Mass Sensing Based on Deterministic and Stochastic Responses of

NANO LETTERS

Mass Sensing Based on Deterministic and Stochastic Responses of Elastically Coupled Nanocantilevers

2009 Vol. 9, No. 12 4122-4127

Eduardo Gil-Santos,† Daniel Ramos,† Anirban Jana,‡ Montserrat Calleja,† Arvind Raman,§ and Javier Tamayo*,† Instituto de Microelectro´nica de Madrid, IMM-CNM (CSIC), Isaac Newton 8 (PTM), Tres Cantos, 28760 Madrid, Pittsburgh Supercomputing Center, Carnegie Mellon UniVersity, 300 S. Craig Street Pittsburgh, PennsylVania 15213, and The Birck Nanotechnology Center and School of Mechanical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 Received July 22, 2009; Revised Manuscript Received September 9, 2009

ABSTRACT Coupled nanomechanical systems and their entangled eigenstates offer unique opportunities for the detection of ultrasmall masses. In this paper we show theoretically and experimentally that the stochastic and deterministic responses of a pair of coupled nanocantilevers provide different and complementary information about the added mass of an analyte and its location. This method allows the sensitive detection of minute quantities of mass even in the presence of large initial differences in the active masses of the two cantilevers. Finally, we show the fundamental limits in mass detection of this sensing paradigm.

Micro- and nanomechanical resonators such as vibrating microcantilevers have emerged as key elements in a wide variety of techniques and devices, such as dynamic atomic force microscopy,1,2 mass sensors,2-6 and radio frequency filters.7 The main virtues of nanomechanical resonators are their high quality factor, tiny size, and low spring constant that make their resonance frequency extremely sensitive to external forces and adsorption phenomena. Moreover, nanomechanical resonators can be fabricated at the wafer scale and at low cost by adopting semiconductor industry protocols. The continuous advancement in micro- and nanofabrication tools has made possible increasingly smaller nanomechanical resonators capable of detecting, based on the resonance frequency shift, adsorbed masses and external forces with subattogram2,3,8-10 and subattonewton resolution,1,2 respectively. In all these applications, the dynamics of a single nanomechanical resonator driven by different techniques including self-excitation schemes and subject to linear and non linear interactions have been thoroughly investigated.11-14 In the area of mass sensing, the performance of isolated micro/nanocantilevers sensors could be significantly enhanced by using an array of elastically coupled nanome* To whom correspondence should be addressed. E-mail: jtamayo@ imm.cnm.csic.es. † IMM-CNM. ‡ Carnegie Mellon University. § Purdue University. 10.1021/nl902350b CCC: $40.75 Published on Web 09/23/2009

 2009 American Chemical Society

chanical resonators of identical size. In these systems, each individual resonance frequency for a single cantilever system splits into N frequencies for the array of N coupled cantilevers, and elastic waves consisting of entangled eigenstates easily propagate through the array.15-19 The addition of a small amount of mass on one of the cantilevers leads to the localization of the vibration in the array in a similar way as the Anderson’s localization. The use of coupled microcantilevers and the associated vibration localization due to mass disorder has been recently used by Spletzer et al.20,21 to develop a mass sensor. However, further development of this kind of device has been hindered by the initial disorders (that is, initial differences in the individual masses or stiffnesses) produced in the coupled system from manufacturing errors that break the required initial vibration delocalization. To keep the initial disorders relative to the overall mass and stiffness small, the overall sizes of the mass sensors based on coupled cantilevers have so far been quite large enabling only picogram-level mass detection.20,21 Another major challenge associated with coupled cantilever arrays is that piezoelectric base excitation applied to the ensemble of the cantilevers cannot efficiently excite many of the entangled coupled eigenstates thus limiting the sensitivity of such arrays. A third disadvantage of the existing method with respect to resonance frequency based mass sensors, is the need of an accurate calibration of the readout technique to calculate the amplitudes. Finally, the fundamental physics

Figure 1. (a) Scanning electron micrograph of a system of coupled cantilevers. The cantilevers were fabricated in low stress silicon nitride. The length, width, and thickness of the cantilevers were 25, 10, and 0.1 µm, respectively. The gap between the cantilevers is 20 µm. The structural coupling between the cantilevers arises from the overhang connecting the cantilevers at the base, which is about 8 µm long. (b) Symmetric and (c) antisymmetric mode of vibration of this coupled array. (c) A lumped parameter model for this coupled array.

