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

2.3 Electron transport in one-dimensional systems with Rashba effect 43
(a)
PSfrag replacements
(b)
(b)
PSfrag replacements
+B 0
x
(a)
+B 0
x
− a 2
+ a 2
− a 2
+ a 2
Figure 2.11: Models of magnetic barriers: rectangular (a) and sinusoidal (b).
In the following, we show results for the transport through magnetic barriers
with rectangular and sinusoidal modulation. Both cases show a commensurability
effect when the modulation period becomes comparable to the SO-induced spin
precession length. Here, we restrict ourselves to Fermi energies much higher than
the gap in the dispersion in Fig. 2.8. Following the experimental motivation of
magnetic superlattices, we assume weak magnetic modulation (∼25mT).
First, we consider a simple rectangular barrier (Fig. 2.11a) of width a where
{ +B0 ê z for |x| > a/2,
B =
−B 0 ê z
for |x| < a/2.
(2.67)
(The SO coupling strength and electrostatic potential is assumed to be constant.)
Although the flip of the magnetic field direction in the barrier does not change
eigenenergies and wave vectors, it does change the eigenstates, see Eq. (2.44).
Therefore, when injecting right-moving electrons with spin σ, in perfect analogy
to the standard spinFET [42], the probability to find the same spin directly behind
the barrier is a periodic function of the barrier width, corresponding to spin precession
with period a 0 = 2π/|k + −k − |, by Eq. (2.47) and (2.48). If the parameters
κ and k SO are small at the scale of Fermi energy this period becomes a 0 ≈ π/2k SO .
In Fig. 2.12 the numerically calculated coefficients for spin-dependent transmission
are shown for a single barrier and a modulation with 10 rectangular periods
in series. For the used parameters with weak SO coupling the spin precession
period corresponds to a 0 ≈ 0.33µm, being within range of state-of-the-art experimental
technique. For the case a of single barrier (panel a+b) the effect of spin
precession is seen as a reduction of the spin conserving transmission T σσ when
the barrier width becomes commensurate with half of the spin precession length.
This commensurability effect is most clearly seen in the narrow dip in the
spin-conserving transmission. At a barrier width such that ak SO /π = 1/4 injected
electrons with polarisation “±” are rotated to “∓” while propagating through the
barrier and thus not matching the wavefunction of the region behind the barrier.