**4.1.2 Quantum Point Contact Operation**

QPCs work by applying a negative voltage to them, this negative voltage produces an electric field that penetrates into the 2DEG causing electrons to be repelled at or around the QPC region; this can be seen if Fig 7(c). Here the black region represents the metallic electrode while the blue region represents the electric field and the green region represents the 2DEG or substrate. It can be taken as a good estimate that the electric field penetrates into the 2DEG from the QPC electrode at a 45° angle. As a larger negative voltage is applied to the QPC the effective screening electric field becomes larger causing the electron path within the 2DEG to become more constricted until single electron transport is achieved; this leads to a Coulomb Blockade, see Sec. 4.2. Then finally, the current is completely pinched off, see Figs 9 and 10.

Fig. 7. Two SEM images of the same DQD structure on top of a suspended tube section of an acoustic waveguide. The DQD is made up of five pairs of QPCs. The material is a GaAs/AlGaAs heterostructure with an embedded 2DEG about 40 nm below the surface. (a) Top view with a scale bar of 200 nm, (b) angled view with a scale bar of 100 nm. (c) schematic view of QPC with applied voltage, the blue area represents the electric field surrounding the QPC and penetrating into the 2DEG (green). (d) Tunneling of electrons across potential barrier created by applied voltage from the QPC

The QPC creates a tunneling barrier which separates the source and drain regions of the sample. As the voltage magnitude becomes larger the strength, or height, or the tunneling barrier is increased and the width of the barrier becomes larger. When plotted a step like

QPCs work by applying a negative voltage to them, this negative voltage produces an electric field that penetrates into the 2DEG causing electrons to be repelled at or around the QPC region; this can be seen if Fig 7(c). Here the black region represents the metallic electrode while the blue region represents the electric field and the green region represents the 2DEG or substrate. It can be taken as a good estimate that the electric field penetrates into the 2DEG from the QPC electrode at a 45° angle. As a larger negative voltage is applied to the QPC the effective screening electric field becomes larger causing the electron path within the 2DEG to become more constricted until single electron transport is achieved; this leads to a Coulomb Blockade, see Sec. 4.2. Then finally, the current is completely pinched

Fig. 7. Two SEM images of the same DQD structure on top of a suspended tube section of an

GaAs/AlGaAs heterostructure with an embedded 2DEG about 40 nm below the surface. (a)

The QPC creates a tunneling barrier which separates the source and drain regions of the sample. As the voltage magnitude becomes larger the strength, or height, or the tunneling barrier is increased and the width of the barrier becomes larger. When plotted a step like

acoustic waveguide. The DQD is made up of five pairs of QPCs. The material is a

Top view with a scale bar of 200 nm, (b) angled view with a scale bar of 100 nm. (c) schematic view of QPC with applied voltage, the blue area represents the electric field surrounding the QPC and penetrating into the 2DEG (green). (d) Tunneling of electrons

across potential barrier created by applied voltage from the QPC

**4.1.2 Quantum Point Contact Operation** 

(a) (b)

(c) (d)

off, see Figs 9 and 10.

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 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**

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 account, due to thermal errors in measurement).

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

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,

Surface Acoustic Waves and Nano–Electromechanical Systems 649

oscillations are due to state transitions, both propagating and reflecting (Maaø & Galperin, 1997). The SAW may reflect from the QPC and create interference patterns; these patterns will affect the electron transmission probability through the QPC simply by changing the potential landscape. This mechanism is also sensitive to scattering events near the QPC, where energy quanta are emitted and absorbed between two waves. Others have

Fig. 9. QPC pinch-off of acoustoelectric current: (a) pinch-off with varying source-drain voltages applied at the SAW, changes are in 1 mV increments (Talyanskii et al., 1997) Copyright (1997) by The American Physical Society. (b) Pinch-off with different SAW power levels, the left most trace is 7 dBm and decrements by 0.2 dBm (Talyanskii et al., 1997)

acoustoelectric current for different gate voltages and SAW powers; *f* = 1007.426 MHz,

Fig. 10 shows the oscillations of the acoustoelectric current as it is being driven through a QPC (orange trace) as it approaches pinch-off. It can be seen that the step like features observed in Fig. 9 are have been replaced by oscillations. These oscillations are negative, or on the lower cycle, when the current from the source-drain bias is flat or non-changing. We then see that as the current starts to decrease the acoustoelectric current has a positive value, or is on the upper cycle of the oscillation. The inset graph of Fig. 10 shows the entire sweep of the QPC gate voltage. As one conduction channel pinches off at the Vg = -0.65 V we see that there is a large, nearly single, oscillation in the acoustoelectric current. Since SAWs are

Copyright (1997) by The American Physical Society. (c) A negative anomalous

Vds = 0 V, T = 1.7 K, (Song et al., 2010)

contributed to the theory and more is being added.

