Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Modeling the electrical double layer at solid-state electrochemical interfaces

Nature Computational Science volume 1, pages 212–220 (2021)Cite this article

Subjects

Abstract

Models of the electrical double layer (EDL) at electrode/liquid-electrolyte interfaces no longer hold for all-solid-state electrochemistry. Here we show a more general model for the EDL at a solid-state electrochemical interface based on the Poisson–Fermi–Dirac equation. By combining this model with density functional theory predictions, the interconnected electronic and ionic degrees of freedom in all-solid-state batteries, including the electronic band bending and defect concentration variation in the space-charge layer, are captured self-consistently. Along with a general mathematical solution, the EDL structure is presented in various materials that are thermodynamically stable in contact with a lithium metal anode: the solid electrolyte Li7La3Zr2O12 (LLZO) and the solid interlayer materials LiF, Li2O and Li2CO3. The model further allows design of the optimum interlayer thicknesses to minimize the electrostatic barrier for lithium ion transport at relevant solid-state battery interfaces.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schematic comparing solid–liquid and solid–solid electrochemical interfaces.
Fig. 2: General solution for a Li/LiF interface.
Fig. 3: Absolute band alignments of SSB materials, calculated using the vacuum slab model.
Fig. 4: Potential profile and space-charge density at the LLZO interface.
Fig. 5: Potential profile showing the EDL at the Li/LiPON SSB interface with a variety of interlayers.

Similar content being viewed by others

Data availability

The first-principles computational results on charged point defects and band alignments that were used to find the input parameters for this model are available in the NOMAD repository71 at https://doi.org/10.17172/NOMAD/2021.02.12-1.

Code availability

The python script PFD_solution.py constructs the analytic approximations ϕ1 and ϕ2. It may be used to reproduce the results in Tables 1 and 2, or to extend the method to another material. PFD_solution.py is available at72 https://github.com/mwswift/PFD_solution and https://doi.org/10.5281/zenodo.4538867. The Mathematica notebooks used to find the numerical solutions and generate the space-charge layer profiles and potential profiles are also available in this repository, as well as in the Wolfram cloud at https://www.wolframcloud.com/obj/swift/Published/PFD_Interlayers.nb and https://www.wolframcloud.com/obj/swift/Published/PFD_LLZO.nb.

References

  1. Janek, J. & Zeier, W. G. A solid future for battery development. Nat. Energy 1, 16141 (2016).

    Article  Google Scholar 

  2. Manthiram, A., Yu, X. & Wang, S. Lithium battery chemistries enabled by solid-state electrolytes. Nat. Rev. Mater. 2, 16103 (2017).

    Article  Google Scholar 

  3. Randau, S. et al. Benchmarking the performance of all-solid-state lithium batteries. Nat. Energy 5, 259–270 (2020).

    Article  Google Scholar 

  4. Bard, A. & Faulkner, L. Electrochemical Methods: Fundamentals and Applications 2nd edn (Wiley, 2000).

  5. Helmholtz, H. Ueber einige gesetze der vertheilung elektrischer ströme in körperlichen leitern mit anwendung auf die thierisch-elektrischen versuche. Ann. Phys. 165, 211–233 (1853).

    Article  Google Scholar 

  6. Stern, O. Zur theorie der elektrolytischen doppelschicht. Z. Elektrochem. Angew. Physik. Chem. 30, 508–516 (1924).

    Google Scholar 

  7. Schmickler, W. Interfacial Electrochemistry (Oxford Univ. Press, 1996).

  8. Nakamura, M., Sato, N., Hoshi, N. & Sakata, O. Outer Helmholtz plane of the electrical double layer formed at the solid electrode–liquid interface. ChemPhysChem 12, 1430–1434 (2011).

    Article  Google Scholar 

  9. Otani, M. & Sugino, O. First-principles calculations of charged surfaces and interfaces: a plane-wave nonrepeated slab approach. Phys. Rev. B 73, 115407 (2006).

