1. Introduction

Tin Hypothiodiphosphate (Sn₂P₂S₆, SPS) possesses impressive optical nonlinearities, especially a pronounced photorefractive effect in the red and near infrared spectral domains [1, 2]. From the first publications on photorefractive gratings recording with Nd³⁺ laser light (λ = 1.06 μm) in SPS it is known that quite fast amplification of a signal wave (in millisecond range) is followed by much slower (up to few minutes) decrease of its intensity until the certain steady state level is reached. A similar type of transient beam coupling has been reported earlier for sillenite crystals [3–7]; it was detected later in several doped SPS crystals (see, e.g., [8]). “Close relatives” of this effect are the thermal fixing process first observed in LiNbO₃ [9] and charge compensation at elevated temperatures first observed in KNbO₃ [10]. The formation of two complementary gratings in Sn₂P₂S₆ has been successfully used for transient gain enhancement [11] and for design of a coherent optical oscillator with particular temporal dynamics, “optical multivibrator” [12,13].

Such temporal dynamics of two-beam coupling with a transient peak is explained by the presence of two types of movable charge carriers, holes and electrons. They both contribute to the formation of the space charge grating, but their particular contributions are exactly out-of-phase and tend to compensate each other. The necessary condition to observe transient coupling is a considerable difference of the characteristic build-up times τ_f and τ_s of the two contributions (subscripts f and s for fast and slow, respectively), which is related in part to a pronounced difference of the electron conductivity from the hole conductivity.

Practically in all cases mentioned in [1–8] the fast grating is developed by the optically excited charge carriers, while the compensation grating appears because of a redistribution of the thermally excited (optically inactive) carriers of opposite sign. This conclusion follows from the independence of the characteristic decay time of compensation grating τ_s of the recording (erasing) wave intensity (see, e.g., [7,8]). The characteristic time of the fast grating τ_f depends on the dielectric relaxation time and is therefore inversely proportional to the sample conductivity and light intensity.

An atypical behavior of the compensation grating that consists of an intensity dependence of the slow grating characteristic decay time is reported in this paper for antimony doped SPS crystals. This particularity may be attributed to the thermal excitation of electrons from the optically populated impurity level Sb³⁺/Sb²⁺ located within the bandgap close to the conduction band. The excessive density of free electrons in the conduction band being in quasiequilibrium with optically recharged antimony ions (Sb²⁺) is linearly dependent on recording (erasing) light intensity. The free-electron component of the dielectric relaxation time ${\tau }_{di}^e$ has therefore a part which is inversely proportional to the intensity of light, thus affecting the slow component characteristic time τ_s.

We describe in this article two-beam coupling dynamics in SPS:Sb with the aim to characterize the crystal and identify the origin of the unusual behavior of the compensation grating.

2. Experimental results

The Sn₂P₂S₆ crystals studied in the experiment were grown in Uzhgorod National University, Ukraine, by a chemical vapor transport technique. All samples have been cut along the crystallographic axes, and their xy-faces were optically finished. The main part of the data shown below was obtained with 1% antimony doped sample K33 (6.0 × 4.5 × 8.0 mm³), while some data were measured using another SPS:Sb(1%) sample K24 (2.0 × 3.5 × 2.4 mm³). The results obtained for SPS:Sb were compared with the data measured in this work for nominally undoped Type I SPS [8] (9.0 × 4.5 × 9.0 mm³). Two-beam coupling was studied in a standard experimental set-up. The copropagating pump and signal beams from the same He-Ne laser impinged upon the SPS sample in the xz-plane, with their bisector aligned along the sample z-axis; the transmitted signal beam was directed to the detector.

Figure 1(a) shows an example of the temporal dynamics of the amplified signal wave intensity for SPS:Sb. The input signal wave with the intensity $I_s^0$, which is much weaker than the pump intensity I_p, is switched-on at the moment t = 0. In the particular case shown, the output intensity of the signal wave increases 23 times within a few milliseconds and then a compensation grating develops during a few seconds resulting in a steady-state gain that is roughly half of its peak value. To reduce the coupling strength and remain well within the undepleted pump conditions the polarization of the two interacting waves was chosen to be normal to the interaction plane. A Pockels coefficient r₂₂₁ which is roughly two times smaller than the largest coefficient for SPS crystal (r₁₁₁) defines the maximum two-beam coupling gain in this case. It should be mentioned that just before measuring the dynamics of two-beam coupling similar to Fig. 1(a), the sample was preexposed to a strong light wave to optically sensitize it; this process is described in reference [14]. When measuring the intensity dependences of the characteristic times, the intensity of preexposure was always the same as the intensity of the pump wave.

