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4.1 Spin Relaxation Matrix Elements

The surface roughness induced spin relaxation matrix elements, normalized to the intrasubband scattering matrix elements at zero strain, are expressed between the wave functions with opposite spin projections [129]

[ ] dΨi--σ(z)dΨjσ(z) Ms,ij = -dΨ--d(zz)dΨ--(zd)z----- , (--idσz----idσz--)εxy=0 z=± t 2
(4.1)

where σ=±1 is the spin projection to the [001] axis.

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Figure 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.

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Figure 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.

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Figure 4.3: The normalized intersubband spin relaxation matrix element for εxy=0.2% is shown, with the Fermi distribution at 300K.

The up-spin and the down-spin wave functions are orthogonal to each other in every subband which makes the intrasubband relaxation matrix elements zero. The corresponding normalized and squared surface roughness induced intersubband relaxation matrix element (|MS|2) is depicted in Figure 4.1. One finds that, |MS|2 is characterized by very sharp peaks when

2 D εxy - ℏ-kxky-= 0 M
(4.2)

condition is satisfied, which is also known as the spin hot spots. At this condition, the value of δ in Equation 3.6 attains its minimum. The subband splitting is at its minimum at the spin hot spots, which signifies a maximum mixing between up- and down-spin eigenstates. These hot spots should be contrasted with the spin hot spots appearing in the bulk system along the same directions at the edge of the Brillouin zone [71, 67]. The origin of the hot spots in thin films lies in the unprimed subband degeneracy, which effectively projects the bulk spin hot spots from the edge of the Brillouin zone to the center of the 2D Brillouin zone. As soon as the intersubband splitting becomes larger than the spin-orbit interaction strength, the mixing of states caused by this spin-orbit interaction is reduced, signifying the reduction of spin relaxation.

Figure 4.2 shows |MS|2 for an unstrained film. The hot spots are along the [100] and [010] directions. Figure 4.3 describes how the shear strain pushes the spin hot spots to the higher energies outside of the states occupied by carriers. This leads to a reduction of the surface roughness induced spin relaxation.

Influence of Arbitrary Spin Orientation

The position of the spin hot spots in the surface roughness induced normalized and squared intersubband relaxation matrix element (|MS|2) with respect to the kinetic energy of the conducting electrons is shown in Figure 4.4, which again shows that increasing εxy pushes the scattering peak to higher energy states. It has already been described how the spin injection orientation defined by the angle Θ influences the subband wave functions in the section 3.4. Under the same conditions, |MS|2 is observed to be decreasing with increasing Θ, reaching its minimum when Θ=π 2 (i.e. when spin is injected along the OX-direction). The dependence of |MS|2 on Θ and (k x, ky) is evaluated to be [149]

( )2 |MS |2 ∝ 1 + kx- ⋅ cos2Θ. ky
(4.3)

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Figure 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).

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Figure 4.5: The variation of the normalized intersubband spin relaxation matrix elements with Θ (c.f. Figure 3.5) and ϕ1 (with tan(ϕ1) = -ky kx) is described. The domain for Θ is choosen to be (0, π2) as it is repeated in the rest of the domain (π 2, π).

Then one can proceed further to write,

( )2 2 1 + kkx ⋅ cos2Θ --|MS--|(Θ-)---= -------y(---)-----. |MS |2(Θ = 0) 1 + kx- 2 ky
(4.4)

Rewriting Equation 4.4 with tan ϕ1 = -ky kx leads to

|MS |2(Θ ) -----2--------= sin2ϕ1 + cos2 ϕ1 ⋅ cos2Θ. |MS | (Θ = 0)
(4.5)

Figure 4.5 describes the variation of |MS|2 with Θ and tan ϕ 1 = -ky kx. |MS|2 oscillates with respect to ϕ1, but steadily decreases when the spin injection orientation is drawn from perpendicular OZ-axis towards in-plane OX axis. This shows that the spin scattering rate must decrease, when the spin injection orientation is drawn gradually towards in-plane.

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