A plane SAW is propagated along the surface of an anisotropic piezoelectric substrate occupying the half space as shown in Figure 2. The wave propagates along the x-axis. If the substrate rotates around the x-axis with a constant angular rate, particles of the wave are subjected to additional inertial forces, including the Coriolis and centrifugal forces. With the counter-clockwise rotation around the x-axis, the Coriolis force acts on the vibrating particles and induces a pseudo running wave shifted by a quarter of a wavelength, and coupling with the initial SAW, resulting in the change of trajectory of the wave particles, and hence, the acoustic wave displacement was deviated, leading to the acoustic wave velocity shift. Consequently, a frequency variation proportional to the input rotation according to the relationship among the frequency was expected. The theoretical analysis of the SAW gyroscopic effect over the whole depth is given in the following section.

In this section, the SAW gyroscopic effect is described by solving the piezoelectric medium equations of motion and surface effective permittivity method. Consider an anisotropic and piezoelectric medium occupying a half-space (x₃ ≤ 0) with no mechanical load about the plane (x₃ = 0) and rotating at a constant angular rate (Ω_i) about the x_i-axis (i = 1,2,3), as schematically illustrated in Figure 2. The dynamic equations of linear piezoelectricity with Coriolis and centrifugal forces contribution take the following form in this coordinate system mentioned in Figure 1: (1) { ρ ∂ 2 u i / ∂ t 2 − C ijk l ∂ 2 u k / ∂ x j ∂ x l − e kij ∂ 2 ϕ ∂ x k ∂ x j + 2 ρ E ijk Ω j ∂ u k ∂ t + ρ ( Ω i Ω j u j − Ω j 2 u i ) = 0 , i , j , k , l = 1 , 2 , 3 e jkl ∂ 2 u k ∂ x j ∂ x l − ε jk ∂ 2 ϕ ∂ x j ∂ x k = 0where x₁, x₂, x₃, denote the Cartesian coordinates x, y, z respectively, E_ijk is the Levi-civita symbol, and we denote by u_i the mechanical displacements and by ϕ the electric potential. c_ijkl, e_kij, and ε_ij stand for the elastic, piezoelectric and dielectric constants, and ρ for the mass density of the substrate, respectively. The summation convention for repeated tensor indices and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index are adopted. The indices i, j, k, and l range from 1 to 3. With the compressed matrix notation [16], the material constants c_ijkl and e_ijk in Equation (1) can be represented by matrices c_pq and e_ip, with the convention that p, q = 1, 2, 3, , , 6. Similarly, the strain tensor S_ij and the stress tensor T_ij can be represented by S_p and T_q. The first three terms in the left-hand side of the Equation (1) are the equation of the motion related to the inertia, the elasticity and piezoelectricity. The fourth and the fifth terms are due to the Coriolis force and the centrifugal force, respectively.

