A polaron approach to photorefractivity in Fe : LiNbO₃

In the following we consider the simplest experimental situation of a Fe:LN sample illuminated with a uniform strip of light (see figure 1). The c axis is supposed to be parallel to the input surface and perpendicular to the illuminated strip, to realize a quasi 1-D configuration. In these conditions a refractive index change is observed as a consequence of the buildup of a light-induced space charge field and of the electro-optic effect [3, 15]. Thus, the main quantities of interest in describing the photo—transport in Fe:LN are essentially three: (i) the photogalvanic current initiating the buildup of the field; (ii) the material's conductivity under illumination; (iii) the stationary space charge field, proportional to the maximum refractive index change. Those quantities will be briefly discussed in the next sessions in the context of polaron transport. We developed an all-optical setup, described in the Supporting Information, which allow for a measure of those three quantities without the need for electrical connections and as a function of temperature. The macroscopic observables were then systematically measured on a set of six samples presented in forthcoming paragraphs.

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2.1.1. Photogalvanic current

In the case of the 1D problem described above, the photogalvanic current is directed essentially along the c axis and can be written in scalar form [1, 10] as:

$$\begin{eqnarray}&&{j}_{{PG}}=\kappa \alpha I={qsN}\displaystyle \frac{I}{h\nu }\phi {L}_{{PG}}\end{eqnarray} \tag{ 1 }$$

where q is the charge of the photo-excited carriers, α = sN is the absorption coefficient (with s the cross section and N the concentration of the photogalvanic centres, respectively), I is the light intensity, hν the photon energy and L_PG the photogalvanic length. ϕ is the photo-generation efficiency factor, i.e. the probability to create a charge contributing to the current from an absorbed photon, appearing also in the expression for the photoconductivity (see next section). Since all the parameters involved in the photo-generation rate $G={qsN}\tfrac{I}{h\nu }$ are known, it is convenient to normalize the experimentally measured j_PG with respect to G, obtaining what we may call the effective photogalvanic length:

$$\begin{eqnarray}&&{L}_{{PG}}^{{eff}}=\displaystyle \frac{{j}_{{PG}}}{G}=\phi {L}_{{PG}}\end{eqnarray} \tag{ 2 }$$

It should be noted that usual experiments provide access to this quantity, but not to ϕ and L_PG separately.

2.1.2. Photoconductivity

The photoconductivity σ_ph is generally expressed [3] as:

$$\begin{eqnarray}&&{\sigma }_{{ph}}=q\mu n={qsN}\displaystyle \frac{I}{h\nu }\phi \mu \tau \end{eqnarray} \tag{ 3 }$$

In this equation μ is the drift mobility of the light—induced carriers, n their concentration and τ their lifetime. The last equivalence is derived from a band-like description using a rate-equation system [3]. It is important to note that this expression requires two fundamental assumptions (i) the charge carriers are normally diffusing particles characterized by a time-independent mobility; (ii) the carrier decay time is described by an equation of the kind ${dn}/{dt}=-\tfrac{1}{\tau }{{nN}}^{+}$, where N⁺ is the concentration of ionized donors acting also as deep traps, [Fe³⁺] in our case. Both those assumptions are fully reasonable in a band model, but when dealing with polaron transport they need more careful verification. Normal diffusion laws may not be verified [16] for a polaron hopping on a defective lattice with a distance- and energy- dependent attempt frequency. On the other hand, a rate equation such as the one used in band models entails the mono-exponential decay of a carrier population, while it is known (see e.g. [2]) that polaron decays are rather described by stretched exponential functions (Kohlrausch—Williams- Watts law) owing to a heavy-tailed distribution of trapping times.

