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Nano-Mechanical Infrared Detectors

Dragoslav GrbovicNaval Postgraduate SchoolPhysics Department

833 Dyer Road, Bldg 232Monterey, CA 93943Tel: 831-656-3884Fax: (831) 656 2834Email: dgrbovic@nps.edu

N.V. LavrikOak Ridge National Laboratory, Oak Ridge TN 378311 Bethel Valley Rd, MS 6054Oak Ridge, TN 37831-6054Tel: 865-241-2636

Fax: 865-574-9407Email: lavriknv@ornl.gov

S. RajicOak Ridge National Laboratory, Oak Ridge TN 378311 Bethel Valley Rd, MS 6054Oak Ridge, TN 37831-6054Tel: 865-574-9416Fax: 865-574-9407Email: rajics@ornl.gov

S.R. HunterOak Ridge National Laboratory, Oak Ridge TN 378311 Bethel Valley Rd, MS 6054Oak Ridge, TN 37831-6054Tel: 865-241-3995Fax: 865-574-9407Email: huntersr@ornl.gov

P.G. DatskosOak Ridge National Laboratory, Oak Ridge TN 378311 Bethel Valley Rd, MS 6054

Oak Ridge, TN 37831-6054Tel: 865-574-6205Fax: 865-574-9407Email: datskospg@ornl.gov

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Table of Contents

1. Introduction.................................................................................................................................31.1 Infrared Radiation.................................................................................................................31.2 Infrared Detectors................................................................................................................4

1.3 Applications..........................................................................................................................51.4 Basic Principles....................................................................................................................51.5 Transduction Mechanisms for Nanomechanical Detectors.................................................6

2. Figures of Merit..........................................................................................................................82.1 Responsivity..........................................................................................................................92.2 Noise Equivalent Power......................................................................................................102.3 Normalized Detectivity.......................................................................................................112.4 Noise Equivalent Temperature Difference.........................................................................122.5 Response Time....................................................................................................................13

3. Theoretical Background for Nanomechanical Infrared Detectors...........................................143.1 Bimetallic Effect.................................................................................................................14

3.2. Thermal Conductance.........................................................................................................163.3. Microcantilever Responsivity.............................................................................................174. Noise Sources and Fundamental Limits .................................................................................20

4.1. Temperature Fluctuation Noise...........................................................................................224.2. Background Fluctuation Noise...........................................................................................254.3. Thermo-Mechanical Noise..................................................................................................264.4 Readout Noise.....................................................................................................................30

4.4.1 Piezoresistive Readout................................................................................................304.4.2. Capacitive Readout.....................................................................................................324.4.3 Optical Readout...........................................................................................................33

4.4.3.1 PSD Readout.........................................................................................................34

4.4.3.2 CCD Readout........................................................................................................364.5 Total Noise...........................................................................................................................374.6 Summary..............................................................................................................................37

5. State of the Art.........................................................................................................................396. Conclusion................................................................................................................................427. References.................................................................................................................................438. Table Captions and Tables........................................................................................................489. Figure Captions and Figures.....................................................................................................52

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1. Introduction

1.1 Infrared Radiation

Infrared radiation (IR) is electromagnetic radiation with wavelengths between 0.7 m and 100m. It extends from visible light to THz waves. Because fundamentally different phenomenacan be observed within the IR region, four sub-bands are usually distinguished: near-IR (NIR),mid-wave-IR (MWIR), long-wave-IR (LWIR) and very long-wave-IR (VLWIR). Althoughsomewhat different definitions exist in literature, wavelengths from 0.7 m to 2.5 m belong toNIR, from 2.5 m to 8 m belong to MWIR, from 8 m to 14 m belong to LWIR andwavelengths above 14 m belong to VLWIR. The IR photon energies range from 1.77 eV for0.7 m photons to 0.0124 eV for 100 m photons.

The significance and practical applications of IR detectors are related to two distinct phenomena:emission of electromagnetic waves by all objects at T> 0 K and interaction of electromagneticwaves with vibrational modes of molecular bonds. Thermal imaging and molecularspectroscopy are, respectively, the two major fields that critically depend on the ability to detectIR radiation.

According to the blackbody radiation principle, every object with non-zero temperature emitsradiation composed of infrared photons of various wavelengths. The photon distribution bywavelength depends on object temperature. The distribution of photons emitted by a blackbodywith respect to their wavelength is governed by the Planck Radiation relationship [1]

u l( ) = 8pchl 5

1ech/l kBT - 1

(1)

where u()dis the volume density within the spectral region between and + dof photonsof a given wavelength , kB is Boltzmanns constant and Tis the temperature of object emittingthe radiation. As can be seen in Figure (1), each distribution curve has a peak. This means thatfor each temperature, photons of a certain wavelength dominate the distribution. Thewavelength of the dominant photons is given by the Viens law:

=2.9 mm

T(K)

(2)

According to Equation (2), the majority of photons emitted by objects near room temperaturehave wavelengths between 8 and 14 m (300K). The emission of blackbody radiation allowsdetection and imaging of objects by detecting the radiation they emit without requiring anyexternal illumination.

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1.2 Infrared Detectors

Detection of infrared (IR) radiation is very important for a variety of activities in bothcommercial and defense areas. Infrared detectors are transducers, which absorb IR radiation andproduce a measurable output proportional to the absorbed energy. Low atmospheric absorption

of photons with wavelengths in bands from 3 to 5 m and 8 to 14 m makes those bandsparticularly important [2, 3]. However, the high cost of IR detectors has limited their present useto primarily military applications. Recent advances in low cost uncooled thermal detectors makea number of commercial applications feasible [4]. Infrared radiation detectors [5, 6] can beclassified broadly as either quantum (electro-optic) [7] or thermal detectors, such as pyroelectric[8], thermoelectric and thermoresistive transducers (bolometers) [9-12], and microcantileverthermal detectors [3, 13-21].

Quantum IR detectors are based on semiconductor materials with narrow bandgaps, g< hc/or metalsemiconductor structures (Schottky barriers) with appropriately small energy barriers, < hc/. One of the drawbacks of these detectors is that they have a cutoff wavelengthabove which they cannot detect photons, as those photons are unable to overcome the energybarrier. In addition the dark current, which contributes to the noise in these detectors, dependsexponentially on temperature due to thermally generated charge carriers. This necessitatescooling of quantum IR detectors. The thermal resolution of cooled quantum IR detectors,however, can be very high, typically in the few mK range. On the other hand, thermal IRdetectors are based on measuring the amount of heat produced in the detector upon theabsorption of IR radiation and can operate at, or even above, room temperatures since thermalnoise in thermal detectors varies as T1/2 [10]. Since they can operate at or even slightly aboveroom temperature, they are often referred to as uncooled IR detectors.

The first type of uncooled IR detector was the bolometer. It converts the energy of incoming

photons into heat, which in turn induces a change in the electrical resistance of the detector. Amore recent type of uncooled IR detector is a nanomechanical transducer, which converts theenergy of incoming photons into mechanical deformations by utilizing thermally sensitivebimaterial structures. This type of detector has emerged as a viable competing IR detector due torecent advances in micro-electromechanical systems (MEMS) design and fabrication techniques.Often, in the technical literature, these detectors are referred to as MEMS uncooled detectors.

Single element IR detectors are used to measure IR radiation intensity and are commonlyreferred to as spot detectors, and used primarily in infrared spectroscopy applications. Althoughspot detectors can be combined with scanning optical components in order to accomplish IRimaging [20], focal plane arrays (FPA) of IR detectors are increasingly the predominant type of

IR detectors for imaging applications. One of the main advantages of nanomechanical IRdetectors is their excellent compatibility with large FPA formats.

Initial imaging applications utilized quantum IR detectors. By the end of the last century, largefocal plane arrays of resistive bolometers and ferroelectric devices with 320 x 240 pixels wereavailable [4]. The reported thermal resolution values for these FPAs are as low as 23 mK [22,23]. In the past several years, nanomechanical IR detector FPAs with up to 256 x 256 pixelshave been reported [14, 24, 25] with measured thermal resolutions of 40 mK [26] to a few

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of thermal conductance G [36]. The absorbing element converts the incident IR radiation intoheat. Heat induced by the incident powerPcauses an increase in detector temperature at a ratedTb/dt = P/C. The temperature approaches the limiting value Ts +P/G with time constant

th=

C

G(3)

When no additional radiation falls on the detector, the detectors temperature relaxes back to Tswith the same time constant.

The principal parts of a typical infrared detector include an absorber, a supporting substrate/heatsink, and a thermal connector [36] (Figure 2). These components are required to meet certaincriteria for the detector to work effectively. The absorber has to be large enough to adequatelyintercept the incoming IR flux. The absorber must also have a low heat capacity in order tomaximize the increase in temperature for a given amount of absorbed energy, as well as highabsorption efficiency in the wavelength band of interest. The substrate has to have a high heat

capacity and a large thermal conductivity to remain isothermal during operation. The thermalconnector, linking the absorber and the heat sink, has to have a low heat capacity and a very lowthermal conductance for the absorbed heat to be retained in the absorber a sufficiently long timeto be detected. Equation (3) however, shows that there is a tradeoff between the requirement offast response times (low ) and low thermal conductance.

