The role of self-trapped excitons in polaronic recombination processes in lithium niobate

In figure 2(a), measurements of the transient absorption on the Fe:LN sample are shown at $T=293$ K at the probing wavelengths 445 nm, 488 nm, 633 nm, and 785 nm. The first two are monitoring the nearly coinciding Fe$_{\rm Li}^{2+}$ and trapped hole absorption bands and the last one roughly corresponding to the maximum of the Nb$^{4+}_{\rm Li}$ band, all being strongly overlapping broad asymmetric features characteristic for polarons [18]. Following the incident pump pulse at $t=0$ s, a transient absorption appears in the near-infrared spectral range and vanishes almost completely in the microsecond range at RT. In the blue spectral range (445 nm and 488 nm), the transient absorption increases on a time scale up to several microseconds and vanishes to zero only after a few seconds.

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Table 1. Parameters of the KWW-function used to describe the TA of the 0.1 mol% Fe:LN sample at $T=293$ K and 198 K exemplarily for the data set depicted in figure 2.

	293 K
	445 nm	488 nm	633 nm	785 nm
$\alpha_{{{\rm li}},1}^0$ (m$^{-1}$)	−72 ± 10	−25 ± 5	184 ± 10	191 ± 10
$\tau_1$ ($\mu$s)	5 ± 3	2 ± 1	6 ± 2	4 ± 2
$\beta_1$	0.39 ± 0.04	0.30 ± 0.03	0.32 ± 0.03	0.29 ± 0.03
$\langle\tau_1\rangle$ ($\mu$s)	18 ± 16	19 ± 17	42 ± 30	43 ± 42
$\alpha_{{{\rm li}},2}^0$ (m$^{-1}$)	99 ± 10	41 ± 5	21 ± 5		—
$\tau_2$ (s)	4 ± 2	7 ± 4	9 ± 4		—
$\beta_2$	0.61 ± 0.06	0.64 ± 0.06	0.62 ± 0.06		—
$\langle\tau_2\rangle$ (s)	5.9 ± 3.0	9.7 ± 6.0	13 ± 6.5		—
	198 K
	445 nm	488 nm	633 nm	785 nm
$\alpha_{{{\rm li}},1}^0$ (m$^{-1}$)	−41 ± 10	−16 ± 5	227 ± 10	238 ± 10
$\tau_1$ ($\mu$s)	(20 ± 10) ⋅ 103	(9 ± 4)  ⋅ 103	121 ± 30	40 ± 10
$\beta_1$	0.52 ± 0.05	0.69 ± 0.07	0.22 ± 0.02	0.18 ± 0.01
$\langle\tau_1\rangle$ ($\mu$s)	(37 ± 3) ⋅ 103	(12 ± 6) ⋅ 103	(68 ± 62) ⋅ 103	(13 ± 10) ⋅ 103
$\alpha_{{{\rm li}},2}^0$ (m$^{-1}$)	82 ± 10	37 ± 5	14 ± 5		—
$\tau_2$ (s)	20 ± 10	26 ± 10	25 ± 10		—
$\beta_2$	0.9 ± 0.1	0.9 ± 0.1	0.8 ± 0.08		—
$\langle\tau_2\rangle$ (s)	21 ± 11	27 ± 11	28 ± 15		—

In comparison, at $T = 198$ K (figure 2(b)), the TA signals show clearly decelerated decay dynamics affecting much stronger the blue than the red spectral range, the maximum for 445 nm and 488 nm being preceded by an almost flat stage up to the millisecond time regime. The blue absorption signal vanishes to zero only after a duration of tens of seconds. This behavior suggests a pronounced time delay between the decaying red and increasing blue transients and can, in fact, be observed for various reduction and oxidation pre-treatments, i.e. for differently adjusted [Fe$^{2+}$]/[Fe$^{3+}$] ratios between 0.01 and 0.19.

