| MOLECULAR FOUNDRY |
| Understanding NANOSCALE SYSTEMS at the Molecular Foundry
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The Molecular Foundry is a DOE User Facility charged with providing support to nanoscience researchers in academic, government, and industrial laboratories around the world. Researchers there are using theory and computation to elucidate details of how nanoscale systems function. Three computational nanoscience research areas at the Molecular Foundry include nanoscale charge transport, spectroscopy, and self-assembly.
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Electron Transport in Single-Molecule Circuits
For the past 35 years, active element density in integrated circuits has doubled roughly every 18 months, a progression embodied in Moore's law. If this progress continues, the size of conducting elements in common electrical components, many of which are already at the nanoscale, will soon need to be reduced to atomic dimensions. Because conventional silicon-based devices no longer function at these length scales, new technology will be required for the scaling implicit in Moore's law to continue. |
| Alternatives to silicon-based technology are currently being explored. Several paradigms for nanoelectronics and optoelectronics have been proposed, including molecular electronics, spintronics, photonics, quantum computing, and biologically inspired architectures, such as nerve cells and neural networks. Understanding the nature of electron flow through nanoscale conducting elements is critical to the realization of any of these nascent technologies. |
| The Molecular Foundry, located at Lawrence Berkeley National Laboratory (LBNL), is a DOE User Facility charged with providing support to nanoscience researchers (figure 1). Research in nanoscience is focused on understanding how the properties of conducting elements are modified as their size is reduced to the atomic level. For more than a decade, researchers have been trying to manipulate and "wire up" individual molecules, perhaps the smallest resistive elements imaginable, and determine their conductance. However, routine formation of high-quality electrical contacts, or "alligator clips," between bulk-like electrical leads and nanostructures is challenging. This stems from the ultra-small dimensions of metal molecule contact. Consisting of just a few atoms, the important structure and chemistry at this nano-interface stubbornly lies beyond the detection limits of today's experimental probes. |
If progress continues to follow Moore's law, the size of conducting elements in common electrical components, many of which are already at the nanoscale, will soon need to be reduced to atomic dimensions. |
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| Illustration: A. Tovey; Source: J. Neaton, LBNL |
| Figure 1. The range of length or time scales accessible with computational approaches provided at the Molecular Foundry Theory Facility and the corresponding accuracy of the techniques appropriate to each. |
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| With the advent of scanning tunneling microscopy (STM) in the early 1990s, it became possible to manipulate atoms one at a time at low temperatures and under high vacuum conditions by placing a fine metal wire, called an STM tip, in close proximity to a conducting surface. An applied voltage between tip and sample causes electrons to tunnel through the junction and, if the tip-adsorbate attraction is comparable to the surface corrugation barrier, adsorbate atoms and molecules can be moved. However, because the STM tip is never allowed to make full contact with the molecule, the tunneling barrier between tip and molecule dominates the measured resistance (sidebar "Physical Origin of Resistance," p24). |
| How then can a single molecule be wired up and its resistance be determined? Several strategies have been pursued, but one of the most successful makes use of an STM, although not in the conventional manner described above. Instead, success was achieved with a modified STM tip that is repeatedly brought into full contact with a metal surface in the presence of a solution of molecules. |
| This technique, pioneered by N. J. Tao at Arizona State University, produced a statistical distribution of conductance values measured from an ensemble of single-molecule junctions created by breaking and reforming metal tip-surface contacts. With this approach, a team of researchers at Columbia University led by Latha Venkataraman recently discovered that molecules with amine link chemistry in contact with gold electrodes can be reliably and reproducibly measured. In other words, measurements of the conductance of these molecules result in statistical distributions with a reproducible mean and width, and the mean is the average molecular junction conductance. Their initial breakthrough led to subsequent demonstrations that internal degrees of freedom, functionalization, and end-group linkers systematically shifted the average single-molecule junction conductance. |
Working with experimentalists through the Molecular Foundry User Program, Theory Facility scientists have begun to address questions and develop an understanding of the nature of quantum electron transport. |
Under the auspices of a Molecular Foundry User Project, Venkataraman's experiments were explained and used, along with a new analysis strategy, to develop a theoretical approach for the conceptual and quantitative understanding of the electrical conductance of individual molecules (such as benzene diamine and bipyridine) and to advance the theory of single-molecule conductance.
