New Mass Limit for White Dwarfs: Explains Super-Chandrasekhar Type Ia Supernovae

Upasana Das (left) and Banibrata Mukhopadhyay (right)







Authors: Upasana Das and Banibrata Mukhopadhyay

Affiliation: Dept of Physics, Indian Institute of Science, Bangalore, India

Background:

Extremely luminous explosions of white dwarfs, known as type Ia supernovae [1], have always been in the prime focus of natural science. Generally, they are believed to result from the violent thermonuclear explosion of a carbon-oxygen white dwarf, when its mass approaches the famous Chandrasekhar limit of 1.44M_⊙ [2], with M_⊙ being the mass of Sun. The observed luminosity is powered by the radioactive decay of nickel, produced in the thermonuclear explosion, to cobalt and then to iron. The characteristic nature of the variation of luminosity with time of these supernovae (see Figure 1) -- along with the consistent mass of the exploding white dwarf -- allows these supernovae to be used as a ‘standard’ for measuring far away distances (standard candle) and hence in understanding the expansion history of the universe.

Figure 1: Variation of luminosity as a function of time of a type Ia supernova [image courtesy: Wikipedia]

Observation and study of this very feature of distant supernovae led to the Nobel Prize in Physics in 2011 for the discovery of the accelerated expansion of the universe [3, 2Physics report]. Also, mainly because of the discovery of the limiting mass of white dwarfs, S. Chandrasekhar was awarded the Nobel Prize in Physics in 1983.

Chandrasekhar, by means of a remarkably simple calculation, was the first to obtain the maximum mass for a (non-magnetized, non-rotating) white dwarf [2]. So far, observations seemed to abide by this limit. However, the recent discovery of several peculiar type Ia supernovae -- namely, SN 2006gz, SN 2007if, SN 2009dc, SN 2003fg [4,5] -- provokes us to rethink the commonly accepted scenario. These supernovae are distinctly over-luminous compared to their standard counterparts, because of their higher than usual nickel mass. They also violate the ‘luminosity-stretch relation’ and exhibit a much lower velocity of the matter ejected during the explosion. However, these anomalies can be resolved, if super-Chandrasekhar white dwarfs, with masses in the range 2.1-2.8M_⊙, are assumed to be the mass of the exploding white dwarfs (progenitors of these peculiar supernovae). Nevertheless, these non-standard ‘super-Chandrasekhar supernovae’ can no longer be used as cosmic distance indicators. However, there is no estimate of an upper limit to the mass of these super-Chandrasekhar white dwarf candidates yet. Can they be arbitrarily large? Moreover, there has been no foundational level analysis performed so far, akin to that carried out by Chandrasekhar, in order to establish a super-Chandrasekhar mass white dwarf.

Our result at a glance:

We establish a new and generic mass limit for white dwarfs which is 2.58M_⊙ [6]. This is significantly different from that proposed by Chandrasekhar. Our discovery naturally explains the over-luminous, peculiar type Ia supernovae mentioned above. We arrive at this new mass limit by exploiting the effects of the magnetic field in compact objects. The motivation behind our approach lies in the discovery of several isolated magnetized white dwarfs through the Sloan Digital Sky Survey (SDSS) with surface fields 10⁵-10⁹ gauss [7,8]. Hence their expected central fields could be 2-3 orders of magnitude higher. Moreover, about 25% of accreting white dwarfs, namely cataclysmic variables (CVs), are found to have magnetic fields as high as 10⁷-10⁸ gauss [9].

Underlying theory:

We first recall the basic formation scenario of white dwarfs. In order to do so, we have to understand the properties of degenerate electrons. When different states of a particle correspond to the same energy in quantum mechanics, they are called degenerate states. Moreover, Pauli’s exclusion principle prohibits any two identical fermions (in the present context: electrons) to occupy the same quantum state. Now, when a normal star of mass less than or of the order of 5M_ʘ exhausts its nuclear fuel [10], it undergoes a collapse leading to a small volume consisting of a lot of electrons. Being in a small volume, many such electrons tend to occupy the same energy states, and hence they become degenerate, since the energy of a particle depends on its momentum which is determined by the total volume of the system. Hence, once all the energy levels up to the Fermi level, which is the maximum allowed energy of a fermion, are filled by the electrons, there is no available space for the remaining electrons in a small volume of the collapsing star. This expels the electrons to move out leading to an outward pressure. If the force due to the outward pressure is able to balance the inward gravitational force, then the collapse halts, forming the compact star white dwarf.

