Electronic structure theory is an essential tool in condensed matter research. It has been implemented in modeling tools such as quantum Monte Carlo, density functional theory (DFT), and tight-binding. It is widely used in modeling materials that span the periodic table. However, in materials where the lattice, charge, orbital, and spin degrees of freedom are strongly correlated, standard electronic structure theories fail. This is partially because most electronic structure modeling is based on the Schrodinger equation, and so cannot fully take into account relativistic effects. Another aspect is that standard techniques have a simplistic treatment of electron-electron correlations. For heavier atoms, actinides especially, this is a significant disadvantage. There are several ways around these issues: we can include a Coulomb parameter for the electron interactions, use mean-field theory models to capture correlations and spectral properties, and we can reformulate the
mathematics in terms of the Dirac equation to better understand relativistic effects (there have been efforts in this direction using the RSPt code described by John Wills et al. in Full-Potential Electronic Structure Method published by Springer in 2010). The challenge of accurately modeling the electronic behavior of actinides has had significant attention from a small community of dedicated researchers due to its importance in quantum and nuclear materials science. In our research, we have explored several tools for addressing electronic behavior in actinides, and we have compared methods of modeling strongly correlated electrons for accuracy and efficiency. Which tool is the best? Ultimately the answer to this depends on the questions we are asking.
The plutonium challenge
The mysteries around f electron behavior have stimulated numerous contributions to literature over the years and produced even more questions. This is particularly true of plutonium, which has been the focus of a great number of studies due its complexity, exotic physics, and central importance in key areas of nuclear science and engineering. Plutonium has six allotropic phases, each of which present challenges to the task of theoretical modeling. The 25% volume contraction between the α and δ phases, the questions around magnetic ordering, and the quantum entanglements between localized and itinerant electrons are a few of the frequently examined questions surrounding this exotic material. By pursuing the goal of describing plutonium metal and its alloys, researchers have made considerable advances in electronic structure modeling for f electron materials and compounds over nearly four decades.
A compendium of techniques
In 1985, Wienberger used a relativistic KKR-Greens function technique to examine the electronic structure of δ-plutonium and several plutonium alloys. In the 1990s, pioneering DFT calculations began a long sequence of studies that would explore the effectiveness of various approximations including exchange correlation functionals, orbital polarization, and Coulomb interactions. The 2000s was dominated by investigations into the magnetic ordering of plutonium’s electrons. Though no long-range magnetic ordering has been observed for any phase of plutonium, theoretical calculations performed using DFT, within either the local density approximation (LDA) or generalized gradient approximation (GGA), can only reproduce the characteristic volume expansion for δ-plutonium by including artificial orbital polarization. This is also the case for DFT with the addition of a Coulomb parameter (LDA/GGA+U) to simulate static electron-electron interactions (see Further Reading for more information).
Figure 1. Electronic structure methods: Conceptually each of the methods are represented by strainers with increasingly fine meshes. The finer the mesh, the more sophisticated the method’s treatment of electronic correlations. The fastest method is DFT then the increasingly sophisticated methods are DFT+U (shorthand for LDA/GGA+U), the GWA, and DMFT. The more information the method can calculate, the more time it requires.
Around the same time, mean-field theory methods showed promising results modeling quantum entanglements between the localized and itinerant electrons by treating the electron-electron interactions as dynamic fluctuations. For example, dynamical mean-field theory (DMFT) is able to describe the electronic characteristics of plutonium, including its volume contraction, in good agreement with photoemission data as well as predict the valence fluctuations validated by neutron spectroscopy without resorting to adding orbital polarization. This has led to two debated explanations of the role of magnetism in plutonium. The first is the “disordered local moment” described by Niklasson et al. in “Modeling the actinides with disordered local moments” published in Physical Review B in 2003. This is also known as the “static” model, which describes the orbitals as individual localized magnetic moments that are spatially and temporally disordered such that any long-range magnetic ordering is obscured by averaging over time. The second is the “valence fluctuation,” or dynamic, model as captured by DMFT and notably described in “The valence fluctuating ground state of plutonium” by Janoschek et al. in Science Advances in 2015. This method predicts quantum entanglements between the localized magnetic moments and the itinerant conduction electrons resulting in valence fluctuations that effectively screen the magnetic ordering below the material’s Kondo temperature. Effectively, the mean-field theory approach more accurately describes the physics of strongly correlated electrons, which is a key piece of the plutonium mystery that many studies have been exploring.
Figure 2. Fermi surface comparisons: Fermi surfaces were calculated for δ-plutonium using four electronic structure methods. Each calculation was performed with spin-orbit coupling and without orbital polarization. The calculations performed with the GWA were comparably accurate compared to DMFT and they required roughly 40% less computational time. A full analysis of the results can be found in Tutchton, et al., Phys. Rev. B 101, 245156 (2020).
