Application of wavelet analysis on transient reflectivity in ultra-thin films

S. Yousefi Sarraf; R. Trappen; S. Kumari; G. Bhandari; N. Mottaghi; C. Y. Huang; G. B. Cabrera; A. D. Bristow; M. B. Holcomb

doi:10.1364/OE.27.014684

Introduction

The interaction of an ultrafast laser pulse with solids excites quasiparticle interactions between the electrons, spins, phonons, magnons, etc [1]. Taking advantage of quasiparticle dynamics and interactions is at the heart of multiple technologies, including transistors, solar cells, photonics and spintronics. A critical gap in our understanding of these interactions is the lack of information we have about the aforementioned interactions in the ultra-thin regime as well as at the surface and interfaces of the solids, which are very different than their bulk counterparts.

Surface sensitive time-resolved optical measurements, such as second harmonic generation, is one of the methods to study the quasiparticle interactions at the surface and interfaces [2]. However, this method is surface sensitive only when the bulk symmetry is centrosymmetric, which results in no second harmonic signal. On the other hand, transient reflectivity (TR) measurements, using an ultrafast optical setup, can provide significant information about the quasiparticle interactions in the solid, however this method is not specifically a surface sensitive method. Generally, TR signal consists of a decay dynamic due to relaxation of charge carriers to lower energy levels, superimposed by multiple sinusoidal quasiparticle interactions (if they exist), such as phonons, magnons, surface plasmon polaritons, etc [3–6].

This sinusoidal behavior in TR measurements is commonly analyzed by Fourier transforms or fitting algorithms [3–6]. However, these traditional methods will not provide all the information possible to extract from the data, especially the abrupt changes happening at the surface and interface. Moreover, in films with thickness of a few nanometers even the bulk related phenomena happen and damp out quickly, which result in failure of traditional methods to obtain information. The reason for this problem is that Fourier transforms are defined as the sum of sine waves which are not localized in time or space. Although, windowed Fourier transforms (WFT) have been introduced to study the signal in the frequency-time domain, they fail to function in time scales much less or much higher than the window width, since it introduces a scale into the analysis [7]. On the other hand, wavelet transforms, which use scalable localized wavelets in time and space as the window function, are perfect tools to analyze abrupt changes in the signals [8]. Wavelet analysis has been commonly used for signal analysis [9] and image processing [10]. While there are some studies regarding the application of wavelet analysis in materials science [11–14], their applications in TR measurements of ultrathin film heterostructures has not been well studied yet. In this paper, wavelet analysis has been investigated and applied to the sinusoidal signal to extract information related to different vibrational modes, especially at the surface and interfaces which cause abrupt changes in the TR signal. We show that the choice of wavelet can extract different information from the same TR signal. In the case of a low noise TR signal, this method will provide more information about the interaction between different vibrational modes in ultra-thin films, compare to the conventional analyzing method.

Wavelet analysis

In signal processing, information about the signal in the time-frequency domain is a matter of great importance. The first proposed approach for this purpose was WFT, which use a weight or window function, $g (t),$ to localize the signal in time. Thus, for a real signal of $W (t),$ WFT is defined as [7]:

\tilde{W} (ω, T) = \int_{- \infty}^{+ \infty} e^{- 2 π i ω T} \bar{g} (t - T) W (t) d T .

By defining

g_{ω, T} (t) = e^{- 2 π i ω T} \bar{g} (t - T),

WFT can be expressed as the inner product of

g_{ω, T}

and

W .

Any feature in WFT is under-localized for the times smaller and over-localized for the times much higher than the width of the window [7]. In other words, WFT fail to isolate the location of each frequency, if the frequency occurrence time is much smaller or much higher than the width of the window.

To solve the scaling problem, wavelet transforms take advantage of a fully scalable modulated window as an alternative windowing approach in WFT. The window function in the case of wavelet analysis is called the mother wavelet, $ψ (t)$ . The scalable modulated form of the window is called daughter wavelet, defined as [7]:

ψ_{s, τ} (t) = s^{- (1 / 2)} ψ (\frac{t - τ}{s}),

with

s

and

τ

defined as the scale and translation factor, respectively. Continuous wavelet transforms are then given as [7]:

W (τ, s) = \int_{- \infty}^{+ \infty} W (t) ψ_{s, τ}^{*} (t) d t .

