In the present paper we have directly computed the Berry curvature terms relevant for Graphene in the presence of an \textit{inhomogeneous} lattice distortion. We have employed the generalized Foldy Wouthuysen framework, developed by some of us \cite{ber0,ber1,ber2}. We show that a non-constant lattice distortion leads to a valley-orbit coupling which is responsible to a valley-Hall effect. This is similar to the valley-Hall effect induced by an electric field proposed in \cite{niu2} and is the analogue of the spin-Hall effect in semiconductors \cite{MURAKAMI, SINOVA}. Our general expressions for Berry curvature, for the special case of homogeneous distortion, reduce to the previously obtained results \cite{niu2}. We also discuss the Berry phase in the quantization of cyclotron motion.
We describe a method for measuring the Berry curvature from the wave-packet dynamics in perturbed arrays of evanescently coupled optical waveguides with honeycomb lattice structure. To disentangle the effects of the Berry curvature and the energy dispersion we utilize a difference measurement by propagating the wave packet under the influence of a constant external force back and forth. In this way a non-vanishing Berry curvature is obtained for photonic graphene with small sublattice bias or strain, where the relative error between the exact Berry curvature and the one derived from the semicl…
The Berry phase of π in graphene is derived in a pedagogical way. The ambiguity of how to calculate this value properly is clarified. Its connection with the unconventional quantum Hall effect in graphene is discussed.
We consider gated graphene nanoribbons subject to Berry-Mondragon boundary conditions in the presence of weak impurities. Using field--theoretical methods, we calculate the density of charge carriers (and, thus, the quantum capacitance) as well as the optical and DC conductivities at zero temperature. We discuss in detail their dependence on the gate (chemical) potential, and reveal a non-linear behaviour induced by the quantization of the transversal momentum.
The phase of a quantum state may not return to its original value after the system's parameters cycle around a closed path; instead, the wavefunction may acquire a measurable phase difference called the Berry phase. Berry phases typically have been accessed through interference experiments. Here, we demonstrate an unusual Berry-phase-induced spectroscopic feature: a sudden and large increase in the energy of angular-momentum states in circular graphene p-n junction resonators when a small critical magnetic field is reached. This behavior results from turning on a $π$-Berry phase associated wit…
The Berry phase and the group-velocity-based traversal time have been calculated for an asymmetric non-contacted or contacted graphene structure, and significant differences have been observed compared to semiconductor heterostructures. These differences are related to the specific, Dirac-like evolution law of charge carriers in graphene, which introduces a new type of asymmetry. When contacted with electrodes, the symmetry of the Dirac equation is broken by the Schrodinger-type electrons in contacts, so that the Berry phase and traversal time behavior in contacted and non-contacted graphene d…
Since its discovery, Berry phase has been demonstrated to play an important role in many quantum systems. In gapped Bernal bilayer graphene, the Berry phase can be continuously tuned from zero to 2pi, which offers a unique opportunity to explore the tunable Berry phase on the physical phenomena. Here, we report experimental observation of Berry phases-induced valley splitting and crossing in moveable bilayer graphene p-n junction resonators. In our experiment, the bilayer graphene resonators are generated by combining the electric field of scanning tunneling microscope tip with the gap of bila…
Ever since the novel quantum Hall effect in bilayer graphene was discovered, and explained by a Berry phase of 2pi [K. S. Novoselov et al., "Unconventional quantum Hall effect and Berry's phase of 2pi in bilayer graphene", Nature Phys. 2, 177 (2006)], it has been widely accepted that the low-energy electronic wavefunction in this system is described by a non-trivial Berry phase of 2pi, different from the zero phase of a conventional two-dimensional electron gas. Here, we show that (i) the relevant Berry phase for bilayer graphene is not different from that for a conventional two-dimensional el…
