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FUNDAMENTAL LTMTTS OF NANOTECHNOLOGY: HOW FAR DOWN IS THE BOTTOM?

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Status, Challenges, and Frontiers of Silicon CMOS Technology JACK HERGENROTHER IBM T. J. Watson Research Center Yorktown Heights, New York For more than three decades, continued improvements in silicon (Si) transis- tor density, integration, switching speed and energy, and cost per electronic func- tion have driven the $160-billion semiconductor industry, one of the most dy- namic industries in the world. These faster and cheaper technologies have led to fundamental changes in the economies of the United States and other countries around the world. The exponential increase in transistor count that has occurred over the past few decades was accurately predicted by Gordon Moore in 1965. To continue to power the information technology economy, however, the Si industry must remain on the Moore's Law trajectory. Because Moore's Law has accurately predicted the progress of Si technol- ogy over the past 38 years, it is considered a reliable method of predicting future trends. These extrapolated trends set the pace of innovation and define the nature of competition in the Si industry. Progress is now formalized each year in an annual update of the International Technology Roadmap for Semiconductors (ITRS), also known as "The Roadmap" (ITRS, 2003~. Table 1 shows the status and key parameters in the 2002 update. ITRS extends for 15 years, but there are no guarantees that the problems confronting Si technology will be solved over this period. ITRS is simply an assessment of the requirements and technological challenges that will have to be addressed to maintain the current rate of exponen- tial miniaturization. At the current pace, it is widely believed that the industry will reach the end of conventional scaling before the end of 2016, perhaps as early as 2010. The device that supports Moore's Law, known as the metal-oxide semicon- ductor field-effect transistor (MOSFET), comes in both e-channel and p-channel flavors depending on whether the primary current is carried by electrons or 27

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28 FRONTIERS OF ENGINEERING TABLE 1 2001 Status and Summary of Key Parameters from the 2002 ITRS Update MPU ASIC Gate Gate Logic SRAM DRAM Technology Length Length Density Density Density Year Node (nary) (nary) (Mtransistors/cm2) (Mtransistors/cm2) (Gbit/cm2) 2001 "130 nm" 65 90 39 184 0.55 2004 "90 nm" 37 53 77 393 1.49 2007 "65 nm" 25 32 154 827 3.03 2010 "45 nm" 18 22 309 1718 6.10 2013 "32 nm" 13 16 617 3532 18.4 2016 "22 nm" 9 11 1235 7208 37.0 Source: ITRS, 2003. holes, respectively. The types and placement of dopants in a MOSFET deter- mine whether it is e-channel or p-channel. The basic structure of the nMOSFET (Figure 1) consists of a moderately p-doped channel formed near the top surface of a polished Si wafer between two heavily e-doped source and drain regions. On top of the channel is a thin insulating layer of silicon dioxide or oxynitride, which separates the heavily e-doped polysilicon gate from the channel. For gate voltages above the threshold voltage VT (typically 0.3 to 0.5 V), electrons are attracted to the gate but remain separated from it by the insulating gate oxide. These electrons form an "inversion layer" that allows a significant electron cur- rent to flow from the source to the drain. The magnitude of this "drive current" ID is a key MOSFET performance metric. In digital circuits, MOSFETs essentially behave like switches, changing the current by as much as 12 orders of magnitude via the gate voltage. To enable low-power chips, MOSFETs must be excellent switches and have very small suicide suicide n+ source Gate gate oxide 1 Cop - type \ channel 14 G silicide n+ drain \ - FIGURE 1 Basic structure of a traditional planar nMOSFET.

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STATUS, CHALLENGES, AND FRONTIERS OF ICON CMOS TECHNOLOGY 29 subthreshold leakage currents (Iota. This is the undesirable current that leaks from the source to the drain when 0 V is applied to the gate. In today's state-of- the-art MOSFETs, metal silicides, such as cobalt silicide (CoSi2) and nickel silicide (NiSi), are used to increase the electrical conductivity of the source, drain, and gate. Individual MOSFETs are electrically separated by silicon diox- ide deposited in shallow trenches and connected by many levels of miniature copper wire interconnects. nMOSFETs and pMOSFETs are combined in circuits to form comple- mentary MOS (CMOS) technology. The general principle of traditional CMOS scaling is the reduction of the horizontal and vertical MOSFET dimensions as well as the operating voltage by the same factor (typically 0.7x per technology generation every two to three years) to provide simultaneous improvements of 2x in areal transistor density, 0.7x in switching time, and 0.5x in switching energy. Figure 2 shows the intrinsic switching time of nMOSFETs plotted against their physical gate lengths LG. The switching time is essentially the time it takes a first transistor carrying a current ID to charge the input gate capacitance CG of an identical transistor to the supply voltage VDD. Note that, although the fastest microprocessors being produced today have clock frequencies in the range of 2 GHz = 1/~500 ps), the intrinsic switching time A _, _. con _. B con ~ A .~ LO (elm) FIGURE 2 The trend of intrinsic switching time CGVDJID versus physical gate length. The fastest Si MOSFETs produced to date have intrinsic switching times well under 500 fs. Source: Bohr, 2002. Reprinted with permission.