Those observations indicate that in the polaron context it is challenging knowing or even defining μ and τ separately. However, as already observed by Sturman et al [17], the above mentioned problems can be overcome considering that what is needed here is not μ, nor τ in isolation but the quantity that in a band model is given by the product μτ i.e. the average distance $\langle z\rangle$ run by a polaron under a unitary electric field from its creation to its trapping. Differently from μ and τ in isolation, this quantity can be operatively defined also in our context:

$$\begin{eqnarray}&&{\rm{\Lambda }}=\langle z\rangle /E\end{eqnarray} \tag{ 4 }$$

and can be straightforwardly computed by a Monte Carlo approach. It should be noted that the $\langle z\rangle$ displacement of a hopping particle under electric field and after a given time remains proportional to the electric field magnitude under very general conditions, including anomalously diffusing particles [16] so that the definition is robust.

The photoconductivity (normalized for the photo-generation rate G) is thus given in terms of Λ:

$$\begin{eqnarray}&&{\rm{\Sigma }}=\displaystyle \frac{{\sigma }_{{ph}}}{G}=\phi {\rm{\Lambda }}\end{eqnarray} \tag{ 5 }$$

2.1.3. Space charge field

Under homogeneous cw-light exposure, charge carriers tend to move along the c axis due to the photogalvanic effect. Charge accumulation outside illuminated areas gives rise to the build-up of a space charge field in the material. Assuming that diffusion currents and dark conductivity are negligible, as it is appropriate in the experimental conditions adopted in this work, the saturation value of the electric field E_sat is attained when, after an initial transient, the photogalvanic current, equation (1), is counterbalanced by the drift current j_drift = σ_phE. From equations (1), (3) one has:

$$\begin{eqnarray}&&{E}_{{sat}}=\displaystyle \frac{{j}_{{PG}}}{{\sigma }_{{ph}}}=\displaystyle \frac{{L}_{{PG}}}{{\rm{\Lambda }}}\end{eqnarray} \tag{ 6 }$$

The initial transient of the space charge field evolution, for our simple 1D geometry, follows a simple exponential law [3, 15]:

$$\begin{eqnarray}&&E(t)={E}_{{sat}}(1-{e}^{-t/{\tau }_{d}})\end{eqnarray} \tag{ 7 }$$

The time constant τ_d in equation (7) is the so-called dielectric relaxation time, given by:

$$\begin{eqnarray}&&{\tau }_{d}=\displaystyle \frac{\varepsilon {\varepsilon }_{0}}{{\sigma }_{{ph}}}\end{eqnarray} \tag{ 8 }$$

and provides a direct access to the sample photoconductivity σ_ph.

Two series samples were grown for this work. The first one is composed of congruent samples ($\mathrm{Nb}/\mathrm{Li}$ ratio equal to 0.94) but with different iron concentrations (Series sample A), grown by Czochralski technique at the University of Padova (Italy). This series of samples is conceived to study the effect of different trap concentrations. The Fe contents in the melt was chosen equal to $0.02\,\mathrm{mol} \%$, $0.05\,\mathrm{mol} \%$ and $0.1\,\mathrm{mol} \%$. The second sample series (referred as B) is the one having a fixed deep trap concentration but a different amount of shallow traps (i.e. niobium antisites). In particular three samples, doped with Fe $0.11\,\mathrm{mol} \%$, with different stoichiometry were grown at the Institute for Physical Research in Ashtarak (Armenia). High-purity compounds of ${\mathrm{Nb}}_{2}{{\rm{O}}}_{5}$ from Johnson-Mattley, and ${\mathrm{Li}}_{2}{\mathrm{CO}}_{3}$ from Merck, in powder form, were used as the starting materials for sintering of the lithium niobate charges of different composition, via solid state reaction. The appropriate amount of iron was added to the initial charges of LN in the form of ${\mathrm{Fe}}_{2}{{\rm{O}}}_{3}$ oxide (Merck) and thoroughly mixed. This kind of growth is known to produce samples with a composition which is not constant along the growth direction. Therefore the precise composition of the final samples cannot be assumed to be equal to the one of the melt and needs to be characterized. All the samples were x-ray oriented, cut and optically polished in parallelepipeds with typical sizes of some millimeters per side.

