Download Interplay between valley-orbit couplings at donor atoms and

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)