micro- and nano systems based on vibrating structures

8/6/2019 Micro- And Nano Systems Based on Vibrating Structures

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978-1-4244-1882-4/08/$25.00 2008IEEE

PROC. 26th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2008), NI, SERBIA, 11-14 MAY, 2008

Micro- and Nanosystems Based on Vibrating Structures

Z. Djuri

Abstract - Invention of the scanning force microscope in1986 and development of micro and nanofabrication technologies

yielded a new generation of vibration based miniature sensors of physical, chemical and biological parameters, with sensitivitywhich could not be achieved before. This promoted again the

research of micro and nanoelectromechanical systems (MEMSand NEMS) in the fields that have been dominated by

semiconductor electronics for more than a half of century.In this paper the principles of operation of the miniature

vibrating structures are given. A particular attention is given to

the fact that the mass of these structures is of the order of tens of picograms, and that their dimensions can be in the sub-

micrometer range, where the effects of Brownian motion ofparticles in the surroundings become significant. These effects are

expressed as thermomechanical noise which in most casesdetermines the ultimate sensor performances. Also, adsorptionand desorption (AD) of particles (atoms, molecules) on the

surface of miniature vibrating structures generate the AD noise.Vibrating micro- and nanostructures are important not only

for the new sensor components, but also for a multitude of other

applications. As an illustration, I will mention a new generationof MEMS oscillators, which successfully replaces the traditional

quartz oscillators in some contemporary applications. Also, a newgeneration of processors which utilize built-in miniature

mechanical structures is envisioned.

I. INTRODUCTION

The idea to exploit the basic operating principles ofatomic force microscope (AFM) for sensing devices

appeared immediately after invention of AFM [1-10].During the past ten years or so, based on the platform of

micro- and nanocantilevers, a new generation of sensorsemerged for measurement of physical, chemical, biophysical and biochemical parameters with very highsensitivity, low energy consumption and high reliability.Most of these sensors operate in the oscillation mode. Thismode establishes either as a result of an external excitation

or due to the self-oscillation effect. In both cases themeasured parameter affects the amplitude, phase or

frequency of oscillation of the vibrating structure.The vibrating structures can be made either of one

material only or of several different materials. This factexpands the range of application of vibrating structures.

For example, if a cantilever is made of two materials ofdifferent coefficients of thermal expansion, a bimaterial

effect occurs (bending of the cantilever due to change of

the temperature), that can be utilized for temperaturemeasurement, for detection of infrared radiation [10] etc.

There has been an explosion in the use of micro/nano-cantilevers for sensing of various biological species [3-5].The presence of a surrounding fluid significantly influencesthe vibration characteristics of such devices. For example,the cantilever resonant frequency decreases by an order ofmagnitude when it is moved from the air in a liquid, while

the Q-factor decreases by two orders of magnitude. Thedetection of single molecules in a liquid is the grand goalof microcantilever biosensors and could have a significantimpact in the field of genomics and proteomics [3].

Vibrating microstructures are used as frequencydetermining components of MEMS oscillators [11]. In

modern wireless telecommunication equipment there is apressing need for substitution of the quartz oscillators withthe smaller MEMS oscillators, which are manufacturedusing silicon technologies and thus can be integrated on thesame chip with other electronic components. This enablesfurther miniaturization of the electronic devices. Thecommercial use of MEMS oscillators has recently begun.

During the last several years great research effortshave been made to utilize the vibrating nanostructures as basic components of a new generation of computers, theso-called nanomechanical computers (NMC) [12, 13]. The

basic element of NMC is a nanoelectromechanical singleelectron transistor, which contains a nanocantilever.In the next chapter a theoretic analysis of an

oscillating structure will be given, which enables us to

describe the operating principle of sensors which are usedin the dynamic mode, in gaseous and liquid environments.The same theory is applicable to the analysis of vibratingstructures which are used as MEMS oscillators, and alsothose intended for the future nanomechanical computers.The theory of the main noise mechanisms in vibrating

micro- and nanostructures will also be presented. It is usedfor determination of the ultimate performances of suchdevices. A short overview of the MEMS oscillators will be

given, including their advantages and the problems that hadto be solved during the years of development of thesecomponents. Finally, the nanomechanical computers will

be presented as an example of another interesting andpromising application of vibrating nanostructures.

