Depolarization current density within the symmetric double-well potential model for solid ionic conductors

An approximate analytic expression for the depolarization current density has been derived within the symmetric double-well potential model. How to relate the depolarization current density response measured under iso- or nonisothermal conditions with the distribution of energy barriers has been demonstrated. A hyperbolic heating regime has been suggested to enhance the role of high-energy barriers in the thermally stimulated depolarization current response.

Figure 1

(Color online) The isothermal regime of dipole relaxation. (a) The barrier distribution obtained according to Eq. (13) (dashed line) compared with the exact barrier distribution given by Eq. (14) (solid line) with $\delta E=0.2\mathrm{eV}$, ${\overline{E}}_1=0.6\mathrm{eV}$, ${\overline{E}}_2=1.0\mathrm{eV}$, ${\sigma }_1={\sigma }_2=0.1\mathrm{eV}$, and weights $p_1=0.2$ and ${\alpha }_1=0.6$. (b) The relative depolarization current, $j\left(t\right)∕j_0$ [$j_0=j(t=0)$ stands for the initial current density], versus time according to Eq. (3) (solid line) and Eq. (5) (dashed line) (the difference between two lines is hardly recognizable on the scale of the graph). The following values of the parameters have been used to obtain the data, ${\tilde{\omega }}_0=2\times {10}^{12}{\mathrm{s}}^{-1}$ and $T_0=300\mathrm{K}$, i.e., $k{T}_0\simeq 0.026\mathrm{eV}$.

Figure 2

(Color online) The dependence of the demarcation energy, $E_d$, versus temperature for the linear heating regime for two values of the heating rate, $q=0.003\mathrm{K}∕\mathrm{s}$ (the top set of curves) and $q=0.3\mathrm{K}∕\mathrm{s}$ (the bottom set of curves), and with ${\tilde{\omega }}_0=2\times {10}^{12}{\mathrm{s}}^{-1}$ and $T_0=100\mathrm{K}$ for all the curves. The bold solid lines represent the exact numerical solution of Eq. (18), while the other curves are obtained according to approximate relation (22) for different values of typical energy $E_t$ (as marked).

Figure 3

(Color online) The linear heating regime of a dipole relaxation. (a) The barrier distributions obtained according to Eq. (19) with the demarcation energy numerically found by solving Eq. (18) and using expression (23) (dashed line), and according to Eq. (26) with approximate expression (22) (dot-dashed line) compared with the exact barrier distribution given by Eq. (14) (solid line) obtained for the same values of parameters used in Fig. 1 and for $T_0=100\mathrm{K}$ and $q={10}^{-1}\mathrm{K}∕\mathrm{s}$. (b) The relative depolarization current, $j\left(t\right)∕j_0$ versus time according to Eq. (16) (solid line), from Eq. (17) with a numerically found demarcation energy by solving Eq. (18) (dashed line), and using approximate Eqs. (25, 22) (dot-dashed line).

Figure 4

(Color online) The dependence of the demarcation energy, $E_d$, versus temperature for the hyperbolic heating regime for two values of the heating rate, $q=0.0008\mathrm{K}∕\mathrm{s}$ (the top set of curves) and $q=0.08\mathrm{K}∕\mathrm{s}$ (the bottom set of curves), and with the same values of other parameters as in Figs. 1, 2. The bold solid lines represent the exact numerical solution of Eq. (28), while the other curves are obtained according to approximate relation (29) for different values of typical energy $E_t$ (as marked).

Figure 5

(Color online) The hyperbolic heating regime of dipole relaxation. (a) The barrier distributions obtained according to Eq. (19), with the demarcation energy numerically found by solving Eq. (28) and using expression (31) (dashed line), and according to Eq. (33) with approximate expression (22) (dot-dashed line) compared with the exact barrier distribution given by Eq. (14) (solid line), which is obtained for the same values of parameters used in Figs. 1, 2 and $q=0.08\mathrm{K}∕\mathrm{s}$. (b) The relative depolarization current, $j\left(t\right)∕j_0$, versus time according to Eq. (16) (solid line), from Eq. (17) with a numerically found demarcation energy by solving Eq. (18) (dashed line), and using approximate Eqs. (25, 22) (dot-dashed line).

