Non-destructive investigation of solid-state batteries: Overview of impedance spectroscopy (EIS)

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What is impedance spectroscopy? What is it used for? How does it work? What are the special features of the investigation of solid-state batteries? This article provides answers.

For the investigation of batteries, non-destructive measurement methods are an important tool for making statements about the condition of the system. One of these measurement methods is impedance spectroscopy. This is based on AC technology and has been in use for lithium-ion batteries for several decades. For solid-state batteries, however, it is a relatively new field of application. This article describes the investigations for which impedance spectroscopy is used for solid-state batteries, how it differs from measurements of Li-ion batteries, and what future applications are conceivable for solid-state batteries.

Note: The first two sections of this article explain the general basics of impedance spectroscopy and the associated modeling. Those, who are already familiar with this topic can skip to the third section, which then explains the details of impedance spectroscopy specifically for solid-state batteries.

Basics of Electrical Impedance Spectroscopy

Impedance spectroscopy is a non-destructive measurement technique that can be used to examine a battery or parts of a battery (e.g. a half-cell, i.e. only the anode). Figure 1 (a) shows the basic measurement principle. For the measurement, the cell is connected at its two poles to an impedance spectroscopy meter. These are offered by various manufacturers such as Biologic or Gamry Instruments. There are different measurement variants, but one of the standard measurement methods is galvanostatic impedance spectroscopy. In this, a sinusoidal alternating current is applied to the cell. A low current is set, since only then does the battery exhibit linear behavior to a first approximation [1]. The alternating current pulse then leads to a phase-shifted voltage response. The behavior follows the basic equations of AC theory. With the current amplitude I0   it results in the following AC time signal:

I(t) = I0*sin(wt)

The voltage of the cell responds to this excitation and accordingly the following phase-shifted AC voltage time signal can be measured:

U(t) = U0sin(wt + ϕ)

With these data it is now possible to calculate the complex resistance (also called impedance) which results from Ohm’s law (cf. Figure 1b):

Z = (U0sin(wt + ϕ))/(I0sin(wt)) = Z0*(sin(wt + ϕ))/(sin(wt))

The frequency of the alternating current can be freely selected and, depending on the measuring instrument, is usually in the kilohertz to microhertz range. Usually, many frequencies are measured in investigations, so that a whole frequency spectrum results – this also explains the name of impedance spectroscopy. This impedance spectrum is then often displayed in a so-called Nyquist diagram (Figure 1 (c)) and is used as the basis for evaluations. In the Nyquist diagram, each point corresponds to a specific frequency. The complex resistance is split into its real and imaginary parts and plotted on the x and y axes.

Figure 1(a): Measurement principle for measuring galvanostatic impedance; (b) Conversion of the measured values into impedance; (c) Representation of the measurement results in the Nyquist plot, Own illustration.

Electrochemical equivalent circuit, physical models and their evaluation

The evaluation of Nyquist plots is usually not very simple and there are different approaches. The simplest approach is to compare the impedance curves with reference measurements in order to draw conclusions about the cell under investigation. The impedance of a cell, for example, is very sensitive to temperature fluctuations and changes over several orders of magnitude in the typical operating range of a battery from -20°C to +60°C. By comparing the impedance curve with a look-up table, it is relatively easy to determine the temperature inside a cell (methods based on this are described in the literature, e.g. by Richardson [2]). The advantage of this approach is that one usually does not need to know exactly about the chemical structure inside a battery, which makes the implementation much easier. The disadvantage of these methods is usually that the accuracy of the parameters studied is not high, especially when predictions are made about states that are not represented in the look-up table.