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30 FRONTIERS OF ENGINEERING of nMOSFETs is only about 1.5 ps, about 300 times shorter than the inverse clock time. The difference is a consequence of several basic elements of CMOS design architecture: (1) transistors must often drive more than one input gate capacitance; (2) interconnect capacitances must also be charged; (3) typically, about 20 logic stages are cascaded and have to be switched within one clock cycle; and (4) the clock frequency must accommodate the slowest of these cas- caded paths. The intrinsic switching time is the critical metric (as opposed to the inverse clock frequency) to keep in mind when comparing the performance of Si MOSFETs with alternative nanodevices intended for logic applications. It is likely that today's research devices with intrinsic switching times under 500 Is will be produced in volume before the end of the decade. Figure 3 shows the switching energy CGVDD2, a simple metric that leaves out second-order effects. Note that the fundamental limit on switching-energy transfer during a binary switching transition is kBTln2 ~ 3 x 10-2i J (Meindl and Davis, 2000~. This is roughly five orders of magnitude smaller than the switch- ing energy of Si devices currently being manufactured, indicating that this truly fundamental limit is not an imminent concern. When discussing state-of-the-art CMOS devices, one must be careful to distinguish research devices from the devices at the cutting edge of volume L`G (~m) FIGURE 3 The trend of switching energy CGVDD2 versus physical gate length. Si MOSFETs with less than 30 aJ switching energies have been fabricated. Source: Bohr, 2002. Reprinted with permission.

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STATUS, CHALLENGES, AND FRONTIERS OF SILICON CMOS TECHNOLOGY 31 manufacturing. Manufactured devices must be so finely tuned that they are able to pass a long list of stringent performance, yield, and reliability tests. Depend- ing on the structural, material, or process innovation in question, it may take anywhere from 5 to 15 years of significant, and in some cases worldwide, invest- ment to prepare a new research concept for volume manufacturing. State-of-the- art, bulk Si (Thompson et al., 2002) and Si-on-insulator (SOI) technologies (Khare et al., 2002), which are very near manufacturing, feature gate lengths of 50 nm or smaller and physical gate oxide thicknesses of 12 A (just a few Si-Si atomic bond lengths) and can pack a six-transistor static RAM (SRAM) cell within a 1.0 ~J=2 area. These devices are interconnected with as many as 10 layers of Cu wiring. The critical features are patterned in engineering marvels known as scanners that use deep ultraviolet light with a wavelength of 193 nm. The entire process is carried out on 300-mm diameter wafers in fabrication lines that cost about $3 billion each to build. Note that in recent years physical gate lengths have been shrunk faster than the technology node would suggest. This "hyperscaling" of the gate length, achieved via photolithographic enhancements that are suitable for the reduction of isolated feature sizes, as well as controll- able linewidth reduction ("trimming") techniques, has accelerated performance improvements. Super-scaled bulk MOSFET research devices (in which there is not as high a premium placed on line-width control and optimized performance) have been demonstrated with gate lengths as small as 15 nm (Hokazono et al., 2002; Yu et al.,2001~. Although these devices have record-breaking intrinsic switching times and switching energies, they must still undergo significant optimization over the next five to seven years to have a shot at meeting the ITRS performance targets. Showstoppers to continued scaling have been predicted for three decades, but a history of innovation has sustained Moore's Law in spite of these chal- lenges. However, there are growing signs today that MOS transistors are begin- ning to reach their traditional scaling limits (ITRS, 2003; Meindl, 2001~. The most notable sign is the ever increasing subthreshold leakage current Ions which results from thermionic emission over the source barrier. The subthreshold leak- age dictates that the exponential decay of the subthreshold current with gate voltage can be no faster than kBT/q, or about 60 mV/decade at room temperature. In practice, MOSFETs do not turn off as abruptly as suggested by the funda- mental 60 mV/decade limit because of capacitive divider effects and, more im- portant, various short-channel effects that become significant for sub-100 nm channel lengths. As gate and channel lengths continue to shrink, it is becoming increasingly difficult to design devices with proper electrostatic scalability, that is, reasonable immunity to these short-channel effects. The short-channel effects that impact Io~ fall into three categories: (1) threshold voltage roll-off, the de- crease of threshold voltage with decreasing channel length; (2) drain-induced barrier lowering (DIBL), the decrease of threshold voltage with increasing drain voltage; and (3) degradation of the subthreshold swing with decreasing channel

