Noise Processes in Nanomechanical Resonators A

JOURNAL OF APPLIED PHYSICS VOLUME 92, NUMBER 5 1 SEPTEMBER 2002

Noise processes in nanomechanical resonators A. N. Clelanda) Department of Physics and iQUEST, University of California at Santa Barbara, Santa Barbara, California 93106 M. L. Roukes Department of Physics, California Institute of Technology, Pasadena, California 91125 ͑Received 15 February 2002; accepted for publication 18 June 2002͒ Nanomechanical resonators can be fabricated to achieve high natural resonance frequencies, approaching 1 GHz, with quality factors in excess of 104. These resonators are candidates for use as highly selective rf ﬁlters and as precision on-chip clocks. Some fundamental and some nonfundamental noise processes will present limits to the performance of such resonators. These include thermomechanical noise, Nyquist–Johnson noise, and adsorption–desorption noise; other important noise sources include those due to thermal ﬂuctuations and defect motion-induced noise. In this article, we develop a self-contained formalism for treating these noise sources, and use it to estimate the impact that these noise processes will have on the noise of a model nanoscale resonator, consisting of a doubly clamped beam of single-crystal Si with a natural resonance frequency of 1 GHz. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1499745͔

I. INTRODUCTION thermal drifts that can be controlled using oven-heated packaging, similar to that used for high-precision quartz clocks. Nanomechanical resonators are rapidly being pushed to Resonators fabricated from polycrystalline materials, such as smaller size scales and higher operational frequencies, partly those including poly-Si and silicon nitride, are also expected due to potential applications as on-chip high-Q filters and to demonstrate noise from anelastic noise processes caused clocks. Such resonators would have the potential for replac- by grain-boundary and point defect motion.13 ing bulk quartz crystals and surface-acoustic wave resonators At present, there does not exist a single self-contained in technological and precision measurement applications, formalism for describing the resonance and noise properties which require extensive separate circuitry and space require- of nanomechanical resonators. In the first part of this work, ments. High-frequency resonators have been fabricated from we therefore develop such a formalism, based on the well- bulk Si,1 silicon-on-insulator,2 silicon carbide,3 silicon known Euler–Bernoulli theory of beams. We hope that this nitride,4 and from polycrystalline Si ͑poly-Si͒.5,6 High reso- will provide a clear and useful framework for future devel- nance frequencies can be achieved using submicron lithog- opments in the field. In the latter part of the work, we use raphy to define doubly clamped beams with relatively large length-to-thickness ratios of L/tϳ10– 20. Smaller aspect ra- this formalism to calculate the effects of the most significant tios, with L/tϳ2 – 5, allow high frequencies to be achieved and fundamental, classical sources of noise on resonator per- with less stringent demands on lithographic capability. For formance. The importance of thermomechanical noise, aris- these smaller aspect ratios, however, thermoelastic damping ing from the nonzero dissipation and temperature of a reso- begins to become an important source of energy loss and nator, has been recognized for some time, and its effects have noise, ultimately limiting the quality factor and noise been included in previous noise analyses of mechanical 9,14 performance.4,7 resonators. Other noise sources have also been included in The resonance frequency of a mechanical structure in more recent analyses, as mentioned herein. However, our general scales as 1/L, where L is the scale of the resonator. results are not in agreement with the results of these more As the size scales are reduced and frequencies increased, recent works, in particular, in terms of the magnitude of the however, the short-term stability of the resonator will be lim- impact of the noise, as well as the method of analysis of ited by certain fundamental noise processes.8 These noise some of the noise sources, in particular, that of the effect of processes include the thermomechanical noise generated by temperature fluctuations. We have also included a discussion the internal loss mechanisms in the resonator,9 Nyquist– of defect noise, that to our knowledge has not previously Johnson noise from the readout circuitry,10 and adsorption– been considered. desorption noise from residual gas molecules in the resonator We do not consider noise or physical limitations pro- 11 packaging. Another noise source is due to temperature fluc- duced by particular transducer implementations. Electrostati- tuations caused by the finite thermal conductance of the cally driven and detected resonators suffer from surface 12 resonator; these fluctuations are fundamental to any object charge motion; magnetomotive approaches require large with finite heat capacity, and are distinct from environmental stable ambient magnetic fields; optical approaches require stable sources of monochromatic light. We are more con- a͒Electronic mail: [email protected] cerned with the limitations set by the physics of resonator

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TABLE II. Parameters for the beam in this calculation.

␯ LtwM 1 k1 ͑␮m͒͑␮m͒͑␮m͒͑fg͒͑GHz͒ (␮mϪ1)

0.66 0.05 0.05 3.84 1.00 7.17

E t ␯ ϭ⍀ ␲ϭ ͱ ͑ ͒ 1 1/2 1.027 , 4 ␳ L2 FIG. 1. Doubly clamped beam with length L, width w and thickness t.The ␯ ␯ ϭ end supports are assumed inﬁnitely rigid. and the higher modes are n / 1 2.756, 5.404, and 8.933 for nϭ2, 3, and 4. ͑ ͒ The eigenfunctions Y n in Eq. 2 are mutually orthogo- behavior, and transducer approaches should be evaluated nal, and we normalize them to the beam length, so that separately from these. L ͵ ͑ ͒ ͑ ͒ ϭ 3␦ ͑ ͒ Y n x Y m x dx L mn . 5 0

II. DOUBLY CLAMPED BEAM RESONATORS The corresponding coefficients C1n and C2n are listed in Table I. An arbitrary solution Y(x,t) to undriven or driven A. Euler–Bernoulli theory motion can be written In Fig. 1, we show the structure forming the basis of our ϱ calculations: A doubly clamped beam of length L, width w, ͒ϭ ͒ ͒ ͑ ͒ Y͑x,t ͚ an͑t Y n͑x , 6 and thickness t, oriented along the x axis, driven into flexural nϭ1 resonance with displacement along the y axis. where the amplitudes a are dimensionless. The dynamic behavior of a flexural beam is most easily n The fundamental frequency ␯ is a function of the ma- treated using the Euler–Bernoulli theory, which applies to 1 terial parameters E and ␳ as well as the beam dimensions t beams with aspect ratios L/tӷ1.15 For an isotropic material, and L. High frequencies can be achieved by reducing the the transverse displacement Y(x,t) of the beam centerline overall resonator scale, by choosing stiffer and lighter mate- ͑along the y direction͒, obeys the differential equation rials, and by reducing the aspect ratio L/t; all three of these -2Y approaches are being used, and at present the highest reץ 2ץ 2Yץ ␳A ͑x,t͒ϭϪ EI ͑x,t͒, ͑1͒ ported frequency is 0.63 GHz, for a SiC beam.16 For the 2 ץ 2 ץ 2 ץ t x x purposes of this article, we will focus on a single-crystal Si where ␳ is the material density, Aϭwt is the cross-sectional beam with dimensions as given in Table II; the relevant area, E is Young’s modulus, and Iϭwt3/12 is the bending physical properties for Si are given in Table III, all at room moment of inertia. The clamped ends, at xϭ0 and xϭL, temperature. Our calculations are for the fundamental reso- impose the boundary conditions Y(0)ϭY(L)ϭ0 and nance nϭ1. YЈ(0)ϭYЈ(L)ϭ0. The solutions have the form In the next section, we discuss the anelastic processes that result in a finite quality factor Q for the beam resonance, Y ͑x,t͒ϭ͑C ͑cos k xϪcosh k x͒ n 1n n n described within the context of Zener’s model for anelastic ϩ Ϫ ͒͒ Ϫ ⍀ ͒ ͑ ͒ C2n͑sin kn sinh knx exp͑ i nt , 2 solids. In that section, we will describe how the Zener model ϭ is included in the formalism described so far. In later sec- with eigenvectors kn satisfying cos knL cosh knL 1. The first ϭ tions, we discuss other types of noise that are not described four eigenvectors are given by knL 4.730 04, 7.8532, ⍀ within the context of the Zener model; these have to do with 10.9956, and 14.1372. The angular frequencies n are given parametric changes in the physical properties of the resona- by tor, such as its mass and length, which cause the natural EI resonance frequency of the resonator to change, but do not ⍀ ϭͱ k2 . ͑3͒ n ␳A n necessarily involve energy dissipation. Any single parametric change can be associated with a change in the resonator The fundamental eigenfrequency is given by