Previously, Berben et al [26] and Herth et al [25] reported that a sum of stretched-exponential functions (Kohlrausch–Williams–Watts function, KWW)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \alpha_{\rm li}(t,\lambda,T)=\sum_{i=1}^{N}\alpha_{{{\rm li}},i}^0(\lambda,T)\cdot {{\rm exp}} \left [-\left (\frac{t}{\tau_i(\lambda,T)}\right)^{\beta_i(\lambda,T)} \right], \label{KWW_funktion} \nonumber \end{align} \tag{ 1 }$$

is a reasonable ansatz for the description of the temporal behavior of the transient absorption in Fe:LN. Accordingly, equation (1) has been fitted to the entire experimental data set as represented in figure 2 by the continuous yellow lines. For the absorption at 785 nm a single KWW-function is used to describe the data ($N=1$), while $N=2$ is chosen for all other wavelengths. The obtained values of the fitting parameters providing a fair description of the results are presented in table 1 for both temperatures and all four probe wavelengths. The mean relaxation and build-up times for the various components defined by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \langle\tau_i(\lambda,T)\rangle &= \int_0^{\infty} {{\rm exp}}\left [-(t/\tau_i(\lambda,T))^{\beta_i(\lambda,T)}\right] {\rm d}t \nonumber \\ &=\tau_i(\lambda,T)\Gamma \left (\frac{1}{\beta_i(\lambda,T)}+1\right) \label{eq:taumean} \nonumber \end{align} \tag{ 2 }$$

are also included in table 1. Here, $\Gamma$ denotes the gamma function. It should be noted that the error intervals for $\langle\tau_i(\lambda,T)\rangle$ can be close to the order of the calculated value itself due to the combined error of $\beta$ and $\tau$. Still, the activation energies given in section 5 can be estimated very well, as we deal with different orders of magnitude of the decay time $\langle\tau_i(\lambda,T)\rangle$ at different temperatures. Furthermore, a large number of separate measurements over a broad temperature range is available.

At RT, all transients seem to be in accordance with the model proposed by Herth et al [25]. In the red spectral range, the initial absorption change $\alpha_{\rm li}^0$ is attributed to the formation of Nb$_{\rm Li}^{4+}$ bound electron polarons excited either from the valence band or Fe$^{2+}_{\rm Li}$ to the conduction band by a two-photon or one-photon process, respectively. The initial absorption change in the blue spectral range can be assumed to originate from various sources. There is a negative contribution, i.e. induced transparency due to the transformation of the Fe$^{2+}_{\rm Li}$ (part or all) into non-absorbing Fe$^{3+}_{\rm Li}$ centers [1, 18]. A positive contribution comes from valence band holes trapped next to Li vacancies as $\text{O}^{-}-\text{V}_\text{Li}$ hole polarons together with a much smaller positive contribution coming from the overlapping absorption tail of the Nb$_{\rm Li}^{4+}$ bound electron polarons formed during the parallel electron trapping event.

According to the Herth model, the hopping motion of electron polarons may lead to the formation of stable or intermediate Fe$^{2+}_{\rm Li}$ centers, resulting in the decrease of red absorption and to the simultaneous appearance of the rising component in the blue on the microsecond timescale, which is in agreement with the coincidence of the values $\tau_1$, $\beta_1$ and $\langle \tau_1\rangle$ measured at RT for different wavelengths simultaneously. Some of the Fe$_{\rm Li}^{2+}$ centers might be metastable (e.g. due to their specific charge compensation), whereby electron trapping and release at such iron centers could occur repeatedly while electrons recombine with all available pump-induced trapped-hole centers; this process could lead to a decrease of the blue absorption of both trapped holes and metastable Fe$_{\rm Li}^{2+}$ centers, contributing to some (rather) long TA component. This model neglects the formation of Nb$_{\rm Li}^{4+}$ electron polarons between possible repeated trapping events on iron which can be justified by Nb$_{\rm Li}^{4+}$ lifetimes shorter by several orders of magnitude than those of metastable Fe$_{\rm Li}^{2+}$ centers (a similar argument explains the neglect of Nb$_{\rm Nb}^{4+}$ free polarons compared to Nb$_{\rm Li}^{4+}$ bound ones at faster stages).

In contrast, cooling the sample by $\approx$$100$ K (figure 2(b)) results in clear changes that are no longer consistent with Herth's model assuming the same $\tau_1$ and $\beta_1$ for all probe wavelengths. In particular significant deviations appear in the first few milliseconds in this low temperature regime (dashed red lines in figure 2). The non-zero blue absorption remains remarkably constant during most of the Nb$_{\rm Li}^{4+}$ decay, prior to its rise to an interim maximum value. Adopting for $\tau_1(445/488~{{\rm nm}},198~{{\rm K}})$ and $\beta_1(445/488~{{\rm nm}},198~{{\rm K}})$ the values $\tau_1(785~{{\rm nm}},198~{{\rm K}})$ and $\beta_1(785~{{\rm nm}},198~{{\rm K}})$ obtained at 785 nm as fixed parameters, i.e. applying the Herth model, the data set cannot be reconstructed by a converging fit. The only way to reach a very good fit quality is by treating the $\tau_1$ and $\beta_1$ parameters at various wavelengths as independent free parameters in the fitting procedures (see yellow lines in figure 2). The decay time in the near-infrared and the rise time in the blue deviate by a factor of 100 and only the mean relaxation/build-up times $\langle\tau_1(\lambda,198~{{\rm K}})\rangle$ have approximately the same values. The bound polarons thus seem to disappear without having recombined simultaneously with holes or having promptly filled Fe$^{3+/2+}_{\rm Li}$ traps, and reappear only later as Fe$_{\rm Li}^{2+}$ centers. This trend is already indicated for temperatures below $\approx$$250$ K.

The considerable displacement of the TA signal towards longer times and its changed shape at low temperatures have to be described by strongly increased $\tau_1$ and $\beta_1$ values in the blue ($\lambda=445/488$ nm) compared to a moderately rising $\tau_1$ and decreasing $\beta_1$ in the red region. The increased stretching factor $\beta_1(488~{{\rm nm}},198$ K$)=0.69$ in the blue has to be contrasted with its decreased value $\beta_1(785~{{\rm nm}},198$ K$)=0.18$ in the red (table 1). The slow decreasing component shows an even larger $\beta$ value in the blue: $\beta_2(445/488~{{\rm nm}},$198 K$)=0.9$. The weakly defined mean values $\langle\tau\rangle$ show much less variation which is also due to the fact that different combinations of $\tau$ and $\beta$ can, by coincidence, lead to the same value of $\langle\tau\rangle$.

Taking into account the widely accepted interpretation of $\beta$ [33–35] including the attempt of Merschjann et al [36] to relate $\tau$ and $\beta$ directly to characteristic parameters of the hopping transport (time of a single hopping event, number of hopping events, etc), means that the blue transients have a much weaker dependence on hopping history before annihilation than the respective transients in the red. While the electron polaron decay for lower temperatures is increasingly characterized by slow hopping, the blue transients become nearly mono-exponential.

These data point to a clear failure of the Herth model and support the previous reports on inconsistencies in Fe:LN crystals under rather different conditions. In particular at low temperatures and with probing wavelengths in the blue and near-infrared spectrum, we are faced by two striking experimental observations: (i) the decay of bound polaron absorption in the near-infrared is not correlated with the increase of the Fe$_{\rm Li}^{2+}$ absorption anymore, and (ii) there is a strong discrepancy of the determined $\beta$ value with the state-of-the-art knowledge for small, strong-coupling polaron dynamics in LN [26, 35, 37] to be further discussed in section 5. At the same time, these data are not sufficient to conclude a general failure of the Herth model, i.e. that Herths' model also fails for the description of TA in LN crystals of different stoichiometry and/or with/without different dopants (including non-photorefractive defect centers). In what follows, we intend to exclude that the failure is solely related to the dominating existence of Fe$_{\rm Li}^{2+/3+}$ by means of equivalent TAS studies with over-threshold Mg:LN.

