From MEMS to NEMS: Scaling Cantilever Sensors

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From MEMS to NEMS: Scaling

Cantilever Sensors

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op woensdag 2 mei 2012 om 10:00 uur

door Chung-Kai YANG

Master of Science in Electrical Engineering

geboren te Port-Au-Prince, Haiti.

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Copromotor Dr.ir. E.W.J.M van der Drift

Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. P.J. French, Technische Universiteit Delft, promotor Dr. E.W.J.M van der Drift, Technische Universiteit Delft, copromotor Prof. E. Yeatman, Imperial College London

Prof. P.M. Sarro, Technische Universiteit Delft Prof. U. Staufer, Technische Universiteit Delft Prof. A. van Keulen, Technische Universiteit Delft Prof. K. Zhang, Technische Universiteit Delft

This thesis work is supported by Dutch Ministry of Economic Aﬀair under the frame-work of NanoNed (Project code: TQVB46).

ISBN 978-94-6191-283-1 c

⃝ Chung-Kai Yang

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Abbreviations

MEMS Micro Electro Mechanical Systems NEMS Nano Electro Mechanical Systems SEM Scanning Electron Microscopy SPM Scanning Probe Microscopy STM Scanning Tunneling Microscopy AFM Atomic Force Microscopy

CNT Carbon Nano Tubes

CPD Critical Point Drying

MWCNT Multi Walled Carbon Nano Tubes SWCNT Single Walled Carbon Nano Tubes EPI Electrostatic Pull-in Instability GDQ Generalized Diﬀerential Quadrature FWHM Full Width Half Maximum

FEM Finite Element Method SNR Signal-to-Noise Ratio

MRI Magnetic Resonance Imaging IBID Ion Beam Iinduced Deposition

NA Numerical Aperature

FE Fowler Nordheim

UHV Ultra High Vacuum

PzR Piezoresistance SiNW SiliconNanoWires

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Symbols

Greek variables

αT linear thermal expansion 1/K

γ ﬁeld emission enhancement factor dimensionless

γG Gruneisen’s constant N m/N m

ϵ strain m/m

ν Poisson ration m/m

ϕ local work function of the emitting surface material eV

πp piezoresistive coeﬃcient (N/m2)−1

ρ density Kg/m3

ρe electrical resistivity Ω m

σ stress Pa (N/m2)

σe electrical conductivity S(Siemens)/m

ςF 1/4 Wheatstone bridge force sensitivity V/N

ςx 1/4 Wheatstone bridge displacement sensitivity V/m

τR relaxation time constant s

τσ stress relaxation time constant s

τϵ strain relaxation time constant s

Roman Variables

A area m2

Ab dynamic nonlinearity bifurcation amplitude m

Ac dynamic nonlinearity critical amplitude m

as signal amplitude m

C heat capacitance J/K

Cn resonance mode dimensionless

DT thermal diﬀusion coeﬃcient m2/s

E Young’s Modulus Pa (N/m2₎

˜

E eﬀective Young’s Modulus Pa (N/m2)

Ee electric ﬁeld V/m

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g gap m h tip-anode distance m hp Plank’s constant Js Ie electrical current A k spring constant N/m L beam length m l cantilever length m lp PR device length m

lT thermal diﬀusion length m

M bending moment Nm m mass kg me electron mass kg mef f eﬀective mass Kg P point force N p doping concentration cm−3 Q quality factor Hz/Hz Q−1 internal friction Hz/Hz Re electrical resistance Ω RM mechanical resistance N s/m

Rs speciﬁc resistance or sheet resistance Ω/

r bending radius m

S scaling factor m/m

S1/f 1/f spectra noise (Hooge noise) V/Hz

SJ N 1/4 electrical thermal spectra noise V/Hz

T temperature K

t cantilever thickness m

tp PzR device thickness (or doping depth) m

Ve electrical voltage V

Vf shear force N

Vp pull-in voltage V

w cantilever width m

wp PzR device width m

⟨KEn⟩ average kinetic energy per resonant mode n J

⟨an⟩ average noise driven amplitude m

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Chapter 1

Introduction

Touch. The stretched ﬁnger with the impression and feeling of touch, symbolised the injection of life, spirit and wisdom from the creator to the created (The Creation of Adam, by Michelangelo Buonarroti, Sistine Chapel).

1.1 Study of sensor scaling

In life, we interact with the environment by receiving information and respond to it. We see, hear, smell, taste, feel and learn the world because we have senses to perceive;

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but sometimes our senses are not enough because we want to sense beyond our sensible range, and therefore we build sensors to sense more. Besides our need to perceive more, we also build sensors into machinery for precise control and automation. This propels to makings of even more sophisticated instrumentation. As result, sensor improves our lives, and further facilitates advance of other technologies.

For past few decades, microelectromechanical systems (MEMS) has emerged as an important player in sensor technology and commercialised devices [1–5]. In more recent years however, some of researches in MEMS sensor technology have gradually shifted to nanoelectromechanical systems (NEMS) [6–10]. This scaling from micrometre devices down to nanometre, is mainly fuelled by the need for better sensor performances such as higher sensitivity, faster response, better integration and lower energy consumption [8].

The transition from MEMS to NEMS however introduces many challenges as well, par-ticularly because many size-dependent effects and properties are most sensitive in those dimensions. As result, the scaling of sensors may not always be advantageous. The effects can influence, for example: quality factors, effective Young’s Modulus and noise. Such influences can be caused by surface contamination, mass-stiffness decoupling and sensor readouts to name a few. These size-dependent effects are of great interests in the NEMS community, and the main purpose of this work is to try to identify the dominating size-effects that influence sensor applications.

In order to carry out the study of scaling from MEMS to NEMS, we focus particularly on the scaling of microcantilevers to nanocantilevers. This is mainly because they are the basic mechanical building blocks in both MEMS and NEMS, which facilitate a more fundamental analysis of the scaling eﬀects. Furthermore, cantilever is an important transducer and has many well established sensor applications. Typical applications of micro and nano cantilevers are:

• Force sensing, such as in AFM [11] and in MRI [12].

• Mass sensing, such as in general mass and gas spectrometry [13,14].

• Stress sensing, such as in label-free biochemical detections [15].

• Flow sensing [16].

• Temperature sensing [17].

• Material research, such as Young’s modulus size-dependency [18,19].

Due to these reasons, cantilever makes a suitable representation and studying subject for sensor scaling.

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1.2 Deﬁning sensors and transducers

Sensors are devices that respond to physical stimuli and transform them into signals that can be readily processed and interpreted. The transformation, also known as transduction, typically involves changes in properties and energies between diﬀerent physical domains. It is the key concept in sensor systems: converting properties that are not easily detected or measured, into other properties that can be distinctively quantiﬁed and analysed.

In the fields of instrumentation, transduction are defined more specifically by trans-ferring signals between energies from radian, mechanical, thermal, electrical, magnetic and chemical domains [20]. The concept of signal domain transduction can be better illustrated in Table1.1, where the input signal from one domain is output to another via a corresponding transduction effect.