    Article  Google Scholar 

  10. Jinnouchi, R. & Anderson, A. B. Electronic structure calculations of liquid-solid interfaces: combination of density functional theory and modified Poisson–Boltzmann theory. Phys. Rev. B 77, 245417 (2008).

    Article  Google Scholar 

  11. Nattino, F., Truscott, M., Marzari, N. & Andreussi, O. Continuum models of the electrochemical diffuse layer in electronic-structure calculations. J. Chem. Phys. 150, 041722 (2019).

    Article  Google Scholar 

  12. Swift, M. W. & Qi, Y. First-principles prediction of potentials and space-charge layers in all-solid-state batteries. Phys. Rev. Lett. 122, 167701 (2019).

    Article  Google Scholar 

  13. Tateyama, Y., Gao, B., Jalem, R. & Haruyama, J. Theoretical picture of positive electrode–solid electrolyte interface in all-solid-state battery from electrochemistry and semiconductor physics viewpoints. Curr. Opin. Electrochem. 17, 149–157 (2019).

    Article  Google Scholar 

  14. de Klerk, N. J. J. & Wagemaker, M. Space-charge layers in all-solid-state batteries; important or negligible? ACS Appl. Energy Mater. 1, 5609–5618 (2018).

    Google Scholar 

  15. Fingerle, M., Buchheit, R., Sicolo, S., Albe, K. & Hausbrand, R. Reaction and space charge layer formation at the LiCoO2–LiPON interface: insights on defect formation and ion energy level alignment by a combined surface science-simulation approach. Chem. Mater 29, 7675–7685 (2017).

    Article  Google Scholar 

  16. Braun, S., Yada, C. & Latz, A. Thermodynamically consistent model for space-charge-layer formation in a solid electrolyte. J. Phys. Chem. C 119, 22281–22288 (2015).

    Article  Google Scholar 

  17. Haruyama, J., Sodeyama, K., Han, L., Takada, K. & Tateyama, Y. Space-charge layer effect at interface between oxide cathode and sulfide electrolyte in all-solid-state lithium-ion battery. Chem. Mater 26, 4248–4255 (2014).

    Article  Google Scholar 

  18. Marcicki, J., Conlisk, A. & Rizzoni, G. A lithium-ion battery model including electrical double layer effects. J. Power Sources 251, 157–169 (2014).

    Article  Google Scholar 

  19. Kasamatsu, S., Tada, T. & Watanabe, S. Parallel-sheets model analysis of space charge layer formation at metal/ionic conductor interfaces. Solid State Ion. 226, 62–70 (2012).

    Article  Google Scholar 

  20. Morgan, B. J. & Madden, P. A. Effects of lattice polarity on interfacial space charges and defect disorder in ionically conducting AgI heterostructures. Phys. Rev. Lett. 107, 206102 (2011).

    Article  Google Scholar 

  21. Maier, J. Ionic conduction in space charge regions. Prog. Solid State Chem. 23, 171–263 (1995).

    Article  Google Scholar 

  22. Aizawa, Y. et al. In situ electron holography of electric potentials inside a solid-state electrolyte: effect of electric-field leakage. Ultramicroscopy 178, 20–26 (2017).

    Article  Google Scholar 

  23. Gittleson, F. S. & El Gabaly, F. Non-Faradaic Li+ migration and chemical coordination across solid-state battery interfaces. Nano Lett. 17, 6974–6982 (2017).

    Article  Google Scholar 

  24. Masuda, H., Ishida, N., Ogata, Y., Ito, D. & Fujita, D. Internal potential mapping of charged solid-state-lithium ion batteries using in situ kelvin probe force microscopy. Nanoscale 9, 893–898 (2017).

    Article  Google Scholar 

  25. Luntz, A. C., Voss, J. & Reuter, K. Interfacial challenges in solid-state Li ion batteries. J. Phys. Chem. Lett. 6, 4599–4604 (2015).