Here,

ψ_{s, τ}^{*} (t)

is the complex conjugate of the daughter wavelet. By applying a collection of daughter wavelets with different scale and translation factors, the result will be a time dependent depiction of oscillation frequencies in the signal. Thus, as a result of wavelet analysis, one can have information about the different oscillatory modes at different times in a transient, or at different positions if the analysis is applied to TR on heterostructures.

In Eq. (2), the basis function is not defined. Unlike a Fourier transforms which has trigonometric polynomial basis functions delocalized in time and space, the wavelet transforms are based on different families of mother wavelets which are localized in time and space. Besides being localized in time and space, a function can be a mother wavelet if it has a mean of zero. Different types of mother wavelet families have been proposed to suit different types of data analysis, such as Gabor-Morlet [15], Log Gabor [16], Bessel [17], Gaussian [18], Morse wavelet [19], Mexican hat [20], etc. Since implementation of different mother wavelets on the same set of data yield different results, the biggest challenge in wavelet analysis is to find the optimum mother wavelet for a specific signal. Mother wavelets are defined by their orthogonality, number of vanishing moments and regularity [21]. The number of vanishing moments and regularity are complex mathematical concepts [21]. In short, the number of vanishing moments is related to the number of terms in the Taylor expansion of the wavelet transforms at $t = 0$ that go to zero. Regularity sometimes refers to how differentiable a function is. Generally, to find the optimum mother wavelet, the similarity between the signal and the mother wavelet is considered [22]. Abrupt changes in the signal are more distinguishable in the derivatives of the signal. Hence, to detect abrupt changes in the signal, the mother wavelet should be sufficiently regular, that is sufficient number of continuous derivatives for the function. While different mother wavelets show regularity, one of the most regular mother wavelets, which is also similar to the sinusoidal signal, is the Morse wavelet. Based on these criteria, the Morse wavelet, which is a super family of wavelets, has been used as the optimum mother wavelet for detection of abrupt changes in the following data. The Morse wavelet, in the frequency domain, is defined as [19]:

ψ_{β, γ} (ω) = \int_{- \infty}^{+ \infty} ψ_{β, γ} (t) e^{- i ω t} d t = U (ω) a_{β, γ} ω^{β} e^{- ω γ},

where

U

is a unit step function and

a

is a normalizing factor.

β

and

γ

are the compactness and symmetry parameters, respectively. The latter two are main key considerations for the Morse wavelets, because changing

β

and

γ

can manipulate the Morse wavelets and make it suitable for abrupt changes in the signal. By employing different compactness and symmetry parameters, we indicated some suitable Morse wavelets which can be applied to TR signal to obtain information regarding the surface, interfaces and very short-lived oscillatory modes. The Gabor limit in signal analysis introduces the trade-off between frequency and time [21]. Thus, if either frequency or time are more important for the scientific goal, some sacrifice in the resolution for the other may be required.

Figure 1 is an indication of what information is obtainable by using different wavelets on the same residual of TR data. Here we compare the results of the wavelet analysis using a generalized Morse wavelet with $γ = 3$ and $β = 20$ for the left column and $γ = 60$ and $β = 2$ for the right column. Figures 1(a) and 1(b), present the real and imaginary parts of the wavelet in time, to show the effect of changing $β$ and $γ$ on the Morse wavelets. Figures 1(c) and 1(d) provide the associated frequency-time distribution of the Morse wavelets shown in Figs. 1(a) and 1(b). The area of concentration in the frequency-time domain [Figs. 1(c) and 1(d)] is obtained by reconstruction of the signal, based on the resolution of identity formalism, explained in the reference [23]. These Figs [Figs. 1(a)-1(d)] show that by decreasing $β$ and increasing $γ$ the resolution in time increases, while the resolution in frequency decreases. Thus, by changing the wavelet from a symmetrical form to a more time-compact form in the frequency-time domain, one can obtain information regarding the time and position of more abrupt phenomena happening in the signal. The comparison of the two wavelets is shown in Figs. 1(e) and 1(f). In the latter, the region highlighted with the red ellipse confirms that better time resolution is achieved by the more compact wavelet. In this case abrupt signatures are seen arising from both the surface and interface.

