Hybrid
systems consisting of a semiconductor quantum well and a layer of quantum dots
or rings are studied as a part of materials design for electronics and
photonics devices, due to their photoconductive properties and unique
magnetoresistance and carrier transport properties [1-7]. The examples are, the
change of conductivity under illumination, the negative refractive index for
certain wavelengths, oscillations of magnetoresistance. Such systems also
exhibit topological properties that can be controlled by electric and magnetic
fields - the Aharonov-Bohm effect and other quantum interference effects
arising from scattering of electrons on an array of quantum structures in a
magnetic field.
The
simulation presented in this work is also carried out for explanation of the
results of the experimental paper [1], which investigates the electronic and
optical properties of hybrid systems with quantum rings (Figure 1). The
scattering of electrons on quantum rings and quantum dots of different shapes
is simulated and visualized. The two-dimensional Schrödinger equation for
the stationary and non-stationary problem is solved by different methods. The
square of the modulus of the wave function is constructed for several potential
forms, and the reduced 2D scattering cross section and the probability current
are calculated. To perform the calculations and visualize the results (such as
wave functions and probability currents) original programs were written by the
authors, using R and Rust programming languages.
Some
of the existing scientific works experimentally investigate the properties of
arrays of quantum rings and dots. The second group of papers studies mainly the
calculation of the optical properties of quantum rings when electrons scatter
on them in a transverse magnetic field [6-8]. Some papers investigate the
motion along a channel that splits at a certain point to form a ring [9, 10].
The
[9] demonstrates the solution of the scattering problem using the
finite-difference method. The potential was represented in the form of a
locally partitioned channel; the main task was to investigate two variants of
open boundary conditions.
The work
[10]
investigates the wave packet scattering. The ring-like channels of various
shapes were used: circular, rectangular, semicircular. There was also an
additional channel connecting the ring at two more points. The main task was to
study the dependence of the wave packet dynamics on the width of the additional
channel.
There
are also papers investigating two-dimensional scattering on a rigid disk [11]
or solving the problem of scattering and searching for energy levels [12, 13].
Figure 1 – Example of a hybrid
system consisting of a quantum well and a quantum dot.
In this paper we study hybrid systems consisting of a semiconductor
quantum well and a layer of quantum rings or quantum dots. Quantum rings and
dots are zero-dimensional structures, quantum well is two-dimensional. This
means that for the scattering problem it is necessary to solve the
Schrödinger equation.
Figure 2 – Comparison of
computational model and experimental structure.
In the usual experimental set up the quantum ring represents a local
expansion of the potential well, i.e. it is necessary to consider the three-dimensional
motion of particles, which is quite difficult to model [1,5]. In our
computational model, the change in the depth of the well is considered, while
the width remains constant, which allows us to solve the two-dimensional
problem. A visual comparison of the two approaches is presented in Figure 2.
The model is qualitatively consistent with the experiment. Calculation of
two-dimensional scattering is much easier than three-dimensional scattering,
and thanks to this approach it is possible to consider many different
configurations.
The main purpose of this work is to solve and simulate the problem
of scattering of electron flux on a potential in 2D. Two different formulations
of this problem are considered. Stationary (1) and time-dependent one (2),
where
is the Hamiltonian operator.
|
|
(1)
|
where
is Planck's constant,
the effective mass of the electron in the quantum well material,
is imaginary unit,
is the Laplace operator,
are the polar coordinates: radius and polar angle,
is the wave function,
is the potential function,
is the electron energy,
is the scattering amplitude. The phase multiplier
is necessary to agree with the optical theorem [11].
|
|
(2)
|
where
– time,
is the time partial derivative,
is the Cartesian coordinates,
is the initial function whose form and meaning we will consider
later,
are the boundaries of the computational domain.
The probability current
along the x-axis was calculated using formula (3). The electric current density
depends on the probability current density, namely:
, where
is the electric current density.
|
|
(3)
|
To solve the stationary problem,
at first Born approximation had been considered; in the articles found, it was
used mainly to solve the inverse scattering problems.
However this method is not suitable in the present case because of
the values of the parameters for which the calculation was carried out (Table
1) - the condition of small potential is not satisfied:
.
The second method for solving the stationary problem is based on
expanding the desired wave function into a Fourier series (4), that defines the
methods name. Substituting the series into the Schrödinger equation, we
obtain the equation for the expansion coefficients. Boundary conditions (5) are
also expanded (6).
|
|
(4)
|
In this paper we consider the solution of the equation for local
axially symmetric potentials, which can be approximately partitioned into constant-value
steps (7). The proposed method can be used for potentials with no symmetry as
well, but it would require to solve a large system of equations.
|
|
(7)
|
For one constant value of the potential
we obtain equation (8). The solution of this equation is a linear
combination of Bessel functions of the first and second kind (9).
