**3. Surface Acoustic Waves in Two Dimensional Electron Gases**

### **3.1 Surface Acoustic Wave Basics**

When applying an RF signal to the IDTs an alternating electric potential is seen by the electrodes, this will create an electric field distribution within the piezoelectric substrate, see Fig. 3a. This field distribution, in turn, will cause a mechanical deformation of the material through the inverse piezoelectric effect. As stated before, this mechanical deformation will propagate away from the IDTs and continue to move along the surface of the device, see Fig. 3d. Here we neglect other forms of waves being produced such as bulk waves.

Once the RF signal of the IDT is coupled into the piezoelectric substrate a Rayleigh-wave is produced. The elliptical wave propagating (Fig. 3b, d) along the surface can be described quite accurately by a traveling wave:

$$\mathbf{U} = \mathbf{U} \mid \mathbf{e}^{\left(\mathrm{at} \cdot \mathrm{kx}\right)} \mathbf{e}^{\cdot \mathrm{k} \left| \mathbf{y} \right|} \tag{4}$$

The electric potential being created by the propagating wave can also be described in a like manner:

$$\Phi = |\Phi\rangle\langle\mathbf{e}^{(\text{out-kx})}\mathbf{e}^{\text{-k|y|}}\tag{5}$$

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*.

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 SAW to propagate

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

$$\mathbf{E}\_{\text{i}} = -\frac{\partial \Phi}{\partial \mathbf{x}\_{\text{i}}} \tag{6}$$

$$\mathbf{T}\_{\mathsf{i}\rangle} = \mathbf{c}\_{\mathsf{i}\#\mathsf{i}} \mathbf{S}\_{\mathsf{k}\mathsf{l}} + \mathbf{e}\_{\mathsf{n}\mathsf{i}} \mathbf{E}\_{\mathsf{n}} \tag{7}$$

$$\mathbf{S}\_{\rm kl} = \frac{1}{2} \frac{\partial \mathbf{u}\_{\rm k}}{\partial \mathbf{x}\_{\rm l}} + \frac{\partial \mathbf{u}\_{\rm l}}{\partial \mathbf{x}\_{\rm k}} \tag{8}$$

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 displacement.

### **3.1.1 Attenuation**

640 Acoustic Waves – From Microdevices to Helioseismology

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

When applying an RF signal to the IDTs an alternating electric potential is seen by the electrodes, this will create an electric field distribution within the piezoelectric substrate, see Fig. 3a. This field distribution, in turn, will cause a mechanical deformation of the material through the inverse piezoelectric effect. As stated before, this mechanical deformation will propagate away from the IDTs and continue to move along the surface of the device, see Fig.

Once the RF signal of the IDT is coupled into the piezoelectric substrate a Rayleigh-wave is produced. The elliptical wave propagating (Fig. 3b, d) along the surface can be described

The electric potential being created by the propagating wave can also be described in a like

i( t kz) - -k|y| U |U|e e <sup>ω</sup> = (4)

i( t kz) - -k|y| | |e e <sup>ω</sup> Φ=Φ (5)

**3. Surface Acoustic Waves in Two Dimensional Electron Gases** 

3d. Here we neglect other forms of waves being produced such as bulk waves.

(a) (b)

(c) (d)

waveguide

manner:

**3.1 Surface Acoustic Wave Basics** 

quite accurately by a traveling wave:

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

$$\Gamma = \mathbf{k} \frac{\mathbf{K}^2\_{\rm eff}}{2} \frac{\mathbf{\sigma}\_s \;/\, \sigma\_\mathbf{M}}{1 + \left(\mathbf{\sigma}\_s \;/\, \sigma\_\mathbf{M}\right)^2} \tag{9}$$

$$\frac{\Delta \mathbf{v}}{\mathbf{v}} = \mathbf{k} \frac{\mathbf{K}\_{\text{eff}}^2}{2} \frac{1}{1 + \left(\sigma\_s / \sigma\_\text{M}\right)^2} \tag{10}$$

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.

