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Interplay between valley-orbit couplings at donor atoms and interfaces in silicon nanostructures Mykhailo Klymenko, Françoise Remacle Department of Theoretical Physical Chemistry, B6c, University of Liege, Belgium e-mail: [email protected] 1. Introduction and theoretical model Recent progress in silicon nanoelectronics allows designing devices whose operational principles are based on charge, spin and energy quantization effects. The envelope function approximation implemented in the framework of the effective mass theory is a very efficient tool for studying energy spectra of inhomogeneous semiconductor media [2]. However, direct applications to doped silicon nanostructures face significant difficulties due to the valleyorbit (VO) coupling effect at donor atoms [3] and interfaces [4]. In spite of a large number of papers devoted to its treatment, the accurate description of VO coupling at a single donor atom subject to several interfaces and of VO coupling in a cluster of donor atoms remain a challenge for theory and modeling. In this work, we study interplay of VO couplings caused by different crystal lattice defects located next to each other. All our computations are based on a general approach for the description of VO coupling based on the unification of the multi-valley envelope-function developed in [5] for donor atoms and the contact-potential approach developed in [4] for Si/SiO2 interfaces. The resulting theoretical model is based on the effective mass approximation and involves four fitting parameters: three for the donor atom and one for the interface. The parameters are defined in terms of atomistic pseudopotentials and periodic Bloch functions. They do not depend on the applied electrostatic fields, the confinement potential and the number of impurities. 2. Results In order to assess the validity of our methodology we compared our results with those given by the tight-binding method [6] for the case of a single donor atom subject to a single Si/SiO2 interface and applied electrostatic fields. The comparison evidences a good agreement for the ground state as well as excited states over a wide range of field strengths. The results shown in Fig 1a highlight the interplay between the VO couplings caused by the central cell potential of the donor atom and the VO coupling at the interface as function of the applied electrostatic field strengths. At small strengths the VO coupling at donor atom dominates, and at large fields the VO coupling at the interface takes over. In Fig. 1 b and Fig. 1 c we show the computed energy spectrum for a single donor atom subject to two perpendicular interfaces. This system is of interest since it can serve as a physical model of a Fin-FET single-atom transistor. Of particular interest is the point A, where the donor atom is located in a corner of a silicon nanobox at equal distances from both interfaces X and Z. In this point, a high level of degeneracy for the first excited state is observed. In Fig. 1 d, we present the results of the energy spectrum computations for two interacting donor atoms. In this case, both atoms contribute to the VO splitting. However, due to the interference between their wave functions, their contribution could be enhanced or suppressed depending on the internuclear distance. Depending on the valley population of each state, some potential curves in Fig. 1 d oscillate fast with the internuclear distance, while others are characterized by a smooth dependence on the internuclear distance without oscillations. In the case of the fast oscillating curves, the corresponding wave functions are composed of bulk silicon states whose wave vectors have large nonzero components along the internuclear axis of the donor molecule. The absence of oscillations indicates that the corresponding wave function is characterized by a large population in the valleys oriented perpendicular to the internuclear axis of the donor molecule. The results evidence that each defect of the crystal lattice causes its own kind of the VO coupling. When several defects are next to each other, they produce cooperative contribution to the VO splitting which is strongly dependent on the geometrical parameters of the system. Taking advantage of the combined effects of VO coupling allows to engineer the energy spectrum of doped silicon nanostructures getting desired spacing between energy levels. 3. Acknowledgment This work is supported by the proactive collaborative project TOLOP (318397) of the Seventh Framework Programme of European Commission. FR acknowledges support from Fonds National de la Recherche Scientique, Belgium. a) d) b) c) Figure 1 - Interplay of the VO coupling at the donor atoms and interfaces: a) the dependence of the energy spectrum on the applied electrostatic field for the donor atom subject to the single Si/SiO 2 interface; b) and c) the dependence of the energy spectrum on the distance to the Z and X interfaces correspondingly (the dashed lines stand for results computed neglecting the VO coupling), d) the energy spectrum of diatomic donor molecular ion in silicon oriented along [100] crystallographic axis 4. References [1] “Single-atom nanoelectronics,” Eds. E. Prati, T. Shinada, CRC Press (2013) [2] G. Bastard. Wave Mechanics Applied to Semiconductor Heterostructures, Wiley: New York (1991) [3] A. Debernardi, A. Baldereschi, and M. Fanciulli. “Computation of the stark effect in p impurity states in silicon,” Phys. Rev. B, Vol. 74, p. 035202 (2006) [4] Mark Friesen and S. N. Coppersmith. “Theory of valley-orbit coupling in a Si/SiGe quantum dot,” Phys. Rev. B, Vol. 81, p. 115324 (2010) [5] M.V. Klymenko and F. Remacle. “Electronic states and wavefunctions of diatomic donor molecular ions in silicon: multi-valley envelope function theory,” J. Phys.: Condens. Matter, Vol. 26, p. 065302 (2014) [6] R. Rahman, et al, “Orbital Stark effect and quantum confinement transition of donors in silicon,” Phys. Rev. B, Vol. 80, p. 165314 (2009)