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Register a new userOrbital Magnetism of Bloch Electrons III. Application to Graphene
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Authors
Masao Ogata
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The orbital susceptibility for graphene is calculated exactly up to the first order with respect to the overlap integrals between neighboring atomic orbitals. The general and rigorous theory of orbital susceptibility developed in the preceding paper is applied to a model for graphene as a typical two-band model. It is found that there are contributions from interband, Fermi surface, and occupied states in addition to the Landau--Peierls orbital susceptibility. The relative phase between the atomic orbitals on the two sublattices related to the chirality of Dirac cones plays an important role. It is shown that there are some additional contributions to the orbital susceptibility that are not included in the previous calculations using the Peierls phase in the tight-binding model for graphene. The physical origin of this difference is clarified in terms of the corrections to the Peierls phase.
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We derive an exact formula of orbital susceptibility expressed in terms of Bloch wave functions, starting from the exact one-line formula by Fukuyama in terms of Greens functions. The obtained formula contains four contributions: (1) Landau-Peierls susceptibility, (2) interband contribution, (3) Fermi surface contribution, and (4) contribution from occupied states. Except for the Landau-Peierls susceptibility, the other three contributions involve the crystal-momentum derivatives of Bloch wave functions. Physical meaning of each term is clarified. The present formula is simplified compared with those obtained previously by Hebborn et al. Based on the formula, it is seen first of all that diamagnetism from core electrons and Van Vleck susceptibility are the only contributions in the atomic limit. The band effects are then studied in terms of linear combination of atomic orbital treating overlap integrals between atomic orbitals as a perturbation and the itinerant feature of Bloch electrons in solids are clarified systematically for the first time.
Orbital susceptibility for Bloch electrons is calculated for the first time up to the first order with respect to overlap integrals between the neighboring atomic orbitals, assuming single-band models. A general and rigorous theory of orbital susceptibility developed in the preceding paper is applied to single-band models in two-dimensional square and triangular lattices. In addition to the Landau-Peierls orbital susceptibility, it is found that there are comparable contributions from the Fermi surface and from the occupied states in the partially filled band called intraband atomic diamagnetism. This result means that the Peierls phase used in tight-binding models is insufficient as the effect of magnetic field.
The spatial distribution of electric current under magnetic field and the resultant orbital magnetism have been studied for two-dimensional electrons under a harmonic confining potential $V(vecvar{r})=m omega_0^2 r^2/2$ in various regimes of temperature and magnetic field, and the microscopic conditions for the validity of Landau diamagnetism are clarified. Under a weak magnetic field $(omega_clsimomega_0, omega_c$ being a cyclotron frequency) and at low temperature $(Tlsimhbaromega_0)$, where the orbital magnetic moment fluctuates as a function of the field, the currents are irregularly distributed paramagnetically or diamagnetically inside the bulk region. As the temperature is raised under such a weak field, however, the currents in the bulk region are immediately reduced and finally there only remains the diamagnetic current flowing along the edge. At the same time, the usual Landau diamagnetism results for the total magnetic moment. The origin of this dramatic temperature dependence is seen to be in the multiple reflection of electron waves by the boundary confining potential, which becomes important once the coherence length of electrons gets longer than the system length. Under a stronger field $(omega_cgsimomega_0)$, on the other hand, the currents in the bulk region cause de Haas-van Alphen effect at low temperature as $Tlsimhbaromega_c$. As the temperature gets higher $(Tgsimhbaromega_c)$ under such a strong field, the bulk currents are reduced and the Landau diamagnetism by the edge current is recovered.
The phase diagram of graphene decorated with magnetic adatoms distributed either on a single sublattice, or evenly over the two sublattices, is computed for adatom concentrations as low as $sim1%$. Within the framework of the $s$-$d$ interaction, we take into account disorder effects due to the random positioning of the adatoms and/or to the thermal fluctuations in the direction of magnetic moments. Despite the presence of disorder, the magnetic phases are shown to be stable down to the lowest concentration accessed here. This result agrees with several experimental observations where adatom decorated graphene has been shown to have a magnetic response. In particular, the present theory provides a qualitative understanding for the results of Hwang et al. [Sci. Rep. 6, 21460 (2016)], where a ferromagnetic phase has been found below $sim30,text{K}$ for graphene decorated with S-atoms.
The quantum anomalous Hall (QAH) effect - a macroscopic manifestation of chiral band topology at zero magnetic field - has only been experimentally realized by magnetic doping of topological insulators (1 - 3) and delicate design of Moire heterostructures (4 - 8). However, the seemingly simple bilayer graphene without magnetic doping or Moire engineering has long been predicted to host competing ordered states with QAH effects (9 - 11). Here, we explore states in bilayer graphene with conductance of 2 e2/h that not only survive down to anomalously small magnetic fields and up to temperatures of 5 K, but also exhibit magnetic hysteresis. Together, the experimental signatures provide compelling evidence for orbital magnetism driven QAH behavior with a Chern number tunable via electric and magnetic fields as well as carrier sign. The observed octet of QAH phases is distinct from previous observations due to its peculiar ferrimagnetic and ferrielectric order that is characterized by quantized anomalous charge, spin, valley, and spin-valley Hall behavior.
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