(a) (b)

(c)

again, a 10º difference between the SAW direction and the (110) GaAs crystal orientation (Kuok et al., 2001).

### **4.2 Coulomb Blockade of Acoustoelectric Current**

When looking over Fig. 9, we see that single steps are observed just before the total acoustoelectric current becomes zero. Focusing on Fig. 9a, we see that the last step is at about 0.5 nA. When solving with Eq. 11 the first step yields an answer of n = 1, i.e. a single electron is being transferred. We can look at all of the remaining steps and solve for them as well which will reveal integer numbers for n and will increase by 1 for each step, as is expected. Fig. 9a also shows that an applied voltage across the source and drain contacts, or ohmics, has very little effect on the final quantized acoustoelectric current. We do see, however, that a larger, or smaller, gate voltage is required for final pinch-off in the system but this is due to the small offset in the Fermi energy because of the applied bias. The only real difference is the shift from negative current to a positive current which is easy explained by the fact that the source-drain bias is producing a current that is opposite in direction, when Vds < 0 mV, to that of the acoustoelectric current. Another aspect is the RF power dependence on the quantized current. When viewing Fig. 9b there is a similar trend to that of Fig. 9a. The small change in RF power has only a small effect on the acoustoelectric current. This is because the RF power is directly proportional to the SAW amplitude and we do not really identify any significant difference in the number of electrons being transferred, or Coulomb steps, until a drop of about 2 dBm.

So far an acoustoelectric current behaves in the same way as a standard source-drain bias current when a QPC is near pinch off. There is, however, another effect which can arise. As seen in Fig 9c a negative current arises and still exhibits the step like behavior. This negative current is the negative anomalous acoustoelectric current. This is said to be produced as an effect of SAW reflections, this is mostly seen in a two IDT system. The second IDT acts as a reflector much as the same way a reflection gradient is used for a unidirectional SAW, see Fig 1b. This can cause a standing wave to occur in the sample in such a way that the reflections effectively reduce the potential of the SAW and cause fewer electrons to be transported.

With slight phase shifts added the standing wave with respect to the QPC a net negative potential in the energy landscape can exist on one side of the QPC which will cause the current to change direction. These steps are best observed for QPCs that are long when compared to the SAW wavelength. This makes it possible to observe the acoustoelectric step current without applying a magnetic field (Shilton et al, 1996). As mentioned earlier, the high mobility of the 2DEG will screen the acoustoelectric current but this effect is not as prevalent when inside of a long QPC channel. The current screening is reduced around the QPC region and thus electrons can be transported through the channel. A long channel can allow ballistic transport of the electron causing it to shuttle from one side of the QPC to another. Since the screening is minimal, the electron will be "dragged" through the channel and the current steps of Fig. 9 can be observed.

Another phenomenon is oscillations in the acoustoelectric current once approaching the QPC pinch-off limit. As described in Shilton et al, 1996, and Maaø & Galperin, 1997, the acoustoelectric current oscillations occur at the same positions as the Coulomb steps. This oscillation is described as interference effects near the QPC at high frequencies which may be attributed to impurities in the 2DEG channel. The theory presented suggests that these

again, a 10º difference between the SAW direction and the (110) GaAs crystal orientation

When looking over Fig. 9, we see that single steps are observed just before the total acoustoelectric current becomes zero. Focusing on Fig. 9a, we see that the last step is at about 0.5 nA. When solving with Eq. 11 the first step yields an answer of n = 1, i.e. a single electron is being transferred. We can look at all of the remaining steps and solve for them as well which will reveal integer numbers for n and will increase by 1 for each step, as is expected. Fig. 9a also shows that an applied voltage across the source and drain contacts, or ohmics, has very little effect on the final quantized acoustoelectric current. We do see, however, that a larger, or smaller, gate voltage is required for final pinch-off in the system but this is due to the small offset in the Fermi energy because of the applied bias. The only real difference is the shift from negative current to a positive current which is easy explained by the fact that the source-drain bias is producing a current that is opposite in direction, when Vds < 0 mV, to that of the acoustoelectric current. Another aspect is the RF power dependence on the quantized current. When viewing Fig. 9b there is a similar trend to that of Fig. 9a. The small change in RF power has only a small effect on the acoustoelectric current. This is because the RF power is directly proportional to the SAW amplitude and we do not really identify any significant difference in the number of electrons being transferred,

