Nanoscale machines in nature

Part two of the Biology brief in The Economist is Cells and how to run them: All life is made of cells, and cells depend on membranes

A few of the main ideas are the following. Cells are either prokaryotic (bacterium) or eukaryotic (animals). Cell membranes are made of lipids that spontaneously form structures due to an interplay between hydrophobic and hydrophilic interactions. The boundary of prokaryotic cells is the membrane. Eukaryotic cells are more complex, containing many organelles (mitochondria), whose boundary are membranes.

Cells are little factories that can multiply themselves and perform distinct biological functions. It requires energy to maintain the cell shape and for it to manufacture new things. Inside and out is maintained by a difference in the concentration of protons (hydrogen ions) across the membrane. There are two aspects to this. First, the electron transport chain produces the protons. Second, a specific protein in the membrane, ATP synthase, pumps protons across the membrane.

The electron transfer chains are driven either by respiration or photosynthesis. 

Energy for processes in the cell is provided by breaking ATP down to ADP. The reverse process is driven by the kinetic energy of rotation (at about 6000 rpm) of the part of the ATP synthase protein.  ATP is Adenosine triphosphate.

To me the amazing/awesome/cool/miraculous thing is what the hardware can do. These are nanoscale chemical machines and factories. The video below shows a simulation of the ATP synthase protein that is located within cell membranes. It acts as a proton pump to maintain the concentration imbalance between the outside and inside of the cell and to convert ADP to ATP.

I learnt from this how the ATP synthase spins in only one direction and the rotation corresponds to sequential conformational changes in the protein subunits.

There is a beautiful discussion of the underlying physics in a chapter in Biological Physics by Phil Nelson. I have written a brief summary here.

The underlying quantum chemistry is explored in

Electrostatic origin of the mechanochemical rotary mechanism and the catalytic dwell of F1-ATPase

Shayantani Mukherjee and Arieh Warshel

The challenge of non-equilibrium thermodynamics

This week I am in Telluride at the bi-annual workshop on Condensed Phase Dynamics. I really enjoyed the talks today. A common topic was that of non-equilibrium thermodynamics, particularly in nanoscale systems.

Abe Nitzan began his talk mentioning a recent PRL, Quantum Thermodynamics: A Nonequilibrium Green’s Function Approach, which unfortunately, is not valid because the expressions it gives do not give the correct result in the equilibrium limit. This is shown in

Quantum thermodynamics of the driven resonant level model 
 Anton Bruch, Mark Thomas, Silvia Viola Kusminskiy, Felix von Oppen, and Abraham Nitzan

What is striking to me about both papers is that they consider a non-interacting model, i.e. the Hamiltonian is quadratic in fermion operators and exactly soluble.
This shows just how far we are from any sort of theory of a realistic system, i.e. one with interactions and which is not integrable.

Phil Geissler gave a nice introduction to different theorems for fluctuations in the dissipation (defined as the difference between the entropy change and heat/temperature). The most general theorem is that due to Gavin Crooks and implies the Jarzynski inequality, the fluctuation theorem, and the second law of thermodynamics.
A key question is what sorts of non-equilibrium processes (protocols) minimise the dissipation and whether the distribution is Gaussian (it often is).
He then described near optimal protocols to invert the magnetisation in a two-dimensional Ising model.

Suri Vaikuntanathan talked about coupled (classical) master equation models for biomolecular networks that have mathematical similarities to an electronic Su-Schrieffer-Heeger model which is an one-dimensional example of a topological insulator.
The work is described  in a preprint with A. Murugan,  "Topologically protected modes in non-equilibrium stochastic systems".
This is potentially  important because it may provide  "a framework for how biochemical systems can use non equilibrium driving to achieve robust function."

David Limmer gave a nice talk which considered thermodynamics as a large deviation theory and how that can even have meaning out of equilibrium and there is a notion of an entropy, a "free energy" and a "temperature". His slides are here.
A key notion is to focus on ensembles of trajectories rather than a probability distribution function. There are two alternative computational strategies: transition path sampling and diffusion Monte Carlo (the cloning algorithm).
He considered several concrete examples, such as thermal conductivity in carbon nanotubes, and electrochemical processes at electrode-water interfaces.