 

Fig. 1 Temporal dynamics of (a) two-beam coupling and (b) grating erasure in SPS:Sb sample K33. The output intensity of signal beam I_s in (a) is normalized to its initial value in the absence of the pump wave $I_s^0$. The diffraction efficiency in (b) is normalized to its saturation value. At t = 0 the signal wave is switched on in panel (a) and switched off in panel (b). The two recording waves from the He-Ne laser impinge upon the z-cut sample symmetrically in the xz-plane and are both y-polarized. The intensities of the signal and pump waves are 2 mW/cm² and 2 W/cm², respectively, and the grating spacing is 8 μm.

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The presence of two space charge gratings manifests itself also in the temporal dynamics of the optical erasure of the recorded grating (Fig. 1(b)). If the input signal wave is terminated abruptly at t = 0, the interference pattern that records the grating no longer exists and the grating starts to decay. At t > 0 the detector monitors a part of the incident wave intensity which is diffracted from the grating in the direction of the signal wave. It can be seen that the diffracted signal drops quickly to zero and then increases reaching a maximum value and finally vanishes. This behavior has been explained in [1] as follows. The grating resulting from the fast recording process in Fig. 1(a) decays more quickly than the compensation grating. At a certain point the amplitudes of the two out-of-phase gratings become equal, which is when full compensation occurs, and the diffracted intensity goes to zero. As time increases, both contributions to the grating continue to decay with their own characteristic times; at this time interval the amplitude of the compensation grating is greater and contributes more to the diffracted signal. A maximum of the diffraction is reached when the amplitude of the fast grating is already small compared to that of compensation grating; for longer times we observe only a decrease of the diffracted intensity.

This interpretation is supported by an independent experiment, which proves that the diffracted wave phase changes to π at the moment when its intensity passes zero value (see Fig. 1(b)). Images of the fringe pattern formed at the output face of the sample by the transmitted readout (pump) wave and the diffracted wave have been taken before and after the moment of exact compensation. The obvious inversion of fringe contrast has been detected, thus proving that the diffraction occurs from the complementary gratings which are shifted exactly by half a period.

The diffraction efficiency of a grating, being proportional to the square of the index modulation, diminishes two times faster than the signal intensity when two interacting waves are present at the crystal input face (see, e.g., [15]. This can be clearly seen from a comparison of panels (a) and (b) in Fig. 1 (note that the time scales in (a) and (b) are not the same).

The two-beam coupling that results in amplification of the signal wave at the expense of the pump wave intensity is usually characterized by the gain factor $\mathrm{\Gamma }\simeq -(1/\ell )\text{ln}\left(I_s/I_s^0\right)$. In the considered case of SPS:Sb the gain factor has two components, fast and slow, given by:

Figure 2 shows the intensity dependences of τ_f and τ_s for two different grating spacings, 8 μm and 1 μm. The decrease of the both fast and slow grating lifetimes with the increasing I is evident (shown as a rise in reciprocal time). Within the intensity range from zero to 2 W/cm², a relative change of τ_f is greater for larger grating spacings; the opposite is true for τ_s. The characteristic time of the slow grating is nearly independent of grating spacing, but it does depend on the recording light intensity, which is contrary to that of nominally undoped SPS [8].

 

Fig. 2 Intensity dependences of reciprocal (a) fast and (b) slow times at ambient temperature for grating spacing 8 μm (filled symbols) and 1.0 μm (open symbols) for K33 SPS:Sb.

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Using a Peltier element the temperature of the sample could be controlled over the range from −25 to +50°C. The temperature dependence of τ_s was measured within this range and an Arrhenius plot for the slow grating recording time is shown in Fig. 3. The activation energy for formation of the compensation grating extracted from this plot is ΔE ≃ 0.42 eV. Similar measurements with different light intensities up to 2 W/cm² did not show any regular intensity dependence for ΔE that remained within a range (0.44 ± 0.02) eV. The data for ΔE obtained from the temperature dependences of grating optical erasure (grating lifetime) are also in close agreement.

