Itinerant Spin Dynamics in Structures of ... - Jacobs University

Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 43
where 〈...〉 denotes the average over the direction of v F and k which we rewrite using
Eq.(C.28) for the given geometry as
(−i∂ y +2e(A S ) y
)C
(
x,y = ± W )
= 0, ∀x, (3.71)
2
where n is the unit vector normal to the boundary ∂S and x is the coordinate along the
wire. The transverse zero-mode Q y = 0 does not satisfy this condition. Therefore, it
is convenient to perform a non-Abelian gauge transformation,[AF01, MC00] so that the
transformed problem has Neumann boundary conditions, and the transformed Cooperon
Hamiltonian can therefore be diagonalized in zero-mode approximation for quantum wires.
Since in quantum wires these boundary conditions apply only in the transverse direction,
a transformation acting in the transverse direction is needed: Ĉ → ˜Ĉ = U A ĈU † A , with
U A = exp(i2e(A S ) y
y). Then, the boundary condition simplifies to −i∂ y ˜C(x,y = ±W/2) =
0, ∀x, and the Hamiltonian changes to
˜H c = Q 2 −2Q SO Q x [cos(Q SO y)S y −sin(Q SO y)S z ]
+Q 2 SO [cos2 (Q SO y)Sy 2 +sin2 (Q SO y)Sz
2
−sin(Q SO y)cos(Q SO y)(S y S z +S z S y )] (3.72)
= (Q+2eàs ) 2 . (3.73)
where the effective vector potential A S , as introduced in Eq.(3.43),
A S = m e
e ˆαS = m e
e
⎛
⎝ 0 −α 2 0
α 2 0 0
⎛ ⎞
⎞ S x
⎠⎜
⎝ S y
S z
⎟
⎠ , (3.74)
is transformed to the effective vector potential às after the transformation U A has been
applied to the Hamiltonian
à s ≡ m e
e ˜ˆα(y)S
⎛
= m e
e
⎝ 0 −α 2cos(Q SO y) −α 2 sin(Q SO y)
0 0 0
⎛
⎞
⎠⎜
⎝
⎞
S x
S y
⎟
⎠ , (3.75)
S z
which varies with the transverse coordinate y on the length scale of L SO . Now, we can
see already that for narrow wires W < L SO , this vector potential varies linearly with y,