# A general procedure for thermomechanical calibration of nano/micro-mechanical resonators

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<ul><li><p>Annals of Physics 339 (2013) 181207</p><p>Contents lists available at ScienceDirect</p><p>Annals of Physics</p><p>journal homepage: www.elsevier.com/locate/aop</p><p>A general procedure for thermomechanicalcalibration of nano/micro-mechanicalresonatorsB.D. Hauer, C. Doolin, K.S.D. Beach , J.P. Davis Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2G7</p><p>h i g h l i g h t s</p><p> Model micro- and nanomechanical resonators displaced by their own thermal motion. Review the theoretical framework for describing thermomechanical systems. Present a recipe for measurement calibration on devices of arbitrary shape. Point out and correct inconsistencies in the existing literature. Provide an authoritative guide and reference for practitioners in this area.</p><p>a r t i c l e i n f o</p><p>Article history:Received 17 May 2013Accepted 4 August 2013Available online 22 August 2013</p><p>Keywords:Thermomechanical calibrationMEMSNEMSClassical resonatorDamped Brownian motion</p><p>a b s t r a c t</p><p>We describe a general procedure to calibrate the detection of anano/micro-mechanical resonators displacement as it undergoesthermal Brownian motion. A brief introduction to the equations ofmotion for such a resonator is presented, followed by a detailedderivation of the corresponding power spectral density (PSD)function, which is identical in all situations aside from a system-dependent effective mass value. The effective masses for a numberof different resonator geometries are determined using both finiteelement method (FEM) modeling and analytical calculations.</p><p> 2013 Elsevier Inc. All rights reserved.</p><p>Contents</p><p>1. Introduction.................................................................................................................................................... 1822. Motion of a resonator .................................................................................................................................... 1833. Power spectral density .................................................................................................................................. 1834. Analytical theories of simple resonators ...................................................................................................... 186</p><p>4.1. Analytical theory of beams................................................................................................................ 186</p><p> Corresponding author. Tel.: +1 7804927176; fax: +1 7804920714. Corresponding author. Tel.: +1 7802481410; fax: +1 7804920714.</p><p>E-mail addresses: bhauer@ualberta.ca (B.D. Hauer), kbeach@ualberta.ca (K.S.D. Beach), jdavis@ualberta.ca (J.P. Davis).</p><p>0003-4916/$ see front matter 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.aop.2013.08.003</p><p>http://dx.doi.org/10.1016/j.aop.2013.08.003http://www.elsevier.com/locate/aophttp://www.elsevier.com/locate/aophttp://crossmark.crossref.org/dialog/?doi=10.1016/j.aop.2013.08.003&domain=pdfmailto:bhauer@ualberta.camailto:kbeach@ualberta.camailto:jdavis@ualberta.cahttp://dx.doi.org/10.1016/j.aop.2013.08.003</p></li><li><p>182 B.D. Hauer et al. / Annals of Physics 339 (2013) 181207</p><p>4.1.1. Cantilevers........................................................................................................................... 1874.1.2. Doubly clamped beams ...................................................................................................... 188</p><p>4.2. Analytical theory of strings ............................................................................................................... 1904.3. Analytical theory of membranes....................................................................................................... 190</p><p>4.3.1. Rectangular membranes .................................................................................................... 1904.3.2. Circular membranes ........................................................................................................... 192</p><p>5. Effective mass................................................................................................................................................. 1945.1. Effective mass of beams .................................................................................................................... 196</p><p>5.1.1. Cantilevers........................................................................................................................... 1965.1.2. Doubly clamped beams ...................................................................................................... 198</p><p>5.2. Effective mass of strings .................................................................................................................... 1995.3. Effective mass of membranes ........................................................................................................... 201</p><p>5.3.1. Rectangular membranes .................................................................................................... 