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Challenges to Theory and Computation
Computing electrical conductance of a single-molecule junction presents several challenges to theory. For example, resistance is determined by electron tunneling through a barrier with the length scale of a single molecule. Determining the barrier requires precise knowledge of the identities and average positions of all atoms that make up the junction and a self-consistent description of their bonding. At present, this atomic-scale information cannot be extracted from experiments. |
Moreover, the problem of nonequilibrium electron transport through molecular devices—intrinsically open systems, with electrons entering and leaving the device through long leads held at fixed potentials—turns out to be unworkable with standard quantum chemistry and electronic structure schemes. Although computing electronic properties of periodic bulk materials is routine with contemporary methods and calculating accurate electronic properties of individual molecules involves a different, but also standard, set of quantum chemical methods that take advantage of their molecular size, neither approach is entirely sufficient to treat open, aperiodic systems. Fully atomistic first-principles methods have only recently been formulated for treating nanostructures under finite bias. Still in their infancy, questions remain concerning the reliability of these techniques and their range of applicability. Working with experimentalists through the Molecular Foundry User Program, Theory Facility scientists have begun to address these questions and develop an understanding of the nature of quantum electron transport.
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Computing Conductance at Molecular Junctions
To compute the conductance of molecular junctions, Molecular Foundry scientists and users have been using and developing a first-principles scattering-state approach based on density functional theory. This atomic-scale computational approach has the ability to predict measurable properties of materials with good accuracy from scratch (for example, to solve the quantum mechanics of a system of interacting electrons in a field of nuclei). In recent years these methods have emerged as a reliable nanoscopic probe of material properties. From a computational point of view, these calculations involve repeated linear algebraic operations—matrix multiplications, diagonalizations, and inversions—on large matrices.
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Applications to Single-Molecule Circuits
In collaboration with the Venkataraman group at Columbia University, Molecular Foundry researchers investigated the conductance of benzenediamine (BDA) between two gold contacts in an effort to correlate theory and experiments with the modified STM. BDA can adsorb to the surface in a variety of ways at different sites on the contacts and several chemical binding configurations were computed using first-principles methods. The conductance for a given contact geometry was predicted using quantum mechanical transport calculations. These theoretical predictions were then compared to a histogram dataset of tens of thousands of experimental values obtained by the Venkataraman group. A relatively narrow statistical spread (half width) of close to 50% was measured, about the average conductance value. The spread of the calculated conductances for several different geometries was also about 30% of the average value, which was in good agreement with the experiment. The relative insensitivity of the conductance to the contact geometry was understood by direct calculation and attributed to the flexibility of the amine-to-gold bonding. However, the current model overestimates the conductance of BDA by a factor of seven, so some refinement of the theoretical approach was still required. |
| Better understanding of this discrepancy came through a separate collaboration. Molecular Foundry user Dr. Mark Hybertsen, now at Brookhaven National Laboratory, extended the standard approximations in density functional theory to explicitly account for many-electron interactions in order to examine how the highest-occupied and lowest-unoccupied molecular orbitals of aromatic molecules are modified at metal-organic interfaces. When an aromatic molecule is weakly bound to a metal surface, strong "image charge" effects—nonlocal electrostatic correlations between an added electron or hole in the molecule, and the metal surface—shift the frontier orbitals closer to the electrode Fermi level by a large amount (~1 eV for benzene on graphite), substantially narrowing the fundamental energy gap. |
Insights have been gained to understand, identify, and design optimal metal-molecule contacts at the ultimate size limit of organic mechanical and optoelectronic devices, such as switches and solar cells. |
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| Illustration: A. Tovey; Source: J. Neaton, LBNL |
| Figure 2. Schematic diagram of an electronic transistor and its single-molecule analog, highlighting the challenges in device performance of molecular electronics. |
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Building on these calculations performed with Hybertsen, the Theory Facility developed and applied a model that could accurately account for electron correlation effects that affect molecular level alignment in the junction. This new general approach resulted in a predicted average value of the BDA conductance in excellent agreement with the experiments. Insights have been gained to understand, identify, and design optimal metal-molecule contacts at the ultimate size limit of organic mechanical and optoelectronic devices, such as switches and solar cells.
