Electronic band structure in a periodic magnetic field.

We analyze the energy band structure of a two-dimensional electron gas in a periodic magnetic field of a longitudinal antiferromagnet by considering a simple exactly solvable model. Two types of states appear: with a finite and infinitesimal longitudinal mobility. Both types of states are present at a generic Fermi surface. The system exhibits a transition to an insulating regime with respect to the longitudinal current, if the electron density is sufficiently low. PACS number(s): 75.50.Rr, 75.70.Ak To appear in Phys. Rev. B ’96 E-mail: [email protected] The interest in the magnetoconductance properties of the two-dimensional electron gas in spatially periodic lateral magnetic fields has been further stimulated by the recent experimental availability of such systems [1, 2]. In the work of Carmona et al. [1] spatial modulation of a magnetic field was produced by means of equidistantly located superconducting stripes where magnetic vortices were trapped by impurities resulting in periodic inhomogeneity of the external magnetic field, while in the work of of Ye et al. [2] it was produced by deposition of ferromagnetic microstructures on top of the high-mobility two-dimensional (2D) electron gas. Vast theoretical efforts on 2D electron gas in an inhomogeneous external magnetic field range from the theory of momentum-dependent tunneling through a magnetic barrier [3] to properties of electronic states and transoprt in a weakly spatially modulated magnetic field [4–7]. In this short paper we will be concerned with the one-electron energy band structure of the 2D electron gas under a periodic lateral magnetic field of an antiferromagnet, which is a limiting case of a strong periodic modulation. We will show that two types of states appear: with a finite and infinitesimal longitudinal mobility. Both types of states are present at a generic Fermi surface. The system exhibits a transition to an insulating regime with respect to the longitudinal current if the electron density is sufficiently low. The effect of a uniform magnetic field on energy bands produced by the periodic (electric) potential is well known [8]. The impact of the slightly inhomogeneous magnetic field on the Landau levels of a free electron was considered by Müller [4]. He showed that the energy bands exhibit a pronounced asymmetry in the lateral direction. For a spatially modulated magnetic field a common theoretical model [5] employs magnetic field perpendicular to the plane of the two-dimensional electron gas which has a “carrier” field B0 with a periodic modulation on top of it: B = (B0 +B cosKy)ẑ . (1) In the work of Peeters and Vasilopoulos [5] the effect of a periodic electric and weakly modulated magnetic field (B ≪ B0) was considered. They showed that the broadening of the Landau levels is roughly proportional to the modulation amplitude B. 1 The “Hofstadter-like” spectrum was obtained by Wu and Ulloa [6], and collective excitations were analyzed in [7] by the same authors. In this work we will deal with an electron gas confined to a plane in a perpendicular periodic magnetic field without a “carrier” field. In other words, in (1) we take B0 = 0. This corresponds to the extreme case of the other limit, B0 ≪ Bm. Such a periodic field will create an energy band structure of its own. We see this type of arrangement experimentally realizable by bringing a two-dimensional electron gas in close contact with a mesoscopic longitudinal antiferromagnetic (sandwich) structure, without an external magnetic field. When dealing with the one-electron spectrum it is useful to have some exactly solvable models (potentials), as they elucidate the whole structure of the energy bands [9]. Below we show that the energy bands can be obtained exactly in a simple way for a reasonably idealized periodic magnetic field. We present full band picture for both electron with spin and spinless, and discuss the topology of the Fermi surface. The Hamiltonian for a free spinless electron in a magnetic field is: