Thermal and Elastic Characterization of Nanostructured Fe₂O₃ Polymorphs and TiO₂-Coated Fe₂O₃ Using Open Photoacoustic Cell

Abstract

α-Fe₂O₃ and ε-Fe₂O₃ thin-film hematite polymorphs, as well as Fe₂O₃ polymorphs’ thin films coated with TiO₂ nanolayer, all deposited on Si substrate plates, were investigated by an open photoacoustic cell set-up within the 20 Hz–20 kHz modulation frequency range. Theoretical analysis of experimental results was based on a two-layer model which includes both substrate and thin-film parameters. For different combinations of the substrate and polymorphic layer, noticeable differences in the total photoacoustic signals are observed at high-frequency range (> 1 kHz), mainly due to elastic bending effect. It was also observed that an ultra-thin TiO₂ surface layer deposited on polymorphs influences the signal of the whole structure, which is attributed to the changed quality of sample surfaces. Presented experimental results and theoretical analysis demonstrate the suitability of open PA cell technique to determine thermal parameters of thin two-layer materials and detect minute differences in their structures originating either from the native structure of the material or from the deposition of ultra-thin coatings. High reproducibility of the photoacoustic measurements and good agreement with values reported in literature and with previous experiments were found.

Appendices

Appendix 1: Temperature Distributions in the Two-Layer Sample

In the case of the two-layer sample consisting of a first layer (thin film) having the thickness l₁ and second layer (substrate) having the thickness l₂, a two-layer theoretical model is developed [8, 12] to study the dependence of photoacoustic response on optical, thermal, elastic and carrier transport properties. Layers are considered to be made of different materials: thin film as a transparent dielectric and substrate as a semitransparent semiconductor of p-type. Due to simplicity, the analyzed sample is assumed to be a thin circular plate having a cylindrical symmetry around the z-axis, with the thickness much lower than its radius. All of these simplifications allow one to reduce heat transfer within the sample to the 1D problem. Furthermore, the thickness of the thin film is taken to be much smaller than the substrate thickness: l₂≫ l₁. The principal scheme of such a two-layer system is depicted in Fig. 6.

Fig. 6

The principal scheme of an investigated two-layer system

When the two-layer sample (Fig. 6) is exposed to the periodic optical excitation using the light source tuned to modulation frequencies f, the component of the periodic temperature distribution T₁ (z,f) in the first layer can be obtained solving 1D diffusion equation [8, 12, 27]:

$$\frac{{\partial^{2} T_{1} \left( {z,f} \right)}}{{\partial z^{2} }} - \frac{i\omega }{{D_{T1} }}T_{1} \left( {z,f} \right) = - \frac{1}{{k_{1} }}\beta_{1} \left( {1 - R_{1} } \right)I_{0} e^{{ - \beta_{1} \left( {z + l_{1} } \right)}} ,$$

(A1)

where ω =2πf is the angular modulated frequency, I₀ the incident light intensity, R₁ the film reflection coefficient, D_T1 the film thermal diffusion coefficient, k₁ the film heat conduction coefficient, and β₁ the film absorption coefficient. The general solution of Eq. 1 can be written in the form:

$$T_{1} \left( {z + l_{1} ,f} \right) = A_{1} e^{{\sigma_{1} \left( {z + l_{1} } \right)}} + A_{2} e^{{ - \sigma_{1} \left( {z + l_{1} } \right)}} + A_{3} e^{{ - \beta_{1} \left( {z + l_{1} } \right)}} ,$$

(A2)

where $\sigma_{1} = \sqrt {i\omega /D_{T1} }$ is the film complex thermal diffusivity, and constant A₃ is given as:

$$A_{3} = - \frac{{\beta_{1} I_{0} (1 - R_{1} )}}{{k_{1} \left( {\beta_{1}^{2} - \sigma_{1}^{2} } \right)}}$$

The component of the periodical temperature distribution T₂ (z,f) in the second layer can be obtained solving the 1D diffusion equation given in the form [8, 12, 27,28,29,30]:

$$\frac{{\partial^{2} T_{2} \left( {z,f} \right)}}{{\partial z^{2} }} - \sigma_{2}^{2} T_{2} \left( {z,f} \right) = - \frac{{\varepsilon_{g} }}{{k_{2} \tau_{2} }}n_{e} \left( {z,f} \right) - \frac{{\beta_{2} I}}{{k_{2} }} \cdot \frac{{\varepsilon - \varepsilon_{g} }}{\varepsilon }e^{{ - \beta_{2} z}} ,$$

(A3)

where $I = \left( {1 - R_{1} } \right)\left( {1 - R_{2} } \right)e^{{ - \beta_{1} l_{1} }} I_{0}$, $n_{e} \left( {z,f} \right)$ is the minority carrier density along the z-axes, $\varepsilon_{g}$ the substrate energy gap, $\varepsilon$ the incident light photon energy, $R_{2}$ the substrate reflection coefficient, $\sigma_{2} = \sqrt {i\omega /D_{T2} }$ the substrate complex thermal diffusivity, D_T2 the substrate thermal diffusion coefficient, k₂ the substrate heat conduction coefficient and β₂ the substrate absorption coefficient.