1.5 Transduction Mechanisms for Nanomechanical Detectors

In this section, we will focus on nanomechanical infrared detectors. Nanomechanical (MEMS)IR detectors are a subgroup of thermal IR, which utilize the effect of structural changes withinthe detector which occurs when its temperature is changed. This structural change, proportionalto the magnitude of temperature change, can be manifested in the form of a shift in resonantfrequency or a structural deformation, and can be detected using various transductionmechanisms usually referred to as readouts. The readouts used to quantify thedetector deformation demonstrated to date include the following: quartzmicro-resonator [37], piezoresistive [17], capacitive [15], electron-tunneling[38], and optical [3, 14, 39].

Based on the type of readout utilized to quantify the deformation, nanomechanical IR detectorsare classified in different categories: a) quartz micro-resonator, where the temperature changeinduces the shift in the easily detected resonant frequency [37]; b) piezoresistivenanomechanical detector, where the deformation is quantified by the change in the resistance ofthe detector [37]; c) capacitive nanomechanical detectors, where the detector and the substrateform a capacitor whose capacitance changes with deformation of the detector due to change incapacitor-plate separation [34, 40, 41]; d) pneumatic nanomechanical detectors [7] have achamber, also known as the Golay cell, with enclosed air whose pressure changes with changesin temperature, deforming the membrane. Membrane motion is usually quantified by thechanges in capacitance [7] or by electron tunneling [38, 42]; e) optically-probed nanomechanical

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detectors, which gauge the deformation using the principle of cantilever readout similar to thereadout utilized in Atomic Force Microscopy (AFM). In the AFM, the laser light is reflected offthe tip of the cantilever and the cantilever deformation is detected by observing the motion of thereflected beam of light by a position sensitive detector.

Since bolometers were developed earlier, methods for measuring the resistance change were themost convenient to use in early nanomechanical infrared detectors [17]. Another method ofchoice was to measure changes in capacitance [41]. Electrical connections for both of thosereadout techniques require electrically conductive links between the detector and the substrate.This potentially means higher thermal conductivity to the substrate, but the latest capacitivenanomechanical detectors use materials with good thermal isolation [41]. Optical readout offersa good alternate solution to this problem since it does not require electrical connections betweenthe detector and the substrate. In this approach a single detector, or an array, is illuminated withvisible light, which is reflected into either a position sensitive detector (such as a quad cell) or acharge coupled device CCD (such as the one used in conventional digital cameras). This type ofreadout allows good thermal isolation between detector and substrate.

In the case of quartz micro-resonators, the temperature change induces a shift in resonantfrequency, and the detector is connected to the circuitry monitoring the frequency. Thefrequency shift is proportional to the change in temperature.

In the case of pneumatic detectors, electron tunneling was utilized to quantify detectordeformations. In the case of piezoresistive readout, the temperature change induces deformationof a detector. Since one of the bimetallic layers is made of piezoresistive material, the resistivityof the material, and therefore the resistance of the whole detector, will change. The resistancechange is proportional to the amount of deformation, which is proportional to the change intemperature. The detector is electrically connected to the bias voltage and a change in resistancewill induce a measurable change in current. In case of a capacitive readout, the temperaturechange induces the deformation of the detector plate. Part of the detector, usually the absorbingelement on the detector plate, acts as a movable electrode of a variable plate capacitor [41]. Theother capacitor plate is usually embedded into the substrate. Designs have been fabricatedfeaturing two parallel bimorph microcantilever arms connected to the absorber plate, and these inturn are connected to the substrate through additional thermal isolation arms [34, 40]. When thedetector bimorph arms deform, the distance between the capacitor plates changes, changing thepixel capacitance. The detector is connected to circuitry in the substrate that measures thecapacitance. The measured change in capacitance is proportional to the increase in detectortemperature.

Optical readout is a commonly used method for two different sensing techniques. One techniqueis based on the approach developed for the atomic force microscopy (AFM) and is used forreading out single microcantilever detectors. The other technique simultaneously probesmultiple microcantilever detectors arranged in an array. Figure 3 illustrates the configuration ofa single-detector optical readout. A laser beam is focused on the tip of the cantilever andreflected into the position sensitive detector (PSD). If the cantilever deflects, it will steer thereflected beam and the spot on the PSD will move. Based on the PSD output voltage, the exactdisplacement can be extracted down to sub-angstrom accuracy [43].

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In a configuration for simultaneously probing multiple detectors, a light source illuminates theentire detector array. Figure 4 illustrates a simplified representation of the optical-detector-arrayreadout. The light is reflected off the front (reflective) side of M N array of detectors and isprojected onto the CCD via Lens 2. IR radiation from the target being imaged, is projected onto

the backside of the array via the IR lens, which is made out of IR-transmitting material such asGe. Depending on the shape and temperature distribution of the target, the magnitude of thedeformation of the individual detectors will differ corresponding to different temperatures on thetarget. The aperture will block a portion of light reflected off the deformed detectors and theamount of light reaching the corresponding CCD pixels will decrease. These changes inintensity will translate into areas of different brightness intensities on the reconstructed thermalimage. The different brightness levels will corresponds to different temperatures on the imagedobject. Based on the brightness of image pixels recorded by the CCD, the relative level ofdeformation of each corresponding detector can be determined.

For both types of optical readout, the absorber of IR radiation, defined in Figure 2, also serves as

a reflector of the probing (visible) light. For the remainder of this discussion, the terms absorberand reflector are used interchangeably depending on the context of the description.

During the last several years, optically probed nanomechanical (MEMS) uncooled infrareddetectors and imagers have drawn substantially increased attention [3, 4, 8, 14, 15, 21, 24, 25,44, 45]. Indeed, focal plane arrays (FPAs) of optically probed nanomechanical detectors offer anattractive alternative to other, well-established, uncooled infrared detectors such asmicrobolometers. This is largely due to the significantly simpler microfabrication and, in turn,potential for high yield and low cost. It is worth noting that the need for on-chip electronicscombined with exotic materials increases microfabrication complexity, leading to highfabrication cost and a low yield for microbolometer FPAs. Several groups have already

demonstrated optically probed, MEMS-based infrared imaging devices [3, 4, 8, 14, 15, 21, 24,25, 44, 45] that address this challenge.

In this chapter, we discuss possible improvements to nanomechanical uncooled IR detectors thatcan be used to improve their performance to make them comparable in performance to othercontemporary uncooled IR imagers. This improved performance can be achieved byimplementing new ideas in detector geometry, choice of materials and modifications to thedetector substrate.

2. Figures of Merit

There exists a need to characterize and compare performance of different types of infrareddetectors such as quantum detectors, bolometers and other types of thermal IR detectors. In thelast several decades, a number of different figures of merit have beendefined [10, 11, 46-50]. Being a new and evolving field, the field of IRdetectors uses the parameters and figures of merit that are evolving as well.New parameters are emerging while some become outdated. In this text, wediscuss the figures of merit currently accepted and used by the IR

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community [46, 51].

Earlier work [13] indicated that although the need for figures of merit is driven by the desire tocompare different detectors, it is important to keep in mind that different assumptions aresometimes made in defining and measuring these parameters. When evaluating the performance

of various IR detectors, especially those utilizing uncooled thermal detectors, the parameters ofmajor importance are 1) responsivity, R; 2) noise equivalent power (NEP); 3) normalizeddetectivity, D*; 4) noise equivalent temperature difference (NETD); 5) minimum resolvabletemperature difference (MRTD); and 6) response time [46, 48, 51]. The definitions of theseparameters and their fundamental limits in the case of uncooled thermal IR detectors arediscussed in detail. There are a number of additional parameters that can be used for a moredetailed and comprehensive characterization of IR detectors. These include linearity of response,cross-talk between detector elements in an FPA, dynamic range, and modulation transferfunction (MTF). The linearity of response, cross-talk, and dynamic range are basic parametersapplicable to a whole variety of analog devices and transducers, and their definitions are readilyavailable from a number of sources [50, 52]. MTF is traditionally used in testing the

performance of lenses, imaging systems and their components, and describes how the outputcontrast changes as a series of incrementally smaller features are imaged [52, 53].

2.1 Responsivity

Responsivity R, which is applicable to all infrared detectors, describes thegain of the detector and is defined as the output signal (typically voltage orcurrent) of the detector produced in response to a given incident radiantpower falling on the detector [46, 48]. The responsivity is expressed as:

RV =Vs

P0or RI =

Is

P0 (4)

where Vs is the output voltage (V), Is is the output current (A), and P0 is theradiant input power (W). If the definition of responsivity is expanded toinclude the frequency dependence and the wavelength (spectral)

dependence [46], it is then referred to as the spectral responsivity, R(, f).Datskos and Lavrik [13], emphasize that in the case of quantum and thermalIR detectors, very distinct factors define the characteristic features ofspectral responsivities. Quantum IR detectors exhibit a cut-off in the spectral

responsivity above a certain characteristic wavelength that is related to thephoton energy sufficient to generate additional charge carriers (free

electrons or electron-hole pairs). Hence, R(, f) has a long-wavelength cut-off defined by the bandgap energy of the semiconductor or the energybarrier at the metal-semiconductor interface used in the detector. In thecase of thermal IR detectors, however, the far-IR range is readily accessiblesimply by using appropriate detector absorbing areas and materials withhigh-absorptivity (either direct or resonant absorption) in this region [53].

