Spin Hot Spots and Spin Precession

4.2 Spin Hot Spots and Spin Precession

As the spin hot spot condition is characterized by a strong increase of the mixing of the up- and down-spin states, the equivalent subband splitting at the spin hot spots is purely determined by the effective spin-orbit term, which is in turn linear with △SO ∘ -2----2-
kx + ky , where k_x and k_y are the components of the in-plane electron wave vector. This linear dependence of the splitting is similar to the Zeeman splitting in a magnetic field [152]. Therefore, the spin-orbit interaction term △SO with =(k_x, -k_y) can be interpreted as an effective magnetic field known as spin-orbit field (SOF), while the pairs of states (X₁, ↑), (X_2^′, ↓) and (X_2^′, ↑), (X₁, ↓) it couples have similarities with the Zeeman up- and down-spin states split because of the effective field [152]. It is now obvious that the angle ϕ₁ defined in the section 3.4 represents the direction of the SOF. The spin injection orientation impacts the spin precession at the spin hot spots and some of the related effects are now studied.

4.2.1 Spin Expectation Value

One can estimate the spin expectations which depend on the spin injection orientation. The spin Pauli matrices for the two [001] valleys can be written as,

⌊ 0 1 0 0 ⌋ | | σx = | 1 0 0 0 | , ⌈ 0 0 0 1 ⌉ 0 0 1 0

(4.6a)

⌊ ⌋ 0 - i 0 0 || i 0 0 0 || σy = ⌈ 0 0 0 - i ⌉ , 0 0 i 0

(4.6b)

⌊ ⌋ 1 0 0 0 | 0 - 1 0 0 | σz = |⌈ 0 0 1 0 |⌉ . 0 0 0 - 1

(4.6c)

Now one can express the expectation values of the spin projection on the coordinate axes, denoted as ⟨Sn,p⟩ (n=1,2 and p ∈ x,y,z),

∫ t ⟨Sn,p⟩ = Ψ†nσp Ψndz. 0

(4.7)

The total spin expectation can be represented as,

∘ ∑---------- ⟨Sn⟩ = ⟨Sn,p⟩2. p

(4.8)

When spin is injected along OZ-axis (Θ=0), ⟨Sn,x⟩ = ⟨Sn,y⟩ =0 but ⟨Sn,z⟩ =1. It is spotted that at spin hot spots, the value also drops to zero. When spin is injected along OX-axis (Θ = π
2 , Φ=0), ==0 but =1. On the contrary at the spin hot spots [153, 154],

2( (kx )) ⟨Sn,x⟩ = sin arctan k-- , y

(4.9a)

( ( kx) ) ⟨Sn,y⟩ = - 0.5sin 2 arctan k , y

(4.9b)

⟨Sn,z⟩ = 0,

(4.9c)

thus making ⟨Sn⟩ = sin arctan kx
ky . When spin is injected along an arbitrary direction on the XZ-plane, one can obtain at the spin hot spots,

2 ( (kx-)) ⟨Sn,x⟩ = sin arctan k ⋅ sin Θ, y

(4.10a)

( ( kx) ) ⟨Sn,y⟩ = - 0.5 sin 2 arctan k ⋅ sinΘ, y

(4.10b)

⟨Sn,z⟩ = 0,

(4.10c)

and hence

( ( kx) ) ⟨Sn⟩ = sin arctan --- ⋅ sin Θ. ky

(4.11)

Figure 4.6: The dependence of the total spin expectation (S_OZ) over a certain (k_x, k_y) pair is shown, when spin is injected along the OZ-direction (εxy=0.5%).

Figure 4.7: The dependence of the total spin expectation (S_OX) over a certain (k_x, k_y) pair is shown, when spin is injected along the OX-direction (εxy=0.5%).

Figure 4.6 and Figure 4.7 portray the values for ⟨Sn⟩ over a range of (k_x, k_y) pairs, when spin is injected along OZ- and OX-directions respectively. The value of always remains to one, but its value drops to zero when the spin is injected along OZ-direction and when the spin hot spot condition is reached. On the contrary when the spin is injected along OX-direction, ⟨Sn⟩ = sin arctan kkxy at the spin hot spots. When Θ is increased from zero to the maximum, ⟨S ⟩
n gradually increases at the spin hot spots. This indicates that the spin relaxation rate (lifetime) is expected to decrease (increase), when the injection orientation is drawn towards in-plane.

4.2.2 Spin Precession

Figure 4.8: The precession of the injected spin (along OX- and OZ-directions) around the existing spin-orbit field (SOF) is portrayed (tan ϕ₁ = - ky
kx

The effective spin-orbit field SOF is given by △SO ˜
k and lies on the XY -plane along the (k_x, -k_y) direction. The electron spin starts precessing around the SOF when injected along any direction. One can see that (c.f. Figure 4.8), when the spin is injected along OZ-direction, the average spin projection on the OX- and the OY -axes will always be zero. On the contrary when the spin injection is gradually drawn towards the OX-direction, the spin projection value gradually increases. This phenomenon is correlated with the already obtained spin expectation values at the spin hot spots. Because of the zero spin expectation value at any pair of (k_x, k_y) resulting in maximal spin randomization, the spin relaxation rate (time) is predicted to be strongest (weakest) for perpendicular-plane spin injection.

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