AN EXPERIMENTAL APPROACH TO
NANOMECHANICAL BUCKLING AND
SNAP-THROUGH PHENOMENON
a thesis submitted to
the graduate school of engineering and science
of bilkent university
in partial fulfillment of the requirements for
the degree of
master of science
in
mechanical engineering
By
Utku Hatipo˘glu
August 2019
ABSTRACT
AN EXPERIMENTAL APPROACH TO
NANOMECHANICAL BUCKLING AND
SNAP-THROUGH PHENOMENON
Utku Hatipo˘glu
M.S. in Mechanical Engineering Advisor: M. Selim Hanay
August 2019
Buckling has received little attention as a valuable resource for engineering ap-plications since it is regarded as a type of failure in civil and mechanical engi-neering. Nevertheless, buckling has a great potential in nanoelectromechanical systems(NEMS) field as a bistable process that has rich and complex dynam-ics. Here, we explore post buckling dynamics of a nano-beam experimentally by employing various probing techniques. By employing an all-electronic architec-ture, we precisely control the buckling amount as well as buckling direction of the nano-beam which eventually gives us the ability to control a two-level me-chanical system with high precision and speed. A full control over the potential energy landscape of the system is demonstrated with different techniques such as Scanning Electron Microscopy operated in three different modes and microwave coupling method. During proof of concept experiments, left and right buckling, large deflection buckling, nonvolatility – which is an indication of pure bistable states – and snap-through phenomenon is demonstrated. Further steps of the study focused on the snap-through phenomenon that is the interstate transitions of the buckling beam after bifurcation. During these experiments, more involved relations are investigated such as mechanical bias and effect of plastic deforma-tion as well as the effect of actuadeforma-tion scheme on interstate jumps. Moreover, to obtain a better grasp of post-buckling dynamics, quantitative measurements are carried out which reveal the reaction speed of the system and time scale of inter-state jumps. Lastly, oscillatory snap-through motion is observed in some special conditions that can be beneficial to understand noise dynamics of the system and it has a potential to contribute energy harvesting applications.
Keywords: Buckling, Post-buckling, Nanoelectromechanical systems, NEMS, Snap-through, Bistability, Two-level system.
¨
OZET
NANOMEKAN˙IK BURKULMA VE AN˙I GEC
¸ ˙IS¸
FENOMEN˙INE DENEYSEL B˙IR YAKLAS¸IM
Utku Hatipo˘glu
Makine M¨uhendisli˘gi, Y¨uksek Lisans Tez Danı¸smanı: M. Selim Hanay
A˘gustos 2019
Burkulma, makine ve in¸saat m¨uhendisli˘ginde ka¸cınılan bir durum olmasından kaynaklı olarak, m¨uhendislik uygulamalarında ¨onemli bir yer kazanamamı¸stır. Buna ra˘gmen burkulma, iki durumlu do˘gası, zengin ve karma¸sık dinamikleri sayesinde nanoelektromekanik sistemler alanında ¨onemli bir konuma sahip olma potansiyeli ta¸sımaktadır. Bu tez, nano boyutlardaki bir kiri¸sin burkulma sonrası dinamiklerini ke¸sfetmek i¸cin yapılan ve bu ama¸cla ¸ce¸sitli ¨ol¸c¨um tekniklerinden yararlanılan deneysel ¸calı¸smaları konu almaktadır. Nano boyuttaki kiri¸sin burkulma miktarı ve y¨on¨u, tamamen elektronik bir mimari kullanılarak kontrol edilerek iki kademeli bir mekanik sistemi y¨uksek hızda ve kesinlikte kontrol etme kabiliyeti olan bir sistem geli¸stirilmi¸stir. Potansiyel enerji da˘gılımının kapsamlı bir bi¸cimde kontrol edili¸si, ¨u¸c ayrı taramalı elektron mikroskobu modu ve mikro-dalga ba˘gla¸sımı olmak ¨uzere d¨ort farklı y¨ontem ile g¨osterilmi¸stir. Konsept ispatı niteli˘gindeki deneylerde, sa˘g/sol burkulma, y¨uksek genlikli burkulma, iki durum-luluk i¸cin kanıt niteli˘ginde olan u¸cucu olmayan veri analizi ve ani ge¸ci¸s fenomeni g¨osterilmi¸stir. C¸ alı¸smanın sonraki a¸samalarında, ani ge¸ci¸s adı verilen burkulma sonrası kararlı durumlar arası ge¸ci¸sler derinlemesine incelenmi¸stir. Bu deneylerde, mekanik y¨onelimin, plastik deformasyonun ve farklı tahrik kalıplarının kararlı du-rumlar arası atlamalara olan etkisi incelenmi¸stir. Dahası, burkulma sonrası di-namiklerin daha iyi anla¸sılması a¸cısından ¨onemli olan nicel ¨ol¸c¨umler yapılmı¸s, sistem reaksiyon hızları ve durumlar arası ge¸ci¸slerin zamanlamaları ¨ol¸c¨ulm¨u¸st¨ur. Son olarak bazı ¨ozel durumlarda titre¸simsel ani ge¸ci¸s durumu g¨ozlemlenmi¸stir. Bu g¨ozlemlerin detaylı analizi sonucu edinilecek bilgiler, g¨ur¨ult¨u dinamiklerini anla-mak ve nano boyutta enerji hasadı uygulamaları i¸cin potansiyel ta¸sıanla-maktadır.
Anahtar s¨ozc¨ukler : Burkulma, Burkulma-sonrasi, Nanoelektromekanik sistemler, NEMS, Ani ge¸ci¸s, ˙Iki durumluluk, ˙Iki kademeli sistem .
Acknowledgement
I would like to thank my advisor Dr. Selim Hanay for his endless support and opportunities that he provided throughout my MS study. His deep knowledge in the field and unique perspective for science and technology guided me and I learnt a lot from him. I also thank my former advisor Dr. Barbaros C¸ etin for introducing me the research environment and letting me express my creativity.
I would like to thank my friends in our research group; Sel¸cuk O˘guz Erbil, Mehmet Kelleci, Hande Aydo˘gmu¸s, Levent Aslanba¸s, Ezgi Orhan, Mert Y¨uksel, Arda Se¸cme, H. Dilara Uslu, Mahyar Ghavami, Mohammed AlKhaled, Hadi Sedaghat Pisheh, Berke Demiralp, R. Tufan Erdo˘gan, Hakan Karakurt, Berk K¨u¸c¨uko˘glu. Special thanks to Sel¸cuk, for teaching me nanofabrication techniques with great patience and his valuable friendship. I want to thank Berke for accom-panying me during fabrication processes and experiments. I want to thank Dr. Cenk Yanık for his support for nanofabrication and Atakan B. Arı for his valuable process recipes. I would like to thank UNAM and METU MEMS members for their help and support during nanofabrication processes, especially Ahmet Mu-rat Ya˘gcı, Semih Bozkurt, MuMu-rat G¨ure, Mustafa G¨uler, ˙Isa Murat C¸ alık, ¨Ov¨un¸c Karakurt and Akın Aydemir. I want to thank Prof. Michael Roukes, Dr. Ewa Rej, Dr. Matt Matheny, Dr. Warren Fon, Jarvis Li and Jonathan White for valuable discussions.