II. THEORETIC CONSIDERATION

A. Micro- and nanocantilever vibrations

Our introductory theoretic consideration will beginwith the basics of the oscillating mechanical structures

Z. Djuri is with the IHTM Institute of Microelectronic

Technologies and Single Crystals, University of Belgrade,Njegoeva 12, 11000 Belgrade, Serbia,

E-mail: [email protected]


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theory. As an illustrative example, we will use a miniaturecantilever of the rectangular cross-section A=Wh and thelengthL, shown in Fig. 1.

For small amplitudes, the mechanical properties of thebeam structures can be analytically described by the Euler-Bernoulli theory [14, 15]

txF

t

txw

t

txwA

x

txwN

x

txwEI ,

,,,,2

2

2

2

4

4

(1)

Here, w(x,t) is the flexural deflection, EIis the flexuralstiffness, A is the mass per unit length of the resonatorbeam,Nis the axial force, is the damping coefficient andF is a driving force per unit length. In micromechanicaldevices, very often the beam width is large, i.e. W5h, and

thus it is necessary to use E/(1-2) instead ofEin Eq. (1),

where is the Poisson ratio of the beam material.

The solutions of Eq. (1) are usually obtained by usingthe method of separation of variables [15]. Consequently,

the displacement w(x,t) of the beam can be split into timeand position dependent components, and the result can beexpressed as a sum of the motion in each mode n,

xtqtxw nn , (2)

Here n(x) is the mode shape function based on the

coordinate along the beam, and qn(t) is the time dependentamplitude of motion for mode n. By substituting the Eq.

(2), in Eq. (1), we obtain the equation for the mode shapefunction, and the time dependent classical oscillatorequation for the modal function qn(t). The beam deflectionin an arbitrary pointx0 (wn(t)=n(x0)qn(t)), corresponding to

the mode n, can be obtained from

tFtwkdttdwdttwdm nnnnnn )/()/)((22

(3)

where the modal parameters are: the effective mass mn, thedamping factorn=mnn/Qn, the resonant frequency n, theQ-factorQn, and kn=n

2mn, the stiffness constant. These

parameters are functions of the parameters from Eq. (1) andalso of the particular mode shape functions obtained for themechanical resonator with the corresponding boundaryconditions. On the left side of the Fig. 1 the first threemodes are shown for the cases of a homogenous (SiO2) and

a bimaterial (SiO2/Ni) cantilever clamped at one end. Onthe right side of Fig. 1 the corresponding oscillating modesare shown for a SiO2 cantilever clamped at both ends.

The dissipation of the mechanical vibration energy ofa microbeam, which determines the Q-factor, can occur dueto internal structural damping, the support loss and theviscous losses in the surrounding fluid.

A change in any of the coefficients that stand by thespatial or time derivatives in Eqs. (1) and (3) causechanges of the oscillation amplitude, phase or resonantfrequency of the beam structure. The principle of operation

of MEMS/NEMS sensors with vibrating sensing element isbased on this influence.

Fig. 1 The first three oscillating modes for a cantilever clamped at

one end (left) and at both ends (right).

As a first example, we will consider the influence of

the axial force, N, applied along a beam clamped at bothends. This is the principle of operation of vibrating pressuresensors. This example is also important because thefollowing theory can be utilized for determination of boththe influence of thermomechanical stress on the resonantfrequency and the MEMS oscillator resonant frequency

fluctuations due to temperature fluctuations.In the case of a double-clamped beam, the exact

expression for the nth mode resonant frequency, ensuring

that the solution for the mode shape functions satisfies theboundary conditions, is

222 /1/)/)(3/1( nnn zzELhf , (4)

where =3NL/(Eh2W), andzn are solutions of the equation

22 /1/1 zztghzztg , (5)

while an approximate expression forn=2fn is [24]