We derive a semiclassical expression for the Green's function in graphene, in which the presence of a semiclassical phase is made apparent. The relationship between this semiclassical phase and the adiabatic Berry phase, usually referred to in this context, is discussed. These phases coincide for the perfectly linear Dirac dispersion relation. They differ however when a gap is opened at the Dirac point. We furthermore present several applications of our semiclassical formalism. In particular we provide, for various configurations, a semiclassical derivation of the electron's Landau levels, ill…
We show that in gapped bilayer graphene, quasiparticle tunneling and the corresponding Berry phase can be controlled such that it exhibits features of single layer graphene such as Klein tunneling. The Berry phase is detected by a high-quality Fabry-Pérot interferometer based on bilayer graphene. By raising the Fermi energy of the charge carriers, we find that the Berry phase can be continuously tuned from $2π$ down to $0.68π$ in gapped bilayer graphene, in contrast to the constant Berry phase of $2π$ in pristine bilayer graphene. Particularly, we observe a Berry phase of $π$, the standard val…
Hamiltonian and eigenstate problem is formulated for a bilayer graphene in terms of Clifford's geometric algebra \textit{Cl}$_{3,1}$. It is shown that such approach allows to perform analytical calculations in a simple way if geometrical algebra rotors are used. The measured quantities are express through spectrum and rotation half-angle of the pseudospin that appears in geometric algebra rotors. Properties of free charge carriers -- pseudospin, velocity and Berry phase -- in a bilayer graphene are investigated in the presence of the external voltage applied between the two layers.
Recent experiments have measured local uniaxial strain fields in twisted bilayer graphene (TBG). Our calculations found that the finite Berry curvature generated by breaking the sublattice symmetry and the band proximity between narrow bands in these TBG induces a giant Berry dipole of order 10\,nm or larger. The large Berry dipole leads to transverse topological non-linear charge currents which dominates over the linear bulk valley current at experimentally accessible crossover in-plane electric field of $\sim 0.1 {\rm mV} / μ\rm{m}$. This anomalous Hall effect, due to Berry dipole, is strong…
We construct the kinetic theory in ($1+2d$)-dimensional phase space and time when all abelian and nonabelian phase-space Berry curvatures are nonzero. Then we calculate anomalous transports induced by the Berry curvatures on the basis of the kinetic theory. As an example, we study anomalous charge and spin transports induced by the SU($2$) Berry curvatures. We also derive the topological effective theory to reproduce the transports in insulators calculated from the kinetic theory. Such an effective theory is given by the nonabelian phase space Chern-Simons theory.
We have observed the quantum Hall effect (QHE) and Shubnikov-de Haas (SdH) oscillations in highly disordered graphene at magnetic fields up to 65 T. Disorder was introduced by hydrogenation of graphene up to a ratio H/C $\approx 0.1\%$. The analysis of SdH oscillations and QHE indicates that the topological part of the Berry phase, proportional to the pseudo-spin winding number, is robust against introduction of disorder by hydrogenation in large scale graphene.
The magnetic effects in ferromagnetic graphene basically depend on the principle of exchange interaction when ferromagntism is induced by depositing an insulator layer on graphene. Here we deal with the consequences of non-uniformity in the exchange coupling strength of the ferromagnetic graphene. We discuss how the in- homogeneity in the coordinate and momentum of the exchange vector field can provide interesting results in the conductivity analysis of the ferromagnetic graphene. Our analysis is based on the Kubo formalism of quantum transport.
Berry phase plays an important role in determining many physical properties of quantum systems. However, a Berry phase altering energy spectrum of a quantum system is comparatively rare. Here, we report an unusual tunable valley polarized energy spectra induced by continuously tunable Berry phase in Bernal-stacked bilayer graphene quantum dots. In our experiment, the Berry phase of electron orbital states is continuously tuned from about pi to 2pi by perpendicular magnetic fields. When the Berry phase equals pi or 2pi, the electron states in the two inequivalent valleys are energetically degen…
This talk presents a study of massless relativistic Dirac fields in three Euclidean dimensions, at finite temperature and density, in the presence of a uniform electromagnetic background. Apart from explaining the behavior of Hall's conductivity for graphene, our results show a direct relationship between the selection of a phase for the Dirac determinant and the generation (or lack thereof) of Berry's phases and Chern-Simons terms.