Figure 2: Landau quantization in presence of magnetic field B. [image courtesy: Warwick University, UK ]

For the current purpose, we have to also recall the properties of degenerate, relativistic electrons under the influence of a strong magnetic field, neglecting any form of interactions. The energy states of a free electron in a magnetic field are quantized into what is known as Landau orbitals [11]. Figure 2 shows that how the continuous energy levels split into discrete Landau levels with the increase of magnetic field in the direction perpendicular to the motion of the electron. Larger the magnetic field, smaller is the number of Landau levels occupied. Recent works [12-14] establish that Landau quantization due to a strong magnetic field modifies the equation of state (EoS), which relates the pressure (P) with density (ρ), of the electron degenerate gas. This should influence significantly the mass and radius of the underlying white dwarf (and hence the mass-radius relation). The main aim here is to obtain the maximum possible mass of such a white dwarf (which is magnetized), and therefore a (new) mass limit. Hence we look for the regime of high density of electron degenerate gas and the corresponding EoS, which further corresponds to the high Fermi energy (E_F) of the system. This is because the highest density corresponds to the lowest volume and hence, lowest radius, which further corresponds to the limiting mass [2]. Note that the maximum Fermi energy (E_Fmax) corresponds to the maximum central density of the star. Consequently, conservation of magnetic flux (technically speaking flux freezing theorem, which is generally applicable for a compact star) argues for the maximum possible field of the system, which implies that only the ground Landau level will be occupied by the electrons.

Generally the EoS can be recast in the polytropic form of P=Kρ^Γ, when K is a constant and Γ (=1+1/n) is the polytropic index. At the highest density regime (which also corresponds to the highest magnetic field regime), Γ=2. Now combining the above EoS with the condition of magnetostatic equilibrium (when net outward force is balanced by the inward force), we obtain the mass and radius of the white dwarf to scale with its central density (ρ_c) as M ∝ K^(3/2) ρ_c^(3-n)/2n and R ∝ K^(1/2) ρ_c^(1-n)/2n respectively [6]. For Γ = 2, which corresponds to the case of limiting mass, K ∝ ρ_c^(-2/3) and hence M becomes independent of ρ_c and R becomes zero. Substituting the proportionality constants, for Γ = 2 we obtain exactly [6]:

where h is the Planck’s constant, c the speed of light, G Newton’s gravitation constant, μ_e the mean molecular weight per electron and m_H the mass of hydrogen atom. For μ_e=2, which is the case for a carbon-oxygen white dwarf, M≈2.58M_⊙. To compare with Chandrasekhar’s result [2], we recall the limiting mass obtained by him as

which for μ_e =2 is 1.44M_⊙.

Figure 3: Mass-radius relation of a white dwarf. Solid line – Chandrasekhar’s relation; dashed line – our relation.

For a better reference, we include a comparison between the mass-radius relation of the white dwarf obtained by Chandrasekhar and that obtained by us in Figure 3.

Justification of high magnetic field and its effect to hold more mass:

The presence of magnetic field in a white dwarf creates an additional outward pressure apart from that due to degenerate electrons, which is however modified in presence of a strong field in it. On the other hand, the inward (gravitational) force is proportional to the mass of the white dwarf. Hence, when the star is magnetized, a larger outward force can balance a larger inward force, allowing it to have more mass.

However, the effect of Landau quantization becomes significant only at a high field B ≥ B_c = 4.414×10¹³ gauss. How can we justify such a high field in a white dwarf? Let us consider the commonly observed phenomenon of a magnetized white dwarf attracting mass from its companion star (called accretion). Now the surface field of an accreting white dwarf, as observed, could be 10⁹ gauss (≪ B_c) [7]. Its central field, however, can be several orders of magnitude higher ∼ 10¹² gauss, which is also less than B_c. Naturally, such a magnetized CV, still follows the mass-radius relation obtained by Chandrasekhar. However, in contrast with Chandrasekhar’s work (which did not include a magnetic field in the calculations), we obtain that, a nonzero initial field in the white dwarf, however ineffective for rendering Landau quantization effects, proves to be crucial in supporting the additional mass accumulated due to accretion.