 

Fig. 3 Arrhenius plot for the reciprocal slow time measured with sample K24; grating spacing Λ = 6 μm, light intensity I = 0.4 W/cm².

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Figure 4 shows the spatial frequency dependences of the reciprocal times 1/τ_f and 1/τ_s for antimony doped SPS. With the same experimental set-up we measured similar dependences for a nominally undoped SPS crystal to get a reference for comparison. All these measurements lead to several qualitative and quantitative conclusions. One general conclusion is the confirmation of the transport lengths ratio: in the antimony doped crystal the Debye screening length is smaller than the diffusion length. This is a general rule for all doped and nominally undoped SPS crystals (including SPS:Sb [16]) with only one exception: the diffusion length is smaller than the Debye screening length in the so called “modified” or “brown” SPS crystal [16, 17]. It can be stated also, that the values of the fast grating times τ_f are very close for nominally undoped and antimony doped crystals, while τ_s may differ by several orders of magnitude for these crystals depending on the grating spacing. The slow grating time τ_s is distinguishable in SPS:Sb by its very weak dependence on grating spatial frequency.

 

Fig. 4 Grating-spacing dependence of the reciprocal characteristic decay times normalized to the total intensity of two recording beams. Squares/diamonds and circles mark τ_f and τ_s, respectively. Half-filled (red) symbols are used for the antimony doped sample K33, while open symbols denote the nominally undoped reference sample. The lifetime for the compensation grating in the undoped sample is measured with I = 3 W/cm². Dashed and solid lines are the best fits of theoretical dependences as explained in text.

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From the comparison with theory for two-beam coupling in crystals with compensation gratings [18], it is possible to evaluate the characteristic transport lengths and get estimates for the effective hole trap density and free electron density. The relationship between characteristic time τ_s and grating spatial frequency K was derived in [18] within the approximation of compensating charges excited only thermally and is therefore unapplicable in our case. This approximation does not affect, however, the relationship between τ_f and K because the fast grating saturates well before the slow grating develops in full (τ_f ≪ τ_s).

 

Fig. 5 Energy level diagram for antimony doped SPS. The absorption of a He-Ne laser photon creates a free hole in the valence band VB that is captured by a hole trap X and recharges the antimony ion Sb³⁺ → Sb²⁺. A portion of excessive electrons is thermally excited from the antimony level to the conduction band and thus decreases the dielectric relaxation time.

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The dashed gray lines and solid red line in Fig. 4 show the results of the best fits of the measured characteristic relaxation times τ_f,s with calculated grating spacing dependences (given by Eqs. (10) of publication [18]). The details of characterization of nominally undoped crystals will be a topic of a separate publication, here we intend to compare macroscopic parameters for photoexcited holes such as dielectric relaxation time ${\tau }_{di}^h=∊{∊}_0/{\sigma }^h$, Debye screening length ${\ell }_s^h=\sqrt{∊{∊}_0{k}_{B}T/N_{\mathit{eff}}^h{e}^2}$, and diffusion length ${\ell }_D^h=\sqrt{\mu {\tau }_c{k}_{B}T/e}$ (with low frequency dielectric susceptibility ∊₀, total hole conductivity σ^h, effective hole trap density $N_{\mathit{eff}}^h$, Boltzmann constant k_B, absolute temperature T, and electron charge e):

It should be mentioned that ${\tau }_{di}^{h}I$ values are taken from data in Fig. 4 in the range of small K where τ_s is already K-independent, while the ratios ${\ell }_D^h/{\ell }_s^h$ are extracted from the comparison of τ_s at very large and very small spatial frequencies K. The screening length ${\ell }_s^h$ in SPS:Sb is known to be dependent on the intensity of the recording beams and on sample preillumination [14]; the value shown in Table 1 is a saturated value measured at large intensities above 1 W/cm². (This is justified below, when discussing the results of Fig. 6(a)).

Table 1. Parameters extracted from dynamics for nominally undoped and Sb-doped crystals

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Fig. 6 Intensity dependence of (a) the Debye screening length ${\ell }_s^h$ and (b) the effective hole trap density $N_{\mathit{eff}}^h$ SPS:Sb 1% (K33). The solid red line shows the best fit of Eq. (2) to the experimental data.

