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 2<br />
<strong>Sp<strong>in</strong></strong> <strong>Dynamics</strong>: Overview and<br />
Analysis <strong>of</strong> 2D Systems<br />
2.1 Short Rem<strong>in</strong>der on the Orig<strong>in</strong> <strong>of</strong> <strong>Sp<strong>in</strong></strong> Orbit Coupl<strong>in</strong>g<br />
The<strong>in</strong>teraction whichmakes sp<strong>in</strong>tronicdevices so<strong>in</strong>terest<strong>in</strong>g is arelativistic effect,<br />
theSOC.The<strong>in</strong>tr<strong>in</strong>sicdegree<strong>of</strong>freedomsp<strong>in</strong>isadirectconsequence<strong>of</strong>theLorentz<strong>in</strong>variant<br />
formulation <strong>of</strong> quantum mechanics. Expand<strong>in</strong>g the relativistic Dirac equation <strong>in</strong> the ratio<br />
<strong>of</strong> the electron velocity and the speed <strong>of</strong> light up to the order (v/c) 2 (the derivation can be<br />
found <strong>in</strong> standard text books like Ref.[Sak67]) one gets<br />
( p<br />
2<br />
2m e0<br />
−eϕ− p4<br />
8m 3 e0 c2 −<br />
e<br />
4m 2 e0<br />
c2σ<br />
·(∇ϕ×p)−<br />
e<br />
)<br />
8m 2 e0 c2∆ϕ ψ = (E −m e0 c 2 )ψ,<br />
with the electrostatic potential ϕ, and the free electron mass m e0 . Our <strong>in</strong>terest concerns<br />
the so called Thomas term −e/(4m 2 e0 c2 )σ·(∇ϕ×p). In atomic physics we assume that the<br />
electric field is a central field, E(r) = −(dϕ/dr)e r which leads to<br />
with ŝ = σ/2.<br />
− e<br />
e<br />
4m 2 ·(∇ϕ×p) = −<br />
e0c2σ (− 1 4m 2 e0 c2 r<br />
e 1<br />
= −<br />
2m 2 e0 c2 r<br />
)<br />
dϕ<br />
σ ·(r×p) (2.1)<br />
dr<br />
dϕ<br />
(2.2)<br />
drŝ·L<br />
≡ λŝ·L, (2.3)<br />
Duetothelattice-periodic potential <strong>in</strong>acrystall<strong>in</strong>esolidthiseffect canhavestrong<strong>in</strong>fluence<br />
6