117978-1-4799-5296-0/14/$31.00 © 2014 IEEE

PROC. 29th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2014), BELGRADE, SERBIA, 12-14 MAY, 2014

RF MEMS and NEMS Components and Adsorption-Desorption Induced Phase Noise

I. Jokić, M. Frantlović, Z. Djurić

Abstract – Radio frequency micro- and nanoelectro-mechanical systems (RF MEMS and RF NEMS) and technologies have a great potential to overcome the constraints of conventional IC technologies in realization of fully integrated transceivers of next generation wireless communications systems. During the last two decades a considerable effort has been made to develop RF MEMS/NEMS resonators so that they could replace conventional bulky off-chip resonators in wireless transceivers. In MEMS, and especially in NEMS resonators, additional noise generating mechanisms exist that are characteristic for structures of small dimensions and mass, and high surface to volume ratio. One such mechanism is the adsorption-desorption (AD) process that generates the resonator frequency (phase) noise. In the first part of this paper a short overview of RF MEMS resonators is given, including comments on the necessary improvements and the direction of future research in this field (especially having in mind the need for NEMS resonators), with the intention to optimize RF MEMS and NEMS components according to requirements of both current and future systems. The main part of the paper presents a comprehensive theory of AD noise in MEMS/NEMS resonators. Apart from having a theoretical significance, the derived models of AD noise in multiple different cases of adsorption are also a useful tool for the design of optimal performance RF MEMS and NEMS resonators. The model of the MEMS/NEMS oscillator phase noise that takes into account the influence of AD noise is presented for the first time.

I. INTRODUCTION

In the development of modern wireless terminals a

convergence exists towards a universal mobile personal device which unites the functions of a telephone, a computer with Internet access, a radio navigational device, a multimedia center etc. It is expected from new generations of mobile terminals to operate in a greater number of frequency ranges (multiband) according to a greater number of communication standards (multi-standard) and achieving a much higher data throughput. Realization of the high frequency part (i.e. the front end) of the transceiver of such a device becomes a challenging task, especially when it has to be affordable, small and

energy efficient, because it requires high performance analog circuits and passive components (filters, duplexers, switches etc.). Analog RF circuits of adequate performance cannot be fabricated by using the latest nm-CMOS processes, while the mentioned high-quality RF passives are off-chip components and therefore an obstacle for RF front-end miniaturization which remains unaffected by the IC technology scaling. Addition of a set of analog circuits and passive components for every new band and/or standard significantly increases the RF transceiver complexity, dimensions and cost. As the number of supported RF bands and standards continues to increase, this approach will no longer be viable. There is a growing need for a RF transceiver reconfigurable analog part that would enable wideband signal processing according to different standards, with a minimal number of recon-figurable (tunable) circuits and suitable for a higher level of integration. However, such requirements are a serious challenge for traditional microelectronic technologies.

Radio frequency micro- and nanoelectromechanical systems (RF MEMS and RF NEMS) and technologies have a great potential to overcome the constraints of conventional IC technologies in realization of fully integrated transceivers of next generation wireless communications systems [1, 2]. They offer new possibilities for realization of high-performance reconfigurable RF circuits, utilizing fabrication techniques compatible with traditional IC technologies, thus enabling monolithic integration with active electronics. A new level of miniaturization, lower energy consumption, lower cost and simpler architecture of the transceiver RF part are the advantages of RF MEMS and NEMS components, necessary for future wireless systems.

A group of very promising components are RF MEMS/NEMS resonators which are being developed with the intention to replace large off-chip (discrete) ceramic and SAW RF filters and quartz resonators in radio transceivers [3-6]. In the first part of this paper we present a short overview of RF MEMS resonators, including classifications and principles of operation. Typical values of basic parameters relevant for wireless transceiver applications are given, as well as advantages compared to solutions based on conventional technologies. We also comment on the necessary improvements and the direction of future research in this field (especially having in mind the need for NEMS resonators), with the intention to optimize RF MEMS and NEMS components according to requirements of both current and future systems. Of

I. Jokić, and M. Frantlović are with the Institute of Chemistry, Technology and Metallurgy - Center of Microelectronic Technologies, University of Belgrade, Njegoševa 12, 11000 Belgrade, Serbia, E-mail: ijokic@nanosys.ihtm.bg.ac.rs

