Surface acoustic wave

Why does a surface acoustic wave (SAW) propagate only in the surface leyer of the solid, let's assume a semiconductor? One can create such a SAW through an interdigital transducer (IDT) as I read, but why does this IDT create a SAW and not an acoustic wave that propagates in the whole volume of the semiconductor? And how had one to modify his approach so that volume acoustic waves are created?

The Rayleigh waves in question are a kind of evanescent wave. At least this is what happens in a surface acoustic wave filter - a neat device that converts electrical signals into acoustic ones and then back again by the piezo-electric effect, allowing the designer the exploit the low acoustic velocity to realise complex filter responses from the wave interference effects of the surface acoustic waves.

This evanescence arises because the spatial frequency of the interdigital launching structure is chosen to be higher than the wavenumber of the acoustic wave at that frequency. This means that, if $x$ is in the direction along the surface, and $z$ the direction into the surface, then $k_x > k$, where $k_x$ is the $x$-component of the wavevector and therefore, since the components of the acoustic wave fulfil Helmholtz's equation $(\nabla^2 +k^2)\psi=0$, we have that $k^2 = k_z^2 + k_x^2\Rightarrow k_z^2 = k^2-k_x^2<0$. Therefore, $k_z$ is imaginary, and the Fourier components of the wave vary in the homogeneous acoustic material like:

$$\psi(x,\,z) = \psi_0\,e^{i\,k_x\,x} \,e^{-\sqrt{k_x^2-k^2}\,z}$$

and so the waves scoot along the surface, and dwindle exponentially with depth into the substrate.

The way of make the transducer excite waves throughout the volume of the substrate is to run it at frequencies high enough that $k_x$, set by the interdigital transducer period, is less than the wavenumber $k = 2\pi/\lambda = \omega/c_a$, where $\omega$ is the source's frequency and $/c_a$ the acoustic velocity for Rayleigh waves (which are combined shear and longitutinal waves).

Evanescence through excitation boundary conditions is also the mechanism whereby total internal reflexion of light (or of acoustic waves) happens: I discuss this more fully in my answer here.

• And there are other solutions that occur near surfaces, like Airy waves (if I'm remembering correctly). A good book on elasticity and acoustics with a chapter on earthquake waves will provide more solutions than you can shake a stick at! Oct 20, 2014 at 13:27
• @JonCuster Very true, but the question was about SAW devices with (presumably) subwavelength interdigital transducers. Oct 21, 2014 at 1:23
• I understand - I always found the plethora of solutions to the equations for elastic waves to be a reminder that even supposedly simple equations can have a number of interesting answers. Oct 21, 2014 at 1:28
• @JonCuster I agree. Even the Helmholtz equation's uniqueness and existence conditions are very complicated. No-one has proven existence or uniqueness of solutions of the general Navier Stokes equation and we really have little intuition for the potential solutions of thus simple equation, which very clearly states conservation of momentum. If you can make significant progress on this one, you'd be in the running for a Clay Mathematics prize. Oct 21, 2014 at 2:09

Bulk acoustic waves - like the one in the quartz piezo-electric in watches or on computer chips are typically launched perpendicular to the surface by a bulk transducer.

More generally, however, In a material with one or more parallel, flat surfaces, the modes of the system can all be classified by their frequencies $\omega$ and in plane two-component wavevector $\vec k_{pl}$. Surface Acoustic Waves are launched (often) by making a device that chooses appropriate values of $\omega$ and $\vec k_{pl}$ for the Surface Wave. The Interdigitated Transducer has a period in space that is $2\pi /\mid k_{pl}\mid$ e.g. The Surfaces waves for any $\vec k _{pl}$ typically have very different (lower) frequencies than any bulk wave. As bulk waves are usually characterized by an additional component of the wavevector $k_\perp$ there will be a range of frequencies for bulk waves with any given $\vec k _{pl}$, however all such frequencies are typically substantially higher that for the surface wave. Thus bulk waves can probably generically be produced simply by driving a SAW producing interdigitated transducer at a higher frequency. However, I would want to know, prior to suggesting this, why you want a bulk wave which is not traveling perpendicular to the surface. I doubt such waves move terribly deep into the system. They may have useful applications but I don't think of one immediately.