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 81<br />
similar manner as has been done previously <strong>in</strong> Ref.[KM02] for wires without SOC, which<br />
showed the crossover <strong>of</strong> the magnetic phase shift<strong>in</strong>g rate, which had been known before <strong>in</strong><br />
the diffusive and ballistic limit, only.<br />
T<strong>of</strong><strong>in</strong>dthespectrum<strong>of</strong>theCooperonHamiltonianwithboundaryconditionsas<strong>in</strong>Sec.4.2.2,<br />
we stay <strong>in</strong> the 0-mode approximation <strong>in</strong> the Q space and proceed as before: Accord<strong>in</strong>g to<br />
Eq.(4.49), the non-Abelian gauge transformation for the transversal direction y is given by<br />
( [ ( ) )<br />
U = exp −i 2α 2 S x +2 α 1 −γ D v 2f 3<br />
S y<br />
]y . (4.51)<br />
f 1<br />
To concentrate on the constant width <strong>in</strong>dependent part <strong>of</strong> the spectrum we extract the<br />
absolute m<strong>in</strong>imum at Q = 0, Fig.(4.4) and Fig.(4.5). A clear reduction <strong>of</strong> the absolute<br />
m<strong>in</strong>imum is visible. Due to the factor f 3 /f 1 <strong>in</strong> the transformation U, the decrease <strong>of</strong> the<br />
m<strong>in</strong>imal sp<strong>in</strong> relaxation depends also on the ratio <strong>of</strong> Rashba and l<strong>in</strong>ear Dresselhaus SOC.<br />
E m<strong>in</strong> /ǫ 2 F γ2 D<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0.<br />
0.5<br />
1.<br />
1.5<br />
2.<br />
2.5<br />
3.<br />
0.2<br />
0.0<br />
5 10 15 20 25 30<br />
N<br />
Figure 4.4: The lowest eigenvalues <strong>of</strong> the conf<strong>in</strong>ed Cooperon Hamiltonian Eq.(4.49), equivalent<br />
to the lowest sp<strong>in</strong> relaxation rate, are shown for Q = 0 for different number <strong>of</strong> modes<br />
N = k F W/π. Different curves correspond to different values <strong>of</strong> α 2 /q s .<br />
From Eq.(4.49) it is clear, that not only the H γD is affected by the reduction<br />
<strong>of</strong> the number <strong>of</strong> channels N but also the shift <strong>of</strong> the l<strong>in</strong>. Dresselhaus SOC, α 1 , <strong>in</strong> the<br />
orbital part. A model to extract the ratio <strong>of</strong> Rashba and l<strong>in</strong>. Dresselhaus SOC developed<br />
<strong>in</strong> Ref.[SANR09] by Scheid et al. did not show much difference between the strict 1D case<br />
and the non-diffusive case with wire <strong>of</strong> f<strong>in</strong>ite width. The results presented here should allow<br />
for extend<strong>in</strong>g the model to f<strong>in</strong>ite cubic Dresselhaus SOC. Deduc<strong>in</strong>g from our theory, the<br />
direction <strong>of</strong> the SO field should change with the number <strong>of</strong> channels due to the mentioned<br />
N dependent shift.