The dynamics of the transient absorption in Mg:LN at 445 nm, 488 nm, 785 nm, and 1310 nm at RT (the latter approximately centered on the broad absorption band of Nb$_{\rm Nb}^{4+}$ free polarons) is depicted in figure 3, showing, in addition to a fast decreasing transient on the same time scale for all wavelengths, a long-lived transient blue absorption as previously reported by Conradi et al [24]. However, in comparison to former studies a new probe at 445 nm was applied and we were able to observe an unexpected feature, i.e. a clear increase in the absorption at surprisingly long timescales. The newly measured transient local maximum for 445 nm is observed after a significant period (few milliseconds) of constant absorption that resembles the dominant feature of the low-temperature TA dynamics of Fe:LN. This corresponds to a small rising component (denoted by the index 1b) with a time constant $\tau_{\rm 1b}(445~{{\rm nm}},293~{{\rm K}})=(70\pm20)$ ms. The fitting parameters derived for all KWW components in Mg:LN are summarized in table 2, essentially coinciding at RT with those reported in [24]. The fast component on the microsecond timescale (denoted by the index 1a) is decreasing in the blue region at variance from the case of Fe:LN. The slow component extends to hundreds of seconds in the visible and is absent in the infrared as in Fe:LN.

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Table 2. Parameters of the KWW-function used to describe the TA of the 6.5 mol% Mg:LN sample at $T=293$ K and $T=100$ K (figure 3).

	293 K												100 K
	445 nm			488 nm			785 nm			1310 nm			488 nm			785 nm
$\alpha_{{{\rm li}},{{\rm 1a}}}^0$ (m$^{-1}$)	90	$\pm$	10	71	$\pm$	10	177	$\pm$	10	248	$\pm$	20	100	$\pm$	5	81	$\pm$	10
$\tau_{\rm 1a}$ ($\mu$s)	1.0	$\pm$	0.2	2.3	$\pm$	0.5	0.4	$\pm$	0.1	0.2	$\pm$	0.1	(10	$\pm$	5) $\cdot 10^{6}$	(1.7	$\pm$	1) $\cdot 10^{6}$
$\beta_{\rm 1a}$	0.45	$\pm$	0.1	0.62	$\pm$	0.1	0.44	$\pm$	0.1	0.45	$\pm$	0.1	0.19	$\pm$	0.01	0.19	$\pm$	0.01
$\langle\tau_{\rm 1a}\rangle$ ($\mu$s)	2.5	$\pm$	1.5	3.3	$\pm$	1.1	1.0	$\pm$	0.6	0.5	$\pm$	0.3	(1890	$\pm$	1200) $\cdot 10^6$	(321	$\pm$	200) $\cdot 10^6$
$\alpha_{{{\rm li}},{{\rm 1b}}}^0$ (m$^{-1}$)	$-15$	$\pm$	5		—			—			—			—			—
$\tau_{\rm 1b}$ (ms)	70	$\pm$	20		—			—			—			—			—
$\beta_{\rm 1b}$	0.8	$\pm$	0.2		—			—			—			—			—
$\langle\tau_{\rm 1b}\rangle$ (ms)	79	$\pm$	40		—			—			—			—			—
$\alpha_{{{\rm li}},2}^0$ (m$^{-1}$)	58	$\pm$	10	51	$\pm$	5	6	$\pm$	2		—			—			—
$\tau_2$ (s)	10	$\pm$	3	6	$\pm$	2	5	$\pm$	3		—			—			—
$\beta_2$	0.45	$\pm$	0.05	0.32	$\pm$	0.04	0.25	$\pm$	0.1		—			—			—
$\langle\tau_2\rangle$ (s)	25	$\pm$	10	42	$\pm$	24	120	$\pm$	110		—			—			—

The dynamics in Mg:LN at RT should be compared to scenarios in Fe:LN taking into account the changed defect structure of Mg:LN: (i) Nb$_{\rm Li}$ antisites and most Fe$_{\rm Li}$ centers are missing as indicated by the vanishing photorefractive effect in over-threshold crystals [38], (ii) a portion of Mg$^{2+}$ dopant ions and some of the background Fe impurities on the ppm level are incorporated on Nb sites [39, 40], where their charge state remains fixed.

