1. Introduction

Photorefractive LiNbO₃:Fe crystals have been extensively investigated for applications in all-optical devices where beam coupling established via a refractive index grating is the dominant process [1–5]. Although these holographic gratings are very efficient and long-lived at room temperature [6–8], their efficiency and decay rate are governed by dark conductivity and therefore strongly affected by the temperature, impurity concentration (extrinsic or intrinsic defects), growth methods/conditions, annealing, etc.

In two-beam coupling (TBC) applications, where a transfer of power from one beam to another occurs, different configurations can be used. One particular configuration is when both the pump and signal beams counter-propagate in the recording of reflection gratings, this is referred to as counter-propagating TBC [9]. In this configuration, the signal beam can be generated internally through a Fresnel reflection of the pump beam from the rear of the crystal (self-pumped), or from two external beams. These methods of TBC in LiNbO₃:Fe, particularly the case of the self-pumped counter-propagating TBC, have been shown to have high TBC efficiencies (large gain and fast recording speeds) [10]; this has been attributed to the large photovoltaic field (>100 kV/cm) that is a dominating effect in beam coupling in LiNbO₃:Fe [11,12]. The concentrations of Fe²⁺, Fe³⁺, and other impurities can greatly influence the conductivity and overall performance of the material (TBC efficiency, response time, and storage time) [7,8,13].

Several papers found in literature measure the grating storage time (decay time); these values were used to deduce the dark conductivity levels in LiNbO₃:Fe [7,8]. In this work we directly measure the dark conductivity over a broad temperature range in several crystals, which contain a typical Fe concentration as those materials used for holography. The crystals were subjected to different annealing treatments resulting in the redistribution between Fe²⁺ and Fe³⁺ concentrations, as well as the variation of other impurity concentrations (e.g. H⁺, etc.). As a result of the dark conductivity measurements, the ionization energy (ΔE) of Fe²⁺ has been directly determined and is in agreement with Born’s principle. A correlation between the Maxwell relaxation times and the temperature-dependent dynamics of TBC, particularly the onset of the reduction in the TBC efficiency has been observed. Although this reduction in efficiency was referred to as a “slow instability” in Ref. [14], it is more accurately described as a “slow transient feature preceding the steady state”.

2. Experimental procedure

A series of three congruently grown LiNbO₃:Fe crystals (10 × 10 × 2 mm³, where the c-axis was 2 mm thick), with an Fe concentration in the melt of 0.05 molar %, were cut from the same boule and subjected to different conventional annealing procedures: one was oxidized in an oxygen flow, one was reduced in a nitrogen flow, and one was untreated (as-grown). The conventional annealing process for the crystals used in this study is described in Ref. [15], which contains the absorption spectra and concentrations of Fe²⁺ and Fe³⁺ for the same samples used in this work. Table 1 lists the values for the concentrations of Fe²⁺, Fe³⁺, and Fe_total in the three samples. Note the measured concentration of total Fe in the as-grown sample does not match the amount introduced into the melt (Table 1); this was not the case for the two annealed samples. This difference is likely due to the presence of an optically silent Fe species [15]. The absence of the optically silent Fe in thermally treated crystals (annealed) is a topic of a future paper.

TABLE I:. The values for the concentrations are in terms of 10¹⁸ cm⁻³ [15]. The term as-grown is synonymous with air-grown as found in the reference.

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The temperature dependence of the dark conductivity was measured directly over a broad temperature range, 300–500 K. The sample was mounted inside a vacuum chamber, and the temperature was varied using a cartridge heater with temperature controller. The x-surfaces of the sample (the faces perpendicular to the x-axis) were coated with a conducting material (Aquadag). Electrical leads applying 1kV were attached to both x-surfaces, and the resistivity across the crystal (10 mm) was measured with an electrometer (Keithley 6517A). This geometry was chosen in order to avoid false conductivity signals that could arise from the pyroelectric effect when detecting the signal along the c-axis.

For the case of measuring the self-pumped counter-propagating TBC efficiencies for LiNbO₃:Fe, the samples were placed with their polished faces (z-surfaces) at normal incidence, with the c-axis parallel to the k-vector of the incident light, i.e. the input beam was propagated along the positive c-direction, and the photorefractive gain is obtained along the negative c-direction. The crystals were held in a sample holder consisting of two cartridge heaters, which were controlled by a temperature controller (Omega CN76000). Temperature dependent TBC efficiency measurements were conducted over a large temperature range, from room temperature up to ~500 K. Each crystal was heated to the appropriate temperature and held there with an accuracy of ±0.1 K. The power of the transmitted beam, the depleted pump beam, was measured using a power meter and detector (Newport models 2832-C and 818-SL).

