k-Linear Spin Splitting in III-V Semiconductors With Reduced Symmetry
Andrzej Skierkowski and Jacek A. Majewski
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Hoża 69,
PL-00-681 Warszawa, Poland
An ability to control the spin transport in semiconductors is essential issue in
semiconductor spintronics and information processing. One of the methods to achieve control
of the electron spin degree of freedom involves the intrinsic spin-orbit interaction (SOI). In
semiconductors the spin-orbit effect appears as an interaction of the electron spin with an
effective magnetic field, whose direction and magnitude depend on the electron momentum.
The specific form of this dependence is determined by the crystal symmetry. Therefore, the
form of SOI in semiconductors can be modified by crystal deformation and can influence the
spin current. Such effect of strain-induced SOI on spin transport has been demonstrated
experimentally [1]. On the other hand, the strain can be associated with mechanical motion of
solid, in particular, with oscillations in nanomechanical systems. This effect, in turn,
facilitates the tuning of SOI with nanomechanical oscillations and allows the design of
interesting devices [2]. Therefore, detailed knowledge of the strain induced SOI in
semiconductors is essential for understanding, modeling, and design of future functional
devices.
Traditionally, the strain effects on the spin splitting caused by SOI are described by the
Hamiltonian introduced many years ago by Pikus and Titkov [3]. However, the microscopic
analysis of the strain induced spin splitting is still lacking. In this paper, we study the spin
splitting in the strained zinc-blende semiconductors by means of ab initio calculations in the
framework of the density functional theory (DFT) with non-scalar relativistic effects taken
non-perturbatively into account. We employ the standard local density approximation (LDA)
as well as more sophisticated LDA+U scheme. Our calculations for typical III-V
semiconductors show that the strain induces linear-k spin splitting in the conduction and
valence bands in the neighborhood of Γ point. Our studies reveal also new features of the SOI
that are not included in the traditionally employed empirical Hamiltonian [3]. In particular, we
find out that the SOI depends on the atomistic structural details of the strained semiconductor,
and therefore cannot be described by the phenomenological Hamiltonian. In contrary, the
first-principles calculations reveal the physical mechanisms that determine the strain induced
linear-k spin splitting. They show that terms quadratic in strain play important role and
neglecting them leads to not correct structure of spin splitting.
On the basis of the first principles calculations, we construct the effective Hamiltonian
that takes effects of the internal strain on spin splitting into account. We determine the
material constants in the effective Hamiltonian by comparing the spin splitting obtained from
the ab initio calculations with the analytic solutions of the effective Hamiltonian. It turns out
that for the conduction band, heavy hole, light hole, and split-off bands the effective
Hamiltonian describing the k-linear spin splitting does not include terms responsible for
mixing of the valence bands. From ab initio calculation of the spin splitting, we are able to
determine also the third order in k terms in the effective Hamiltonian that has been previously
obtained through group analysis [4].
[1] Y. K. Kato, et al., Phys. Rev. Lett. 93, 176601 (2004).
[2] A. G. Mal’shukov et al. , Phys. Rev. Lett. 95, 107203 (2005).
[3] G. E. Pikus and A. N. Titkov, in Optical Orientation, edited by F. Meier and B. P. Zakharchenya
(North Holland, Amsterdam, 1984).
[4] X. Cartoixa, L.-W. Wang, D. Z.-Y. Ting, and Y.-C. Chang, cond-mat/0511538