Revealing trap depth distributions in persistent phosphors with a thermal barrier for charging

The performance of persistent phosphors under given charging and working conditions is determined by the properties of the traps that are responsible for these unique properties. Traps are characterized by the height of their associated barrier for thermal detrapping, and a continuous distribution of trap depths is often found in real materials. Accurately determining trap depth distributions is hence of importance for the understanding and development of persistent phosphors. However, extracting the trap depth distribution is often hindered by the presence of a thermal barrier for charging as well, which causes a temperature-dependent filling of traps. For this case, we propose a method for extracting the trap depth distribution from a set of thermoluminescence glow curves obtained at different charging temperatures. The glow curves are first transformed into electron population functions via the Tikhonov regularization, assuming first-order kinetics. Subsequently, the occupation of the traps as a function of their depth, quantified by the so-called filling function, is obtained. Finally, the underlying trap depth distribution is reconstructed from the filling functions. The proposed method provides a substantial improvement in precision and resolution for the trap depth distribution compared with existing methods. This is hence a step forward in understanding the (de)trapping behavior of persistent and storage phosphors.

Figure 1

The procedure for a thermoluminescence (TL) experiment. The phosphor was charged by ultraviolet (UV) light at variable charging temperature $T_{\text{ch}}$ with a fixed charging duration $t_{\text{ch}}$. After lowering the temperature to $T_0\le T_{\text{ch}}-30\text{K}$, the phosphor will be heated to temperature $T_{\text{max}}$ at a fixed heating rate $\beta$ (0.5 K ${\mathrm{s}}^{-1}$). Here, $T_{\text{ch}}=[T_{\text{ch}0}:\mathrm{\Delta }{T}_{\text{ch}}:T_{\text{chm}}]$, with $\mathrm{\Delta }{T}_{\text{ch}}=5\text{K},T_{\text{ch}0}=223\text{K},\text{and}{T}_{\text{chm}}=393\text{K}$.

Figure 2

Flowchart of the method. The thermoluminescence (TL) curves are first converted into electron population functions, and the relative filling functions are approximated. Two methods can be chosen to calculate the trap depth distribution $N\left(E_t\right)$.

Figure 3

The electron population function $n(E_t,q,t_c)$. (a) The kernel $K(E_t,T)$ maps the thermoluminescence (TL) curve $I\left(T\right)$ into the electron population function $n(E_t,q,t_c)$ via the discrete regularization method in Eqs. (4a)–(4c) ($T_{\text{ch}}=243\text{K}$). For variable charging temperature $T_{\text{ch}}$, (b) the TL curves can thus turn into (c) the electron population functions $n(E_t,q,t_c)$, from which an envelope can be constructed accordingly (the orange line). Note ${\nu }_r={10}^{10}{\text{s}}^{-1}$.

Figure 4

The existence of $\mathrm{\Delta }E$. (a) The total number of trapped electrons per volume $n_t(q,t_c)$ as a function of $T_{\text{ch}}$ reaches a peak at $T_{\text{ch}}\approx 263\text{K}$. (b) As $T_{\text{ch}}$ decreases, the corresponding difference $\mathrm{\Delta }{n}_t(q,t_c)$ turns from positive into negative at $T_{\text{ch}}\approx 263\text{K}$.

Figure 5

Local model for trapping and recombination. The isolated-pair approximation is assumed for trapping and recombination. (a) The trapping process takes place between an isolated luminescent activator and an empty trap, i.e., an activator-trap pair. (b) Recombination takes place between an isolated filled trap and a hole, i.e., an electron-hole pair. Parameters are displayed for important electron transitions. The trap depth distribution $N\left(E_t\right)$ and electron population function $n(E_t,q,t_c)$ are shown as a gray curve and a blue filled area, respectively.

Figure 6

The filling function in the case without a thermal barrier for charging ($\mathrm{\Delta }E=0.0$ eV). (a) The filling function $f(E_t,\mathrm{\Delta }E,q,t_c)$ in the $T_{\text{ch}}\times E_t$ plane ($q\to T_{\text{ch}}$) suggests the magnitude function $f_m\left(E_t\right)$ is independent of charging temperature $T_{\text{ch}}$. The gray line indicates the filling function for $q\to 295\text{K}$. (b) Indeed, the magnitude function $f_m\left(E_t\right)$ is independent of charging temperature $T_{\text{ch}}$ (orange line). An individual filling function (e.g., the one for $q\to 295$ K) characterizes a magnitude of filling $f_0$ and a characteristic trap depth $E_o$. Note that $t_{\text{ch}}=0.01\text{s}$.

