2.2 Bloch sphere with different qubit states corresponding to the axes is shown. The position of an arbitrary state (χ) can be uniquely represented by the polar (Θ) and the azimuthal angles (Φ).
2.3 Schematic of the Elliot-Yafet and D’yakonov-Perel’ spin relaxation mechanisms.
3.1 Sketch of the diamond crystal lattice is portrayed. Colors gray and red represent A′ atoms and color blue represents A′′ atoms. For silicon, both atoms are identical.
3.2 The first Brillouin zone of the relaxed silicon lattice is shown. The valley positions and high symmetry points are shown as well.
3.3 The silicon band structure calculated by the pseudopotential method (CB is the conduction band, and VB is the valence band) is described. The VB edges are located exactly at the Γ-point and the minimum of the lowest CB lies on the symmetry Δ line close to the X-point. The lowest two CBs degenerate exactly at the X-point.
3.4 Left: constant energy surface of unstrained silicon (six-fold degeneracy) is shown, right: conduction band splitting under shear tensile strain on (001) plane is shown. The red (green) color-fill signifies high (low) electron concentration.
3.5 Sketch showing the spin injection into a (001) thin silicon film of thickness t in an arbitrary direction, described by the polar angle Θ.
3.6 The large component of the wave function of the lowest unprimed subband in an unstrained film located in the valley centered at k0 is shown. Θ =
(c.f. Figure 3.5) is maintained. 3.7 The absolute value of the large (small) component of the spin wave functions reduces (increases), when the spin injection changes gradually from OZ- to OX-direction. kx, ky, and εxy are set to be 0.4nm-1, 0.4nm-1, and 0.5% respectively.
3.8 Intersubband splitting is shown as a function of shear strain εxy for different values of the sample thickness t, and for kx=0.25nm-1 and ky=0.25nm-1.
4.1 The normalized and squared intersubband spin relaxation matrix element (|MS|2) is shown as function of the shear strain ε xy, and for an arbitrary (kx,ky) pair (sample thickness t=2nm). Spin is oriented along OZ-direction (Θ=0, c.f. Figure 3.5). The splitting of the lowest subbands (valley splitting) is also shown.
4.2 The normalized intersubband spin relaxation matrix element (c.f. Equation 4.1) for an unstrained sample is shown, with the Fermi distribution at 300K.
4.3 The normalized intersubband spin relaxation matrix element for εxy=0.2% is shown, with the Fermi distribution at 300K.
4.4 The variation of the normalized intersubband spin relaxation matrix elements with the kinetic energy of the conduction electrons in [110] direction is depicted. The influence of the spin injection direction is also shown (t=1.36nm).
4.5 The variation of the normalized intersubband spin relaxation matrix elements with Θ (c.f. Figure 3.5) and ϕ1 (with tan(ϕ1) = -
) is described. The domain for Θ is choosen to be (0,
) as it is repeated in the rest of the domain (
, π). 4.6 The dependence of the total spin expectation (SOZ) over a certain (kx, ky) pair is shown, when spin is injected along the OZ-direction (εxy=0.5%).
4.7 The dependence of the total spin expectation (SOX) over a certain (kx, ky) pair is shown, when spin is injected along the OX-direction (εxy=0.5%).
4.8 The precession of the injected spin (along OX- and OZ-directions) around the existing spin-orbit field (SOF) is portrayed (tan ϕ1 = -
). 4.9 The four phonon modes are found in elemental semiconductors: (a) longitudinal acoustic, (b) transversal acoustic, (c) longitudinal optical, and (d) transversal optical modes.
4.10 The dependence of the spin lifetime including the surface roughness (SR), the longitudinal acoustic (LA) phonon, and the transversal acoustic (TA) phonon mediated components on the temperature and for different values of the electron concentration (εxy=0, t=1.36nm) is shown.
4.11 The dependence of the spin lifetime and its surface roughness and acoustic phonon induced components over a wide range of εxy is shown. The film thickness is t=2.5nm, T=300K, and the electron concentration is NS=1012cm-2.
4.12 The dependence of the spin lifetime on the shear strain εxy is depicted for a film thickness of t=1.36nm, and an electron concentration of NS=1012cm-2.
4.13 The variation of the spin lifetime with its inter- and intrasubband components with εxy is shown. The film thickness is t=2.1nm, T=300K, and the electron concentration is NS=1012cm-2.
