1. Introduction

Ferroelectric thin films have demonstrated potential for a wide range of applications such as electro-optic devices, high dielectric-constant and charge-density capacitors, un-cooled infrared detectors [1,2], memory devices [3,4] and optical cloaks [5], all of which are highly dependent on the film dielectric properties. Indeed, it is well known that strain in ferroelectric films plays an important role in influencing their dielectric response [6] and ferroelectric transition temperature [7]. This strain can result from lattice misfit or thermal expansion mismatch between the film and the underlying substrate, as well as oxygen vacancies. There have been a number of reported observations of strain relaxation in ferroelectric thin films using reflection high-energy electron diffraction (RHEED) [8], high-resolution transmission electron microscopy (HRTEM) [9] and high-resolution X-ray diffraction (HRXRD) [10]. For example, RHEED has been used for real-time monitoring of strain relaxation in SrTiO₃ (STO) thin films and a critical thickness for complete strain relaxation of 50 – 250 nm was extracted depending on the substrate and deposition temperature [8]. HRTEM has been used to study the growth dynamics and strain relaxation mechanism in BaTiO₃ thin films and revealed a multilayer structure, each layer with a different morphology, residual strain and defect density [9]. Although these techniques are capable of directly observing strain evolution in ferroelectric thin films at the nanoscale and even at atomic level, they are not able to evaluate the effect of strain on the dielectric response or optical constants. These, however, are the most important parameters for a number of opto-electronic applications. Here we show that in addition to revealing the optical constants of the as-grown films, spectroscopic ellipsometry serves as a non-destructive technique for extracting the strain profile of epitaxially-grown ferroelectric thin films.

Spectroscopic ellipsometry is widely used as a non-destructive technique to measure film thicknesses and optical constants. There have been, for example, ellipsometric studies of the optical properties of polycrystalline Ba_xSr_1−xTiO₃ films on metal-coated substrates [11, 12] and glass [13]. In these cases, the films were deposited by either RF magnetron sputtering, metal-organic chemical vapor deposition or the sol-gel method. Strain is absent in the resulting polycrystalline Ba_xSr_1−xTiO₃ thin films, but the refractive index was found to depend on the film composition, film thickness, deposition temperature and post-annealing conditions. To our knowledge, however, there have been no reports on the effects of strain on the optical properties of ferroelectric thin films prepared on single-crystal substrates. In this work, we use variable angle spectroscopic ellipsometry to extract the optical constants of the strained and relaxed layer as well as their respective thicknesses, as a function of the total growth thickness. Moreover, we show that the complicated strain and surface profile obtained preclude a single-layer model for the optical constants of Ba_0.5Sr_0.5TiO₃ (BSTO) thin films and illustrate some of the potential pitfalls associated with single-wavelength ellipsometry, reflectance and other optical techniques that cannot assess film homogeneity [14, 15].

From ellipsometric modeling of different thickness films grown on (001) MgO substrates, we extract a critical thickness above which the established in-plane tensile strain collapses and relaxes completely and a bulk-phase layer with a smaller index of refraction than that of the strained layer starts to grow. The observed critical thickness is comparable to that found in previous RHEED studies [8]. Further studies were made for BSTO films grown on different single crystal substrates: (001) LaAlO₃ (LAO), (001) STO and (001) (La_0.3Sr_0.7)(Al_0.65Ta_0.35)O₃ (LSAT). These studies reveal that for films grown on different substrates, the refractive index of BSTO decreases for substrates showing higher in-plane compressive strain. Indeed, we show that strain engineering obtained by tuning the lattice mismatch can be used to control the film optical properties.