When a gap of tunable size opens at the conic band intersections of graphene, the Berry phase does not vanish abruptly, but progressively decreases as the gap increases. The phase depends on the reciprocal-space path radius, i.e., for a doped system, the Fermi wave vector. The phase and its observable consequences can thus be tuned continuously via gap opening --by a modulating potential induced by strain, epitaxy, or nanostructuration-- and doping adjustment.
In a three-dimensional Fermi liquid, quasiparticles near the Fermi surface may possess a Berry curvature. We show that if the Berry curvature has a nonvanishing flux through the Fermi surface, the particle number associated with this Fermi surface has a triangle anomaly in external electromagnetic fields. We show how Landau's Fermi liquid theory should be modified to take into account the Berry curvature. We show that the "chiral magnetic effect" also emerges from the Berry curvature flux.
Brillouin zones of graphene systems possess Dirac points, where band degeneracies occur. We study the variety of (and large magnitude) phases that the electronic states can acquire when a uniform time-dependent electric field carries the electrons around one or more Dirac points in a non-concentric fashion. An experimentally accessible determination of excess Berry phases is proposed involving the Zitterbewegung of electronic current near an orthogonality point in adiabatic motion.
We propose an experimental scheme to probe the Berry curvature of solids. Our method is sensitive to arbitrary regions of the Brillouin zone, and employs only basic optical and terahertz techniques to yield a background free signal. Using semiconductor quantum wells as a prototypical system, we discuss how to inject Berry curvature macroscopically, and probe it in a way that provides information about the underlying microscopic Berry curvature.
High-order terahertz (THz) sideband generation (HSG) in semiconductors is a phenomenon with physics similar to high-order harmonic generation but in a much lower frequency regime. It was found that the electron-hole pairs excited by a weak optical laser can accumulate Berry phases along a cyclic path under the driving of a strong THz field. The Berry phases appear as the Faraday rotation angles of the emission signal under short-pulse excitation in monolayer MoS$_2$. In this paper, the theory of Berry phase in THz extreme nonlinear optics is applied to biased bilayer graphene with Bernal stack…
It has been recently established that optoelectronic and non-linear transport experiments can give direct access to the dipole moment of the Berry curvature in non-magnetic and non-centrosymmetric materials. Thus far, non-vanishing Berry curvature dipoles have been shown to exist in materials with substantial spin-orbit coupling where low-energy Dirac quasiparticles form tilted cones. Here, we prove that this topological effect does emerge in two-dimensional Dirac materials even in the complete absence of spin-orbit coupling. In these systems, it is the warping of the Fermi surface that trigge…
Klein tunneling in gapless bilayer graphene, perfect reflection of electrons injecting normal to a pn junction, is expected to disappear in the presence of energy band gap induced by external gates. We theoretically show that the Klein effect still exists in gapped bilayer graphene, provided that the gaps in the n and p regions are balanced such that the polarization of electron pseudospin has the same normal component to the bilayer plane in the regions. We attribute the Klein effect to Berry phase $π$ (rather than the conventional value $2 π$ of bilayer graphene) and to electron-hole and tim…
We have found that the Berry phase of bilayer graphene becomes from 2$π$ estimated by Shubnikov-de Haas oscillations when the A-B stacked pristine bilayer graphene experiences the Li-intercalation and sequential Li-desorption process in ultrahigh vacuum. Furthermore, the mobility of such processed bilayer graphene increases around four times larger, ~ 8,000 cm2/V$\cdot$s, than that of the pristine bilayer graphene. This is mainly due to increment of the scattering time and decrement of the cyclotron mass, which can be interpreted as a result of the change of the stacking structure of bilayer g…
