Compositional phase stability of correlated electron materials within DFT+DMFT

TL;DR

This paper demonstrates that DFT+DMFT provides a more accurate and reliable prediction of phase stability and intercalation voltages in correlated electron materials like LixCoO2, outperforming DFT+U by capturing dynamical correlations.

Contribution

It introduces the application of DFT+DMFT to study phase stability in correlated materials, highlighting its advantages over DFT+U in accurately predicting energetics and avoiding spurious states.

Findings

DFT+DMFT dampens the energy increase with U, capturing dynamical correlations.

DFT+DMFT correctly predicts phase stability without spurious charge ordering.

Intercalation voltage predictions from DFT+DMFT decrease relative to DFT, aligning better with experiments.

Abstract

Predicting the compositional phase stability of strongly correlated electron materials is an outstanding challenge in condensed matter physics, requiring precise computations of total energies. In this work, we employ the density functional theory plus dynamical mean-field theory (DFT+DMFT) formalism to address local correlations due to transition metal d electrons on compositional phase stability in the prototype rechargeable battery cathode material LixCoO2, and detailed comparisons are made with the simpler DFT+U approach (i.e. the Hartree-Fock solution of the DMFT impurity problem). Local interactions are found to strongly impact the energetics of the band insulator LiCoO2, most significantly via the Eg orbitals, which are partially occupied via hybridization with O p states. We find CoO2 and Li1/2CoO2 to be moderately correlated Fermi liquids with quasiparticle weights of 0.6-0.8 for the T2g states, which are most impacted by the interactions. As compared to DFT+U, DFT+DMFT considerably dampens the increase in total energy as U is increased, which indicates that dynamical correlations are important to describe this class of materials despite the relatively modest quasiparticle weights. Unlike DFT+U, which can incorrectly drive LixCoO2 towards spurious phase separating or charge ordered states, DFT+DMFT correctly captures the system's phase stability and does not exhibit a strong charge ordering tendency. Most importantly, the error within DFT+U varies strongly as the composition changes, challenging the common practice of artificially tuning U within DFT+U to compensate the errors of Hartree-Fock. DFT+DMFT predicts the average intercalation voltage decreases relative to DFT, opposite to the result of DFT+U, which would yield favorable agreement with experiment in conjunction with the overprediction of the voltage by the strongly constrained and appropriately normed DFT functional.

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Full text

I Introduction

Strongly correlated materials, for which density functional theory (DFT) calculations often break down due to strong electron-electron interactions, are a challenging class of condensed matter systems relevant to several important technologies Kotliar and Vollhardt (2004); Morosan et al. (2012). One example is Li ion rechargeable batteries. These electrochemical cells rely critically on a cathode material that can reversibly intercalate Li ions Whittingham (2004). Since cathode materials typically are based on transition metal oxides to accommodate changes in oxidation state, they have an open $d$ electron shell and are susceptible to strong correlation physics.

Currently, the dominant cathode materials are based on LixCoO2 (LCO), a layered compound in which Li ions are intercalated between layers of edge-sharing Co–O octahedra, as shown in Fig. 1(d) Mizushima et al. (1980). Several early theoretical studies that revealed significant insight into the electronic structure and phase diagram of LCO Czyżyk et al. (1992); Aydinol et al. (1997); Wolverton and Zunger (1998); Van der Ven et al. (1998) were based on DFT Hohenberg and Kohn (1964); Kohn and Sham (1965), the de facto standard for first-principles calculations in solid-state physics and chemistry. It is not uncommon, however, for DFT to fail to capture the physics of correlated materials due to the approximation for the exchange-correlation functional [e.g., local density approximation (LDA) or generalized gradient approximation (GGA)].

While DFT calculations in many ways reliably characterizes LCO, there are deficiencies in their description. DFT (GGA) underestimates the intercalation voltage by around 0.8 V Chevrier et al. (2010). In addition, using a plane-wave basis set and ultrasoft pseudopotentials, Van der Ven et al. found that LDA overestimates the order-disorder transition temperature for $x=1/2$ by 100 °C Van der Ven et al. (1998). We note that the LDA linear augmented plane wave results of Wolverton and Zunger Wolverton and Zunger (1998) do not show the same overestimation, though this study uses only roughly one third as many DFT calculations to parameterize the cluster expansion, in addition to performing slightly restricted structural relaxations.

One widely utilized approach to go beyond DFT is the DFT+$U$ method Liechtenstein et al. (1995), in which an explicit on-site Coulomb interaction $U$ is added to account for the strong interactions in the $d$ shell along with a simple mean-field ansatz for the energy functional. However, DFT+$U$ does not fully remedy the shortcomings of DFT and in some cases hurts the description more than it helps. DFT+$U$ still underestimates the voltage by 0.3 V, and it can overestimate the order-disorder transition temperature by as much as several hundred degrees Isaacs and Marianetti (2017). DFT+$U$ drives LiCoO2 towards a high-spin transition Andriyevsky et al. (2014) not observed in experiments van Elp et al. (1991); Ménétrier et al. (1999, 2008) and, unless spurious charge ordering is permitted to occur, incorrectly predicts phase separation Isaacs and Marianetti (2017); Seo et al. (2015). Moreover, DFT+$U$ finds CoO2 to be an insulator in disagreement with experiment Zhang et al. (2004). DFT+$U$ clearly is problematic in the context of LCO.

