# Mathis on Graphene? Any hints?

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## Re: Mathis on Graphene? Any hints?

Also... a quantum style explanation:

https://en.wikipedia.org/wiki/Van_Hove_singularity

The optical absorption spectrum of a solid is most straightforwardly calculated from the electronic band structure using Fermi's Golden Rule where the relevant matrix element to be evaluated is the dipole operator A → ⋅ p → {\displaystyle {\vec {A}}\cdot {\vec {p}}} {\vec {A}}\cdot {\vec {p}} where A → {\displaystyle {\vec {A}}} {\vec {A}} is the vector potential and p → {\displaystyle {\vec {p}}} {\vec {p}} is the momentum operator. The density of states which appears in the Fermi's Golden Rule expression is then the joint density of states, which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy. The optical absorption is then essentially the product of the dipole operator matrix element (also known as the oscillator strength) and the JDOS.

The divergences in the two- and one-dimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like graphite (quasi-2D) and Bechgaard salts (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities.

https://en.wikipedia.org/wiki/Van_Hove_singularity

**A Van Hove singularity is a singularity (non-smooth point) in the density of states (DOS) of a crystalline solid**. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. For three-dimensional crystals, they take the form of kinks (where the density of states is not differentiable). The most common application of the Van Hove singularity concept comes in the analysis of optical absorption spectra. The occurrence of such singularities was first analyzed by the Belgian physicist Léon Van Hove in 1953 for the case of phonon densities of states.[1]**Experimental observation**The optical absorption spectrum of a solid is most straightforwardly calculated from the electronic band structure using Fermi's Golden Rule where the relevant matrix element to be evaluated is the dipole operator A → ⋅ p → {\displaystyle {\vec {A}}\cdot {\vec {p}}} {\vec {A}}\cdot {\vec {p}} where A → {\displaystyle {\vec {A}}} {\vec {A}} is the vector potential and p → {\displaystyle {\vec {p}}} {\vec {p}} is the momentum operator. The density of states which appears in the Fermi's Golden Rule expression is then the joint density of states, which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy. The optical absorption is then essentially the product of the dipole operator matrix element (also known as the oscillator strength) and the JDOS.

The divergences in the two- and one-dimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like graphite (quasi-2D) and Bechgaard salts (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities.

**Van Hove singularities play a significant role in understanding optical intensities in single-walled carbon nanotubes (SWNTs) which are also quasi-1D systems. The Dirac point in graphene is a Van-Hove singularity that can be seen directly as a peak in electrical resistance, when the graphene is charge-neutral. Twisted graphene layers also show pronounced Van-Hove singularities in the DOS due to the interlayer coupling.[6]****Cr6**- Admin
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## Re: Mathis on Graphene? Any hints?

I feel like the theory is really solid, Cr6. I just have a harder time visualizing it than I should. As a matter of potentials it makes sense. I really hope I can attack it visually at some point, or of course Nevyn's method is preferred. But it seems like a pretty simple variance, shifting the input energy from electric to magnetic?

**Jared Magneson**- Posts : 427

Join date : 2016-10-11

## Re: Mathis on Graphene? Any hints?

Here's more observed "weirdness" that Mathis at least explains by destroying the Quantum Hall Effect:

http://milesmathis.com/hall.pdf

http://milesmathis.com/stark.pdf

Scientists from Columbia University have reportedly proven a 30-year-old theory called "the even-denominator fractional quantum Hall state" and established bilayer graphene as a promising platform that could lead to quantum computation.

The team observed an intensely studied anomaly in condensed matter physics—the even-denominator fractional quantum Hall (FQH) state—via transport measurement in bilayer graphene. “Observing the 5/2 state in any system is a remarkable scientific opportunity, since it encompasses some of the most perplexing concepts in modern condensed matter physics, such as emergence, quasi-particle formation, quantization, and even superconductivity,” the team says. “Our observation that, in bilayer graphene, the 5/2 state survives to much higher temperatures than previously thought possible not only allows us to study this phenomenon in new ways, but also shifts our view of the FQH state from being largely a scientific curiosity to now having great potential for real-world applications, particularly in quantum computing.”

First discovered in the 1980s in gallium arsenide (GaAs) heterostructures, the 5/2 fractional quantum hall state remains the singular exception to the otherwise strict rule that says fractional quantum hall states can only exist with odd denominators. Soon after the discovery, theoretical work suggested that this state could represent an exotic type of superconductor, notable in part for the possibility that such a phase could enable a fundamentally new approach to quantum computation. However, confirmation of these theories has remained elusive, largely due to the fragile nature of the state; in GaAs it is observable only in the highest quality samples and even then appearing only at milikelvin temperaures (as much as 10,000 times colder than the freezing point of water).