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1^ FRONTIERS OF ENGINEERING length. All three effects are related in that they are the simple result of two- dimensional electrostatic effects, and they can dramatically increase the sub- threshold leakage current. Because there is a fundamental limit to subthreshold swing at room temperature, short-channel effects put a lower limit on the thresh- old voltage of the target device. Until recently, VT has been scaled downward in proportion to the supply voltage VDD. The floor on VT has a significant impact of the MOSFET drive current, which depends on the gate overdrive VDD_VT. To maintain the drive current performance dictated by The Roadmap, subthreshold leakage targets have been significantly relaxed. For example, in the 0.25 ,um generation that en- tered manufacturing circa 1997, Ions was maintained at approximately 1 nA/ ,um. In the 90 nm generation soon to be ready for manufacturing, Ions for high- performance devices are typically about 40 nA/,um. If the leakage goes much higher, it is unlikely that the power generated by chips with hundreds of millions of these transistors can be tolerated. In fact, difficulty in controlling the subthreshold leakage current has already led to a scaling bifurcation be- tween high-performance and low-power transistors that has been formally adopted in the ITRS. Another significant challenge to continued scaling is the rapid increase of the quantum mechanical tunneling current that passes through the gate oxide as it is progressively thinned. This gate current is rapidly approaching the size of the subthreshold leakage. Current 90 nm generation high-performance devices soon to be in manufacturing have 12 A SiO2-based gate oxides that are within at most 2 A of the limit imposed by gate leakage. Although this gate leakage does not destroy the functionality of digital circuits, in the absence of a solution, chip power-dissipation levels will be unacceptably high. Lithography, which has been one of the key enablers of scaling, has recently emerged as one of the key challenges to scaling. The Rayleigh equation defines the half-pitch resolution as R=k'(\lNA) where k' is a coefficient that accounts for the effectiveness of resolution enhancement techniques, ~ is the wavelength of the light used, and NA is the numerical aperture of the lens. In the past, the evolution of lithography occurred primarily through the adoption of successively shorter exposure wavelengths. More recently, because of increasing difficulty of moving to shorter wavelengths, progress in lithography has depended somewhat more heavily on resolution enhancement techniques (such as the use of phase- shift masks and customized illumination techniques), and the development of higher NA lenses. At the 90 nm node, 193 nm wavelength lithography is used for critical mask levels, whereas more mature 248 nm lithography is widely used for other mask levels. Although 157 nm lithography was originally targeted for introduction at the 65 nm node (mainstream volume manufacturing in 2007), significant un- foreseen problems have arisen that have delayed its availability. As a result, lithographers are facing the challenge of extending 193 nm lithography to the

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STATUS, CHALLENGES, AND FRONTIERS OF ICON CMOS TECHNOLOGY 33 65 nm node. The lithography required for the 45 nm node (2010) will have to be a very mature 157 nm lithography or a next-generation lithography (NGL) technique such as extreme ultraviolet (EUV) lithography (Attwood et al., 2001) or electron-projection lithography (EPL) (Harriott, 1999~. However, the step from 157 nm to 13.5 nm (EUV) is huge, because virtually all materials absorb rather than transmit light at 13.5 nm wavelengths; thus, the only option is re- flection optics. With such a short wavelength, EUV allows the use of conserva- tive k' and NA values, but it faces significant challenges: (1) source power and photoresist sensitivity; (2) the production of defect-free reflection masks; and (3) contamination control in reflection optics. Even if advanced lithographic techniques can be developed to pattern the required ultrafine features in photoresist, these features must be transferred reli- ably and with high fidelity into the materials that make up the transistors and interconnect. This will require significant advances in Si process technology, in particular in etching and film deposition. A wide range of processes are under development to meet these future needs (Agnello, 2002~. Among them are self- limited growth and etch processes, such as atomic-layer deposition, that have the potential to provide atomic-level control (Ahonen et al., 1980; Suntola and Antson, 1977~. In MOSFETs at the limit of scaling, it is also critical to provide precise dopant-diffusion control while also enabling the high levels of dopant activation required for low parasitic resistances. The frontiers of research on Si technology feature a wide spectrum of work aimed at enhancing CMOS performance, solving (or at least postponing) some of the major issues, such as gate leakage, and providing options in case the traditional scaling path falters. One of the most active areas of research is the use of strain to enhance carrier-transport properties (Hwang et al., 2003; Rim et al., 2002a, 2002b; Thompson et al., 2002; Xiang et al., 2003~. Si under biaxial tensile strain is well known to exhibit higher electron and hole mobilities than unstrained Si. Devices built with strained-St channels have shown drive currents as much as 20 percent higher than in unstrained-St channels (Hwang et al., 2003; Rim et al., 2002a; Thompson et al., 2002; Xiang et al., 2003~. A variety of approaches have been used to provide strained-St channels. The dominant ap- proaches include: (1) the formation of a thin, strained-St layer on top of a carefully engineered SiGe, relaxed graded buffer layer (Fitzgerald et al., 1992~; and (2) the formation of a strained-St layer on top of a relaxed SiGe on insulator (SGOI) (Huang et al., 2001; Mizuno et al., 2001~. In both cases, the larger lattice constant of SiGe constrains the lattice of Si, essentially pulling it into tensile strain. The strained-St layer must be quite thin to prevent strain relief from occuring through dislocations. Currently intense efforts are being made in several areas: (1) studying how much mobility enhancement translates to an improvement in drive in short-channel devices; (2) reducing the defect densities of strained-St starting materials; and (3) solving integration issues (e.g., rapid arsenic diffusion) that result from the presence of SiGe in the active area of the