TABLE III. Properties for Si at room temperature. TABLE I. Numerical solutions for a doubly clamped beam. Density ␳ 2330 kg/m3 ϫ 11 2 nϭ1234 Young’s modulus E 1.69 10 N/m Thermal conductivity ␬ 1.48 W/cm K 3 knL 4.730 04 7.8532 10.9956 14.1372 Speciﬁc heat CV 1.64 J/cm K ␯ ␯ n / 1 1 2.756 5.404 8.933 Sound speed cs 5860 m/s Ϫ Ϫ Ϫ Ϫ C1n /L 1.0000 1.0000 0.9988 1.0000 Phonon mean-free path l 50 nm ␣ ϫ Ϫ6 C2n /L 0.9825 1.0008 0.9988 1.0000 Thermal expansion T 2.8 10 /K

Downloaded 21 May 2003 to 128.111.14.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 2760 J. Appl. Phys., Vol. 92, No. 5, 1 September 2002 A. N. Cleland and M. L. Roukes

energy, depending on where in the oscillation cycle the Equation ͑8͒ implies that the stress ␴ will include a com- change occurs, but given events that occur randomly over the ponent that is 90° out of phase with the strain ⑀, which oscillation period, on average, the energy change is zero. The causes energy loss at a rate proportional to ⌬. For small ⌬, noise sources we discuss include adsorption–desorption we define the quality factor Q as the ratio of the imaginary to noise due to molecules around the resonator, temperature the real part of E: fluctuations that change the length and longitudinal stress in ␻␶ the resonator, and defect motion within the resonator. The QϪ1ϭ ⌬. ͑10͒ latter can to some extent be included in the Zener model, but 1ϩ␻2␶2 some modes of defect motion will not generate intrinsic dis- We then use the effective Young’s modulus E in the sipation but instead give rise to parametric changes. eff Euler–Bernoulli formula, Eq. ͑1͒ at frequency ␻, For these sources of parametric noise, an instantaneous 4Yץ measurement of the response of a resonator, as a function of i ␻2␳ ͑ ͒ϭ ͑␻͒ ͩ ϩ ͪ ͑ ͒ ͑ ͒ frequency, would indicate the actual dissipation associated AY x Eeff I 1 x . 11 x4ץ Q with material losses, while a measurement that takes a nonzero time to complete allows the resonator frequency to fluc- The spatial solutions Y(x) are the same as for Eq. ͑1͒, but the ⍀Ј tuate over the period of the measurement, and would give a dispersion relation giving the damped eigenfrequencies n ⍀ response curve that appears to be associated with a higher in terms of the undamped frequencies n is rate of dissipation than is actually present, due to the spread i of resonance frequencies that appear over the course of the ⍀Јϭͩ ϩ ͪ ⍀ ͑ ͒ n 1 n , 12 measurement. Separating these two effects experimentally is 2Q a very challenging but intriguing problem. Ϫ1 ⍀Ј for small dissipation Q . The imaginary part of n implies that the nth eigenmode will decay in amplitude Ϫ⍀ as exp( nt/2Q). B. Dissipation in mechanical resonators The most significant mechanism for energy loss in a na- C. Driven damped beams nomechanical resonator is through intrinsic losses in the We now add a harmonic driving force F(x,t) beam material, which can be treated using Zener’s model for ϭ f (x)exp(i␻ t), where f (x) is the position-dependent force anelastic solids.13 Other important loss terms include ther- c 7 per unit length. The force is uniform across the beam cross moelastic processes, which are negligible for the resonator ␻ section and directed along y, and the carrier frequency c is geometry and dimensions given here, and through the trans- close to ⍀ . The equation of motion is15 duction mechanism,17 which can usually be minimized 1 4ץ 2ץ through design considerations. Y Y ␻ ␴ ␳A ϩEA ϭ f ͑x͒ei ct. ͑13͒ x4ץ t2ץ In Zener’s model, the Hooke’s stress–strain relation ϭE⑀, relating the stress ␴ to the strain ⑀, is generalized to ⍀ ӷ allow for mechanical relaxation in the solid: We solve this equation for long times, 1t/Q 1, so any transients damp out. The solution then has the form Y(x,t) d␴ d⑀ i␻ t ␴ϩ␶ ϭ ͩ ⑀ϩ␶ ͪ ͑ ͒ ϭY(x)e c . The amplitude Y(x) may be complex, so that ⑀ E ␴ , 7 dt R dt the motion is not necessarily in phase with the force F. Ex- panding the displacement in terms of the eigenfunctions Y n , where ER is the relaxed value of Young’s modulus. Loads ϱ ϱ 4Yץ ,applied slowly generate responses with the relaxed modulus Ϫ␻2␳ ͑ ͒ϩ n ϭ ͑ ͒ ͑ ͒ while rapidly varying loads involve a different value for the c A ͚ anY n x EA͚ an f x . 14 x4ץ nϭ1 nϭ1 modulus. We consider harmonic stress and strain variations, ␴(t) Using the defining relation for the eigenfunctions, Eq. ͑1͒, ␻ ␻ ϭ␴ei t and ⑀(t)ϭ⑀ei t. At low frequencies ␻␶Ӷ1, this be- the dispersion relation, Eq. ͑12͒, and the orthogonality rela- ϭ comes the standard Hooke’s law relation with E ER .At tions, Eq. ͑5͒, this can be written ␻␶ӷ ϭ high frequencies 1, the modulus becomes E EU ϭ ␶ ␶ 1 L ( ␴ / ⑀)ER , the unrelaxed Young’s modulus. For interme- ͑⍀Ј2Ϫ␻2͒ ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ m c am Y m x f x dx, 15 diate frequencies, Young’s modulus is complex, of the form ␳AL3 0 ␻ ⍀ i␻␶ for each term m in the expansion. For c close to 1 , only EϭE ͑␻͒ͩ 1ϩ ⌬ ͪ , ͑8͒ the mϭ1 term in Eq. ͑15͒ has a significant amplitude, given eff ϩ␻2␶2 1 by 1/2 with mean relaxation time ␶ϭ(␶␴␶⑀) , fractional modulus ⌬ϭ Ϫ 1 1 L difference (EU ER)/ER , and effective Young’s modu- a ϭ ͵ Y ͑x͒f ͑x͒dx, ͑16͒ 1 ␳ 3 ⍀2Ϫ␻2ϩ ⍀2 1 lus AL 1 c i 1/Q 0 Ϫ1 1ϩ␻2␶2 for small dissipation Q . E ϭ E . ͑9͒ We now take a uniform force, f (x)ϭ f . The integral in eff ϩ␻2␶2 R 1 ⑀ Eq. ͑16͒ is then