The Columbia team has observed this same state in bilayer graphene and appearing at much higher temperatures - reaching several Kelvin. “While it’s still 100 times colder than the freezing point of water, seeing the even-denominator state at these temperatures opens the door to a whole new suite of experimental tools that previously were unthinkable,” says the team. “After several decades of effort by researchers all over the world, we may finally be close to solving the mystery of the 5/2.”

“We needed a new platform,” say the researchers. “With the successful isolation of graphene, these atomically thin layers of carbon atoms emerged as a promising platform for the study of electrons in 2D in general. One of the keys is that electrons in graphene interact even more strongly than in conventional 2D electron systems, theoretically making effects such as the even-denominator state even more robust. But while there have been predictions that bilayer graphene could host the long-sought even-denominator states, at higher temperatures than seen before, these predictions have not been realized due mostly the difficulty of making graphene clean enough.”

The Columbia team managed to improve the quality of graphene devices, creating ultra-clean devices entirely from atomically flat 2D materials: bilayer graphene for the conducting channel, hexagonal boron nitride as a protective insulator, and graphite used for electrical connections and as a conductive gate to change the charge carrier density in the channel. A crucial component of the research was having access to the high magnetic fields tools available at the National High Magnetic Field Laboratory in Tallahassee, Fla., a nationally funded user facility with which Hone and Dean have had extensive collaborations. They studied the electrical conduction through their devices under magnetic fields up to 34 Tesla, and achieved clear observation of the even-denominator states.

https://www.graphene-info.com/columbia-researchers-observe-exotic-quantum-particle-bilayer-graphene

http://milesmathis.com/hall.pdf

http://milesmathis.com/stark.pdf

**Columbia researchers observe exotic quantum particle in bilayer graphene**Scientists from Columbia University have reportedly proven a 30-year-old theory called "the even-denominator fractional quantum Hall state" and established bilayer graphene as a promising platform that could lead to quantum computation.

**Columbia team observes exotic quantum particle in graphene image**The team observed an intensely studied anomaly in condensed matter physics—the even-denominator fractional quantum Hall (FQH) state—via transport measurement in bilayer graphene. “Observing the 5/2 state in any system is a remarkable scientific opportunity, since it encompasses some of the most perplexing concepts in modern condensed matter physics, such as emergence, quasi-particle formation, quantization, and even superconductivity,” the team says. “Our observation that, in bilayer graphene, the 5/2 state survives to much higher temperatures than previously thought possible not only allows us to study this phenomenon in new ways, but also shifts our view of the FQH state from being largely a scientific curiosity to now having great potential for real-world applications, particularly in quantum computing.”

First discovered in the 1980s in gallium arsenide (GaAs) heterostructures, the 5/2 fractional quantum hall state remains the singular exception to the otherwise strict rule that says fractional quantum hall states can only exist with odd denominators. Soon after the discovery, theoretical work suggested that this state could represent an exotic type of superconductor, notable in part for the possibility that such a phase could enable a fundamentally new approach to quantum computation. However, confirmation of these theories has remained elusive, largely due to the fragile nature of the state; in GaAs it is observable only in the highest quality samples and even then appearing only at milikelvin temperaures (as much as 10,000 times colder than the freezing point of water).

The Columbia team has observed this same state in bilayer graphene and appearing at much higher temperatures - reaching several Kelvin. “While it’s still 100 times colder than the freezing point of water, seeing the even-denominator state at these temperatures opens the door to a whole new suite of experimental tools that previously were unthinkable,” says the team. “After several decades of effort by researchers all over the world, we may finally be close to solving the mystery of the 5/2.”

“We needed a new platform,” say the researchers. “With the successful isolation of graphene, these atomically thin layers of carbon atoms emerged as a promising platform for the study of electrons in 2D in general. One of the keys is that electrons in graphene interact even more strongly than in conventional 2D electron systems, theoretically making effects such as the even-denominator state even more robust. But while there have been predictions that bilayer graphene could host the long-sought even-denominator states, at higher temperatures than seen before, these predictions have not been realized due mostly the difficulty of making graphene clean enough.”