Fig. 1 (a) and (b) Real, imaginary and absolute parts of the Morse wavelets. (c) and (d) The frequency-time domain representative of a and b with concentration area of 2 (Figs only depict the positive frequencies). (e) and (f) Wavelet transforms of different Morse wavelets on the same residual TR signal. Morse parameters are γ =3, β = 20, for a, c, and e, and γ = 60, β = 2 for b, d, and f.

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Figures 1(a)–1(d) are plotted using a freely available MATLAB toolbox called JLAB, available at http://www.jmlilly.net. The generalized Morse wavelets [Figs. 1(a) and 1(b)] are implemented with morsewave, while the frequency-time domain plots [Figs. 1(c) and 1(d)] are computed in the function morseregion.

After defining the Morse wavelet, frequency-time spectra of the signal [Fig. 1(e) and 1(f), wavelet transforms plots] has been plotted using the cwt function in MATLAB. In the wavelet transforms plot [Figs. 1(e) and 1(f)], X and Y axes are the time of the occurrence and the frequency of the oscillations, respectively. The amplitude of the oscillations is defined by the color bar.

Results and discussion

Measurements are performed on a La_0.7Sr_0.3MnO₃ (LSMO) thin film with the thickness of d = 43nm, grown on a 001-cut SrTiO₃ (STO) substrate in a pulsed laser deposition system at 750°C and in oxygen pressure of 100 mTorr. Details about the growth can be found in the prior work [24]. Moreover, as a further illustration on a more complicated system, the same measurements are performed on the multiferroic heterostructure of BaTiO₃ (BTO)/LSMO (ferroelectric/ferromagnetic) grown on an STO (001) substrate. In this case, both BTO and LSMO layers were grown at 750°C and in oxygen pressure of 150 mTorr.

TR measurements have been performed using a degenerate pump-probe setup with a Ti:Sapphire femtosecond laser tuned to 810 nm. The optical setup is explained in more detail in the prior work [25]. Our observations indicate that to utilize wavelet transforms with the resolution of $Δ f / f \approx 0.5,$ where $f$ is the center frequency of an oscillatory mode and $Δ f$ is its bandwidth, the residual TR signal should have a signal to noise ratio of greater or equal to 83.5 dB. For LSMO thin films which have very weak TR signals, this high signal to noise ratio has been achieved by repeating the scan from a few hours to a few days, corresponding to 50 to 300 scans. Depending on the sample, the strength of the oscillations in TR signal and the time span of the scan, the number of repetitions can be different.

Figure 2(a) provides a schematic of the experiment and different oscillatory modes which eventually result in the main sinusoidal reflectivity signal which is due to the interference of reflection from different vibrational modes. The TR result for a 43 nm LSMO thin film on an STO substrate is shown in Fig. 2(b). The TR signal consists of a sharp negative rise [25] due to the band filling effect [26], which indicates excitation of the carriers to the higher energy levels. The negative rise in the TR signal is followed by multiple exponential decay processes with different decay time constants (τ). Electron-phonon recombination (τ_e-p < 1 ps), phonon-assisted spin-lattice recombination (τ_p-s ~100 ps), thermal diffusion and radiative recombination (τ > 1 ns) are responsible for relaxation of the carriers back to lower energy levels. Moreover, Fig. 2(b) indicates weaker sinusoidal oscillations as the carriers recombine via the multi-exponential decay process.

Fig. 2 (a) Schematic of the TR experiment and the oscillatory signal in LSMO/STO sample. (b) TR signal for LSMO thin film indicating multiple decay components and oscillatory modes. (c) Residual of the TR signal after fitting as the input of wavelet analysis.

