# Battery Characterization via Eddy-Current Imaging with Nitrogen-Vacancy Centers in Diamond
## 4 Battery Measurement  
The dimensions of the solid state battery measured in this work are \(4.0\times 3.0\times 1.0\,\mathrm{mm^{3}}\), and are shown in Figure 3a [17]. This comprises Li-based-ceramic-oxide inner electrodes (noted as 2), a Li-based-ceramic-oxide electrolyte (denoted as 3), a copper charge collector (indicated as 1), and external electrodes made of aluminum. Figure 3b shows a photograph of the solid battery, which has an artificial 1 mm-length defect at the right external electrode. The battery sample was placed at a distance of 0.1 mm to the diamond at the perpendicular direction (\(x\)-direction), and scanned over the transverse plane (\(y\)-\(z\) plane). Figure 3g-j show the spatially resolved phase signals of the LIA at the corresponding modulation frequencies of 1 kHz, 5 kHz, 10 kHz and 40 kHz, respectively. The external electrodes including the defect are distinctly visible under the modulation frequency of 1 kHz as shown in Figure 3g. As the modulation frequency increases, the phase \(\theta\) shows an image of the battery structure between the electrodes. There the induced field from different portions of the battery can be differentiated, as shown in Figure 3h-j. Figure 3c-f show the spatially resolved amplitude signals of LIA at the corresponding modulation frequencies of 1 kHz, 5 kHz, 10 kHz and 40 kHz. In order to investigate the ability of the diamond-based magnetometer to identify structural anomalies inside batteries, we introduced two kinds of artificial anomalies. First, an iron-containing brass cylinder with a height of 1 mm and diameter of 1 mm was placed inside the battery, serving as an impurity. The results are shown in Figure 4a-d, displaying the spatially resolved amplitude and phase signals of the LIA for the solid battery at frequencies of 5 kHz and 20 kHz. This impurity is visible at both frequencies in the phase signal, indicated with red dashed circles in Figure 4. The impurity image is absent in the amplitude signal at frequencies lower than 5 kHz, this is because the amplitude variations caused by the impurity, are smaller than the background shifts caused by temperature changes, for the amplitude signal. The other anomaly is made by cutting the battery in half and then attaching the halves back together, leaving no defect visible to the naked eye. The results are shown in Figure 5, where (a)-(d) display spatially resolved amplitude and phase signals of LIA for the solid battery at frequencies of 5 kHz and 20 kHz. The crevice is distinctly visible in the middle of the battery in the phase signals at both frequencies, indicated by the yellow ellipses in (b) and (d). In the amplitude signals the crevice is absent again. The primary field produced at the center of the coil where the battery is located is given by:  
\[B_{\mathrm{primary}}=\frac{\mu_{0}\times N\times I}{2r_{\mathrm{coil}}}, \tag{1}\]  
where \(\mu_{0}=4\pi\times 10^{-7}\,\mathrm{T}\times\mathrm{m}/\mathrm{A}\) is the permeability of free space, \(N=4\) is the number of turns of the RF coil, \(I=0.8\,\mathrm{A}\) is the amplitude of the alternating current through the coil, and \(r_{\mathrm{coil}}=4.2\,\mathrm{mm}\) is the radius of the coil. To perform the LIA detection scheme the modulated field amplitude was selected to be 0.48 mT, confirmed experimentally by the GSLAC PL scan and theoretically by Equation (1). For AC measurements, the magnetic susceptibility \(\chi_{m}\) yields two quantities [18; 19]: the magnitude and phase of the susceptibility. Alternatively, we can express it in the form of complex susceptibility with the real component \(\chi_{r}\) and imaginary component \(\chi_{i}\),  
\[\chi_{m}=\chi_{r}+i\chi_{i} \tag{2}\]  
where \(\chi_{r}\) is the slope of the direct current (DC) magnetization curve \(M(H)\) of the battery, and \(\chi_{i}\) indicates the dissipative processes in the battery. The secondary field produced by the battery also depends on the skin depth in the conducting material:  
\[\delta=\sqrt{\frac{2}{\sigma\omega\mu}}, \tag{3}\]Figure 3: **(a)** Battery schematic.