FLUCTUATION PHENOMENON IN A NONLINEAR MICROELECTROMECHANICAL OSCILLATOR

Transcription

1 FLUCTUATION PHENOMENON IN A NONLINEAR MICROELECTROMECHANICAL OSCILLATOR By COREY ALAN STAMBAUGH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 c 2009 Corey Alan Stambaugh 2

3 I dedicate this to those that sacrificed to provide me with the opportunity at a better life 3

4 ACKNOWLEDGMENTS I would like to first thank my advisor Prof. Ho Bun Chan. When I first started working in Prof. Chan s laboratory he had only been at the University of Florida for one semester. Nonetheless, he was able to provide me with the dissertation project presented here. Over the last five years, Prof. Chan has provided me with the support I have needed to succeed. This includes taking me on as a Graduate Research Assistant, providing me with all the equipment needed to perform my research, and providing the financial support needed to attend several APS conferences. I would also thank him for the years of mentoring which were and will be invaluable in my scientific pursuits. I would like to thank Prof. Dykman of Michigan State University. Prof. Dykman provided the theoretical basis for the majority of my experiments. More importantly, he provide his time over the last five years to help me understand his work. Finally, he provided an additional voice of guidance that I am very grateful for. I thank all of my committee members, Prof. Stanton, Prof. Hebard, Prof. Tanner, and Prof. Arnold, for taking the time to help me fulfill my requirements for graduation. I owe a deep amount of thanks to the technical staff. The machine shop members Marc Link, Bill Malphurs and Ed Storch provided me both their time and skills in constructing the precision parts needed for several experiments. Also the cryogenics lab, Greg Labbe and John Graham, for constantly and quickly providing filled dewars of liquid nitrogen and liquid helium. To both of these groups a special thanks for the extra time given to help me learn all the little things. I wish to thank my officemates over the years, Yiliang Bao and Zsolt Marcet, for making the hard days bearable and my other lab members Kostas Ninios and Jhe Zou for their constant help. Finally, a thanks to my family for accepting and supporting my decision to pursue this career, even though it means being so far away. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF FIGURES ABSTRACT CHAPTER 1 FLUCTUATIONS IN A NONLINEAR SYSTEM Introduction Interstate Switching in Multistate Systems Kramers Rate for a System in Thermal Equilibrium Activated Escape in a Nonlinear System Driven Out of Equilibrium Universal Features in a Nonlinear System Most Probable Switching Path Critical Exponent Fluctuations Near a Kinetic Phase Transition Fluctuation Relations Summary THE EXPERIMENTAL SYSTEM Fabrication of MEMS Device Preparation and Electrical Connections Torsional Oscillator Excitation and Detection Schemes Summary DEVICE RESPONSE Linear Device Response Duffing Oscillator Parametric Oscillator Summary PATHS OF FLUCTUATION INDUCED SWITCHING: MPSP Introduction Qualitative Picture and Preview of the Results Theory of the Switching Path Distribution Micromechanical Torsional Oscillator Device Characteristics Transformation to Slow Variables and Parametric Resonance Determination of Device Parameters

6 4.5 The Switching Path Distribution Experiment Measured Switching Path Distribution Generic Features of the Switching Path Distribution Lack of Time Reversal Symmetry in a Driven Oscillator Conclusions SCALING OF THE ACTIVATION BARRIER Introduction Critical Exponent Near a Spinodal Bifurcation Critical Exponent Near a Pitchfork Bifurcation Summary THE KINETIC PHASE TRANSITION Introduction Kinetic Phase Transition Spectral Densities of Fluctuations Frequency Mixing Summary FLUCTUATION THEOREM Introduction Device and Experimental Setup Theory Results Summary SUMMARY APPENDIX A PROCEDURE USED TO ANALYZE RESULTS A.1 Trajectory A.1.1 Theory A.1.2 Experiment A.2 Activation Barrier Scaling A.2.1 Extracting Switching Times for Duffing Oscillator A.2.2 Extracting Switching Times for Parametrically Driven Oscillator A.2.3 Determining Critical Exponent A.3 Fluctuation Spectrum A.4 Work Fluctuations B DETERMINATION OF SYSTEM PARAMETERS B.1 Linear Oscillator B.2 Duffing Oscillator

7 B.3 Parametric Oscillator REFERENCES BIOGRAPHICAL SKETCH

8 Figure LIST OF FIGURES page 1-1 Hysteresis in ferromagnet Hysteresis in Duffing oscillator Basins of attraction Cubic potential Quartic potential Basic steps in MEMS fabrication SEM of MEMS torsional oscillator Capacitance dependence on d.c. voltage Cross-sectional schematic of oscillator and electrical setup Electrical setup for heterodyne measurement Response of linear oscillator Response of Duffing oscillator Response of Duffing oscillator to noise Response of parametric oscillator Time-translation symmetry in response to parametric drive Response of parametric resonance to noise Phase portrait of a two-variable system with two stable states Switching probability distribution Harmonic and parametric resonances Measured averaged velocity along the MPSP Conservation of the stationary probability current Comparison of the MPSP and the dissipation-reversed path Duffing oscillator Switching in Duffing Oscillator Activation energy for a Duffing oscillator

9 5-4 Scaling of actibation barrier Response of oscillator to parametric drive Switching in a parametrically driven oscillator The occupation in phase space at four different driving frequencies Transistion rate in parametrically driven oscillator Scaling of the activation barrier Kinetic phase transition Dependence of the intensity of the supernarrow spectral peak on the driving frequency Power spectral density of fluctuations, view I Power spectral density of fluctuations, view II The supernarrow peak, view I The supernarrow peak, view II Response and switching in Duffing oscillator Distribution of states in Duffing oscillator Power spectral density of fluctuations Frequency mixing Switching at optimal noise level Stochastic resonance Work variance for linear oscillator Independence of work variance on driving frequency Work variance and noise intensity in Duffing oscillator Exponential dependence of work variance noise intensity

10 Chair: H. B. Chan Major: Physics Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FLUCTUATION PHENOMENON IN A NONLINEAR MICROELECTROMECHANICAL OSCILLATOR By Corey Alan Stambaugh May 2009 This dissertation details my research on the effects of fluctuations on vibrational systems driven out of equilibrium. Under the right conditions such systems will exhibit multiple stable states. When the fluctuations are large enough, the systems may overcome an activation barrier and switch from one stable state to another. The transitions lead to large changes in the system s vibrational amplitude. These changes can have a large impact on the performance and behavior of the system. The activation barrier plays a major role in determining how the system reacts to noise. We study its dependence on the system parameters, driving amplitude and driving frequency. Previous theoretical work has predicted that the barrier should display specific dependences, on these parameters, that are independent of the system. In this dissertation we test these predictions in our system, along with examining other features that systems possessing multiple stable states and driven out of equilibrium are expected to show. The system used in this study is a micro-electromechanical torsional oscillator. The device is fabricated using a standard commercial process that is reliable and customizable. The device provides a robust system that can be operated at different temperatures, pressures and in different resonant modes. When the damping of the system is small enough and the drive is large enough, the system displays multistable states. The micro-electromechanical torsional oscillators used in these studies are ideal apparatus 10

11 for studying the transitions that occur between states of a multistable system when fluctuations are present. 11

12 CHAPTER 1 FLUCTUATIONS IN A NONLINEAR SYSTEM 1.1 Introduction Over the last few decades, micro- and nano-technology has revolutionized nearly all of the experimental sciences. Indeed, the development of small scale systems has had far reaching effects in biology, chemistry, applied physics, and engineering [1 3]. The trend in these disciplines is toward the development of systems whose sizes are on the atomic scale. However, as devices become smaller, the role that fluctuations play in their behavior becomes increasingly important. Fluctuations, commonly referred to as noise, play an important role in a wide variety of systems. Noise could arise from either dissipative processes that connect to a thermal reservoir or the discreteness of particles that make up the flow of charge. For example, Johnson noise results from the thermal motion of the electrons across a resistor. In this case the resistor is the dissipative element. Noise resulting from thermal fluctuations exists in all systems at finite temperature. Small systems, in general, are particularly susceptible to these fluctuations because the energy associated with the fluctuations can often be on the same scale as the energy of the system. Even at temperatures of a couple of Kelvin, the fluctuations in voltage introduced by the thermal motion can be large enough to make precision measurements difficult. On the other hand, noise can be advantageous in certain cases. For example, in the phenomenon of stochastic resonance [4] increases in the noise can actually lead to better signal-to-noise ratios. The role of fluctuations in systems possessing two or stable states is an area of interest in a variety of fields. For systems in thermal equilibrium a stable state is a local or global minimum of the free energy. A very simple example would be a particle placed in a deep well. Clearly, here the stable state is the bottom or minimum of the well. At finite temperatures, the particle fluctuates about the minimum of the well due to thermal motion. If the disturbance becomes large enough, however, the particle can escape out of 12

13 the well in a process called fluctuation induced escape. This phenomena, which has been extensively studied for equilibrium systems [5], includes such processes as protein and RNA folding [6], chemical reactions, and nucleation [7]. Quite often systems displaying two more stable states also display hysteresis. In hysteretic systems, the response depends on the history of the system. A well known example of such a system is the magnetization of a ferromagnet. Here, the magnetization is dependent on the applied magnetic field. When the field H is increased past a critical value, the total magnetization saturates at M. As the field is decreased the magnetization does not decrease in a proportional manner. Instead, the magnetization remains finite, even when the applied field becomes zero. At a negative critical applied field the magnetization saturates at -M. The response, shown in Fig. 1-1, has two states which can be accessed by correctly adjusting the applied field. The hysteretic dependence of the M on H is indicated by the arrows in the figure. The on/off or ±M behavior seen here is utilized in the storage industry for magnetic tapes and hard drives since the system can, in a controllable manner, be moved from one state to the other. In recent years, attention has turned to nonlinear systems which, when driven by a sufficiently strong periodic modulation, display hysteresis and thus possess two or more stable states. An example of the hysteresis typically seen in this type of driven system, called a Duffing oscillator, is shown in Fig As in the case of the magnetization of a ferromagnet, a control parameter like the applied magnetic field exist. For the Duffing oscillator this is typically the driving frequency. The parameter can be tuned to move the system around the hysteretic response, providing access to the different stable states. Within the hysteresis region, fluctuations in the system can induce it to switch from one stable state to another without having to first move along the hysteresis loop. These switches would be analogous to thermal fluctuations inducing the magnetization, in the previous example, to switch from M to -M while H=0. So while the fluctuations may, on average be weak, the transitions can lead to large changes in the response. These 13

14 M H Figure 1-1. Hysteretic response of a ferromagnet to an applied magnetic field. Depending on the history of the applied magnetic field, the magnetization at zero field can be ±M. large changes can produce unwanted behavior in the system, such as a loss of storage in magnetic memories. However, they can also be used in an advantageous manner. As already mentioned, this is a prime example of a system where stochastic resonance could be utilized, leading to possible sensing applications in the presence of noise. The type of nonlinear, periodically driven system mentioned is far from equilibrium and its behavior is not as well understood. While theoretical and simulation work does exist dealing with the variety of features this type of system exhibits [8 13, 13, 14], only recently, with advances in micro- and nano-scale fabrication, have serious experimental studies been pursued. The experimental systems include atoms in a magneto-optical trap [15], electrons in a penning trap [16], micro- and nano-mechanical oscillators [17 19] and rf-driven Josephson junctions [20]. These system are used to investigate fundamental science and applications in a wide range of fields. Nevertheless, they share the common 14

15 Amplitude ω/ω 0 Figure 1-2. Hysteretic response of the Duffing oscillator, a periodic modulated system with a cubic nonlinearity. The amplitude of the oscillatory response depends on the history of the driving frequency ω. characteristic that when an external periodic modulation is applied, they all display hysteresis. The hysteresis they exhibit can be described by the same equations of motion and according to theory, they are expected to exhibit specific universal properties near their critical points. In general, a critical point is where a phase boundary in a system ceases to exist, for example a bifurcation is a critical point where the number of states changes. The behavior near the critical points is expected to be system independent; a remarkable prediction given the apparent differences in these systems. In this dissertation we focus on one particular device, the micro-electromechanical torsional oscillator. Devices of similar designs have been used in a wide range of experiments, including the measurement of the Casimir force [21] and the measurement of the magnetization of novel materials [22]. The torsional oscillator provides a very 15

16 robust platform to test several theoretical predictions for the following reasons. Its system parameters (mass, spring constant,...) can be experimental determined by making rather basic measurements. They can also be calculated to a reasonable degree of accuracy. The system can be operated in a variety of conditions allowing experiments to take place in different temperature environments. The fabrication process provides the ability to make devices with a wide range of system parameters. Also, the fabrication process is a batch process that produces nearly identical devices. This allows for experiments to be repeated for consistency checks. In addition, the batch process and small scale of the devices provides a possible cost effective design procedure for the implementation of the devices for commercial applications. From a practical standpoint, understanding the hysteretic response and the role that fluctuations play in the oscillator is essential in order to exploit the nonlinear oscillator for sensing applications. For instance, the bifurcation in the Josephson junction [23] is already being used to design amplifiers for the reading of quantum bits. As will be discussed in Ch. 6, a stochastic resonance like response was seen in our torsional oscillator. Possible applications could include tunable narrow band filtering [24] or frequency mixing [25]. As we will find in Ch. 5, the probability of switching out of a stable state is exponentially dependent on an activation barrier. The activation barrier has a strong dependence on the system parameters. This could be taken advantage of in detection schemes based on shifts in resonant frequency like mass sensing [26]. Here a very small change in mass, which would shift the resonant frequency, would produce an exponentially sensitive change in the probability of switching. The work performed through out this dissertation lays the ground work for a host of possible uses. A better understanding of these types of systems will also benefit the science community in general, since the behavior we study is expected to be system independent. In the next section of this chapter the groundwork for understanding fluctuation induced escape is laid out in more detail. In the final section of this chapter we introduce 16

17 the different features that are studied in the dissertation. These include experimental results that are consistent with several theoretical predictions for systems that display two or more coexisting states when driven far from equilibrium. All of these results, to the best of our knowledge, are the first experimental verification of the predictions outside of circuit models. The results include the mapping of the most probable switching path (MPSP), the determination of critical exponents for switching in different types of bifurcations, and the analysis of several phenomena near the kinetic phase transition of the system. 1.2 Interstate Switching in Multistate Systems In the presence of fluctuations, transitions between coexisting states can occur. A classic problem in diffusion theory, the calculation of the rate of transitions between stable states separated by a barrier, was done by Kramers for the case of a Brownian particle [5]. Here the barrier height needed to be overcome for a transition to occur is typically given by the free energy of the system. In this dissertation we focus on the more difficult problem of transitions when the potential of the system has a periodic time dependence and the barrier height can no longer be characterized by the free energy of the system. In this case the system is far from equilibrium and lacks detailed balance [8]. Detailed balance [27] is a condition that system in thermal equilibrium satisfy. An in depth explanation of the principle can be found in Ref. [27]. In the case of a system which possess two stable states, the two attractors (A 1 and A 2 ) are separated in phase space by a separatrix, see Fig Here, attractor refers to a stable state. The word attractor is used to mean a point to which the system is attracted and to which the system returns after a disturbance. The attractor need not be a point. It could also be a path the system moves along in a periodic fashion. As shown in Fig. 1-3, the separatrix (dashed line) is a boundary that defines the basins of attraction. A basin of attraction is comprised of a set of points (q 1, q 2 ) from which a system evolves into an attractor. For example, in Fig. 1-3 all points that lead to attractor A 1 are shaded 17

18 1 0.5 q 2 0 S 0.5 A 1 A q 1 Figure 1-3. The basins of attraction for a system possessing two stable states. The coordinates q 1 and q 2, introduced in Sec , are the in and out of phase coordinates of the system in the rotating frame. The white region indicates the basin of attraction, where A 2 is the attractor. In the absence of noise a system prepared within the white region will relax to A 2. The gray region represents the other basin of attraction. The basins are separated in phase space by the separatrix (dashed line). The saddle node S lies on the separatrix. During a fluctuation induced escape the system is most likely to pass through the saddle during the transition to the other state. in gray. In the absence of fluctuations, the system relaxes to either attractor A 1 or A 2. On this line is a saddle node S. The saddle node is the location of the unstable state. As will be shown later, during a fluctuation induced escape the system goes over the separatrix at the saddle node as it switches into another state. If the system is initially set within the gray region of Fig. 1-3, in the absence of fluctuations, the system will move to the attractor A 1 in a characteristic relaxation time t r. This time is related to the dissipation in the system. Now, in the presence of weak fluctuations the trajectory the systems takes to the attractor is no longer deterministic but on average will be the same path. Once near the attractor, the system will fluctuate about it for a time much greater than t r. This type of fluctuation will be referred to as an intrastate fluctuation, since it involves only fluctuations associated with moving within the state. Given a sufficient length of time, a large enough fluctuation will occur moving the system up towards the saddle. Ultimately, it will either fall back to the initial attractor or continue over the saddle; where it will 18

19 relax over a time t r into the other attractor. The system undergoes a fluctuation induced escape when it continues over the saddle and into another state. The fluctuations in the response associated with the switching event are known as interstate fluctuations. In the rest of this section we consider the rate Γ at which transitions occur. Weak fluctuations are considered so that t r Γ 1. The probability of switching between states is not always equal and the ratio of the individual rates provides the relative occupation between the states. In Sec a Brownian particle is considered. Here the distribution is determined by the Boltzman formula. In Sec , the more complex problem of a periodically driven, nonlinear potential is used to address the issue of transition rates in the case of weak noise Kramers Rate for a System in Thermal Equilibrium We start with a simple model, a Brownian particle in a one dimensional static cubic potential. We assume here and throughout this dissertation that all the systems discussed are classical. Therefore we will not be considering tunneling between states as a viable mechanism for switching between states. The motion of the particle can be described by the Hamiltonian with a unit mass: H = 1 2 p2 + U(q), where U(q) = 1 3 q3 + ηq. (1 1) Here q and p are the coordinate and momentum of the oscillator. The variable η represents a control parameter of the system. By adjusting η the potential U(q) can be changed. In Fig. 1-4 the cubic potential, with a Brownian particle sitting at an attractor, is plotted. If the system is connected to a thermal bath there will be both dissipation and fluctuations. The thermal noise causes the particle to move about the minimum of the well. In order for the particle to escape from the stable state it must overcome a barrier U which, for systems close to thermal equilibrium, is typically the free energy barrier. Occasionally, at a rate Γ ij = const e U/k BT (1 2) 19