Table 1.1: Examples of transduction of the six signal domains, based on [21]. Some transduction may have more than one eﬀects but not listed.

Input Output

Radiation Mechanical Thermal Electrical Magnetic Chemical Radiation ﬂuorescence radiation

pressure radiation heating photo-voltaic effect photo-magnetic effect photo-chemical effect Mechanical photoelastic effect conservation of momen-tum friction piezoelectric, piezoresis-tive Villari ef-fect mechano-chemical effect Thermal incande-scence themal ex-pansion thermal conduc-tion Seebeck, pyro-electric effect Currie Weiss effect endothermic reactions Electrical electro- lumine-scence piezoelectric effect Peltier ef-fect pn junc-tion effect Ampere’s law electrolysis Magnetic Faraday effect magneto-striction Etting Hausen effect

Hall eﬀect magnetic induction Chemical chemo- lumine-scence explosive reaction exothermic reaction Volta eﬀect chemical reaction

In principle, transducers can be operated to sense or to actuate. Sensors on the other hand are one-way transducers and typically conveys a relation to human senses. Throughout the course of this thesis, transducers will be used to express a device that simply converts signals from one form to another; while sensors will be used to describe a device that can sense for human needs and output measurants in electrical domain which can be easily processed and interpreted.

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1.3 Deﬁning scaling and scalability

The action to change the size of an object by enlargement or shrinkage at fixed ratio, is to scale. Size matters because the physics at large scales is different to that of small scales. The only force that is strong enough to keep the planets orbiting the sun is the gravitational force, but at smaller dimensions the dominating force that keeps electrons around the nucleus becomes the electromagnetic force. The scaling of MEMS to NEMS happens at a critical dimension where the surface-dependent effects starts to dominate due to the increasing surface-volume ratio. At this transition region, competition between different effects and forces becomes complex, and is what we would like to gain better knowledge about.

Forces typically undergo what is called the “scaling law” as they act upon a scaled device. The concept of scaling law can be best represented by the Trimmer’s notation [22], where an indexed parameter Sx _{is used as scaling factor, and a vertical bracket} system is applied to compare scaling of diﬀerent forces. In our case, the interests are not in comparing forces, but in sensor performances such as sensitivity, response time and dynamic range etc. Nevertheless we can still use Trimmer’s scaling factor S notation to derive and study the scaling eﬀect of cantilever sensors (See appendixA for more explanation on Trimmer’s notations).

When discussing scaling, it is important to note that some scaling are fundamental, while others are “quasi-fundamental” [23]. The diﬀerence is that in “quasi-fundamental” scaling, the determination of the factors are dependent on extra conditions and assump-tions. For example volume scales directly to the third power of the length (S3_{) while}

mass scales to S3_{only if the mass-density is constant. Scaling of the mass is therefore}

a quasi-fundamental scaling. Table 1.2 shows a few more examples of scaling factor expressions, and the diﬀerences between fundamental and quasi-fundamental scalings. Many of the scaling factors that will be discussed are quasi-fundamental, therefore sensing conditions and assumptions will play important roles in our later analysis.

From the Trimmer’s notation, how well a parameter scales with size can be easily deter-mined by looking at the index of its scaling factor. The “scalability”1_{of a parameter is} determined by whether its eﬀect is dominant or signiﬁcant at the dimension of interest (otherwise considered unscalable when it becomes inconsequential or negligible). Sim-ilarly, it is also possible to determine the scalability of a transduction method, and call

1_{scalability is an inﬂected form of the verb scale, which means to weigh in a scale or to measure.}

Strictly speaking scalability is deﬁned as: capable of being weighted or measured, but it is often used in telecommunication and computer science literature to indicate the ability of a network system to expand or shrink. Here we deﬁne it as the action to enlarge or shrink a device’s physical size.

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it “scalable” if its output signal is measurable and its system can be realised. Therefore we would for example, refer to a force cantilever sensor “scalable,” if it can transduce force into a measurable signal with some increased performances, and can be further realised and applied.

Table 1.2: Examples of fundamental and quasi-fundamental scaling with Trimmer’s notation. Adapted from [23].

Parameter Scaling factor Notes and assumptions

Length S1 _Fundamental

Area S2 Fundamental

Volume S3 _Fundamental

Surface-volume ratio S1 _Fundamental

Mass S3 _{Size-independent density}

Max inertial stress S−1 For ﬁxed acceleration spring-mass sys-tems

Resonant frequency S−1 Size-independent modulus and density Electric resistance S−1 Size-independent resistivity coeﬃcient Electrostatic force S2 _{Size-independent permittivity and}

di-electric

Electromagnetic force S4 _{Size-independent current density}

Magnetic force S3 _{Size-independent magnetic strength}

Piezoelectric force S2 Size-independent piezoelectric coeﬃ-cient

Thermal dissipation S2 _{Size-independent hear transfer}

coeﬃ-cient Characteristic

diﬀusion time S

1/2 _{Size-independent diﬀusion coeﬃcient}

Hydraulic resistance S−3 Size-independent dynamic viscosity

1.4 Establishing the baseline: MEMS and NEMS

sensors

MEMS and NEMS are engineering of mechanical structures coupled to electronics in micrometre and nanometre scales. Since many of our physical measurants/actuation are mechanical and signal processing are electrical, this mechanical-electrical coupling becomes important for many sensor and actuator applications [24].

The development of MEMS owes much to the rise of integrated circuit (IC) technology and micro processes in the 1960s. The maturity of the IC technology inspired people

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to fabricate moving mechanical structures on silicon-compatible technologies2_{, which} enabled the integration of mechanical devices with electrical circuits. As result, MEMS was created and lead to realisation of smart, multi-functional mechanical devices on chips [1]. The invention of MEMS set the foundation for a whole new generations of mechanical sensors and actuators such as: accelerometers [25,26], gyroscopes [27], pressure sensors [28], micromirrors [29], ﬁlters [3, 30], biosensors [4, 5], micro-pumps [31] and micro-motors [32].

The rise of NEMS in comparison, is still at its early research phase. The dimensions of NEM devices are sometimes loosely deﬁned and overlap with that of MEMS. In this dissertation, a device is considered to be NEMS if its critical, or characteristic, dimension is within 100 nm. Therefore a cantilever with dimensions of 60 µm×1 µm× 0.1 µm (length × width × thickness) and a readout transducer, could be seen as a NEMS device.

Although NEMS is considered as the scaled-down version of MEMS, it is not a mere duplication of MEMS. Not all MEMS materials and processes can be scaled into NEMS, and also not all NEMS technologies are applicable on MEMS. The innovation of MEMS is really about miniaturisation and batch production technologies, to make minute me-chanical structures that can integrate with signal processors on chips for diﬀerent ap-plications [33]. NEMS on the other hand, is not to shrink micromirrors into nanometre size, but to exhibit the extra functions and diﬀerent characteristics that are related to its small size [34]. Among NEMS devices, simple beam structures made from carbon nanotubes [35], crystalline nanowires [36] and beams [9], are particularly interesting for state of the art researches including electrometry [37] and other quantum applications [10, 38]. Shown in Figure 1.1 are some comparison of the complex MEMS, and the simple but much smaller NEMS structures.

