MSMechanical SciencesMSMech. Sci.2191-916XCopernicus PublicationsGöttingen, Germany10.5194/ms-8-111-2017A two-dimensional electron gas sensing motion of a nanomechanical cantileverShevyrinAndreyshevandrey@isp.nsc.ruPogosovArthurRzhanov Institute of Semiconductor Physics SB RAS, Novosibirsk 630090, RussiaNovosibirsk State University, Novosibirsk 630090, RussiaAndrey Shevyrin (shevandrey@isp.nsc.ru)12May2017811111151November201620February201712April2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://ms.copernicus.org/articles/8/111/2017/ms-8-111-2017.htmlThe full text article is available as a PDF file from https://ms.copernicus.org/articles/8/111/2017/ms-8-111-2017.pdf
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.
Introduction
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 and fractional quantum Hall
effects, weak localization , conductance quantization
, the Coulomb blockade etc. The usually studied
low-dimensional electron systems are embedded in a semiconductor
heterostructure being a part of a massive bulk. However, it is also possible
to create micro- and nanomechanical resonators containing low-dimensional
electron systems. Several papers report on non-trivial properties of hybrid
GaAs / AlGaAs-based electromechanical systems combining the mechanical
resonators and such mesoscopic devices as a single-electron transistor
and a quantum point contact
. Experiments show that
the 2DEG conductivity is sensitive to vibrations of a resonator containing
it, and it has been recently shown that this coupling is of piezoelectric
origin . However, there is still no common physical model
describing the electromechanical coupling and predicting the magnitude of the
effect. In the present paper, we propose a quantitative physical model that
gives an opportunity to estimate the vibration-induced change in the electron
density and reveals some important details of the electromechanical coupling
in the nanoelectromechanical systems based on GaAs / AlGaAs
heterostructures.
Model description
Consider a model nanoelectromechanical system representing a cantilever with
a 2DEG (see Fig. 1a) similar to that experimentally studied in
. The cantilever, having thickness t=166 nm, consists of the
layers schematically shown in Fig. 1b. The 2DEG resides in the 13 nm-thick
GaAs layer buried 77 nm below the surface. The cantilever studied in
is created by means of selective etching of
Al0.8Ga0.2As sacrificial layer from under the top layers of the
heterostructure. The etching front boundary forms the clamping. To avoid the
stress singularity in our calculations, we consider a model system with a
fillet of the radius R introduced in the corner between the remained
sacrificial layer and the cantilever. Let the cantilever have length L≫t
(L=3µm for the system studied in ). We
introduce a coordinate system with the x axis directed along the cantilever
towards its free end and the y axis directed perpendicularly to the top
surface and pointing upwards. Let x=0 at the vertical boundary of the
remained sacrificial layer, and y=0 at the middle plane of the cantilever.
For simplicity, we restrict ourselves to the plane strain model assuming that
displacements in the z direction are zero and no parameters are changed
along this axis. Let the x axis coincide with piezoelectric-active [110]
crystallographic direction and the y axis be directed along the [001]
direction.
(a) Schematic image of the model cantilever and its
orientation with respect to the crystallographic axes. L is the cantilever
length, t is thickness, R is the fillet radius. (b) The
heterostructure with a two-dimensional electron gas (2DEG) from which the
model cantilever is created.
Let the cantilever perform flexural vibrations at the first eigenmode. The oscillating cantilever, being an electromechanical system, obeys the motion equation
∂σij∂xj=-ρΩ02Ui
and the Gauss's law
∂Di∂xi=0.
Here σij is stress tensor, ρ is mass density, Ω0 is
angular eigenfrequency, Ui is the displacement of a given point and Di
is electric displacement. The stress and the electric displacement can be
expressed in terms of Ui and electric potential ϕ using the
constitutive equations:
σij=Cijkl∂Uk∂xl+ekij∂ϕ∂xk,Di=eijk∂Uj∂xk-ϵϵ0∂ϕ∂xi,
where Cijkl is stiffness tensor, eijk is piezoelectric tensor,
ϵ is relative dielectric constant and ϵ0 is the vacuum
dielectric permittivity. Since tensors Cijkl and eijk are
symmetric, it is convinient to write them in the Voigt notations
as 6×6 and 6×3 matrices, respectively:
Cij=C11+C12+2C442C12C11+C12-2C442000C12C11C12000C11+C12-2C442C12C11+C12+2C442000000C44000000C11-C122000000C44,eij=e140000011/20-1/2000000-100.