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A derivative of responsivity, known as blackbody responsivityR(T, f), is defined as an additionalparameter and includes the dependence of the detector output signal on the temperature, T, of theblackbody-type source.

The responsivity is a useful design parameter that describes the magnitude of output responsecaused by a given level of IR radiation coming from an object at a given temperature andemissivity. Although a good indicator of an IR detector performance, the responsivity does notdescribe the level of any intrinsic noise in the detector and, therefore, provides little or noinformation about detectors threshold sensitivity. This means that an IR detector characterizedwith high responsivity is not necessarily able to detect low-level IR radiation or to distinguishbetween different IR sources with nearly the same temperature. Previous studies concluded thatknowing the detector responsivity is important during the IR detection system design, whilecomparative evaluation of different detectors should rely on other figures of merit [53].

In the case of nanomechanical IR detectors, which mechanically deform in response to the

incoming IR power, it is natural to define the responsivity in terms of the mechanical response ofthe detector, i.e., displacement,zs, per unit of absorbed power,P0, in units of meters per watts, as

Rz =zs

P0. (5a)

Similarly, a spectral responsivityRz(T, f ) and a blackbody responsivityRz(, f ) can be defined.Recently, the level of deformation of nanomechanical IR detectors has been described by thedeformation angle of the reflector [45, 54]. Using the angle to quantify the deformation isconvenient, especially for systems with an optical readout. This is because, in most cases, theoptical readout is sensitive to the change of the angle of the reflector. In addition, measurement

of the tip displacement can be deceptive since smaller relative deformations of a large structureyields larger tip deflections. Angle of deformation can be scaled to, and is independent of, thedetector size. The parameter describing the angle of deflection per unit of temperature increase(discussed below) requires a responsivity defined in terms of the angle of the detectordeflection per unit of absorbed power,P0, in units of radians per watt.

Rq =qs

P0. (5b)

2.2 Noise Equivalent Power

Noise Equivalent Power, NEP, is defined as the incident radiant power that produces a signalequal to detectors root mean square (rms) noise [46, 48]. This parameter represents a convenientway to characterize the sensitivity of an IR detector. By this definition, NEP includesinformation about both gain and noise parameters of the detector and can be related to thedetector responsivity,RV,RI,Rz orR, and the rms detector noise [46]

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NEP =V

N

RVor NEP =

IN

RIor NEP =

zN

Rzor NEP =

qN

Rq(6)

where VN (IN,zN orN) is the rms noise voltage (current, displacement or angle) measured withinthe operating bandwidth. As pointed out earlier [13], sinceNEPdepends onR, it also dependson the photon wavelength as well as on the modulation frequency of the IR power and, therefore,can be referred to asNEP(, f). These authors further state thatNEPcan also be specified as afunction of detector temperature, i.e., NEP(TD, f) [46]. The frequency dependence ofNEP isdetermined by the detector thermal response time, , and by the spectral density (i.e., frequencydependence) of the detector noise. It is important to note that even if the noise amplitude isfrequency independent (white noise), the rms noise spectral density exhibits a square rootdependence on the frequency. NEP(, f) and NEP(TD, f) refer to a 1-Hz bandwidth and haveunits of W Hz-1/2;NEPwithout specifying the frequency may have an ambiguous interpretation.The units ofNEPimply either a full operational bandwidth or a 1-Hz bandwidth [53].

2.3 Normalized Detectivity

Even though the NEP is sufficient for adequate evaluation and comparison of performance ofindividual (spot) IR detectors by predicting the minimum detectable power, its magnitude isinversely proportional to the detector performance, i.e. a higher value implies lower quality.Therefore the need rose for a parameter that would be directly proportional to detectorsperformance. Starting with a parameter known as detectivity,D, which is defined as the inversevalue ofNEP and taking into account the detector absorbing (active) area, Ad, and the signalbandwidth, B, one can define a specific (or normalized) detectivity,D*, as [46]

D*= AdBNEP

(7)

According to Equations (4) - (6), the normalized (or specific) detectivity D* isthe detector output signal-to-noise ratio at 1 W of input IR radiationnormalized to a detector with a unit active area and a unit bandwidth. Theunits ofD* are in Jones; 1 Jones = 1 cm Hz1/2 W-1. In earlier work [13], it wasnoted that the definition of specific detectivity, D*, was originally proposed forquantum detectors, in which the noise power is always proportional to thedetector area, and noise signal (Vn or In) is proportional to the square root ofthe area. However, the noise in thermal IR detectors does not always obey

this scaling trend. In fact, neither temperature fluctuations nor thermo-mechanical noise (see the next section) scales up with the detector area.Therefore, D* should be very cautiously interpreted when applied to thermalIR detector. In fact, D* tends to overestimate the performance of largerabsorbing area thermal detectors and underestimates the performance ofsmaller ones. Normalized detectivity, D*, (even in the case of quantumdetectors), generally ignores the significance of smaller detector size forhigh-resolution FPAs [53].

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2.4 Noise Equivalent Temperature Difference

Noise Equivalent Temperature Difference (NETD) is a parameter that

describes the low-signal performance of thermal imaging systems and ismore applicable to FPAs used in IR cameras, rather than to individualdetectors. Over the years, this figure-of-merit has emerged as one of themost commonly used parameters for describing the performance of IRimagers. NETD is defined as the temperature of a target above (or below)the background temperature that produces an output signal equal to the rmsdetector noise [46, 48, 55]. NETD can be represented as a histogram ofvalues for each individual detector element, or can be averaged for alldetector elements in an array. Alternatively, NETD can be defined as thedifference in temperature between two blackbodies, which corresponds to asignal-to-noise ratio of unity [48]. The image produced by an IR imager is a

map of detected temperature variations across a scene or an object and theresulting images are also affected by the emissivity of the objects in thescene. Small values ofNETD reflect the ability of an IR imager to distinguishslight temperature or emissivity differences of objects in the scene. Therelationships used for predicting NETD have been described elsewhere [10,11, 46, 55]. NETD can also be determined experimentally for a givendetector area, detector absorptivity, optics used, and output signalbandwidth [55] by

NETD =VN

VS

(Tt - TB ) or (8a)

NETD =IN

IS(Tt - TB ) or (8b)

NETD =zN

zS(Tt - TB ) (9)

where VN (IN orzN) is the voltage (current or deflection) rms noise, Vs (Is orzs)is the voltage (current or deflection) signal, Tt is the temperature of theblackbody target, and Tb is the background temperature. The NETD ofoptimized thermal IR detectors is limited by temperature fluctuation noise,while background fluctuation noise imposes an absolute theoretical limit onthe NETD of any IR detector [53]. The factors affecting the temperaturefluctuation noise, background fluctuation noise and thermo-mechanical noiseare discussed in more detail below.

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2.5 Response Time

IR detectors exhibit characteristic transient behavior when the input powerchanges abruptly, as in case of other sensors or transducers. A general

definition of the response time, , for any sensor system is the time required

for a transient output signal to reach 0.707 (

2 ) of its steady-state change.The expressions of the responsivity in the time and frequency domains aregiven by [52]

R t( ) = R t= ( ) 1- e-

t

t

(10)

R f( ) =R0

1+ 2pft( )2 (11)

where R0 is the static responsivity. In the case of photo-electronic (quantum)detectors whose transduction of the absorbed IR energy into the outputsignal is based on photo-electronic processes, the intrinsic response time canbe less than a nanosecond [56]. Even though the impedance of theelectronic readout often limits their response times, the response times ofcomplete imaging systems are still shorter than 1 ms, which is more thenenough to meet the requirements of most real-time imaging applications.On the other hand, the overall response time for nanomechanical detectorsis directly related to their much longer intrinsic response times and not bythe readout. Long intrinsic response times (mostly in the range of 1 to 100ms) are the consequence of the transduction mechanism, which requiresaccumulation of heat in the detector active area. The response time of a

thermal IR detector, th [Equation (3)], is calculated as the ratio of heatcapacity of the detector to the effective thermal conductance between theactive area of the detector and its support structure (i.e. a heat sink).

Equation (3) is a convenient tool for predicting the response times of alltypes of thermal IR detectors, including nanomechanical IR detectors. InEquation (3), the heat capacitance, C, is the total capacitance, whichcombines the heat capacitances of each layer in the active area of the

detector. The heat capacitance of each layer is calculated as a product of thespecific heat capacitance of the layer material and its mass. The thermalconductance, G, needs to include all of the heat loss mechanisms in thedetector (i.e. conductive, convective and radiative losses).