3. Results and discussion

From the measured resistivities as a function of temperature for all three samples, the conductivity was calculated and shown in an Arrhenius plot in Fig. 1. The activation energies were determined from the following equation:

where σ_d is the dark conductivity, k is the Boltzmann constant, and T is the temperature. The activation energy, ΔE, was obtained from the slope of the log_e(σ_d) as a function of 1/kT. A single activation energy (ΔE = 1.1 eV over the full temperature range, 380 – 500 K) was measured in the as-grown and oxidized samples, whereas the reduced sample yielded at least two significantly different values for activation energy (ΔE = 0.5 eV measured for the low temperature region, 300 – 440 K, and 1.1 eV measured for the high temperature region, 440 – 500 K).

 

Fig. 1. Arrhenius plots of conductivity data for three LiNbO₃:0.05mol.% Fe crystals: oxidized (open red squares), as-grown (solid black circles), and reduced (solid blue triangles).

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An activation energy of 1.1 eV, attributed to the hydrogen impurity, falls within the broad range of published values [16]; when considering crystals with Fe, the value of ΔE falls in the range of 0.97 -1.17 eV [8,17]. In Fig. 2, it can be seen that the absorption of the OH⁻ strongly varies between samples, where the maximum differences measured is between the as-grown sample (α = 1.7 cm⁻¹) and the reduced sample (no detectable absorption). However, there is no significant difference in the measured dark conductivity of all three samples in the higher temperature region (see Fig. 1). A plausible explanation has been given in Ref. [18], which suggests that Li ions may be the dominate charge carrier having a similar activation energy in the range of 1.1 -1.3 eV [18–20]. Since there is no observed difference in dark conductivity over the high-temperature range, it is assumed that the source of dark conductivity is approximately constant in all three samples. Since the concentration of H⁺ is considerably different in all samples, while the dark conductivity remains fairly constant, it is possible that our samples have the same type of conductivity (Li⁺) as found in Ref. [18].

Only in the reduced sample is there a second activation energy measured, ΔE = 0.5 eV (low-temperature range). The key difference in the reduced sample with respect to the other samples is the concentration of Fe²⁺ ions, where the reduced sample has nearly an order of magnitude more Fe²⁺ than the as-grown and approximately 40 times more Fe²⁺ than the oxidized sample. For this reason, we attribute the 0.5 eV activation energy to the ionization of Fe²⁺. This value of ΔE and its assignment is in agreement with Born’s principle in thermodynamics, which states that the sum of the ionization energy and charge transfer energy of the Fe²⁺/Fe³⁺ level must equal the band gap energy (See Fig. 3).

 

Fig. 2. OH⁻ absorption spectra of as-grown (black line), oxidized (red line), and reduced (blue line) LiNbO₃:Fe.

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With a measured charge transfer (CT) energy (valence band to the Fe³⁺) of 3.1 eV and an estimated band gap (E_g) value of approximately 3.72 eV [21], Clark, et al., calculated the threshold for photoionization to be approximately 0.62 eV. Our value of 0.5 eV is in reasonable agreement with the predicted value from the Born’s principle calculation used in Ref. [21]. Furthermore, the published spectrum of LiNbO₃:Fe from Ref. [22] reveals an absorption band in the infra-red ranging from 0.6 to 1.5 eV. Although this band was attributed to an intra-ion d-d transition in Fe²⁺ [21,22], an absorption associated with the photoionization of Fe²⁺ may also be contributing. The Franck-Condon principle may be strongly affecting the spectral shape of the photoionization part of the absorption, but the long wavelength threshold must be the same as that determined from the ionization of Fe²⁺.

 

Fig. 3. (Left) Energy level diagram of the Fe²⁺/Fe³⁺ level in LiNbO₃:Fe. (Right) Absorption spectrum (digitized from Ref. [22]) indicating the charge transfer and d-d transition in Fe²⁺ (⁵A醒⁵E) absorption bands. In this paper, we suggest the feature starting at 0.6 eV may be the threshold of Fe²⁺ photoionization. The asterisks indicate photoionization and charge transfer of the Fe²⁺/Fe³⁺ level.

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When comparing our results with literature data, one has to keep in mind a very important fact: the samples used in this work are Fe doped, and both charge states of Fe (Fe²⁺ and Fe³⁺) are present in all samples (see Table 1). This means that the Fermi level is pinned to the Fe^2+/3+ level in the bandgap of LiNbO₃, and ionization of only Fe²⁺ can govern dark conductivity on the low temperature side – unlike the case of undoped reduced samples where a polaronic conductivity is believed to be dominating [23].

In order to determine the effect that dark conductivity has on counter-propagating TBC, theoretical and experimental beam coupling results are compared to the temperature dependence of conductivity (resistivity) and Maxwell relaxation time experimental data. The effect dark conductivity has on counter-propagating TBC can be readily calculated by modifying the standard coupled intensity equations to include a dark erasure term, I_dark [24]:

where I_p,s are the pump and signal beam intensities respectively, α is the absorption coefficient, Γ is the beam coupling gain coefficient, and L is the optical path length of the crystal. I_dark is the equivalent optical intensity which would erase the grating at the same rate as the natural thermal erasure rate in the absence of any external light source. Although I_dark may be dominated by dark conductivity, it contains any mechanism that can erase the grating, i.e. scattering, etc. Figure 4 shows how the dark conductivity affects the TBC process for three different ΓL products. As the dark conductivity increases, the steady state transmission through the crystal increases. The threshold for this process depends on the value of the ΓL product (i.e. on the total optical gain), but the general trend holds true for all gain values. In the limit, if the dark conductivity is sufficiently large compared with the pumping intensity (large I_dark/I_p), grating formation can be entirely suppressed.