I would like to thank my colleagues; Dr. G¨une¸s Kibar, Muhammed Bilgin, Levent Dilavero˘glu, Cem Ayg¨ul, G¨ok¸ce ¨Ozkazan¸c, Furkan G¨u¸c, Didem F. Demir, Tamer Ta¸skıran, Atakan Atay, O˘guz Altunka¸s, Nima Mahkam, B¨u¸sra Sarıaslan, Emre Eraslan, Alp Aksın, Shari Gamaniel, Alper Topuz, Kaan Karaca, Hammam Mohamed, Mustafa ¨U¸senmez.
vi
I would like to extend my gratitude to some special friends; Yi˘git Oskay, Cem Aydo˘gan, D¨undar Dedekargıno˘glu, Mert Y. C¸ am, ˙Ilhami I. Aydo˘gdu, O. Berkay S¸ahino˘glu, Efe Tuncay, Cem Bilalo˘glu, Egehan ¨Oktem, Alper Karasar, Onur U¸canok, Damla ˙Ilba¸s.
I would like to thank my mom, dad and sister for supporting me in every possible way throughout my journey and for their unconditional love that has no substitute. I like to thank my sister Yalın for her effort for teaching me addition, subtraction and multiplication at early ages that eventually induces an inescapable love for mathematics. I want to thank my dearest Tanya, her presence is more than enough for me to be happy. I would also thank my whole family, my cousins, aunts, uncles, grandparents. Special thanks to Can for endless scientific discussions, to Yi˘gitcan for imposing me the joy of creating something, to Cem for believing me to become a mad scientist someday...
Last but not least, I would like to thank Richard P. Feynman for reminding me that I am doing science not for money, fame or prestigious prizes but for the priceless pleasure of finding things out.
This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) with project number 115E833.
Contents
1 Introduction 1
1.1 Bistability, Bifurcation and Buckling . . . 3
1.2 Thesis Outline . . . 5
2 Design and Fabrication 6 2.1 System Overview . . . 6
2.2 Fabrication . . . 11
3 Experimental methods 15 3.1 Scanning Electron Microscopy (SEM) . . . 15
3.1.1 Full frame scanning mode . . . 16
3.1.2 Spot mode . . . 18
3.1.3 Line mode . . . 19
CONTENTS viii
4 Proof-of-concept experiments 23
4.1 Right/Left buckling . . . 23
4.1.1 Full frame SEM . . . 24
4.1.2 Line mode SEM . . . 25
4.1.3 Spot mode SEM . . . 27
4.1.4 Microwave coupling . . . 28
4.2 Nonvolatility . . . 30
4.2.1 Full frame SEM . . . 30
4.3 Large displacement buckling . . . 31
4.3.1 Full frame SEM . . . 32
4.4 Snap-through . . . 33
4.4.1 Full frame SEM . . . 35
4.4.2 Spot mode SEM . . . 35
4.4.3 Line mode SEM . . . 36
5 Snap-through analysis 38 5.1 Mechanical bias and memory effect . . . 39
5.2 Effect of actuation scheme . . . 41
5.3 Reaction speed and timescale of interstate jumps . . . 42
CONTENTS ix
6 Conclusion 48
6.1 Future Directions . . . 49
A Data 56
B Code 57
B.1 Line mode post processing . . . 57
List of Figures
1.1 Visual demonstration of stable and unstable states of the system. Potential barrier between states is showed as the energy difference between the stable states and the unstable state located at the local maximum. . . 4
1.2 Two energetic regions before and after bifurcation. Before bifur-cation there is only one stable state that is shown by a single well. After bifurcation two new stable states appear and first stable state becomes unstable. . . 4
2.1 System elements. Comb drive, crab legs, beam and side gates are showed with the input voltages of the system. P stands for the compressive force supplied by the comb drive structure. Scale bar: 20µm . . . 6
2.2 Comb drive structure. a) Tilted view of the left comb drive pairs. b) Front view right comb drive pairs. Scale bars: 10µm . . . 7
2.3 Modified crab leg structure and the spring mechanism that con-nects the comb frame to the anchoring point. Scale bar: 10µm . . 9
LIST OF FIGURES xi
2.4 JG design layout. a) A general view with contact pads and inter-connections. b) Detailed device layer layout. There are two side gates each located slightly under the maximum deflection point of
the beam and 4µm away from the neutral position. . . 10
2.5 MW design layout. a) A general view with contact pads and inter-connections. b) Detailed device layer layout. There are four side gates, two of them are bigger and located at the center while other two are smaller and located near the bottom anchoring point. . . 10
2.6 HF vapor etcher. a) All subparts of the system assembled on the wet bench. b) Process surface with the mounted chip that is going to be processed. . . 12
2.7 WestBond branded wire bonder. a) A general view of the wire bonder. b) Golden wire coil and bonding head is zoomed in. . . . 13
2.8 Final view of the nanomechanical buckling device. Scale bar: 20µm 14 2.9 a) Final view of the whole system mounted on a specially designed and made PCB with SMA connections. b) Chip holder is zoomed in. A chosen device among the pack of nine is bonded to the chip holder’s bonding pads. . . 14
3.1 Device PCB is mounted in the SEM chamber and feedthrough connections were made. . . 16
3.2 Full frame scan schematic. . . 17
3.3 Spot mode schematic . . . 19
LIST OF FIGURES xii
3.5 Microwave readout schematic. a) Coupling between bucking device and the microstripline resonator is showed. b) Drive and readout circuit of the system. . . 21
3.6 Multimode microwave resonator schematic. First and second mode modeshapes are overlapped and nodes/antionodes are showed. . . 22
4.1 Left and right buckling under SEM. False colored, scale bars: 20µm 24
4.2 a) The wide view of the system. b) Zoomed region of the scene. Left and right gates are included in for the purpose of referencing. Scan line is shown in indigo. c) The raw signal acquired by scan-ning the designated line. Left and right gates appear at the two edge while beam appears as a peak at the center. All frame takes 180µs to acquire. Scale bar: 10µm . . . 25
4.3 Line scan raw data for left and right buckled beams. As showed in the plot, for the left buckled beam signal, beam peak appears closer to the left gate while for the right buckled case, beam peak appears closer to the right gate. Gate signals are also different since their charge alters the electron beam reading (lower appearance indicates more positively charged surface). . . 26
4.4 Extracted beam position data is plotted from raw line mode data. a) Left buckling in line mode. At the end during releasing the force and beam turns back to the neutral position, overshoots are observed. b) Right buckling in line mode. Every increment corre-sponds to 1V. . . 27
4.5 Spot mode buckling data. Gate voltage is fixed, buckling achieved by altering comb voltage. It is observed some peaks on the gate voltage data as a response to mechanical motion or electrical charg-ing of the beam. . . 28
LIST OF FIGURES xiii
4.6 Microwave readout data. In each buckling event the frequency of the microwave resonator shifts approximately 3 kHz. . . 29
4.7 Frames from non-volatility video. For left and right buckling cases, first guide is activated then beam buckles to the chosen side then guiding voltage is retained. As it can be seen for both cases, after deactivating guide voltages, beam is still in the buckled state. . . 31
4.8 From beginning to the final shot, with two of the intermediate frames, large buckling full frame SEM scans. . . 32
4.9 Post processed large buckling data. . . 33
4.10 a) Potential energy landscape deepening (or shallowing) with re-spect to comb voltage. b) Potential energy landscape asymmetry with respect to side gate voltage difference. . . 34
4.11 a) Energy landscape for fixed 52.5V comb drive voltage with dif-ferent side gate voltage difference values ranging from 0V to 20V. After 15V side gate voltage difference, two stable states collapse to the right stable state. b) Energy landscape for fixed 53V comb drive voltage with different side gate voltage difference values rang-ing from 0V to 20V. For this case, even 20V side gate voltage dif-ference does not disrupt the bistability of the system even though symmetry is drastically broken. . . 35
4.12 Snap-through in spot mode SEM. Every side gate voltage change triggers a jump from one stable state to another. Peaks after a change are indications of jumps. . . 36
4.13 a) Snapthrough jumps in SEM line mode after post processing for beam locating. b) A close-up view of two jumps with highlighted data points. Left to right jump occurs very fast compared to right to left jump. . . 37
LIST OF FIGURES xiv