First, we assume a general solution of the Equation (1), the particle displacement and electrical potential, in the form: (2) { u i = A i exp { j k s ( x 1 + α x 3 ) − j ω t } , i = 1 , 2 , 3 ; φ = A 4 exp { j k s ( x 1 + α x 3 ) − j ω t }where k_s and ω are the wave number in the x₁ direction and the time frequency, respectively. α_s is a decay constant along the x₃ direction. A_j (j = 1, 2, 3) and A₄ are wave amplitudes. Substitution of Equation (2) into Equation (1) leads to a four linear algebraic equations (Cristoffel equation) for A_j and A₄ as follows: (3) [ Γ 11 − ρ V s 2 Γ 12 Γ 13 Γ 14 Γ 12 Γ 22 − ρ V s 2 Γ 23 Γ 24 Γ 13 Γ 23 Γ 33 − ρ V s 2 Γ 34 Γ 14 Γ 24 Γ 34 Γ 44 ] [ A 1 A 2 A 3 A 4 ] = 0where: Γ 11 = c 55 α 2 + 2 α c 15 + c 11 ;     Γ 12 = c 45 α 2 + α ( c 14 + c 56 ) + c 16 + 2 j ρ V s 2 ( Ω 3 / ω ) Γ 13 = c 35 α 2 + α ( c 13 + c 55 ) + c 15 − 2 j ρ V s 2 ( Ω 2 / ω ) Γ 14 = e 35 α 2 + α ( e 15 + e 31 ) + e 11 ;     Γ 21 = c 45 α 2 + α ( c 14 + c 56 ) + c 16 − 2 j ρ V s 2 ( Ω 3 / ω ) Γ 22 = c 44 α 2 + 2 α c 46 + c 66 ;     Γ 23 = c 34 α 2 + α ( c 36 + c 45 ) + c 56 + 2 j ρ V s 2 ( Ω 1 / ω ) Γ 24 = e 34 α 2 + α ( e 14 + e 36 ) + e 16 ;     Γ 31 = c 35 α 2 + α ( c 13 + c 55 ) + c 15 + 2 j ρ V s 2 ( Ω 2 / ω ) Γ 32 = c 34 α 2 + α ( c 36 + c 45 ) + c 56 − 2 j ρ V s 2 ( Ω 1 / ω ) Γ 33 = c 33 α 2 + 2 α c 35 + c 55 ;     Γ 34 = e 33 α 2 + α ( e 13 + e 35 ) + e 15 Γ 44 = − ε 33 α 2 − 2 α ε 13 − ε 11 ;     Γ 41 = Γ 14 ; Γ 42 = Γ 24 ; Γ 43 = Γ 34

Then, for nontrivial solutions of A_j and/or A₄, the determinant of the coefficient matrix of the linear algebraic equations must vanish, and this leads to a polynomial equation of degree eight for α_s. The coefficients of this polynomial equation are generally complex. To ensure the decrease in the displacement u_i and the potential ϕ into the substrate, the generally complex constant α_s must have a negative imaginary part. Thus, we select four eigenvectors with negative imaginary part denoted by α_s(m), and the corresponding eigenvectors by A_j(m), A₄(m), m = 1, 2, 3, 4. Thus, the general wave solution to Equation (1) in the form of Equation (2) can be written as: (4) { u i = ∑ m = 1 4 C ( m ) A ' i ( m ) exp { jk s ( x 1 + α ( m ) x 3 ) − j ω t } ,       i = 1 , 2 , 3 φ = ∑ m = 1 4 C ( m ) A ' 4 ( m ) exp { jk s ( x 1 + α ( m ) x 3 ) − j ω t }where C_m(m = 1, 2, 3, 4) are the weight factors.

The solutions of the motion equation satisfy both the mechanical boundary condition, and the electrical boundary condition respectively.

The mechanical boundary condition: (5) T i 3 ( x 1 , x 3 ) | x 3 = 0 = c i 3 k l ∂ u k ∂ x l + e ki 3 ∂ ϕ ∂ x k = 0 i , k , l = 1 , 2 , 3where T_i3 is the stress on SAW propagating direction.

The electrical boundary conditions were considered in case of free surface and metallic surface.

For a free surface, the effect of the electric field in the surrounding space can be considered by requiring the electrical potential cross the interface continuously: (6) { φ ′ | x 3 = 0 + = φ | x 3 = 0 − φ ′ ( x 1 ,   x 3 ) | x 3 > 0 = φ ( x 1 , 0 ) exp ( − 2 π f | s | x 3 )also, the surface charge density σ(x₁) created by the interdigital transducers (IDT) electrodes equal to the incontinuity of the normal component D₃ of the dielectric displacement vector cross the interface: (7) D 3 ( x 1 ,   x 3 ) | x 3 = 0 + − D 3 ′ ( x 1 , x 3 ) | x 3 = 0 − = σ ( x 1 )

