Theory of Friedel oscillations in monolayer graphene and group-VI dichalcogenides in a magnetic field

Friedel oscillations (FO) of electron density caused by a delta-like neutral impurity in two-dimensional (2D) systems in a magnetic field are calculated. Three 2D cases are considered: free electron gas, monolayer graphene and group-VI dichalcogenides. An exact form of the renormalized Green's function is used in the calculations, as obtained by a summation of the infinite Dyson series and regularization procedure. Final results are valid for large ranges of potential strengths $V_0$, electron densities $n_e$, magnetic fields $B$ and distances from the impurity $r$. Realistic models for the impurities are used. The first FO of induced density in WS$_2$ are described by the relation $\Delta n(\vec{r}) \propto \sin(2\pi r/T_{FO})/r^2$, where $T_{FO} \propto 1/\sqrt{E_F}$. For weak impurity potentials, the amplitudes of FO are proportional to $V_0$. For attractive potentials and high fields the total electron density remains positive for all $r$. On the other hand, for low fields, repulsive potentials and small $r$, the total electron density may become negative, so that many-body effects should be taken into account.