The ${\mathrm{Fe}}^{2+}$ absolute concentration is obtained from the optical absorption at $\lambda =532\,\mathrm{nm}$ as proposed by Berben et al [18]. Assuming that, for weak doping levels as the ones used here, the total Fe concentration in the crystals is the same as in the melt, the amount of Fe³⁺ traps present in the samples is obtained by difference. The concentration of Nb_Li antisite defect can be estimated by Raman spectroscopy [19–21] since a linear relationship between the Li deficiency and the broadening of Raman peaks was established. All the samples of the series B were measured in the X(zz)X backscattering configuration using a LabRAM Aramis Micro-Raman spectrometer, in order to obtain the ${{\rm{A}}}_{1}({\mathrm{TO}}_{1})$ modes, corresponding to $\mathrm{Nb}/{\rm{O}}$ vibrations in x-cut samples [22]. The modes were fitted by Lorentzian functions in order to measure their FWHM Γ and from the equation [21]:

$$\begin{eqnarray}&&{X}_{C}=69.39-0.123\times {{\rm{\Gamma }}}_{{A}_{1}({{TO}}_{1})}\end{eqnarray} \tag{ 9 }$$

the molar concentration ${X}_{C}=\tfrac{[{Li}]}{[{Li}]+[{Nb}]}$ can be obtained. However, in this work a different instrument with respect to the one mentioned in [21] was used, so the formula 9 has to be checked. The spectral line profile used in this analysis is Lorentzian which width is due to the 'true' Raman line shape convoluted with a 'instrumental' function, which depends on the instrument used, on the wavelength etc. As the convolution of two Lorentzian functions is again a Lorentzian whose width is equal to the sum of the widths of the two functions, it can be considered that a change in the setup may affect only the intercept of equation (9), while the slope may be considered as accurate. To estimate the correct intercept, a reference sample is needed. The congruent composition is by definition the one in which the crystal composition is equal to the melt composition and can be expected to be the one with the highest compositional uniformity. In the following therefore the samples grown from the congruent melt are considered as reference and their composition is assumed by default.

The results of those characterizations are reported in table 1. In the following, to indicate a sample we adopt the labelling convention: Series/Deep Traps (in ${10}^{25}\,{{\rm{m}}}^{-3}$)/Shallow Traps (in ${10}^{25}\,{{\rm{m}}}^{-3}$). Thus for example, the label A/0.34/19.0 indicates the sample of the series A with a Fe³⁺ concentration of $0.34\times {10}^{25}\,{{\rm{m}}}^{-3}$ and a ${\mathrm{Nb}}_{\mathrm{Li}}$ antisite defect content of $19.0\times {10}^{25}\,{{\rm{m}}}^{-3}$.

Table 1.  Summary of compositional parameters for the two groups of samples investigated. In the last column the reduction ratio $R=[{\mathrm{Fe}}^{2+}]/[{\mathrm{Fe}}^{3+}]$ is also reported.

Sample name	XC%	$[{\mathrm{Nb}}_{{Li}}]$	$[{\mathrm{Fe}}_{{tot}}]$	$[{\mathrm{Fe}}^{2+}]$	$[{\mathrm{Fe}}^{3+}]$	R%
		${10}^{25}\,{{\rm{m}}}^{-3}$	${10}^{25}\,{{\rm{m}}}^{-3}$	${10}^{25}\,{{\rm{m}}}^{-3}$	${10}^{25}\,{{\rm{m}}}^{-3}$
A/0.34/19.0	48.45	19.0	0.37	0.04 ± 0.01	0.34 ± 0.1	10.9 ± 0.3
A/0.82/19.0	48.45	19.0	0.95	0.12 ± 0.01	0.82 ± 0.02	14.3 ± 0.3
A/1.56/19.0	48.45	19.0	1.89	0.33 ± 0.01	1.56 ± 0.07	21.1 ± 0.4
B/1.84/19.0	48.45	19.0	20.9	0.24 ± 0.01	1.84 ± 0.01	13.08 ± 0.09
B/1.78/17.5	48.58 ± 0.03	17.5	20.9	0.31 ± 0.01	1.78 ± 0.05	17.2 ± 0.5
B/1.94/6.4	49.49 ± 0.03	6.4	20.9	0.15 ± 0.01	1.94 ± 0.02	7.6 ± 0.1