5.1 a) Snap-through jumps with triangular wave actuation. During initial four cycles of interstate jumps, transition voltage shifts. Af-ter four cycles, beam stays at its final state even if the system is kept being actuated. b) The method to find jump voltages. First jump occurs at -1V while the consecutive jump occurs at -11V. . . 40
5.2 Snap-through transitions between bistable states, induced by side gate voltage difference. (a) Bifurcation diagram under a slight lateral bias. (b) Lateral voltage vs. displacement curve. Snap-through jumps occur when the slope approaches infinity. Orange arrows show sudden jumps between left and right states. (c, d) Experimental data for 52.5V and 53V comb voltage difference. By applying 3 different triangular waveforms to the left gate - chang-ing from 0V to 20V with 8s, 12s and 16s periods – snap-through transition voltages are recorded. As the dwell time increases, the amount of hysteresis decreases. . . 42
5.3 Response of the beam (as a snap-through jump) to the side gate activation voltage. Between the first moment the signal is given and the beam motion there is 7.12ms. A strange discharging occurs after -8V and until the beam passes the neutral position, it keeps discharging until -6V. After beam jump, side gate voltage continues to decrease. . . 43
5.4 A closer look to two events. Inset: Zoomed in a jump. Reaction time of the snap through jump is 16.6ms after the side gate is activated. . . 44
5.5 Zoomed snapthrough jump from left to right. Between -1.7µm to +0.75µm deflection, there is no data point. It means that the jump occurs in less than 180µs which is the smaller time step that the method can resolve with current configurations. . . 45
LIST OF FIGURES xv
5.6 A snapshot from observed oscillations. Beam appears both on the right and left sides (ghosting) since the full frame scan cannot provide more than 15 FPS. . . 46
5.7 A potential energy landscape with an asymmetry induced by side gate voltage. Deflection amounts are 500nm. When the symmetry is provided again, the potential barrier is on the order of hundreds of kBT which is comparable with the noise level of the system. . . 47
List of Tables
Chapter 1
Introduction
Nanoelectromechanical systems (NEMS) can be classified as downscaled, elec-tronically controllable/detectable mechanical structures that comprise at least one sub-micron feature [1, 2]. NEMS first introduced in 1996 [3] and ever since it gained a great importance in the fields ranging from mass sensing [4-6] to quantum computing [7]. Thanks to their high operation frequencies, small mass and dimensions, NEMS devices have the ability to weight molecules as heavy as zeptograms [8], operate at GHz frequencies [9] and having quality factors on the order of billions [10]. NEMS structures are also suitable for very large scale integration (VLSI) with their small volumes. These abilities put NEMS one step ahead the older (and bigger) brother Microelectromechanical Systems(MEMS) which is also highly integrated in our daily life with MEMS accelerometers, gyros and other numerous sensors located in smartphones. Today NEMS research is expanding into various other applications and areas such as inertial imaging [11], force sensing [12], sensitivity enhancement [13], gas sensing [14-17], nanomechan-ical computation [18-26] and quantum information science [27].
To achieve control and mobility in NEMS, different approaches have been used including thermal, magnetomotive, optomechnical, capacitive or piezoelec-tric actuation most of the times to promote bending in the system. So far, so little attention has been payed to the buckling dynamics which actually is a rich
and promising phenomenon [28-35]. Among recent studies about buckling at nanoscale, sensors with higher resolution, more robust relays, energy harvesters [36] and configurable smart materials [37] can be considered. It is also started to be used as a resource to understand the cellular interfaces and cerebellum development [38].
In a very recent study, full electronic control over buckling instability at nanoscale has been showed [39]. Buckling with controllable direction and de-flection amount at nanoscale opens many doors for other applications and it is also an important resource to study buckling dynamics itself. The present study, by using readily available system developed in the mentioned study, aims to de-velop and implement detection techniques to effectively detect and measure the buckling deflection of the system and by employing these techniques, to under-stand the internal dynamics with the leading factors responsible for structural changes. Throughout the study, device is examined in terms of bistable potential energy landscape after the critical load and jumps between these stable states separated with an energy barrier according to the buckling amplitude so called snap-through transitions. Snap-through is a remarkable phenomenon since it is a way to change the state of the system without transitioning to the neutral po-sition which makes it a fast and direct process [40]. This tranpo-sition phenomenon has the potential to be an alternative method for applications that require fast and high displacement actuation. One can probe the transition energies of the buckling beam to reconstruct the potential energy landscape real time between jumps. This way the change in the potential energy landscape during jumps can be studied and information about the structural change in the crystal structure of the system can be harvested.
In the study, I implemented 3 different methods to probe the snap-through transition conditions of the buckling beam. By implementing these techniques, I observed that system has a highly dynamical potential energy landscape which is presumably affected by the changes in the crystal structure of the beam itself. For different actuation patterns that have different frequencies and waveforms, system response is examined. For a narrow band of parameters, oscillatory snap-through motion is observed which can be interpreted as noise induced continuous
snap-through jumps.
1.1
Bistability, Bifurcation and Buckling
A bistable system (or a process) is a system that can have two stable states for a fixed parameter space. Bistable systems have a potential energy landscape with two potential wells. It can be taught as two pits placed side by side, if one let a ball roll from the intersection of two pits, it lands and settles in one of the two pits naturally. In this example, the chosen state is decided by the initial conditions given the system which correspond to the initial position of the ball. Such a system must have an unstable point that is not a local minima as in the case of pits but a local maxima just like the point located between two pits where the ball is released. Since this is an unstable point, the ball cannot maintain its presence on that spot and it falls to find a stable point to settle in.
This simple example of bistable system can be represented mathematically;
ds
dt = (s− η)(γ − s
2) (1.1)
Where γ is real and √γ > η. In such a system, there are three steady states two of which are stable while one of them is an unstable state as in the given example.
ss1 =−√γ, ss2 =√γ, su = η (1.2)
Connecting this mathematical model to the physical example, ss1 and ss2
cor-respond to local minima of the pit, while su is the middle point between pits
(Figure 1.1).
A system that starts with a single stable solution can later divide into two stable solutions as the system parameter changes. The critical point where the
Figure 1.1: Visual demonstration of stable and unstable states of the system. Potential barrier between states is showed as the energy difference between the stable states and the unstable state located at the local maximum.
single stable solutions becomes two different stable solutions is called bifurcation point. This is what happens in buckling case; before the critical buckling force, the beam stays straight in the neutral position however when the applied force exceeds critical buckling force, it buckles either left or right that depicts two different stable state different than the pre-bifurcation stable state. Moreover, the neutral state becomes an unstable steady state after the bifurcation (see Figure 1.2).
