Effective Coulomb interaction in transition metals from constrained random-phase approximation

Abstract

The effective on-site Coulomb interaction (Hubbard U ) between localized d electrons in 3d, 4d, and 5d transition metals is calculated employing a parameter-free realization of the constrained random-phase approximation using Wannier functions within the full-potential linearized augmented-plane-wave method. The U values lie between 1.5 and 5.7 eV and depend on the crystal structure, spin polarization, d electron number, and d orbital filling. On the basis of the calculated U parameters, we discuss the strength of the electronic correlations and the instability of the paramagnetic state toward the ferromagnetic one for 3d metals. 3 However, in these models the Coulomb and also the one-particle hopping matrix elements are typically empirical parameters that are determined such that the employed model reproduces experimental results of interest.For a long time, DFT-LDA and many-body model Hamiltonian methods have been separate and complementary approaches. This has drastically changed with the advent of the dynamical mean-field theory (DMFT), 4 which merged with the LDA to become a novel computational method referred to as LDA + DMFT, 5 which developed into a modern many-body approach for treating correlated electron materials. In retrospect, the so-called LDA + U method, 6 an early attempt to correct the LDA functional by introducing a simple mean-field-like Hubbard U term for localized d or f states, and today routinely applied to a broad spectrum of systems, can be regarded as its static limit. Both LDA + U and LDA + DMFT as well as other approaches not mentioned here rely on the Hubbard U as an additional parameter. Frequently, the exact value of U is unknown, which impedes the predictive power of these approaches.The problem of calculating the parameter-free Hubbard U for transition metals (TMs), i.e., from first principles, has been addressed by several authors.7-11 A number of different approaches have been proposed. Among them, the constrained local-density approximation (cLDA) 9 is the most popular. However, the cLDA is known to give unreasonably large U values for late TMs due to difficulties in compensating for the self-screening error of the localized electrons.10 Furthermore, the frequency dependence of U is unattainable. On the other hand, the constrained random-phase approximation (cRPA), though numerically much more demanding, does not suffer from these difficulties. In contrast to the cLDA, it also allows access to individual Coulomb matrix elements, e.g., on-site, off-site, intra-orbital, interorbital, and exchange.The aim of this Rapid Communication is to present a systematic study of the effective on-site Coulomb interaction (Hubbard U) between localized d electrons in TMs determined by means of first-principles calculations. Previous cRPA studies of U in TMs have focused only on the nonmagnetic (NM) state of the 3d series, and the results appeared to be strongly dependent on the parameters used in the cRPA schemes. 10,11 In the present work, we propose an alternative simple parameter-free c...

Screened Coulomb interaction of localized electrons in solids from first principles

“…Except for a few extreme cases, most correlated materials involve entangled states of localized and delocalized (e.g., d and sp) electrons, and the separation of the intra-channel polarizability is not straightforward. 11,13,19 In this work, we use a wave function projection technique to calculate the intra-channel polarizability: (12) where C i nk = φ di |nk is the projection of a Bloch wave function |nk onto a localized orbital φ di .…”

Theoretical investigation into the possibility of very large moments in Fe16N2

“…25,34 The cRPA approach offers an efficient way to calculate the effective Coulomb interaction U and allows to determine individual Coulomb matrix elements, e.g., on-site, off-site, inter-orbital, intraorbital, and exchange as well as their frequency dependence. We use the FLAPW method as implemented in the FLEUR code 27 with the PBE exchange-correlation potential 28 for ground-state calculations.…”

“…The PBE+U spin moment for bcc Fe was calculated using the interaction parameters computed in Ref. 25-U = 3.16 eV and J = 0.68 eV. We use the fullylocal (FLL) double counting correction in the calculation of both the bcc Fe and the Fe 16 N 2 moments.…”

“…The PBE+U results were obtained using the cRPA-obtained interaction parameters U = 3.99, 3.12, and 3.52 eV for the 4d, 4e, and 8h sites, respectively (J = 0.64, 0.59, and 0.61 eV), except for bcc Fe, for which we used U = 3.16 eV and J = 0.68 eV as in Ref. 25. GW (s, p val) denotes a VASP-GW calculation in which the Fe 3s and 3p electrons are treated on the same level as the 3d and 4s.…”

“…19 In the present work, we perform an extensive search for the proposed large magnetization; we calculate the hyperfine field at the three Fe sites and compare with published Mössbauer spectra; we search for additional energy minima at moments away from the theoretical prediction as a function of tetragonal distortion; we apply the HSE06 hybrid-functional method 20 and the GW approximation 21 as implemented in VASP 22 to α ′′ -Fe 16 N 2 , testing the two methods on bcc Fe to ensure that any enhancement of the moment we obtain is gen-uine. Further, we compute the effective on-site Coulomb interaction (Hubbard U ) between localized 3d electrons employing the constrained random-phase approximation (cRPA) [23][24][25] (as implemented in the SPEX 26 extension of the FLEUR 27 code), allowing us to provide for the first time first-principles predictions for the U and J parameters. Finally, we present new PBE+U 28 calculations using these parameters and discuss their implications for existing models.…”