LN hosts four different kinds of intrinsic small polarons [23]: the free polaron (FP) ${\mathrm{Nb}}_{\mathrm{Nb}}^{4+}$, the bound polaron (GP) ${\mathrm{Nb}}_{\mathrm{Li}}^{4+}$, the bipolaron (BP) ${\mathrm{Nb}}_{\mathrm{Nb}}^{4+}:{\mathrm{Nb}}_{\mathrm{Li}}^{4+}$, and the hole polaron (HP) ${{\rm{O}}}^{-}-{{\rm{V}}}_{{Li}}$. Electrons bound to ${\mathrm{Fe}}_{\mathrm{Li}}^{3+}$ may also be described in the framework of the strong-coupling-polaron picture. Hole polarons are produced at visible wavelengths by high intensity pulsed beams by two-photon processes and will not be considered here, as in our Fe doped samples and at cw intensities the electron generation from Fe²⁺ is by far predominant. Moreover, in Fe-doped samples, bipolarons are not observed because the system tends to relax to the more stable Fe defect, so in the following we will disregard also this centre. In conclusion, we will assume that in the dark the sample does not contain free or bound polarons and that all the charges are stored in Fe²⁺ centres. When a photon in the visible range is absorbed by a Fe²⁺ donor, a small electron polaron is created. It performs a certain number of hops either on Nb_Nb or on Nb_Li sites until it is re-trapped at a deep ${\mathrm{Fe}}_{\mathrm{Li}}^{3+}$trap, which defines the final distance run by the particle.

According to the Marcus—Holstein polaron hopping model [11–13], the non-adiabatic hopping frequency for a ($i\to f$) hop is:

$$\begin{eqnarray}&&{w}_{i,f}(r,T)=\displaystyle \frac{1}{2}{\left(\displaystyle \frac{\pi }{{kT}{\lambda }_{i,f}}\right)}^{\tfrac{1}{2}}\displaystyle \frac{{I}_{i,f}^{2}}{{\hslash }}\exp \left(-\displaystyle \frac{r}{{a}_{i,f}}-\displaystyle \frac{{U}_{i,f}}{{kT}}\right)\end{eqnarray} \tag{ 10 }$$

In equation (10), r is the distance between initial and final sites and kT the absolute temperature (in energy units). λ_i,f is the reorganization energy of Marcus' theory corresponding to the energy paid to rearrange the lattice, here equal to $({E}_{i}+{E}_{f})$, sum of the elastic energies of the two polarons; a_i,f is an orbital parameter describing the overlap between the electronic wave functions at site i and f. The 1/2 factor is due to the fact that equation (10) expresses the individual rate to a given final site, and is thus one half of the total rate in Holstein's molecular chain [13]. ${U}_{i,f}\ne {U}_{f,i}$ is the hopping barrier, given by [24, 25]:

$$\begin{eqnarray}&&{U}_{i,f}=\displaystyle \frac{{(2{E}_{i}+{\varepsilon }_{i}-{\varepsilon }_{f})}^{2}}{4({E}_{i}+{E}_{f})}\end{eqnarray} \tag{ 11 }$$

with ε_i and ε_f the pre-localization energies of the electron at zero deformation. When the hop occurs between sites of the same type (i = f), U_i,i = E_i/2, recovering the standard result that the hopping activation energy is one half of the polaron energy [2, 23, 26]. The pre-exponential factor I_i,f describes the intrinsic hopping rate between the two sites and is determined by the choice of the (i, f) combination. According to this model, each of the three centers here considered (two Nb- based polarons, Nb_Li and Nb_Nb and the ${\mathrm{Fe}}^{2+/3+}$ defect) and the hopping combinations between them are described by a set of parameters. According to some recent results [14], the parameters choice reported in table 2 provides a fair description of all the possible hopping processes in Fe:LN, and will be used throughout this work.

