Towards Optimised Cell Design of Thin Film Silicon-Based Solid-State Batteries via Modelling and Experimental Characterisation

The SE is described by concentrated solution theory ¹⁹ and with Li⁺ ions assumed to be the only mobile species present. This is a reasonable assumption as the transference number has been experimentally measured to be close to unity for LiPON. ^6,9 The Nernst-Plank equation is used to model ion conduction in the bulk SE region

$$\begin{eqnarray}&&{J}_{{\rm{b}}}=-{D}_{{\rm{b}}}{\rm{\nabla }}{c}_{{\rm{b}}}+\,\displaystyle \frac{{z}_{i}F}{RT}{D}_{{\rm{b}}}{c}_{{\rm{b}}}{\rm{\nabla }}{\varphi }_{{\rm{b}}}\end{eqnarray} \tag{ 1 }$$

where ${J}_{{\rm{b}}}$ is the flux across the bulk electrolyte, ${z}_{i}$ is the species charge, F is Faraday's constant (96,485 C mol⁻¹), D_b is the diffusion coefficient, ${c}_{{\rm{b}}}$ is the concentration of the bulk SE and ${\varphi }_{{\rm{b}}}$ is the electric potential across the medium. Given that the electrolyte is a solid body, convection terms were neglected. In addition, the system must obey charge conservation

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {{\boldsymbol{i}}}_{{\bf{b}}}=0\end{eqnarray} \tag{ 2 }$$

where${{\boldsymbol{i}}}_{{\bf{b}}}$ is the current in the bulk SE (bold symbols represent vector quantities). The stress in the SE obeys mechanical equilibrium such that body forces are neglected

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {{\boldsymbol{\sigma }}}_{{\bf{b}}}=0\end{eqnarray} \tag{ 3 }$$

where σ_b is the Cauchy stress tensor. The SE is assumed to obey Hooke's Law: σ_b = C_b × ε_b, where C_b is the stiffness matrix, and in this instance, we consider the model to be isotropic with Young's modulus, E_b and Poisson's ratio, ν_b. A small strain formulation is considered, where the total strain ε _b is obtained by solving for the displacement field, u in the SE

$$\begin{eqnarray}&&{{\boldsymbol{\varepsilon }}}_{{\bf{b}}}=\frac{1}{2}\left({\left({\rm{\nabla }}{\boldsymbol{u}}\right)}^{T}+{\rm{\nabla }}{\boldsymbol{u}}\right)\end{eqnarray} \tag{ 4 }$$

Like the SE, the LCO positive electrode is assumed to observe mechanical equilibrium (Eq. 3), while a similar small strain formulation and isotropic Hooke's law are employed, with stress σ _p, a total strain ε _p, stiffness matrix C _p, Young's modulus E_p and Poisson's ratio, ν_p. Here, there is diffusion induced strain ${{\boldsymbol{\varepsilon }}}_{p}^{ch},$ and consequently we must decompose the total strain ${{\boldsymbol{\varepsilon }}}_{{\rm{p}}},$ into an elastic component ${{\boldsymbol{\varepsilon }}}_{{\rm{p}}}^{{\rm{e}}},$ and a diffusion-related component: ${{\boldsymbol{\varepsilon }}}_{{\rm{p}}}={{\boldsymbol{\varepsilon }}}_{{\rm{p}}}^{{\rm{e}}}+{{\boldsymbol{\varepsilon }}}_{{\rm{p}}}^{{\rm{ch}}},$ where ${{\boldsymbol{\varepsilon }}}_{{\rm{p}}}^{{\rm{ch}}}$ is given as

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{p}}}^{{\rm{ch}}}=\displaystyle \frac{1}{3}{{\rm{\Omega }}}_{{\rm{p}}}\left({c}_{{\rm{p}}}-{c}_{{\rm{p}}0}\right){\bf{I}}\end{eqnarray} \tag{ 5 }$$

where c_p is the Li concentration, c_p0 is the initial Li concentration, Ω_p is the partial molar volume and I is the identity tensor. Solid-state diffusion in the positive electrode is modelled using Fick's first law with an additional contribution due to diffusion-induced swelling (hydrostatic) stresses

$$\begin{eqnarray}&&{{\boldsymbol{J}}}_{{\bf{p}}}=-{D}_{{\rm{p}}}{\rm{\nabla }}{c}_{{\rm{p}}}+\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{p}}}{c}_{{\rm{p}}}}{RT}{\rm{\nabla }}{{\boldsymbol{\sigma }}}_{{\rm{p}},{\rm{H}}}\end{eqnarray} \tag{ 6 }$$

where σ _p,H = tr[ σ ]/3, D_p is the diffusion coefficient, and ${{\boldsymbol{J}}}_{{\bf{p}}}$ is the flux, where the subscript p detonates these parameters to relate to the positive electrode region. Fick's second law describes the transient transport of Li in the electrode

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{{\rm{p}}}}{\partial t}={\rm{\nabla }}\cdot {{\boldsymbol{J}}}_{{\bf{p}}}\end{eqnarray} \tag{ 7 }$$