1.5 Small size, major challenge.

The challenge comes in many aspects when scaling MEMS to NEMS. The main chal-lenges aspects are: (1) fabrication of the device, (2) characterisation of the device and (3) readout of the device.

2_{Today most of MEMS devices are silicon based, but the use of III-V materials, metals and}

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a b c

d e f

5 µm

Figure 1.1: Examples of MEMS and NEMS, from top-left to bottom-right are: MEMS SiC rotation device [39], MEMS pull-in accelerometer [40], MEMS oscillator [41], carbon nanotubes [42], selft-assembly Si nanowires [36] and single crystal Si

double-clamped nanobeams [9].

1.5.1 Fabrication of the device

The realisation of NEMS can be achieved by the top-down and bottom-up approaches (More detail in section2.1). In the top-down approach, the increase of the resolution, accuracy and precision in the deposition, lithography and etching processes should all be highlighted. For example it has been found out that the initial stabilisation time of a typical ﬁne-tuned deposition process in MEMS [43], could have already caused signiﬁcant thickness error and material characteristics variations for nanocantilever applications [44]. Hence processes that were considered precise and reliable in micro fabrications would have to be re-characterised and optimised.

In the bottom-up approach, controllable self-synthesis and its patterning is crucial for reproducibility of the anticipated mass fabrication [45]. For NEMS structures in particular, the bottom-up approach will have to be able to integrate with the existing top-down electronic processes to realise a full functioning system. This integration challenge will prove to be crucial, as on-chip electronics will be necessary for achieving low-noise readout of the NEMS that typically has very small output response signals.

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1.5.2 Characteristics of the device

Another main challenge in scaling is to understand the origin and the influence of size-effects. For example the explanations to the drop of Q-factors in nanocantilevers [7, 46,47] and the decrease of the effective Young’s modulus of silicon [18, 19,48–50] are still debatable. These effects can influence the design and operation of the sensor device by changing its sensitivity and stiffness. Similar size-dependencies can also be found in other aspects such as noise, dynamic range and non-linearity of cantilever sensor.

Apart from the device characteristics, there are also practical aspects about scaling. Due to the scaling, the total area available for functionalisation decreases. It then affects the sensitivity and response time of the electromechanical sensors. Also, as the sensor structure become smaller, the decoupling of mass and stiffness and the influence of surface contamination become important. Some of these effects used to be negligible in devices such as microcantilevers, but new measurement considerations and operation now need to be applied on NEMS sensing.

1.5.3 Readout of the device

Finally, for a mechanical sensor such as cantilever, another scaling challenge lies at the additional mechanical-electrical transduction that is needed to readout the cantilever responses. When we scale down a cantilever, the decrease in its response time and out-put signals typically mean that a faster and more sensitive transducer is also needed, in order to turn the mechanical responses into electrical signals. Therefore the scala-bility of mechanical-electrical transducers becomes an important issue. These readout transducers that were used in microcantilever will also need to be characterised to ﬁt applications of nanocantilever.

There are also many aspects involved in the discussion of scaling readout transduc-ers. Firstly, one needs to know if the transducer is fundamentally capable of sensing a nanocantilever, and what would be the scaling law of its performances such as sensi-tivity and response time. Another aspect is to know if the transducer is applicable to sense a nanocantilever, for example considering the disturbance of the readout action on the sensing device. Finally, in some cases where the readout transducer physically scales with the cantilever, such as in piezoresistive sensing, scaling eﬀects and material changes also applies on it. It is then important to also study the scaling of such readout transducers in parallel to the cantilever scaling.

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1.6 Study of sensor scaling: this thesis

Early work on scaling related discussions have shown its broadness in multiple study ﬁelds. Work such as Trimmer’s invention of scaling factors [22], Feynman’s speech on miniaturisation [51], Thornell’s examples of everyday-life scaling [52], Spearing’s observations on MEMS scaling [23], Roukes’ introduction to NEMS [8], Bell’s summary in sensor selections [33] and Agrawal’s study in multiscale experiments [53], to name a few, all agreed on its importance to future technologies. Despite the consensus however, the discussions on scaling does not specify the scaling factor of important sensing parameters and their application consequences. It is therefore the interest of this thesis to investigate the scaling eﬀects of the sensor parameters.

In this thesis, particular interest is focused on cantilever sensors. The results concluded from studying cantilever scaling can then be generalised into scaling of other mechanical structures, and finally into MEMS and NEMS. Unlike previous studies though, this thesis looks in-depth at the fundamental limitations of both physical properties and practical applications of a device as a sensor, and shows both the benefits and pitfalls of scaling. The thesis addresses and summarises important effects of cantilever scaling, which will help assessing the feasibility, and provide guidelines to the design of future miniaturised electromechanical sensors. It helps answering the questions: can we gain benefits from scaling a electromechanical sensor device, and what are the consequences?

While the 1st scaling challenge, the device fabrication, is eminent and immediate, challenges in the characterisation and readout of the device are less obvious. These two challenges will require deeper investigations into the sensor properties, as well as good understanding of the application. They will also be the core of this work and form the backbone of this thesis. The structure of the thesis and the purposes of each chapter are outlined as the following:

Chapter 2: Fabrication, mechanics and readout/actuation of cantilevers This chapter aims at giving detailed background information necessary for later dis-cussions. It includes fabrication of silicon cantilever and their basic mechanics to help understand their sensing response and scaling. This chapter also presents a general introduction to typical readout and actuation methods for cantilever sensors.

Chapter 3: Scaling of silicon cantilever characteristics This chapter focuses on the 2nd challenge of scaling, the characterisation. The challenge is further di-vided into two parts: the practical aspect and the physical aspect. The practical

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aspect concerns the actual usage of the sensor, investigating the upcoming prob-lems in measurements as the result of its size decrease. It includes derivations of sensitivity-area relation, experiments on mass-stiﬀness decoupling and analysis on surface contamination and stabilisation. The physical aspect on the other hand, concerns the material and intrinsic properties of the device. It includes original work on Young’s modulus size dependency, as well as author’s interpre-tations and expansions based on prior-arts in dissipation, noise and non-linearity. The chapter in essence answers the question: How does size aﬀect the properties and the usages of this sensor, and what are the limitations?

Chapter 4: Scalability of readouts This chapter focuses on the 3rd challenge of scaling, the readout scaling. Although the readout systems themselves don’t necessarily scale in size, their transduction capability does in accordance to the size of the cantilever. This chapter will start by assessments of some popular readout techniques for their applicability on detecting down-scaling cantilevers, and compared in a table. Amongst the detection methods, laser deﬂection, hard contact, ﬁeld emission and piezoresistive detections are selected for elaborated investigations in each sections.