Figure 7

The filling function in the case of a thermal barrier for charging ($\mathrm{\Delta }E=0.255\text{eV}$). (a) The filling function $f(E_t,\mathrm{\Delta }E,q,t_c)$ (top panel) and the normalized filling function $f(E_t,\mathrm{\Delta }E,q,t_c)/f_m\left(E_t\right)$ (bottom panel) in the $T_{\text{ch}}\times E_t$ plane ($q\to T_{\text{ch}}$) suggest an optimal range of charging temperature for a given trap depth. (b) The magnitude function $f_m\left(E_t\right)$ is tangent to the filling function $f(E_t,\mathrm{\Delta }E,q,t_c)$. (c) The magnitude function $f_m\left(E_t\right)$ can be approximated by the characteristic trap depth $E_i$ and the magnitude $f_0(\mathrm{\Delta }E,q)$ of the filling function. Note $t_{\text{ch}}=100$ s.

Figure 8

Approximating the magnitude function. (a) The relative filling function $R(E_t,q,t_c)$ (bottom panel, $q_r\to 223\text{K}$) approximates the filling function $f(E_t,\mathrm{\Delta }E,q,t_c)$. The discrete $[E_i,R_0(\mathrm{\Delta }E,q)]$ pair and the magnitude function $R_m\left(E_t\right)$ are displayed as blue dots and an orange line, respectively (bottom panel). The simulated $E_i$ agrees with the experimental ones within a constant difference (top panel). (b) The trap depth distribution $\tilde{N}\left(E_t\right)$ is calculated from the envelope $n_{\text{env}}(E_t,q,t_c)$ by using Eq. (19). Note $n_{\text{env}}(E_t,q,t_c)$ has been scaled to the same magnitude of $\tilde{N}\left(E_t\right)$.

Figure 9

Extracting $N\left(E_t\right)$ via Eq. (20). (a) The area between two normalized electron population functions for $q$ and $q_1$ characterizes $\mathrm{\Delta }{\tilde{n}}_t(q,t_c)$ (gray area, bottom panel). Here, $R_n(E_t,q,t_c)=R(E_t,q,t_c)/R_0(\mathrm{\Delta }E,q)$. (b) The trap depth distribution $\tilde{N}\left(E_t\right)$ according to Eq. (20) is represented by a histogram, together with $\mathrm{\Delta }{\tilde{n}}_t(q,t_c)/\delta E$ for $q\to 263\text{K}$ as a blue bar.

Figure 10

Trap depth distribution of ${\mathrm{BaSi}}_2{\mathrm{O}}_2{\mathrm{N}}_2$:2%${\mathrm{Eu}}^{2+}$. (a) The trap distribution $\tilde{N}\left(E_t\right)$ extracted by Eq. (24) (blue) agrees with that extracted from Eq. (20) (histogram). (b) The electron population function at low charging temperature, e.g., $q_1\to 233\text{K}$, can approximate the trap depth distribution $N\left(E_t\right)$ but may fail in some ranges.

Figure 11

The influence of the recombination frequency factor ${\nu }_r$. (a) Changing the frequency ${\nu }_r$ to $a{\nu }_r$ by multiplying a positive scalar $a$ will compress ($a<1$) or stretch ($a>1$) the trap depth distribution and shift it along the $E_t$ axis. (b) The value of trap depth $E_m$ increases almost linearly with increasing ${\text{log}}_{10}\left[{\nu }_r\left({\text{s}}^{-1}\right)\right]$. Note the trap depth distribution can be approximated by the normalized electron population function $n(E_t,q,t_c)/n(E_m,q,t_c)(q\to 233\text{K})$, with $n(E_m,q,t_c)\text{and}{E}_m$ being the maximum of $n(E_t,q,t_c)$ and the corresponding trap depth, respectively.