3.3 Surface and guided waves

A sound wave can also cling to a surface, like a ripple on water, with its energy within about one wavelength of the top. On LN such a surface acoustic wave (SAW) travels at about 3.4 to 4.0 km/s, 75,000 to 88,000 times slower than light, so a 1 GHz signal becomes a ripple only 4 µm long that a pair of comb-shaped electrodes can launch and catch. Some crystal cuts carry a faster surface wave that leaks a little energy into the bulk as it travels; on the right cut that leak is tiny.

On 128° Y-X the surface atoms trace ellipses and the motion dies within about a wavelength; switch to 64° Y-X and the motion turns mostly horizontal shear, with a partial wave that carries energy down into the crystal.

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Rayleigh wave: traction-free surface wave on an elastic half-space, displacement in the sagittal plane, decaying with depth (Rayleigh 1885 S). On a piezoelectric substrate the partial waves also carry an electric potential that extends into the vacuum; the wave is slower when the surface is shorted by a thin metal film.

Partial-wave method (Campbell & Jones 1968 S): fields ∝ exp[ik(x₁ + αx₃ − vt)]; for each trial v the extended 4×4 Christoffel system in (u₁, u₂, u₃, φ) gives an 8th-order polynomial in α; keep the 4 roots that decay into the substrate (or radiate away from the surface, for leaky waves); the SAW velocity is the root of the 4×4 boundary determinant (three traction components plus D₃ continuity with the vacuum, or φ = 0 for a shorted surface) derived.

Coupling as defined by Ingebrigtsen (Ingebrigtsen 1969 S). Leaky SAW (pseudo-SAW): a predominantly shear-horizontal surface wave faster than the slow shear bulk wave; it radiates into the bulk and its velocity is complex (Yamanouchi & Shibayama 1972 treat leaky surface waves in LN S).

Standard cuts, Kushibiki et al. 1999 constants

Computed with the partial-wave solver from the Kushibiki et al. 1999 constants P derived. Vendor column: Fomos-Materials knowledge base V; its coupling percentages (5.04, 5.7, 10, 17%) refer to an undefined quantity and are not compared. Losses of the leaky waves from the imaginary part of the complex root (first order).

The 36 to 41° free-surface leaky roots sit 0.7 to 0.8 m/s below the fast-shear bulk velocity along X (4750.1 m/s); they are isolated complex roots with very small radiation loss (41°: about 2×10⁻⁴ dB per wavelength free, 0.016 shorted), so the leaky wave on these cuts is almost a true surface wave until the surface is metallized.

Guided waves in thin films

Love-type and index-guided waves. A film whose shear velocity is lower than the substrate's guides acoustic waves without suspension; LN on sapphire uses this “phononic analogue of index guiding” (Mayor et al. 2021, abstract reported).

Plate modes in suspended films (Lamb and shear-horizontal): SH0, S0, A0 and thickness-shear overtones such as A1. A 400 nm ZY-LN platelet excited by lateral fields (XBAR) resonates near 4.8 GHz with coupling about 25% and Q about 300 (Plessky et al. 2019 reported). The bulk shear velocity along Z, 3590 m/s, over twice the thickness gives 3590/(2 × 400 nm) = 4.5 GHz (derived; ignores stiffening and electrode loading). Thin-film numbers are reported values plus this estimate; no plate-mode dispersion is computed.

Wavelength-scale mechanical waveguides (1 µm wide) on suspended TFLN and on LN on sapphire: 3.7.

Kushibiki et al. 1999 P; Ogi et al. 2002 c, e, ρ 4636 P with the Kushibiki et al. ε^S (blended sensitivity case; Ogi et al. report no ε^S); Warner et al. 1967 c (Weis & Gaylord 1985 Table 6 P-comp), e (Ogi et al. 2002 Table III S), ε^S 44.3/27.9 (Weis & Gaylord 1985 Table 1), ρ 4700 kg/m³ assumed (the value Weis & Gaylord use in their AO figures). All computed derived.

The older Warner set gives about 17% and 11% for 41 and 64°; the Kushibiki et al. 1999 and Ogi et al. 2002 sets give 15.6 to 15.7% and 10.3 to 10.4%. A k² quoted for a cut is meaningful only with its constant set named.

Root selection. Below the limiting velocity (1/max s₁ on the sagittal-plane slowness curves, which can lie below the slowest bulk velocity along x₁ for rotated cuts) all 8 roots are complex; keep Im α < 0 (substrate at x₃ < 0). For leaky waves keep the real roots whose time-averaged Poynting component points into the substrate, then continue the selection to complex velocity. Nondimensionalize (c by 10¹¹ Pa, potential by √(c₀/ε_ref)) or the problem is ill-conditioned and the Y-Z root is lost in numerical noise.

Spurious minima. The determinant has false minima at bulk thresholds (e.g. 3590.4 m/s on Y-Z, 4750.1 m/s on X-propagating cuts); a 1 m/s window around each bulk velocity along x₁ is excluded; a minimum that ends at a window edge is searched again inside a 0.05 m/s window, which resolves the 36 to 41° leaky roots 0.7 to 0.8 m/s below the fast-shear velocity.

k² definitions differ. 2Δv/v (Ingebrigtsen) vs electrode-admittance or effective-permittivity definitions; vendor sheets often do not state theirs.

Temperature coefficient of frequency. Vendor values −95 (Y-Z), −72 (128° Y-X), −70 (64 and 41° Y-X) ppm/K V. A computed TCF needs temperature coefficients of c, e, ε and thermal expansion as one consistent set; Smith and Welsh 1971 has c and s coefficients (Weis & Gaylord 1985 Tables 5-6 P-comp); the thermal expansion along c is 4.1 ppm/K in the Volk & Wöhlecke 2008 compilation (Table 1.1 P-comp). No computed TCF is shown.