Graphene Brillouin Zone Research Articles

Emergence of Different Replica Dirac Cones and Intra- and Intervalley Scatterings in Short-Wavelength Graphene Superlattices Modulated by an Atomic-Scale Sharp Potential

Here, we calculate the unfolded band structure of short-wavelength graphene superlattices modulated by the atomic-scale sharp potential of a semiconductor surface, using density functional theory. We show that in the case of the superlattice graphene Brillouin zone (SBZ) center folded into the primitive graphene Brillouin zone (GBZ) corner, the emergence of different replica Dirac cones and their scattering behaviors are driven by the substrate-induced potential or atomic-scale disorders as well as by the correlation between the rotation angle of the superlattice and the trigonal warping orientation of the Dirac cone because of the quantum interference associated with the scattering at the SBZ edge. In particular, a superlattice with a rotational angle of φ = 30° (G–N√3 × N√3–R30°, N > 1) facilitates intervalley scattering, whereas an unrotated superlattice (G–N3 × N3) is favorable to intravalley scattering. For a superlattice with a rotational angle in the range of 0 < φ < 30°, the intervalley and intravalley scatterings may be comparable to each other, leading to the emergence of mixed inequivalent replica cones at the SBZ center (or GBZ corner). Interestingly, such inequivalent replica cones facilitate a stronger intervalley scattering at the GBZ corner, compared to the G–N√3 × N√3–R30° and G–N3 × N3 superlattices, thus opening a larger energy gap at the primitive Dirac point. In addition to the inversion symmetry breaking in graphene, we show that intervalley scattering can also generate an energy gap at the secondary Dirac point.

Clarifying the apparent flattening of the graphene band near the van Hove singularity

Graphene band renormalization near the van Hove singularity (VHS) has been investigated by angle-resolved photoemission spectroscopy (ARPES) on Li-doped quasifreestanding graphene on a cobalt (0001) surface. The absence of graphene band hybridization with the substrate, the doping contribution well represented by a rigid energy shift, and the excellent electron-electron interaction screening ensured by the metallic substrate offer a privileged point of view for such an investigation. A clear ARPES signal is detected along the $KMK$ direction of the graphene Brillouin zone, giving rise to an apparent flattened band. By simulating the graphene spectral function from the density functional theory calculated bands, we demonstrate that the photoemission signal around the $M$ point originates from the ``tail'' of the spectral function of the unoccupied band above the Fermi level. Such an interpretation puts forward the absence of any additional strong correlation effects near the VHS, reconciling the mean-field description of the graphene band structure even in a highly doped scenario.

Symmetry-enforced nodal lines in the band structures of vacancy-engineered graphene

We elaborate that single-layer graphene with periodic vacancies can have a band structure containing nodal lines or nodal loops, opening the possibility of graphene-based electronic or spintronic devices with novel functionalities. The principle is that by removing carbon atoms such that the lattice becomes nonsymmorphic, every two sublattices in the unit cell will map to each other under glide plane operation. This mapping yields degenerate eigenvalues for the glide plane operation, which guarantees that the energy bands must stick together pairwise at a boundary of the Brillouin zone. Moving away from the Brillouin zone boundary causes the symmetry-enforced nodal lines to split, resulting in accidental nodal lines caused by the crossings of the split bands. Moreover, the density of states at the Fermi level may be dramatically enhanced if the nodal lines crosses the Fermi level. The nodal lines occur a variety of vacancy configurations even in the presence of Rashba spin-orbit coupling. Finally, our theory also explains the nodal loops surrounding the entire Brillouin zone of a chevron-type nanoporous graphene fabricated in a recent experiment.

Resonance Raman enhancement by the intralayer and interlayer electron\u2013phonon processes in twisted bilayer graphene

Twisted bilayer graphene is a fascinating system due to the possibility of tuning the electronic and optical properties by controlling the twisting angle theta between the layers. The coupling between the Dirac cones of the two graphene layers gives rise to van Hove singularities (vHs) in the density of electronic states, whose energies vary with theta. Raman spectroscopy is a fundamental tool to study twisted bilayer graphene (TBG) systems since the Raman response is hugely enhanced when the photons are in resonance with transition between vHs and new peaks appear in the Raman spectra due to phonons within the interior of the Brillouin zone of graphene that are activated by the Moiré superlattice. It was recently shown that these new peaks can be activated by the intralayer and the interlayer electron–phonon processes. In this work we study how each one of these processes enhances the intensities of the peaks coming from the acoustic and optical phonon branches of graphene. Resonance Raman measurements, performed in many different TBG samples with theta between 4^{circ } and 16^{circ } and using several different laser excitation energies in the near-infrared (NIR) and visible ranges (1.39–2.71 eV), reveal the distinct enhancement of the different phonons of graphene by the intralayer and interlayer processes. Experimental results are nicely explained by theoretical calculations of the double-resonance Raman intensity in graphene by imposing the momentum conservation rules for the intralayer and the interlayer electron–phonon resonant conditions in TBGs. Our results show that the resonant enhancement of the Raman response in all cases is affected by the quantum interference effect and the symmetry requirements of the double resonance Raman process in graphene.

Fermi Velocity Reduction of Dirac Fermions around the Brillouin Zone Center in In2 Se3 -Bilayer Graphene Heterostructures.

Emergent phenomena such as unconventional superconductivity, Mott-like insulators, and the peculiar quantum Hall effect in graphene-based heterostructures are proposed to stem from the superlattice-induced renormalization of (moiré) Dirac fermions at the graphene Brillouin zone corners. Understanding the corresponding band structure commonly demands photoemission spectroscopy with both sub-meV resolution and large-momentum coverage, beyond the capability of the current state-of-the-art. Here the realization of moiré Dirac cones around the Brillouin zone center in monolayer In2 Se3 /bilayer graphene heterostructure is reported. The renormalization is evidenced by reduced Fermi velocity (≈23%) of the moiré Dirac cones and the reshaped Dirac point at the Γ point where they intersect. While there have been many theoretical predictions and much indirect experimental evidence, the findings here are the first direct observation of Fermi velocity reduction of the moiré Dirac cones. These features suggest strong In2 Se3 /graphene interlayer coupling, which is comparable with that in twisted bilayer graphene. The strategy expands the choice of materials in the heterostructure design and stimulates subsequent broad investigations of emergent physics at the sub-meV energy scale.

Raman Spectroscopy of Twisted Bilayer Graphene

When two periodic two-dimensional structures are superposed, any mismatch rotation angle between the layers generates a Moiré pattern superlattice, whose size depends on the twisting angle θ. If the layers are composed by different materials, this effect is also dependent on the lattice parameters of each layer. Moiré superlattices are commonly observed in bilayer graphene, where the mismatch angle between layers can be produced by growing twisted bilayer graphene (TBG) samples by CVD or folding the monolayer back upon itself. In TBG, it was shown that the coupling between the Dirac cones of the two layers gives rise to van Hove singularities (vHs) in the density of electronic states, whose energies vary with θ. The understanding of the behavior of electrons and their interactions with phonons in atomically thin heterostructures is crucial for the engineering of novel 2D devices. Raman spectroscopy has been often used to characterize twisted bilayer graphene and graphene heterostructures. Here, we review the main important effects in the Raman spectra of TBG discussing firstly the appearance of new peaks in the spectra associated with phonons with wavevectors within the interior of the Brillouin zone of graphene corresponding to the reciprocal unit vectors of the Moiré superlattice, and that are folded to the center of the reduced Brillouin Zone (BZ) becoming Raman active. Another important effect is the giant enhancement of G band intensity of TBG that occurs only in a narrow range of laser excitation energies and for a given twisting angle. Results show that the vHs in the density of states is not only related to the folding of the commensurate BZ, but mainly associated with the Moiré pattern that does not necessarily have a translational symmetry. Finally, we show that there are two different resonance mechanisms that activate the appearance of the extra peaks: the intralayer and interlayer electron–phonon processes, involving electrons of the same layer or from different layers, respectively. Both effects are observed for twisted bilayer graphene, but Raman spectroscopy can also be used to probe the intralayer process in any kind of graphene-based heterostructure, like in the graphene/h-BN junctions.