Figure 1.2: Two energetic regions before and after bifurcation. Before bifurcation there is only one stable state that is shown by a single well. After bifurcation two new stable states appear and first stable state becomes unstable.
1.2
Thesis Outline
Chapter 1 briefly introduces the concept of NEMS and explains the place of nano-electromechanical buckling device among other nanomechanical systems. Device properties and capabilities are discussed briefly and an introductory theoretical background is given by relating bistability and bifurcation concepts.
Chapter 2 is the part where the device is introduced to the reader. Device fab-rication highlights are also included in this chapter along with working principle of each sub-element.
Chapter 3 explains all experimental methods used in this study to detect buck-ling motion. The principles behind all techniques are explained in detail to create a deep understanding about the detection process of each method.
Chapter 4 demonstrates proof of concept experiments which either proves the functionality of the device is as expected and also works as a validation of ex-perimental methods. In this chapter, all proof of concept experiments conducted by employing as many method as possible to cross check different methods are presented in detail.
Chapter 5 presents a more involved experimental set which mainly focuses on snap-through phenomenon. All demonstrations are made by using the most convenient technique for the study in question. Obtained results are presented with detailed explanations and interpretations.
Chapter 2
Design and Fabrication
2.1
System Overview
Figure 2.1: System elements. Comb drive, crab legs, beam and side gates are showed with the input voltages of the system. P stands for the compressive force supplied by the comb drive structure. Scale bar: 20µm
System consists of four main parts to achieve reversible, bidirectional, control-lable buckling (See Figure 2.1). First and the most important sub element is the beam which is being buckled according to the actuation voltage supplied. This is a very slender beam with 150nm to 250nm cross section and 40µm length that is
anchored from one end and attached to a comb structure to the other end which is the second subpart of the system.
To exert force on the beam a capacitive actuation scheme with a comb-like shape so called comb drive mechanism is used. Comb drive mechanism has two sets of combs, one of which is the stationary comb structure and the other is movable. By applying different charges to the comb structure, it is possible to create an electric field between two sets and eventually they pull each other thanks to the capacitive force created [41-43]. Comb structure is composed of interdigitated fingers which are 4µm long and 500nm wide and the separation between each finger is also 500nm (See Figure 2.2).
FC = 1 2 ∂C ∂xV 2 = N 0t g V 2 (2.1)
Where FC is the comb force, C is the capacitance between fingers, x is the
displacement, V is the voltage difference between fingers, 0 is the permittivity
of vacuum, N is the finger count and t is the spacing between fingers[43].
Figure 2.2: Comb drive structure. a) Tilted view of the left comb drive pairs. b) Front view right comb drive pairs. Scale bars: 10µm
One considerable advantage of the comb structure is the inherent stabilization mechanism which provides a symmetrical actuation. This is originated from
evenly and symmetrically distributed interdigitated fingers that compensate small asymmetric deflections. Another big advantage of the comb drive structure is the ability to apply force with a large travel, again thanks to interdigitated fingers that are kept at a constant distance throughout the travel. Force generated by the comb mechanism can be calculated with Eq. 2.1. For further details see Ref.[44]
Thirdly, a spring mechanism which is basically two doubly clamped beams that are attached to the movable combs from two ends (see Figure 2.3). Springs link the movable combs to the anchoring points on the contact pad with a modified crab leg structure that is reinforced to withstand out of plane bending as well as inward pull coming from the deflection of the comb frame. There is a delicate balance for springs since they must let the movable combs move downward when the compression is applied while they have to be stiff enough to recover the system to its first state when the forces are released. The bending stiffness of the spring beams can be calculated via Eq. 2.2.
k = 24EI1 L3 1 L1I2+ L2I1 L1I2 + 4L2I1 (2.2)
Where k is the bending stiffness, E is Young’s Modulus, I is the geometric moment of inertia, L1 is the horizontal beam length while L2 is the vertical
anchoring beam length. For the modified crab leg design, L2 can be taken 0 since
the diagonally placed support beam does not allow any bending on the vertical anchoring beam[41].
Last subpart of the nano electromechanical buckling paradigm is guiding gate that is responsible for the symmetry breaking and also fine tuning of the potential energy landscape. Guiding gates are simple islands located near the buckling beam to apply lateral forces again by capacitive coupling. By using guides, it is possible to preset the buckling direction of the beam as well as pulling the beam from one stable state to another.
Figure 2.3: Modified crab leg structure and the spring mechanism that connects the comb frame to the anchoring point. Scale bar: 10µm
or in a way that creates a voltage difference even if applied charges are not in opposite sign – they exert a pulling force to each other. Stationary combs are not free to move so they act as anchors while movable combs are moving towards them. Since the buckling beam is attached to the movable set from one end and anchored from the other, this applied force compresses the beam. When applied force exceeds the critical force defined in terms of beam geometry and structural properties, beam starts to buckle. (see Eq. 2.3)
PC =
4π2EI
L2 (2.3)
Where PC is the critical buckling force and L is the beam length.
If the process starts with no intentional guiding coming from guiding gates, beam chooses one side to buckle according to mechanical bias originated from fabrication defects or other factors that are responsible for the noisy environment such as the measurement method used. However, it is possible to guide the beam prior to the buckling by simply creating a lateral force and then compressing the beam beyond its critical buckling load. This pre-guiding can be performed by employing small voltages such as 5 volts or even 0.5 volts.
After one state is chosen, it is still possible to change buckling direction by using side gate voltages, this time with more voltage difference. After buckling, system potential energy landscape has two distinct stable states which are sep-arated by a potential barrier. Applying a lateral force can alter the height of the potential and a noise induced jump to the other state can be observed. It is also possible to destroy the bistability by applying excessive lateral forces which terminates the buckling and creates a pure bending scenario.
Figure 2.4: JG design layout. a) A general view with contact pads and intercon-nections. b) Detailed device layer layout. There are two side gates each located slightly under the maximum deflection point of the beam and 4µm away from the neutral position.
Figure 2.5: MW design layout. a) A general view with contact pads and inter-connections. b) Detailed device layer layout. There are four side gates, two of them are bigger and located at the center while other two are smaller and located near the bottom anchoring point.
During the experiments, 2 main device designs are used that are identical in terms of beam, comb and spring structures but differ by guide gate configurations.
First design (code name JG) has single gate on both sides of the beam and it is mainly for conducting simple left/right buckling experiments as well as snap-through experiments (see Figure 2.4). In the second design (code name MW), there are two different gates on both sides of the beam. The upper gates are designed for microwave coupling experiments (will be discussed in Chapter 3) and lower ones are for guiding the beam to left or to right (see Figure 2.5). This design is also suitable for other experiments once lower gates are grounded to prevent floating.
2.2
Fabrication
Nanoelectromechanical buckling device is fabricated by using Silicon-on-Insulator (SOI) wafer composed of a 500µm silicon (Si) handling layer, 3µm silicon dioxide layer (buried oxide or BOX) and on top of that, a 250nm p-doped silicon layer. During the fabrication process, a top-down method is utilized to form designed features. Every set of new chips are fabricated over 1.8mm by 1.8mm square diced wafer pieces.