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3. Theoretical Background for Nanomechanical Infrared Detectors

Since most of the work done in the field of nanomechanical uncooled IR detectors, such as thosewith piezoresistive, capacitive and optical readouts, is performed with bimaterial cantileverdetectors, only these types of detectors will be discussed in detail. Figure 5 displays a cartoon

representation of a typical nanomechanical IR detector. A nanomechanical IR detector typicallyconsists of the absorber or an active part, which undergoes deformation when the detectortemperature changes, and two supporting beams (legs) that provide structural support for theabsorber as well as thermally connect the absorber to the substrate. Most of the structure isusually made out of the same material taking advantage of already well-establishedmicrofabrication methods developed for the Si-based microelectronics industry. Therefore thestructures analyzed in the literature are built mostly of microfabrication-friendly materials suchas silicon or silicon nitride. The absorber is also usually a bimaterial region, which consists ofthe silicon, based structural material and another material usually a metal. Some designs [13,24] separate the absorbing section of the detector into the deforming region and reflecting region,and usually employed in optically-probed systems. As discussed above, the legs are designed to

provide optimal thermal isolation between the absorber and the heat sink, which in most cases isthe substrate on which the detectors are fabricated.

3.1 Bimetallic Effect

When heated, the active part of the detector undergoes a deformation due to the bimetallic effect.This effect was first explained by Timoshenko [57], and it states that the structure will deformwhen heated if it consists of two layers with materials of different coefficients of thermalexpansion (CTE) (Figure 6). This deformation occurs because the two layers expand at differentrates for the same temperature change. Deformations of microstructures are of the order of tens

to hundreds of nanometers per Kelvin temperature change. Once readouts that can quantifymicrostructure deflections, reached sensitivities sufficient for measuring even sub-angstromdeflections [43], optically probed bimetallic structures became sensitive enough to be applied toIR detection.

The deflection, z, of the tip of a cantilever consisting of two material layers, due to temperatureincrease ofTis given by [57, 58]

z =3l

b

2

t1 + t2

1+t1

t2

2

3 1+t1

t2

2

+ 1+t1

t2

E1

E2

t12

t22+

t2

t1

E2

E1

a 1 - a 2( )DT (12a)

where lb is the microcantilever bimaterial length, t1 and t2 are the thickness of the coating and themicrocantilever substrate respectively, 1 and 2 are the thermal expansion coefficients of thecoating and the microcantilever and E1 and E2 are the Youngs moduli of the coating and the

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microcantilever. Ifn is defined as n = t1/t2 and =E1/E2 andK= 4 + 6n + 4n2 + n3+1/ n thisexpression can be simplified to:

z = 3 a 1 - a 2( )t1 + t2

t22K

lb

2DT. (12b)

Similarly, the angle of deformation is expressed as [44]:

= 6 a 1 - a 2( )t1 + t2

t22K

lbDT. (13)

It can be seen from Equations (12a) - (13) that with optimized detector geometry and properlyselected materials, zcan be maximized. One would think that better deflection automaticallyimproves sensitivity and responsivity of the detector. However, it will be shown thatmaximizing z alone does not necessarily improve the overall performance of the detector.

Many parameters are interconnected and improving one of them may be to the detriment ofanother. For example, increasing cantilever length improves z, but increase also increases thethermomechanical noise.

Typical dimensions of the detector structure range from a few tens to few hundred m. Themain structure thickness varies from few hundred nm to few m. Taking into account thematerial properties of the SiNx (or SiO2) and Al (Table 1), the reported thermo-mechanicalsensitivities, z/T, range from ~50 nm/K [14] to ~500 nm/K [26].

Once z/Tis defined, we need to consider the temperature increase, T, of the detector due to

photon absorption. As a first approximation, it can be assumed that the main heat lossmechanism is the conductance of the thermal isolation region (legs). In addition to heatdissipation through the legs, the heat can be dissipated through convection in the surrounding gasas well as through radiation. However, the convection of the gas surrounding the detector ispressure dependent [59] and is minimized by operating the detectors at very low pressures. Theheat loss through radiation is negligible in comparison to the conductance loss through thethermal isolating region (legs). With these assumptions in mind, the solution to the heat flowequation yields [11]:

T=h P0

G 1+w2t2(14)

whereP0 is the radiant power falling on the cantilever, is the absorbance (absorbed fraction) ofthe radiant power, G is the thermal conductance of the principal heat loss mechanism, is theangular frequency of the modulation of the radiation, and is the thermal response timedescribed by Equation (3).

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3.2. Thermal Conductance

The value of the thermal conductance of the thermal isolation region, Glegs, is a product ofthermal conductivity,g, of the microcantilever legs and their cross-sectional area divided by theirlength

Glegs = 2 gwleg tleg

lleg, (15)

wheregis the thermal conductivity of the material constituting the legs. The factor of 2 comesfrom the fact that most geometries feature two identical legs as illustrated in and Figure 5. Theterm (wleg . tleg) is the cross sectional area of the legs. For microcantilever detectors with complexleg structures [13], the thermal conductance Glegs will have a more complex expression.

The thermal conductance of the legs, however, is not the only path for energy dissipation. Figure

7 shows the thermal conductance as a function of the thermal isolation length calculated usingthe parameters listed in Table 1, assuming a leg thermal isolation width wleg of 1.5 m with athickness tleg 0.6 m. The dashed-dotted lines in Figure 7 represent the thermal conductance dueonly to conduction through the legs. Most reported structures have the legs with thermalconductance of the order of ~ 10-7 W/K [13, 14, 25]. From Figure 7, we can see that using SiO2as the base material of the cantilevers can provide structures with lower thermal conductance[41]. This is also evident from Table 1, which shows that SiO2 has much lower thermalconductivity than SiNx. The lower dashed and dotted line (Grad) represents the radiativecomponent of the total conduction, where the energy is dissipated by radiation. The radiativecomponent is expressed as [53]

Grad = 4 emetal +estructure( )Ads TT3 , (16)

where metal and structure are emissivities of the metal coating in the absorbing element and themain material respectively, Ad is the area of the detector, T is the Stephan-Boltzmann constantand T is the detector temperature. For most SiNxbased structures and dimensions, theconductance is of the order of ~10-7 W/K at room temperature when taking into account thedetector area of the order of ~10-9 m2 and the StephanBoltzmann constant, T= 5.67 x 10-8 W-2

K-4. This calculation also assumes that the emissivity of the metal and structural material side ofthe detector is, respectively, metal ~ 10-2 and structural ~ 10-1. Under ambient pressure andtemperature, the thermal conductance through air is of the order of ~10 -5 W/K [13] in the case of

gaps between the detector and the substrate of several m (see the two upper dashed line markedGair in Figure 7). This value assumes the thermal conductivity of air at standard temperature and

pressure to be at 2.5 x10-2 Wm-1 K-1 [12]. At temperatures of 200 K and 400 K, the thermal

conductivity of air is 1.9 x10-2 Wm-1 K-1 and 3.2 x10-2 Wm-1 K-1, respectively.

As discussed earlier [13], it is worth noting that uncooled IR detectors rely on both intrinsicphoton absorption and resonant cavity effects in order to increase the overall photon absorption.

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Tuning the absorption maximum of the resonance cavity to a wavelength of 10 m requires a 2.5m (for front side illumination) or 4.5 m (for back side illumination) gap between the substrateand the microcantilever detector. However, when the detector is positioned at these distancesabove the substrate in air, gas convection limits its performance. Depending on the distancebetween the detector and the substrate, this may yield a thermal conductance larger than the

thermal conductance through the legs of a typical microcantilever. Therefore, heat convectionthrough air is likely to be a dominant heat dissipation mechanism when a microcantileverdetector operates in an atmospheric pressure environment, and in close proximity to the substrate[13]. Djuric et al. discuss the dependence of the thermal conductance and NETD on pressure indetail [59]. In order for the assumption that the heat is dissipated through the legs only to bevalid, the conductance through the other two mechanisms has to be negligible. To minimize theconvection through the air surrounding the microcantilever detector, the detector must be placedin an environment with a substantially reduced pressure, such as an evacuated package. Thethermal conductance of nanomechanical detectors, in their normal operating conditions, isusually of the order of 10-7 W/K.

The only frequency-dependent term in Equations (12) - (13) is T, which involves the thermaltime constant [Equation (14)]. As the modulation frequency increases, T and the detectorresponse remains constant as long as the product is small compared to unity. At higherfrequencies, Tbegins to decrease, which causes the response to decrease [13]. This explainsthe observed roll-off in the response of the cantilever detector shown in Figure 8 (curve A)above ~60 Hz. The frequency at which the slope of the response changes on a log-log plot canbe used to experimentally evaluate the response time as it occurs at a frequency where the product becomes equal to unity (Bode plot method). The response time can therefore beestimated as 1/fc, where fc is the frequency at which the slope on the graph changes. The datashown in Figure 8 were recorded using a setup similar to the one described elsewhere [17]. It isimportant to emphasize that microcantilevers also exhibit mechanical resonances. The curve (B)

in Figure 8 [13] shows that although the signal increases at the resonance, so does the noise.Operating detectors at their resonance frequency would, therefore, not provide a significantadvantage for IR detection.