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At first glance the presented data might look unexpected. The antimony doped SPS crystal usually features larger values of the gain factor compared to those of nominally undoped SPS. This seemingly contradicts the stronger charge screening effect in SPS:Sb, which shows a larger Debye screening length ${\ell }_s^h=0.33\hspace{0.17em}\mu \text{m}$. The stronger screening really reduces the gain factor, but this reduction is overcompensated by the enhanced effective Pockels coefficient as it follows from the data in Table 1 of reference [17]. This larger effective Pockels coefficient might be the consequence of a close to perfect single-domain state for SPS:Sb, comparing to nominally undoped crystals that usually possess a considerable amount of small domains with 180° orientation of the spontaneous polarization. The estimated diffusion lengths do not differ much for these crystals, but a larger value of dielectric susseptibility (280 for SPS:Sb [16, 17] instead of 230 for nominally undoped crystal [8]) makes its characteristic response time τ_s slightly longer.

Before starting the discussion of the presented results, we would like to underline that one needs to be very careful when extracting τ_s and τ_f from the simple relationship given by Eq. (1). Apart from the two processes of the fast and slow grating formation considered here, there is in addition one more dynamic process, the self-sensitizing of the SPS:Sb sample exposed to the red He-Ne laser light [14]. The lifetime of secondary centers responsible for optical sensitizing is ≈ 14 s at ambient temperature [14], but the initial generation rate of these centers after beginning of exposure is obviously intensity dependent. For particular experimental conditions the characteristic build-up time of the compensation grating might become comparable, within the time of measurement, to the effective time of sensitizing. This may lead to wrong estimates for τ_s and even for wrong impression that τ_s changes its sign (see, e.g. Fig. 4 of Ref. 16). In addition, the influence of the sensitizing process is grating spacing dependent because the ratio τ_s/τ_f depends on the spatial frequency (see Fig. 4). While at large grating spacings the manifestation of the compensation grating is clearly visible (see Fig. 1) it becomes hardly detectable at small spacings (see, e.g., Fig. 3 of Ref. 14). All measurements described above were conducted taking into consideration this danger of misinterpretation. The data for τ_s that become unreliable in critical range of high spatial frequencies are not shown in Fig. 4.

3. Discussion

The experimental data presented above show that a distinction of SPS:Sb from nominally undoped material consists of an unusually small formation time of a compensation grating (on the order of one second and shorter), which in addition appears to be intensity dependent.

To reveal the origin of an unusual behavior of compensation grating in SPS:Sb it is necessary to consider the role of antimony in this doped material. According to EPR data [19] trivalent antimony ions substitute for divalent tin ions in the crystal lattice. They remain EPR silent when the sample is kept in the dark, i.e., with no light (or no preillumination) all antimony ions are in trivalent state. If the sample is illuminated with blue (0.44 μm, He-Cd laser) light trivalent antimony gains an electron and becomes divalent, Sb³⁺ + e → Sb²⁺ [19]. This divalent state has a limited lifetime, at room temperature it returns to the trivalent state roughly within 10–20 s as determined from the lifetime of the sensitized state in [14] and confirmed further in [20]. The recharging of the antimony ion mentioned above is possible not only with high energy quantum within a range of the fundamental absorption, it occurs also with excitation by He-Ne laser light.

With the isothermal annealing technique of photogenerated EPR active centers (Sb²⁺) [21], the activation energy of the relaxation process Sb²⁺ → Sb³⁺ was estimated to be about 0.42 ± 0.01 eV. Taking into account that the energy of He-Ne laser light photon is equal to 1.95 eV and the bandgap of nominally undoped SPS crystal is estimated to be 2.35 eV (see, e.g., [22]), the Sb³⁺/Sb²⁺ level can be positioned 0.42 eV below the conduction band as it is shown in Fig. 5. This can be explained by the Born-Haber cycle, as described in detail in reference [23]. It is in agreement also with the fact that even for slightly antimony doped (0.5%) SPS the absorption edge is considerably shifted to the larger wavelengths [16,17].