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Figure 2(c) is the residual of the exponential fit to the TR signal, indicating multiple vibrational modes lasting for hundreds of picoseconds. These vibrational modes are mainly due to carriers’ excess energy transferred to the lattice via electron-phonon scattering. Considering the half-metallic behavior of LSMO thin films, LSMO can be used as an optoacoustic metal transducer [27], transferring the ultra-short laser pulse to a strain pulse, traveling longitudinally through the film with the speed of sound. Since this strain pulse is traveling in the longitudinal direction, at a specific time, the pulse will hit the interface between the thin film and the substrate, resulting in Brillouin oscillation traveling through the substrate. Due to structural differences between the thin film and the substrate, the secondary strain pulse, which is due to lattice vibrations in the substrate, will have a different frequency compared to that from the thin film. Since the Fourier transforms fail to identify which particular region of the heterostructure is responsible for each oscillatory mode, the wavelet transforms have been used to address this issue.

As an example, Fig. 3 demonstrates some of the physical properties that can be inferred from the result of the wavelet analysis on the fit residual of the TR signal [Fig. 2(b)] from 43 nm LSMO thin film. In the acquisition of Fig. 3 a Morse wavelet with parameters of $γ = 3$ and $β = 20$ , has been used. The wavelet transforms’ plot of the LSMO thin film on an STO substrate indicates a strong, short-lived, vibrational mode starting as the laser pulse hits the surface of the thin film at 100 GHz and a long-lived vibrational mode at 45 GHz in the substrate [3].

Fig. 3 Wavelet analysis on residual of TR signal for 43 nm LSMO thin film using a Morse wavelet with $γ = 3$ and $β = 20$ , indicating multiple short and long-lived oscillatory modes (log scale). Inset (linear scale), which is a zoomed version of the main Fig from −30 ps to 50 ps, is to emphasize the abrupt change at the surface and the interface.

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When the strain pulse propagates from one medium to another medium, both the amplitude and the frequency of the oscillations will change. The strain pulse travels through each medium with the respective speed of sound in the medium which can be calculated with:

V_{s} = \frac{f . λ_{p r o b e}}{2. n . \cos θ},

Where

f

is the frequency of the lattice vibrations and

λ_{p r o b e}

,

θ

and

n

are the wavelength, incident angle of the probe beam and the refractive index of the medium, respectively. Considering the long-lived oscillations at 45 GHz, the speed of sound in the substrate (STO with

n = 2.34

[27] at

λ_{p r o b e} = 810 n m

and probe incident angle of 15°) is

8200 m / s

which is in good agreement with published literature [28]. This result indicates that the long-lasting oscillations at 45 GHz are the acoustic phonons propagating in the substrate. The refractive index of the LSMO thin film used in this experiment is confirmed to be 2.3 by ellipsometry. By analogy, the sound velocity for the LSMO thin film with the thickness of 43 nm from the short-lived oscillation at around 100 GHz, can be calculated as

\approx 18000 m / s

, which has not been reported for LSMO ultra-thin films.

In the wavelet analysis of the TR measurements of heterostructures one should not confuse the long-lived oscillatory modes in the thin film with the modes from the substrate. One method to examine whether the oscillatory mode is long-lived or from the substrate, is to check the abrupt oscillatory modes, in the time domain, which happen at specific times. The inset of Fig. 3 shows a reduced time range of −30 to 50 ps, emphasizing two vibrational modes that broaden in frequency, and occurring at around 0 and 7 ps. These modes are examples of abrupt changes in the signal, which is the main advantage of using Morse wavelet analysis. These two modes can be due to higher energy oscillatory modes such as surface and interface lattice vibrations, high energy short-lived magnons, or echoes of the Brillouin oscillations from surface or interface. These echoes from the interface can be observed in delay times equal to multiples of $t_{e c h o} = 2 d / V_{s}$ [29], where d is the thickness of the thin film. Considering $V_{s} \approx 18000 m / s$ and $t_{e c h o} \approx 7 p s$ for the LSMO sample used in this experiment, this will correspond to 63 nm of LSMO thin film, indicating that these two modes are related to the surface and interface. Here, the difference in the actual (43 nm) and calculated (63 nm) thickness is due to reduced accuracy when $λ_{P h o n o n} ≫ d .$ Higher energy modes occurring at random times are likely due to the noise in the signal which can be reduced with more advanced optical techniques such as asynchronous operation and frequency combs.