5.0325.022 ))/(1()/))(12/(( EWhNLELh nnn .(6)

f0=38.4 kHz

f1=240.2 kHz

f2=673.8 kHz

f0=245.3 kHz

f1=676.1 kHz

f2=1327 kHz

L

W h

1MHz

SiO2 bridge

Amplitude[a.u

]

SiO2/Ni

1MHz100k10k

SiO2


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Numeric calculations show that for the first and thesecond mode 1=4.730,1=0.295, 2=7.853,2=0.145.

The axial force along the beam of homogenous cross-section can be represented by the mechanical stress, , asN=A. For low stress , the frequency change is

proportional to the stress and the squared ratio of the lengthand thickness of the beam.

It is already said in the introduction that at this time agreat interest exists for development of highly sensitive

biological sensors (for detection of certain kinds of cells inhuman physiological liquids and early diagnostics ofdiseases) [3]. Therefore a short introduction will be givento the theory of oscillations of microcantilevers in liquids.

At the microscale, viscous losses in fluids aretypically two to three orders of magnitude greater thanother losses. The surrounding fluid also affects the naturalfrequencies of the microbeam due to so-called added mass

of the surrounding fluid to the microbeam.

In interaction with an oscillating structure, thesurrounding fluid causes two effects: inertial anddissipative. Hence, fluidic drag force can be expressed as

tvmvF fff / (7)

where v=dw/dt is the velocity, and f and mf are theparameters modelling the dissipative and inertial effects of

the fluid, respectively. The total external force F on thestructure in the fluid can be defined as a contribution ofboth the drive forceF0 and fluidic drag, so thatF=F0-Ff.

In vacuum F=F0. Hence, based on eq. (3), the modeequation of motion in the complex domain is

0220 / Fwmim , (8)

where m is the oscillator effective mass, 0 is the resonantfrequency, and =0m/Q is the damping factor of thecorresponding mode. Using (7), the equation of motion canbe reorganized for fluid as [9]

022 / Fwmim FFFF (9)

where the equivalent parameters are mF=m+mf=m(1+f),F=

0(1+f)

-1/2and QF=Fm(1+ f)/(+f).

Determination of the coefficients mf and f requiressolving of the complex Navier-Stokes equations for a fluidthat surrounds the oscillating cantilevers [6, 16]. The issuespertinent to a vibrating microbeam in a fluid can be broadly

divided into three parts: (i) those that deal with a singlemicrobeam vibrating in fluid, (ii) those that deal with single

microbeam vibrating close to a surface, and (iii) those thataddress the hydrodynamic coupling between microbeamsin fluids or microbeams bounded by a confined liquid [9].As an illustration, an approximate solution will be given for

incompressible fluids, obtained in [16] by curve fitting ofthe numerical results for the case (i)

)065.0Re4.4)(2/( 3/2 Wm ff (10)

)1Re8.2)(2/( 2/12 Wff (11)

Here the Reynolds number Re=fW2/(4) describes the

ratio of inertia to friction terms, and f and are the fluiddensity and viscosity, respectively.

B. Fluctuation phenomena

Fluctuations of parameters of miniature MEMS and NEMS vibration structures increase as their physicaldimensions decrease [15]. These fluctuations determine both the ultimate performances of the sensors and the

minimal power required for transition between the states 0and 1 in digital systems. In oscillating systems, the

amplitude of these fluctuations is mostly determined by theintensity of the dissipation processes. So, for example, the

thermomechanical noise (fluctuation of the deflection)increases for several orders of magnitude when the

cantilever is brought from vacuum into a liquid. Thus thesensitivity of a BioNEMS sensor is significantly reduced.

In the following text the basic mechanisms that generatethe main noises in MEMS and NEMS vibrating structuresare analysed, with the objective to achieve the optimalperformance of the devices based on such structures.

