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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