Here, we revisit the electronic structure, voltage, and phase stability of LCO using more sophisticated DFT plus dynamical mean-field theory (DFT+DMFT) calculations Kotliar et al. (2006) based on GGA. In this framework, the many-body DMFT approach captures the dynamical local correlations of Co $d$ electrons embedded in the crystal, whereas only the static effects are described within DFT+$U$. Total energy DFT+DMFT calculations have become an important tool for understanding structural stability of materials with electronic correlations Savrasov et al. (2001); Held et al. (2001); Dai et al. (2003); McMahan (2005); Amadon et al. (2006); Pourovskii et al. (2007); Leonov et al. (2008); Di Marco et al. (2009); Leonov et al. (2010); Kuneš et al. (2009); Aichhorn et al. (2011); Leonov et al. (2012); Grieger et al. (2012); Bieder and Amadon (2014); Zaki et al. (2014); Park et al. (2014a, b); Chen et al. (2015); Haule and Birol (2015); Delange et al. (2016); Haule and Pascut (2017); Haule (2018); Han et al. (2018), and our work extends this exploration to the realm of compositional phase stability.

We find that DFT+DMFT describes LiCoO2 as a band insulator with modest shifts and broadenings of the low-energy spectrum, most prominently via the $E_{g}$ levels partially occupied via hybridization with O $p$ states. CoO2 and Li1/2CoO2 are Fermi liquids whose $T_{2g}$ states are most strongly affected by the interactions, with quasiparticle weight of around 0.6–0.7. DFT+DMFT, unlike DFT+$U$, does not strongly stabilize charge ordering in Li1/2CoO2, nor does it predict insulating behavior for CoO2 or Li1/2CoO2; in other words, DFT+DMFT substantially improves the description of the electronic structure. Dynamical correlations significantly dampen the impact of $U$ on the total energy of LCO, but more substantially for CoO2 than LiCoO2, leading to a reduction in voltage as compared to DFT, whereas DFT+$U$ yields the qualitatively opposite behavior. Given the more accurate strongly constrained and appropriately normed (SCAN) DFT functional overestimates the experimental voltage, such a decrease in the predicted voltage is expected to lead to agreement between experimental and predicted voltage for DFT+DMFT based on the SCAN functional. Similar to the voltage behavior, the $x=1/2$ formation energy prediction is significantly affected by dynamical correlations: unlike DFT+$U$, DFT+DMFT only weakly influences the formation energy of $x=1/2$ as compared to DFT. Our results demonstrate the importance of dynamical correlations, missing in DFT+$U$, to accurately describe the electronic structure and energetics of correlated electron materials.

II Computational Details

We perform single-site paramagnetic DFT+DMFT total energy calculations using the formalism of Ref. 36 based on the spin-independent GGA of Perdew, Burke, and Ernzerhof (PBE) Perdew et al. (1996) and the projector augmented wave (PAW) method Blöchl (1994); Kresse and Joubert (1999) as implemented in the vasp code Kresse and Hafner (1994, 1993); Kresse and Furthmüller (1996a, b). Select calculations are also performed using the SCAN functional Sun et al. (2015). Given that LCO exhibits no long range magnetic order in experiment Motohashi et al. (2009), the paramagnetic state is justified and we do not search for long-range magnetic order. The structures are fixed to the fully relaxed spin-dependent DFT ground state structures with O3 layer stacking Van der Ven et al. (1998), corresponding to a band insulator for $x=1$ and ferromagnetic low-spin metals for $x=0$ and $x=1/2$. Except where otherwise noted, calculations are performed using the fixed non-spin-polarized DFT charge density, i.e., they are non-charge-self-consistent (NCSC); this is done for reasons of computational efficiency. We characterize the magnitude of the error associated with charge self-consistency by directly comparing NCSC and charge-self-consistent (CSC) calculations within DFT+$U$, demonstrating that the error is sufficiently small for the trends we are studying in this paper. A 500 eV energy cutoff and $k$-point meshes of $k$-point density corresponding to $9\times 9\times 9$ for the rhombohedral LiCoO2 primitive cell and $19\times 19\times 19$ for the bulk Li primitive cell are employed. The ionic forces and total energy are converged to 0.01 eV/Å and 10*-6* eV, respectively.