Current flow was modelled using Ohm's law

$$\begin{eqnarray}&&{{\boldsymbol{i}}}_{{\rm{p}}}=-{K}_{{\rm{p}}}\,{\rm{\nabla }}{\varphi }_{{\rm{p}}}\end{eqnarray} \tag{ 8 }$$

where ${{\boldsymbol{i}}}_{{\rm{p}}}$ and ${K}_{{\rm{p}}}$ are the current and electronic conductivity across the positive electrode respectively. Finally, charge conservation was observed

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \,{{\boldsymbol{i}}}_{{\rm{p}}}=0\end{eqnarray} \tag{ 9 }$$

It is known that thin film electrodes at a given capacity do not display a difference in stress (across the thickness of the electrode) unless the electrode materials experience plastic deformation. ²⁰ If only elastic deformation of the electrodes occurs, then hysteresis would be not observed as the loading and unloading during cycling would occur along the same stress path. It is important to highlight that other stress contributions could affect the hysteresis loop, such as the Li concentration gradients at the electrode interfaces. However, as LCO is not expected to plastically deform due to its higher Young's modulus and hardness than a-Si (even when fully lithiated) ²¹ it is reasonable to assume that voltage hysteresis (as observed experimentally in the electrochemical testing section) is primarily due to plastic deformation of a-Si occurring during discharge.

Upon lithiation Si can exhibit nominal strains up to 300%, thus it is appropriate to adopt a viscoplastic-type yield model. ^15,20 The following approach was adapted from Di Leo et al. ¹⁶ who experimentally validated their electro-mechanical model against half-cell curvature data using mechanical measurements of an a-Si electrode and a liquid electrolyte. ^15,16 To the best of our knowledge, analogous experiments have not been reported for a-Si with a solid electrolyte. The method is summarised as follows: we consider finite deformation kinematics with large elastic-plastic strains and multiplicative decomposition of the deformation gradient, F = F ^e F ^p F ^ch, where the superscripts represent the elastic, volume-preserving plastic, and lithiation-induced deformation gradients. The lithiation-induced deformation gradient is given as ${{\boldsymbol{F}}}^{{\bf{c}}{\bf{h}}}={\left(1+{\rm{\Omega }}\left({c}_{n}-{c}_{n0}\right)\right)}^{1/3},$ such that $\bar{c}={c}_{n}/{c}_{n,max},$ where c_n is the Li concentration and c_n,max is the maximum Li concentration in the negative electrode. The plastic deformation evolves as

$$\begin{eqnarray}&&\dot{{{\boldsymbol{F}}}^{{\rm{p}}}}=\dot{{\varepsilon }_{{\rm{eq}}}^{{\rm{p}}}}\left(\displaystyle \frac{3{\boldsymbol{\sigma }}}{2{\sigma }_{{\rm{eq}}}}\,\right){{\boldsymbol{F}}}^{{\rm{p}}}\,\end{eqnarray} \tag{ 10 }$$

where $\dot{{\varepsilon }_{{\rm{eq}}}^{{\rm{p}}}}\gt 0$ is the equivalent plastic strain rate and ${\sigma }_{{\rm{eq}}}=\sqrt{3/2}\left|{\boldsymbol{\sigma }}\right|$ is the equivalent stress. During plastic flow, we take the equivalent strain rate to be

$$\begin{eqnarray}\dot{{{\boldsymbol{\varepsilon }}}_{\mathrm{eq}}^{p}}=\left\{\begin{array}{ll}0 & \,if\,{\sigma }_{\mathrm{eq}}\leqslant {\sigma }_{{\rm{Y}}}(\bar{c})\\ {\dot{\varepsilon }}_{0}{\left(\frac{{\sigma }_{\mathrm{eq}}-{\sigma }_{{\rm{Y}}}\left(\bar{c}\right)}{{\sigma }_{* }}\right)}^{m} & \,if\,{\sigma }_{eq}\gt {\sigma }_{Y}(\bar{c})\end{array}\right.\end{eqnarray} \tag{ 11 }$$

where${\sigma }_{* }$ is a stress-based constant, $\dot{{\varepsilon }_{0}}$ is a reference plastic strain-rate and m is a strain-rate related fitting parameter. The concentration-dependent yield stress, ${{\boldsymbol{\sigma }}}_{{\rm{Y}}}\left(\bar{c}\right)$ is given as

$$\begin{eqnarray}&&{\sigma }_{{\rm{Y}}}\left({\bar{c}}_{{\rm{n}}}\right)={\sigma }_{\mathrm{sat}}+\left({\sigma }_{0}-{\sigma }_{\mathrm{sat}}\right){e}^{-\frac{{\bar{c}}_{n}}{{\bar{c}}_{* }}}\end{eqnarray} \tag{ 12 }$$

where ${\sigma }_{0},$ ${\sigma }_{sat},$ ${c}_{* }$ are positive-valued stress-related fitting parameters. As with the positive electrode, diffusion of Li within the negative electrode is captured using Fick's Law and an additional swelling term, analogous to Eq. 6

$$\begin{eqnarray}&&{{\boldsymbol{J}}}_{{\bf{n}}}=-{D}_{n}{\rm{\nabla }}{c}_{n}+\,\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{n}}}{c}_{{\rm{n}}}}{RT}{\rm{\nabla }}{\sigma }_{{\rm{n}},{\rm{H}}}\end{eqnarray} \tag{ 13 }$$