2. Preparation and characterization of BSTO films

The BSTO thin films used in this study were prepared on a variety of 5 × 5 mm² single crystal substrates by single-target pulsed laser deposition (PLD). A KrF excimer laser 248 nm with a fluence of 0.8 J/cm² and a repetition rate of 8 Hz was used to deposit BSTO thin films at an oxygen pressure of 300 mTorr. During growth, the substrate temperature was kept at 740 °C and the total film thickness was controlled by fixing the number of laser pulses hitting the target. After growth, the films were slowly cooled to room temperature in an oxygen rich atmosphere (760 Torr) [16]. The surface morphology and crystal structure of the prepared films were characterized by atomic force microscopy (AFM) and X-ray diffraction (XRD), respectively. These BSTO films have been found to exhibit Stranski-Krastanov growth and their epitaxial nature was confirmed using the azimuthal angle dependence of the grazing incidence diffraction pattern [17].

Figure 1 shows the AFM surface profile for two BSTO films of different thicknesses deposited on MgO. As can be seen from the images, both films are dense and fine-grained with a uniform grain size, indicating that BSTO films have a well-defined microstructure. The root mean square (RMS) and peak-to-valley roughnesses measured over a 1.5 × 1.5 μm² area increase from 1.9 nm to 5.6 nm and from 16.0 nm to 66.2 nm, respectively, for the 37 nm and 220 nm-thick films shown in Fig. 1. The peak-to-valley values measured for all of the BSTO films grown in this study were consistent with the surface layer thicknesses used in the ellipsometric modeling.

 

Fig. 1 AFM surface morphology images of 37 nm (a) and 220 nm (b) BSTO films. The scale bar in both images corresponds to 500 nm. Note the difference in color scales. They are 0–20 nm in (a) and 0–60 nm in (b), respectively, chosen to show the best contrast.

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Figure 2(a) shows the Bragg-Brentano XRD patterns for BSTO films of different thicknesses grown on MgO substrates. Only peaks corresponding to the (00l) crystallographic planes are observed (l=1 to 4). In Fig. 2(a) the 2θ range shown is chosen to emphasize the diffraction peak shift and the emergence of a second peak for increasing film thickness. We find that the 68 nm and 110 nm-thick BSTO films have a single symmetric (002) peak that can be well fit to a Gaussian function, while the peaks for thicker films exhibit a substaintial shift to smaller angle and appear significantly broadened. These peaks appear to form a doublet and can be fit using a superposition of two Gaussians with one peak centered close to the thinner film position and the other located at a smaller angle. This is emphasized in the inset of Fig. 2(a) showing a magnified version of the (002) diffraction peaks for the 68 nm and 220 nm-thick films. The c-parameter for each film was calculated from the measured (00l) peak positions [17] and is shown in Fig. 2(b) as a function of film thickness. In the figure, BSTO films thinner than 171 nm are represented with one lattice parameter c₁, while thicker films are represented with two effective lattice-spacings c₁ and c₂ with c₂ > c₁. This proposition agrees with our ellipsometry measurements, and serves as a guide for interpretation of our ellipsometry model.

 

Fig. 2 XRD patterns of BSTO films of different thickness (a) and the calculated c-parameter as a function of film thickness (b). The inset in (a) shows an enlarged view of (002) peaks for 68 nm and 220 nm BSTO films.

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3. Spectroscopic ellipsometry measurements

Variable angle ellipsometry measurements were made in the standard reflection geometry with a rotating analyzer configuration. In all cases, measurements were carried out for three incident angles (65°, 70° and 75°) in the wavelength range 400–1600 nm, with a 10 nm step. The lower wavelength of 400 nm was chosen to avoid spectral overlap with the absorption band of BSTO [11–14]. This allows the refractive index to be well described by a Cauchy dispersion model. A least squares fit to the data was performed and a mean squared error (MSE) was used to evaluate the quality of the fit. Although, in principle, strain is expected to lead to weak uniaxial anisotropy, isotropic models were used for all of the layers. Inclusion of anisotropy was not found to substantially improve the fit. Therefore, the reported refractive indices here can be assumed to be more closely related to the in-plane refractive index of BSTO. The refractive indices of the substrates were measured separately and modeled using a Cauchy dispersion profile. A constant refractive index was assumed for the surface layer to account for surface roughness and surface relaxation. All data analysis was performed using WVASE32 software [18].

