SUPPEMENTAL INFORMATION
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
𝜌𝑆(𝑥)
∂2 𝑤𝑛 (𝑥,𝑡)
∂𝑡 2
∂2
+ ∂𝑥 2 (
𝐸𝐼(𝑥) ∂2 𝑤𝑛 (𝑥,𝑡)
𝐿4
∂𝑥 2
)=0
(1)
where 𝑤𝑛 (𝑥, 𝑡) is the 𝑛𝑡ℎ order mode contribution to the nanowire displacement at the
normalized longitudinal position 𝑥 and time t (actual position 𝑋 is normalized to the NW
length, 𝑥 = 𝑋/𝐿), and 𝑆(𝑥) and 𝐼(𝑥) are the cross-section area and second moment of area,
1
respectively and both depend on the longitudinal position 𝑥 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:
𝑤𝑛 (𝑥, 𝑡) = 𝑤𝑛 (𝑥)𝑒 𝑖𝜔𝑛𝑡
(2)
where 𝜔𝑛 = 2𝜋𝑓𝑛 is the angular resonance frequency of the 𝑛𝑡ℎ 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:
𝑑2
𝐸𝐼(𝑥) 𝑑2 𝑤𝑛 (𝑥)
−𝜌𝑆(𝑥)𝜔𝑛2 𝑤𝑛 (𝑥) + 𝑑𝑥 2 (
𝐿4
𝑑𝑥 2
)=0
(3)
A general solution for this equation for any tapered beam-like structure can be obtained if
𝑆(𝑥) and 𝐼(𝑥) are expressed as:
𝑆(𝑥) = 𝑆0 (1 − 𝛼𝑥)𝑚
𝐼(𝑥) = 𝐼0 (1 − 𝛼𝑥)𝑚+2
(4)
(5)
where 𝑆0 and 𝐼0 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:
𝑅(𝑥) = 𝑅0 (1 − 𝛼𝑥)
(6)
where 𝛼 is the tapering parameter given by
2
𝛼=
𝑅0 −𝑅𝐹
𝑅0
(7)
and equations (4) and (5) are written using 𝑚 = 2.
After these considerations, the beam equation can be rewritten as:
−𝑘𝑛4 𝑤𝑛 (𝑥) + 12𝛼 2
𝑑2 𝑤𝑛 (𝑥)
𝑑𝑥 2
+ (1 − 𝛼𝑥) (−8𝛼
𝑑3 𝑤𝑛 (𝑥)
𝑑𝑥 3
+ (1 − 𝛼𝑥)
𝑑4 𝑤𝑛 (𝑥)
𝑑𝑥 4
)=0
(8)
where 𝑘𝑛 is related to the resonance frequency of the 𝑛𝑡ℎ order mode by:
𝑘 4 𝐸𝐼
𝜔𝑛2 = 𝑆𝑛𝐿4 𝜌0
0
(9)
It can be shown that the solution to equation (8) can be expressed as:
4
𝑤𝑛 (𝜁) = 𝛼2 𝜁 2 [𝐴𝑛 𝐾2 (𝑘𝑛 𝜁) + 𝐵𝑛 𝐽2 (𝑘𝑛 𝜁) + 𝐶𝑛 𝑌2 (𝑘𝑛 𝜁) + 𝐷𝑛 𝐼2 (𝑘𝑛 𝜁)]
(10)
where 𝐽2 is the Bessel function of first specie and second order, 𝑌2 is the modified Bessel
function of second order and specie, 𝐼2 is the modified Bessel function of first specie and
second order and 𝐾2 is the modified Bessel function of second specie and second order. The
variable 𝜁 is related to 𝑥 by:
𝜁=
2(1−𝛼𝑥)1/2
𝛼
(11)
3
and this expression is only valid for 𝛼 > 0, i.e for a nanowire with a radius that decreases
from the clamp to the free-end. The constants 𝐴𝑛 , 𝐵𝑛 , 𝐶𝑛 , 𝐷𝑛 and 𝑘𝑛 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:
𝐴𝑛 𝐾2 (
2𝑘𝑛
−𝐴𝑛 𝐾3 (
2𝑘𝑛
) + 𝐵𝑛 𝐽2 (
𝛼
2𝑘𝑛
𝛼
) + 𝐵𝑛 𝐽3 (
2
4(1−𝛼)𝑘𝑛
𝐴𝑛 (√1 − 𝛼 (
𝐵𝑛 (√1 − 𝛼 (
𝛼2
2
4(1−𝛼)𝑘𝑛
𝛼2
2
4(1−𝛼)𝑘𝑛
𝐶𝑛 (√1 − 𝛼 (
𝛼2
2
4(1−𝛼)𝑘𝑛
𝐷𝑛 (√1 − 𝛼 (
𝐴𝑛 ((
𝛼2
2𝑘𝑛
) + 𝐶𝑛 𝑌2 (
𝛼
2𝑘𝑛
𝛼
) − 𝐶𝑛 𝑌3 (
+ 24) 𝐾0 (
− 24) 𝐽0 (
𝛼
− 24) 𝑌0 (
𝛼2
𝐷𝑛 (
𝛼2
𝛼
)+
−(
𝛼
)=0
4(1−𝛼)𝑘2
𝑛 +6)𝐾 (2𝑘𝑛 √1−𝛼)
1
𝛼
𝛼2
4𝛼(
)+
𝑘𝑛
4(1−𝛼)𝑘2
𝑛 −6)𝐽 (2𝑘𝑛 √1−𝛼)
1
𝛼
𝛼2
4𝛼(
)+
𝑘𝑛
4(1−𝛼)𝑘2
𝑛 −6)𝑌 (2𝑘𝑛 √1−𝛼)
1
𝛼
𝛼2
4𝛼(
)+
𝑘𝑛
4(1−𝛼)𝑘2
𝑛 +6)𝐼 (2𝑘𝑛 √1−𝛼)
1
𝛼
𝛼2
4𝛼(
𝛼
)=0
𝑘𝑛
2𝑘𝑛 √1−𝛼
)+
− (−
+ 384) 𝐽1 (
𝛼2
𝛼
)+
)+
𝛼
4(1−𝛼)𝑘2
𝑛 −72)
𝛼2
2𝑘𝑛 √1−𝛼
(12)
4(1−𝛼)𝑘2
𝑛 +16)𝐾 (2𝑘𝑛 √1−𝛼)
24√1−𝛼𝑘𝑛 (
0
𝛼
𝛼2
2(
4(1−𝛼)𝑘𝑛
+ 384) 𝑌1 (
4(1−𝛼)𝑘2
𝑛 +16)𝐼 (2𝑘𝑛 √1−𝛼)
0
𝛼
𝛼2
24√1−𝛼𝑘𝑛 (
𝛼
)−
+ 384) 𝐾1 (
𝛼
4(1−𝛼)𝑘2
𝑛 −72)
𝛼2
𝛼
2𝑘𝑛 √1−𝛼
+ 24) 𝐼0 (
)+
)+
2𝑘𝑛 √1−𝛼
4(1−𝛼)𝑘2
𝑛 −16)𝐽 (2𝑘𝑛 √1−𝛼)
0
𝛼
𝛼2
𝐶𝑛 ((
𝛼
)=0
𝛼
2𝑘𝑛
) − 𝐷𝑛 𝐼3 (
2𝑘𝑛 √1−𝛼
2
2 (4(1−𝛼)𝑘𝑛 +72)
4(1−𝛼)𝑘𝑛
𝛼2
2(
4(1−𝛼)𝑘𝑛
𝛼
2𝑘𝑛 √1−𝛼
24√1−𝛼𝑘𝑛 (
𝐵𝑛 (
2𝑘𝑛
) + 𝐷𝑛 𝐼2 (
𝛼
2𝑘𝑛
2𝑘𝑛 √1−𝛼
𝛼
)) +
4(1−𝛼)𝑘2
𝑛 −16)𝑌 (2𝑘𝑛 √1−𝛼)
0
𝛼
𝛼2
24√1−𝛼𝑘𝑛 (
)+
𝛼
4(1−𝛼)𝑘2
𝑛 +72)
𝛼2
𝛼2
2(
4(1−𝛼)𝑘𝑛
2𝑘𝑛 √1−𝛼
− 384) 𝐼1 (
𝛼
)) = 0
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 𝑘𝑛 obtained by
4
numerically solving the eigenvalue equation depends on 𝛼. 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 𝑘𝑛 to polynomial
functions of 𝛼. The two polynomial functions obtained from such fitting are:
𝑘1 (𝛼) = 1.150𝛼 4 − 1.115𝛼 3 + 0.697𝛼 2 + 0.336𝛼 + 1.875
𝑘2 (𝛼) = 21.337𝛼 7 − 59.437𝛼 6 + 66.091𝛼 5 − 36.654𝛼 4 + 10.501𝛼 3 − 1.512𝛼 2 − 0.440𝛼 + 4.694
(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:
𝑓2
𝑓1
≈ 6.267 − 4.103𝛼
(11)
After imposing the determinant to be zero, the system of equations 12 is undetermined. This
means that the coefficients 𝐵𝑛 , 𝐶𝑛 and 𝐷𝑛 will be proportional to 𝐴𝑛 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 𝐵𝑛 , 𝐶𝑛 and 𝐷𝑛 and thus we have
the complete expressions for the vibration mode shapes given by eq (10).
5
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.
6
Figure 2. Numerical solution (red symbols), analytical approach (red line) from eq (13) and
linear fitting for the ratio f2/f1 in the range 0<<0.9.
7