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Computational Matching of Spectral Fingerprints with Material Suspects
Spectroscopic measurements, such as optical or X-ray absorption (sidebar "X-rays Ideal for Probing Nanomaterials"), involve the interaction of radiation or particles with matter as a function of energy and can be used to fingerprint a given material sample or device. Such measurements are widely employed in all fields of science to characterize samples and classify them by comparison with known spectra in preexisting databases. Spectra of numerous nanoscale materials and assemblies can be difficult to understand, because no corresponding database yet exists for many new materials. Computational spectroscopy provides a means of fingerprinting new materials and can establish a new list of "usual suspects" to aid experimentalists in interpreting future measurements. |
| The Theory Facility of the Molecular Foundry is making significant strides toward the development of new tools for computational spectroscopy. With synthesis and characterization capabilities in the same building, staff and users have increased access to the latest measurements of novel nanostructures and the opportunity to drive experimental efforts to establish a library linking known nanostructures with their spectral fingerprints. Other nearby LBNL user facilities such as the National Center for Electron Microscopy and the Advanced Light Source (ALS)—a third-generation X-ray synchrotron—provide access to local researchers as well as a broad international user community, which contributes to and benefits from the growth of simulated spectroscopy at the Molecular Foundry. |
The first-principles, parameter-free modeling capability of the Theory Facility is vital to understanding and predicting these fundamentally quantum mechanical interactions. |
Predicting spectroscopy requires theoretical models that describe the energetic coupling of light or charged particles with matter, or more specifically with electronic or vibrational degrees of freedom. The first-principles, parameter-free modeling capability of the Theory Facility is vital to understanding and predicting these fundamentally quantum mechanical interactions. Furthermore, the computational expense associated with accurate, first-principles approaches is borne by the regularly upgraded, in-house computing cluster and support provided by DOE's National Energy Research Scientific Computing (NERSC) Center, also at LBNL.
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Predicting Core-Level Spectra
First-principles approaches to simulating core-level spectra can accurately predict the X-ray absorption spectrum of a range of materials, as measured using such standard techniques as near-edge X-ray absorption fine structure, X-ray absorption near-edge structure, X-ray Raman spectroscopy, or inner-shell electron energy loss spectroscopy. Initial calculations using this technique established the origin of the X-ray spectral differences between liquid water and ice at ambient pressures. |
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| Illustration: A. Tovey; Source: D. Prendergrast, LBNL |
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| Figure 3. An electron energy level diagram indicating the relative positions of core and valence electrons and the possible transitions probed using core-level spectroscopy. |
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| In collaboration with Molecular Foundry user Richard Saykally and his group at the Department of Chemistry at the University of California-Berkeley, the Theory Facility is currently providing predictions of the core-level spectra of organic molecules in aqueous environments in an effort to establish a well-interpreted spectral library of the building blocks of biological nanostructures, such as polypeptides (proteins) or nucleic acids, such as ribonucleic acid (RNA) and deoxyribonucleic acid (DNA). |
| Core-level excitations can be bound or unbound to the target atom (if the excited electron is detached), and describing both scenarios within the same framework can be challenging. In the first case, the electronic wavefunction is localized in space, while in the second case, a more delocalized description is necessary. Traditional approaches, using a linear combination of localized functions to represent the wavefunctions, work quite well for ground state properties of molecular systems. However, they can prove insufficient to accurately describe unbound excited states. Typically, ad hoc numerical broadening of calculated spectra is used to compensate for shortcomings and to improve agreement with measurements. In an approach developed by Molecular Foundry scientists, planewaves are used to describe the electronic structure, providing uniform convergence in accuracy. The resulting simulated spectra show an unbiased prediction of core-level excitations, bound or unbound, and so provide a truly predictive capability. |
Ultimately, a single spectrum may require hundreds of individual, expensive first-principles calculations, which makes extensive studies of very large molecules in solution prohibitive without the help of large computing resources, such as those provided by NERSC. |
A comparison of Molecular Foundry simulations with measurements by the Saykally group at the ALS resulted in an extremely detailed understanding of the "spectroscopic fingerprints" of molecules in solution. Furthermore, detailed interactions of core-level excitations with intrinsic electronic and vibrational degrees of freedom contribute in unexpected ways to the observed spectral lineshapes characteristic of transitions in these biological nanostructures. Details cannot be separated within experiments, and simulations are now providing new interpretations.