Surface Acoustic Waves and Nano–Electromechanical Systems 643

is shifted between the two IDTs as the current is measured. At a phase difference of 180° the

(d)

Fig. 5. Results for a suspended beam that is 1.2 μm long and 300 nm wide. (a) Anomalous acoustoelectric current as a result of phase adjusting two counter propagating SAWs. (b) Acoustoelectric current vs. applied frequency of the IDTs. There is always a negative current value to the left and is indicative of the anomalous current. (c) Formation of a shock wave seen by its Sinc(x) shape. The device's response jumps to a higher order mode and returns to Sin(x). (d) Derived anomalous acoustoelectric current amplitude as a result of the calculated

SAW amplitude. All images are taken from Beil et al., 2008, Copyright (2008) by The

an zzzy zy GaAs <sup>e</sup> <sup>j</sup> (L /2) ( S (L /2) ) <sup>2</sup> <sup>σ</sup> =− Π ε

zy t 3 2 SAW 6z 6z S (z,t) 2A ( ) Cos( /2) Cos(k L / 2) L L

= ϕ+ ϕ

This zero current occurs because the two SAWs interfere destructively and create a nearly smooth surface with no deformation. At 0°, and also 360°, the current is at a maximum since the two SAWs interfere constructively allowing a maximum in the surface deformation. The anomalous acoustoelectric current can be used to probe the SAW amplitude of a suspended beam or nanostructure. The accompanying graphs show the acoustoelectric current from the device. It has been shown that the anomalous acoustoelectric current through the device in relation to the SAW amplitude can be described by Eqs. 12-14; where Eq. 12 is the general equation for an anomalous acoustoelectric current, 13 is the derived

0 z4 2


(13)

(14)

current tends towards zero.

(c)

American Physical Society

equation, and 14 is the strain equation of the beam.

(a) (b)

Fig. 4. (a) The SAW attenuation in units of *Keff2*, see Eq. 9. (b) The change in SAW velocity in units of *Keff2*, see Eq. 10

Where *k* is the SAW wave vector, *K2 eff* is the effective piezoelectric coupling coefficient, *σs* is the 2DEG sheet conductivity, and *σM = v0(ε1 + ε2)*. Again, where *v0* is the sound velocity and *ε1* and *ε2* are the dielectric constants of the piezoelectric substrate and half space above it.

### **3.2 Acoustoelectric Current**

As the SAW propagates across the material it creates two types of currents, one is the normal acoustoelectric current and the other being the anomalous acoustoelectric current. The normal acoustoelectric current is created by electrons being "dragged" across the material and can be described by Eq. 11 where *I* is the current, *n* is the number of electrons, *e* is the charge of an electron, and *f* is the frequency of the SAW or RF signal to the IDT. This current always flows in the direction of the SAW and is produced as a DC current despite an oscillating RF signal being applied to the IDTs. Eq. 11 shows that at higher frequencies the normal acoustoelectric current becomes quantized.

$$\mathbf{I} = \mathbf{r} \mathbf{e} \mathbf{f} \tag{11}$$

$$\mathbf{j(z)} = \frac{\mathbf{a}\mathbf{0}}{2\pi} \int\_0^{2\pi/\alpha} \sigma\_{z\mathbf{z}}(\mathbf{z}, \mathbf{t}) \mathbf{E}\_x(\mathbf{z}, \mathbf{t}) d\mathbf{t} \tag{12}$$

Once this deformation occurs the energy bands in the material bend as well causing the electrons to fall into the created quantum wells and are dragged along with the SAW, see Fig. 3d. As the frequency increases the wavelength, and pitch of the IDT, decreases causing fewer energy states to be available within the wells. This idea is also implemented to generate QDs within the SAW, (Barnes et al., 2000).

The anomalous acoustoelectric current is produced from the deformation of the material and flows as a DC current. The difference is that the anomalous current always flows in one direction regardless of which IDT, left or right, produced the SAW. The current is smaller than the normal acoustoelectric current and is detected by different methods. The anomalous acoustoelectric current can be obtained by sweeping the RF signal of the IDTs and typically appears at an off-center frequency. Since the normal acoustoelectric current is much smaller, the anomalous current can be more easily detected, as shown in Fig. 5b. One such method for direct detection of the anomalous acoustoelectric current is to apply an RF signal to both the left and right IDTs, while phase locking the two signals (Beil et al., 2008). By phase locking the IDT signals a standing SAW can be created. Thus, a surface deformation is produced without a net direction of propagation. As seen in Fig. 5a the phase

Fig. 4. (a) The SAW attenuation in units of *Keff2*, see Eq. 9. (b) The change in SAW velocity in

the 2DEG sheet conductivity, and *σM = v0(ε1 + ε2)*. Again, where *v0* is the sound velocity and *ε1* and *ε2* are the dielectric constants of the piezoelectric substrate and half space above it.

