is 3 nm (29), which is consistent with the grain
boundary widths (which correspond to atomic-scale features) seen in our plasmon energy maps
(fig. S1). However, the sample does not support a
temperature gradient for separations smaller than
the electron mean free path ‘e because electrons
are ballistic over distances less than ‘e. Thus, ‘e
describes the smallest thermal feature size that
can exist in continuous aluminum. Similarly, because phonons generate thermal expansion, temperature cannot produce different densities at
separations smaller than a phonon mean free
path ‘ph. We estimate ‘e ≲ 4 to 15 nm and ‘ph ≲ 2
to 5 nm in our temperature range (table S1).
For Lpl smaller than ‘ph or ‘e, PEE T achieves the
maximum possible spatial resolution; temperature
differences do not exist on length scales smaller
than the larger mean free path.
PEET is applicable to many other technologically important metals and semiconductors.
Tungsten, silver, silicon, gallium arsenide, and
gallium nitride all have sufficiently sharp plasmon resonances (29). [The width of the plasmon
resonance limits PEET’s precision, so decreasing
the zero loss peak width (30) gives only a small
sensitivity improvement.] Because the product
of the thermal expansion coefficient a with the
melting temperature is a Tm 0.02 for many materials (31), one will generally trade high sensitivity for a large accessible temperature range, or
vice versa, depending on the application. Ideally,
the system to be measured serves as its own thermometer, without requiring the introduction of
thermometric materials that might compromise
the thermal behavior or device function.
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This work has been supported by National Science Foundation
award DMR-1206849 and in part by Function Accelerated
nanoMaterial Engineering (FAME), one of six centers of
Semiconductor Technology Advanced Research network
(STARnet), a Semiconductor Research Corporation program
sponsored by the Microelectronics Advanced Research Corporation
and Defense Advanced Research Projects Agency. Data
presented in this article were acquired at the Center for Electron
Microscopy and Microanalysis at the University of Southern
California. Work at the Molecular Foundry was supported by the
Office of Science, Office of Basic Energy Sciences, of the U.S.
Department of Energy under contract DE-AC02-05CH11231.
Materials and Methods
Figs. S1 to S8
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6 November 2014; accepted 12 January 2015
Asynchronous rotation of Earth-mass
planets in the habitable zone of
Jérémy Leconte,1,2,3 Hanbo Wu,1,4 Kristen Menou,2,5 Norman Murray1,4
Planets in the habitable zone of lower-mass stars are often assumed to be in a state of tidally
synchronized rotation, which would considerably affect their putative habitability. Although
thermal tides cause Venus to rotate retrogradely, simple scaling arguments tend to attribute
this peculiarity to the massive Venusian atmosphere. Using a global climate model, we show
that even a relatively thin atmosphere can drive terrestrial planets’ rotation away from
synchronicity. We derive a more realistic atmospheric tide model that predicts four asynchronous
equilibrium spin states, two being stable, when the amplitude of the thermal tide exceeds a
threshold that is met for habitable Earth-like planets with a 1-bar atmosphere around stars
more massive than ~0.5 to 0.7 solar mass. Thus, many recently discovered terrestrial planets
could exhibit asynchronous spin-orbit rotation, even with a thin atmosphere.
As we experience in our everyday life, atmo- spheric temperatures oscillate following the diurnal insolation cycle. This in turn cre- ates periodic large-scale mass redistribution inside the atmosphere—the so-called thermal atmospheric tides. But as we all also have
experienced, the hottest moment of the day is
actually not when the Sun is directly overhead,
but a few hours later. This is due to the thermal
inertia of the ground and atmosphere that
creates a delay between the solar heating and
thermal response (driving mass redistribution), causing the whole atmospheric response to
lag behind the Sun (1).
Because of this asymmetry in the atmospheric
mass redistribution with respect to the subsolar
point, the gravitational pull exerted by the Sun
on the atmosphere has a nonzero net torque that
tends to accelerate or decelerate its rotation, de-
pending on the direction of the solar motion (2, 3).
Because the atmosphere and the surface are usu-
ally well coupled by friction in the atmospheric
boundary layer, the angular momentum trans-
ferred from the orbit to the atmosphere is then
transferred to the bulk of the planet, modifying
its spin (4).
On Earth, this effect is negligible because we
are too far away from the Sun, but the atmospheric torque due to thermal tides can be very
powerful, as seen on Venus. Indeed, although tidal friction inside the planet is continuously trying
to spin it down to a state of synchronous rotation,
thermal tides are strong enough to drive the planet out of synchronicity and to force the slow
retrograde rotation that we see today (2–6). Very
simple scaling arguments predict that the amplitude of the thermal tide is proportional to the
ratio of the atmospheric mean surface pressure
over its scale height (1). Everything else being
equal, one would thus expect the thermal tide to
be ~50 times weaker if Venus had a less massive,
cooler Earth-like atmosphere. Whether this scaling really holds and how massive the atmosphere
1Canadian Institute for Theoretical Astrophysics, 60 St
George Street, University of Toronto, Toronto, ON M5S3H8,
Canada. 2Center for Planetary Sciences, Department of
Physical and Environmental Sciences, University of Toronto
Scarborough, Toronto, ON M1C 1A4, Canada. 3Laboratoire de
Météorologie Dynamique, Institut Pierre Simon Laplace, 4
Place Jussieu, BP 99, 75252 Paris, France. 4Department of
Physics, University of Toronto, 60 St George Street, Toronto,
ON M5S 1A7, Canada. 5Department of Astronomy and
Astrophysics, University of Toronto, Toronto, ON M5S 3H8,
*Corresponding author. E-mail: email@example.com