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34 FRONTIERS OF ENGINEERING device. In spite of these problems, strained Si is one of the most promising nontraditional technology options. A much more radical method to improve carrier-transport properties in- volves the use of germanium as the channel material itself (Chui et al., 2002; S hang et al., 2002~. This is primarily driven by its significantly higher bulk mobilities for electrons (2x) and holes (4x). One of the primary challenges is the lack of a stable native Ge oxide, which makes it difficult to passivate the surface of Ge. Renewed interest in Ge MOSFET is due in large part to recent advances in the understanding of oxynitrides and high-k dielectrics (initially aimed at Si- based transistors), suggesting that these materials might be used effectively to passivate the surface of Ge. Work on Ge MOSFETs is still in a very early phase. At the time of this writing, only Ge pMOSFETs have been demonstrated, with several groups attempting to build the first Ge nMOSFETs. Returning to the imminent challenges facing Moore's Law, one possible solution to the difficult gate-leakage problem is the introduction of high-k (high dielectric constant) gate dielectrics (Wallace and Wilk, 2002~. Many different high-k materials, such as Si3N4, HfO2, ZrO2, TiO2, Ta2O5, La2O3, and Y2O3 have been studied for use as a MOSFET gate dielectric. The materials that have received the most attention recently are hafnium oxide (HfO2) (Gusev et al., 2001; Hobbs et al., 2003), hafnium silicate (HfSiXOy) (Wilk et al., 2000), and nitrided hafnium silicate (HfSiXOyNz) (Inumiya et al., 2003; Rotondaro et al., 2002~. Because the main objective of scaling the gate oxide is to increase the capacitance density of the gate insulator, high-k dielectrics are attractive because the same capacitance density can be achieved with a physically thicker (and if the new material's barrier height is sufficient, a lower gate-leakage) layer. How- ever, after 40 years, the Si/SiO2 system has almost ideal properties, many of which have yet to be satisfied by a new high-k dielectric. Replacing SiO2 is not as easy as it appears. Among these requirements is material-bulk and interface properties comparable to those of SiO2. The new material must also exhibit thermal stability during the temperature cycles required in Si processing, low dopant-diffusion coefficients, and sufficient reliability under constant voltage stress. Although rapid progress has been made, no group to date has demon- strated a sufficiently thin high-k dielectric that preserves the transport properties (i.e., carrier mobilities) of the Si/SiO2 system. An interesting fact about high-k gate dielectrics is that they will be required for moderate performance, low- power applications (e.g., hand-held communication devices) before they are needed for high-performance applications (e.g., microprocessors). Another approach to increasing the capacitance density of the doped polySi/ gate dielectric/St stack is to replace the doped polySi in the gate with a metal (Kedzierski et al., 2002; Lee et al., 2002; Ranade et al., 2002; Samavedam et al., 2002; Xiang et al., 2003~. This would eliminate undesirable polysilicon-depletion effects, which typically add an effective thickness of 4 to 5 A to the thickness of the gate oxide. With physical gate-oxide thicknesses currently in the 12 A range,

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STATUS, CHALLENGES, AND FRONTIERS OF ICON CMOS TECHNOLOGY 35 there is a significant degradation of capacitance density. One of the challenges to metal-gate technology is that the work function of the gate material is used to set the threshold voltage of the MOSFET. For bulk MOSFETs, a suitable pro- cess must be devised that allows the formation of tunable or dual work function metal gates (Lee et al., 2002; Ranade et al., 2002~. In recent years, double-gate MOSFETs have received a good deal of atten- tion, and a variety of approaches to building them have been investigated (Guarini et al., 2001; Hisamoto et al., 1998; Wong et al., 1997) (Figure 4~. These devices provide several key advantages over traditional planar MOSFETs (Won" et al., 2002~: (1) enhanced electrostatic scalability because two gates are in close proximity to the current-carrying channel; (2) reduced-surface electric fields that can significantly improve the carrier mobilities and the CGVDD/ID performance; and (3) because the electrostatics are controlled with geometry (rather than with doping as in the planar MOSFET), they can be built with undoped channels. This eliminates the problems caused by statistical fluctuations in the number of dopants in the device channel, which is becoming an increasing concern for planar MOSFETs. Double-gate MOSFETs present difficult fabrication challenges. The most significant of these is that the Si channel must be made sufficiently thin (no thicker than about 2/3 LG) and must be precisely controlled. In addition, the two gates must be aligned to each other as well as to the source-drain doping to minimize the parasitic capacitance that could otherwise negate the advantage derived from enhanced electrostatic scalability. The most thoroughly studied double-gate MOSFET is known as the FinFET (Hisamoto et al., 1998; Kedzierski et al., 2001; Yu et al., 2002) and in slightly modified forms as the tri-gate transistor (Doyle et al., 2003) or omega-FET FIGURE 4 Structure of a representative double-gate MOSFET. LG Gate dielectric