So far an acoustoelectric current behaves in the same way as a standard source-drain bias current when a QPC is near pinch off. There is, however, another effect which can arise. As seen in Fig 9c a negative current arises and still exhibits the step like behavior. This negative current is the negative anomalous acoustoelectric current. This is said to be produced as an effect of SAW reflections, this is mostly seen in a two IDT system. The second IDT acts as a reflector much as the same way a reflection gradient is used for a unidirectional SAW, see Fig 1b. This can cause a standing wave to occur in the sample in such a way that the reflections effectively reduce the potential of the SAW and cause fewer electrons to be

With slight phase shifts added the standing wave with respect to the QPC a net negative potential in the energy landscape can exist on one side of the QPC which will cause the current to change direction. These steps are best observed for QPCs that are long when compared to the SAW wavelength. This makes it possible to observe the acoustoelectric step current without applying a magnetic field (Shilton et al, 1996). As mentioned earlier, the high mobility of the 2DEG will screen the acoustoelectric current but this effect is not as prevalent when inside of a long QPC channel. The current screening is reduced around the QPC region and thus electrons can be transported through the channel. A long channel can allow ballistic transport of the electron causing it to shuttle from one side of the QPC to another. Since the screening is minimal, the electron will be "dragged" through the channel

Another phenomenon is oscillations in the acoustoelectric current once approaching the QPC pinch-off limit. As described in Shilton et al, 1996, and Maaø & Galperin, 1997, the acoustoelectric current oscillations occur at the same positions as the Coulomb steps. This oscillation is described as interference effects near the QPC at high frequencies which may be attributed to impurities in the 2DEG channel. The theory presented suggests that these

(Kuok et al., 2001).

transported.

**4.2 Coulomb Blockade of Acoustoelectric Current** 

or Coulomb steps, until a drop of about 2 dBm.

and the current steps of Fig. 9 can be observed.

oscillations are due to state transitions, both propagating and reflecting (Maaø & Galperin, 1997). The SAW may reflect from the QPC and create interference patterns; these patterns will affect the electron transmission probability through the QPC simply by changing the potential landscape. This mechanism is also sensitive to scattering events near the QPC, where energy quanta are emitted and absorbed between two waves. Others have contributed to the theory and more is being added.

Fig. 9. QPC pinch-off of acoustoelectric current: (a) pinch-off with varying source-drain voltages applied at the SAW, changes are in 1 mV increments (Talyanskii et al., 1997) Copyright (1997) by The American Physical Society. (b) Pinch-off with different SAW power levels, the left most trace is 7 dBm and decrements by 0.2 dBm (Talyanskii et al., 1997) Copyright (1997) by The American Physical Society. (c) A negative anomalous acoustoelectric current for different gate voltages and SAW powers; *f* = 1007.426 MHz, Vds = 0 V, T = 1.7 K, (Song et al., 2010)

Fig. 10 shows the oscillations of the acoustoelectric current as it is being driven through a QPC (orange trace) as it approaches pinch-off. It can be seen that the step like features observed in Fig. 9 are have been replaced by oscillations. These oscillations are negative, or on the lower cycle, when the current from the source-drain bias is flat or non-changing. We then see that as the current starts to decrease the acoustoelectric current has a positive value, or is on the upper cycle of the oscillation. The inset graph of Fig. 10 shows the entire sweep of the QPC gate voltage. As one conduction channel pinches off at the Vg = -0.65 V we see that there is a large, nearly single, oscillation in the acoustoelectric current. Since SAWs are

Surface Acoustic Waves and Nano–Electromechanical Systems 651

When the QD needs to be populated, the center plunger gate voltage value is lowered; causing the Fermi energy to lower below the Fermi energy of the surrounding 2DEG. The gate electrodes are still held at a high voltage value so the electrons cannot tunnel in or out of the QD, see Fig. 11e. The IDT is pulsed and as the SAW enters the QD region it changes the tunneling barrier created by the electrodes. The barrier is changed just enough that a single electron can tunnel into the dot, this is population; see Figs. 11a-b. The IDT is pulsed many times since the tunneling barrier is so high that the additional pulses are needed to

Fig. 11. Results from the paper Kataoka et al., 2007, Copyright (2007) by The American Physical Society. (a) and (b) The population of the QD, the –e line is an electron entering the

