How a New Angle on the Imaging of Nanoscale Metamaterials Resulted in the Discovery of a Novel Crystal Structure

[Top left] Wiel Evers, [top right] Daniel Vanmaekelbergh, [bottom] Mark Boneschanscher

Authors:
M.P. Boneschanscher¹, W.H. Evers² and D. Vanmaekelbergh¹

Affiliation:
¹Condensed Matter and Interfaces, Debye Institute for Nanomaterials Science, Utrecht University, Netherlands.
²Opto-electronic Materials, Delft University of Technology, Netherlands.

In the field of nanotechnology, we are concerned with materials that are larger than atoms, but significantly smaller than what we call ‘bulk’ materials. As the name indicates these nanoparticles have at least one dimension in the nanometre regime. Interestingly, materials within this size regime have physical properties that are not only determined by its chemical constituents (which atoms make up the structure?) but also by their size and shape. The classical rules of physics are not applicable anymore. Instead quantum mechanics dictates how size, shape and chemical composition influence the materials’ properties like adsorption, emission, conduction and magnetism [1].

It is nice that we can design materials at the nanometre scale to have certain desired properties, but in the end we would of course like to use these materials in devices. The trick is to assemble the nanoscale materials in such a way that the engineered material properties due to the nanometre scale of the building blocks are maintained. This process has very aptly been compared to the assembly of LEGO bricks into a LEGO building: the building bricks are still identifiable within the superstructure [2]. The way this assembly is commonly achieved is by means of self-assembly: a suspension containing the nanoparticles is dried under specific circumstances, in order to let the nanoscale building blocks crystallize into a superstructure (see Figure 1). In this superstructure the properties of the nanoscale building blocks are maintained. Moreover, different building blocks can be combined into one superstructure giving rise to synergetic properties [3].

Figure 1: Formation of the nanoparticle superstructures. a, Schematic of the evaporation process during which the nanoparticle superstructures form. b, TEM image of the resulting superstructure. Even though this is a transmission image of a 3D superstructure, the different building blocks (nanoparticles) are still visible.

Since the material properties depend on the way that they can couple to each other in the superstructure, it is of major importance that the exact crystal structure of the superstructure is known. Fully resolving these superstructures however is not an easy task. The usual way this is done in the field is the comparison of transmission electron microscopy (TEM) images (2D projection information) to crystal structures known from the atomic world. This approach has a number of drawbacks, since all information along the 3rd dimension (along the electron beam) is lost. We have addressed this issue before, when we used electron tomography to study superstructures and their defects [4].

In the latest edition of Nano Letters we describe how we have used this technique once more to study a superstructure with a crystal structure not observed in nature before [5]. But let us first have a look into electron tomography. As stated before, TEM creates 2D projection images of our 3D superstructure. Using electron tomography we take multiple TEM images while rotating our sample under the electron beam. In this way we create a tilt series, e.g. a series of transmission images at different angles. Movies of these tilt series can be found here (files si_002 to si_004 are tilt series on different samples). If we now combine all transmission images along the angles under which they were obtained we can reconstruct a 3D image of our superstructure. Movies where we slice through the 3D image can be found here (files si_005 to si_007 are the 3D images corresponding to the tilt series si_002 to si_004). After this it is a question of (rather difficult) computer-aided image analysis to extract the exact coordinates of the nanomaterials making up the superstructure. See Figure 2 for an overview of this process.

Figure 2: Electron tomography process. a, transmission images at different angles are taken and combined into a tilt series. b, after back projection of these transmission images along their respective angles a 3D image reconstruction is acquired, which can then be used for computer-aided image analysis. c, the final crystal structure of the nanoparticle superstructure as extracted from the 3D reconstructed image.

Our particular interest in the structure presented in this work was raised due to the formation process behind these structures. We established in earlier work that in most cases the formation of nanoparticle superstructures can be modelled using hard spheres [6]. This hard sphere model worked very well in predicting the crystal structures of the nanoparticle superstructures formed by a combination of two differently sized nanoparticles. However, there is a certain window of size ratios between the two different nanoparticles where the hard sphere model does not predict any superstructures at all. Therefore we were quite surprised to find that actually quite some different superstructures formed when we combined nanoparticles with just such a size ratio. That is, we did observe quite some different TEM images of what appeared to be different superstructures. The TEM patterns could not be attributed to any projection of a known crystal structure.

Figure 3: Three totally different TEM pictures originating from one and the same crystal structure. a, the [PbSe]₆[CdSe]₁₉ structure imaged along the [0001] direction. b, another crystallite where there is a planar stacking fault. Half of the TEM contrast is caused by the one orientation, the other half by the other orientation. c, yet another crystallite where the crystal growth appeared with the [0001] direction tilted 43^o with respect to the TEM grid.

Therefore we took 3 of such superstructures and performed tomography on them. The TEM images of these 3 structures all appeared to be very different (Figure 3). However, using electron tomography we resolved all three and found that they, in fact, resemble the same crystal structure. This however was a crystal structure not observed in nature before, nor in any artificial crystal structure (although we found an alloy having a similar – but not the same – crystal structure). Its unit cell is made up by 6 PbSe nanocrystals and 19 CdSe nanocrystals, and has a hexagonal symmetry (no. 178 P6m2). The different TEM pictures that we observed appeared to be caused by different orientations of that particular crystal structure and by one structure with a planar defect (Figure 3). This is once more a warning to the field that what you see in TEM is not always what you get.

Finally this discovery will have some implications in the field. First, the structure itself is built from a kagomé lattice, a type of symmetry that is of high interest in the field of condensed matter for its curious spin properties. Secondly the fact that this crystal structure is found in a size regime where no crystal structure was expected implies that either the hard-sphere model is not always valid for nanoparticles, or that this particular crystal structure is overlooked in the hard sphere modelling community. And finally, we hope to have set a new standard in the research to these nanocrystal superstructures: the use of electron tomography to fully resolve structures of interest.

References:
[1] Celso de Mello Donegá, "Synthesis and properties of colloidal heteronanocrystals", Chemical Society Reviews, 40, 1512-1546 (2011). Abstract.
[2] Dmitri V. Talapin, "Is it possible to form much larger ordered nanocrystal assemblies and to transfer them to different substrates?" ACS Nano 2, 1097-1100 (2008). Abstract.
[3] D. Vanmaekelbergh, "Self-assembly of colloidal nanocrystals as route to novel classes of nanostructured materials". Nano Today 6, 419-437 (2011). Abstract.
[4] Heiner Friedrich, Cedric J. Gommes, Karin Overgaag, Johannes D. Meeldijk, Wiel H. Evers, Bart de Nijs, Mark P. Boneschanscher, Petra E. de Jongh, Arie J. Verkleij, Krijn P. de Jong, Alfons van Blaaderen and Daniel Vanmaekelbergh, "Quantitative Structural Analysis of Binary Nanocrystal, Superlattices by Electron Tomography". Nano Letters, 9, 2719-2724 (2009). Abstract.
[5] Mark P. Boneschanscher, Wiel H. Evers, Weikai Qi, Johannes D. Meeldijk, Marjolein Dijkstra, and Daniel Vanmaekelbergh, "Electron Tomography Resolves a Novel Crystal Structure in a Binary Nanocrystal Superlattice". Nano Letters, 13, 1312-1316 (2013). Abstract.
[6] Wiel H. Evers, Bart De Nijs, Laura Filion, Sonja Castillo, Marjolein Dijkstra and Daniel Vanmaekelbergh, "Entropy-Driven Formation of Binary Semiconductor-Nanocrystal Superlattices". Nano Letters, 10, 4235-4241 (2010). Abstract.