Figure 3 shows the measured ellipsometry data, at an incident angle of 65°, for 37 nm, 68 nm and 110 nm-thick BSTO films grown on MgO. The two parameters given in the figure, Ψ and Δ describe, respectively, the amplitude ratio and phase change of the incident light polarization upon reflection from the sample. The fit, which is also shown in the figure, was created using an optical model consisting of one homogeneous layer modeled by a Cauchy dispersion profile and a surface layer to account for surface roughness and surface relaxation (see the inset of Fig. 3(b)). To reduce parameter correlations and to obtain a suitable fit confidence, a single set of Cauchy parameters for each layer in a given configuration (two or three layers) was fit simultaneously to all of the relevant data. In addition, the resulting thicknesses were confirmed by profilometer measurements suggesting that our approach is indeed justified.

 

Fig. 3 Measured (dashed lines) and fit (solid lines) ellipsometry data Ψ (and Δ in the insets) for BSTO films of different thicknesses. The inset in (b) shows the optical model used to do the fittings. The model consists of a strained layer implemented with a Cauchy dispersion profile and a surface layer characterized by a constant refractive index.

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As is evidenced by the good agreement between the measurement and the modeling in Fig. 3(a), an optical model consisting of one Cauchy layer and a surface layer is able to account perfectly for the measured data for the three measured films, resulting in a small MSE value equal to 9.8. We emphasize that the inclusion of a low-index surface layer is essential to obtain a good agreement for all three thicknesses, regardless of whether a global or individual fit is performed. This points to a potential pitfall of using optical techniques that cannot accurately assess the homogeneity of each layer. XRD measurements highlighted the fact that thin films (<171 nm) consist of a single strained layer. Indeed, this is confirmed by the quality of the fits shown in Fig. 3(a). Figure 4(a) shows the refractive index obtained for thinner films (37 nm, 68 nm and 110 nm) extracted by the simultaneous fitting of nine sets of Ψ and Δ (three films measured at three incident angles) based on the model described above (see the inset).

 

Fig. 4 Refractive indices obtained by simultaneously fitting ellipsometry data for three thin films of BSTO on MgO (a) and for the strained layer (thickness t₁) and the relaxed layer (thickness t₂) obtained by fitting data for thicker BSTO films (b). The two different optical models used are shown in the insets. The tables summarize the thickness of each layer extracted from the models.

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For film thicknesses above 171 nm, however, the modeled data (both Ψ and Δ) show a large deviation from the measured values as depicted in Fig. 3(b), resulting in a global MSE equal to 73. This suggests that a single Cauchy layer and surface layer are no longer sufficient to describe the optical properties of thick BSTO films. The optical properties must be graded in some fashion, or in a simplified model consist of two or more layers with different refractive indices. As suggested by XRD observations for thicker films, we can construct a three-layer model to describe these films in which a relaxed layer with lattice parameter c₂ is sandwiched between a strained layer with lattice parameter c₁ and the surface layer. Figure 4(b) displays the refractive indices for the strained layer and relaxed phase obtained by performing a global fit of the ellipsometric data for thicker BSTO films (171 nm, 220 nm and 265 nm) to such a model, resulting in a significantly reduced global MSE equal to 24. As can be seen from the figure, the index of the strained layer is substantially larger than that of the relaxed layer, but close to the one obtained for thinner films. The tables in Fig. 4 summarize the thickness of each layer extracted from the models. From the fits, we find that the strain present in the thicker films is constrained to approximately the first 60 nm.

4. Strain evolution with film thickness

Since the ellipsometric modeling is capable of accurately determining the thickness of each layer in the film, we can now confidently take advantage of the results to study the strain evolution with film thickness. Figure 5 shows the thickness of the strained layer, for growth on MgO, as a function of the total BSTO film thickness. We find that the strained layer thickness initially increases with the total growth thickness, collapses above a critical thickness of 170 nm, and then eventually saturates. The critical thickness we have identified here is comparable to the value determined for STO films by RHEED measurements in Ref. [8]. In contrast to RHEED, however, the ellipsometer measurements also allow us to determine the strained layer thickness after relaxation.

