Self-assembly of block copolymers under non-isothermal annealing conditions as revealed by grazing-incidence small-angle X-ray scattering

Synopsis The self-assembly kinetics of a lamellar block copolymer is accurately characterized in situ and at real time using GISAXS. A novel modelling methodology is implemented for the determination of self-assembly kinetic parameters by properly considering the initial annealing conditions for both isothermal and non-isothermal processes


Introduction
The self-assembly of block copolymer (BCP) thin films has gained interest during the last decade due to its application as a complementary process for nanostructuring surfaces. (Jeong et al., 2013;Bates et al., 2014) Particularly, BCP-based large-area patterning can reliably achieve sub-20 nm resolution at low cost and with process simplicity. Pursuing its ultimate development in terms of resolution, defectivity and process integration would allow its implementation in the next technological nodes.
BCPs are composed of two (or more) homopolymer chains, whose blocks are chemically distinct while linked through a covalent bond. (Bates, 1990) When the repulsive interaction between blocks is high enough, a phase separation can be induced by thermal or solvent annealing. (Jung & Ross, 2009) Different structural nanodomains, such as lamellas, cylinders or spheres, can be obtained depending on the degree of interaction among the polymer chains, which is quantified by the Flory Huggins interaction parameter, χ, plus the number of monomers, N, and the volume fraction of each block, f. However, self-assembled nanopatterns usually present a random orientation and short order domains.
To overcome this limitation, directed self-assembly (DSA) combines standard lithography techniques to create topographical (graphoepitaxy (Gottlieb, Kazazis et al., 2018)) or chemical (chemoepitaxy (Fernandez-Regulez et al., 2014;Evangelio et al., 2019)) guiding patterns, so that specific orientation and position of the block copolymer self-assembled structures can be obtained. (Hu et al., 2014) DSA of BCPs is still considered by the semiconductor industry as a complementary approach to conventional photolithography and patterning towards improving parameters such as resolution, throughput or line edge roughness. (IRDS, 2018) Nevertheless, its incorporation in fabrication processes requires the consolidation of multiple parameters such as defectivity, compatibility of self-assembly process conditions, kinetics and throughput among others. (Marencic & Register, 2010;Tseng & Darling, 2010;Li & Müller, 2016) The understanding of the parameters governing the kinetics of BCP self-assembly process is of key importance to model the evolution of the system as it can be crucial to predict the final order and determine the processing conditions to obtain defect-free patterned areas (Harrison et al., 2002;Murphy et al., 2015;Kim et al., 2014). Both, the self-assembly kinetics and obtained morphology, have been widely investigated as a function of different physical/experimental parameters (Ji et al., 2011) such as annealing conditions, (Berry et al., 2007;Sanz et al., 2011;Majewski & Yager, 2015;Gu et al., 2014) BCP thickness (Black et al., 2017) and BCP/substrate interaction.  Typically, the characterization of BCP thin films is addressed by using microscopy techniques. In particular, scanning electron microscopy (SEM), (Harrison et al., 2004;Ferrarese Lupi et al., 2013;Perego et al., 2014) atomic force microscopy (AFM) (Magerle, 2000;Hahm & Sibener, 2001;Berry et al., 2007;Chandra et al., 2019) or transmission electron microscopy (TEM) (Segal-Peretz et al., 2015) have been extensively used. However, despite their high spatial resolution, they only inform about the material surface, like in the case of SEM and AFM, and they are typically limited to surveying small areas. Therefore, conventional topographical characterization of selfassembled BCP layers taken at different sample positions offers limited possibility for statistical analysis of pattern quality and defectivity, and its evaluation is time consuming. Moreover, in situ and real time characterization of the time evolution of the system is limited because of the time resolution inherent to microscopy methods and the incompatibility of equipment with the required harsh conditions involved in the sample processing. In this scenario, Grazing-Incidence Small-Angle X-ray Scattering (GISAXS) with synchrotron radiation plays a pivotal role to complement the evaluation provided by standard microscopy techniques, (Müller-Buschbaum, 2016;Soccio et al., 2014;Rueda et al., 2012;Gottlieb, Rösner et al., 2018) due to the fact that GISAXS probes a sample area that can be several orders of magnitude larger, typically square millimeters, than the one explored by microscopy techniques, thanks to the glancing X-ray beam and hence large footprint.. Moreover, the collected information is not limited to the polymer surface (Ferrarese Lupi et al., 2017) but also includes certain sample volume because of the X-ray penetration into the material, which can be controlled by the incidence angle. This feature opens the possibility for statistical analysis of large areas while having access to information from the entire depth of the film. (Müller-Buschbaum, 2016) In addition, GISAXS set-ups often allow the experiments to be performed in a wide range of sample environments and configurations, e.g. high temperature, vacuum, controlled atmosphere conditions, etc., making possible to perform in situ and real-time experiments such as solvent and thermal annealing of block copolymers.
Remarkably, due to the recent progress in synchrotron instrumentation for GISAXS, sub-millisecond time-resolution measurements can be performed, thus, enabling an accurate study of the kinetics of block copolymer self-assembly process. However, most of the reported in situ and real-time studies using synchrotron radiation were performed on solvent-annealed samples. (Papadakis et al., 2008;Di et al., 2010;Gu et al., 2014;Zhang et al., 2014;Sinturel et al., 2014;Berezkin et al., 2018;Lee et al., 2019) In general, the characterization of thermally annealed samples is mainly performed on already self-assembled ex situ samples by taking snapshots during the process of interest, (Ferrarese Lupi et al., 2017) while only few studies have been performed during the thermal annealing. (Yager et al., 2009;Sepe et al., 2011;Maret et al., 2014;Samant et al., 2016) In this paper, we report on the self-assembly process of lamellar block copolymer poly (styrene-blockmethylmetacrylate), PS-b-PMMA thin films. The spin-coated/deposited block copolymer thin films were investigated by means of synchrotron GISAXS during thermal annealing (i.e. in situ) which consisted on either isothermal or non-isothermal conditions ,and in particular during heating ramps at different heating rates. Real-time monitoring has allowed us to investigate the evolution of the pitch as well as the block copolymer correlation length, thus making possible to determine basic kinetics mechanism of the self-assembly process for the complete set of investigated samples. In fact, different from classical modeling, normally limited to isothermal annealing, our study has been extended to and provides predictions for systems with diverse initial sample conditions including: i) unknown previous annealing history and ii) pre-annealing evaporation of the solvent at low temperature. In addition, dedicated data analysis enables us to predict the evolution of the system under arbitrary nonisothermal annealing conditions. The model successfully predicts the evolution of the correlation length during the heating ramp. When the final target temperature has been reached after the heating ramp, the evolution during the isothermal annealing is modelled considering the previous evolution of the self-assembly.
Samples were prepared on 2x2 cm 2 substrates from silicon wafers (p-type doped, <100> oriented and 4-40  cm -1 ). First, the substrates were conditioned by exposure to an oxygen plasma at 500W for 10 min. Then, a thin layer of PS-r-PMMA was spin-coated (2000 rpm for 30s) on the surface and annealed at 230 ºC in air for 10 min for grafting. The un-grafted polymer chains were removed by dipping the sample on an ultrasonic bath with PGMEA solvent for 1 min. Finally, the block copolymer, PS-b-PMMA, was spin-coated at 2500 rpm and immediately placed on the GISAXS sample holder to perform its in situ thermal annealing characterization to monitor the self-assembly process. Due to the employed annealing conditions and the surface brush functionalization, perpendicular oriented lamellar structures with a period of 38 nm were formed. (Gottlieb et al., 2017;Lorenzoni et al., 2017)