For this reason, a widely used approach is to develop an empirical equivalent circuit of the cell based on the impedance curve of the cell. Typically, this is composed of a combination of passive elements (i.e. resistors, capacitors and possibly coils). In the empirical approach, it is common to assemble the model so that the equivalent circuit has a minimum deviation from the measured impedance spectrum. For the assembly of the entire equivalent circuit, one can use the individual elements from Figure 2 as a guide. In the impedance spectrum, there are usually semicircles which can be represented by a parallel RC network and an upstream ohmic resistor (Figure 2a). If several semicircles are found in the impedance spectrum, this can be implemented with several RC links (Figure 2b). The diameter of the semicircle corresponds to the real resistance of the RC element. The capacitance of the RC element can be read out from the impedance spectrum by a simple formula (see the work of Choi, W., Shin, H., et al. [3]).

Since the passive components resistance, capacitance and coil are not sufficient to replicate an impedance spectrum with sufficient accuracy, the Warburg impedance has been added (Figure 2c), which makes it possible to simulate the straight line that occurs at low frequencies. As a further element, the capacitance can be replaced by a constant phase element (CPE). As a single element, this corresponds to an ideal capacitor, which, however, does not point with the angle 90° in the direction of the negative imaginary axis (Figure 2d, blue arrow), but corresponds to the angle (n*90°). n can assume values between 0 and 1 (Figure 2d, green arrow). For n=1 the CPE corresponds to an ideal capacitor, for n=0 to an ohmic resistor. The CPE thus represents a hybrid of these two components. In a parallel element of resistor and CPE, the CPE causes the semicircle to be tilted [4]. The tilted semicircle is typical for many impedance characteristics, which is why it is often used.

Figure 2: Illustration of different simple equivalent circuits in the Nyquist diagram. (a) RC element with additional resistor; (b) Series connection of two RC elements and one internal resistor; (c) RC element with Warburg impedance and internal resistor; (d) R|| CPE element with internal resistor, Own representation based on Choi, W., Shin, H., et al.[3]

The empirical models already allow more accurate statements about the behavior of a cell, but the parameterization is often done relatively arbitrarily, without being able to be sure that the selected equivalent circuit can actually represent the correct behavior of the cell.

The most reliable models used for the evaluation of impedance spectroscopy are based on the physicochemical processes taking place in the battery. However, these presuppose that the internal structure of the battery is precisely known. Model building is sometimes very complex in individual cases. The physical-chemical concepts are mostly based on the porous electrode theory as developed by John Newman and William Tiedemann in 1975 (cf. [5]). For the models, all processes are considered that have an influence on the behavior of the overall cell. The porous electrode theory is based on the assumption that the electrodes (anode and cathode) consist of a large number of interconnected pores filled with electrolytes. The size, shape and type of the pores are taken into account.

Figure 3 (a) shows the structure of a battery. The anode and cathode each consist of many small grains of active material (for the anode here graphite, for the cathode here LCO) immersed in the liquid electrolyte (For Li-ion batteries). During the discharge process, the lithium moves within a grain at its grain boundary. The lithium deintercalates from the graphite grains and splits into a positively charged lithium ion and an electron. The lithium ion migrates through the porous separator to the cathode with the help of the conducting salt (usually LiPF6). The electron uses conductive soot (carbon black), and migrates via the negative terminal and the connected load to the positive terminal and reaches the cathode this way. At the grain boundaries of the active cathode particles, the Li-ion and electron react with the cathode. The lithium is then deposited between the cobalt oxide layers [6], [7].

Figure 3: Movement of the ions/electrons during discharge using the example of a lithium-ion battery with graphite anode and LCO cathode.

In order to generate an (approximately) correct chemical-physical equivalent circuit of the battery, the individual influence parameters of the chemical discharge process must be modeled in an equivalent circuit. Nevertheless, simplifications are still used here and, for example, the influence of the SEI layer on the surface of the anode is not taken into account. Figure 4 shows the equivalent circuit for the anode and the separator for a single reaction path.

The movement of the Li-ions, the electrons and also the anions of the conducting salt is not loss-free, which is why these are taken into account with ohmic resistances (in each case in the anode and for Li-ion and anion in the separator). The transition from the anode to the current collector is also not loss-free and is included in the model. The interaction between Li-ion and anion is expressed by an additional capacitor.