Chapter 5: Conclusions and future works This chapter will state the future trend of scaling sensors, looking at what still have to be investigated, and what are actu-ally worth to be investigated. This chapter also summarises the thesis, ﬁnalising the discussions and concludes the essential question: why do we scale?

The main purpose of this thesis is to gather current state-of-art developments and present the results in the form of a multi-subtopic review. The scientific significance for such kind of review study, lies in the fact that considerations in many researches over scaling effects are often insufficient. Researches engaged in cantilever sensors are typically directed to limited scaling aspects of the device, such as sensitivity, frequency response and detection, but neglected other effects’ roles in the process. For example, it is common to decrease the size of a microcantilever to increase its mass sensitivity and use it as a mass detector. Such scaling advantage propelled numerous researches competing for the smallest cantilever and the ultimate mass sensing resolution. How-ever it is only in recent years that the scaling effect of stiffness is also addressed to be influential to mass detection, and challenged the validity of some of the previous mass sensing results. An overview of scaling effects can then help researchers under-stand the benefits and pitfalls of scaling their device, and prevent them from neglecting important aspect as the result of scaling.

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This thesis is written with the intention of presenting a complete story. Each subtopic is introduced with explanation of its effect origin and latest findings. The subtopic is then extended to the context and analysis of scaling. All analyses related to scaling are origin work. In some subtopics however, the basic analysis is based on prior arts that focused on specific discussions other than scaling. The results from these prior arts, are re-interpreted or modified to explain the scaling effects. Such is in the cases of discussions on Q-factor (Section3.3.1), noise (Section3.3.3), nonlinearity (Section

3.3.4), detection comparisons (Section4.2) and PzR (Section4.6). In other subtopics,

analyses and experiment results are from original work intended for scaling discussions. Such is in the cases of sensitivity (Section3.2.1), decoupling mass and stiﬀness (Section

3.2.2), surface contamination (Section 3.2.3), Young’s modulus (Section 3.3.2), laser

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Chapter 2

Fabrication, mechanics and

readout/actuation of

cantilevers

2.1 Fabrications of micro and nanocantilevers

Structures in nanometre size can be approached mainly in two ways, the top-down and the bottom-up approaches (see Figure2.1). In the top-down approach, structures typically start big, their sizes are gradually reduced by removing materials until ﬁnally reaching the desired shape and size. Such an approach takes the advantage of using external-controlled tools to shape structures from big to small, and from rough to detail. Bottom-up approach by contrast, relies on small building blocks, such as atoms and molecules, to aggregate, order and form the ﬁnal and bigger device. One of the best examples of such approach is the self-assembly of crystals [54] and the growth of biological species [55]. The combination of the top-down and bottom-up approaches together formed the nanotechnology fabrication of what is known today.

For micro and nanocantilevers, due to the necessity of coupling with electronics, the structures are usually attached onto a top-down patterned semiconductor substrate for handling or for electronic readout reasons. The cantilever itself however need not be top-down fabricated, it can be bottom-up grown on the substrate directly, or pre-synthesised and transferred onto substrate [56]. Such is the case for many nanowires

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b. a.

Figure 2.1: Two approaches to realise nanostructures. a). the top-down approach, which rely on removing materials, and b). the bottom-up approach, which is based

on adding materials.

and CNT [42, 57]. Nevertheless most of the semiconductor nanocantilevers till now, have been fabricated directly on the substrate using top-down technologies inherited from the microcantilever fabrication [9,58], simply because the process is mature and more controllable.

The top-down fabrication of micro and nanocantilevers mainly rely on two processes: (1) the bulk micromachining [59], which is based on selective etchings of the bulk silicon substrate to shape out the mechanical part, and (2) the surface micromachining [2], which is characterised by deposition, etch and release of thin-ﬁlm layers on the surface of substrates to form the mechanical parts. While microcantilever fabrications have been successfully developed in both processes, nanocantilevers are mostly made by surface micromachining, mainly because of its higher accuracy and wider material selections.

In this work, single crystal silicon cantilevers are fabricated using surface microma-chining process based on the Smartcut ⃝R SOI (Silicon on insulator) wafers. These

wafers have a silicon substrate, a buried thermal oxide sacriﬁcial layer and a top silicon device layer, as shown in Figure 2.2a. The top-silicon layer is further grown using epitaxy or thinned-down by oxidation and stripping processes until the desired thick-ness is obtained. Then the top silicon layer is patterned by lithography masking and subsequently etched in plasma etcher until the buried oxide layer. The buried oxide is then isotropically etched using HF solution. Finally the releasing of the cantilever is completed by cleaning and drying of the HF solution. Due to stiction eﬀects (Figure

2.3) from water surface tensions however, techniques such as the solid-gas sublimation of the freeze-drying [60], and liquid-gas transition of CPD (critical point drying) [61],

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Photoresist (Mask layer) Silicon (Mechanical layer)

Silicon oxide (Sacrificial layer) Silicon (Substrate)

a

Starting with a sacrificial layer and a mechanical layer on the silicon substrate.

b

deposite a layer of photoresist and use lithographic techniques to pattern the resist layer

c

Anisotropic etching to transfer patterns onto the mechanical layer. Using the resist layer as a protective mask. Over-etching is sometimes needed.

d

Striping the resist layer

e

Isotropic etching of the sacrificial layer using high selectivity process and releasing the mechanical structure.

Figure 2.2: Typical process steps of a surface micromachining cantilever process. Starts with a). formation of a sacrificial layer and mechanical layer, b). deposit and pattern photoresist using lithography techniques, c). etching of the device on the mechanical layer using the photoresist as masking layer, d). stripping the photoresist layer and finally e). sacrificial etching the under layer and releasing of the structure.

are used during the drying step to assist on the full release of the cantilevers. Examples of fully released silicon cantilevers used in this work are shown in Figure2.4.

2.2 Mechanics of cantilever and its scaling

In order to understand how cantilevers are used as transducers in sensor applications, it is important to look at the fundamental mechanics of a cantilever. The mechanics in discussion is part of the classic mechanical theories which treats solids as continuous

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a b

Figure 2.3: a). Stiction during fabrication, residual ﬂuid capillary force. b). Operation stiction. the cantilevers are originally released, the electrostatic pull-in operation initiated the surface contact, in which the minuscus capillary and the van der Waal forces overcome the mechanical restoration, holding the cantilever in its

adhesion position.

a. b.

Figure 2.4: a). Single crystal silicon cantilevers of 340 nm thickness, and b). single crystal silicon cantilevers of 57 nm thickness.

materials and not as an assembly of discrete atoms. Fortunately it has been shown that this continuum model only fails at cantilever size of a few nanometres, which corresponds to a few tens of atoms [62]. Therefore the continuum model is still valid over a wide range of nanocantilevers and used to approximate our cantilevers.