The lack of Nb$_{\rm Li}$ antisites results in an increase of the electron polaron hopping frequency, now occurring in the regular Nb sublattice, leading to a shortening of $\tau_1$ by nearly an order of magnitude, compared to Fe:LN, so that $\tau_1(785/1310$ nm, RT$) \approx 0.3~\mu$s. Accordingly, there are even longer delays (compared to Fe:LN) between this fast decay of free polarons and the slower components in the blue. Due to the small number of Fe$^{2+}_{\rm Li}$ centers, a significant influence on the TA can be excluded as supported also by the monotonous decrease of the 488 nm TA signal in the entire observed temporal region. Only the small rising component $\tau_{\rm 1b}$ at 445 nm extending into the 70 ms range has to be attributed to background Fe impurities, based on the analogy with component $\tau_1$ in Fe:LN. The fast blue TA component in Mg:LN, temporarily locked with polaron decay monitored in the near-infrared, can be attributed to a direct process of polaron recombination with trapped-hole centers. For $T<100$ K, TA signals in Mg:LN extend to several minutes both in the near-infrared and the blue region due to the slow hopping of polarons and delays in the recombination processes. Therefore, the three KWW-components in the blue spectral range cannot be satisfactorily distinguished anymore due to similar decay times of several overlapping TA components. A single-component fit at $T=100$ K and $\lambda=488$ nm yields $\tau\approx 10$ s and $\beta\approx 0.19$, and therefore $\langle\tau\rangle\approx 1.9\cdot 10^{3}$ s. The latter is much longer in comparison to the value determined in the near-infrared, thus indicating the presence of the RT processes at low temperatures, as well. The values obtained for 785 nm and 488 nm are summarized in table 2.

Apart from the latter process, the RT TA-data in Mg:LN thus clearly resemble the findings in Fe:LN, and further stress the failure of the original microscopic model for polaronic recombination. Similarly to the case of Fe:LN the model cannot be corrected by simply considering some process of more complex type but involving only single-electron transfers. Instead, a more complex electronic center, the STE, having clear fingerprints for its identification, will be discussed in more detail below.

Figure 4(a) shows the TRPL of Mg:LN upon ns-pulse exposure at 355 nm ($I_{\rm P}\approx 100$ GW m$^{-2}$, extraordinary polarization) for the emission wavelength of 460 nm for a series of temperatures in the broad range from 15 K to 200 K in a semi-logarithmic plot. For comparison, all data have been normalized to the maximum luminescence signal at $t=10^{-7}$ s. With increasing temperature the luminescence decay time is found to decrease from the milli- to the microsecond time range as expected from the studies of Powell and Freed, and Fischer et al [21, 23]. For a quantitative analysis, and according to Zatryb et al [41], the evolution rate of excited emitters has to be considered, so that the first time derivative of the stretched-exponential KWW function is used for the fitting procedure:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I(t,T)=I_0(T) \cdot t^{\left(\beta_{\rm PL}(T)-1 \right)} \cdot {{\rm exp}} \left [-\left (\frac{t}{\tau_{\rm PL}(T)}\right)^{\beta_{\rm PL}(T)} \right]. \label{KWW_derivative} \nonumber \end{align} \tag{ 3 }$$

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As shown by the black lines in figure 4(a), very good correspondence could be achieved between equation (3) and the experimental data using only a single KWW component with two independent fitting parameters, the stretching coefficient $\beta_{\rm PL}(T)$ and the decay time $\tau_{\rm PL}(T)$, giving further support to the unambiguous (STE-related) origin of the PL observed. The dependence $\beta_{\rm PL}(T)$ depicted in figure 4(b) is in full accordance with a trend recently recognized in a smaller temperature interval by Kämpfe et al [19]. The change of $\beta_{\rm PL}(T)$ is very significant and represents a transition from unity for a mono-exponential function at $T\approx 25$ K to the stretched case $\beta_{\rm PL}\approx 0.6$ at $T\approx 180$ K (see figure 4(b)). The decay time $\tau_{\rm PL}(T)$ remains nearly constant for temperatures below 100 K with a value of $\tau\approx10^{-4}$ s whereas an Arrhenius-like behavior with an activation energy $E_{\rm A}\approx 0.09$ eV is found for $T\gg{{\rm 100~K}}$.