Z. Djurić is with the Institute of Technical Sciences SASA, Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11000 Belgrade, Serbia, E-mail: zoran.djuric@itn.sanu.ac.rs

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particular significance in this sense is the analysis of noise generation mechanisms, specific for these components. Understanding of these mechanisms enables optimization of components' design, as well as devices and systems and the development of methods for minimization of signal degradation. As an illustration of MEMS/NEMS components specific properties compared to microelectronic components and macroscopic mechanical components in terms of noise generation, in the second part of this paper we present the analysis of adsorption-desorption noise in RF MEMS and NEMS electromechanical resonators. Apart from having theoretical significance, the analysis is also a useful tool for the design of optimal performance RF MEMS and NEMS resonators. Finally, we present a theoretical model of MEMS/NEMS oscillator phase noise, which takes into account the influence of the resonator AD noise.

II. RF MEMS AND NEMS COMPONENTS

Passive components (inductors, capacitors, resistors, varactors, resonators, switching elements etc.) are used for realization of wireless transceiver RF circuits such as impedance matching networks, filters, reference oscillators, frequency synthesizers, antenna switches etc. Switches and tunable passives are the key components for transceiver reconfigurability. For example, tunable RF matching circuits are necessary for realization of wideband, reconfigurable low-noise amplifiers (LNAs) and power amplifiers (PAs).

Imperfections of passive components can degrade transceiver performance such as receiver sensitivity and selectivity, noise figure, transmission power etc. High-quality passives are therefore necessary for achieving high RF performance. Their significance is especially high for multiband multistandard operation which imposes more stringent requirements for RF front-end performance in every next generation. The lack of high-quality integrated passive components, especially reconfigurable ones, fabricated by traditional Si technologies, necessitates the use of many off-chip passives (RF SAW filters and duplexers, quartz resonators etc.).

At the end of the 1980s the development began of RF MEMS components (variable capacitors, inductors, resonators, tunable filters, switches). High RF performance (even of tunable components), micrometer and sub-micrometer dimensions, integrability with active CMOS circuits, low power consumption, and mass production, make them perfect candidates for application in wireless transceivers. Integrated RF MEMS components can directly replace off-chip SAW and crystal resonators used as RF filters and crystal oscillator references in conventional transceiver architectures (Fig. 1a), as well as other passives and traditional RF switches. Furthermore, RF MEMS components can be used to enable front-end reconfigurability in multiband multistandard transceivers (Fig. 1b), significantly reducing its complexity.

Fig. 1. Application of RF MEMS components in multiband multistandard wireless transceivers: a) direct replacement of off-chip conventional components with integrated MEMS components, b) introduction of reconfigurability. A. RF MEMS Resonators

Resonators are widely used in wireless transceivers: in filters and duplexers, in VCOs as tunable tanks, in frequency references, frequency synthesizers etc. Required basic parameters values depend on resonator application. For example, the Q of a resonant circuit in VCO of super-heterodyne receiver can be 30–50, but realization of RF bandpass filters, with the central frequencies 0.8–5.5 GHz, requires Q~500–10000. In frequency references the required Q can be >105. In mobile terminals the frequency stability of frequency references better than ±10 ppm in the range 0–70 ºC is needed [5]. For frequency synthesis the maximum acceptable resonator frequency variation is ±2 ppm in the same temperature range. In RF filters the temperature coefficient of frequency ~10 ppm/ºC can be tolerated. Resonator long-term stability should be better than 3 ppm/year [6]. The equivalent series resistance needs to be low enough to allow impedance matching of conventional RF circuits (typically 50 Ω).

In modern wireless transceivers resonators (LC, SAW – Surface Acoustic Wave, BAW – Bulk Acoustic Wave) are fabricated using conventional technologies.

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Electromechanical resonators (SAW, BAW) are off-chip components, and have better performance (Q~103–106, better frequency stability, lower phase noise) than electro-magnetic (LC) ones. SAW resonators are used in high-performance RF filters and duplexers, with central frequencies up to 2 GHz. The nominal frequencies of crystal (BAW) resonators in wireless transceivers are ~10 kHz–10 MHz. They are highly reliable and technologically mature. However, none of the mentioned resonator types can simultaneously meet the following requirements: high Q at ~1 GHz, small dimensions, low power consumption, high frequency stability, low cost and integrability (especially monolithic) with CMOS circuits.

