# Monthly Archives: June 2015

## Staying cool this summer

Many green energy solutions originate as condensed matter physics problems. Prominent examples include electricity production via solar cells, brighter and more efficient lighting via LEDs, and thermoelectric materials for converting waste heat into electricity.  Another promising but less well-known approach is using metamaterials for passive—no power input required– daytime radiative cooling.  This may one day supplement or replace air conditioners or be incorporated in clothing.  Just what we need with summer upon us.

A recent Science paper illustrated how a species of ants in the Sahara Desert uses special triangular hairs on the top part of its body to cool themselves in the middle of the day in temperatures up to 158° F [1].  This work was also highlighted in a New York Times article.

Left: SEM image of Sahara silver ant head.
Right: SEM image of ant hairs. Notice that they are corrugated on the top surfaces, have a triangular cross-section, and are of different sizes. From Ref. [1].

The engineering requirements for daytime radiative cooling are high reflectivity at visible wavelengths, where the solar spectrum is peaked, and high-emissivity (corresponding to low reflectivity, aka high absorptivity) at mid-infrared wavelengths, where the blackbody spectrum of a human or a hot building are peaked (~9 microns).  This originates from Kirchhoff’s law of thermal radiation, which states that the absorptivity ($\alpha_\lambda$) and emissivity ($\epsilon_\lambda$) must be equal at a given wavelength ($\lambda$) in order to maintain equilibrium.  However, the overall absorption and emission is weighted by the solar spectrum ($I_{\lambda,sun}$) and the blackbody spectrum of the hot body ($I_{\lambda,body}$), respectively.

Total absorptivity: $\alpha =\frac{ \int^\infty_0 \alpha_{\lambda,body}I_{\lambda,sun} \mathrm{d}\lambda}{\int^\infty_0 I_{\lambda,sun}\mathrm{d}\lambda}$

Total emissivity: $\epsilon = \frac{\int^\infty_0 \epsilon_{\lambda,body}I_{\lambda,body} \mathrm{d}\lambda}{\int^\infty_0I_{\lambda,body} \mathrm{d}\lambda}$

The values above might not be equal, even though emissivity and absorptivity must be equal at a given wavelength.  An object which is highly reflective at visible wavelengths is not effectively heated up by absorbing energy from the sun.  Meanwhile, an object which has high emissivity (aka high absorptivity aka low reflectivity) at mid-IR wavelengths can effectively radiate heat, provided the atmosphere has sufficient transparency, which it does for radiation with wavelengths between 8 and 13 microns.  An example of the latter is a plant leaf: their high emissivity at mid-IR wavelengths allows them to cool below ambient temperature at night, which is why you find dew condensed on the grass in the morning.  If a single material or structure has high reflectivity at visible wavelengths and high emissivity at mid-IR wavelengths, there is the possibility of passively cooling below ambient temperature in the middle of the day.

These engineering requirements are generally hard to achieve in bulk materials, and this is where metamaterials—composites or structures engineered to have properties not found in bulk materials– come in.  Recently, passive daytime radiative cooling has been predicted [2] and achieved [3] in engineered photonic structures.

Returning to the desert ants, their hairs are an example of a natural photonic metamaterial which passively cools them in the middle of the day so they can venture out for a snack while their predators are hiding from the heat.  The hairs’ shape, size, and surface all conspire to reflect visible light.  The triangular cross-section permits total internal reflection for a range of incidence angles (see figure below).  The cross-sectional size of the hairs is comparable to the wavelength of visible light, which allows for the trapping and re-radiation of light via a process called Mie scattering.  A given cross-sectional area will give enhanced reflection at a single wavelength, but the polydisperse distribution of hair sizes allows for broadband coverage.  Finally, surface roughness allows for diffuse scattering of visible light at the surface of the hair.  As for mid-IR absorption, the paper says that the hairs act as an antireflective coating at these wavelengths, though they do not specify the mechanism.  Most likely, the sub-wavelength size of the hairs present an effective medium with an effective index of refraction to incoming mid-IR light, and the triangular shape helps produce a gradient of the refractive index which minimizes reflection due to index mismatch with air.  Another factor might be that the size of the hairs (and they appear to grow in a single layer) is roughly comparable to ¼ of the wavelength of mid-IR light, and quarter-wave-thick layers are frequently used in antireflective coatings.   And it really works.  Both in vacuum and in air, the ant is able to stay 10° C cooler with the hairs than without.

Total internal reflection happens for a range of incidence angles, and is one mechanism by which the ants’ triangular hairs reflect visible light.

It sounds too good to be true—cooling a body or an object passively in full sunlight—but the Sahara Desert ants (Ref. [1]) and the experiments in Ref. [3] show that it is really possible to achieve this feat by manipulating the optical properties of materials.

Thermal images of the ant head show that radiative cooling works both in vacuum and in air, and that the hairs are the source of this phenomenon.