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64 FRONTIERS OF ENGINEERING back the bits from the media. Alternative technical strategies have been pro- posed (perpendicular magnetic recording, patterned media, heat assisted mag- netic recording), but all of these approaches offer only marginal improvements over current are al densities based on incremental advances in recording effi- ciency. For all practical purposes, the most cost-effective means of increasing value for the data storage consumer is the same as for other commodity items- manufacturing finesse, improved reliability, and marketing. Nevertheless, it should be appreciated that magnetic recording technology has reached nanoscale dimensions in spite of the mechanical components that have always been considered a hindrance; bit dimensions are almost at 35 nm (length) x 180 nm (width) x 10 nm (thickness). By any measure, such numbers are nothing short of astonishing. However, moving to even smaller bit dimen- sions will require more than improvements in engineering. It will require a greater understanding of the fundamental physics underlying the phenomena of hysteresis and the thermodynamics of the ferromagnetic phase. Such under- standing might enable the discovery of new physics that would allow for new modalities of the recording process. It is worth the time to consider this last point in more detail. Hysteresis in a spin system, at its most elementary level, is an astonishing effect. It should be kept in mind that the switching of magnetic bits involves transitions between nearly degenerate energy levels. Recording a bit does not involve the storage of energy in the archival medium, but rather the expenditure of work to cause a thermodynamically irreversible alternation in the magnetization state of the me- dium. Thus, the magnetic bit is essentially an indelible record that some fraction of the electromagnetic work provided by the recording head has been converted to heat via the spin system of the ferromagnet, resulting in a net increase in entropy, /\S = KUV/T; the more entropy generated in the writing of a bit, the more stable the recorded mark. Such a process stands in marked contrast to our basic understanding of spin dynamics at the quantum level. One need only consider that the moment of a ferromagnetic volume is composed of electron spins, each of which would have a very short memory time if isolated from all the other spins in the ferromagnetic system, even at low temperatures. Electron spin resonance measurements typi- cally yield decoherence times ranging from nanoseconds to microseconds in most conductors. The ferromagnetic phase, enabled by the exchange interaction between the uncompensated electron spins, is stabilized against such relaxation processes, but at a cost. Switching the magnetization in a stable fashion requires the irreversible conversion of work into heat. Thus, the effort to increase recording densities is essentially a struggle to find increasingly efficient means of converting the same amount of work into heat per bit, even though the size of the bits is shrinking. Current recording technology relies upon intrinsic relaxation processes induced by large applied magnetic fields. Borrowing terminology from nuclear magnetic resonance, large

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LIMITS OF STORAGE IN MAGNETIC MATERIALS 65 Zeeman splittings induced by the recording heads drives spin reversal via longi- tudinal relaxation processes. The rephrasing of the recording process in the context of spin resonance techniques suggests one possible strategy for increas- ing recording densities. It is well known that resonant techniques (e.g., adiabatic fast passage) are far more efficient means of manipulating spin states in para- magnetic systems. Perhaps similar methods could be developed for the im- proved focus of electromagnetic energy to yet smaller spin volumes in a ferro- magnetic medium. REFERENCES Abraham, E. and C. Kittel. 1952. Spin lattice relaxation in ferromagnets. Physical Review 88(5): 1200. Chandler, D. 1987. Introduction to Modern Statistical Mechanics. New York: Oxford University Press. Helstrom, C.W. 1984. Probability and Stochastic Processes for Engineers. New York: Macmillan. Korenman, V. and R.E. Prange. 1972. Anomalous damping of spin waves in magnetic metals. Physical Review B 6(7): 2769-2777. Landau, L., and E. Lifshitz. 1935. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Physik. Z. Sowjetunion 8: 153-169. Lu, P.-L., and S.H. Charap. 1994. Magnetic viscosity in high-density recording. Journal of Applied Physics 75(10): 5768-5770. Lu, B., D. Weller, A. Sunder, G. Ju, X. Wu, R. Brockie, T. Nolan, C. Brucker, and R. Ranjan. 2003. High anisotropy CoCrPt(B) media for perpendicular magnetic recording. Journal of Applied Physics 93(10): 6751-6753. Morrish, A.H. 2001. The Physical Principles of Magnetism. New York: IEEE Press. Neel, L. 1949. Theorie du trainage magnetique des ferromagnetiques en grains fins avec applica- tions aux terres cuites. Annual Geophysics 5: 99-136. Pathria, R.K. 1996. Statistical Mechanics, 2nd Ed. New York: Butterworth-Heinemann. Rizzo, N.D., M. DeHerrera, J. Janesky, B. Engel, J. Slaughter, and S. Tehrani. 2002. Thermally activated magnetization reversal in submicron magnetic tunnel junctions for magnetoresistive random access memory. Applied Physics Letters 80(13): 2335-2337. Rizzo, N.D., T.J. Silva, and A.B. Kos. 1999. Relaxation times for magnetization reversal in a high coercivity magnetic thin film. Physical Review Letters 83(23): 4876-4879. Silva, T.J., C.S. Lee, T.M. Crawford, and C.T. Rogers. 1999. Inductive measurement of ultrafast magnetization dynamics in thin-film permalloy. Journal of Applied Physics 85(11): 7849- 7862. Smith, N., and P. Arnett. 2001. White-noise magnetization fluctuations in magnetoresistive heads. Applied Physics Letters 78(10): 1448-1450. Sparks, M. 1965. Ferromagnetic Relaxation Theory. New York: McGraw-Hill. Stoner, E.C., and E.P. Wohlfarth. 1948. A mechanism of magnetic hysteresis in heterogeneous alloys. Philosophical Transactions of the Royal Society of London, Series A 240(826): 599- 642. Street, R.C., and J.C. Wooley. 1949. A study of magnetic viscosity. Proceedings of the Physical Society, Section A 62(9): 562-572. Stutzke, N., S.L. Burkett, and S.E. Russek. 2003. Temperature and field dependence of high- frequency magnetic noise in spin valve devices. Applied Physics Letters 82(1): 91-93. Suhl, H. 1998. Theory of the magnetic damping constant. IEEE Transactions on Magnetics 34(4): 1834-1838.

