Following Ashcroft and Mermin, the conduction electrons (“electrons” or “holes”) are assumed to move as wave packets. Dirac’s theorem states that the quantum wave packets representing massive particles always mov...Following Ashcroft and Mermin, the conduction electrons (“electrons” or “holes”) are assumed to move as wave packets. Dirac’s theorem states that the quantum wave packets representing massive particles always move, following the classical mechanical laws of motion. It is shown here that the conduction electron in an orthorhombic crystal moves classical mechanically if the primitive rectangular-box unit cell is chosen as the wave packet, the condition requiring that the particle density is constant within the cell. All crystal systems except the triclinic system have k-vectors and energy bands. Materials are conducting if the Fermi energy falls on the energy bands. Energy bands and gaps are calculated by using the Kronig-Penny model and its 3D extension. The metal-insulator transition in VO2 is a transition between conductors having three-dimensional and one-dimensional k-vectors.展开更多
The Wigner-Seitz unit cell (rhombus) for a honeycomb lattice fails to establish a k-vector in the 2D space, which is required for the Bloch electron dynamics. Phonon motion cannot be discussed in the triangular coordi...The Wigner-Seitz unit cell (rhombus) for a honeycomb lattice fails to establish a k-vector in the 2D space, which is required for the Bloch electron dynamics. Phonon motion cannot be discussed in the triangular coordinates, either. In this paper, we propose a rectangular 4-atom unit cell model, which allows us to discuss the electron and phonon (wave packets) motion in the k-space. The present paper discusses the band structure of graphene based on the rectangular 4-atom unit cell model to establish an appropriate k-vector for the Bloch electron dynamics. To obtain the band energy of a Bloch electron in graphene, we extend the tight-binding calculations for the Wigner-Seitz (2-atom unit cell) model of Reich et al. (Physical Review B, 66, Article ID: 035412 (2002)) to the rectangular 4-atom unit cell model. It is shown that the graphene band structure based on the rectangular 4-atom unit cell model reveals the same band structure of the graphene based on the Wigner-Seitz 2-atom unit cell model;the π-band energy holds a linear dispersion (ε−k ) relations near the Fermi energy (crossing points of the valence and the conduction bands) in the first Brillouin zone of the rectangular reciprocal lattice. We then confirm the suitability of the proposed rectangular (orthogonal) unit cell model for graphene in order to establish a 2D k-vector responsible for the Bloch electron (wave packet) dynamics in graphene.展开更多
文摘Following Ashcroft and Mermin, the conduction electrons (“electrons” or “holes”) are assumed to move as wave packets. Dirac’s theorem states that the quantum wave packets representing massive particles always move, following the classical mechanical laws of motion. It is shown here that the conduction electron in an orthorhombic crystal moves classical mechanically if the primitive rectangular-box unit cell is chosen as the wave packet, the condition requiring that the particle density is constant within the cell. All crystal systems except the triclinic system have k-vectors and energy bands. Materials are conducting if the Fermi energy falls on the energy bands. Energy bands and gaps are calculated by using the Kronig-Penny model and its 3D extension. The metal-insulator transition in VO2 is a transition between conductors having three-dimensional and one-dimensional k-vectors.
文摘The Wigner-Seitz unit cell (rhombus) for a honeycomb lattice fails to establish a k-vector in the 2D space, which is required for the Bloch electron dynamics. Phonon motion cannot be discussed in the triangular coordinates, either. In this paper, we propose a rectangular 4-atom unit cell model, which allows us to discuss the electron and phonon (wave packets) motion in the k-space. The present paper discusses the band structure of graphene based on the rectangular 4-atom unit cell model to establish an appropriate k-vector for the Bloch electron dynamics. To obtain the band energy of a Bloch electron in graphene, we extend the tight-binding calculations for the Wigner-Seitz (2-atom unit cell) model of Reich et al. (Physical Review B, 66, Article ID: 035412 (2002)) to the rectangular 4-atom unit cell model. It is shown that the graphene band structure based on the rectangular 4-atom unit cell model reveals the same band structure of the graphene based on the Wigner-Seitz 2-atom unit cell model;the π-band energy holds a linear dispersion (ε−k ) relations near the Fermi energy (crossing points of the valence and the conduction bands) in the first Brillouin zone of the rectangular reciprocal lattice. We then confirm the suitability of the proposed rectangular (orthogonal) unit cell model for graphene in order to establish a 2D k-vector responsible for the Bloch electron (wave packet) dynamics in graphene.