1 L sponse of the resonator, the noise spectral density takes on a ␩ ϭ ͵ ͑ ͒ ϭ ͑ ͒ 1 Y 1 x dx 0.8309. 17 somewhat different form. We will only treat the high- L2 0 ӷប␻ temperature limit, kBT , for the resonator noise. The amplitude is then A. Dissipation-induced amplitude noise ␩ f The displacement of a forced, damped beam driven near ϭ 1 ͑ ͒ a1 , 18 ͑ ͒ ⍀2Ϫ␻2ϩi⍀2/Q M its fundamental frequency is given by Eq. 18 . In the ab- 1 c 1 sence of noise, this solution represents pure harmonic motion where Mϭ␳AL is the mass of the beam, and the correspond- ␻ at the carrier frequency c . As discussed, the nonzero value ϭ ␻ Ϫ1 ing displacement of the beam is Y(x,t) a1Y 1(x)exp(i ct). of Q and temperature T necessitates the presence of noise, If the force distribution f (x) is instead chosen to be pro- from the fluctuation–dissipation theorem. Regardless of the ͑ ͒ portional to the eigenfunction Y 1(x), the integral Eq. 16 is origin of the dissipation mechanism, it acts to thermalize the ␩ ͑ ͒ unity, so that 1 in Eq. 18 is replaced by the number 1. motion of the resonator, so that in the presence of dissipation We point out that the response function, Eq. ͑18͒, while ͑ ͒ only no driving force , the mean energy ͗En͘ for each mode similar to that of a damped, one-dimensional harmonic oscil- ϭ n of the resonator will be given by ͗En͘ kBT, where T is lator, differs slightly in the Q-dependent denominator, but the the physical temperature of the resonator. This noise term has difference is only apparent for small values of Q: For values been considered by a number of authors.9,14 of Q greater than 15, the fractional difference at any fre- The thermalization occurs due to the presence of a noise quency is less than 1%. force f N(x,t) per unit length of the beam. Each point on the beam experiences a noise force with the same spectral density, but fluctuating independently from other points; the III. NOISE IN DRIVEN DAMPED BEAMS noise at any two points on the beam is uncorrelated. The Systems that dissipate energy are necessarily sources of noise be written as an expansion in terms of the eigenfunc- noise; the converse is also often true. This is the basic state- tions Y n(x), ment of the fluctuation–dissipation theorem, and is best ϱ 1 ͒ϭ ͒ ͒ ͑ ͒ known in relation to electrical circuits, where it is termed the f N͑x,t ͚ f N ͑t Y n͑x , 20 Nyquist–Johnson theorem. An electrical circuit element with L nϭ1 n an electrical impedance Z(␻) that has a nonzero real part, where the force f N associated with the mode n is uncorre- R(␻)ϭRe Z(␻), will be a source of noise, that is, of fluc- n lated with that for other modes nЈ; the factor 1/L appears tuations in the voltage V(t) across the impedance Z,or because of the normalization of the Y . equivalently in the current I(t) through Z. A voltmeter placed n The noise force f (t) has a white spectral density across the circuit element will measure an instantaneous Nn S (␻), and a Gaussian distribution with a zero mean. The voltage that fluctuates with a Gaussian distribution in ampli- f n magnitude of the spectral density S may be evaluated by tude, with zero average value, and a width that is determined f n only by R(␻) and the temperature T. A useful way to quan- requiring that it achieve thermal equilibrium for each mode tify the noise is to use the average spectral density of the n. The spectral density of the noise-driven amplitude an of noise in angular frequency space, defined for a noise voltage the nth mode is given by V(t)by S ͑␻͒ 1 f n ϱ S ͑␻͒ϭ . ͑21͒ S ͑␻͒ϭͳ ͵ V2͑t͒cos͑␻t͒dtʹ . ͑19͒ an ͑⍀2Ϫ␻2͒2ϩ͑⍀2 ͒2 2 V /Q M Ϫϱ n n The SI units for S are (N/m)2/͑rad/s͒ϭkg2/͑s3 rad). Those Here the angle brackets ͗ ...͘ indicate that a statistical en- f n for S are 1/͑rad/s͒, because a is dimensionless. semble average, over many equivalent systems, is to be an n taken. The spectral density is proportional to the electrical The kinetic energy KEn of the nth mode associated with the spectral density S is given by noise power in a unit bandwidth. The Nyquist–Johnson theo- an rem states that this quantity is given by S (␻) V ϱ L ϭ ␲ ␻ ប␻ ប␻ 1 2 (2/ )R( ) coth( /kBT). At high temperatures or low ͗ ͘ϭ ͵ ͵ ␳ ␻2 ͑␻͒ ͑ ͒ ␻ KEn A Sa Y n x dxd ӷប␻ 2 0 0 n frequencies, such that kBT , this approaches the classical limit S (␻)→2k TR(␻)/␲. The spectral noise density V B ϱ ␲ 2 S ͑␻͒ ϭ␻ ␲ 1 QL f n SV( f ), as a function of frequency f /2 , is given in the ϭ ͵ ␳AL3␻2S ͑␻͒d␻Ϸ , ϭ ␲ ␻ ϭ an ⍀ high-temperature limit by SV( f ) 2 SV( ) 4kBTR( f ). 2 0 4 n M 2 The metric units of SV( f ) are V /Hz. The corresponding ͑ ͒ ϭ 2 22 current spectral noise density is SI( f ) SV( f )/R ( f ) → Ϫ1→ 4kBT/R( f ), in the high temperature limit, with units of where the last equality becomes exact in the limit Q 0. A2/Hz. The error in Eq. ͑22͒ for finite Q is less than 1% for Q The fluctuation–dissipation theorem applies to mechani- Ͼ10. cal resonators with nonzero dissipation, i.e., with finite Q, In order that this yield thermal equilibrium, the kinetic energy is ͗KE ͘ϭ 1k T, so the spectral density S must be and ensures that the mechanical resonator will also be a n 2 B f n source of noise, but due to the resonant nature of the re- given by

Downloaded 21 May 2003 to 128.111.14.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 2762 J. Appl. Phys., Vol. 92, No. 5, 1 September 2002 A. N. Cleland and M. L. Roukes

The phase variation at ␻ generates sidebands spaced Ϯ␻ Ϯ ␾ from the carrier, with amplitude a0 0/2. The lower sideband is phase coherent with the upper sideband, with the opposite sign; this is characteristic of phase noise. Indepen- dent sideband signals can be generated by adding an amplitude noise source M(t) to the phase noise ␾(t), so that the amplitude is written as ͒ϭ ϩ ͒͒ ␻ ϩ␾ ͒ϩ␪͒ ͑ ͒ a͑t a0͑1 M͑t sin͑ ct ͑t . 27 We consider a single component at ␻ for both the phase and amplitude modulation, so that ͒ϭ ␻ ͒ M͑t M 0 sin͑ t FIG. 2. Frequency spectrum of a driven beam in the presence of noise, ͑28͒ showing both the central driven peak as well as the noise sidebands. ␾ ͒ϭ␾ ␻ ͒ ͑t 0 sin͑ t . Again assuming small variations, this can be written as ⍀ 2kBTM n ͑ ͒ϭ ͑␻ ϩ␪͒ϩ 1 ͑ ϩ␾ ͒ ͑͑␻ ϩ␻͒ ͒ S ͑␻͒ϭ . ͑23͒ a t a0 sin ct 2 a0 M 0 0 sin c t f n ␲ 2 QL ϩ 1 ͑ Ϫ␾ ͒ ͑͑␻ Ϫ␻͒ ͒ ͑ ͒ 2 a0 M 0 0 sin c t . 29 The term L2 appears in Eq. ͑23͒ because f is the force per Nn ϭ␾ unit length of beam. An equivalent derivation for a one- Setting the amplitude M 0 0 , the lower sideband disap- dimensional simple harmonic oscillator yields the force den- pears and we are left with the independent upper sideband ␻ ϭ ⍀ ␲ term, sity SF( ) 2kBTM / Q. We can write the spectral density of the thermally driven amplitude as ͒ϭ ␻ ϩ␪͒ϩ ␾ ␻ ϩ␻͒ ͒ ͑ ͒ a͑t a0 sin͑ ct a0 0 sin͑͑ c t . 30 ⍀ n 2kBT ϭϪ␾ S ͑␻͒ϭ . ͑24͒ Choosing the opposite sign relation M 0 0 allows the an ͑⍀2Ϫ␻2͒2ϩ͑⍀2 ͒2 ␲ 2 n n/Q ML Q lower sideband to be chosen. A noise signal at a frequency offset from the carrier can When superposed with a driving force with a carrier fre- be created from the superposition of a phase and an ampli- quency ␻ ϭ⍀ , the amplitude noise power consists of a c 1 tude modulation of the original carrier. In applications where ␦-function peak at the carrier superposed with the Lorentzian the resonator is to be used as a frequency source or a clock, given by Eq. ͑24͒, as sketched in Fig. 2. the amplitude modulation is unimportant: Use of a zero- crossing detector, or a perfect limiter, eliminates the effects B. Dissipation-induced phase noise of the amplitude modulation. We therefore ignore this noise source. This is equivalent, from the arguments leading to Eq. ͑ ͒ The form in Eq. 24 represents frequency-distributed ͑26͒, to limiting the noise to that which is phase coherent amplitude noise. Equivalent expressions can be written for between the upper and lower sidebands, with amplitudes the phase noise, the fractional frequency noise, and the Allan ␻ ϩ␻ ϭϪ ␻ Ϫ␻ 18 a( c ) a( c ). Noise which has the opposite variance, which are useful for time-keeping and ﬁlter ap- ␻ ϩ␻ ϭϩ ␻ Ϫ␻ phase relation, with a( c ) a( c ), is due to am- plications. We note that the different expressions are all plitude modulation. Noise associated with only one sideband equivalent ways of expressing the same noise, and do not consists of the sum or difference of these two ‘‘modes’’. represent additional sources of noise. The resonator is driven Dissipation-induced noise, of the form given by Eq. ⍀ by the carrier signal near its resonance frequency 1 , and in ͑24͒, is intrinsically phase incoherent on opposite sides of the addition by dissipation-induced noise. The time-dependent ␻ carrier signal at c . Half of the noise power is therefore amplitude is then associated with amplitude modulation, and half with phase ͒ϭ ␻ ϩ␾ ͒ϩ␪͒ ͑ ͒ a͑t a0 sin͑ ct ͑t , 25 modulation; the phase noise power is therefore half the original total noise power. where ␾(t) represents a phase variation from the carrier at ␻ Ϸ⍀ ␪ We can evaluate the dissipation-induced phase noise for frequency c 1 ; the amplitude a0 is constant, and is a a resonator driven at its fundamental resonance frequency. phase offset. Following Robins,19 we pick one frequency ␻ ␾ ϭ␾ ␻ We drive the resonator with a force f per unit length, at the component at for the phase variation, (t) 0 sin( t). ␻ ϭ⍀ ␾ frequency c 1 . The amplitude for the response is given Assuming small maximum deviation 0 , the amplitude may by Eq. ͑18͒, be written ␩ ␾ 1 Qf ͑ ͒ϭ ͑␻ ϩ␪͒ϩ 0 ͑͑␻ ϩ␻͒ ͒ a ϭϪi . ͑31͒ a t a0 sin ct a0 sin c t 1 ⍀2 M 2 1 ␾ 0 The amplitude lags the force by 90°, and includes the mul- Ϫa sin͑͑␻ Ϫ␻͒t͒. ͑26͒ 0 2 c tiplicative factor Q. Dissipation generates incoherent noise,