To define the correlated subspace, we utilize the maximally-localized Wannier function (MLWF) basis Mostofi et al. (2008) for the full $p$-$d$ manifold and perform a unitary rotation of the $d$ orbitals to minimize the off-diagonal hoppings Park et al. (2014a). The Slater-Kanamori (SK) interaction with $J_{\mathrm{SK}}$ set to 0.7 eV is employed, and we use the numerically exact hybridization expansion continuous-time quantum Monte Carlo (CTQMC) solver for the 5-orbital impurity problem Haule (2007); Gull et al. (2011) at temperature $T=290$ K. For DMFT, we perform calculations using density-density interactions. In order to assess the impact of terms beyond density-density interactions, we also perform calculations augmenting density-density interactions with the off-diagonal $J$ terms within the $E_{g}$ manifold, in which the impact of off-diagonal $J$ is expected to be most important due to the partial filling of $E_{g}$. For comparison, we also perform DFT+$U$ calculations using the projector (corresponding to projection onto spherical harmonics within the PAW spheres Bengone et al. (2000)) correlated subspace in vasp (ldautype=4) and present all our results in terms of the $U$ and $J$ corresponding to this interaction model via $U=U_{\mathrm{SK}}-8J_{\mathrm{SK}}/5$ and $J=7J_{\mathrm{SK}}/5$ Pavarini et al. (2011). Although it has limitations such as lack of normalization Haule et al. (2010); Schüler et al. (2018), we consider the projector correlated subspace since it is widely used. It should be noted that $J$ is fixed in all calculations, even though a range of different $U$ is explored. We employ the fully-localized-limit (FLL) form of the double counting Anisimov et al. (1993). The total energy is converged to within $\sim 10$ meV/f.u. for the DMFT self-consistency condition.

III Results and Discussion

III.1 Electronic structure of CoO2 and LiCoO2

We begin by studying the basic electronic structure of LiCoO2 and CoO2 as a function of $U$, allowing for a direct comparison between DFT+DMFT and DFT+$U$. Corresponding results for Li1/2CoO2 are included, but these are not directly discussed until Section III.3. In LCO, the ability of the oxygens to relax in the out-of-plane direction slightly distorts the CoO6 octahedra and results in a symmetry lineage of $T_{2g}\rightarrow A_{1g}+E_{g}^{\prime}$ relative to cubic symmetry, though we will still sometimes refer to this manifold as $T_{2g}$ for brevity. The DFT density of states is shown in Fig. 1(a) and 1(c) for CoO2 and LiCoO2, respectively. Within DFT, LiCoO2 is a band insulator with nominally filled $T_{2g}$ and empty $E_{g}$ states, whereas CoO2 is metallic with a hole in the $T_{2g}$ manifold. The density of states from the Wannier correlated subspace for the full $p$-$d$ manifold, shown in the dashed red lines, is numerically identical to that of DFT by construction. The Wannier functions are well localized with values for the spread $\langle(\mathbf{r}-\bar{\mathbf{r}})^{2}\rangle$ of around 0.42 and 0.45 Å2 for the individual Co $d$ orbitals of CoO2 and LiCoO2, respectively.

An essential quantity in Green function based approaches is the self-energy, which is central to computing the total energy and determining the low-energy properties. The electronic self-energy $\Sigma$ on the imaginary (Matsubara) frequency axis obtained via the CTQMC solver is shown for CoO2 and LiCoO2 in Fig. 2 for density-density interactions. The noise in the self-energy stems from the stochastic nature of the CTQMC solver, and for frequencies above 20 eV there is no noise since we utilize the analytic form of $\Sigma$ in the high-frequency limit. We note that the self-energy is well converged, particularly for low frequency.

For both CoO2 and LiCoO2, $\mathrm{Im}(\Sigma)$ goes to 0 at low frequency, consistent with well-defined quasiparticles and a band insulator, respectively. This indicates that CoO2 can be described as a Fermi liquid and is not a Mott insulator, consistent with experiments on CoO2 de Vaulx et al. (2007); Motohashi et al. (2007); Kawasaki et al. (2009), whereas past DFT+$U$ studies predict an insulating state Zhang et al. (2004); as do our DFT+$U$ results in the present study. Therefore, DFT+DMFT is providing an improved description of the electronic structure of LCO. As a function of $U$, the magnitude of $\mathrm{Im}(\Sigma)$ increases. The imaginary part of the self-energy is essentially identical for the $E_{g}^{\prime}$ and $A_{1g}$ states, which indicates the symmetry breaking within the $T_{2g}$ manifold is small. The overall magnitude of $\mathrm{Im}(\Sigma)$ is moderately larger for CoO2 than for LiCoO2. For CoO2, the imaginary part of the self-energy of the $E_{g}^{\prime}$ and $A_{1g}$ states are larger in magnitude than those of the $E_{g}$ states below $i\omega\approx 10$ eV, which is reasonable given their respective occupancies. The opposite trend is found for LiCoO2 with a larger magnitude of $\mathrm{Im}(\Sigma)$ for the $E_{g}$ states for the full range of frequency shown. This suggests that for LiCoO2 the correlations have a larger impact on the nominally-unoccupied $E_{g}$ states since they are partially occupied via hybridization with O $p$ states, whereas the $E_{g}^{\prime}$ and $A_{1g}$ are filled.

For LiCoO2, in the high-frequency limit, $\mathrm{Re}(\Sigma)$ is typically negative for $E_{g}^{\prime}$ and $A_{1g}$ and positive for $E_{g}$. This indicates that the static part of the correlations tend to push $E_{g}^{\prime}$ and $A_{1g}$ down in energy and $E_{g}$ up in energy as is observed using DFT+$U$. The $U=1.9$ eV case is an exception as $J$ is likely too large relative to $U$ in this case. The real part of the self-energy increases at lower frequency for $E_{g}^{\prime}$ and $A_{1g}$, whereas it decreases for $E_{g}$. This leads to a higher $\mathrm{Re}(\Sigma)$ for $E_{g}^{\prime}$ and $A_{1g}$ than $E_{g}$ towards zero frequency. Overall, the magnitude of the changes in $\mathrm{Re}(\Sigma)$ with $U$ are significantly larger for $E_{g}^{\prime}$ and $A_{1g}$ than for $E_{g}$.

