Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

2.4 Summary 45
For weak SO coupling the dominating Fourier component of the transmission
coefficients can be approximated in lowest order by
T ++ ≈ T −− ≈ 1 − ¯κ 2 [1 − cosa(k − − k + )], (2.68)
T +− ≈ T −+ ≈ ¯κ 2 [1 − cosa(k − − k + )], (2.69)
where ¯κ := κ/k F . The oscillations in the reflection probabilities R σσ ′ (not shown
here) are orders of magnitude smaller and carry also higher Fourier components
(e.g. periods π/k σ ) as a signal of standard Fabry–Perot like interference of the
orbital wavefunction. In the chosen regime with weak SO coupling and magnetic
field, apart from the small deviations due to reflection, the barrier is almost perfectly
conducting since T ++ +T +− =T −− +T −+ ≈1 by Eq. (2.68) and (2.69). The
oscillating features in the spin-resolved transmission in Fig. 2.12a+b clearly originate
from the interference of the spinors which is driven by the spin precession.
We remark that in order to actually resolve the commensurability effect in the
transmission a spin-dependent measurement is needed. Without being sensitive to
the spin, the total transmission T := ∑ σσ ′ T σσ ′ shows only standard Fabry–Perot
like interference on the scale of the Fermi wavelength, 1/k F ≪a 0 .
We point out that due symmetry properties of the scattering matrix of the system
no polarisation effect in the transmitted electron flux can be expected. For a
detailed symmetry analysis see Ref. [58].
In the context of magnetic superlattices, a sinusoidal modulation matches
more realistically the experimental situation. Figure 2.13a+b shows the spindependent
transmission coefficients for a cosine-shaped magnetic barrier (see
Fig. 2.11b) of width a where the magnetic field is
⎧
⎨ +B 0 ê z for |x| > a/2,
B =
⎩−B 0 ê z cos2π x for |x| < a/2.
a
(2.70)
In contrast to the result for a rectangular barrier, the reduction of the spin-conserving
transmission is not a periodic feature with increasing barrier width due to the
non-constant modulation. In panel c+d the results for transmission through 10
sinusoidal periods are shown. Again, the reduction of T σσ is strongly enhanced by
cumulating a small effect when successively passing consecutive barriers.
2.4 Summary
In this chapter, the interplay of spin-orbit (SO) coupling and geometrical confinement
in quasi-one-dimensional quantum wires (QWRs) in perpendicular magnetic