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

TL;DR

This paper provides an exact theoretical analysis of Friedel oscillations in various 2D materials under magnetic fields, revealing how impurity potential strength and magnetic field influence electron density patterns.

Contribution

It introduces an exact Green's function approach to calculate Friedel oscillations in monolayer graphene and dichalcogenides, extending understanding of impurity effects in 2D materials under magnetic fields.

Findings

Friedel oscillations in WS₂ follow a sinusoidal decay proportional to 1/r².

Oscillation amplitude scales with impurity potential strength V₀.

High magnetic fields and attractive potentials keep electron density positive everywhere.

Abstract

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.