4.14 The variation of the surface roughness mediated spin lifetime with its inter- and intrasubband components as a function of εxy at two distinct values of temperature is depicted. The film thickness is t=1.36nm, and the elctron concentration is NS=1012cm-2. The variation of the Fermi levels and the minimum energies (kx=ky=0) of the lowest unprimed subbands with εxy is shown (inset).
4.15 The variation of the surface roughness and the phonon mediated components of the spin lifetime with εxy is shown, when the spin injection orientation (represented by the angle Θ) is used as a parameter. The film thickness is t=1.36nm, T=300K, and the electron concentration is NS=1012cm-2.
4.16 The variation of inter- and intrasubband components of the spin lifetime with εxy is depicted, when Θ is used as a parameter (c.f. Figure 4.15).
4.17 The variation of the spin lifetime with the spin injection orientation angle Θ at any fixed value for εxy is shown. The analytical expression can be found at Equation 4.37.
4.18 The normalized and squared intersubband scattering matrix element is shown as function of εxy and for several pairs of (kx, ky). The film thickness is set to t=2nm.
4.19 The subband energies for the first and the minimum of the second subband (i.e. at kx=ky=0) as a function of εxy with the same conditions as in Figure 4.18 is shown. The points where the two cross are highlighted by arrows.
4.20 The normalized and squared intrasubband scattering matrix elements are shown as function of εxy (kx=ky=0.25nm-1, and t=2.7nm).
4.21 The variation of the momentum relaxation time with εxy is shown with its surface roughness (SR) and phonon (Ph) mediated components at two distinct temperatures. The film thickness is t=1.36nm, and the electron concentration is NS=1012cm-2.
4.22 The variation of the momentum relaxation time with εxy is shown with its inter- and intrasubband components corresponding to Figure 4.21.
4.23 The variation of the normalized and squared intersubband spin relaxation matrix elements as function of εxy is depicted, where kx=0.3nm-1, k y=1nm-1, and for different values of ΔΓ (t=2.72nm).
4.24 The variation of the normalized and squared intersubband spin relaxation matrix elements with εxy is shown, when the spin injection direction is taken as a parameter (ΔΓ=5.5eV, t=1.36nm).
4.25 The variation of the surface roughness (SR) and the phonon (Ph) mediated spin lifetime with εxy is shown, when all possible values for ΔΓ (c.f. Equation 4.45) are considered. The sample thickness is t=2.72nm, T=300K, and the electron concentration is NS=1012cm-2.
4.26 The energies of the two lowest unprimed subbands with εxy at two distinct ΔΓ values are shown. The sample thickness is t=2.72nm, T=300K, and the electron concentration is NS=1012cm-2.
4.27 The variation of the spin lifetime and its inter- and intrasubband components with εxy are depicted, when t=2.72nm, the electron concentration NS=1012cm-2, and Δ Γ=5.5eV.
4.28 The variation of the spin lifetime and its surface roughness (SR), the longitudinal (LA) and the transversal (TA) acoustic phonon mediated components with εxy is shown. ΔΓ=5.5eV (c.f. Equation 4.45), the sample thickness t=1.36nm, T=300K, and the electron concentration NS=1012cm-2 are used.
4.29 The variation of the spin lifetime and its components (along with the optical phonon Op mediated component) with εxy is shown when t=2.72nm and the other parameters are as given in Figure 4.28.
4.30 The variation of the spin lifetime and its components with εxy is shown when t=4.34nm and the other parameters are as given in Figure 4.28.
4.31 The dependence of the minimum energies for primed and unprimed subbands as well as the Fermi energy on strain is shown for Figure 4.30.
4.32 The prediction of the spin lifetime with the valley splitting results (c.f. Figure 4.26) is highlighted.
4.33 The variation of the spin lifetime with the valley splitting is described, when ΔΓ=5.5eV. The sample thickness is t=1.36nm, T=300K, and the electron concentration is NS=1012cm-2. The spin injection orientation is used as a parameter.
5.1 The two-dimensional representation of the Voronoi box is shown. The domain ˜ Vi contains the mesh point i and is surrounded by six adjacent mesh points. Ai,j represents the boundary between the box around the ith node and its neighbor j. d i,j implies the distance between the ith and the jth nodes.
5.2 The schematic shows the simulation set up for the spin injection in Si from a ferromagnetic semiconductor (FMS). The left boundary (x = -
) is grounded, and the right boundary (x = 
) is
under the voltage bias (Uc). The interface is shown as a dotted line. The
doping (effective density of states) in the Si side is ND (NC), and in the
FMS is K1ND (K2NC). 5.3 The analytically calculated spin densities in the channel are shown, when the applied voltage (Uc) is used as a parameter. The bulk spin polarization in the ferromagnetic semiconductor (FMS) is P=20%.