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Computational Challenges for Spectroscopy at the Nanoscale
Apart from the large computational cost associated with first-principles estimates of X-ray spectra for specific atoms, there are multiplicative factors that render complete studies of complex systems computationally demanding. The X-ray probe typically consists of a beam of photons approximately one micrometer in diameter and illuminates the sample in experiments for approximately one second. On the Angstrom length scale and attosecond time scale of individual core-level excitations, measurements incorporate extensive sampling of the ensemble of possible spatial and temporal variations in a given material. |
| In the case of the aqueous systems mentioned above, the ensemble was simulated using molecular dynamics of a small system component—perhaps only one solvated biomolecule. Sampling involved taking uncorrelated molecular configurations or "snapshots" from a long molecular dynamics trajectory (figure 4). This system should be ergodic; in other words, a time-average of molecular configurations is equivalent to a spatial average at a given time. Ultimately, a single spectrum may require hundreds of individual, expensive first-principles calculations, which makes extensive studies of very large molecules in solution prohibitive without the help of large computing resources, such as those provided by NERSC. |
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| Illustration: A. Tovey; Source: D. Prendergrast, LBNL |
| Figure 4. Using snapshots from molecular dynamics trajectories to sample the ensemble of atomic configurations probed in typical X-ray spectra, and the manifold contributions to X-ray spectra of nanostructures which we can resolve and interpret using first-principles simulations. |
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| For nanoscale materials, spatial variations can be quite large, with X-rays probing multiple surfaces and interfaces, as well as atoms embedded in bulk-like environments. Providing meaningful and predictive spectra to experiments requires either nanoscale spatial resolution for the X-ray probe, or extremely uniform and monodisperse samples, which can be modeled effectively using a single nanoparticle. Excellent spatial resolution is possible using scanning tunneling X-ray microscopy (STXM)—available on certain beamlines at the ALS—while synthesis techniques are achieving greater control of nanoparticle size distributions. However, in the absence of these experimental advances, simulations can still provide invaluable interpretations of nanoscale X-ray spectra. By cataloging characteristic spectra of the various interfaces involved, they can be superposed to verify experimental measurements. Furthermore, it is often true that the synthesized interfaces are not well understood or are of ambiguous structure or composition. A thorough characterization of relevant interfaces using computational spectroscopy is vital to accurate assessment of the composition of nanomaterials—particularly those synthesized using wet chemistry. |
Scientists are interested in understanding the physics underlying self-assembly and developing algorithms to permit efficient computer simulation of nanoscale self-assembling components. |
Future work in this area will extend the existing X-ray absorption simulations to other related spectroscopy, such as X-ray emission, Auger recombination, and emission. The access to nanoscale details provided by these simulations will be vital to the interpretation of new experimental advances in obtaining measurements at nanoscale resolution and on ultrafast time scales.
