**2. Fabrication**

The focus for generating SAWs in this chapter will involve the fabrication of interdigitated transducers (IDT). An IDT is simple in concept but can be very involved when fine tuning a structure for engineering applications. Such topics as electronic impedance matching to RF lines, effects of bulk waves in contrast to SAWs, and increasing bandwidth will not be covered; though, this is simply a shortened list of things to consider when designing a proper IDT for engineering applications, they do fall outside the scope of this chapter. Nevertheless, another fabrication step we will consider is the use of acoustic waveguides for acoustic impedance matching of IDTs to nanomechanical devices.

### **2.1 Interdigitated Transducer Design**

The main equation to consider when designing an IDT is:

Surface Acoustic Waves and Nano–Electromechanical Systems 639

Another detail of fabrication is the use of waveguides for the SAW. A waveguide for SAWs is a pattern that is fabricated onto the device so the SAWs can be guided into a certain region allowing a stronger SAW signal or amplitude; it is analogous to a coplanar waveguide for RF signals. When fabricating IDTs, and a device as a whole, using waveguides does not have much of a use in the areas of RFID, cellular delay lines, sensors, and other non-region specific devices. However, this chapter will focus on NEMS and the use of SAWs to interact with these devices. Since the NEMS device is orders of magnitude smaller than the IDT aperture very little SAW power can interact with the NEMS structure; it can be beneficial to include a waveguide to focus the SAW onto the

Waveguides can be modeled using standard acoustic, or horn, waveguides with some minor tuning. There are several shapes which can be used and each shape offers its own benefits regarding a particular application. We will start by first mentioning the base equation for hyperbolic horns used in speaker systems, these horns can also be referred to as hyperbolic-


*cc c*

= +

Eq. 2 gives the wave front area expansion of the wave as propagating through the waveguide or horn. *At* is the area of the throat, or base, of the waveguide, *T* is a factor describing the shape of waveguide; for T = 1 the waveguide becomes exponential in shape, and becomes canonical as T → ∞, *k* is given as *(2πfc)/v*; where *v* is the velocity of sound, and *fc* is the cutoff frequency of the waveguide. For hyperbolic and exponential waveguides there is no transmission below the cutoff frequency, *fc*, since at this point the impedance is purely reactive. Eq. 3 gives the impedance of the waveguide; all variables are defined

Fig. 2 contains three SEM images of an exponential waveguide with a tube waveguide in the center. A tube waveguide is simply a small or narrow SAW delay line. The material is a GaAs/AlGaAs heterostructure containing a two-dimensional electron gas (2DEG) and a sacrificial layer. We use a 2DEG because the high electron mobility makes this an ideal candidate for nanostructures, such as quantum dot (QD) and DQD systems. The waveguide was defined using electron-beam lithography to open up an etch region through the PMMA. The material was then wet etched with H3PO4:H2O2:H20 into the sacrificial layer. Then diluted HCl was used to remove the sacrificial layer which resulted in a suspended region in the center and around the waveguide. This suspended region is visible as a darkened shadow indicating a height difference due to strain relaxation of the material. The suspended region forms the actual nanomechanical device. Since this device contains a DQD in the center region of the structure it is beneficial to use a waveguide to ensure a higher SAW power density coupled into that region. Since the IDT aperture of this device is

*f fT f T z i A ff f*

(1 ) 1 1

2 1 2 2

<sup>2</sup> A A [Cosh(kx) T Sinh(kx)] = +⋅ t (2)

2


(3)

**2.2 Acoustic Waveguide Design** 

exponential horns or hypex horns (Salmon, 1946).

0

ρ

*t*

previously with the addition of *ρ0* which is the density of the material.

50 μm and the DQD region is roughly 600 nm at the widest QPCs.

*v*

nanostructure.

$$\mathbf{v} = \lambda \mathbf{f} \tag{1}$$

Where *v* is the velocity of sound in the material, *λ* is the SAW wavelength or pitch of the IDT, and *f* is the frequency of the propagating SAW. The pitch of the IDT fingers is the same as the SAW wavelength, *λ*, which will propagate across the sample. Fig. 1 shows a simple IDT schematic along with a scanning electron micrograph (SEM) image and optical image.