Here C11=(11.88+0.14χ)×1010 Pa,
C12=(5.38+0.32χ)×1010 Pa and C44=(5.94-0.05χ)×1010 Pa are AlχGa1-χAs elastic
constants, and e14=-0.16-0.065χ C m-2 is the piezoelectric
constant .
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 Ω0, displacements Ui and potential ϕ. The
vacuum surrounding the cantilever can be modeled as a media characterized by
negligibly small stiffness, the zero piezoelectric tensor and dielectric
constant ϵ=1. We apply Dirichlet conditions Ui=0, ϕ=0 at the
boundary that is far from the cantilever. A special area is the layer
containing the 2DEG. When the cantilever bends, the 2DEG density changes in
such a way as to maintain constant electrochemical potential. However, to
simplify the calculations, we use the pure electrostatic screening model
and consider the 2DEG as a media having a constant electrical
potential. As shown in , this simplification is reasonable if
the cantilever thickness far exceeds the Bohr radius
aB=4πϵϵ0ℏ2m-1e-2≈13 nm (here
m is an electron effective mass and e is the elementary charge). Thus, we
neglect the chemical energy and model the 2DEG as a metal, or, equivalently,
as a material with a dielectric constant far exceeding the constants of all
other materials involved in the problem. Once the displacements Ux,Uy
and the potential ϕ have been calculated, the electric displacement
Di can be found using Eq. (4), and the electron density change resulting
from the cantilever bending can be found as δn(x)=-e-1(Dy1-Dy2), where Dy1 and Dy2 are Dy values
near the top and bottom 2DEG boundaries. The calculated electron density
change, in the diffusive transport regime, should lead to a proportional
change in the 2DEG conductivity δσ=δn×eμ, which can
be measured experimentally (here μ is the electron mobility). Notice
that, under the assumptions made, the calculated δn does not depend
on the equilibrium electron density of a non-vibrating 2DEG.
Results and discussion
Figure 2a, b show the bending-induced changes in the electron density δn(x) normalized to the deflection of the cantilever free end Uy(L). The
curves are calculated at the fillet radii R=3 nm (panel a) and
R=1µm (panel b). Figure 2c, d show the corresponding absolute
stress |σxxUy-1(L)| calculated near the upper surface of the
cantilever. It can be seen that the stress reaches its maximum near the
clamping point and rapidly decreases to the left from the maximum. This
stress drop can be roughly fitted by an exponent
|σxxUy-1(L)|=Aexp(αx). The density change is most
prominent near the maximal stress point. However, δn(x)Uy-1(L)
decreases with increasing x much faster than the stress. Moreover, despite
the maximal stess is higher at R=1µm, the density response is
lower in this case than at R=3nm. This shows that the density
change at a given point is not determined by the stress at this point.
Instead, it is a functional of the spatial stress distribution and it is
largely determined by the stress gradient near the considered point.
(a, b) The change in electron density δn(x)
normalized to deflection of the cantilever free end Uy(L) calculated at
the fillet radii R=3nm(a) and
R=3µm (b). (c, d) Corresponding mechanical
stress σxx normalized to the free end deflection Uy(L)
calculated near the upper surface of the cantilever. Red lines show
exponential fits to the data. (e) The change in electron density
calculated at various fillet radii R (displayed in the figure). The curves
are vertically offset by 1.6×1011 cm-2µm-1
with respect to each other.
Figure 2e shows the δn(x)Uy-1(L) curves obtained at various
fillet radii R. It can be seen that each of the obtained curves
demonstrates two main peaks, one of which is negative (Peak 1) and the other
is positive (Peak 2). At small R values, there is also a small region to
the near left of Peak 1, where the electron density change is positive. This
additional small peak is seemingly caused by the features associated to the
stress concentration near the corner with a small curvature radius. It can be
seen that, when R increases, the main peaks shift towards the free end of
the cantilever, and the distance between them increases. Positions of the
peaks and the neutral point, where δn(x)=0, are shown in Fig. 3a as
the functions of fillet radius R.