2015.3.2. Circular membranes ........................................................................................................... 201</p><p>6. Torsional resonators ...................................................................................................................................... 2027. Conclusion ...................................................................................................................................................... 206</p><p>Acknowledgments ......................................................................................................................................... 206References....................................................................................................................................................... 206</p><p>1. Introduction</p><p>Nano/micro-mechanical resonators are fascinating devices, having microscopic dimensions andhigh quality factors (Q s) that allow them to be used in precise experiments. They are thereforeideal formanymeasurement applications, including biosensing [14], chargemeasurements [5],masssensing [69], accelerometry [10], temperature sensing [11,12], torque or force transduction [1315],single spin detection [16], and observation of qubits [17]. In addition, nano/micro-mechanicalresonators provide an excellent platform for investigating scientific phenomena, such as vortices[18,19], quantum zero-pointmotion [2022],molecular structure [23], gravity [24], superfluidity [25],and the Casimir force [26].</p><p>As nano/micro-mechanical resonator systems become capable of performing ever more preciseexperiments, the sensitivity of the detection methods used to observe the devices must increasecommensurately. The current state of the art techniques used for this detection include capacitivemeasurements [27], free-space optics [2830], piezoresistivity [31], and optomechanics [13,3235].These methods offer extremely sensitive transduction, achieving sub-attometer displacement sensi-tivity [36], yoctogrammass resolution [9], and force measurements on the order of attonewtons [15].</p><p>In order to quantify such sensitive measurements, it is crucial that the calibration of thesedevices be performed accurately. As the dimensions of nano/micro-mechanical resonators continue todecrease, this can prove to be a difficult task. However, there exists a powerful method by which themotion of these resonators can be calibrated via observation of their random thermal motion. Thisis known as thermomechanical calibration and is performed by invoking the equipartition theorem,which establishes a relationship between the thermal energy of a device and its mean squaredamplitude of motion [37]. Many of the detection methods mentioned above are sensitive enough tomeasure the thermalmotion of devices on themicro- or nanoscale. In addition, it has been shown thatthis calibration scheme agreeswell with othermethods, such as physically pushing on the device [38],but has the advantage of being noninvasive.</p><p>Thermomechanical calibration has been used to measure the motion of many systems,including cantilevers [29,30,33,3843], doubly clamped beams [10,4446], strings [28,32], torsionalresonators [5,13,4749], and membranes [50,51]. To the best of our knowledge, though, no singlestandalone document exists that describes themethod of thermomechanical calibration in its entirety.We have noticed that a number of mutually inconsistent approaches to performing such a calibrationcan be found in the literature. This manuscript intends to elucidate the process of thermomechanicalcalibration by providing a full, self-consistent derivation of the procedure and ensuring that allpertinent equations are explained and defined. In this way, a general process is developed bywhich nano/micro-mechanical resonators can be calibratedthe only difference between varying</p></li><li><p>B.D. Hauer et al. / Annals of Physics 339 (2013) 181207 183</p><p>geometries being reduced to a single parameter, the devices effectivemass. In this paper, the effectivemass is calculated both analytically and numerically, for a number of common device geometries.These calculations are then benchmarked against the well-known methods of Sader et al. [52] andCleveland et al. [53], which have been developed to calibrate atomic force microscopy cantilevers. Inaddition, we extend the methods of Sader et al. and Cleveland et al. to encompass other resonatorgeometries, such as doubly clamped beams and torsional resonators.</p><p>This document, which includes both a general overview of the process and specific examples,provides a clear, straightforward recipe for carrying out thermomechanical calibration for anynano/micro-mechanical resonator system, regardless of its complexity.