Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University Itinerant Spin Dynamics in Structures of ... - Jacobs University
40 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems which has its cause in the suppression of the triplet modes in Eq.(3.46), is indeed a direct measure of the spin relaxation. Mathematically, there exists a unitary transformation H c = U CD H SD U † CD, (3.60) ⎛ ⎞ U CD = ⎜ ⎝ − 1 √ 2 i √ 2 0 0 0 1 1√ 2 i √ 2 0 ⎟ ⎠ , (3.61) with the according transformation between spin-density components s i and the triplet components of the Cooperon density ˜s, 1 √ 2 (−s x +is y ) = ˜s ⇈ , (3.62) s z = ˜s ⇉ , (3.63) 1 √ (s x +is y ) = ˜s . 2 (3.64) This is a consequence of the fact that the four-component vector of charge density ρ = (ρ + + ρ − )/2 and spin-density vector S are related to the density vector ˆρ with the four components 〈ψ † αψ β 〉/ √ 2, where α,β = ±, by a unitary transformation. Relation to the Diffuson The classical evolution of the four-component density vector ˆρ is by definition governed by the diffusion operator, the Diffuson. The Diffuson is related to the Cooperon in momentum space by substituting Q → p−p ′ and the sum of the spins of the retarded and advanced parts, σ and σ ′ , by their difference. Using this substitution, Eq.(3.48) leads thus to the inverse of the Diffuson propagator H d := ˆD −1 = Q 2 +2Q SO (Q y˜Sx −Q x˜Sy )+Q 2 D (˜S SO y 2 + ˜S x 2 ), (3.65) e with ˜S = (σ ′ −σ)/2, which has the same spectrum as the Cooperon, as long as the timereversal symmetry is not broken. In the representation of singlet and triplet modes the diffusion Hamiltonian becomes ⎛ √ 2Q 2 SO +Q 2 2Q SO Q − 0 − √ ⎞ 2Q SO Q + √ 2Q SO Q + Q 2 SO H d = +Q2 0 0 . (3.66) ⎜ ⎝ 0 0 Q 2 0 ⎟ − √ ⎠ 2Q SO Q − 0 0 Q 2 SO +Q 2
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 41 Comparing Eqs. (3.49) and Eq.(3.66), we see that diagonalization leads to Eqs. (3.50)- (3.52). Influence of Dresselhaus SOC It can be seen from Eqs. (3.58) and (3.59) that in the case of a homogeneous Rashba field, the spin-density has a finite decay rate as we pointed out in Sec.2.3.4. However, if we go beyond the pureRashba system and include a linear Dresselhaus coupling, the first term in Eq.(3.33), we can find spin states which do not relax and are thus persistent. The spin relaxation tensor, Eq.(3.56), acquires nondiagonal elements and changes to ⎛ ⎞ 1 2 1 (k F ) = 4τkF 2 α2 −α 1 α 2 0 ⎜ 1 ˆτ s ⎝ −α 1 α 2 2 α2 0 ⎟ ⎠ , (3.67) 0 0 α 2 with α = √ α 2 1 +α2 2 . For Q = 0 and α 1 = α 2 = α 0 , we find indeed a vanishing eigenvalue with a spin-density vector parallel to the spin-orbit field, s = s 0 (1,1,0) T . Moreover there are two additional modes which do not decay in time but are inhomogeneous in space: the persistent spin helices,[BOZ06, LCCC06, OTA + 99, WOB + 07, KWO + 09] ⎛ ⎞ 1 ( ) S = S 0 ⎜ ⎝ −1 ⎟ 2π ⎠ sin (x−y) L SO 0 ⎛ ⎞ 0 ( ) √ +S 0 2 ⎜ ⎝ 0 ⎟ 2π ⎠ cos (x−y) , (3.68) L SO 1 (Fig.3.5) and the linearly independent solution, obtained by interchanging cos and sin. Here, L SO = π/m e √ 2α0 . One has to keep in mind that this solution is not an eigenstate anymore in a quantum wire. However, we will show that there exist also long persisting solutions in a quasi-1D case. It is worth to mention that in the case where cubic Dresselhaus coupling in Eq.