A Single-Molecule Label-Free Identification of Single-Nucleotide Colorectal-Cancer-DNA Polymorphism Using Impedance Spectroscopy of Self-Redox-Active Decorated Carbon Nanotubes

A novel label-free impedance assay which is based on self-redox active carbon nanotube arrays decorated by nanocyclic organometallic complex monolayer is offered to discriminate single-nucleotide polymorphism of colorectal-tumor genome. The assay is based on the following effects. A mass transfer for the multiwalled carbon nanotubes (MWCNTs) is testified by a Cole–Cole plot with Warburg impedance in dielectric spectra. Charging K(K')-point of graphene Brillouin zone is associated with the mass transfer for MWCNTs that leads to arising of charged MWCNT-end states. Plasma oscillations of charge carriers shield electric field of charged electrodes. Complementary hybridization of target deoxyribonucleic acid (DNA) with probe DNA on the MWCNT surface facilitates the penetration of single-stranded target DNA in the nanopores. Dielectric-band appearance and shielding effects testify the duplex formation.

Interacting valley Chern insulator and its topological imprint on moiré superconductors

One salient feature of systems with Moir\'{e} superlattice is that, Chern number of "minibands" originating from each valley of the original graphene Brillouin zone becomes a well-defined quantized number because the miniband from each valley can be isolated from the rest of the spectrum due to the Moir\'{e} potential. Then a Moir\'{e} system with a well-defined valley Chern number can become a nonchiral topological insulator with $U(1) \times Z_3 $ symmetry and a $\mathbb{Z}$ classification at the free fermion level. Here we demonstrate that the strongly interacting nature of the Moir\'{e} system reduces the classification of the valley Chern insulator from $\mathbb{Z}$ to $\mathbb{Z}_3$, and it is topologically equivalent to a bosonic symmetry protected topological state made of local boson operators. We also demonstrate that, even if the system becomes a superconductor when doped away from the valley Chern insulator, the valley Chern insulator still leaves a topological imprint as the localized Majorana fermion zero mode in certain geometric configuration.

General dynamic properties of conduction electron within the first Brillouin zone of graphene

Dispersion law of an electron (as a quasiparticle in the graphene conduction band) is sufficiently complex to construct its generalized analytic dynamics. This is due to such a “mixture” of the wave vector components that the dispersion law (which is also the quasiparticle Hamiltonian) cannot be “factored” into two summands containing only one component of the wave vector. It was established that within the first Brillouin zone of graphene there is a sufficiently large region, $ \vert k_{x}\vert \le \pi/3a\sqrt{3}$ ; $ \vert k_{y}\vert \le \pi/3a$ , in which it is possible to construct an approximate dispersion law. Based on this, the corresponding energy range of electron injection was estimated for graphene: $ E \le 5$ eV. This range completely “covers” the entire range used in applications for graphenes, and the estimation itself became possible due to the generalized de Broglie relation between the components of the wave and mechanical momenta. In this case, the velocity and the effective mass of the injected electron were determined to determine the components of the mechanical momentum. A Lagrangian for an electron was also determined, as a quasiparticle, with respect to its wave description. Such a Lagrangian is a summand in the phase of electron wave function at its quantum description.

Low-frequency phase diagram of irradiated graphene and a periodically driven spin- 12 XY chain

We study the Floquet phase diagram of two-dimensional Dirac materials such as graphene and the one-dimensional (1D) spin-1/2 $XY$ model in a transverse field in the presence of periodic time-varying terms in their Hamiltonians in the low drive frequency ($\omega$) regime where standard $1/\omega$ perturbative expansions fail. For graphene, such periodic time dependent terms are generated via the application of external radiation of amplitude $A_0$ and time period $T = 2\pi/\omega$, while for the 1D $XY$ model, they result from a two-rate drive protocol with time-dependent magnetic field and nearest-neighbor couplings between the spins. Using the adiabatic-impulse method, we provide several semi-analytic criteria for the occurrence of changes in the topology of the phase bands of such systems. For irradiated graphene, we point out the role of the symmetries of $H(t)$ and $U$ behind such topology changes. Our analysis reveals that at low frequencies, phase band topology changes may also happen at $t= T/3, 2T/3$ (apart from $t=T$). We chart out the phase diagrams at $t=T/3, 2T/3,\, {\rm and }\, T$ as a function of $A_0$ and $T$ using exact numerics, and compare them with the prediction of the adiabatic-impulse method. We show that several characteristics of these phase diagrams can be analytically understood from results obtained using the adiabatic-impulse method and point out the crucial contribution of the high-symmetry points in the graphene Brillouin zone to these diagrams. Finally we study the 1D $XY$ model with a two-rate driving protocol using the adiabatic-impulse method and exact numerics revealing a phase band crossing at $t=T/2$ and $k=\pi/2$. We also study the anomalous end modes generated by such a drive. We suggest experiments to test our theory.