5.4 The spin current density (α0) and spin density (β0) injection efficiencies at the Si interface are shown as a function of the applied electric field ˜ E. Lines→ theory and dots→ simulation (P=10%).
5.5 The spin current density (α0) and spin density (β0) injection efficiencies at the Si interface as a function of the applied electric field ˜ E (c.f. Figure 5.4) is shown (P=50%).
5.6 A plot of Mup = V T ln (
) and Mdown = V T ln (
)
through the bar (P=10%, 
=2μm-1 where ˜ E is the applied electric
field) is depicted, showing a discontinuity at the junction, which gives
the term G (c.f. Equation 5.35a and Equation 5.35b). n↑eql (n
↓eql) is the
up(down)-spin concentration at the thermal equilibrium. 5.7 The spin density accumulation near the junction over a channel of 4μm is shown, when the current density (Jn) is fixed to 23.4MA/m2 with P=20%. K 1 is used as a parameter (c.f. Table 5.1). λD represents the Debye length (c.f. Equation 5.30).
5.8 The variation of the electric potential through the channel is described, related to Figure 5.7.
5.9 The spin current density through the channel is shown, with the same conditions as in Figure 5.7. With the notations as described in Equation 5.16b and Equation 5.19b, the direction of the spin current is from the ferromagnetic semiconductor FMS towards Si.
5.10 The spin density accumulation through the channel is depicted, with the same conditions as in Figure 5.7. Both K1 and K2 are used as parameters (c.f. Table 5.1).
5.11 The spin density and the spin current density injection efficiencies (βD and αD respectively) are shown, taken at a Debye length away from the interface towards Si, with the same conditions as in Figure 5.7.
5.12 The schematic portrays the simulation set up for the spin injection in a Si bar. The left boundary (x=0) is grounded, and the right boundary (x = W) is under a voltage bias (Uc).
5.13 The analytically (c.f. Equation 5.44) calculated spin densities through the channel are shown. The boundary spin density polarization β0=50%. The channel length is 3μm.
5.14 The analytically (c.f. Equation 5.47) calculated spin current densities are shown, corresponding to Figure 5.13. The directions of the drift and the diffusive components of the spin current density are highlighted.
5.15 The variation of the spin current density in the channel with the boundary spin density polarization β0 is shown (Uc=0, W=3μm).
5.16 The schematic depicts the spin extraction phenomenon from the Si bar towards the left boundary (x=0). The left boundary is under zero-bias, and the right boundary (x = W′, where W′ ≫ L i with Li as the intrinsic spin diffusion length) is under a negative bias (Uc<0). The direction of the electric field is also shown.
5.17 The normalized critical current density is shown for spin-blockade as a function of the boundary current density spin polarization.
5.18 The variation of the electron current density with MCh (c.f. Equation 5.56) and the bias voltage Uc is depicted, when the channel length is 4μm.
5.19 The electron concentration is shown near the charge injection boundary (c.f. Equation 5.56). The current density is 11.9MAm-2, and the channel length 4μm. λD represents the Debye length.
5.20 The variation of the spin current density is delineated with the same conditions as in Figure 5.19.
5.21 The spin current density is shown for up to 5 times the Debye length λD from the left boundary, when spin is injected from a charge neutral and charge accumulated source. The carrier current is absent. The left boundary for (1) is set with n↑0=900ND and n↓0=100ND, and for (2) c.f. Equation 5.57.
5.22 The charge and the spin distribution over the channel are shown under the charge accumulation and with the boundary condition Equation 5.57 (MCh=100mV, Uc=-300mV). The inset figure shows the spin density polarization β near the spin injecting interface. The current density 11.9MAm-2 is maintained over a 4μm channel.
5.23 The spin and spin current densities in depletion (MCh=-100mV, Uc=-140mV) and for charge neutrality (Uc=-204mV) with the boundary as in Equation 5.57 are shown. The inset figure shows the spin density polarization β. The current density 11.9MAm-2 is maintained over a 4μm channel.
5.24 The spin current densities with their up(down)-spin components are depicted, when the channel is in depletion (MCh=-100mV) with the boundary as in Equation 5.57. The current density 7.9MAm-2 is maintained.
5.25 The spin current density for accumulation, charge neutrality, and depletion with the boundary condition Equation 5.57 are shown. The current density 7.9MAm-2 is maintained. The channel length is 3μm.