Spin-orbit coupling in periodic systems with broken time-reversal symmetry: Formal and computational aspects

Abstract

We discuss the treatment of spin-orbit coupling (SOC) in time reversal symmetry broken periodic systems for relativistic electronic structure calculations of materials within the generalized noncollinear Kohn-Sham density functional theory (GKS-DFT). We treat SOC self-consistently and express the GKS orbitals in a two-component spinor basis. Crucially, we present a methodology (and its corresponding implementation) for the simultaneous self-consistent treatment of SOC and exact non-local Fock exchange operators. The many advantages of the inclusion of non-local Fock exchange in the self-consistent treatment of SOC, as practically done in hybrid exchange-correlation functionals, are both formally derived and illustrated through numerical examples: i) it imparts a local magnetic torque (i.e. the ability of the two-electron potential to locally rotate the magnetization with respect to a starting guess configuration) that is key to converge to the right solution in non-collinear DFT regardless of the initial guess for the magnetization; ii) because of the local magnetic torque, it improves the rotational invariance of non-collinear formulations of the DFT; iii) it introduces the dependence on specific pieces of the spinors (i.e. those mapped onto otherwise missing spin-blocks of the complex density matrix) into the two-electron potential, which are key to the correct description of the orbital-and spin-current densities and their coupling with the magnetization; iv) when space-inversion symmetry is broken, it allows for the full breaking of timereversal symmetry in momentum space, which would otherwise be constrained by a sum-rule linking the electronic band structure at opposite points in the Brillouin zone (k and -k). The presented methodology is implemented in a developmental version of the public Crystal code. Numerical tests are performed on the model system of an infinite radical chain of Ge2H with both spaceinversion and time-reversal symmetries broken, which allows to highlight all the above-mentioned effects.

Electronic Structure and Magnetism of Pristine, Defected, and Strained Ti2N MXene

“…The origin of gap opening in the Dirac cones are due to hybridization of the electronic wave function between the top and bottom surfaces with the same quantum numbers [71]. In our case, the hybridization causes the time reversal symmetry breaking under SOC [72] and the Dirac cones behave massive states [73,74]. Similarly, linear gapped Dirac cone (i) and distorted gapped Dirac cone (j) are found near the Fermi level (above and below) in region K-Γ (FIG.…”

“…6B). The gap opening results from the time reversal symmetry breaking [72] and hybridization [71] of the opposite spin states (FIG. 6B).…”

Unraveling Electronic Structure, Magnetic States, and Topological Phenomena in Pristine, Defected, and Strained Ti2N MXene

“…31,35 The origin of gap opening in the Dirac cones are due to hybridization of the electronic wave function between the top and bottom surfaces with the same quantum numbers. 53 In our case, the hybridization causes the time reversal symmetry breaking under SOC 54 and the Dirac cones behave massive states 55,56 . Similarly, linear gapped Dirac cone (i) and distorted gapped Dirac cone (j) are found near the Fermi level (above and below) in region K-Γ (Figure 4Q).…”

Third order non-linear optical susceptibilities (χ3) of yttria stabilized cubic hafnium(IV) oxide

“…Spin-orbitcoupling effects (SOC) on the band structures of pure and doped hafnia using richer basis sets [21,22,23,24]. for all atoms have been considered by applying a newly developed scheme in CRYSTAL based on two-component spin-current density functional theory [25,26]. In brief For Y and Hf we used a fully relativistic effective core potential (ECP) including both the scalarrelativistic AREP and the spin-orbit SOREP parts, developed by Peterson et al [27] and Figgen et al [28].…”