</p><p>2. Motion of a resonator</p><p>We begin by describing the motion of an extended resonator structure in the linear regime. Ingeneral, the complete, three-dimensional motion will be given by a displacement function R(x, t)that can be decomposed into an infinite number of independent modes. To each mode we ascribe alabel, n, allowing for the displacement function to be expressed as [44,54]</p><p>R(x, t) =n</p><p>an(t)rn(x), (1)</p><p>where rn(x) is the three-dimensional mode shape of the nth mode, and an(t) is a function thatdescribes the time dependence of the resonators motion. Here we choose to normalize rn(x) suchthat themaximum value of |rn(x)| is unity. This choice of normalization ensures that an(t) has units ofdistance (as the normalization leaves rn(x) unitless) and is in 11 correspondencewith the resonatorstrue physical displacement at the point of measurement. It should be noted that with this conventionthe modes are orthogonal to one another but not orthonormal.</p><p>Many devices are described by models simple enough that rn(x) can be solved analytically. Whenthe devices motion is more complicated, however, we turn to FEM modeling to determine the modeshape of the structure. In this paper, these twomethods are compared in a number of situationswherethe mode shapes of the resonator have an analytical form.</p><p>The function an(t) is determined by mapping each of the resonators independent modes to adamped harmonic oscillator of the form [42]</p><p>an +n</p><p>Qnan + 2nan =</p><p>F(t)meff,n</p><p>. (2)</p><p>Here, n, Qn, andmeff,n are the angular frequency, quality factor, and effective mass for the nth modeof the resonator. F(t) is a time-dependent driving force. An effective spring constant for the nth modecan also be defined and is given by keff,n = meff,n2n [42]. A general expression formeff,n, along with itsdetermination for a number of simple models, will be outlined in Section 5.</p><p>Often a nano/micro-mechanical resonators motion can be reduced to a one-dimensional displace-ment function</p><p>z(x, t) =n</p><p>zn(x, t) =n</p><p>an(t)un(x), (3)</p><p>where un(x) describes the one-dimensional mode shape for the nth mode of the resonator as a func-tion of position x, and an(t) is unchanged from Eq. (1). As in the three-dimensional case, we chooseto normalize un(x) such that the maximum value of |un(x)| is unity. This equation will be used whenthe displacement of a nano/micro-mechanical resonator is simple enough to be modeled by a one-dimensional function. Eqs. (1) and (3) fully describe the motion of these devices in the linear regimeand will be used when explicating the thermal calibration process.</p><p>3. Power spectral density</p><p>The various methods used to detect the motion of a nano/micro-mechanical resonator share onemajor similarity: the motion of the resonator in question is encapsulated by some time-varyingsignal, typically a voltage. Generally, this measured signal is transformed into the frequency domain,</p></li><li><p>184 B.D. Hauer et al. / Annals of Physics 339 (2013) 181207</p><p>providing a spectral representation of the resonatorsmotion that is peaked (up to small corrections in1/Q ) at its resonance frequency. By squaring this spectrum and dividing by the resolution bandwidthof the instrument, themeasured one-sidedpower spectral density Szz()of the signal is produced [50].</p><p>When thermally calibrating a mechanical resonator, we must be able to relate the devices timeaveraged motion to its thermal energy. To do this, the equipartition theorem is used to expressthe devices mean-square amplitude of motion an2(t) in terms of its thermal noise PSD. In thissection, we derive a theoretical expression for the PSD corresponding to a damped harmonic oscillatorundergoing random thermal motion. This function can be used to properly calibrate thermal PSD datafor any nano/micro-mechanical resonator.</p><p>We introduce the autocorrelation function Rzz(t), which describes how the signal an(t ) is relatedto itself at a later time t + t [55]:</p><p>Rzz(t) = limT0</p><p>1T0</p><p> T00</p><p>dt an(t )an(t + t). (4)</p><p>The two-sided power spectral density function Pzz(), can then be obtained from the Fouriertransform of the autocorrelation function [55]</p><p>Pzz() =</p><p>dt Rzz(t)eit . (5)</p><p>Likewise, the autocorrelation function can be produced by performing the inverse Fourier transform</p><p>Rzz(t) =12</p><p>d Pzz()eit . (6)</p><p>Pzz() is defined such that it spans all frequencies, both positive and negative, and will give the totalpower of the signal if integrated over this entire frequency range. For real-valued signals, as is the casein physical situations, Pzz() is an even function [56]. Therefore, we introduce Szz(), which is definedon the frequency interval from 0 to such that it produces the power in the signal when integratedover all positive frequencies. From these two definitions, we find that Szz() and Pzz() are related bySzz() = 2Pzz() [56]. This relation allows the total power in the signal to be conserved when eachspectral density function is integrated over the frequency interval on which it is defined. In the rest ofthe derivation, Szz() will be used, as it provides a theoretical spectral density that can be fit directlyto physical data.</p><p>We define the quantity known as the mean-square amplitude of our signal as [55]</p><p>an2(t) =1T0</p><p> T00</p><p>dt [an(t)]2. (7)</p><p>Upon inspection, we see that this quantity is equal to Eq. (4) at zero time shift, provided that we havesampled th...</p></li></ul>

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