Firstly, metallized contact pad areas which are used for feeding the device by applying voltages are patterned on the silicon layer by using electron beam lithog-raphy (EBL). This process can also be completed by a simpler photolithoglithog-raphy process which is quicker for large volume fabrication processes. After coating the wafer with a special polymer called polymethyl methacrylate (PMMA), de-sired parts of the coating are exposed to controlled electron bombardment which changes the chemical structure of the PMMA. After developing, patterned PMMA layer is used as a mask to deposit thermally evaporated gold on the silicon. After lift-off process used to get rid of excess gold, one more EBL process is applied to the wafer with proper alignment, this time to pattern silicon device features. This is not an optional case which photolithography is also applicable because silicon device design contains sub 200nm features that are challenging to obtain with a photon-based lithography process. Until this phase all processes are performed in Sabancı University Nanotechnology Research and Application Center (SUNUM)
within the collaboration between Hanay Group and Dr. Cenk Yanık.
After the second patterning with EBL, wafer is coated with SiO2 to form a
mask layer for silicon device layer. After the lift-off process, the square wafer which contains 9 device groups is diced into pieces to separate these 9 device groups. After dicing process, chips are etched in an inductively coupled plasma etching system (ICP) by employing Cl2 plasma. This process etches silicon
anisotropically and does not damage SiO2 mask layer. In the final etching step,
hydrofluoric acid (HF) vapor is used to etch the mask as well as the BOX layer to suspend the silicon structures. Chips that contain suspended devices are wire bonded through their contact pads to a PCB with SMA connectors for making it possible to make electrical connections.
Further details related to these fabrication steps can be found in Ref.[44] Chap-ter 2.
Figure 2.6: HF vapor etcher. a) All subparts of the system assembled on the wet bench. b) Process surface with the mounted chip that is going to be processed.
Two of the aforementioned fabrication steps are changed throughout the project to be able to sustain the nanomechanical buckling device fabrication in-house. For the second half of the experiments, instead of using METU MEMS HF vapor etch system, we used a simpler setup produced by IDONUS company (see Figure 2.6). This setup contains a HF container, a reaction chamber and a heater. Sample is attached to the heater plate and placed onto the reaction chamber. HF is released by a simple handle lift which raise the container and
transfer all HF to the reaction chamber thanks to gravity. Etching is performed at 60◦C during 165 – 180 mins to achieve 2.6 - 3µm BOX layer etching.
Figure 2.7: WestBond branded wire bonder. a) A general view of the wire bonder. b) Golden wire coil and bonding head is zoomed in.
Second fabrication step that is changed is the wire bonding. An in-house wire bonder is a requirement for a quick prototyping and multiple experiments. For this purpose, a WestBond branded wedge-wedge wire bonder with 25µm golden wire owned by Advance Research Labs at Bilkent University is utilized (see Figure 2.7). This change is made at the end of the project yet it is very important for sustainability of further experiments.
After all fabrication steps, obtained chip has nine devices on it which are ready to be tested (see Figure 2.8). A specifically designed PCB is fabricated via micromachining and NEMS chip is mounted to the PCB on a chip holder to realize contacts between SMA connections and device bonding pads (see Figure 2.9).
Figure 2.8: Final view of the nanomechanical buckling device. Scale bar: 20µm
Figure 2.9: a) Final view of the whole system mounted on a specially designed and made PCB with SMA connections. b) Chip holder is zoomed in. A chosen device among the pack of nine is bonded to the chip holder’s bonding pads.
Chapter 3
Experimental methods
3.1
Scanning Electron Microscopy (SEM)
Scanning electron microscopy is one of the most useful techniques when smallest feature size of the sample is less than a micron. After a limit, it is very hard if not impossible to monitor all features in detail because the wavelength of the visible light is hundreds of nanometers which is comparable or smaller than the smallest feature size of the sample. When it is NEMS, use of electron microscope is inevitable at least for characterization and inspection. For nanoelectromechanical buckling experiments, it is also a good method for conducting experiments since it is a sophisticated monitoring system that can be implemented dynamical post buckling experiments.
During the experiments, it is crucial to be able to actuate the system with an external voltage source. For this purpose, an SEM with feedthrough connections – 4 co-axial wires connecting the vacuum chamber to outside electrically – is used (see Figure 3.1).
There are two different detection schemes which use two types of electrons, secondary electrons and backscattered electrons. Secondary electrons are results
Figure 3.1: Device PCB is mounted in the SEM chamber and feedthrough con-nections were made.
of inelastic collisions between electron beam and the sample surface. Backscat-tered electrons are the ones scatter back after elastic collisions with the sample nuclei. After collecting electrons, they are transformed into photons and ampli-fied by the help of photon multiplier. Throughout the experiments, secondary electrons are used to gather information about the chip surface since they can create more contrast when the device/substrate structure is considered. Even the main working principle – bombarding with electrons and collecting back the product of the electron-surface collisions – is same, SEM can process this infor-mation in a versatile fashion. For buckling experiments, three types of monitoring schemes are utilized; scanning mode which gives an actual image of the sample, spot mode which only gathers the data coming from one designated spot and line mode which gives data along a designated line.
3.1.1
Full frame scanning mode
In the scanning mode, the output of the detection is an actual grey scale image. This is a very informative technique when it is important to collect qualitative data such as buckling motion, direction or movement of comb frame. It is also
suitable to postprocess the data with image processing to extract quantitative information as well.
In this method, electron beam scans the focused region first left to write, then same scanning is repeated for the next line towards the bottom end of the region. This way collected data is recombined and a picture of the current scene appears (see Figure 3.2).
Figure 3.2: Full frame scan schematic.
This technique is used usually along with video recording to demonstrate real-time operation. This is important for proof of concept experiments which are basic presentation of device operation.
One downside of this technique is low sampling rate that is originated from data collected for unnecessary parts of the sample. With this technique, it is possible to get information nearly 1 million points on the sample, however this much data can be collected only for 15 times a second. This sampling rate may be adequate for some applications but when considering getting quantitative data for fast varying systems, it is quite slow. If the sampling rate is not critical, it is possible to increase resolution of the image obtained which gives more precision while post processing.
For measuring the deflection amount, a high-resolution image is imported to MATLAB. After thresholding the image with a threshold value that is enough to distinguish the beam structures and the substrate, pixels are counted along a line crossing the maximum deflection point until a fixed point. According to a prior measurement that tells the length of each pixel, the distance between the fixed point and the beam can be calculated and by using the initial length, it is possible to calculate the deflection amount with approximately 10nm precision (see Appendix B Full frame post processing).
3.1.2
Spot mode
In spot mode, electron beam is focused on one point on the device that all data is collected from (see Figure 3.3). Collected electrons are amplified again by photomultiplier then the information is collected from the BNC output of the electron detector. In this mode a separate data acquisition system is required to collect data. After acquiring the data, it is post processed to extract jump information or to find state of the buckling beam. Since the electron beam is focused on one point, this method can provide faster information about that point. This is crucial to obtain quantitative data about fast jumps between states or oscillations.
The downside of this technique is binary nature of the information acquired since there is no data about deflection amplitude or direction. That’s why spot must be placed strategically to properly capture the motion of beam. This tech-nique is susceptible to errors if there is any drifting originated from mechanical motion of the chip in the SEM chamber or surface charging.
Collected data is filtered and stored for further post processing. It is possible to collect related actuation voltages along with the electron detector data in real time. This is a perfect way to investigate the relation between actuation and the mechanical reaction of the system.
