Analysis of Dispersion Characteristics of Rayleigh Waves in Nanostructured Thin Films

The structures of thin films range from single thin film structures to laminated structures like semiconductors. Most thin films have multilayered structures with a finite number of layers above a substrate, as shown in Fig. 1. Thin films having a multilayered structure that generates different acoustic characteristics depending on the material constituting each layer. In Fig. 1, P and S denote upward longitudinal and transverse wave components, and P′ and S′ denote downward longitudinal and transverse wave components. Because each layer has different physical properties, it affects the propagation of Rayleigh waves and shows different propagation characteristics depending on the shape of the thin film structure. Typical examples are slow-on-fast and fast-on-slow thin film structures. The former refers to the case where the acoustic wave speed in the substrate is faster than the acoustic wave speed in the thin film, and the latter refers to the opposite case. These two thin film structures have opposite dispersion characteristics. In this study, the dispersion characteristics were calculated and experimentally verified by considering both slow-on-fast and fact-on-slow thin film structures.

As mentioned above, because Rayleigh waves have velocity dispersion in a multilayered structure, the dispersion characteristics according to the frequency and thin film thickness were analyzed through a dispersion diagram using the transfer matrix method [6,7].

The displacement can be expressed in vector form through an analysis of the three-dimensional wave equation as follows [8]:

where u is the displacement vector in the x, y, or z directions, λ and μ are the Lame’s constants of the material, and ρ is the density of the material. The solutions to Eq. (1) can be divided into equations for longitudinal and transverse waves, respectively. If the above equation is expressed by the scalar function ϕ and vector ψ function by the Helmholtz equation, the following equation is obtained [8]:

Furthermore, ϕ and ψ can be expressed as follows:

where k is the wavenumber vector, w is the angular frequency, and x and z are coordinates parallel and perpendicular to the surface, respectively, as shown in Fig. 1. It is assumed that the component in the y direction does not exist because in Rayleigh waves, only the propagation direction (x) and the depth direction (z) are considered. Eqs. (1) and (3) can be rearranged with the displacement vectors for longitudinal and transverse waves as follows, where u_x, u_y and u_z are the displacement vectors in the x, y, and z directions, respectively.

Bulk waves have the same components according to Snell’s law. The wavenumber vector k is parallel to the medium. The equation for displacement and stress can be represented by F as follows:

The wavenumber vector for the z direction is as follows, where α and β denote the sound velocities of longitudinal and transverse waves, respectively:

Eq. (6) can be substituted into Eq. (4), and the displacement and stress of the layer can be rearranged into an equation for the amplitude as follows by omitting the common component F [4]:

For each layer of the thin film, the downward direction is represented by ＋ and the upward direction by −. In Eq. (7), C_α, C_β, g_α, g_β and B are random abbreviations for convenience and are defined as follows:

The 4×4 matrix in Eq. (7) can be used to define a coefficient matrix [D] and can be rewritten as follows:

In a multilayered structure of the type shown in Fig. 1, the amplitude vector of the top layer can be expressed as the inverse matrix of the coefficient matrix [D] as follows:

where the subscript l1 denotes the layer of the sequence, and t denotes the top layer. The relational equation for the top and bottom of the first layer can be expressed as follows:

where the matrix [L] is a coefficient matrix representing the relationship between the top and bottom interfaces of a layer. Assuming that the index of the last layer is n, the relationship between the surface the bottom layer is defined as follows:

where matrix [S] is

In order to solve Eq. (12), the boundary condition for Rayleigh waves must be applied. The stress components σ_zz and σ_xz at the top of the first layer are zero because according to the free stress condition. As there are no longitudinal and transverse wave components reflected from the half-space, the amplitude vectors A_(L-) and B_(S-) are zero. Therefore, the equation for a semiinfinite substrate in free space can be expressed as follows:

The 0 items in the left term can be removed and rearranged as follows:

For the above equation to be valid, the determinant of matrix [S] must equal 0. This is known as the characteristic equation of the Rayleigh wave.

The dispersion diagram is formed when the frequency and phase velocity satisfying the above characteristic equation are derived and expressed along the x and y axes, respectively.

Fig. 2 illustrates the generation process of the V(z) curve between the acoustic lens of the scanning acoustic microscope and specimen.