Oxide Heterostructures for Efficient Solar Cells

Elias Assmann (left) and Giorgio Sangiovanni (right)













Authors: Elias Assmann¹, Peter Blaha², Robert Laskowski², Karsten Held¹, Satoshi Okamoto³, Giorgio Sangiovanni^1,4

Affiliation: 
¹Institute of Solid State Physics, Vienna University of Technology, Austria
²Institute of Materials Chemistry, Vienna University of Technology, Austria
³Materials Science and Technology Division, Oak Ridge National Laboratory, USA
⁴Institut fuer Theoretische Physik und Astrophysik, Universitaet Wuerzburg, Germany

Sunlight is a very promising source of renewable energy, but a number of  "structural'' upper bounds limit the efficiency of solar cells in principle and in practice. The most important effect concerns the band gap of the absorber material: Those photons with energy smaller than the gap cannot be absorbed and are wasted. At the same time, relaxation effects come into play, and energy in excess of the gap is released to the lattice by the electrons and holes excited by photons. Including further losses from smaller effects like black-body radiation and radiative recombination, the efficiency of single-junction solar cells is limited to about 34%, a number known as the Shockley-Queisser limit [1,2].

However, considering the scale of solar cell production and of global photovoltaic installations [3], even small improvements on the final efficiency of the devices have a huge economical and environmental impact and are intensively pursued by industry. In particular, a big effort concentrates on the reduction of the so-called Shockley-Read-Hall electron-hole recombination due to "trap'' states provided by lattice defects and, in general, of any other intrinsic mechanism opposing electron-hole separation.

Working to achieve higher efficiencies, the community is on the one hand trying to further optimize established technologies, such as those based on crystalline silicon, but on the other hand the search for new, more efficient absorber materials goes on.

Figure 1

In a recent publication[4], we have proposed an unexplored alternative, namely to build photovoltaic cells from oxide heterostructures. Lanthanum vanadate, LaVO₃ grown on strontium titanate, SrTiO₃ (shown in Fig. 1) has a direct band gap in the optimal range for photovoltaics, and can be engineered to have an intrinsic electric field. This field can help electron-hole separation and reduce the recombination rate, one of the key problems of solar cells. A third feature is the conducting interface between LaVO₃ and SrTiO₃ and the ultra-thin conducting surface layer, providing for naturally built-in contacts for the solar cell. Furthermore, oxide heterostructures make it possible to design multijunction cells in unprecedented, extremely flexible ways.

Oxide heterostructures are artificial crystals combining different materials. The excitement about them stems from the fact that, using techniques such as pulsed laser deposition (PLD) or molecular beam epitaxy (MBE), scientists can combine a wide range of different compounds with atomic precision like Lego blocks. A ground-breaking discovery was made a decade ago by Ohtomo and Hwang [5,6]: The interface between the insulators SrTiO₃ and LaAlO₃ can be conducting. Thereafter, an intense research concentrated on layered oxide heterostructures. The aim has been not only to engender new physical effects that are absent in the constituent bulk materials, but also to design tailor-made materials and tune specific properties for desired functionalities.

As pointed out by one of us [7], if materials with partially filled d-shells are involved, this flexibility becomes even stronger and therefore potentially more useful. Electrons in d and f shells are strongly correlated, i.e. their mutual interaction effects cannot be described within the standard framework of band-structure theory. Materials containing electrons of this kind often display very pronounced responses to external perturbations, even if these are small. This makes them ideal for use in tailor-made devices. For instance, it is well known that in some bulk oxide materials, like for example V₂O₃, a tiny change in external pressure triggers a first-order transition between a metallic and a Mott insulating phase, i.e. a phase in which electrons are localized because of strong mutual repulsion.

Lanthanum vanadate, the active material in the LaVO₃|SrTiO₃ heterostructure that we consider here, is such a strongly correlated material. Our proposal rests on four key points, which we back up in the paper [4] by ab-initio calculations based on Density Functional Theory (DFT):

1) The heterostructure intrinsically develops a large electric field (about 0.3 eV per atomic layer, or in more familiar units: 8 million Volts per centimeter) which will drive negatively charged electrons and positively charged holes in opposite directions, inhibiting their recombination and the losses associated therewith.

2) The electrical contacts to extract the electrons and holes are provided by the material in a natural manner. First, the interface between SrTiO₃ and LaVO₃ becomes conducting above a certain "critical thickness'' of the (ultra-thin) LaVO₃ film. This conducting layer (or "two-dimensional electron gas, 2DEG'') will serve as a back-contact for the solar cell. Second, our calculations show that the LaVO₃ surface should similarly develop a conducting layer. However, this conducting surface has to date not been observed in experiments (this could be due to localization by surface defects, for example). If it cannot be realized, an additional metallic layer such as strontium vanadate may be an expedient.

3) LaVO₃ (in bulk or in the heterostructure) has a band gap of 1.1 eV, which is near the optimum for single-junction solar cells. In fact, this is the same value as for silicon, but silicon has a so-called indirect band gap. This means that the valence band maximum and the conduction band minimum do not occur at the same k-vector ("momentum'' of the electrons or holes). Therefore, and because photons carry very little momentum, the optical absorption is strongly suppressed (the lattice has to provide the missing momentum in the form of a phonon), and silicon solar cells have to be rather thick. On the contrary, the gap of LaVO₃ is  "direct'', and its optical absorption is large; in fact, it is large not only compared to silicon, but also to the direct-gap absorbers cadmium telluride CdTe and gallium arsenide GaAs, two industry standard materials for thin-film solar cells, see Fig. 2.

Figure 2

4) The most important way of surpassing the Shockley-Queisser limit quoted above is to combine different materials to create a "tandem'' or multijunction solar cell. As mentioned before, oxide heterostructures can be grown very flexibly; in terms of solar cells, this will allow designing a multijunction cell layer by layer.

These four points are also illustrated in Fig. 3.

Figure 3

Our theoretical study suggests that layered oxide heterostructures hold great promise as absorber materials for solar cells. We have shown that they can provide a new concept for designing efficient solar cells, including gap grading and reduction of electron-hole recombination. We have reached this conclusion using calculations from first principles.

While important theoretical work remains to be done (such as a careful analysis of the excitonic properties and correlation effects), the crucial next step will be experimental work. In order to test whether or not our proposal actually works, experiments have necessarily to be performed. Experimental colleagues from the University of Wuerzburg are already working on that, using existing and new samples of layered heterostructures. We are looking forward to the first results of these experiments.

References:
[1] William Shockley and Hans J. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells". Journal of Applied Physics, 32, 510 (1961). Abstract.
[2] Albert Polman & Harry A. Atwater. "Photonic design principles for ultrahigh-efficiency photovoltaics". Nature Materials, 11, 174-177 (2012). Abstract.
[3] EPIA report, "Global Market Outlook for Photovoltaics until 2016". Full Report.
[4] Elias Assmann, Peter Blaha, Robert Laskowski, Karsten Held, Satoshi Okamoto and Giorgio Sangiovanni, "Oxide Heterostructures for Efficient Solar Cells". Physical Review Letters, 110, 078701 (2013). Abstract.
[5] A. Ohtomo, D. A. Muller, J. L. Grazul and H. Y. Hwang, "Artificial charge-modulation in atomic-scale perovskite titanate superlattices". Nature, 419, 378-380 (2002). Abstract.
[6] A. Ohtomo and H. Y. Hwang, "A high-mobility electron gas at the LaAlO₃|SrTiO₃ heterointerface". Nature 427, 423-426 (2004). Abstract.
[7] Satoshi Okamoto & Andrew J. Millis, "Electronic reconstruction at an interface between a Mott insulator and a band insulator". Nature 428, 630-633 (2004). Abstract.
[8] Naoyuki Nakagawa, Harold Y. Hwang & David A. Muller, "Why some interfaces cannot be sharp". Nature Materials, 5, 204-209 (2006). Abstract.
[9] Guneeta Singh-Bhalla, Christopher Bell, Jayakanth Ravichandran, Wolter Siemons, Yasuyuki Hikita, Sayeef Salahuddin, Arthur F. Hebard, Harold Y. Hwang & Ramamoorthy Ramesh, "Built-in and induced polarization across LaAlO₃|SrTiO₃ heterojunctions". Nature Physics, 7, 80-86 (2011). Abstract.
[10] Y. Hotta, Y. Susaki and H. Y. Hwang, "Polar Discontinuity Doping of the LaVO₃|SrTiO₃ Interface". Physical Review Letters, 99, 236805 (2007). Abstract.