͒ϭ ␻ ϩ ␦␻ ␻͒ ␻ ͒ϩ␪͒ distributed about the carrier with noise power given by Eq. a͑t a0 sin͑ ct ͑ 0 / sin͑ t ͑24͒. The phase noise power density S␾(␻) at frequency ␻ ␦␻ 1 0 from the carrier frequency is then given by ϭa sin͑␻ t͒ϩ a sin͑͑␻ ϩ␻͒t͒ 0 c 2 0 ␻ c ͑⍀ ϩ␻͒ 1 Sa 1 1 ␦␻ S␾͑␻͒ϭ 1 0 2 ͉ ͉2 Ϫ a sin͑͑␻ Ϫ␻͒t͒, ͑40͒ a1 2 0 ␻ c ⍀ 1 kBT a result very similar to that for phase variations, Eq. ͑26͒. ϭ . ͑32͒ ͑ ⍀ ␻ϩ␻2͒2ϩ͑⍀2 ͒2 ␲͉ ͉2 2 The arguments leading to the spectral density, Eq. ͑34͒, 2 1 1/Q a1 L MQ may be reworked to yield the equivalent expression for the ␻ For frequencies that are well off the peak resonance, frequency variation noise density S␦␻ . A more useful quan- ӷ⍀ 1 /Q, but small compared to the resonance frequency, tity is the fractional frequency variation, deﬁned as y ␻Ӷ⍀ , we may approximate the denominator as ϭ␦␻ ␻ 1 / c . The noise density for y is related to that for the phase noise density by kBT S␾͑␻͒Ϸ 2 2 y ␻ץ ␲⍀ ␻2͉ ͉2 2 4 1 a1 L MQ ͑␻͒ϭͩ ͪ ͑␻͒ϭͩ ͪ ͑␻͒ ͑ ͒ ␾ S␾ ␻ S␾ , 41ץ Sy c k T⍀ Ϸ B 1 ⍀ Ӷ␻Ӷ⍀ ͒ ͑ ͒ ␻ ͑ 1 /Q 1 . 33 where we use the fact that modulation at generates side- 8␲E Q␻2 Ϯ␻ ␻ ͑ ͒ c bands at from the carrier at c . From Eq. 34 , we then have Here, we deﬁne the energy Ec at the carrier frequency, Ec ϭ ⍀2 2͉ ͉2 M 1L a1 /2. This can also be written in terms of the kBT power P ϭ⍀ E /Q needed to maintain the carrier ampli- S ͑␻͒Ϸ ͑␻ /QӶ␻Ӷ␻ ͒. ͑42͒ c 1 c y ␲ 2 c c tude, i.e., that needed to counter the loss due to the nonzero 8 PcQ value of 1/Q: In the frequency domain, this is ⍀ 2 kBT 1 ␲ ͑␻͒Ϸ ͩ ͪ ͑⍀ Ӷ␻Ӷ⍀ ͒ ͑ ͒ kBT S␾ 1 /Q 1 . 34 ͑ ͒Ϸ ͑␯ Ӷ Ӷ␯ ͒ ͑ ͒ ␲ 2 ␻ Sy f c /Q f c . 43 8 PcQ 2 4PcQ We can also write this expression in terms of frequency f ϭ2␲␻,

k T ␯ 2 D. Allan variance ͑ ͒Ϸ B ͩ 1 ͪ ͑␯ Ӷ Ӷ␯ ͒ ͑ ͒ S␾ f 1 /Q f 1 . 35 4P Q2 f A third useful quantity, commonly used to compare fre- c ␴ ␶ 20,21 quency standards, is the Allan variance A( A). The phase and frequency noise are defined in the frequency do- C. Frequency noise main; the Allan variance is defined in the time domain, as the The phase fluctuations can also be viewed as frequency variance over time in the measured frequency of a source, ␶ fluctuations, where the amplitude a(t) has a time depen- each measurement averaged over a time interval A , with dence zero-dead time between measurement intervals. The defining

t expression for the square of the Allan variance is a͑t͒ϭa sinͩ ͵ ␻͑tЈ͒dtЈϩ␪ͪ . ͑36͒ 0 N Ϫϱ 1 1 ␴2 ͑␶ ͒ϭ ͑¯ Ϫ¯ ͒2 ͑ ͒ A A ͚ f m f mϪ1 , 44 2 NϪ1 ϭ The time-dependent frequency ␻(t) is related to the carrier 2 f c m 2 ␻ ␾ frequency c and the phase (t)by ¯ where f m is the average frequency measured over the mth ␻ ϩ␾ ͒͒ ␾ d͑ ct ͑t d time interval, of length ⌬tϭ␶ , and f is the nominal carrier ␻͑t͒ϭ ϭ␻ ϩ . ͑37͒ A c dt c dt frequency. The squared Allan variance is related to the phase noise density by21 We deﬁne time-dependent frequency variation ␦␻(t)as 2 2 ϱ ␾͑ ͒ ␴2 ͑␶ ͒ϭ ͩ ͪ ͵ ͑␻͒ 4͑␻␶ ͒ ␻ ͑ ͒ d t A A 2 ␻ ␶ S␾ sin A/2 d , 45 ␦␻͑t͒ϭ␻͑t͒Ϫ␻ ϭ . ͑38͒ c A 0 c dt where ␻ ϭ2␲ f and ␻ is the modulation frequency. We consider a single-phase modulation component, so that c c The Allan variance can be calculated for various func- ␾(t)ϭ␾ sin(␻t). The frequency variation 0 tional forms for the phase noise density. For a fractional ␦␻ ͒ϭ␦␻ ␻ ͒ϭ␻␾ ␻ ͒ ͑ ͒ ␻ ͑t 0 cos͑ t 0 cos͑ t 39 frequency noise that has a 1/f component, so that Sy( ) ϭ ␻ ␻ A( c / ), where A is a scale factor, the phase noise den- represents a sinusoidal variation of the frequency, with am- 3 sity is S␾(␻)ϭA(␻ /␻) , and the Allan variance is ␦␻ ϭ␻␾ ␻ c plitude 0 0 , modulated at . The time-dependent ͑ ͒ ␴ ␶ ͒ϭͱ ␻ ͑ ͒ amplitude in Eq. 36 is A͑ A 2 loge 2A c. 46