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Thermodynamics of Nanosystems CHRISTOPHER JARZYNSKI Los Alamos National Laboratory Los Alamos, New Mexico The field of thermodynamics was born of an engineering problem. Reflec- tions on the Motive Power of Fire, an analysis of the efficiency of steam engines published by the French engineer Sadi Carnot in 1824 led ultimately to the elucidation of the second law of thermodynamics, the unifying principle that underlies the performance of all modern engines, from diesel to turbine. Today engineers are excited about nanosystems, including machines that operate at molecular length scales. If history is any guide to the present, a grasp of thermo- dynamics at the nanoscale is essential for the development of this field. With this in mind, the focus of this brief talk is on the following question: If we were to construct an engine the size of a large molecule, what fundamental principles would govern its operation? To put it another way, what does thermodynamics look like at the nanometer length scale? It is perhaps not immediately obvious that thermodynamics should "look different" at the microscopic scale of large molecules than at the macroscopic scale of car engines. Of course, at the microscopic level, quantum effects might play a significant role, but in this talk we will assume the effects are negligible. For present purposes, the difference between the macro- and the microworld boils down to this. On the scale of centimeters, it is safe to imagine that matter is composed of continuous substances (e.g., fluid or solid, rigid or elastic, etc.) with specific properties (e.g., conductivity, specific heat, etc.~. On the scale of nanometers, by contrast, the essential granularity of matter (i.e., that it is made up of individual molecules and atoms) becomes impossible to deny. Once we consider systems small enough that we can distinguish individual molecules or atoms, thermal fluctuations become important. If we take a large system, such as a rubber band, and stretch it, its response can accurately be 67

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68 FRONTIERS OF ENGINEERING predicted from knowledge of the properties of the elastic material of which it is made. But, imagine that we take a single RNA molecule, immerse it in water, and stretch it using laser tweezers. As a result of continual bombardment by the surrounding water molecules, the RNA molecule jiggles about in a way that is essentially random. Moreover, because the RNA is itself a very small system, these thermal motions are not negligible; the noise-to-signal ratio, so to speak, is substantial. Thus, the response of the RNA strand has a considerable element of randomness in it, unlike the predictable response of an ordinary rubber band. This naturally leads us to suspect that the laws of thermodynamics familiar in the context of large systems might have to be restated for nanosystems in a way that accounts for the possibility of sizeable thermal fluctuations. Admittedly, machines the size of single molecules have not yet been con- structed, but the issues I have raised are not merely speculative. The RNA- pulling scenario outlined above was carried out by an experimental group at University of California, Berkeley that measured the resulting fluctuations stretching the RNA strand (Liphardt et al., 2002~. Independently, researchers at the Australian National University in Canberra and Griffith University in Brisbane have used laser tweezers to drag microscopic beads through water, with the explicit aim of observing microscopic "violations" of the second law of thermodynamics caused by thermal fluctuations (Wang et al., 2002~. In a sense, the research is happening in reverse order to the research that gave rise to nine- teenth-century thermodynamics. We do not yet have a nanomachine (a twenty- first century analogue of a steam engine), but we have begun to play with its potential components, with the aim of teasing out the fundamental principles that will govern the device when (or if) we ultimately construct one. The study of the thermal behavior of tiny systems is by no means a new field. A century ago, Einstein's and Smoluchowski's quantitative explana- tions of Brownian motion the spastic movement of pollen particles first observed by the British botanist Robert Brown in 1827 not only helped clinch the atomic hypothesis, but also led to the puctuation-dissipation theo- rem, a remarkably simple relationship between friction and thermal noise. In the context of the RNA experiments mentioned above, the fluctuation- dissipation theorem predicts that: (Wdiss)= NEW (1) where (Weiss) is the average amount of work that is dissipated when we stretch the RNA; c,2w = (W2> - (Vf/)2 is the variance in work values; T is the temperature of the surrounding water; and kB = 1.38 x 10-23 J K-i is Boltzmann's constant. Here the words average and variance refer to many repetitions of the pulling experiment; because of thermal noise, the exact amount of work performed in stretching the RNA molecule differs from one repetition to the next, as illus- trated in Figure 1. By dissipated work, we mean the amount by which the total work performed (during a single realization) exceeds the reversible work,