Cantilevers typically interact with the environment in two ways, by changing its ge-ometric shape (usually bending) and by shifting its resonance frequency when in vi-bration (see Figure2.5). Therefore it is possible to operate a cantilever transducer in two ways, the quasi-static bending approach and the dynamic resonance approach. We focus on the ﬁrst bending and the ﬁrst resonance mode in both approaches.

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External disturbance External disturbance

a b

Figure 2.5: a). Direct bending response. The amplitude of the bending is related to the strength of the disturbance. b). Resonant response. The resonance frequency shifts according to the strength of the disturbance. The disturbance can be forces,

additional stress, mass, temperature, damping etc.

2.2.1 Quasi-static mode: Cantilever bending

When a hyperelastic material such as silicon is under force in its linear regime, it displays an elastic property that can be described as:

σ = Eϵ (2.1)

where σ is the stress, deﬁned as σ = F/A = F orce/Area, measured as N/m2. ϵ is the strain, has no unit and is deﬁned as ∆l/l where l is the dimension of the structure in which stress is applied upon (In the case of a cantilever, it is usually along its length). Finally, E is the modulus of elasticity.

dx

P

dθ

υ

ds

M₁ M₂ x r s s s L’ dx y dA y M s s dx L’ M y r Cross section Bending section dx y x

Figure 2.6: Deﬂection of a cantilever with a point force P at the free-end, assuming small bending under linear elastic regime.

When a cantilever is under bending, as illustrated in Figure2.6, we deﬁne the curvature of the cantilever κ to be the reciprocal of the bending radius r. Knowing also that

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ds = rdθ, and that ds≈ dx for small dθ, we obtain: κ = 1 r = dθ ds ≈ dθ dx (2.2)

If we zoom into the bending section dx, as shown in Figure2.6, it is possible to deter-mine a plane ss with a length dx before bending and stays unchanged after bending. This plane is called the neutral surface of the beam. Considering the change of length before and after the bending, we can obtain this (assuming negligible shear deforma-tion):

L′ = (r− y)dθ = dx −y

rdx L′− dx = −y

rdx

therefore, strain and stress on the x-axis should be:

ϵx=−

y

r =−κy and σx= Eϵx=−Eκy (2.3)

For each small cross-section of the beam dA at y distance away from the neutral surface, the bending moment is then:

M = ∫ σxydA =−κE ∫ y2dA =−κEI κ =−M EI (2.4)

where I is deﬁned as the integral of y2 _{and is referred to as the moment of inertia.}

Now, for small deformation and small θ, ds≈ dx and θ ≈ tanθ = dυ/dx. Therefore together with Eq. 2.2and Eq. 2.4

κ = dθ ds = dθ dx = d2υ dx2 =− M EI (2.5)

and from the relationship of shear force and bending moment (see AppendixB), we obtain the derivative expression of a cantilever deﬂection:

d3_υ dx3 =− V EI and d4_υ dx4 =− q EI (2.6)

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Finally now for a section x distance away from the ﬁx end, and with a point force P at the free end, consider Eq. 2.5(with sign conversion applied on the moment M ):

d2_υ dx2 =− M EI = P (L− x) EI = P L EI − P x EI dυ dx = P L EIx− 1 2 P x2 EI + C1 υ = 1 2 P L EIx 2₋1 6 P x3 EI + C1x + C2

boundary condition υ′(0) = 0 and υ(0) = 0 results in C1 = C2 = 0, and the total

deﬂection at the free end x = L is expressed as:

υ(L) = 1 2 P L3 EI − 1 6 P L3 EI = P L3 3EI (2.7)

Which 3EI/L3_{is the stiﬀness of the cantilever. If now we want to know the maximum}

stress applied on the cantilever, consider combining Eq. 2.3and Eq. 2.5, we obtain:

M I = σx y σx= M y I (2.8)

For a symmetrical cross section, the maximum compressive and tensile stress are situ-ated at the furthest point away from the neutral axis ymax and we deﬁne a so called section moduli N to be:

N = I ymax

σx=

M

N (2.9)

And therefore considering Figure2.6 and Eq. 2.9we have:

σx(0) = M N = P (L− x) N = P L N (2.10)

For a cantilever of rectangular cross section with width w and thickness t, the moment of inertial and section modulus are known to be:

I =wt

3

12 N =

wt2

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Replacing Eq. 2.11 into Eq. 2.7 and Eq. 2.10 and express in Trimmer’s scaling notation, then: υ(L) = 4 wt3 P L3 E ∝ S −1 _(2.12) σx(0) = 6 wt2P L ∝ S −2 _(2.13)

From Eq. 2.12and Eq. 2.13, it is obvious that under same point force, scaling down cantilevers generally result in more deﬂection on the tip end and more stress on the ﬁx end. It is therefore advantageous to scale a cantilever down to obtain higher force and stress sensitivities.

2.2.2 Dynamic mode: Cantilever resonance

In addition to static deflections, a cantilever can also change its resonance frequency due to changes in the environment. To grasp the concept of a resonant cantilever, we look at a simplified “free-vibration” model where we consider the cantilever system as a spring-mass system with an effective mass mef f, displacement x and stiffness k:

mef f d2x dt2 =−kx d2x dt2 + k mef f x = 0 (2.14)

The effective mass is used instead of total mass m because unlike a lumped spring-mass model, not all the spring-mass on the cantilever is under the same vibration and for a rectangular cross section cantilever, mef f is ∼ 0.24m (see Appendix B for more detailed derivation). This differential equation is a typical harmonic oscillation system without damping and external forces. Considering the stiffness found in Eq. 2.7 and

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the moment of inertial in Eq. 2.11, the angular frequency ω of the system becomes: ωi= √ k mef f (2.15) ωi= Ci √ 3EI mL3 = Ci √ 3E (ρwtL)L3 √ wt3 12 = Ci t 2L2 √ E ρ (2.16)

Where Ci is a constant that considered the eﬀective mass of the ith bending model vibration, and ρ is the density. The scaling eﬀect of resonance frequency can then be represented by Trimmer’s notation as:

ω = Ci t L2 √ E ρ ∝ S −1 _(2.17)

hence the resonance frequency increases linearly as both t and L decrease. The increase of resonance frequency is very beneﬁcial for sensing. For example, from Eq. 2.15we know that the resonance is sensitive to changes in mass and stiﬀness shown as:

ω′− ω =1 2ω( k′− k k − m′− m m )

which can also be expressed as:

∆ω(∆m=0)= 1 2 ∆k k ω0 ∝ S −2 _(2.18) ∆ω(∆k=0)=− 1 2 ∆m m ω0 ∝ S −4 _(2.19) (2.20)

Where ∆m = m′− m and ∆k = k′− k. Here we see that scaling increase resonance sensitivity to changes in mass and in stiﬀness.

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2.3 Isometric and anisometric cantilever scaling

When a cantilever decreases its dimension equally in all direction and retaining its shape (geometric ratios), it is referred as isometric scaling. In some cases however, scaling is used to express the change in only one dimension. Such can be for exam-ple, varying a cantilever length while keeping its width and thickness constant. It is important to specify anisometric scaling apart from isometric, because changes in the shape of cantilever may result to changes in its analysis and response that is unrelated to size-eﬀects. Three kinds of cantilever anisometric scaling and their consequences are compared in the following.