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Toward Simulating Self-Assembly in Nanoscale Components
Experiments in nanoscale self-assembly range from the design of foldable artificial proteins to the synthesis of nanoparticles able to self-organize into superlattices. Scientists are interested in understanding the physics underlying self-assembly and developing algorithms to permit efficient computer simulation of nanoscale self-assembling components. |
| "Self-assembly" describes processes in which autonomous components interact to form stable patterns or aggregates. Interactions as diverse as weak covalent bonds and capillary forces drive self-assembling components that range in size from angstroms to centimeters. Self-assembly is familiar from everyday contexts, such as the patterns formed by mixtures of soap, oil, and water. Self-assembly occurs in inorganic settings—nanoparticles, for example, form patterns under a variety of conditions—and pervades biology: proteins fold into functional apparatuses; cell membranes organize from their lipid constituents; and the genetic material of a virus is protected by a self-assembled protein shell. |
| Mimicry of nature's self-assembly offers a promising route to the inexpensive and efficient synthesis of functional inorganic and biological materials. This proposition is especially attractive on the nanometer scale, where direct fabrication of patterned materials such as computer circuitry is very difficult and expensive. This realization underlies the Foundry's emphasis on the supramolecular assembly of nanostructured materials. |
Understanding the assembly behavior of specific systems aids the understanding of how self-assembly works generically. |
| However, mastering self-assembly is a formidable task. While traditional covalent chemistry and polymerization have been refined (to the extent that complicated multi-stage syntheses of, for instance, naturally occurring drugs are routine), the associated body of knowledge provides little guidance on how to extend this success to the nanoscale. |
Persuading nanoscale components to form ordered structures requires a mastery of weak, chiefly non-covalent forces, whose free energies of association may be comparable to the thermal energy, kBT. This weakness is both a blessing and a curse. It confers upon the assembly process a degree of reversibility, which permits components to explore many potential bound morphologies and so assemble with little error. Reversibility also renders assembly highly adaptive, allowing considerable variation in the architecture of stable structures in response to subtle environmental changes. But weak interactions are easily disrupted by thermal fluctuations, and similar slight changes in environment may lead to the failure of overall assembly. Every second a typical biological cell receives a thermal impact force equal to its own weight. Put in this context, it is easy to appreciate the challenge facing nanoscale components attempting to organize in the face of thermal buffeting and the challenge to experimentalists who must carry out reproducible experiments involving products poised on the brink of falling apart.
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Simulations Guide Experimental Research
Faced with such difficulties, computer simulations of self-assembly provide a crucial guide for experiments. Simulations of model systems can identify which features of component-component interactions stabilize desired structures and can quantify the dynamics through which components associate. The picture that emerges from such studies is one of competition: successful self-assembly depends as much upon the dynamics of how components come into contact as it does upon the free energetic stability of the desired aggregate with respect to its components. The stability of a desired targeted structure may be ensured by equipping its constituents with strong and highly directional interactions (note that such interactions may still be considered "weak" in terms of free energies of binding). However, such requirements run counter to the principles by which components, initially randomly mixed and oriented, organize themselves as the target structure. |
| Strong interactions impair structural relaxation and "freeze" into place defective binding configurations. Directional attractions may facilitate the ordering of components into formations of a desired symmetry. They also make such interactions too specific in an angular sense and productive collisions between components will not happen rapidly enough for structures to grow. These competing requirements of thermodynamic stability and dynamic accessibility in general permit high-fidelity assembly only in a relatively small "Goldilocks zone" of model or experimental parameters, an idea illustrated in figure 5 (p30). |
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| Source: S. Whitelam, LBNL |
| Figure 5. The Goldilocks zone of experimental parameters permitting high-fidelity self-assembly of nanoscale components. |
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| Given the importance of dynamics to the success or failure of assembly, computer simulations of self-assembly must preserve the essential features of physically realistic dynamics. Molecular dynamics algorithms, which evolve components by integrating Newton's equations of motion, are natural choices for particle systems. Given that most nanoscale components operate under viscous conditions and are frequently many times larger than the molecules that solvate them, it is common to neglect inertia and explicit solvent and perform simulations according to a Langevin or Brownian dynamics. Such algorithms preserve the notion that particles move with a drift velocity proportional to the force they experience and account for the collective modes of motion through which self-organizing molecular and colloidal systems can evolve. Specifically, components can aggregate to form clusters that serve as building blocks for subsequent stages of assembly. |
However, nanoscale components frequently possess interactions of maximal strength much greater than kBT, and range much less than their girth. Colloids of diameter 100 nm, for instance, might interact via strong electrostatic forces attenuated on the single-digit nanometer scale or via hydrophobic or depletion forces of similar range. In addition, nanoparticle-nanoparticle interactions can be anisotropic (mediated, say, by magnetic dipoles) and vary significantly in strength upon relatively modest changes in binding orientation. Interactions that vary rapidly with angle or distance place stringent limits on the maximum integration time step able to preserve numerical stability. Given the formation of target structures may involve the concerted motion of many components over distances many times their girth, integrating equations of motion under such conditions can become very time consuming.