Doped carbon nanotubes as a model system of biased graphene

Albeit difficult to access experimentally, the density of states (DOS) is a key parameter in solid state systems which governs several important phenomena including transport, magnetism, thermal, and thermoelectric properties. We study DOS in an ensemble of potassium intercalated single-wall carbon nanotubes (SWCNT) and show using electron spin resonance spectroscopy that a sizeable number of electron states are present, which gives rise to a Fermi-liquid behavior in this material. A comparison between theoretical and the experimental DOS indicates that it does not display significant correlation effects, even though the pristine nanotube material shows a Luttinger-liquid behavior. We argue that the carbon nanotube ensemble essentially maps out the whole Brillouin zone of graphene thus it acts as a model system of biased graphene.

Controlled formation of isolated miniband in bilayer graphene on almost commensurate 3×3 substrate

We investigate theoretically the interplay between the effects of a perpendicular electric field and incommensurability at the interface on the electronic properties of a heterostructure of bilayer graphene and a semiconducting substrate with a unit cell almost three times larger then that of graphene. It is known that the former introduces an asymmetry in the distribution of the electronic wave function between the layers and opens a band gap in the electronic spectrum. The latter generates a long wavelength periodic moir\'{e} perturbation of graphene electrons which couples states in inequivalent graphene Brillouin zone corners and leads to the formation of minibands. We show that, depending on the details of the moir\'{e} perturbation, the miniband structure can be tuned from that with a single band gap at the neutrality point and over-lapping minibands on the conduction/valence band side to a situation where a single narrow miniband is separated by gaps from the rest of the spectrum.

Nontrivial spin structure of graphene on Pt(111) at the Fermi level due to spin-dependent hybridization

The electronic and spin structure of a graphene monolayer synthesized on Pt(111) has been investigated experimentally by angle- and spin-resolved photoemission with different polarizations of incident synchrotron radiation and using density functional theory calculations. It is shown that despite the observed total quasifreestanding character of the dispersion of the graphene $\ensuremath{\pi}$ state remarkable local distortions and breaks in the dispersions take place due to hybridization between the graphene $\ensuremath{\pi}$ and Pt $d$ states. Corresponding spin-dependent avoided-crossing effects lead to significant modification of the spin structure and cause an enhanced induced spin-orbit splitting of the graphene $\ensuremath{\pi}$ states near the Fermi level in the region of the $\overline{\mathrm{K}}$ point of the graphene Brillouin zone (BZ) with a magnitude of 80--200 meV depending on the direction in the BZ. Using $p$, $s$, and elliptical polarizations of the synchrotron radiation, the contributions of the graphene $\ensuremath{\pi}$ and Pt $d$ states were separated and their intersection at the Fermi level, which is important for effective spin injection between these states, was shown. Moreover, analysis of the data allows us to conclude that in the region of the Dirac point the spin structure of the system cannot be described by a Rashba splitting, and even a spin-orbit gap between lower and upper Dirac cones is observed.

Valley filter from magneto-tunneling between single and bi-layer graphene

We consider tunneling transport between two parallel graphene sheets where one is a single-layer sample and the other one a bilayer. In the presence of an in-plane magnetic field, the interplay between combined energy and momentum conservation in a tunneling event and the distinctive chiral nature of charge carriers in the two systems turns out to favor tunneling of electrons from one of the two valleys in the graphene Brillouin zone. Adjusting the field strength enables manipulation of the valley polarization of the current, which reaches its maximum value of 100% concomitantly with a maximum of the tunneling conductance.

Raman spectroscopy of twisted bilayer graphene

Twisted bilayer graphene (tBLG) is a two-graphene layers system with a mismatch angle θ between the two hexagonal structures. The interference between the two rotated layers generates a superlattice with a θ-dependent wavevector that gives rise to van Hove singularities in the electronic density of states and activates phonons in the interior of the graphene Brillouin zone. Here we review the use of Raman spectroscopy to study tBLG, exploring the θ-dependent effects, corroborated by independent microscopy analysis. The phonon frequencies give a Raman signature of the specific tBLG, while their linewidths provide a straightforward test for tBLG structural homogeneity. Rich resonance effects, including single and multiple-resonances, intra- and inter-valley scattering events make it possible to accurately measure the energy of superlattice-induced van Hove singularities in the electronic joint density of states, as well as the phonon dispersion relation in tBLG, including the layer breathing vibrational modes.