1. Thermomechanical noise. If the temperaturearound our mechanical oscillator is finite and if the system

is in thermodynamic equilibrium, then the mechanicaloscillator must encompass some level of randommovement. These stochastic vibrations basically form

thermomechanical (TM) noise. The magnitude of theserandom vibrations depends on the damping level in theoscillator. Namely, to avoid breaking the second law ofthermodynamics, the model of a damped harmonicoscillator must be supplemented by a generator of noise

force with a sufficient amplitude to keep the level ofstochastic vibrations dictated by the system temperature.Without this generator of noise force, the system dampingwill stop any oscillations, which would mean that thesystem is at zero temperature. This would contradict theassumed condition of thermal equilibrium at a temperature

different from zero.For a linear dissipative system the generalized form

of the Nyquist theorem was given by Callen and Welton in

1951 17. This general theorem determines the relation between the impedance and the fluctuations of thegeneralized force. The mathematical formulation of thisso-called fluctuation-dissipation theorem has the form

dTERF ,)/2(2 (12)

whereR() is the real part of the impedanceZ()=F/v, andE(,T) is the oscillator's mean energy at a temperature Twith a frequency


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11)/exp(2, TkTE B (13)

At higher temperatures (kBT, E(,T)kBT, wherekB is the Boltzmann constant) spectral distribution ofthermal noise force is

TRkZTkdfFdF BBTMN 4}Re{4/22

, .(14)

he spectral density of displacement is

YTkS Bw Re)/4(2 , (15)

where ReY( is the real part of the mechanical

admittance Y(=Z-1(. From (3) we obtain Y( andusing (15) the power spectral density of displacement is

12

0222

00 )))(/(4( QmQTkS Bw . (16)

2. Frequency noise. The performance of the micro-cantilever dynamic sensors depends on various noise

generation mechanisms. These mechanisms include noisein readout circuits (where the most important noise is the

noise arising from a cantilever deflection sensor) andintrinsic frequency cantilever noise. The recent studies [18]have shown that the spectral density of the deflectionsensor noise can be one order of magnitude lower than thecantilever intrinsic noise for the frequency offset (f-f0),which is less than the practically used bandwidth B

(typically less than 1kHz). Due to this, we will analyse theintrinsic cantilever noise dominated by two basic

independent noise generation mechanisms. The firstmechanism is related to the induced stress in the cantileverdue to spontaneous fluctuations of its temperature duringthe heat exchange with the ambient. These stressfluctuations generate the resonant frequency fluctuations

24222202, /)2/( NTiTN ThLfff (17)

where T is the coefficient of temperature expansion, f0 isthe stress-free cantilever resonant frequency and fi is the

resonant frequency of the cantilever with intrinsic stress i.The frequency fi equals i/(2), where i is given by the

expression (6), in which N=iWh.According to [15], the temperature fluctuations are

BRTkT ththBn12222 )1(4 , (18)

where Rth is the thermal resistance, and th is the thermaltime constant.

The second noise mechanism is related to the casewhen the proposed sensor is the self-oscillating system

with positive feedback. It is well known that self-sustained

oscillators universally exhibit linewidth broadening ofvarying degree in their output power spectra. Thislinewidth broadening, often referred to as phase noise, is

caused by noise inherent to the oscillator and is a measureof spectral purity of the oscillator signal. In a short form,the physics of the phase nose is as follows. The trajectory

of the steady-state oscillation in the deflection-velocitystate space is a closed curve due to periodicity and is calledthe limit cycle. In the presence of noise, the fluctuations

would remain small in the radial (amplitude) direction dueto the tendency of the state to return to the limit cycle.Fluctuation in the direction along the limit cycle does notexperience restoring force to return the phase to its originalvalue. That means that, in the presence of noise, the state point experienced Brownian motion, or the phaseundergoes diffusion process, with the diffusion constantD. To determine the spectral density of the phase noise, itis necessary to solve the Langevin or the correspondingFocker-Planck equation.