And then, substituting Equation (4) into the boundary conditions Equations (5–7) yields the following four linear algebraic equations for C(m): (8) [ r 11 r 12 r 13 r 14 r 21 r 22 r 23 r 24 r 31 r 32 r 33 r 34 r 41 r 42 r 43 r 44 ] [ C 1 C 2 C 3 C 4 ] = [ 0 0 0 − j σ ¯ ( s ) / ( 2 π f s ) ]where: { r 1 n = ( c 15 + c 55 α n ) A 1 ( n ) + ( c 56 + c 45 α n ) A 2 ( n ) + ( c 55 + c 35 α n ) A 3 ( n ) + ( e 15 + e 35 α n ) A 4 ( n ) r 2 n = ( c 14 + c 45 α n ) A 1 ( n ) + ( c 46 + c 44 α n ) A 2 ( n ) + ( c 45 + c 34 α n ) A 3 ( n ) + ( e 14 + e 34 α n ) A 4 ( n ) r 3 n = ( c 13 + c 35 α n ) A 1 ( n ) + ( c 36 + c 34 α n ) A 2 ( n ) + ( c 35 + c 33 α n ) A 3 ( n ) + ( e 13 + e 33 α n ) A 4 ( n ) r 4 n = ( e 31 + e 35 α n ) A 1 ( n ) + ( e 36 + e 34 α n ) A 2 ( n ) + ( e 35 + e 33 α n ) A 3 ( n ) + ( ε 31 + ε 33 α n + j ε 0 ) A ( 4 ) ( n ) , n = 1 , 2 , 3 , 4

Here, the ε₀ is the dielectric constant in vacuum. Usually, after solving Equation (8), the C(m) can be determined. However, due to the inhomogeneity of Equation (8), the C(m) is difficult to extract directly. So, referring to the surface effective permittivity method [12], the acoustic field in the piezoelectric substrate can be easily described. The surface effective permittivity constant ε_s(s) of the piezoelectric substrate is represented as: (9) ε s ( s ) = σ ¯ ( s ) 2 π | s | ϕ ¯ ( s , x 3 ) | x 3 = 0where, ϕ̄(s, x₃) is the electrical potential distribution in slowness domain of the piezoelectric substrate. s is the slowness, and s =1/V_s.

Then, the weight factors C(m) can be deduced as following by the surface effective permittivity: C n = Δ 4 n Δ 0 σ ¯ ( s ) j 2 π f sthus, the effective permittivity constant ε_s(s) was obtained by: (10) ε s ( s ) = js | s | Δ 0 ∑ n = 1 4 Δ 4 n A 4 ( n )where Δ₀ is the determinant value of the matrix in left side of Equation (8), Δ_4n is the algebraic complement of matrix elements r_4n in the matrix of [r_in]. The ε_s values in case various slowness s and normalized rotation (Ω_i/ω, i = 1, 2, 3) can be extract. Hereof, after the analysis from the ε_s curves in case the slowness, the SAW velocity can be determined under different rotation. Moreover, the gyroscope gain factor β relating to the frequency sensitivity of the gyroscope can also be determined by: (11) Δ f i / f 0 ≈ Δ V i / V s = β i × Ω i / ω , i = 1 , 2 , 3here, Δf and ΔV are the frequency and velocity shifts due to the normalized rotation Ω_i/ω.

As a numerical example, the gyroscopic effect of the SAW propagating along some common piezoelectric substrates like ST-X quartz, 128°YX LiNbO₃ and X-112°Y LiTaO₃ was considered. The mechanical parameters like the density, elastic constants, piezoelectricity and dielectric constants of the above mediums are listed in Table 1 [16].

Using Equations (8–10), the surface effective permittivity constant ε_s(s) functioned as slowness (s) on the X-112°Y LiTaO₃ substrate in the case of a normalized rotation of 2,000 ppm (Ω/ω) around the x₁-axis was depicted as shown in Figure 3. From the picture, we can find that there are several discontinuous points in the curve of the real part of ε_s(s), indicating different acoustic wave modes. The Rayleigh wave mode occurs at lower acoustic wave velocity (slowness of ∼3.2 × 10⁻⁴ s/m), and the image part of ε_s(s) was equal to zero due to non-dissipation in acoustic energy. Meanwhile, there are pairs of zero point and pole points in the regime of Rayleigh wave mode. Obviously, the value of |ε_s(S_mR)| tends to infinity at the pole point of S_mR, corresponding the situation of a metallic surface with limited charge distribution and zero surface potential. Thus, the S_mR represents the slowness of Rayleigh wave propagating along the metallic surface with no attenuation. Additionally, the value of ε_s(S_oR) was equal to zero at the zero point of S_oR, which chracterizes the situation of a free surface with limited electric potential and zero charge distribution, and it describes the Rayleigh wave propagating along the free surface with no attenuation. Usually, the phase velocity of the Rayleigh SAW under given rotation, V_s, can be described by: (12) V s = ( 1 / S OR + 1 / S m R ) / 2

