Electron diffraction by periodic arrays of quantum antidots J.-P. Leburton and Yu. B. Lyanda-Geller Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ~Received 16 May 1996! Electron diffraction by a periodic array of repulsive d barriers is an analytically solvable quantum- mechanical problem. In this geometry, bearing some analogy with single-barrier tunneling, incident electrons are perpendicular to the periodic barrier of antidots. In contrast to conventional quasi-one-dimensional tunnel- ing, which conserves the component of the electron wave vector transverse to the current, electron diffraction occurs through multiple channels characterized by the transverse electron wave vectors differing by the recip- rocal lattice vector of the periodic array. For a one-dimensional ~1D! array of two-dimensional ~2D! d poten- tials we predict highly nonlinear characteristics in the vicinity of Fermi energies when a new channel for diffraction opens up. Two lines of 1D arrays reveal a rich resonant diffraction structure. @S0163-1829~96!04644-9# I. INTRODUCTION The possibility of realizing lateral superstructures by modulating the electric potential in a two-dimensional ~2D! electron gas with the expectation of novel electronic proper- ties has been anticipated by Sakaki 1 twenty years ago. In the meantime, with the continuous development of nanostructure technology, a wide class of superstructures has been pro- posed for the investigation of novel quantum transport ef- fects and their applications in high functional devices. 2 Pres- ently, many phenomena resulting from the periodic modulation of the electron gas have been observed at low temperature or in the mesoscopic regime, mainly because of the difficulty of confining or modulating the electron gas over short distances in more than one direction. Recent progresses in self-assembled microstructures with nanometer size features and the manipulation of single atoms by scan- ning tunneling microscopy have created new opportunities for realizing nanostructures with strong confinement of the order of the de Broglie wavelength at room temperature. 3–7 It becomes therefore possible to generate three-dimensional ~3D! configurations of molecular scale structures with quantum-mechanical properties and transport phenomena not yet envisioned. The simplest configurations of periodic nanostructures are short-period arrays of quantum antidots or quantum anti- wires, which act as diffraction centers for incident electrons perpendicular to the plane of the arrays ~see Fig. 1!. From a physical point of view, this problem bears some analogy with the diffraction of light by a lattice of small apertures, but also with the von Laue diffraction of x rays by crystals. Aside from this analogy, the problem is also interesting from a transport viewpoint since the geometrical configuration is reminiscent of tunneling configuration across a ~single or double! potential barrier. However, because of the periodic- ity in the plane ~the direction perpendicular to the current!, the transverse component of the electron wave vector is no longer conserved for coherent transport processes. Formally, the problem cannot be treated within a one-dimensional ~1D! model by separation of variables as would be, for instance, the case in a tunneling problem across a periodic potential such as the Kroenig-Penney model, nor automatically treated with a perturbative technique using a scattering formalism, since the potential does not vanish at the infinity in the array plane. In the present paper, we consider a class of 2D and 3D problems in diffraction geometry for which the periodicity of quantum antidots or antiwires permits exact analytical quantum-mechanical solutions, and provides the wave func- tions in the whole space. We find that in contrast to quasi-1D tunneling, which conserves the transverse component of the wave vector, electron diffraction occurs through multiple channels characterized by transverse electron wave vectors that differ from each other and from the wave vector of the incident electron by reciprocal lattice vectors of the periodic arrays as one can expect from the von Laue and the Wolf- Bragg formula for x-ray diffraction in crystals. As we will see by using the periodicity of the arrays it becomes possible to solve 2D or 3D Schro ¨ dinger equations analytically in cases when variables are inseparable and the problem cannot be reduced to 1D equations. Meanwhile, we will also show that the opening of diffraction channels results in highly non- linear tunneling characteristics of two distinct conductance regimes separated by a sharp transition at the Fermi energy corresponding to the half of the reciprocal lattice vector of the periodic arrays. We proceed as follows: In Sec. II we describe the diffrac- tion geometry and the electron scattering model; Sec. III deals with solutions of the Schro ¨ dinger equation for several periodic barriers. Finally, in Sec. IV we calculate the tunnel- ing current for these barriers. II. DIFFRACTION GEOMETRIES AND MODEL In this section we consider several configurations of quan- tum antidot and antiwire arrays for the 2D or 3D diffraction of electrons. In our search for analytical solutions of the Schro ¨ dinger equation, we model the repulsive potential of the quantum structures by a d function. This approximation is justified if the geometrical dimensions of the diffraction center are relatively small, but its potential strength relatively important. PHYSICAL REVIEW B 15 DECEMBER 1996-II VOLUME 54, NUMBER 24 54 0163-1829/96/54~24!/17716~8!/$10.00 17 716 © 1996 The American Physical Society