Table 2.  Parameters used for the simulation (see text).

Parameter	Value	Unit	References	Note
EFP	0.545	eV	[23]	Free polaron energy
EGP	0.75	eV	[14]	Bound polaron energy
${E}_{\mathrm{Fe}}$	0.7	eV	[23]	Fe defect energy
${\varepsilon }_{{FP}}$	0	eV		Free polaron pre-localization energy
${\varepsilon }_{{GP}}$	0.2	eV	[14]	Bound polaron pre-localization energy
${\varepsilon }_{\mathrm{Fe}}$	1.22	eV	[23]	Fe defect pre-localization energy
${a}_{{FP},{FP}}={a}_{{GP},{GP}}={a}_{{GP},{FP}}={a}_{{FP},{GP}}=a$	1.6	Å	[14]	Hopping parameter
${a}_{{FP},\mathrm{Fe}}={a}_{{GP},\mathrm{Fe}}=c$	1.3	Å	[14]	Trapping parameter
${I}_{{FP},{FP}}={I}_{{FP},{GP}}={I}_{{GP},{FP}}={I}_{{GP},{GP}}=I$	0.1	eV	[27]	Transfer integral pre-factor

The effect of the space-charge field E on the hopping frequency is described by adding in equation (11) the term $q({{\bf{r}}}_{i}-{{\bf{r}}}_{f})\cdot {\bf{E}}$ inside the parenthesis at the numerator, with r_i(j) the position vectors of the starting (final) site. It is important that the field is not too strong to maintain the character of random diffusion in our simulation. In other words, the potential energy $q({{\bf{r}}}_{i}-{{\bf{r}}}_{f})\cdot {\bf{E}}$ gained by the polaron must remain small with respect to kT. Our field is therefore set at $5\times {10}^{6}\,{\rm{V}}/{\rm{m}}$ for all the temperatures, a compromise between the above -mentioned requirement and the necessity to observe a measurable displacement. The field is directed along the crystallographic c direction to mimic our experimental geometry.

It should be recalled that the hopping formula (10) is valid only in the non-adiabatic approximation [11] which holds whenever the transfer integral ${J}_{i,f}(r)={I}_{i,f}^{2}\exp (-r/2{a}_{i,f})$ is much smaller than the reorganization energy ${\lambda }_{i,f}=({E}_{i}+{E}_{f})$. By using the values reported in table 2 we can check that this condition is well satisfied for all the hop types considered in our model. Moreover the approximation requires the lattice to possesses sufficient thermal energy to follow the charge motion and, as a rule of thumbs, is considered safe for temperatures above ${{\rm{\Theta }}}_{D}/2$ where ${{\rm{\Theta }}}_{D}\approx 503\,{\rm{K}}$ is the Debye temperature for lithium niobate [28]. However, experimental results obtained by Faust et al [29] show that the the Arrhenius behaviour of the mobility in antisite-free LN is preserved till $150\,{\rm{K}}$, indicating that the non-adiabatic approximation in LN holds at least down to this temperature.

The simplest numerical approach consists in simulating one single electron at time. This situation corresponds to a low electron density system where interaction between electrons can be ignored. The electron, randomly placed at an initial Nb_Nb site, performs a walk in a structure reproducing the LN lattice. The code generates a random defect configuration in a 80 × 80 × 80 super-cell of the LN structure with periodic boundary conditions, in which the polaron is launched and the position at which it is captured by a Fe³⁺ trap is recorded. The different defect concentrations of the various samples are set in accordance with table 1. The code computes the rest time on each visited site and the next destination site according to equations (10), (11) by a classical Gillespie algorithm, as explained in [25]. The probability to hop from a site i to a site f is computed as:

$$\begin{eqnarray}&&{p}_{i,f}=\displaystyle \frac{{w}_{i,f}}{{\displaystyle \sum }_{f}{w}_{i,f}}\end{eqnarray} \tag{ 12 }$$

where the summation at the denominator goes on all the possible destination sites inside a suitably defined volume surrounding the starting site. The simulation is repeated for a sufficient number of times, in order to compute the average distance $\langle z\rangle$ along the field direction run by the particles before being captured. From equation (4) the value of Λ is calculated for the different samples and for all the experimental temperatures.

