A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle(electron-hole)Schrodinger equation including Coulomb attraction is presented.As an example,we analyze the energetically low...A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle(electron-hole)Schrodinger equation including Coulomb attraction is presented.As an example,we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material.The eigenvalues of the corresponding Schrodinger equation,i.e.,the onedimensional exciton Wannier equation with disorder,correspond to the energies of excitons in the quantum wire.The wavefunctions,in turn,provide information on the optical properties of the wire.We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric,continuous,and coercive.The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers.The numerical solution of this problem is based on adaptive wavelets.Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates.Numerical examples demonstrate the behavior of the smallest eigenvalue,the ground state energies of the exciton,together with the eigenstates depending on the strength and spatial correlation of disorder.展开更多
High harmonic generation(HHG)from solids shows great application prospects in compact short-wavelength light sources and as a tool for imaging the dynamics in crystals with subnanometer spatial and attosecond temporal...High harmonic generation(HHG)from solids shows great application prospects in compact short-wavelength light sources and as a tool for imaging the dynamics in crystals with subnanometer spatial and attosecond temporal resolution.However,the underlying collision dynamics behind solid HHG is still intensively debated and no direct mapping relationship between the collision dynamics with band structure has been built.Here,we show that the electron and its associated hole can be elastically scattered by neighboring atoms when their wavelength approaches the atomic size.We reveal that the elastic scattering of electron/hole from neighboring atoms can dramatically influence the electron recombination with its left-behind hole,which turns out to be the fundamental reason for the anisotropic interband HHG observed recently in bulk crystals.Our findings link the electron/hole backward scattering with Van Hove singularities and forward scattering with critical lines in the band structure and thus build a clear mapping between the band structure and the harmonic spectrum.Our work provides a unifying picture for several seemingly unrelated experimental observations and theoretical predictions,including the anisotropic harmonic emission in MgO,the atomic-like recollision mechanism of solid HHG,and the delocalization of HHG in ZnO.This strongly improved understanding will pave the way for controlling the solid-state HHG and visualizing the structure-dependent electron dynamics in solids.展开更多
基金supported in part by the Institute for Mathematics and its Applications(IMA)at the University of Minnesota with funds provided by the National Science Foundation(NSF)supported by the Deutsche Forschungsgemeinschaft(DFG).
文摘A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle(electron-hole)Schrodinger equation including Coulomb attraction is presented.As an example,we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material.The eigenvalues of the corresponding Schrodinger equation,i.e.,the onedimensional exciton Wannier equation with disorder,correspond to the energies of excitons in the quantum wire.The wavefunctions,in turn,provide information on the optical properties of the wire.We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric,continuous,and coercive.The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers.The numerical solution of this problem is based on adaptive wavelets.Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates.Numerical examples demonstrate the behavior of the smallest eigenvalue,the ground state energies of the exciton,together with the eigenstates depending on the strength and spatial correlation of disorder.
基金supported by the National Natural Science Foundation of China(Grant No.12074240,No.91950101,and No.11774215)the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)(project number 231447078 TRR 142)(project A07)+2 种基金the Sino-German Mobility Programme(Grant No.M-0031)the Department of Education of Guangdong Prov-ince(Grant No.2018KCXTD011)the Open Fund of the State Key Laboratory of High Field Laser Physics(SIOM).
文摘High harmonic generation(HHG)from solids shows great application prospects in compact short-wavelength light sources and as a tool for imaging the dynamics in crystals with subnanometer spatial and attosecond temporal resolution.However,the underlying collision dynamics behind solid HHG is still intensively debated and no direct mapping relationship between the collision dynamics with band structure has been built.Here,we show that the electron and its associated hole can be elastically scattered by neighboring atoms when their wavelength approaches the atomic size.We reveal that the elastic scattering of electron/hole from neighboring atoms can dramatically influence the electron recombination with its left-behind hole,which turns out to be the fundamental reason for the anisotropic interband HHG observed recently in bulk crystals.Our findings link the electron/hole backward scattering with Van Hove singularities and forward scattering with critical lines in the band structure and thus build a clear mapping between the band structure and the harmonic spectrum.Our work provides a unifying picture for several seemingly unrelated experimental observations and theoretical predictions,including the anisotropic harmonic emission in MgO,the atomic-like recollision mechanism of solid HHG,and the delocalization of HHG in ZnO.This strongly improved understanding will pave the way for controlling the solid-state HHG and visualizing the structure-dependent electron dynamics in solids.