As in the studies of Sethuraman et al. ²⁰ and Di Leo et al., ¹⁶ we neglect quadratic, higher-order stress-related terms by assuming their negligible influence on the overall response. As before, Fick's second law provides a description of transient diffusion

$$\begin{eqnarray}&&\frac{\partial {c}_{{\rm{n}}}}{\partial t}={\rm{\nabla }}\cdot {{\boldsymbol{J}}}_{{\bf{n}}}\,\end{eqnarray} \tag{ 14 }$$

As before, Ohm's law describes current flow

$$\begin{eqnarray}&&{{\boldsymbol{i}}}_{{\rm{n}}}=-{K}_{{\rm{n}}}\,{\rm{\nabla }}{\varphi }_{{\rm{n}}}\end{eqnarray} \tag{ 15 }$$

with ${{\boldsymbol{i}}}_{{\rm{n}}}$ and ${K}_{{\rm{n}}}$ being the current and electronic conductivity, across the negative respectively, and charge conservation is observed

$$\begin{eqnarray}&&{\rm{\nabla }}.\,{{\boldsymbol{i}}}_{{\rm{n}}}=0\end{eqnarray} \tag{ 16 }$$

It is necessary to impose boundary conditions at the interfaces of the SE and electrode in order to accurately solve model equations. By doing so the charge transfer reactions and the additional stress overpotential required for lithiation to proceed are captured. The charge transfer rate is commonly expressed using a Butler-Volmer type equation

$$\begin{eqnarray}&&{i}_{\mathrm{BV}}={i}_{0}\left(\exp \left(\frac{{\alpha }_{n}F{\eta }_{{\rm{n}}}}{RT}\right)-\exp \left(-\frac{{\alpha }_{p}F{\eta }_{{\rm{p}}}}{RT}\right)\right)\,\end{eqnarray} \tag{ 17 }$$

where ${\alpha }_{n}$ and ${\alpha }_{p}$ are the negative and positive charge transfer coefficients respectively. The local exchange current density, i₀ is dependent on both Li and bulk electrolyte concentrations

$$\begin{eqnarray}&&{i}_{0}=F{\left({k}_{{\rm{p}}}\right)}^{{\alpha }_{{\rm{n}}}}{\left({{\rm{k}}}_{n}\right)}^{{\alpha }_{{\rm{p}}}}{\left({c}_{i}-{c}_{i,max}\right)}^{{\alpha }_{{\rm{n}}}}{\left({c}_{{\rm{i}}}\right)}^{{\alpha }_{{\rm{p}}}}{\left(\displaystyle \frac{{c}_{b}}{{c}_{b,ref}}\right)}^{{\alpha }_{{\rm{n}}}}\end{eqnarray} \tag{ 18 }$$

where the subscript i is n or p depending on the interface, c_b,ref is a reference SE concentration, ${c}_{{\rm{\max }}}$ is the maximum SE concentration and the rate constants of the positive and negative electrodes are ${k}_{{\rm{p}}}$ and ${k}_{n}$ respectively. The total overpotential ${\eta }_{{\rm{i}}},$ is expressed as

$$\begin{eqnarray}&&{\eta }_{{\rm{i}}}={\varphi }_{{\rm{i}}}-{\varphi }_{{\rm{b}}}-{U}_{{\rm{i}}}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{i}}}{\sigma }_{{\rm{H}},{\rm{i}}}}{F}\,\end{eqnarray} \tag{ 19 }$$

where U_i is the open circuit voltage, and additional overpotentials due to diffusion-induced hydrostatic stress ${\sigma }_{{\rm{H}},{\rm{i}}}$ and the initial partial molar volume ${{\rm{\Omega }}}_{{\rm{i}}},\,$are incorporated via the final term on the right-hand side of Eq. 19. ²²

Consider the coordinate system (x, y, z) in Fig. 1. At the positive current collector (CC), we apply a current density ${{\boldsymbol{i}}}_{{\rm{cc}}}\cdot {{\boldsymbol{n}}}_{{\rm{cc}},{\rm{p}}}={i}_{{\rm{in}}}$ at x = (t_cc,n + t_ne + t_sep + t_pe + t_cc,p), where n _cc,p is the unit normal vector pointing outwards from the positive CC surface in the positive z-direction. The applied current density i_in at 1C discharge, is calculated based on the sub volume of the cell. Since the maximum accepted quantity of Li for the given electrode material, c_p,max : the current density is given as i_in = c_p,max FV_p /t₀ A, where the area is in the dimensions of the cell in the z, y coordinates, A = zy and t₀ = 3600 s. At the negative current collector, a potential of ${\varphi }_{{\rm{p}}}\,$= 0 V is applied at x = 0.

