Modeling the electrical double layer at solid-state electrochemical interfaces

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. Access through your institution Buy or subscribe This is a preview of subscription content, access via your

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institutional subscriptions * Read our FAQs * Contact customer support SIMILAR CONTENT BEING VIEWED BY OTHERS NON-LOCAL INTERACTIONS DETERMINE LOCAL STRUCTURE AND LITHIUM DIFFUSION IN SOLID

ELECTROLYTES Article Open access 15 February 2025 MOLECULAR-SCALE INSIGHTS INTO THE ELECTRICAL DOUBLE LAYER AT OXIDE-ELECTROLYTE INTERFACES Article Open access 26 November 2024 UNLOCKING THE

SECRETS OF IDEAL FAST ION CONDUCTORS FOR ALL-SOLID-STATE BATTERIES Article Open access 19 July 2024 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 _ϕ_1 and _ϕ_2. 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 * Janek, J. & Zeier, W. G. A solid future

for battery development. _Nat. Energy_ 1, 16141 (2016). Article  Google Scholar  * Manthiram, A., Yu, X. & Wang, S. Lithium battery chemistries enabled by solid-state electrolytes.

_Nat. Rev. Mater._ 2, 16103 (2017). Article  Google Scholar  * Randau, S. et al. Benchmarking the performance of all-solid-state lithium batteries. _Nat. Energy_ 5, 259–270 (2020). Article 

Google Scholar  * Bard, A. & Faulkner, L. _Electrochemical Methods: Fundamentals and Applications_ 2nd edn (Wiley, 2000). * 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  * Stern, O. Zur theorie der

elektrolytischen doppelschicht. _Z. Elektrochem. Angew. Physik. Chem._ 30, 508–516 (1924). Google Scholar  * Schmickler, W. _Interfacial Electrochemistry_ (Oxford Univ. Press, 1996). *

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  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * Maier, J. Ionic conduction in space charge regions. _Prog. Solid State Chem._ 23, 171–263 (1995). Article

  Google Scholar  * 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  * 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  * 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  * 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  * 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  * Xiao, Y. et al. Understanding interface stability in solid-state batteries. _Nat. Rev. Mater_ 5, 105–126 (2020).

Article  Google Scholar  * 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  * 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  * Early, J. M. Effects of space-charge layer widening in junction transistors. _Proc. IRE_ 40, 1401–1406 (1952). Article 

Google Scholar  * 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  *

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  * 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  * 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  * Lupetin, P., Gregori, G. & Maier, J. Mesoscopic charge carriers chemistry in

nanocrystalline SrTiO3. _Angew. Chem. Int. Ed._ 49, 10123–10126 (2010). Article  Google Scholar  * Freysoldt, C. et al. First-principles calculations for point defects in solids. _Rev. Mod.

Phys._ 86, 253–305 (2014). Article  Google Scholar  * 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  * 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  * 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  * Grundmann, M. in _The Physics of Semiconductors: An Introduction

Including Nanophysics and Applications_ 525–527 (Springer, 2010). * Johnson, R. A. Use of Fermi–Dirac statistics for defects in solids. _Phys. Rev. B_ 24, 7383–7384 (1981). Article  Google

Scholar  * 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  * 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  * 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  * Luo, J. Let thermodynamics do the interfacial

engineering of batteries and solid electrolytes. _Energy Storage Mater._ 21, 50–60 (2019). Article  Google Scholar  * 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  * 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  * 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  * 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  * 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  * 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  * 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  * Wang, Y. et al. Design principles for solid-state lithium superionic conductors. _Nat. Mater._ 14, 1026–1031 (2015).

Article  Google Scholar  * 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  * 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  * 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). * 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  * 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  * Huggins, R. _Advanced Batteries: Materials Science Aspects_ (Springer, 2008). * 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

  * 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  *

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  * 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  *

Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. _Phys. Rev. Lett._ 77, 3865–3868 (1996). Article  Google Scholar  * Blöchl, P. E. Projector

augmented-wave method. _Phys. Rev. B_ 50, 17953–17979 (1994). Article  Google Scholar  * Jain, A. et al. A high-throughput infrastructure for density functional theory calculations. _Comput.

Mater. Sci._ 50, 2295 – 2310 (2011). Article  Google Scholar  * Jain, A. et al. Commentary: the materials project: a materials genome approach to accelerating materials innovation. _APL

Mater._ 1, 011002 (2013). Article  Google Scholar  * 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  * 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  * 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  * 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). * 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 * Department of Chemical Engineering and Materials Science, Michigan State

University, East Lansing, MI, USA Michael W. Swift & Yue Qi * Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ, USA James W. Swift * School of

Engineering, Brown University, Providence, RI, USA Yue Qi Authors * Michael W. Swift View author publications You can also search for this author inPubMed Google Scholar * James W. Swift

View author publications You can also search for this author inPubMed Google Scholar * Yue Qi View author publications You can also search for this author inPubMed 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 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

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