Theoretical background

3.1 Love wave

LW is an elastic wave that propagates in a layered structure consisting of a substrate and a guiding layer on top of it. LW has pure shear horizontal vibrations with particle movement parallel to the surface and perpendicular to the wave propagation direction (perpendicular to the sagittal plane) [104]. The propagation of LW is possible only if the shear velocity in the layer is less than the shear velocity in the substrate. Thus, LW is a type of shear horizontally polarized wave that can be produced in an SSBW device or leaky SAW device when an overlayer with an acoustic shear velocity less that in bulk is deposited on the surface of the substrate [68]. The guiding layer slows down the propagating acoustic shear mode, thus decreasing the penetration depth and confining the energy of the wave in the layer.

The variation in the power flow of the fundamental LW mode as a function of the guiding layer thickness is shown in Fig. 3.1. LW device consists of a piezoelectric substrate with IDT fabricated on the surface. A guiding layer of thickness h is coated on top of the device. When the thickness of the guiding layer is much smaller than the wavelength (hλ), most of the wave energy is located in the piezoelectric substrate, and the LW propagates with a velocity close to the velocity in the substrate. When thicker layers are applied, with thickness still smaller than the wavelength (h < λ), the energy is concentrated in the overlayer, and the velocity of LW tends towards the velocity in the layer. At a certain thickness, the wave is maximally confined in the layer, with a large normalized wave amplitude (normalized to the total energy of the wave) at the surface [149], [150]. This thickness corresponds to the maximum mass sensitivity of the device. The guiding layer not only confines the wave energy to the surface and increases mass sensitivity but also serves to lower the insertion loss of the device. In addition, the guiding layer shields the metal electrodes from the liquid medium typically used for biosensing. SiO₂, ZnO, gold and polymers are often used as guiding layer in LW sensors.

Piezoelectric substrates like 36^◦-YX LiTaO₃, 41^◦-YX LiNbO₃, 90^◦- ST and AT quartz are some of the crystals that generate SH wave [9]. Quartz substrates generate pure SH wave, provide good temperature stability and high mass sensitivity, but they suffer from lowK² and high insertion losses. LiNbO₃ has high coupling coefficient but has poor thermal stability. 36^◦- YX LiTaO3 provides better thermal stability than LiNbO3, sufficiently high K² (higher than quartz but lower than LiNbO₃), and lower insertion loss in comparison to quartz substrates [77], therefore it is often preferred for designing LW sensors.

3.2. Theoretical background

Figure 3.1: Schematic representation of LW device showing the power flow in the guiding layer and piezoelectric substrate as a function of the guiding layer thickness (adapted from [149]).

Figure 3.2: The configuration of the Love wave device showing the guiding layer and mass loading bio-layer (adapted from [150]).

tion of phase velocity, electromechanical coupling coefficient and mass sensitivity of a layered structure with SiO₂ on quartz substrate. Abdollahiet al. [127] evaluated the mass sensitivity of SAW devices considering different piezoelectric materials by calculating wave propagation speeds and energy distributions through 3D FE simulation. Calculation of mass sensitivities of LW devices using wave velocity and displacements obtained from FE simulation have also been reported in [21], [136].

The configuration of the Love wave device is shown in Fig. 3.2. It shows a semi-infinite substrate of density ρ_s overlaid with a guiding layer of thickness h and density ρ_l. They axis is normal to the surface of the substrate, and the wave propagates in thex direction with SH displacement in thezdirection. The phase velocity and mass sensitivity of the Love wave device depend on the density and thickness of the guiding layer present on the substrate. The dispersion equation of Love wave is given as [151]

tan(β_lh) = µs

µ_l

s1−(v_p²/v_s²)

(v_p²/v_l²)−1 (3.1)

whereµ_sandv_sare the shear modulus and shear velocity of the substrate whileµ_landv_lare the

Figure 3.3: Geometry used for 3D FE simulation of Love wave resonator.

corresponding values of the guiding layer. The phase velocity of the LW and angular frequency of the propagating wave in the layered media are denoted by vp and ω, respectively. βl and β_s are the transverse propagation constant of the waveguiding layer and substrate respectively given as

β_l= sω²

v²_l −ω²

v_p² β_s= sω²

v²_p −ω²

v_s² (3.2)

The required condition for LW mode propagation is thatβ_l andβs should be real andvs≥vp ≥ vl. Once the phase velocity of Love wave is obtained by solving the dispersion equation, the group velocity v_g is calculated by the relation [141]

v_g =v_p

1 +h/λ v_p

dv_p d(h/λ)

(3.3) In the sensor configuration shown in Fig. 3.2, a uniform thin film representing a chemical or biological species of thickness l, density ρb and shear velocity vb present on the waveguiding layer causes mass loading of the device. Mass sensitivityS_f is obtained from the relative change in frequency as a result of the deposition of incremental mass per unit area ∆m uniformly distributed on the sensor surface [67] as

S_f = lim

∆m→0

∆f /f0

∆m

. (3.4)

wheref₀is the unperturbed frequency of the device and ∆f=f−f₀is the change in the frequency due to the mass loading of the thin film. If the thickness of mass loading layer is much less than the SAW wavelength, then first-order perturbation theory can be applied by ignoring the elastic effects of the thin mass loading layer (v_b = 0), to get the theoretical mass sensitivity of the