Standing waves near the zigzag and armchair edges, and their Berry's phases are investigated. It is suggested that the Berry's phase for the standing wave near the zigzag edge is trivial, while that near the armchair edge is non-trivial. A non-trivial Berry's phase implies the presence of a singularity in parameter space. We have confirmed that the Dirac singularity is absent (present) in the parameter space for the standing wave near the zigzag (armchair) edge. The absence of the Dirac singularity has a direct consequence in the local density of states near the zigzag edge. The transport prop…
Chiral quasiparticles in Bernal-stacked bilayer graphene have valley-contrasting Berry phases of 2π. This nontrival topological structure, associated with the pseudospin winding along a closed Fermi surface, is responsible for various novel electronic properties, such as anti-Klein tunneling, unconventional quantum Hall effect, and valley Hall effect1-6. Here we show that the quantum interference due to intervalley scattering induced by atomic defects/impurities provides further insights into the topological nature of the bilayer graphene. The scattered chiral quasiparticles between distinct v…
We analyse the Kitaev honeycomb model, by means of the Berry curvature with respect to Hamiltonian parameters. We concentrate on the ground-state vortex-free sector, which allows us to exploit an appropriate Fermionisation technique. The parameter space includes a time-reversal breaking term which provides an analytical headway to study the curvature in phases in which it would otherwise vanish. The curvature is then analysed in the limit in which the time-reversal-symmetry-breaking perturbation vanishes. This provides remarkable information about the topological phase transitions of the model…
Electronic band structures dictate the mechanical, optical and electrical properties of crystalline solids. Their experimental determination is therefore of crucial importance for technological applications. While the spectral distribution in energy bands is routinely measured by various techniques, it is more difficult to access the topological properties of band structures such as the Berry phase γ. It is usually thought that measuring the Berry phase requires applying external electromagnetic forces because these allow realizing the adiabatic transport on closed trajectories along which qua…
We consider the anomalous thermoelectric transport in gapped single and bilayer graphene where the gap may be due to broken inversion symmetry. In the presence of the gap, non-trivial Berry phase effects can be shown to mediate a transverse thermoelectric voltage in response to an applied temperature gradient even in the absence of a perpendicular magnetic field. This spontaneous, anomalous Nernst effect is nonzero for non-uniform chemical potential in the two inequivalent valleys in the graphene band structure. Conversely, the Nernst response can be used to create a valley-index polarization …
When electrons are confined in two-dimensional (2D) materials, quantum mechanically enhanced transport phenomena, as exemplified by the quantum Hall effects (QHE), can be observed. Graphene, an isolated single atomic layer of graphite, is an ideal realization of such a 2D system. Here, we report an experimental investigation of magneto transport in a high mobility single layer of graphene. Adjusting the chemical potential using the electric field effect, we observe an unusual half integer QHE for both electron and hole carriers in graphene. Vanishing effective carrier masses is observed at Dir…
The intrinsic spin Hall conductivity and the anomalous Hall conductivity of ferromagnetic L1$_0$-CoPt are studied using first principle calculations of the spin Berry and Berry curvatures, respectively. We find that the Berry curvature and the spin Berry curvature exhibit different symmetry with respect to that of the band structure. The Berry curvature preserves the $C_{4v}$ crystal rotation symmetry along the c-axis whereas the symmetry of the spin Berry curvature reduces to $C_{2v}$. Contributions to the Berry curvature and the spin Berry curvature are classified by the spin character of ba…
For a long period of time, we have been seeking how Berry curvature influnces the transport properties in materials breaking time-reversal symmetry. In time-reversal symmetric material, there will be no thermoelectric current induced by Berry curvature in linear regime. However, the nonlinear Hall current can be shown in non-magnetic and non-centrosymmetric materials, where Berry curvature dipole plays an important role. Most studies are developed from semi-classical Boltzmann equation. Here we show the quantum kinetic theory for nonlinear Nernst effect and introduce a new type of Berry curvat…
