Valley Splitting in Unstrained Films

4.6 Valley Splitting in Unstrained Films

The [001] equivalent valley coupling through the Γ-point results in a subband splitting in confined electron structures [87]. This subband splitting is not limited to strained structures and thus must be taken properly into account for the calculation of the spin lifetimes in relaxed structures as well. The valley splitting in a silicon quantum well at zero strain as a function of the quantum well width has been studied for a long time. The expression for the valley splitting can be written as [55]

-2π2Δ-Γ-- ΛΓ = (k ⋅ t)3 ⋅ |sin(k0Γ t)|. 0Γ

(4.45)

where ΔΓ is the splitting at Γ-point, t is the film thickness, and the other parameters are listed in Table 3.2.

The values of the valley splitting obtained from a 30-band k ⋅ p model can be found in [167]. This method was developed around the Γ-symmetry point of the Brillouin zone for strained silicon, germanium, and SiGe alloys [145]. On the other hand, a theory based on the localized-orbital approaches has been developed to describe the valley splitting observed in silicon quantum wells in the limit of low electron density by using a sp³d⁵s^* spin-orbit coupled tight-binding model [168]. The use of sp³s^* tight binding model has also been investigated [169]. Nevertheless, the values of the valley splitting obtained from a 30-band k ⋅ p model and the tight binding models are summarized and analyzed in [170]. By using the simplified analytical expressions,

ΔΓ=2eV for the 30-band k ⋅ p model, and
ΔΓ=5.5eV for the tight-binding model,

one can reproduce the corresponding data. Both methods reproduce the features of the conduction and valence band equally well, but require additional experimental verification at higher energies where discrepancies appear.

Once ΛΓ is known, one can modify the dispersion equation Equation 3.5 as shown below.

┌ ----------------- │ ( )2 ℏ2k2z- ℏ2(k2x-+-k2y) │∘ ℏ2kzk0- 2 E (k ) = 2m + 2m ± m + δc, l t l

(4.46)

with

∘ -----(---)6---- δc = δ2 + -k0- Δ2Γ , k0Γ

(4.47)

where δ is given as in Equation 3.6.

4.6.1 Spin Relaxation Matrix Elements

Figure 4.23: The variation of the normalized and squared intersubband spin relaxation matrix elements as function of εxy is depicted, where k_x=0.3nm^-1, k_y=1nm^-1, and for different values of Δ Γ (t=2.72nm).

Figure 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).

The surface roughness induced normalized and squared intersubband spin relaxation matrix element |MS|2, with and without the Δ Γ term, are compared in Figure 4.23. It is noted that the spin relaxation hot spot peaks are greatly diminished to become smoother with increasing values of ΔΓ. It has also been mentioned that the difference in the matrix elements’ values calculated with and without the ΔΓ term can reach two orders of magnitude [150]. As like before, the peaks can be correlated with the unprimed subband splitting minima. However, the peaks remain well pronounced, and still attain the maximum at the spin hot spots. Figure 4.23 also depicts how the minimum of the valley splitting increases when the ΔΓ term increases.

Figure 4.24 describes how |MS|2 is reduced with increasing spin injection angle Θ, for any pair of (k_x, k_y). This behavior is consistent with Equation 4.4, indicating that this equation is general and it is applied in both bulk silicon and thin silicon films (with or without presence of the unstrained subband splitting term ΔΓ). Therefore, the spin relaxation rate can be suppressed with increasing Θ.

4.6.2 Spin Lifetime Calculations

Figure 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 N_S=10¹²cm^-2.

The variation of the surface roughness (SR) and the phonon (Ph) mediated spin relaxation components of the total spin lifetime τs with the shear strain εxy, for different ΔΓ is shown in Figure 4.25. One can observe that the increase of τs is less pronounced, when the value of ΔΓ increases. However, even when ΔΓ is taken to be 5.5eV, the value of τs increases by two orders of magnitude. It can also be seen that at high εxy, the total spin lifetime and its components converge to the same value independent of ΔΓ [171].

Figure 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 N_S=10¹²cm^-2.

Figure 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 N_S=10¹²cm^-2, and Δ Γ=5.5eV.

In order to understand this behavior, the spin-flip caused by the intra- and intervalley transitions needs to be analyzed. Figure 4.26 shows the energy levels of the two lowest unprimed subbands, which are primarily responsible for the spin relaxation at the different ΔΓ values. The unprimed subbands are degenerate at zero strain without the ΔΓ term. The ΔΓ term lifts the degeneracy even at zero strain. The figure also shows how the increasing εxy inflicts the subband splitting between the unprimed valley pair. With increasing εxy, the influence of ΔΓ vanishes as is obvious from Equation 4.46. Figure 4.27 depicts the inter- and intrasubband scattering components of τs at two distinct temperatures. The intersubband spin-flip process remains dominant in determining the total spin lifetime τs. However, with increasing strain its influence becomes weaker, in accordance with Figure 4.26. At high valley splitting, the spin relaxation is determined by intrasubband scattering which does not depend on ΔΓ.

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