Sample characterization
In situ GISAXS measurements were carried out at the NCD-SWEET beamline at ALBA synchrotron (Cerdanyola del Vallès, Spain) during two separate campaigns/sets of experiments. An X-ray beam of 12.4 keV was set using a Si (111) channel cut monochromator. Then, the beam was collimated using an array of Be lenses, obtaining a beam size of 475 × 340 μm 2 (H × V) at the sample position. The scattered radiation was recorded with a Pilatus3 S 1M detector (Dectris ®, Switzerland), which 5 consists of an array of 981 × 1043 (H × V) pixels of 172 x 172 μm 2 (H × V). The sample to detector distance was 4046 mm for a first set, and 6036 mm for the second round of experiments. The sample to detector distance, as well as the reciprocal space, was calibrated using a silver behenate sample for calibration. The X-ray incident angle employed was 0.15 o , which corresponds to the critical angle of the Si substrate for the set energy, ensuring minimum penetration and high surface sensitivity.
An adapted Linkam® TMS600 films heating stage (accuracy 0.01 ºC) for grazing-incidence experiments was employed for the in situ annealing experiments. Sample temperature was corrected by a calibration curve previously obtained. A protective N2 atmosphere was used during the performed experiments. For the isothermal annealing, the samples were first placed at room temperature on the heating stage and then heated-up to the target annealing temperature (between 160 ºC and 200 ºC) using a heating ramp of up to 50 ºC/min without being exposed to X-rays. GISAXS patterns during the isothermal experiments were collected at the (final) target temperature. On the other hand, samples for the study of non-isothermal treatments were annealed from room temperature to 180 ºC using different heating rates (5, 10 and 20 ºC/min), followed by a dwell time at the maximum temperature for 10 min. In this case, GISAXS patterns were periodically taken during both transient phases, the heating ramp, and the steady phase, i.e. the isothermal annealing.

