Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
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Chapter 4: Direction Dependence <strong>of</strong> <strong>Sp<strong>in</strong></strong> Relaxation and Diffusive-Ballistic Crossover 77<br />
4.3.2 With cubic Dresselhaus SOC<br />
If cubic Dresselhaus SOC cannot be neglected, the absolute m<strong>in</strong>imum <strong>of</strong> sp<strong>in</strong><br />
relaxation can also shift to k x m<strong>in</strong><br />
= 0. This depends on the ratio <strong>of</strong> Rashba and l<strong>in</strong>.<br />
Dresselhaus SOC:<br />
If q 2 /q 1 ≪ 1, we f<strong>in</strong>d the absolute m<strong>in</strong>imum at k x m<strong>in</strong><br />
= 0,<br />
E m<strong>in</strong>1 = ˜q 3 + ˜q 2 1 +q2 2<br />
2<br />
−∆ c + 1 12 ∆ c(q 2 W) 2 , (4.44)<br />
with<br />
∆ c = 1 2<br />
√<br />
(˜q 3 + ˜q 2 1 )2 +2(˜q 2 1 − ˜q 3)q 2 2 +q4 2 . (4.45)<br />
If q 2 /q 1 ≫ 1, we f<strong>in</strong>d the absolute m<strong>in</strong>imum at k x m<strong>in</strong><br />
≈ ± ∆ 24 (24−(q 2W) 2 ),<br />
) (˜q<br />
E m<strong>in</strong>2 = kx 2 2<br />
m<strong>in</strong><br />
−k x m<strong>in</strong><br />
q 1 ˜q 3<br />
2 2<br />
q2<br />
2 +2 −<br />
16k x m<strong>in</strong><br />
q 2<br />
+∆ 2 + ˜q )<br />
3<br />
(˜q<br />
2<br />
1<br />
+1<br />
2 q2<br />
2 (˜q3˜q 1<br />
2 −<br />
12 − ˜q 3 2q 2<br />
+ q4 2<br />
24 − (˜q<br />
2<br />
1<br />
24 + q2 2<br />
12<br />
3072kx 3 − q2 2<br />
m<strong>in</strong><br />
24 (˜q 3 − ˜q 1)<br />
2<br />
)<br />
q 2 k x m<strong>in</strong><br />
− q ((<br />
2 ˜q<br />
2<br />
3<br />
k x m<strong>in</strong><br />
128 + ˜q 3˜q 1<br />
2<br />
192<br />
)<br />
− ˜q 3q2<br />
2 ))<br />
W 2 . (4.46)<br />
96<br />
We can conclude that reduc<strong>in</strong>g wire width W will not cancel the contribution due to cubic<br />
Dresselhaus SOC to the sp<strong>in</strong> relaxation rate.<br />
4.4 Weak Localization<br />
In Ref.[Ket07] and the previous chapter the crossover from WL to WAL due<br />
to change <strong>of</strong> wire width and SOC strength was expla<strong>in</strong>ed <strong>in</strong> the case <strong>of</strong> a (001) system.<br />
Whether WL or WAL is present depends on the suppression <strong>of</strong> the triplet modes <strong>of</strong> the<br />
Cooperon. The suppression <strong>in</strong> turn is dom<strong>in</strong>ated by the absolute m<strong>in</strong>imum <strong>of</strong> the spectrum<br />
<strong>of</strong> the Cooperon Hamiltonian H c . The f<strong>in</strong>d<strong>in</strong>gs presented <strong>in</strong> Sec.4.2.2 therefore po<strong>in</strong>t out<br />
that e.g. the crossover width, at which the system changes from WL to WAL, can shift<br />
with the wire direction θ. Recently experimental results on WL/WAL by J. Nitta et al.,<br />
Ref.[Nit06], seem to show a strong dependence on growth direction. Our presented results