We prescribe the thin film SSB to be fixed in all directions to the positive current collector surface, i.e. ${{\boldsymbol{u}}}_{{\rm{cc}}}\cdot {{\boldsymbol{n}}}_{{\rm{cc}},{\rm{p}}}=0$ at x = (t_cc,n + t_ne + t_sep + t_pe + t_cc,p), whilst the negative CC surface remains unconstrained. Given that a small sub volume of the electrode is modelled in the y-direction, it is appropriate to apply symmetry boundary conditions for species fluxes, displacements and potentials.

At the interface between the separator and the electrodes, we specify that the electronic current flow must be zero: ${{\boldsymbol{i}}}_{{\rm{i}}}\cdot {{\boldsymbol{n}}}_{{\rm{sep}},{\rm{i}}}=0,$ where n _sep is the unit normal vector to the interface between the electrode and the separator, pointing in the direction away from the electrodes. This ensures that only the ionic current is permitted across this interface. At the electrolyte-electrode interface we observe a flux of Li into the electrode, or Li⁺ ions into the electrolyte as a result of the charge transfer reaction. The fluxes are as follows: ${{\boldsymbol{J}}}_{{\rm{b}}}\cdot {{\boldsymbol{n}}}_{{\rm{sep}}}=-{i}_{{\rm{BV}}}/F$ and ${{\boldsymbol{J}}}_{{\rm{i}}}\cdot {{\boldsymbol{n}}}_{{\rm{i}}}=-{i}_{{\rm{BV}}}/F,$ where n _i is the normal vector pointing from the electrolyte to the electrodes. We also prescribe a current density at this interface: ${{\boldsymbol{i}}}_{{\rm{b}}}\cdot {{\boldsymbol{n}}}_{{\rm{sep}}}=-{i}_{{\rm{BV}}}$ and ${{\boldsymbol{i}}}_{{\rm{i}}}\cdot {{\boldsymbol{n}}}_{{\rm{i}}}=-{i}_{{\rm{BV}}}.$ An initial Li concentration in the electrodes, c_i0, is prescribed, whilst the initial concentration in the electrolyte is given by c_b0. The electrode and all associated constituent domains are assumed to be in an initially unstressed state.

To summarise, the strain type simulated in the SE and positive electrode domains is linear elastic whereas for the negative electrode (a-Si) elastic-viscoplastic behavior is modelled. The three types of strains occurring within the negative electrode are elastic, volume-preserving plastic, and lithiation-induced deformation gradients. The electrode and SE are not expected to plastically deform due to their greater Young's moduli and hardness at all stages of lithiation.

The mechanical and electrochemical parameters used in the model were either previously reported values from the literature or experimentally determined in this study (Table I). LCO and LiPON were assumed to be isotropic linear-elastic solids, whereas a-Si was treated as an isotropic elastic-viscoplastic solid with a Li concentration-dependent Young's modulus and yield strength as defined in Eq. 12. The elastic properties of a-Si vary with state of lithiation (SoL) during cell cycling. The Young's modulus and Poisson's ratio vary between the elastic limits of pure Li and a-Si depending on SoL. This produces a non-linear trend which is captured in the model (a detailed explanation of these parameters can be found in the study by Leo et al. ¹⁶ ). The current collectors were assumed to be electronically conductive, linear elastic solids with Young's moduli of ≈100 GPa. Finally, the universal gas constant, R was taken as 8.314 J mol⁻¹ K⁻¹, and all simulations and experiments were carried out at a temperature, T of 298 K.

Table I. Model parameters.

	Parameter	Units	Value	Source
Electrochemical	Db	m2 s−1	1.7 × 10−16	Ref. 23
	cb0	mol m−3	1000	—
	cp,max	mol m−3	5.19 × 104	Ref. 24
	cn,max	mol m−3	1.55 × 105	Calculated
	cn0	mol m−3	0.05 × cSi,max	—
	cp0	mol m−3	0.95 × cLCO,max	—
	αn, αp	1	0.5	—
	Kb	S cm−1	2.3 × 10−6	Electrochemical impedance spectroscopy (EIS)
	Kn	S cm−1	0.33	Ref. 25
	Kp	S cm−1	1 × 10−5	Averaged from Ref. 21
Elastic	Ep	GPa	191	Ref. 21
	En	GPa	f(cn /cn,max)	Ref. 16
	Eb	GPa	77	Ref. 26
	υp	1	0.24	Ref. 21
	υb	1	0.25	Ref. 26
	υn	1	f(cn /cn,max)	Ref. 16
	Ωn	m3 mol−1	8.8 × 10−6	Ref. 27
	Ωp	m3 mol−1	−1 × 10−7	Ref. 12
Plastic	σY0	GPa	0.9	Ref. 17
	σsat	GPa	0.4	Ref. 17
	$\dot{\varepsilon }$	1/s	2.3 × 10−3	Ref. 17
	m	1	2.94	Ref. 17
Rate kinetics	kn, kp	mol m−2 s−1	1.3 × 10−5	Obtained from 1C cycling data
	Up	V	f(cn /cn,max)	Ref. 24