Figure 6

(Color online) The comparison of the approximately hyperbolic and hyperbolic heating regimes of dipole relaxation. (a) The barrier distributions obtained according to Eq. (33) with the demarcation energy found numerically by solving Eq. (28) and with the current density numerically obtained for the approximately hyperbolic heating regime [$N=10$ (thin solid line) and $N=30$ (dot-dashed line)] compared with the exact barrier distribution given by Eq. (14) (thick solid line) as obtained for the same values of parameters used in Figs. 1, 2 ($q=0.08\mathrm{K}∕\mathrm{s}$ and $t_{\mathrm{exper}}={10}^3\mathrm{s}$). (b) The relative depolarization current, $j\left(t\right)∕j_0$, versus time according to Eq. (16), which is calculated both for exactly hyperbolic (the thick solid line) and approximately hyperbolic [$N=10$ (thin solid line) and $N=30$ (dot-dashed line)] heating regimes. The inset in (b) shows the dependence of temperature versus time for the approximately hyperbolic heating regime (for $N=10$).

Figure 7

(Color online) The temperature dependence of the thermally stimulated currents in ${\mathrm{Na}}^{+}$ mordenite for four rates in the linear heating regime [(a)–(d)] as marked in the figure. The solid curves describe the experimental data. The dashed and dot-dashed lines were obtained by using Eq. (16) with the distribution of barriers evaluated by a numerical fit to the experimental data as described in Ref. 21 (see the solid line in Fig. 8) and by using Eq. (26) within the approximate scheme described in Sec. 4A (see the dashed line in Fig. 8), respectively.

Figure 8

(Color online) The distribution of energy barriers, $\rho \left(E\right)$, in ${\mathrm{Na}}^{+}$ mordenite. The solid line was obtained by a numerical fit to the experimentally measured current density versus temperature at $q=2\mathrm{K}∕\min \simeq 3.3\times {10}^{-2}\mathrm{K}∕\mathrm{s}$ [see the solid line in Fig. 7c] by using Eqs. (16, 4) assuming that $\rho \left({\omega }_0\right)=\delta ({\omega }_0-{\tilde{\omega }}_0)$ with ${\tilde{\omega }}_0=4\times {10}^{12}{\mathrm{s}}^{-1}$ (see Ref. 21 for more details). The dashed line was obtained according to Eq. (26) by using the same experimental data for $j\left(T\right)$ and the same value of ${\tilde{\omega }}_0$ as for the solid line $(E_t=0.757\mathrm{eV})$. The inset demonstrates the collapse of the distributions of barriers obtained from Eq. (26) for different quench heating rates: $q=0.5\mathrm{K}∕\min \simeq 8.3\times {10}^{-3}\mathrm{K}∕\mathrm{s}$ (the solid line), $q=1\mathrm{K}∕\min \simeq 1.67\times {10}^{-2}\mathrm{K}∕\mathrm{s}$ (the dashed line), $q=2\mathrm{K}∕\min \simeq 3.3\times {10}^{-2}\mathrm{K}∕\mathrm{s}$ (the dot-dashed line that is indistinguishable on the scale of the graph from the solid one), and $q=5\mathrm{K}∕\min \simeq 8.3\times {10}^{-2}\mathrm{K}∕\mathrm{s}$ (the double dot-dashed line).

Figure 9

(Color online) Thermally stimulated current response in the hyperbolic heating regime $(q=0.08\mathrm{K}∕\mathrm{s})$ for three different distributions of attempt frequencies: (i) $\delta$-functional distribution (the solid line), (ii) the box distribution (the dashed line), and (iii) the exponential distribution (the dot-dashed line). The distribution of the energy barriers was modeled by Eq. (14) with the same values of parameters used in Fig. 1. All the other parameters are the same as in Figs. 1, 2.

Figure 10

(Color online) The distribution of energy barriers in ${\mathrm{Na}}^{+}$ mordenite obtained from $j\left(T\right)$, as shown in Fig. 7c (solid line), by using approximate relation (26) for different values of the characteristic frequency ${\tilde{\omega }}_0$ as marked in the figure.

Figure 11

(Color online) The distribution of attempt frequencies ${\omega }_0$ (measured in ${\mathrm{s}}^{-1}$) in ${\mathrm{Na}}^{+}$ mordenite obtained from $j\left(T\right)$ shown in Fig. 7 according to Eq. (35). Different lines correspond to different quench rates: $q=0.5\mathrm{K}∕\min$ (solid line), $q=1\mathrm{K}∕\min$ (dashed line), $q=2\mathrm{K}∕\min$ (dot-dashed line), and $q=5\mathrm{K}∕\min$ (double dot-dashed line). The inset shows the temperature dependence of the demarcation frequency [see Eq. (36)] for $q=0.5\mathrm{K}∕\min$ (solid line) and $q=5\mathrm{K}∕\min$ (dashed line). The value of $\tilde{E}=0.757\mathrm{eV}$ has been used for all of the curves.