In figures 2(a), (c) are reported the experimental results (dots) for the saturation space charge field value E_sat as a function of temperature for sample series A and B. The field shows a monotonous increase by decreasing the temperature.

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The central result of our simulation is the drift coefficient Λ, i.e. the mean distance covered by a polaron under a unitary electric field from its birth to its trapping. In figures 2(b), (d) the simulation results obtained in the same conditions as in (a), (c) are shown. The Λ value becomes smaller and smaller by decreasing the temperature, indicating that at low T the polaron is able to cover on average a smaller distance. By increasing the trap concentration in sample series A, Λ is decreased as well, as it can be expected (figure 2(b)). The effect of the antisite concentration in sample series B (2(c)) is somehow less evident, due to the high Fe concentration that in the temperature range here explored keeps always Λ below $3\times {10}^{-17}\,{{\rm{m}}}^{2}/{\rm{V}}$ (note the different ranges in the ordinates of (b) and (c)). However a trend is visible indicating that a higher antisite concentration hinders the charge motion or, conversely, by increasing the Li content towards stoichiometry Λ is increased.

In figure 3 is shown the correlation plot between the reciprocal of the measured saturation space charge field 1/E_sat and the Λ values simulated for the corresponding experimental conditions. A linear correlation is recovered for almost all the samples, as it can be expected by the reciprocal of equation (6), ${E}_{{sat}}^{-1}={\rm{\Lambda }}/{L}_{{PG}}$, and by the observation that L_PG is an intrinsic parameter of the Fe:LN system which should be almost independent from temperature and composition [10]. Thus, the fact that almost all our data converge on a unique line can be considered as a self-consistency test. However a nonzero Λ₀/L_PG intercept is clearly evident from the graph. This can be interpreted as a parasitic contribution to the sample conductivity. Since this contribution appears to add up to to all the samples and for all the temperatures, this feature is likely to be an extrinsic contribution e.g. some unwanted leakage perhaps facilitated by the contact with the metallic sample holder inside our cryostat system. We note here, however, that few points relatives to the sample A/0.34/19.0 at high temperatures appear to deviate from the linear behaviour. Our data do not allow at the moment deciding whether this is an artifact or the indication that in a range of experimental conditions wider than the one here considered the description becomes more complicated. As the large majority of our data is in agreement with the simple description of equation (6), we will stick here to this model, with the caveat that further work is needed to confirm their validity out of the conditions here explored.

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From the linear fit of figure 3 we retrieve ${L}_{{PG}}=(1.44\pm 0.05)\,\mathring{\rm A}$ and ${{\rm{\Lambda }}}_{0}=(1.69\pm 0.07)\times {10}^{-17}\,{{\rm{m}}}^{2}/{\rm{V}}$. From those values and from the results of the simulations, the saturation space charge field was re-computed by equation (6) for all the experimental conditions here examined and reported in figures 2(a) and (c) as solid lines. To our knowledge, this is the first direct comparison between temperature- and composition- dependent data in photorefractive Fe:LN and the calculations obtained from a microscopic polaron model in which no free parameters besides L_PG and ${{\rm{\Lambda }}}_{0}$ were used.

