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The development of theoretical models for crystals has led to the evolution of computational methods with which much more thorough investigations than previously possible can be done, including studies of the nonlinear optical properties. There has recently been a rise in interest in 2-dimensional materials; unfortunately, measurements of the nonlinear susceptibility of these materials in the wavelength range of the order of hundreds of nanometers by traditional methods are difficult. Studies of second-harmonic generation (SHG) from the transition-metal dichalcogenides (TMDCs), MoS
_{2} and MoSe
_{2}, have been reported; however, SHG from other typical van der Waals crystals such as GaSe and other transition metal monochalcogenides (TMMCs) has rarely been studied under the same conditions. In this study, the 211 (i = 2, j = 1, k = 1) elements in the susceptibility matrices of GaSe, InSe, MoS
_{2} and WS
_{2} were calculated and compared. A tendency for the SHG intensity to weaken as the wavelength increases from 500 nm to 1000 nm was observed for GaSe and InSe, and, apart from some periodic fluctuations, no clear increase could be seen for these two materials in the SHG response curve in the near infrared. By comparison, MoS
_{2} and WS
_{2} have obvious peaks in both the visible and infrared bands. Calculations of the SHG response show peaks at around 500 nm (for GaSe), 570 (for InSe), 660 nm, 980 nm (for MoS
_{2}) and 580 nm, 920 nm (for WS
_{2}). Moreover, similarities between the SHG curves for GaSe and InSe and for MoS
_{2} and WS
_{2} were revealed, which may be due to the similarities found for these two groups of crystals.

Because of their high mobility and thermal conductivity, and their good mechanical properties, the interest in van der Waals crystals has increased in recent decades. In addition to graphene, chalcogenide compound semiconductors with two-dimensional sublayer structures have been prepared and studied [_{2} as an example, a strong resonant SHG coefficient was obtained which was attributed to electronic structural changes at the layer edges [

Other nonlinear optical effects are the intensity of the SHG response with respect to the polarization of the light incident on the crystal in a particular electric field; thus, the SHG response is considered to be a function of the incident light frequency. It is important to find the optimal nonlinear optical crystal for a particular frequency band. Although efforts have been made on investigating nonlinear optical properties, these have been limited by the high expenditure required for crystal growth and the construction of the optical system; thus, experimental investigations of a huge number of crystals in order to compare the nonlinear coefficients of each material is costly and inefficient. In the last decade, due to the development of theoretical models and computational methods, some results, such as the susceptibilities of TMDCs, have been obtained from first principles calculations. More specifically, calculations for MoS_{x}Se_{2−x} were reported by Hu et al. and an interpretation of the SHG disparity at 810 nm was put forward [_{2}, WS_{2}). Considering the practical applications, we chose the bulk form of typical van der Waals crystals to do the calculation, which is also different from previous research [

This study theoretically predict the SHG response of crystals in the 500 to 1100 nm wavelength range by using the calculations, which are mainly based on density functional theory (DFT) and density functional perturbation theory (DFPT) with ab initio application of the 2n + 1 theorem [_{2} and WS_{2} were chosen as representatives not only because they are typical van der Waals crystals but also because their structural similarities allow them to be divided into two groups—MX and MX_{2}, where M represents the metal (Ga, In, Mo) and X is the chalcogen (S, Se). With our calculations, it is possible to filter out inappropriate candidates, and find the best optical crystal needed for a given frequency band.

According to the Hohenberg and Kohn theorem, all the physical properties of an electronic system can be uniquely determined by its ground-state charge density distribution [

E [ n ] = 2 ∑ N = 1 N / 2 ϵ n − e 2 2 ∫ n ( r ) n ( r ′ ) | r − r ′ | d r d r ′ + E x c [ n ] − ∫ n ( r ) v x c ( r ) d r (1)

