Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices

Authors: Vincent Mourik¹, Kun Zuo¹, Sergey Frolov¹, Sébastien Plissard², Erik Bakkers^1,2, Leo Kouwenhoven¹

Affiliation:
¹Kavli Institute of Nanoscience, Delft University of Technology, Netherlands.
²Dept of Applied Physics, Eindhoven University of Technology, Netherlands.

Particle Predictors:
Paul Dirac was the very first particle predictor. In 1927, Dirac developed a formula that linked two new theories: Einstein’s special theory of relativity and quantum mechanics. Dirac’s equation, however, had several solutions. The first solution described the familiar electron: a particle with a negative charge holding a certain amount of positive energy. Another solution actually constituted its very opposite: a positively charged particle holding a certain amount of negative energy. Rather than ignoring the contradiction raised by his additional solution, Dirac surmised that there must be a particle in nature with a positive electrical charge and negative energy [1]. This particle therefore has properties that would exactly mirror the properties of an electron. Several years later, this particle was indeed found, and was named positron. Together, the electron and the positron form a particle and antiparticle pair.

A pure genius, Paul Dirac was utterly convinced of the veracity of his formula. If his equation offered a certain solution, then a corresponding particle simply had to exist in nature. Since that time, numerous other particles have been predicted and identified this way. For example, the ongoing search for the Higgs boson is set up just like that, based on a prediction from the Standard Model.

Ettore Majorana was a physicist and a contemporary of Dirac. Majorana had an enigmatic biography that formed the topic of many books and films in Italy. At some point in the 1930s, Majorana was playing around with Dirac’s equation and after slightly modifying it he found a new solution: a particle that is identical to its antiparticle. And something can only be identical to its counterpart if it has properties that are all zero. Ettore Majorana, too, had a firm belief in formulas and in 1937 he published a paper [2] predicting his new particle, which has since become known as the Majorana fermion.

For decades the Majorana particle received little attention, but in the 1970s the search began afresh. Using large accelerators and detectors, scientists started hunting for neutrino particles with Majorana properties. Indeed, these elementary Majorana particles might even solve the mystery of dark matter that fills our Universe. So far, the elementary Majorana particles have remained elusive, but this important quest is still being pursued by CERN in Geneva.

Particle Creators:

In addition to elementary particles, composite or collective particles (see box) also exist in the world of condensed matter physics. We know of heat particles (phonons), electron density waves (plasmons), magnetic waves (magnons) and a long list of other collective particles. These collective particles are particularly convenient for making the physics of materials a lot simpler. Materials hold a distinct place in physics because by combining materials we can create objects that did not exist before. Technology, for instance, abounds with remarkable material combinations, such as silicon and silicon oxide forming the backbone of electronics. But material combinations can also be used in fundamental physics to create something new. This prompted a number of theoretical physicists to reflect on whether we could combine materials in such a way that the collective particles inside them will acquire the properties of Majorana fermions.

The one-dimensional lattice proposed by Alexei Kitaev in 2001 was still highly mathematical and abstract [3]. A number of propositions then followed based on (p-wave superconducting) materials that did not yet exist. In 2008, Liang Fu and Charles Kane’s theory [4] was the first to be based on existing materials, but was still difficult to put into practice. The year 2010 saw the publication, in Physical Review Letters of two similar theories by two groups of theorists, independently of each other, which for the first time looked feasible in practice. One of the publications [5] came from theorists at the University of Maryland (Roman Lutchyn, Jay Sau and Sankar Das Sarma); the other [6] was a collaborative effort between theorists at the Weizmann Institute in Israel, California Institute of Technology of USA and the Free University of Berlin in Germany (Yuval Oreg, Gil Rafael and Felix von Oppen). The importance of the aforementioned theoretical developments was that it shifted the focus from what is found in nature to the artificial creation of Majorana particles.