    Article  Google Scholar 

  26. Haruta, M. et al. Negligible ‘negative space-charge layer effects’ at oxide-electrolyte/electrode interfaces of thin-film batteries. Nano Lett. 15, 1498–1502 (2015).

    Article  Google Scholar 

  27. Xiao, Y. et al. Understanding interface stability in solid-state batteries. Nat. Rev. Mater 5, 105–126 (2020).

    Article  Google Scholar 

  28. Zhao, Q., Stalin, S., Zhao, C.-Z. & Archer, L. A. Designing solid-state electrolytes for safe, energy-dense batteries. Nat. Rev. Mater 5, 229–252 (2020).

    Article  Google Scholar 

  29. Tan, D. H. S., Banerjee, A., Chen, Z. & Meng, Y. S. From nanoscale interface characterization to sustainable energy storage using all-solid-state batteries. Nat. Nanotechnol. 15, 170–180 (2020).

    Article  Google Scholar 

  30. Early, J. M. Effects of space-charge layer widening in junction transistors. Proc. IRE 40, 1401–1406 (1952).

    Article  Google Scholar 

  31. van Heek, H. Hall mobility of electrons in the space-charge layer of thin film CdSe transistors. Solid State Electron. 11, 459–462 (1968).

    Article  Google Scholar 

  32. Belisle, R. A. et al. Interpretation of inverted photocurrent transients in organic lead halide perovskite solar cells: proof of the field screening by mobile ions and determination of the space charge layer widths. Energy Environ. Sci. 10, 192–204 (2017).

    Article  Google Scholar 

  33. Vollman, M. & Waser, R. Grain boundary defect chemistry of acceptor-doped titanates: space charge layer width. J. Am. Ceram. Soc. 77, 235–243 (1994).

    Article  Google Scholar 

  34. De Souza, R. A. The formation of equilibrium space-charge zones at grain boundaries in the perovskite oxide SrTiO3. Phys. Chem. Chem. Phys. 11, 9939–9969 (2009).

    Article  Google Scholar 

  35. Lupetin, P., Gregori, G. & Maier, J. Mesoscopic charge carriers chemistry in nanocrystalline SrTiO3. Angew. Chem. Int. Ed. 49, 10123–10126 (2010).

    Article  Google Scholar 

  36. Freysoldt, C. et al. First-principles calculations for point defects in solids. Rev. Mod. Phys. 86, 253–305 (2014).

    Article  Google Scholar 

  37. Yang, J., Youssef, M. & Yildiz, B. Predicting point defect equilibria across oxide hetero-interfaces: model system of ZrO2/Cr2O3. Phys. Chem. Chem. Phys. 19, 3869–3883 (2017).

    Article  Google Scholar 

  38. Bowes, P. C., Wu, Y., Baker, J. N., Harris, J. S. & Irving, D. L. Space charge control of point defect spin states in AlN. Appl. Phys. Lett 115, 052101 (2019).

    Article  Google Scholar 

  39. Pan, J., Zhang, Q., Xiao, X., Cheng, Y.-T. & Qi, Y. Design of nanostructured heterogeneous solid ionic coatings through a multiscale defect model. ACS Appl. Mater. Interfaces 8, 5687–5693 (2016).

    Article  Google Scholar 

  40. Grundmann, M. in The Physics of Semiconductors: An Introduction Including Nanophysics and Applications 525–527 (Springer, 2010).

  41. Johnson, R. A. Use of Fermi–Dirac statistics for defects in solids. Phys. Rev. B 24, 7383–7384 (1981).

    Article  Google Scholar 

  42. Das, T., Nicholas, J. D. & Qi, Y. Long-range charge transfer and oxygen vacancy interactions in strontium ferrite. J. Mater. Chem. A 5, 4493–4506 (2017).

    Article  Google Scholar 

  43. Sallaz, V., Oukassi, S., Voiron, F., Salot, R. & Berardan, D. Assessing the potential of LiPON-based electrical double layer microsupercapacitors for on-chip power storage. J. Power Sources 451, 227786 (2020).

