Because of the paramount importance of silicon on insulator (SOI) and FinFET 3D technology for the technology nodes from 22nm to 14nm and beyond in order to achieve a tighter confinement and thus better electrostatic control to support miniaturization as discussed earlier, it is expected that spin relaxation will further increase in the conducting channels [55]. A systematic investigation of the electron spin relaxation time in silicon was conducted a long time ago [56]. The estimate for the spin lifetime at room temperature obtained is of the order 0.1-10ns [44], which corresponds to a spin diffusion length maximum of 2μm. A long spin transfer distance of conduction electrons through an undoped 350μm thick silicon wafer has been shown in a ground breaking experiment [57], and hence the spin propagation up to such a distance combined with the possibility of injecting spin at room [58] or even at elevated temperature [59] makes the fabrication of spin-based switching devices made of silicon quite plausible in the near future. In contrast, a large experimentally observed spin relaxation in electrically-gated silicon structures is an obstacle for a successful realization of spin-driven devices [60, 61]. Therefore, more research on all of the different spin relaxation mechanisms and also means to improve the spin lifetime in silicon films is needed.
The spin lifetime is determined by the spin-flip processes, and several spin relaxation mechanisms in a semiconductor can be attributed [62]. The two most significant relaxation mechanisms in metals and in semiconductors are the Elliot-Yafet [63, 64] (EY) mechanism and the D’yakonov-Perel’ [65] (DP) mechanism, and this will be explained in detail in the following chapters.
It is well known that the conduction band in silicon consists of three pairs of equivalent valleys, with their energy minima located close to the corresponding X-points of the first Brillouin zone. The theory of spin relaxation in thin silicon films must account for the most relevant scattering mechanisms which are due to the electron-phonon (Ph) interaction and the surface roughness (SR) scattering [66, 63, 67, 68]. The surface roughness at the two interfaces is assumed to be equal and statistically independent. It is described by a mean and a correlation length [69]. The spin relaxation due to the electron-phonon interaction is taken care of in the deformation potential approximation [70]. The electron-phonon interaction can further be distinguished into two sub-categories. The acoustic-phonon mediated relaxation, i.e. the longitudinal(LA)- and the transversal(TA)-acoustic phonons, and the optical phonons. The most prominent contribution to the spin relaxation in bulk silicon has been identified to be due to the optical (Op) phonon scattering between the valleys residing at different crystallographic axes [53, 71]. Nevertheless, in order to investigate the impact of the surface roughness and the acoustic phonon mediated spin-flip, only the relevant [001] oriented valley pair with spin degree included must be considered, as it produces the low energy unprimed subband ladder under the confinement potential [72]. The unprimed subbands in the unstrained (001) film are degenerate, without spin-orbit effects included [72]. An accurate inclusion of the intrinsic spin-orbit interaction results in a large mixing between the spin-up and spin-down states, resulting in spin hot spots characterized by strong spin relaxation. To remove this spin mixing and thus increase the spin lifetime one has to introduce mechanical stress into such structures.
The influence of stress on the carrier (both electron and hole) transport in semiconductors [73] has been known for over half a century. That the application of stress may influence the intrinsic electron mobility in silicon was predicted by Hall and Bardeen in 1951 [74]. The development of the generalized deformation potential theory can be accredited to [75, 76] explaining this effect. Nowadays, stress is regularly used in industry to efficiently increase the transistor drive current by enhancing the mobility of carriers in the channel [77, 78], as well as in non-classical CMOS structures [79, 80]. The strain εxy produced by an uniaxial tensile shear stress in [110] direction is known to lift the degeneracy between the unprimed subbands in the [001] oriented valley pair [72], thus it is predicted to decrease the mentioned spin mixing in such a way that the intersubband spin relaxation rate is reduced and the spin lifetime gets enhanced.
Now in order to calculate the spin lifetime, one needs to know the spin relaxation matrix elements. To estimate the matrix elements, the subband wave functions and also the eigenenergies must be known [81, 72]. One way to calculate the subband wave functions is to use the k ⋅ p method.
The k ⋅ p method was first introduced by Seitz [82] and later extended to study semiconductor band structures [83, 84, 85]. This method in combination with a Hamiltonian including strain [86] has been a reliable and computationally inexpensive method to study the stress-induced valence band modification. An effective two-band k ⋅ p based method for the relevant [001] oriented valley pair written at the vicinity of the X-point suitable to describe the electron subband structure in the presence of the shear strain εxy [72] is generalized to include the spin degree of freedom [71]. The height of the potential barrier corresponds to the potential energy barrier at the semiconductor-oxide interface for a silicon-on-insulator (SOI) structure.
In this work the subband wave functions and the eigenenergies of the unprimed valley pair are scrutinized, and the corresponding matrix elements are shown. The spin lifetime calculation steps are explained. A giant enhancement of the spin lifetime with all its components as a function of the shear strain is elucidated. The role of inter- and intrasubband transitions in determining the spin lifetime is explained, and the role of the former is observed to be dominant. In contrast and under the same conditions, the momentum relaxation time is found to be solely determined by the intrasubband transitions. The subband energies of the primed valley pair is taken into account to incorporate the influence of the optical phonons. The increase of the optical phonon mediated spin lifetime is not found to be as severe as the other mechanisms. The unprimed subband splitting in a relaxed silicon structure [87] has also been a prominent area of research for a long time, and this is incorporated to estimate the splitting between the equivalent subbands, the matrix elements, and finally the spin relaxation time.
The spin injection direction sensitive spin lifetime model is investigated in detail. It is observed that an arbitrary spin injection direction modifies the subband wave functions, the intersubband spin relaxation matrix element, and henceforth all the distinct spin relaxation mechanisms.