 

Fig. 5 Plot of the strained layer thickness as a function of the total film thickness obtained from ellipsometry.

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The thickness dependence of the strained layer paints a comprehensive picture of the growth dynamics of epitaxial BSTO thin films on lattice mismatched single crystal substrates. During the growth process, the BSTO film undergoes tensile strain in the plane parallel to the MgO substrate surface due to the fact that the in-plane lattice constant of MgO (a = b = 4.212 Å) is larger than that of BSTO bulk phase (4.004 Å). Note that the in-plane lattice constant of the cubic BSTO bulk phase is assumed to be the same as the lattice parameter measured by XRD of the source material (the PLD target). The in-plane tensile strain in BSTO films caused by the MgO substrate reduces the out-of-plane lattice parameter as observed in Fig. 2. This tensile strain persists with increasing film thickness, but further growth creates additional dislocations and defects which relax the residual strain in the structure. As the film thickness reaches a critical value (170 nm in our case), the defect density is sufficient for the strain to fully relax. This process results in a sudden decrease of the strained layer thickness (see Fig. 5) and a significant increase of the out-plane c-parameter (see Fig. 2) for thicker films. Further deposition results in a thickening of the relaxed layer corresponding to the second peak at a smaller angle observed in the XRD patterns and the new lattice spacing c₂ discussed above.

5. Strain relaxation monitored by the change in optical constants

The observations for BSTO grown on MgO substrates suggest that strain engineering is a flexible means to modify and control the optical response of epitaxial BSTO films. For this purpose, three pairs of BSTO films with thickness below the critical thickness were deposited on single crystal substrates with different lattice parameters: LAO (a = b = 3.791 Å), LSAT (a = b = 3.872 Å) and STO (a = b = 3.905 Å). For all of these, in-plane compressive strain is expected. The out-of-plane lattice parameter for each film was measured by XRD. Strain relaxation is expected to dominate in BSTO films deposited on LAO due to the large lattice mismatch of 5.96% (see Fig. 6) while films on STO substrates are likely to show a lower strain relaxation due to the small lattice mismatch. As for the case of thin BSTO films on MgO, a global fit to the ellipsometry data was performed for each film/substrate combination (two films per substrate) using the simplified model of Fig. 3(b).

 

Fig. 6 Refractive indices for BSTO films grown on LAO, LSAT and STO single crystal substrates. The percentage values quoted are calculated by the expression (α_film – α_sub)/α_sub. α_film and α_sub are the in-plane lattice constants of the bulk BSTO and the different substrates, respectively. The refractive index is found to decrease for increasing in-plane compressive strain.

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Figure 6 shows the refractive index obtained for BSTO films grown on the different substrates. It can be seen that the BSTO refractive index decreases with increasing lattice mismatch between substrate and film (i.e. with an increase of the compressive strain). This trend is consistent with the observation that, on MgO, the refractive index of the relaxed layer is smaller than that of the (tensile) strained layer. In addition, a larger lattice mismatch reduces the total thickness required for the film to relax and the range of validity of the single-layer model. Note that our observation that the refractive index decreases with increasing in-plane compressive strain is consistent with previous studies of the microwave permittivity of strained BSTO films [19].

6. Conclusion

This study has demonstrated that spectroscopic ellipsometry can be used as a sensitive and non-destructive probe to monitor strain relaxation in ferroelectric BSTO thin films and assess the effect of strain on the film dielectric response. We have identified a critical thickness of 170 nm for strain relaxation in films grown on MgO and found that the refractive index of the strained layer is substantially larger than that of the relaxed phase. Our results have shown that ellipsometry can accurately model the complex strain evolution in epitaxial BSTO thin films, and extract the strained layer thickness for post-relaxation, but that a model consisting of a minimum of three distinct layers is required. In addition, studies of films grown on different lattice-mismatched substrates have revealed that the BSTO refractive index decreases with increasing lattice mismatch, suggesting that strain engineering is a flexible means to control the film dielectric response.