Microscopic attribution of the higher-frequency semicircle in electron-electron impedance spectra of all-solid-state electrode mixture layers

All-solid-state lithium batteries use dense composite ('mixture layer') positive electrodes in which ionic transport through a solid electrolyte network and electronic transport through a cathode-active-material network proceed in parallel, and their effective conductivities are commonly separated by electrochemical impedance spectroscopy analyzed with transmission-line models (TLMs) on blocking-electrode cells. The 2016 study of Siroma et al. introduced an ion-blocking 'electron-electron' cell configuration whose Nyquist intercepts yield both the effective ionic and electronic conductivities even when the electronic resistance is not negligible, but left open the microscopic origin of the higher-frequency semicircle observed in these spectra: the feature admits at least two distinct microscopic interpretations (electronic-path heterogeneity, such as particle-particle contact capacitance, versus ionic-path structure, such as grain-boundary ionic resistance) that the equivalent-circuit fits cannot discriminate. The same TLM-with-constant-phase-element analysis idealizes the interfacial branch as purely capacitive and neglects finite-length diffusion and state-of-charge-dependent interfacial processes, which restricts the established conductivity separation to high-charge-transfer-resistance, static blocking-cell conditions. Subsequent work shows substantial progress - scanning spreading resistance microscopy now demonstrates that interparticle electronic contact resistance is real and dominant locally, and distribution-of-relaxation-times-assisted decomposition has advanced component separation - yet no study uniquely assigns the impedance feature to one microscopic origin, and 2026 work still lists contact, grain-boundary, and transport contributions as tentative for high-frequency features. The attribution question therefore remains open.

Background

All-solid-state lithium-ion batteries (ASSLBs) replace the flammable liquid electrolyte of conventional lithium-ion cells with an inorganic solid electrolyte (SE), e.g. a sulfide glass such as $75\,\mathrm{Li_2S}\cdot25\,\mathrm{P_2S_5}$ or the argyrodite $\mathrm{Li_6PS_5Cl}$. The positive electrode must then be a dense composite layer - a 'mixture layer' or composite cathode - of cathode active material (CAM, e.g. $\mathrm{LiNi_{0.5}Co_{0.2}Mn_{0.3}O_2}$, NCM) particles and SE particles, sometimes with carbon additive. Because the SE conducts lithium ions but is an electronic insulator while the CAM conducts electrons, cell performance is governed by two coupled percolating networks: an ionic path through the SE and an electronic path through the CAM/carbon. Separately quantifying the two effective conductivities of such a layer is a prerequisite for rational electrode design and for diagnosing transport-limited degradation.

Electrochemical impedance spectroscopy (EIS) measures a cell's complex impedance as a function of frequency; in a Nyquist plot (negative imaginary versus real part) distinct relaxation processes appear as semicircles at characteristic frequencies. The standard physical description of a porous or composite electrode is a transmission line model (TLM): two resistive rails - per-unit-length ionic resistance $r_{\mathrm{ion}}$ and electronic resistance $r_{\mathrm{e}}$ - running along the electrode thickness and coupled everywhere through an interfacial branch $z_{\mathrm{B}}$ (a charge-transfer resistance in parallel with a double-layer capacitance, usually generalized to a constant-phase element, CPE, a non-ideal capacitive element used to fit depressed arcs). Symmetric 'blocking' cells terminate one or both rails on electrodes that pass only one carrier: ion-blocking, electron-conducting substrates (e.g. stainless steel) force the measured current through the layer electronically, while electron-blocking SE layers with reversible electrodes force it ionically.

