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news & views a b E c 1 W(t) E B0 0 Burst time, t Figure 1 | Electrically manipulating spins in quantum dots. a, Band diagram of the experimental quantum dot system. An applied oscillating electric field induces spin-flips via EDSR, which manifest as oscillations in the transport characteristics owing to the spin-blockade phenomenon. b, A localized electron spin interacts with a nuclear spin bath, leading to decoherence. c, Time evolution of the probability W(t) of the dot’s electron spin being in the ‘up’ state when the mediating mechanism is due to SO/inhomogeneous magnetic field (solid line) or a gradient of the Overhauser field (dashed line). transverse components of the nuclear spins’ magnetization with respect to the externally applied static magnetic field. The details of the averaging process turn out to have direct consequences for experimental observations. For example, analytical formulas for W(t) in the asymptotic limit for the SO coupling and magnetic-field-gradient mediated EDSR scenarios show oscillatory behaviour 4, a manifestation of Rabi oscillations observed in experiments1,5. More importantly, the theory shows that W(t) is damped at a rate inversely proportional to the square-root of the burst time, t, with a universal phase shift with respect to the expected sinusoidal squared oscillation (solid line in Fig. 1c). This characteristic damping behaviour has also been observed experimentally 5. Its origin is explained by the theory in terms of ‘decoherence’ owing to fluctuations in only the longitudinal component of the nuclear spins’ magnetization. In contrast, for the case of an Overhauser-field-mediated interaction between the electron spin and an applied electric field, fluctuations in both the longitudinal and the transverse components of the nuclear spin magnetization are important. As a characteristic signature differentiating it from the SO and magnetic cases, W(t) was theoretically shown to have a monotonic increase with burst time (dashed line in Fig. 1c.) in agreement with the experimental observations by Laird et al.3. The presented mean-field theoretical approach seems to explain the experimentally observed EDSR transport characteristics in W(t) very well. However, there are still challenges that remain to be addressed. One important aspect relates to the quantum nature of the nuclear spins, which was not considered fully in the presented theoretical framework, but would become important for large external magnetic fields6. In order to clarify the importance of hyperfine interactions with the nuclear spins on the electron spins’ coherence, further experimental and theoretical studies are needed. Theoretical analyses along these lines would be challenging because the hyperfine coupling strength is not uniform over the nuclear spins involved (about 105–106) owing to the localized nature of electron wave functions. There seem to be at least two ways to avoid decoherence due to the nuclear spins. One is to manipulate the nuclear spin magnetization and to reduce its fluctuations by a controlled interaction with electron spins7,8. The dynamics in such a composite system would be an exciting field of future research. The other possible, but technologically challenging, avenue is to switch to other systems, such as p-type GaAs, Si/Ge or carbon nanotubes, which have much weaker interaction strengths with the nuclear spins9. ❐ Yasuhiro Tokura is at the Basic Research Laboratories, NTT Corporation and Quantum Spin Information Project, ICORP–JST, Atsugi‑shi, 243‑0198, Japan. e‑mail: [email protected] References 1. Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypen, L. M. K. Science 318, 1430–1433 (2007). 2. Pioro-Ladrière, M. et al. Nature Phys. 4, 776–779 (2008). 3. Laird, E. A. et al. Phys. Rev. Lett. 99, 246601 (2007). 4. Rashba, E. I. Phys. Rev. B 78, 195, 302 (2008). 5. Koppens F. H. L. et al. Phys. Rev. Lett. 99, 106803 (2007). 6. Coish, W. A., Loss, D., Yuzbashyan, E. A. & Altshuler, B. L. J. Appl. Phys. 101, 081715 (2007). 7. J. Baugh, Y. Kitamura, K. Ono & S. Tarucha, Phys. Rev. Lett. 99, 096804 (2007). 8. J. R. Petta et al. Phys. Rev. Lett. 100, 067601 (2008). 9. Schroer, M. D. & Petta, J. R. Nature Phys. 4, 516–518 (2008). CONDENSED MATTER How do your crystals grow? More than 100 years ago, Wilhelm Ostwald predicted that crystalline structures would grow from the melt via a series of unstable states — now this cascade has been observed directly in an inorganic semiconductor. Simon J. L. Billinge C rystals nucleate out of molten material and then grow to make the beautiful, perfect, final product. How this happens is surprisingly complex and difficult to understand despite being studied for centuries. On page 68 of this issue1, Chung and colleagues show for the first time the formation of a single crystallite through a cascade of intermediate forms on the way to the final thermodynamically nature physics | VOL 5 | JANUARY 2009 | www.nature.com/naturephysics © 2009 Macmillan Publishers Limited. All rights reserved stable state — an observation that supports Ostwald’s ‘rule of stages’. Below the freezing-temperature, groups of atoms spontaneously congregate and separate from the melt, rather like clusters 13 news & views 5 nm Figure 1 | It happens in stages. This ‘movie’ of a single crystallite as it cascades through different states towards its stable form (left to right) was created by Chung et al.1 from high-resolution electron micrographs. Atomic displacements are shown taking place over a period of four minutes. of people conversing at a crowded party. As a group forms, others join and the group grows, but it is still unstable — it can dissolve and dissipate again — until it reaches a certain size, the critical radius. At this point, everyone at the party realizes that this is the place to be and the group grows rapidly, limited