3.3. Microcantilever Responsivity

As mentioned in section 3.1, the responsivity, R, of an IR detector is defined as the output signalproduced by a unit of incident radiant power. Earlier work has shown [17] that the signal(relative response) of a microcantilever is linear with incident radiant power over a large range.Therefore, it is valid to determine an expression for responsivity that is independent of incidentradiant power. Furthermore, the gain of the optical readout is anticipated to be independent ofzin the case of relatively small cantilever deformations [13].

Since microcantilever thermal detectors can be regarded as oscillators, and the incomingradiation can be regarded as a non-DC dynamic stimulus, the analysis for driven harmonicoscillators can be applied to cantilevers. Therefore, the responsivity, R, of a nanomechanicalinfrared detector with rectangular bimaterial region similar to that shown in Figure 6, can be

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calculated by using Equations (12a) and (14) regarding the cantilever as a driven harmonicoscillator:

R w( ) =

Dz

P0= h

3lb2

(t1 + t2 )

1+t1

t2

2

3 1+t1

t2

2

+ 1+t1

t2

E1

E2

t12

t22+

t2

t1

E2

E1

a 1 - a 2( )

1

GT 1+ w2t2 1-

w2

w02

2

+w

2

w02Q

2

= hDz

DT

1

GT

1

1-w

2

w02

2

+w

2

w02Q

2

(17a)

where is the angular frequency of modulation of the radiation, 0 is the resonance frequencyof the cantilever, and Q is the quality factor. The units ofRare meters (of deflection) per Watt(of incident power). Alternatively, using Equations (12b) and (14) or Equations (13) and (14),we obtain respectively:

R w( ) =Dz

P0= 3h a 1 - a 2( )

t1 + t2

t22K

lb

2 1

GT 1+w2t2 1-

w2

w02

2

+w2

w02Q2

= hDz

DT

1

GT

1

1-w2

w02

2

+w 2

w02Q2

(17b)

and

Rq w( ) =Dq

P0= 6h a 1 - a 2( )

t1 + t2

t22

K

lb

1

GT 1+w2t2 1-

w2

w02

2

+w

2

w02Q

2

= hDq

DT

1

GT

1

1-w

2

w02

2

+w

2

w02

Q

2

(18)

If a function () is defined to describe the normalized resonance curve, and is expressed as

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( ) =1

1-w

2

w02

2

+w

2

w02Q

2

,

(19)

then the responsivity of a thermal detector can be expressed asR() = R0()() whereR0() isresponsivity for steady irradiation. Equations (17) - (18) also define other parameters commonlyused, such as the thermomechanical sensitivities defined as z/Tand /T.

It is not trivial to obtain the value of the absorbance, since it depends on a number of factors,such as material properties and detector geometry. Furthermore, as pointed out in earlier studies[13], it is possible to improve the absorption in the wavelength region of interest and henceincrease the value of,by utilizing the resonant cavity effects. By designing a resonant opticalcavity into the active (absorber) area, it is possible to maximize the infrared absorbance and, inturn, the thermally induced deformation of the detector. For simpler microcantilever designs,however, the value of= 0.6 is a reasonable assumption for most cases [13], although valuesin the range 0.8-0.95 have been measured for other microcantilever IR detectors [60, 61]. Thedependence ofR on the wavelength has been shown to be closely related to the wavelengthdependence of the absorbance bands of the detector structural material and/or the bands of theoptical cavity designed to increase the photon absorption [13].

Figure 9 [13] shows the measured microcantilever responses as a function of wavelength forabsorbed photons in the wavelength range from 2.5 m to 14.5 m. Solid line shows thedetector spectral response normalized by the incident radiation power. The observed increasedin responsivity around a wavelength of 4 m can be attributed to the effect of an opticalresonance cavity formed by a 550 nm thick SiNx film on an Al mirror. The dashed curve in

Figure 9 shows the independently measured absorptivity of a SiNxmembrane, which was used inthe fabrication of the cantilever detector. The correlation between the two indicates that thehigher detector responsivity in the region of 8 to 14 m is mostly due to the intrinsic absorptionof SiNx.

It should be noted that Equation (17) defines the responsivity in terms of the magnitude ofdeflection of the detector, z, and therefore, characterizes the nanomechanical detector only.The total system-level responsivity should be expanded to also include the differential signal ofthe photodetector used in the optical readout. The respective expression and the values obtainedusing this expression will be discussed later in this chapter.

It is also useful to know the relationship between the deflection of a nanomechanical infrareddetector and the change in the temperature of the target, Tt. The change in target temperaturewill induce a corresponding change in the temperature of the detector. It is obvious that theimaging optics play an important role in this and it should be included in the equation. Assumingthe emissivity of the target T = 1, the power incident on the nanomechanical infrared detector isgiven by [13]

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P0 = t0Ad

4F2dP

dT

l 1- l 2

Tt, (19)

whereAd is the surface area of the detector,Fis the f-number of the optics, 0 is the transmissionof the optics, and (dP/dT)1-2 is the slope of the functionP = f(Tt) wherePis power radiated bya blackbody target within the spectral band from 1 to 2. Combining Equations (17) and (19),the change in microcantilever deflection can be expressed as

z = t0Ad

4F2R w( )

dP

dT

l 1- l 2

Tt (20)

where R is the microcantilever responsivity. This expression shows the relation of changes inthe microcantilever deflection to changes in the temperature of the target. For most reportedstructures, the above-defined z is of the order of few per 1 K temperature change of thetarget. If the ratio of temperature change in the detector to the temperature change of the target

is obtained, it gives the transfer function [13, 25]. The transfer function can be obtained bycombining Equations (14) --in static state-- and Equation (19) and is given by:

H=h t0Ad

4F2G

dP

dT

l 1- l 2

. (21)

The transfer function usually varies between 1/30 and 1/300. This means that in case of transferfunction of 0.01, for each degree K change of temperature of the target, the temperature of thedetector will change by 10 mK.

4. Noise Sources and Fundamental Limits

This chapter analyzes the noise sources impacting the mechanical IRdetectors and the extent of the influence of those sources on detectorperformance. This analysis is needed in order to be able to compare theperformance of uncooled mechanical IR detectors to that of cooled detectorsand uncooled non-mechanical detectors. Noise in nanomechanical IRdetectors can come from the detector itself, from the detector-environmentinteraction or from the readout. Microfabrication allows for batch fabrication

of highly efficient transducers converting small temperature differences orheat fluxes into easily measurable output signals. While the reduced sizeand heat capacitances of thermal detectors improve the image resolutionsensitivity, their usefulness as IR and THz detectors is governed by theinfluence of various noise sources. The detector noise characteristics imposefundamental limitations to their performance. Limitations such astemperature fluctuation limited and background limited noise are applicableto all thermal IR and THz detector types and arise from the fact that all

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objects, depending on their thermal mass and degree of heat exchange withthe environment, are subject to thermal fluctuations. These fluctuations,negligible in macroscopic objects, become significant noise sources for highlythermally isolated microscopic detectors such as microbolometers andnanomechanical thermal IR detectors.

An additional limitation, applicable only to nanomechanical IR and THzdetectors, is spontaneous microscopic mechanical oscillations of suspendedstructures due to their thermal energy (kBT). For any of the potentialreadouts, these oscillations are indistinguishable from thermally induceddeformations and as such directly contribute to detector noise.

The fundamental limits to the performance of the nanomechanical thermal detectors are intrinsicproperties of the detectors and are therefore, readout-independent. The fundamentalperformance limits are the background thermal fluctuation limit and the temperature fluctuationlimit. They arise from the fluctuations in the detector temperature that exist because of the

dynamic nature of heat exchange between the detector and its environment. An ideal noiselessreadout amplifies both the useful signal and the detectors intrinsic noise without distorting themor changing the signal-to-noise ratio. In practice, there are no ideal noiseless readout methods; inbest case, the readout decreases the inherent signal-to-noise ratio of the microcantilevers onlyminimally.

In order to obtain analytical expressions for the noise limited figures-of-merit NEP, D* andNETD, we need to consider the expressions for those figures-of-merit defined above. FromEquations (5a) and (6), we obtain the expression for noise limitedNEPN

NEPN

=zN

DzP0

, (22)

where zN is the amplitude of the deflection for a particular noise mechanism, and z is theamplitude of the deflection due to the signal. From Equations (7) and (22) we obtain theexpression for noise limited normalized detectivityD*N

DN* =

Dz

zN

AdB

P0, (23)

where z/zN is signal to noise ratio for a particular noise mechanism. In order to obtain noise

limitedNETDN, we will consider the signal-to-noise ratio, which can be obtained from Equation(20)

z

zN=

t0AdR w( )dP

dT

l 1- l 2

4F2zN

Tt(24)

IfNETD is defined as the target temperature for which the signal to noise ratio equals unity and

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Equation (24) is used, the following expression is obtained

NETDN =4F

2zN

t0AdR w( )dP

dT

l 1- l 2

(25)

The Equations (22), (23) and (25) are generalized expressions for noise limited NEP, D* andNETD, and take into account the tip displacementfluctuations resulting from the contributionsfrom different noise sources. These expressions are used in the subsequent sections to evaluatehow different noise mechanisms influence the figures-of-merit.