The Columbia team managed to improve the quality of graphene devices, creating ultra-clean devices entirely from atomically flat 2D materials: bilayer graphene for the conducting channel, hexagonal boron nitride as a protective insulator, and graphite used for electrical connections and as a conductive gate to change the charge carrier density in the channel. A crucial component of the research was having access to the high magnetic fields tools available at the National High Magnetic Field Laboratory in Tallahassee, Fla., a nationally funded user facility with which Hone and Dean have had extensive collaborations. They studied the electrical conduction through their devices under magnetic fields up to 34 Tesla, and achieved clear observation of the even-denominator states.

“By tilting the sample with respect to the magnetic field, we were able to provide new confirmation that this FQH state has many of the properties predicted by theory, such as being spin-polarized,” says the paper’s lead author. “We also discovered that in bilayer graphene, this state can be manipulated in ways that are not possible in conventional materials.”

“By tilting the sample with respect to the magnetic field, we were able to provide new confirmation that this FQH state has many of the properties predicted by theory, such as being spin-polarized,” says the paper’s lead author. “We also discovered that in bilayer graphene, this state can be manipulated in ways that are not possible in conventional materials.”

https://www.graphene-info.com/columbia-researchers-observe-exotic-quantum-particle-bilayer-graphene

**Cr6**- Admin
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## Re: Mathis on Graphene? Any hints?

“After several decades of effort by researchers all over the world, we may finally be close to solving the mystery of the 5/2.”

I think we have confirmation of Miles' theory though with that last paragraph. Tilt was the key.

I still remain beyond skeptical when these people tell us that it's "atomically thin", though. They barely know what an atom is, and have no way to tell if it's one atom thin or ten. Their scales and radii are off by a great deal, if Miles' analysis is sound. The Bohr radius itself isn't what it was supposed to be.

**Jared Magneson**- Posts : 427

Join date : 2016-10-11

## Re: Mathis on Graphene? Any hints?

I agree. Looks like they are putting the publication cart before their theoretical horse.

----------

This article was invited by Rolf J Haug.

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energies. We take into account five tight-binding parameters of the Slonczewski–Weiss–McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.

http://iopscience.iop.org/article/10.1088/0034-4885/76/5/056503/pdf

-----------

Kyounghwan Kim, Ashley DaSilva, Shengqiang Huang, Babak Fallahazad, Stefano Larentis, Takashi Taniguchi, Kenji Watanabe, Brian J. LeRoy, Allan H. MacDonald, and Emanuel Tutuc

PNAS March 28, 2017. 114 (13) 3364-3369; published ahead of print March 14, 2017.

https://doi.org/10.1073/pnas.1620140114

Accurately controlled, very long wavelength moiré patterns are realized in small-twist-angle bilayer graphene, and studied using electron transport and scanning probe microscopy. We observe gaps in electron transport at anomalous densities equal to ±8 electrons per moiré crystal unit cell, at variance with electronic structure theory, and the emergence of a Hofstadter butterfly in the energy spectrum in perpendicular magnetic fields. These findings open up an avenue to create artificial crystals by manipulating the relative angle between individual layers in a heterostructure.

According to electronic structure theory, bilayer graphene is expected to have anomalous electronic properties when it has long-period moiré patterns produced by small misalignments between its individual layer honeycomb lattices. We have realized bilayer graphene moiré crystals with accurately controlled twist angles smaller than 1° and studied their properties using scanning probe microscopy and electron transport. We observe conductivity minima at charge neutrality, satellite gaps that appear at anomalous carrier densities for twist angles smaller than 1°, and tunneling densities-of-states that are strongly dependent on carrier density. These features are robust up to large transverse electric fields. In perpendicular magnetic fields, we observe the emergence of a Hofstadter butterfly in the energy spectrum, with fourfold degenerate Landau levels, and broken symmetry quantum Hall states at filling factors ±1, 2, 3. These observations demonstrate that at small twist angles, the electronic properties of bilayer graphene moiré crystals are strongly altered by electron–electron interactions.

