A quantitative physical model, describing the piezoelectric electromechanical coupling in nanomechanical resonators with a two-dimensional electron gas, is developed. Numerical calculations of the change in density of a two-dimensional electron gas contained in a vibrating cantilever are performed using the model and are shown to be consistent with the experiment. The obtained data show that the vibration-induced electron density modulation is localized near the clamping point and that it is related to a rapid spatial change in the mechanical stress near this point. It is shown that details of the clamping geometry significantly affect the magnitude of the effect.

The low-dimensional electron systems, such as a two-dimensional electron gas
(2DEG), quantum wires and quantum dots, have been intensively studied for
decades, and these studies have led to discovery of bright phenomena
including the integer

Consider a model nanoelectromechanical system representing a cantilever with
a 2DEG (see Fig. 1a) similar to that experimentally studied in

Let the cantilever perform flexural vibrations at the first eigenmode. The oscillating cantilever, being an electromechanical system, obeys the motion equation

The substitution of Eqs. (3) and (4) in Eqs. (1) and (2) gives a system of
partial differential equations. We solve this system using the finite element
method on the full stack of the material shown in Fig. 1 and obtain
eigenfrequency

Figure 2a, b show the bending-induced changes in the electron density

Figure 2e shows the

The change in fillet radius

Figure 3d shows the absolutized vibration-induced changes in the electron
density (deviations from a uniform equilibrium value charactersistic for a
resting cantilever), corresponding to the main peaks observed in Fig. 2e as
functions of the rate of stress decay

The calculation results can be compared to the experimental results reported
in

So far we have discussed a cantilever with a naked surface. However, the
geometry of the mesoscopic devices, such as single-electron transistors and
quantum point contacts, is often determined by metal gates partially covering
the surface. Obviously, the presence of the metal affects electron density,
and the gated systems should be considered separately. In the following, we
limit ourselves to the consideration of a cantilever similar to the system
considered above, but with its top surface entirely covered with a thin metal
layer. It is clear that this should lead to the cantilever stiffening, but we
deliberately exclude this side effect from consideration by tending the metal
elastic constants to zero. Thus, at the stage of the problem formulation, the
appearance of the metal leads only to the zeroing electrical potential at the
top surface. Figure 4a, b show the calculated

The proposed physical model shows that the vibrations of a piezoelectric nanomechanical cantilever with a two-dimensional electron gas should lead to a change in the electron density. If the cantilever is not covered with a metal, such as a Schottky gate, then this change is prominent only near the clamping point and drops much faster than the stress with the increasing distance from this point. It is shown that the magnitude of the effect is determined mainly by the rate of the stress decay that occurs with the movement from the clamping point into the bulk, to which the cantilever is attached. It is demonstrated that the microscopic details of the clamping significantly affect the magnitude of the effect and should be taken into account. In the case of a cantilever covered with a metal, the considered localized change in the electron density is superimposed on a signal approximately proportional to the stress, but these components have comparable magnitudes.

All datasets used in the manuscript can be requested from the corresponding author.

The authors declare that they have no conflict of interest.

The work is supported by the Russian Foundation for Basic Research grants 16-32-60130, 15-02-05774 and 16-02-00579. Part of this work related to the data interpretation was funded by the State Programme (grant No. 0306-2016-0015). Edited by: M. K. Ghatkesar Reviewed by: three anonymous referees