Figure 3.3: Spot mode schematic
3.1.3
Line mode
Line mode is rather a hybridization of two previous methods since it can give both quantitative and qualitative data with fast sampling. It is basically scanning a line on the device surface instead of scanning the whole frame or capturing a single spot (see Figure 3.4). It is important for eliminating the unnecessary scanning which cuts down the sampling rate whilst gathering enough data to extract beam state and deflection amount. This method also makes use of external BNC output coming out of the electron detector and just like the spot mode, again there must be an external data acquisition system to gather the data.
Line positioning must be done properly to extract as much data as possible while having an adequate spatial resolution. A line perpendicular to the buckling beam which also crosses side gates can be a perfect candidate to achieve high resolution fast deflection detection since it makes the detection drift-proof by fixing the travel span. The two golden side gates appear shiny on the SEM screen since they emit more secondary electrons than silicon substrate. Doped silicon device layer also emits more electrons than substrate since it is negatively charged during the experiment. These settings are perfect fit for line mode to make post buckling analysis.
Figure 3.4: Line mode schematic
3.2
Microwave coupling
Microwave resonators are being used widely for various technologies such as an-tennas and substantial amount of research is being conducted nowadays to widen the abilities of microwave resonators. In biological research, microwave imaging of cells is one of the most important examples which microwave resonators are used to detect cells according to the permittivity difference between the cells and their medium [45]. Microwave resonators are also used to obtain ground state quantum mechanical vibrations by the help of side band cooling and they are used to detect vibrations at quantum level[46]. Today substantial part of circuit quantum electrodynamics (circuit QED) research is also conducted on the basis of superconducting microwave circuits[48, 47].
Microwave resonators have plenty of different kinds such as micro stripline, co-planar waveguide, ring resonator etc. In this project, micro stripline resonators are utilized to detect buckling motion. These resonators are composed of an FR4 substrate layer that is sandwiched between a copper signal line and a copper ground surface. Signal line has two coupling gaps located between the strip and the two feed SMA connections. These gaps are forming the boundary conditions
for the resonator as well as their geometric structure is used to tune the quality factor and insertion loss of the resonator. Any change near the strip – especially between the strip and the ground surface – such as an approaching metal piece or a microscopic object that is different than the surrounding medium in terms of permittivity can cause a shift at the resonance frequency of the device since it changes the electric field distribution which is related to the inductance and capacitance of the circuit. This way microwave resonator can respond to any change that causes a fluctuation in the electric field forming around the signal line (see Figure 3.5).
Figure 3.5: Microwave readout schematic. a) Coupling between bucking device and the microstripline resonator is showed. b) Drive and readout circuit of the system.
Their high responsiveness, makes use of microwave resonators a good way to detect tiniest capacitance changes. This is exactly what is observed during a buckling process; when the slender beam buckles to one side, the capacitance value between the beam and the gate changes. When coupled to a microwave resonator by a simple wire bond, this change can be tracked to extract deflection information. Various methods can be utilized to drive the microwave resonator at the resonance frequency such as a simple drive with an open loop control system that we keep track of phase difference between drive and the device response or a phase locked loop system can be employed to be able to gather accumulative data.
When driven at resonance frequency, according to the mode that is driven, a standing wave with its characteristic waveform sits between two coupling gaps. This characteristic waveform is a non-ideal sine with half, full, twice or more periods. So it can be inferred that different points on the microwave resonator can have different responsiveness according to the mode shape. For example, a doubly grounded resonator driven at its first mode has two nodes and one antinode. If one touches middle of the strip with a metal pin, ideally there must be a huge frequency shift while touching two ends does nothing to the readout frequency. This is because the electric field intensities are different at those three points on the resonator. The response will change at higher modes since the mode shape will be different which eventually leads a different electric field distribution along the strip.
Figure 3.6: Multimode microwave resonator schematic. First and second mode modeshapes are overlapped and nodes/antionodes are showed.
With this method, unlike others, multiple buckling beams can be detected by coupling each of them at different points on the strip (see Figure 3.6). This kind of coupling provides different frequency shifts for different buckling beams which can be interpreted as an addressing system. Thus it can be a good way to use for device arrays.
Chapter 4
Proof-of-concept experiments
Throughout the thesis work project, the main purpose was to control the na-noelectromechanical buckling device as desired while monitoring the dynamical response of the system. First of all, it is crucial to show main device functions by using proposed methods. This is important for both proving that device works as expected and for making sure about the functionality of proposed experimental methods. Since these are some of the simplest proof of concept experiments, they are also useful to characterize the system as a whole.
4.1
Right/Left buckling
First and the most important proof of concept experiment is controllable left and right buckling. For this purpose both MW and JG devices can be employed. This experiment proves that system has two stable states after a threshold value is exceeded. These stable states must be adopted controllably and repeatedly. To achieve bidirectional buckling, a three-channel voltage supply is utilized. Comb drive mechanism is charged with opposite signs to create enough potential dif-ference between interdigitated fingers. After reaching the critical voltage value before observing buckling, desired side gate is activated by charging the gate with
a positive voltage while the opposite guiding gate is being kept grounded. This can cause two outcomes; if the beam (and the movable fingers) is charged nega-tively, a positive charged side gate will attract the beam more than the grounded side gate which eventually pulls the beam towards the charged side gate. Oth-erwise, if the beam is charged positively, the attraction between the positively charged guiding gate and the beam will be smaller because of the diminished voltage difference. This time the beam will move towards the grounded side gate. Theoretically it is possible to choose buckling direction just by controlling the voltage applied to a single gate while keeping the other always grounded. After buckling to one side, it is possible to unbuckle the beam by removing the comb actuation and recharge the combs after activating the opposite side gate which leads a buckling in the opposite direction.
4.1.1
Full frame SEM
Full frame scanning mode is very useful to capture left and right buckling with various side gate voltages. Since this is just for demonstration purposes, it is enough to show consecutive left and right buckling motion repeatedly. While recording a video of the current screen, left and right buckling motions are realized (see Figure 4.1).
4.1.2
Line mode SEM
Line mode can also be used for this purpose and it is possible to capture faster transitions. As a matter of fact, this is more of a validation for line mode method than a proof of concept experiment for buckling. In this example, it can be concluded that line mode can be very beneficial to detect buckling motion with a great speed and resolution. Data is needed to be post processed for further analysis but for a qualitative point of view, even the time trace is explanatory.
Figure 4.2: a) The wide view of the system. b) Zoomed region of the scene. Left and right gates are included in for the purpose of referencing. Scan line is shown in indigo. c) The raw signal acquired by scanning the designated line. Left and right gates appear at the two edge while beam appears as a peak at the center. All frame takes 180µs to acquire. Scale bar: 10µm
screen, and a scanning line is defined passing through the maximum deflection point of the beam. Then the line mode is activated and data is collected from the BNC output of the electron detector. In the data, the peak corresponds to the beam and two blocks located left and right sides are the side gates. These side gate sections are useful to fix the frame which makes the system less susceptible to any mechanical or electrical drift. Even side gate voltages can be understood just by analysing the block heights according to the contrast and brightness settings.
Figure 4.3: Line scan raw data for left and right buckled beams. As showed in the plot, for the left buckled beam signal, beam peak appears closer to the left gate while for the right buckled case, beam peak appears closer to the right gate. Gate signals are also different since their charge alters the electron beam reading (lower appearance indicates more positively charged surface).