We interpret both findings by assuming that at low $T$ the decay time of the luminescence is defined by a localized process where hopping is suppressed leading to a single exponential decay $(\beta_{\rm PL} = 1)$. As the temperature is increased above 100 K, the steep decrease of both $\tau$ and the stretching exponent $\beta$ suggests that a hopping process becomes dominant. To explain the extremely low activation energy for luminescence decay, we assume that hopping of the STE as a whole is involved. In comparison, for free polaron hopping a clearly larger activation energy $E_{\rm A} \approx 0.26$ eV can be derived from the observed temperature dependence of the mean relaxation time $\langle\tau_1(785~{{\rm nm}},T)\rangle$ which also follows an Arrhenius behavior for $T>100$ K in our Mg:LN sample; practically the same value has been estimated earlier as one fourth of the optical excitation energy $\approx$1 eV of Nb$^{4+}_{\rm Nb}$ [12, 42]. Accordingly the temperature dependence of $\beta_{\rm PL}$ has to be attributed to the hopping of STEs and their subsequent localized recombination.

Comparing $\beta_{\rm PL}(T)$ with TA measurements of $\beta_1(785~{{\rm nm}},T)$ on Mg:LN displaying the behavior expected for polaron hopping (see figure 4(b)), it is immediately clear that $\beta_{\rm PL}(T)$ exhibits a reversed temperature dependence. We chose $\lambda=785$ nm as a probe wavelength where the largest contribution, in the absence of antisites, can be attributed to Nb$_{\rm Nb}^{4+}$ free polarons. Both types (TRPL and TA) of temperature behavior in figure 4(b) can be described phenomenologically, e.g. by an error-function with a common inflection point at $T_\beta^{\rm Mg} = (110 \pm 10)$ K indicating the temperature where the processes due to hopping become dominant. Thus, the given reverse type of $\beta_{\rm PL}(T)$ dependence, can be regarded as a fingerprint of STEs, in contrast to the $\beta(T)$ behavior observed for small electron polarons; this is supported by the fact that the activation energies obtained for luminescence and red/near-infrared absorption decay are different requiring a second hopping entity, STEs. Note, that even though the transients in luminescence and absorption are on different timescales, a comparison between the $\beta(T)$ behaviors may be justified as the stretching-factor is a good indicator for the type of transport and annihilation processes involved.

In this section detailed temperature-dependent measurements of the red and blue TA on Fe:LN are presented with the goal to uncover similar fingerprints. The obtained behavior of the various stretching-factors $\beta_i(\lambda,T)$ is shown in figure 5. Results in the red and blue spectral range differ in two major points: (i) with decreasing temperatures $\beta_1(T)$ is observed to decrease in the red spectral range, compared to the opposite behavior of both $\beta_1(T)$ and $\beta_2(T)$ in the blue (see figure 5, but also table 1). Both types of temperature behavior in figure 5, again, can be described by an error-function (orange and blue lines in figure 5) where $T_\beta^{\rm Fe}=(260\pm10)$ K is discovered as a common inflection point. (ii) At higher temperatures, the mean decay times $\langle \tau_i(\lambda,T)\rangle$ follow an Arrhenius behavior, however, for temperatures $T<T_\beta^{\rm Fe}$ the mean decay times in the blue range become nearly temperature-independent. The obtained activation energy $E^1_{\rm A}(785~{{\rm nm}})\approx 0.37~{{\rm eV}}$ apparently corresponds to polaron hopping in the presence of antisites (to be compared to one fourth of the optical excitation energy $\approx$1.6 eV of Nb$_{\rm Li}^{4+}$ [12]). It should be added that the most recent value of the polaron stabilization energy was determined to $\approx 0.75$ eV [37]. The activation energies derived for the blue region, $E^1_{\rm A}(488~{{\rm nm}})\approx 0.71~{{\rm eV}}$ and $E^2_{\rm A}(488~{{\rm nm}})\approx 0.78~{{\rm eV}}$ are correspondingly higher and will be discussed in section 6.