Fig. 1. Schematics and scanning electron microscope images of IDTs. (a) Schematic of bidirectional IDT, (b) schematic of unidirectional IDT through the use of a reflection grating, (c) SEM image of a bidirectional IDT on GaAs with a pitch of 4 μm and a center frequency of ~715 MHz. The scale bar is 10 μm. (d) Optical image of device with an IDT on both ends and a nanomechanical device placed in the center

In Fig. 1 the arrows indicate the SAW propagation directions. It can be easily seen that the SAW will propagate in both directions away from the IDT. A singly propagating SAW direction can be achieved by placing a reflection grating on one end of the SAW. The reflection grating will have the same geometry as the IDT; that is, the same finger spacing and width. The grating distance from the IDT is ~ *λ*/2. The IDT fingers are typically chosen with evenly spaced fingers, where the spacing between the fingers is equal to the finger width. The finger width and finger spacing is *λ*/4 in this scenario and gives a metallization ratio of η = 0.5, which will generate only odd harmonics with no response of the third harmonic (Campbell, 1998). The bandwidth of an IDT is defined as *BW% = f / NPairs*. Where *f*, again, is the center frequency and *NPairs* is the number of finger pairs of the IDT. The IDT in Fig. 1a has five pairs of fingers and is symmetric about a center point along the axis of SAW propagation. It is typical to fabricate two IDTs, one on each end of the device, so that their propagating waves can interfere either constructively or destructively across the center region of the device; this is typically the region where a Quantum Point Contact (QPC) or other structures resides. This can be seen in Fig. 1d where in the center a double quantum dot (DQD) is placed and the IDTs are placed to the left and right.

### **2.2 Acoustic Waveguide Design**

638 Acoustic Waves – From Microdevices to Helioseismology

 vf = λ (1) Where *v* is the velocity of sound in the material, *λ* is the SAW wavelength or pitch of the IDT, and *f* is the frequency of the propagating SAW. The pitch of the IDT fingers is the same as the SAW wavelength, *λ*, which will propagate across the sample. Fig. 1 shows a simple IDT schematic along with a scanning electron micrograph (SEM) image and optical image.

Fig. 1. Schematics and scanning electron microscope images of IDTs. (a) Schematic of bidirectional IDT, (b) schematic of unidirectional IDT through the use of a reflection grating, (c) SEM image of a bidirectional IDT on GaAs with a pitch of 4 μm and a center frequency of ~715 MHz. The scale bar is 10 μm. (d) Optical image of device with an IDT on both ends and

In Fig. 1 the arrows indicate the SAW propagation directions. It can be easily seen that the SAW will propagate in both directions away from the IDT. A singly propagating SAW direction can be achieved by placing a reflection grating on one end of the SAW. The reflection grating will have the same geometry as the IDT; that is, the same finger spacing and width. The grating distance from the IDT is ~ *λ*/2. The IDT fingers are typically chosen with evenly spaced fingers, where the spacing between the fingers is equal to the finger width. The finger width and finger spacing is *λ*/4 in this scenario and gives a metallization ratio of η = 0.5, which will generate only odd harmonics with no response of the third harmonic (Campbell, 1998). The bandwidth of an IDT is defined as *BW% = f / NPairs*. Where *f*, again, is the center frequency and *NPairs* is the number of finger pairs of the IDT. The IDT in Fig. 1a has five pairs of fingers and is symmetric about a center point along the axis of SAW propagation. It is typical to fabricate two IDTs, one on each end of the device, so that their propagating waves can interfere either constructively or destructively across the center region of the device; this is typically the region where a Quantum Point Contact (QPC) or other structures resides. This can be seen in Fig. 1d where in the center a double quantum

(a) (b)

(c) (d)

a nanomechanical device placed in the center

dot (DQD) is placed and the IDTs are placed to the left and right.

Another detail of fabrication is the use of waveguides for the SAW. A waveguide for SAWs is a pattern that is fabricated onto the device so the SAWs can be guided into a certain region allowing a stronger SAW signal or amplitude; it is analogous to a coplanar waveguide for RF signals. When fabricating IDTs, and a device as a whole, using waveguides does not have much of a use in the areas of RFID, cellular delay lines, sensors, and other non-region specific devices. However, this chapter will focus on NEMS and the use of SAWs to interact with these devices. Since the NEMS device is orders of magnitude smaller than the IDT aperture very little SAW power can interact with the NEMS structure; it can be beneficial to include a waveguide to focus the SAW onto the nanostructure.