Electrochemical properties often vary as a function of composition. The diffusion coefficient was experimentally estimated as a function of state of charge (SoC) for a-Si and LCO using the galvanostatic intermittent titration technique (GITT). Diffusion through the SE was given by D_b, reported by Raijmakers et al. ²³ The initial concentration of Li⁺ ions, c_b0 was parametrised in the study to give the best fit to the experimental 1C data. The maximum concentration ${c}_{Si,\,\max }$ of Li in Li_y Si was estimated using the equation: ${c}_{Si,\,\max }=y{\rho }_{Si}.$ Here y represents the Li stoichiometry in Li_y Si and ${\rho }_{Si}$ is the theoretical maximum molar density of the hosting material. By analysing the quantity of Li extracted from LCO during charging, the amount of Li alloyed with the a-Si electrode was calculated. The upper voltage limit was 4 V and by extrapolation to the open circuit voltage (OCV) of LCO, the amount of Li extracted from LCO, $y$ was quantified. It should be noted that the value of y must be normalised by the LCO thickness.

The 2D electro-chemo-mechanical model was created using the finite element modelling software package, COMSOL Multiphysics (v5.6, Sweden). The 2D geometry in this study is used to model the mechanical properties in the x-y plane. The mesh consisted of approximately 4,000 quadratic elements with 94,000 degrees of freedom, while the solutions were found to be mesh independent. The Parallel Direct Sparse Solver (PARDISO) was used to solve the discretised transport, electrode kinetics and deformation kinematics equations. A segregated approach was used, which involved solving the coupled field variables in a sequential staggered manner. Time stepping was handled using 2nd order backward Euler differentiation, whilst time step sensitivity analysis was performed.

A commercial thin-film SSB was supplied by Ilika Technologies Ltd (Southampton, UK). The cell had a capacity of 250 μAh, comprising an a-Si negative electrode, a LiPON SE and a crystalline LCO positive electrode sputtered on top of a substrate using vacuum processing methods. The SSB cell was mounted on a printed circuit board with embedded electrical connections and housed inside a thermal chamber. A potentiostat (Biologic EC-lab) was used to execute cell cycling protocols and a thermocouple was attached to the cell to monitor its temperature which was recorded using a data logger (PicoLog TC-08).

The cell was formed using 5 charge/discharge cycles at C/5 and differential capacity ($\tfrac{dQ}{dSoC}$) analysis performed to reveal electrode processes. Since the $\tfrac{dQ}{dSoC}$ analysis of the full cell contains information from both electrodes, comparison with half-cell data from literature was used to assign the peak contributions from the a-Si and LCO. ^21,28 The $\tfrac{dQ}{dSoC}$ data were compared during the first formation cycle and after subsequent cycling at 1C.

The open circuit voltage (OCV) as a function of SoC was determined for the full cell after relaxation for 24 h at 10% capacity increments. A pseudo-OCV was measured using a small cycling current of C/30. Electrochemical impedance spectroscopy (EIS) measurements were taken at each 10% SoC interval after 24 h relaxation during charge and discharge. During EIS, a 5 mV voltage perturbation and a frequency range of 1 mHz to 1 MHz were used. Nyquist plots (-Im(Z) vs Re(Z)) were fitted with an equivalent circuit model (ECM) using ZView software (Scribner Associates).

When conducting linear EIS, two conditions must be obeyed: (i) the form of the input and output functions must be the same, and (ii) must be linear to ensure that higher harmonic terms are avoided, as these represent irreversible electrochemical changes to the system. ²⁹ In order to ensure these conditions were obeyed, a Kramers-Kronig (K-K) relation was applied to test the linearity, stability and causality of the EIS data. From the impedance spectra, the ionic conductivity, (${K}_{{\rm{b}}}$) of LiPON was calculated as an input for the electro-chemo-mechanical model using:

$$\begin{eqnarray}&&{K}_{{\rm{b}}}=\displaystyle \frac{{{\rm{t}}}_{{\rm{sep}}}}{{A}_{{\rm{sep}}}{R}_{{\rm{b}}}}\end{eqnarray} \tag{ 20 }$$

where, ${A}_{{\rm{sep}}}$is the surface area of the SE, and ${R}_{{\rm{b}}}\,$is the bulk resistance of the SE taken from the ECM of the EIS data.

To separate polarisation contributions from the various cell components and identify all time processes in the system, a Fourier transform of the EIS data was performed for distribution of relaxation times (DRT) analysis by, ^24,30

$$\begin{eqnarray}&&Z\left(w\right)={R}_{{\rm{ohmic}}}+{Z}_{p{\rm{ol}}}\left(w\right)={R}_{{\rm{ohmic}}}+\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{R}_{{\rm{pol}},{\rm{k}}}}{1+j\omega {\tau }_{k}}\end{eqnarray} \tag{ 21 }$$

where ${R}_{{\rm{ohmic}}}$ is the Ohmic resistance of the SSB and is independent of frequency, while ${Z}_{p{\rm{ol}}}\left(w\right)$ accounts for the polarisation resistance, ${R}_{pol,k}$ and is a function of frequency. This deconvolution is possible since the different cell processes have characteristic frequencies, and therefore time constants, associated with specific processes (Table II). h MATLAB code by Wan et al. ³⁰ was used to perform DRT analysis. The K-K residual plots of EIS data were $\pm$1% (Figs. S1 and S2 in the Supporting Information (SI)).

