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

112 List of Figures
3.12 The relative magnetoconductivity ∆σ(B) − ∆σ(B = 0) in units of 2e 2 /2π,
with the same parameters and number of modes as in Fig.3.11. . . . . . . . 53
3.13 Width of wire WQ SO at which there is a crossover from negative to positive
magnetoconductivity as function of the lower cutoff 1/D e Q 2 τ SO ϕ. . . . . . . 54
3.14 Spectrum in the case of adiabatic boundaries. Except for states which are
polarized in z direction, the relaxation rates for all states diverge at small
wire widths, Q SO W ≪ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.15 (a) Real and (b) imaginary parts of the spectrum of the 2D Cooperon with
Zeeman term of strength gµ B B/D e Q 2 = 0.4. E SO B,2D,1 (black), E B,2D,2 (red
dashed), E B,2D,3 (green), E B,2D,4 (blue dashed). Dashed vertical lines are
located at K x = ±1/ √ 2, the wave vector where the triplet mode E T− and
the singlet mode E S are crossing each other (without loss of generality K y = 0). 59
3.16 (a) Real and (b) imaginary parts of the spectrum of the 2D Cooperon with
Zeeman term of the strength gµ B B/D e Q 2 SO
= 0...2 in steps of 0.25 (w.l.o.g
K y = 0). The B independent mode E T0 is not shown.. . . . . . . . . . . . . 61
3.17 (a) Real and (b) imaginary parts of the spectrum of the Cooperon with
Zeeman term of the strength gµ B B/D e Q 2 SO
= 0...0.8 in steps of 0.1 in a
finite wire of the width Q SO W = 0.5. The B independent mode E t0 is not
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.18 The magnetoconductance ∆σ(B) in a magnetic field perpendicular to the
quantum well, where its coupling via the Zeeman term is considered by exact
diagonalization while its effect on the orbital motion is considered effectively
by the magnetic phase-shifting rate 1/τ B (W) for wire width Q SO W = 1,
dephasingrate 1/τ ϕ = 0.06D e Q 2 SO, andelastic-scattering rate 1/τ = 4D e Q 2 SO.
ThestrengthofthecontributionoftheZeemantermisvariedbythematerialdependent
factor ˜g = gµ B H s /D e Q 2 SO
in the range ˜g = 0...1.5 in steps of 0.5:
The system changes from positive magnetoconductivity (˜g = 0, green) to
negative one (˜g = 0.5...1.5, continuous decrease of the absolute minimum). 64
3.19 (a) Change of crossover width W c with g factor: The magnetic field is included
as an effective field 1/τ B and in the Zeeman term. The strength of the
contribution of the Zeeman term is varied by the material dependent factor
˜g = gµ B H s /D e Q 2 . (b) Change of crossover width W SO c with Zeeman field:
To calculate W c , we fix the Zeeman field to a certain value, horizontal axis,
while we vary the effective field independently and calculate if negative or
positive magnetoconductivity is present. For different Zeeman fields B Z /H s ,
we find thereby a different width W c . Here, we set g/8m e D e = 1. In (a)
and (b), the cutoff due to dephasing is varied: 1/D e Q 2 SOτ ϕ ∈
= 0.04,0.06,0.08
(lowest first). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Dependence of the W 2 coefficient in Eq.(4.27) on the lateral rotation (θ).
The absolute minimum is found for α x1 (θ = 0) = −q 2 (here: q 1 /q 2 = 1.63)
and for different SO strength we find the minimum at θ = (1/4+n)π, n
if q 1 < (q s3 / √ 2) (dashed line: q 1 = (q s3 / √ 2)) and at θ = (3/4+n)π, n
else. Here we set q s3 = 0.9. The scaling is arbitrary. . . . . . . . . . . . . . 73