4.1. Temperature Fluctuation Noise

All IR detectors that operate as transducers of incoming IR radiation into

output signal are affected by temperature fluctuation noise due to continuousheat exchange at the microscopic level (see Kruse [11, 48, 49]). In the case ofnanomechanical IR detectors, temperature fluctuation noise manifests itself as fluctuations of thedetector tip displacement due to the bimetallic effect. As discussed previously by Kruse [11, 48],the mean square magnitude of the fluctuations in detector temperature can be derived from thefluctuation-dissipation theorem and is given by:

T2 =kBT

2

C, (26)

where kB is Boltzmanns constant and T is the temperature of the detector. The temperaturefluctuation of Equation 26 is the integration over all frequencies f, wheref= /2. Thefrequency spectrum for temperature fluctuations is given by[11]

Tf2 =

4kBT2B

GT 1+w2t2

( ), (27)

where B is the measurements bandwidth and GT is the total thermal conductance between theabsorber and the environment. From Equation (27), it follows that the root mean square (rms)temperature fluctuation can be expressed as

T21/ 2

=4kBBT

GT1/ 2 1+w2t2

(28)

Figure 10 shows exemplary temperature fluctuation spectra calculated for a typical IR sensitivenanomechanical detector using Equation (28).

It is important to note that the frequency dependence of temperature increases the useful signal

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[Equation (14)] and corresponds to the frequency dependence of temperature fluctuation due tothe noise [Equation (28)]. Frequency dependence of fluctuations in the displacement of thedetector tip, z, due to temperature fluctuation noise is influenced by both the thermal andmechanical response of the detector. The expression for spontaneous fluctuations indisplacement of the microcantilever caused by temperature fluctuations can be obtained from

Equations (14), (17) and (28), where Tis substituted with 1/2

zTF2

1/2

=R w( )T 4kBBGT

h(29)

It can be seen from Equations (12a), (12b) and (13) that, due to the linear dependence ofzandT, the signal-to-noise ratio, z/1/2, can be calculated as T/1/2,

z

dzTF

21/2

=h P0

T 4kBBGT(30)

From Equation (30), the total thermal conductance, GT, defined as a sum of principal heat lossmechanisms, is the main design parameter affecting the ratio of the signal to the temperaturefluctuation noise in nanomechanical infrared detectors. In practice, the smallest total thermalconductance achievable is the radiative heat exchange between the detector and its surroundings.

It should also be noted that the signal-to-noise ratio as defined by Equation (30) is frequency-independent, even at > 1. Assuming the mechanical parameters provided in Table 2., thecalculated spectral densities of spontaneous microcantilever fluctuations, 1/2, due totemperature fluctuations in vacuum are plotted in Figure 11 (solid line marked TF).

Using the expressions for amplitude of fluctuations due to the temperature fluctuation noise1/2 [Equation (29)] and corresponding signal to noise ratio z/1/2 [Equation(30)], together with Equations (22), (23) and (25), we obtain the expressions for temperaturefluctuation limited noise equivalent powerNEPTF,

NEPTF =T 4kBBGT

h, (31)

the normalized detectivityDTF*,

DTF* = h AdB

T 4kBBGT, (32)

and the noise equivalent temperature differenceNETDTF,

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NETDTF =4F2T 4kBBGT

t0h AddP

dT

l 1- l 2

.(33)

Figure 12 shows the plots of temperature fluctuation noise limit, NETDTF, for the detectoranalyzed in reference [13] and calculated using Equation (33) for three different detectortemperatures 200, 300, and 400 K. The values of Grad plotted in Figure 12 as vertical linescorrespond to the conductance due to only radiative heat loss mechanism at three temperatures of200, 300, and 400 K. These values ofGrad represent the ultimate thermal isolation limit (for thecorresponding temperatures). Figure 13 illustrates the dependence ofD*TF on Tand GT plottedon different detector geometries. Plots in both Figures 12 and 13b emphasize the importance ofdesigning detectors with a low thermal conductance G. However, in case of nanomechanicaldetectors suspended over a substrate, the conductance through air, Gair, dominates the heat lossmechanisms. As illustrated in Figure 12, values of thermal conductance close to the upper limitobserved when operating at the atmospheric pressure, Gair, correspond to high values ofNETDTF.

From Equations (14) and (27), it is apparent that one cannot separate the temperature fluctuationnoise from the signal since both may have the same frequency components. Equation (29)suggest that the rms amplitude of the oscillations in the detector temperature maybe beminimized by operating the detector at higher frequencies. However, consistent with thepreviously observed experimental data [13], no improvement in the signal-to-noise ratio can begained by operating at the microcantilever resonant frequency since both the noise and the signalpeak at the resonance (see Figure 8) [14].

Equation (30) shows that it is crucial to minimize the heat exchange (small value ofGT) betweenthe detector and the environment to decrease the temperature fluctuation noise and maximize thesignal-to-noise ratio. If operating at atmospheric pressures, the dominant heat-loss mechanism is

likely to be convection through the air. For this reason, traditional nanomechanical IR detectorsare kept at reduced pressure in an evacuated package.

Equation (33) further indicates that NETDTF can be improved with increased thermal isolation(lowerGT). It reaches its minimum for ideally isolated detectors, i.e. in case of purely radiativeheat exchange. It may appear that improving the thermal isolation, i.e. decreasing the totalthermal conductance, GT, also decreases the detector performance as it increases the magnitudeof temperature fluctuations 1/2, i.e. noise, which is inversely proportional to GT1/2 [Equation(28)]. However, lower thermal conductance also improves detector sensitivity z/T, which isinversely proportional to GT, to an even higher degree, hence improving the overall performance[Equation (17)].

To offset the noise in nanomechanical IR detectors originating from ambient temperaturefluctuations, techniques like self-leveling reported previously [24, 26, 61, 62] and damping themechanical energy by converting it into the AC current via an RC circuit formed by thedetectors capacitance and damping resistor have been developed [41].

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4.2. Background Fluctuation Noise

Designing thermally well-isolated detectors is the key to optimizing their performance.However, the degree of thermal isolation is always limited by the radiative heat exchangebetween the detector and its environment, since this type of heat dissipation cannot be prevented.

As in case of other heat dissipation mechanisms, the temporal fluctuations in the radiative heatexchange cause temporal fluctuations in the detector temperature. Such fluctuations create thenoise level commonly referred to as the background fluctuation noise. The backgroundfluctuation limit, in the case of microcantilever thermal detectors, can be quantified in terms ofthe microcantilever tip displacement, i.e. oscillations in z. Conventional uncooled detectorscurrently in use can only remotely approach the background fluctuation noise limit.

Since the background fluctuation noise can be regarded as the manifestation of temperaturefluctuation noise when the radiative conductance is the principal heat loss mechanism, theexpressions derived forNEPTF,D*TF andNETDTF are still applicable. The only difference is thatthe total conductance GT is replaced by the radiative heat conductance Grad.

Grad is obtained using the first derivative with respect to temperature of the Stephan-Boltzmannfunction

Grad = 4Ahs T(TB3 +TD

3 ), (34)

where TB is the temperature of the background and TD is the temperature of the detector. UsingEquation (34), together with Equations (31), (32) and (33), we obtain the following expressionsforNEPBF,

NEPBF =16kBAdBs T TB

5

+TD5

( )

h1/ 2

, (35)

andD*BF

DBF

* =h 1/ 2

16kBs T TB5 +TD

5( )

(36)

andNETDBF

NETDBF = 4F

2

16kBBs T TB5

+TD5

( )

t0 AdhdP

dT

l 1- l 2

. (37)

Background fluctuation noise is the lower noise limit, as it cannot be affected by designoptimization.

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4.3. Thermo-Mechanical Noise

In addition to temperature fluctuation noise and background fluctuation noise, there exist noisesources unique to nanomechanical IR detectors. A unique feature of nanomechanical detectors isthat, unlike other types of uncooled IR detectors, they are mechanical structures (oscillators) that

can accumulate and store mechanical energy. When a nanomechanical detector is in equilibriumin a thermal bath, there exists a continuous exchange of the mechanical energy accumulated inthe detector and thermal energy of the environment. This exchange, governed by the fluctuationdissipation theorem [63, 64], results in spontaneous oscillation of the microcantilever so that theaverage mechanical energy per mode of cantilever oscillation is defined by thermal energy kBT.Sarid [43] has described this noise source as thermally induced lever noise. [13].

An analysis is provided by Sarid [43], which involves the Q-factor of a vibratingmicrocantilever, its resonant frequency, 0 , and spring constant, k. While the Q-factor can bedefined empirically as the ratio of the resonance frequency to the resonance peak width, it is alsoimportant to know the exact mechanisms of microcantilever damping for evaluation of the

thermomechanical noise spectrum. The analyses of several groups who have attempted todevelop analytical models to describe thermo-mechanical noise and energy dissipation innanomechanical and nanomechanical resonators has been reviewed and compared [65]. It is nowknown that the intrinsic losses in the microcantilever material such as viscoelastic losses [65]represent an important mechanism of the mechanical energy dissipation. These losses have beenevaluated [65] using the Zener model for elastic solids [66-68]. Phonon scattering within thenanomechanical system due to defects within the solid or at the interface between the solid andvacuum has also been considered [13, 69].

