8.5 Scattering in the Ripple Waveguide
263
configuration, we only open the left wall of the ripple billiard, we add more ripples,
and attach the left side to an infinitely long straight lead with hard walls. In this
second case, electrons enter from the left and scatter back to the left.
The Schrodinger equation, which describes propagation of a particle wave,
y, t), for all times t is given by
i ¯
h
∂∂
∂t
= −
¯
h 2
2m
∂ 2
∂x 2 +
∂ 2
∂y 2
+ V (x, y)),
(8.88)
where ¯
h is Planck’s constant. The potential V(x,y) is determined by the shape of the
various walls of the ripple waveguide, and will be different for each of the two cases
considered.
8.5.1 Scattering Resonances in a Ripple Waveguide
The ripple billiard in Sect. 7.5.3 can be converted to a waveguide if the left and
right walls are removed and the cavity is attached to infinitely long leads on the
left and right sides of the billiard (Lee and Reichl 2006, 2008, 2009) Then we have
an infinitely long waveguide with a short segment containing a ripple in the wall, as
shown in Fig. 8.6a. We will assume that the length of the cavity along the x-direction
is L = 300.0 Å. The ripple has peak amplitude a with a = 0.0462W = 14.324 Å
and the leads have width W = d − a, where d = 0.1576W = 47.747 Å. Electrons
can only propagate along the straight leads if they have enough energy, E≥ ¯
h 2 π 2
2m ∗ W 2 ,
to set up a transverse state in the leads. The first propagating mode opens at the
incident energy E 1 = ¯
h 2 π 2
2m ∗ W 2 = 0.503 eV, for the parameters given in Fig. 8.6.
We can use Wigner–Eisenbud scattering theory to study the scattering properties
of the system. The Hamiltonian H QQ can be constructed using the analytic
expression of the Hamiltonian matrix that can be obtained using the coordinate
transformation described in Sect. 7.5.3. The waveguide shown in Fig. 8.6a has
straight hard walls at y = 0, and an upper hard wall with shape
y(x) = d − acos
2πx
L
,
(8.89)
The two infinite leads are attached at x = 0 and x = L. The cosine billiard has a
fixed point at the center of cavity whose stability depends on the parameters as well
as two unstable fixed points at x = 0 and L, which we call the outer fixed points.
Figure 8.6b shows a Poincare surface of section for the classical electron
dynamics in the cavity. We use the Birkhoff coordinates (x, p x ), when the particle
hits the bottom boundary (at y = 0). There are the large KAM islands surrounded
by chaotic sea and small islands.
263
configuration, we only open the left wall of the ripple billiard, we add more ripples,
and attach the left side to an infinitely long straight lead with hard walls. In this
second case, electrons enter from the left and scatter back to the left.
The Schrodinger equation, which describes propagation of a particle wave,
y, t), for all times t is given by
i ¯
h
∂∂
∂t
= −
¯
h 2
2m
∂ 2
∂x 2 +
∂ 2
∂y 2
+ V (x, y)),
(8.88)
where ¯
h is Planck’s constant. The potential V(x,y) is determined by the shape of the
various walls of the ripple waveguide, and will be different for each of the two cases
considered.
8.5.1 Scattering Resonances in a Ripple Waveguide
The ripple billiard in Sect. 7.5.3 can be converted to a waveguide if the left and
right walls are removed and the cavity is attached to infinitely long leads on the
left and right sides of the billiard (Lee and Reichl 2006, 2008, 2009) Then we have
an infinitely long waveguide with a short segment containing a ripple in the wall, as
shown in Fig. 8.6a. We will assume that the length of the cavity along the x-direction
is L = 300.0 Å. The ripple has peak amplitude a with a = 0.0462W = 14.324 Å
and the leads have width W = d − a, where d = 0.1576W = 47.747 Å. Electrons
can only propagate along the straight leads if they have enough energy, E≥ ¯
h 2 π 2
2m ∗ W 2 ,
to set up a transverse state in the leads. The first propagating mode opens at the
incident energy E 1 = ¯
h 2 π 2
2m ∗ W 2 = 0.503 eV, for the parameters given in Fig. 8.6.
We can use Wigner–Eisenbud scattering theory to study the scattering properties
of the system. The Hamiltonian H QQ can be constructed using the analytic
expression of the Hamiltonian matrix that can be obtained using the coordinate
transformation described in Sect. 7.5.3. The waveguide shown in Fig. 8.6a has
straight hard walls at y = 0, and an upper hard wall with shape
y(x) = d − acos
2πx
L
,
(8.89)
The two infinite leads are attached at x = 0 and x = L. The cosine billiard has a
fixed point at the center of cavity whose stability depends on the parameters as well
as two unstable fixed points at x = 0 and L, which we call the outer fixed points.
Figure 8.6b shows a Poincare surface of section for the classical electron
dynamics in the cavity. We use the Birkhoff coordinates (x, p x ), when the particle
hits the bottom boundary (at y = 0). There are the large KAM islands surrounded
by chaotic sea and small islands.