Downloaded 21 May 2003 to 128.111.14.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 2764 J. Appl. Phys., Vol. 92, No. 5, 1 September 2002 A. N. Cleland and M. L. Roukes

␴ FIG. 3. Dependence of the Allan variance A on the dimensionless time FIG. 4. Allan variance ␴ as a function of measurement interval ␶ ,cal- ␻ ␶ A A interval c ; the Allan variance has been scaled to remove the overall culated for the model 1 GHz resonator for a range of values of Q from 103 dependence on Q and on drive energy ␧ . The full dependence ͓from Eq. 6 c to 10 . The drive power level Pc was chosen so that the amplitude of motion ͑32͔͒ is plotted as a solid line, while the approximate form ͓Eq. ͑50͔͒ is ϭ a1 t, the beam thickness, For comparison, we also display the variance for plotted as a dotted line. a 10 MHz quartz crystal frequency standard, the HP10811D; ﬁgures taken from manufacturer’s speciﬁcations.

Hence a 1/f fractional frequency noise yields an Allan vari- frequency-lock applications, one is typically interested in ance that is independent of the measurement time interval. times much longer than this, so the approximate form is quite For a source that displays frequency drift, the fractional adequate. ␻ ϭ ␻ ␻ 2 frequency noise will have the form Sy( ) B( c / ) ; the We now turn to examining what the implications are for ␻ ϭ ␻ ␻ 4 phase noise density is then S␾( ) B( c / ) , and the Al- the fundamental sources of noise in our model resonator. We lan variance is will focus on calculating the predicted dependence of the ␲ Allan variance. ␴ ͑␶ ͒ϭͱ B␻2␶ . ͑47͒ A A 3 c A E. Model resonator Q-dependent noise Finally, for a white fractional frequency noise density ␴ We calculate the dependence of the Allan variance A on S (␻)ϭC, the phase noise density has the form S␾(␻) y the time interval and Q for our model 1 GHz resonator, de- ϭC(␻ /␻)2, and the Allan variance is c scribed in Table II. The results of the calculation are shown ␲C in Fig. 4. The drive power level P is chosen so that the ␴ ͑␶ ͒ϭͱ ͑ ͒ c A A ␶ . 48 amplitude of motion a in Eq. ͑18͒ is equal to the beam A 1 thickness t, approximately the amplitude for the onset of In particular, for the approximate form for the phase noise nonlinearity. The calculated temperature rise due to dissipa- ͑ ͒ density Eq. 34 , the Allan variance is tion in the resonator from this level of drive power is very small, of order 0.1 K. We also display for comparison the k T ␴ ͑␶ ͒ϭͱ B ͑ ͒ Allan variance for an oven-controlled 10 MHz quartz crystal A A . 49 2␶ 22 8PcQ A oscillator, the HP10811D. For quality factors higher than ␧ 105, the calculated thermally induced fluctuations are com- Defining the dimensionless drive energy c as the ratio of ␧ parable to or better than those of the bulk quartz resonator. drive energy per cycle to the thermal energy, c ϭ ␲ ␻ 2 Pc / ckBT, we have IV. TEMPERATURE FLUCTUATIONS 1 ␲ ␴ ͑␶ ͒ϭ ͱ . ͑50͒ We now turn to a discussion of the effects of finite ther- A A Q 4␧ ␻ ␶ c c A mal conductance and heat capacitance on the Allan variance. We see that the Allan variance falls inversely with the square The small dimensions associated with our model resonator, ␻ ␶ root of the product c A , and it is also proportional to the and of nanoscale resonators in general, imply that the heat dissipation QϪ1. Other things being equal, increasing the capacity of the resonator is very small. The corresponding ␻ resonator frequency c lowers the Allan variance. thermal fluctuations are proportionally larger, and these may In Fig. 3, we display the approximate result Eq. ͑50͒ as a in turn produce significant frequency fluctuations, due to the ␻ ␶ function of c A , scaled to remove the dependence on Q temperature dependence of the resonator material parameters ␧ and on c ; we also show the full result obtained from inte- and geometric dimensions. Here, we present a simple model grating Eq. ͑32͒, for values of QϾ100; for values of Q less through which the magnitude of these effects can be esti- than this, the calculated value for the scaled variance falls mated. below that plotted. A heat capacitance c, connected by a thermal conduc- We see that the approximate expression given by Eq. tance g to an infinite thermal reservoir at temperature T, will ͑ ͒ ␶ ϭ 50 works quite well for averaging times A more than a few have an average thermal energy ͗Ec͘ cT in the absence of ␲ ␻ tens of the oscillation period 2 / c ; in time-keeping or any power loads. Changes in the temperature relax with the

FIG. 6. Thermal model for doubly clamped beam, consisting of a series

connection of heat capacitances cn and thermal conductances gn , each associated with a cross-sectional slice of the beam of length ⌬x. Each thermal FIG. 5. Thermal circuit with a ﬁnite thermal conductance g and a ﬁnite heat conductance is associated with a power noise source pn . The ends are capacitance c, including a power noise source p. assumed clamped at the reservoir temperature T. There are a total of N ϭL/⌬x elements.