During the last two decades a considerable effort is made in the development of RF MEMS resonators so that they could replace conventional bulky off-chip resonators in wireless transceivers [3-6]. Using the MEMS technology, both electromagnetic and electromechanical (EM) resonators have been fabricated. MEMS EM resonators are smaller than electromagnetic ones of equal resonance frequency (f0), and have a similar or higher Q. Their main advantages are: integration with CMOS ICs; small dimensions enabling fabrication of monolithically integrated MEMS resonator arrays in multiband transceivers (versatility and programmability on a small surface); high Q×f0 (Q comparable to conventional, f0 up to ~1 GHz); low cost mass production. Operation of EM resonators is based on mechanical oscillations of a resonant structure, actuated by the input electrical signal and converted in the output electrical signal. The actuation is based on the action of electrostatic (ES) or magnetic force, piezoelectric effect, thermal expansion etc., while the conversion of mechanical energy into the electrical signal can be capacitive, piezoelectric (PE), piezoresistive etc. The most common are: 1. capacitive (ES) actuation and capacitive detection of mechanical oscillations, and 2. PE actuation and detection. Capacitive resonators can oscillate in flexural or torsional modes, and then they are called mechanical resonators. There are also capacitive resonators with bulk oscillation modes (extensional, contour (radial), wine-glass, Lamé). Resonant structure of capacitive resonators is in the shape of a beam, quadratic plate, disc, ring or comb. PE (SAW or BAW) resonators contain a layer of piezoelectric material (ZnO, AlN, PZT). SAW resonators oscillates in the Rayleigh mode, and BAW in bulk extensional mode (FBAR – thin Film Bulk Acoustic Resonator) in the direction of piezolayer thickness, or in lateral extensional mode (LBAR – Lateral BAR). Modes in which f0 depends on lateral dimensions (the most of capacitive bulk modes, SAW, LBAR) are suitable for realization of monolitically integrated RF filter banks. Capacitive resonators are compatible with Si IC technologies, their f0 can be tuned by the actuation voltage (for temperature compensation or filter tunability), and filter selection from a bank of filters can be done without switches. Compared to capacitive, PE resonators have lower Q and Q×f0, but also lower series resistance and

better linearity (better power handling). Fig. 2 shows some of RF MEMS EM resonators.

Based on the achieved values of f0 and Q, dimensions and CMOS compatibility, RF MEMS resonators are considered as a solution for realization of fully integrated RF systems. Significant results achieved after the year 2000 (better temperature stability, better long term stability, improved packaging) enabled commercialization of both PE and capacitive MEMS resonators. Remaining problems that prevent wider practical application are: high series resistance, insufficient power handling, and significant dependence of Q on pressure (necessitates vacuum packaging).

Fig. 2. Examples of RF MEMS EM resonators: mechanical (a) [7] (with permission of the author), b) [8], d) [9], e) [10], f) [11]), and bulk-mode resonators (c) [12], g) [13], h) [14] © IOP Publishing. Reproduced with permission. All rights reserved, i) [15], j) [16]). © 2000, 2008, 2012, 2010, 2010, 2004, 2004, 2005 IEEE. Reprinted, with permission, from [8], [9], [10], [11], [12], [13], [15], [16].

The development of MEMS resonators with f0~1 GHz and Q>104 would enable implementation of new and compact multiband multistandard transceiver architectures [17]. Frequency references in the GHz range are also desirable in future mobile systems. Resonators with GHz resonance frequencies have nanometer dimensions and are therefore fabricated using NEMS technologies. However, some solutions used in MEMS do not scale well into the NEMS domain [18]. That includes efficient energy conversion mechanisms and coupling between NEMS and other components and circuits. Solutions are also needed for reduction of the series resistance of NEMS resonators, improvement of their power handling, and reduction of noise caused by physical effects that become pronounced as the dimensions decrease. In MEMS, and especially in NEMS resonators, additional noise generating mehanisms exist that are characteristic for structures of small dimensions and mass, and high surface to volume ratio. It is therefore necessary to investigate their influence on the

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resonator performance as a function of dimensions of the structures and the operating conditions.