As an above-mentioned magnetized white dwarf first gains mass due to accretion, its total mass increases which in turn increases the gravitational power and hence the white dwarf contracts in size due to the increased gravitational pull. However, the total magnetic flux in a white dwarf is understood to be conserved, which is magnetic field times the square of its radius. Therefore, if the white dwarf shrinks, its radius decreases and hence magnetic field increases. This in turn increases the outward force balancing the increased inward gravitational force (due to increase of its mass), which leads to a quasi-equilibrium situation. As the accretion is a continuous process, above process of shrinking the white dwarf, increasing the magnetic field and holding more mass, goes in a cycle. This continues until the gain of mass becomes so great that total outward pressure is unable to support the gravitational attraction. This finally leads to a supernova explosion, which we observe as a peculiar, over-luminous type Ia supernova, in contrast to their normal counter parts.

Punch lines:

More than 80 years after the proposal of Chandrasekhar mass limit, this new limit perhaps heralds the onset of a paradigm shift. This discovery has several consequences as briefly described below.

The masses of white dwarfs are measured from their luminosities assuming Chandrasekhar's mass-radius relation, as of now. These results may have to be re-examined based on the new mass-radius relation, at least for some peculiar objects (e.g. over-luminous type Ia supernovae). Further, some peculiar known objects, like magnetars (highly magnetized compact objects, supposedly neutron stars, as of now) should be examined based on the above considerations, which could actually be super-Chandrasekhar white dwarfs.

This new mass limit may also lead to establishing the underlying peculiar supernovae as a new standard candle for cosmic distance measurement. Hence, in order to correctly interpret the expansion history of the universe (and then dark energy), one might need to carefully sample the observed data from the supernovae explosions, especially if the peculiar type Ia supernovae are eventually found to be enormous in number. However, it is probably too early to comment whether our discovery has any direct implication on the current dark energy scenario, which is based on the observation of ordinary type Ia supernovae.