␶ ϭ thermal time constant T c/g. The energy Ec will however fluctuate, as the fluctuation–dissipation theorem applies to One might expect that the most accurate model would finite thermal conductances in a manner similar to the use slices with differential lengths ⌬xϭdx→0. However, dissipation-induced mechanical noise.23 The thermal circuit once the slices become shorter than the phonon mean-free therefore includes a power noise source p with spectral den- path l , the temperature in a slice is no longer well defined. ␻ ϭ 2 ␲ ͑ ͒ ⌬ ϭ ϭ sity S p( ) 2kBT g/ see Fig. 5 . The instantaneous en- We therefore choose slices with a length x l 50 nm, so ϭ ϭ ϫ ϫ 3 ergy of the heat capacitance can then be written Ec(t) cT that each element has a volume V 50 50 50 nm . The ϩ␦ ϭ ϭ ϫ Ϫ16 E(t), where the spectral density of the energy fluctua- corresponding heat capacity is c CVV 2 10 J/K, and tions ␦E(t) can be derived from the thermal circuit, the thermal links have gϭ␬l ϭ7.4ϫ10Ϫ6 W/K. The ther- ␶ ϭ 2 2 mal time constant is T 30 ps, corresponding to thermal 2 kBT c /g ϳ S ͑␻͒ϭ . ͑51͒ frequencies 35 GHz, well outside the range of frequencies E ␲ ϩ␻2␶2 ⍀ Ӷ␻ 1 T of interest for resonator phase fluctuations, 1 /Q Ӷ⍀ . For the purposes of this calculation, therefore, we can We can interpret the energy fluctuations as temperature fluc- 1 treat the thermal fluctuations in the low-frequency limit. tuations ␦T (t), if we define the temperature as T ϭE /c. c c c Consider only the nth power source of the conductance The corresponding spectral density of the temperature fluc- ␻ Ϫ pn . If we take the frequency component at , the (n 1)th tuations is given by ␻ and nth slices have temperature variations TnϪ1( ) and 2 ␻ 2 k T /g Tn( ) given by ␻͒ϭ B ͑ ͒ ST͑ . 52 ␲ 1ϩ␻2␶2 p T ϩ ␻␶͒ ϭϪ n ϩ ϩ ͑2 i TnϪ1 TnϪ2 Tn At low frequencies ␻, below that of the thermal frequency g 1/␶ , the temperature fluctuations ␦T follow those driven by ͑53͒ T p the noise source p, while at higher frequencies the nonzero ϩ ␻␶͒ ϭ n ϩ ϩ ͑2 i Tn TnϪ1 Tnϩ1 . heat capacitance acts as a filter. g For a resonator with the geometry shown in Fig. 1, there The corresponding equation for the (nϩm)th slice is given is no clear separation of the structure into a distinct heat by capacitance and thermal conductance. Instead, we divide the ⌬ ϩ ␻␶ ͒ ϭ ϩ ͑ ͒ resonator into slices of length x and cross-sectional area ͑2 i T Tnϩm TnϩmϪ1 Tnϩmϩ1 . 54 Aϭwϫt, so that the nth slice has heat capacity c n ␻␶ Ӷ ϭC A⌬x, where C is the specific heat per unit volume. Taking the limit T 1, we find that the power source v v ␻ The (nϪ1)th and nth slices are connected to one another by pn( ) driving the nth slice generates a temperature variation ␻ ϭ ␻ the thermal conductance g ϭ␬A/⌬x, with thermal conduc- T( ) pn( )/2g uniformly across the beam. The corre- n Ϫ ␻ Ϫ ␬ ␬ϭ sponding anticorrelated source pn( ) driving the (n 1)th tivity given by the classical formula, (1/3)CVcsl (l is the phonon mean-free path and c the sound speed͒. The slice generates an equal but opposite temperature variation. s ␻␶ Ӷ Hence, in the limit T 1, the fluctuations driven by con- thermal conductances gn are associated with noise power ␻ ϭ 2 ␲ ductances within the beam have no net effect. sources pn , with spectral density S p ( ) 2kBT gn / . Fi- n The other source of temperature fluctuations comes from nally, the temperatures at the ends of the beam, where the the conductances at the beam ends, g and g ϩ . These also beam is mechanically clamped, are assumed to be given by 1 N 1 drive the beam uniformly, but as the energy that appears in the reservoir temperature T; see Fig. 6. the first and last elements does not have an adjacent anticor- In this model, energy fluctuations in the slices nϪ1 and related source, there is now a net effect. The final result from n are anticorrelated through the shared conductance g :An n this model is that the temperature of all the elements in the energy ␦E driven into the nth slice by p corresponds to the n beam fluctuate uniformly, with spectral density S (␻) given same energy taken from the (nϪ1)th slice. These energy T by the incoherent sum of the two end sources, fluctuations then relax through conductance into adjacent slices, and so on through the beam length, so that there is 2 4 kBT /g some correlation between the fluctuations in all slices, al- S ͑␻͒ϭ ͑␻␶ Ӷ1͒. ͑55͒ T ␲ ϩ␻2␶2 T though the correlations get weak for distant slices. 1 T

Downloaded 21 May 2003 to 128.111.14.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 2766 J. Appl. Phys., Vol. 92, No. 5, 1 September 2002 A. N. Cleland and M. L. Roukes

␶ We have kept the frequency dependence of the thermal cir- the same magnitude and dependence on averaging time A as cuit in order to retain the high-frequency cutoff in the fluc- that due to mechanical dissipation for a resonator with Q of tuations. about 104, and is a significant source of fluctuation. Ways to The effect these temperature fluctuations have on the reduce the size of this source of variance include the use of resonator behavior involves changes in the density and elas- materials with larger thermal conductance, such as AlN ͑3.20 tic modulus, which affect the frequency directly, and through W/cm K͒25 and sapphire ͑4.50 W/cm K͒, or better changes in the overall resonator length. The density and elas- temperature-compensated materials, such as quartz. tic modulus determine the resonator frequency through the ͱ ␳ combination E/ , which is the sound speed cs ; this quantity has the fractional temperature dependence V. OTHER SOURCES OF NOISE ϭϪ ϫ Ϫ5 24 ץ ץ (1/cs) cs / T 5 10 /K for pure Si. The corresponding temperature dependence of the resonator frequency is There are a number of other sources of noise that can given by affect resonator performance. We discuss here two such sources, one due to adsorption–desorption noise of residual ץ ⍀ץ 1 n 1 cs ϭ ͑sound speed dependence͒. ͑56͒ gas molecules, and the other due to defect motion within the Tץ T cץ ⍀ n s resonator structure. The changes in resonator length generate longitudinal A. Adsorption–desorption noise stress in the resonator, as the ends are assumed rigidly clamped. The resonator length L changes with temperature Adsorption–desorption noise has been discussed in some ϭ␣ 12 11 ץ ץ due to the thermal expansion of Si, with (1/L) L/ T T detail by Vig and Kim and Yong and Vig. The resonator ϭ2.8ϫ10Ϫ6/K. A temperature change ⌬T therefore induces environment will always include a nonzero pressure of ␶ϭϪ ␣ ⌬ a longitudinal extensional stress E T T. This in turn surface-contaminating molecules. These molecules, when ⍀ ␶ causes a change in the nth resonator frequency n( ), which they adsorb on a site on the resonator surface, mass load the for small extensional stress ␶ is given by15 resonator, and thereby change its resonance frequency. As the molecules adsorb and desorb due to their finite binding ␶ ⍀2͑␶͒ϭ⍀2͑0͒ϩk2 , ͑57͒ energy and nonzero temperature, the resulting changes in n n n ␳ frequency translate to a source of phase or fractional frequency noise. As discussed herein, this type of noise does in terms of the beam eigenvectors kn and eigenfrequencies ⍀ ϭͱ ␳ 2 not fit into the Zener formalism, as the adsorption– n EI/ Akn . The fractional frequency dependence due to length change is therefore desorption cycle is not intrinsically a dissipative one: As the arrival and departure times of the atoms are random, they do ⍀ 2ץ 1 n 1 E kn not on average change the energy of the resonator, but cause ϭϪ ␣ ͑length dependence͒. ͑58͒ T 2 ␳ ⍀2 T its frequency to change in a discontinuous fashion, leavingץ ⍀ n n the quality factor unchanged. This type of parametric noise Hence, the spectral density of fractional frequency fluc- ͑where the overall resonator mass is fluctuating͒ is therefore ␻ tuations Sy( ) caused by the combined temperature depen- not described by the lossy stress–strain relation developed dence on sound speed and beam length is given by by Zener. The frequency change due to a single adsorbed mol- ⍀ 2ץ 1 ͑␻͒ϭͩ ͪ ͑␻͒ ecule, ⌬⍀, is proportional to the ratio of the molecule to Sy ST T resonator total mass, m/M; smaller mass resonators are moreץ ⍀ 2 sensitive than larger ones. Furthermore, as the resonator size ␲ 2 ץ 2 2 cs kn 2 cs kBT / g ϭͩ Ϫ ␣ ϩ ͪ . ͑59͒ scale is reduced, the number of adsorption sites Na on the T ϩ␻2␶2ץ ⍀2 T c n s 1 T resonator surface grows in proportion to the number of total number of resonator atoms: The surface-to-volume ratio For the fundamental mode, we have k ϭ4.73/L, and insert- 1 grows inversely to the size scale. Hence, nanoscale resona- ing the model resonator parameters, we find tors are more susceptible to adsorption–desorption noise 2 ϫ Ϫ21 than larger, bulk mechanical resonators. Ϫ kBT /g 2.7 10 1 S ͑␻͒ϭ͑1.6ϫ10 8/K2͒ ϭ . We use a simple model to estimate the noise from this y ϩ␻2␶2 ϩ␻2␶2 rad/s 1 T 1 T source. We assume a single molecular species with mass m, ͑60͒ surface binding energy Eb , and pressure P. Expressions may ͑ ͒ From the fractional frequency noise, Eq. 60 , we calcu- be derived for the adsorption and desorption rates ra and rd late the Allan variance: at any given surface site;11 with sticking coefficient s, the adsorption rate at any site is given by ␴ ␶ ͒ϭ ϫ Ϫ4 2͒ͱ 2 ␶ A͑ A ͑2.25 10 /K kBT /g A 1 2 P Ϫ11 r ϭ s. ͑62͒ ϭ9.3ϫ10 , ͑61͒ a 5 ͱ ͱ␶ mkBT A ␶ ӷ␶ 26 in the limit A T . The contribution of thermal fluctuations The sticking coefficient is typically temperature dependent. to the Allan variance for our model resonator is therefore of Once bound to the surface, a molecule desorbs at a rate