III. PHASE NOISE IN RF MEMS/NEMS

Mechanical response of an RF MEMS or NEMS structure, caused by an electrical actuation signal, can directly influence the electrical properties of the system that contains it, for example, by changing the circuit topology (in the case of an RF MEMS switch) or by changing some electric parameter value (e.g. the capacitance of a variable capacitor). In MEMS/NEMS resonators a certain mechanism (PE, capacitive, etc.) is utilized to convert the mechanical response into the electrical signal. EM RF MEMS and NEMS components can be integrated with electronic components (e.g. integrated MEMS oscillators containing a MEMS resonator and CMOS circuits on the same chip). It is obvious that the operation of EM RF MEMS and NEMS components is based on the interaction between the mechanical and the electrical domain. Thus, apart from the noises inherent to electrical and electronic devices and circuits (generated in EM transducer circuits, amplifiers and other electronic parts of MEMS/NEMS systems), noise analysis of MEMS/NEMS systems has to include noise generating mechanisms in the mechanical domain [19]. Internal mechanical noises in MEMS and NEMS components are the consequence of the stochastic nature of physical processes occurring inside the component or in the boundary zone where the mechanical structure has a contact with its environment. They result in stochastic fluctuations of displacement of a micro/nanomechanical structure and/or of its mechanical resonance frequency, thus causing fluctuations of the electrical output signal. In mechanical structures of micrometer or sub-micrometer dimensions and minuscule mass, whose displacement is in nanometer range, the effects of physical phenomena that are negligible in the macroscopic world become significant. Characteristic fundamental mechanical noises of MEMS and NEMS EM resonators are: thermo-mechanical (TM) noise, noise due to temperature fluctuations (TF), and adsorption-desorption (AD) noise. Their contribution to the total noise increases and may become dominant as the dimensions decrease.

A significant parameter of MEMS/NEMS EM resonators is the short-term frequency stability which is determined by the phase noise. For example, the oscillator phase noise that degrades the signal transmission quality originates not only from phase fluctuations caused by noise generation mechanisms in the oscillator's electronic circuit, but also from the resonator noise [20]. The mentioned fundamental mechanical noises are indeed those that cause inevitable stochastic phase and frequency fluctuations of a resonant structure and determine the lowest (fundamental) limit of the MEMS/NEMS resonator noise. This makes TM, TF and AD noise a measure of the MEMS/NEMS resonator ultimate performance. In the existing literature

TM, TF and to a lesser extent AD noise in MEMS/NEMS resonators were considered [19, 21-24]. However, there is still a need for new, more comprehensive models of noise in MEMS/NEMS resonators and oscillators. Further in this section a theoretic model of AD noise in RF MEMS/NEMS EM resonators and of phase noise in MEMS/NEMS oscillators will be presented. A. Adsorption-Desorption noise in MEMS/NEMS resonators

Stochastic adsorption-desorption processes of surrounding gases spontaneously and inevitably occur on surfaces of all solid bodies at temperatures higher than 0 K, and at pressures above 0 Pa. Stochastic nature of both the instantaneous adsorption rate of gas particles on the surface of a mechanical resonant structure and the instantaneous desorption rate from the surface results in stochastic fluctuations of the number of adsorbed particles, ∆N(t) [21, 23, 25, 26]. This means that the total adsorbed mass fluctuates (Δm(t)), causing fluctuations of the mechanical resonance frequency of the MEMS/NEMS structure, Δν(t), i.e. causing the resonator AD frequency and phase noise. The mean AD frequency noise power of a resonator in the bandwidth 1 Hz at the offset-frequency f from the nominal frequency ν0, expressed in dBc/Hz, L(f), and the power spectral density (PSD) of resonator AD frequency noise, SΔν(f), are determined by the PSD of the adsorbed mass fluctuations , SΔm(f) (m is the resonator mass)

( ) ( ))2/()()2/(log10)2/()(log10 220

2 ffSmffSL m∆∆ == νν . (1)

The approach based on the analogy of AD processes on the surface of MEMS structures and generation-recombination (GR) processes of charge carriers in semiconductors was applied for the first time in the analysis of stochastic AD induced fluctuations in the case of single gas single-layer adsorption, described in [24], and then in [26]. Subsequently, theoretic models of AD noise were devised for different cases of gas adsorption on the surface of micro/nanostructures: adsorption of an arbitrary number of gases [27-29], adsorption in an arbitrary number of layers [30, 31], and adsorption coupled with mass transfer [32]. All of them are based on the established analogy between AD processes and the coresponding GR processes. This approach is general for statistical analysis of a much broader group of stochastic processes, known as the gain-loss processes. In all off the mentioned cases of gas adsorption the starting equation(s) has (have) the general Langevin form

inieniei NNNrNNNgdtdN ξ+−= ),...,,(),...,,(/ 2121 (2)

i=1,2,...n, describing the change in time of the number of particles of the ith gas or of the number of particles in the ith adsorbing layer (which are not covered by (i+1)th layer), as a function of: 1. the equivalent rate of "generation" of particles of the type i (i.e. of increase of their number) and