The galvanostatic intermittent titration technique (GITT) was used to analyse the total Li⁺ ion diffusion of the full cell at different SoCs. Figure S3 in the SI shows the schematic of a typical GITT pulse procedure, where ${\rm{\Delta }}{V}_{t}$ is the voltage response due to the applied current pulse (calculated after subtracting the initial IR drop due to internal cell resistance) and the subsequent voltage relaxation, ${\rm{\Delta }}{V}_{s}.$ This method was used to estimate the cell diffusion coefficient, ${D}_{{\rm{cell}}}$ using the equation:

$$\begin{eqnarray}&&{D}_{{\rm{cell}}}=\displaystyle \frac{4}{\pi {\tau }_{pulse}}{{t}_{{\rm{elec}}}}^{2}{\left(\displaystyle \frac{{\rm{\Delta }}{V}_{{\rm{s}}}}{{\rm{\Delta }}{V}_{{\rm{t}}}}\right)}^{2}\end{eqnarray} \tag{ 22 }$$

where ${\tau }_{{\rm{pulse}}}$ is the time period of the GITT pulse and ${t}_{{\rm{elec}}}$ is the electrode thickness. This equation is valid for ${\tau }_{{\rm{pulse}}}$ << $\displaystyle \frac{\,{{t}_{{\rm{elec}}}}^{2}}{{D}_{{\rm{cell}}}}\,.$ ³² In these experiments, 20 s was used for the pulse duration, followed by a relaxation to OCV. The GITT pulses were carried out at 10% SoC intervals and the diffusion coefficients were calculated at these points.

In this configuration, the different electrode contributions are convoluted. To estimate the individual electrode contributions, the positive and negative electrode contributions were scaled by their respective charge transfer resistances, which were determined using DRT analysis (Eq. 21). These estimates of the diffusion coefficients in each electrode were used as parameters in the model.

Hybrid pulse power characterisation (HPPC) testing was performed to probe the dynamic cell behaviour over usable voltage ranges of the cell. The full HPPC protocol is illustrated in Fig. S4 in the SI and yielded three discharge and charge datasets. These were used to parameterise a first order Thevenin ECM, containing a resistor in series with two parallel resistor/capacitor (RC) pairs. Separate sets of parameters were developed for discharge and charge. The equivalent circuit is shown in Fig. S5 in the SI.

Parameter extraction was conducted using a script developed in-house, which uses the MATLAB curve-fitting toolbox to fit the ECMs response (i.e. simulated terminal voltage) to the experimental voltage data through a least-squares optimisation method. The methods for extraction follow well established processes, as set out by Ahmed et al. ³³ and Jackey et al. ^34,35 ECM parameters vary considerably with cell SoC and therefore 5% SoC windows (i.e., 100%–95%, 95%–90%, etc.) were used to extract unique parameters which describe the cell's behaviour for specific SoC ranges.

The simulated terminal voltage (${V}_{t}$) at any timestep was described by Eq. 23,

$$\begin{eqnarray}&&{V}_{t}=U-\,{R}_{0}I-\displaystyle \sum _{n}^{i=1}{U}_{i}{U}_{RC,}\end{eqnarray} \tag{ 23 }$$

where $U$ is a function a given SoC, R₀ is the series (Ohmic) resistance due to the bulk solid electrolyte impedance R_bulk, ${U}_{i}$ is the voltage drop across the $i$th RC pair, $n$ is the total number of RC pairs ($n\,$= 2 for the purposes of the present study where, R_elec is the lumped charge transfer resistance of the electrodes and R_diff is the diffusion resistance) and $I$ is the magnitude of current flow.

The voltage profiles of the 5 initial charge cycles at C/5 showed a sharp "knee-point" at ≈3.65 V (Fig. 2a). Using differential capacity analysis (Fig. 2b), this feature was attributed to a characteristic LCO phase transformation observed previously ²¹ and a broad peak due to a-Si lithiation to Li₂Si could also be assigned, ²⁸ consistent with a-Si not being fully lithiated to Li₁₅Si₄ over the voltage range used here. We observed a decrease in the peak area attributed to the a-Si electrode on discharge, likely due to side reactions consuming Li during the alloying reaction with a-Si on formation, as has been observed in a previous full cell study. ⁷ Conversely, the differential capacity analysis for 1C cycling post-cell formation (Fig. 2c) showed the peak areas attributed to LCO and a-Si to be near equal between charge and discharge—indicative of stable cycling.

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Experimental tests consisting of OCV and low current (C/30) cycling measurements post-formation were used to determine the SoC-OCV relationship and cell capacity respectively. During the OCV testing, a voltage hysteresis was observed between charge and discharge (Fig. 3a), which is commonly seen for a-Si-based half-cells. ³⁶ An increased voltage hysteresis was observed during low current (C/30) cycling (Fig. 3b) compared with OCV and was attributed to the hydrostatic stress caused by diffusion induced strains (Eq. 6) as a result of lithiation and delithiation of the a-Si electrode. This was consistent with the expected viscoplastic behavior.