References:
[1] D. Andrew Howell, “Type Ia supernovae as stellar endpoints and cosmological tools”, Nature Communications, 2, 350 (2011). Abstract.
[2] S. Chandrasekhar, “The highly collapsed configurations of a stellar mass (Second Paper)”, Monthly Notices of the Royal Astronomical Society, 95, 207 (1935). Article.
[3] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and The Supernova Cosmology Project, “Measurements of Omega and Lambda from 42 high-redshift supernovae”, The Astrophysical Journal, 517, 565 (1999). Article.
[4] D. Andrew Howell, Mark Sullivan, Peter E. Nugent, Richard S. Ellis, Alexander J. Conley, Damien Le Borgne, Raymond G. Carlberg, Julien Guy, David Balam, Stephane Basa, Dominique Fouchez, Isobel M. Hook, Eric Y. Hsiao, James D. Neill, Reynald Pain, Kathryn M. Perrett and Christopher J. Pritchet, “The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star”, Nature, 443, 308 (2006). Abstract.
[5] R. A. Scalzo, G. Aldering, P. Antilogus, C. Aragon, S. Bailey, C. Baltay, S. Bongard, C. Buton, M. Childress, N. Chotard, Y. Copin, H. K. Fakhouri, A. Gal-Yam, E. Gangler, S. Hoyer, M. Kasliwal, S. Loken, P. Nugent, R. Pain, E. Pécontal, R. Pereira, S. Perlmutter, D. Rabinowitz, A. Rau, G. Rigaudier, K. Runge, G. Smadja, C. Tao, R. C. Thomas, B. Weaver, and C. Wu, “Nearby supernova factory observations of SN2007if: First total mass measurement of a super-Chandrasekhar-mass progenitor”, The Astrophysical Journal, 713, 1073 (2010). Article.
[6] Upasana Das & Banibrata Mukhopadhyay, “New mass limit for white dwarfs: Super-Chandrasekhar type Ia supernova as a new standard candle”, Physical Review Letters, 110, 071102 (2013). Abstract.
[7] Gary D. Schmidt, Hugh C. Harris, James Liebert, Daniel J. Eisenstein, Scott F. Anderson, J. Brinkmann, Patrick B. Hall, Michael Harvanek, Suzanne Hawley, S. J. Kleinman, Gillian R. Knapp, Jurek Krzesinski, Don Q. Lamb, Dan Long, Jeffrey A. Munn, Eric H. Neilsen, Peter R. Newman, Atsuko Nitta, David J. Schlegel, Donald P. Schneider, Nicole M. Silvestri, J. Allyn Smith, Stephanie A. Snedden, Paula Szkody, and Dan Vanden Berk, “Magnetic white dwarfs from the Sloan Digital Sky Survey: The first data release”, The Astrophysical Journal, 595, 1101 (2003). Article.
[8] Karen M. Vanlandingham, Gary D. Schmidt, Daniel J. Eisenstein, Hugh C. Harris, Scott F. Anderson, Patrick B. Hall, James Liebert, Donald P. Schneider, Nicole M. Silvestri, Gregory S. Stinson, and Michael A. Wolfe, “Magnetic white dwarfs from the SDSS. II. The second and third data releases”, The Astronomical Journal, 130, 734 (2005). Article.
[9] D. T. Wickramasinghe and Lilia Ferrario, “Magnetism in isolated and binary white dwarfs”, Publications of the Astronomical Society of the Pacific, 112, 873 (2000). Article.
[10] S.L. Shapiro and S.A. Teukolsky, “Black holes, White dwarfs and Neutron stars: The physics of compact objects” (John Wiley & Sons Inc, 1983).
[11] Dong Lai and Stuart L. Shapiro, “Cold equation of state in a strong magnetic field – Effect of inverse beta-decay”, The Astrophysical Journal, 383, 745 (1991). Abstract.
[12] Upasana Das and Banibrata Mukhopadhyay, “Strongly magnetized cold degenerate electron gas: Mass-radius relation of the magnetized white dwarf”, Physical Review D, 86, 042001 (2012). Abstract.
[13] Upasana Das and Banibrata Mukhopadhyay, “Violation of Chandrasekhar mass limit: The exciting potential of strongly magnetized white dwarfs”, Int. J. Mod. Phys. D, 21, 1242001 (2012). Abstract.
[14] Aritra Kundu and Banibrata Mukhopadhyay, “Mass of highly magnetized white dwarfs exceeding the Chandrasekhar limit: An analytical view”, Modern Physics Letters A, 27, 1250084 (2012). Abstract.

Relativistic Heavy Ion Collider (RHIC) and the Puzzle of Proton’s “missing” Spin.

Stony Brook University/RHIC physicist Barbara Jacak at RHIC's PHENIX detector [photo courtesy: Brookhaven National Laboratory, USA]

The refrigeration system at the Relativistic Heavy Ion Collider (RHIC) began humming to life last week, beginning cool-down of the magnets in the 2.4-mile-circumference accelerator ring at Brookhaven Lab. Temperatures inside the magnets will ultimately reach a frigid four degrees Kelvin (-452 degrees Fahrenheit) as Run 13 at RHIC gets under way. When collisions begin this Monday, scientists from Brookhaven and around the world will collect data from particles emerging from the particle smashups to try to solve one of the biggest mysteries of the basic building blocks of matter: the puzzle of the proton’s “missing” spin.

“Recent data from RHIC show for the first time that gluons carry some of the proton’s spin; we now want to find out whether the same is true for antiquarks. RHIC has the unique capability for doing this,” said Berndt Mueller, who was recently named the Associate Laboratory Director for Nuclear and Particle Physics.

Run 13 is scheduled to continue for 15 weeks, with funding provided by a continuing resolution set to expire in March. More time may be added, depending on how the next federal budget is resolved.

RHIC is the only particle collider operating in the United States, and the only collider in the world where scientists can collide polarized protons—bunches of 100 billion protons all spinning like gyroscopes with their axes aligned in a particular direction. Collisions between two beams of these polarized protons are key to the quest to understand the subatomic components that make up the proton—quarks and gluons—and how those pieces contribute to the proton’s overall spin. RHIC operators will spend most of Run 13 colliding these polarized protons at 255 billion electron volts (GeV) for proton spin research.

