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

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 lower, ≤ 4.2 K.

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

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

### **4.1.1 Quantum Point Contact Fabrication**

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. 4.1.2.

Surface Acoustic Waves and Nano–Electromechanical Systems 647

feature can be seen which corresponds to a single conductance step which has a value of G = 2e2/h, where h is Plank's constant and e is the charge of an electron. Now the temperature must be low so the thermal energy, E = kbT, of the background is smaller than the tunneling energy needed for the electrons to "jump" across the barrier, where kb is the Boltzmann constant. As seen in Figs 9 and 10 this step like feature can be seen by doing an I-V measurement. By changing the temperature of the system the phonon energy is increased and causes scattering events to increase, or increase the electron-phonon interaction, and the

The use of QPCs offers a benefit of determining which SAW mode(s) are propagating in the sample. Different SAW modes, such as bulk, longitudinal, and transverse with propagate at different frequencies due to the fact that they have different sound velocities, see Eq. 1. Another factor, which affects the sound velocity, is the propagation direction of the SAW with respect to the crystal orientation of the material. In Fig. 8 a QPC had an applied voltage of -0.8 V, which puts the QPC into pinch-off mode. Since it is in pinch-off higher RF power is required to create a sufficiently strong SAW that will overcome the potential barrier. As the power is increased from -18 dBm to -10 dBm, three peaks emerge as transferring current through the tunnel barrier. From Eq. 7 we can calculate the electron count to be 6, 3, and 2 for RF powers of -10 dBm, -12 dBm, and -14 dBm, respectively (some rounding is taken into

Fig. 8. A frequency sweep of varying RF powers while the center QPC of the sample in Fig. 4 is held at -0.8 V. The first peak is at 840 MHz with a current of 540 pA and a velocity of 3,368 m/s, the second peak at 1.005 GHz with a current of 472 pA and a velocity of 4,020 m/s, the

Now the three peaks represent different SAW modes. The highest frequency peak of Fig. 8 of 1.095 GHz represents a longitudinal wave with an acoustic velocity of 4,380 m/s and an angle of about 10º off from the (110) direction (Kuok et al., 2001). This small angle variation is due to a small misalignment during the lithography process. When viewing the lower peak of 840 MHz at a velocity of 3,368 m/s, this coincides with a fast transverse wave with,

third peak is at 1.095 GHz with a current of 1.098 nA and a velocity of 4,380 m/s

steepening or smoothing of the step like feature is a direct measure of this.

**4.1.3 Usage of Quantum Point Contacts and Surface Acoustic Waves** 

account, due to thermal errors in measurement).