20 given by Kramers [5], the particle will switch out of the stable state i and into another state j. For the cubic potential U = U(q s ) U(q a ) = 4 3 (η η c) 3/2 with q a = η 1 2 and q s = η 1 2. (1 3) The parameter η c represents a critical point in the system. From Eq. 1 3 it is evident that changing the value of the control parameter η will alter the height of the barrier U. As η approaches η c the height of the barrier decreases and i j transitions are more probable. When η = η C the well no longer exists. For the cubic type potential the merging of the unstable and stable state occurs, while the other state stays far way. By far away, we mean that the other state does not also merge with the unstable state. This type of bifurcation is called a spinodal bifurcation. Looking back at Eq. 1 2 for the cases of η > η c, we consider the situation of placing many non-interacting particles into the well of the cubic potential. By increasing the temperature, the fluctuation of each particle also increases. Therefore on average, more particles can escape during a given period of time. The probability of escape for the particles is a Poisson process, where each switch is independent of another switch and switches do not occur simultaneously. A familiar example of such a process is the decay of radioactive atoms. The example examined here was for a particle switching out of a state i and into another state j. This other state may be another stable state where the probability of switching back is low, such that Γ ij Γ ji and transitions from j i can be considered not to occur on the time scales of interest. Several of the experiments we performed satisfy this requirement, where the probability of switching from i j is much larger than switching from j i. Another potential of interest is the quartic potential: U(q) = 1 4 q4 ηq 2 with q a = ±η 1 2 and q s = 0. (1 4) 20

21 η η c = 0.2 η η c = 0.6 η η c = 1.0 q s U(q) 0 U q a Figure 1-4. A Brownian particle trapped in a cubic potential. The control parameter η changes the depth of the well and the height of the barrier U. As η approaches η c the two states q s and q a also approach each other. When η = η c, the two states merge and a spinodal bifurcation occurs. Thermal fluctuations can induce the particle to jump out of the stable state by overcoming the barrier (Eq. 1 3). 0 q In this potential (see Fig. 1-5) there are two identical stable states separated by the barrier. The barrier heights are identical for both states as long as η > η c, so Γ ij = Γ ji. This is in contrast to the case seen for the cubic potential where Γ ij Γ ji. Because changing η alters the barrier height for both wells in the quartic potential equivalently, as η approaches η c the two attractors q a,1 and q a,2 move towards the unstable state. At η = η c all three states merge together in a pitchfork bifurcation. 21

22 η η c = 0.2 η η c = 0.6 η η c = 1.0 U(q) 0 q s U q a,1 q a,2 Figure 1-5. A Brownian particle trapped in a quartic potential. The control parameter η changes the depth of the well and the height of the barrier U. As η approaches η c the three states q s, q a,2, and q a,1 also approach each other. When η = η c, the three states merge and a pitchfork bifurcation occurs. Thermal fluctuations can induce the particle to jump out of the stable state by overcoming the barrier (Eq. 1 4). 0 q The different types of bifurcations seen in these two models will play an important role in later chapters. In addition, the qualitative description of the stable states as potential wells and the free energy as a barrier to be overcome during a switch provides an intuitive picture of fluctuation induced escape. Strictly speaking, this is true even in the cases that will be examined in this dissertation, where the system is far from equilibrium and the barrier height can no longer be described by the free energy. As a note, the stable states in the torsional oscillator manifest themselves as distinct mechanical amplitudes 22

23 of oscillation at different phases. Still, the intuitive picture of the stable states as wells is applicable Activated Escape in a Nonlinear System Driven Out of Equilibrium In the previous section transitions out of a stable state for a system in thermal equilibrium were examined. Here a more complex problem is analyzed, transitions between states of a system driven out of equilibrium. We consider a periodically driven, nonlinear oscillator commonly referred to as a Duffing oscillator. The Duffing oscillator is a good choice as it can be used to describe the general behavior of most of the systems mentioned in the introduction. The Hamiltonian for a such a system is H = 1 2 (p2 0 + ω 2 0q 2 0) γq4 0 qf 0 cos ωt. (1 5) Here q 0 and p 0 are the coordinate and momentum, ω 0 is the resonant frequency, and γ describes the strength of the nonlinearity. We assume that there is phenomenological damping λ in the system, which could result from a variety of sources, and that it is much less than the driving and resonant frequencies; therefore we can transform to slow variables in the rotating frame. Introducing the in and out of phase dimensionless coordinates q 1 and q 2, we can express the variables q 0 and p 0 as where C = const g(q 1, q 2 ) where q 0 (t) = C [q 1 cos(ωt) q 2 sin(ωt)] p 0 (t) = Cω [q 1 sin(ωt) + q 2 cos(ωt)] 3 γ. In these variables the Hamiltonian can be written as H 2ω ω ω 0 = (1 6a) (1 6b) g(q 1, q 2 ) = 1 4 (q2 1 + q 2 2 1) 2 βq 1 (1 7) and β is the effective amplitude squared. Using the conventional averaging method [28] terms of exp inωt, where n 2, were ignored. Now the Langevin equation, which 23

24 describes the Brownian like motion in our potential, can be written as q 1 = K q1 + f(τ) with K q1 = ηq 1 + q2 g (1 8a) q 2 = K q2 + f(τ) with K q2 = ηq 2 + q1 g. (1 8b) Here the dissipation has been accounted for using the dimensionless friction parameter η = λ/ ω ω 0, τ = ω ω 0 t is the dimensionless time, and the applied noise f is Gaussian white noise or f(τ)f(0) = 2λDδ(τ). The term D is a dimensionless noise parameter, where for weak noise D 1. In the absence of fluctuations, the vector K determines the system dynamics and thus the attractors can be found by setting K q1 = 0 and K q2 = 0. These types of nonlinear systems, which are driven out of equilibrium, generally lack detailed balance. Therefore Kramers equation no longer directly applies. It has been shown [16, 19, 20, 29 32], however, that the rate of escape still maintains an activation type behavior: Γ ij e Ri/D. (1 9) The transition rate depends on the noise D which can originate from many sources including thermal, electrical and mechanical. Typically, the fluctuations have a small amplitude. In order for the system to go over the activation barrier R i a large fluctuation is needed. The determination of R i involves finding the optimal noise that is most likely to induce a transition. A detailed derivation can be found in Ref. [8, 33]. Essentially though, by changing the problem from one depending on the noise to one depending on the trajectory, the activation barrier can be written as R i = min dτl( q, q; τ), L = 1 4 ( q K)2, K = (K q1, K q2 ), and q = (q 1, q 2 ). (1 10) This change can be done using the Feynman path integral formulation [34] where each realization of the noise has a path associated with it. Thus the optimal noise should 24

25 have a corresponding optimal path that is taken during a transition. Here the problem of finding R i is reduced to a variational problem that involves finding the optimal path that minimizes the integral of the Lagrangian L( q, q; τ). The path is the uphill portion of the most probable switching path (MPSP). In this section an equation for the activation barrier was found. The solution is not trivial, unlike the case in Section Through out this dissertation the activation barrier and its effect on the probability of a transition occuring is examined. While the problem is complex we find that the intuitive model given in Section can be applied to gain a qualitative understanding for the different behaviors being studied. 1.3 Universal Features in a Nonlinear System In this dissertation specific features predicted to appear in systems driven far from equilibrium, possessing multiple stable states, are examined. In the following sections the different general characteristics analyzed are presented Most Probable Switching Path When a system switches from one attractor to another, it follows an optimal path in phase space. In a switching event, the system moves from one attractor, over the saddle point separating the stable states, and into the other attractor. While each switching event does not necessarily exactly follow the optimal path; in general, the trajectory taken does center around it. The optimal path is also crucial in finding the solution for the activation barrier (see Eq. 1 10), a nontrivial problem for systems lacking detailed balance. This result is expected to be independent of the type of system and should only rely on a couple of determinable system parameters. In Chapter 4 we show results experimentally verifying the existence of the MPSP. Here we calculate the MPSP, then perform switching experiments to determine if the actual trajectory taken during a switch coincides with the calculated MPSP. We find our experimental results, using no adjustable parameters, to be in excellent agreement with the calculated path. This experiment marks the first experimental proof, for a 25

26 multivariable system, of the MPSP. Several other generic properties are also identified. Finally, we conclude by demonstrating that the system is out of equilibrium, in that it lacks detailed balance, by showing that the fluctuation induced uphill path does not coincide with the time-reversed, deterministic downhill path. For systems near equilibrium the uphill and downhill path are expected to coincide Critical Exponent Analytical solutions for the activation barrier of a system driven out of equilibrium are difficult to find. Unlike systems near equilibrium where the free energy can be used, here numerical calculations of the variational problem (Eq. 1 10) need to be done in order to determine the activation barrier. This is typically a difficult problem. Instead, scaling laws have been identified which predict how the activation barrier should depend on general system parameters. The activation barrier in these cases can written as R = φ(η η c ) ξ, (1 11) where ξ is the critical exponent and η c represents a critical point in the system. While generally the prefactor φ depends on the particular system, near bifurcation points the exponent is expected to be system independent. In the first part of Ch. 5 the scaling of the activation barrier near a spinodal bifurcation is considered. A spinodal bifurcation occurs when a stable and unstable state merge together. This case occurs, for example, in the cubic potential (shown in Fig. 1-4) when η approaches η c. Just prior to merging the motion in the rotating frame [33] is characterized as overdamped. Here the effective potential becomes shallow and the particles motion is slowed. Theoretical analysis has established that the activation barrier for fluctuation induced escape should scale near a spinodal bifurcation point [33, 35] with critical exponent ξ = 3/2. This scaling relationship is supposed to be system-independent. In Chapter 5 experimental results are shown for the first observation of this scaling in a periodically driven system outside of analog circuit simulations. 26

27 Another critical exponent is expected when two stable states merge with an unstable state. This pitch-fork bifurcation is seen in the quartic potential (see Fig. 1-5). Like the previous case, this system is also far from equilibrium, in that multiple stable states only exist when the system is strongly driven. Unlike the previous experiment however, all three states merge together at a critical point and a different critical exponent is expected. Theoretical analysis predicts that the critical exponent, near the critical points, will be ξ = 2. This exponent is expected regardless of whether the motion is overdamped or underdamped [36]. Additional system dependent scalings away from the two bifurcation frequencies are also investigated. The critical exponent of the system will be discussed along with experimental results in Chapter Fluctuations Near a Kinetic Phase Transition For a particularly small set of system parameters the occupation between states is expected to be comparable within the hysteresis region of the Duffing oscillator. This indicates that the activation barriers must be approximately the same since the ratio of the rate of transitions between the two states is an exponential of the difference between the activation barriers [37] or w 1 w 2 = const e (R 1 R 2 )/D, w 1 + w 2 = 1. (1 12) In Eq the probability of finding the system in state i is given by w i = Γ ji /(Γ ij + Γ ji ). This particular region is identified as the kinetic phase transition [37]. The term is used to describe a nonequilibrium phase transition [38], partially because the phenomena exhibits generic properties that share similarities with the phase transition of a thermal equilibrium system. The first is that in a thermal equilibrium system the system will usually be in one of two phases, such as water, or another, vapor. Only at the phase transition will the two phases coexist. The other indication of a phase transition in a thermal equilibrium system is the appearance of large fluctuations. In a thermal 27

28 equilibrium system this occurs, for example, at the phase transition of water and vapor and is referred to as opalescence [39]. In Chapter 6 the region where the two states are equally likely to be occupied is investigated. Here the spectral density of fluctuations, which describes how the fluctuations are distributed with frequency, is analyzed and a large super narrow peak in the distribution is identified at the kinetic phase transition. Additionally, the constructive role noise can play in the mixing of frequency is covered. By adding a small secondary modulation signal, at a frequency near the driving frequency, we find that the switching between states becomes synchronized for a range of noise intensities. The synchronization leads to an enhanced signal-to-noise ratio for non-zero noise levels; a phenomenon associated with stochastic resonance Fluctuation Relations The final topic, covered in Chapter 7, involves fluctuation relations for systems out of equilibrium. The work done on a driven system is calculated for two cases: (a) when a weak drive is applied, so that the response is linear, (b) for a strong drive so the system response is nonlinear. Noise is applied and the ratio of the variance of the work over the mean work is calculated. The linear case is shown to be consistent with the fluctuation-dissipation relation. However for systems displaying bistability the relation no longer holds; yet, universal features are still expected. We examine the ratio of the variance to the mean work near the kinetic phase transition and find an exponential dependence on the driving frequency, a result consistent with theoretical predictions. 1.4 Summary In this chapter the impact that micro- and nano-technology has had on science was discussed in the context of nonlinear systems and fluctuations. An area of particular interest, periodically driven systems possessing multiple stable states, was introduced. When fluctuations are added to such systems they can be induced to switch between the available states. The rate of transitions is given by Eq. 1 9 and primarily involves 28

29 the intensity of the fluctuation and the height of the activation barrier separating the states. Additionally, the solution of the activation barrier for a system driven out of equilibrium is non-trivial. However, in certain situations it scales with the parameters in a system-independent manner. Several universal features are expected to exist in all systems near critical points. In the following chapters detailed descriptions of these features and experiments performed to analyze them are discussed. 29

30 CHAPTER 2 THE EXPERIMENTAL SYSTEM All experiments in this dissertation were performed using micro-electromechanical systems (MEMS). MEMS are structures, fabricated on wafers, with typical sizes on the order of tens to hundreds of microns and are used to perform tasks such as sensing, actuating, and signal acquisition. Over the years MEMS have grown to encompass a large category of small systems including those involving thermal, magnetic, fluidic and optical domains [40]. One common feature among MEMS is that they act as transducers converting energy from one domain to another, for example thermal to mechanical. In the case of this dissertation, electrical energy is converted to mechanical and then back to electrical. The small scale of MEMS allow for studying realms otherwise difficult to investigate on the macroscale; this includes studying the effect of small fluctuations on small systems. In this chapter the fabrication of MEMS is discussed. The type of device used to carry out the experiments performed in this dissertation is presented and different schemes for actuating and sensing the oscillations of the device are examined. 2.1 Fabrication of MEMS MEMS are typically fabricated using photolithography based techniques. Most of the processes are borrowed from the microelectronics industry and fine tuned for the micromachining process. The batch processing found in the microchip industry also carries over to MEMS devices. This allows production of many, nearly identical, devices whose performance are similar up to fabrication and processing errors. All of the devices used in the dissertation were fabricated at the commercial foundry MEMSCAP R, using a surface micromachining process known as PolyMUMPS [41]. The PolyMUMPS process begins with a silicon wafer which has a thin (600 nm) silicon nitride layer deposited on it (Fig. 2.1a). The ground layer, composed of polysilicon, is patterned using standard photolithography techniques. This involves, first, depositing a layer of polysilicon using low pressure chemical vapor deposition (Fig. 2.1b). Next, a 30

31 layer of photoresist is spun (Fig. 2.1c). A photomask is then placed above the photoresist and ultraviolet light is shined through the mask and onto the photoresist (Fig. 2.1d). The photomask determines which areas of the photoresist are exposed to the light. Depending on the type of photoresist used the exposed areas either become more soluble or less soluble. The appropriate developer is then used to wash away the undesired regions of photoresist. Next a reactive-ion etch (RIE) is done to etch away the exposed regions of the polysilicon (Fig. 2.1e). The polysilicon comprises the bottom conductive structural layer. The remaining photoresist is then removed (Fig. 2.1f) and the first structural layer is complete. The process is then repeated, except now a sacrificial layer of phosphosilcate glass (PSG) is patterned. Again, the process is repeated for the second layer of polysilicon, the second layer of PSG, and a third layer of polysilicon. Finally, a gold film is patterned for electric connections to the device. The final product involves layers of polysilicon and PSG. In this state the sacrificial layers of PSG keep the layers of polysilicon, which may be intended to be able to move freely, in a fixed position until the product reaches the consumer. 2.2 Device Preparation and Electrical Connections After receiving the dies from MEMSCAP R several steps need to be performed in the laboratory before the device is ready for use. Each die is composed of several chips which contain numerous devices. Using a dicing saw the large chip is cut into smaller segments each measuring 2.5 cm 2. The small chip, which contains several devices, is then soaked in acetone to remove any residual photoresist. Next, the sacrificial PSG layer is etched away using 49% concentrated hydrofluoric acid. This wet etch is isotropic and typically etches around 40 µm every 5 minutes [42]. After the removal of the PSG, the suspended polysilicon structures are free to move. If the chip is allowed to dry in air these layers, which have a gap of 2µm between them, may stick together as a result of the capillary effect [40]. To avoid this problem a critical point dryer is used. The device, after the wet etch, is submerged into methanol and placed into the chamber of the critical point dryer. 31

32 (a) (b) (c) Nitride Silicon (d) PolySi Ultraviolet Light Photoresist (e) E (f) Figure 2-1. The following steps are repeated for each structural and sacrificial layer in the PolyMUMPS process. (a) Start with silicon (red box) with a 600 nm thick layer of nitride (green box). (b) A layer of polysilicon (orange box) is deposited using LPCVD. (c) A layer of photoresist (blue box) is spun on top. (d) Shine ultraviolet light through a photomask, exposing only certain areas of photoresist and use developer to wash away unexposed regions of photoresist. (e) A reactive-ion etch is performed to remove unwanted regions of polysilicon (orange box). (f) Remaining photoresist can be removed, leaving desired structure. 32

33 After the chamber is cooled and pressurized the methanol is replaced by liquid carbon dioxide. The temperature and pressure are subsequently raised and the liquid undergoes a transition to a gas by going around the critical point; thus avoiding the phase transition. This process avoids the stiction problem which results from the capillary effect that occurs when the liquid evaporates. After drying, the device is mounted in a 16 pin ceramic package using silver proxy. Using a wire bonder, gold wires are bonded between the bonding pads on the device and gold pads on the package. The package is then inserted into a receptacle located at the bottom of a probe. The probe is made of a long steel tube ( 2 m) which has shielded wires run down the inside of it. At the bottom of the probe the wires are soldered to the receptacle holding the package. At the top of the probe the wires are soldered to several BNC feedthroughs. Using these feedthroughs the electrical connections can be made to the device at the bottom of the probe. A vacuum tight seal between the bottom of the probe and cap is maintained by a rubber o-ring or an indium o-ring [43]. In this state the sample space can be pumped down to less than 10 6 Torr. Using the indium seal the probe can also be submerged in liquid nitrogen (77 K) or liquid helium (4 K). A main advantage for measuring at low temperatures is the reduction of temperature fluctuations. The fixed temperature removes temperature dependent drift in the resonant frequency of the oscillator. The vacuum reduces viscous damping so the oscillator is underdamped. 2.3 Torsional Oscillator The MEMS used in this experiment was a torsional oscillator. The oscillator, shown in Fig. 2-2, was fabricated using the PolyMUMPS process described in Section 2.1. This particular device consists of three main parts: a movable top plate, two electrodes positioned directly beneath the top plate, and two torsional springs which are used to suspend the top plate 2 µm above the electrodes. The movable top plates used have dimensions of either 500 µm 500 µm 3.5 µm or 200 µm 200 µm 3.5 µm. The mass of the plate is given by m = ρv, where V is the volume of the movable plate and ρ, the 33