The precipitation of lithium from the grains of the active material typically has a large influence on the measurable impedance. The actual chemical reaction of the Li battery takes place at the grain boundaries. The following reduction equation applies to graphite anodes:

LiC6 → Li+ + C6 + e

The influence of this intercalation reaction into the cobalt oxide layers is parameterized by a charge transfer resistance  RCT in the model. During the reaction, an enrichment of both the lithium and the conducting salt residue occurs at the surface of the active material grain, because the charge transfer reaction proceeds significantly faster than the diffusion (=movement) of the lithium ions through the system. The accumulation of these compounds in the model is achieved by a double-layer capacitance  CDL   parallel to the charge transfer resistance
RCT represented.

In addition to the diffusion of the Li-ions through the electrolyte, diffusion also takes place within the active material grains. Due to the reduction of the LiC6  to  Li+ a concentration gradient occurs at the grain boundaries, which causes lithium ions to move from the center to the edge within the active material particle (so-called solid-state diffusion). This generally takes place very slowly and is therefore taken into account in the model via the Warburg impedance ZW[8].

Figure 4 only describes the equivalent circuit for one Li reaction path. For all other parallel Li reactions, the model would have to be extended accordingly. In addition, only a half-cell consisting of anode and separator is considered. The equivalent circuit of the cathode is qualitatively similar to the anode equivalent circuit and is therefore not described in detail here. For those interested, the complete model of a complete cell can be found in [8] .

Figure 4: Equivalent circuit path for a single reaction path from anode to separator (cathode path analogous). Representation of a graphite anode. For a complete equivalent circuit, all other reaction paths would have to be connected in parallel and the capacitances between them would have to be taken into account. Own illustration according to [8].

A complete electrochemical model is usually too complex to be used to investigate actual processes in the battery. Therefore, a much used approach is to simplify the model and to summarize or omit parameters that have only a minor influence on the equivalent circuit. A well-known simplification is the De Levie model, which completely neglects the effects of diffusion of ion and anion in separator, anode and cathode. In principle, the model can be further simplified. For example, it may be possible to completely neglect the ohmic resistances in the anode and cathode. This results in a final equivalent circuit as shown in figure 5.

However, the assessment of which parameters of the complete physical equivalent circuit can be neglected is not trivial. It must always be specifically checked in each individual case which simplifications are permissible without distorting the result. For example, if only high frequencies are investigated, ignoring diffusion in the separator, anode and cathode is perfectly justifiable, but for other investigations this adjustment may not be permissible [9]. In other words, the De Levie model may have excellent accuracy in one case and completely fail for another cell in a different set of experiments.

Figure 5: Simplified De Levie model of a complete cell. The admissibility of the simplification is not given as a general rule, but must be verified by suitable procedures, Own illustration according to [9].

This is the reason why the validation of the created models has an important meaning. Already for the measured EIS raw data it makes sense to check their quality. With the help of the Kramers-Kronig analysis – a procedure to check whether the measured data are linear, causal and stable – a first preliminary investigation can already be carried out [1]. The Distribution of Relaxation Times (DRT) method can help in the creation of the physico-chemical based electrical equivalent circuits. It is often difficult for an impedance spectrum to find all the intrinsic RC members, especially when the respective semicircles partially overlap. With DRT it is easier to find all time constants and thus to establish correct equivalent circuits. An explanation of the method can be found e.g. on the homepage of Biologic [10]. Last, it is always useful (no matter if empirical or physico-chemical equivalent circuits are involved) to validate the model more deeply, if possible. One way to do this is to change individual parameters such as the layer thickness of the anode and then check whether the model correctly reflects this change [1].

Solid-state batteries and impedance spectroscopy

After giving a general introduction to impedance spectroscopy in the first two sections and then explaining the procedure for forming empirical and electrochemical equivalent circuit models, this section now deals specifically with impedance spectroscopy of solid-state batteries. First, a brief overview of the current applications of impedance spectroscopy is given, followed by a discussion of the specifics of equivalent circuit modeling of solid-state batteries. Finally, an overview of other possible research fields of impedance spectroscopy for solid-state batteries is presented.

