A mechanical nanosensor in the gigahertz range: where mechanics meets electronics

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surface science letters ELSEVIER

Surface

Science

Letters

301 (1994) L224-L228

Surface Science Letters

A mechanical nanosensor in the gigahertz range: where mechanics meets electronics Vu Thien Binh ‘, N. Garcia h-c, A.L. Levanuyk ” ~~p~~rfcrn~nf de h Departamento

’

P~~~si~ue des ~~ilt~ri~~, GA CNRS, Unil~~r~i~~ Cfuude ~~~n~r~~Lyon I, 69622 Vi~~eurb~F~?~e~ France de Fisica de la Muteria Condensudu, Unir,ersidcrd Autonoma de Madrid. 28049 Mudrid, Spain ’ FISINTEC, Ruperto Chap 19. Alcohendus, Mudrid 28100, Spain (Received

13 September

1993; accepted

for publication

27 September

19931

Abstract To overcome the limits which are intrinsic to standard macroscopic cantilevers and to have structures possessing, in concomitance, mechanical resonance frequencies of the order of the GHz and spring constants around the N/m, we propose the use of tips ending with a nanoscale filiform constriction and a solid drop. We show that such tips were obtained by a surface diffusion mechanism in a reproducible manner. Using a model to describe such a structure, which is a nanocantilever, the mechanical characteristics were calculated with the known bulk parameters. Estimations using known equations show then the possibility to use such nanocantilevers as mechanical sensors to measure mass variations of few atoms.

Theoretical [l] and experimental studies [2,3] have shown that, by the heating of fine wires terminated with sharp tips, it is possible to have at the tip the formation of a solid drop that is connected to the rest of the wire by a narrow neck. Imagine that the realisation of the following system is possible: a ball of 100 nm size connected to the shank by a neck of 10 nm diameter and a length of 100 nm, i.e., a filiform neck. Elasticity theory shows that this nanocantilever would have resonance frequencies between 0.01 and 1 GHz and spring constants between 0.01 and 10 N/m. This is fascinating because these mechanical frequencies approach those obtained with electronic devices. Theoretical estimates show that these nanocantilevers

should have Q factors larger than 10’ for an environment pressure < 0.1 Torr. With these parameters, it should be possible to measure lo-*’ g of mass variations 141, i.e., the mass of a few atoms. We present in this article the experimental realisations of W and Ni nanocantilevers. New possibilities and applications in mechanical devices should be feasible, as for example the recentIy proposed atomic-scale nuclear magnetic resonance imaging [5] or an atomic sensitive mass sensor. Controlled surface diffusion can be used for manipulating atoms and fashioning structures of nanometer dimensions that are thermodynamically stable. The fashioning of nanostructures by this mechanism means a morphological modifica-

0167.2584/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI Olh7-2584(93)E0215-F

Vu Thien Binh et al. /Surface Science Letters 301 (1994) L224-L228

tion by mass transport of surface atoms. The surface diffusion transport flux, J, is described by the Nernst-Einstein equation [6]: J=

-

where n is the concentration of surface diffusing atoms, D is the surface diffusion coefficient, k, is the Boltzmann constant, 7’ is the temperature and Al,is the chemical potential. This equation shows how the flux can be controlled by the temperature and tip geometry. The temperature gives the concentration of adatoms, R, and influences D through the Arrhenius law. For isotropic surface energy the driving force, grad&), which controls the direction of the atomic flow, can be described by [61:

acts)

grad(p) = yfl-g---’

(a) _ w

OZ!P

where y is the surface tension, fI is the atomic volume and C(s) is the local cu~ature along the curve s on the tip. Using surface diffusion we have fabricated nanocantilevers which we estimate below to have GHz resonance frequencies in combination with very low spring constants. Eqs. (1) and (2) say that, as T increases, the surface atoms move from regions of large curvature towards regions of smaller curvature in order to minimise the total energy. Numerical simulations have shown that for conical tips of angle larger than N 6”, the surface diffusion biunts the tip. However, for tips with smaller conical angle, a constriction will develop just under the apex region which evolves with the heating time leading eventually to the detachment of a solid drop. This behaviour was predicted by theory [l] and confirmed experimentally [2]. By stopping the heating just before the detachment of the solid drop, a tip ending in a

0.5 firn

0.5 pm

(C) - Ni

Fig. 1. Scanning electron microscopy images. (a) W nanocantilever. (b) The same W nanocantilever with a higher magnification showing the filiform neck. (c) Ni nanocantilever. Notice the presence of facets on the Ni ball that formed during the heat treatment.