34 Figure 2-2. (a) Scanning electron microscope image of a micro-electromechanical torsional oscillator. The large square paddle is suspended above two electrodes by two torsional springs. The three smaller square pads allow attachment of gold wires for electrical connections to the top plate and the two electrodes. On the top plate, the small dots are etch holes to allow the wet etchant to remove the PSG effectively. (b) A magnified view of one of the torsional springs. In this device the spring is connected to the second layer of polysilicon. density of polysilicon, is 2330 g m 3. The moment of inertia for the plate is I = 1 12 mb2, where b is the effective moment arm. Springs of different sizes were used; a typical size was 40 µm 4 µm 3.5 µm. The springs are anchored to the substrate and provide an electrical connection to the top plate. The two electrodes beneath the movable plate have an area approximately half that of the top plate. A small gap, directly beneath the axis of rotation, isolates the electrodes from each other. Electrical leads run from each electrode and from the anchor of one the springs to the three gold bonding pads. As mentioned in the previous section, gold wires connect these pads to the package. Basic Operation. A basic operation of the torsional oscillator is the measurement of capacitance. For small angles the capacitance can be approximated by C = ɛ oa d bθ, (2 1) 34

35 where for θ = 0 the formula for a parallel plate capacitor is obtained. In Eq. 2 1 A is the area of the overlapping plates, d is the fixed gap between the plates, θ is the angle of rotation and ɛ o is the electrical permittivity of free space. For the devices of interest b θ 1 so the parallel plate approximation is reasonable. d given by The energy E stored in a capacitor when a d.c. voltage V dc is applied across it is and can be related to an applied electrostatic torque T dc through E = 1 2 CV 2 dc (2 2) E = T dc dθ. (2 3) Solving for the torque gives T dc = ɛ oav 2 dc 2(d bθ) 2. (2 4) By putting a d.c. voltage on the bottom electrode, an attractive torque can be applied to the movable top plate. This attractive electrostatic torque is balanced by the restoring torque of the torsional springs: T r = k θ θ, (2 5) where k θ is the torsional spring constant. In Fig. 2-3 a plot of the capacitance versus the applied d.c. voltage is shown for a 500 µm 500 µm 3.5 µm device. In this figure the d.c. voltage is applied to one electrode while the capacitance is measured, using an Andeen-Hagerling capacitance bridge, between the top plate and the other electrode. As the plate is attracted toward the electrode with a d.c. voltage it tilts away from the plate where the capacitance is being measured. This leads to a decreasing capacitance for an increasing voltage. In Fig. 2-3 it is shown that above a certain voltage the capacitance drops suddenly and stays at a fixed value for higher voltage levels. If the voltage is decreased back down the capacitance no longer returns to its initial value. This result, known as the pull-in effect [44], occurs when the plate tilts such that bθ d/3. This 35

36 Capacitance (pf) d.c. voltage Figure 2-3. The capacitance between the movable top plate and a bottom electrode is shown as a function of the applied d.c. voltage to the other bottom electrode. At V d = 1.45 V the plate succumbs to the pull-in effect. For higher d.c. voltages the capacitance shows a small increase as the middle section of the plate is pulled slightly down. As the voltage is reduced ( ) the plate stays stuck in the tilted position. coincides with the electrostatic torque exceeding the restoring capabilities of the torsional spring, causing the plate to snap down. After the voltage is removed the movable plate is often left stuck in the tilted position, possibly as a result of static charge between the two plates. Using a small glass capillary and a micromanipulator the device can be manually released and returned to it starting position. A second method to free a stuck device involves applying a large a.c. voltage to one of the electrode or the top plate for a short period of time, typically less than 2 s. While not always successful, this method can be used even when the device is in the probe. This helps eliminate the need to dismantle the entire experiment whenever the device gets stuck. 36

37 θ Lock in Amp R Pre Amp V Noise V DC1 V DC2 Figure 2-4. Cross-sectional schematic of oscillator (not to scale) and electrical setup. A d.c. bias is applied between both electrodes and the top plate. The drive signal is applied to one of the electrodes, while the other is used to measure the response capacitively. 2.4 Excitation and Detection Schemes In this dissertation we focus on the dynamic resonant features of the oscillator. During my doctoral program, the excitation and detection scheme used to monitor the system at the appropriate frequencies was modified and improved. Below is a description of the original detection scheme and any improvements that were made to it. In Fig. 2-4 a cross-sectional schematic of the oscillator with electrical connections and measurement circuitry is shown. A driving voltage V d = V dc + V ac cos ω d t is applied to one electrode setting up a periodic electrostatic torque, see Ch. 3. An SRS 345 function generator was used to create the a.c. voltage while a low noise d.c. source, custom built by the University of Florida s Physics Electronic Shop, was used to produce a steady d.c. voltage. To ensure that the applied torque (see Eq. 3 2) is linearly related to the 37

38 a.c. voltage, the applied d.c. voltage is much greater than the applied a.c. voltage. The a.c. voltage is applied through a blocking capacitor onto the electrode. The d.c. voltage is connected through a resistor. The periodic electrostatic torque excites mechanical oscillations of the top plate. The detection electrode is connected to a Amptek 250 charge sensitive pre-amplifier, which is located on the outside of the probe. As the plate oscillates the capacitance changes. The virtual ground from the pre-amplifer fixes the voltage across the top plate and the detection electrode. Since Q = CV the charge must vary in time with the capacitance. The pre-amplifier detects the varying charge and outputs an amplified time varying voltage. The output is fed into a lock-in amplifier (SRS 830) which measures the amplitude B and phase δ of the response at the reference frequency. The measured response can be related to the in and out phase quadratures X = B cos(δ) and Y = B sin(δ). The measured signals X and Y are proportional to the coordinates q 1 and q 2 introduced in Sec The lock-in amplifier essentially filters out signals at frequencies away from the reference frequency and amplifies those at the reference frequency. This isolates the response signal from the electronic noise. The major drawback of this setup is that capacitive coupling between then detection and excitation electrodes can lead to offsets in the measured signal which do not represent mechanical motion. Here the applied excitation voltage leaks through to the detection electrode and gets added into the detected mechanical response. Typically this is of little consequence since the measured mechanical response is much larger than the leaked signal, however for particular experiments it is necessary that the scheme be altered to minimize this background signal. The easiest way to improve the signal to noise ratio is to reduce the stray capacitance by amplifying the signal closer to the device. This can be achieved by connecting a high electron mobility transistor (a FHx35X GaAs HEMT was used) to the device at 38

39 Lock in Amp θ R V Noise V DC1 V DC2 Figure 2-5. Cross-sectional schematic of oscillator (not to scale) and electrical setup. A d.c. bias is applied between both electrodes and the top plate. The drive signal is applied to one of the electrodes along with a high frequency carrier signal; the other electrode also has a high frequency carrier signal shifted by 180 applied to it. The response is measured at the top plate through a transistor. the package level. This was incorporated into the detection scheme for the experiments performed in Ch. 4 and 7. Heterodyne Detection. The work in Ch. 7 requires measurement of the mechanical oscillation with a high signal-to-noise ratio and minimal background. To achieve this a heterodyne measurement technique was used. The approach is similar to the previous method, except that the detected signal is modulated into a higher carrier frequency, see Fig This method involves adding a carrier signal ω c to each of the bottom electrodes. The frequency of the carrier signal is chosen such that ω c τ 1 lock in,1 ω d where τ lock in,1 is the time constant of the lock-in amplifier. The applied carrier signals are 180 out of phase and their amplitudes are approximately equal. Without the excitation signal ω d, 39

40 this method is commonly used in differential capacitance measurements [44]. The output voltage on the top plate is related to the applied carrier voltage V c ac by V out = Vac c C 1 C 2, (2 6) C 1 + C 2 where C 1 and C 2 are the capacitances between each electrode and the top plate. For simplicity, we assume that both carrier signals have the same amplitude V c ac. The output voltage is monitored using a lock-in amplifier which is referenced at the frequency of the carrier wave. If the plate is not moving the measured voltage V out is fixed because the lock-in measures the amplitude of the periodic voltage V out. If a d.c. voltage is applied the resulting torque tilts the plate and the signal on the lock-in changes by an amount proportional to the change in capacitance. Now, if the excitation signal is added in, the device will oscillate and the capacitance will be modulated at the the excitation frequency. The total output signal from the device is comprised of signals at several frequencies: ω c, ω c ± ω d, and ω d. The ω c ± ω d components result from the mixing of the carrier signal and the mechanical response at the excitation frequency: sin ω c t sin ω d t = [cos(ω c ω d )t + cos(ω c + ω d )t] /2. (2 7) The ω d component is the excitation signal that leaked through the device and manifests itself as a stray capacitance. The first lock-in amplifier is referenced at the carrier frequency with the time constant τ lock in,1. The time constant is chosen such that the ω c τ 1 lock in,1 ω d so that frequencies outside of ω c ± ω d are filtered out. The output signal from the first lock-in is the mechanical response at ω d. This output is fed into a second lock-in, referenced at the excitation frequency, which measures the amplitude and phase of the mechanical oscillation of the device. By mixing the mechanical response of the oscillator into the carrier frequency, the background problem in the previous detection scheme is eliminated. This is because the stray capacitance, which occurred at ω d, is no longer present since it was filtered out at the first lock-in. The same goes for the flicker 40

41 noise that enters into the signal in the low frequency range. Also, by balancing the signal using Eq. 2 6 such that V out = 0 for the static oscillator, the sensitivity of the lock-in can be increased to avoid the problem of digitization in the recorded signal. Noise. In Section the dimensionless parameter D was defined to describe the noise intensity needed to induce a transition over the activation barrier. Typically the parameter depends on the temperature of the environment. As will be shown, for our system thermal fluctuations are not large enough to induce a transition. Therefore an amplified Johnson noise source was used. A 50 Ω resistor is used as the primary noise source. The root mean square noise voltage or Johnson noise is v n = 4k B T R f, (2 8) where R is the resistance of the dissipative element and f is the frequency bandwidth across which the noise is applied. At room temperature (300 K) over a 1 Hz bandwidth the 50 Ω resistor produces a noise voltage of 0.9 nv Hz 1/2. This noise voltage, in general, is too small to induce a transition on the time scale of the experiments. Therefore two SRS 560 low noise amplifiers were used in series to amplify the primary noise source into a range of 10 to 1000 µv Hz 1/2. The amplifier was also used to filter the noise. The bandwidth of the filtered noise was selected to be much larger than the width of the resonance peak so that about the frequency range of interest the noise can be considered white. In some experiments an additional digital filter was used to filter the noise; again within the frequency range of interest, the noise remains white. The filtering was performed to remove frequency components of the noise that could interact with other modes of the oscillator. This was verified by both measuring the frequency distribution using a network analyzer and performing fast Fourier transforms on time traces of the noise signal. When noise is injected into the driving voltage V d = V dc + V ac cos ω d t + V noise. The applied amplified and filtered noise is V noise. From the Taylor expansion in Eq. 3 5 the 41

42 resultant noise torque can be written as N(t) = bɛ oav dc V noise. In this dissertation the 2Id 2 applied noise will be referred to in terms of a noise intensity I N. The noise intensity I N D = const N(t) 2. The arbitrary unit given be the constant factor has no impact on any of the results in this dissertation, because the generic features of interest are, in general, dimensionless. To ensure that the intrinsic noise due to damping does not exceed the applied noise, a comparison of the noise torque for both situations can be done. Assuming a minimum applied noise voltage of 10 µv Hz 1/2, the associated noise torque for a device with a 500 µm 500 µm area and V dc = 0.2 V is τ noise, N m Hz 1/2. The result can be compared to the mechanical noise torque associated with the dissipation [45] τ noise,2 = 4k B T (2λI), (2 9) where I is the moment of inertia and λ is the damping coefficient (see Ch. 3). In order for the mechanical noise in this situation to compare with the minimal applied noise the temperature would have to be at least T 10 3 K. However, all experiments are carried out at a temperature of either 4 K or 77 K, so an external noise source with a much larger effective temperature is used to induce transitions. 2.5 Summary Using a commercial fabrication process we create a torsional oscillator whose size is on the order of hundreds of microns. The excitation and detection schemes are described for exciting mechanical oscillations and detecting the response. The rotation of the plate is measured capacitively and the oscillation amplitude and phase of the oscillator is measured under variety of conditions. To provide a stable environment the experiments were performed in vacuum and under a constant temperature by submerging the probe into either liquid helium or liquid nitrogen. Finally, the size of the noise needed to induce transitions was shown to be much larger than that provided by thermal fluctuations. 42

43 CHAPTER 3 DEVICE RESPONSE In this chapter the response of the torsional oscillator to a periodic excitation is examined. The driving torque and excitation frequency are shown to have a dramatic effect on the system response. When the driving torque exceeds a critical value the oscillator response becomes hysteretic. Within the hysteresis multiple stable states coexists. Each stable state represents an actual mechanical oscillation amplitude and its phase. In the presence of a sufficiently high noise intensity, the system can be induced to switch between the stable states. Examples of this behavior, known as interstate switching, will be given. 3.1 Linear Device Response For small driving torques the response of the torsional oscillator can be described, after dividing through by the moment of inertia I, by the familiar expression for a damped, driven linear oscillator θ + 2λ θ + ω 2 0θ = T ac cos ω d t, (3 1) where θ is the angular rotation of the top plate, λ is the damping coefficient, ω 0 = k θ /I is the natural frequency of the oscillator, and T ac = bɛ 0AV dc V ac I(d bθ) 2 (3 2) is the amplitude of the driving torque. Also, A is the overlapping area across which the voltage is applied, d = 2µm is the initial plate separation, and b is the effective moment arm. The applied torque, as mentioned in Sec. 2.4, actually has a total driving voltage of V d = V dc + V ac cos ω d t. In general the torque depends on the square of the voltage V d where V 2 d = V 2 dc + 2V dc V ac cos ω d t + V 2 ac cos 2 ω d t. (3 3) 43

44 Amplitude Frequency (Hz) Figure 3-1. The response of the oscillator for small drives is linear and well-fitted by the square root of a Lorentzian. The response was measured on a device with a top plate measuring 500 µm 500 µm 3.5 µm and each torsional spring measuring 40 µm 4 µm 2 µm. In the experiments covered in this dissertation V dc V ac, therefore V 2 ac cos 2 ω d t provides a negligible d.c. component and is ignored in this analysis. In the linear response regime V 2 dc leads to a shift in the resonant frequency and 2V dc V ac cos ω d t is the periodic drive. In this dissertation the idea of control parameters affecting the system is discussed; an example of two control parameters are T ac and ω d. Of particular interest is how the response of the system depends on the two parameters. Following Landau [28] a solution is sought for the amplitude B and the phase δ such that θ = Be i(w d+δ)t. By assuming ω 0 λ a solution for the amplitude and phase can be written: B = T ac λ and δ = tan. (3 4) 2ω 0 (ωd ω 0 ) 2 + λ 2 ω d ω 0 44

45 Equation 3 4 provides a clear indication of how the amplitude of the oscillator response depends on specific control parameters. The overall magnitude of the response increases linearly as the torque is increased. As the driving frequency is swept near the resonant frequency the response becomes large and is a maximum where ω d = ω 0. The damping coefficient keeps the denominator from going to zero. The damping coefficient is one-half the width of the the response B 2. Typical values for our experiments range from 0.1 Hz to 1 Hz. Damping. The damping coefficient determines the amount of energy dissipated during a given oscillation cycle. When the damping constant is small, less energy is dissipated away and the resonant response becomes very large. As the damping coefficient increases the amount of energy dissipated also increases and thus the response decreases. The quality factor Q is a dimensionless parameter which describes the energy dissipated away in the system during the time the system takes to oscillate through the one radian. Using the notation in this dissertation Q = ω 0 2λ. The damping coefficient is the inverse of the relaxation time of the system. Defined as t r = 1/λ, it is the characteristic timescale over which the system returns to the attractor after a disturbance. So, if the driving frequency is changed suddenly, a corresponding change in amplitude will occur over a timescale equivalent to the relaxation time. As a result, experimental difficulties arise when the relaxation time is greater than a couple of seconds. Here any measurement being made, where parameters are changed, must take into account the relaxation time. In Chapters 4-7 the effect of noise on the system is analyzed. Here the relaxation time plays a major part in both the analysis and experimental procedure. This is because every time a fluctuation agitates the system it is moved away from the attractor. When the noise is removed it can take up to several times the relaxation time for the system to settle back down. The damping in the system results from two main sources: squeeze film damping [46] and thermoelastic damping [47]. The easiest way to change the first source is to 45

46 change the surrounding pressure. The sample space is pumped using a two stage vacuum setup. The first stage utilizes a Varian dry scroll pump to provide a rough vacuum around 10 3 Torr. The second uses a Varian turbo pump, which further reduces the pressure to under 10 7 Torr. For pressures larger than 10 5 Torr, viscous damping decreases with pressure. It is difficult to maintain the pressure at a constant value. Instead, the sample is typically maintained at the base pressure. In the case of thermoelastic damping it may be possible to alter the design of the springs and there anchors. This could be an interesting avenue to pursue in order to study different damping dependent features. 3.2 Duffing Oscillator As previously mentioned, the response of the oscillator for small driving torques is given by Eq The squared response is well fitted by a Lorentzian. In Fig. 3-1 the response is fitted using Eq. 3 4, which is the square root of the Lorentzian lineshape. As the driving torque is increased Eq. 3 1 no longer completely describes the motion of the oscillator because of the strong, nonlinear distance dependences in the electrostatic torque. It is necessary to perform a Taylor expansion of the electrostatic torque about bθ d. The Taylor expansion is T d = bɛ oavd 2 1 2Id 2 (1 = bɛ oavd 2 (1 2 bθ bθ d )2 2Id 2 d + 3(bθ d )2 4( bθ d )3 +...). (3 5) Again the terms containing V 2 ac cos 2 ω d t are ignored. Inserting the result into the right hand side of Eq. 3 1 gives θ + 2λ θ + [ω 2 0 η]θ + αθ 2 + βθ 3 = T 0 cos(ωt), (3 6) where α = 3b3 ɛ 0 A V 2 2Id 4 dc1, β = 2b4 ɛ 0 A V 2 Id 5 dc1, η = b2 ɛ 0 A V 2 Id 3 dc1, and T 0 = bɛoav dcv ac. As seen in 2Id 2 Eq. 3 6, the first order term from the Taylor expansion of the torque leads to a shift in the oscillator s natural frequency, while the higher order terms lead to nonlinear contributions to the restoring torque. These nonlinear effects arising from α and β can be characterized by a constant [28] κ = 3β/8ω 0 5α 2 /12ω 3 0. This coefficient κ relates the squared amplitude 46