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THERMODYNAMICS OF NANOSYSTEMS 69 FIGURE 1 Distribution of work values for many repetitions (realizations) of a thermo- dynamic process involving a nanoscale system. The process might involve the stretching of a single RNA molecule or, perhaps, the compression of a tiny quantity of gas by a microscopic piston. The tail to the left of AF represents apparent violations of the second law of thermodynamics. The distribution p(W) satisfies the nonequilibrium work theorem (Eq.6), which reduces to the fluctuation-dissipation theorem (Eq.l) in the special case of near-equilibrium processes. WdisSWWrev (2) (As discussed below, this is equivalent to W- /\F, where AF is the free energy difference between the initial and final states of the system.) The fluctuation-dissipation theorem, along with most subsequent work in this field, pertains to systems near thermal equilibrium. (Eq.1 will not be satis- fied if we stretch the RNA too rapidly!) In the past decade or so, however, a number of predictions have emerged that claim validity even for systems far from equilibrium. These have been derived by various researchers using a vari- ety of theoretical approaches and more recently have been verified experimen- tally. It is impossible to do justice to a decade of work in such a short space, but I will attempt to convey the flavor of this progress by illustrating one result on a toy model. Consider that familiar workhorse of thermodynamic pedagogy, a container filled with a macroscopic quantity of gas closed off at one end by a piston. Suppose we place the container in thermal contact with a heat bath at a tempera- ture T. and we hold the piston fixed in place, allowing the gas to relax to a state

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70 FRONTIERS OF ENGINEERING of thermal equilibrium. Then we vary the position of the piston so as to com- press the gas, from an initial volume VA to a final volume VB. In doing so we perform external work, W. on the gas. If we carry out this process reversibly (slowly and gently), then the gas stays in equilibrium at all times, and the work performed on it is equal to the net change in its free energy: i Wrev = FB FA AF (3) If instead we carry out the process irreversibly (perhaps even rapidly and violently), the work, W. will satisfy the inequality W > /\F (4) with AF defined exactly as above. Eq.4 is simply a statement of the second law of thermodynamics, when we have a single heat bath. To understand what Eq.3 and 4 look like for microscopic systems, imagine that we scale our system down drastically, to the point that our container is measured in tens of nanometers and contains only a handful of gas molecules. Again we place it in contact with a heat bath, and we consider a process whereby we move a tiny piston so as to compress the volume of the container from an initial volume VA to a final volume VB. (I do not mean to suggest that a realistic nanomachine will be composed of nanopistons; this familiar ex- ample is used only for illustration). Suppose we repeat this process very many times, always manipulating the volume in precisely the same way. During each repetition, or realization, we perform work as the gas molecules bounce off the moving piston. However, because the precise motion of the molecules differs each time we carry out the process, so does the precise value of W. In effect, the thermal fluctuations of the gas molecules give rise to statistical fluctuations in the value of W from one realization to the next. After very many repetitions, we have a distribution of work values, p(W), where p(W)dW is the probability that the work during a single realization will fall within a narrow window dW around the value W. What is the relationship between this distribution and the free-energy difference /\F = FB - FA between the equilib- rium states corresponding to the initial and final volumes? Based on our knowl- edge of macroscopic thermodynamics (Eq.3 and 4), we might guess that the average work value is no less than OF, i.e., (W) _ ~ dW p(W)W 2 /\F (5) Recall that the free energy associated with a state of thermal equilibrium, at temperature T. is given by F = E- ST, where E and S are, respectively, the internal energy and entropy of that state. In Eq.3, FA and FB are associated with the equilibrium states corresponding to container volumes VA and VB.

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THERMODYNAMICS OF NANOSYSTEMS 71 with the equality holding only if the process is carried out reversibly. However, this inequality does not rule out the possibility that p(W) has a small "tail" extending into the region W < AF; see Figure 1. If we were to observe a value of W less than AF for a process carried out with a macroscopic piston, this would be an extraordinary event a true viola- tion of the second law of thermodynamics. But in a system with relatively few degrees of freedom, this would not be such a shocking occurrence. We will refer to such an event (W < AF in a microscopic system) as an apparent violation of the second law. Eq.5 is correct, but we can do better. Instead of considering the average of W over many repetitions of our thermodynamic process, let us consider the aver- age of exp(-W/kBl) over these realizations. It turns out that this average obeys a simple and somewhat surprising equality: ~dW p(W)exp(-W/kBl) = exp(-AF/ IBM, or more succinctly ( ) (6) This is the nonequilibrium work theorem, which remains valid no matter how gently or violently we force the piston in changing the volume from VA to VB. This result has been derived using various theoretical approaches (Crooks, 1998, 1999; Hummer and Szabo, 2001; Jarzynski, 1997a,b) and confirmed by the Ber- keley RNA-pulling experiment discussed above (Liphardt et al., 2002~. Eq.5 and 6 both make predictions about the probability distribution of work values, p(W), but the latter is a considerably stronger statement than the former. For instance, Eq.6 immediately implies that p(W) cannot be located entirely in the region W > AF. (If that were the case, then e-W/ksT would be less than e-~F/ksT for every realization of the process, and there would be no way for Eq.6 to be satisfied.) Thus, if the typical value of W is greater than AF, as will be the case for an irreversible process, then there must also be some realizations for which W < /\F. Eq.6 not only tells us that p(W) must have a tail extending into the region W < AF, but it also establishes a tight bound on such apparent violations of the second law, namely, | dW p(W) 0. This inequality can be stated in words: the probability that we will observe a work value that falls below AF by at least n units of kBT is bounded from above by e-n. This means that substantial violations of the second law (n >> 1) are extremely rare, as expected. (The derivation of Eq.7 from Eq.6 is left as an exercise for the reader.) If perchance we change the volume of the container relatively slowly so that the gas remains close to thermal equilibrium at all times, then p(W) ought