E ϭ␯ ͩ Ϫ b ͪ ͑ ͒ rd d exp , 63 kBT ␯ where d is the desorption attempt frequency, typically of 13 order 10 Hz, and Eb is the desorption energy barrier. The ϭ ϩ average occupation f of a site is then f ra /(ra rd); the ␴2 ϭ variance in the occupation probability is occ rard /(ra ϩ 2 rd) . An expression may then be derived for the phase noise S␾(␻) resulting from the statistics of the adsorption– desorption process:11

␴2 ␶ ␲ ⌬⍀2 2Na occ A / S␾͑␻͒ϭ . ͑64͒ FIG. 7. Allan variance fora1saveraging interval as a function of package ϩ␻2␶2 ␻2 1 r pressure for two resonator temperatures, 300 and 500 K. The contaminant molecule has a binding energy E ϭ10 kcal/mol, with one adsorption site ␶ b Here, r is the correlation time for an adsorption–desorption every 0.25 nm2, and a sticking coefficient of 0.1. Also shown is the frac- ␶ ϭ ϩ cycle, 1/ r ra rd . A simple mass-loading formula may be tional noise for the HP10811D 10 MHz quartz resonator, fora1saveraging ⌬⍀ϭ ⍀ interval. assumed, (m/2M) 1 , where M is the resonator mass ⍀ and 1 is the fundamental frequency of the resonator. A corresponding expression for the fractional frequency noise is then given by B. Defect motion ␴2 ␶ ␲ 2 Na occ r/2 m S ͑␻͒ϭ ͩ ͪ . ͑65͒ The last source of noise we would like to consider is that y ϩ␻2␶2 M 1 r caused by defect motion within the resonator volume. For the single-crystal resonators we have been considering, de- We calculate the Allan variance, fect levels are very low in the base material. Statistically, ␶ ␴ Na r occm there are of order 0 to 1 defects within the volume of the ␴ ͑␶ ͒ϭͱ , ͑66͒ A A 2␶ M resonator. We therefore do not expect this to provide a seri- A ous source of noise in these resonators. ␶ ӷ␶ in the limit A r . However, there has been extensive work on developing ␴2 The occupation variance occ , and therefore the noise, is polycrystalline materials for resonators, especially using sili- maximum when the site occupation probability is f ϭ0.5, con nitride and poly-Si.5,6 These materials include a large i.e., when the adsorption and desorption rates are equal. The density of grain boundaries, point defects, and some voids, noise is minimized when the occupation probability is either and the motion of these defects can cause phase and fre- near zero or near unity. For typical packaged pressures, mo- quency noise in high-Q resonators. Amorphous Si created by lecular sticking coefficients, and binding energies, occupa- implantation has defect densities of order 1%;27 similar lev- tion probabilities are quite small; we therefore try to mini- els are expected for chemical vapor deposition-grown poly- mize the occupation to reduce the fluctuation variance. The silicon. The quality factors of resonators fabricated from this exponential dependence of the desorption rate rd on tempera- material are comparable, at room temperature, to those made ture provides a useful approach; heating the resonator, using from single-crystal materials such Si and GaAs, but defect an on-chip heating element, causes significant increases in processes may still play a very important role. the desorption rate and therefore in the occupation variance Point defects in a solid can be treated as elastic dipoles, ␴2 occ . with symmetry different from that of the underlying crystal In Fig. 7, we plot the Allan variance fora1saveraging ͑see e.g., Nowick and Berry13͒. In the relaxed state, the de- interval, as a function of package pressure, for two resonator fects are randomly oriented; reorientations occur due to ther- ⌫ ϭ␯ Ϫ⌬ temperatures, 300 and 500 K. We have chosen a contaminant mally induced motion, at a rate d 0 exp( g*/kBT), with ϭ ␯ ϳ 12 molecule with an binding energy of Eb 10 kcal/mol, with an attempt frequency 0 10 Hz and free energy barrier one adsorption site every 0.25 nm2, and a sticking coefficient ⌬g*, typically of order 0.1–1 eV. If we consider a defect of 0.1, typical values for gas molecules adsorbing on metal with two possible orientations, with equal energies, the oc- surfaces.11,26 Note that the sticking coefficient is typically cupation probabilities are given by the Gibbs distribution and temperature dependent,26 but here we have taken it as con- are each equal to 1/2; the variance in the mean occupation is ␶ ϭ ⌫ stant. We show for comparison the overall Allan variance for equal to 1/4, with mean reorientation time d 1/ d . De- the HP 10811D 10 MHz quartz oscillator, again fora1s fects in which the possible orientations have different ener- averaging interval. gies will have smaller variances in the occupation, and there It is clear from Fig. 7 that this source of noise is ex- will be a range of reorientation times as well. tremely important. Great care must be taken to passivate the A single defect moving or reorienting itself can cause a resonator surface, thus reducing the sticking coefficient, re- change in the local Young’s modulus E; if the defect has two duce the pressure, for instance by including getters in the possible configurations, Ϫ and ϩ, the corresponding local ϩ ϩ package, and possibly raising the ambient temperature. modulus changes from Es EϪ to Es Eϩ , where Es is the

Downloaded 21 May 2003 to 128.111.14.162. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp 2768 J. Appl. Phys., Vol. 92, No. 5, 1 September 2002 A. N. Cleland and M. L. Roukes

TABLE IV. Expressions for the various spectral noise densities worked out for different representations of the same noise, as well as for noise associated with different fundamental processes.

Type of noise Symbol Expression Equation

⍀ ␻ 2kBTM n ͑ ͒ Force noise, mode n S f ( ) 23 n ␲QL2

⍀ 2k T Amplitude noise, mode n S (␻) n B ͑24͒ an ͑⍀2Ϫ␻2͒2ϩ͑⍀2 ͒2 ␲ 2 n n/Q ML Q

2k TQ On-resonance amplitude noise, mode n S (⍀ ) B ͑24͒ an n ␲ 2⍀ ML n

⍀ 2 kBT 1 Off-resonance phase noise, mode nϭ1 S␾(␻) ͩ ͪ ͑34͒ ␲ 2 ␻ 8 PcQ

ϭ ␻ kBT ͑ ͒ Off-resonance fractional frequency noise, n 1 Sy( ) 42 ␲ 2 8 PcQ

k T Allan variance, nϭ1 ␴ (␶ ) ͱ B ͑49͒ A A 2␶ 8PcQ A

4 k T2/g Temperature ﬂuctuations S (␻) B ͑55͒ T ␲ ϩ␻2␶2 1 T

Allan variance, temperature ﬂuctuations ␴ (␶ ) ϫ Ϫ4 2͒ͱ 2 ␶ ͑61͒ A A ͑2.25 10 /K kBT /g A 2N ␶ ␴ m Allan variance, adsorption desorption ␴ (␶ ) ͱ a r occ ͑66͒ A A ␶ A M