Likewise, as shown in Figs. 11c-d, the QD is depopulated by a similar mechanism. Here the plunger gate voltage is raised, made to be more negative, which causes the Fermi energy of the QD to be raised in comparison to the 2DEG Fermi energy outside of the dot, Fig. 11e. At this time the IDT is pulsed and the SAW enters the QD region. The SAW alters the tunneling barrier created by the gate electrodes when the potential of the SAW is superimposed onto the barrier, see Eq. 5. When the barrier is raised nothing happens since this will simply ensure the electron stays in the system, but when the negative cycle of the SAW

dot. (c) and (d) Depopulation of the QD. (e) Schematic of SAW for population and

superimposes with the barrier it decreases it which causes the electron to tunnel out.

This method of operation has proven, quite nicely, that SAWs and high potential QPCs offer a more robust method of single electron population and depopulation. As shown in Figs. 11a-d the QD can maintain the electron confinement for a long period of time which can be difficult in a traditional setup due to noise reduction of the measurement electronics. The IDTs can be accessed from an outside system such as an antenna, for example, and interact with the QD allowing the system another degree of freedom from the traditional closed electronics. However, this may come at the cost of additional noise reduction equipment

This chapter has shown all of the basic properties and uses of SAWs in nano-structures and nano-systems. The reader is shown the parameters required for fabrication, theory of operation, real results, and application. The use of SAWs is quite limitless in the area of

increase the tunneling probability of the electron.

depopulation of the QD

and filtering.

**5. Conclusion** 

very sensitive to 2DEGs we get what seems to be an amplified signal when compared to a source-drain bias. This can be used to identify information that may otherwise have been too weak or nearly washed out from thermal effects.

Fig. 10. Acoustoelectric current oscillations as the QPC nears pinch-off. The data plot has been normalized to show the relationship between the acoustoelectric current oscillations (orange trace) and the pinch-off when using a voltage bias across the source-drain electrodes (red trace). The acoustoelectric current trace has been normalized to -8.832 nA and the source-drain bias trace has been normalized to 4.692 nA. Measurements were taken at 4.2 K with the sample shown in Fig. 7. Inset plot shows entire gate voltage sweep range

### **4.3 Population and Depopulation of Quantum Dots**

SAWs have been used widely as a mechanism to control the population and depopulation of QDs and DQDs. The high speed, or frequency, of operation combined with quantized current production make SAWs suitable for use with quantum systems. In a traditional QD system the gate and source-drain voltages are changed to allow single electron population and depopulation. This has several drawbacks; one being it is hard to decouple the dot from the surrounding environment due to the precise control of the voltages. There is always some noise either in the system or from the electronics. Another problem is if one gate is changed then the others must be changed to ensure a constant electron count in each dot, if applicable.

A SAW can be used as an alternative and has gotten more attention lately for the use in a QD or DQD system. The biggest reason for the attention is that SAWs can be used to interact with QD systems at much higher frequencies then what has been previously achieved by pulsing the gate electrodes. Since the acoustoelectric current can be well defined a QD system can use very high gate voltages to ensure no leakage current. We will take a look at the work done in Kataoka et al., 2007. This work describes very well the use of SAWs as a way to populate and depopulate a QD.

very sensitive to 2DEGs we get what seems to be an amplified signal when compared to a source-drain bias. This can be used to identify information that may otherwise have been

Fig. 10. Acoustoelectric current oscillations as the QPC nears pinch-off. The data plot has been normalized to show the relationship between the acoustoelectric current oscillations (orange trace) and the pinch-off when using a voltage bias across the source-drain electrodes (red trace). The acoustoelectric current trace has been normalized to -8.832 nA and the source-drain bias trace has been normalized to 4.692 nA. Measurements were taken at 4.2 K

SAWs have been used widely as a mechanism to control the population and depopulation of QDs and DQDs. The high speed, or frequency, of operation combined with quantized current production make SAWs suitable for use with quantum systems. In a traditional QD system the gate and source-drain voltages are changed to allow single electron population and depopulation. This has several drawbacks; one being it is hard to decouple the dot from the surrounding environment due to the precise control of the voltages. There is always some noise either in the system or from the electronics. Another problem is if one gate is changed then the others must be changed to ensure a constant electron count in each dot, if

A SAW can be used as an alternative and has gotten more attention lately for the use in a QD or DQD system. The biggest reason for the attention is that SAWs can be used to interact with QD systems at much higher frequencies then what has been previously achieved by pulsing the gate electrodes. Since the acoustoelectric current can be well defined a QD system can use very high gate voltages to ensure no leakage current. We will take a look at the work done in Kataoka et al., 2007. This work describes very well the use of SAWs as a

with the sample shown in Fig. 7. Inset plot shows entire gate voltage sweep range

**4.3 Population and Depopulation of Quantum Dots** 

way to populate and depopulate a QD.

applicable.

too weak or nearly washed out from thermal effects.

