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

Abstract: The effective on-site Coulomb interaction (Hubbard $U$) between localized \textit{d} electrons in 3\textit{d}, 4\textit{d}, and 5\textit{d} transition metals is calculated employing a new 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, \textit{d} electron number, and \textit{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 towards the ferromagnetic one for 3\textit{d} metals.

• The paper introduces a cRPA method using maximally localized Wannier functions within FLAPW to compute Hubbard U without empirical parameters.
• It reports Hubbard U values from 1.5 to 5.7 eV, demonstrating a dependence on crystal structure, electron filling, and magnetic state, especially in 3d metals.
• The approach improves electronic structure models such as LDA+U, offering enhanced predictions for magnetic properties and electron correlations.

Effective Coulomb Interaction in Transition Metals from Constrained Random-Phase Approximation

The paper authored by Ersoy Şaşıoğlu, Christoph Friedrich, and Stefan Blügel presents an innovative approach to calculating the effective on-site Coulomb interaction (Hubbard $U$) for transition metals, utilizing a parameter-free method via the constrained random-phase approximation (cRPA). The study addresses an essential aspect of understanding electronic correlations, critical for accurately describing the electronic properties of materials with intermediate and strong electron correlations.

Methodology and Approach

The authors employ a novel realization of cRPA, integrated with Wannier functions within the full-potential linearized augmented-plane-wave (FLAPW) method. This approach allows for the calculation of the Hubbard $U$ between localized d electrons in 3d, 4d, and 5d transition metals, without relying on empirical parameters. The cRPA method is particularly advantageous as it can isolate specific screened interactions and distinguish between different types of Coulomb matrix elements, such as on-site, off-site, intra-orbital, inter-orbital, and exchange terms.

Understanding the applicability of LDA and its limitation in scenarios where $U/W$ (where $W$ is the bandwidth) is equal to or greater than one, the study lays out the necessity of precisely determining Hubbard $U$. Historically, the derivation of these parameters was based on empirical fitting, thereby limiting predictive accuracy. The study demonstrates that by employing maximally localized Wannier functions (MLWFs), the complexity of entangled bands is addressed, to construct the polarization matrix $P_d$ and derive the effective $U$.

Results

The research finds that the Hubbard $U$ values for transition metals, calculated using the devised method, span from 1.5 to 5.7 eV. These values vary depending on factors such as crystal structure, spin polarization, number of d electrons, and d orbital filling. For the 3d transition metals, a clear dependency on the d electron number and orbital filling is observed. Specifically, the results indicate a higher $U$ for ferromagnetic states compared to non-magnetic ones in elements like Fe, Co, and Ni.

The paper emphasizes the correlation between the electron correlation strength and the magnetic properties, where the $U/W$ ratio is a determining factor. It also illustrates enhanced correlation effects in 3d TMs compared to their 4d and 5d counterparts, due to the narrower bands in the former, confirming that elements like Fe, Co, and Ni exhibit conditions favorable for ferromagnetism based on the Stoner criterion.

Implications

The work holds significant implications for theoretical and applied material science, especially in improving the reliability of electronic structure calculations in models like LDA+U and LDA+DMFT. By better constraining the Hubbard $U$ values, the predictive power of these models in describing correlated electron materials is enhanced. The research also paves the way for more accurate simulations of magnetic properties in transition metals, crucial for developing novel materials with tailored electronic characteristics.

Future Directions

Future research could explore extending this methodology to a wider range of materials, including transition metal oxides and complex systems where electron correlations play a critical role. Further development and refinements in computational techniques are imperative for enhancing the scope of first-principles calculations in capturing correlated electron dynamics accurately.

This paper represents a substantial contribution to the field of condensed matter physics, providing a robust framework for calculating Hubbard $U$ using first-principles methods, and deepening the understanding of electron correlations in transition metals.