CHAPTER 4. FEMTOSECOND TIME-RESOLVED
4.2 A NALYSIS OF TIME - RESOLVED DATA
4.2.4 Simulation of transient-transmission spectra
Contributions to the dielectric function of ZnO:
The dielectric function of ZnO was modelled with the help of the model dielectric function suggested by Adachi. The model takes into account transitions corresponding to the direct-bandgap, and those between the lowest conduction band and the components A, B and C of the valence band in ZnO. In addition to that, excitonic interactions in the neighborhood of the lowest- direct band edge and core-level transitions are also considered. Thermal effects were incorporated into the calculation through a temperature dependent band-gap and exciton broadening [84].
The fundamental contributions to the real and imaginary part of the dielectric function of ZnO together with the overall contribution are plotted in Figure 4.10 (a) and (b) respectively.
Figure 4.10: Overall and individual contributions to the real (a) and imaginary (b) part of the dielectric function of ZnO
Contribution to dielectric function from discrete exciton states:
The discrete series of exciton states can be expressed as a Lorentzian line shape given by equation (4.13), where Anα is the discrete excitonic strength parameter and Γnα the excitonic broadening. Enα (the discrete-exciton energy) is given by equation (4.14) where G is the 3D exciton Rydberg energy.
∑
, ,∑
(4.13)(4.14) Three discrete exciton absorptions at ~ 375 nm, 373 nm and 366 nm correspond to transitions from the three valence bands A, B and C respectively to the exciton states.
The oscillator strength Anα is proportional to n-3, i.e. the oscillator strength is considerably reduced for higher order excitons as compared to the first or second ordered ones. The exciton states having n ≥ 3 occur at the same spectral position as
(a) (b)
Femtosecond time-resolved spectroscopy on ZnO and BTO/ZnO thin films
41 n = 2, but with subsequently diminishing amplitudes as shown in Figure 4.11 (a).
Thus, a factor 2 was added in the numerator of the third term of equation (4.13) to approximately incorporate the contribution from all higher energy excitonic states.
The factor also takes into account the contributions which might arise due to exciton-phonon coupling in the sample. The overall contribution from discrete exciton A is shown in Figure 4.11 (b).
Figure 4.11: (a) Contribution from higher order states and (b) Overall contribution from n=1 and higher order states of discrete exciton A
Contribution from Continuum excitons and free electrons:
The contribution of the continuum excitons to the dielectric function ( ) is given by equation (4.15) where Γα(λ) represents the broadening energy of the band gap E0λ and Gα is the exciton Rydberg energy. The line shape of free electron contribution ( ), given by equation (4.16) shows a square root dependence on the frequency.
∑
, , (4.15)
∑
, , (4.16)(4.17)
Contribution from core-level transitions:
1 (4.18)
A constant term 1off was added in the model dielectric function to take into account higher lying gaps and core excitonic transitions. In addition a wavelength dependent (a) (b)
Femtosecond time-resolved spectroscopy on ZnO and BTO/ZnO thin films
42 factor (second term in equation (4.18)) was added to incorporate contributions from transitions in the UV region. These core-level transitions mainly contribute to the real part of the complex refractive index of ZnO.
Fitting of transient spectra
The strongest peak corresponding to negative OD at 375 nm could be simulated by decreasing the amplitude of discrete exciton A (AnA). A decrease in AnA results in minor modulations (on the order of ~ 10-3) in the spectral range of 390 nm to 600 nm, in addition to the main bleaching peak at 375 nm. The bleaching peak is asymmetric with higher order excitons (n ≥ 2) contributing to the extended blue wing as clearly visible in Figure 4.12 (a).
Figure 4.12 (a) Transmission increase at 375 nm due to decrease in amplitude of discrete exciton-A; (b) Contributions from the different parameters which generate
modulations between 400 nm and 600 nm
On the other hand, variation of each of the parameters Apol, Epol, continuum excitons-A and B and the free electrons contributions excitons-A, B, C yield modulations between 400 nm to 600 nm in the simulated spectra. For a detailed study of the contribution of these parameters, all parameters were varied by 1% and plotted in Figure 4.12(b) for comparison. It is apparent that all parameters do not contribute equally to the change in optical density. The most physically plausible parameter for generating the modulations would be the one which has to be varied the least. The parameter ε1off
contributes most significantly and hence has to be varied lesser as compared to the parameters. This renders it the most suitable parameter for simulating the modulations. However, it also results in a positive component in the UV-region which is not observed for the experimental spectra. To counterbalance this contribution, amplitude of continuum exciton A (which contributes the highest among the negative UV-component generating parameters) was varied in conjunction to ε1off. This procedure demonstrates the strategy of choosing the strongest parameters that yields the lowest sum of the residuals for the entire spectral range of interest.
Although this fitting procedure simulates most part of transient spectra, none of the parameters in the dielectric function could account for the transmission increase at around 391 nm and the absorption increase at 382 nm. To take these contributions
(a) (b)
Femtosecond time-resolved spectroscopy on ZnO and BTO/ZnO thin films
43 into account, Gaussian curves centred at ~ 391 nm and ~ 382 nm were added into the model. It should be mentioned here that, because of the close proximity of the different spectral contributions, their spectral positions and widths were allowed to vary over a narrow range of values (based on experimental observation).This allows for the experimental fluctuations, but helps in avoiding the tendency of Gaussian curves to align at the same spectral position and increase (in amplitude) with every iterative step which in turn generates infinitely large and physically meaningless values for the amplitudes and the spectral widths of the contributions. Increasing the temperature in the expression for temperature-dependent band-gap and excitonic broadening, results in absorption increase at around 377 nm. This corresponds to the absorption increase in the transient observed later in the time sequence. However, the amplitude of the experimental peak is comparatively lower than the simulated peak corresponding to the expected rise in temperature (due to heating of the sample).
This point is addressed and discussed later in section 5.3.5.
. . .
Figure 4.13 Experimental and simulated transient spectra for Z-365 at delay times 0.5 ps (a) and 1.1 ps (b)
From the dynamical point of view, what is most important is the time behaviour of the different contributions. And only those contributing features which have a reasonable time evolution can be included in the self-consistent picture. The differential transmission spectra corresponding to each delay time (in the measured time range) was analysed using the same fitting procedure. As an example, the simulated transient spectra at delays of 0.5 ps and 1.1 ps along with the corresponding contributions are presented in Figure 4.13 (a) and (b) respectively.
The goodness of the fits in terms of the sum of squared residuals is of the order of
~10-3. Based on above considerations and mathematical treatments, a comprehensive analysis of the charge carrier dynamics in ZnO and BTO/ZnO heterostructures will be presented in the succeeding chapters.
.
(a) (b)
44
Ultrafast Dynamics of ZnO
45