Structured sparsity for building predictive models in chemically complex systems

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Electric vehicles are gaining ground worldwide. According to the International Energy Agency, in 2021 alone, more than six million electric vehicles (EVs) were sold globally, up from 120,000 EVs back in 2012 [1]. However, this rapid increase in EV sales has been seriously tested by surges in price of essential battery electrode components, namely cobalt and nickel, which are highly limited resources. Furthermore, they are categorized as critical minerals for their uneven geological distribution around the world and subsequent vulnerability to supply chain disruption.

 Since Mn is far more earth-abundant, high-energy density manganese (Mn)-based lithium (Li)-ion electrodes are sustainable and inexpensive alternatives for electrochemical energy storage. In fact, a new class of partially disordered Mn-spinel (PDS) electrode materials has been experimentally reported to be superior in specific energy and in specific power compared to both state-of-the-art layered nickel-rich cathodes (e.g. NMC811) and other, emerging high-energy density cathodes [2].  

 The PDS material is a clear departure from normal spinel, a well-studied Li intercalation material pioneered by Thackeray, David, Bruce, and Goodenough in 1982 [3]. The need to understand PDS at the atomic level motivates using a lattice cluster expansion (CE)[4] model. A CE model  can  exhaustively describe atomic occupancies and resulting properties in various states of order, i.e. from ordered spinel to the fully disordered configuration.  

 PDS contains complex physics because Li and Mn cations can occupy one of three cation sites (one octahedral and two tetrahedral) per oxygen or fluorine, as shown in Figure 1. Furthermore, the Mn ion can behave quite differently depending on its divalent, trivalent, or tetravalent nature. For this reason, we need to differentiate among these three types of Mn species in our model. As a result, the CE lattice model for PDS considers quinary disorder on octahedral sites (with species: Li+, Mn2+, Mn3+, Mn4+, and Vacancy), ternary disorder on tetrahedral sites (with species: Li+, Mn2+, and Vacancy), and binary disorder (with species: oxygen and fluorine) on anion sites. To our knowledge, this is the largest, multicomponent, coupled-cluster expansion [5] ever constructed; and doing so has required new practical advances to address several technical difficulties. For instance, we apply black-box optimization to optimally assign transition metal charge states and new methods for structural mapping to increase data efficiency.  

Figure 1. Complexity of the partially disordered Li-Mn-O-F spinel with respect to the conventional spinel unit cell [7]. The Z=3/8 plane as an example is shown. In partially disordered spinel, experimental refinement indicates partial disorder on normally occupied Li sites (the 8a sites shown in green, hatched or filled), Mn sites (the 16d sites in purple, hatched or filled), the normally vacant 16c sites (filled yellow), which are now partially filled with Mn (hatched yellow), and the normally vacant 48f or 8b sites (hatched blue) which can now also host Li+ or Mn2+. Anion disorder (92.5% O) is also present. The primitive cell of the lattice model is outlined in red. The configuration space of the primitive cell is 90.

We also discover that our ionic CE model, based on a cubic close-packed (ccp) anion lattice, suffers from configurational under-sampling due to proximity of the cation sites. This challenge can be addressed by applying the sparse group lasso regularization approach to reduce overfitting, by  grouping fully decorated sets of basis functions that act over the same symmetrically equivalent clusters. Furthermore, we find that regularization that results in structured-sparsity [6], compared to non-structured sparsity, results in greater predictability  measured by a lower out-of-sample root-mean-squared error. Lastly, we suggest, through examples of lower-dimensional fits of our data, that predictability of multicomponent models should scale with chemical complexity.

 Models to study the configurational thermodynamics of bulk systems can in principle be broadened to any degree of chemical complexity. In practice, insights into how to build these complex models should also be openly discussed and developed. Our findings and discussions may help future modeling and design of disordered ionic systems for next-generation battery materials and beyond.

 For more details on this work, please see:


  1. IEA (2022), Global EV Outlook 2022, IEA, Paris
  2. Ji, H., Wu, J., Cai, Z., Liu, J., Kwon, D.-H., Kim, H., Urban, A., Papp, J. K., Foley, E., Tian, Y., Balasubramanian, M., Kim, H., Clément, R. J., McCloskey, B. D., Yang, W., and Ceder, G., "Ultrahigh power and energy density in partially ordered lithium-ion cathode materials." Nature Energy, 5(3), 213–221 (2020).
  3. Thackeray, M. M. , David, W. I. F. , Bruce, P. G., and Goodenough, J. B., “Lithium insertion into manganese spinels.” Mater. Res. Bull., 18, 461 (1983). 
  4. Sanchez, J. M., Ducastelle, F., and Gratias, D., "Generalized cluster description of multicomponent systems." Physica A: Statistical Mechanics and Its Applications, 128(1), 334–350, (1984).
  5. Tepesch, P. D., Garbulsky, G. D., and Ceder, G., "Model for Configurational Thermodynamics in Ionic Systems. "Phys. Rev. Lett., 74(12), 2272–2275, (1995).
  6. Yang, J.H., Chen, T., Barroso-Luque, L., Jadidi, Z., and Ceder, G., "Approaches for handling high-dimensional cluster expansions of ionic systems." npj Comput Mater 8, 133 (2022).
  7. Sickafus, K. E., Wills, J. M., and Grimes, N. W., "Structure of Spinel." Journal of the American Ceramic Society, 82(12), 3279–3292, (1999).

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