A Happy Accident and Subsequent Insight: A New X-Ray Imaging Technique Yields Unprecedented View of Nanoworld

Physicists Kevin Yager (left) and Ben Ocko reviewing their paper on the cover of the Journal of Applied Crystallography [Photo Courtesy: Brookhaven National Laboratory]

Photographers rely on precision lenses to generate well-focused and crystal-clear images. These high-quality optics—readily available and produced in huge quantities—are often taken for granted. But as scientists explore the details of materials spanning just billionths of a meter, engineering the nanoscale equivalent of a camera lens becomes notoriously difficult.

Xinhui Lu, lead author of the GTSAXS study [Photo Courtesy: Brookhaven National Laboratory]

Instead of working with polished glass, physicists must use ingenious tricks, including shooting concentrated beams of x-rays directly into materials. These samples then act as light-bending lenses, and the x-ray deflections can be used to deduce the material's nanostructures. Unfortunately, the multilayered internal structures of real materials bend light in extremely complex and unexpected ways. When scientists grapple with this kind of warped imagery, they use elaborate computer calculations to correct for the optical obstacles found on the nanoscale and create detailed visual models.

Now, owing to a happy accident and subsequent insight, researchers at the US Department of Energy's (DOE) Brookhaven National Laboratory have developed a new and strikingly simple x-ray scattering technique—detailed in their paper in the Journal of Applied Crystallography — to help draw nanomaterials ranging from catalysts to proteins into greater focus.

"During an experiment, we noticed that one of the samples was misaligned," said physicist Kevin Yager, a coauthor on the new study. "Our x-ray beam was hitting the edge, not the center as is typically desired. But when we saw how clean and undistorted the data was, we immediately realized that this could be a huge advantage in measuring nanostructures."

This serendipitous discovery at Brookhaven's National Synchrotron Light Source (NSLS) led to the development of a breakthrough imaging technique called Grazing-Transmission Small Angle X-ray Scattering (GTSAXS). The new method requires considerably less correction and a much simpler analysis, resulting in superior images with profound implications for future advances in materials science.

"Conventional scattering produces images that are 'distorted'—the data you want is there, but it's stretched, compressed, and multiply scattered in complicated ways as the x-rays enter and exit the sample," said physicist and coauthor Ben Ocko. "Our insight was that undistorted scattering rays were emitted inside the sample—but they usually get absorbed as they travel through the substrate. By moving the sample and beam near the edge of the substrate, we allow this undistorted scattering to escape and reach the detector."

The Brookhaven Lab collaboration was not the first group to encounter the diffraction that occurs along a material's edge, but it was the first to reconsider and harness the unexpected error.

This rendering shows the high-intensity x-ray beam striking and then traveling through the gray sample material. In this new technique, the x-ray scattering—the blue and white ripples—is considerably less distorted than in other methods, producing superior images with less complex analysis [Image Courtesy: Brookhaven National Laboratory].

"Until now, no one bothered to dig into the details, and figure out how to use it as a measurement technique, rather than as a misalignment to be corrected," added Xinhui Lu, the lead author of the study.

GTSAXS, like other scattering techniques, offers a complement to other imaging processes because it can measure the average structure throughout a sample, rather than just pinpointing selected areas. Scattering also offers an ideal method for the real-time studies of nanoscale changes and reactions such as the propagation of water through soft nanomaterials.

"This technique is broadly applicable to any nanostructure sitting on a flat substrate," said study coauthor Chuck Black. "Lithographic patterns, catalytic nanoparticles, self-assembled polymers, etc.—they can all be studied. This technique should be particularly powerful for very thin films with complicated three-dimensional structures, which to date have been difficult to study."

Brookhaven's NSLS supplies the intense x-ray beams essential to this technique, which requires extremely short wavelengths to interact with nanoscale materials. At NSLS, accelerated electrons emit these high-energy photons, which are then channeled down a beamline and focused to precisely strike the target material. When the next generation light source, NSLS-II, opens in 2014, GTSAXS will offer even greater experimental potential.

"We look forward to implementing this technique at NSLS-II," Yager said, with Ocko adding: "The excellent beam focusing should enable us to probe the near-edge region more effectively, making GTSAXS even more robust."

Reference:
[1] Xinhui Lu, Kevin G. Yager, Danvers Johnston, Charles T. Black and Benjamin M. Ocko, "Grazing-incidence transmission X-ray scattering: surface scattering in the Born approximation", Journal of Applied Crystallography, 46, 165-172 (2013). Abstract.

[This report is written by Justin Eure of Brookhaven National Laboratory, USA]

Nanostructuring Improves Vortex Pinning in Superconductors at Elevated Temperatures and Magnetic Fields

Photos of all authors -- ordered as the author list below, from top left to bottom right.

Authors:
R. Córdoba^1,2, T. I. Baturina^3,4, J. Sesé^1,2, A. Yu. Mironov³, J. M. De Teresa^2,5, M. R. Ibarra^1,2,5, D. A. Nasimov³, A. K. Gutakovskii³, A.V. Latyshev³, I. Guillamón^6,7, H. Suderow⁶, S.Vieira⁶, M. R. Baklanov⁸, J. J. Palacios⁹ & V.M.Vinokur⁴

Affiliation:
¹Instituto de Nanociencia de Aragón, Universidad de Zaragoza, Spain
²Departamento de Física de la Materia Condensada, Universidad de Zaragoza, Spain
³A.V. Rzhanov Institute of Semiconductor Physics SB RAS, Novosibirsk, Russia
⁴Materials Science Division, Argonne National Laboratory, Illinois, USA
⁵Instituto de Ciencia de Materiales de Aragón, Universidad de Zaragoza-CSIC, Facultad de Ciencias, Spain
⁶Laboratorio de Bajas Temperaturas, Departamento de F´ısica de la Materia Condensada, Instituto de Ciencia de Materiales Nicol´as Cabrera, Facultad de Ciencias, Universidad Autónoma de Madrid, Spain
⁷H.H. Wills Physics Laboratory, University of Bristol, United Kingdom
⁸IMEC, Leuven, Belgium
⁹Departamento de Física de la Materia Condensada, Instituto de Ciencia de Materiales Nicolás Cabrera, Facultad de Ciencias, Universidad Autónoma de Madrid, Spain

Corresponding author: Hermann.suderow@uam.es

A recent collaboration of the US, Russian and Spanish researchers finds a new method to improve current carrying capability of superconductors. Usually, superconducting vortices induced by the magnetic field move under the applied current and dissipate the energy degrading thus the ability of superconductors to carry electrical current with zero resistance. To recover superconductivity one has to pin vortices down stopping their motion[1]. However all pinning mechanisms known so far become inefficient at technologically important temperatures and magnetic fields, and this constitutes the major problem restricting applications of superconductivity[2,3,4]. The international team demonstrates the method to immobilize vortices at elevated temperatures and magnetic fields, reversing the deleterious effect of vortex motion as the applied magnetic field is increased[5].