The RF input signal (tone burst mode) applies a voltage to a ZnO piezoelectric transducer through an electrode. The transducer propagates the plane acoustic wave through the buffer rod by the piezoelectric effect. When this acoustic wave propagates through a contact medium, specularly reflected acoustic waves and Rayleigh waves are generated. Rayleigh waves are generated when the incident waves with an angle greater than the critical angle propagate between the surface of the specimen and the contact medium due to the shape of the acoustic lens. After propagating for a certain distance along the interface, the Rayleigh waves leak into the contact medium layer before propagating again in the reverse direction.

The specularly reflected acoustic waves also propagate in the reverse direction after being reflected from the material surface and output a voltage owing to the inverse piezoelectric effect. If a tone burst wave is used, the output voltage changes periodically owing to the changing distance between the acoustic lens and specimen caused by the interference of these two acoustic waves. The V(z) curve graphically represents this relationship and can be used to calculate the propagation velocity of Rayleigh waves. This velocity is calculated using the interval (Δz) between the valleys in the interval where the periodic voltage changes occur in the V(z) curve, and is calculated using Eq. (17), where V_R is the velocity of the Rayleigh waves, and v_w is the acoustic wave velocity in water.

Slow-on-fast and fast-on-slow thin film specimens were fabricated to evaluate the dispersion characteristics. The slow-on-fast structure consists of an Ni thin film deposited on an Si substrate using e-beam evaporation. A 6-inch Si wafer was used as the substrate of the Ni/Si thin film. Before deposition, ultrasonic cleaning was performed with alcohol, acetone, and distilled water, in that order. Then the surface was fully dried with nitrogen gas. To verify the dispersion characteristics according to the specimen thickness, specimens were fabricated by controlling the deposition time and the thin film thickness after calculating the deposition rate through preliminary deposition. Ni thin films with thicknesses of 100, 200, 400, 600, 800, and 1000 nm were deposited on the cleaned Si wafers. The fabricated specimens were diced to a size of 1 cm × 1 cm to measure the velocity of Rayleigh waves and the actual deposited thickness using a SEM. For the fast-on-slow structure, Si₃N₄ was deposited on a GaAs substrate using plasma-enhanced chemical vapor deposition (PECVD).

A 2-inch GaAs substrate and mixed gas of 5% silane/helium (SiH₄/He) and ammonia (NH₃) were used to fabricate an Si₃N₄/GaAs thin film, where the chamber temperature was maintained at 250℃ during deposition. Furthermore, the deposition rate was calculated and the deposition time was controlled through preliminary deposition in the same manner as for the Ni/Si thin-film fabrication process. The resulting Si₃N₄/GaAs thin-film specimens had thicknesses of 100, 400, 800, 1200, 1600, and 2000 nm. For the two types of fabricated specimens, the actual thickness was accurately measured using an SEM. Figs. 3 and 4 show the cross-sectional images of the thin films measured using an SEM, and Tables 1 and 2 show the measured thicknesses of the thin films.

Table 1.

Thickness of Ni thin film [unit: nm]

	100	200	400	600	800	1000
1	135.0	217.5	303.7	472.5	731.2	946.9
2	129.4	217.5	318.7	476.2	750.0	960.0
3	131.2	215.6	311.2	457.5	751.9	946.9
Avg.	131.8	216.8	311.2	486.7	744.3	951.2

Table 2.

Thickness of Si₃N₄ thin film [unit: nm]

	100	400	800	1200	1600	2000
1	87.5	328.1	768.8	1046.9	1506.3	1812.5
2	90.6	326.1	762.5	1056.3	1493.8	1815.3
3	90.6	328.1	765.6	1059.4	1053.1	1812.5
Avg.	89.6	327.4	765.6	1,054.2	1,501.1	1,813.4

To theoretically calculate and experimentally verify the dispersion characteristics of Rayleigh waves in this study, the acoustic velocity of Rayleigh waves in the thin films was determined by measuring the V(z) curve with a scanning acoustic microscope (Olympus; UH3).

In line with the section describing the velocity dispersion of Rayleigh waves, spherical probes with operating frequencies of 400 MHz and 200 MHz were used for the Ni and Si₃N₄ thin films, respectively. Probes with a 120° lens angular aperture were used to generate Rayleigh waves. Deionized water was used as the contact medium between the specimen and lens. The temperature was maintained at 21.5 ℃ using a heater in the sample stage to attempt to maintain a constant propagation velocity, regardless of the temperature of the contact medium. The velocity was measured in each specimen three times at random positions, and the velocity of the Rayleigh waves was calculated using the average value.