As the SAW propagates across the material it creates two types of currents, one is the normal acoustoelectric current and the other being the anomalous acoustoelectric current. The normal acoustoelectric current is created by electrons being "dragged" across the material and can be described by Eq. 11 where *I* is the current, *n* is the number of electrons, *e* is the charge of an electron, and *f* is the frequency of the SAW or RF signal to the IDT. This current always flows in the direction of the SAW and is produced as a DC current despite an oscillating RF signal being applied to the IDTs. Eq. 11 shows that at higher frequencies the

I nef = (11)

<sup>a</sup> j(z) (z,t)E (z,t)dt <sup>2</sup>

Once this deformation occurs the energy bands in the material bend as well causing the electrons to fall into the created quantum wells and are dragged along with the SAW, see Fig. 3d. As the frequency increases the wavelength, and pitch of the IDT, decreases causing fewer energy states to be available within the wells. This idea is also implemented to

The anomalous acoustoelectric current is produced from the deformation of the material and flows as a DC current. The difference is that the anomalous current always flows in one direction regardless of which IDT, left or right, produced the SAW. The current is smaller than the normal acoustoelectric current and is detected by different methods. The anomalous acoustoelectric current can be obtained by sweeping the RF signal of the IDTs and typically appears at an off-center frequency. Since the normal acoustoelectric current is much smaller, the anomalous current can be more easily detected, as shown in Fig. 5b. One such method for direct detection of the anomalous acoustoelectric current is to apply an RF signal to both the left and right IDTs, while phase locking the two signals (Beil et al., 2008). By phase locking the IDT signals a standing SAW can be created. Thus, a surface deformation is produced without a net direction of propagation. As seen in Fig. 5a the phase

zz z

2 /

π ω ω = σ

0

*eff* is the effective piezoelectric coupling coefficient, *σs* is

<sup>π</sup> (12)

(a) (b)

units of *Keff2*, see Eq. 10

Where *k* is the SAW wave vector, *K2*

normal acoustoelectric current becomes quantized.

generate QDs within the SAW, (Barnes et al., 2000).

**3.2 Acoustoelectric Current** 

is shifted between the two IDTs as the current is measured. At a phase difference of 180° the current tends towards zero.

Fig. 5. Results for a suspended beam that is 1.2 μm long and 300 nm wide. (a) Anomalous acoustoelectric current as a result of phase adjusting two counter propagating SAWs. (b) Acoustoelectric current vs. applied frequency of the IDTs. There is always a negative current value to the left and is indicative of the anomalous current. (c) Formation of a shock wave seen by its Sinc(x) shape. The device's response jumps to a higher order mode and returns to Sin(x). (d) Derived anomalous acoustoelectric current amplitude as a result of the calculated SAW amplitude. All images are taken from Beil et al., 2008, Copyright (2008) by The American Physical Society

This zero current occurs because the two SAWs interfere destructively and create a nearly smooth surface with no deformation. At 0°, and also 360°, the current is at a maximum since the two SAWs interfere constructively allowing a maximum in the surface deformation. The anomalous acoustoelectric current can be used to probe the SAW amplitude of a suspended beam or nanostructure. The accompanying graphs show the acoustoelectric current from the device. It has been shown that the anomalous acoustoelectric current through the device in relation to the SAW amplitude can be described by Eqs. 12-14; where Eq. 12 is the general equation for an anomalous acoustoelectric current, 13 is the derived equation, and 14 is the strain equation of the beam.

$$\mathbf{j}\_{\rm an}(\mathbf{L} \text{ / 2}) = -\frac{\mathbf{G}\_0 \mathbf{e}\_{z4}}{2\varepsilon\_{\rm GaAs}} (\Pi\_{z\rm xxy} \mathbf{S}\_{xy} (\mathbf{L} \text{ / 2})^2) \tag{13}$$

$$\mathbf{S}\_{xy}(\mathbf{z}, \mathbf{t}) = 2\mathbf{A}\_{\mathbf{t}} \left( \frac{6\mathbf{z}^2}{\mathbf{L}^3} \cdot \frac{6\mathbf{z}}{\mathbf{L}^2} \right) \left( \text{Cos}(\boldsymbol{\upphi} \,/\, \mathbf{2}) \cdot \text{Cos}(\mathbf{k}\_{\text{SAW}}\,\mathbf{L} + \boldsymbol{\upphi} \,/\, \mathbf{2}) \right) \tag{14}$$