Utilizing the definition of phase noise for givenfrequency offset =2(f-f0) and the mentioned solution ofthe stochastic equation, we obtain for the phase noise [19]

122 ))((2)(

DD (19)

The diffusion constant is given by the expression

D=kBT0/(A02keffQeff), where keff is the effective stiffness

constant, A0 is the oscillation amplitude and Qeff is not aconventional quality factor. Qeff, defined above, is a direct

measure of the amount of noise in the oscillator as itincludes every noise source in the detector system. If a

noise source has the thermal origin, for example a thermalvibration of the cantilever which is the most dominantnoise source in our case, Qeff becomes a conventionalquality factorQ.

The relation between the spectral densities of thefrequency and phase noise is =()()2. After

integration within the bandwidth B, the power spectraldensity of the frequency noise is obtained in the form

BxarctgxCf BN ))((2

, (20)

where C=D2/(23B) andx=B/D.

It is interesting to note that for large x (x>>1), i.e. for>>D, the well known T.R. Albrecht et al. expression[20] for the TM frequency noise

)2/( 2002

, kQATBkff BBN (21)

can be obtained from (20). This expression is universallyused for the frequency noise analysis for the AFM.

Since the two dominant intrinsic noise generationmechanisms are mutually independent, the power spectraldensity of the total frequency noise equals the sum of the

components and .


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3. AD processes. In the case of structures of smalldimensions and mass, and such are the micro- andnanostructures, adsorption of particles has a significantinfluence on their mechanical characteristics. Due to

adsorption on vibrating micro/nanostructures, the changesin oscillation parameters occur as a consequence of both

the added (adsorbed) mass and the change in the stiffnessconstant. The AD process can be undesirable in someMEMS and NEMS resonant structures. Fluctuations of thenumber of adsorbed particles due to the random nature of

the AD process cause the change of the resonator mass and,consequently, the unwanted parasitic changes of its

resonant frequency (AD frequency noise). Hence the ADprocess adversely affects the performances of the vibratingMEMS and NEMS structures.

We analysed first the physical adsorption of particlesof one gas. Assuming that adsorption occurs in one layerand that the AD process in thermodynamical equilibrium

can be described by the Langmuirs isotherm, we derived

an exact expression for the spectral density of the AD phase noise in micromechanical resonant structures. Thisnoise is generated by instantaneous differences in the ratesof adsorption and desorption of molecules to and from theresonator surface, which cause mass fluctuations andconsequentially the resonator frequency fluctuations. Weused the analogy between AD processes in resonant

structures and generation-recombination processes insemiconductors. We presented in [21] a step-by-step exactderivation of the AD fluctuations-induced phase noise, andhere will be given only the expression we obtained for thespectral density of normalized frequencyy (y=f/f0)

)1/()/(4/ 222202

022

02

mMNCffS ay , (22)

where m0 and f0 are the mass and resonant frequency,respectively, of the microcantilever without the adsorbed

particles, C2=1/, =01exp(Ed/RT) is the average

time the particle spends in the adsorbed state (0 is the

adatom period of thermal vibrations normal to the surface),=/(1+bp) is the AD process time constant, which

determines the rate of reaching the equilibrium value ofadsorbed mass,Ed is desorption energy,R is gas constant,pis the pressure, b=SC1/(C2Nm), where S is the adhesion

coefficient, C1=(2MakBT)-1/2

, Nm is the maximum number

of adatoms per unit area, N0 denotes the number of theadsorbed particles in the stationary state, and Ma is themass of a single particle. The single sideband spectraldensity of phase fluctuations is S(f)=0.5(f0

2/f2)Sy(f), andfinally the expression for the AD fluctuations-inducedphase noise is (f)=10logS(f) [dBc/Hz].

Our numerical results show that the AD induced phasenoise is comparable to other sources of noise inmicromechanical resonant structures and that it prevailswhen the resonator dimensions are very small. Analysis ofAD induced resonator frequency fluctuations and of thecorresponding phase noise shows that there is the strong

dependence between the resonator performance and theenvironmental conditions.

