Momentum Relaxation Time Simulations

4.5 Momentum Relaxation Time Simulations

In the previous sections one has seen how intersubband transition predominantly determines the total spin lifetime τs. The shear strain εxy can remove the degeneracy between the equivalent valleys to cause the reduction of intersubband spin-flip and hence the giant increment in τs. As the momentum scattering times for all the individual components can also be calculated by the thermal averaging as as described in Equation 4.12, one must calculate the total momentum scattering time τm as a function of εxy to investigate how the inter- and intrasubband transitions impact τm.

4.5.1 Momentum Scattering Matrix Elements

The simplified expression for the surface roughness SR induced momentum scattering matrix elements, normalized to the intrasubband scattering matrix elements at zero strain, is calculated as [129]

[ dΨiσ(z)dΨjσ(z) ] M = ------dz----dz----- , m,ij (dΨiσ(z) dΨiσ(z))εxy=0 t dz dz z= ±2

(4.40)

where Ψ_i,j are the subband eigenfunctions as obtained earlier. i, j=1, 2 are the subband indices, and σ=± 1 is the spin projection to the [001] axis.

Figure 4.18: The normalized and squared intersubband scattering matrix element is shown as function of εxy and for several pairs of (k_x, k_y). The film thickness is set to t=2nm.

Figure 4.19: The subband energies for the first and the minimum of the second subband (i.e. at k_x=k_y=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.

Figure 4.20: The normalized and squared intrasubband scattering matrix elements are shown as function of εxy (k_x=k_y=0.25nm^-1, and t=2.7nm).

Figure 4.18 shows the variation of the normalized and squared intersubband scattering matrix elements with εxy, for an arbitrary in-plane projection of the k vector. In contrast to Figure 4.1 no peak is noticed in this case. The intersubband scattering attains its minimum when Equation 4.2 is met [163]. In each case, the intersubband scattering is noticed to be vanishing after a certain strain point. To explain this behavior, the variation of the corresponding energies of the first lowest subband with εxy, and the minimum energy of the second subband are shown in Figure 4.19. When former goes below the latter, no intersubband scattering would be possible as because the kinetic energy is not sufficient to occupy the states after scattering in the second subband. Figure 4.20 shows the respective normalized and squared intrasubband scattering matrix elements, for a certain (k_x, k_y) pair. The stress-induced unprimed subband splitting is illustrated (c.f. right hand side). The valley splitting minimum is observed to be around εxy=0.145% in agreement with Equation 4.2.

4.5.2 Calculation of the Momentum Relaxation Rates

Analogous to Equation 4.16, the surface roughness SR induced momentum relaxation rate is

1 2π ∑ ∫ 2π 1 ℏ4 |k | --------- = ------- dφ ⋅ π △2L2 ⋅-2----------⋅---2 ⋅|---2--| τi,SR (k1) ℏ(2π )2j=1,2 0 ϵij|k2 - k1 | 4m l ||∂E∂(kk2)|| ( ) 2 [(dΨik1σ )*( dΨjk2σ) ]2 - |k2 - k1 |2L2 ⋅ ------- ------- t ⋅ exp -------------- dz dz z= ±2 4 ⋅ θ(E (k ) - E (0)). j 2 j

(4.41)

k2 is the in-plane wave vector after scattering, φ is the angle between the k1 and k2 vectors, ϵ_i,j is the dielectric permittivity, L is the autocorrelation length, and △ is the mean square value of the surface roughness fluctuations [164]. σ = ±1 is the spin projection to the [001] axis. This scattering rate is modeled according to [81], who adapted the original approach of [87] for scattering at two interfaces.

Analogous to the calculation of the spin-flip rate, the phonon induced momentum scattering rate can be written as [69]

1 2πK T ∑ ∫ 2π dφ 1 |k | --------- = ----B2--- ---⋅ -----|---2--| τi,PH (k1) ℏρ νPH j 0 2π (2π)2||∂E∂(kk2)|| | | 2 [ ||∂E-(k2)||f(E (k2))] ⋅ 1 - |--∂k2-|--------- ||∂E-(k1)||f(E (k )) ∫ ∂k1 1 t [ † PH ]*[ † PH ] ⋅ 2π dz ψjk2σ (z )M ψik1σ(z) ψ jk2σ(z)M ψik1σ(z) 0 ⋅ θ(Ej(k2 ) - E (j0)).

(4.42)

Here,

νPH = 2νTA-+-νLA-, 3

(4.43)

where ν_TA=5300 m-
s is the transversal phonon velocity, ν_LA=8700 m-
s is the longitudinal phonon velocity, ρ=2329 Kmg3- is the silicon density [158], t is the film thickness, and M^PH can be described as

⌊ ⌋ Ξ 0 0 0 P H || 0 Ξ 0 0 || M = ⌈ 0 0 Ξ 0 ⌉ , 0 0 0 Ξ

(4.44)

with Ξ=12eV [165] as the acoustic deformation potential.

Figure 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 N_S=10¹²cm^-2.

Figure 4.22: The variation of the momentum relaxation time with εxy is shown with its inter- and intrasubband components corresponding to Figure 4.21.

The dependence of the total momentum relaxation time (τm) and its surface roughness and phonon mediated components on the applied shear stress strength εxy is shown in Figure 4.21 for two different temperatures. It is clear that for a film thickness of t=1.36nm the surface roughness scattering effect dominates the total momentum relaxation time. This can be seen in Figure 4.21 for T=300K as well as T=153K. While the surface roughness component is not much affected by temperature, the phonon mediated momentum relaxation rate (time) increases (decreases) strongly with increasing temperature. For this reason, τm drops with increasing temperature. Within the εxy range of 0 to 1.8%, the increment of τm is around 130% at T=300K, and 150% at T=153K for the film thickness of t=1.36nm.

In order to explain the observed behavior, the inter- and intrasubband components of the calculated momentum relaxation time is shown in Figure 4.22. It is revealed that intrasubband scattering solely determines the momentum relaxation time over a wide range of shear strain and at any temperature range. This is in agreement with the selection rule that the elastic processes result in strong intrasubband momentum relaxation [166, 129].

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