Classic TLM analysis of composite electrodes assumes the electronic rail is negligible ($r_{\mathrm{e}}\approx 0$); the ionic resistance then follows from the low-frequency intercept via a 'one-third' rule in suitable configurations. Siroma et al. (J. Power Sources 316 (2016) 215) showed this assumption fails for electrode mixture layers with low electronic conductivity: applying the conventional intercept rules then systematically distorts the extracted ionic resistance, with a calculable error $\alpha(K)=(1-K+K^2)/(1+K)$ depending on $K=r_{\mathrm{e}}/r_{\mathrm{ion}}$. They analyzed two 'connections' of symmetric blocking cells. In the electron-electron connection - a pressed pellet of the electrode mixture sandwiched directly between two ion-blocking, electron-conducting substrates - the TLM gives the real-axis limits

$$\lim_{\omega\to 0} Z = R_{\mathrm{e}} = r_{\mathrm{e}}L, \qquad \lim_{\omega\to \infty} Z = \frac{R_{\mathrm{ion}}R_{\mathrm{e}}}{R_{\mathrm{ion}}+R_{\mathrm{e}}},$$

with $L$ the layer thickness, so both effective conductivities follow algebraically from the two Nyquist intercepts. For graphite-rich mixtures (small $K$) the conventional ion-electron analysis remained quantitatively adequate. For NCM-based layers, whose effective electronic conductivity is low, the electron-electron spectra showed two features: a lower-frequency semicircle, assigned - explicitly tentatively - to the TLM 'teardrop' response from which $r_{\mathrm{ion}}$ and $r_{\mathrm{e}}$ were extracted (for the 50 vol% NCM sample: $r_{\mathrm{ion}}\approx 11230\ \Omega\,\mathrm{cm}$, $r_{\mathrm{e}}\approx 31490\ \Omega\,\mathrm{cm}$), and a higher-frequency semicircle whose microscopic origin the analysis could not assign. Fitting three plausible circuit topologies - an unstructured TLM, an electronic-path-structured variant (bulk electronic resistance in parallel with an electronic-path/contact capacitance), and an ionic-path-structured variant (separate bulk and grain-boundary ionic resistances) - changed the integrated electronic resistance by at most about 2% and the ionic resistance by up to about 22%, so the extracted conductivities were fairly robust while the attribution remained undetermined. The same analysis idealizes the interfacial branch as purely capacitive (CPE) and assumes no finite-length diffusion (Warburg) processes and a high charge-transfer resistance, which holds only for fully lithiated CAM in blocking cells, not for electrodes under operation.

This problem sits at the center of a now-large effort to characterize coupled ionic/electronic transport in solid-state composite electrodes: subsequent groups adopted blocking-cell TLM analysis with ion-blocking and electron-blocking configurations (Minnmann et al. 2021; Puls et al. 2025), microstructure tomography and simulation revealed unresolved particle-particle contact resistances and grain boundaries (Minnmann et al. 2024), scanning spreading resistance microscopy (SSRM, a conductive atomic-force-microscopy technique mapping local electronic resistance) demonstrated that interparticle electronic contact resistance dominates local transport and grows on charging (Gamo et al. 2024-2026), and distribution-of-relaxation-times (DRT) analysis - deconvolving the impedance into a spectrum of relaxation times - is being combined with TLMs to separate overlapping components (Choi et al. 2024; Kim et al. 2025; Kim et al. 2026). Yet the attribution implied by the source study has not been established.

Problem Statement

Left unresolved by the subsequent literature, the problem for all-solid-state electrode mixture layers (composite electrodes of cathode active material and solid electrolyte, in the spirit of the NCM/$75\,\mathrm{Li_2S}\cdot25\,\mathrm{P_2S_5}$ layers originally measured) is the following.

Attribution of the higher-frequency semicircle. In electron-electron (ion-blocking) impedance spectra, where the lower-frequency feature is assigned to the transmission-line 'teardrop' response used to extract the effective ionic and electronic conductivities, the additional higher-frequency semicircle(s) can be reproduced equally well by at least two families of microscopic interpretation: an electronic-path structure (e.g. particle-particle contact capacitance in the electronically conducting network) or an ionic-path structure (e.g. separate bulk and grain-boundary ionic resistances in the electrolyte network). Determine which microscopic process, or combination of processes, actually generates the higher-frequency feature, on the basis of independent or correlated evidence that distinguishes these interpretations, since structurally different equivalent circuits can reproduce identical impedance spectra and fit quality alone cannot discriminate.

The objective is to make impedance-based separation of ionic and electronic transport in solid-state composite electrodes physically interpretable, with each spectral feature assigned to an identified microscopic process.

References

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