Zener et al. demonstrated the effects of thermo-elastic internal friction in the late 1930s bymeasuring the frequency response spectrum of a copper reed over a wide frequency range [66-

68]. The responses were observed to be adiabatic at high frequencies and isothermal at lowfrequencies. The measured internal friction was also observed to be the highest at a frequencyf= (/2)(D/w2) where D is the thermal diffusivity and w is the reed width. These authors alsodemonstrated that the internal friction was related to the heat flow across the reed.

If we assume isotropic solid and diffusive thermal phonons, the interaction between the acousticmode and the thermal phonon bath can be described by the coefficient of thermal expansion ofthe material. Thermo-elastic solid, when excited into motion, reaches a nonequilibrium state andthe coupling of the strain field to the temperature field gives rise to an energy dissipationmechanism that allows the system to relax back to equilibrium [70]. Lifshitz and Roukes [70]investigated the thermo-elastic damping process as a dissipation mechanism in small scale

resonators and concluded that the thermo-elastic damping is a significant source of dissipationdown to the nanometer scale.

Yasumura et al. [71] have investigated the Q-factors in small-scale silicon and silicon nitrideresonators whose thicknesses varied from a few m to a few hundred nm. They have determinedthat the Q-factor decreased monotonically as thickness decreased and that the effect ofthermoelastic dissipation became negligible. Houston et al. [72], however, found that, at roomtemperature, thermo-elastic dissipation becomes a significant loss mechanism.

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The discrepancy with the conclusion of Lifshitz and Roukes [70] can be explained by the factthat the thermo-elastic dissipation is frequency dependent [13]. For a microcantilever withthickness t, the frequency,fTE, that maximizes thermo-elastic dissipation is given by [71]

fTE = p2

gr Ct

2, (38)

wheregis the thermal conductivity of the material and is the density. If the thickness of themicrocantilever is varied while keeping the length constant, the resonant frequency will shift.This shift in frequency will, in turn, maximize the thermo-elastic dissipation [72] and, therefore,the ratio t/lwill become significant to the point it has to be taken into account. In fact, increasingthe resonance frequency of the microcantilever by decreasing its the length causes the thermo-elastic dissipation to be shifted into a frequency region dominated by other processes (phononphonon limit) [72].

White and Pohl [73] have measured the low-temperature internal friction of thin a-SiO2 filmswith thicknesses ranging from 0.75 to 1000 nm in an attempt to determine whether the spectraldistribution of the low energy excitations, believed to exist in all amorphous solids, is caused bystrong interactions between defects. Their findings indicated that the low temperature internalfriction of these films is nearly identical to that of bulk a-SiO2, and they concluded thatinteractions, limited to distances less than 0.75 nm, can be viewed as intermolecular forces [13].

Sarids thermo-mechanical model assumes a viscous nature for the damping in microcantileveroscillations [43]. The assumption of predominantly viscous damping is valid formicrocantilevers operating in air or water and, therefore, justified for nanomechanical structuresutilized as probes in scanning probe microscopy. In the case of a microcantilever operating in aviscous medium, such as air or water, the damping force is proportional to the linear velocity ofmicrocantilever. The resulting noise spectrum can be expressed as [74]

zTM2

1/ 2

=4kBTB

Qk

w0

3

w0

2 - w2( )2

+w 2w0

2

Q2(39)

According to the Equation (39), the noise density is frequency-independent for frequencies wellbelow the mechanical resonance frequency, 0(i.e.

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zTM2

1/ 2

=4kBTBQ

kw0. (41)

Equation (39) is commonly used to estimate thermo-mechanical noise limits of ananomechanical IR detector [3, 25, 75].

Since nanomechanical IR and THz detectors usually operate in vacuum (at pressures below 10mTorr) one should consider the dependence of both Q and 0 on pressure. In fact, Q typicallychanges from about 1020 in air to 100 or more in vacuum. Figure 11 shows the spectral densityof the thermomechanical noise (plot TM). These plots have been obtained using Equation (39)and the mechanical properties in Table 2.

As emphasized in [13], the higherQ-factors of microcantilevers operating in vacuum are definedmostly by the mechanisms of intrinsic friction and inelastic damping [74, 76] in themicrocantilever material. Additionally there are a number of mechanisms which include thermo-

elastic dissipation, motion of defects, and phononphonon scattering [69, 77], and can be sourcesof internal friction. However, while the internal friction is a bulk effect, surface effects maydominate in nanometer thick structures [73]. Therefore, there is a question about the accuracy ofthe model of thermomechanical noise based on the assumption of viscous damping in the case ofmicrocantilevers with microscopic dimensions, such as those used as IR detectors [74, 76].

Consequently, the model discussed by Majorana et al. [74], who investigated an alternativemodel of thermo-mechanical noise that accounts for internal friction processes rather thanviscous damping, may be applicable. In this model, the thermo-mechanical noise spectrum isexpressed as [74, 76]:

zTM'2 1/ 2 = 4kBTB

Qkww0

4

w02- w

2( )

2

+w0

4

Q2

(42)

It can be seen from Equation (42) that the thermo-mechanical noise density follows a 1/f1/2

dependence below the resonant frequency when the damping is due to intrinsic frictionprocesses. This is also apparent in Figure 11, (plot TM) which is plotted using Equation(42) and the mechanical detector properties from Table 2. An analysis of Equations (39) and(42) shows that regardless of the dissipation mechanism, the off-resonance thermomechanicalnoise is lower in the case of microcantilevers with a higherQ-factor and higherk. It should beemphasized that while predictions based on Equation (39)are often reported in the literature [3,

75, 77], the noise density calculated according to the two alternative models may substantiallydiffer from each other at low frequencies [64, 74, 76]. Furthermore, the intrinsic friction modelpredicts the noise at low frequencies to be independent of the microcantilever resonant frequencyassuming a fixed stiffness, k. By contrast, the viscous damping model predicts that the lowfrequency noise of a microcantilever detector can be decreased by increasing its resonancefrequency, even without changing its stiffness. Therefore, it is critical to know the actualmechanisms of the mechanical dissipation in the microcantilever detector for analyzing thermo-mechanical noise of a nanomechanical detector in the frequency range relevant to real-time IR

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imaging [13].

The expression for the rms amplitude of tip displacement Equation (40) (in case of viscousdamping), along with Equations (17), (22), (23) and (25) can be used to obtain the thermo-mechanical noise limited values ofNEP,D* andNETD. NEPTM becomes:

NEPTM =1

R w( )

4kBTB

Qkw0, (43)

D*TMbecomes:

DTM* = R w( )

Qkw0Ad

4kBT, (44)

andNETDTM becomes:

NETDTM =4F2

t0AdR w( )dP

dT

l 1- l 2

4kBTB

Qkw0.

(45)

Using the alternative model [Equation (42)] that takes the damping caused by intrinsic frictioninto account, the frequency-dependent rms noise can be predicted. This frequency dependence,somewhat complicates the estimate of the correspondingNETDTM. In addition to a measurementbandwidth, an assumption should be made with respect to the frequency. Off resonance, thevalue ofNEPTMbecomes:

NEPTM

' =1

R( )

4kBTB

Qk, (46)

D*TMbecomes:

DTM'* = R w( )

QkwAd

4kBT(47)

andNETDTM becomes:

NETDTM' =4F2

t0AdR w( )dP

dT

l 1- l 2

4kBTB

Qkw (48)

In practice, it is very important to determine which of the two thermomechanical noise modelsapplies to a particular type of nanomechanical thermal detector. Nevertheless, despite the

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differences, both models agree on the importance of having stiffer microcantilevers (higherk)and higherQ-factors.

Figure 11 displays the cantilever noise spectra. Noise originating from both temperaturefluctuation (solid lines) and thermo-mechanical noise (dashed lines) is included. The graph

features theoretical predictions for both alternative ways of calculating thermo-mechanical noise.Those include assuming the viscous damping (dotted line) and assuming the energy is lost due tointernal friction processes (dashed line).

4.4 Readout Noise

A portion of the total noise can be attributed to the readout used to quantify the deformations ofthe mechanical infrared detectors. Some of the readouts implemented include piezoresistivereadout, capacitive readout and optical readout (both using the photodetector and CCD).

4.4.1 Piezoresistive Readout

The following analysis will consider a simple, rectangular, bimaterial piezoresistive detectorwith geometry as defined in [78] (Figure 14). A piezoresistive IR detector measures thermallyinduced stress in the structure. Through the piezoresistive effect, the stress is converted tochange in detector resistance R, which is in turn, is converted to a voltage signal through aWheatstone bridge. The two main noise sources in a piezoresistive readout are Johnson Noiseand Hooge (or flicker) Noise. Johnson noise is caused by the random motion of mobile carriersin resistive materials at finite temperature T. It is independent of frequency and depends only onthe temperature and resistance of the piezoresistive element. The voltage fluctuations are given

by:

vJN2 = 4kBTRB (49)

where T is the temperature of the detector, R is the resistance of the detector and B is thebandwidth. At low frequencies, Hooge noise is dominant and it follows 1/fdependence.