In figures 4(a) and (c) are reported the experimental values of the specific photoconductivity Σ (dots). Combining them with the determination of L_PG and from the measured values of E_sat it is possible to estimate the photo-generation efficiency ϕ as a function of temperature. In fact, from equations (2), (5), (6) one gets easily $\phi ={j}_{{PG}}/{{GL}}_{{PG}}={E}_{{sat}}{\rm{\Sigma }}/{L}_{{PG}}$. The results are reported in figures 4(b) and (d). All the data are coherent among the different samples: ϕ appears to decrease by cooling the sample from a value of 10%–15% at room temperature to about 5% at low T.

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By comparison, we recompute Σ according to equation (5) using the obtained values of ϕ, the simulated values for Λ and the correction for the leakage Λ₀, i.e. ${\rm{\Sigma }}=\phi ({\rm{\Lambda }}+{{\rm{\Lambda }}}_{0})$. The results are plotted in figures 4(a) and (c) as solid lines, showing that again all the description is consistent.

The interpretation in terms of polaron hopping of the temperature and composition dependence of the photorefractive phenomena opens up a rich scenario, in which the interplay between the different defect centers determines the drift coefficient Λ and ultimately the main photorefractive observables.

Our simulation provides access to Λ. The decrease of this parameter with temperature (see figures 2(b), (d)) is due to the fact that, as a general rule, by decreasing the temperature the polarons perform a smaller number of hops to get trapped by a Fe ion [25]. It should be noted that in a hypothetical material with no Nb_Li antisites and vanishing Fe concentration this would not be the case. In this situation the only charge carriers would be free polarons performing a given number of hops before being trapped by a deep trap. The conduction would be ensured by thermally activated hopping among equal sites characterized by a single activation energy. Therefore the temperature term of the Marcus-Holstein frequency, equation (10), would be the same for all hopping processes and could be simplified out from equation (12). This proves that the random walk performed by the polaron in this case is not affected by temperature, which would only speed up or slow down the hopping frequency, but would not change the number of hops and the distance run under the field before being trapped. The situation is different for a material with defects. In this case the starting and destination sites may be of different kind so that the hopping barriers may not be the same for all the hopping processes. The thermal part of the hopping frequency therefore cannot be simplified in equation (12) which becomes temperature dependent. Therefore, in material containing defects a change in temperature, besides modifying the speed of the hopping process, influences also the random walk of the particle. In particular, at sufficiently low temperatures and for the Fe concentrations used in this work, the hopping frequency of the trapping processes ${\mathrm{Nb}}_{{Nb}}({\mathrm{Nb}}_{{Li}})\to \mathrm{Fe}$ becomes dominant with respect to the others, leading to an increase of the trapping probability upon decreasing T.

For a given sample composition we may therefore distinguish two limiting hopping regimes, upon cooling down from room temperature (or above): (i) Multi-hop, mixed transport regime: the thermal budget is so high that all the hopping processes occur more or less frequently and the polaron performs several hops before being captured by a Fe³⁺ trap. In this regime the drift coefficient Λ depends strongly on the relative amount of the different hop types; in particular, at high temperatures and/or low antisite concentrations the free polaron jumps ${\mathrm{Nb}}_{{Nb}}\to {\mathrm{Nb}}_{{Nb}}$ dominate the transport [14]. (ii) Trapping regime: at sufficiently low temperatures, the hopping frequency towards Fe traps becomes higher than for all other processes, so that direct trapping becomes the most likely event. The two situations are illustrated in figure 5 where the average percentage of the different hop types over the total hop number has been computed from our simulation code for sample A/0.82/19.0 as an example.

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The simulations show that in these conditions the trapping can occur even if the trap is quite far away from the newborn polaron, i.e. several tens of Angstroms. A useful way to visualize this process is the concept of trapping radius firstly introduced in [25]. The trapping probability can be estimated by comparing the hopping frequency towards the Fe donor against the frequency towards any other center. As the hopping frequency (equation (10)) depends both on distance and on temperature, one can ask what is the distance d(T) from a neighboring Fe center under which the polaron is more likely to be trapped than to hop away, for a given T. By some simple considerations detailed in [25], it turns out that this distance obeys an equation of this kind:

$$\begin{eqnarray}&&d(T)=A+B\displaystyle \frac{U}{{kT}}\end{eqnarray} \tag{ 13 }$$

and increases from about 20 to $40\,\mathring{\rm A}$ by decreasing T from room temperature to $150\,{\rm{K}}$. The onset of the trapping regime can thus be visualized by imagining that each trap is surrounded by a finite 'capture volume' with a trapping radius that increases by cooling down the sample. At low T the trapping volumes start to fill most of the available space, so that the polarons are captured within few hops.