References:

[1] N. N. Shi et al. Science (2015), Advance online publication

[2] E. Rephaeli et al. Nano Lett.,  13 (4), pp 1457–1461 (2013)

[3] A. Raman et al. Nature515 pp 540–544 (2014)

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## Merchants of Doubt

I watched a documentary yesterday entitled Merchants of Doubt, which is based on a non-fictional book by Naomi Oreskes and Erik M. Conway. It centers around public relations (PR) specialists who play “experts” on TV, radio and other forms of media. These PR specialists are hired by corporations (e.g. ExxonMobil, Philip Morris, etc.) to undermine scientific consensus in the public domain in a field of study where they have no formal expertise.

For instance, Philip Morris would hire many of these specialists to appear on TV as “authorities” to convince the public that there was no scientific consensus regarding the health effects of tobacco. When pitted against actual scientific experts on TV, these “authorities” are often aggressive and argumentative, thereby seeming to subvert the scientist’s message in the eyes of the public.

One of the most startling revelations from this documentary (at least to me), was the role played by two prominent physicists, Fred Singer and Frederick Seitz. Both physicists have aided in legitimizing the claims against anthropogenic climate change. Strangely, both figures had also played a role “in helping the tobacco industry produce uncertainty concerning the health impacts of smoking”. According to Wikipedia, Singer has also publicly questioned “the link between UV-B and melanoma rates, and that between CFCs and stratospheric ozone loss”.

Clearly, for these two physicists, there is a political element to these decisions, which cannot be based on sound scientific reasoning. It is deeply disturbing for me to know that the building in which I have worked for the previous few years, the Frederick Seitz Materials Research Laboratory, while an historic research facility, is named after a man who has purposefully eroded the public’s trust in the scientific consensus on anthropogenic climate change (paywall).

## A Little More on LO-TO Splitting

In my previous post, I addressed the concept of LO-TO splitting and how it results from the long-ranged nature of the Coulomb interaction. I made it a point to emphasize that while the longitudinal and transverse optical phonons are non-degenerate near $\textbf{q}=0$, they are degenerate right at $\textbf{q}=0.$ This scenario occurs because of the retarded nature of the Coulomb interaction (i.e. the finite speed of light).

What exactly goes on? Well, it so happens that in a very narrow momentum window near $\textbf{q}=0$, the transverse optical phonon is strongly coupled to light and forms a polariton. This is a manifestation of the avoided crossing or level repulsion principle that I have blogged about previously. Since light is a transverse wave, it interacts with the transverse optical phonon (but not the longitudinal one).

In a tour-de-force experiment at Bell Labs by Henry and Hopfield (the same Hopfield of Hopfield neural networks), Raman scattering was conducted at grazing incident angles to measure the dispersion of the lower polariton branch as shown below:

The dispersing solid lines represent the transverse optical (TO) phonon interacting with light. The straight solid line is the unaffected longitudinal optical (LO) phonon branch. The dotted line labelled with the angles are the incident beam angles in the Raman experiment. The remaining dotted lines represent the non-interacting TO phonon and the non-interacting light dispersion.

Usual Raman measurements are taken in a backscattering as opposed to a grazing incidence geometry, hence the momentum transfers are ordinarily too high to observe the low-$\textbf{q}$ dispersion. Because of this, the authors mentioned that some Raman exposures in this experiment took up to seven hours!

The takeaway from the plot above is that the transverse optical phonon at $\textbf{q}=0$ is degenerate with the longitudinal one and “turns into a photon” at higher momenta, while the photon branch at $\textbf{q}=0$ “turns into the transverse optical phonon” at higher momenta.

Unfortunately, the paper does not contain their raw data, only the dispersion. Publishing standards seem to have been different back then. Nonetheless, this is a very clever and illuminating experiment.

## Plasmons, the Coulomb Interaction and a Gap

In a famous 1962 paper entitled Plasmons, Gauge Invariance and Mass (pdf!), P.W. Anderson described the relationship between the gap in the plasmon spectrum and the idea of spontaneous symmetry breaking. It is an interesting historical note that Higgs cites Anderson’s paper in his landmark paper concerning the Higgs mechanism.

While there are many different formulations, including Anderson’s, of why the plasmon is gapped at zero momentum in a 3D solid, they all rely on one crucial element: the long-range nature of the Coulomb interaction (i.e. the electrons are charged particles). Of these formulations, I prefer one “cartoon-y” explanation which captures the essential physics well.

Before continuing, let me stress that it is quite unusual for a fluid medium (such as the electrons in a metal) to possess no zero frequency excitations at long wavelengths. For instance,  the dispersion relation for surface gravity waves on water (pdf!) is:

$\omega^2(k)=gk \tanh kh$.

Now, in 3D and in the long-wavelength limit, the plasmon sets up opposite charges on the surfaces of the solid as pictured below:

The long-wavelength plasmon therefore sets up the same electric field as in a capacitor. The electric field for a capacitor is $\textbf{E} = \frac{\sigma\hat{x}}{\epsilon_0}$. This expression is surprisingly independent of the distance separating the surfaces of the solid. Therefore, it takes a finite amount of energy to set up this electric field, even in the limit of infinite distance. This finite energy results in the gapping of the plasmon.