Seeking definitive experimental signatures of a Weyl semimetal

Weyl and Dirac semimetals are getting quite a bit of attention. Part of this interest is because of the possible solid state realisation of the chiral anomaly from quantum field theory. One proposed signature is negative longitudinal magnetoresistance. A different, arguably more definitive, signature is in the following paper.

Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals 
Andrew C. Potter, Itamar Kimchi, and Ashvin Vishwanath

In a thin slab of material there are "Fermi arc" states on the top and bottom surfaces. When a magnetic field is applied perpendicular to the slab, there are unusual closed orbits (shown below) where an electron can move around the arc on the  top surface, tunnel via a bulk chiral state to the bottom surface, move around the arc on the bottom surface, and then tunnel back to the top surface.

The resulting Shubnikov de Haas oscillations have some unique signatures such as the periodicity and the dependence of the phase of the oscillations on the thickness of the sample.

There is a very nice set of experiments to test these ideas.

Chirality transfer dynamics in quantum orbits in the Dirac semi-metal Cd3As2 
Philip J.W. Moll, Nityan L. Nair, Tony Helm, Andrew C. Potter, Itamar Kimchi, Ashvin Vishwanath, James G. Analytis

The main finding of this study is directly evident in the raw data: while parallel [magnetic] fields lead to a single SdH frequency , an additional higher frequency component Fermi surface associated with the surface oscillations appears for fields perpendicular to the surface. This high frequency is clearly distinguishable from higher harmonics of the low frequency Fermi surface.

Focused Ion Beams were used to prepare samples with different geometries, rectangular and triangular, shown below. In the latter the oscillations associated with chirality are washed out by destructive interference due to the dependence of phase on the slab thickness.

Aside: In Figure 3 they show experimental signatures of hydrodynamic flow.

I thank James Analytis for helpful discussions about this work.

There is a also very nice theory paper.
Axial anomaly and longitudinal magnetoresistance of a generic three dimensional metal 
 Pallab Goswami, J. H. Pixley, S. Das Sarma

It is quite pedagogical, comprehensive in scope, and contains some important new insights. One particularly significant one is that one can get negative magnetoresistance without a Weyl or Dirac metal. Furthermore, in a system with a cylindrical Fermi surface, near the Yamaji angles [normally associated with semi-classical Angle Dependent Magnetoresistance Oscillations (AMRO)] one can have only one partially full Landau level, leading to negative longitudinal magnetoresistance.

Hopefully once I have digested this paper  more I will write something. I am particularly curious as to whether this theory can explain the unusual angle-dependent interlayer magnetoresistance seen in a diverse set of strongly correlated electron metals.

Emergent length scales in quantum matter

Only last week I realised that an important and profound property of emergent quantum matter is the emergence of new length scales. These can be mesoscopic - intermediate between microscopic and macroscopic length scales. Say, very roughly between 100 nanometers and 100 microns.

Previously, I highlighted the emergence of new low energy scales in quantum many-body physics. The energy and length scales are often related.
In or near broken symmetry phases, some emergent length scales can be related to the rigidity of the order parameter such as the spin stiffness.

An example is the superconducting coherence length, xi which determines the minimum thickness of a thin film required to sustain superconductivity and the size of vortices in a type II superconductor. It is roughly given by  xi ~ hbar v_F/Delta ~ a E_F/Delta
where v_F is the Fermi velocity, Delta is the energy gap, a is a lattice constant, and E_F is the Fermi energy. Since in a weak-coupling BCS superconductor Delta is much less than E_F, the length xi can be orders of magnitude larger than a lattice constant, and so is mesoscopic.
In BCS theory, the coherence length can be interpreted roughly as the "size" of a Cooper pair.

In neutral superfluids, such as liquid 4He or bosonic cold atomic gases, the corresponding length scale is sometimes known as the healing length, and defines the size of quantised vortices. In 4He this length scale is microscopic, being of the order of Angstroms, but in cold atoms it can be mesoscopic.

A second independent emergent length scale associated with superconductivity is the London penetration depth, that determines the scale on which magnetic fields penetrate the superconductor, or are expelled, i.e. the Meissner effect.

