

nextnano^{3}  Tutorialnext generation 3D nano device simulator1D TutorialTriangular wellAuthor: Stefan Birner If you want to obtain the input file that is used within this tutorial, please contact stefan.birner@nextnano.de. Triangular wellA triangular well consists of a potential with a constant slope that is bound at one side by an infinite barrier.
The Schrödinger equation for the transverse component of the electronic wave function has the following form inside the well:
Usually one applies Dirichlet boundary conditions at x = 0 nm so that psi(x=0) = 0 in order to represent an infinite barrier, i.e. the high barrier prevents significant penetration of electrons into the barrier region. In our case, we apply Neumann (or Dirichlet) boundary conditions at x = 10 nm and x = 150 nm and let the infinite barrier be represented by the conduction band offset of 100 eV. Both boundary conditions lead to the same eigenenergies for the lowest eigenstates. The Schrödinger equation can be simplified by introducing suitable new variables and thus reduces to the Stokes or Airy equation. Its solutions, the socalled Airy functions, are discussed in most textbooks, see for example:
One can see that the distance between the energy levels decreases with
increasing n because the quantum well width gets larger for higher energies. The eigenvalues of the Airy equation can be calculated using the formula:
The nextnano³ eigenvalues for the lowest four eigenstates are in very good agreement with the analytic results:
The triangular potential is not symmetric in x, thus the wave functions lack the even or odd symmetry that one obtains for the infinitely deep square well. The triangular well model is useful because it can be used to approximate the (idealized) triangularlike shape near a heterojunction formed by the discontinuity of the conduction band and an electrostatic field of electrons or remote ionized impurities.