For CoO2, the self-energy of the $E_{g}$ states has a small real part (at most 0.21 eV), which decreases and becomes negative at low frequency. The magnitude is substantially larger for $E_{g}^{\prime}$ and $A_{1g}$ than $E_{g}$ with a maximum magnitude of 1.2 eV for $U=5.9$ eV. For these states, like in the LiCoO2 case, the values are negative at high frequency (except for very low $U$) and become positive at low frequency. As opposed to the imaginary part, the real part of the self-energy has smaller magnitude for CoO2 than for LiCoO2.

From the low-frequency behavior of $\mathrm{Im}(\Sigma)$, we compute the quasiparticle weight $Z=[1-\partial\mathrm{Im}(\Sigma)/\partial i\omega|_{i\omega\rightarrow 0}]^{-1}$, shown in Fig. 3. This quantity is unity for $U=J=0$ [$\mathrm{Im}(\Sigma)=0$] and is inversely proportional to the effective mass arising from electron interactions. All the values decrease with $U$, as expected, in a roughly linear fashion. $Z$ is always larger for LiCoO2 than CoO2, consistent with the fact that LiCoO2 is a band insulator. This effect is pronounced in the $E_{g}^{\prime}$ and $A_{1g}$ states, for which the CoO2 values are 0.14–0.20 lower than those of LiCoO2. For the $E_{g}$ states, the disparity is smaller, with differences of only 0.01–0.04. For CoO2, $Z$ is larger and decreases less rapidly for the $E_{g}$ orbitals compared to the $E_{g}^{\prime}$ and $A_{1g}$ orbitals. From $U=1.9$ to 5.9 eV, $Z$ of the $E_{g}^{\prime}$ and $A_{1g}$ orbitals of CoO2 goes from 0.79 to 0.57 and that of the $E_{g}$ orbitals goes from 0.88 to 0.73. For LiCoO2, over the same range of $U$, $Z$ of the $E_{g}^{\prime}$ and $A_{1g}$ orbitals goes from 0.93 to 0.77 and that of the $E_{g}$ orbitals goes from 0.89 to 0.77. Here, $Z$ is smaller and decreases less rapidly for the $E_{g}$ states such that $Z$ is the same for all the orbitals at $U=5.9$ eV.

III.2 Atomic configurations and $d$ occupancies

To further understand the detailed electronic configuration of CoO2 and LiCoO2, in Fig. 4 we plot the probabilities of the different atomic configurations sampled by the CTQMC solver in terms of the number of $d$ electrons ($N_{d}$) and the spin projection $S_{z}$. The results for $U=4.9$ eV are shown as a representative example. We note that the probability distribution is symmetric about $S_{z}=0$ since our DFT+DMFT calculations are paramagnetic (i.e., there is no long-range magnetic order).

Although CoO2 and LiCoO2 are nominally $d^{5}$ and $d^{6}$, respectively, the probability distribution is centered at higher values of $N_{d}$ for both cases due to the appreciable hybridization with O $p$ states. For example, for LiCoO2 there is substantial time in the Monte Carlo simulation in which an electron from an O $p$ state has hopped into an $E_{g}$ orbital, leading to a $d^{7}$ state. There are substantial fluctuations in $N$ as well as $S_{z}$ for the Co site in both systems. For CoO2 the spin fluctuations are moderately larger than in LiCoO2; there is even probability of $S_{z}=3/2$ states. We note that these fluctuations of the Co site highlight why both DFT and DFT+$U$ struggle to capture all the physics in this system.

It is also useful to examine the behavior of $N_{d}$ versus $U$ for all the methodologies employed in this work in Fig. 5. This has been shown to give insight into the behavior and impact of the double counting correction, which is purely a function of $N_{d}$ Wang et al. (2012); Dang et al. (2014); Park et al. (2014a). Within DFT, one can observe that $N_{d}$ is larger for the projector correlated subspace than the Wannier correlated subspace. The difference is moderate for LiCoO2 (0.09), but significantly larger for CoO2 (0.26). LiCoO2 has 0.08 (0.26) more $d$ electrons than CoO2 in the projector (Wannier) correlated subspace. These values are much smaller than the nominal value of unity, which is indicative of the strong $p$–$d$ rehybridization in this system Wolverton and Zunger (1998); Marianetti et al. (2004).

$N_{d}$ typically decreases with $U$. When treating LiCoO2 with DFT+$U$ in the Wannier correlated subspace, we find a transition to high-spin Co at $U$ of around 4 eV. Since Co in LiCoO2 is not high-spin in experiments van Elp et al. (1991); Ménétrier et al. (1999, 2008), we only consider the low-spin Co state to facilitate comparison with DFT+DMFT, for which we do not find this spurious high-spin state. For LiCoO2 in the Wannier correlated subspace (the band insulator state), the decrease in $N_{d}$ with $U$ for DFT+$U$ is around 0.25 electrons. The inclusion of dynamical correlations (DFT+DMFT) substantially dampens this decrease to a only 0.06 electrons. Unlike in the Wannier case, for DFT+$U$ in the projector correlated subspace, the decrease in $N_{d}$ is small in magnitude (around $0.03$ electrons) and including charge self-consistency leads to even smaller changes on the order of 0.006 electrons.