47 1.5 x 10 3 Oscillation Amplitude (rad) f 1 f 2 Frequency (Hz) Figure 3-2. As the drive strength is increased the response of the oscillator becomes asymmetric as the peak shifts to lower frequencies. At a critical amplitude a discontinuity appears in the response, for larger drives the response is hysteretic. The arrows indicate the direction in which the frequency was swept. The bifurcation frequencies are f 1 = ω b1 2π and f 2 = ω b2 2π. of the response to the second order term in the approximation of the applied frequency [28]. For our device geometry κ is dominated by the β term and thus in the absence of fluctuations the device can be regarded as a Duffing oscillator. Also, including the shift from the d.c. voltage the resonant frequency is redefined as ω 2 0 = ω2 0 η. Following Landau [28], the amplitude dependence of the Duffing oscillator on the system parameters is [ ((ωd B 2 ω 0 ) + κb 2) ] 2 + λ 2 = T 0 2. (3 7) 4I 2 ω0 2 47

48 When the drive is small, terms in amplitude of a second power can be ignored. In this case Eq. 3 7, is simplified to Eq As the drive increases, however, these terms cannot be ignored and the resonant lineshape becomes asymmetric and the peak begins to shift to lower frequencies as expected for a negative β. The negative β, known as a soft spring nonlinearity [48], is associated with an external nonlinearity which results from the high order terms in the applied electrostatic torque. Intrinsic nonlinearity, which leads to positive β, tilts the peak to the right. This type of nonlinearity, known as a hard spring nonlinearity, can result from the geometry of the spring and the boundary conditions of its connection to the anchor and the top plate. When the peak of the response reaches a frequency of ω = ω 0 3λ a discontinuity appears in the response. For larger values of the driving torque the response is hysteretic and the oscillator exhibits three dynamic states: two stable and one unstable. The transition from a linear response to a hysteretic response is shown in Fig. 3-2 and is well described by Eq The two oscillation amplitudes, which exist between the lower bifurcation frequency ω b1 2π and the upper bifurcation frequency ω b2, are stable in the 2π absence of fluctuations. The history of the oscillator determines which state the system resides. For a constant driving amplitude, increasing the driving frequency past ω b1 2π places the oscillator in the lower oscillation amplitude of the system. The system will stay in this state until the driving frequency surpasses ω b2, where it abruptly switches to the upper 2π amplitude state. When the driving frequency is swept back down the oscillator resides in the larger oscillation amplitude state until the driving frequency passes ω b1. At this 2π frequency, the upper amplitude state ceases to exist and the system switches back to the lower amplitude state. Interstate Transitions. When noise is injected into the driving voltage the system can be induced to switch from one of the stable oscillation amplitudes to the other, assuming the driving frequency lies within the hysteresis loop. Figure 3-3a shows an example of the system switching from one oscillation amplitude to the other and back 48

49 2 x 104 (a) 1 (b) Counts/Bin Norm. Osc. Amp Norm. Osc. Amp Time (s) Figure 3-3. (a) Histogram of oscillation amplitude showing the relative occupation of the two amplitude states. (b) The oscillator switching between the two distinct oscillation amplitudes states of the system in the presence of noise. for a particular driving frequency and driving amplitude. By taking a longer time trace of the oscillation response and binning the amplitude of the response in a histogram, as shown in Fig. 3-3b, the relative occupation between the two states can be inferred. In this case, the system favors the upper oscillation amplitude. This indicates that the activation barrier needed to be overcome by the noise intensity, when going from the lower state to the upper state, is smaller than the barrier separating transitions from the upper state to the lower state. 3.3 Parametric Oscillator In contrast to the previous section where the oscillator was driven at ω d ω 0, here the spring constant is modulated at ω d 2ω 0. This parametric pumping gives rise to a resonant signal at ω d /2 ω 0. The idea is similar to the parametric pumping seen in the case of a child on a swing 1 [49], where the child modulates his moment of inertia at twice 1 In this example a child wants to swing higher, but unfortunately the child is alone. To achieve the desired effect the child crouches on the swing and begins the initial motion. As the swing moves forward from the minimum of its trajectory the child stands up and sits 49

50 1.2 x B (rad) ω b2 2ω 0 ω b1 2ω (ω d 2ω 0 )/2π (Hz) Figure 3-4. Parametric resonance in a torsional oscillator. The amplitude of the oscillation is plotted versus the driving frequency. The response is at half the driving frequency. Above ω b1 only one stable state, with a zero amplitude, exists. Between ω b2 and ω b1 the zero amplitude state becomes unstable, however, a monotonically decreasing amplitude appears which possesses two identical states that are out of phase by π rad. Finally at ω b2 the zero-amplitude state becomes stable again and three states coexist. the swing frequency, resulting in the swing reaching a larger height. In this section we discuss how the same idea is achieved using a torsional oscillator, the only difference being the effective spring constant is modulated instead of the moment of inertia. Following Landau [28], the equation of motion for a response at ω d /2 when a drive of ω d 2ω 0 is applied can be written as θ + 2λ θ + ( ω0 2 + k ) e cos ωt θ + αθ 2 + βθ 3 = 0, (3 8) I back down by the time the swing reaches the peak height of its forward motion. As the swing returns to the minimum the child repeats the up and down motion. By continually raising and lower, the child effectively is modulating the moment of inertia of the swing at twice the swinging frequency. This acts to parametrically excite oscillations. 50

51 where k e = C (θ 0 )V dc V ac is the amplitude of the spring modulation, C (θ 0 ) is the second derivative of the capacitance between the driving electrode and the top plate with respect to θ evaluated at the equilibrium angular position θ 0. The response at ω 2ω 0 is negligible due to the high Q of the oscillator. The solution to Eq. 3 8 is given by ( B 2 = 1 1 ) 2 ατ κ 2 (ω 2ω 0 ) ± λ 6ω (3 9) For driving amplitudes below a critical amplitude, given by the condition that the determinant of Eq. 3 9 is zero, the system shows no resonant response. When the determinant of Eq. 3 9 is real, two bifurcation frequencies appear at frequencies ω b2 and ω b1 given by ( ατ ω b1,b2 = 2ω 0 ± 3ω 3 0 ) 2 4λ 2. (3 10) As shown in Fig. 3-4, when the oscillator is driven at frequencies ω d /2 > ω b1 only a single stable state with zero amplitude exist. We refer to this state as the zero amplitude state. At ω b1 a pitch-fork bifurcation occurs and two stable states emerge and the zero state becomes unstable. The stable states are period-two vibrations that are are out of phase by π rad and have the same amplitude which increases as the frequency is decreased. Again the curving of the response toward lower frequencies is a result of the negative coefficient of nonlinearity. At the lower bifurcation frequency ω b2 a second pitchfork bifurcation occurs as the unstable zero state becomes stable again and two unstable states corresponding to period-two vibrations emerge. The high amplitude state, possessing the stable period-two vibrations, still exists here. The two unstable period-two vibrations separate the stable period-two vibrations from the stable zero amplitude state. Much like the Duffing oscillator case, the response is hysteretic. As the frequency is increased past ω b2 the system jumps up to the high amplitude state. When the frequency is swept back down past ω b2 it will stay in the upper amplitude state. The case of the stable period-two vibrations is different from the Duffing oscillator. When the system 51

52 ω d ω d 2 ω d 2 + π Figure 3-5. The system is driven at ω d while the response is at ω d /2. Shifting the response by π leads to an equivalent and valid solution to the equation of motion. These two solutions have an identical amplitude but represent two unique states. crosses ω b1 from the high frequency side the two coexisting states are occupied with equal probability; this results from a time-translation symmetry in the response, where shifting the response by π rad gives another valid solution. Ultimately, the system diffuses out of the unstable state and into one the stable states due to the background thermal noise. In Fig. 3-5 a plot of the driving frequency is shown with the ω d 2 response. A second response at ω d 2 + π is also shown. Both responses are seen to be equivalent solutions to the equation of motion. Interstate Transitions. When noise is injected into the driving voltage the system can be induced to switch from one of the stable oscillation amplitudes to the other and between the period-2 vibrations, depending on the value of the driving frequency. Figure 3-6a shows a histogram resulting from the system switching between the period-2 vibrations at a frequency ω b2 < ω d /2 < ω b1. In Fig. 3-6b a histogram of the amplitude of the period-2 vibrations is also shown. The amplitude fluctuates about a fixed value. 52

53 Relative Occupation Phase (rad) Oscillation Amplitude (rad) x 10 3 Figure 3-6. (a) When noise is applied, the system is induced to transition between the two states. For the period-2 vibrations the probability of switching is 50/50. A histogram showing this occupation is plotted. (b) The amplitude for this type of transition only changes when the system is switching from one state to the other. When a switch occurs the system moves though the zero amplitude unstable state. For frequencies below ω b2 the zero state also becomes stable. Similar to the case of the Duffing oscillator the amplitude can switch between two distinct values. In this region the two stable period-2 states still exist. 3.4 Summary In this chapter we looked at the different resonant responses that can occur in the torsional oscillator. For small drives the response of the system was shown to be that of a driven, damped linear oscillator. Upon increasing the driving strength the nonlinearity begins to play a major part in the system response and it can actually become hysteretic. 53

54 In this region, the bistability that is central to this dissertation, is seen. An example was given of the system switching between the available stable states. We also showed another driving scheme called parametric resonance. Here, it is possible to actually have three stable states for a single set of system parameters. Again, transitions between the different states were demonstrated. The oscillation responses presented make up the different systems that are studied in the following chapters. These systems are examined near their critical features, i.e. bifurcations. Here we expect to find features, resulting from the fluctuations and transitions, that should be present in any type of system or device that shows similar resonant type behavior. 54

55 CHAPTER 4 PATHS OF FLUCTUATION INDUCED SWITCHING: MPSP The results and derivations in this chapter can be found in the articles, Paths of Fluctuation Induced Switching, H. B. Chan, M. I. Dykman, and C. Stambaugh, Physical Review Letters 100, (2008) Copyright 2008 by the American Physical Society and Switching-path distribution in multidimensional systems, H. B. Chan, M. I. Dykman, and C. Stambaugh, Physical Review E 79, (2008) Copyright 2008 by the American Physical Society. 4.1 Introduction Fluctuations lead to switching between coexisting states. In different switching events, the system, in general, follows a different trajectory in phase space due to the random nature of activated switching. A basic, although somewhat counterintuitive, physical feature of large infrequent fluctuations leading to switching is that, in such fluctuations, the system is most likely to move along a certain path in its phase space. This path is known as the most probable switching path (MPSP). For low fluctuation intensity, the MPSP is obtained by the solution of a variational problem. This problem also determines the switching activation barrier [10, 37, 50 58]. Despite its central role for the understanding of fluctuation-induced switching and switching rates, the idea of the MPSP has not been tested experimentally in multivariable systems. The question of how the paths followed in switching are distributed in phase space has not been asked either. Perhaps closest to addressing the above issues was the experiment on dropout events in a semiconductor laser with optical feedback [59]. In this experiment the switching path distribution in space and time was measured and calculated. However, the system was characterized by only one dynamic variable, and thus all paths lie on one line in phase space. In another effort, electronic circuit simulations [60] compare the distribution of fluctuational paths to and relaxational paths from a certain point within one basin of 55

56 attraction; the data refer to the situation where switching does not occur. The methods [59, 60] do not apply to the switching path distribution in multivariable systems. In this Chapter we introduce the concept of switching path distribution in phase space and the quantity that describes this distribution, calculate this quantity, and report the direct observation of the tube of switching paths. A more detail account of the calculation can be found in Ref. [61]. The experimental and theoretical results on the shape and position of the path distribution are in excellent agreement, with no adjustable parameters. The results open the possibility of efficient control of the switching probability based on the measured narrow path distribution. In Sec. 4.2 we provide the qualitative picture of switching and give a preview of the central theoretical and experimental results. Sec. 4.3 presents a theory of the switching probability distribution in the basins of attraction to the initially occupied and initially empty stable states as well as some simple results for systems with detailed balance. In Sec. 4.4 the system used in the experiment, a micromechanical torsional oscillator, is described and quantitatively characterized. Section 4.5 presents the results of the experimental studies of the switching path distribution for the micromechanical oscillator, with the coexisting stable states being the states of parametrically excited nonlinear vibrations. Generic features of the distribution are discussed, and the lack of time-reversal symmetry in switching of systems far from thermal equilibrium is demonstrated for the first time. Section 4.6 contains concluding remarks. 4.2 Qualitative Picture and Preview of the Results We consider a bistable system with several dynamical variables q = (q 1,..., q N ). The stable states A 1 and A 2 are located at q A1 and q A2 respectively. A sketch of the phase portrait for the case of two variables is shown in Fig For low fluctuation intensity, the physical picture of switching is as follows. The system prepared initially in the basin of attraction of state A 1, for example, will approach q A1 over the characteristic relaxation time t r and will then fluctuate about q A1. We assume the fluctuation intensity to be 56

57 small. This means that the typical amplitude of fluctuations about the attractor (the characteristic diffusion length) l D is small compared to the minimal distance q between the attractors and from the attractors to the saddle point q S. Even though fluctuations are small on average, occasionally there occur large fluctuations, including those leading to switching between the states. The switching rate Γ 12 from state A 1 to A 2 is much less than the reciprocal relaxation time t 1 r, that is, the system fluctuates about A 1 for a long time, on the scale of t r, before a transition to A 2 occurs. In the transition the system most likely moves first from the vicinity of q A1 to the vicinity of q S. Its trajectory is expected to be close to the one for which the probability of the appropriate large rare fluctuation is maximal. The corresponding trajectory is illustrated in Fig From the vicinity of q S the system moves to state A 2 close to the deterministic fluctuation-free trajectory. These two trajectories comprise the MPSP. For brevity, we call the sections of the MPSP from q A1 to q S and from q S to q A2 the uphill and downhill trajectories, respectively. The terms would literally apply to a Brownian particle in a potential well, with A 1,2 corresponding to the minima of the potential and S to the barrier top (see Sec. 1.2). We characterize the switching path distribution by the probability density for the system to pass through a point q on its way from A 1 to A 2, p 12 (q, t) = dq f ρ(q f, t f ; q, t q 0, t 0 ). (4 1) Ω 2 Here, the integrand is the three-time conditional probability density for the system to be at points q f and q at times t f and t, respectively, given that it was at q 0 at time t 0. The point q 0 lies within distance l D of q A1 and is otherwise arbitrary. Integration with respect to q f goes over the range Ω 2 of small fluctuations about q A2 ; the typical linear size of this range is l D. 57

58 1.5 q 2 0 S A 2 A q 1 Figure 4-1. Phase portrait of a two-variable system with two stable states A 1 and A 2. The saddle point S lies on the separatrix that separates the corresponding basins of attraction. The thin solid lines show the downhill deterministic trajectories from the saddle to the attractors. A portion of the separatrix near the saddle point is shown as the dashed line. The thick solid line shows the most probable trajectory that the system follows in a fluctuation from A 1 to the saddle. The MPSP from A 1 to A 2 is comprised by this uphill trajectory and the downhill trajectory from S to A 2. The plot refers to the system studied experimentally, see Sec We call p 12 (q, t) the switching probability distribution. Of utmost interest is to study this distribution in the time range Γ 1 12, Γ 1 21 t f t, t t 0 t r. (4 2) Here, t r is the Suzuki time [62]. It differs from t r by a logarithmic factor log[ q/l D ]. This factor arises because of the motion slowing down near the saddle point. The time t r is much smaller than the reciprocal switching rates, and the smaller the fluctuation intensity the stronger the difference, because the dependence of Γ ij on the fluctuation 58

59 intensity is of the activation type. If the noise causing fluctuations has a finite correlation time, t r is the maximum of the Suzuki time and the noise correlation time. For t t 0 t r, by time t the system has already forgotten the initial position q 0. Therefore the distribution ρ(q f, t f ; q, t q 0, t 0 ), and thus p 12, are independent of q 0, t 0. On the other hand, if the system is on its way from A 1 to A 2 and at time t is in a state q far from the attractors, it will most likely reach the vicinity of A 2 over time t r and will then fluctuate about q A2. This will happen well before the time t f at which the system is observed near A 2, and therefore p 12 is independent of t f. It is clear from the above arguments that, in the time range given by Eq. 4 2, the distribution p 12 (q, t) for q far from the attractors is formed by switching trajectories emanating from the vicinity of A 1. It gives the probability density for these trajectories to pass through a given point q at time t. In other terms, the distribution p 12 (q, t) is formed by the probability current from A 1 to A 2 and is determined by the current density. The Shape of the Switching Probability Distribution. The peak of p 12 (q, t) is Gaussian transverse to the MPSP for q q A1,2, q q S l D, p 12 (q, t) = Γ 12 v 1 (ξ )Z 1 exp ( 12 ) ξ Qξ, (4 3) where ξ and ξ are coordinates along and transverse to the MPSP, and v(ξ ) is the velocity along the MPSP. The matrix elements of matrix ˆQ = ˆQ(ξ ) are l 2 D, and Z = [(2π) N 1 / det ˆQ] 1/2. It follows from Eq. (4 3) that the overall probability flux along the MPSP is equal to the switching rate, dξ p 12 (q, t)v(ξ ) = Γ 12. We have observed a narrow peak of the switching path distribution in experiment. The results are shown in Fig They were obtained using a micro-electromechanical torsional oscillator described in Sec. 4.4 and Ch. 2. The path distribution displays a sharp ridge. We demonstrate that the cross-section of the ridge has Gaussian shape. As seen 59

60 Figure 4-2. (a) Switching probability distribution in a parametrically driven micro-electromechanical oscillator. The probability distribution p 12 (X, Y ) is measured for switching out of state A 1 into state A 2. (b) The peak locations of the distribution are plotted as black squares and the theoretical most probable switching path is indicated by the green line. All trajectories originate from within the cyan circle in the vicinity of A 1 and later arrive at the green circle around A 2. The portion of the distribution outside the maroon lines is omitted. from Fig. 4-2, the maximum of the ridge lies on the MPSP which was calculated for the studied system. In App. A.1 the code necessary to generate the MPSP is given. Equation (4 3) is written for a generally nonequilibrium system, but the system is assumed to be stationary. In the neglect of fluctuations its motion is described by equations with time-independent coefficients. In this case p 12 (q, t) is independent of time t. 60