Figure 1: Resistance as a function of the magnetic field in perforated nanostructures.

To achieve this, the authors have carved patterns in superconductors using advanced nanofabrication tools. They have revealed geometrical structures, which impede vortex motion just when it is most harmful for applications, at high magnetic fields and temperatures. The work provides a new avenue for research on blocking vortex motion using nano-patterns[7,8]. The science involved brings new concepts to light: vortices confined on a row dig for themselves a deep potential well which suppresses their capability to move. Being tightly squeezed together vortices join into large clusters so that even the combined action of temperature and current fails to destroy them and move vortices. The result is truly surprising: the resistance drops down when increasing the magnetic field, even if temperature is high and close to the critical one, and remains zero over a broad range. It is exactly opposite to what the conventional wisdom in superconductors would have expected.

Figure 2: Magnetic field dependence of the resistance in a nanowire with a single vortex row.

The to-do list of researchers includes now imaging these immobile clusters and developing a quantitative theory of the effect in order to achieve complete understanding and fully utilize the potential technological promise of their discovery. One of the directions of the future work is the extension of the novel approach to pinning to other materials including high-temperature superconductors[4], where nanopatterning is expected to bring a dramatic improvement of their performance[8]. For example, while many researchers are optimistic about synthesizing the room temperature superconductors, they remain skeptical about their usefulness for applications, since at elevated temperatures mobile vortices would anyway destroy the ability of superconductors to carry current without resistance. The novel approach developed by the team promises to meet this challenge of pinning vortices at high temperatures thus breaking ground for ‘quantum leap’ of superconducting materials into industrial and technological applications.

Figure 3: Perforated superconducting thin film.

References:
[1] P.W. Anderson & Y.B. Kim. "Hard superconductivity-theory of motion of Abrikosov flux lines". Review of Modern Physics, 36, 39-43 (1964). Abstract.
[2] V.V. Moshchalkov, R. Wördenweber and W. Lang, "Nanoscience and engineering in superconductivity" [Springer, ISBN: 9783642151361, 2010].
[3] A.M. Campbell & J.E. Ivetts, "Critical Currents in Superconductors - Monographs on Physics" [Taylor & Francis Ltd., London, 1972].
[4] David Larbalestier, Alex Gurevich, D. Matthew Feldmann & Anatoly Polyanskii. "High-Tc superconducting materials for electric power applications". Nature 414, 368-377 (2001). Abstract.
[5] R. Córdoba, T.I. Baturina, J. Sesé, A. Yu. Mironov, J.M. De Teresa, M.R. Ibarra, D.A. Nasimov, A.K. Gutakovskii, A.V. Latyshev, I. Guillamón, H. Suderow, S. Vieira, M.R. Baklanov, J.J. Palacios and V.M. Vinokur. "Magnetic field-induced dissipation-free state in superconducting nanostructures". Nature Communications, 4, 1437 (2013). Abstract.
[6] M. Baert, V. V. Metlushko, R. Jonckheere, V. V. Moshchalkov, and Y. Bruynseraede. "Composite flux-line lattices stabilized in superconducting films by a regular array of artificial defects". Physical Review Letters, 74, 3269-3272 (1995). Abstract.
[7] J. I. Martín, M. Vélez, A. Hoffmann, Ivan K. Schuller, J. L. Vicent, "Temperature dependence and mechanisms of vortex pinning by periodic arrays of Ni dots in Nb films". Physical Review B, 62, 9110-9116 (2000). Abstract.
[8] A. Llordés, A. Palau, J. Gázquez, M. Coll, R. Vlad, A. Pomar, J. Arbiol, R. Guzmán, S. Ye, V. Rouco, F. Sandiumenge, S. Ricart, T. Puig, M. Varela, D. Chateigner, J. Vanacken, J. Gutiérrez, V. Moshchalkov, G. Deutscher, C. Magen and X. Obradors. "Nanoscale strain‐induced pair suppression as a vortexpinning mechanism in high‐temperature superconductors". Nature Materials, 11, 329 (2012). Abstract.

Quantum Ratchet in Graphene: One-way Electron Traffic at Atomic Scale

S.D. Ganichev (Left) and S.A. Tarasenko (Right)

Authors: 
S.D. Ganichev¹ and S.A. Tarasenko²

Affiliation:
¹Terahertz Center, University of Regensburg, Germany
²Ioffe Physical-Technical Institute, St. Petersburg, Russia

A mechanical or electronic system driven by alternating force can exhibit a directed motion facilitated by thermal or quantum fluctuations. Such a ratchet effect occurs in systems with broken spatial inversion symmetry. Canonical examples are the ratchet-and-pawl mechanisms in watches, electric current rectifying diodes and transistors in electronics, and Brownian molecular motors in biology [1,2]. The ratchet suggests one-way traffic. One can pull it back and forth, but it moves predominantly in a certain direction. Therefore, the effect has fascinating ramifications in engineering and natural sciences.

Now an international consortium consisting of research groups from Germany, Russia, Sweden, and the U.S. has demonstrated that electronic ratchets can be successfully scaled down to one-atom thick layers [3]. Specifically, it has been shown that graphene layers support a ratchet motion of electrons when placed in a static magnetic field. The ac electric field of terahertz radiation [4] was applied to push the Dirac electrons back and forth, while the magnetic field acted as a valve letting the electrons move in one direction and suppressing the oppositely directed motion. The resulting magnetic quantum ratchet transforms the ac power into a dc current, extracting work from the out-of-equilibrium Dirac electrons driven by undirected periodic forces.

Graphene, a one-atom-thick layer of carbon with a honeycomb crystal lattice [5], is usually threated as a spatially symmetric structures, as far as its electric or optical properties are concerned. Driven by a periodic electric field, no directed electric current can be expected to flow. However, if the space inversion symmetry of the structure is broken due to the substrate or chemisorbed adatoms on the surface, an electronic ratchet motion can arise. In Ref.[3], we and our colleagues report on the observation and experimental and theoretical study of quantum ratchet effects in single-layer graphene samples, proving and quantifying the underlying spatial asymmetry.

Figure 1: Alternating electric field drives a ratchet current in graphene.

The physics behind the magnetic quantum ratchet effect in graphene is illustrated in Fig. 1. The alternating electric field E(t) drives Dirac electrons back and forth in the graphene plane. Due to the Lorentz force, the applied static magnetic field B deforms the electron orbitals such that the right-moving electrons have their centre of gravity shifted upwards, while the left-moving electrons are shifted downwards. (In quantum mechanical consideration, the shift is caused by the magnetic-field-induced coupling between σ- and π-band states). For spatial symmetric systems the net dc current would vanish. However, in a graphene layer with spatial asymmetry, e.g., caused by top adsorbates, the electrons shifted upwards feel more disorder and exhibit a lower mobility than the electrons shifted downwards and moving in the opposite direction. This difference in the effective mobility for the right- and left-moving carriers results in a net dc current. The current scales linearly with the magnetic field, changes a sign by switching the magnetic field polarity, and proportional to the square of the amplitude of the ac electric field. The linear dependence on B comes from the Lorentz force. The electric field appears twice: on the one hand, it causes the oscillating motion of carriers in the plane, and on the other, the Lorentz force itself is proportional to the electron velocity.