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the equivalent rate of their "recombination" (i.e. of decrease of their number), gie and rie, respectively, and 2. the stochastic source function ξi, while n is the total number of gases in the surroundings or the number of adsorbed layers. gie and rie take into account the influences of all the processes relevant for the change of Ni, and their forms differ depending on the analyzed case of adsorption. For small fluctuations (∆Ni) of the number of adsorbed particles around the corresponding equilibrium value (Nie), so that ∆Ni<<Nie, based on Eqs. (2) the Langevin equations can be written in the matrix form

ξNKN +∆⋅−=∆ dtd /)( (3)

where ∆N=[∆N1 ∆N2 ... ∆Nn]T, ξ=[ξ1 ξ2 ... ξn]T, K is the n×n matrix of elements Kij=-(∂gie/∂Nj-∂rie/∂Nj)N=Ne, N=[N1 N2 ... Nn]T, and Ne=[N1e N2e ... Nne]T. The steady state values Nie, i=1,2,...n, are obtained from the steady state conditions gie(N1e,N2e,...Nne)=rie(N1e,N2e,... Nne). By performing Fourier analysis and using the results of the theory of stochastic GR processes [33] according to which the PSD of the ith source function equals Sξi=4gie(N1e, N2e, ..., Nne)=4gie,E, a square n×n matrix S∆N2(ω) is obtained whose elements (i,j) are single-sided spectral and cross-spectral densities, S∆Ni∆Nj*(f) (Sξ is a diagonal n×n matrix of elements Sii=Sξi, and I is the unity n×n matrix, ω=2πf)

11 )()()(2−−

∆ −⋅⋅+= IKSIKS TN ωωω ξ jj (4)

The total fluctuation of the adsorbed mass is

Δm=M1∆N1+M2∆N2+…+Mn∆Nn. Mi is the mass of a single particle of the type i (number of such particles at the surface is Ni), i=1, 2, ...n, M=[M1 M2 ... Mn]. Therefore, the PSD of the adsorbed mass fluctuations is

TN MSM ⋅⋅= ∆∆ )()( 2, ωωnmS (5)

and it has the general form

+

= ∏∑

=

−

=∆

n

ii

n

iinm kS

1

221

0

2, )1(/)()( τωωω (6)

where ki and τi are the constants obtained analytically for the particular case of adsorption. The resonator AD phase and frequency noise are now obtained using Eq. (1), where SΔm(f) is given by Eq. (5) or (6).

By applying the described method in the case of single gas single-layer adsorption, the PSD of the adsorbed mass fluctuation is obtained in the form [24,26]

21

2

2111

21

1, )2(1)(4

)(τπτ

fNgM

fS eem +

=∆ (7)

where τ1=1/K11=-(∂g1e/∂N1-∂r1e/∂N1)-1N1=N1e, and N1e is obtained from the steady state condition g1e(N1e)=r1e(N1e).

The PSD of the mass adsorbed in a single layer on the surface of a resonator operating in a two-gas atmosphere is given as the simplest example of multiple gas adsorption [27-29]

)1)(1()1)((

4)( 22

221

2

22,2

22,1

21

2

22

21

2, τωτωτω

τττ

ω++

++=∆

zEeEe

zm

gMgMS (8)

where τ1,2=2{K11+K22±[(K11+K22)2-4(K11K22-K12K21)]1/2}-1, τz=(M1

2g1e,E+M22g2e,E)[(M1K22-M2K21)2g1e,E+(M1K11-M2K12)2

g2e,E)]-1. The expression for the PSD of adsorbed mass fluctuations for the case of two-layer adsorption has the same form, but τ1,2 and τz depend on different parameters because the functions gie and rie are different [30].

Mass transfer processes of particles in a resonator chamber can also influence the magnitude of fluctuations of the number of adsorbed particles, especially for low-pressure (low-concentration) environments [32]. In a single-gas environment the PSD of the adsorbed mass fluctuations for single-layer case, considering te mass transfer process, is given by Eq. (7) with τ1,MT=τ1(1+g1e,ERT/(Akmp)) instead of τ1 (p is gas pressure, T is temperature, A is the resonator surface area and km is the mass transfer coefficient) [32].