Since the atmosphere around the resonator is amixture of gases in a majority of cases, we expanded our

analysis to address the case of the simultaneous adsorptionof particles of two or more different gases on the resonator

surface [22]. The exact expressions are derived for thepower spectral density of the fluctuations of the number ofadsorbed particles for each gas from the mixture, as well asfor the total adsorbed mass fluctuation, using the analytical

Langevin approach. As an illustration of the presentedtheory, we determined the adsorbed mass fluctuations onthe surface of a sensor with a silicon micro- ornanocantilever in the atmosphere of three gases (Fig. 2).

Fig. 2 Adsorbed mass fluctuations in the case of a mixture of

three gases, whose pressures are p1= p2=10-3 Pa, p3=100 Pa. The

cantilever dimensions are 200 m 50 m 2 m.

The theoretic models of an AD process are useful forboth qualitative and quantitative analysis of the AD noiseand also for estimation of its contribution to the total noise

in MEMS/NEMS sensors and oscillators. Based on thesemodels, it is possible to determine both the minimal

detectable signal and sensitivity of the micro/nanosensors,and to perform the analysis of the influence of the adsorbedparticles number and mass fluctuations on performance ofvarious MEMS and NEMS devices. Thus the theory of ADprocesses is useful for optimization of their parameters andworking conditions. It is concluded that the influence of

AD induced fluctuations on both oscillator and sensor

performance becomes particularly significant as thedimensions of the structures scale down to the order ofhundreds of nanometers and lower.

The possibility of developing the method foridentification of the gases in mixtures based on the ADnoise spectral density can also be considered [22]. While

gas identification is possible in the case of a single gasatmosphere, the number and the position of knees in thenoise spectrum (Figs. 2 and 3 a) could be misleading whenthe number, type and amount of gases in the mixture are to be determined (i.e. when identification of gases in amixture is considered). However, the analysis of the AD

2m

]/[ Hzkg

f [Hz]

gas 2

gas 3

mixture

gas 1

100

102

104

106

108

10-2

10-26

10-25

10-24

10-23

10-22

10-21

10-20

10-19


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noise spectrum in gas sensors can be used in order todistinguish between different gas mixtures.

The resulting adsorbed mass fluctuations in the case of

the mixture can be lower than in the case of the single gasat the same pressure (Fig. 2).

Fig. 3b shows the range of both pressure of gas 1 and

the frequency where a reduction of fluctuations exists (thepressure of the second gas is assumed to be constant). Inthe meshed part the adsorbed mass fluctuations in the caseof a gas mixture decrease by an order of magnitude

compared to the case of single-gas adsorption, while thedark shaded part denotes the area ofp1 and fwhere the

reduction is of the lower order.The diagrams in Figs. 3 a and b enable determination

of the amount of additional gas, which should be added tothe present gas in order to achieve the reduction of theadsorbed mass fluctuations. This is useful for optimizationof the working conditions of the MEMS/NEMS oscillators,

by choosing the mixture of the surrounding gases which

enables minimization of the AD and total phase andfrequency noise and better accuracy of oscillator frequency.

Fig. 3. (a) The power spectral density of the adsorbed massfluctuations for the two-gas mixture, assuming a constant pressure

of gas 2 (p2=103 Pa). (b) The range of both pressure of gas 1 and

the frequency where the adsorbed mass fluctuations in the case of

a gas mixture are lower than in the case of single-gas adsorption.

In the available literature only a little amount of datacan be found about the AD process parameters on the

surface of micro/nanostructures, that is useful forquantitative determination of the influence of the ADprocess on both the response and the ultimate performancesof MEMS/NEMS sensors and oscillators. In the methodsdescribed in the literature, the data about the AD process

dynamics are obtained by measuring the resonant

frequency change in time. Such methods are applicableonly if the adsorption process is slow in comparison with

the response rate of the oscillator itself. In contrast to these,the method for AD process parameter determination basedon time domain analysis of experimentally obtained

oscillator transient response in the presence of adsorption[23] is also applicable to very fast processes. The methodoriginated from the analysis of the solutions of a