Just in case on thinks the emergence of new length scales is trivial in the sense that it is really the same as the emergence of new energy scales, consider the case of the Kondo effect. Clearly there are many experimental and theoretical signatures of the Kondo temperature, T_K.
One can easily construct a "Kondo length",  L_K ~ hbar v_F / k_B T_K, and can identify this with the size of the "Kondo screening cloud."
Yet this has never been observed experimentally.
Ian Affleck has a nice review article discussing the relevant issues.

Partly as an aside I include below a nice graphic about length scales, taken from a UK Royal Society report on nanotechnology, from 2004.

Quantum of thermal conductance

Here are a couple of things I find surprising about the electronic transport properties of materials.

1. One cannot simply have materials, particularly metals, that have any value imaginable for a transport coefficient. For example, one cannot make the conductance or the thermopower as large as one wishes by designing some fantastic material.

2. Quantum mechanics determines what these fundamental limits are. Furthermore, the limiting values of transport coefficients are often set in terms of fundamental constants [Planck's constant, Boltzmann's constant, charge on an electron].

The fact that this is profound is indicated by the fact that this was not appreciated until about 25 years ago. A nice clean example is the case of a quantum point contact with N channels. The conductance must be N times the quantum of conductance, 2e^2/h. This result was proposed by Rolf Landauer in 1957 but many people did not believe it until the first experimental confirmation in 1988.

The thermal conductance through a point contact should also be quantised. The quantum of thermal conductance is

Asides:
1. note that the Wiedemann-Franz ratio is satisfied.
2. this sets the scale for the thermal conductivity of a bad metal.

A paper in Science this week reports the experimental observation of this quantisation.

Is meso the new nano?

There is an interesting article Emergent Physics the Mesoscale: Report from the special Kavli session at the 2012 APS March meeting by Sam Bader on The Back Page of the May edition of the American Physical Society News.

It sounds like there was a fascinating and contrasting series of talks by Bob Laughlin, Bill Phillips, Angela Belcher, Bill Bialek, and George Whitesides. Apparently Phillips "gave short shrift of the concept of emergence, discarding it mercilessly."

I would be interested to see copies of the talks. Has anyone seen them online?

Overall, this latest focus on the "meso" seems to be driven by hopes of a new burst of funding like what happened with nanotechnology in 2000. [See this brief piece in Science] In the end I think that initiative was a big disappointment scientifically. I feel the whole field was hijacked by people who just relabelled whatever they were doing as nanoscience or nanotechnology. To me it should have been all about control and manipulation at the nanoscale, e.g., single molecule electronics.

Nanotechnology: from the fourth century A.D. to the Middle ages

Today there was an interesting Quantum science seminar by Ulrik L. Andersen (Technical University of Denmark) Quantum Plasmonics: Controlled Coupling of a Single Nitrogen-Vacancy Center to a Silver Nanowire.

A question came up about plasmons in gold  nano-particles and how the surface plasmon frequency is renormalised downwards (i.e. blue-shifted) compared to the frequency in the bulk. Gerard Milburn pointed out that this is illustrated by The Lycurgus Cup in the British Museum. Coincidentally, an article by Mark Stockman in this months Physics Today states:

The resonant properties of plasmonic metal nanoparticles are readily apparent to the naked eye because the excitations absorb and scatter light at optical frequencies. The most ancient example is the famous fourth-century CE Lycurgus cup from the British museum, whose glass looks green in reflected light but ruby red in transmitted light. Those colors are complementary, evidence that there is little optical loss inside the glass. Investigation has shown that the dichroic glass contains nanocrystals of a gold–silver alloy at a fraction of less than 1%.

Such colloidal suspensions of gold and silver have been widely used in stained glass since the Middle Ages. Transmission through a silver colloid yields yellow light and transmission through gold yields ruby red. The magnificent colored light from the stained glass of the Sainte Chapelle in Paris is assumed to be largely due to the nanoplasmonic resonances.

Unlike glass-staining metal ions such as iron, chromium, copper, and cobalt, metallic nanoparticles, which both absorb and scatter, transmit light with an intensity that strongly depends on the incident and viewing angles. The Sainte Chapelle dramatically exploits the effect: At sunset, the grazing-angle scattering of light by gold nanoparticles in the windows creates a pronounced red glow that appears to slowly move downward, while intensities of blue tints from ions of copper or cobalt remain the same. The artistic impression, probably intended, suggests a stream of blood slowly flowing downward. [See the photo below].

BTW: I can't see the stream of blood.