61 4.3 Theory of the Switching Path Distribution The Model of a Fluctuating System. We derive Eq. 4 3 for a system described by the Langevin equation of motion q = K(q) + f(t), f n (t)f m (t ) = 2Dδ nm δ(t t ). (4 4) Here, the vector K determines the dynamics in the absence of noise; K = 0 at the stable state positions q A1, q A2 and at the saddle point q S. We assume that q S lies on a smooth hypersurface that separates the basins of attraction of states A 1 and A 2 (see Fig. 4-1). The function f(t) in Eq. 4 4 is white Gaussian noise; the results can be also extended to colored noise. The noise intensity D is assumed small. The dependence of the switching rates Γ nm on D is given by the activation law, log Γ nm D 1 [10, 37, 50 54]. This is also the case for noise-driven continuous systems, cf. Ref. [63 65] and papers cited therein. There exists extensive literature on numerical calculations of the switching rate and switching paths, cf. Ref. [66 70] and papers cited therein. In the model (4 4), the characteristic relaxation time t r and the characteristic diffusion length l D are t r = max Re λ k 1, l D = (Dt r ) 1/2, (4 5) k where λ k are the eigenvalues of the matrix K m / q n calculated at q A1, q A2 and q S. For a white-noise driven system (see Eq. 4 4), the three-time probability distribution ρ(q f, t f ; q, t q 0, t 0 ) in Eq. (4 1) can be written as a product of two-time transition probability densities, ρ(q f, t f ; q, t q 0, t 0 ) = ρ(q f, t f q, t)ρ(q, t q 0, t 0 ), (4 6) which simplifies further analysis. The analysis is done separately for the case where the observation point q lies within the attraction basins of the initially empty attractor A 2 and the initially occupied attractor A 1. 61

62 From conservation of the stationary probability current it follows that the distribution p 12 (q, t) should sharply increase near the saddle point. Indeed, the velocity v(ξ ) = 0 for q = q S. The current close to q S is due to diffusion. In the general case of nonequilibrium systems the shape of the switching probability distribution near the saddle point is complicated; its analysis is beyond the scope of this paper. Systems with Detailed Balance. An explicit solution for p 12 (q, t) near the saddle point can be obtained for systems with a gradient force K = q U(q). Such systems have detailed balance. The uphill section of the MPSP in this case is literally the uphill path that goes from the local minimum of the potential U(q) at A 1 to the saddle S and is given by equation q = q U(q) [71]. In contrast to systems without detailed balance [72, 73], where for smooth U(q) the MPSP near the saddle point is described by an analytic function of coordinates and ˆξ is perpendicular to the separating hypersurface. 4.4 Micromechanical Torsional Oscillator Device Characteristics The switching probability distribution is measured using a micro-electromechanical torsional oscillator, with a high quality factior (Q = 9966), driven into parametric resonance (see Sec. 3.3). The movable top plate (200 µm 200 µm 3.5 µm) is suspended by two torsional rods (4 µm 2 µm 36 µm and k θ = Nm). There are two fixed electrodes on the substrate, one on each side of the torsional rod. The 2 µm gap underneath the movable plate is created by etching away a sacrificial silicon oxide layer. As described in Chapter 3 torsional oscillations of the movable top plate are excited by applying a driving voltage V d = V dc + V ac cos (ω d t) + V noise (t) to one of the lower electrodes while the top plate remains electrically grounded. The driving frequency ω d = 2ω 0 + ε is close to twice the natural frequency ω 0. The dc voltage V dc (1 V) is much larger than the amplitude V ac (141 mv) of sinusoidal modulation and the random noise 62

63 voltage V noise. The equation of motion for the angle θ counted off from θ 0 is θ + 2λ θ + [ ω k e cos (ω d t) ] θ + αθ 2 + βθ 3 = N(t). (4 7) Torsional oscillations of the top plate are detected capacitively by the other electrode. This electrode is connected to a d.c. voltage source through a large resistor. A high electron mobility transistor is placed in close proximity to the device to measure the oscillating charge on the detection electrode induced by motion of the top plate. The output of the transistor is connected to a lock-in amplifier referenced at half the driving frequency ω d. For the chosen time constant of 300 µs, the measurement uncertainty is 80 µrad, about 0.6% of the full scale in Fig. 4-2 and much smaller than the width of the path distributions. The oscillation amplitudes in-phase (X) and out-of-phase (Y) with the reference frequency were recorded every 2 ms. All measurements were performed at 77 K and < 10 6 Torr Transformation to Slow Variables and Parametric Resonance Since the oscillator is strongly underdamped (λ/ω ) and the modulation is almost resonant ( ω 2ω 0 ω), we analyze the motion of the oscillator in the rotating frame, with slow dimensionless variables q 1 and q 2 and dimensionless time t k e t/2ω d (note that even though the oscillator has one degree of freedom, its motion is characterized by two dynamical variables). In this case it is convenient to introduce the slow variables as ( 2ke θ(t) = 3 γ ( dθ ω 2 dt = d k e 6 γ ) 1/2 [ q 1 cos ) 1/2 [ q 1 sin ( ) ( ωd t ωd t q 2 sin 2 2 ( ) ( ωd t ωd t + q 2 cos 2 2 )], )]. (4 8) The quadratures q 1 and q 2 are directly proportional to the signal components X and Y measured with the lock-in amplifier, with the proportionality constant E determined by the measuring apparatus, q 1 = EX, q 2 = EY. (4 9) 63

64 Substituting Eq. (4 8) into Eq. (4 7) and neglecting fast oscillating terms, we can write the equations of motion for q = (q 1, q 2 ) in the form (4 4). The function K in dimensionless time is given by K = ζ 1 q + ˆε g. (4 10) Here ζ = k e /2ω d λ, µ = ω d (2ω 0 ω d )/k e, and g = q 4 /4 (1 + µ)q1/2 2 + (1 µ)q2/2, 2 where ˆε is the permutation tensor. Equation q = K gives the downhill section of the MPSP of the oscillator. The uphill section of the MPSP can be calculated by solving the Hamiltonian equations of motion that follow from the Hamilton-Jacobi equation for an auxiliary particle [61] Determination of Device Parameters We first consider motion of the device in the absence of fluctuations. When the amplitude of the spring modulation is sufficiently strong (ζ > 1), the oscillator response exhibits period doubling [28]. Oscillations are induced at half the modulation frequency in a range close to ω 0. Between the two bifurcation frequencies ω b1 ( rad/s) and ω b2 ( rad/s) there exists two stable states of oscillations at frequency ω d /2. They differ in phase by π but have identical amplitude. Both states are stable solutions of Eq. (4 7). Their basins of attraction in the rotating frame are separated by a separatrix that goes through the unstable stationary state, which in the laboratory frame has zero vibration amplitude at frequency ω d /2. The phase portrait in the rotating frame is illustrated in Fig The driving frequency is chosen to be rad/s for measurement of the switching path distribution. We note that parametric resonance in nano- and micro-electromechanical systems has attracted considerable attention [19, 74 78]. Since here we are interested in the studies of the principal features of noise-induced switching, we chose the simplest nontrivial regime where the system has only two stable states, which occurs for ω b1 < ω d /2 < ω b2. The modulation frequency ω d is chosen to be close to 2ω b1 so that the motion in the rotating 64

65 θ (mrad) Oscillation frequency (rad s 1 ) Figure 4-3. Harmonic and parametric resonances of the micromechanical torsional oscillator. For resonant driving ( ), the oscillation amplitude is plotted as a function of the oscillation frequency. The thin line is a fit to the harmonic oscillator response. It gives device parameters λ and ω 0. For parametric resonance ( ), the driving frequency is twice the oscillation frequency. The fit (thick line) yields κ nonlinear and the effective parametric modulation amplitude k e. frame is underdamped, which is advantageous for studying a generic feature of fluctuations in systems far from thermal equilibrium, the breaking of time reversal symmetry. Calculation of the MPSP requires a number of device parameters including λ, ω 0, the parametric modulation amplitude k e, and the nonlinear constant κ nonlinear = 3γ/8ω 0 [79]. These parameters are obtained from the linear and nonlinear responses of the device. When the device is resonantly driven with small amplitude at frequency close to ω 0, it responds as a harmonic oscillator. From the resonance line shape (Fig. 4-3), λ and ω 0 are determined to be 6.99 rad/s and rad/s respectively. The remaining two parameters are extracted from the parametric resonance of the oscillator for ω d close to 2ω 0. Specifically, the parametric modulation amplitude k e is determined from the bifurcation frequencies ω b1,2 = 2ω 0 ω p, where ω p = (ke 2 (4ω 0 λ) 2 ) 1/2 /2ω 0. This gives k e = s 2. The nonlinear parameter κ nonlinear ( s 1 ) is obtained from the proportionality constant between the square of the parametric oscillation amplitude 65

66 θ A and the detuning from bifurcation frequency close to the bifurcation frequency seen in Fig See Appendix B for a more detailed analysis. Using these measured device parameters, the dimensionless constants contained in K in Eq. (4 10) and in Eq. (4 9) can be directly calculated to be E = , ζ = 4.968, and µ = The theoretical optimal escape path in Fig. 4-2 is calculated with the above parameter values. No adjustable parameters are used. 4.5 The Switching Path Distribution Experiment Measured Switching Path Distribution When white noise is added to the excitation voltage, the system can occasionally overcome the activation barrier and switch from one stable state to the other. The noise intensity is chosen to ensure that the mean residence time in each state ( 10 s) is much larger than the relaxation time (t r 1 s) of the system. Transitions are identified when the oscillator begins in the vicinity of A 1 (within the left cyan circle Ω 1 in Fig. 4-2a) and subsequently arrives at state A 2 (within the right cyan circle Ω 2 ). Figure 4-2 shows the switching probability distribution derived from more than 6500 transitions. While in each transition the system follows a different trajectory, the trajectories clearly lie within a narrow tube. The maximum of the distribution gives the MPSP. In Fig. 4-2b, the location of this maximum is plotted on top of the MPSP obtained from theory. The oscillator is underdamped not only in the laboratory frame, but also in the rotating frame. Therefore both the uphill and downhill sections of the MPSP are spirals. On the uphill section, the MPSP emerges clockwise from A 1 and spirals toward the saddle point at the origin. Upon exit from the saddle point, it makes an angle and, on the downhill section, continues to spiral clockwise toward A 2. The agreement between the measured peak in the probability distribution and the MPSP obtained from theory is excellent. There are no adjustable parameters since all device parameters are accurately determined from the harmonic and parametric 66

67 resonances of the oscillator without noise in the excitation as described in the previous section. Close to the stable states the peaks of the distribution at successive turns of the MPSP overlap, preventing the accurate determination of the MPSP. The plot in Figs. 4-2a and 4-2b has excluded the portions of trajectories prior to escaping from the initial state A 1 and upon arriving at the final state A 2, which are bound by the two maroon lines. Such cutoff also eliminates the large peaks of the distribution centered at A 1 and A 2, which arise because the oscillator spends most of its time fluctuating about A 1 and A 2. These peaks are not relevant to switching dynamics. Figure 4-4 compares the measured and predicted velocity along the MPSP. Here, again, the good agreement is demonstrated with no adjustable parameters. As expected, the measured velocity decreases near the saddle point, ξ = 0. However, it does not become equal to zero, in agreement with the argument that the total probability current remains constant. Motion near the saddle point is dominated by diffusion Generic Features of the Switching Path Distribution The switching probability distribution in our multi-variable system displays important generic features. Figure 4-5a shows the distribution cross-section along the blue line transverse to the MPSP in Fig. 4-2b. It is well-fitted by a Gaussian. Gaussian distributions with different height and area are observed also in other cross-sections except close to the saddle point. Figure 4-5b plots the area under the Gaussian distribution versus the reciprocal measured velocity on the MPSP, for different cross-sections. The linear dependence agrees with Eq. 4 3 and indicates that the probability current from the initially occupied attractor to the empty one is constant. This current gives the switching rate Γ 12 [5]. We find that the probability current concentrates within a narrow tube deep into the basins of attraction of A 1 and A 2. In the basin of attraction to A 2 but not too close to A 2, much of the probability distribution carries the switching current. However, the 67

68 Velocity (rad/s) ξ (mrad) Figure 4-4. Measured averaged velocity along the MPSP ( ) and the velocity predicted by theory (line). The velocity decreases to zero at the saddle point (ξ = 0). overall quasistationary probability distribution deep inside the basin of attraction of A 1 is largely associated with fluctuations about A 1 that do not lead to switching. The part of the distribution responsible for the switching current is an exponentially small fraction of the total distribution. Nevertheless our formulation makes it possible to single out and directly observe this fraction. The slowing down near the saddle point shown in Fig. 4-4 leads to strong broadening and increase in height of the switching probability distribution seen in Fig. 4-2a. Because motion near the saddle is diffusive, switching paths loose synchronization. In other words, the distribution of times spent by the system near the saddle point is comparatively broad. This is why it is advantageous to study the distribution of switching paths in the space of dynamical variables rather than in time. 68

69 50 40 (a) 1.5 (b) P 12 (rad 2 ) Velocity (rad/s) ξ (mrad) /Area (rad) Figure 4-5. (a) The cross-section along the blue line in Fig. 4-2 transverse to the MPSP. The solid line is a Gaussian fit. (b) Velocity on the MPSP vs. inverse area under cross-sections of the switching probability distribution. The solid line is a linear fit forced through the origin Lack of Time Reversal Symmetry in a Driven Oscillator Another generic feature of the observed distribution is characteristic of systems far from thermal equilibrium. For equilibrium systems, the most probable fluctuational path uphill, i.e., from an attractor to the saddle point, is the time reversal of the fluctuation-free downhill path from the saddle point back to the attractor. More precisely, it corresponds to the change of the sign of dissipation term in the equation of motion [71, 80], i.e., to replacing γ with γ in Eq. (4 7). In overdamped equilibrium systems with detailed balance, these two paths coincide in space (but are opposite in direction). Our parametric oscillator is driven far from thermal equilibrium. Therefore the uphill section of the MPSP does not simply relate to the deterministic trajectory with reversed sign of dissipation. This section of the MPSP, i.e., the most probable fluctuational path from A 1 to the saddle point at the origin is plotted as the thick solid line in Fig Upon sign reversal of the dissipation, the attractor becomes a repeller, as in the case of systems in thermal equilibrium described earlier. However, in contrast to equilibrium 69

70 Y (mrad) 0 4 A 1 S 4 A X (mrad) 4 0 Figure 4-6. Comparison of the MPSP and the dissipation-reversed path. The section of the most probable switching path from A 1 to S is shown as a thick blue line. Upon changing the sign of dissipation, the attractor is shifted to a new location A 1 and becomes a repeller. The fluctuation-free path with reversed dissipation from A 1 to S is shown as the thin orange line. systems, it is also shifted away from its original location (from A 1 to A 1 in Fig. 4-6). The dissipation-reversed path is shown as the thin solid line in Fig In addition, Fig. 4-1 allows one to compare the uphill section of the MPSP with the deterministic downhill path from S to A 1. Our data show that the uphill section of the MPSP, which is formed by fluctuations, the dissipation-reversed path, and the downhill noise-free path from the saddle to the stable state are all distinct. The time irreversibility of the switching paths is directly related to the lack of detailed balance of our driven oscillator, distinguishing it from bistable systems in thermal equilibrium. 4.6 Conclusions In this chapter we have studied the phase space distribution of paths followed in activated switching between coexisting stable states. The analysis refers to systems 70

71 with several dynamical variables. We introduced a quantity, the switching probability distribution, that gives the probability density of passing a given point in phase space during switching. The distribution is defined in a way that makes it experimentally accessible. No a priori knowledge of the system dynamics is required except the positions of the stable states, which can be immediately determined, since the system spends most of the time near these states. The switching probability distribution was shown theoretically to have a shape of a narrow ridge in phase space. Far from the stationary states, the cross-section of the ridge is Gaussian. The maximum of the ridge lies on the most probable switching path (MPSP). Experimental studies of the switching path distribution were done using a high-q micromechanical torsional oscillator. All parameters of the oscillator, including the nonlinearity constant, were directly measured. The oscillator was driven into parametric resonance, where it had two coexisting vibrational states that differ in phase. The paths followed in switching between these states were accumulated and their distribution in the space of the two dynamical variables (the oscillation quadratures) was obtained. It was found that the distribution has indeed the shape of a Gaussian ridge. There is excellent agreement between the experimental and theoretical results, with no adjustable parameters. The measured maximum of the switching path distribution lies on top of the theoretically calculated MPSP. The measured velocity of motion along the MPSP as a function of the position on the MPSP also quantitatively agrees with the theory. An important property of the path distribution is the total current conservation: the product of the velocity of motion along the MPSP and the cross-section area of the path distribution remains constant. This conservation of probability current was demonstrated experimentally. In addition, we observed, for the first time, that the lack of detailed balance leads to the difference between the uphill section of the MPSP and the noise-free path with reversed sign of dissipation. 71

72 The observation of the most probable switching path reported here provides, in some respects, an experimental basis for the broadly used concept of a reaction coordinate, which can be associated with the coordinate along this path. Our method does not rely on the specific model of the fluctuating system but only on the characteristics accessible to direct measurement. It applies to systems far from thermal equilibrium as well as to equilibrium systems. Measuring the switching trajectories can help to determine the model globally, far from the stable states. It can also provide an efficient way of controlling the switching rates by affecting the system locally on the most probable switching path. 72

73 CHAPTER 5 SCALING OF THE ACTIVATION BARRIER The results in this chapter can be found in Noise activated switching in a driven, nonlinear micromechanical oscillator, C. Stambaugh and H. B. Chan, Physical Review B, 73, (2006) Copyright 2006 by the American Physical Society and Activation Barrier Scaling and Crossover for Noise-Induced Switching in Micromechanical Parametric Oscillators, H. B. Chan and C. Stambaugh, Physical Review Letters, 99, (2007) Copyright 2007 by the American Physical Society. 5.1 Introduction In both equilibrium and non-equilibrium systems, as a system parameter η approaches a bifurcation value η c, the activation barrier decreases to zero and the number of stable states of the system changes. In general, the activation barrier is determined by the device parameters and depends on the specifics of the system under study. However, at parameter values close to the bifurcation point, the activation barrier is expected to exhibit universal scaling. The activation barrier varies as φ η η c ξ with a critical exponent ξ that is system independent. While the prefactor φ might be different for each system, ξ is universal for all systems and depends only on the type of bifurcation [33]. For instance, in a Duffing oscillator resonantly driven into bistability, spinodal bifurcations occur at the boundaries of the bistable region. One stable state merges with the unstable state while the other stable state remains far away in phase space (see Fig. 1-4). Theory predicts [33, 35] that the activation barrier scales with critical exponent 3/2 near spinodal bifurcations in driven systems. On the other hand, a different critical exponent of 2 is expected at a pitchfork bifurcation [36] in systems where all three states merge (see Fig. 1-5). Such bifurcation commonly takes place in parametrically driven systems where period doubling occurs. For instance, fluctuation-induced phase slips were observed in parametrically driven electrons in Penning traps [16] between two coexisting attractors and transitions between three attractors were studied in modulated magneto-optical 73