In this picture of a proton-proton collision, the spin of the particles is shown as arrows circling the spherical particles. The red and green particles represent reaction products from the collision which will be "seen" and analyzed by RHIC detectors [image courtesy: Brookhaven National Laboratory, USA]

Budgets permitting, the operators will also spend a few weeks at the end of the run colliding gold ions at energies of about 15 GeV. These relatively low-energy heavy ion collisions will provide a set of data that should be useful in learning more about how ordinary matter changes to quark-gluon plasma—a phase change that may be similar to how water changes from a liquid to ice or steam under certain conditions.

Protons, Quarks and Gluons Spin—But the Numbers Don’t Add Up:

New detector components recently inserted into the heart of STAR will track "heavy flavor" quarks and W bosons to reveal subtle details of conditions created in particle collisions [photo courtesy: Brookhaven National Laboratory, USA]

The early thought that protons get their spin from their three constituent quarks was proved to be over simplistic many years ago when fixed target experiments revealed that these building blocks account for only about 30 percent of total spin. Scientists have been looking for the missing source of proton spin ever since.

“Protons’ quarks, antiquarks, gluons and other pieces all contribute fractions of the proton’s spin,” explained Jamie Dunlop, a deputy spokesperson for the STAR collaboration, one of the two experiments at RHIC. “If you add everything up, including the motion of the quarks, antiquarks, and gluons, they have to add up to the whole of the proton’s spin. But we don’t know what fraction is in the spin of the antiquarks and gluons, and in the internal motion of all these particles inside the proton.”

RHIC’s polarized proton collisions were the first to probe the gluons’ role. From these experiments it appears that gluons make a significant contribution, but still not enough to account for all the missing spin. So the search goes on.

“New detectors at both STAR and PHENIX give us the ability to track particles called W bosons that emerge from collisions,” Dunlop said. “These W bosons can be used as probes to quantify spin contributions from a proton’s antiquarks and from different ‘flavors’ of quarks.”

Teasing apart these subtle contributions is essential to help reveal the complexity that resides within one of the most seemingly simple objects on Earth, explained Dave Morrison, a co-spokesperson for the PHENIX collaboration at RHIC.

“Protons are the most simple of all stable states of QCD matter,” he said, referring to matter made of quarks and gluons whose interactions are described by a theory called quantum chromodynamics (QCD). “The equation for QCD can be written in one line, but it’s taken us 40 years of theory and experimentation to get to the point we’re at today,” he said.

“You, I, and the coffee we drink are all made of protons and the quarks and gluons inside them. QCD is not some distant thing that only happens far off in the cosmos. It just takes an amazingly complex machine like RHIC to enable us to see how these components work together.”

Tracking Particles at STAR and PHENIX:

Using muon detectors contained inside the funnel-shaped sides of the PHENIX experiment, collaborators will study the production of W bosons and learn about how up and down quarks contribute to the spin of the proton [photo courtesy: Brookhaven National Laboratory, USA]

During Run 13, the STAR collaboration will track W bosons with a forward GEM tracker that was tested during Run 12 and is now ready for serious use. GEM stands for gaseous electron multiplier. The state-of-the-art detector relies not on wires, but sheets of plastic film coated with copper with holes punched in it (like Gore-Tex) to amplify the path and charge of collision debris with accuracy of 100-150 microns—about the width of a hair.

Additionally, muon telescope detector trays at STAR will look for lower-momentum muons produced from decays of other subatomic debris—upsilon and J/psi particles, which offer clues about collision conditions in the heart of STAR. And collaborators will also begin commissioning the Heavy Flavor Tracker, a $15 million major upgrade designed to track heavy quarks.

Meanwhile, the PHENIX collaboration will use silicon-based forward vertex detectors, tested during Run 12, to identify short-lived particles that are produced and decay within microns of the primary collision. New electronics were recently installed along with these new detectors to help PHENIX rapidly select those rare collisions that contain muons. Some of these muons come from the decay of W bosons. Using the PHENIX muon detectors, contained inside the funnel-shaped sides of the experiment, collaborators will study the production of W bosons and learn about how up and down quarks contribute to the spin of the proton.

“The new detectors and electronics we tested last year should work extremely well in Run 13,” said Morrison. “The particle signatures we’re looking for are fairly rare, so we have to accumulate a lot of data to do the physics we want to do. With the improvements the Collider-Accelerator Department [C-AD] made last year, we have what we need to take data like crazy from every bit of beam sent our way.”