The ratchet motion implies that the particle flow depends on the orientation of the ac force with respect to the direction of built-in spatial asymmetry. In the case of magnetic quantum ratchets, where the asymmetry stems from the magnetic field, the relevant parameter is the angle β between the ac electric field E(t) and the static magnetic field B. Shown in Fig. 2 is the measured dependence of the dc current on the angle β. The current reaches a maximum for the perpendicular electric and magnetic fields and remains finite for the co-linear fields. The whole angular dependence is well described by the equation

j_x(β) = j₁ Cos(2β) + j₂

with two contributions j₁ and j₂, which is in agreement with the developed theory [see Ref.3]. It has also been shown that the ratchet transport can be induced by a force rotating in space. By exciting the graphene samples with a clockwise or counterclockwise rotating in-plane electric field E(t), the dc current is detected. Interestingly, the current measured along the static magnetic field turns out to be sensitive to the radiation helicity being of the opposite sign for the clockwise and counterclockwise rotating fields.

Figure 2: Dependence of the ratchet current on the orientation of ac electric field. Experimental data (dots) are obtained for an epitaxial graphene on SiC at temperature 115K, magnetic field 7T, and electric field amplitude 10 kV/cm. Solid line is a theoretical fit.

Graphene may be the ultimate electronic material, possibly replacing silicon in electronic devices in the future. It has attracted worldwide attention from physicists, chemists, and engineers. The discovery of the ratchet motion in this purest possible two-dimensional system known in nature indicates that the orbital effects may appear and be substantial in other two-dimensional crystals such as boron nitride, molybdenum dichalcogenides and related heterostructures. The measurable orbital effects in the presence of an in-plane magnetic field provide strong evidence for the existence of structure inversion asymmetry in graphene.

References:
[1] R. P. Feynman, R. B. Leighton, and M. Sands, "The Feynman Lectures on Physics, Vol. 1" (Addison-Wesley, 1966).
[2] Peter Hänggi and Fabio Marchesoni, "Artificial Brownian motors: controlling transport on the nanoscale", Review of Modern Physics, 81, 387 (2009). Abstract.
[3] C. Drexler, S. Tarasenko, P. Olbrich, J. Karch, M. Hirmer, F. Müller, M. Gmitra, J. Fabian, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Wang, R. Vajtai, P. Ajayan, J. Kono, and S.D. Ganichev: "Magnetic quantum ratchet effect in graphene", Nature Nanotechnology 8, 104 (2013). Abstract.
[4] S.D. Ganichev and W. Prettl, "Intense Terahertz Excitation of Semiconductors" (Oxford Univ. Press, 2006).
[5] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, "The electronic properties of graphene". Review of Modern Physics, 81, 109 (2009). Abstract.

Surmounting The Great Silica Integration Problem

The Team [From Left to Right]: 
Masood Naqshbandi, John Canning, Brant Gibson, Melissa Nash, Maxwell Crossley

Authors: John Canning¹, Brant Gibson²

Affiliation:
¹interdisciplinary Photonics Laboratory (iPL), School of Chemistry, The University of Sydney, Australia,
²School of Physics, The University of Melbourne, Australia

The global interconnectedness we all share today, primarily via the optical internet, comes from one material: silica. It makes up the global web that carries communication to most corners containing bipeds. Only silica, with a sprinkling of germanium and a few trace dopants, has offered both an extraordinary toughness, chemical inertness and a super-extraordinary transparency that enables this to happen, aided by silicate-based amplifiers lightly doped with erbium to kick these signals along. So it’s no wonder a Nobel Prize was awarded to acknowledge this impact, literally the backbone of the Glass Age [1]. And its existence continues to fire the imagination and yearning for something even greater – an intelligent web that can emulate a living organism with sensors along its paths providing data on the environment and a distributed intelligence analysing, sifting and even making decisions based on this data [2].

These sensors promise to span remote intelligent energy monitoring, machinery control, remote telemedicine and teaching and more. But in order to do this it has become clear that the very properties that enabled the global village – robustness, stability and reliability – are the same that limits the transformation of the optical web from a passive transport medium to an active, dynamic intelligent network. For this, silica must have added functionality at least in places along the grid. For the time being, all solutions explore externalisation onto alternative material platforms, the majority of which are simply incompatible making integration with the web one of the most prolific research areas across all disciplines.

This begs the question: why cannot silica be functionalised? The answer comes down to simple thermodynamics involved with the traditional chemical bond – silica processing involves extraordinarily high temperatures: 1900 °C plus before it can be melted and drawn into fibre. Very little material can survive these conditions – some of the rare earths are fortunately, enabling critical amplification to exist; but organic or carbon systems, which increasingly underpin sensors and new technologies such as diamond based photonics [3], are not.

At the interdisciplinary Photonics Laboratories at The University of Sydney, Australia, we believe we have come up with a solution: using nanoparticles held together by intermolecular forces [4]. Unlike chemical bonds, intermolecular forces are universal with most cases being attractive at the same temperature, namely 25 °C. To exploit and demonstrate the potential of this approach, we take a novel fabrication avenue that nearly everyone has some experience with, albeit often without realising it – evaporative self-assembly.

Figure 1: A batch of self-assembled wires approximately 10 μm by 5 cm long fabricated from 20 nm silica nanoparticles on a glass substrate. An aspect ratio > 50 000 is easily demonstrated.

Watching a coffee drop evaporate [5] can be like sticking needles into one’s eyes but if you watch carefully enough you’ll see a relatively wonderful example of physics in action – convective flow directing particles to the outer rim so that the brown spot becomes clear in the middle. When intermolecular forces are thrown in by replacing the coffee with silica nanoparticles, packing constraints take place and the steadily shrinking drop experiences very high radial stresses – cracks form, and after a couple of bifurcations, uniform cracking is obtained to produce silica wires (Figure 1). In further work, by controlling the evaporation conditions using laser processing, a very high degree of directionality is possible improving uniformity of wires and more [6]. Intermolecular forces are often seen as weak and at the molecular level this is often the case – but unlike the chemical bond they are additive so the more of it there is the stronger a material.

Since these forces are universal and operate at room temperature – this means we can mix in almost anything and have done so using organic dyes to dope the wires during their fabrication. Importantly, in collaboration with the School of Physics at the University of Melbourne, mixed nano-particle self-assembly has allowed a wire to be fabricated containing nanodiamonds. Some of these nanodiamonds themselves have nitrogen vacancy defect sites which emit single photons (Figure 2). With little blinking observed and good thermal stability, diamond is one of the most ideally suited material systems for single photon generation. We have now been able to integrate this into silica itself, clearly demonstrating the potential of our approach for enabling the tools for quantum techniques into the global web.

Figure 2. (a) A scanning confocal map of the photoluminescence from nitrogen-vacancy (NV) defect centres within the nanodiamond-embedded silica microwire sample (image is taken from the top surface of the microwire; scale bar corresponds to 10 μm). (b) Single photon emission detected from the particle shown in the zoomed region of (a) where the scale bar corresponds to 2 μm. The inset shows the photostable emission from the single emitter.

The silica nanoparticle platform, held together by intermolecular forces, allows total integration of new materials into existing silica communications and sensor networks. This hybrid material has the potential to open up a vast field for compositional control of other organic, inorganic and biological molecules and species within silica waveguides (or any other nanoparticle platform) for applications in opto-electronics (for example, graphite), photovoltaics (for example, customised porphyrins and metals), plasmonics and metamaterials (for example, metals) and novel optical circuitry (for example, magnetic materials).