Raw time trace data is nothing but a topology map of a single line passing through the beam and gates(see Figure 4.2). To extract a beam deflection data from the time trace, one should post process the time trace periods and according to the appearance of the beam peak in the period, it is possible to find the current deflection of the beam. Such a signal processing code is implemented to the post processing (see Appendix B Line mode post processing). Beam finder code chops the time trace data according to the new line start points then it finds the fixed distance between two side gates. After finding the travel span of the beam, code
searches for the beam peak and the exact peak location is stored as a time data which will be converted to deflection amount by employing scan speed of the system and the distance between gates. Time versus deflection amount is plotted for both left and right buckling experiments in Figure 4.4.
In Figure 4.4, it is possible to see the effect of comb voltage stepping. As the voltage increases, beam buckles towards the activated side gate. It is possible to extract buckling information with a 120nm resolution – which is less then the beam width – for the scan that is performed. An interesting point is the overshoot peaks observed in the left buckling case. When the voltage is decreased fast to recover the beam, in each decreasing step an overshoot towards the neutral state is observed (see Figure 4.4a inset). These overshoots can be beneficial to study inertial effects in the buckling system. It can be considered that these overshoots originated not from the beam but the movable comb frame which is bulkier.
Figure 4.4: Extracted beam position data is plotted from raw line mode data. a) Left buckling in line mode. At the end during releasing the force and beam turns back to the neutral position, overshoots are observed. b) Right buckling in line mode. Every increment corresponds to 1V.
4.1.3
Spot mode SEM
When the detection spot is placed on the beam’s buckled position, it is possible to detect the beam dwelling at that point. For this purpose, the spot can be
placed on the right (or left) buckled positions and if the beam buckles to either sides it is possible to detect these transitions (see Figure 4.5). It is not possible to detect right and left buckling with the same spot placement.
Figure 4.5: Spot mode buckling data. Gate voltage is fixed, buckling achieved by altering comb voltage. It is observed some peaks on the gate voltage data as a response to mechanical motion or electrical charging of the beam.
4.1.4
Microwave coupling
Microwave coupling is the most compact method when considering other three possible solutions that make use of an electron microscope. This is also an ex-periment to prove the functionality of the method more than a device-oriented proof of concept experiment.
Same procedure is used to actuate the device and this time, a coupling wire bond connects one side gate designed specifically for this purpose to the stripline placed on the PCB. Stripline resonator is driven in the PLL mode and frequency
is measured during consecutive buckling motions. It is shown that buckling cre-ates enough capacitance change to shift the resonator frequency in a detectable amount (see Figure 4.6).
Figure 4.6: Microwave readout data. In each buckling event the frequency of the microwave resonator shifts approximately 3 kHz.
A validation test is done where the comb voltage is increased with the same amount but without any mechanical effect on the beam. This experiment showed that without a mechanical motion the resonator frequency does not shift at all.
4.2
Nonvolatility
Nonvolatility stands for the permanence of the adopted stable state by the beam even before the initial conditions are removed. In the buckling system these gen-eral terms correspond to staying at the chosen buckling state even after removing the side gate voltage that gives the system a discernible asymmetry that helps the beam to chose one of the stable states. If the beam can be persistently buckled to one direction after the lateral force is removed, it means that this is not just a bending but really a buckling state which solely driven by the compressive force supplied from comb drive mechanism.
To realize this experiment, comb drive mechanism is charged to reach the critical limit. Before applying more force, a side gate is activated to break the symmetry between states and beam is directed towards the activated side gate. After that point, comb mechanism is charged to exceed the threshold which yields to buckling. The crucial part of this experiment is what comes next; activated side gate is deactivated after the beam buckles. Beam is expected to maintain its buckling direction even after the side gate voltage removal. Buckling amplitude may be smaller since the attraction coming from the side gate has also an effect on the buckling amplitude. This process should be applied for both buckling di-rections and if the state is maintained for both cases, it can be inferred that this is a nonvolatile process that the beam buckling is originated only from longitudinal force and does not depend on side gate attraction.
4.2.1
Full frame SEM
Since this experiment is slow and qualitative the best way of demonstration is the full frame scanning mode. First the comb mechanism is charged and then one side gate is activated. After that, beam buckled to the activated side. When the activated side gate is deactivated, beam position is observed. This is done for both directions while a real-time video is being recorded (see Figure 4.7). Actuation of both comb drive and side gates are done in a programmed fashion.
Figure 4.7: Frames from non-volatility video. For left and right buckling cases, first guide is activated then beam buckles to the chosen side then guiding voltage is retained. As it can be seen for both cases, after deactivating guide voltages, beam is still in the buckled state.
4.3
Large displacement buckling
Thanks to the design and actuation mechanism utilized, one can create a control-lable force that can be applied for long travels. This aspect of the paradigm lets the system can buckle largely without deforming or fracturing. This is important because of characterization purposes. To test this aspect, comb mechanism is charged until the critical voltage that corresponds to the bifurcation point of the buckling. After that point, voltage is increased gradually and in every voltage value deflection data is obtained. This is a more quantitative measurement when compared with previous two experiments. At the end of the measurement, a voltage difference vs. displacement graph is plotted.
4.3.1
Full frame SEM
Large buckling demonstration is not a fast process since the voltage values are altered once for every snapshot. To be able to get a precise measurement, snap-shots are taken at maximum resolution. Comb voltage difference is augmented 100mV per snapshot. After obtaining all snapshots (see Figure 4.8), they are post processed with a simple image processing algorithm.
Figure 4.8: From beginning to the final shot, with two of the intermediate frames, large buckling full frame SEM scans.
This algorithm first thresholds the image with a proper thresholding value to distinguish the substrate and the device layer then by using dilation and erosion of pixels it forms a decent binary image. After the binary image generation, a vertical line starts scanning the scene from a fixed point (the end point of the nearest side gate) and scans until hitting to a white island with adequate pixels. Algorithm saves the scanning distance and convert it into micrometers according to a tuning constant which defines the pixel size that is supplied beforehand. This
process is repeated for every snapshot and obtained travel values are subtracted from the initial travel value to obtain the net displacement of the beam (see Figure 4.9). This method provides a precision of 12.5nm.
Figure 4.9: Post processed large buckling data.
4.4
Snap-through
Snap-through is a less trivial dynamical process when compared with other proof of concept experiments yet it has a great importance to have richer device dynam-ics and to study indirect problems related to buckling. Snap-through is simply the name for interstate jumps that occurs when potential energy landscape is altered or the current state is pumped with some sort of noise source which eventually leads to change of state in the system. For buckling this can be interpreted as jumping between left and right buckling states without dwelling in the neutral state. This is an important phenomenon since it is basically a two-level system with tunable state parameters and it is possible to change state energies or even get rid of one state by an extreme intervention that ends up with a single stable state.
Figure 4.10: a) Potential energy landscape deepening (or shallowing) with respect to comb voltage. b) Potential energy landscape asymmetry with respect to side gate voltage difference.
Here, the comb voltage difference tunes the depth of two stable states in terms of potential energy while the side gate voltage is responsible for the symmetry of the two states (see Figure 4.10). If the beam is largely buckled, stable states sit deeper in the energy landscape with a larger energy barrier between them, other-wise a slightly buckled system has shallower stable wells that are separated with a smaller energy barrier. So, it can be inferred that a largely buckled beam hardly jumps from the current state to the opposite state while the slightly buckled beam is more susceptible to jump between states (see Figure 4.11)1.