that describes the dynamic behavior of coupled nanomechanical resonators have not been fully established especially with regard to their stochastic response. Thus, the ultimate limits of mass detection that these devices can achieve are still uncertain. In this article, coupled nanocantilever systems were fabricated with effective masses of 17 pg, more than 3 orders of magnitude smaller than in previous studies.20,21 It is shown that the stochastic response of such coupled systems provides an important transduction scheme for observing analyte massinduced localization of all entangled eigenmodes bypassing many limitations of base excitation. We explain our experimental observations on the basis of the theory of coupled harmonic oscillators and the fluctuation dissipation theorem (FDT).17,22 We also quantify the important role of fabrication imperfections when the size of the resonators is small in comparison with the lithography tolerance. Based on this description, we propose a new paradigm for mass sensing using a pair of elastically coupled cantilevers, in which initial disorders are no longer required to be near zero, thus removing a huge obstacle in the further development of this kind of sensor. Moreover, the stringent need of calibration of the amplitudes is eliminated. We also suggest applying this method to thermomechanical fluctuations which do not suffer from the poor excitation of the antisymmetric mode. Finally, we present the theoretical limits of mass detection using eigenmode localization in coupled nanomechanical oscillator arrays. Figure 1a shows a scanning electron micrograph of the device, which consists of two nominally identical nanocantilevers, coupled by an overhang at their bases. A number of these cantilever pairs were fabricated from low stress Nano Lett., Vol. 9, No. 12, 2009

silicon nitride by optical lithography. The fabrication details will be described elsewhere.23 The length, width, and thickness of each cantilever is nominally 25, 10, and 0.1 µm, respectively. The gap between the cantilevers is 20 µm. The overhang connecting the cantilevers at the base is 8 µm long and is responsible for the elastic coupling. Such a coupled system exhibits two fundamental vibration modes: a symmetric mode (Figure 1b) and an antisymmetric mode (Figure 1c). For our mass-sensing experiments, the above coupled cantilever pair was housed in a vacuum chamber whose internal pressure was maintained at 10-5 mbar. The frequency spectra of the vibrations of the coupled cantilevers were measured using a homemade Michelson interferometer with a He-Ne laser with a spot size of about 3 µm. To add a small mass on one of the cantilevers (hereinafter labeled as 2), the cantilevers were transferred to a field emission electron microscope (FE-SEM) where we focused the electron beam on the cantilever’s free end. The electron beam produced the deposition of an amorphous carbon layer on the tip of cantilever 2 through the dissociation of organic species present in the vacuum chamber.24 A homemade cantilever holder compatible with the vacuum chambers of the optical readout and the FE-SEM was used to minimize cantilever manipulation during the transfer between both chambers. This is crucial to avoid contamination that arises when cantilevers are manipulated with tweezers. A future improvement to achieve the ultimate limits in mass sensing shall be to minimize the contamination that it may come up when the cantilevers are exposed to air by depositing the mass in the readout vacuum chamber itself.25 The added mass was calibrated by adding mass to the free end of a single cantilever and deducing the mass by using the Euler-Bernoulli beam theory.26 Notice that changes in the cantilever stiffness can be ruled out as the mass was deposited on the cantilever tips.26 Two distinctly different types of vibration spectra were measured: (i) the amplitude spectra of vibrations externally excited by applying a voltage across a piezoelectric bimorph placed near the cantilevers’ bases, shown in Figure 2a, and (ii) the spectra of inherent thermomechanical fluctuations present in the system even in the absence of any external excitations, shown in Figure 2b. Figure 2 shows that, before the addition of mass, both vibration spectra exhibit one resonance peak at ∼133.7 kHz, corresponding to the symmetric mode, and another resonance peak at ∼146.6 kHz, corresponding to the antisymmetric mode. The quality factors of these resonance peaks are about 3500. However, for the base-excited vibrations, the height of the antisymmetric peak is ∼50 times smaller than that of the symmetric peak. Indeed, the antisymmetric peak is hard to detect in this case. This is because the forcing on the two cantilevers arising from external base excitation are in phase. On the contrary, for the thermomechanical noise spectra, the height of the antisymmetric peak is only slightly lower (about 10%) than that of the symmetric peak before the addition of mass. The above observations suggest that ultrasensitive mass sensing schemes using coupled arrays could measure the thermomechanical fluctuations, instead of base-excited vibrations 4123

Figure 2. Vibration spectra of a coupled array before (black, broken line) and after (red, solid line) the deposition of 170 fg of mass on cantilever 2. (a) Deterministic response driven by external base excitation using a piezoelectric actuator. (b) Stochastic response driven by ambient thermal excitation. All the spectra have been normalized such that the height of the symmetric peak equals 1.The insets show a zoom of the relative change in the amplitude of the antisymmetric mode. The amplitude of the symmetric mode in the driven oscillation was 90-100 times larger than the one observed in the natural thermal response.