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72 FRONTIERS OF ENGINEERING to be a normal (Gaussian) distribution. (I will not try to justify this expectation here beyond saying that it follows from the Central Limit Theorem.) It takes a few lines of algebra to show that if p(W) is a Gaussian distribution that satisfies Eq.6, then its mean and variance must be related by Eq.1. Thus, although the nonequilibrium work theorem (Eq.6) is valid regardless of whether we move the piston slowly or quickly, in the former (near-equilibrium) case it reduces to the well-known fluctuation-dissipation theorem (Eq.l). Eq.6 is closely related to another result, which can be stated as follows. Suppose we perform many repetitions of the process described above, in which we change the volume of our tiny container of gas from VA to VB, according to some schedule for moving the piston. Then suppose we perform many repeti- tions of the reverse process, in which we move the piston according to the re- verse schedule, thus changing the container volume from VB to VA. For both sets of experiments, we observe the work performed during each realization of the process, and in the end we construct the associated distributions of work values, PA - B (W) and PB - A (We. Then these two distributions satisfy PA - B (W) = e(W~F)/kBT PB - A ~ W) (8) where AF = FB - FA as before (Crooks, 1998, 1999~. This is remarkable because the response of the gas during one process, VA ~ VB, is physically quite different from the response during the reverse process, VB ~ VA. For example, when we push the piston into the gas, a modest shock wave (a region of high density) forms ahead of the piston. For the reverse case, a region of low density trails the piston as it is pulled away from the gas. Nevertheless, the two distributions are related by a simple equality, Eq.8 above. Eq.6 and 8 are two examples of recent results pertaining to microscopic systems away from thermal equilibrium. They make very specific predictions about the fluctuations of an observed quantity, in this case the work we perform on our system. The fluctuation theorem is a collective term for another set of such results (Evans and Searles, 2002), originally derived in the context of sys- tems in nonequilibrium steady states and recently confirmed by the Australian bead-dragging experiment mentioned earlier (Wang et al., 2002~. All of these predictions have potentially important design implications for nanomachines, for which thermal fluctuations are bound to play an important role. For instance, if we design a microscopic motor fueled by the transfer of molecules from high to low chemical potential, then it is of interest to estimate how often random fluc- tuations might cause this motor to run backward, by moving a molecule or two from low to high chemical potential, in flagrant (if brief) violation of the second law. Eq.7 addresses this sort of issue. I don't want to overstate the case here, but it seems reasonable to assume that the better we understand the laws of thermo- dynamics at the nanoscale, the more control we will have over such systems.

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THERMODYNAMICS OF NANOSYSTEMS 73 CONCLUSIONS When considering the thermodynamics of systems at the scale of nano- meters, ;f~uctuations are important. If we carefully repeat an experiment many times, we will obtain noticeably different results from one repetition to the next. The take-home message of this brief talk is that although these fluctuations originate in thermal randomness, they obey some surprisingly stringent and gen- eral laws (such as Eq.6 and 8), which are valid even far from equilibrium. More- over, these laws are fundamentally microscopic; they do not follow by educated guess from our knowledge of macroscopic thermodynamics. However, the story is not yet complete. So far, we have a jumble of related predictions, as well as experimental validation, but nothing like the beautifully coherent structure of macroscopic thermodynamics. Perhaps as nanotechnology develops and the need for a better understanding of such phenomena becomes more pressing, a more complete and satisfying picture will emerge. REFERENCES Crooks, G.E. 1999. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Physical Review E 60(3): 2721-2726. Crooks, G.E. 1998. Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. Journal of Statistical Physics 90(5/6): 1481-1487. Evans, D.J., and D.J. Searles. 2002. The fluctuation theorem. Advances in Physics 51(7): 1529- 1585, and many references therein. Hummer, G., and A. Szabo. 2001. From the cover: free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proceedings of the National Academy of Sciences 98 (7): 3658-3661. Jarzynski, C. 1997a. Nonequilibrium equality for free energy differences. Physical Review Letters 78(14): 5018-5035. Jarzynski, C. 1997b. Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach. Physical Review E 56(5): 5018-5035. Liphardt, J., S. Dumont, S.B. Smith, I. Tinoco, and C. Bustamante. 2002. Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's Equality. Science 296(5574): 1832-1835. Wang, G.M., E.M. Sevick, E. Mittag, D.J. Searles, and D.J. Evans. 2002. Experimental demonstra- tion of violations of the Second Law of Thermodynamics for small systems and short time scales. Physical Review Letters 89(5): 050601-1-050601-4.

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