The behavior is similar for CoO2. Here, DFT+DMFT gives a small decrease in $N_{d}$ of 0.04 electrons, whereas the decrease is much more substantial (0.3 electrons) for DFT+$U$ in the Wannier correlated subspace, which describes CoO2 as an insulator except for the smallest $U$ considered. For DFT+$U$ in the projector case, in which there is a discontinuous decrease in $N_{d}$ corresponding to a metal-insulator transition, charge self-consistency dampens the change in $N_{d}$ from around 0.1 to 0.02 electrons.

The behavior of $N_{d}$ versus $U$ can further be understood by decomposing $N_{d}$ into the components from the $T_{2g}$ ($E_{g}^{\prime}$ and $A_{1g}$) and $E_{g}$ orbitals, as shown in Fig. 6. We do not show the individual $E_{g}^{\prime}$ and $A_{1g}$ occupancies for brevity, but they are included in the Supplemental Material 111See Supplemental Material for crystal structure details and a breakdown of $N_{d}$ into the $E_{g}^{\prime}$, $A_{1g}$, and $E_{g}$ occupancies.. Within DFT, the Wannier correlated subspace leads to higher (lower) occupancy of $T_{2g}$ ($E_{g}$) by 0.13–0.23 (0.32–0.39) electrons compared to the projector case. In DFT+$U$, the LiCoO2 $T_{2g}$ occupancy increases with $U$, whereas the $E_{g}$ occupancy decreases more rapidly; this leads to the overall decrease in $N_{d}$. For the Wannier case, the $T_{2g}$ occupancy increases more rapidly with $U$ at lower $U$ compared to the projector case; for larger $U$, the occupancy begins to saturate close to the nominal value of 6. Similarly, the decrease in $E_{g}$ occupancy is more substantial in the Wannier case compared to the projector case. Including charge self-consistency has a very small effect on the occupancies of LiCoO2 in the projector case. The trends in the occupancies are the same for DFT+DMFT as in DFT+$U$, but the magnitude of the changes in occupancy are much smaller.

Within DFT+$U$, the CoO2 $T_{2g}$ occupancy is relatively constant with $U$ in the Wannier correlated subspace, whereas the $E_{g}$ occupancy decreases by 0.22 electrons. The CoO2 $T_{2g}$ occupancy also does not change very much with $U$ in the projector case, though there is a discontinuity leading to a decrease in occupancy through the metal-insulator transition. In this case, charge self-consistency serves to slightly enhance the $T_{2g}$ and $E_{g}$ occupancies. As in the case of LiCoO2, the changes in occupancy within DFT+DMFT are smaller than those of DFT+$U$ with both $T_{2g}$ and $E_{g}$ occupancies slightly decreasing by 0.003 and 0.04, respectively, over the range of $U$. We note that the DFT+DMFT occupancies are fairly similar to those of DFT for both CoO2 and LiCoO2.

III.3 Electronic structure of Li1/2CoO2

Here, we discuss the electronic structure of Li1/2CoO2, which warrants extra attention due to the issue of spurious charge ordering which occurs in DFT+$U$ but is not seen experimentally Takahashi et al. (2007). This known structure has an in-plane ordering of Li corresponding to a primitive unit cell with two formula units Reimers and Dahn (1992); Van der Ven et al. (1998); Wolverton and Zunger (1998). We perform DFT+DMFT calculations in two different ways. First, we enforce the symmetry between the two structurally-equivalent Co sites, i.e., only a single impurity problem is solved. However, the aforementioned approach does not allow for charge ordering to spontaneously break point symmetry. Therefore, we also use a second approach where two impurity calculations are employed (i.e., one for each Co atom in the unit cell), and we only execute this at $U=3.9$ eV due to computational expense. Using the second approach, we do find a stable charge-ordered state, with a difference in $N_{d}$ on the two sites of 0.4 electrons, but it is slightly higher in energy than the non-charge-ordered state (by 3 meV/f.u.). Although this is a quite small energy difference, it it clear that DFT+DMFT removes the strong tendency for spurious charge ordering that is produced by DFT+$U$, and we proceed with our analysis of the symmetric solution.

Within DFT, Li1/2CoO2 is metallic with half a hole in the $T_{2g}$ manifold, as can be seen from the density of states in Fig. 1(b). For DFT+$U$ in the Wannier correlated subspace, like in the LiCoO2 case (discussed above), we ignore states containing high-spin Co found at large $U$; instead, we consider the low-spin ground state, a charge-ordered insulator (see Figs. 5(b), 6(b), and 6(e)). Within DFT+$U$ in the projector correlated subspace, Li1/2CoO2 is in a symmetric low-spin state at low $U$ that transitions to a low-spin charge ordered insulator at larger $U$. The DFT+DMFT self-energy in Fig. 2 illustrates that Li1/2CoO2 is, like CoO2, a Fermi liquid with correlations most significantly affecting the $T_{2g}$ manifold. We find a Fermi liquid up to the highest $U$ value considered, in agreement with the metallic behavior observed in experiment Motohashi et al. (2009); Miyoshi et al. (2010); Ou-Yang et al. (2012). The $T_{2g}$ quasiparticle weights, in Fig. 3 are significantly lower than those of LiCoO2 and slightly larger than those of CoO2. This reflects that the electronic structure of Li1/2CoO2 is closer to that of CoO2 than LiCoO2.