Possible applications of impedance spectroscopy of solid-state cells

The use of impedance spectroscopy is currently limited mainly to the material analysis of solid-state batteries, which is not surprising since this technology is not yet in series production. Therefore, often not full cells are investigated, but only half cells, in order to investigate partial phenomena of the system. Figure 6 lists some examples of research work that has investigated solid-state batteries with impedance spectroscopy.

Figure 6: Exemplary three research papers that have performed impedance spectroscopy measurements on solid-state batteries and based on them also built empirical or simplified physical equivalent circuit models respectively, Own illustration with information from [11],[12],[13].

Solid state battery equivalent circuit diagrams

Although the basic structure of a solid-state is predominantly similar to previous lithium-ion batteries, it differs quite significantly from previous systems in some areas. In the following section, the differences are discussed and the influence of these differences on the modeling of the equivalent circuit is analyzed.

  • Separator/electrolyte: The most obvious difference between the solid-state battery and the lithium-ion battery lies in its solid electrolyte, which at the same time takes over the task of the separator. In lithium-ion models, the separator is often modeled with an RC network (or R-CPE network for surpressed semicircles) and the electrolyte with an ohmic resistor (cf. [14].) Vadvha, Hu et al [15] suggest that such a simple network for the solid-state solid electrolyte may not be sufficient to adequately account for the microstructural properties of the material. For sulfide-based LGPS electrolytes, for example, modeling with an R-CPE element, Warburg impedance, and two ohmic resistors (which can, however, be combined into one resistor) would be feasible. One part of the ohmic resistance would then correspond to the resistance inside the polycrystalline grains of the separator and the other resistance to the interfacial resistance at the surface of the grains.
  • Anode|electrolyte and cathode|electrolyte interface: The interface between the solid electrolyte and the active material of the anode or cathode is also a major challenge in solid-state batteries. This is especially true for the Li anode-electrolyte interface in sulfide-based electrolytes, where degradation products (similar to the SEI layer in Li-ion batteries) are deposited and lead to a significant increase in the interface resistance [16], which is then also noticeable by another semicircle in the Nyquist plot (cf. [13]). Since solid-state batteries are currently still under development, the final configuration for series cells is not yet foreseeable. One possibility to solve interface problems in particular (such as those with Li anode|sulfur electrolyte) is the integration of additional protective layers or artificial SEI. These would then have to be taken into account accordingly for the equivalent circuit generation and would then also be visible, if necessary, through further semicircles or a real shift in the Nyquist diagram.
  • Anode and cathode: While the cathode side is expected to remain the same for solid-state batteries and can therefore continue to work with the previous equivalent circuit, the anode with Li metal instead of graphite differs significantly from today’s configurations. There are few papers to date that have addressed the Li anode of solid-state batteries. One of the exceptions is the work of Lee, Choi et al [17] who studied AgF as a lattice structure for anodeless anodes. Here, the impedance spectrum at 0 and 100% SOC was analyzed at cycles 0, 1, 3, and 10, respectively. The corresponding equivalent circuit of the full cell (NMC|SE|AgF) was defined with two R-CPE resonant circuits, Warburg impedance, and ohmic resistance (similar to Figure 5). Thus, the model is not fundamentally different from Li-ion cells with graphite cells. In a study by Cheng, Kushida et al [18], the aging of a solid-state battery with LLZO electrolyte, Li anode and LCO cathode coated with LBO (Li3 BO3 ) was investigated. Here, the equivalent circuit was constructed with three resonant circuits to account for the extra LBO layer. Otherwise, however, the model does not differ from models for Li-ion batteries.