Scaling in length As the cantilever becomes shorter, its stiﬀness scales by S3 _and

its resonance frequency scales by S2 (Eq. 2.7 and Eq. 2.16). This means although the cantilever becomes more responsive and sensitive due to frequency increase, its stiffness makes deflections extremely difficult. Furthermore, as the length-thickness ratio decreases to less than 20:1, the shear deformation of the beam becomes more pronounced where more complex models such as the Timoshenko model (instead of the Euler-Bernoulli model) should be used to describe the beam bending. Table

2.1 shows the calculated 1st mode resonance frequency of cantilevers with diﬀerent length-thickness ratio, using ﬁnite element method (FEM) and Euler-Bernoulli (EB) theory. Comparison of the two methods shows large variation for cantilevers with low length-thickness aspect ratio. The calculation of EB theory is based on Eq. 2.16with

Ci= 0.1613, E = 169 GPa and ρ = 2330 kg/m3. Computation of the FEM is carried out in COMSOL’s eigenfrequency analysis1_{, assuming silicon as an isotropic material} with E = 169 GPa. The isotropic assumption is to simplify the coordination of the simulations and the computed diﬀerence to an anisotropic settings is small [63].

Table 2.1: Comparison of first mode resonant frequency of different anisometric scaling cantilever, calculated via finite element method (FEM) simulations and by

Euler-Bernoulli (EB) beam theory

l× w × t(µm) ratio (l:thk) FEM (kHz) EB (kHz) (FEM-EB)/EB

100× 10 × 1 100:1 139.5 137.4 1.52 %

100× 10 × 2 50:1 278.4 274.8 1.27 %

100× 10 × 5 20:1 699.8 687.1 1.82 %

20× 10 × 5 4:1 16465.3 17176.8 -4.32 %

5× 10 × 5 1:1 175710.2 274829.9 -56.41 %

Besides the length-thickness ratio, the length-width ratio also plays an important role in the resonance characteristics. As the ratio approaches unity, torsional deformation

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becomes more likely. Shown in Figure 2.7a is the FEM simulation of a cantilever under length scaling. Transverse bending is preferred for the 1st and 2nd modes on all cantilevers, but for shorter cantilevers torsional resonance replaces bending at the 3rd mode. Hence if one were to study size effect based on the 3rd resonant mode, he would have to switch his analysis model from transverse to lateral mode if the cantilever scaling were an anisometric length scaling. Change of the resonant modes means direct comparison of these cantilevers becomes difficult, because many size ef-fects are also mode-dependent. Such mode change would not have happened if the cantilever scaling were isometric. The simulations in Figure2.7 are computed using COMSOL’s eigenfrequency analysis module, which normalises the deformation in or-der to facilitates easy comparisons on the mode-shapes of different resonance modes, but does not simulate the absolute displacement values. The greyscale indicates where the maximum and minimum bending occur on the cantilever.

Scaling in width As the cantilever width becomes narrower, both stiffness and mass decrease linearly. Therefore its resonance frequency (to the first order approximation) is not affected. This has an advantage of having a softer cantilever without sacrificing the frequency. However when width-thickness ratio becomes 1, the cantilever acts more like a beam and the resonance in both lateral direction and torsional are equally preferable to the transverse. Further decrease in the width will finally result to a lateral cantilever (See Figure2.7b). It is also important to point out that, for a wide beam (defined as w≥ 5t), the Young’s modulus should be corrected to plate modulus

E/(1− ν2), due to the plane-strain condition (ν is the Poisson’s ratio) [64]. Taking into account that the Poisson’s ratio of silicon is about 0.27 [65], the resulting modulus with and without the plane-strain correction is about 8% in diﬀerence, and leads to roughly 2.8% error in frequency calculations.

Scaling in thickness Finally as the cantilever becomes thinner, the stiﬀness drops by S3_{and the frequency by S. Ultrathin cantilevers have been successfully developed for}

attonewton force sensing [66], but ultimately the anisometric scaling will result in a structure too soft to operate, diﬃcult to fabricate and behave like a plate rather than a beam. Finally, similar to other anisometric scaling, the resonance characteristic changes at higher modes as shown in Figure 2.7c; while lateral bending is preferred at thicker cantilevers, the ultra-thin cantilever changed to transverse bending at the 2nd mode resonance.

Note on the surface-to-volume ratio In isometric scaling, the surface-to-volume ratio of a cantilever roughly follows the expected S2-to-S3 relation, and the ratio is dependent on the scaling factor S−1. However for an anisometric scaling, the ratio is

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1st mode 1st mode 2nd mode 2nd mode t=2 t=1 l=100 w=10 t=5 1st mode 1st mode 2nd mode 2nd mode l=100 w=10 t=5 w=5 w=1 1st mode 3rd mode 1st mode 3rd mode l=20 l=5 l=100 w=10 t=5 a. Scaling in length l b. Scaling in width w c. Scaling in thickness t no deform max deform

Figure 2.7: a). Scaling in length, note that resonance mode turns to torsional for shorter cantilevers on the higher mode. b). Scaling in width, note the transition of a normal cantilever to a square-section beam and ﬁnally to a lateral cantilever. c). Scaling in thickness. Note the change in the bending from lateral to vertical at the

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highly dependent on the shape of the structure. Figure2.8 illustrates the relation of the surface-to-volume ratio and the change in the dimensions of a cantilever. Take a typical cantilever of 100× 10 × 1 µm (length × width × thickness), and calculates its surface-to-volume ratio as one of its dimension decreases 90%, 80%, 70% and so forth. From the Figure we see that scaling a cantilever down isometrically will drastically in-crease its surface-to-volume ratio. However, while scaling its thickness anisometrically gives a similar eﬀect, decreasing its width or length result to very limited inﬂuences on the surface-to-volume ratio, even when the dimensions are decreased by more than 50%.

Percentage of the original size (%)

40 50 60 70 80 90 100 Su rface-to-volume ratio ( µ m -1) 2.0 2.5 3. 0 3. 5 4. 0 4. 5 5. 0 5. 5 6. 0 Anisometric scaling of length Anisometric scaling of width Anisometric scaling of thickness Isometric scaling Cantilever of 100x10x1um (length x width x thickness)

Figure 2.8: The surface-to-volume ratio of anisotropic scaling is dependent on the geometrical aspect ratio changes. The x-axis represents the scaling proportion of the length, width and thickness of a cantilever. The thickness change has the

most inﬂuential eﬀect on the ratio, and is most similar to isotropic scaling.

In short conclusion, when considering anisometric scaling, one has to account for the consequences of changing aspect ratio and structural shapes. In isometric scaling, the original long slender cantilever stays slender after scaling, however in anisometric scaling, the cantilever eventually become a slab, a plate, a block or a wire. In which case, the fundamental mechanics, modes and analysis methods change accordingly. For this reason, it is maybe more proper to treat anisometric scaling as “structural change” or “structure optimisation”, and not to be further discussed in this study of scaling eﬀects, unless otherwise stated.