as in earlier attempts. Furthermore, being inherent in all systems, thermomechanical excitation has the added advantage of not requiring any excitation circuitry, thereby reducing the complexity and cost of implementation. In the sensing scheme proposed here, the detection of adsorbed mass is based on the change in relative heights of the symmetric and antisymmetric peaks in the vibration spectra. Figure 2 shows the change in the vibration spectra when a 170 fg mass is deposited on cantilever 2. To ease the observation of how the relative amplitudes of the eigenmodes change, the spectra have been normalized such that the height of the symmetric peak equals 1. We see that the change in the relative height of the antisymmetric peak is intimately dependent on whether the cantilevers are excited by external forces or random thermal forces. With base excitation, the antisymmetric peak increases in height relative to the symmetric peak by about 2.5% after the mass deposition. This behavior occurs on both cantilevers. However, the thermal resonance peak amplitudes respond in a different manner to mass adsorption. In this case, while the antisymmetric peak rises relative to the symmetric peak on cantilever 1 (ca. 7%), the opposite behavior is observed on cantilever 2. To understand the above observations, we model the cantilever array as the lumped parameter system shown in Figure 1d. In this model, m1 and m2 represent the active masses of the two cantilevers, k1 and k2 represent their individual bending stiffnesses, and c1 and c2 represent their structural dissipations. The structural coupling of the cantilevers due to the overhang is modeled by the coupling spring of stiffness k12. For our coupled cantilevers, we estimate that k ≡ k1 ) k2 ) (0.012 ( 0.004) N m-1, κ ≡ k12/k ) (0.10 ( 0.01), c ≡ c1 ) c2 ) (4.10 ( 0.05) × 10-12 kg s-1, m ≡ m1 ) (17 ( 2) pg, and m2 ) m(1+δ), where δ ) (m2 - m1)/m1 represents the relative differential mass between the two cantilevers. For realistic cantilever arrays, δ * 0 even before any mass deposition because of fabrication imperfections. 4124

As a first step, we calculate the deterministic response of the cantilever array subject to external driving forces. Let x(t) ) [x1(t), x2(t)]T be the vector of (deterministic) cantilever tip displacements when subjected to external driving forces fd(t) )[fd1(t), fd2(t)]T. The deterministic dynamics of the above coupled system is governed by the system of differential equations Mx¨ + Cx˙ + Kx ) fd

(

(1)

)

1 0 , the damping where the mass matrix M ) m · 0 1+δ 1 0 matrix C ) c · , and the stiffness matrix K ) k · 0 1 1+κ -κ . The susceptibility matrix, χ(ω), is defined to -κ 1+κ linearly relate the Fourier transform of the cantilevers’ response xˆ(ω) to that of the driving forces fˆd(ω) as xˆ(ω) ) χ(ω)fˆd(ω). By taking the Fourier transform of eq 1, it can be shown that

(

)

( )

χ(ω) ) (-ω2M + iωC + K)-1

(2)

where the Fourier transform of a time series y(t) is defined ∞ y(t)e-iωt dt. as yˆ(ω) ) ∫-∞ In the case of external base excitation, ˆfd(ω) ) F0·[1 1]T. The stochastic response of the cantilevers due to the random thermal forces can be related to their deterministic, externally driven response via the Fluctuation Dissipation Theorem (FDT) as follows. Let S(ω) be the matrix of power spectral densities of the thermomechanical displacement fluctuations of the cantilever array at equilibrium. Then according to the FDT22 S(ω) ) -

2kBT S (χ + iχAR ) πω I

(3)

Nano Lett., Vol. 9, No. 12, 2009

Figure 3. Variation of the amplitude of the antisymmetric mode normalized to the amplitude of the symmetric mode as a function of the disorder coefficient, η, when the cantilevers are (a) base excited, and (b) thermally excited. The data for cantilevers 1 and 2 are plotted in violet and orange, respectively. Solid lines represent the numerical calculations based on our theory, the dashed lines represent the analytical approximations (eq 4) and the symbols the experimental data.

