nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Triangular well
Author:
Stefan Birner
> 1DGaAs_triangular_well_nn3.in / *_nnp.in  input file for the nextnano^{3} and nextnano++ software
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Triangular well
A triangular well consists of a potential with a constant slope that is bound
at one side by an infinite barrier.
For x < 0 nm we have an infinite barrier. In our case it is represented by
a conduction band offset of 100 eV.
For x > 0 nm we have a linear potential of V(x) = e F x.
V(x) describes a charge e in an electric field F where the product e F is
assumed to be positive.
The Schrödinger equation for the transverse component of the electronic wave
function has the following form inside the well:
[  h_{bar}²/(2m*) d²/dx² + e F x ] psi(x) = E psi(x)
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:
The Physics of LowDimensional Semiconductors  An Introduction
John H. Davies
Cambridge University Press (1998)

The figure shows the conduction band edge (black
line) which is represented by a triangular potential well V(x) = e F x.
Also shown are the four lowest energy levels and corresponding wave
functions.
The electric field that has been applied is F = 5 MV/m, i.e. 0.05 V / 10 nm.
The effective electron mass has the value 0.067m_{0} (GaAs). 
One can see that the distance between the energy levels decreases with
increasing n because the quantum well width gets larger for higher energies.
Note that in a parabolic well, the energy levels are equally spaced whereas in
an infinitely deep square well, the energy level separation increases with
increasing energy.
The eigenvalues of the Airy equation can be calculated using the formula:
E_{n} = c_{n} [ (e F h_{bar})² / (2m*)]^{1/3}
(The units of E_{n} in this equation are [J].)
The lowest eigenvalue has the value c_{1} = 2.338.
For large n, c_{n} can be approximated by the following equation
which can be derived from WKB theory (named after Wentzel, Kramers and
Brillouin):
c_{n} ~ [ 3/2 pi (n  1/4) ]^{2/3}
The nextnano³ eigenvalues for the lowest four eigenstates are in very
good agreement with the analytic results:

nextnano³ eigenvalue (eV) 
calculated eigenvalue (eV) 
c_{n} (exact) 
c_{n} (approximated) 
n = 1 
0.05644 
0.05664 
c_{1} = 2.338 
c_{1} = 2.320251 
n = 2 
0.09882 
0.09889 
c_{2} = 
c_{2} = 4.081810 
n = 3 
0.13351 
0.13365 
c_{3} = 
c_{3} = 5.517164 
n = 4 
0.16416 
0.16435 
c_{4} = 
c_{4} = 6.784455 
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.
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[at] nextnano.com .