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2.4 Detection of cantilevers

Cantilever is a mechanical transducer, it transforms external disturbances into bending and resonance shifting. In order to completely interpret the cantilever responses, a mechanism is needed to “read-out” and turn them into electrical signals. Therefore, the detection of cantilever sensor becomes an important issue in the application of the device. In the following subsections, cantilever detection techniques and their scaling eﬀects are introduced and discussed in detail.

As shown in Figure 2.9, the combination of a cantilever and an electromechanical transducer, now named as χ, forms a complete sensor system. For example in the temperature experiment developed by Sadeghian et al. [67], the thermal measurand is related to the mechanical stiffness which is measured by the silicon cantilever and transformed into electrical voltage. The cantilever relates temperature measurand into stiffness measurand, and the EPI detection relates the stiffness measurand into voltage measurand. Electric signals in either time or frequency domain Disturbance Bending of cantilever Resonant

Static Optical interferromtry

Optical reflection Piezoelectric Magnetomotif Magnetic Piezoresistive Hard contact Capacitive etc... stive act e environment input mechanical transducer mech-electro transducer signal output Temperature disturbance Temperature-s!ffness change

S!ffness - Pull-in voltage change

Example:

signal output Can!lever sensor

Figure 2.9: A typical cantilever sensor consist of 2 transducers, a cantilever trans-ducer that responds to the disturbance measurand, and a mechanical-electrical transducer, χ, that detects the cantilever response and outputs electrical signal for further processing. The second transducer has various options and most of them

are scale dependent.

The choice of χ can be many, as long as it measures the bending displacement or the bending stress of the cantilever, and transforms the measurands into electrical signals accordingly. A few common cantilever detection methods are listed in Table

2.2. The list provides a general overview of the popular methods and shows the wide possibilities available. It is however not exhaustive, as diﬀerent and new varieties of techniques also exist. For example in optical techniques, existing variations include

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lever deﬂection, interferometry [68], scattering [69] and other detections that monitor laser light’s intensity, phase, wavelength, position, frequency or polarisation changes [70]. Yet not all these optical methods are easily implemented. Hitherto, the lever deﬂection technique is the most successful and widely implemented optical technique for cantilever sensing.

In many cases, the χ transducer consists of extra devices such as an external laser source for the optical method, counter electrodes for the coulomb method and mag-netic ﬁeld generator for the magnetomotive method. Yet it is sometimes possible to physically combine and integrate the χ transducer into the cantilever itself. For ex-ample if we make a cantilever out of high piezoresistive material (such as doped Si), then the bending stress at its base can produce enough resistive changes for electrical measurement. Similarly, if the cantilever is made of, or covered, with piezoelectric ma-terial (such as ZnO), then a voltage diﬀerence is created and can be directly measured. In these two cases, the materials (Si and ZnO) act as transducer χ, but are physically combined with the cantilever and become a single integrated sensor system. This is also often referred as a self-sensing or self-detection sensor.

When choosing among different cantilever detections, sensor properties and applica-tion restricapplica-tions play crucial roles in the decision making. The sensor properties include internal sensor parameters such as resolution, precision and sensitivity. Application re-strictions on the other hand, are external conditions such as measurement environment (vacuum, air or liquid), power consumption, cost or device size (on-chip integration or room-size setup). Based on the priority of the restrictions, a specific detection can be determined. For example in AFM systems, a fast, easy and robust setup is needed. But most of all, the cost of the disposable cantilever and probes should be minimised, and allowed to be made from different materials. Therefore optical level method is chosen in most of the commercialised AFM’s, providing cheap, sensitive, robust measurement and relative easy setup, while sacrificing integration and power consumption which are less important in laboratory environments.

One particular restriction in which we are interested but seldom mentioned elsewhere, is the scaling restriction of the detection methods due to cantilever size. Namely we ask: is the detection method still valid and practical when we size down microcantilevers into nanometre? For example, the capacitive detection method relies on the surface and the gap of the electrodes. Assuming the gap is already small and cannot be further reduced due to short-circuiting, then as the cantilever scales from micron to nano, the total capacitance follows the surface area and reduces by a factor of S2_{, to the ﬁrst order}

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Table 2.2: Diﬀerent MEMS and NEMS cantilever detection techniques. Some of the techniques are developed and mainly used in double clamped beams, but are also applicable to cantilevers.

Techniques Description Ref.

Optical lever

Focus laser beam applied onto an optical reflective resonator, the laser light is then deflected onto a set of photodiodes. Cantilever motion is sensed by detecting the displacement of the deflected laser light. This method is one of the most popular method in microcantilevers, and widely used in AFM systems.

[71, 72]

Capacitive

Capacitive cantilever sensing involves a moving cantilever and a fixed counter-electrode. Cantilever motion is sensed by monitoring capacitance change due to displacement in the gap. This method is very popular in MEMS devices in the form of interdigitated structures. When used in cantilever sensing, good controls in thermal compensation and parasitic capacitance are needed.

[73]

Single- Electron-Transistors (SET)

Single electron transistor is a quantum effect device involving tunnelling and coulomb blockade on an isolated Coulomb island. The sensing technique is based on using SET as a high sensitive electrometer/amplifier, for monitoring the capacitance or voltage changes induced by a displacement transducer. This sense-and-amplify configuration increases SNR by direct amplification of the charge signals near the source, and with high sensitivity, high response. Typically the SET is linked to a cantilever with a capacitive readout, yet other readout transducers such as piezoelectric also exists.

[74– 77]

Piezoresisitive

Piezoresisitive sensing involves electric resistance change due deformation. The resistance change is a combinational effect of both material property (gauge factor) and geometric change, and therefore exists in all kinds of material. Cantilever motion is senses by monitoring the amount of deformation near the base/anchoring point. This is usually done by depositing an extra layer on one side of the cantilever.

[78– 81]

Piezoelectric

Piezoelectric sensing involves special material which produces electric charges when subjected to force deformation. Cantilever motion is sensed by monitoring the total charge accumulated along the deformed cantilever. This method requires the piezoeletric material to be sandwiched between electrodes, and this usually results to fabrication complications and multi-layered cantilevers. A particular advantage of integrating piezoelectric element on cantilever is that it can provide both sensing and actuation.

[82, 83]

Magnetomotive

Magnetomotive involves running an AC electric current path though the cantilever in a magnetic field. The electromagnetic field creates a Lorentz force that actuate the cantilever into motion. Yet in return, the displacement of the conducting current in the magnetic field induces an electric potential on the cantilever that can be measured and referred back to the motion. Similar to piezoelectric technique, magnetomotive can be used as a combined cantilever sensing and actuation.