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Toward More Efficient Simulations
Overcoming these limitations constitutes one focus of theoretical work at the Molecular Foundry, a program that will be advanced in-house and via collaboration through the User Program. One route to more efficient simulation may be to employ coarse-grained dynamical procedures that move particles in an approximately physical fashion according to the forces they experience, without explicitly integrating equations of motion. The Monte Carlo technique offers a means of doing so, provided that the possibility of moving collections of particles in concert is allowed. |
| One way to enforce collective motion within a Monte Carlo protocol is with an approach that approximates over-damped dynamics. Future improvements in this area might involve the adaptation to structured components of very efficient "diffusion free" algorithms developed to treat particles that aggregate irreversibly upon contact. Theoretical work at the Foundry also seeks to understand the microscopic underpinnings of "good assembly." By augmenting a physical dynamics with unphysical, nonlocal moves, the fidelity of assembly in simulations of model systems may be improved. Quantifying the effect of such improvements using generalized fluctuation-dissipation relations reveals key characteristics of good assembly. For example, how frequently must a component make and break bonds before it attains a "well assembled" configuration? This work is performed in external collaboration with Robert Jack at the University of Bath, United Kingdom. |
Theory and computation help guide experimental work and help researchers go beyond the physical limits of experimentation. |
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| Source: S. Whitelam, LBNL |
| Figure 6. Determination of a self-assembly phase diagram. |
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A second focus of the Molecular Foundry's research in this area is to provide theoretical support for experimentalists, both in-house and from the User Program, who wish to use self-assembly as a means of synthesizing new materials. Theoretical work in this vein typically involves developing a simplified model of the experimental system, followed by its simulation using molecular dynamics or collective-move Monte Carlo algorithms. A key aim of such work is to determine an "assembly phase diagram," a characterization of the thermodynamically stable and kinetically accessible products of self-assembly in a space of model and experimental parameters. Figure 6 highlights two examples of work of this nature, derived from external collaborations. A major focus of Foundry research on self-assembly is to develop a flexible set of algorithms and routines to permit efficient simulation of a diverse range of nanoscale components. Such components may possess arbitrary shapes and be equipped with interactions broadly distributed in range (from very short to very long) and of different character (for example, pairwise or multibody). Understanding the assembly behavior of specific systems aids the understanding of how self-assembly works generically.
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Conclusion
Within the scientific community, computational science has been heralded as the "third leg" of science, complementing theory and experiment. At the Molecular Foundry, such interaction is driving leading-edge nanoscale research as theory and computation help guide experimental work and help researchers go beyond the physical limits of experimentation.
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Contributors: Jeffrey B. Neaton, David Prendergast, Steven Whitelam, and Steven G. Louie—all at the Molecular Foundry, Lawrence Berkeley National Laboratory.
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