Volume 10, Issue 1
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Staff and Contributors
T R O T T ' S C O R N E R
In the 9:3 Corner, I discussed the sound of a fractal drum. More technically speaking, I calculated selected point spectrum eigenvalues and eigenfunctions of the Helmholtz equation of a complicated, bounded 2D domain with homogeneous Dirichlet boundary conditions. In this Corner, I will treat a problem similar in spirit--I will calculate some eigenfunctions of the continuous spectrum of an unbounded 2D domain with homogeneous Dirichlet boundary conditions. More concretely, we will study the transmission of a wave through a 2D quantum dot in the stationary regime. Here is a sketch of the domain under consideration: (the -direction runs to the left and the -direction runs upwards).
The arms have lateral width and infinite horizontal extension, and the center disk has radius . To make calculations simpler, we take the arms to extend up to the circular boundary of the center disk.
We will solve the elliptic eigenvalue problem with the boundary condition . We will assume an incoming wave from the left with unit strength of the form . We will use the mode-matching technique to solve this Helmholtz equation. That is, we will construct within neighboring regions locally parametrized complete families of solutions and join them smoothly along certain curves by fixing the values of the parameters. We will naturally choose the left arm, the center disk, and the right arm for the regions.
A practical realization of such structures are quantum dots [1-8]. The Helmholtz equation in this case is identical to the time-independent Schrödinger equation with energy . We will use this model as the physical interpretation in our calculations.
Because the domain is symmetrical, we can construct eigenfunctions with a definite parity with respect to reflections in the -axis. We will always consider symmetric states. The construction of antisymmetric states is completely analogous.
Inside the left arm, we can write the general solution of the Helmholtz equation in the form
Here the inhomogeneous term has strength 1. We truncate the infinite sum because the contributions of the higher states decay exponentially with for energies in the range (the lowest sub-band in the -direction is occupied). The lateral wave number is, for a given energy , defined through . The first term in the expansion contains the reflection coefficient . The form of the -dependent part automatically ensures the fulfillment of the Dirichlet boundary conditions in the lower and upper boundaries of the left arm.
Similarly, in the right arm, we use the expansion
Again, we truncate the infinite sum. This time, the first term in the expansion contains the transmission coefficient and is defined as in the left arm.
Inside the center disk, we use a polar coordinate system. In polar coordinates, any solution of the Helmholtz equation can be written in the form
Here the are Bessel functions.
The unknown coefficients , , and are determined through the continuity of , the first derivative of along curves joining the left and right arms with the center disk, and the fulfillment of the Dirichlet boundary condition along the arcs between the arms and the center disk. We will unite the three sets of coefficients , , and into the coefficients and use the following convention:
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