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

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2.3 Electron transport in one-dimensional systems with Rashba effect 35
E
x = 0
E
E F
I
II
E F
y
x
Figure 2.7: Sketch of interface between two quasi-1D wires with different widths
(centre) and their dispersion relations (left and right). At the Fermi energy
(dashed) only the lowest two subbands are propagating.
This is done by “matching” the transverse modes (and their probability flux ‡ ) at
the boundaries between the sections. We illustrate this with the basic example of
a discontinuity of a quantum waveguide which is shown in Fig. 2.7. In contrast to
Sec. 2.1 and 2.2, in the following, we change the geometry such that the quasi-1D
system is parallel to the x-axis. We are looking for the stationary-state solution for
the energy E F . Expansion into transverse modes yields
ψ I (x,y) =
[1ex]ψ II (x,y) =
∞
∑
n=0
∞
∑
n=0
(
) A n e ikI nx + B n e −ikI nx
φ I n(y), (2.33)
(
) C n e ikII n x + D n e −ikII n x
φ II
n (y), (2.34)
with modes {φ I(I)
n } of region I(I), resp. The wave vectors follow from the energy
dispersions, En(k I n) I = En II (kn II ) = E F . Depending on E F , modes are either propagating
if min(E n ) < E F with real k n , or evanescent if min(E n ) > E F , leading to
imaginary k n and thus to exponentially decaying (or growing) wavefunctions. At
the interface (x=0) the usual boundary conditions have to be satisfied for all y,
ψ I (x = 0,y) = ψ II (x = 0,y), (2.35)
ˆv x ψ I (x,y) ∣ ∣
x=0
= ˆv x ψ II (x,y) ∣ ∣
x=0
, (2.36)
requiring a continuous wavefunction and the continuity equation for the probability
density, ∂ t ρ=−∂ x J x , to be fulfilled. The probability current is given by
J x = 1 2[
ψ∗ ˆv x ψ + ψ( ˆv x ψ) ∗] , (2.37)
‡ Note that for SO-interacting systems the continuity condition at the interface affects the probability
flux; only without SO coupling this reduces to the continuity of the derivative of the wavefunction.