The electronic dispersion of a graphene bilayer is highly dependent on rotational mismatch between layers and can be further manipulated by electrical gating. This allows for an unprecedented control over electronic properties and opens up the possibility of flexible band structure engineering. Here we present novel magnetotransport data in a twisted bilayer, crossing the energetic border between decoupled monolayers and coupled bilayer. In addition a transition in Berry phase between pi and 2pi is observed at intermediate magnetic fields. Analysis of Fermi velocities and gate induced charge c…
We study the influence of a magnetic flux tube on the possibility to electrostatically confine electrons in a graphene quantum dot. Without magnetic flux tube, the graphene pseudospin is responsible for a quantization of the total angular momentum to half-integer values. On the other hand, with a flux tube containing half a flux quantum, the Aharonov-Bohm phase and Berry phase precisely cancel, and we find a state at zero angular momentum that cannot be confined electrostatically. In this case, true bound states only exist in regular geometries for which states without zero-angular-momentum co…
In a magnetic field few-layer graphene supports, at the lowest Landau level, a multiplet of zero-mode levels nearly degenerate in orbitals as well as in spins and valleys. Those pseudo-zero-mode (PZM) levels are generally sensitive to interactions and external perturbations, and have a crossing among themselves or with other higher Landau levels when an external field is swept over a certain range. A close study is made of how such PZM levels evolve when they are gradually brought from empty to filled levels under many-body interactions. It is pointed out that the level spectra generally avoid…
Transverse current due to Berry curvature in phase space is formulated based on the Boltzmann equations with the semiclassical equations of motion for an electron wave packet. It is shown that the Hall effect due to the phase space Berry curvature is absent because the contributions from "anomalous velocity" and "effective Lorentz force" are completely cancelled out.
We show that the charge conductivity for impurity free multi band electronic systems can be expressed in terms of the diagonal and non-diagonal elements of the Zitterbewegung amplitudes while the Berry curvature and the Chern number is related only to the diagonal elements. Thus, the phenomenon of the Zitterbewegung can no longer be viewed just as an interesting consequence of quantum physics but it has also an experimental relevance. Moreover, through several examples we demonstrate how efficient our approach is in the analytical calculation of the charge conductivity.
The geometric phase is known to play a role both in the rotation of the Foucault pendulum and in the anomalous Hall effect (AHE) due to the Berry curvature. Here, we show that a 2D harmonic oscillator with AHE induced by Berry curvature behaves exactly like the Foucault pendulum: in both, the plane of the oscillations rotates with time. The rotating pendulum configuration enhances the AHE, simplifying its observation and allowing high-precision measurements of the Berry curvature. We also show how the non-adiabaticity and anharmonicity determine the maximal rotation angle and find the optimal …
The Berry curvature dipole is a physical quantity that is expected to allow various quantum geometrical phenomena in a range of solid-state systems. Monolayer transition metal dichalcogenides provide an exceptional platform to modulate and investigate the Berry curvature dipole through strain. Here we theoretically demonstrate and experimentally verify for monolayer MoS$_\rm{2}$ the generation of valley orbital magnetization as a response to an in-plane electric field due to the Berry curvature dipole. The measured valley orbital magnetization shows excellent agreement with the calculated Berr…
We demonstrate that flat bands with local Berry curvature arise naturally in chiral (ABC) multilayer graphene placed on a boron nitride (BN) substrate. The degree of flatness can be tuned by varying the number of graphene layers N. For N = 7 the bands become nearly flat, with a small bandwidth of 3.6 meV. The two nearly flat bands coming from the K and K' valleys cross along lines in the reduced zone. Weak intervalley tunneling turns the bandcrossing into an avoided crossing, producing two nearly flat bands with global Chern number zero, but with local Berry curvature. The flatness of the band…