    Article  Google Scholar 

  44. Zhu, Y., He, X. & Mo, Y. Origin of outstanding stability in the lithium solid electrolyte materials: insights from thermodynamic analyses based on first-principles calculations. ACS Appl. Mater. Interfaces 7, 23685–23693 (2015).

    Article  Google Scholar 

  45. Luo, J. Let thermodynamics do the interfacial engineering of batteries and solid electrolytes. Energy Storage Mater. 21, 50–60 (2019).

    Article  Google Scholar 

  46. Zhang, L., Zhang, K., Shi, Z. & Zhang, S. LiF as an artificial SEI layer to enhance the high-temperature cycle performance of Li4Ti5O12. Langmuir 33, 11164–11169 (2017).

    Article  Google Scholar 

  47. Wang, A., Kadam, S., Li, H., Shi, S. & Qi, Y. Review on modeling of the anode solid electrolyte interphase (SEI) for lithium-ion batteries. npj Comput. Mater. 4, 15 (2018).

    Article  Google Scholar 

  48. Jurng, S., Brown, Z. L., Kim, J. & Lucht, B. L. Effect of electrolyte on the nanostructure of the solid electrolyte interphase (SEI) and performance of lithium metal anodes. Energy Environ. Sci. 11, 2600–2608 (2018).

    Article  Google Scholar 

  49. Bhattacharya, S., Riahi, A. R. & Alpas, A. T. Electrochemical cycling behaviour of lithium carbonate (Li2CO3) pre-treated graphite anodes—SEI formation and graphite damage mechanisms. Carbon 77, 99–112 (2014).

    Article  Google Scholar 

  50. Eijima, S. et al. Solid electrolyte interphase film on lithium metal anode in mixed-salt system. J. Electrochem. Soc 166, A5421–A5429 (2019).

    Article  Google Scholar 

  51. Wang, M. et al. Tailoring lithium deposition via an SEI-functionalized membrane derived from LiF decorated layered carbon structure. Adv. Energy Mater. 9, 1802912 (2019).

    Article  Google Scholar 

  52. Murugan, R., Thangadurai, V. & Weppner, W. Fast lithium ion conduction in garnet-type Li7La3Zr2O12. Angew. Chem. Int. Ed. 46, 7778–7781 (2007).

    Article  Google Scholar 

  53. Wang, Y. et al. Design principles for solid-state lithium superionic conductors. Nat. Mater. 14, 1026–1031 (2015).

    Article  Google Scholar 

  54. Xie, H., Alonso, J. A., Li, Y., Fernández-Díaz, M. T. & Goodenough, J. B. Lithium distribution in aluminum-free cubic Li7La3Zr2O12. Chem. Mater. 23, 3587–3589 (2011).

    Article  Google Scholar 

  55. Cussen, E. J. The structure of lithium garnets: cation disorder and clustering in a new family of fast Li+ conductors. Chem. Commun. 4, 412–413 (2006).

    Article  Google Scholar 

  56. O’Callaghan, M. P. & Cussen, E. J. Lithium dimer formation in the Li-conducting garnets Li5+xBaxLa3xTa2O12(0 < x ≤ 1.6). Chem. Commun. 20, 2048–2050 (2007).

  57. Tian, H.-K., Xu, B. & Qi, Y. Computational study of lithium nucleation tendency in Li7La3Zr2O12 (LLZO) and rational design of interlayer materials to prevent lithium dendrites. J. Power Sources 392, 79–86 (2018).

    Article  Google Scholar 

  58. Pinson, M. B. & Bazant, M. Z. Theory of SEI formation in rechargeable batteries: capacity fade, accelerated aging and lifetime prediction. J. Electrochem. Soc. 160, A243 (2013).

    Article  Google Scholar 

  59. Huggins, R. Advanced Batteries: Materials Science Aspects (Springer, 2008).

  60. Pan, J., Cheng, Y.-T. & Qi, Y. General method to predict voltage-dependent ionic conduction in a solid electrolyte coating on electrodes. Phys. Rev. B 91, 134116 (2015).