 

Fig. 4. The normalized steady state transmission as a function of I_dark/I_p derived from the modified coupled wave theory for three values of ΓL: (a) 5, (b) 10, and (c) 15.

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The Maxwell relaxation time (the time required for the majority carriers to respond to a disturbance in the electric field or charge redistribution) is directly proportional to the crystal resistivity. If we assume strong beam coupling, the optical modulation depth within the region of energy exchange between the counter-propagating beams will be large. Under these conditions, we may expect the net conductivity to be dominated by the dark conductivity since a large modulation depth essentially introduces a periodic near-zero photoconductivity between adjacent light fringes. Assuming the electrical permittivity remains approximately constant over our experimental temperature range (T≪Curie temperature), we may expect to observe a correlation between the Maxwell relaxation time and the beam coupling efficiency. For the case of weak beam coupling, the optical modulation depth becomes small, and the photoconductivity between adjacent light fringes is no longer negligible.

The relationship between the experimental results of the Maxwell relaxation time and the coupling efficiency is also considered. Both the resistivity (ρ) and the corresponding Maxwell relaxation time (τ = ε₀ερ, where ε₀ is the permittivity in free space, ε = 28 is the permittivity in LiNbO₃ [25, 26], and ρ is the resistivity in LiNbO₃ [27]) as a function of temperature is shown in Fig. 5. A correlation between the Maxwell relaxation times and the onset of a temperature-dependent reduction in two-beam coupling efficiency is observed (See Fig. 6).

 

Fig. 5. Measured resistivity and calculated Maxwell relaxation times as a function of temperature in the as-grown (open black circles) and oxidize (solid red circles) LiNbO₃:Fe crystals. The holographically-measured Maxwell relaxation times (open black triangles) are plotted against the right vertical axis only.

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Fig. 6. Transmitted power as a function of time (two-beam coupling efficiencies) for various temperatures: (top to bottom) 473 K (red line), 448 K (green line), 423 K (blue line), 398 K (orange line), and 294 K (black line).

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In Fig. 6, the transmitted power as a function of temperature is shown for the as-grown crystal. The resolution of the temperature was 0.1 K. The minor oscillations correspond to the oscillation in temperature (±0.1 K); this effect was greater with larger temperature fluctuations. In these measurements, the transmitted power is a measure of the TBC efficiency, i.e. the greater the photorefractive gain the greater the reduction in the transmitted power in the self-pumped counter-propagation TBC geometry (where the incident pump beam is propagating along the +c axis). After the initial reduction in transmitted power, there is a subsequent increase in transmitted power; the time where the minimum power occurs is referred to as the onset of the reduction in TBC efficiency (or simply the occurrence of a slow transient feature to the steady state [14, 28]). The onset of this slow transient feature corresponds with the Maxwell relaxation time as shown in Fig. 5. The results of the measured minima of this transient feature as a function of temperature are in good agreement with the calculated Maxwell relaxation times and display the same characteristics as predicted by the modified coupled wave theory (Eqs. (2), (3) and Fig. 4). This also agrees with theory found in Ref. [29], which states that the coupling efficiency increases with time until the dielectric (Maxwell) relaxation time is reached, in which case the efficiency is reduced until a steady state is reached (See also Ref. [14]).

For temperatures >450 K, there is a deviation between calculated and holographically measured Maxwell relation times. This is likely related to the modulation depth, where at larger temperatures the coupling efficiency is weak, thus the optical modulation depth is small. Under this condition, both dark conductivity and photoconductivity are present between adjacent light fringes during beam coupling measurements, whereas for the resistivity measurements only dark conductivity is present.

Conclusion

We have observed two activation energies in LiNbO₃:0.05mol.%Fe: ΔE = 1.1 eV is measured in all samples regardless the concentrations of Fe²⁺ and Fe³⁺, and ΔE = 0.5 eV is measured only in the sample with a larger concentration of Fe²⁺ (i.e. reduced crystal). It is believed that the activation energy of 1.1 eV corresponds to Li⁺ ions, rather than H⁺ ions with supporting evidence in the OH⁻ absorption spectra. The value of ΔE = 0.5 eV is attributed to the ionization energy (photoionization threshold) of Fe²⁺, which is in agreement with Born’s principle and directly supports to the calculated photoionization threshold found in Ref. [21]. Calculated Maxwell relaxation times as a function of temperature from measured resistivity data correspond to the onset of a reduction in the TBC efficiency, which agrees with the modified coupled wave and published theories.