Bulk photovoltaic effect of LiNbO${}_3$:Fe and its small-polaron-based microscopic interpretation

Based on recent experimental evidence on the electronic and optical properties of Fe${}_{\mathrm{Li}}^{2+}$ and Nb${}_{\mathrm{Nb}}^{4+}$ in LiNbO${}_3$:Fe, both strongly determined by their small polaron character, a microscopic model is presented accounting for the main features of the bulk photovoltaic effect (BPVE) in this material. The relative sizes of the components of the photovoltaic tensor are explained on an atomic basis. Optical small polaron transfer from Fe${}_{\mathrm{Li}}^{2+}$ to Nb${}_{\mathrm{Nb}}^{5+}$ conduction band states and the subsequent coherent bandlike electron transport, terminated by the formation of Nb${}_{\mathrm{Nb}}^{4+}$ free small polarons within about ${10}^{-13}$ s, characterize the first steps of the BPVE. These free polarons, transported by thermally activated incoherent hopping, are then trapped by deeper defects such as Nb${}_{\mathrm{Li}}^{5+}$ and Fe${}_{\mathrm{Li}}^{3+}$ impurities. The model allows us to explain the strong increase of the ionization probability of Fe${}_{\mathrm{Li}}^{2+}$ and the coherent transport length with photon energy. The low mobility of the Nb${}_{\mathrm{Nb}}^{4+}$ conduction polarons appears to be the reason for the high open-circuit photovoltaic fields attainable in LiNbO${}_3$.

Figure 1

Arrangement of Fe${}_{\mathrm{Li}}^{2+}$ and neighboring Nb${}_{\mathrm{Nb}}^{5+}$ ions in LiNbO${}_3$, assuming atomic positions unchanged with respect to the defect-free crystal. The lengths of the interionic vectors ${\mathbf{d}}_i$ are given in Å. (a) Cut along a $\mathit{yz}$ glide mirror plane. For three sites, as examples, the axial structure (symmetry $A$${}_1$ in the 3$m$ group) of the involved orbitals of (3$z$${}^2$-$r$${}^2$) type are indicated. For Fe${}_{\mathrm{Li}}^{2+}$ (3$d$${}^6$), shown hatched, the orbital of the last electron is labeled $|I\rangle$. For the Nb${}_{\mathrm{Nb}}^{4+}$ neighbors 1 and 7, they are called $|F_1\rangle$ and $|F_7\rangle$. (b) Projection on a $\mathit{xy}$ plane (perpendicular to $c$); the trace of the glide mirror plane, along $y$ [part (a)] is indicated by the dashed-dotted line. In this figure, the positions of the ${\mathrm{O}}^{2-}$ ions next to the shown cations and their distances to the neighboring cations are included. The thicker the connecting lines, the shorter the cation-oxygen bonds. The Fe${}_{\mathrm{Li}}^{2+}$ ion is shown hatched. The positions of the Nb and O ions relative to the Fe${}_{\mathrm{Li}}^{2+}$ position, i.e., above and below the $\mathit{xy}$ plane, are indicated by the respective shadings (see legend).

Figure 2

Optical absorption of Fe${}_{\mathrm{Li}}^{2+}$ in LN. The experimental data (dots) are reproduced (Ref. 17) (full line) at energies up to the peak near 2.6 eV by the small polaron approach [Eq. (4)]. The deviation between this small polaron prediction and experiment at energies higher than the peak energy is attributed (Ref. 17) to excitations to extended states lying within the conduction band. In this range, there is also a superposition with transitions from the valence to the conduction band, rising steeply. We note that a small peak is superimposed on the broad band; it is caused by Fe${}^{3+}$ spin forbidden transitions (Ref. 20).

Figure 3

Comparison of the energy dependences of the absorption $\alpha (\hslash \omega )$ (Ref. 17) and the Glass factor $\kappa (\hslash \omega )$ (Ref. 48) for LN:Fe${}_{\mathrm{Li}}^{2+}$. The maximum of $\kappa (\hslash \omega )$, about 3.1 eV, lies at energies where $\alpha$ is likely to be attributed to optical excitations to delocalized final states (Ref. 17). This suggests that ionization into such bandlike states is at the root of the BPVE as discussed in the text.

Figure 4

Scheme of the Fe-Nb potential sheets, allowing us to indicate the small polaron optical excitation from Fe${}_{\mathrm{Li}}^{2+}$ to Nb${}_{\mathrm{Nb}}^{5+}$. $a$$\to$$b$: Here the electron is moved forward to a potential sheet centered around Nb${}_{\mathrm{Nb}}^{5+}$, while the lattice is fixed. $b$$\to$$c$: The Franck-Condon excited lattice adapts to the new electronic state Nb${}_{\mathrm{Nb}}^{4+}$, accompanied by multiphonon emission. At $c$, the electron crosses to the potential sheet centered around Fe${}_{\mathrm{Li}}^{3+}$ (backward movement). $c$$\to$$a$: The lattice adapts to the newly formed Fe${}_{\mathrm{Li}}^{2+}$, again emitting many phonons. For the derivation of this scheme, see Appendix .

Figure 5

Scheme of the Franck-Condon intervalence transitions from Fe${}_{\mathrm{Li}}^{2+}$ to Nb${}_{\mathrm{Nb}}^{5+}$ final states and of the processes following this initial step. At low photon energies, the transitions are dominated by the local Fe${}_{\mathrm{Li}}$ defect potential, and hybridization with the conduction band states is then low. Ionization from such low-lying initial excited states is weak and the Glass factor $\kappa (\hslash \omega )$ is accordingly small (Fig. 4). At higher energies, the localizing influence of the defect is less decisive; ionization, mediated by a transfer integral $J_b$, and tunneling farther into the crystal is more pronounced and the Glass factor is thus expected to rise continuously with photon energy. It is presumed that the coherent tunneling movement occurs in the form of large polarons; the coupling to the lattice accompanying the extended electronic tunneling is symbolized by truncated parabolae. In such optically excited states, the carriers are not thermalized. They can lose their surplus energy by forming free Nb${}_{\mathrm{Nb}}^{4+}$ small polarons having ground state energies slightly below the rigid lattice conduction band continuum. From there, the polarons can be transferred, by thermally activated hopping, to lower defect states, provided, for instance, by Nb${}_{\mathrm{Li}}$ defects and further Fe${}_{\mathrm{Li}}$ ions.

Figure 6

(a) Short-circuit situation, supporting the discussion (see text) of the direction of the incoherent, diffusive current as related to the coherent one. For details, see text. (b) Open-circuit situation: the coherent forward current is compensated to zero by the incoherent backward drift current, driven by the open-circuit field $E_{\mathrm{phv}}$. Also, in this case of a zero net current, a photovoltaic Hall effect is measured (Ref. 46): essentially, only the coherent part contributes to it because of the very low mobility of the incoherent part.

Figure 7

Potentials related to two small polarons forming at two equivalent sites, as plotted over the relative coordinate $q=Q_1-Q_2.$