[9]

Hard contact

The hard contact technique is based on the on-off switch mechanism, where a biased cantilever ‘turns-on’ the electrical path when it bends and touches the counter electrode, and ‘turns-off’ when it returns to the original position. The technique can generate large electrical signals comparing to most of other readouts, and therefore potentially simplifies the electronics design. Due to the on-off nature of the technique, it cannot be used for quasi-static bending measurements. However, it can produce ‘digitized’ output that enables dynamic monitoring of the cantilever by counting switching cycles. Since the technique requires contact, stiction prevention, current induced micro-welding, contact wear, shock and overall mechanical reliability is significant in its application.

[84]

Direct tunnelling

Direct tunnelling sensing is a solid-solid quantum tunnelling that involves a counter electrode at the proximity of the moving cantilever. Under slight electric potential difference, electrons will jump from one electrode to the other at small distance. The probability, or the rate, of jumping is dependent on the distance. Cantilever motion is therefore sensed by monitoring the tunnelling rate/current.

[85, 86]

Field emission

Field emission sensing is a solid-surface quantum tunnelling that involves a counter electrode, typically pointed shape, under high electric tension. The sharp electrode is used as a cathode that shoots electrons and the cantilever collects the electrons. The emission current is a function of the distance, therefore the cantilever motion is sensed by monitoring the current changes.

[87– 90]

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detection may become smaller than the noise and the parasitic capacitance from the readout electronic circuits, making the extraction of the response signal extremely diﬃcult and not practical. Although there are techniques to alleviate this problem, the capacitance detection under the assumptions is nonetheless considered not scalable or at least limited by practical issues. Therefore, not all methods are applicable to scaled cantilevers. A cantilever transducer has no practical value if its response cannot be read and transformed into valid and processable electrical signals. The scaling of the diﬀerent readout techniques will be covered in more detail in Chapter4.

2.5 Actuation of cantilevers

When a cantilever is used for sensing, it sometimes needs to be mechanically stimulated. The stimulation, or actuation, can be quasi-static or resonant. Either way, it is scale-dependent and falls into our discussion of scaling transducers.

Unlike cantilever detections which have diﬀerent varieties of techniques, actuation of cantilever is in principle limited to only a few, namely electrostatic, piezoelectric, ther-mal and magnetic. A short description of each is given in the following:

Electrostatic actuation

The electrostatic actuation is the reverse of the capacitive sensing. It is based on the Coulomb force induced by accumulating opposite electric charges on two conductive surfaces. The strength of the force can be roughly described as:

Fc = ϵe

lw

2g2Vbias

where ϵe is the permittivity and g is the gap between the cantilever and its actuation electrode. It is then clear that Fc∝ S2 when the gap is kept constant. Fortunately in most cases, the gap can be designed independently to the dimension of the nanocan-tilever, hence in reality the force scales less than S2_{, and furthermore on an already}

softened and highly sensitive nanocantilever, the eﬀect of the decreasing actuation force is not so crucial anymore. Henceforth, despite less favourable scaling for the electro-static capacitance sensing, its actuation is in fact very promising in NEM cantilever applications.

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One of the main advantages of the electrostatic actuation compared to other actuation methods, is its wide bandwidth. Electrostatic actuation can operate at static mode, as well as in the resonant mode up to very high frequencies, making it a very popular and widely applied actuation technique. Moreover, as the area of the cantilever decreases, the capacitance decreases and results in less capacitance time constant, and a possible higher frequency response. One of the major fall-back however, is that the actuation requires cantilevers to be adequately conductive. Silicon nanocantilevers would need to be doped or deposited with a layer of conductive material for a more eﬃcient actuation. This may slightly increase the diﬃculties in nanocantilever fabrication, as more control over the doping, deposition thickness and quality, as well as avoidance to pre-bending, is needed.

Piezoelectric actuation

Another popular actuation technique, is to use the piezoelectric materials as transduc-ers and transform electric voltage into mechanical displacement. The applied voltage is directly transformed into a related displacement which moves the cantilever. When an AC signal is applied at the right frequency, piezoelectric actuators can be used to resonate cantilevers. Yet as the resonant frequency increases, the piezoelectric device will eventually reach its cut-oﬀ frequency and stop its actuation. The cut-oﬀ frequency of piezoelectric devices is material and geometry dependent, but typically limited to MHz ranges.

The force and displacement of a piezoelectric actuator is linearly proportional to its di-mension, therefore for an integrated piezoelectric actuator, the scaling is not favourable. Finally, the integration of piezoelectric material in cantilever and CMOS devices has been done before [81], but at costs of complicated steps and special treatments. How-ever for applications that does not need individual cantilHow-ever control, the piezoelectric actuator can be simply put close and ’shake’ the cantilever. Therefore the scaling of the cantilever does not particularly aﬀect the integration of the actuation technique in some resonance applications.

Thermal actuation

Thermal actuation is essentially transformation of thermal energy into mechanical en-ergy. There are mainly two approaches for actuating cantilevers using thermal eﬀects:

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1). through thermal expansion on a bimorph cantilever, and 2) through thermal me-chanical noise.

In the first case, temperature change is created on the bimorph cantilever made of materials with different thermal expansions. The change in the temperature, which is usually created by integrated resistive heaters or external lasers, will then induce expansion mismatching within the cantilever and cause it to change shape. The force and displacement is proportional not only to the temperature, but also to the size, and therefore the actuation decreases when scaling down. In terms of response, smaller cantilever has less thermal capacitance and higher heat dissipation due to its large surface-to-volume ratio, hence the response of this actuation increases when scaled down. The thermal time constant of typical NEMS structures has been estimated to range from pico- to nano-seconds [91], and actuation of a few MHz has been achieved [92]. However the actuation requires two or more layers on the nanocantilever, which may pose difficulties in fabrication and avoidance of pre-bending.

In the case of thermal mechanical noise, the cantilever actuation is achieved by the random ﬂuctuation that originates from the equipartition of energy to all frequencies. Therefore the actuation is very suitable for nanocantilever applications working in high frequency ranges. The actuation requires no additional device or transducers, hence it has no integration or fabrication concerns. The main limitation is its relatively small eﬀective force and displacement. Furthermore it is not applicable to quasi-static actuation due to its random characteristics. For these reasons, it is only suitable for soft cantilevers with sensitive readout transducers.

Magnetic actuation

Finally the last widely used actuation technique, is to use the magnetic force to actuate the cantilevers. To achieve magnetic actuation, a magnetic ﬁeld is required around the cantilever, meaning that one of the greatest challenge in this technique is the ability to setup controllable magnetic ﬁelds in an integrated way. The magnetic actuation can be further divided into two main methods: the direct magnetic actuation and the magnetomotive actuation.

In the first case, direct magnetic force is used by depositing a magnetic material on the cantilever, and actuate the cantilever with a varying magnetic field. The force magnitude of magnetic actuation can be seen as analogous to the electrostatic force, using the magnetic charge model. In this case, the force is proportional to the field strength and therefore scales with the size of the cantilever. The frequency response of

From MEMS to NEMS: Scaling Cantilever Sensors