Letters 301 (19941 L224-L228

‘Hall” witl- a cantilever formed by the constriclion, i.e.. ;I “ball-tip” was obtained. As an examplc SL’CRc ‘, [3] where W and Cu ball-tips were pruscntcd. We have proposed 131that such balltips can be used as weIl-defined probes in atomic force micrc scopy. The analysis below shows that these tips Lre cantilevers having more than one GHz reson mce frequencies with spring constants > w 100 N ‘m. What is extremely interesting is to have cantilevers wit 1 resonance frequencies of the order of the GH. but with a spring constant around 1 N/m. This means an increase in the length of the neck regio I, a filiform profile. Such tips were successfull! fabricated for example with the surface diffusi In process in the presence of a differcntial evaporation in the neck region [2]. For W tips for ex lmple, this differential evaporation is obtained b’ heating small angle tips at - 1800 K in 10-“-IC -’ Torr of oxygen [2]. This process is governed b i thermodynamical laws and therefore the format: 3n of filiform necks is reproducible. It must be nc ted that with electron microscopy one can easily ‘ollow the morphological evolution of the nanoc; ntilevers which can be therefore tailored to sllecific requirements. Figs. la and lb show scanr ing electron microscopy @EM) images of a typic;.1 W cantilever having the following dimensions: ball radius of w 150 nm, cantilever diameter i nd length of N 10 and - 200 nm, respective! s. In order to characterise the mechanic behaviour of the obtained nanocantilevers, we have estimated .he resonance frequencies wg, spring constant k and quality factor Q by calculations using elasticity theory [7]. A fuller description of the relatio IS and estimations given below will be presented in a forthcoming paper [S]. The nanocantill.:ver tip is modelled as a ball of radius R, with a cylindrical constriction of Iength L and radius r, connected to a larger cylinder of length L, = 1OL ,.md having the same radius R as the ball. This ensemble is prolonged by a cylinder with a dianleter a, = IOOR and a length 5OL. The origin of x is taken at the connection point between the constriction and the cylinder L,. See Fig. 2 for a sketch of the nanocantilever model considered in the calculation.

a) approximate real shape of the lip _.-----

b) nanocantilever

Fig. 2. (a) Sketch of the real profile of a nanocantilever Model of the nanocantilever used for calculations mechanical characteristics (r = R/10).

The lateral displacement erned by: EI-

d4z dx4

= pSo2z,

tip. (h) of the

z of a beam is gov-

(3)

where E, I, p, S are, respectively, the Young modulus, the massless moment of inertia, the density and the cross section of the cantiiever; w is the frequency of an elastic wave moving along the cantilever at the distance x. To estimate the spring constant and resonance frequency of the system, we can use the foIlowing boundary conditions: (a) At x = 0, we have z = 0 and dz/dx = 0. (b) At x = L, we have d2z/dx2 = 0 and mbmZz = EZ d3z/dx3, where mb is the mass of the ball. It can be shown that the lowest resonance frequency of the cantilever o0 is given by: 3EI &);Z.Z-

mL” *

(4)

with m = mb + (15/14fm, and m, is the mass of the cantilever. Typical values for nzb and m, are lo-“” g and 10-l” g, respectively. Note that Eq. (4) is valid when &!+ < 1. This is true in the present case because the wave vector is described by K2 = w(~S/EI)‘/~, giving then KL - lo--“.