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Electrochemical impedance spectroscopy (EIS) showed two semi-circles in the high and mid-to-low frequency regions on a Nyquist plot (a representative plot at 50% SoC is shown in Fig. 4a). There was very little variation in the Nyquist spectra during charge and discharge (Fig. S6 in the SI), thus minimal information could be determined about the internal cell impedances from EIS alone. As the latter semi-circle was substantially depressed, we investigated the possibility that this feature was made up of multiple polarisation processes using DRT analysis.

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From the DRT spectrum (Fig. 4b), several distinct polarisation processes were identified. The peak observed at ≈10⁻⁶ s was attributed to ion migration in the LiPON SE and was largely invariant during operation as expected. At the other end of the spectrum, the peak at ≈0.1 s was assigned to relatively slow diffusion processes and exhibited a complex dependance with SoC. As low frequency EIS measurements can be unreliable, time-domain experiments were used to deconvolute and quantify individual electrode contributions to the diffusion polarisation in these cells (galvanostatic intermittent titration technique section).

Intermediate polarisation processes in the range 10⁻⁵ to 10⁻² s were attributed to charge transfer contributions at the various cell interfaces. Based on previous reports, a single LiPON∣LCO time constant was expected to occur at ≈10⁻³ s (Table II), ^24,31 thus, the peaks at ≈10⁻⁵ and ≈10⁻⁴ s were attributable to the a-Si electrode. The exact meaning of these processes is not known, but we speculate they may result from the Li_y Si alloy at the SE∣ a-Si interface and the additional interface between lithiated Li_y Si and unlithiated a-Si in the electrode bulk. This picture is consistent with partial a-Si lithiation during operation and the differential capacity analysis which hinted at irreversible Li loss after the first formation cycle (formation cycle section).

During charge and discharge, the SE∣LCO polarisation was approximately constant, as was the a-Si electrode process centred around ≈10⁻⁵ s (Fig. 5). In contrast, the a-Si contribution at ≈10⁻⁴ s was invariant on delithiation but increased during lithiation. The observed increase in charge transfer resistance may be due to the swelling of a-Si as it lithiates to Li_y Si. Therefore, we make the following preliminary assignments: the faster interfacial process (≈10⁻⁵ s) represents the SE∣Li_y Si interface, while slower charge transfer (≈10⁻⁴ s) occurs between Li_y Si and bulk a-Si.

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The five polarisation processes revealed by DRT analysis were incorporated into an ECM that fit the EIS data well as shown in Fig. 4c. The ionic conductivity of LiPON was calculated using the value of R₁ obtained from the ECM fit and inputted into Eq. 20, which was in good agreement with literature values. ²⁴ DRT analysis also allowed quantification of the electrode resistances during charge and discharge, which were used to estimate Li diffusion coefficients for individual electrodes from full-cell galvanostatic intermittent titration technique data in the following section.

From experiments using the galvanostatic intermittent titration technique (GITT) the total cell diffusion was extracted. The Li diffusion coefficients, D_GITT in the a-Si and LCO electrodes were estimated for charge and discharge as explained in pulsed electrochemical tests section. The D_GITT value in a-Si was determined to be ≈2 orders of magnitude smaller than the D_GITT value in LCO (Fig. 6).

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For the charge case at low SoC, D_GITT was fairly constant but gradually increased from mid to high SoC. A similar trend was found for diffusion on discharge, though a reduction in D_GITT was more pronounced in the low SoC region at the end of discharge. This behaviour could be due to hindered extraction of Li⁺ ions from a-Li_y Si with low Li content. Similarly, the reduced D_GITT value in LCO at low SoC on discharge relative to charge can be explained by the intercalation of Li⁺ ions into LCO being impeded by a high Li concentration at the LCO∣SE interface.

Hybrid pulse power characterisation (HPPC) was conducted, and the resultant voltage profile was fit to two parallel RC units in series. The following R values were extracted as a function of SoC: R_bulk, R_elec and R_diff.. Figures 7a and 7b shows their behaviour during charge and discharge respectively. The charge profile was diffusion-limited especially at high SoC and the resistance associated with the SE, R_bulk, remained approximately constant over the SoC range whereas the charge transfer resistance, R_elec had a complex dependence on SoC. However, during discharge there was high diffusion resistance at the SoC extremes but was overall limited by the slow electrode kinetics (higher R_elec) with R_bulk remaining approximately constant over the SoC range. This observation is in contrast with the EIS results at steady state which showed the charge transfer resistances to be near equal at different SoC points (Fig. 5). If the increase in interfacial resistance R_elec was due to irreversible side reactions or a stable decomposition layer, then this should have been apparent in the EIS data (which were measured after HPPC pulsing). Rather, these results suggest that the increase in diffusion impedance, R_diff during pulsing was a transient cell response that was due to diffusion limitation in the electrodes (increase in R_diff) and mechanical stress due to the build-up of concentration gradients at the SE∣electrode interface (increase in R_elec).

