By performing systematic $ab$ $initio$ density functional calculations, here we study two relativity-induced properties of atomically thin ferromagnetic (FM) Cr$_2$Ge$_2$Te$_6$ films [monolayer (ML), bilayer (BL) and trilayer (TL) as well as bulk], namely, magnetic anisotropy energy (MAE) and magneto-optical (MO) effects. Competing contributions of both magneto-crystalline anisotropy energy (C-MAE) and magnetic dipolar anisotropy energy (D-MAE) to the MAE, are computed. Calculated MAEs of these materials are large, being in the order of $sim$0.1 meV/Cr. Interestingly, we find that the out-of-plane magnetic anisotropy is preferred in all the systems except the ML where an in-plane magnetization is favored because here the D-MAE is larger than the C-MAE. Crucially, this explains why long-range FM order was observed in all the few-layer Cr$_2$Ge$_2$Te$_6$ except the ML because the out-of-plane magnetic anisotropy would open a spin-wave gap and thus suppress magnetic fluctuations so that long-range FM order could be stabilized at finite temperature. In the visible frequency range, large Kerr rotations up to $sim$1.0 deg in these materials are predicted and they are comparable to that observed in famous MO materials such as PtMnSb and Y$_3$Fe$_5$O$_{12}$. Moreover, they are $sim$100 times larger than that of 3$d$ transition metal MLs deposited on Au surfaces. Faraday rotation angles in these 2D materials are also large, being up to $sim$120 deg/$mu$m, and are thus comparable to the best-known MO semiconductor Bi$_3$Fe$_5$O$_{12}$. These findings thus suggest that with large MAE and MO effects, atomically thin Cr$_2$Ge$_2$Te$_6$ films would have potential applications in novel magnetic, MO and spintronic nanodevices.

Published : "arXiv Mesoscale and Nanoscale Physics".

We study spectra of long wavelength plasma oscillations in a system of two energy splitted one-dimensional (1D) massless Dirac fermion subbands coupled by spin-orbit interaction. Such a system may be formed by edge subbands in semiconducting transition metal dichalcogenide monolayers. Intrasubband transitions of massless Dirac fermions give rise to optical and acoustic gapless branches of intrasubband 1D plasmons. We reveal that the optical branch is of quantum character with group velocity being inverse proportional to square root of the Planck constant, whereas the acoustic branch is classical one with group velocity proportional to geometric mean of the edge subband velocities. Spin-orbit interaction, allowing intersubband transitions in the system, results in emergence of two branches of intersubband 1D plasmons: upper and lower ones. The upper and lower branches are gapped at small wave vectors and evolve with positive and negative group velocities, respectively, from energy splitting of the edge subbands at Fermi-level. The both intersubband branches adjoin intersubband single particle excitation continuum from above, while in case of the edge subbands with unequal velocities the lower one experiences Landau damping at small wave vectors. In addition, the lower branch, attaining zero frequency at a non-zero wave vector, alters its group velocity from negative to positive one.

Published : "arXiv Mesoscale and Nanoscale Physics".