Orientation Dependence Of Even-order Harmonics Generation In Biased Bilayer Graphene

The high-order harmonic generation（HHG）is an important nonlinear optical phe-nomenon in strong-field physics,which has been extensively studied due to its great potential in detecting extreme ultraviolet and soft X-ray light sources.Despite research on HHG in gases is relatively mature,the low atomic density results in low conversion efficiency,which greatly limits its development and application.Thanks to the develop-ment of mid-infrared laser pulse technology,solid HHG has become a research hotspot in the fields of strong-field and condensed matter physics.It not only provides a new approach to attosecond photonics,but also brings a powerful tool for the detection of structures and ultrafast dynamic processes in condensed matters.Recently,graphene and graphite films have been widely discussed due to their unique electronic and optical properties.Compared with graphene,the band gap of biased bilayer graphene（BLG）can be controlled by an applied electric field which is perpendicular to the graphene plane.This makes biased BLG the only known semiconductor with a tunable energy gap up to 250 me V.Currently,clear even-order harmonics have been reported in biased BLG.However,there is still a lot of controversy over the mechanism of even-order har-monic generation.For example,Berry curvature,phase of interband transition matrix elements,and multiple-path quantum interference can all lead to the generation of even-order harmonics.Based on this,this paper investigates the mechanism of even-order harmonic generation in biased BLG by focusing on the orientation dependence of even harmonics.The specific research content is as follows:First,the orientation dependence of even-order harmonics in biased BLG is studyed by numerically solving the semiconductor Bloch equations in the velocity gauge.It is shown that the yield of even-order harmonics has a perfect periodicity of 60°with the rotation of the crystal orientation.The even-harmonic intensity reaches the maximum when the laser field is polarized along with the nearest-neighbor atom.Futher,it is found that the even harmonics are mainly polarized along with the polarization of the laser field by investigating the crystal orientation dependence for the parallel and perpendicular components of even harmonics,unlike monolayer Mo S₂,the Berry curvature plays an insignificant role in the even-harmonic generation of biased BLG.Then,it is found that interlayer coupling plays an important role in the generation of even-order harmonics in biased BLG.It is shown that when the biased voltage exists,the asymmetric electron density between in-plane adjacent atoms caused by polariza-tion is responsible for the generation of even harmonics in biased BLG.In addition,a Y-shaped molecular model is also constructed,which provides an intuitive and clear physical image for the generation of even-order harmonics.Finally,the ellipticity dependence of even harmonics in biased BLG is studyed when the elliptical principal axis of the laser pulse is along with the in-plane nearest-neighbor atoms.As expected,the harmonic yields decrease with the ellipticity.Since the component amplitude of the laser field which is along with the in-plane nearest-neighbor atoms decreases with the laser ellipticity,it leads to the reduction of the coupling be-tween the two atoms（with asymmetrical electron density distribution）.This paper provides an in-depth investigation into the generation of even-order harmonics in biased BLG and establishes a clear physical image,which contributes to a deeper understanding of the orientation dependence of even harmonics in biased BLG.This has significant implications for detecting internal information in crystals.