While DMFT offers more accurate results than DFT, it is a computationally expensive and time-consuming calculation (Fig. 1). An alternative mean-field theory approach uses the Gutzwiller wavefunction approximation (GWA) first described in 1965 by Martin C. Gutzwiller in Physical Review. Like DMFT, this method has been used to successfully calculate the volume dependence in plutonium phases without the addition of artificial orbital polarization. The combination LDA (or GGA) and GWA formulation has the same mathematical structure as LDA combined with DMFT, with the difference that it assumes infinite quasiparticle lifetimes. This makes the GWA method less accurate for calculating quantum entanglement effects than DMFT, however it is significantly less computationallydemanding and does capture important physics of the electron-electron correlation effects. However, the GWA method is still significantly more time consuming than DFT or the LDA+U methods that require orbital polarization to capture key features of plutonium. There is, therefore, a need to consider strategies to perform calculations on plutonium and other actinide materials. It is relatively common for researchers to use a DFT method with an anti-ferromagnetic configuration for plutonium phases as an approximation for the correlation effects. This technique is essentially understood to give the correct volume expansion for the wrong reason, but it is much more flexible for complex simulations where supercell structures are required to examine alloys, surfaces, lattice dynamics, etc. It can also be used for DFT-based molecular dynamics calculations that may help to understand details of interatomic interactions.
Figure 3. Beyond electronic structure: The flexibility of the GWA method allows us to extend our calculations beyond basic electronic structure calculations. On top is a diagram of the computational workflow from the Cygutz GWA implementation described in Lanatà, et al., Phys. Rev. X 5, 011008 (2015) through the exact diagonalization process to calculate the X-ray absorption spectra for PuB4, shown below. The details of this calculation are fully explored in Chiu, et al., Phys. Rev. B 102, 085150 (2020).
Choosing the right method
The question, then, is which tool or method is appropriate for the questions we are trying to answer. With the more comprehensive techniques we may be able to get more complete data, but the price is time and computational resources, so we need to be strategic and choose the best method based on the requirements of the material and what we want to know about it. For heavier elements, plutonium especially, DFT will not be able to capture a significant scope of the electronic behavior, but if a large supercell is necessary to model an alloy, defect, or any dynamics, DFT may be the only practical option. Adding a Coulomb interaction with a DFT+U method and/or orbital polarization may be useful approximations that improve the overall results, but we will still miss important physics. With the GWA and DMFT, we start to see results that match experimental observations. Using the GWA, we sacrifice some of the spectral information by approximating infinite particle lifetimes but we save time compared to DMFT (Fig. 1). So, depending on what we need to know for a given calculation, the GWA can be a strategic choice for plutonium and other strongly correlated materials.
As an example, each of the discussed methods has been applied to δ-plutonium in “Electronic correlation induced expansion of Fermi pockets in δ-plutonium,” published in Physical Review B in 2020. In this work, we examined the Fermi surfaces to study the electronic correlations in momentum space (Fig 2). We included spin-orbit coupling in all methods, but no orbital polarization. Both the GWA and DMFT were able to capture the electronic correlations, but the GWA was roughly 40% faster than DMFT. This is also applicable to other strongly correlated materials including actinide and lanthanide compounds and alloys. Another advantage of the GWA is that because the computational expense is less intense, we can use the results to go beyond the electronic structure to model more complex material characteristics such as X-ray absorption spectra. We showed this with PuB4 in the 2020 Physical Review B article “Hybridization effect on the x-ray absorption spectra for actinide materials: Application to PuB4” by Chiu et al., where the end result can be directly compared with experimental results (Fig. 3). There are additional opportunities to use the GWA to explore novel quantum materials in which strong electronic and spin correlations play key roles in superconductivity.
Summary
The methods discussed here are not the only first principles modeling techniques that can be used to calculate the electronic structures of strongly correlated materials. The challenge of these types of complex materials has and will continue to require a full exploration of the capabilities of many different techniques and tools. DMFT is a powerful method that is able to capture the physics of strong electronic correlations, but it is also computationally demanding and limited to structures that can be represented by relatively small unit cells. The GWA is slightly less accurate but allows for much more efficiency in terms of computational resources. This creates space for some flexibility in the types of physics and structures that may be studied. DFT, while not able to capture much of the physics of the electron-electron correlations, is still a useful tool to address a great deal of the groundwork research that still needs to be performed for the actinides. Rather than discarding the results we have obtained using DFT, we can use what we learn from the compendium of theory techniques and interpret the data from each method according to its limitations. The tools for truly accurate modeling of f-electron materials and strongly correlated systems are still being formed. It is necessary to use and expand the tools that we have to study the electronic structures of these materials and take the next step in accurate first principles modeling.
Acknowledgments
Thanks to Wei-Ting Chiu, Jian-Xin Zhu, R. C. Albers, G. Kotliar, and, F. Ronning, Giacomo Resta, Tsung-Han Lee, Eric D. Bauer, Richard T. Scalettar, Q. Si, J. Singleton, P. Wolfle, N. Harrison, J. M. Wills, and G. Zwicknagl for helpful discussions. This work was carried out under the auspices of the U.S. Department of Energy (DOE) National Nuclear Security Administration under Contract No. 89233218CNA000001. The Fermi-surface topology analysis work was supported by the LANL LDRD Program. The DFT + DMFT simulation work at high temperatures was supported by the NNSA Advanced Simulation and Computing Program. It was, in part, supported by the Center for Integrated Nanotechnologies, a DOE BES user facility, in partnership with the LANL Institutional Computing Program for computational resource.