Thus, the velocity shift induced by the external rotation ΔV_s is represented by: (13) Δ V s = V s − V 0 , V 0 = ( 1 / S OR 0 + 1 / S m R 0 ) / 2here, the S_OR0 and S_mR0 are the slowness under no applied rotation. And then, the gyroscopic gain constant β can also be deduced by Equation (11).

Using the surface effective permittivity method described as above and the corresponding mechanical parameters listed in Table 1, the gyroscopic gain constant β and expected sensitivity were calculated and are listed in Table 2. From the calculated results, we find that the calculated β values agree well with the values reported in previous literatures [6–8], it indicates the validity of surface effective permittivity constant method. Also, from the calculated results, larger sensor response will be expected from the rotated X-112°Y LiTaO₃ piezoelectric substrate.

EWC/SPUDT and a combed transducer were utilized to structure the SAW delay lines on X-112°Y LiTaO₃ substrate to improve the frequency stability of the oscillator. The SAW velocity on the X-112°Y LiTaO₃ substrate with 110 nm Al metallization was 3,295 m/s. The operation frequency of the SAW delay line is specifically given by 160 MHz, thus, the wavelength λ of the SAW is given by 20.6 μm. Two delay line patterns with opposite direction were fabricated on a same LiTaO₃ wafer by standard photolithography techniques. Each delay line consists of a launching transducer and readout transducers; the length of the launching transducer was set to 195λ with four groups, which was about 80% of the center-to-center distance between the launching and readout transducers. The distance between the transducers was 135λ. In order to limit the total number of Al fingers in each transducer to about 60, the launching transducer was thinned into a comb structure. To keep the uniformity of the acoustic velocity, many pseudo-fingers are distributed between the SPUDT cells of the comb transducer. It means each group was composed of 15 SPUDT cells and some pseudo-fingers with length of 45λ, as shown in Figure 4(a). A large aperture of ∼1 mm (50λ) was used. The fabricated dual delay line is shown in Figure 4(a).

Using the HP 8753D network analyzer, the amplitude and phase response of the SAW delay lines were measured under matched conditions, as shown in Figure 5. Low insertion loss of 9 dB and linearity phase response in 3-dB frequency bandwidth were observed.

Next, the fabricated SAW device chip was loaded into a standard metal base, as shown in Figure 4(a). As the feedback of oscillators, the launching and read transducers of the fabricated SAW delay lines were connected by an oscillator circuit which was made of discrete elements (amplifier with a gain of 25 dB, phase shifter, mixer and LPF and so on) on a printed circuit board (PCB) as shown in Figure 4(a). The output of the amplifier was mixed in order to obtain a difference frequency in the MHz range. This technique allows us to double the detection sensitivity and reduce the influence of the thermal expansion of the substrate. The output of the oscillator was monitored and recorded by an oscilloscope and a programmable frequency counter. It is well-known that the frequency stability of the oscillator affects the threshold limit of detection and stability of the sensor directly. Thus, an experiment was performed to evaluate the frequency stability of the fabricated SAW oscillator using the programmable frequency counter under room temperature (20 °C). The oscillation was also modulated at the frequency point with lowest insertion loss by a strategically phase modulation. The measured frequency stability is shown in Figure 6. The shorted frequency stability (min) is up to 0.05 ppm as shown in the insert of Figure 6. The typical long-term stability (h) of the oscillator was measured as 0.4 ppm (Figure 6). From the measured results, excellent frequency stability was observed from the fabricated oscillator.