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Since the publication of famous Hofstadter paper [1], research interest in the spectral properties of twodimensional periodic arrays in a magnetic field has greatly increased. The fractal structure of the spectrum has been a theoretical result for long time, but subsequently, experimental confirmation was obtained. Electron tunneling through periodic array of quantum dots in a homogeneous magnetic field has been intensively investigated over the last few years because it can be relevant for nanotechnology applications [2–4]. There are several different approaches for building models of quantum dot arrays. In this paper, the zero-range potential model [5,6] based on the theory of self-adjoint extensions of symmetric operators is used (see, e.g., [7–12] and references in [13]). One of its benefits is that the model is explicitly solvable. In double-layer lattices, the layer number provides an additional degree of freedom, which leads to some interesting experimentally measurable physical effects [14]. We suggest several explicitly solvable models for double two-dimensional layer of quantum dots using the approach described in [15]. Because of high quality of nanostructure devices, large Fermi wave length (i.e. the de Broglie wavelength of electrons with energy close to the Fermi energy) (4×10−8 m) and long mean free path of electron (10−5 m) can be observed. Therefore, one deals with the ballistic regime of electron transport. In this case, the Landauer-Buttiker formalism can be used to derive the conductivity σ for the nanostructure with several leads from the transmission coefficient T (E)|E=EF (here EF is the Fermi energy). In the simplest case (one incoming and one outgoing channel), the Landauer formula has the form σ = e T h̄(1−T ) [16]. Here, e is the electron charge, h̄ is the Planck’s constant. The geometry of a nanostructure can influence transmission coefficient significantly [17]. In this article we, consider tunneling in the system consisting of double two-dimensional periodic array (with square or hexagonal (honeycomb) lattices in each layer) of quantum dots with two connected semi-infinite leads orthogonal to the plane of the array. We study the influence of the magnetic field and tunneling electron energy on the transmission coefficient and compare our results with the tunneling through the corresponding singlelayer periodic arrays studied earlier [18, 19]. The main ideas of the present paper was described in an extended abstract [20]. We start from the Hamiltonian of a single particle in constant homogeneous magnetic field B. Let us assume that particle has mass m and electric charge e. Since the space is R, we choose a standard basis i, j,k such that B is collinear to k: B = Bk, B 6= 0. Then, the particle can move freely along z axis, and the state space of the model is L(R), where R is the plane based on vectors i, j. The Hamiltonian H0 has the following form:

Model of tunnelling through double quantum layer in a magnetic field