If all the influences on the electrical equivalent circuit of a solid-state battery described above are taken into account, an equivalent circuit can be created on this basis. This equivalent circuit with the corresponding theoretical Nyquist plot is shown in Figure 7. For better representation, it was assumed that the individual semicircles do not overlap, even though this is to be expected for the charge transfer processes in particular. It should be noted that this (approximately) complete equivalent circuit is much too complicated for practical applications, especially for investigations on full cells, and it must be checked which components can be neglected in practice. The equivalent circuit diagrams of the studies in Figure 6 provide a clue in this respect, even if there is no clear picture here as to which semicircle is to be assigned to which electrochemical cause and an oversimplification must rather be assumed here.

Figure 7: (a) Proposed equivalent circuit for solid-state batteries, taking into account the relevant physicochemical parameters. (b) Theoretical Nyquist plot resulting from the equivalent circuit model of (a), assuming no overlap of the semicircles (which usually does not correspond to reality), Own illustration.

Future applications of EIS

To date, the research environment of impedance spectroscopy for solid-state batteries is still relatively straightforward and there is relatively little work systematically addressing this area. Future topics that are now more widely applied to impedance spectroscopy for Li-ion batteries are still largely uncharted territory for solid-state batteries and offer avenues for further research. Vadhva, Hu et al [15] lists some of these fields:

This includes, for example, further work on standardizing the evaluation and modeling of measurement data. To date, there is no standard procedure on how to systematically form models with high quality. Also, checking the quality of measurement data using the Kramers-Kronig procedure is still not standard in all ongoing research. However, there is work that aims to fill this gap and describes an accurate approach for quantitative analysis of impedance measurements. In particular, the increased use of Distrbution of Relaxation Times (DRT) analysis may provide improvements in this area[12].

The evaluation of solid-state battery EIS data using machine-learning algorithms is another field that still leaves a lot of potential open for further research. Especially when first manufacturers start mass production of solid-state batteries in the next years and large amounts of field data are available for evaluation, the automatic evaluation of measurement data offers a lot of potential for optimization.

In-situ (i.e. on-site) measurement of impedance spectroscopy during cell operation represents another growth area. An example application here is the measurement of impedance during the operation of an electric car, for example, while it is being charged or driven. Here, the impedance measurement would serve to monitor the cells and could, for example, detect when individual cells are damaged. For solid-state half cells, it has already been shown that it is potentially possible to detect Li plating and thus the formation of dendrites [19].

Conclusion

In this article, a measurement method was presented with which it is possible to examine a cell or parts of a cell without damaging them. Impedance spectroscopy is a measurement technique that has been successfully used for many years for the investigation of lithium-ion batteries and which provides insights into the physical-chemical structure of the cell. The measurement principle is based on applying an alternating electrical signal to the cell and then measuring the signal response. By varying the signal length, an entire spectrum with different frequencies can be recorded.

The evaluation of the spectra can be done qualitatively by the change in comparison to look-up-tables or by electro-technical or physico-chemical models. The physicochemical models generally achieve the highest accuracy, but are very difficult to develop and validation is also not easy. The most common physicochemical model for Li-ion batteries is based on the theory of porous electrodes and takes into account all relevant chemical reactions and interfaces that have an influence on the system.

The application of impedance spectroscopy for solid-state batteries represents a rather young field of research and therefore there are relatively few publications around this topic so far. Previous works mainly use simple empirical equivalent circuits to evaluate the impedance of solid-state batteries. The main difference in the impedance spectra of solid-state batteries compared to Li-ion batteries lies mainly in the interfaces between separator and electrodes, which are still often a problem today. While these can often be neglected for Li-ion batteries, this is usually not the case for solid-state batteries.

In this article, a proposal was made – analogous to proposals that also exist for Li-ion batteries – as to how a more complete equivalent circuit for solid-state batteries could look. The known factors influencing the impedance, which have been discovered in other works, have been taken into account. The presented equivalent circuit is only intended as an idea for further investigations. Up to now, there is no scientific work on how a physical-chemical model based on the theory of porous electrodes could look like.

 

Impedance spectroscopy for solid-state batteries is overall a field in which there is still much need for development. However, due to the increasing commercialization of solid-state batteries, it can be assumed that this gap will be increasingly closed in the coming years. 

Sources

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