Spin Lifetime Simulations

4.4 Spin Lifetime Simulations

Figure 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.

Figure 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 N_S=10¹²cm^-2.

Figure 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 N_S=10¹²cm^-2.

The observed spin lifetime in unstrained thin films is much different compared to bulk samples. Experiments show a dramatic reduction of the spin relaxation time in Si/SiO₂ interfaces [159]. Figure 4.10 describes how the total spin lifetime (τs) and its surface roughness and phonon mediated components vary with the temperature in an unstrained silicon thin-film, for different electron concentrations N_S. With increasing temperature, the number of hot spot points which lie in the energy range determined by the term f(E)(1 - f(E)) increases [160]. In combination with the Fermi level lowering this results in the reduction of the spin lifetime. The opposite trend is observed when N_S increases. Increasing the electron concentration shifts the Fermi level upwards and thus increases the spin lifetime. The surface roughness mediated component of τs becomes prominent for a sample thickness of as low as t=1.36nm. Thus, τs in unstrained silicon film is significantly affected by the electron concentration and temperature. It is further noticed that with increasing temperature, the difference between the spin lifetimes for different values of the electron concentration becomes less pronounced.

4.4.1 Spin Lifetime Enhancement with Shear Strain

Figure 4.11 shows the contributions of the surface roughness (SR), longitudinal acoustic (LA), and transversal acoustic (TA) induced spin lifetime for a sample thickness of t=2.5nm at room temperature. The spin injection orientation is perpendicular (Θ=0) to the (001) surface (c.f. Figure 3.5). Several orders of magnitude enhancement is noticed with increasing shear strain εxy for the total spin lifetime τs and its all individual components. The variation of τs with εxy at different temperatures is shown in Figure 4.12. An orders of magnitude increase of τs is noticed for all four evaluated temperatures, and τs is highly sensitive to the operating temperature at each stress point. One can confirm that, at higher temperatures the electron-phonon scattering rate significantly increases causing the reduction of τs.

4.4.2 Inter- and Intrasubband Components

Figure 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 N_S=10¹²cm^-2.

Figure 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 N_S=10¹²cm^-2. The variation of the Fermi levels and the minimum energies (k_x=k_y=0) of the lowest unprimed subbands with εxy is shown (inset).

In order to elucidate the spin relaxation mechanism, the spin-flip caused by the intra- and intersubband scattering must be analyzed. The corresponding components of the spin lifetime at the room temperature (RT) for a sample thickness of t=2.1nm is shown in Figure 4.13. It is revealed that the major contribution to the spin relaxation time τs comes from the intersubband processes. This dependence of τs on intersubband transition is attributed to the presence of the spin hot spots. It is further noted that, for a thickness t=2.1nm when εxy ≥ 1.4%, the intrasubband component also turns out to be non-negligible.

Figure 4.14 delineates the surface roughness induced spin relaxation time with its inter- and intrasubband components at two distinct temperatures and at a very low sample thickness. One can see that, when the temperature decreases the Fermi level energy increases (c.f. Figure 4.14 inset), the intersubband scattering becomes less efficient already at lower values of the shear strain component εxy, and the total intrasubband component of the spin relaxation time starts playing the significant role at higher εxy. It is also revealed that at higher εxy, the surface roughness scattering intrasubband components become close to each other for different temperatures [161]. Therefore, the increase of the total spin lifetime (τs) is a consequence of the fact that εxy introduces a splitting between the usually degenerate unprimed subbands. This lifting of the degeneracy is the crucial factor for the spin lifetime enhancement, as the splitting pushes out the regions of large mixing between the spin-up and spin-down states to higher energies outside of the occupied states (c.f. Figure 4.4). The degeneracy between equivalent valleys has been a longstanding problem in silicon spintronics [55].

4.4.3 Effect of Spin Injection Orientation

Figure 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 N_S=10¹²cm^-2.

Figure 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).

Figure 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.

Figure 4.15 shows how the surface roughness and the phonon mediated components of the total spin lifetime (denoted as τs) depend on the spin injection direction over a wide range of shear strain εxy. They increase with the spin injection polar angle Θ with equal proportionally factor. This happens as the inter- and intrasubband components of τs are equivalently sensitive to Θ, as those increase with increasing Θ. This phenomenon is described in Figure 4.16. Accordingly, τs increases with Θ for all values of the shear strain εxy. τs attains the maximum value for an in-plane spin injection. This result can be correlated with the earlier findings that the spin randomization decreases with increasing Θ at the spin hot spots, and accordingly the spin relaxation rate decreases and the spin lifetime increases (c.f. the section 4.2).

Now, one needs to investigate the dependence of the total spin lifetime τs (and the inter- and intrasubband components) on Θ at a fixed stress point. Figure 4.17 highlights the ratio of τs for an arbitrary Θ value compared to a perpendicular-plane injection. An analytical expression describing this dependence can be deduced by averaging |MS|2 (c.f. Equation 4.3) over the in-plane momentum vector and can be expressed as

--1--- 2 τ(Θ ) ∝ 1 + cos Θ. s

(4.37)

In such a condition,

--τs(Θ-)-- = -----2----, τs(Θ = 0) 1 + cos2Θ

(4.38)

and therefore,

( ) τs Θ = π- = 2 ⋅ τs(Θ = 0). 2

(4.39)

So, the total spin lifetime τs is increased by a factor of two when injected in-plane (OX-direction) compared to the perpendicular-plane (OZ-direction). It is mentioned here that this increase of spin lifetime by the factor of two for an in-plane injection has also been mentioned for bulk silicon [67, 162].

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