We study 1-loop effects for massless Dirac fields in two spatial dimensions, coupled to homogeneous electromagnetic backgrounds, both at zero and at finite temperature and density. In the case of a purely magnetic field, we analyze the relationship between the invariance of the theory under large gauge transformations, the appearance of Chern-Simons terms and of different Berry's phases. In the case of a purely electric background field, we show that the effective Lagrangian is independent of the chemical potential and of the temperature. More interesting: we show that the minimal conductivity…
There are known two distinct types of the integer quantum Hall effect. One is the conventional quantum Hall effect, characteristic of two-dimensional semiconductor systems, and the other is its relativistic counterpart recently observed in graphene, where charge carriers mimic Dirac fermions characterized by Berry's phase pi, which results in a shifted positions of Hall plateaus. Here we report a third type of the integer quantum Hall effect. Charge carriers in bilayer graphene have a parabolic energy spectrum but are chiral and exhibit Berry's phase 2pi affecting their quantum dynamics. The L…
The Berry curvature is a fundamental concept describing topological order of quantum systems. While it can be analytically tractable in non-interacting systems, numerical simulations are necessary in interacting systems. We present a formulation to calculate the Berry curvature in lattice QCD.
Within the semiclassical Boltzmann transport theory, the formula for Seebeck coefficient $S$ is derived for an isotropic two-dimensional electron gas (2DEG) system that exhibits anomalous Hall effect (AHE) and anomalous Nernst effect (ANE) originating from Berry curvature on their bands. Deviation of $S$ from the value $S_0$ estimated neglecting Berry curvarture is computed for a special case of 2DEG with Zeeman and Rashba terms. The result shows that, under certain conditions the contribution of Berry curvature to Seebeck effect could be non-negligible. Further study is needed to clarify the …
We study the energy spectrum of a graphene bilayer in the presence of transverse electric and magnetic fields. We find that the resulting Landau levels exhibit a nonmonotonic dependence on the electric field, as well as numerous level crossings. This behavior is explained using quasiclassical quantization rules that properly take into account the pseudospin of the quasiparticles. The pseudospin generates the Berry phase, which leads to a shift in energy quantization and results in a pseudo-Zeeman effect. The latter depends on the electric field, alternates in sign among the two valleys, and al…
Single-layer carbon, or graphene, demonstrates amazing transport properties, such as the minimum conductivity near $\frac{4e^2}{h}$ independent of shapes and mobility of samples. This indicates there exist some unusual effects due to specific Dirac dispersion relation of fermion in two dimensions. By deriving fermion-lattice interaction Hamiltonian we show that Berry phases can be produced in fermion states around two Dirac points by relative rotations of two sublattices. The Berry phases in turn remove the degeneracies of energies for states near the Fermi surface, leading to a dynamical inst…
The electronic band structure of ABC-stacked multilayer graphene is studied within an effective mass approximation. The electron and hole bands touching at zero energy support chiral quasiparticles characterized by Berry's phase N pi for N-layers, generalizing the low-energy band structure of monolayer and bilayer graphene. We investigate the trigonal-warping deformation of the energy bands and show that the Lifshitz transition, in which the Fermi circle breaks up into separate parts at low energy, reflects Berry's phase N pi. It is particularly prominent in trilayers, N=3, with the Fermi circ…
Magnetotransport such as the giant magnetoresistance and Hall effect lies at the heart of fundamental physics and technologies. Recently, some experiments have clearly demonstrated linear magnetotransport (LMT) proportional to magnetic field but the underlying physical mechanism is still unclear. In this work, we show that Berry curvature effect is a new mechanism dominating the LMT. The Berry-curvature-induced LMT widely exists in 66 out of 122 magnetic point groups. For typical magnetic Weyl semimetals Co$_3$Sn$_2$S$_2$ and ferromagnetic MnBi$_2$Te$_4$, Berry curvature induces LMT conductivi…
