SUPPEMENTALINFORMATION
Tapered Silicon Nanowires for Enhanced Nanomechanical Sensing
O. Malvar1, E. Gil-Santos1, J.J. Ruz1, D. Ramos2, V. Pini1, M. Fernandez-Regulez3, M. Calleja1, J. Tamayo1 and A. San Paulo1,3
Tapered Si NW Growth:
Si NWs were grown in a CVD reactor at 800 ºC and atmospheric pressure with 10% H2/Ar as both diluent and carrier gas. A fixed flow rate of 270 s.c.c.m. was used for the diluent gas line (direct to growth chamber). The carrier gas was passed through a refrigerated (0ºC) liquid SiCl4 bubbler in order to deliver the precursor gas to the growth chamber. Variations in the carrier gas flow rate in the range 40-90 s.c.c.m. were introduced in different growth sessions in order to obtain NWs with different degree of tapering. Au colloidal nanoparticles (NP) with a diameter of 50 and 80 nm where used as metal catalyst.
Theoretical model
The Euler-Bernoulli equation for a beam with non-uniform cross section is given by
(1)
where is the order mode contribution to the nanowire displacement at the normalized longitudinal position and time t (actual position is normalized to the NW length, ), and and are the cross-section area and second moment of area, respectively and both depend on the longitudinal position in order to account for the tapered geometry. Assuming a harmonic oscillator response for each mode, the solution to eq. (1) can be written as:
(2)
where is the angular resonance frequency of the mode. Then, the temporal and spatial parts of the equation can be separated so that the spatial part of the equation can be writen as:
(3)
A general solution for this equation for any tapered beam-like structure can be obtained if and are expressed as:
(4)
(5)
where and are the area and second moment of area at the clamp. In the case of a tapered nanowire which diameter decreases linearly from the clamp to the free-end, the radius can be written as:
(6)
where is the tapering parameter given by
(7)
and equations (4) and (5) are written using .
After these considerations, the beam equation can be rewritten as:
(8)
where is related to the resonance frequency of the order mode by:
(9)
It can be shown that the solution to equation (8) can be expressed as:
(10)
where is the Bessel function of first specie and second order, is the modified Bessel function of second order and specie, is the modified Bessel function of first specie and second order and is the modified Bessel function of second specie and second order. The variable is related to by:
(11)
and this expression is only valid for , i.e for a nanowire with a radius that decreases from the clamp to the free-end. The constants and must be calculated by imposing the boundary conditions for a singly clamped beam which are zero slope and deflection at the clamp and zero bending moment and shear force at the free end.
Applying these boundary conditions the following system of equations is obtained:
(12)
Since this system of equations is homogeneous, in order to have a non-trivial solution the determinant of the matrix of the system must be zero, condition from which the eigenvalue equation is obtained. It is impontant to notice that every eigenvalue obtained by numerically solving the eigenvalue equation depends on . This dependence is plotted on figure 1 for the first two modes of vibration. We can provide a simple analytical expression for the resonance frequencies by fitting the numerical solutions for to polynomial functions of . The two polynomial functions obtained from such fitting are:
(13)
Interestingly, for 0<α<0.9, the ratio of the square of these polynomial functions,i.e f2/f1 , gives a linear dependence on α. Figure 2 shows the linear fitting of this ratio with α. The ratio f2/f1 can thus be approximated by the simple expression:
(11)
After imposing the determinant to be zero, the system of equations 12 is undetermined. This means that the coefficients and will be proportional to and thus our new system will consists of just three equations, but this time the system will not be homogeneous. Once we solve the new system of equations we have the values of and and thus we have the complete expressions for the vibration mode shapes given by eq (10).
Figure 1. Numerical solution (red symbols) and polynomial fitting (blue line) for the eigenvalues kn for the first (left) and second (right) modes. The fitting are used to provide simple analytical expressions for the resonance frequencies in the range.
Figure 2. Numerical solution (red symbols), analytical approach (red line) from eq (13) and linearfitting forthe ratio f2/f1 in the range 0<<0.9.
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