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The temperature dependence of the stretching exponents $\beta_1(T)$ and $\beta_2(T)$ in the blue of the TA of Fe:LN (figure 5) qualitatively shows the same fingerprint behavior as $\beta_{\rm PL}(T)$ obtained from time-resolved luminescence measurements on Mg:LN (figure 4(b)). The similar dynamics of the considered blue transients in emission and absorption strongly suggests their common origin, namely STEs. In Mg:LN the temperature $T_\beta^{\rm Mg} \approx 110$ K equally characterizes the onset of polaronic and excitonic hopping processes observed in transient absorption and luminescence decay, respectively. In Fe:LN, the $T_\beta^{\rm Fe} \approx 260$ K value reflects the smaller mobility of the polarons and a parallel change in electron–phonon coupling due to the presence of antisite Nb defects, as compared to Mg:LN where antisites are absent. These findings will be used in the following section to reconsider the microscopic polaronic recombination model in LN.

STEs may be assumed to have a significant hopping mobility as confirmed by the presence of a thermally activated process ($E_{\rm A}\approx 0.09$ eV) of the luminescence decay. At higher temperatures and in defective materials, STEs formed at early stages of pumping are expected to get pinned on defects, similarly to the formation and trapping of single-site small polarons. Resonant excitonic/energy transfer can be discarded due to the large Stokes shifts. In particular, STEs may be captured, among others, by the same defects as free holes (e.g. by lithium vacancies or Mg$_{\rm Nb}^{2+}$ defects). The final recombination site should play a decisive role, especially at lower temperatures, when localization sets in and $\tau$ gets nearly temperature-independent. The exact optical properties depend on the presence and type of defects adjacent to the niobium-oxygen octahedron containing the STE as shown by luminescence studies in LN with various stoichiometric compositions [16] including also crystals with ilmenite type cationic stacking order [17]. While the presence of Mg up to the $\approx$5 mol% threshold, i.e. the elimination of antisites, leads to increased emission [21], Fe doping seems to lead to luminescence quenching comparable to the case of Cr doping [23], indicating that both intrinsic defects and Fe form pinning centers for STEs where luminescence is suppressed. Non-radiative decay should be strongly preferred for STEs pinned on defects with high pinning/stabilizing energies leading to the longer absorption decay times and higher activation energies as observed for TA in the blue. In this case, the decreasing $\beta_{\rm PL}(T)$ function characterizes the temperature dependent pinning history of the STE influencing its decay. The existence of a second, slow component, with activation energies slightly larger than in our case (0.14 eV and 0.11 eV) valid above somewhat higher temperatures 140 K and 160 K for the respective luminescence components, obtained in 7.5 mol% Mg-doped samples by Kämpfe et al [19], may then be attributed to minor deviations from the crystal structure optimal for luminescence. It should be noted that pinned STEs might also be formed as interim states during the recombination of electron polarons with trapped-hole centers.

STEs, which have been established in various materials, e.g. alkali halides, beside their capability to decay radiatively, have absorption bands similar to those of small polarons as demonstrated by Williams et al [5, 45]. As pointed out by these authors, the hole and electron, which a STE consists of, can be excited optically and show a near ultraviolet and a near-infrared absorption, respectively, corresponding to optically triggered jumps of the respective constituent to equivalent neighboring sites. In LN, due to the transfer of the hole constituent to an equivalent or non-equivalent neighboring oxygen site, STEs are also expected to have an absorption in the blue/near-ultraviolet region. However, a separate contribution of the radiatively decaying STEs to the TA at the measured wavelengths was not observed even in Mg:LN at low temperatures although the luminescence and TA decay times deviate by a factor of at least 1000 from each other and a TA of such STEs—if present—should have been clearly visible. Furthermore, to the best of our knowledge, evidence for the presence of excitonic infrared/ultraviolet absorption in LN is not reported in literature, so far.

Concerning the absorption properties of pinned STEs, we have no experimental indication for the presence of a near-infrared absorption. The 'quenching' of infrared-absorption may be understood by taking into account proposed dynamic or static models of trapped hole O⁻ centers containing a unique Nb neighbor in a strongly preferred position for electron trapping [46, 47]. In such cases electron hopping may be hampered or requires much higher energy. The blue/ultraviolet absorption of pinned STEs, though present, might be nearly indistinguishable from the absorption of an O⁻ trapped hole.