The source of it is currently still not fully understood. However, an empirical formula derived byHooge [79] indicates that

vHN2 = a V

2

BNf

(50)

where is Hooges factor which is not constant, but a dimension-independent parameter rangingfrom 10-7 to 10-3 [79], Vis the bias voltage of the piezoresistive element, andNis the number ofcarriers.

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Harley et al. [78] plotted a typical piezoresistive cantilever readout noise spectrum, which isdisplayed in Figure 15. It is apparent that at lower frequencies, the noise follows the 1/fdependence of Hooges noise and asymptotically approaching the constant Johnson noise athigher frequencies.

In order to utilize the above-defined formulas for noise limited figures-of-merit, the displacementof detectors tip, equivalent to above defined noise fluctuations, need to be defined. Hansen etal. [80] and Harley et al. [78] define a transfer function of a piezoresistive detector SPD whichrelates the tip displacement zto the voltage signal vPD generated by the piezoresistive detectorsuch that vPD = SPD z. The transfer function is given by

SPD =

6kp effV l -lleg

2

wt2

,(51)

where kis the spring constant of the detector, eff is the effective piezoresistive coefficient, Visthe voltage across the detector, l is the detector total length and lleg is the length of thepiezoresistive legs of the detector, w is the detector width and tis its thickness.

Using the Equations (49), (50) and (51), the following expressions are obtained:

zJN2 =

1

SPD

dvJN

2 =2wt

2kBTRB

3kp effV l -lleg

2

(52)

and

zHN2

=1

SPDdvHN

2=

wt2a VB

6kp effNf l -lleg

2

(53)

From here, using the Equations (17), (22), (23) and (25), we can obtain the piezoresistive readoutnoise-limited NEP, D* and NETD. We will derive expressions for both Johnson noise andHooge noise, having in mind that the former will be dominant at higher frequencies and latter atlower frequencies.

In case of Johnson noise, the following expressions define

NEPJN =1

R w( )

2wt2kBTRB

3kp effV l -lleg

2

,(54)

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DJN

*= R w( )

3kp effVl l -lleg

2

2t2kBTR

(55)

and

NETDJN

=4F2

t0R w( )dP

dT

l 1- l 2

2t2kBTRB

3kp effV l -lleg

2

wl

2 (56)

whereAd has been approximated by lw. In the case of Hooge noise, the following expressionsdefine

NEPHN = 1R w( )

wt2

a VB

6kp eff Nf l -lleg

2

,(57)

DHN* = R w( )

6kp effNfl l -lleg

2

t2a V

(58)

and

NETDHN =4F2

t0R w( )dP

dT

l 1- l 2

t2a VB

6kp effNfwl2

l -lleg

2

. (59)

4.4.2. Capacitive Readout

Capacitive readout utilizes the change of capacitance of detectors due to their deformation inresponse absorption of the IR radiation. When detectors temperature increases, the detector willbe deformed due to the bimetallic effect, as the other types of nanomechanical infrared detectors.

Since a part of the detector is at the same time one of the capacitors plates, this deformation willchange the distance between the plates, inducing the change in capacitance. This capacitancechange will be detectable and proportional to the temperature change [41, 81]. The capacitancewill increase or decrease with increasing IR absorbance (increasing paddle temperature)depending on whether the higher coefficient of thermal expansion material layer lies below [26,41, 61] or above the lower CTE material [15, 16, 60].

Figure 16 shows a schematic diagram of a capacitive bridge readout circuit demonstrating the

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operation of the capacitive bridge detector and the cantilever motion damping circuit [26, 41,61]. The microcantilever sensor forms part of the two capacitor bridge circuit. The sensor isenergized by applying symmetric, oppositely phased voltage pulses, VS, to the cantilever andbridge reference capacitors (CS and CR respectively) around a reference voltage, VRef. If thecantilever and bridge capacitances are the same size and VS and VRare of opposite sign (i.e. CSVS

= -CRVR), then the voltage appearing at the common node between the capacitors is zero. Duringoperation, when the microcantilever sensor is exposed to the IR radiation, the paddle moves up,increasing the capacitor gap, thereby decreasing the sensor capacitance and generating an offsetvoltage, Vg, at the common node and at the input to the gain and integrator circuits (Figure 16).

There are several noise sources associated with capacitively read microcantilever based IRimagers, some present in all IR sensing techniques, while others specific to micromechanicalbased systems as outlined above. Noise in a capacitively sensed nanomechanical infrareddetector is introduced in the signal chain at the detector, the electronics and from other camerabased noise sources. The same fundamental sources of noise present in all IR radiation sensingsystems are temperature or shot noise, and thermal conduction or fluctuation noise. In addition tothese thermal noise sources, there are the thermally driven noise sources that are specific tomicrocantilever sensors as outlined in the previous section. The final sources of noise, and theones which usually dominate capacitively sensed thermal imaging systems performancelimitations, are the readout and camera electronics noise sources. These noise sources includeJohnson and 1/f noise in the on-pixel amplification and integration circuitry, kBT/C noise in thecharge integration capacitors, and various correlated on-chip switching noise sources and in theoutput signal chain. Using the responsivity, RV defined by Equation 17, these noise sources addin quadrature as follows:

NETDROIC = bDV

n

2

RV

(60)

The ROIC noise terms were modeled by Hunter et al. [41, 61] for a 50 m sized detector pixelreadout circuit, which was used to control and sense the operation of the microcantilever sensorarray. The 1/f and Johnson noise terms were modeled using Spice simulation techniques withprocess parameters supplied by the CMOS foundry. The kBT/C noise was calculated from thequadrature sum of a noise electron analysis for the charge integration and feedback capacitorsused in the ROIC circuit. Switching and other noise terms were neglected in this model. Thebackground thermal noise contribution to the total detectorNEDT from these calculations was0.9 mK, the temperature fluctuation noise was 4.6 mK, the thermomechanical noise duringoperation was 0.7 mK, the 1/f and white noise in the ROIC was 9.7 mK and the ROIC kT/C

noise was 15.1 mK. These gave a total NEDT 18 mK. These results show that the readoutnoise dominates the noise sensitivity calculations and that the capacitively read nanomechanicalinfrared detector has a performance comparable or better than state-of-the-art microbolometer IRcameras [26, 61].

4.4.3 Optical Readout

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As we mentioned above, the term optical readout refers to two different configurations. Oneconfiguration is the optical cantilever readout identical to the one utilized in Atomic ForceMicroscopy [43] and is the most suitable for single (spot) detectors. The second configuration isthe optical readout using a charge-coupled device (CCD) utilized for simultaneously probing

deflections of multiple detectors in two-dimensional arrays and is mostly suitable for infraredimaging. We will address some of the noise issues for both of them.

4.4.3.1 PSD Readout

In this approach, a laser beam is focused on a microcantilever tip. This configuration is verysimilar to that employed in the atomic force microscope [43]. The reflected beam deflects inaccordance to microcantilever bending. The magnitude of microcantilever deformation(bending) can be quantified by directing the reflected spot onto a PSD that consists of twophotodiodes: PD1 and PD2. An extensive analysis of the noise introduced into the system whichuses this type of readout can be found in previous work [13]. They have considered the

displacement, z, of the bimaterial microcantilever tip with length ldue to thermal IR radiationincident upon it. The additional input parameters for this calculation include the probing laserbeam power, Pl and the distance between the microcantilever and the photodetector L.Microcantilever deflection causes a redistribution of the laser beam power incident upon each ofthe two photodiodes. If a square (i.e., non-Gaussian) spatial distribution of power within thelaser beam is assumed, the difference between the power of light fallen upon each of the twohalves of the photocell can be approximated as [43]

Pl = Pl4L

ldDz (61)

where d is the diameter of the light spot projected on the photocell. This difference inilluminations of the two photodiodes leads to a difference, i, between photo-currents of the twohalves of the photocell, [43]

i = gPl4L

ldDz (62)

where is the photocell responsivity in A/W. Equation (62)shows that the gain of the opticalread-out is proportional to the laser power, Pl , as well as the geometric factor of the photocell = 4L/l, and inversely proportional to the diameter, d, of the light spot projected on the

photocell. A differential amplifier is connected to the two halves of the bi-cell [13]. Theamplifier produces the output current, iS, proportional to z, [43]

iS = gPl4L

ldDz =

4gLPl

ldP0R (63)

Therefore, the thermal infrared power P0 falling on the microcantilever can be obtained

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indirectly by measuring the photocurrent iS, if the photocell responsivity , the distance L, thelaser powerPl, the microcantilever length l, the diameter of the light spot projected on thephotocell d and of the microcantilever responsivity R are known. If the responsivity, R, isdetermined independently using a calibrated source, the incident infrared power,P0, is

P0 = iSld

4gLPlR(64)

We can define the total responsivity of the optically-probed nanomechanical IR detector as thecurrent measu