2␴2 ␶ Allan variance, defect motion ␴ (␶ ) ͱ ␻ ͱ d ͑68͒ A A 2 ␶ ͗⍀͘ A

defect-free modulus. Typical values for EϮ are in the range For this distribution, the corresponding Allan variance has a Ӷ 0.01– 0.1Es . A mole fraction Cd 1 of such defects, all si- similar functional form to that shown in Fig. 3. The defect Ϫ ϩ ␶ ␮ multaneously reorienting from to , will cause the effec- reconfiguration time d can range from s to minutes or 13 ϩ ϩ tive modulus to change from Es CdEϪ to Es CdEϩ .If hours at room temperature, with an exponential temperature ␶ ␶ we consider a total mole fraction of identical defects C0 dependence. For averaging times A much larger than d , Ӷ1, that reorient independently between configurations the approximate form for the Allan variance is Ϫ and ϩ with equal free energies, so that the two configu- ␴2 ␶ rations are equally likely, the average elastic modulus for the 2 ␻ d ␴ ͑␶ ͒ϭͱ ͱ . ͑68͒ ͗ ͘Ϸ ϩ ϩ A A 2 ␶ solid will be E Es C0(EϪ Eϩ)/2. The variance in the ͗⍀͘ A ␴2 elastic modulus is given by the Poisson formula, E 2 ϭ ϷC (EϩϪEϪ) /4. Additional noise sources can appear If we assume a defect mole fraction C0 0.001, and modulus 0 ϭϮ ␴ Ϸ from the defect motion itself; the defect may resonate at a changes EϮ 0.1Es , the frequency variance is ⍀ 1 ϫ Ϫ6⍀ ␶ ϭ frequency near the resonator natural frequency, causing ad- 10 . For a defect reconfiguration time d 1 ms, we ␶ ϭ ϫ Ϫ8 ditional dissipation and additional noise terms. We ignore find an Allan variance at A 1 s of 5 10 , a quite large such effects here. contribution compared to those we have been considering. Applying this discussion to resonator frequency fluctua- Clearly, active defect concentrations C0 of less than 1 part in 5 tions, if we assume the defects all have equal impact on the 10 are needed to achieve Allan variances competitive with resonator frequency ⍀, the mean resonator frequency ͗⍀͘ those of single-crystal resonators. will be that calculated from Eq. ͑3͒ using the average modu- We note that a typical solid will include a range of re- ␶ lus ͗E͘. The frequency will have a mean variance given by orientation times d , which when superposed generically ␴2 ϭ ⍀ Ϫ⍀ 2 ⍀ produce 1/f noise through the Dutta–Dimon–Horn model.28 ⍀ (C0/8)( ϩ Ϫ) , where Ϯ are calculated from Eq. ͑ ͒ ϩ In that case, we can write 3 using the moduli Es C0EϮ . These fluctuations, occurring with a single reorientation ␻ time ␶ , will generate fractional frequency noise with the ͑␻͒Ϸ c D͑¯ ͒ ͑ ͒ d Sy A ␻ kBT E , 69 spectral density ␴2 ␶ where D(¯E) is the density of defect states at energy ¯E 2 ⍀ d S ͑␻͒ϭ . ͑67͒ ϭϪk T log (␻/␯ ) and A is a scale factor. In this case the y ␲ ͗⍀͘2 ϩ␻2␶2 B e 0 1 d Allan variance works out to be ͓see Eq. ͑46͔͒

␴ ͑␶ ͒Ϸͱ2 log 2A␻ ͱk TD͑¯E͒. ͑70͒ mine whether the suggested applications are indeed viable. A A A e c B better understanding of the role of defects and molecular Here, we see the Allan variance independent of averaging adsorption and desorption is also needed to evaluate the ef- ␶ time A , but with a magnitude comparable to that of Eq. fect these have on frequency stability. ͑68͒. Such processes, which play an important role in single- ACKNOWLEDGMENTS crystal and polycrystalline metals, and for which much infor- mation in metals exists, need to be investigated in silicon The authors acknowledge the financial support provided nitride and poly-Si to determine whether these play an im- by the National Science Foundation XYZ-On-A-Chip Pro- portant role in limiting resonator performance. gram, Contract No. ECS-9980734, by the Army Research Office, and by the Research Corporation through a Research VI. CONCLUSIONS Innovation Award. We have described a formalism for treating the resonance behavior, loss processes, and resulting frequency and 1 A. N. Cleland and M. L. Roukes, Appl. Phys. Lett. 69, 2653 ͑1996͒. phase noise in nanoscale resonators. We then applied this 2 A. N. Cleland and M. L. Roukes, Nature ͑London͒ 320, 160 ͑1998͒. 3 formalism to evaluate the role of a number of fundamental Y. T. Yang, K. L. Ekinci, X. M. H. Huang, L. M. Schiavone, M. L. Roukes, C. A. Zorman, and M. Mehregany, Appl. Phys. Lett. 78,162 and material-dependent noise sources, and how these sources ͑2001͒. affect the frequency stability of a model 1 GHz nanome- 4 K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Keeny, B. Stipe, and D. chanical resonator. For practical applications, the relevant Rugar, J. Microelectromech. Syst. 9,117͑2000͒. 5 comparison has been made with an industry standard, the C. Nguyen and R. Howe, Proc. IEEE Intl. Freq. Control Symp. 48,127 ͑1994͒. oven-controlled high-precision quartz crystal. We find that 6 K. Wang, A. Wong, and C. Nguyen, J. Microelectromech. Syst. 9,347 the anticipated resonator noise is predominantly from ther- ͑2000͒. momechanical noise, temperature fluctuations, and 7 R. Lifshitz and M. Roukes, Phys. Rev. B 61, 5600 ͑2000͒. 8 adsorption–desorption noise. The noise levels from these F. Walls and J. Vig, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 576 ͑1995͒. sources are comparable in magnitude to that of the quartz 9 T. Gabrielson, IEEE Trans. Electron Devices 40, 903 ͑1993͒. crystal, provided some care is taken to minimize certain im- 10 L. Cutler and C. Searle, Proc. IEEE 54, 136 ͑1966͒. portant loss processes. In Table IV, we have tabulated the 11 Y. Yong and J. Vig, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 36, ͑ ͒ various expressions for the noise, from different treatments 452 1989 . 12 J. Vig and Y. Kim, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, and from different noise sources. 1558 ͑1999͒. The results we have calculated here are for a doubly 13 A. Nowick and B. Berry, Anelastic Relaxation in Crystalline Solids ͑Aca- clamped, flexural resonator. Any resonator whose basic equa- demic, New York, 1972͒. 14 ¨ tions of motion can be reduced to those of a linear simple T. Albrecht, P. Grutter, D. Horne, and D. Rugar, J. Appl. Phys. 69,668 ͑1991͒. harmonic oscillator, driven by a force term, will have results 15 S. Timoshenko, D. Young, and J. W. Weaver, Vibration Problems in En- of the form shown here. The results for a cantilevered beam, gineering ͑Wiley, New York, 1974͒. 16 a torsional resonator, and a longitudinal wave resonator will X. Huang, K. Ekinci, Y. Yang, C. Zorman, and M. L. Roukes, ͑unpub- lished͒. therefore all be identical to these, except that the resonance 17 A. N. Cleland and M. L. Roukes, Sens. Actuators A 72,256͑1999͒. frequencies and the mode shapes are different, so that the 18 W. Egan, IEEE Trans. Instrum. Meas. 37, 240 ͑1988͒. numerical prefactors will be somewhat different. 19 W. Robins, Phase Noise in Signal Sources ͑Peter Peregrinus Ltd., London, For practical applications, nanoscale resonators can be 1982͒. 20 D. Allan, Proc. IEEE 54, 221 ͑1966͒. fabricated on chip with electronics needed to provide preci- 21 W. Egan, Frequency Synthesis by Phase Lock ͑Wiley, New York, 1981͒. sion frequency control. This would obviate the need for an 22 Hewlett-Packard 10811D/E application note. externally packaged and controlled quartz crystal, and enable 23 L. Landau, E. Lifshitz, and L. Pitaevskii, Statistical Physics, 4th ed. ͑ ͒ integrated fabrication. Pergamon, Oxford, 1980 . 24 W. Mason and T. Bateman, Phys. Rev. A 134, 1387 ͑1964͒. Clearly, there are gaps in the available data for evaluat- 25 G. S. et al., J. Phys. Chem. Solids 48, 641 ͑1987͒. ing whether the noise performance calculated here can be 26 A. Zangwill, Physics at Surfaces ͑Cambridge University Press, New York, achieved in fact. 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