Here N is the number of electrons, −e is the electron charge, n(r) is the charge density at position r , E_{xc} is the exchange-correlation energy, ν_{xc} is defined as ν ≡ δ E / δ n ( r ) and ϵ_{n} is the Kohn Sham eigenvalue corresponding to the nth state. In the formula we have divided the system energy into three parts, and the exchange-correlation term, E_{xc} is known to be the smallest, but cannot be ignored. Since an exact expression for the exchange-correlation energy is unknowable, Kohn and Sham proposed a reasonable approximation for it. It is considered that the exchange-correlation energy for a real electron is equal to the E_{xc} in a homogeneous electron gas with the same charge-density distribution, and the assumption is known as the local density approximation (LDA) [

The second-harmonic generation corresponds to the third-order derivative of the system energy, but due to the 2n + 1 theorem, the first-order derivative of the wave function is sufficient to calculate the SHG response, and the susceptibility is as follows:

χ ( 2 ) = 3 i e Ω 2 π E 0 2 ∑ m , n = 1 N ∫ B Z d k × 〈 u m k | ∂ ∂ k ( | u n k 〉 〈 P c u n k ( 1 ) | ) | P c u m k ( 1 ) 〉 (2)

where Ω is the dimensions of the unit cell and u is the periodic part of the Bloch wave function, E_{0} is the external electric field, and P_{c} is the state projection operator [

Details of the structural parameters of GaSe, InSe, MoS_{2} and WS_{2} used in the calculations are shown in

Norm-conserving Fritz-Haber pseudopotentials with a cutoff energy of 100Ry were used in our calculations, which was deemed to cover the states needed to calculate the SHG response. The sampling point densities of the GaSe, InSe, MoS_{2} and WS_{2} Brillouin zones were set to 10 × 10 × 10, 7 × 7 × 7, 8 × 8 × 8, 8 × 8 × 8 respectively.

GaSe | InSe |
---|---|

t 1 = ( − a , 0 , 0 ) t 2 = ( − a / 2 , a 3 / 2 , 0 ) t 3 = ( 0 , 0 , c ) | t 1 = ( − a , 0 , 0 ) t 2 = ( − a / 2 , a 3 / 2 , 0 ) t 3 = ( 0 , 0 , c ) |

a = 3.755 Å , c = 15.94 Å | a = 4.05 Å , c = 16.64 Å |

Se I : ( 1 / 3 , 2 / 3 , 0.1003 ) | Se I : ( 1 / 3 , 2 / 3 , 0.0907 ) |

Ga I : ( 0 , 0 , 0.1701 ) | In I : ( 0 , 0 , 0.1677 ) |

Ga II : ( 0 , 0 , 0.3299 ) | In II : ( 0 , 0 , 0.3233 ) |

Se II : ( 1 / 3 , 2 / 3 , 0.3997 ) | Se II : ( 1 / 3 , 2 / 3 , 0.4093 ) |

Se III : ( 2 / 3 , 1 / 3 , 0.6003 ) | Se III : ( 2 / 3 , 1 / 3 , 0.5907 ) |

Ga III : ( 1 / 3 , 2 / 3 , 0.6701 ) | In III : ( 1 / 3 , 2 / 3 , 0.6677 ) |

Ga IV : ( 1 / 3 , 2 / 3 , 0.8299 ) | In IV : ( 1 / 3 , 2 / 3 , 0.8323 ) |

Se IV : ( 2 / 3 , 1 / 3 , 0.8997 ) | Se IV : ( 2 / 3 , 1 / 3 , 0.9093 ) |

MoS_{2} | WS_{2} |

t 1 = ( − a , 0 , 0 ) t 2 = ( − a / 2 , a 3 / 2 , 0 ) t 3 = ( 0 , 0 , c ) | t 1 = ( − a , 0 , 0 ) t 2 = ( − a / 2 , a 3 / 2 , 0 ) t 3 = ( 0 , 0 , c ) |