References:
[1] 2009 Nobel Prize in Physics, Charles Kao. Link to Nobel Prize 2009.
[2] J. Canning, “Optical sensing: the last frontier for enabling intelligence in our wired up world and beyond”, Photonic Sensors, SpringerOpen, 2 (3), 193-202, (2012). Abstract.
[3] Mark P. Hiscocks, Kumaravelu Ganesan, Brant C. Gibson, Shane T. Huntington, François Ladouceur, and Steven Prawer, “Diamond waveguides fabricated by reactive ion etching”, Optics Express, 16, 19512-19519 (2008). Abstract.
[4] Masood Naqshbandi, John Canning, Brant C. Gibson, Melissa M. Nash & Maxwell J. Crossley, “Room temperature self-assembly of mixed nanoparticles into photonic structures”, Nature Communications, 3, 1188 (2012). Abstract.
[5] Robert D. Deegan, Olgica Bakajin, Todd F. Dupont, Greb Huber, Sidney R. Nagel and Thomas A. Witten, “Capillary flow as the cause of ring stain from dried liquid drops”, Nature 389, 827–829 (1997). Abstract.
[6] J. Canning, H. Weil, M. Naqshbandi, K. Cook, and M. Lancry, “Laser tailoring surface interactions, contact angles, drop topologies and the self-assembly of optical microwires”, To appear in Opt. Mat. Express (2013).

Evidence of Majorana States in an Al Superconductor – InAs Nanowire Device

[From left to right] Moty Heiblum, Yuval Oreg, Anindya Das, Yonathan Most, Hadas Shtrikman, Yuval Ronen

Authors: Yuval Ronen, Anindya Das, Yonatan Most, Yuval Oreg, Moty Heiblum, and Hadas Shtrikman

Affiliation: Dept. of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel

When a bridge between fields in physics is created, exciting physics can emerge. In 1962 Anderson walked on a bridge connecting condensed matter physics with particle physics, by introducing the Anderson mechanism in superconductivity to explain the Meissner effect. A similar idea was used on the other side of the bridge by Higgs in 1964, to explain the mechanism that generates the mass of elementary particles known as the Higgs mechanism. Nowadays, another bridge is formed between these two fields emanating from an idea first originated by Ettore Majorana in 1937 – where spin 1/2 particles can be their own anti-particles[1]. Back then, Majorana suggested the neutrino as a possible candidate for his prediction, and experiments such as double-beta decay are planned to test his prediction.

A link between Majorana’s prediction of new elementary particles and the field of condensed matter physics was formed already more than a decade ago. Quasi-particle excitations, which are equal to their anti-quasi-particle excitations, are predicted to be found in the solid. Specifically, in vortices that live in an esoteric two-dimensional P-wave spinless superconductor. Moreover, these excitations are expected to be inherently different from their cousins the elementary particles: they have non-abelian statistics. The non-abelian statistics is one of the beautiful triumphs of the physics of condensed matter.

This so far unobserved quasi-particle, that has non-abelian statistics, has for a while been a ‘holy grail’ in the fractional quantum Hall effect regime; with filling factor 5/2 being the most promising candidate for its observation. Lately, another realization of Majorana quasi-particles is pursued. It follows a 1D toy model presented by Kitaev in 2001, showing how one can isolate two Majorana states at two widely separated ends of a 1D P-wave spinless superconductor [2]. These two Majorana states are expected to sit in the gap of the superconductor (at the Fermi energy) for a wide range of system parameters. Seven years later, Fu and Kane [3] found that a P-wave spinless superconductor can be induced by an S-wave superconductor in proximity to a topological insulator, occurring in a semiconductor with an inversion gap. It was thus not long before two theoretical groups [4,5] provided a prescription for how to turn a 1D semiconductor nanowire into an effective Kitaev 1D spinless P-wave superconductor.

The prescribed system is a semiconductor nanowire, with strong spin-orbit coupling, coupled to an S-wave superconductor (a trivial superconductor, with Cooper pairs in a singlet state). Electrons from the semiconductor undergo Andreev reflections, a process which induces S-wave superconductivity in the nanowire. The induced superconductivity opens gaps in the nanowire spectrum around the Fermi energy, at momentums k=0 and k=k_F (the Fermi momentum), due to the two spin bands being separated by spin-orbit coupling. An applied magnetic field quenches the gap at k=0 while hardly affecting the gap at k_F (the Zeeman splitting competes with superconductivity at k=0, where spin-orbit coupling, being proportional to k, plays no role), creating an effective gap different from the one induced by superconductivity. A gate voltage is used to tune the chemical potential into the effective gap. When the Zeeman energy is equal to the induced superconducting gap, the effective gap at k=0 closes; it then reopens upon further increase of the magnetic field, bringing the nanowire into a so called ‘topological phase’. Kitaev’s original toy model of a 1D P-wave superconductor is then implemented (Fig. 1).

Figure 1: Energy dispersion of the InAs nanowire excitations (Bogoliubov-de Gennes spectrum), in proximity to the Al superconductor. Heavy lines show electron-like bands and light lines show hole-like bands. Opposite spin directions are denoted in blue and magenta (red and cyan) for the spin-orbit effective field direction (perpendicular direction), where a relative mixture denotes intermediate spin directions. (a) Split electronic spin bands due to spin-orbit coupling in the InAs wire. Spin-orbit energy defined as Δ_so, with the chemical potential μ measured with respect to the spin bands crossing at p=0. (b) With the application of magnetic field, B, perpendicular to the spin-orbit effective magnetic field, B_so a Zeeman gap, E_z= gμ_BB/2, opens at p=0. (c) Light curves for the hole excitations are added, and bringing into close proximity a superconductor opens up superconducting gaps at the crossing of particle and hole curves. The overall gap is determined by the minimum between the gap at p=0 and the gap at p_F, while for μ=0 and E_z close to Δ_ind the gap at p=0 is dominant. (d) As in (c) but E_z is increased so that the gap at p_F is dominant. (e) B is rotated to a direction of 30^o with respect to B_so. The original spin-orbit bands are shifted in opposite vertical directions, and the B component, which is perpendicular to B_so is diminished. (f) The evolution of the energy gap at p=0 (dotted blue), at p_F (dotted yellow), and the overall energy gap (dashed black) with Zeeman energy, E_z, for μ=0. The overall gap is determined by the minimum of the other two, where the p=0 gap is dominant around the phase transition, which occurs at E_z=Δ_ind. At high E_z the p_F gap, which is decreasing with E_z, becomes dominant.

Seventy five years after Majorana’s monumental paper, we may be close to a realization of a quasi-particle that is identical to its anti-quasi-particle, possessing non-abelian statistics. Several experimental groups [6,7,8] follow the prescribed recipe for a 1D P-wave spinless superconductor[4,5], with our group being one of them. A zero energy conductance peak, at a finite Zeeman field, had been seen now in InSb and InAs nanowires in proximity to Nb and Al superconductors, respectively. This peak is considered a signature for the existence of a Majorana quasi-particle, since the Majorana resides at the Fermi energy.

Figure 2: Structure of the Al-InAs structures suspended above p-type silicon covered with 150nm SiO₂. (a) Type I device, the nanowire is supported by three gold pedestals, with a gold ‘normal’ contact at one edge and an aluminum superconducting contact at the center. The conductive Si substrate serves as a global gate (GG), controlling barrier as well as the chemical potential of the nanowire. Two narrow local gates (RG and LG), 50nm wide and 25nm thick, displaced from the superconducting contact by 80nm, also strongly influence the barrier height as well as the chemical potential in the wire. (b) Type II device, similar to type I device, but without the pedestal under the Al superconducting contact. This structure allows control of the chemical potential under the Al contact. (c) SEM micrograph of type II device. A voltage source, with 5 Ohm resistance, provides V_SD, and closes the circuit through the ‘cold ground’ (cold finger) in the dilution refrigerator. Gates are tuned by V_GG and V_RG to the desired conditions. Inset: High resolution TEM image (viewed from the <1120> zone axis) of a stacking faults free, wurtzite structure, InAs nanowire, grown on (011) InAs in the <111> direction. TEM image is courtesy of Ronit Popovitz-Biro. (d) An estimated potential profile along the wire. The two local gates (LG and RG) and global gate (GG) determine the shape of the potential barriers; probably affect the distance between the Majoranas.