A detailed analysis of the MEMS/NEMS resonator AD noise for single-gas single-layer adsorption is performed in [26, 34]. It is observed that this noise has a low magnitude at near-atmospheric pressures, but becomes significant at lower pressures. This should be considered when operating conditions for MEMS/NEMS resonators are to be chosen. In a majority of studies TM noise of resonators is analyzed and optimization of the operating conditions is performed in order to minimize it. TM noise can be reduced by choosing low pressure values, because energy dissipation due to the influence of the surrounding medium is then decreased. However, at typical pressure values existing in evacuated MEMS/NEMS resonator packages, AD noise is at its maximum, so it can become dominant [26, 34]. Therefore, optimization of operating conditions in order to minimize the total RF MEMS/NEMS resonator noise must be performed based on the analysis of all the fundamental noises dependent on the ambient pressure and temperature. It was also shown that AD noise increases with the increase of the resonance frequency, i.e. as the resonator's dimensions decrease [34].

The presence of multiple gases or multiple layer adsorption can considerably affect resonator AD noise [28-31]. Fig. 3a shows the AD phase noise of a MEMS/NEMS Si resonator (f0=50 MHz, m=8.2·10-16 kg) in a two-gas atmosphere, as a function of both the gas 1 pressure (the gas 2 pressure is p2=103 Pa, T=300 K) and the offset frequency, obtained by using the presented theoretic model (Eqs. (1) and (8)). As can be seen in Fig. 3b, there is a decrease in the AD noise spectrum in a certain pressure and frequency range for two-gas adsorption, as opposed to the AD noise PSD for the case of one gas. However, since the decrease does not exist at all frequencies, the effect of the

122

gas mixture composition on the total AD noise should be observed. The presented theory enables performing the analysis that yields the optimal gas mixture composition at which the resonator AD noise is minimized.

a)

b) Fig. 3. a) AD phase noise of an MEMS/NEMS resonator in a two-gas atmosphere, as a function of the pressure of the gas 1 (the pressure of the gas 2 is p2=103 Pa). b) Ratio of PSDs of AD noise for single gas adsorption (gas 1) and for two-gas adsorption, shown in both the pressure range of the gas 1 and the frequency range where reduction of AD noise is achieved in the case of a mixture. B. Phase noise in RF MEMS/NEMS oscillators

Modeling of phase noise of RF oscillators in modern wireless communication systems is very useful because phase noise effects on system performances become increasingly destructive with the introduction of new standards based on advanced modulation schemes [35]. The theory of phase noise is being constantly improved [35, 36].

While the signal of an ideal oscillator is perfectly sinusoidal, a signal generated by a real oscillator can be described by

)(0 0)2/()( tjtj eeAtu φω= (9)

A0 is the amplitude (usually considered as constant [35]), ω0=2πf0, f0 is the carrier frequency, and φ(t) is the resulting

phase stochastic fluctuation caused by noise generating mechanisms in the oscillator constituting components. The spectrum of the signal u(t) is shaped by θ(t)=ejφ(t), and is located around the frequency ω=ω0. When considering only the spectrum shape, it is sufficient to observe the spectrum translated to the baseband, which is then the spectrum of the signal θ(t). If we denote the PSD of θ(t) with Sθ(f), the oscillator phase noise in dBc/Hz is [35]

))(log(10)( fSfLosc θ= (10) Sθ(f) is obtained by using the Wiener-Khinchin theorem

∫∞

∞−

−= ττ τπθθ deRfS fj2)()( (11)

where the autocorrelation function of θ(t) (considering the Gaussian distribution for ∆φ) is [35]

2/)(2)( τσ

θφτ ∆−= eR (12)

Since φ(t) is the integration result of stationary frequency noise, the variance of the phase increments (or the phase jitter variance) σ2

∆φ(τ) is related to the PSD of frequency noise S∆ν(f) according to the expression [35, 37]

∫

∫∞

∞−∆

∞

∞−

∆∆

=

−=

dff

ffS

dfff

fS

2

2

22

)()(sin)(

))2cos(1()2(

)(1)(

πτπ

τπππ

τσ

ν

νφ

(13)

A MEMS oscillator most often consists of a MEMS mechanical resonator as a frequency selective element, and sustaining electronics. Therefore, the total phase fluctuations are caused by noise generation mechanisms in the oscillator's electronic circuit, but also by the resonator mechanical frequency noises. Assuming that these noise sources are not correlated, the total variance of the phase increments equals the sum of the components that correspond to each of the frequency noise sources. The corresponding autocorrelation function equals the product of the autocorrelation functions of all noise contributors (Eq. 12). The PSD of θ(t), i.e. Sθ(f), is the convolution of the individual power spectrums in the frequency domain.