As for the endmembers ($x=0$ and $x=1$), for Li1/2CoO2, $N_{d}$ (Fig. 5) is larger in the projector correlated subspace. Within DFT+DMFT, $N_{d}$ decreases very mildly with $U$ similar to the endmember behavior. Here, dynamical correlations only slightly reduce $N_{d}$ (on the order of 0.02 electrons). Within NCSC DFT+$U$, the magnitude of the charge ordering is substantial (as much as $\sim 0.8$ electrons) in the Wannier correlated subspace. It is significantly smaller for the projector case, and is further dampened by charge self-consistency. Similar effects are found for the individual $T_{2g}$ and $E_{g}$ occupancies are shown in Fig. 6.

In summary, DFT+DMFT can properly describe the electronic structure of Li1/2CoO2. Unlike DFT+$U$, which predicts a charge ordered insulator, DFT+DMFT properly describes the electronic structure as a Fermi liquid without a strong tendency for charge ordering.

III.4 Total energy of LiCoO2 and CoO2

Having documented the basic electronic structure, we proceed to explore the total energy of LiCoO2 and CoO2 as a function of $U$, allowing for a direct comparison between DFT+DMFT and DFT+$U$. The total energy of LCO is shown as a function of $U$ for several methodologies in Fig. 7. The two vertical dashed lines indicate the values of $U$ for CoO2 and LiCoO2 as computed from first principles via linear response Zhou et al. (2004). The total energies increase with $U$, as expected. We note that extrapolating any of the results to $U=0$ does not approach the DFT result (large purple diamond), which is simply due to the fact that we used a fixed value of $J$ for all calculations. The magnitude of the increase in total energy with $U$ is generally greater for CoO2 than LiCoO2, which makes sense since the impact of the on-site interaction is expected to be larger for the system for which $T_{2g}$ is partially filled (nominally). For NCSC DFT+$U$ in the Wannier correlated subspace, for example, over the full range of $U$ shown the increase in energy of CoO2 is 2.4 eV as compared to only 1.8 eV for LiCoO2. For the same set of calculations using the projector correlated subspace, we find the same trend with energy increases of roughly 2.7 eV for CoO2 and 2.3 eV for LiCoO2. We note that the individual total energies from methods utilizing these different correlated subspaces (projector and Wannier) are not directly comparable, but the behavior of the total energy with $U$ is similar.

We estimate the magnitude of the error associated with neglecting charge self-consistency via DFT+$U$ calculations in the projector correlated subspace. Here, for LiCoO2, we find only very small differences (at most 9 meV/f.u.) between the NCSC and CSC total energies. For CoO2, the situation is similar with differences in total energy of at most 20 meV/f.u.. The small impact of changes in charge density on the total energies suggest the fixed charge density should be a reasonable approximation for DFT+DMFT. More importantly, we have a clear guideline on the magnitude of the effect for charge self-consistency within DFT+$U$, and we expect this is an upper bound for DFT+DMFT calculations in which the impact will likely be dampened.

Within DFT+DMFT, we find very little impact of including the off-diagonal $J$ interaction terms within the $E_{g}$ manifold in addition to the density-density interactions. The magnitude of the differences is typically only around 5–15 meV/f.u. for LiCoO2 and 3–9 meV/f.u. for CoO2. This suggests that density-density interactions are likely sufficient to describe this class of systems. In all of the following results, we find no significant difference in employing these two interaction forms. The DFT+DMFT results, which employ the Wannier correlated subspace, appear to merge with the corresponding DFT+$U$ results in the limit of small $U$, as should be the case. We note again that in this limit, neither the DFT+$U$ nor the DFT+DMFT results recover the DFT values (large purple diamonds in Fig. 7) simply because we take have taken a fixed finite $J$ value, whereas the DFT values correspond to $J=0$.

We find the general impact of dynamical correlations on the energetics is to dampen the magnitude of the increase in total energy with $U$ as compared to the static Hartree-Fock treatment in DFT+$U$. When $U$ is increased from 1.9 to 5.9 eV, the total energy of LiCoO2 increases by 1.8 eV within DFT+$U$ as opposed to only 1.3 eV within DFT+DMFT. For CoO2, the magnitude of these energies is substantially larger with an increase of 2.4 eV for DFT+$U$ and 1.6 eV for DFT+DMFT. By this measure, dynamical correlations decrease the energy penalty of $U$ by 26% for LiCoO2 and 33% for CoO2. Therefore, dynamical correlations have a larger impact on CoO2 than LiCoO2. This corresponds to very large absolute differences in the energies predicted by DFT+$U$ and DFT+DMFT. For CoO2, for example, around the linear response values of $U$ the difference in energy is around 1 eV. This strongly suggests dynamical correlations, missing in the DFT+$U$ approach, are important for accurate total energies.