74 ω /2 π b Normalized Oscillation Amplitude Frequency (Hz) Figure 5-1. Frequency response of sample A scaled by the excitation voltage amplitudes of 47 µv ( ) and 450 µv ( ). The red line represents a fit to the data at smaller excitation using the response of a damped harmonic oscillator. For the large excitation, two dynamical states coexist from Hz to Hz. The blue line fits the data to a damped oscillator with cubic non-linearity [28], yielding β = rad 2 s 2. traps [15]. To our knowledge, the activation barriers have not been measured over a wide enough parameter range in these parametrically driven systems to demonstrate the universal scaling at driving frequencies near the two critical points and the crossover to system-specific dependence at large frequency detuning. In Sec. 5.2 we report our investigation of noise-activated switching in systems far from equilibrium. A well-characterized system, an underdamped micromechanical torsional oscillator periodically driven into nonlinear oscillations. Here the oscillator has two stable dynamical states with different oscillation amplitude within a certain range of driving frequencies. We induce the oscillator to escape from one state into the other by injecting noise in the driving force. By measuring the rate of random transitions as a function 74

75 of noise intensity, we demonstrate the activated behavior for switching and deduce the activation barrier as a function of frequency detuning. Close to the bifurcation frequency the activation barrier is predicted by variational calculations and asymptotic scaling theory to display system-independent scaling [12, 33, 81, 82]. Here a critical exponent of 3/2 is expected. In Sec. 5.3, we report measurements of the activation barrier for fluctuation-induced switching in parametrically-driven micromechanical torsional oscillators, a system that is far from thermal equilibrium. The spring constant of our device is modulated electrostatically near twice the natural frequency. Under sufficiently strong parametric modulation, two pitchfork bifurcation points exist. At the supercritical bifurcation, there emerge two stable oscillation states that differ in phase by π. At the subcritical bifurcation, an additional, stable state with zero oscillation amplitude appears. Noise induces transitions between the coexisting attractors. By measuring the rate of random transitions as a function of noise intensity, we deduce the activation barrier for switching out of each attractor as a function of frequency detuning. Near both bifurcation points, the activation barriers are found to depend on frequency detuning with critical exponent of 2, consistent with the predicted universal scaling in parametrically driven systems [36]. Away from the immediate vicinity of the bifurcation point, universal scaling relationships for the activation barrier no longer hold. We find that in our parametric oscillator, the dependence of the activation barrier on frequency detuning changes from quadratic to 3/2 th power. 5.2 Critical Exponent Near a Spinodal Bifurcation Device and Experimental Setup. In this experiment, measurements were performed on two micromechanical oscillators (samples A and B). Each oscillator consists of a movable poly-silicon plate (500 µm 500 µm 3.5 µm) that is supported by two torsional rods (40 µm 4 µm 2 µm). The other ends of the torsional rods are anchored to the silicon substrate. The torsional spring constants are N m rad 1 and 75

76 N m rad 1 for samples A and B respectively. Beneath the top plate, there are two fixed electrodes (500 µm 250 µm) on each side of the torsional springs. One of the electrodes is used for exciting the torsional oscillations while the other electrode is used for detecting the oscillations. More details about the oscillators can be found in Ch. 2. In Fig. 2-4 a cross-sectional schematic of the oscillator is shown with its electrical connections and measurement circuitry. The application of a periodic voltage with d.c. bias V dc1 to one of the electrodes leads to an electrostatic attraction between the grounded top plate and the electrode. Torsional oscillations of the top plate are excited by the periodic component of the electrostatic torque. The detection electrode is connected to a d.c. voltage V dc2 through a resistor R. As the plate oscillates, the capacitance between the plate and the detection electrode changes. The detection electrode is connected to a charge sensitive preamplifier followed by a lock-in amplifier that measures the signal at the excitation frequency. Measurements for both samples were performed at pressure of less than Torr. Samples A and B were measured at liquid nitrogen and helium temperatures respectively. The main effect of decreasing the temperature from liquid nitrogen to liquid helium is the reduction of the damping constant, yielding quality factors Q of about 4, 000 for sample A and 16, 000 for sample B. The excitation voltage V consists of three components: V d = V dc + V ac sin(ωt) + V noise (t). (5 1) The three terms on the right side of Eq. 5 1 represent the dc voltage, periodic ac voltage with angular frequency ω and random noise voltage respectively. As described in Ch. 3, V dc is chosen to be much larger than V ac and V noise. V noise is Gaussian with a bandwidth of 100 Hz about the natural frequency of the oscillator. The strong distance dependence of the electrostatic attraction between the top plate and the electrode leads to nonlinear contributions to the restoring torque. A Taylor expansion of the electrostatic torque leads 76

77 to Eq. 3 6 with the addition of a noise source N(t): θ + 2λ θ + [ω 2 0 η]θ + αθ 2 + βθ 3 = T 0 cos(ωt) + N(t). First, we focus on the response of the oscillator with no injected noise in the excitation. We show in Fig. 5-1b the frequency response of sample A at two different oscillation amplitudes. Both responses have been scaled by their respective excitation voltages. For the smaller excitation, the resonance peak is fitted well by the red line that corresponds to the response of a damped harmonic oscillator. As the periodic excitation is increased, the cubic term in Eq. 3 6 leads to nonlinear behavior in the oscillations. The resonance curve becomes asymmetric with the peak shifting to lower frequencies, consistent with a negative value of β. At a high enough excitation, hysteresis occurs in the frequency response, as shown by the squares in Fig. 5-1b. Within a certain range of driving frequencies, there are two stable dynamic states with different oscillation amplitude and phase. Depending on the history of the oscillator, the system resides in either the high-amplitude state or low-amplitude state. In the absence of fluctuations, the oscillator remains in one of the stable states indefinitely. As described in Sec , when sufficient noise is applied in the excitation, the oscillator is induced to escape from one dynamic state into the other. Since this driven, bistable system is far from thermal equilibrium and cannot be characterized by free energy, calculation of the escape rate is a non-trivial problem. Theoretical analysis [12, 33] suggests that the rate of escape Γ at a particular driving frequency depends exponentially on the ratio of an activation barrier E a to the noise intensity I N : Γ = Γ 0 e Ea/I N. (5 2) The noise intensity I N was introduced in Sec. 2.4 and in this chapter has arbitrary units. Likewise E a R i, where the units of E a can be assumed to be the same as the units of 77

78 I N. The arbitrary units will have no effect on the final results as we are only concerned with the power law dependence of E a on frequency. Close to the bifurcation frequency where the high-amplitude state disappears the activation barrier is expected to display system-independent scaling: E a ( ν) ξ, (5 3) where the frequency detuning ν is the difference between the driving frequency and the bifurcation frequency. The frequency parameter ν is used here in lieu of the generic system parameter η. The activation barrier is predicted [12, 33] to increase with frequency detuning with critical exponent ξ = 3/2. This scaling relation is generic and is expected to occur in a number of non-equilibrium systems. We describe below our comprehensive experimental investigation of activated switching from the high-amplitude to the low-amplitude state for two micromechanical oscillators with different resonant frequencies and damping coefficients. The critical exponents measured for both samples were in good agreement with theory. Results. In this experiment, we induce transitions from the high-amplitude state to the low-amplitude state by injecting noise in the excitation with a bandwidth of 100 Hz centered about the resonant frequency. The bandwidth of the noise is much larger than the width of the resonance peak. We chose a sufficiently large sinusoidal excitation so that the hysteresis loop exceeds twice the resonance peak width. Figure 5-2a shows typical switching events at an excitation frequency of Hz for sample A where the oscillator resides in the high-amplitude state for various durations before escaping to the low-amplitude state. Due to the random nature of the transitions, a large number of switching events must be recorded to determine the transition rate accurately. During the time interval between switching events in Fig. 5-2a, the oscillator is reset to the high-amplitude state using the following procedure. First, the noise is turned off and the driving frequency is increased beyond the range of frequencies where bistability 78

79 Norm. Osc. Amp (a) Number of Transistions (b) Time (s) Residence Time (s) Figure 5-2. (a) In the presence of noise in the excitation, the oscillator switches from the high-amplitude state to the low-amplitude state at different time intervals. The system is reset to the upper amplitude state between switching events. The detuning frequency is ν = 0.05 Hz. (b) Histogram of the residence time in the upper state before switching occurs, at a detuning frequency of 0.05 Hz for sample A. The line is an exponential fit using the decay rate equation. occurs (> Hz as shown in Fig. 5-1b). The driving frequency is then decreased slowly towards the target frequency so that the oscillator remains in the high-amplitude state. Once the target frequency is reached, the noise is turned back on and the time for the oscillator to escape from the high-amplitude state is recorded. This process is then repeated multiple times to accumulate the statistics for switching. Such a procedure is necessary because the energy barrier for transitions from the low-amplitude state back to the high-amplitude state is much larger than the barrier for transitions in the opposite direction. Thus, noise induced transitions from the low-amplitude state to the high-amplitude state will fail to occur in the duration of the experiment and the oscillator must be reset to the high-amplitude state using the steps described above. Figure 5-2b shows a histogram of the residence time in the high-amplitude state before a transition occurs. The exponential dependence on the residence time indicates that the transitions are random and follow Poisson statistics as expected. 79

80 logγ (s 1 ) /I N Figure 5-3. Logarithm of the transition rate from the high-amplitude state as a function of inverse noise intensity at a detuning frequency ν of 0.25 Hz for sample A. The slope of the linear fit yields the activation barrier. To determine the activation barrier for a particular detuning frequency, we record a large number of transitions for multiple noise intensities I N. The average residence time at each noise intensity is extracted from the exponential fit to the corresponding histograms. Figure 5-3 plots the logarithm of the average transition rate as a function of inverse noise intensity. The transition rate varies exponentially with inverse noise intensity, demonstrating that escape from the high-amplitude state is activated in nature. According to Eq. 5 2, the slope in Fig. 5-3 yields the activation barrier for escaping from the high-amplitude state at the particular detuning frequency. We repeat the above procedure to determine the activation barrier for other detuning frequencies ν ( ν is the difference between the driving frequency and the bifurcation frequency at which the high-amplitude state disappears) and show the results in Fig. 5-4 for sample A and sample B. All the detuning frequencies chosen for sample A are smaller 80

81 log E a 6 log E a log ω (rad s 1 ) log ω (rad s 1 ) A B Figure 5-4. Dependence of the activation barrier on detuning frequency for sample A and sample B. The solid lines are power law fits, yielding critical exponents of 1.38 ± 0.15 and 1.4 ± 0.15 respectively. than its resonance peak width while the maximum detuning frequency for sample B is about 4 times its resonance peak width. Fitting the activation energies with a power law dependence on the detuning frequency yields critical exponents of 1.38 ± 0.15 and 1.4 ± 0.15 for samples A and B respectively. Despite the different resonant frequencies and a factor of 4 difference in damping, the critical exponents obtained for both samples are in good agreement with theoretical predictions [12, 33, 81, 82]. Such scaling behavior near a spinodal point is expected to be universal in all systems far from thermal equilibrium. Apart from periodically driven micromechanical oscillators [83], a critical exponent of 3/2 was recently observed in rf-driven Josephson junctions [84]. Other non-equilibrium systems such as nanomagnets driven by polarized current [85] and double barrier resonant tunneling structures [86] are also expected to obey the same scaling relationship. Recently Aldridge and Cleland [19] measured noise-induced switching between dynamical states in a nanomechanical beam. They found a quadratic dependence of the activation barrier on the distance to the critical point where the two stable states of forced vibrations and the unstable periodic state all merge together. Such quadratic dependence 81

82 2 Amplitude (mrad) ω b2 ω b ω (rad/s) d Figure 5-5. Response of sample A at ω d /2 versus the frequency of parametric driving ω d. The solid and dashed lines represent the stable attractors and the unstable oscillation states respectively. arises [82] when the parameters are changed along the line where the populations of the two stable states are equal to each other. This requires changing simultaneously both the amplitude and the frequency of the driving field. In contrast, in our experiment, we approach a bifurcation point where a stable large-amplitude state and the unstable state merge together, while the stable small-amplitude state is far away. We vary only one parameter, the detuning frequency, while maintaining the periodic driving amplitude constant. We found that the activation barrier for escape is reduced to zero with critical exponent of 3/ Critical Exponent Near a Pitchfork Bifurcation Device and Experimental Setup. The micromechanical torsional oscillators in this experiment consist again of a movable polycrystalline silicon plate (500µm 500µm) suspended by torsional springs. Details can be found in Chapter 2. Two electrodes are 82

83 located underneath the top plate. A periodic driving voltage V d = V dc + V ac cos ω d t is applied to one of the electrodes, where the driving frequency ω d = 2ω 0 + ɛ is close to twice the natural frequency ω 0 ( rad s 1 and rad s 1 for samples C and D respectively). The top plate is therefore subjected to a periodic electrostatic torque, the angular gradient of which modulates the spring constant. The equation of motion with the addition of a noise source N(t) is given by Eq. 3 8: θ + 2γ θ + ( ω0 2 + k ) e I cos ω dt θ + αθ 2 + βθ 3 = N(t). Torsional oscillations of the top plate are detected capacitively by the other electrode. All measurements were performed at 77 K and < 10 6 Torr. The Q of the oscillators are sufficiently high ( 7500) so that the response of the oscillator at ω d 2ω 0 is negligible. When the amplitude of the spring modulation exceeds a threshold value k T = 4ω 0 γi, period doubling occurs in the oscillator response. Oscillations are induced at half the modulation frequency in a range close to ω 0. As shown in Fig. 5-5, there are three ranges of frequencies with different number of stable attractors, separated by a supercritical bifurcation point ω b1 = 2ω 0 + ω p and a subcritical bifurcation point ω b2 = 2ω 0 ω p, where ω p = ke 2 kt 2 /2Iω 0. In the first region (ω > ω b rad s 1 ), no oscillations take place, as the only stable attractor is a zero-amplitude state. At ω b1, there emerge two stable states of oscillations at frequency ω d /2 that differ in phase by π but are otherwise identical, because of the symmetry with respect to a translation in time by 2π/ω d. These two stable states are separated in phase space by an unstable state with zero oscillation amplitude (dashed line in Fig. 5-5). At frequencies below ω b2 ( rad s 1 ), the zero-amplitude state becomes stable, resulting in the coexistence of three stable attractors. These stable states are separated in phase space by two unstable states indicated by the dashed line in Fig The presence of noise allows the oscillator to occasionally overcome the activation barrier and switch between the different attractors. Since the parametrically driven 83

84 oscillator is far from equilibrium and is not characterized by free energy, the transition rate cannot be determined from the height of a free energy barrier. Theoretical analysis indicates that the transitions remain activated in nature [33] (see Eq. 5 2). In general, the activation barrier E a is determined by the device parameters such as the damping constant, nonlinearity coefficients and the driving frequency. Near the bifurcation points, the system dynamics is characterized by an overdamped soft mode and E a decreases to zero according to ω d 2ω b ξ, where the critical exponent ξ is universal and depends only on the type of bifurcation. In a parametric oscillator, the supercritical and subcritical bifurcations involve the merging of two stable oscillation states and an unstable zero-amplitude state (at ω b1 ) and the merging of two unstable states and a zero-amplitude stable state (at ω b2 ) respectively. When three states merge together in such pitchfork bifurcations, the critical exponent is predicted to be 2 [36]. Away from the bifurcation points, the scaling relationship no longer holds and different exponents were obtained depending on the nonlinearity and damping of the system. In order to investigate the transitions between stable states in our parametric oscillator, we inject noise with a bandwidth of 600 rad s 1 centered at ω 0. Figures 2a and 2c show respectively the oscillation amplitude and phase at a driving frequency in the range of two coexisting attractors. Transitions can be identified when the phase slips by π. The two oscillation states have the same amplitude. These two attractors can also be clearly identified in the occupation histograms in Figs. 5-7a and 5-7b. Figures 5-6b and 5-6d show switching events at a driving frequency with three attractors, where the zero-amplitude state has also become stable. In contrast to Fig. 5-6a, the oscillator switches between two distinct amplitudes. At high amplitude, the phase takes on either one of two values that differ by π. When the oscillator is in the zero-amplitude state, there are large fluctuations of the phase as a function of time. The coexistence of three attractors in phase space is also illustrated in Figs. 5-7c and 5-7d for two other driving frequencies. 84

85 Amplitude (mrad) 1 (a) (b) (c) 7 6 (d) Phase (rad) Time (s) Time (s) Figure 5-6. Oscillation amplitude (a) and phase (c) for ω d = rad s 1. For ω b2 < ω d < ω b1, transitions occur when the phase slips by π. (b) When ω d (= rad s 1 ) is lower than ω b2, transitions involve jumps in the amplitude. (d) In the high amplitude state, the oscillation phase takes on either one of two values that differ by π. The phase fluctuates when the oscillator is in the zero-amplitude state. 85

86 Y (mrad) 1 (a) (b) (c) (d) X (mrad) Figure 5-7. The occupation in phase space at four different ω d s. X and Y denote the two quadratures of oscillation. The grey scale represents the number of times that the oscillator is measured to lie within a certain location in phase space. (a) ω d = rad s 1. A pair of oscillation states emerges near ω b1. (b) ω d = rad s 1. As ω d decreases, the two states move further apart in phase space. (c) ω d = rad s 1. When ω d < ω b2, an addition attractor at the origin appears. (d) ω d = rad s 1. With further decrease in ω d, the occupation of the zero-amplitude state increases. Results. Following along in the same manner as Sec. 5.2 we identify the residence time in each state before a transition occurs and plot the results as a histogram for one of the oscillation states (see Fig. 5-8a). The exponential dependence on the residence time indicates that the transitions are random and follow Poisson statistics as expected. From the exponential fit to the histograms, the transition rate out of each state is extracted. The transition rates out of the two oscillation states are measured to be identical to within experimental uncertainty at all noise intensities. Figure 5-8b plots the logarithm of the transition rate as a function of the inverse noise intensity. The transition rate varies exponentially with inverse noise intensity, demonstrating that the switching is activated in nature. Using Eq. 5 2, we obtain the corresponding activation barriers at a particular driving frequency from the slope in Fig. 5-8b. Transitions out of the zero-amplitude state are also found to be activated and follow Poisson statistics in a similar manner. The above procedure is repeated to determine the activation barriers at other driving frequencies. Figure 5a shows the driving frequency dependence of the activation barriers E a1 and E a2 for switching out of the oscillation states and the zero-amplitude state 86