Vu Thien Bid ef al./Surface Science Letters301 (1994) L224-I-228

The spring constant k is given by: 3a Er”

k=qyp

(5)

where r is the cantilever radius. By using the above relations, we find 0.06 GHz and 0.3 N/m for the resonance frequency and spring constant respectively for the nanocantilever of Figs. la and lb. In Fig. lc we show a SEM image of a Ni nanocantilever, having values of 250, 20 and 400 nm for ball radius, constriction radius and length, respectively. The related resonance frequency and spring constant are then appro~mately 0.02 GHz and 3 N/m, respectively. The Q factor of the system is estimated considering three relevant contributions to the losses @I: (1) The viscosity of the cantilever. For W, this contribution gives a value for the Q factor of N 10’. (2) The energy transfer to the rods supporting the cantilever. This gives a lower estimation Q factor of - 10”. (31 The viscosity or friction with the surrounding atmosphere in which the cantilever is immersed: this is governed by a Stokes-like formula when the mean free path of the gas is larger than the ball dimension. A nice solution to this problem can be seen for example in Ref. [9], which estimates the Q factor to be wO/P, with ~tf* in s-i and P the surrounding pressure in dyn/cm2. This gives, for the nanocantilevers, Q > lo5 when P < 0.1 Torr. It is important to mention that the fabrication of these nanostructures obey thermodynamic laws and therefore allow the reproducibIe realisation of cantilevers having frequencies and spring constants in the range of 0.01-l GHz and 0.01-10 N/m, respectively. They are controlled by the choice of the material and the various dimensions of the nanocantilever. Among the different possible applications is the use of this nanocantilever as a mechanical nanosensor to measure the mass variation. By using the above values for the masses, the resonance frequencies and the Q factors, we can estimate the variation of the mass of the ball, 6m,

which could be detected by measuring the variation of the nanocantilever resonance frequency 141.This is given by the following relation: 6m = 2m,Q-‘.

(6)

Eq. (6) shows that for mb = IO-‘” g and a Q factor of N 105, it is possible to detect a variation of mass of N lo- ” g with such a nanocantilever. This latter value represents *u3000 of W atoms deposited on the ball. When measuring a frequency change due to a mass change, the experimental procedure will involve vibrating the cantilever and detecting the frequency change directly (FM detection) or detecting a change in oscillation amplitude (slope detection) if the drive frequency is constant [lo]. Let’s assume that FM detection is used, the minimum detectable frequency change is [lo]:

where A is the amplitude of oscillation and B is the bandwidth. Assuming A = 10-s m, w0 = 2~ x60 MHz, T=300 K, B=l Hz, k=0.3 N/m and Q = 105, then the minimum detectable frequency change is 6w = 0.7 Hz. The corresponding change in mass is 6m = 2m, 6w/wo, which gives a value of 3.7 x 10Pz2 g with mh = lo-” g. This is equivalent to a mass variation of about 1.2 tungsten atoms. Both estimations are rather crude for they do not take into account all the experimental parameters, such as thermal noise, measurement bandwidth, etc., that could put a threshold for the minimum detectable frequency change. However, these values give orders of magnitude that could be reached by using the nanocantilever tip as a mechanical sensor. In conclusion, we have presented here the experimental realisation of nanocantilevers with estimated resonance frequencies of 0.01 to 1 GHz, spring constants of 0.01 to 10 N/m and Q factors of N 10”. These nanocantilevers could be used for measuring mass variations approaching that of one atom. Moreover, these cantilevers can open new fields as mechanical sensors and magnetic detectors, as for example in NMR imaging at the atomic scale [5], with all its ensuing appli-

he life science studies. This could be ‘or studies of chemical reactions in th the atomic-scale structure of its iurface; one can imagine an experi3mic force microscopy using one of :antilevers in which the rate of mass d the topography of the supported Id be measured concomitantly. In genexperiments which seemed an utopia rendered possible by the use of these vers. This represents a non-negligible d.

El

Vu Thien

Binh and R. Uzan, Binh,

Drechslcr.

J. Phys. E: Scientific

131N.

Garcia

and Vu Thien

k has been supported by the Spanish I government agencies. We thank H. discussions and for mentioning to us ity of using these nanocantilevers as .ors. Discussions with S.T. Purcell are iated.

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