Collisions, Intensity and Polarization, Courtesy of C-AD:

C-AD made a number of improvements to the RHIC accelerator complex during Run 12, some that led to a new world record and three world firsts. For Run 13, C-AD is again working toward superlative performance: high beam intensity with the most polarized protons crammed into the smallest area possible; high luminosity with the highest rate of particles colliding; and the highest degree of polarization as protons race around the RHIC ring.

“We have only about 12 weeks for collisions, so we must attain high luminosity as quickly as possible,” said C-AD Chair Thomas Roser. “We are always working at the edge of what’s really possible. If we’re not, we didn’t explore enough and need to push harder.”

Anatoli Zelenski of Brookhaven's Collider-Accelerator Department and the new Optically Pumped Polarized Ion Source (OPPIS), which will pump up the production of polarized protons at RHIC for Run 13 [photo courtesy: Brookhaven National Laboratory, USA]

A new Optically Pumped Polarized Ion Source (OPPIS) will make its debut for Run 13. The OPPIS system uses a laser to polarize negatively charged electrons, which are then attached to protons to which their spin is transferred. The new OPPIS source will produce many more polarized protons than the old one. The new system—designed by Anatoli Zelenski and his team from C-AD and the Budker Institute of Nuclear Physics in Russia—took three years to develop and was only commissioned successfully in the weeks leading up to Run 13.

With uncertain federal budgets in the coming years, the future for research at RHIC is unclear. Run 13 is scheduled to begin this Monday, but people who operate RHIC and rely on the data it provides are already thinking about Run 14 and beyond.

Collisions will be put on hold every other Wednesday during the run for accelerator research and development, including testing new technologies such as electron lenses that mitigate the detrimental effects of beam-beam interactions.

“We work constantly to increase luminosity and polarization, so research and development will even continue during the run,” explained C-AD Accelerator Division Head Wolfram Fischer. “There are new, important questions to answer, not to mention records to set and break.”

[The Text is written by Joe Gettler of Brookhaven National Laboratory, USA]

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.

Origin of Cosmic Magnetic Fields: From Small Random Aperiodic to Ordered Large-Scale Structures

Author: Reinhard Schlickeiser

Affiliation:
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, Germany
and
Research Department: Plasmas with Complex Interactions, Ruhr-Universität Bochum, Germany.

Magnets have practically become everyday objects. Permanent ferromagnetism is a property of only a few densely packed materials, such as iron, in which the spin exchange interactions of individual atoms naturally line up in the same direction and create a residual persistent magnetic field. In the early universe, before iron and other magnetic materials had been created inside stars, such permanent magnetism did not exist.

Scientists have long wondered[1,2] where the observed cosmic magnetization came from, given that the fully ionized gas of the early universe contained no ferromagnetic particles. Many astrophysicists believe that galactic magnetic fields are generated and maintained by dynamo action[3,4], whereby the energy associated with the differential rotation of spiral galaxies is converted into magnetic field energy. However, the dynamo mechanism is only a means of amplification and dynamos require seed magnetic fields. Neither the dynamo process nor plasma instabilities[5] generate magnetic fields out of nothing: they need finite seed fields to start from.

Before the formation of the first stars, the luminous proto-interstellar matter consisted only of a fully ionised gas of protons, electrons, helium nuclei and lithium nuclei which were produced during the Big Bang. The physical parameters that describe the state of this gas are, however, not constant. Density and pressure fluctuate around certain mean values, and consequently electric and magnetic fields fluctuate around vanishing mean values. This small but finite dispersion in the form of random magnetic fields has now been calculated[6], specifically for the proto-interstellar gas densities and temperatures that occurred in the plasmas of the early universe at redshifts z=4-7 of the reionization epoch, when something, probably the light from the first stars, provided the energy needed to break up the previously neutral gas that existed in the universe. The protons and electrons inside the plasma would have moved around continuously, simply by virtue of existing at a finite temperature. It is the finite variance of the resulting magnetic fluctuations, that subsequently led to the creation of a stronger magnetism across the universe.