differential equation for the first oscillation mode Eq. (3).The mass of the oscillator changes over time due to the AD

process (m(t)=m0+m0(1-exp(-t/)), m0 is the adsorbedmass in steady state). It is shown that the solution of thisequation, obtained using an iterative method, contains

information about AD process kinetics, and therefore it can be used for determination of the AD process parameters.The experimentally obtained oscillator time response,wexp(t), can be fitted with the function Cexp(-(t-ts)/f,exp) fort>ts (ts is the moment when AD process starts), to obtain

the time constant f,exp, which determines the establishmentrate of the steady state in the presence of AD process.

Iff,exp>>o (o=2Q/0 is the oscillator time constant),the AD process time constant determines the responseenvelope, and equals the constant obtained by fitting

(f,exp). However, if f,exp>>o is not valid, the timeconstant determining the AD process kinetics can be

obtained based on Fig. 4. We created a computersimulation which performs both the curve fitting of thefunction w(t) (obtained by solving motion equation) anddetermination off,sim, for various given values of, and fora constant value ofo. When and o are of the same orderof magnitude, and also when is one order of magnitude

lower, can be determined from the diagram, as the valuewhich corresponds to the experimentally obtained f,exp. Thesmaller cantilevers (higher resonant frequency, lower o)enable characterization of fast AD process kinetics (withtime constant as low as 10

-5s) by the described method.

Fig. 4. Time constant f,sim (obtained by computer simulation) as a

function of the AD time constant , for three different values ofo.

As a part of our analysis of noise in EMS/NEMS

devices, we also considered the thermomechanical (TM)noise in the presence of an AD process [24]. The methodwe utilized is based on the Onsager's regression hypothesis[25], which enables determination of the autocorrelation


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function of the deflection, using a deterministic responsew(t), as kBTw(t)/F, whereFis the applied perturbation. Thecosine Fourier transform of the autocorrelation function

yields the TM noise spectrum which is the

experimentally relevant quantity. It is shown that the ADprocess time constant can be determined by analysis of the

oscillator response in the time domain, and the same canalso be done based on the known power spectrum of theTM noise (Fig. 5). This theory allows the investigation ofgas adsorption-desorption kinetics using nanoscale

oscillating structures. Determination of the AD processtime constant is useful for research of both AD andcatalytic processes in general, for estimation of the AD andtotal noise in MEMS/NEMS sensors and oscillators, andalso for development of the methods for characterizationand, possibly, recognition of the adsorbate.

a)t [ms]

0 2 4 6 8 10 12 14 16 18 20

xt

nm

0

0.5

0.4

0.3

0.2

0.1

-0.5

-0.4

-0.3

-0.2

-0.1

= 10-4 s

10-3

s

10-2

s

b)

x

2()nm

2/Hz

742 742.5 743 743.5 744 744.5 745105 kHz

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0

10-5

m0+m0

m0

= 10-4s

10-3

s

10-2

s

Fig. 5. ) The difference in response; b) The TM noise in a nano-oscillator with AD induced variable mass and of a constant mass

oscillator, for three AD processes with different time constants, .

C. MEMS silicon oscillators

An electronic frequency reference or a clock in adigital system is based on an oscillator, which is composedof a resonant tank element and a sustaining circuit whichdrives the resonant element. The characteristics of theresonant element, the resonant frequency, quality factor,temperature sensitivity etc., largely determine the

characteristics of the oscillator output. Difference betweenreference or clock technologies is the resonator. The mostimportant requirement for the frequency reference is thatthe frequency of the output signal should be constant overtime. According to [11] the 21st century begins the erawhen quartz oscillators will be abandoned, for two reasons:

i) Miniaturisation. A modern microprocessor chip mayhave 10 million transistors. To operate, it requires a singlequartz frequency reference that is half of size of the entire

chip! ii) Silicon compatibility. Devices that can bemanufactured with silicon technology are promoted because they can be manufactured cheaply using the

existing silicon batch fabrication capacity, and digitalcircuitry can be integrated into them directly. MEMSresonators have been a topic of research for almost 40

years. Significant advances have occurred in the past 20years, and commercialization efforts begun in earnest in thelast 5 years. The reason for this was a stability problem.