Surface Acoustic Waves and Nano–Electromechanical Systems 645

Special attention has to be given to the acoustic current trace of Fig. 6 around 3 – 4 Tesla. There is a splitting of the peaks that occurs. This information does not appear in the SdH oscillation and is only isolated to a SAW effect. During this splitting the conductivity of the 2DEG is very low and σs becomes much smaller than σM, see Eq. 9. When this happens a maximum in SAW attenuation will occur which results in a reduced SAW amplitude. The center of a SAW current split is a minimum in the 2DEG conductivity; further details are

From Fig. 6 there are two observations that need to be explained. One is the spike-like feature or discontinuity around the 8.5 T mark of the acoustic trace. This is caused from SAW reflections on the sample. This spike-like feature can be seen on other data plots as well with SAW currents. The final feature is the negative acoustic current that was measured. Since at this point the SAW was highly attenuated, causing little or no current to flow, caused the sample itself to heat. The negative current may be a combination of a

SAWs are used to produce a current in low dimensional electronic systems and NEMS. This can be used in various applications and for numerous different designs. The information presented in this section will deal with piezoelectric materials with an embedded two dimensional electron gas (2DEG); mainly GaAs/AlGaAs heterostructures unless otherwise stated. These measurements are carried out at liquid helium temperatures or

SAWs can be used to create a quantized current based upon Eq. 7. The advantage of the SAW inducing a quantized current is that this process can be used to populate and depopulate QDs and DQDs at higher frequencies then what can be used by applying an oscillating signal to the source-drain of the 2DEG. When using a SAW as the current source the acoustoelectric current can be pinched off in a Coulomb blockade just like a standard source-drain current can be. This pinch off is done via a quantum point contact (QPC) or a

Fabricating QPCs is done with the same methods as fabricating IDTs, see Sec. 2. Since QPCs are have small dimensions it is most common to use Electron Beam Lithography, or e-beam lithography, to create the structures. The exact dimensions of the QPC pair depends on what works best for the user and there is no set rules for design like there is for IDTs. When viewing Fig. 7 it can be seen that there are five sets of QPCs. A single QPC is seen as an electrode with another electrode opposite its position. All of the QPCs shown have a few common features; the tip of the QPC is small when compared to the rest of the electrode and the gap between the electrode tips is small as well. The tip is small so the electric field being emitted from the QPC is very localized, and the majority of the electrode is made wider so that it covers a wider portion of the 2DEG so the electrons are repelled. The gap between electrodes is small so pinch-off can be achieved with small voltages, more on this in Sec.

thermal current and a small offset in the measuring equipment.

**4. Surface Acoustic Waves in Quantum Electronics** 

**4.1 Quantum Point Contacts and Low Dimensional Channels** 

set of QPCs, which can be used to form QDs; see Fig. 7.

**4.1.1 Quantum Point Contact Fabrication** 

discussed in Wixworth et al., 1989.

lower, ≤ 4.2 K.

4.1.2.

Where *σ0* is the unperturbed conductivity, *εGaAs* is the permittivity of GaAs, *Π* is a tensor which describes the effect of strain *S*, and *At* is the transverse component of the SAW. An interesting phenomenon that occurs is the formation of shock waves in the suspended

structure. From Figs. 5a and 5c it is seen that the device exhibits a Sin(x) type behavior in the anomalous acoustoelectric current at low RF powers. Once the RF power starts to increase it can be seen that the shape of the current starts to exhibit more of a Sinc(x) shape. This transition from a linear to a non-linear response indicates the formation of a shock wave in the suspended beam. As the power increases, the beam jumps into the next higher order mode and the current trace returns back to a linear response again. The shock wave formation is an indication that the beam will transition from one mode to another.

### **3.3 Surface Acoustic Waves in Magnetic Fields**

We can use SAWs to probe the characteristics of a 2DEG under the presence of a magnetic field. Since a high mobility 2DEG is subject to Shubnikov-de Haas (SdH) oscillations, which creates changes in the conductivity, and SAWs are more sensitive to a conducting plane or surface, and the conductance of that plane, makes this combination an ideal candidate to investigate quantum effects of the 2DEG. The real interesting features are seen when integral Landau level filling factors are observed which causes a drop in conductivity of the 2DEG. At these quantized values the SAW responds strongly to the conductivity, σ.