GISAXS Analysis
Prior to the in situ experiments, the radiation damage induced by the beam on the BCPs was evaluated at different temperatures (see supporting information Figure S1). We determined that a total sample exposure to the X-rays higher than 1.2 s induced significant damage to the sample, especially at temperatures above the polymer glass transition. Therefore, the GISAXS pattern acquisition time was limited to 30 frames per sample with an integration time of 37 ms, limiting the sample beam-damage while obtaining an adequate signal to noise ratio for the recorded GISAXS 2D patterns. To guarantee experimental reproducibility, each experiment was repeated at least three times. detector with respect to the α (out of plane) and ω (in plane) exiting angles. The reciprocal space units (q) represent the scattering wavevector, which depends on the X-ray energy (λ) and the exiting angles (α and ω).
where 0 = 2 / is the wave vector of the incident X-ray beam of wavelength . (Renaud et al., 2009) One example of the obtained 2D-GISAXS patterns during isothermal annealing is shown in Figure 2.
Typically, the patterns exhibit a primary diffraction maximum at a certain position (qy,pos, qz,pos) which is associated with the BCP period in perpendicularly-oriented lamellar structures (L0). The recorded 2D-GISAXS patterns were analyzed by an in house-developed software based on MatLab®. 2D to 1D data reduction consisted of horizontal line cuts (along qy) at the position of the main diffraction maximum (qz,pos, ca. 0.251 nm -1 ), by means of integrating the number of pixels corresponding to a q interval of qz,pos ± 0.01 nm -1 . The main peak qy position (qy,pos) can be directly related with the average center-to-center distances between neighboring domains, while the peak width is inversely proportional to the correlation length defined as the domain long range order distance. (Förster et al., 2005) In fact, the period of the lamellar structures, L0, from the qy,pos can be calculated from the position of the scattering maximum as follows: where m is the order of the reflection (being m = 1 for the first scattering peak). The scattering peak qy maximum position was ca. 0.165 nm -1 which fits with the expected value for a polymer with 38 nm pitch. On the other hand, the grain size (i.e. the order of structure) is inversely proportional to the width of the scattering peak in GISAXS patterns. As the order in the perpendicularly-oriented lamellar structures of the block copolymer film increases, a narrowing of the first order Bragg peak occurs, concurrent with the eventual appearance of higher order peaks. The correlation length of selfassembled block copolymer can be related to the width of the first order scattering peak as follows: where "FWHM" is the full width half-maximum of the peak and k is a correction parameter related to the detector type and the fitting of the peak (see Figure 3). For a 2D detector and a Gaussian fitting, k has been defined as 0.96. (Smilgies, 2009) In this work, the horizontal cuts in a window of qy,pos ± 0.15 nm -1 were fitted to a Gaussian curve using a Matlab® code prior to the extraction of both the main peak position and FWHM.

Figure 3
Calculation of the block copolymer correlation length using a Gaussian fit of the first order scattering peak.