The stability of the signal is the primary performancecharacteristic of a frequency reference. Stability can beclassified into three types based on the time period overwhich the signal is measured: long-term or aging (overhours, days or months), medium term or stability (over

second to hours), and short term or noise (second orless). Usually stability is given in terms of normalized

deviation from target value and measure is the part-per-million (ppm). A silicon MEMS resonator suitable for highfrequency reference applications should have the frequencystability approaching 0.1 ppm. Recently fabricated MEMSoscillator [11] with epi-seal process satisfied this frequency

stability conditions. At the same time it had the powerconsumption less than 20mW and it occupied 0.3 mm

3.During the past 40 years, multiple fabrication

processes have been mastered that enabled the abovementioned results to be achieved. Among the most

important processes are encapsulation of the resonantbeam, highly accurate positioning of excitation electrodesand, finally, ovenization for keeping the resonator at the

exact temperature. As we mentioned earlier, AD processescan have a significant influence on the short-term stabilitythrough the AD noise. As discussed earlier, the resonant

frequency is proportional to the square root of the inverseof its mass. The mass of the resonator is of the order of 100 picograms. So physical contamination, equivalent to asingle atomic layer of additional mass deposition, canchange the frequency by hundreds or thousands of ppm.

D. Nanomechanical computers

Recenty, Robert H. Blick et al. [13] proposed a fullymechanical computer (NMC) based on nanoelectro-

mechanical elements. The main motivation behindconstructing such a computer is threefold: (i) mechanicalelements are more robust to electromagnetic shocks thancurrent dynamic random access memory based purely oncomplementary metal-oxide semiconductor technology(CMOS), (ii) dissipated power can be an order ofmagnitude below CMOS and, (iii) the operatingtemperature of such an NMC can be an order of magnitude

above that of conventional CMOS.Without going into details of NMC design, it is

important to emphasize here that its fundamental element isa nanoelectromechanical single electron transistor (NEM


8/8

SET). In this type of SET [13] the island is movable and itis situated on the top of a cantilever which can be excitedby an AC source-drain voltage. Recent measurements have

shown that self-excitation can be exploited to generatemechanical oscillation without any AC excitation. Thismeans that DC voltage is sufficient to operate the NMC.

Besides its practical value, research of NEMSET is ofgreat importance for fundamental sciences, since itincorporates both the field of single electron devices andthe vibrating nanostructures whose dimensions are such

that they can represent the ideal models for demonstrationof the quantum mechanics laws.

III. CONCLUSION

Vibrating micro- and nanostructures are finding anever-increasing number of applications in a multitude offields: as basic building blocks of the new generation of

sensors with sensitivity of the order of single molecules, aselements of the new kind of MEMS oscillators with low power consumption and fabrication process compatiblewith CMOS, as central components of NEMSETs etc.

In this paper we presented some of the fundamentaltheories upon which the operation of these components is

based. The emphasize was on the fact that in case ofminiature vibrating structures the effects pertinent to theirdimensions, such as Brownian motion, AD processes,mass, temperature and frequency fluctuations etc. becomesignificant. The author hopes that the results of this workcan be utilized for proper design and optimization of

various vibrating micro- and nanocomponents.

As it can be expected in such a propulsive field,several interesting phenomena, such as the quantum effectsin nanovibrating structures, vibrating structures withnanotubes, multilayer adsorption processes on nano-structures etc. had to remain out of scope of this paper.Anyway, research in this field is expected to enable bothbetter understanding of the nature itself and wise utilization

of its resources.

ACKNOWLEDGEMENT

The author wishes to thank Ms. I. Joki, Mr. M.Frantlovi and Dr K. Radulovi for their collaboration andto the Serbian Ministry of Science for their support.

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micro- and nano systems based on vibrating structures

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