where χIS ) (Im(χ) + Im(χ)T)/2 is the imaginary, symmetric part, and χAR ) (Re(χ) - Re(χ)T)/2 is the real, antisymmetric part, of the mechanical susceptibility matrix, kB ) 1.38 × 10-23 J/K is the Boltzmann constant, T is the absolute temperature and i ≡ -1. The diagonal elements of S(ω) are the auto power spectral densities measured on the individual cantilevers. We propose that the key quantity for mass sensing be the ratio of the height of the antisymmetric peak to that of the symmetric peak in the vibration spectra of a cantilever. For external base excitation, this ratio can be defined as aBk ) |xˆk(ω2)/xˆk(ω1)|, and for thermomechanical excitation as aTk ) Skk(ω2)/Skk(ω1). Here ω1 and ω2 are the peak frequencies for the symmetric and antisymmetric mode respectively, and the subscript k ) 1 and 2 for cantilevers 1 and 2. Figure 3 shows the theoretical prediction (solid lines) of how the ratio of peak heights for the two cantilevers vary as a function of the mass disorder coefficient defined as η ) δ/(4κ). As shown later, this normalization provides a more universal description of the response of this kind of nanomechanical systems. Let us first examine the deterministic response due to base excitation (Figure 3a). When η ) 0, the ratio of peak amplitudes aBk ≈ 0 for both cantilevers, Nano Lett., Vol. 9, No. 12, 2009

since the antisymmetric mode is hard to excite via base excitation, especially in lightly damped systems. As η is made more and more positive (for example, by gradually depositing mass on cantilever 2), the ratio aBk initially increases for both cantilevers as observed in the experiments. When η is increased further, the ratio aB2 for cantilever 2 quickly reaches an almost constant value. However, for cantilever 1, the ratio aB1 continues to increase. For the thermomechanical noise response (Figure 3b), when η ) 0, aTk ) 1/(1 + 2κ)1/2 < 1. That is, for perfectly identical cantilevers, the antisymmetric peak, although very prominent, is still slightly lower than the symmetric peak, due to the stiffness of the antisymmetric mode being greater compared to the symmetric mode. In this case, as η is made more and more positive, the ratio aT1 for cantilever 1 increases, whereas the ratio a2T for cantilever 2 decreases. This is exactly in accordance with our experimental observations. We also see that aT2 is less sensitive to mass disorder for η > 0.2. For η < 0 (which can be achieved, for example, by gradually depositing mass on cantilever 1), the roles of the cantilevers get switched (not shown in Figure 3). The theoretical predictions shown above are exactly in accordance with the experimental data shown in Figure 3 (symbols). This data comes from chips with inherent mass disorder due to the microfabrication tolerance. We find that when using base excitation, among the two cantilevers, the one with less active mass in general responds more sensitively to the added mass. Furthermore, as long as the ratio of peak heights is measured on the lower active mass cantilever, a large range of added masses can be accurately measured without a significant dropoff in sensitivity. A consequence of this is that even arrays with large initial mass disorders can sensitively detect minute quantities of adsorbed mass. However, the deterministic response, especially when the initial mass of both cantilevers is very similar, cannot provide information about in which cantilever the mass is added. This makes the measurement of the thermomechanical noise response crucial. The fact that the stochastic response of the two cantilevers respond in an opposite fashion enables one to easily identify to which cantilever the mass is added. Simple analytical expressions for aBk and aTk can be derived from the theory by making the following assumptions: c , (mk)1/2 (very light damping) and δ , 1, δ , κ (small mass disorders). Under these assumptions, it can be shown that aB1 ) √1 + 2κ · η[1 + 2η] aB2 ) √1 + 2κ · η[1 - 2(1 + 2κ)η] aT1 )

1 [1 + 4(1 + κ)η + 4(2 - κ)η2]1/2 √1 + 2κ

aT2 )

1 [1 - 4(1 + κ)η + 4(2 + 3κ)η2]1/2 √1 + 2κ

(4)

Plots of the above formulas are depicted in Figure 3 by dashed lines. For our cantilevers, these approximate analytical 4125

Figure 4. Numerical calculation of the minimum detectable differential mass for the devices shown in Figure 1 as a function of the initial mass difference, δ0 ) (m2 - m1)/m1 set by the fabrication imperfections. The data for cantilevers 1 and 2 are plotted in violet and orange, respectively. The dashed lines represent the analytical approximations (eq 5).

formulas show a good agreement with the full theoretical predictions for η < 0.2. Finally, we estimate the minimum detectable mass for our mass sensing scheme when the cantilever array is base excited, by postulating that the uncertainty in the measured added mass is primarily due to thermomechanical noise contaminating the measured vibration spectra of the base excited cantilevers. This minimal detectable mass is plotted as a function of the initial relative differential mass between the two cantilevers δ0 ) (m2 - m1)/m1 in Figure 4, for a 1 Hz measurement bandwidth of the lock-in amplifier, and an external forcing amplitude F0 ) 1 pN. In these conditions, the minimal detectable mass is just below one femtogram. For identical cantilevers, the fundamental minimum detectable mass approximately is