It should be emphasized that the difference between DFT+DMFT and DFT+$U$ changes substantially as a function of $x$, and this error will therefore strongly affect observables. One strategy to correct errors within DFT+$U$ calculations is to tune $U$ to artificially low values, and this can be successful if $U$ is first calibrated to some experimental observable. However, our work indicates that the errors vary strongly with composition, and therefore DFT+$U$ studies of compositional phase stability would need to tune $U$ as a function of doping, a far more challenging task. Below we explore the average battery voltage, where the composition dependent errors within DFT+$U$ have severe consequences.

III.5 Average intercalation voltage

We turn our attention to the average intercalation voltage of LCO for $0\leq x\leq 1$, plotted in Fig. 8, which is a key observable for a rechargeable battery cathode. The average intercalation voltage $V$ is computed via $eV=E(\mathrm{Li})+E(\mathrm{CoO_{2}})-E(\mathrm{LiCoO_{2}}),$ where $e$ is the elementary charge and body-centered-cubic Li is the reference electrode Aydinol et al. (1997). As has been known, DFT tends to underpredict the experimental voltage Aydinol et al. (1997), in this case by around 0.7 V. The predicted voltage increases with DFT+$U$. In the projector correlated subspace, the value comes close to reasonable agreement with the experimental value (approximately 4.26 V Amatucci et al. (1996)) for the largest $U$ considered, and charge self-consistency has a negligible effect. The voltage also becomes larger than the DFT value for DFT+$U$ in the Wannier case, though the predicted values are appreciably smaller and do not reach agreement with experiment. In stark contrast, within DFT+DMFT, the predicted intercalation voltage is smaller than the DFT value up to the highest computed value of $U=6$ eV. As compared to DFT+$U$, the voltage at fixed $J$ increases much more slowly as a function of $U$. This mainly stems from the dampened increase in energy for CoO2. For the computed values of $U$, the predicted voltage is only 3.39–3.45 V, a moderate amount less than that of pure DFT.

The increase in $V$ within DFT+$U$ and general agreement with experiment was shown previously Zhou et al. (2004) and seemed to suggest that DFT+$U$ is reliable for this class of materials. However, this viewpoint should be carefully scrutinized given that DFT+$U$ is a rather crude theory, in that DFT+$U$ is obtained from DFT+DMFT when the quantum impurity problem is solved within static mean-field theory, neglecting dynamical correlations. Therefore, DFT+DMFT is superior in every respect. Since the voltage curve predicted by DFT+DMFT produces a result smaller than that of DFT, opposite to that of DFT+$U$, dynamical correlations are clearly essential to describe the energetics of LCO. The fact that DFT+$U$ increases the voltage, relative to DFT, and provides more reasonable agreement with experiment appears to be fortuitous.

Given that DFT+DMFT actually worsens the predicted voltage as compared to DFT, we are left with the puzzling question as to why. We explore several possibilities. First and foremost, while DMFT should improve DFT with respect to local physics, it is possible that there are still substantial nonlocal errors within the density functional employed in the DFT+DMFT functional. Given the recent successes of the relatively new SCAN functional Isaacs and Wolverton (2018); Zhang et al. (2018), which contains nonlocal physics via a dependence on the orbital kinetic energy density, an obvious question is how the voltage would change if we replaced the PBE functional with SCAN; we pursue this idea at the level of DFT+$U$ (see Fig. 8). The $U=0$ voltage predicted by SCAN is 4.58 V, which is already greater than the experimental voltage, in stark contrast to LDA and PBE. One expects the trends as a function of $U$ for DFT(SCAN)+$U$ and DFT(SCAN)+DMFT to be unchanged relative to PBE given that the SCAN electronic structure is very similar to that of PBE. As expected, increasing $U$ within DFT(SCAN)+$U$ causes the voltage to further increase, moving away from the experimental value. We do not compute the DFT(SCAN)+DMFT results due to the computational cost, but we anticipate that they should have the same $U$-dependence as DFT(PBE)+DMFT, just as DFT(SCAN)+$U$ and DFT(PBE)+$U$ have a very similar $U$-dependence. If so, the DFT(SCAN)+DMFT voltage should be mildly decreased compared to the DFT(SCAN) voltage, yielding reasonable agreement with experiment. Alternatively, DFT(SCAN)+$U$ only worsens the voltage prediction, and this suggests the reason why DFT(PBE)+$U$ performs well is due to the cancellations of two large and distinct errors. Therefore, our results suggest one cannot expect DFT+$U$ to perform as a predictive tool in the context of compositional phase stability.

Of course, another factor to consider is that the precise value of the predicted voltage will depend on the precise values $U$ and $J$. Allowing for small $U$ differences in the endmembers will result in small changes in the voltage. Similiar arguments might be made for other uncertainties in the methodology, such as the double counting correction, etc. We do not attempt to build a case for the most correct set of parameters in this work. It is worth noting that one early explanation for discrepancies in the DFT predicted voltages was the overestimated magnitude of the cohesive energy of body-centered-cubic Li within LDA Aydinol et al. (1997). However, we find the cohesive energy predicted by PBE ($-1.60$ eV) and SCAN ($-1.59$ eV) are in good agreement with experiment ($-1.63$ eV Kittel (1986)). Therefore, the discrepancy between theory and experiment should stem from the energetics of the cathode material itself.