There have been alternative proposals for cosmic seed magnetic fields. Indeed, as far back as 1950 the German astronomer Ludwig Biermann[7] proposed that the centrifugal force generated in a rotating plasma cloud will separate out heavier protons from lighter electrons, thereby creating a separation of charge that leads to tiny electric and magnetic fields. However, this scheme suffers from a lack of suitable rotating objects everywhere, meaning that it could only ever generate the magnetic fields in a small portion of the medium.

To work out the field-strength variance of the fluctuations, Schlickeiser used a theory developed together with Peter Yoon of the University of Maryland[8]. The fluctuations are aperiodic, which means that, unlike the variations in magnetic and electric fields that give rise to electromagnetic radiation, they do not propagate as a wave. Indeed, their wavelength, the spatial distance over which the fluctuations occur, and their frequency, dictating how long these fluctuations last, are uncorrelated, in contrast to light, for which the values of wavelength and frequency are tied to one another via the wave's velocity. Summed over all possible wavelengths and frequencies for the magnetic fluctuations in a gas at 10,000 K, which would have been roughly the temperature of the proto-interstellar medium at the time of reionization. The calculation revealed magnetic fields variances of about 10^-16 Tesla inside very early-stage galaxies and around about 10^-25 Tesla in the voids between the protogalaxies. These values compare with the roughly 30 millionths of a Tesla of the Earth's magnetic field and 0.01 Tesla typical of a strong refrigerator magnet. The magnetic field in the plasma of the early universe was thus very weak, but it covered almost 100 percent of the plasma volume. Being so weak it could serve providing the seeds of primordial magnetic fields. The seed fields are tied passively to the highly conducting proto-interstellar plasma as frozen-in magnetic fluxes.

Figure 1: Illustration of the hydrodynamical stretching and ordering of cosmic magnetic fields. On the left figure a turbulent random magnetic field pervades the medium between five protostars. The right figure shows the ordering and stretching of the magnetic field as one of the stars explodes as a supernova. The outgoing shock wave compresses and orders the magnetic field in its vicinity [Image courtesy: Stefan Artmann].

Earlier analytical considerations and numerical simulations[9-12] showed that any shear and/or compression of the proto-interstellar medium not only amplifies these seed magnetic fields, but also make them anisotropic. Considering a cube containing an initially isotropic magnetic field, which is compressed to a factor η ≪ 1 times its original length along one axis, these authors showed that the perpendicular magnetic field components are enhanced by the factor &eta^-1. Depending on the specific exerted compression and/or shear, even one-dimensional ordered magnetic field structures can be generated out of the original isotropically tangled field configuration[12].

Hydrodynamical compression or shearing of the IGM medium arises from the shock waves of the supernova explosions of the first stars at the end of their lifetime, or from supersonic stellar and galactic winds. Fig. 1 sketches the basic physical process. The seed magnetic field upstream of these shocks is random in direction, and by solving the hydrodynamical shock structure equations for oblique and conical shocks it has been demonstrated[13], that the shock compression enhances the downstream magnetic field component parallel to the shock, but leaving the magnetic field component normal to the shock unaltered.

Consequently, a more ordered downstream magnetic field structure results from the randomly oriented upstream field. Such stretching and ordering of initially turbulent magnetic fields is also seen in the numerical hydrodynamical simulations of supersonic jets in radio galaxies and quasars[11]. Obviously, this magnetic field stretching and ordering occurs only in gas regions overrun frequently by shocks and winds. Each individual shock or wind (with speed V_s compression orders the field on spatial scales R on time scales given by the short shock crossing time R/V_s, but signifant amplification requires multiple compressions. The ordered magnetic field filling factor is determined by the shock's and wind's filling factors which are large (80 percent) in the coronal phase of interstellar media[14] and near shock waves in large-scale cosmic structures[15].

In cosmic regions with high shock/wind activity, this passive hydrodynamical amplification and stretching of magnetic fields continues until the magnetic restoring forces affect the gas dynamics, i.e. at ordered plasma betas near unity. As a consequence, magnetic fields with equipartition strength are not generated uniformly over the whole universe by this process, but only in localized cosmic regions with high shock/wind activity.

In protogalaxies significant and rapid amplification of the spontaneously emitted aperiodic turbulent magnetic fields results from the small-scale kinetic dynamo process[16,17] generated by the gravitational infall motions during the formation of the first stars[18-20]. Additional gaseous spiral motion may stretch and order the magnetic field on large protogalactic spatial scales.

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