The ground state of two-dimensional electrons in a nonuniform magnetic field
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The Ground State of Two-Dimensional Electrons in a Nonuniform Magnetic Field A. M. Dyugaeva,b and P. D. Grigorieva,* aLandau
Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia bMax-Planck-Institute for the Physics of Complex Systems, Dresden D-01187, Germany *e-mail: [email protected] Received July 9, 2005
Abstract—An exact solution to the Schrödinger equation for the ground state of two-dimensional Pauli electrons in a nonuniform transverse magnetic field H is presented for two cases. In the first case, the field H depends on a single variable, H = H(y), while in the second case, the field is axially symmetric, H = H(ρ), ρ2 = x2 + y2 . The electron density distributions n = n(y) and n = n(ρ) that correspond to a completely filled lower level are found. For quasiuniform fields of fixed sign, the functions n(y) and n(ρ) are locally related to the magnetic field: n(y) = H(y)/φ0 and n(ρ) = H(ρ)/φ0 , where φ0 = hc/|e| is a magnetic flux quantum. Magnetic fields are considered that are periodic, singular, and bounded in the plane xy. Finite electron objects in a nonuniform magnetic field are analyzed. PACS numbers: 73.20.–r, 73.20.At, 75.70.–i DOI: 10.1134/S1063776106010080
1. The spectrum of two-dimensional electrons in a uniform transverse magnetic field H is characterized by a high degree of degeneracy. Each level En, σ is degenerate with multiplicity HS/φ0 , where φ0 = hc/|e | is a magnetic flux quantum and S is the area of the domain that is accessible to electrons, 1 E n, σ = ⎛ n + --- + σ⎞ ω c , ⎝ ⎠ 2
bound by the magnetic field. In other words, the degree of degeneracy is the same as in a uniform field with the same flux H (2). The result of [1] is essentially based on the fact that electrons obey the linear Dirac equation. In the case of doubly periodic magnetic fields, the ground state of two-dimensional electrons was studied in [2]. In the general statement of the problem, which was considered in [1, 2], it is difficult to determine the observables, for example, the electron density distribution n(x, y) that corresponds to the completely filled, saturated ground state E0, –1/2 = 0 in a given nonuniform field H(x, y). 2. In this paper, we consider two simple types of a nonuniform field that admit the determination of the function n(x, y). The first field depends on a single variable, H = H(y), and the second field is axially symmetric, H = H(ρ), where ρ2 = x2 + y2. For these fields, the exact solution to the problem is strikingly simple; therefore, we can immediately present the main results. For a one-dimensional field H = H(y), the wavefunctions ψp(x, y) of the ground state are expressed as (see Section 4 below for the detailed verification)
(1)
He ω c = ---------- . Mc In Eq. (1), σ = ±1/2 is the projection of the electron spin onto the direction of the magnetic field H, ωc is the cyclotron frequency, and e and M are the charge and mass of an electron. In a nonuniform field H = H(x, y), the degeneracy of En, σ is lifted for all le
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