Figure 4 shows wavelet analysis on the TR signal of the BTO/LSMO/STO heterostructure. Since BTO/LSMO is a multiferroic heterostructure (ferroelectric/ferromagnetic), it is expected to show multiple oscillatory modes in the same energy ranges, which are not trivial to decouple.

Fig. 4 Wavelet analysis of a more complicated system of multiferroic heterostructure (BTO/LSMO/STO), indicating the capability of distinguishing between different oscillatory modes with very close energy ranges.

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Indeed, six oscillatory modes are observed in Fig. 4. As previously discussed, area A from 70 to 150 ps, includes acoustic phonon modes in bulk STO at frequency of around 45 GHz [30]. Area B indicates split oscillatory modes between 0 to 70 ps with very close energy ranges of 20 and 60 GHz. These two modes are attributed to acoustic phonons and magnons in the ferromagnetic LSMO layer. The oscillatory mode in area C from 0 to 25 ps at frequency of 120 GHz occurring in the top layer of the heterostructure and is attributed to acoustic phonons in the BTO layer. Moreover, the short-lived modes in area D, at around 800 GHz and 1.3 THz, are attributed to surface/interface phenomena (such as surface phonons and surface plasmons). To distinguish each mode, and provide an accurate physical reason, more measurements in different temperatures and applied magnetic fields are required.

Comparison with Fourier transforms, to further illustrate the strength of the Morse wavelet analysis is shown in Fig. 5. To help improve the results from the Fourier transform, the data has been divided into two parts. The first part (a-c, t < 50 ps) indicates the part of the signal which is dominantly affected by the BTO and LSMO layers, however the second part (b-d, t > 50 ps) is mainly from the substrate (STO). Without this separation, the result is mainly dominated by the substrate section and no information from the thin film section is achievable. Figure 5(d) indicates the information achieved using the Fourier transforms for the substrate section is comparable to the one achieved with the wavelet analysis. However, Fig. 5(c) indicates when the focus is the heterostructure thin film, the Fourier transforms are not as informative as the wavelet analysis in Fig. 4.

Fig. 5 Residual TR signal for BTO/LSMO layers (a) and STO (b). Fourier transforms for BTO/LSMO layers (c) and STO (d).

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Conclusion

In conclusion, wavelet analysis has been applied to the residual of transient reflectivity signals obtained from ultra-thin films. Implementation of the Morse wavelets, with a variety of compactness and symmetry parameters, favors extraction of information in time, position and the frequency domains of the oscillations. This implementation illustrates that this technique is a powerful method for analyzing the residual of transient reflectivity signal.

We suggest this technique to be used on the residual of exponential fits to pump-probe data, when multiple oscillatory modes with the same energy ranges are expected in the heterostructure, as well as studying of abrupt changes or fast decaying oscillatory modes in ultra-thin films.

Some future applications of this technique include the study of acoustic surface and bulk phonons and magnons, magnon-phonon coupling, and surface plasmon polaritons in ultra-thin films. Advanced laser schemes including asynchronous operation [31] and frequency combs [32] can further improve the achievable information from this technique. Moreover, with the emergence of pulsed X-Ray sources found at synchrotrons and free-electron lasers, study of the modes with higher energies such as short-lived THz magnons is conceivable [33].

Funding

American Chemical Society (PRF #56642-ND10), National Science Foundation (DMR-1608656), and US Department of Energy (DE-SC0016176).

Acknowledgments

We acknowledge Lilly, J. M. (2017), jLab: A data analysis package for Matlab, v. 1.6.5, http://www.jmlilly.net/jmlsoft.html, for Figs. 1(a)-1(d).

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Application of wavelet analysis on transient reflectivity in ultra-thin films

Abstract

Introduction

Wavelet analysis

Results and discussion

Conclusion

Funding

Acknowledgments

References

Cited By

Figures (5)

Equations (5)

Optics Express