only by factors such as the ability of the crowd to diffuse over and join it. The critical radius is a tipping point where the bulk energy of the more stable crystalline core overcomes the extra cost of forming an interface. For large clusters the bulk energy always wins, but below the critical radius the cost of making the interface is greater, resulting in an energetic barrier to crystal formation, even in the undercooled or supersaturated state where, thermodynamically, the crystal is preferred. Atoms must be thermally excited to overcome the energy barrier, a stochastic process. It is worth pondering whether there are any intermediate states that can be accessed that allow one to circumvent the high activation barrier. These are not the most thermodynamically stable states, but they can be accessed by thermal excitation with greater probability because the activation barrier to form them is lower. Once the intermediate state is formed, there may be a pathway to the thermodynamically stable state with a lower activation barrier. Kinetically, this will then be the preferred path for the transformation to take. In 1897, Wilhelm Ostwald postulated just such a process for crystal formation2. Crystals are notorious for having multiple forms with nearly equal energy; this is the reason for polymorphism and structural phase transitions at moderate temperatures. Based on observations of the crystallization of a number of systems from supercooled melts and supersaturated solutions, Ostwald postulated that intermediate states form a cascade from the least stable available crystalline form to the most stable. This became known as the ‘Ostwald rule of stages’ (refs 2,3). For many years this rule has been a guiding principle for chemists 14 seeking previously unknown crystal forms, for example in pharmaceutical science. But the rule is quite an extrapolation from observations — is it right? It has taken 111 years to be able to follow the cascade as it takes place in a single crystallite, but Chung et al.1 present the first direct experimental evidence for a process like this actually occurring in inorganic materials. Why did it take 111 years? The experimental difficulties are formidable. The nuclei form and dissolve very rapidly in the melt: how can we capture the dynamic process at all, let alone follow an evolution through multiple intermediate states? Also, many thousands of nuclei are forming at any given time: how can we follow the time-dependent behaviour of a single nucleus out of the whole ensemble? A number of creative experimental insights led Chung et al. to the solution. First, they chose to study recrystallization from an amorphous state, where the dynamics of the process can be slowed down to experimentally accessible timescales. Indeed, the timescale can be controlled to some extent with temperature, provided the temperature stays above that for recrystallization and below that for melting. Second, they used a local probe of structure, electron diffraction, which enables individual nuclei to be isolated and followed with time. It is quite difficult to get quantitative 3D-structure solutions using electrons (although recent progress in this regard has been rapid4), but quantitative determination of the lattice symmetry and parameters enables structural forms to be differentiated (Fig. 1) — all that is required here. The authors were richly rewarded with images and diffraction patterns of a cascade of structural forms in LiFePO4 evident before the stable crystal structure was finally reached (with a wink to history: polymorphism itself was discovered in phosphate/arsenate salts by Mitscherlisch5 in 1822). As a bonus, additional information was also revealed about the mechanism of the transformations between intermediate states. So where do we go from here? The experimental results are certainly tantalizingly supportive of Ostwald’s prescient insight: we now have a movie of a single crystallite cascading through intermediate states on its way to the stable form. Many questions remain, however. Is it a universal law or a rare fortuitous occurrence in this particular material system? Crystallization is a highly complex process and no-one has found any sound theoretical basis for such a general principle as Ostwald’s rule6,7. Is the same behaviour also observed from the melt, as Ostwald originally envisaged it? Is the result experiment-dependent? Studying a crystallite at the very edge of a thin section in vacuum under an intense, highly interacting, electron beam is necessary to make the experiment a success, but these are not exactly optimal conditions for studying near-equilibrium behaviour. Finally, and most tantalizing of all, Ostwald’s prediction was very precise: the cascade is from the least stable accessible crystalline form to the most stable. In this experiment, the existence of intermediate states is established, but Ostwald’s full law of stages remains unverified. We have observed the crowd dynamics at the packed party, but the social dynamics that give rise to this behaviour are still a distant, barely intelligible, conversation. ❐ Simon J. L. Billinge is in the Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA and the Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA. e‑mail: [email protected] References 1. Chung, S.-Y., Kim, Y.-M., Kim, J.-G. & Kim, Y.-J. Nature Phys. 5, 68–73 (2009). 2. Ostwald, W. Z. Z. Phys. Chem. 22, 289–330 (1897). 3. Threlfall, T. Org. Process Res. Dev. 7, 1017–1027 (2003). 4. Zou, X. & Hovmöller, S. Acta Crystallogr. A 64, 149–160 (2008). 5. Mitscherlich, E. Ann. Chim. Phys. 19, 350–419 (1822). 6. Bernstein, J. Polymorphism in Molecular Crystals (Oxford Univ. Press, 2002). 7. ten Wolde, P. R. & Frenkel, D. Science 277, 1975–1978 (1997). nature physics | VOL 5 | JANUARY 2009 | www.nature.com/naturephysics © 2009 Macmillan Publishers Limited. All rights reserved