Spin structure factors of doped monolayer Germanene in the presence of spin-orbit coupling

Farshad Azizi,Hamed Rezania

doi:10.1038/s41598-021-87268-x

Abstract

In this paper, we present a Kane-Mele model in the presence of magnetic field and next nearest neighbors hopping amplitudes for investigations of the spin susceptibilities of Germanene layer. Green’s function approach has been implemented to find the behavior of dynamical spin susceptibilities of Germanene layer within linear response theoryand in the presence of magnetic field and spin-orbit coupling at finite temperature. Our results show the magnetic excitation mode for both longitudinal and transverse components of spin tends to higher frequencies with spin-orbit coupling strength. Moreover the frequency positions of sharp peaks in longitudinal dynamical spin susceptibility are not affected by variation of magnetic field while the peaks in transverse dynamical susceptibility moves to lower frequencies with magnetic field. The effects of electron doping on frequency behaviors of spin susceptibilities have been addressed in details. Finally the temperature dependence of static spin structure factors due to the effects of spin-orbit coupling, magnetic field and chemical potential has been studied.

Highlights

In this paper, we present a Kane-Mele model in the presence of magnetic field and nearest neighbors hopping amplitudes for investigations of the spin susceptibilities of Germanene layer
Novel electronic properties have been exhibited by Graphene layer with a zero band gap which compared to materials with a non-zero energy gap
The different dopants within the Germanene layer gives arise to the sizable band gap opening at the Dirac point an the electronic properties of this material are affected by that[16,17]

Summary

Model Hamiltonian and formalism

The index α implies two inequivalent function ψnσ (k, r) can sublattice atoms A, B in the unit cell, r denotes the position vector of electron, k is the wave function belonging in the first Brillouin zone of honeycomb structure. Such band wave function can be written as ψnσ (k, r) =. The band structures of electrons with spin σ of Germanene described by model Hamiltonian in Eq (2) are obtained by using the matrix form of Schrodinger as follows.

The two matrix elements of neighbor atoms on

Dynamical and static spin structure factors

Numerical results and discussions

Additional information