The photoinduced transition of an electron from the valence band to the Sb level creates a free hole in valence band that could produce a hole polaron [24]. These hole polarons are thermally stable only at low temperatures. At ambient temperature they are quickly trapped (in sub millisecond time range) by not yet identified X-centers, which can remain positively charged much longer time but can release the holes back to VB when absorbing He-Ne light photons. These are presumably the secondary “hole donor” centers that are responsible for optical sensitizing of SPS:Sb which was discovered in [14]. A photogenerated population of positively charged X-centers increases the effective trap density, even in the case where the density of the intrinsic deep hole traps is constant and does not depend on the light intensity.

Below we consider another possible effect of the photorecharging process Sb³⁺ → Sb²⁺ that consists of creating an excessive electron density on the antimony level below the conduction band (see Fig. 5). In part, these electrons are thermally excited to the conduction band; a thermal quasiequilibrium between the antimony level and conduction band establishes at any given moment of time. Thus, the free electron density in CB should linearly follow the density of Sb²⁺ ions which, in turn, increases linearly with the light intensity. As a consequence, the electron conductivity σ^e has an intensity dependent component (with no direct optical excitation of the electrons to the conduction band); this is also true for the dielectric relaxation time ${\tau }_{di}^e=∊{∊}_0/{\sigma }^e$. In such a way the intensity dependence of a slow grating relaxation time (see Fig. 2b) is also explained. (It is quite understandable that such an intensity dependence would not exist if the lifetime of Sb²⁺ centers were smaller than ${\tau }_{di}^e$; in that case the photosensitized state would vanish before the compensation grating is developed.)

The proposed explanation does not contradict the experimental data and model described in [14], which explains the optical sensitizing of SPS:Sb. A conclusion of this last paper [14] is that the effective trap density increases with the intensity of the recording light or with the time of preillumination. Following the explanation given in the present article, one can express the effective hole trap density for SPS:Sb as follows:

One prediction of Eq. (2), which can be checked experimentally, is that with reasonable parameters the effective trap density $N_{\mathit{eff}}^h$ should saturate with the recording light intensity even within the accepted assumption that no saturation occurs for the Sb²⁺ density itself.

To verify this prediction we plotted the intensity dependence of the Debye screening length (Fig. 6(a)), extracted from measurements of angular dependences of the gain factor with different intensities of the recording light. Then, using the definition of the screening length ${\ell }_s^h=\sqrt{∊{∊}_0{k}_{B}T/N_{\mathit{eff}}^h{e}^2}$, the intensity dependence of $N_{\mathit{eff}}^h$ was plotted, as shown by solid squares in Fig. 6(b), with ∊ = 280 [17]. The red line shows the best fit of Eq. (2) to the experimental data. The fit gives the following set of parameters

In formal way the dependences shown in Fig. 6 can be explained well by the expression for $N_{\mathit{eff}}^h$ with a hole trap density which is intensity dependent but the hole donors density remains constant. This explanation is, however, much less probable taking into account that the transport of positively charged species is responsible for space charge formation both in nominally undoped and antimony doped Sn₂P₂S₆.

As it follows from values shown above and the data from Fig. 6(b), the effective hole trap density $N_{\mathit{eff}}^h\approx 4.0\cdot {10}^{15}\hspace{0.17em}{\text{cm}}^{-3}$ remains moderate even when saturated at large intensities. It is smaller, for example, as compared to ${\text{N}}_{\mathit{eff}}^h\approx 2\cdot {10}^{16}\hspace{0.17em}{\text{cm}}^{-3}$ for nominally undoped BaTiO₃ [25] and that of photorefractive semiconductor CdTe (N_eff = 2.4·10¹⁶ cm⁻³ at 1.064 μm and 1.1·10¹⁶ cm⁻³ at 1.55 μm [26]). Finally, it is orders of magnitude smaller than in photorefractive crystals that allow for efficient recording of the reflection-type space-charge gratings, LiNbO₃:Fe ( $N_{\mathit{eff}}^e\approx {10}^{18}\hspace{0.17em}{\text{cm}}^{-3}$ [27]) and reduced KNbO₃:Fe ( ${\text{N}}_{\mathit{eff}}^e\approx {10}^{17}\hspace{0.17em}{\text{cm}}^{-3}$, based on data of [23]).

It seems, for antimony doped SPS the effective trap density is presently limited by the insufficient density of the intrinsic hole traps $N_T^h$. One can expect a considerable improvement of two-beam coupling gain, especially at high spatial frequencies of the recorded gratings if this problem could be solved by advancements in SPS growth technologies.