In the presence of the noise induced by dissipation processes in a resonator and sustaining circuits (called the Brownian motion noise, or white noise), the phase undergoes diffusion process, with the diffusion constant Dφ. Apart from that, the omnipresent 1/f noise of oscillator components also causes phase fluctuations. In the presence of these noises the corresponding variances of the phase increments are [35, 36]

ττσ φφ DB 2)(2, =∆ , 22

/1, )( ττσ φ Kf =∆ (14)

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When the dissipative mechanisms in a resonator dominate, Dφ=kBTω0/(A0

2keffQL) (keff is the effective resonator stiffness constant, QL is the loaded resonator quality factor) [38]. K is related to parameters of the 1/f noise in an oscillator circuit. By using Eq. (13) the variance of the phase increments due to AD noise can be obtained, applying S∆ν(f)=(f0/2m)2S∆m(f), where S∆m(f) is given by Eq. (5) for an arbitrary case of AD process. In the case of single gas single-layer adsorption (Eq. (7)) we obtain

)]1([2

)( 1/1

2,

ττφ τττσ −

∆ −−= ePAD (15)

where P=g1e(N1e)τ1

2M12(f0/2m)2. The corresponding PSDs

of θ, denoted with Sθ,B(f), Sθ,1/f(f) and Sθ,AD(f), are obtained using Eqs. (11), (12), (14) and (15). The total Sθ(f) is their convolution in the frequency domain, and the oscillator phase noise, which includes the influence of the AD noise, is then obtained from Eq. (10). The solution for the term originating from the Lorentzian AD frequency noise is

=

= ∫ −− ),(Re2Re2)( 10

11, az

aedtet

aefS z

aatz

z

a

AD γττθ

where γ(z,a) is the lower incomplete gamma function, a=Pτ1/4, and z=Pτ1/4+jωτ. Fig. 4 shows the calculated PSD Sθ,AD(f) for a single gas atmosphere at different gas pressures (the values are chosen around the pressure where AD noise has its maximum), illustrating the spectral dependence of the component of the oscillator phase noise, which is caused by the AD frequency noise.

Fig. 4 The calculated PSD of the oscillator phase noise constituent caused by AD process of a single gas of pressure p.

III. CONCLUSION

Micro- and nanoelectromechanical systems have the

potential for realization of high performance radio frequency components required to meet the growing

functionality and capacity needs of future mobile communications devices and systems. RF MEMS and NEMS became a competitive alternative to many components used in transceivers of commercial communication devices. MEMS/NEMS resonators have generated significant interest in this field because of their ultra-high resonant frequencies, small size, very low operating power, high quality factors and possibility of integration with silicon IC technologies.

Oscillator phase noise is very important in modern wireless communication systems. In this paper the model of MEMS/NEMS oscillator phase noise is presented for the first time, taking into account the adsorption-desorption noise contribution. The presented models of adsorption-desorption phase noise are significant for development of RF MEMS/NEMS resonators and oscillators whose performances fulfill the requirements of existing and future telecommunication systems. This is especially true having in mind the trend toward higher frequencies where the fluctuating mass change due to AD process may became a dominant mechanism of resonator noise generation. The prediction of the AD noise should be performed during the design of such components in order to identify the dominant noise generating mechanism and then to optimize the resonator parameters and operating conditions in terms of noise minimization.

ACKNOWLEDGEMENT

We would like to express our gratitude to Prof. Dr. Gradimir Milovanović, full member of the Serbian Academy of Sciences and Arts, for his contribution in solving mathematical problems, and to Ms. Milica Ševkušić for her support during the manuscript preparation.

This work was funded by the Serbian Ministry of Education, Science and Technological Development (Project TR 32008) and by the Serbian Academy of Sciences and Arts (Project F/150).

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