Isothermal annealing of block copolymers
The evolution of the correlation length of BCPs as a function of time under isothermal annealing conditions follows the power law expression: (Harrison et al., 2004; = (4) where the growth exponent value, , depends on the polymer material composition as well as the applied annealing conditions. It has a typical value of 0.1-0.3. The parameter AT depends on the temperature through an Arrhenius behaviour, which can be expressed as is an activation energy for the self-assembly which depends on the block co-polymer (composition, molecular weight and morphology).
However, equation (4) can be only applied when the self-assembly is strictly occurring at pure isothermal annealing process or when the target isothermal temperature is reached under rapid annealing conditions, i.e., when a negligible evolution of the self-assembly occurs during the short time required for reaching the isothermal temperature. Quantitatively, Perego and co-workers estimated that, in order to assume a rapid thermal annealing (RTA), a heating ramp faster than 18 ºC/s is needed. (Ferrarese Lupi et al., 2013) Moreover, it has been demonstrated that the final selfassembled film structure of a block copolymer film is affected both by the presence of intermediate states and by the heating rate (Majewski & Yager, 2015). Therefore, the power law dependence for BCP self-assembly in equation (4) is not strictly valid under arbitrary annealing conditions and hence, a correction factor is required to obtain a precise modelling of the self-assembly evolution.
In fact, the isothermal experiments reported in this work show some polymer ordering occurring during the heating time involved in reaching the annealing temperature. Once isothermal temperature is reached, the BCP correlation length, (ξ 0 ), is expected to depend on the previous history of the film (heating rate to reach the target temperature, solvent evaporation kinetics, etc.). In order to include the initial conditions into expression (4), an equivalent time, (teq) has to be introduced as an offset, which corresponds to the time that the polymer would need to reach the initial observed correlation length under a purely isothermal annealing regime: 0 = . Accordingly, equation (4) can be rewritten as: Then, self-assembly kinetics of BCP during thermal annealing depends on three parameters: the growth exponent ( the Arrhenius coefficient AT and the equivalent time (teq), which is directly related with the initial correlation length ( 0 ) as = ( 0 ( =0) ) 1 . A similar approach is used to model the kinetics of the silicon oxide thermal growth (Deal and Grove model) where the time coordinate is used to account for the presence of an initial oxide layer on the sample. (Deal & Grove, 1965) The analysis of the self-assembly process under isothermal annealing was performed at temperatures between 160 ºC and 190 ºC, using a heating rate of 50 ºC/min, well below the minimum value for being considered RTA. For annealing temperatures lower than 160 ºC, the block copolymer selfassembly kinetics is too slow to be efficiently monitored in situ by synchrotron experiments, since several hours are needed to observe some polymer order. On the contrary, at high annealing temperatures (>190 ºC) the self-assembly kinetics is fast enough to produce the complete polymer ordering during the heating ramp. Thus, no significant changes were observed in the correlation length during the isothermal annealing from 200 ºC and above.
During the isothermal annealing, GISAXS patterns correspond to snapshots of the structure and ordering of the block copolymer. Figure 2 presents a selection of 2D GISAXS patterns recorded during isothermal annealing at 180 ºC. GISAXS patterns during isothermal annealing at 160, 170 and 190ºC can be found at supporting document Figure S2 to S4). It can be observed, first, the formation and, then, the evolution of scattering rods at ca. qy= 0.165 nm -1 produced by the arrangement of lamellae perpendicular to the sample. From the recorded patterns, the correlation length evolution could be extracted. Figure 4 depicts the evolution of the qy horizontal cuts at the Yoneda peak position (qz,pos) versus the annealing time for different temperatures. A detailed view of the evolution of the main scattering peak profiles with annealing time is shown in supporting information document, in Figure S5. As observed, the main scattering peak evolves by increasing its intensity and slightly shifting towards lower qy values. This characteristic evolution signals an increment of the scattering domain size, indicated by the narrowing of the scattering peak as well as an improved lamella packing, by the reduction of the qy peak position. As mentioned, to corroborate these findings, the same experiment was repeated several times for each isothermal target temperature (160 ºC, 170 ºC, 180 ºC and 190 ºC), see Figure 5. An additional experiment was performed by directly placing the sample on the heating stage to produce fast heating for thermal annealing, where the equivalent time would be zero (teq0). Isothermal annealing experiments with slow heating rates resulted on a correlation length evolution following the canonical power law (equation (5)). Table 1 compiles the experimental  and values extracted from the in situ data recorded during the annealing experiments. The extracted values (Table   2) were consistent with previously reported investigations for this polymer using SEM defectivity analysis at Claveau´s thesis (Claveau, 2018) and for similar polymers. (Ji et al., 2011;Majewski & Yager, 2015;Black et al., 2017) As expected, value increases following an Arrhenius behaviour, being the activation energy of the polymer 29.7 kJ/mol. Additionally, the values are within the ra nge of previously reported values (Black et al., 2017;Perego et al., 2014;Majewski & Yager, 2015;Ferrarese Lupi et al., 2017). It is worth mentioning the existence of a temperature dependence revealed by our experiments (Figure 6). This dependency can be well described by a linear law, according to the narrow range of temperatures under evaluation: ( ) = 1 + 2 . The temperature dependence of AT and  are shown in Figure 6 and the fitting parameters in Table 2.