a = 3.122 Å , c = 12.32 Å | a = 3.153 Å , c = 12.323 Å |

S I : ( 1 / 3 , 2 / 3 , 0.1250 ) | S I : ( 1 / 3 , 2 / 3 , 0.0907 ) |

Mo I : ( 0 , 0 , 0.2500 ) | W I : ( 0 , 0 , 0.1677 ) |

S II : ( 1 / 3 , 2 / 3 , 0.3750 ) | S II : ( 1 / 3 , 2 / 3 , 0.4093 ) |

S III : ( 2 / 3 , 1 / 3 , 0.6250 ) | S III : ( 2 / 3 , 1 / 3 , 0.5097 ) |

Mo III : ( 1 / 3 , 2 / 3 , 0.7500 ) | W III : ( 1 / 3 , 2 / 3 , 0.6677 ) |

S IV : ( 2 / 3 , 1 / 3 , 0.8750 ) | S IV : ( 2 / 3 , 1 / 3 , 0.9093 ) |

To verify the accuracy of the calculated electronic structures, we compared the band diagrams obtained with ones published in previous papers. Underestimates of the bandgaps were revealed by the comparison. Here, the band structures of GaSe InSe MoS_{2} and WS_{2} are given in

As we know, the second-order susceptibility matrix has 3^{3} = 27 elements. For GaSe, InSe, MoS_{2}, and WS_{2}, the 211 element is the one that concerns us most: most of the other elements are canceled to zero due to the symmetry of these crystals, and coaxial phase matching is non-negligible in nonlinear optical processes. The susceptibility is a complex number, in which the real part represents the light arising, and information about the energy loss is obtained from the imaginary part combined with the real part. The absolute values of the 211 element of the susceptibility matrix, which is proportional to the SHG emergent light intensity is given in

In

In particular, the nonlinear responses of all four crystals in the green to red light region of the visible light band reach significant levels, with InSe, GaSe and MoS_{2} having their highest response efficiency in this region. The results predict that the nonlinear response will give rise to peaks at 500 nm, 570 nm, 660 nm and 570 nm for the four kinds of material. As the wavelength increases, the

Material | Bandgap obtained by calculation | Actual size of Bandgap | Scissor correction |
---|---|---|---|

GaSe | 0.57 eV | 2.01 eV [ | 1.44 eV |

InSe | 0.33 eV | 1.31 eV [ | 0.98 eV |

MoS_{2} | 0.91 eV | 1.23 eV [ | 0.31 eV |

WS_{2} | 0.97 eV | 1.32 eV [ | 0.35 eV |

intensity of the SHG response decreases and is weaker for GaSe and InSe, but we can still see some small periodic peaks in the spectral distribution as shown in _{2} and MoS_{2} are deemed to be better choices if an IR-band SHG device is needed. Specifically, we can expect to see clear SHG peaks at around 980 nm for MoS_{2} and 920 nm for WS_{2}. However, in the DFT and DFPT calculations, the effects of thermal vibrations in the lattice and defects are not included, and these may give rise to discrepancies between our calculations and the results of experiments.

We used first principles calculations to obtain the SHG transition efficiencies of four typical van der Waals crystals, including the TMMCs, MoS_{2} and WS_{2}, and the TMDCs, GaSe and InSe. From our calculations we obtained a linear regression for the scissors correction, and similar SHG response curves were observed for the two groups of materials, which may be due to the similarities between the crystal structures and their electronic states. Our calculations can be used to predict the SHG response of several crystals in the 500 to 1100 nm wavelength range, and so enable unsuitable nonlinear optical materials to be filtered out. SHG from GaSe and InSe in the visible light region is predicted. The calculations predict that high peaks for light emerging at 500 nm (for GaSe) and 570 nm (for InSe) would be observed. Moreover, two high SHG response regions in the visible light and infrared regions for MoS_{2} and WS_{2} are predicted by the calculations. Peaks in the SHG transformation efficiency at 660 nm and 980 nm for MoS_{2} and at 580 nm and 920 nm for WS_{2} were obtained. In the future, the present calculation will be extended to other 2D materials such as graphene, to find their potentials of nonlinear optical application.

This work was partly supported by Grant-in-Aid for JSPS Research Fellow JP19J20564, Japan.

The authors declare no conflicts of interest regarding the publication of this paper.

Chen, M.X., Tang, C., Tanabe, T. and Oyama, Y. (2019) Calculation of the Nonlinear Susceptibility in van der Waals Crystals. Optics and Photonics Journal, 9, 178-188. https://doi.org/10.4236/opj.2019.911016