87 100 (a) (b) # of Transitions log Γ (s 1 ) Residence Time (s) /I N Figure 5-8. (a) Histogram of the residence time in one of the oscillation states (ω d = rad s 1 ) before switching occurs. The solid line is an exponential fit. (b) Logarithm of the transition rate as a function of inverse noise intensity. respectively. At the high frequency end of Fig. 5-9a, only the zero-amplitude state is stable. As ω d is decreased, two stable oscillation states (separated by an unstable state) emerge at ω b1. With increasing frequency detuning ω 1 = ω b1 ω d, the pair of oscillation states move further apart in phase space (Figs. 3a and 3b) and E a1 increases. At ω b2, the zero-amplitude state becomes stable. The appearance of the stable zero-amplitude state is accompanied by the creation of two unstable states separating it in phase space from the two stable oscillation states. E a2 initially increases with frequency detuning ω 2 = ω b2 ω d in a fashion similar to E a1. Close to ω b2, E a1 exceeds E a2 and the occupation of the oscillation states is higher than the zero-amplitude state (Fig. 5-7c). As ω d decreases, E a2 continues to increase monotonically while E a1 remains approximately constant. As a result, E a1 and E a2 cross each other at rad s 1, beyond which the occupation of the zero-amplitude state becomes higher than the oscillation states. The dependence of the occupation on frequency detuning was also observed in parametrically driven atoms in magneto-optical traps [15]. 87

88 E a (arb. units) (a) ω b2 ω b1 log E a (b) ω (rad/s) log ω 2 (rad/s) log E a1 5 (c) log ω 1 (rad/s) log E a (d) log ω 1 (rad/s) Figure 5-9. (a) Dependence of the activation barriers E a1 ( ) and E a2 ( ) on the frequency of parametric modulation for sample C. (b) E a2 vs log ω 2 for sample C. The lines are power law fits to different ranges of ω 2. (c) and (d) log E a1 vs log ω 1 for samples C and D respectively. In general, the activation barriers E a1 and E a2 depend on various parameters of the device. Nonetheless, at frequencies close to the bifurcation points, theoretical analysis indicates that the activation barriers exhibit universal scaling, with E a1,a2 ω b1,2 ω d ξ. For pitchfork bifurcations in a parametric oscillator that involve merging of three states, ξ is predicted to be 2 [36]. Figures 5b and 5c show the dependence of E a1,a2 on frequency detuning ω 1,2 on logarithmic scales. At small detuning, both activation barriers show power law dependence on detuning. The critical exponents 88

89 are measured to be 2.0 ± 0.1 and 2.00 ± 0.03 for E a1 and E a2 respectively for sample C. For sample D, the exponent of E a1 is measured to be 2.00 ± 0.02 (the range of ω 1 is smaller in sample D because near ω b2 where the oscillations are large, the torsional plate occasionally comes into contact with the electrodes in the presence of injected noise). This quadratic dependence of the activation barrier on detuning near the bifurcation points is predicted to be system-independent and is expected to occur in other parametrically-driven, nonequilibrium systems such as electrons in Penning traps [16] and atoms in magneto-optical traps [15, 87]. Away from the vicinity of the bifurcation point, however, the variation of the activation barrier with frequency detuning is device-specific. Figures 5b and 5c show crossovers from the quadratic dependence to different power law dependence with exponents 1.43 ± 0.02 and 1.53 ± 0.02 for E a1 and E a2 respectively. These values are distinct from the exponents obtained in parametrically driven electrons in Penning traps [16] because the nonlinearity and damping are different for the two systems. Recent theoretical predictions indicate that the symmetry in the occupation of the two oscillation states in a parametrically driven oscillator will be lifted when an additional small drive close to frequency ω d /2 is applied [79]. A number of phenomena, including strong dependence of the state populations on the amplitude of the small drive and fluctuation-enhanced frequency mixing, are expected to occur. Further experiments are warranted to test such predictions and reveal other fluctuation phenomena in parametrically driven oscillators. Parametric pumping is widely used to improve the sensitivity of micromechanical detectors by mechanically amplifying a signal [74] or by reducing the resonance linewidth in viscous environments [88]. The sharp jump in oscillation amplitude at the bifurcation points is utilized for accurate determination of device parameters [19, 84, 89]. As the dissipation increases in a resonantly-driven Duffing oscillator, the natural resonance linewidth becomes very broad and the hysteresis region shrinks for comparable oscillation amplitude. Parametrically driven oscillators, on the other hand, maintain the sharp jump 89

90 in response at the subcritical bifurcation point even for large damping. It is not necessary to increase the oscillation amplitude provided that the stronger parametric pumping compensates the energy loss due to damping [90]. Therefore, parametric oscillators are particular useful for sensing in liquid or gaseous environments. Apart from the relevance to other parametrically driven nonequilibrium systems [15, 16, 87], the comprehensive study of the dependence of the transition rate on frequency reported here may prove useful in sensing applications [91] with parametrically driven micromechanical devices. 5.4 Summary The vast majority of micro- and nano-mechanical sensors operate in the linear regime. Typically, sensing is achieved by measuring the d.c. response or by monitoring variations in the resonant frequency due to changes in device parameters. The study of noise-induced switching between stable oscillation states in a strongly driven, nonlinear mechanical oscillator could open up new opportunities for sensing applications. For instance, the switching rate varies exponentially with the noise intensity and the device parameters. When the oscillator resides at the high amplitude state near the bifurcation point, small changes in device parameters lead to large changes in the transition rate. Switching events can be easily detected due to the jump in oscillation amplitude. Such a strong dependence could be exploited for high sensitivity signal detection. In conclusion, we demonstrated the activated behavior of noise induced switching in a nonequilibrium system, a nonlinear, underdamped, micromechanical torsional oscillators modulated by a strong resonant field. The measured critical exponent for the activation barrier near the bifurcation point agrees well with the predicted value of 3/2, consistent with the system-independent scaling of the activation barrier in the vicinity of the bifurcation point. Such scaling relationship also applies to other systems that are far from equilibrium near the spinodal point, including rf-driven Josephson junctions [20, 31], nanomagnets driven by polarized current [85] and double barrier resonant tunneling structures [86]. In contrast, we see that when the system is parametrically driven and 90

91 the type of bifurcation changes, so that 3 states merge at one time, the critical exponent changes. In this chapter results consistent with the scaling of 2 near such bifurcation were shown. 91

92 CHAPTER 6 THE KINETIC PHASE TRANSITION The results in this chapter can be found in articles Supernarrow spectral peaks near a kinetic phase transition in a driven, nonlinear micromechanical oscillator, C. Stambaugh and H. B. Chan, Physical Review Letters, 97, (2006) Copyright 2006 by the American Physical Society and Fluctuation-enhanced frequency mixing in a nonlinear micromechanical oscillator, H. B. Chan and C. Stambaugh, Physical Review B, 73, (2006) Copyright 2006 by the American Physical Society. 6.1 Introduction In this chapter the switching between states in a Duffing oscillator is analyzed. As we have seen, when noise is injected into the system switching is observed. In Ch. 5 it was shown that when the noise is weak, the escape rate Γ i out of state i (i = 1 or 2) depends exponentially on the ratio of an activation barrier R i and the noise intensity D: Γ i e R i/d. (6 1) For clarity, we return here to the dimensionless forms of the noise and activation barrier introduced in Ch. 1. Also R i was shown to depend on one the control parameters of the system: the driving frequency. The activation barrier is also known to depend on the amplitude, as well as the shape of the power spectrum of the noise. The ratio of the populations of the two dynamical states is given by: w 1 /w 2 e (R 2 R 1 )/D. (6 2) As a result of the exponential dependence of the population ratio on the difference in the activation barriers, the system will be found in either state 1 or state 2 with overwhelmingly large probability over most of the parameter space [33, 92]. This was seen in Ch. 5 when the activation barrier was analyzed near the bifurcation frequency. The occupations of the two states are comparable only over a very narrow range of parameters. 92

93 This behavior bears close resemblance to systems in thermal equilibrium with two phases such as liquid and vapor. Such thermodynamic systems are usually in either one of the two phases and only at the phase transition will the two phases coexist. Even though driven, nonlinear systems are in general far from equilibrium, theoretical works predicted that a similar kinetic phase transition would occur under the appropriate conditions [33]. Similar to thermodynamic systems, fluctuations increase significantly when these systems far from equilibrium undergo kinetic phase transition s. A range of generic, system-independent phenomena, including the appearance of a supernarrow peak in both the susceptibility and the spectral density of fluctuations [33, 37], is expected to take place. However, these phenomena have so far only been observed in analog circuit simulations [81]. Additionally near the kinetic phase transition noise is expected to play a constructive role in enhancing the mixing of oscillations at different frequencies, an important use of a nonlinear system. Well-known examples are higher harmonic generation and four-wave mixing in nonlinear optics. Frequency mixing is also of great importance in high sensitivity detection and signal processing. In Sec. 6.2 the system is introduced and data is presented that shows that a region exists where the occupation of the two states is approximately the same. In Sec. 6.3 we measure the spectral densities of fluctuations of an underdamped, nonlinear micromechanical torsional oscillator near the kinetic phase transition. The most prominent feature in the fluctuation spectrum is a narrow peak centered at the frequency of the periodic excitation. We demonstrate that this narrow peak is associated with noise-induced transitions between the two attractors. The width of the peak varies linearly with the transition rate and is more than a factor of 10 smaller than the natural line width of the resonance in our experiment. Away from the kinetic phase transition, the intensity of the peak decreases exponentially. Apart from the narrow peak, we also observe smaller, much broader peaks in the spectrum that are associated with fluctuations 93

94 Norm. Osc. Amp (a) Counts/Bin (b) Time (s) Norm. Osc. Amp. Figure 6-1. (a) Time trace of the normalized oscillation amplitude near a kinetic phase transition demonstrating transitions between the two states. (b) Histogram of the oscillation amplitude of the oscillator near a kinetic phase transition showing approximately equal occupation of the two states. within each attractor. These broad peaks are present for all driving frequencies within the hysteresis loop and their dependence on the noise intensity is distinctly different from the narrow peaks at the kinetic phase transition. In Sec. 6.4 we demonstrate that when a second, weak periodic trial excitation with frequency ω s ω d is applied in addition to the strong periodic drive, fluctuation-induced interstate transitions become synchronous with the beating between the two driving frequencies at certain noise intensities, resulting in enhanced response at both the frequency of the weak excitation and the down-converted frequency. Responses at both frequencies initially increase with noise and subsequently decrease as the noise intensity becomes very strong. The occurrence of such high frequency stochastic resonance is in agreement with theoretical predictions [33, 81]. 6.2 Kinetic Phase Transition A micromechanical oscillator, consisting of a movable polysilicon plate supported by two torsional rods, was used to take the measurements in this experiment. As described in 94

95 Chapter 3 a voltage V is applied to one of the electrodes generating an electrostatic torque that excites the torsional oscillations. The other electrode is used to capacitively detect the oscillations. All measurements were taken at a temperature of 77 K and pressure < 10 7 Torr. The driving voltage V is a sum of a biasing d.c. voltage, a periodic a.c. voltage with frequency f d = ω d 2π and random noise. The oscillator, driven simultaneously by a periodic excitation and random noise, is well-described by Eq. 3 6 with the addition of the noise term N(t). In the absence of noise and at small driving torque, the response of the oscillator corresponds to that of a damped harmonic oscillator with a resonant frequency of 3286 Hz and Q When the driving torque is increased beyond a critical value, the frequency response becomes hysteretic due to the cubic nonlinearity. Within a range of driving frequencies, two stable dynamical states coexist. In the absence of fluctuations, no transitions occur between the two states. When sufficient noise is injected into the driving voltage, the oscillator switches between the two attractors. Over most of the hysteresis loop, the activation barriers R 1,2 for escape from the two states are significantly different. The hollow and solid triangles in Fig. 6-2b shows the occupation of the high-amplitude and low-amplitude states respectively. On the low frequency side of the hysteresis loop, the occupation of the low amplitude state is considerably higher than the high amplitude state and the probability of finding the oscillator in the low amplitude state is close to unity. As the driving frequency decreases, the activation barrier R 1 for switching out of the high amplitude states decreases. At the bifurcation frequency (f 1 = Hz), R 1 goes to zero and the low-amplitude state becomes the only attractor. In Chapter 5, we showed that R 1 depends on the detuning frequency (f d f 1 ) in the vicinity of the bifurcation point with a critical exponent of 3/2 in agreement with theoretical predictions [33]. On the high frequency side of the hysteresis loop, a similar argument applies except that the high-amplitude state is the stable attractor. 95

96 Normalized Amplitude Occupation Peak Intensity (a) (b) (c) f w p low w high f d (Hz) Figure 6-2. (a) Normalized frequency response of the oscillator ( ) fitted to a damped oscillator with cubic nonlinearity (solid line). (b) Occupation of the two states versus frequency. At the kinetic phase transition (f p = Hz), the occupation of the high amplitude state ( ) and the low amplitude state ( ) are comparable. (c) Dependence of the intensity of the supernarrow spectral peak on the driving frequency f d. The intensity falls off exponentially as the driving frequency is moved away from f p. While the oscillator is predominantly in one of the attractors over most of the hysteresis region, there exists a small range of frequencies where the occupation of the two attractors is of the same order of magnitude. Figure 6-1a shows the oscillation amplitude as a function of time at a driving frequency of Hz and clearly illustrates the system switching between the two states. The relative occupation of the two states at this driving frequency is deduced by calculating the area under the two peaks in the histogram of the oscillation amplitude (Fig. 6-1b). 96

97 Even though the nonlinear oscillator is a driven system that is far from equilibrium, the above behavior bears resemblance to thermodynamical systems at phase transitions when two phases coexist. A kinetic phase transition was predicted to occur in systems far from equilibrium when the populations of the two dynamical states are equal [33]. An important feature associated with phase transitions is the large fluctuations arising from transitions between the two states. We perform a systematic study of the fluctuations of the nonlinear oscillator at the kinetic phase transition by examining the power spectral densities of the oscillator response. 6.3 Spectral Densities of Fluctuations To resolve features in the spectral densities of fluctuations, it is necessary to have high resolution in frequency. The maximum number of consecutive data points that can be recorded with our instrument limits the frequency resolution in calculating the fast Fourier transform (FFT) of the angular displacement θ(t). Instead, we record the slowly varying envelope of the oscillations using a lock-in amplifier with a bandwidth of about 30 Hz. The response of the oscillator can be written as θ(t) = X(t) cos(2πf d t) + Y (t) sin(2πf d t), (6 3) where X(t) and Y (t) are the slowly varying amplitudes of oscillations in and out of phase with the driving torque at frequency f d. The spectral density of fluctuations is given by [93] Q(f) = 1 N [ (X(t) ) ( ) iy (t) X(t + τ) + iy (t + τ) e i2π(f f d )τ], (6 4) τ t where N is a normalization constant. In the absence of injected noise, oscillations occur only at the periodic driving frequency f d and the measured spectrum consists of a delta function centered at f d. When noise is added to the excitation, the spectral densities of fluctuations become dramatically different. Figure 6-3 shows the spectral density of fluctuations at 3 different periodic 97

98 2 (a) 250 (b) 12 (c) 1.5 ω d <ω p 200 ω d ω p 10 ω d >ω p Q(ω) ω ω d (rad/s) Figure 6-3. Power spectral density of fluctuations for three different drive frequencies f d : (a) 3284 Hz, (b) Hz, (c) Hz. Notice that the vertical scale of (b) is more than 20 times larger than (a) and (c). driving frequencies. To focus on fluctuations about the ensemble average response, the delta function peak at f d obtained with no injected noise is removed from the spectrum. In other words, the data point at f d is omitted for each panel in Fig Figure 6-3b shows the spectral density of fluctuations at the driving frequency f d f p where the occupations of the two states are equal. The most prominent feature is a very sharp peak centered at the driving frequency. The width of this peak is a factor of 10 smaller than the natural width of the resonance peak. This sharp peak is predicted to arise [33] due to fluctuation-induced transition between the two dynamical states. Figures 6-3a and 6-3c shows the spectral density at two other driving frequencies that are comparatively far away from f p. As the periodic driving frequency is changed so that the oscillator moves 98

99 0.25 (a) ω d <ω p (b) ω d ω p 2.5 (c) ωd >ω p Q( f ) ω ω d (rad/s) Figure 6-4. Power spectral density of fluctuations for three different drive frequencies f d : (a) 3284 Hz, (b) Hz, (c) Hz. Notice that the vertical scale of (b) is more than 20 times larger than (a) and (c). The axes in these figures have been rescaled to reveal the smaller and broader peaks in the fluctuation spectrums of Fig. 6-3(a), (b) and (c) respectively. away from the phase transition point, the sharp peak shifts accordingly to remain centered at the driving frequency. The area under the peak, however, drops significantly. Figure 6-2c plots the area under the narrow peak as a function of periodic driving frequency, clearly demonstrating that the intensity of the supernarrow peak attains maximum at the kinetic phase transition and decreases exponentially as the occupation of one of the states exceeds the other and transitions between the states become less frequent. Figure 6-5a shows the behavior of the supernarrow peak with different noise intensities when the periodic driving frequency is held constant to maintain the oscillator at the kinetic phase transition. The noise intensity differs by a factor of 4 between 99

100 Q( f ) Area I N Width (Hz) 10 5 x Γ (s 1 ) 100 Figure f f d (Hz) The supernarrow peak for two intensities of injected noise at the kinetic phase transition. The injected noise intensity for the hollow circles is 4 times greater than the solid circles. The dotted and solid lines are Lorentzian fits to the data. Left inset: Dependence of the area of supernarrow peak on injected noise intensity. Right inset: The peak width vs. transition rate. solid and hollow circles. Both sets of data are fitted well by Lorentzians [37, 81]. As the noise increases, the peak width increases and the peak height decreases. We found that the area under the peaks remains about constant, changing by less than 10% when the noise intensity changes by more than a factor of four (left inset of Fig. 6-5). The right inset of Fig. 6-5 plots the peak width as a function of the sum of the transition rates Γ i out of each state, where the transition rates are determined directly from the residence time between transitions. The linear dependence of the peak width on the transition rate and the exponential decrease in the area of the peak away from the kinetic phase 100