(3.33) cannot be neglected, the strength of linear Dresselhaus coupling α 1 is shifted[Ket07] to ˜α 1 = α 1 −m e γ D E F /2, as we noted before, Eq.(3.44), and, e.g., in the Q = 0 case, the spin relaxation rate becomes ( 1 α = 2p 2 2 2 − ˜α 2 2 1) F τ s α 2 2 + τ +D e (m 2 eE F γ D ) 2 . (3.69) ˜α2 1
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40 Chapter 3: WL/WAL Crossover and <strong>Sp<strong>in</strong></strong> Relaxation <strong>in</strong> Conf<strong>in</strong>ed Systems<br />
which has its cause <strong>in</strong> the suppression <strong>of</strong> the triplet modes <strong>in</strong> Eq.(3.46), is <strong>in</strong>deed a direct<br />
measure <strong>of</strong> the sp<strong>in</strong> relaxation. Mathematically, there exists a unitary transformation<br />
H c = U CD H SD U † CD, (3.60)<br />
⎛ ⎞<br />
U CD =<br />
⎜<br />
⎝<br />
− 1 √<br />
2<br />
i √<br />
2<br />
0<br />
0 0 1<br />
1√<br />
2<br />
i √<br />
2<br />
0<br />
⎟<br />
⎠ , (3.61)<br />
with the accord<strong>in</strong>g transformation between sp<strong>in</strong>-density components s i and the triplet components<br />
<strong>of</strong> the Cooperon density ˜s,<br />
1<br />
√<br />
2<br />
(−s x +is y ) = ˜s ⇈ , (3.62)<br />
s z = ˜s ⇉ , (3.63)<br />
1<br />
√ (s x +is y ) = ˜s . 2<br />
(3.64)<br />
This is a consequence <strong>of</strong> the fact that the four-component vector <strong>of</strong> charge density ρ =<br />
(ρ + + ρ − )/2 and sp<strong>in</strong>-density vector S are related to the density vector ˆρ with the four<br />
components 〈ψ † αψ β<br />
〉/ √ 2, where α,β = ±, by a unitary transformation.<br />
Relation to the Diffuson<br />
The classical evolution <strong>of</strong> the four-component density vector ˆρ is by def<strong>in</strong>ition<br />
governed by the diffusion operator, the Diffuson. The Diffuson is related to the Cooperon<br />
<strong>in</strong> momentum space by substitut<strong>in</strong>g Q → p−p ′ and the sum <strong>of</strong> the sp<strong>in</strong>s <strong>of</strong> the retarded<br />
and advanced parts, σ and σ ′ , by their difference. Us<strong>in</strong>g this substitution, Eq.(3.48) leads<br />
thus to the <strong>in</strong>verse <strong>of</strong> the Diffuson propagator<br />
H d :=<br />
ˆD<br />
−1<br />
= Q 2 +2Q SO (Q y˜Sx −Q x˜Sy )+Q 2<br />
D (˜S SO y 2 + ˜S x 2 ), (3.65)<br />
e<br />
with ˜S = (σ ′ −σ)/2, which has the same spectrum as the Cooperon, as long as the timereversal<br />
symmetry is not broken. In the representation <strong>of</strong> s<strong>in</strong>glet and triplet modes the<br />
diffusion Hamiltonian becomes<br />
⎛ √<br />
2Q 2 SO<br />
+Q 2 2Q SO Q − 0 − √ ⎞<br />
2Q SO Q +<br />
√ 2Q SO Q + Q 2 SO<br />
H d =<br />
+Q2 0 0<br />
. (3.66)<br />
⎜<br />
⎝ 0 0 Q 2 0 ⎟<br />
− √ ⎠<br />
2Q SO Q − 0 0 Q 2 SO<br />
+Q 2