Non-isothermal annealing of block copolymers
The evolution of the block copolymer lamellar structure under non-isothermal annealing conditions has not been previously studied in as much detail as under isothermal annealing. The modeling of the self-assembly evolution becomes more complex due to the fact that BCP parameters ( ( ), ( )) are changing during the self-assembly process. Consequently, for the block polymer under evaluation, we have extracted the dependence with temperature of both parameters from the isothermal experiments. Thus, we can model their time evolution for any arbitrary thermal treatment for which the temperature is known at each moment. We have implemented an extended model of the selfassembly evolution by dividing the self-assembly process into infinitesimal time intervals, dt, where we can assume that the self-assembly is evolving as under isothermal annealing. A new equivalent time is calculated for each new isothermal interval.
As example, we have evaluated the particular case of an annealing ramp with a constant heating rate, r, as it is depicted in Figure 7.
Then, during the first time interval, ∈ (0, ), the polymer correlation length evolves as: We can recalculate a new equivalent time, 1−2 , at the end of the first time interval (t=dt) for the evaluation of the polymer correlation length, 2 , under next isothermal temperature, 2 = 1 + = 1 + , during next period from ∈ ( , 2 ) by enforcing the continuity of (t):  (11) and (12): the mathematical modeling is equivalent to translate a segment of the isothermal curve (red, green and blue curves) a time equal to teq to fit to the value of the correlation length on the non-isothermal curve (pink curve) at the initial time of the interval dt. In the figure dt is set to 10 s for clarity. When modelling real experiments at a constant heating rate, the interval of time, dt, is taken small enough to assume that during each time the system evolve under isothermal conditions, preventing the introduction of any significant error. For samples analyzed on this paper, the heating ramps is divided into 1000 isothermal steps, what corresponds to dt values between 0.0025 s and 0.01 s. Then, the time correlation length evolution is solved numerically using equations (6) to (10) by implementing a MATLAB routine.

Figure 7
Modelling of the non-isothermal annealing. This figure illustrates the procedure to determine the evolution of the correlation length when the annealing temperature is not constant. To clarify the process, in this example the ramp temperature has a step like form with time steps of 10 seconds, as it is shown in the inset. The heating rate for the model is of 90ºC/min and the initial and final temperature is 180ºC and 215ºC respectively. At each step, the evolution follows that of an isothermal annealing but translated a certain time (teq) to ensure the continuity of the actual correlation length.
The structural evolution of the BCP lamellar structure was obtained for different heating rates (5 ºC/min, 10 ºC/min and 20 ºC/min). Samples were placed after the spin coating step at the heating stage and heated up to the target temperature of 180ºC. Then, after reaching the final temperature of 180 ºC, the polymer was annealed under isothermal conditions during few minutes. The initial temperature of the heating ramp for the studies was chosen to be 150 ºC that is the temperature at which polymer evolution starts to be observed (see supporting document Figure S6).. At lower temperature, the evolution of the correlation length is difficult to evaluate due to low intensity and the width of the peaks which difficulties the Gaussian fit. The block copolymer kinetics is also very slow and the effect of the annealing in the block copolymer order is not observed at low temperatures (see Figure S6 at supporting document). Additionally, the initial order present on the samples at 150 ºC can be mainly attributed to the evaporation of the remaining PGMEA solvent on the film, that take place during the spin-coating and the first minutes of the heating ramps. This effect is especially important in these experiments due to the slow heating rates used during the ramp.  After the mathematical model verification, we are able to predict how the annealing conditions, heating rate and initial correlation length influence the kinetics of the self-assembly process. Figure 9a and c illustrate the evolution of the correlation length during the thermal annealing at 1 ºC/min to 180 ºC and 230 ºC, respectively, as a function of the initial polymer correlation length. Results show that for short isothermal annealing times, and especially at the lower temperatures (180 ºC  For the figures on the right (b and d), the block copolymer film is annealed using different temperature ramp rates (from 0.1 ºC/s to 40 ºC/s and then kept at constant temperature for few min).
The initial correlation length is 90 nm. All ramps are starting at t=0 s. and correlation length during the ramp is represented by solid lines and dashed lines when isotherm is reached.
As a consequence the extraction of block copolymer kinetic parameters using the classic approach, = , is only valid for a pure isothermal annealing, where the target temperature has been quasi instantaneously reached or using a fast thermal annealing. We have compared the calculation of  using the classic approach (see supporting information Figure S7) with the accurate modelling that we have presented (Figure 10). When  is estimated using the classical approach on samples previously annealed with a relatively slow heating rate, the obtained value is lower than the real one, and it is only reached after long annealing time. Consequently, in order to reach an accurate value of  the experiment must consider a long enough annealing time under isothermal conditions. The minimum annealing time under isothermal conditions depends on the annealing temperature and on the heating rate.