101 Area 1 Q( f ) I N 0.01 Figure f f d (Hz) The broad, small peak at two injected noise intensities that differ by a factor of 2, at f d = 3284 Hz. Inset: Linear dependence of the peak area on injected noise intensity. transition clearly identify the supernarrow peak with noise-induced transitions between the attractors. In addition to the supernarrow peak, there are other peaks in the fluctuation spectrum that are weaker and much broader than the supernarrow peak at the kinetic phase transition. These peaks, unlike the supernarrow peak, are present for all excitation frequencies within the hysteresis loop, as shown in Figs. 6-3a and 6-3c when the oscillator is far from the kinetic phase transition and in Fig. 6-3b when the oscillator is at the kinetic phase transition. These smaller peaks represent characteristic frequencies of fluctuations about each dynamical state when the fluctuations are not strong enough to induce a transition over the activation barrier. In contrast to the supernarrow peak, the 101

102 1.5 (a) 1.2 (b) 9 1 A (mrad) A (mrad) ω d (rad s 1 ) Time (s) Figure 6-7. (a) Oscillation amplitude A vs. driving frequency at V d of 460µV ( ) and 51µV ( ). The solid line represents a fit to the data at the smaller excitation using the response of a damped harmonic oscillator. For the large excitation, bistability occurs between rad s 1 and rad s 1. The dashed line fits the data to a damped oscillator with cubic nonlinearity [28]. (b) At driving frequency of rad s 1, the oscillation amplitude switches between two distinct values when noise is injected into the system. shape of these small peaks do not change considerably as the noise intensity increases (Fig. 6-6) and their area varies proportionally with noise intensity (inset of Fig. 6-6). 6.4 Frequency Mixing A different oscillator with the same dimensions as the one used above, was used for the frequency mixing experiment. It has a resonance frequency of rad s 1 and a quality factor of Q The experiment was performed at temperature of 77 K and pressure of < 10 6 Torr. As in the previous case under a sufficiently strong periodic excitation, the oscillator possesses two oscillation states with distinct amplitude and phase over a range of frequencies (solid squares in Fig. 6-7a). The presence of noise enables the system to occasionally overcome the activation barrier and switch between these two states. Figure 6-7b shows that the oscillation amplitude jumps between two values as a function of time when noise is injected into the excitation voltage. The occupations of the two states depend on the driving frequency. As shown in Fig. 6-8a, the oscillator resides 102

103 1 0.8 (a) (b) C Occupation A B ω d (rad s 1 ) V d (µv) Figure 6-8. (a) Occupations of the high-amplitude state ( ) and the low-amplitude state ( ) as a function of driving frequency at V d = 460µV. (b) Occupations as a function of V d at fixed driving frequency of rad s 1. predominantly in the high- or low-amplitude state at the high and low frequency sides of the hysteresis loop respectively. The occupations of the two states are comparable only at a small range of frequencies. As was shown in Sec. 6.3, noise-induced interstate transitions at the kinetic phase transition lead to a narrow peak in the fluctuation spectrum that is centered at the driving frequency (Fig. 6-9a). The width of this peak is inversely proportional to the residence time of the states and its height drops exponentially as the excitation frequency is tuned away from the value at which the occupations of the two states are comparable. If the excitation amplitude is varied with the driving frequency kept constant, the occupations of the two states exhibit a similar behavior: the occupation of the low-amplitude state is high at small driving amplitude and vice versa for large driving amplitude. The occupations of the two states become comparable at some intermediate amplitude (point A in Fig. 6-8b). We investigate the response of the oscillator when a weak periodic drive ( 120µV) at frequency ω s close to ω d is applied on top of the strong periodic drive. Since the nonlinearity of the oscillator mixes the primary and secondary driving frequencies, the weak drive induces oscillations not only at its own frequency ω s, but also at other frequencies. The response to the weak drive is the strongest at the frequencies ω s and 103

104 Power Spectrum (arb. units) x 10 3 (a) (b) (c) ω ω (rad s 1 ) d Figure 6-9. (a) At the driving frequency, where the occupations of the two states are equal, a sharp peak develops in the power spectral density of fluctuations. (b) The addition of a secondary, weak excitation at ω s1 = ω d rad s 1 leads to spectral peaks at ω s1 and 2 ω d ω s1. (c) The frequency of the secondary excitation is changed to ω s2 = ω d rad s 1. 2 ω d ω s. We found that at the kinetic phase transition where the occupations of the two oscillation states are equal, the presence of an optimal amount of noise enhances the mixing, resulting in amplified responses at both ω s and 2 ω d ω s, in agreement with predictions from theoretical analysis [81]. Figures 6-9b and 6-9c show the spectral response of the oscillator when weak periodic drives at two different frequencies (ω s1 = ω d rad s 1 and ω s2 = ω d rad s 1 respectively) were applied in the presence of noise. Spectral peaks at both the weak driving frequencies ω s1,2 and the down-converted frequencies 2ω d ω s1,2 can be clearly identified. The peaks are about a factor of 6 stronger compared to the case when noise is not applied to the excitation. In both cases, the frequency difference ω s1,2 ω d is much smaller than the relaxation rate of the oscillator (ω R /2Q 1.3 s 1 ). As shown in Fig. 6-9c, both spectral peaks diminish when the 104

105 frequency of the weak drive moves further apart from the frequency ω d of the primary drive. Next, we examine the dependence of the spectral peaks on the noise intensity. Figure 6-10(a) shows that the spectral peaks at ω s first increase with noise intensity, attaining a maximum at a noise power of 0.012mV 2 /Hz and subsequently decrease. Figure 6-10b demonstrates a similar behavior for the spectral peak at the down-converted frequency 2 ω d ω s where an optimal amount of noise enhances the frequency mixing. We emphasize that there is no periodic excitation at 2 ω d ω s and the spectral response in Fig. 6-10b arises from the nonlinear mixing of the primary and secondary frequencies. Figures 6-10c and 6-10d plot the signal-to-noise ratio, defined as the spectral responses at ω s and 2 ω d ω s divided by the spectral responses in the absence of the weak periodic excitation (Fig. 6-9a) at the corresponding frequencies. At both ω s and 2 ω d ω s, the signal-to-noise ratios achieve relative maxima at some intermediate noise intensity. The enhancement of the spectral responses at ω s and 2 ω d ω s was predicted to occur when noise-induced interstate transitions become synchronous with the beating between the primary and secondary driving frequencies [81]. For an intuitive understanding, we consider the beating between the primary and secondary drives that leads to modulation of the amplitude of the strong field at frequency ω s ω d. Since the relaxation rate of the oscillator (ω R /2Q 1.3 s 1 ) is much larger than ω s ω d, the oscillation amplitude at the primary frequency varies adiabatically with the driving amplitude and is therefore modulated at the beating frequency (Fig. 6-11a). In the absence of noise, the oscillator remains in the low-amplitude state provided that the modulation envelope is small. When sufficient noise is introduced into the system, the oscillator switches between the two oscillation states (Fig. 6-11b). Slow amplitude modulations of the primary excitation change the probability of fluctuational transitions and hence the occupations of the two states. 105

106 Peak Height (arb. units) 2 x (a) (c) Peak Height (arb. units) 1 x (b) (d) SNR 20 SNR I N (mv 2 /Hz) I N (mv 2 /Hz) Figure (a) The height of the spectral peak at ω s as a function of the noise intensity for ω s1 = ω d rad s 1 ( ) and ω s2 = ω d rad s 1 ( ). (b) The spectral peak at the down-converted frequency of 2 ω d ω s1,2. (c) The signal-to-noise ratios at ω s1,2 as a function of the noise intensity. (d) The signal-to-noise ratios at 2 ω d ω s1,2. As illustrated in Fig. 6-7(e), the occupation probability of the low-amplitude state, for example, decreases as the excitation amplitude increases (point B) and vice versa (point C). The occupation probabilities hence vary periodically as a function of time. At the optimal noise intensity, interstate transitions occur approximately once every half-cycle of the beating (Fig. 6-11b). Therefore, the amplitude modulation of the response becomes significantly larger compared to Fig. 6-11a, resulting in spectral components at both the secondary frequency ω s and the down-converted frequency 2 ω d ω s that are enhanced compared to the case when noise is absent. Upon further increase of the noise intensity, 106

107 A (mrad) (a) (b) (c) Time (s) Figure (a) Oscillation amplitude as a function of time in the absence of noise. (b) Under optimal noise, the oscillation amplitude switches between two stable values as a function of time and in sync with the beating frequency (square wave represented by the solid line) (c) With large intensity of noise, interstate switching loses synchronization with the beating frequency. the transition rate between the two states exceeds the beating frequency and the switching events are no longer synchronous with the modulation in driving amplitude (Fig. 6-11c). As a result, the spectral responses at ω s and 2 ω d ω s decrease. The noise enhancement of spectral responses at ω s and 2 ω d ω s occurs only in the vicinity of the kinetic phase transition where the occupations of the two states are comparable. Figure 6-12 shows the spectral responses at ω s and at 2 ω d ω s as a function of ω d, where the frequency difference between the primary and secondary drives ω s ω d is maintained constant. The spectral responses at both frequencies attain maximum when the occupations of the two states are nearly equal. As the driving frequency is moved away from the kinetic phase transition, the occupation of one of the states become much larger than the other and interstate transitions become significantly less frequent. As a result, the enhancement of spectral responses becomes weaker. Such high frequency stochastic resonance was predicted to occur when a weak, periodic drive is applied to systems where bistability arises as a result of a primary, strong periodic modulation [81]. For ordinary systems with bistable potential, stochastic resonance [1, 4] takes place when the rate of noise-induced transitions becomes comparable 107

108 6 x Peak Height (arb. units) ω d (rad s 1 ) Figure Dependence of the spectral power at ω s1 ( ) and 2 ω d ω s1 ( ) on the driving frequency ω d. The frequency difference between the primary and secondary drives ω s1 ω d is maintained constant. The arrow indicates the frequency at which the occupations of the two states are equal. This frequency is slightly shifted from Fig. 6-7 because the device has been warmed up to room temperature and re-cooled down. to the frequency of the periodic drive. Our oscillator, in contrast, is monostable under weak periodic driving and develops bistability only when it is driven strongly. Both the primary and secondary periodic drives are applied at frequencies much higher that the rate of noise induced transitions. Another important difference from conventional stochastic resonance is that noise-enhanced response also takes place at 2 ω d ω s at which no periodic excitation is applied. Such efficient down-conversion of excitation frequency occurs due to synchronization of the switches in the oscillation amplitude with the beating frequency. 108

109 Apart from the response at 2 ω d ω s, nonlinear mixing at the difference frequency ω d ω s is also expected to occur in our oscillator. At a fixed driving frequency ω d, the oscillations in the high- and low-amplitude states take place about different angular positions (θ 1o and θ 2o ) due to the quadratic nonlinearity (θ 2 term in the restoring torque). As the oscillator switches between the two states, the mean angular position changes between θ 1o and θ 2o. When the noise-induced transitions become synchronous with the beating frequency ω s ω d, the spectral response at ω s ω d attains a maximum value. Such noise-enhanced heterodyning [81] should be observable in our oscillator after modifying the detection circuit to allow measurement of low frequency signals. 6.5 Summary Systems in thermal equilibrium, such as a Brownian particle fluctuating in symmetric double-wells potentials, also exhibit narrow peaks in their fluctuation spectrum [36]. The broadening of these peaks with increasing noise intensity leads to the well known phenomenon of stochastic resonance [1]. Even though the supernarrow spectral peak observed in our experiment also originates from transitions between coexisting states, there are important differences between our nonlinear oscillator and systems in thermal equilibrium. First, noise-induced switching in our oscillator occurs between two oscillation states with different amplitude. Our oscillator is bistable only under strong periodic drive. It is far from equilibrium [1, 11, 94] and is not characterized by free energy. Second, at the kinetic phase transition, the sharp spectral peak is centered at the driving frequency. The lack of time reversal symmetry results in a characteristic asymmetry in the broad peaks of the fluctuation spectrum about the driving frequency (see Fig. 6-11b). In contrast, for bistable systems in thermal equilibrium, the fluctuation spectrum is centered at zero frequency and no such asymmetric features are present. Critical kinetic phenomena such as the emergence of the narrow peak at the driving frequency [33, 81, 95] and noise enhanced frequency mixing reported here occur as a result of the interplay between noise and nonlinearity. Other bistable systems far from equilibrium, including rf-driven 109

110 Josephson junctions [20], nanomagnets driven by polarized current [85] and double barrier resonant tunneling structures [86], are expected to exhibit similar behavior in the parameter range where the two states have comparable occupations. While theoretical analysis [33] considered a Duffing oscillator with cubic nonlinearity, our measurements indicate that the supernarrow peak is robust even when higher order nonlinearities are present. The study of such critical kinetic phenomena could open new opportunities in tunable narrow band filtering and detection using micromechanical and nanomechanical oscillators, or frequency mixing applications. 110

111 CHAPTER 7 FLUCTUATION THEOREM 7.1 Introduction As mentioned in the introduction, thermal fluctuations and dissipation are connected. In Sec. 2.4 Johnson noise was introduced and the relationship between the resistance, a dissipative element, and voltage fluctuations was presented. This result was a direct consequence of the fluctuation-dissipation theorem; a theorem which in general describes a wide range of such relationships. More broadly fluctuation relations seek to identify and describe the general features of all systems [2, 96] that experience fluctuations. While the fluctuation-dissipation relations exist for systems in thermal equilibrium, they fail to account for the effect of fluctuations on systems driven far from equilibrium. A topic of interest within the broader context of nonlinear systems is the role fluctuations play in vibrating systems which exhibit bistability. As was shown in Ch. 3, when the damping of a system is small, weak forces can lead to large vibrational amplitudes. In this region where the nonlinearity can be especially strong, it is possible for two or more stable states to coexist. Switching between the states may occur if the fluctuations in the system are large enough [33]. This leads to dramatic effects on the amplitude of the vibrations even though on average the fluctuations may be small. This switching behavior has been previously studied in several systems including magneto-optical traps [15], electrons in a penning trap [16], micro- and nano-mechanical oscillators [17 19] and Josephson junctions [20]. Recent theoretical work [97] has identified specific features in the work variance of these systems that should occur near critical features relating to the bistability. In this chapter we investigate work fluctuations in a nonlinear oscillator driven out of equilibrium. We begin by operating the system in a linear regime to demonstrate that here the fluctuation-dissipation relation holds. We show that the variance is proportional to the average work and the proportionality coefficient between them is independent 111

112 of the driving frequency. After establishing that the system can be used to study work fluctuations, it is driven out of equilibrium and into a region where bistability is displayed. In this region the fluctuation-dissipation is not expected to be applicable. Here it is shown that the ratio of the variance to the average work has an exponentially large, frequency dependent peak at the kinetic phase transition [33], where the relative occupation of the two states is the same. More on the kinetic phase transition can be found in Ch. 6. Our results are the first experimental evidence of this universal feature. 7.2 Device and Experimental Setup A driven micro-electromechanical torsional oscillator was used as the system for these experiments. The device is composed of a movable plate which is suspended 2 microns above two detection electrodes by two torsional springs. The electrodes and movable plate can be used to excite oscillations and detect the response. Further device design details can be found in Ref. [18, 95, 98]. The system is operated in vacuum at a pressure < 10 6 Torr and at a temperature of 77 K. By applying a driving voltage V = V DC + V AC t + V noise to one of the electrodes a time varying electrostatic torque is setup which excites mechanical oscillations in the movable top plate. The device response is θ + 2λ θ + (2πf 2 0 )θ + βθ 3 = T 0 cos 2πf d t + N(t) (7 1) where f 0 is the resonant frequency, β is the nonlinear coefficient and T 0 is the driving torque. The damping in the system is λ 2π = 0.5 Hz and results from resistive losses [99]. The noise torque is N(t) and was previously defined in Sec. 2.4; about the region of interest the noise can be considered white. The noise torque is related to the noise intensity where I N N(t) 2. As in Ch. 5, I N will has arbitrary units. For small drives the response is linear and its squared amplitude can be fitted by a Lorentzian lineshape. When a sufficiently strong drive is applied the cubic term begins to effect the response; it becomes asymmetric as the peak shifts to a lower frequency [28]. Above a critical 112

113 drive strength the peak tilts over and the response becomes hysteric. Here a region exists where two stable vibrational states i = 1, 2 coexist. Under these conditions the system is considered to be far from equilibrium. 7.3 Theory The work done on a system by a periodic torque over time τ in the presence of fluctuations can be written as τ W W (τ) = dt T 0 (t) θ(t), (7 2) where θ is the response of the system and we assume at time t = 0 the system is in the steady state. The variance of the work distribution can be written as σ 2 = (W W ) 2. For both strong and weak drives the distribution of the work is Gaussian: 0 P (W ) = (2πσ 2 ) 1 2 exp (W W )2 /2σ 2. (7 3) Furthermore, it can be shown that W σ 2 τ. The variance can also be related to the average work and noise intensity: σ 2 = CI N W. (7 4) The parameter C is constant when the system is in the linear response regime [100]. When the system displays bistability, C is found to no longer be constant [97] and the fluctuation-dissipation relation can no longer be applied. In our experimental setup the response is measured using a heterodyne method in order to minimize noise and cross-talk in the response signal. This is achieved by applying 180 out of phase, high frequency (100kHz) sine waves to the two electrodes. One carrier signal is added to the driving signal on one of the electrodes, while the other is added to the other electrode in addition to a d.c. voltage. The overall effect of the high frequency signals on the system is a shift in the resonant frequency. The response is measured at the top plate using a lock-in amplifier, locked to the carrier signal frequency. The signal is demodulated and sent to a second lock-in amplifier which measures at the driving 113

114 (a) (b) σ 2 / <W> τ 10 0 (c) (d) 25 Bin Count 10 1 σ 2 / <W> W / <W> I N Figure 7-1. (a) SEM of MEMS device. (b) The ratio σ 2 / W is plotted versus the integration time of the work for f d = For τ longer than 1/λ the ratio is constant. (c) The work distribution ( ) at f d = for integration times of τ = 0.08 s ( ), 0.28 s ( ), 0.80 s ( ), 1.70 s ( ), 10.0 s ( ). The distribution is fitted (solid line) to Eq The average work and its variance can be extracted from the fit. (d) The ratio σ 2 / W ( ) at f d = and τ = 10 s is plotted versus the noise intensity and is fitted (solid line) to Eq The slope gives the coefficient C. 114