∆mmin ≈4 m

Eth QB κ Ec ω0 1 + 2κ

(5)

where ω0 ) (k/m)1/2, Q ) mω0/c, B is the measurement bandwidth, Eth ) kBT is the thermal energy, and Ec ) Q2F02/k is a measure of the maximum oscillation energy. This approximate expression, indicated in Figure 4 as a dashed line suggests that, the key for achieving the performance limits based on elastically coupled mechanical resonators is to decrease the coupling strength κ. In fact, ∆mmin approximately decreases as κ. This minimum detectable mass differs from that based on resonance frequency shift27 by a factor of ∼Qκ. Thus, depending on the quality factor and elastic coupling strength, one method may have better or worse sensitivity to mass adsorption. Notice that coupled nanomechanical resonators can simultaneously employ both resonance frequency shift and vibration localization for robust detection of adsorbed masses. The former gives the total added mass on the two cantilevers and the latter the differential mass and distribution. It is important to stress that in biological adsorption experiments the limiting source of noise arises from the random nonspecific adsorption events 4126

of the molecules and other possible contaminants coming into play during the incubation process.28 So biological sensing with nanomechanical resonators would largely benefit from the differential measurements provided by coupled nanomechanical resonators. The minimum coupling strength that can be attained will be that for which the amplitude peaks of the antisymmetric and symmetric eigenmodes can be resolved. A basic criteria to establish the minimum coupling strength would be that the frequency separation between both peaks, ∼κminω0, should be larger than the half-width of the resonace peaks, ∼ω0/Q. In this situation, κmin ∼ 1/Q and the minimum detectable mass by the coupled nanomechanical resonators would be similar to that obtained by the resonant frequency method. However, eq 5 is derived using standard perturbation method assuming ∆δ , κ and assuming approximately identical cantilevers δ0 , κ. When κ is extremely small both assumptions can not hold. In particular, small fabrication imperfections produce cases in which δ0 ∼ κ, Thus, a general theoretical derivation of the limiting case when the coupling strength is extremely small requires a singular perturbation analysis that is beyond the scope of this manuscript.29 In conclusion, we have presented a new paradigm for ultrasensitive mass detection using a pair of elastically coupled nanocantilevers, based on the stochastic and deterministic resonant responses. The results were interpreted by developing a theoretical model based on the theory of harmonic oscillators and the fluctuation dissipation theorem. We demonstrate with these sensors and this method information about the added mass at the femtogram level and about its distribution on the two cantilevers. This allows differential mass measurements that are crucial in biosensors for discrimination of specific target binding from contamination and non specific adsorption. Moreover, the mass resolution attained here can be improved dramatically by using smaller resonators and tuning the elastic coupling. This miniaturization approach is attainable with the proposed method due to its relative insensitivity to the initial disorder. Acknowledgment. D.R. acknowledges the fellowship funded by the Autonomous Community of Madrid (CAM). E.G.-S. acknowledges fellowship funded by C.S.I.C. J.T. and M.C. acknowledge financial support by Spanish Ministry of Science under Grant Nos. TEC2006-10316 and CSD 200700010, CSIC-200650F0091, and Autonomous Community of Madrid under Contract S-0505/MAT-0283. All of the authors acknowledge Hien Duy Tong from Nanosens for the fabrication of the devices. References (1) Rugar, D.; Budakian, R.; Mamin, H. J.; Chui, B. W. Nature 2004, 430 (6997), 329–332. (2) Li, M.; Tang, H. X.; Roukes, M. L. Nat. Nanotechnol. 2007, 2, 114– 120. (3) Waggoner, P. S.; Craighead, H. G. Lab Chip 2007, 7, 1238–1255. (4) Braun, T.; Ghatkesar, M. K.; Backmann, N.; Grange, W.; Boulanger, P.; Letellier, L.; Lang, H. P.; Bietsch, A.; Gerber, C.; Hegner, M. Nat. Nanotechnol. 2009, 4, 179–185. (5) Dorrestijn, M.; Bietsch, A.; Ac¸ikalin, T.; Raman, A.; Hegner, M.; Meyer, E.; Gerber, Ch. Phys. ReV. Lett. 2007, 98, 026102. (6) Varshney, M.; Waggoner, P. S.; Tan, C. P.; Aubin, K.; Montagna, R. A.; Craighead, H. G. Anal. Chem. 2008, 80 (6), 2141–2148. Nano Lett., Vol. 9, No. 12, 2009

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