(a) Positions of the peaks in electron desnity change and
the neutal point where the electron density change is zero. (b) The
inverse rate of the stress decay α linearly depends on the fillet
radius R. (c) The eigenfrequency f0 and maximal mechanical
stress σxx normalized to the free end deflection Uy(L) as
functions of the fillet radius. (d) Absolute amplitude of the peaks
normalized to the free end deflection Uy(L) and the eigenfrequency f0
are determined by the stress decay rate α.
The change in fillet radius R leads to spatial redistribution of the
stress. As shown in Fig. 3b, the increase in R leads to the increase in
length α-1, which is the distance characterizing the stress decay.
Besides, it leads to the cantilever stiffening due to shortening of its
effective length and increased thickness near the base. Figure 3c shows that
this stiffening manifests itself in a growth of the eigenfrequency
(2π)-1Ω0 and an increase in the maximal stress σmax
normalized to the deflection Uy(L) of the cantilever free end. Functions
Ω0(R) and σmaxUy-1(L) are similar, because both of
them are roughly proportional to the effective value of tL-2, where t
is the cantilever thickness and L is its length.
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 α. To reveal the effect of the
α change and to separate this effect from the cantilever stiffening,
these peak values are normalized to (2π)-1Ω0Uy(L), rather than
to Uy(L), as in Fig. 2e. It can be seen that the considered peak values
are determined by the rate of the stress decay α, and they increase
with the increasing α. The peak values grow relatively slowly at
α<6µm-1, while, at α>6µm-1,
Peak 1 starts growing much faster, and Peak 2 reaches its saturation. This
featured value of α approximately equals to the inverse cantilever
thickness t=166nm. One can see from Fig. 3b that the same α
value corresponds to the fillet radius R≈t at which an additional
peak appears at the δn(x)Uy-1(L) dependence (see Fig. 2e). We
suppose that the features, observed at the fillet radius being less than the
cantilever thickness, are related to the high stress and the high gradients
of the stress appearing near the fillet with a small radius.
The calculation results can be compared to the experimental results reported
in . The change in the electron density at the distance of
1.3 µm from the clamping point, estimated from the experimental
data, is about δn≈5×108 cm-2µm-1.
The calculated value of δn corresponding to this distance is about
3×108 cm-2µm-1, and it weakly depends on R
at small radii. Thus, the results obtained from the developed model are
consistent with the experiment. Notice that, at the distance of
1.3 µm, the change in the electron density is far below the change
near the clamping point, and it would be desirable to compare the results
obtained from the developed model to the change in the electron density
experimentaly measured in the vicinity of the clamping point. However, to our
knowledge, there are currently no papers reporting on such experimental
results. Notice that the discussed change in the electron density is
proportional to piezoelectric constant e14 and, in principle, it could
be increased more than two times for GaN-based heterostructures.
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 δn(x)Uy-1(L)
dependence obtained at R=0.1µm for the case of a metalized
cantilever and for the case of a naked surface, respectively. Figure 4a shows
also the difference of these two signals. Figure 4c, d show the corresponding
spatial distributions of the stress. It can be seen that, in the case of a
metalized cantilever, the electron density change consists of two components.
One of them decreases with the increasing x as slowly as the stress, and
the other (rapidly changing) is approximately equal to the density change
observed in the case of a non-metalized cantilever. Notice that these two
components have comparable magnitudes. Thus, the electron density change
related to a rapid spatial change in the stress near the clamping point is
significant even in the case of a metalized cantilever. In real mesoscopic
devices, whose surface is only partially covered with metal, the density
response is expected to be more complex and intermediary between the results
of the two considered models.
(a, b) Solid black lines: the change in electron density
δn(x) normalized to deflection of the cantilever free end Uy(L)
calculated for the cantilever covered with a metal (a) and
cantilever with a naked surface (b). The blue dashed line in
(a) shows the difference between the signals in displayed
in (a, b). (c, d) Corresponding mechanical stress
σxx normalized to Uy(L) calculated near the top surface.
Conclusions
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.
Acknowledgements
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
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