) Side view of a buckled GaAs monolayer, where the cations Ga (blue) plane is separated by a distance h from anion As (yellow) plane.The buckling angle θ is defined as the angle between the Ga-As bond and the c-axis direction.

FIG. 3 .
FIG. 3. The buckling height values changed by the electric field with direction parallel and antiparallel to the c axis without SOC.Inclusion of SOC gives same structural change results.

FIG. 4 .
FIG.4.band gap change by the positive and negative external electric fields along the c axis direction.The band structures with SOC show gap closing effect at a critical field strength of about 0.5 eV/Å.Solid symbols are for calculations with SOC, unfilled symbols are for calculations without SOC.

FIG. 6 .
FIG. 6.(a) The spin textures of the GaAs structure in Fig. 1 (As above Ga) and (b) the inverted GaAs structure (Ga above As) for states of VBs (0.18eV below the Fermi level) for the whole Brillouin zone.