Prior knowledge: there are particles and then there are particles…

If you blow into your hand, what you will primarily feel are oxygen and nitrogen molecules. Those molecules are minute, subnanometer-scale particles that are composed, in this case, of two atoms each. In turn, each atom is made up of an atomic nucleus encircled by electrons. The electrons cannot be divided into smaller particles - they are ‘elementary’ particles. However, the protons and neutrons inside the nucleus can be shattered to create even smaller particles. This shattering process is generated in accelerators such as the one at CERN in Geneva, where the search for the Higgs particle continues unabated. Other popular particles are the neutrinos (which for a short period were believed to travel even faster than light) and the Majorana fermions. These Majorana fermions have not yet been found at CERN.The Majorana fermions may well be the key to explaining the dark matter mystery. In the universe, there is five times as much dark matter as ordinary matter, and so Majorana fermions could be the most widespread particles in the universe.

CERN are engaged in the study of fundamental particles. Each of these particles are smaller than the smallest atom, hydrogen. Our world of matter is based on atoms and clusters of atoms that form molecules. The glue that binds these atoms to molecules is described by quantum mechanics. Our bodies, for instance, are chemical factories in which atoms are stuck together with quantum glue. Apart from complex biological materials, there are also crystals that frequently hold the same atoms stacked inside a grid. Even the smallest materials may contain large numbers of atoms. For example, a nanowire with a diameter of 100 nm and a length of 1000 nm (1000 nm = 1 micrometer) alone contains some 10 billion atoms.

Next to fundamental particles and atoms there are also collective particles. The ‘wave’ in a stadium is a good example. The ‘wave’ is simply a group of spectators jumping up and down to create a wave. If we wanted to describe this wave in mathematical terms, we might do that by including everyone in a large formula. Then again, we could also approach it more simply by forgetting about all those individuals and only describe their collective behavior, that is, the wave. And for simplicity’s sake we could call the ‘wave’ a particle, in this case, a collective particle. This reduction to collective particles simplifies matters enormously and is often highly successful. An example: heat in a material is not described in terms of a bunch of vibrating atoms but, far more easily, as heat particles that are known as phonons.

You may think that ‘collective particles’ is a rather imprecise way of describing what actually happens. This may be true of the wave but phonons, for example, can in fact provide us with a very exact, realistic description. What is perhaps the most surprising fact is that collective particles can actually behave in accordance with the laws of quantum mechanics. A phonon can find itself in the superposition of both hot and cold. Such a quantum superposition may sound absurd enough for elementary particles, but is really stretching our imagination where collective particles are concerned.

The Majorana fermions in crystals are not only interesting from a fundamental viewpoint, but also have unique properties that can be used to build a quantum computer. Field medalist Michael Freedman works at Microsoft and has been carrying out active research into topological quantum computers with a team of scientists since 2005. This computer works by moving Majorana particles around each other and forming space-time braids.

The proposals put forward by Lutchyn et al [5] and Oreg et al [6] are both based on bringing semiconducting nanowires into contact with a superconducting material. We had already successfully accomplished this combination in Delft, which resulted in publications in Science (2005) and Nature (2006). Combining these specific materials suddenly made us the experimental specialists in the search for Majorana fermions.

Note that the Majorana quest had already been described at an early stage in the journal 'Science' [7].

Majorana in Delft:

How do you create a Majorana fermion? Based on the condition that the particle is identical to its antiparticle, you can do some reverse engineering. It cannot, for instance, have an electrical charge, nor have energy or spin. The theoretical proposals argue that those properties are created by combining materials consisting of a superconductor with a special semiconductor that has strong spin-orbit coupling. This semiconductor should take the form of a one-dimensional nanowire. If a magnetic field is also applied, the Majorana fermion should appear at low temperatures, just above absolute zero temperature. We combined these materials on a microchip. We developed InSb (Indium Antimonide) nanowires for the semiconductor. InSb has strong spin-orbit coupling. We used a Nb alloy as a superconductor. This material will retain its superconductive properties also in the presence of an external magnetic field. For this material we were granted permission to use the technology available in Teun Klapwijk’s group in Delft. Using nanotechnology, we produced an electronic chip that, admittedly, looks rather messy (top right). Zooming down to sub-micrometre scale, we can see the nanowire thread and the electrical contacts (below right).