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While diffusion limitations at low SoC during discharge were in agreement with the GITT results, those observed during charging at high SoC were not. Thus, the dynamic cell behaviour is highly complex, especially in the case of charging when a-Si strain is increasing. It is possible that the time constants of charge transfer and diffusion processes overlap and therefore R_elec and R_diff cannot be solely attributed to one or the other, rather one may dominate in certain time domains. Two additional HPPC measurements for charge and discharge were conducted and used for model parameterisation and validation against the experimental data as shown in Figs. S7 and S8 in the SI. These followed similar trends, which was expected for pulses at similar C-rates. An acceptable root mean square (rms) error of <15 mV (predicted by the Thevenin model in Figs. S9–S10 in the SI) was obtained.

The electro-chemo-mechanical model was validated using experimental charge and discharge curves, both taken at a 1C rate. Figure 8 shows a comparison between these experimental data and two simulation cases using different values of the Li⁺ ion diffusion coefficient, D for the electrode materials, as elaborated below.

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In the first case (Fig. 8, light blue trace), the D values for LCO and a-Si used in the model were based on half-cell experiments using liquid electrolyte reference data (supplied by the SSB manufacturer, Ilika). Although good agreement between experiment and simulation was observed on charge, significant deviations (>200 mV) were apparent on discharge and the discharge capacity was severely underpredicted. In the second case (Fig. 8, dark blue trace), D_GITT values for LCO and a-Si were estimated using our full-cell GITT data (GITT section), modified by the total electrode charge transfer resistances (quantified by DRT analysis). The inclusion of these parameters markedly improved the agreement with experiment on discharge, accurately predicting cell capacity and highlighting the importance of accurate diffusion coefficient values when modelling these systems.

The remaining discrepancies between model and experiment during cycling were primarily attributed to uncertainties in the OCV values of a-Si and LCO, which were extracted from half-cell data using a liquid electrolyte. When a liquid electrolyte is used, the electrodes can expand freely with minimal constraint. In contrast, the SSB considered in the present study uses a SE that may influence electrode response, e.g. volumetric expansion may be limited by the high Young's modulus of the SE. Given that there will be a change in the stress and strain fields, the OCV is likely to be affected. Representative solid-state half-cell data are therefore highly desirable for future SSB modelling studies.

The model was further validated against charge and discharge during HPPC (Fig. 9). The HPPC simulations followed the profile and general trend of the experimental data well, deviating by a maximum value of ≈100 mV. These offsets can be explained by considering the simulated 1C charge data in Fig. 8, which did not perfectly capture the "knee point" at high SoC. We speculate that this difference has also translated into the HPPC charge profile as a voltage offset. During discharge, the initial pulse was captured as a sharper drop in voltage than observed experimentally, again consistent with the 1C discharge data (Fig. 8). It can be concluded that the model reproduced the pulse behaviour of the cell with further improvement being possible with more accurate experimental parameters extracted from representative half-cell OCV and GITT data.

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Increasing electrode thickness is desirable to maximise cell capacity but may compromise power capability. In order to aid SSB design, we studied the effects of electrode thickness on stresses and strains in a-Si -based, thin-film SSBs. Maps of maximum principal nominal strains and stresses (which both occurred at the SE∣Li_y Si interface during 1C cycling) as a function of relative electrode thickness were created using the developed model.

Figure 10 illustrates the tradeoff that exists in terms of limiting the maximum stress and volumetric expansion. As the electrode thickness ratio (nominally a negative to positive electrode thickess ratio of 1:1) was increased to 4:1, the model predicted a maximum strain of −13% and a maximum stress of +30%. By targeting a thicker a-Si electrode, which can incorporate more Li, the strain is reduced provided the LCO electrode remains relatively thin, i.e. the degree of a-Si lithiation remains relatively low. However, this direction may not be practical from an application perspective, as the cell capacity is limited by the LCO and scales linearly as a function of LCO thickness. For thicker LCO layers, the increase in the total Li available results in a greater degree of lithiation in the a-Si, and hence greater strain. Tailoring the relative electrode thicknesses compared to the nominal 1:1 up to a factor of 1:4 (negative to positive electrode thickess ratio), resulted in increased stress from 0.71 GPa up to 1.17 GPa, corresponding to a volumetric expansion from 69% to 104%. Our model showed that stress and strain variations are heterogenous in the cell, with the SE∣Li_y Si interface experiencing the greatest stress. Thus, a thin a-Si electrode coupled with a thick LCO electrode would be desirable to minimise overall stress. Although this results in a highly lithiated and highly strained a-Si electrode, a lower concentration gradient was produced resulting in more a homogenous stress distribution while maximising cell capacity.

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We note that our model used an unconstrained a-Si electrode for simplicity. In practice this will not be the case, as the top CC and casing will constrain expansion to some extent. As the maximum stress is experienced at the SE∣Li_y Si interface, it is critical that the SE mechanical properties are tailored to reduce the a-Si stress and be sufficiently mechanically strong to withstand the stresses occurring during cycling, otherwise fracture propagation may occur and cause cell failure. Future work will investigate the relationship between external pressure and stress-strain to inform cell design as well as the mechanical properties of suitable SE candidates for a-Si SSBs.