In Fig. 6 it can be seen that the SdH oscillations and the acoustoelectric current oscillations are nearly identical. The SdH measurement was measured using a standard four-point lockin technique, where as the acoustoelectric current was created by a SAW and was taken from two ohmic contacts on opposite sides of the sample. The peaks and valleys of the two measurements line up, for the most part, but there is a small offset. This offset is due to the fact that the SAW attenuation is not a linear with respect to the 2DEG conductivity, or magneto conductivity in this case, see Eq. 9.

Fig. 6. A magnetic field, of maximum value 9 T, was applied perpendicular to the surface of the device. The device is shown in figures 1d and 2a-2d and contains a 2DEG 40 nm below the surface. Upper trace is the Shubnikov-de Haas oscillations measured using standard lock-in techniques and normalized to 713.4 Ω. The lower trace is the acoustoelectric current generated from a SAW with -12 dBm applied to the right IDT at 1.488 GHz and is normalized to 10.47 nA

Where *σ0* is the unperturbed conductivity, *εGaAs* is the permittivity of GaAs, *Π* is a tensor which describes the effect of strain *S*, and *At* is the transverse component of the SAW. An interesting phenomenon that occurs is the formation of shock waves in the suspended structure. From Figs. 5a and 5c it is seen that the device exhibits a Sin(x) type behavior in the anomalous acoustoelectric current at low RF powers. Once the RF power starts to increase it can be seen that the shape of the current starts to exhibit more of a Sinc(x) shape. This transition from a linear to a non-linear response indicates the formation of a shock wave in the suspended beam. As the power increases, the beam jumps into the next higher order mode and the current trace returns back to a linear response again. The shock wave

We can use SAWs to probe the characteristics of a 2DEG under the presence of a magnetic field. Since a high mobility 2DEG is subject to Shubnikov-de Haas (SdH) oscillations, which creates changes in the conductivity, and SAWs are more sensitive to a conducting plane or surface, and the conductance of that plane, makes this combination an ideal candidate to investigate quantum effects of the 2DEG. The real interesting features are seen when integral Landau level filling factors are observed which causes a drop in conductivity of the

In Fig. 6 it can be seen that the SdH oscillations and the acoustoelectric current oscillations are nearly identical. The SdH measurement was measured using a standard four-point lockin technique, where as the acoustoelectric current was created by a SAW and was taken from two ohmic contacts on opposite sides of the sample. The peaks and valleys of the two measurements line up, for the most part, but there is a small offset. This offset is due to the fact that the SAW attenuation is not a linear with respect to the 2DEG conductivity, or

Fig. 6. A magnetic field, of maximum value 9 T, was applied perpendicular to the surface of the device. The device is shown in figures 1d and 2a-2d and contains a 2DEG 40 nm below the surface. Upper trace is the Shubnikov-de Haas oscillations measured using standard lock-in techniques and normalized to 713.4 Ω. The lower trace is the acoustoelectric current generated from a SAW with -12 dBm applied to the right IDT at 1.488 GHz and is normalized to 10.47 nA

formation is an indication that the beam will transition from one mode to another.

2DEG. At these quantized values the SAW responds strongly to the conductivity, σ.

**3.3 Surface Acoustic Waves in Magnetic Fields** 

magneto conductivity in this case, see Eq. 9.

Special attention has to be given to the acoustic current trace of Fig. 6 around 3 – 4 Tesla. There is a splitting of the peaks that occurs. This information does not appear in the SdH oscillation and is only isolated to a SAW effect. During this splitting the conductivity of the 2DEG is very low and σs becomes much smaller than σM, see Eq. 9. When this happens a maximum in SAW attenuation will occur which results in a reduced SAW amplitude. The center of a SAW current split is a minimum in the 2DEG conductivity; further details are discussed in Wixworth et al., 1989.

From Fig. 6 there are two observations that need to be explained. One is the spike-like feature or discontinuity around the 8.5 T mark of the acoustic trace. This is caused from SAW reflections on the sample. This spike-like feature can be seen on other data plots as well with SAW currents. The final feature is the negative acoustic current that was measured. Since at this point the SAW was highly attenuated, causing little or no current to flow, caused the sample itself to heat. The negative current may be a combination of a thermal current and a small offset in the measuring equipment.