    Article  Google Scholar 

  61. Ong, S. P. et al. Python materials genomics (pymatgen): a robust, open-source python library for materials analysis. Comput. Mater. Sci. 68, 314–319 (2013).

    Article  Google Scholar 

  62. Broberg, D. et al. Pycdt: a python toolkit for modeling point defects in semiconductors and insulators. Comput. Phys. Commun. 226, 165–179 (2018).

    Article  Google Scholar 

  63. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    Article  Google Scholar 

  64. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  Google Scholar 

  65. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

    Article  Google Scholar 

  66. Jain, A. et al. A high-throughput infrastructure for density functional theory calculations. Comput. Mater. Sci. 50, 2295 – 2310 (2011).

    Article  Google Scholar 

  67. Jain, A. et al. Commentary: the materials project: a materials genome approach to accelerating materials innovation. APL Mater. 1, 011002 (2013).

    Article  Google Scholar 

  68. Ong, S. P. et al. The materials application programming interface (API): a simple, flexible and efficient API for materials data based on representational state transfer (rest) principles. Comput. Mater. Sci. 97, 209–215 (2015).

    Article  Google Scholar 

  69. Freysoldt, C., Neugebauer, J. & Van de Walle, C. G. Fully ab initio finite-size corrections for charged-defect supercell calculations. Phys. Rev. Lett. 102, 016402 (2009).

    Article  Google Scholar 

  70. Freysoldt, C., Neugebauer, J. & Van de Walle, C. G. Electrostatic interactions between charged defects in supercells. Phys. Status Solidi B 248, 1067–1076 (2011).

    Article  Google Scholar 

  71. Swift, M. W., Swift, J. W. & Qi, Y. Modeling the electrical double layer at solid-state electrochemical interfaces. NOMAD https://doi.org/10.17172/NOMAD/2021.02.12-1 (2021).

  72. Swift, M. W., Swift, J. W. & Qi, Y. Poisson–Fermi–Dirac solution v1.0. Zenodo https://doi.org/10.5281/zenodo.4538867 (2021).

Download references

Acknowledgements

This work was mainly supported by the Nanostructures for Electrical Energy Storage (NEES) centre, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under award number DESC0001160. Y.Q. also acknowledge the support from the Assistant Secretary for Energy Efficiency and Renewable Energy, Vehicle Technologies Office of the US Department of Energy under contract no. award number DE-EE0008863 under the Battery Material Research (BMR) Program.

Author information

Authors and Affiliations

  1. Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI, USA

    Michael W. Swift & Yue Qi

  2. Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ, USA

    James W. Swift

  3. School of Engineering, Brown University, Providence, RI, USA

    Yue Qi

Authors
  1. Michael W. Swift
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James W. Swift
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yue Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization, M.W.S. and Y.Q.; methodology, M.W.S., J.W.S. and Y.Q.; software, M.W.S. and J.W.S.; investigation, M.W.S., J.W.S. and Y.Q.; writing—original draft, M.W.S. and Y.Q.; writing—review and editing, M.W.S., J.W.S. and Y.Q.; supervision, Y.Q.; funding acquisition, Y.Q.

Corresponding authors

Correspondence to Michael W. Swift, James W. Swift or Yue Qi.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Computational Science thanks the anonymous reviewers for their contribution to the peer review of this work. Fernando Chirigati was the primary editor on this article and managed its editorial process and peer review in collaboration with the rest of the editorial team.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–3 and Table 1.

Source data

Source Data Fig. 2

Numerical solution data.

Source Data Fig. 3

Density functional theory calculated data.

Source Data Fig. 4

Numerical solution data.

Source Data Fig. 5

Numerical solution data.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Swift, M.W., Swift, J.W. & Qi, Y. Modeling the electrical double layer at solid-state electrochemical interfaces. Nat Comput Sci 1, 212–220 (2021). https://doi.org/10.1038/s43588-021-00041-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s43588-021-00041-y

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing