Multipole analysis in the radiation field for linearized f(R) gravity with irreducible Cartesian tensors

Abstract

The 1/r-expansion in the distance to the source is applied to the linearized f (R) gravity, and its multipole expansion in the radiation field with irreducible Cartesian tensors is presented. Then, the energy, momentum, and angular momentum in the gravitational waves are provided for linearized f (R) gravity. All of these results have two parts which are associated with the tensor part and the scalar part in the multipole expansion of linearized f (R) gravity, respectively. The former is the same as that in General Relativity, and the latter, as the correction to the result in General Relativity, is caused by the massive scalar degree of freedom, and places an important role in distinguishing GR and f (R) gravity.

Evaluation of Charged Defect Energy in Two‐Dimensional Semiconductors for Nanoelectronics: The WLZ Extrapolation Method

“…It is well known that density functional theory underestimates band gap [54] and the many-body perturbation theory is demonstrated to be reliable in predicting band gap. [55] Hence, the choice of functional would significantly affect the evaluation of defect IE [45,47,56] as it is the energy distance between the defect transition level and the band edge.…”

“…Besides, drawing on the experience of the routine calculation method for charged defects in 3D materials, which is based on the electrostatic energy difference of a model charge in between isolated and periodic boundary conditions, several correction methods for charged defects in 2D and quasi-2D systems have been proposed. [40][41][42][43][44][45] Since the weak and highly anisotropic screening in 2D systems makes the energy difference calculation much trickier, Sundararaman et al [46] and Wu et al [47] unambiguously define an anisotropic dielectric profile, which can be directly derived from regularized density-functional-theory electrostatic potentials, to solve the problem. [46,47] In this work, we focus on the development of the WLZ extrapolation method and review its application in various defect research in 2D materials.…”

“…[40][41][42][43][44][45] Since the weak and highly anisotropic screening in 2D systems makes the energy difference calculation much trickier, Sundararaman et al [46] and Wu et al [47] unambiguously define an anisotropic dielectric profile, which can be directly derived from regularized density-functional-theory electrostatic potentials, to solve the problem. [46,47] In this work, we focus on the development of the WLZ extrapolation method and review its application in various defect research in 2D materials. The WLZ formalism for charged defects in monolayer and non-monolayer 2D materials is first addressed, displaying its plausibility and feasibility.…”

Substrate effects on charged defects in two-dimensional materials

“…The anisotropic dielectric function of the 2D material is also calculated from first principles as described in Ref. 18. (See Supplemental Material for details.)…”

“…16 Several complementary approaches specialized for charged defects in 2D materials [15][16][17] have made it possible to reliably predict charge transition levels and engineer defects in freestanding 2D materials. [18][19][20][21][22][23][24] However, 2D materials in most experiments and device configurations are not free-standing and are instead deposited, grown or transferred onto a substrate. The substrate is typically an integral part of developing and utilizing 2D materials, critical for nucleation during synthesis and mechanical stability in operation, and it is an unavoidable modification introduced to tune its properties.…”