Disentangling small-polaron and Anderson-localization effects in ceria: Combined experimental and first-principles study

By comparison of the electrical conductivity of ceria doped with penta- and hexavalent ions, we separate the total electron localization energy into the two contributions originating from the small polaron effects and the Coulomb interaction with the donor ions. The upper bound of the itinerant small polaron hopping energy is estimated at $66\pm 20$ meV. The binding energy of the ${\mathrm{Ce}}^{3+}–M^{5+/6+}$ defect complex increases from 121 meV for $M={\mathrm{Nb}}^{5+}/{\mathrm{Ta}}^{5+}$ to 243 meV for $M={\mathrm{W}}^{6+}/{\mathrm{U}}^{6+}$. The first-principles simulations are in qualitative agreement with the experimental findings. At low temperatures the $f$ electrons bound to the donor defects show dielectric relaxation with the lowest activation energy of 2.7 and 17 meV for Nb(Ta)- and W-doped ceria, respectively. Remarkably, these energies are significantly smaller than the hopping energy of the itinerant small polarons. While both the electron-lattice and the electron-defect interactions cause the $f$ electron localization in real-case ceria, the latter effects seem to be the dominant.

Figure 1

Schematics of the itinerant SP (a) and the SP bound to donor defect (c) by Coulomb potential (b).

Figure 2

Electrical conductivity of ${\mathrm{Ce}}_{1-x}{\mathrm{M}}_x{\mathrm{O}}_2,M$ = Nb, Ta, and W, as a function of reciprocal temperature. The concentration of ${\mathrm{Ce}}^{3+}$ per formula unit is indicated in the legend. The conductivity of ${\mathrm{Ce}}_2{\mathrm{O}}_3$ adapted from Ref. [19] is included for comparison.

Figure 3

Activation energy of the conductivity of the donor-doped ceria as a function of the effective charge of the donor ion, $\mathrm{\Delta }q$.

Figure 4

(a) Schematic of the ${\mathrm{Ce}}^{3+}–M^{5+}$ dipolar defect in ceria. (b) Energy diagram of the ${\mathrm{Ce}}^{3+}–M^{5+}$ dipole. The degeneracy of the energy levels is indicated in the brackets.

Figure 5

Low-temperature dependence of the dielectric constant (top panel) and dielectric loss, tan $\delta$ (bottom panel) of ${\mathrm{Ce}}_{0.9995}{\mathrm{Ta}}_{0.0005}{\mathrm{O}}_2$ ceramics measured at selected ac frequencies, $f$, from 1 Hz to 1 MHz. The thick solid black lines are the dielectric data for pure ${\mathrm{CeO}}_2$ measured at 70 Hz and 100 kHz.

Figure 6

Top panel: Low-temperature dependence of $\mathrm{\Delta }{\varepsilon }^{\prime }={\varepsilon }_0^{\prime }$ – ${\varepsilon }_{\infty }^{\prime }$, where ${\varepsilon }_0^{\prime }$ was measured at 1 Hz and ${\varepsilon }_{\infty }^{\prime }$ was measured at 1 MHz. (bottom panel) Dielectric loss, tan $\delta$ of ${\mathrm{Ce}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_2$ ceramics measured at $f=1$ Hz.

Figure 7

Frequency dependence of the tan $\delta$ peak as a function of 1/$T$ for ${\mathrm{Ce}}_{0.9995}{\mathrm{Ta}}_{0.0005}{\mathrm{O}}_2$ ceramics.

Figure 8

Low-temperature dependence of the dielectric constant (top panel) and dielectric loss, tan $\delta$ (bottom panel) of ${\mathrm{Ce}}_{0.999}{\mathrm{W}}_{0.001}{\mathrm{O}}_2$ ceramics measured at selected ac frequencies, $f$, from 4 Hz to 5 kHz.

Figure 9

Formation energies of (a) ${\mathrm{Nb}}_{\mathrm{Ce}}$, (b) ${\mathrm{Ta}}_{\mathrm{Ce}}$, and (c) ${\mathrm{W}}_{\mathrm{Ce}}$ in different charge states as a function of Fermi level $\left(E_F\right)$. The blue, green, red, and purple lines indicate neutral charge state $M_{Ce}^0$, neutral complex ${(M_{Ce}^{n+}-n\text{polaron})}^0$, 1+ charge state $M_{\text{Ce}}^{1+}$, and 2+ charge state $M_{\text{Ce}}^{2+}$, respectively.

Figure 10

Calculated spin densities of the favorable configuration of (a) ${({\mathrm{Nb}}_{\mathrm{Ce}}^{1+}-1\text{polaron})}^0$, (b) ${({\mathrm{Ta}}_{\mathrm{Ce}}^{1+}-1\text{polaron})}^0$, (c) ${({\mathrm{W}}_{\mathrm{Ce}}^{2+}-2\text{polarons})}^0$, and (d) ${({\mathrm{U}}_{\mathrm{Ce}}^{2+}-2\text{polarons})}^0$, respectively. The position of the first-nearest-neighbor O atoms surrounding the dopant $M$ is labeled by $i$ =1, 2, 3,…8) as indicated in panel (a). The distance from the dopant $M$ to ${\mathrm{O}}_i$ atom $\left(M\text{−}{\mathrm{O}}_i\right)$ is listed in Table 1.

Figure 11

Density of states (DOS) from spin-polarized calculations of (a) ${({\mathrm{Nb}}_{\mathrm{Ce}}^{1+}-1\text{polaron})}^0$, (b) ${({\mathrm{Ta}}_{\mathrm{Ce}}^{1+}-1\text{polaron})}^0$, (c) ${({\mathrm{W}}_{\mathrm{Ce}}^{2+}-2\text{polarons})}^0$, and (d) ${({\mathrm{U}}_{\mathrm{Ce}}^{2+}-2\text{polarons})}^0$, respectively. The valence band maximum is set to zero and the dashed line represents Fermi level. The localized state (small polaron) is slightly below the conduction band.