III.6 Phase stability of Li1/2CoO2

As another test of the computed DFT+DMFT total energy, we compute the phase stability of Li1/2CoO2. The formation energy $\Delta E$, computed as $E(\mathrm{Li_{1/2}CoO_{2}})-\frac{1}{2}[E(\mathrm{CoO_{2}})+E(\mathrm{LiCoO_{2}})]$, is shown in Fig. 9. We use the difference in average experimental voltage values for $0<x<1/2$ ($V_{-}$) and $1/2<x<1$ ($V_{+}$) to estimate the experimental formation energy for $x=1/2$, via $\Delta E=x(1-x)(eV_{+}-eV_{-})$ Aykol and Wolverton (2014). Using the data of Ref. 74, we compute $\Delta E$ of $-114$ meV/f.u. for Li1/2CoO2. As the formation energy has a significantly smaller scale than the voltage, it is more sensitive to the different methodologies employed in this work. Furthermore, the energetics of NCSC calculations will be less reliable when there is a substantial rearrangement of charge.

We find a DFT value of $-218$ meV/f.u. NCSC DFT+$U$ in the projector correlated subspace significantly decreases the magnitude of the formation energy and nearly reaches agreement with experiment at large $U$. Charge self-consistency tends to destabilize Li1/2CoO2 compared to the endmembers, further lowering the formation energy magnitude to values much smaller than experiment. While the formation energy remains negative (in agreement with the experimentally-known phase stability), we have shown previously that the more common DFT+$U$ methodology based on spin-dependent DFT incorrectly predicts phase separation of Li1/2CoO2 in the absence of charge ordering Isaacs and Marianetti (2017). The formation energy is also significantly affected by $U$ using DFT+$U$ in the Wannier correlated subspace, but in the opposite direction. In this case, the formation energy is pushed more negative to values of 260–270 meV/f.u.

Given that DFT+DMFT prefers not to charge order, the errors associated with NCSC are expected to be comparable to those estimated for the endmembers, which would still be small on this scale. DFT+DMFT shows only a small increase in formation energy with $U$, on the order of tens of meV/f.u. The predicted value in the range of computed $U$ is similar to that of DFT. The DFT+DMFT error may also be associated in part with the DFT exchange-correlation functional. We also plot the formation energy using the SCAN functional, which is approximately 50 meV/f.u. higher than the PBE value. Therefore, we would anticipate the DFT(SCAN)+DMFT results to be shifted by roughly the same amount, which would be much closer to the experimental range. In summary, similar to the voltage behavior, our results suggest that dynamical correlations significantly impact the predicted formation energy and that DFT+DMFT based on SCAN will lead to improved agreement with experiment.

IV Conclusions

We investigate the electronic structure, intercalation voltage, and phase stability of LCO with the many-body DFT+DMFT methodology and compare to DFT and DFT+$U$. In DFT+DMFT, LiCoO2 is a band insulator, while we find that CoO2 and Li1/2CoO2 are moderately correlated Fermi liquids, without a strong tendency for charge ordering in Li1/2CoO2, in agreement with experiments. Dynamical correlations (missing in DFT+$U$) substantially impact the energetics of LCO by dampening the changes in total energy and $N_{d}$ found via the DFT+$U$ approach, especially for CoO2 and Li1/2CoO2. The intercalation voltage behavior of DFT+$U$ and DFT+DMFT are qualitatively different, with the latter decreasing the voltage with respect to DFT; the phase stability of Li1/2CoO2 within DFT+DMFT also differs starkly from DFT+$U$. DFT+DMFT calculations based on the PBE GGA DFT functional underpredict the voltage and overestimate the stability of Li1/2CoO2 compared to experiment; we find evidence that DFT+DMFT based on the more accurate SCAN DFT functional will lead to significantly closer agreement to experiment.

We find that dynamical correlations are important to describe this class of materials despite the relatively modest quasiparticle weights. Our results suggest that the Hartree-Fock treatment of the impurity problem in DFT+$U$ is insufficient to accurately describe the electronic structure and thermodynamics of correlated electron materials. In addition, due to the strong composition dependence of the impact of dynamical correlations, our results challenge the common practice of artificially tuning $U$ within DFT+$U$ to compensate for the errors of Hartree-Fock. Given the significant computational expense of solving the impurity problem in DFT+DMFT, the development of less computationally expensive but still sufficiently accurate impurity solvers will be important future work to enable the study of compositional phase stability of strongly correlated electron materials.

V Acknowledgments

We acknowledge Hyowon Park (UIC/Argonne) for useful discussions. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. E.B.I. gratefully acknowledges the U.S. Department of Energy Computational Science Graduate Fellowship (Grant No. DE-FG02-97ER25308) for support. C.A.M. was supported by the grant DE-SC0016507 funded by the U.S. Department of Energy, Office of Science.