## 3 Results and Discussion
### YbGAGG: Cr\({}^{3+}\) Ratiometric Thermometer Based on the Emission of Yb\({}^{3+}\) (Mode 1)  
The temperature dependence of PL of YbGAGG: Cr\({}^{3+}\) from 100 K to 400 K was measured and shown in Figure 5a. A notable observation is that the intensity of the central PL peak, located near 1026 nm (within the second BTW), decreases as the temperature increases. Furthermore, within the first BTW (650-950 nm), the PL spectra display an inverse temperature dependence compared with the central peak H in the more extended wavelength range. The reason for this phenomenon is the Stark splitting of the \({}^{2}\)F\({}_{5/2}\) level of Yb\({}^{3+}\) in the garnet, resulting in the separation into the lowest Stark sublevel (denoted as \(Z_{\text{low}}\)) and several higher ones (denoted as \(Z_{\text{high}}\), typically containing two levels according to [PERSON]'s work [19]). Due to the Boltzmann distribution, the intensity of the PL from \(Z_{\text{high}}\) exhibits a positive correlation with temperature (corresponding to the short wavelength range before 1000 nm in Figure 5a). To obtain the temperature dependence of the intensity of each PL peak of YbGAGG: Cr\({}^{3+}\), a deconvolution analysis was performed on the PL spectra across a temperature range of 100-400 K. Examples of this analysis process are shown in Figure 5b. Detailed information on the temperature dependence of the peak center position and peak intensity for peaks labeled A-J were derived from deconvolution and are shown in Figure 5c,d. Figure 4: Luminescence decay curves of Cr\({}^{3+}\) in YbGAGG and GAGG. After obtaining the temperature dependence of the position and intensity of each peak, peaks C and H, located in the first BTW and second BTW, respectively, were selected due to their contrasting temperature dependencies. These peaks were then utilized to determine the temperature dependence of the LIR. The temperature dependence of \(LIR=I_{C}/I_{\rm H}\) (Mode 1) was calculated and shown in Figure 6a. The cubic polynomial function was employed to fit the temperature dependence of the LIR. Subsequently, absolute sensitivity (\(S_{\rm a}\)) and relative sensitivity (\(S_{\rm r}\)) were calculated using Equations (1) and (2), respectively. \[S_{\rm a}=\frac{\mathrm{d}(LIR)}{\mathrm{d}T} \tag{4}\]  
\[S_{\rm r}=\frac{1}{LIR}\times\frac{\mathrm{d}(LIR)}{\mathrm{d}T}\times 100\% \tag{5}\]  
The calculated \(S_{\rm a}\) and \(S_{\rm r}\) curves are shown in Figure 6b. Notably, when the temperature reaches 122 K, \(S_{\rm r}\) reaches its maximum value of 1.30% K\({}^{-1}\). This value exceeds the \(S_{\rm r}\) of the Nd\({}^{3+}\)-based luminescence thermometer, which also utilizes Stark sublevels for temperature measurements but has a lower \(S_{\rm r}\) maximum value of 0.7% K\({}^{-1}\)[17]. However, as the temperature increases above 122 K, \(S_{\rm r}\) monotonically decreases, leaving a value of 0.55% K\({}^{-1}\) for \(S_{\rm r}\) at 310 K. This trend indicates that Mode 1 of the YbGAGG: Cr\({}^{3+}\) thermometer has the potential for application in biological temperature measurements and also exhibits higher sensitivity in lower temperature ranges. Figure 5: (**a**) Temperature dependence of the PL spectrum; (**b**) deconvolution analysis of the photoluminescence spectra at 100 K and 400 K; temperature dependence of (**c**) integrated intensity and (**d**) peak position of A–H peaks. Figure 6: (**a**) Temperature dependence of the luminescence intensity ratio, fitted by a cubic polynomial function; (**b**) calculated temperature dependence of \(S_{\rm a}\) and \(S_{\rm r}\) for the YbGAGG: Cr\({}^{3+}\) thermometer. YbGAGG: Cr\({}^{3+}\) Thermometer Based on the Photoluminescence Intensity Ratio of Cr\({}^{3+}\)/Yb\({}^{3+}\) (Mode 2)  
The temperature dependence of the PL spectrum of the YbGAGG: Cr\({}^{3+}\) sample, excited at 455 nm, is shown in Figure 7a. The several PL peaks in the range of 650-800 nm originate the \({}^{2}\)E-\({}^{4}\)A\({}_{2}\) transition of Cr\({}^{3+}\), and their intensity strongly decreases with increasing temperature.