Schirmer et al [10] gave a comprehensive revision of the photogalvanic effect in Fe:LN reinterpreting it within the polaron model. The initial direction along which the charge is emitted as a Bloch wave is determined by the Fermi golden rule, the different matrix elements of the transition being determined by the geometrical arrangement of the ions surrounding the donor centre. In this vision the photogalvanic length L_PG in equation (1) corresponds to the averaging along the c-axis of the distance covered in the different directions prior to the polaron formation. The electron is supposed to move several Nb_Nb − Nb_Nb distances coherently along the c axis before the lattice relaxes around it to form a ${\mathrm{Nb}}_{\mathrm{Nb}}^{4+}$ small polaron state.

Our data show a consistency with the Marcus-Holstein model if a value of ${L}_{{PG}}=(1.44\pm 0.05)\,\mathring{\rm A}$ is assumed. In accordance with microscopic considerations, this parameter appears to be independent on the sample and on the temperature for the range of experimental conditions explored in this work. The quantitative result is in reasonable agreement with previous tentative estimates [10] which gave L_PG in the Angstrom range for Fe excitation with visible light. This value may be compared with the diffusion distance that a newly emitted 'hot' electron travels before condensate into a polaron, which is much larger and estimated around $23\,\mathring{\rm A}$ [10] on average. This means that the photo-excited charge is emitted nearly isotropically and that the nonzero average displacement is only a tiny fraction of the distance run by the electron wave before the self-localization.

The results obtained on different samples are consistent between them and point out that ϕ is between 10 and 15% at room temperature and decrease to 5% at 150 K. In [30] it is reported that ϕ decreases of about 4 times by increasing the wavelength λ from the blue to the green. Since ϕ cannot be larger than 1, in our experiments with $\lambda =543\,\mathrm{nm}$ we must have ϕ < 0.25, in accordance with our data.

According to the model for emission of photo-excited charges mentioned in the precedent paragraph, after the initial thermalization stage the polaron is formed at a certain distance from its donor site according to some probability distribution with a mean equal to L_PG and a width of the order of $20\,\mathring{\rm A}$. The particle has now two possible choices to continue its life: to move away or to jump back to the same initial site it came from, both processes being thermally activated but with a different energy barrier. This provides a microscopic interpretation for the parameter ϕ introduced in equation (1), which denotes here the probability for the polaron to not go back to the starting site [31]. As explained above, at low temperature the trapping become more efficient and this can be visualized by considering that the trapping radius R(T) of the Fe centers increases at low T. The trapping radius is of the order of $20\,\mathring{\rm A}$ at room temperature, i.e. of the same order of magnitude of the diffusion distance of the hot polaron. The low value of the photo-generation efficiency is therefore explained by the fact that a large part of the polarons are formed inside the trapping radius and are thus recaptured by the same Fe donor they came from (see figure 6).

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The effect of the temperature can be easily understood as the trapping radius increases by lowering T, as a consequence of the temperature dependence of the hopping frequencies, so that the chance to escape is smaller and smaller. The increase of the photo-generation efficiency with the wavelength reported in [30] can be easily interpreted in this framework as well, considering that more energetic photons are able to project the photo-emitted charge at a longer distance from the Fe center [10]. It would be interesting to push this analysis to a more quantitative modeling, but for this aim it would be necessary to know the probability distribution function for the formation site of the polarons around the donor center. Further work is in progress to investigate this idea.