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

List of Figures 111
3.3 Exemplification of the second term in Eq.(3.4): Interference of electrons
traveling in the opposite direction along the same path causes an enhanced
back-scattering, the WL effect. (a) Closed electron paths enclose a magnetic
fluxfromanexternalmagneticfield, indicatedastheredarrow, breakingtime
reversal symmetry, breaking constructive interference. (b) The entanglement
of spin and charge by SO interaction causes the spin to precess inbetween
two scatterers around an axis which changes with the momentum vector of
the itinerant electron. This effective field can cause WAL. . . . . . . . . . . 30
3.4 2D spectrum of H c , K i = Q i /Q SO . The physical meaning of the gaps in the
triplet modes is more comprehensible if H c is related to spin diffusion, where
the gaps appear as spin relaxation rates. . . . . . . . . . . . . . . . . . . . . 38
3.5 Persistent spin helix solution of the spin-diffusion equation for equal magnitude
of linear Rashba and linear Dresselhaus coupling, Eq.(3.68). . . . . . 42
3.6 Longpersistingspinhelixsolutionofthespin-diffusionequationinaquantum
wirewhosewidthW issmaller thanthespinprecessionlengthL SO forvarying
ratio of linear Rashba α 2 = αsinϕ and linear Dresselhaus coupling, α 1 =
αcosϕ, Eq.(3.82), for fixed α and L SO = π/m e α. . . . . . . . . . . . . . . . 46
3.7 Dispersion of the triplet Cooperon modes for different dimensionless wire
units Q SO W: (a) Q SO W = 2, (b) Q SO W = 8, (c) Q SO W = 12, (d) Q SO W =
30, plotted as function of K x = Q x /Q SO . For Q SO W ≫ 3, E {t0,0} and E {t−,0}
evolve into degenerate branches for large K x . (For Q SO W = 30, not all
high-energy branches are shown.) . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8 Probability density of the Cooperon eigenmodes in the wire for Q SO W/π =
30. (a) 3D plot, (b) density plot for one of the two lowest branches, showing
their edge mode character. (c) 3D plot and (d) density plot of the density of
the third lowest mode, which shows bulk character. . . . . . . . . . . . . . . 49
3.9 Absolute minima of the lowest eigenmodes E {t0,0} , E {t−,0} , and E {t+,0} plotted
as function of Q SO W/π = 2W/L SO . We note that the minimum of E {t−,0}
is located at ±K x ≠ 0. For comparison, the solution of the zero-mode approximation
E t0 is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.10 Lowest eigenvalues at K x = 0 plotted against Q SO W/π. For comparison, the
global minimum of the Cooperon spectrum for Q SO W 9 is plotted, F 3 .
Curves F 1 [n] are given by 7/16+ ( (n/(Q SO W/π)) √ 15/4 ) 2
, n ∈ N. F2 shows
the energy minimum of the 2D case, F 2 ≡ F 1 [n = 0]. Vertical dotted lines
indicate the widths at which the lowest two branches degenerate at K x = 0.
They aregiven by n/( √ 15/4); consider that the wave vector for theminimum
of the E T− mode is ( √ 15/4)Q SO . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.11 The quantum conductivity correction in units of 2e 2 /2π as function of magnetic
field B (scaled with bulk relaxation field H s ), and the wire width W
scaled with 1/Q SO for pure Rashba coupling and cutoffs 1/D e Q 2 τ SO ϕ = 0.08,
1/D e Q 2 SOτ = 4: Comparison of thezero-mode calculation (grid without shading)
to the exact diagonalization where the lowest 21 triplet branches and
seven singlet branches were taken into account. . . . . . . . . . . . . . . . . 52