For observing snap-through jumps, first the beam is buckled to one side by purpose or randomly. Next, the side gate that is placed on the opposite side of the buckling direction is charged with an opposite charge while the other side gate is kept grounded. This charging can be done abruptly or gradually, and the beam response is expected to change accordingly. Instead of charging oppositely the reciprocal side gate, it is also possible to charge the side gate placed the initial buckling direction with the same sign. As explained before, this configuration creates a lateral attraction towards the grounded side gate with a weaker effect
1Numerical analysis for potential energy landscape of the buckling beam is performed by Mahyar Ghavami, a former member of Hanay Group. For the analysis, beam deflection is modelled with Euler-Bernoulli Beam Theory and solved with Galerkin Method. Original code is modified to generate demonstrated plot.
Figure 4.11: a) Energy landscape for fixed 52.5V comb drive voltage with different side gate voltage difference values ranging from 0V to 20V. After 15V side gate voltage difference, two stable states collapse to the right stable state. b) Energy landscape for fixed 53V comb drive voltage with different side gate voltage dif-ference values ranging from 0V to 20V. For this case, even 20V side gate voltage difference does not disrupt the bistability of the system even though symmetry is drastically broken.
since there is less voltage difference.
4.4.1
Full frame SEM
For snap-through jumps, full frame scanning is a rather slow method and it does not give the full information. Nevertheless, it is a good starting point since it does not depend on any post-processing but a real-time visual feedback is given to the user. While altering side gate voltage with a triangular shaped waveform, jumps are observed and recorded. This method is also used for more evolved experiments to show rate dependent snap-through jumps.
4.4.2
Spot mode SEM
Since this is a binary process (left to right jump or vice versa) it is possible to make detection by placing a spot right at the center, where the beam stays during its neutral state before buckling. This way it is possible to capture the beam jumping one side to the other in a binary fashion. While the beam passing through the neutral axis, it causes more electrons to be collected by the detector. This creates a peak in the stored data which can be further analysed. During the experiment, the side gate actuation voltage is also recorded and this way it is possible to collect information about the reaction speed of the beam compared to the electrical actuation (see Figure 4.12).
Figure 4.12: Snap-through in spot mode SEM. Every side gate voltage change triggers a jump from one stable state to another. Peaks after a change are indi-cations of jumps.
4.4.3
Line mode SEM
Line mode is a very fast method which is capable of supplying information about jumps that are occurring in a couple milliseconds. Since this is a very fast process, using line mode can give various information related to jumps such as reaction time or duration of the jumps. For this example an abrupt side gate voltage is incorporated and resulting jumps are recorded. It is possible to capture the whole transition with 4 or 5 data points thanks to the fast processing of the line scan (see Figure 4.13).
Figure 4.13: a) Snapthrough jumps in SEM line mode after post processing for beam locating. b) A close-up view of two jumps with highlighted data points. Left to right jump occurs very fast compared to right to left jump.
In the processed snap-through data, 4 transitions happening in 5 seconds are demonstrated. In the plot, the asymmetry of the system is visible (see Figure 4.12b). Two main aspects of the plot shows an asymmetry between stable states; deflection amounts and relaxing times. Beam starts buckled approximately 3µm towards the left side and when the side gate voltage is abruptly changed, it passes to the right side but this time settling point is slightly less than 2µm towards right. It shows that the left potential well is farther from the neutral position when compared with the right state well. The second clue is the relaxing time which is smaller for left-to-right jump. For the left-to-right jump there is a slight deflection reduction before the distant jump between states. However
while passing right-to-left, beam is rather moving gradually towards the opposite side (see Figure 4.12b). It may be an indication for the potential barrier and left stable state well characteristics.
Chapter 5
Snap-through analysis
In this chapter, main focus is snap-through phenomenon which is a more involved process since it is directly related with the potential energy landscape of the buck-ling beam. When the compressive force applied to the beam exceeds a critical value – so called the bifurcation point – beam potentially has two stable states when 2D buckling is considered. One can direct the beam to one of these two states by applying a kind of symmetry breaking factor just before exceeding the bifurcation point. However, it is also possible to jump from one stable state to another thanks to dynamical energy landscape control ability of the system and these interstate jumps are called snap-through phenomenon. Snap-through char-acteristics of a beam gives direct information about the current energetic status of stable states and by knowing more about stable states, one can extract more information about crystal structure of the beam. From a different perspective, full control over energy landscape can shed light on the relation between infor-mation and thermodynamics as well as noise induced oscillations of a buckled beam.
5.1
Mechanical bias and memory effect
Even if CAD softwares are highly capable of realizing complex 3D structures in computer environment in an immense precision and symmetry, it is not possible to maintain this level of precision and symmetry after fabrication of designed features. In every fabrication method there are some inevitable sources of imper-fection and this limiting is well applicable to nano fabrication. Given that the buckling beam has a 150nm width which corresponds to roughly 750 atoms from one side to the other for a single layer of atoms – when Si atom diameter is taken roughly 2 ¨A – even one extra atom can affect the system symmetry considerably. By going further, even if the system is fabricated atom-by-atom – which is totally infeasible – to reach a perfect symmetry, it is not possible to maintain that per-fection in normal conditions, without maintaining the environmental conditions strictly controlled. This is the reason why it is not possible to have a perfectly symmetrical system even it is fabricated with state-of-the-art nano fabrication techniques.
These fabrication related errors are extremely effective when the device is be-ing tested for the first time. One defect located at the right side of the beam can direct the beam to the left side by breaking the symmetry between the two potential wells by lowering one energy level while raising the other by introducing a mechanical asymmetry. It is also possible to have an uneven comb frame po-sitioning which also can be a big source of imperfection. All imperfections come together to define an energy landscape that basically defines the post-buckling dynamics of the beam.
Besides imperfections come from the fabrication process, it is possible to ob-serve dynamical changes in the beam originated from the applied forces and experienced strains during buckling. Especially during large buckling scenarios, crystal structure of the beam can be altered drastically which can also affect the mechanical bias. This phenomenon is showed clearly during an experiment (see Figure 5.1).
Figure 5.1: a) Snap-through jumps with triangular wave actuation. During initial four cycles of interstate jumps, transition voltage shifts. After four cycles, beam stays at its final state even if the system is kept being actuated. b) The method to find jump voltages. First jump occurs at -1V while the consecutive jump occurs at -11V.
In the experiment, a triangular wave with a period of 8 seconds and peak to peak 20 volts is applied to the right side gate while the beam is also charged with -32 volts and buckled to the right side. Beam buckles right by a native mechanical bias without any side gate attraction. After approximately 2µm of maximum deflection, triangular wave is given to the system. While the voltage of the right side gate approaches to the beam voltage, grounded left side gate starts to attract the beam more. At one point a right to left jump is observed and while the side gate voltage is approaching 0V, a jump back to the right stable state is observed. This back and forth jumping occurred 4 times before the beam stays at the right side and does not react to the rising side gate voltage. Until the last pass, it is possible to see a shift in the jump voltages which indicates a gradual increase in biasing. This is a clear demonstration of the dynamical biasing created by the applied strains. When considered that silicon is a brittle material, this kind of a plastic deformation is an unexpected phenomenon. Experiment is done by using spot mode SEM to detect jumps and gate voltage is recorded in real-time.