moiré crystalgraphenetwisted bilayermoiré bandHofstadter butterfly

Moiré patterns form when nearly identical two-dimensional (2D) crystals are overlaid with a small relative twist angle (1⇓⇓–4). The electronic properties of moiré crystals depend sensitively on the ratio of the interlayer hybridization strength, which is independent of twist angle, to the band energy shifts produced by momentum space rotation (5⇓⇓⇓⇓⇓⇓–12). In bilayer graphene, this ratio is small when twist angles exceed about 2° (10, 13), allowing moiré crystal electronic structure to be easily understood using perturbation theory (5). At smaller twist angles, electronic properties become increasingly complex. Theory (14, 15) has predicted that extremely flat bands appear at a series of magic angles, the largest of which is close to 1°. Flat bands in 2D electron systems, for example the Landau level bands that appear in the presence of external magnetic fields, allow for physical properties that are dominated by electron–electron interactions, and have been friendly territory for the discovery of fundamentally new states of matter. Here we report transport and scanning probe microscopy (SPM) studies of bilayer graphene moiré crystals with carefully controlled small-twist angles (STA), below 1°. We find that conductivity minima emerge in transport at neutrality, and at anomalous satellite densities that correspond to ±8 additional electrons in the moiré crystal unit cell, and that the conductivity minimum at neutrality is not weakened by a transverse electric field applied between the layers. Our observations can be explained only by strong electronic correlations.

----------

This article was invited by Rolf J Haug.

**Abstract**

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energies. We take into account five tight-binding parameters of the Slonczewski–Weiss–McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.

http://iopscience.iop.org/article/10.1088/0034-4885/76/5/056503/pdf

-----------

**Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene**Kyounghwan Kim, Ashley DaSilva, Shengqiang Huang, Babak Fallahazad, Stefano Larentis, Takashi Taniguchi, Kenji Watanabe, Brian J. LeRoy, Allan H. MacDonald, and Emanuel Tutuc

PNAS March 28, 2017. 114 (13) 3364-3369; published ahead of print March 14, 2017.

https://doi.org/10.1073/pnas.1620140114

**Significance**

Accurately controlled, very long wavelength moiré patterns are realized in small-twist-angle bilayer graphene, and studied using electron transport and scanning probe microscopy. We observe gaps in electron transport at anomalous densities equal to ±8 electrons per moiré crystal unit cell, at variance with electronic structure theory, and the emergence of a Hofstadter butterfly in the energy spectrum in perpendicular magnetic fields. These findings open up an avenue to create artificial crystals by manipulating the relative angle between individual layers in a heterostructure.

**Abstract**According to electronic structure theory, bilayer graphene is expected to have anomalous electronic properties when it has long-period moiré patterns produced by small misalignments between its individual layer honeycomb lattices. We have realized bilayer graphene moiré crystals with accurately controlled twist angles smaller than 1° and studied their properties using scanning probe microscopy and electron transport. We observe conductivity minima at charge neutrality, satellite gaps that appear at anomalous carrier densities for twist angles smaller than 1°, and tunneling densities-of-states that are strongly dependent on carrier density. These features are robust up to large transverse electric fields. In perpendicular magnetic fields, we observe the emergence of a Hofstadter butterfly in the energy spectrum, with fourfold degenerate Landau levels, and broken symmetry quantum Hall states at filling factors ±1, 2, 3. These observations demonstrate that at small twist angles, the electronic properties of bilayer graphene moiré crystals are strongly altered by electron–electron interactions.

moiré crystalgraphenetwisted bilayermoiré bandHofstadter butterfly

Moiré patterns form when nearly identical two-dimensional (2D) crystals are overlaid with a small relative twist angle (1⇓⇓–4). The electronic properties of moiré crystals depend sensitively on the ratio of the interlayer hybridization strength, which is independent of twist angle, to the band energy shifts produced by momentum space rotation (5⇓⇓⇓⇓⇓⇓–12). In bilayer graphene, this ratio is small when twist angles exceed about 2° (10, 13), allowing moiré crystal electronic structure to be easily understood using perturbation theory (5). At smaller twist angles, electronic properties become increasingly complex. Theory (14, 15) has predicted that extremely flat bands appear at a series of magic angles, the largest of which is close to 1°. Flat bands in 2D electron systems, for example the Landau level bands that appear in the presence of external magnetic fields, allow for physical properties that are dominated by electron–electron interactions, and have been friendly territory for the discovery of fundamentally new states of matter. Here we report transport and scanning probe microscopy (SPM) studies of bilayer graphene moiré crystals with carefully controlled small-twist angles (STA), below 1°. We find that conductivity minima emerge in transport at neutrality, and at anomalous satellite densities that correspond to ±8 additional electrons in the moiré crystal unit cell, and that the conductivity minimum at neutrality is not weakened by a transverse electric field applied between the layers. Our observations can be explained only by strong electronic correlations.

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