Resonance Raman spectroscopy in twisted bilayer graphene

In this work we study the Raman spectra of twisted bilayer graphene samples, with different twisting angles, by changing the incident laser energy between 2.54 and 4.14eV. The spectra exhibit a number of extra peaks, classified in different families, each one associated with bilayer graphenes with different twisting rotational angles. We theoretically analyze the laser energy dependence of these extra peaks considering a set of discrete wavevectors within the interior of the Brillouin zone of graphene, which activate special double-resonance Raman processes. Our result show a nice qualitative agreement between the experimental and simulated spectra, demonstrating that these extra peaks are indeed ascribed to an umklapp double-resonance process in graphene systems.

Dominant phonon wavevectors of the 2D Raman mode of graphene

We investigate the dominant phonon wavevectors q* and the associated dominant phonon-assisted electronic transitions implied by the 2D Raman mode of graphene by combining ab initio calculations with a full two-dimensional integration over the graphene Brillouin zone. We find that q* are highly anisotropic and rotate with the polarizer:analyzer condition, providing access to the entire angular extent around K. The resonant electronic transitions do not lie along the line and can be transformed from being apparently “inner” to “outer” with the addition of a reciprocal lattice vector, showing that both are equivalent. We thus invalidate the notion of “inner” and “outer” processes completely.

Interedge magnetic coupling in transition-metal terminated graphene nanoribbons

The magnetic structures and interedge magnetic couplings of Fe, Co and Ni transition-metal terminated graphene nanoribbons with zigzag (ZGNR) and armchair (AGNR) edges are studied by first-principles calculations. Fe-ZGNR is found to show antiferromagnetic (AF) coupling between two edges, while the interedge coupling of Co-ZGNR is ferromagnetic (FM). For Fe-AGNRs and Co-AGNRs, increasing the interedge distance we follow oscillatory transitions from FM to AF coupling with a period of about 3.7 {\AA}. The damped oscillatory behavior indicates a Ruderman-Kittel-Kasuya-Yosida type interedge magnetic coupling and the oscillation period is determined by the critical spanning vector which connects two inequivalent Dirac points in the graphene Brillouin zone. The two edges in Ni-ZGNR are decoupled independent of the ribbon width and Ni-AGNRs are found to be nonmangetic.

Effects of Layer Stacking on the Combination Raman Modes in Graphene

We have observed new combination modes in the range from 1650 to 2300 cm(-1) in single-(SLG), bi-, few-layer and incommensurate bilayer graphene (IBLG) on silicon dioxide substrates. A peak at ∼1860 cm(-1) (iTALO-) is observed due to a combination of the in-plane transverse acoustic (iTA) and the longitudinal optical (LO) phonons. The intensity of this peak decreases with increasing number of layers and this peak is absent for bulk graphite. The overtone of the out-of-plane transverse optical (oTO) phonon at ∼1750 cm(-1), also called the M band, is suppressed for both SLG and IBLG. In addition, two previously unidentified modes at ∼2200 and ∼1880 cm(-1) are observed in SLG. The 2220 cm(-1) (1880 cm(-1)) mode is tentatively assigned to the combination mode of in-plane transverse optical (iTO) and TA phonons (oTO+LO phonons) around the K point in the graphene Brillouin zone. Finally, the peak frequency of the 1880 (2220) cm(-1) mode is observed to increase (decrease) linearly with increasing graphene layers.

Invariant expansion for the trigonal band structure of graphene

We present a symmetry analysis of the trigonal band structure in graphene, elucidating the transformational properties of the underlying basis functions and the crucial role of time-reversal invariance. Group theory is used to derive an invariant expansion of the Hamiltonian for electron states near the K points of the graphene Brillouin zone. Besides yielding the characteristic k-linear dispersion and higher-order corrections to it, this approach enables the systematic incorporation of all terms arising from external electric and magnetic fields, strain, and spin-orbit coupling up to any desired order. Several new contributions are found, in addition to reproducing results obtained previously within tight-binding calculations. Physical ramifications of these new terms are discussed.