Our work, with MBE grown InAs nanowire in proximity to an Al superconductor [8] (Fig 2), demonstrated a zero bias peak and several more interesting features in the parameters' space. First, the closing of the gap at k=0 was clearly visible when the Zeeman energy was equal to the induced gap. Second, splitting of the zero-bias-peak was observed at low and high Zeeman field; likely to result from spatial coupling of the two Majorana states. Third, the zero-bias-peak was found to be robust in a wide range of chemical potential (assumed to be within the k=0 gap). While these observations agree with the presence of a Majorana quasi-particle (though the peak height is much smaller than expected, maybe due to the finite temperature of the experiment), the available data does not exclude other effects that may result with a similar zero bias peak (such as, interference, disorder, multi-bands, Kondo correlation).

Quoting Wilczek: “Whatever the fate of these particular explorations, there is no doubt that Majorana's central idea, which long seemed peripheral, has secured a place at the core of theoretical physics"[9].

References:
[1] Ettore Majorana, "Teoria simmetrica dell’elettrone e del positrone", Il Nuovo Cimento, 171 (1937). Abstract.
[2] A. Yu. Kitaev, "Unpaired Majorana fermions in quantum wires". Physics Uspekhi, 44, 131 (2001). Full Article.
[3] Liang Fu and Charles Kane, "Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator", Physical Review Letters 100, 096407 (2008). Abstract.
[4] Roman M. Lutchyn, Jay D. Sau and Sankar Das Sarma, "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures", Physical Review Letters, 105, 077001 (2010). Abstract.
[5] Yuval Oreg, Gil Refael, and Felix von Oppen, "Helical Liquids and Majorana Bound States in Quantum Wires", Physical Review Letters, 105, 177002 (2010). Abstract.
[6] V. Mourik, K. Zuo, S.M. Frolov, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, "Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices", Science 336, 1003 (2012). Abstract. 2Physics Article.
[7] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, H. Q. Xu, "Observation of Majorana Fermions in a Nb-InSb Nanowire-Nb Hybrid Quantum Device", arXiv: 1204.4130 (2012).
[8] Anindya Das, Yuval Ronen, Yonathan Most, Yuval Oreg, Hadas Shtrikman, Moty Heiblum, "Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions", Nature Physics, 8, 887–895 (2012). Abstract.
[9] Frank Wilczek, "Majorana Returns", Nature Physics, 5, 614 (2009). Abstract.

Inspiration from Nature: Ultralight Fractal Designs for High Mechanical Efficiency

[From Left to Right] Daniel Rayneau-Kirkhope, Robert Farr, Yong Mao



Authors: Daniel Rayneau-Kirkhope¹, Robert Farr^2,3, Yong Mao⁴

Affiliation:
¹Open Innovation House, School of Science, Aalto University, Finland,
²Unilever R&D, Colworth House, Sharnbrook, Bedford, UK
³London Institute for Mathematical Sciences, Mayfair, London, UK
⁴School of Physics and Astronomy, University of Nottingham, UK

Hierarchical design is ubiquitous in nature [1]. Material properties can be tailored by having structural features on many length scales. The gecko, a lizard ranging from 2 to 60 cm in length, has a remarkable ability to walk on vertical walls and even upside-down on ceilings. This ability is brought about through the repeated splitting of the keratinous fibres on the bottom of the gecko’s foot, which increases the contact area so effectively that even the very weak van der Waals interactions can support the entire weight of the gecko [2].

A more specific form of hierarchical design is self-similar design, where one structural feature is found to be repeated on a number of different length scales. A natural example is the trabecular or spongy bone found around the joints in animals [3]. Here, a series of small beams are arranged in such a way that the stiffness and strength requirements are met while using minimal material. Regardless of the level of magnification, the same patterns are found in the structure. Interestingly, the exact configuration of the constituent beams in the trabecular bone is constantly changing: it is the result of a continuous opitimisation process that goes on throughout the lifetime of the bone and responds to change in stress levels [4]. It is found that when the animal’s bones support only small loads, many very slender pillars are present, and when the loading increases, fewer but stouter pillars are employed [5].

In our recently published work [6], we demonstrate that through the use of hierarchical, self-similar design principles, advantageous structural properties can be obtained. We show that the scaling of the amount of material required for stability against the loading can be altered in a systematic manner. A particular structure is fabricated through rapid prototyping, and we obtain the optimal generation number (for our specific structure) for any given value of loading.

Scaling

The volume of material required for stability can be related to the loading through a simple power law relationship. That is, the volume required is given by a dimensionless loading parameter raised to some power with a pre-factor (that is dependent only on material properties and specifics of the geometry). When the loading is small, it is the scaling (power) of the loading that dominates the relationship. Under tension, this power is one and a structure requires an amount of material that is proportional to the loading it must withstand; for a solid beam under compression, due to elastic buckling, the power is one-half. Given, for all realistic applications, the non-dimensional loading parameter is much less than 1, this means more material is required to support compressive than tensional loads. This one-half power law has direct consequences when one considers optimal structure: if a beam is bearing a compressive load, it is more efficient to use one beam rather than two, whereas in the case of tension, due to the linear relationship, splitting a tension member into more than one piece has no effect on the volume required for stability.

Fractal design

Our work centers on a very simple, iterative procedure that can be used to create designs of great complexity. The “generation” of a structure describes the number of iterations used to create the geometry. The simplest compression bearing structure is a solid slender beam. When loaded with a gradually increasing force, the beam will eventually buckle into a sinusoidal shape known as an “Euler buckling mode”. We can suppress this by using a hollow tube, but we introduce a second mode of a local failure of the tube wall – Koiter buckling. After optimizing for tube diameter and thickness, it is found that the scaling power increases to two-thirds, and the volume of material required for stability is reduced.

Figure 1: Showing the iterative process from low generation numbers to higher for structures bearing compression along their longest axis. At each step, all beams that are compressively loaded are replaced by a (scaled) generation-1 frame.

The next step is to replace the hollow beam with a space frame of hollow beams. The space frame used here is made up of n octahedra and two end tetrahedra. Optimising the number of octahedra, the radius and the wall thickness of the component beams (which are all assumed be identical) we find a new power law, and again, an improvement over the hollow beam design.

Continuing this procedure of replacing all beams under compressive load with (scaled) space frames constructed from hollow beams (figure 1), we find that the scaling law is always improved by the increased level of hierarchy. In general, the scaling is described by a (G+2)/(G+3) power-law relating non-dimensional volume to non-dimensional loading. Thus, as the generation number tends to infinity, the scaling relating material required for stability to loading approaches that of the tension member.

3D Printing

Working with Joel Segal, of the University of Nottingham, we fabricated an example of a generation-2 structure with solid beams, shown in figure 2. This was done through rapid prototyping technologies: micrometer-layer-by-micrometer-layer the structure was printed in a photosensitive polymer with each beam a fraction of a millimeter in radius. This structure shows the plausibility of the design and the extent to which modern manufacturing techniques allow for an increased creativity in design geometry. Through a process of 3-d printing and electro less deposition, it is believed that a metallic, hollow tubed structure could be created.