When the QD needs to be populated, the center plunger gate voltage value is lowered; causing the Fermi energy to lower below the Fermi energy of the surrounding 2DEG. The gate electrodes are still held at a high voltage value so the electrons cannot tunnel in or out of the QD, see Fig. 11e. The IDT is pulsed and as the SAW enters the QD region it changes the tunneling barrier created by the electrodes. The barrier is changed just enough that a single electron can tunnel into the dot, this is population; see Figs. 11a-b. The IDT is pulsed many times since the tunneling barrier is so high that the additional pulses are needed to increase the tunneling probability of the electron.

Fig. 11. Results from the paper Kataoka et al., 2007, Copyright (2007) by The American Physical Society. (a) and (b) The population of the QD, the –e line is an electron entering the dot. (c) and (d) Depopulation of the QD. (e) Schematic of SAW for population and depopulation of the QD

Likewise, as shown in Figs. 11c-d, the QD is depopulated by a similar mechanism. Here the plunger gate voltage is raised, made to be more negative, which causes the Fermi energy of the QD to be raised in comparison to the 2DEG Fermi energy outside of the dot, Fig. 11e. At this time the IDT is pulsed and the SAW enters the QD region. The SAW alters the tunneling barrier created by the gate electrodes when the potential of the SAW is superimposed onto the barrier, see Eq. 5. When the barrier is raised nothing happens since this will simply ensure the electron stays in the system, but when the negative cycle of the SAW superimposes with the barrier it decreases it which causes the electron to tunnel out.

This method of operation has proven, quite nicely, that SAWs and high potential QPCs offer a more robust method of single electron population and depopulation. As shown in Figs. 11a-d the QD can maintain the electron confinement for a long period of time which can be difficult in a traditional setup due to noise reduction of the measurement electronics. The IDTs can be accessed from an outside system such as an antenna, for example, and interact with the QD allowing the system another degree of freedom from the traditional closed electronics. However, this may come at the cost of additional noise reduction equipment and filtering.

### **5. Conclusion**

This chapter has shown all of the basic properties and uses of SAWs in nano-structures and nano-systems. The reader is shown the parameters required for fabrication, theory of operation, real results, and application. The use of SAWs is quite limitless in the area of NEMS. As people continue to refine the research through experiment and by continuing to add to the theory more applications will emerge.

### **6. Acknowledgment**

We like to thank the National Science Foundation for support under a NIRT grant (No. ECCS-0708759) and the Air Force Office for Scientific Research for support under a MURI grant (No. FA9550-08-1-0337).

### **7. References**


NEMS. As people continue to refine the research through experiment and by continuing to

We like to thank the National Science Foundation for support under a NIRT grant (No. ECCS-0708759) and the Air Force Office for Scientific Research for support under a MURI

Barnes, C. H. W., Shilton, J. M., & Robinson, A. M. (2000). Quantum computing using

Beil, F. W., Wixforth, A., Wegscheider, W., Schuh, D., Bichler, M., & Blick, R. H. (2008).

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Song, L., Chen, S. W., He, J. H., Zhang, C. Y., Lu, C., & Gao, J. (2010). The anomalous

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Jones, G. A. C., Farrer, I., Ritchie, D. A., & Pepper, M. (2007). Single-Electron Population and Depopulation of an Isolated Quantum Dot Using a Surface-Acoustic-Wave Pulse. *Physical Review Letters*, Vol. 98, No. 4, (January 2007), pp.

(001), (110), and (111) GaAs. *Journal of Applied Physics*, Vol. 89, No. 12, (June 2001),

Ritchie, D. A. (1996). On the acoustoelectric current in a one-dimensional channel. *Journal of Condensed Matter Physics*, Vol. 8, (March 1996), pp. L337-L343, ISSN 0953-

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Ritchie, D. A., & Jones, G. A. C. (1997). Single-electron transport in a onedimensional channel by high-frequency surface acoustic waves. *Physical Review B*,

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add to the theory more applications will emerge.

**6. Acknowledgment** 

**7. References** 

8984

grant (No. FA9550-08-1-0337).