Figure 10
Isothermal annealing time required to achieve a 90% accuracy of the correlation length growth exponent,  when the initial correlation length is not considered. The calculation is performed for four different temperatures and for a range of heating rates from 0.1 to 40 ºC/s during the non-isothermal annealing. The correlation length was estimated using the model described in the article that takes into account the equivalent time, teq. Time equal to zero corresponds to the starting time of the isothermal annealing. All the samples were heated up from 150 ºC to the target temperature (180, 205, 215 and 230 ºC) at different heating rates. The initial correlation length is set to 90 nm for all the curves.

Conclusions
The analysis of the kinetics of the structural order in lamellar structures of block copolymer kinetics using GISAXS allows access to statistical information over relatively large areas as compared to standard microscopy techniques. GISAXS characterization presents the additional advantage to be Heating rate (C/s) compatible with the self-assembly annealing experimental conditions usually employed in DSA processing, allowing for the in-situ and real time characterization of block copolymer ordering.
However, for data interpretation and, in order to take into account non-ideal experimental conditions, the classical modelling for BCP kinetics needs to be adapted. We have introduced a new parameter, the equivalent time, that enables the analysis of the evolution of the correlation length independently of the previous thermal history of the sample, caused for example by solvent evaporation or slow heating ramps. This approach can be used to reproduce the evolution of the correlation length on nonisothermal annealing. The model has been validated during different heating ramps but could be applied to any system where the temporal dependence of temperature is known.
Finally, we have demonstrated that, in order to obtain an accurate estimation of the parameters that determine the block copolymer self-assembly kinetics, the equivalent time should be considered when analysing the experimental data from the real time experiments. This effect is particularly important for samples undergoing a slow heating rate or annealed at low temperature if accurate values of the kinetics parameters need to be obtained.

Figure S1
Effect of the GISAXS beam on the block copolymer sample at room temperature (RT) and at 190ºC. Evolution of Yoneda main peak position (qy, max) at five different acquisition conditions. In the first fourth samples, 100 frames were taken using first 40 ms of exposure time (blue and red circles) and using 10s of exposure time (black and green triangles).As expected longer acquisition times and higher temperature caused more damage on the sample. In the last experiment, the sample was heated at 190ºC and 10 frames with an acquisition time and period of 1s and 100s were taken. The annealing time and temperature of this sample is similar than previous sample but the observed beam damage was minimal.     Figure   4 (main text). The increase of the peak intensity and a reduction of the peak broadness during the thermal annealing arise from an increase of the long range order of the block copolymer pattern.

Figure S6
Intensity and correlation length of the first order peak at Yoneda band plotted as a function of the temperature. The experimental values were taken from GISAXS patterns during a heating ramp at 5ºC/min. At low temperatures, up to 145ºC, there is not evolution on the block copolymer film, both intensity and correlation length remain constant. The initial order at lower temperatures is attributed to solvent evaporation during the spin-coating and at the beginning of the annealing process. represented by a dashed line on each graph. Before isothermal annealing, the samples are annealed from 150ºC to the target temperature, , using a heating ramp of 1ºC/s and a starting correlation length of 90nm. This demonstrate that only at long annealing times, the classical approach can be used for the calculation of the power exponent.