Figure 2: Showing a structure fabricated through rapid prototyping techniques. The inset shows the layering effect of the 3D printing technique. The structure shown in constructed in RC25 (Nanocure) material from envisionTEC on an envisionTEC perfactory machine.

Optimal generations

Although the scaling is always improved by increasing the generation number of the structure, the prefactor isn’t. The optimal structure is then obtained by balancing the scaling relationship with the prefactor in the expression. Generally, as the loading decreases (or the size of the structure increases), the scaling becomes more important and the optimal generation number increases. For large loads (or small structures) it can even be the case a simple, solid, beam is optimal.

Our work also formalises this relationship, for a long time engineers have created chair legs from hollow tubes or cranes out of space frames, Gustave Eiffel used three levels of structural hierarchy in designing the Eiffel tower. We show formally, that the optimal generation number has a set dependence on the loading conditions and allow future structures to be designed with this in mind. A further consequence of the alteration of the scaling law is that the higher the generations, the less difference it makes as to whether you have one structure holding a given load or two structures holding half the load each.

Reference:
[1] Robert Lakes, "Materials with structural hierarchy", Nature, 361, 511 (1993). Abstract.
[2] Haimin Yao, Huajian Gao, "Mechanics of robust and releasable adhesion in biology: Bottom–up designed hierarchical structures of gecko", Journal of the Mechanics and Physics of Solids, 54,1120 (2006). Abstract.
[3] Rachid Jennanea, Rachid Harbaa, Gérald Lemineura, Stéphanie Bretteila, Anne Estradeb, Claude Laurent Benhamouc, "Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection", Medical Image Analysis, 11, 91 (2007). Abstract.
[4] Rik Huiskes, Ronald Ruimerman, G. Harry van Lenthe, Jan D. Janssen, "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone", Nature, 405, 704 (2000). Abstract.
[5] Michael Doube, Michał M. Kłosowski, Alexis M. Wiktorowicz-Conroy, John R. Hutchinson, Sandra J. Shefelbine, "Trabecular bone scales allometrically in mammals and birds", Proceedings of the Royal Society B, 278, 3067 (2011). Abstract.
[6] Daniel Rayneau-Kirkhope, Yong Mao, Robert Farr, "Ultralight Fractal Structures from Hollow Tubes", Physical Review Letters, 109, 204301 (2012). Abstract.

Unconventional Fractional Quantum Hall Sequence in Graphene

Ben Feldman (left) and Amir Yacoby (right)
















Authors: Ben Feldman and Amir Yacoby

Affiliation: Department of Physics, Harvard University, USA

Graphene, a two-dimensional sheet of carbon that is one atom thick, has attracted considerable interest due to its unique and potentially useful physical properties. Like other two-dimensional materials, application of a perpendicular magnetic field leads to the formation of a sequence of flat energy bands called Landau levels (LLs). At high magnetic fields or when samples are very clean, interactions among electrons become important and produce additional energy gaps, even when the LLs are only partially filled. This phenomenon is known as the fractional quantum Hall effect (FQHE), and it leads to striking physical consequences such as excitations with a fraction of an electron charge [1-3].

Graphene provides an especially rich platform to study the FQHE. The low dielectric constant and unique band structure lead to FQH states with energy gaps that are larger than in GaAs at the same field. Moreover, charge carriers in graphene have an overall fourfold degeneracy that arises from their spin and valley degrees of freedom. This means that graphene can support FQH states that have no analogues in more conventional systems. Suspending samples above the substrate or depositing them on boron nitride minimizes disorder, and the FQHE was recently observed in such devices at all multiples of filling factor ν = 1/3 up to 13/3, except at ν = 5/3 [4-7]. The absence of a state at ν = 5/3 might result from low-lying excitations associated with the underlying symmetries in graphene, but alternate scenarios associated with disorder could not be ruled out in prior studies.

Figure 1: Picture of the scanning Single-Electron Transistor (SET) microscope setup

To further explore this behavior, we used a scanning single-electron transistor (SET) to probe a suspended graphene flake [8]. The SET is a unique local probe that is particularly non-invasive. It measures the presence of energy gaps in the electronic spectrum with sensitivity that no other technique can provide and therefore is very well adapted to the exploration of the FQHE. Moreover, we are able to study small regions of a graphene flake, and these local measurements reveal a dramatic improvement in sample quality relative to prior studies of larger-scale areas.

Our measurements show that electron-electron interactions in graphene produce different types and patterns of electronic states from what has been observed in more conventional materials. Although we observe the standard sequence of FQH states between ν = 0 and 1, states only occur at even-numerator fractions between ν = 1 and 2. This suggests that both spin and valley degeneracy are lifted below ν = 1, but one symmetry remains between ν = 1 and 2. The pattern of states that we observe and their corresponding energy gaps indicate an intriguing interplay between electron-electron interactions and the underlying symmetries of graphene.

Figure 2: Schematic of the measurement setup. The scanning single-electron transistor is held about 100 nm above a suspended graphene flake, and it measures the energy cost of adding additional electrons to the system.

Moreover, the scanning technique allows us to study variations in behavior as a function of position. Although all regions of the graphene flake show qualitatively similar behavior, local doping shifts the gate voltage required to observe each FQH state. Global measurements such as transport studies therefore require an especially homogenous sample to observe the delicate effects associated with interactions among electrons, whereas using a local probe allows us to observe especially clean regions and therefore observe more of the intrinsic physics.

Figure 3: Inverse compressibility of graphene as a function of carrier density and magnetic field. Incompressible behavior, which indicates the presence of an energy gap, is labeled at integer and certain fractional filling factors.

In the future, we are interested in continuing to explore the unusual FQH in graphene. In particular, we hope to better understand how the electrons are ordered in the various FQH states. We are also interested in learning more about the FQHE at higher filling factors and in related materials such as bilayer graphene.

References:
[1] D. C. Tsui, H. L. Stormer, A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit”, Physical Review Letters, 48, 1559 (1982). Abstract.
[2] R. B. Laughlin, “Anomalous quantum Hall-effect - an incompressible quantum fluid with fractionally charged excitations”, Physical Review Letters, 50, 1395 (1983). Abstract.
[3] J. K. Jain, “Composite-fermion approach for the fractional quantum Hall-effect”, Physical Review Letters, 63, 199 (1989). Abstract.
[4] Kirill I. Bolotin, Fereshte Ghahari, Michael D. Shulman, Horst L. Stormer, Philip Kim, “Observation of the fractional quantum Hall effect in graphene”, Nature, 462, 196 (2009). Abstract.
[5] Xu Du, Ivan Skachko, Fabian Duerr, Adina Luican, Eva Y. Andrei, “Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene”, Nature, 462, 192 (2009). Abstract.
[6] C. R. Dean, A. F. Young, P. Cadden-Zimansky, L. Wang, H. Ren, K. Watanabe, T. Taniguchi, P. Kim, J. Hone, K. L. Shepard, “Multicomponent fractional quantum Hall effect in graphene”, Nature Physics, 7, 693 (2011). Abstract.
[7] Dong Su Lee, Viera Skákalová, R. Thomas Weitz, Klaus von Klitzing, Jurgen H. Smet, “Transconductance fluctuations as a probe for interaction induced quantum Hall states in graphene”,  Physical Review Letters, 109, 056602 (2012). Abstract.
[8] Benjamin E. Feldman, Benjamin Krauss, Jurgen H. Smet, Amir Yacoby, “Unconventional sequence of fractional quantum Hall states in Suspended Graphene”, Science. 337, 1196 (2012). Abstract.