This cartoon can be extended further to 2D and 1D solids. In the 2D case, the electric field for the 1D “lines of charge” bounding the solid falls off like $\textbf{E}\sim\frac{1}{\textbf{r}}$. Therefore, in the infinite distance limit, it takes no energy to create this electric field and the plasmon is not gapped at $\textbf{q}=0$. Similarly, for the 1D case, the electric field from the points bounding the solid falls of as $\frac{1}{\textbf{r}^2}$, and the plasmon is again gapless.

This reasoning can be applied further to the phenomenon known as LO-TO splitting in a polar solid. Here, the longitudinal optical phonon (LO) and the transverse optical phonon (TO) branches are non-degenerate down to the very lowest (but non-zero!) momenta. Group theory predicts these modes to be degenerate at $\textbf{q}=0$ for the zincblende crystal structure of typical semiconducting compounds. Below is the phonon dispersion for GaAs demonstrating this phenomenon:

Again, the splitting occurs due to the long-ranged nature of the Coulomb interaction. In this case, however, it is the polar ionic degree of freedom that sets up the electric field as opposed to the electronic degrees of freedom. Using the same reasoning as above, one would predict that the LO-TO splitting would disappear in the 2D limit, and a quick check in the literature suggests this to be the case as reported in this paper about mono-layer Boron Nitride.

I very much appreciate toy models such as this that give one enough physical intuition to be able to predict the outcome of an experiment. It has its (very obvious!) limitations, but is valuable nonetheless.

## Neil deGrasse Tyson on NASA Funding

I reiterate a previous post about scientific public intellectuals. There are few scientists as impassioned, articulate and persuasive as Neil deGrasse Tyson as evidenced by this short clip:

Let me also quote his inspirational testimony in front of the United States Senate:

The 2008 bank bailout of \$750 billion was greater than all the money NASA had received in its half-century history.

## Hume’s Uniformity of Nature

The 18th century philosopher David Hume raised issue with inductive logic, which has fundamental ramifications for science. His contention has not been firmly settled, and it is very difficult to foresee a solution to what is now known as “Hume’s Problem” in the near future.

Just as a little bit of background, here is an example of inductive reasoning:

All bodies that have been observed obey Newton’s law of gravitation.

Therefore all bodies obey Newton’s law of gravitation.

We are extrapolating from a small data set (on a cosmic scale) to formulating laws about the universe. Obviously, this reasoning is not perfectly sound, but this is how physicists (and all other scientists) reason. Importantly, there is no sound deductive argument to suggest that all bodies should necessarily obey Newton’s law of gravitation.

Hume calls this assumption the assumption of the Uniformity of Nature (UN). The assertion is that the laws of physics do not change, for example, from object to object.

Only under the assumption of UN does reasoning inductively actually work. However, the UN assumption cannot be proven deductively (at least not yet!), and therefore there is no reason to suppose that the laws of physics might not look totally different, say, in another part of the universe that we have yet to observe.

I think that every practicing physicist is at some level aware of the UN assumption, but it is always illuminating to have ideas explicitly stated. But now that we have articulated it and are plainly aware of its constraints on our reasoning, let us get on with physics.

## Spin Fluctuations in the Cuprates

In the history of the cuprate superconductors, many predictions have been put forth, but there was one non-trivial prediction that has stood out among the rest. This is the prediction of the $d_{x^2-y^2}$ order parameter symmetry from spin-fluctuation models that were put forth before its experimental verification.

The idea is quite simple, and you can read more about it in this set of lecture notes by A.J. Leggett, where he lays out the concepts very well. I summarize the main points below.

In the cuprates, the Fermi surface is usually assumed to look like so, which has been determined by ARPES experiments:

Schematic of Fermi Surface as Determined by ARPES

One also knows that the antiferromagnetic phase in the parent compound looks like so:

(a) Real space representation of the antiferromagnetic parent phase. (b) Reciprocal space representation with Q representing the antiferromagnetic Bragg wavevector.

i)     Now, one can see that the points on the Fermi surface close to ($\pi$, 0) and (0, $\pi$) are the ones connected by the antiferromagnetic Bragg wavevector, Q. One would then predict a singlet pair wavefunction, as those points on the Fermi surface would be expected to exhibit the largest gap.

ii)     The other input is that scattering should not change the sign of the pair wave function, $F_\textbf{k} = orbital\times spin$, which comprises the orbital and spin components. Since the spin part is a singlet, (i.e. $1/\sqrt{2} (\uparrow_1\downarrow_2-\downarrow_1\uparrow_2)$), it will change sign when the pair interacts through a spin-fluctuation. Therefore, to keep $F_\textbf{k}$ invariant, the orbital part must also change sign under the scattering/interaction of wavevector Q.

The two criteria leave $d_{x^2-y^2}$ symmetry as the only option, and hence spin-fluctuation theories explicitly predict this symmetry.

Obviously, this does not mean that spin-fluctuation theories are correct, but it is worth noting that they have made a non-trivial prediction.

While this historical note is well-known to those have been studying high-temperature superconductivity since its discovery, those of us who were born around the same time as the discovery of the cuprates sometimes lose this kind of historical context.

Images are taken from the lecture notes by A.J. Leggett linked above.