In this device, the superconductor is larger than the semiconductor. The diameter of the nanowire is so small that it actually becomes a one-dimensional conductor. A portion of the superconductor is covering the nanowire, which causes the superconductivity to leak into the semiconductor, effectively creating a one-dimensional superconductor. These do not exist in nature but can be induced this way. The strong spin-orbit coupling in the InSb nanowire makes this one-dimensional superconductor particularly unique. It has a so-called p-symmetry, which again has also not been found in nature. This p-superconductor extends across the entire section where the nanowire is in contact with the superconductor. At the end points, where the p-superconductor ends, two Majorana fermions appear, one on each end point.

Image: The microchip used with three different Majorana devices. This chip is cooled down to almost absolute zero point (-273 degrees Celsius). The electrical wires are connected to measuring equipment at room temperature.
 
We can measure the Majorana fermions in the electrical conductivity. From the gold contact we send electrons into the nanowire, towards the lower Majorana fermion. Only when we send electrons inside with precisely zero energy can we measure a current. If we add voltage to the electrons to energise them further, they are reflected at the p-superconductor and we measure zero conductivity. The presence of the Majorana fermion in our system is therefore visible as a conductance peak at a voltage that is precisely zero.

In the 'Science' publication [8] we also included various control experiments, which demonstrate that each single ingredient from the original theory is essential for this observation. The results can only be interpreted if we assume the presence of Majorana fermions. The article is published online on 12 April in 'Science Express' [8].

Image: The nanowire, shown vertically in this photo, is lying flat on a substrate. Hidden in the substrate are different gate electrodes (the horizontal ‘stripes’ below the nanowire and the contacts), which can change the conductivity of the nanowire. The lower electrical contact to the nanowire is madefrom gold, a normal conductor. The contact on the top is covering half of the nanowire. This is the superconductor. The total length of the nanowire is three micrometres. The anticipated positions of two Majorana fermions are indicated with red stars. 

We have since been carrying out new experiments. As the title of our article ‘Signatures of …’ suggests, we also want to demonstrate other unique properties of Majorana fermions. And our Majorana fermions are literally one of a kind. Nature has two types of particles: fermions (such as electrons, positrons, neutrons, etc.) and bosons (photons, Higgs particles, phonons, etc.). Our Majorana particles are likely to have other properties than fermions and bosons. In terms of physics, their behaviour is described by non-Abelian statistics. If we can demonstrate these statistics in our new experiments, we add a completely new chapter to the book of physics. This new round of experiments is based on a highly theoretical approach using new concepts that are not yet quite understood. To translate abstract concepts into experiments we are working with Carlo Beenakker’s theory group from Leiden. The non-Abelian statistics also make Majorana particles useful for a topological quantum computer.

References:
[1] P. A. M. Dirac, "The Quantum Theory of the Electron". Proceedings of the Royal Society of London: Series A 117, 610–624 (1928). Full Article.
[2] Ettore Majorana, "Teoria simmetrica dell’elettrone e del positrone", Il Nuovo Cimento, 171 (1937). Abstract. 
[3] A. Yu. Kitaev, "Unpaired Majorana fermions in quantum wires". Physics Uspekhi, 44, 131 (2001). Full Article. 
[4] Liang Fu and Charles Kane, "Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator", Phys. Rev. Lett. 100, 096407 (2008). Abstract.   
[5] 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.
[6] Yuval Oreg, Gil Refael, and Felix von Oppen, "Helical Liquids and Majorana Bound States in Quantum Wires", Physical Review Letters, 105, 177002 (2010). Abstract.
[7] Robert F. Service, "Search for Majorana Fermions Nearing Success at Last?", Science, 332, 193 (2011). Abstract.
[8] 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 Express, DOI: 10.1126/science.1222360 (Published Online April 12 2012). Abstract.