Waveguides can be modeled using standard acoustic, or horn, waveguides with some minor tuning. There are several shapes which can be used and each shape offers its own benefits regarding a particular application. We will start by first mentioning the base equation for hyperbolic horns used in speaker systems, these horns can also be referred to as hyperbolicexponential horns or hypex horns (Salmon, 1946).

$$\mathbf{A} = \mathbf{A}\_t [\mathbf{Cosh(kx)} + \mathbf{T} \cdot \mathbf{Sink(kx)}]^2 \tag{2}$$

$$z = \frac{\rho\_0^{\circ}}{A\_t} \left( \sqrt{1 \cdot \left( \frac{f\_c}{f} \right)^2 + i \frac{f\_c T}{f}} \left| \left( 1 \cdot \frac{f\_c^{\circ} \left( 1 \cdot T^2 \right)}{f^2} \right) \right|^{\circ} \tag{3}$$

Eq. 2 gives the wave front area expansion of the wave as propagating through the waveguide or horn. *At* is the area of the throat, or base, of the waveguide, *T* is a factor describing the shape of waveguide; for T = 1 the waveguide becomes exponential in shape, and becomes canonical as T → ∞, *k* is given as *(2πfc)/v*; where *v* is the velocity of sound, and *fc* is the cutoff frequency of the waveguide. For hyperbolic and exponential waveguides there is no transmission below the cutoff frequency, *fc*, since at this point the impedance is purely reactive. Eq. 3 gives the impedance of the waveguide; all variables are defined previously with the addition of *ρ0* which is the density of the material.

Fig. 2 contains three SEM images of an exponential waveguide with a tube waveguide in the center. A tube waveguide is simply a small or narrow SAW delay line. The material is a GaAs/AlGaAs heterostructure containing a two-dimensional electron gas (2DEG) and a sacrificial layer. We use a 2DEG because the high electron mobility makes this an ideal candidate for nanostructures, such as quantum dot (QD) and DQD systems. The waveguide was defined using electron-beam lithography to open up an etch region through the PMMA. The material was then wet etched with H3PO4:H2O2:H20 into the sacrificial layer. Then diluted HCl was used to remove the sacrificial layer which resulted in a suspended region in the center and around the waveguide. This suspended region is visible as a darkened shadow indicating a height difference due to strain relaxation of the material. The suspended region forms the actual nanomechanical device. Since this device contains a DQD in the center region of the structure it is beneficial to use a waveguide to ensure a higher SAW power density coupled into that region. Since the IDT aperture of this device is 50 μm and the DQD region is roughly 600 nm at the widest QPCs.

Surface Acoustic Waves and Nano–Electromechanical Systems 641

In both Eqs. 4 and 5, *ω* is the angular frequency and *k* is the phase constant. The SAW penetrates into the depth of the material by about one wavelength, ~*λ*. This value is different in the suspended region. Hence, the actuation of nanomechanical resonators is considerably enhanced. Another property of the SAW is that the electric field created from the induced electric potential does not terminate at the surface of the material but can extend beyond by *λ*.

(a) (b) (c) (d)

Fig. 3. (a) Side view of IDT with an applied voltage, the electric field couples into the piezoelectric substrate causing a deformation. (b) The elliptical motion of the Rayleighwave. (c) Axis used for reference in equations as it applies to the orientation of the piezoelectric substrate. (d) Side view of piezoelectric material as it is deformed causing a

i

E

kl

i

k l

1u u <sup>S</sup> 2x x ∂ ∂ = +

Eq. 6 is the electric field intensity that is produced from the deformed piezoelectric material from the SAW. Eq. 7 is the piezoelectric mechanical stress and Eq. 8 is the linear strain

eff s M

21( / )

2 s M

> 2 s M

<sup>Δ</sup> <sup>=</sup> +σ σ (10)

SAW attenuation can be described by the following equations (Wixworth et al., 1989):

K / <sup>k</sup>

σ σ Γ =

2 eff

vK 1 <sup>k</sup> v 21( / )

Here the attenuation occurs because part of the longitudinal electric field of the propagating wave couples into the electrons of the 2DEG. This not only causes a current to flow but pulls power from the SAW due to ohmic losses. This attenuation is described by Eq. 9. A SAW velocity shift is also observed due to the piezoelectric stiffening of the substrate, see Eq. 10 (Wixworth et al., 1989). Below are the recreated graphs from (Wixworth et al., 1989) to show the relationship of the attenuation and sound velocity shift due to a change in conductivity.

2

l k

∂Φ = − ∂ (6)

∂ ∂ (8)

+σ σ (9)

T c S eE ij = + ijkl kl nij n (7)

x

The set of base equations used for describing SAW phenomena are listed.

SAW to propagate

displacement.

**3.1.1 Attenuation** 

Fig. 2. SEM images of an exponential waveguide with a cutoff frequency of 75 MHz and throat width of ~1.5 μm. The darker region around the waveguide shows that the area is suspended, which was achieved by an HCl etch to remove the sacrificial layer below the 2DEG, (a), (b) and (c). (d) Schematic of waveguide, center region is the throat (pipe waveguide) with width *Wt* and length *Lt*. The outer curves are the exponential portion of the waveguide