The general solution of Eq. A3 can be written in the form:

$$T_{2} \left( {z,f} \right) = B_{1} e^{{\sigma_{2} z}} + B_{2} e^{{ - \sigma_{2} z}} + B_{3} n_{e} \left( {z,f} \right) + B_{4} e^{{ - \beta_{2} z}} ,$$

(A4)

where

$$B_{3} = - \frac{{\varepsilon_{g} }}{{k_{2} \tau_{2} \left( {\sigma_{2}^{2} - \frac{1}{{L^{2} }}} \right)}},{\kern 1pt} \quad B_{4} = - \frac{{\beta_{2} \left( {1 - R_{1} } \right)\left( {1 - R_{2} } \right)e^{{ - \beta_{1} l_{1} }} I_{0} }}{{\varepsilon \left( {\beta_{2}^{2} - \sigma_{2}^{2} } \right)}}\left( {\frac{{B_{3} }}{{D_{e} }} - \frac{{\varepsilon - \varepsilon_{g} }}{{k_{2} }}} \right),$$

and $L = \sqrt {\frac{{D_{e} \tau_{2} }}{{1 + i\omega \tau_{2} }}}$ is the complex minority carrier diffusion length (D_e is the diffusion coefficient of minority carriers).

Constants A₁, A₂, B₁ and B₂ can be found solving the boundary conditions:

• (a) $- k_{1} \left. {\frac{{\partial T_{1} \left( {z,f} \right)}}{\partial z}} \right|_{{z = - l_{1} }} = 0 ,$

• (b) $T_{1} \left( {0,f} \right) = T_{2} \left( {0,f} \right) ,$

• (c) $- k_{2} \left. {\frac{{\partial T_{2} \left( {z,f} \right)}}{\partial z}} \right|_{z = 0} = s_{\text{F2}} n_{e} \left( {0,f} \right)\varepsilon_{g} - k_{1} \left. {\frac{{\partial T_{1} \left( {z,f} \right)}}{\partial z}} \right|_{z = 0}$

• (d)
  $$- k_{2} \left. {\frac{{\partial T_{2} \left( {z,f} \right)}}{\partial z}} \right|_{{z = l_{2} }} = - s_{\text{R2}} n_{e} \left( {l,f} \right)\varepsilon_{g}$$

  (A5)

where $s_{\text{F2}}$ and $s_{\text{R2}}$ are the substrate surface recombination speeds at the front $\left( {z = 0} \right)$ and rear $\left( {x = l_{2} } \right)$ surfaces, respectively, and $\tau_{2}$ is the bulk minority carrier lifetime.

Appendix 2: Theory of Thermoelastic Bending

Following all the assumptions about the circular plate sample presented in Fig. 6, it is possible to use the simple theory of the elastic thin plate to obtain the elastic bending, i.e., the sample surface displacements as the photoacoustic signal generators. In the beginning, we will consider the sample as a one-layer plate. Later, we will distinguish one- and two-layer case following the parameters’ notation from Fig. 6.

The simplified thermoelastic bending theory of a uniform-thickness circular plate being subject to the surface heating with modulated light source describes thermoelastic displacements perpendicular U_r(r,z) (r-axis) and parallel U_z(r,z) (z-axis) to the surface normal for a circular plate (Fig. 7). The plate is mass centered at the origin and has a thickness of l and a radius, R_s. The U_z(r,z) at the back surface, z = l, important in a transmission photoacoustic measurements, can be written in a general form as [8, 27,28,29,30]

Fig. 7

Thermoelastic displacements perpendicular U_r(r,z) (r-axis) and parallel U_z(r,z) (z-axis) to the surface normal for a circular plate

$$U_{z} \left( {r,z} \right) = \frac{{C_{m} }}{2}\left( {R_{s}^{2} - r^{2} } \right), \, m = {\text{I, II}} .$$

where m = I refers to one-layer, and m = II to a two-layer plate.

Following the parameters’ notation for a thin film and substrate (Fig. 1), in the one-layer case (substrate only) the constant C_m= C_I is equal to [6, 27,28,29,30]

$$C_{\text{I}} = \frac{{12\alpha_{T2} }}{{l_{2}^{2} }}M_{T2} ,$$

while in the two-layer case C_m= C_II is equal to [8, 12]

$$C_{\text{II}} = \frac{{6E_{2}^{2} l_{2} \left( {2M_{T2} - l_{2} N_{T2} } \right)\alpha_{T2} + E_{1}^{2} l_{1} \left( {2M_{T1} - l_{1} N_{T1} } \right)\alpha_{T1} + E_{2} E_{1} \left[ {l_{1} \left( {2M_{T2} - l_{1} N_{T2} } \right)\alpha_{T2} + l_{2} \left( {2M_{T1} - l_{2} N_{T1} } \right)\alpha_{T1} } \right]}}{{E_{2}^{2} l_{2}^{4} + E_{1}^{2} l_{1}^{4} + 2E_{2} E_{1} l_{2} l_{1} \left( {2l_{2}^{2} + 3l_{2} l_{1} + 2l_{1}^{2} } \right)}},$$

where E₂ and E₁ are the Young’s modulus of the substrate and film, respectively, and M_T2, M_T1, N_T2 and N_T1 are defined as

$$M_{T2} = \int\limits_{0}^{{l_{2} }} {zT_{2} \left( {z,f} \right){\text{d}}z} ,\quad M_{T1} = \int\limits_{{ - l_{1} }}^{0} {zT_{f} \left( {z,f} \right){\text{d}}z} ,$$

$$N_{T2} = \int\limits_{0}^{{l_{2} }} {T_{2} \left( {z,f} \right){\text{d}}z} ,\quad N_{T1} = \int\limits_{{ - l_{1} }}^{0} {T_{f} \left( {z,f} \right){\text{d}}z} .$$

The back-side displacement U_z(r,z) exhibits parabolic thermoelastic displacement representing bending. Such thermoelastic bending produces a back-side acoustic wave with a pressure

$$\delta p_{\text{TE}} \left( f \right) = \frac{{\gamma p_{0} }}{{V_{0} }}\int\limits_{0}^{{R_{s} }} {2\pi rU_{z} \left( {r,z} \right)} {\text{d}}r,$$

usually detected with a microphone as a photoacoustic thermoelastic signal component, δp_TE(f) (see Eq. 3).