STEs should also be capable to get pinned on iron centers. Most Fe$_{\rm Li}^{3+}$ centers in LN may be assumed to have a nearest neighbor charge-compensating lithium vacancy (V$_{\rm Li}$) suggesting the following scenario: the constituents of a nearby STE are attracted by the constituents of the Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$ dipole leading to an STE pinned on Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$ with the Nb$^{4+}$ and O⁻ sites at nearest neighbor positions of the Fe$_{\rm Li}^{3+}$ and ${{\rm V}}_{\rm Li}$ sites, respectively. A single jump of the electron from Nb$_{\rm Nb}^{4+}$ to the Fe site (along the $c$ axis) transfers the complex into a new state, which is a coupled Fe$_{\rm Li}^{2+}-{{\rm O}}({{\rm V}}_{\rm Li})^-$ defect. Both Fe$_{\rm Li}^{2+}$ and the hole constituent absorb in the blue, their aggregate causing a net increase of the total blue TA with respect to the pinned STE absorption, thus providing a rising TA component. However, this new defect state again has a metastable excitonic character and tends to recombine, restoring the original Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$ defect having no absorption at all. Thus we consider a strongly localized two-step STE recombination process where the electron constituent, starting from a Nb$_{\rm Nb}^{4+}$ state, proceeds to a lower Fe$_{\rm Li}^{2+}$ level before rejoining its O⁻ partner. Since both the pinned STE and the intermediary Fe$_{\rm Li}^{2+}-{{\rm O}}({{\rm V}}_{\rm Li})^-$ complex can be assumed to have metastable ground states with differently relaxed lattice environments, both jumps have to be phonon-induced/assisted, and may involve significant, separate temperature dependent time delays. This two-step STE recombination path, leading to an interim TA maximum with a subsequent steep decay clearly explains the most conspicuous features of the TA at 445 nm, while it seems to be much less pronounced at 488 nm. As the second jump seems to take significantly longer than the first, a description in terms of two separate KWW-functions has to be a good approximation.

The derived activation energies support the two-step relaxation model of the STE pinned on Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$ yielding $E^1_{\rm A}\approx 0.71$ eV for the first and a larger $E^2_{\rm A}\approx 0.78$ eV for the second step, the latter corresponding to the decay of the strongly distorted Fe$_{\rm Li}^{2+}-{{\rm O}}({{\rm V}}_{\rm Li})^-$- state, and roughly equaling one fourth of the strong absorption measured at 445 nm (2.79 eV). A similar ratio may hold for the STE pinned on Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$ in the first step as well, confirming $\propto E_{\rm opt}$/4 as the activation energy characterizing the hopping of a polaron from a given site to an adjacent one [12]. Negligible differences between 488 nm and 445 nm data ($\tau$, $\beta$) in Fe:LN indicate small superpositions from other recombination centers not containing Fe.

At low temperatures the proposed two-step relaxation of the STE pinned on Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$ clearly dominates the 445 nm TA spectrum in Fe:LN for $t>10$ ms (see figure 2(b)). While these two steps seem to explain both the rising and decreasing 445 nm components in Fe:LN, an important contribution to the long decreasing component in Mg:LN may come from STEs pinned on hole-trap defects like (V$_{\rm Li}$)-pinned STEs. Contributions from such pinned STEs correspond to centers with observed lifetimes in the minute range and have been attributed to O⁻  -type centers in 5 mol% Mg-doped LN by Xin et al [40]. In fact, our probe wavelengths are in the central part of the O⁻  and on the very wing of a TA band attributed to O⁻ (Mg$_{\rm Nb}^{2+}$) centers in [40].

Accordingly the first-step lifetime of the (Fe$_{\rm Li}^{3+}-{{\rm V}}_{\rm Li}$)-pinned STE state recombination may correspond to the lifetimes of the rising components $\tau_1$ and $\tau_{1{{\rm b}}}$ observed at 445 nm in Fe:LN at low temperatures and in Mg:LN at RT, respectively. In contrast, much longer second-step lifetime may correspond to $\tau_2$ of the long decreasing component at 445 nm in Fe:LN. The separation of time domains of polaron and pinned-STE decay is rather large in Mg:LN (figure 3). However, there may be a considerable overlap in Fe:LN at RT which resulted in the earlier misinterpretation of the blue TA components discussed above.