Journal of Luminescence 107 (2004) - Department of Physics ...

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$Defects in inorganic photorefractive materials and their investigations$

ARTICLE IN PRESS M. Drobizhev et al. / Journal of Luminescence 107 (2004) 194–202 199 fluorescence spectrum: Fðn; TÞ ¼ KðnÞ½1 þ MðTÞ cosð2pDtn þ DjðTÞÞŠ; ð7Þ where the amplitude MðTÞ and the phase shift DjðTÞ of the grating are given by MðTÞ ¼bf½A þ B cos d 2 þ C cos d 1 þ D cosðd 1 þ d 2 ÞŠ 2 þ½B sin d 2 þ C sin d 1 þ D sinðd 1 þ d 2 ÞŠ 2 g 1=2 ; ð8Þ DjðTÞ B sin d 2 þ C sin d 1 þ D sinðd 1 þ d 2 Þ ¼ arctg A þ B cos d 2 þ C cos d 1 þ D cosðd 1 þ d 2 Þ : ð9Þ Here functions AðTÞ; BðTÞ; CðTÞ; and DðTÞ are expressed through the model parameters as follows: AðTÞ ¼a 1 ðTÞa 2 ðTÞ; ð10Þ BðTÞ ¼½1 a 1 ðTÞŠa 2 ðTÞ f1 þðpDtn 2 Þ 2 g 3=2 ; CðTÞ ¼a 1 ðTÞ½1 a 2 ðTÞŠ f1 þðpDtn 1 Þ 2 g 3=2 ; DðTÞ ¼½1 a 1 ðTÞŠ½1 a 2 ðTÞŠ f½1 þðpDtn 2 Þ 2 Š½1 þðpDtn 1 Þ 2 Šg 3=2 ð11Þ ð12Þ ð13Þ and d 1; 2 ¼ 3 arctgðpDtn 1; 2 Þ: ð14Þ We note here that in the limiting case a 1 ð0Þ ¼ a 2 ð0Þ ¼1 Eq. (9) does give a quantitativelycorrect phase shift, Djð0Þ ¼0: However, for example, for a 1 ¼ a 2 ¼ 0; the correct behavior is reproduced onlyat small pDtn 1; 2 : Dj ¼ 3pDtðn 1 þ n 2 Þ: This last value corresponds to a frequencyshift (Stokes shift) of Dn ¼ 3 2 ðn 1 þ n 2 Þ; equal to a difference between centers of gravityof PWs (5) in absorption and fluorescence. In general case, the values of Dj allowed byEq. (9) are restricted bydefinition of function arctgðxÞ; and, therefore, Eq. (9) is unable to describe a wide range of other real sets of parameters. In our approach we use Eq. (8) for the simulation of the temperature dependence of the amplitude (contrast) of the fringes, but we resort to a simpler model for the simulation of the temperature dependence of the phase. We consider the Stokes shift as a difference between the first moments (centers of gravity) of absorption and fluorescence homogeneous spectra, namely, DnðTÞ ¼ð1 a 2 ðTÞÞn 2 þð1 a 1 ðTÞÞn 1 ; ð15Þ and, therefore, DjðTÞ ¼2pDt½ð1 a 2 ðTÞÞn 2 þð1 a 1 ðTÞÞn 1 Š: ð16Þ 5. Results of simulations and discussion 5.1. Chl in PVB First of all, we simulated the temperature dependence of the fringes in Chl:PVB system by using symmetrical model with x 1 ¼ x 2 ¼ x and Phase shift, ∆ϕ (rad) Modulation amplitude, M (a) (b) 0.3 0.2 0.1 3 2 1 20 40 60 80 100 120 T, K 40 60 80 100 Fig. 5. Temperature dependence of the amplitude M (a) and phase shift Dj (b) of spectral grating observed in Chl:PVB and corresponding fits of these data to Eqs. (8) and (16) in mirrorsymmetrical model.
ARTICLE IN PRESS 200 M. Drobizhev et al. / Journal of Luminescence 107 (2004) 194–202 n 1 ¼ n 2 ¼ n m : Byvarying two adjustable parameters, x and n m ; we obtain the best fit, shown in Fig. 5a, atx ¼ 0:36; (i.e. a 1 ð0Þ ¼a 2 ð0Þ ¼0:70) and n m ¼ 25 cm 1 . Substitution of these parameters into Eq. (16) without anyfurther fitting gives a phase shift, shown in Fig. 5b bysolid line. One can conclude that the model fits well in this case and the obtained parameters are reasonable. The value of n m is slightlylarger than the maximum phonon frequency(18 cm 1 ), but it is veryclose to a center of gravityof PW distribution, which can be a real effective frequencyentering Eq. (6). The limiting Debye–Waller factor a 1 ð0Þ agrees well with the value measured at 7 K (a 1 ¼ 0:55) in Ref. [13]. We then performed the fit procedure of the same data but using the asymmetrical model, where all four parameters x 1 ; x 2 ; n 1 ; and n 2 were varied independently. The best fit of MðTÞ (Fig. 6a), and DjðTÞ (Fig. 6b), do, in fact, describe the experiment quite well, but the limiting Debye–Waller factor a 1 ð0Þ ¼0:86 comes out much larger than the literature data, cited before. The main conclusion, which can be drawn from the simulation obtained for Chl:PVB is that the symmetrical model works quite well for this system. 5.2. SiNc in PVB The simulation of experimental data for SiNc:PVB with mirror-symmetrical model is shown in Figs. 7a and b. Again, the fit procedure was first accomplished for the amplitude and then the best parameters were introduced in Eq. (16) to describe the phase shift. The main inconsistency between the simulation and experiment is observed in the phase behavior, because the model tends to overestimate the rate of the shift with temperature. The obtained best-fit parameters also differ considerablyfrom literature data: the limiting value of Debye–Waller factor ða 1 ð0Þ ¼0:56Þ is waytoo small as compared to that reported previously ða 1 ¼ 1:0 at1:5KÞ [14] and the characteristic Modulation amplitude, M 0.3 0.2 0.1 Modulation amplitude, M 0.2 0.1 (a) 20 40 60 80 100 120 (a) 20 40 60 80 100 120 T, K ∆ϕ (rad) Phase shift, (b) 3 2 1 20 40 60 80 100 120 T, K Fig. 6. Temperature dependence of the amplitude M (a) and phase shift Dj (b) of spectral grating observed in Chl:PVB and corresponding fits of these data to Eqs. (8) and (16) in nonmirror-symmetrical model. Phase shift, ∆ϕ (rad) (b) 5 4 3 2 1 20 40 60 80 100 120 Fig. 7. Temperature dependence of the amplitude M (a) and phase shift Dj (b) of spectral grating observed in SiNc:PVB and corresponding fits of these data to Eq. (8) and (16) in mirrorsymmetrical model.
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