For the first region
the solution always contains only the Bessel function of the first
kind since the solution must be finite at zero.
The finite-difference method has been developed to solve the
time-dependent problem. The electron is replaced by the wave packet (10), where
is the initial position,
is the initial dispersion. These are the additional free
parameters, compared to the stationary case.
There is a grid with variables x, y .
The coordinate derivatives are replaced by the
finite-difference approximation according to the equation (11), where
is the value of the wave function on the
coordinate grid. The probability current was calculated according to the
formula (12), which is a finite-difference representation of (3).
|
|
(11)
|
|
|
|
(12)
|
Figure 3 – Scheme of bypassing
all grid points.
The finite-difference time scheme is based on the asymmetric Sauliev
scheme [14], which on the one hand is explicit, which simplifies the algorithm,
and on the other hand is more stable and leads to less error than other
explicit schemes. This scheme can be proven to be unconditionally stable, but
in its asymmetric form (moving along the grid in one direction) doesn’t
preserve the norm of the wave function. Switching directions at each step as
it’s done here greatly improves the norm stability.
A full description of the finite-difference scheme is given in
Appendix 1.
Table 1 shows the values of the parameters for which the
calculations were carried out. In this work several models of quantum rings and
dots were investigated, 3D image and section of which are shown in Figure 4 (
– characteristic value of the potential).
The parameter values correspond to the experimental samples from the
article [1]. In this study, a quantum well of 6 nm width was investigated, and
the distance between the bottom of the first subband and the top of the quantum
well was 0.08 eV.
Figure 4 – Examples of
investigated potential holes: a) as a Gaussian function, b) the difference of
two Gaussian functions, c) as a cylindrical ring.
Table
1. Values of the parameters for which the calculation was carried out.
|
Quantum well width
, nm
|
6
|
|
Effective mass of the
electron,
(
is
the mass of the electron in a vacuum)
|
|
|
Distance between the
bottom of the first subzone and the top of the quantum well
, eV
|
0.08
|
|
Electron energy
, eV
|
0.01
0.1
|
|
The depth of the
potential pit
, eV
|
-0.2
|
|
Outer ring radius
, nm
|
30
100
|
|
Inner radius of the
ring
, nm
|
(0.3
0.7)
|
To solve the scattering problem with this method, it is necessary to
divide the potential into a set of steps with a constant value (equation (7)).
For a cylindrical ring we get 3 sections with a constant value of the
potential. For the potentials in the form of Gaussian functions significantly
more steps is required. An example of potential partitioning for a ring is
shown in Figure 5. Along the radius-vector axis, the value of the function on
the left boundary is taken at equal intervals. The potential is equated to
zero, when the value of the Gaussian function is less than 1·10-12.
Figure 5 – Gaussian potential
divided into columns with the same value.
The calculation has been carried out for different energy values.
Plots of the probability current dependence on energy for different potentials
are shown in Figure 6. For rings a plateau after a certain value of energy is
noticeable, for the cylindrical form it appears earlier. For a quantum dot a
smooth increase in the probability current is observed. The obtained
dependences qualitatively correspond to the behavior of the probability current
in the case of quantum tunneling.
Figures 7 and 8 show a comparison of the squared modulus of wave
functions and the reduced 2D scattering cross section for reduced potentials
for an energy of 0.5 eV. It can be seen that the cylindrical potential well is
characterized by multiple centers of oscillations. This is due to additional
reflections from the ring boundaries. For the Gaussian potentials this is not
observed, which is due to the smoothness and continuity of the function.
Figure 6 – Plot of probability
current versus energy for different potentials: red line - Gaussian ring with
radii of 30 and 50 nm; green - Gaussian quantum dot with radius of 40 nm;
purple - cylindrical ring with radii of 30 and 50 nm.
Figure 7 – Square of the wave
function modulus for potentials: a - Gaussian ring with radii of 30 and 50 nm;
b - Gaussian point with radius of 40 nm; ñ - cylindrical ring with radii of 30
and 50 nm.
When comparing the scattering patterns, one can see that for a
Gaussian ring there is a movement through both parts of the ring, followed by
interference (Figure 7(a)). For a quantum dot, the wave function is focused at
the center of the nanostructure (Figure 7(b)).
When comparing the scattering cross section, forward scattering
mainly occurs for all three cases. For the Gaussian ring, additional maxima
located close to zero. This corresponds to the interference of the parts of the
wave function coming out of the potential well (Figure 8(a)). For a cylindrical
ring we see smaller maxima at some distance from zero (Figure 8(c)), they
correspond to the parts of the wave function that did not fall into the well.
For the quantum dot the central maximum is greater in intensity than for the
rings, and there are also small additional maxima (Figure 8(b)).
Figure 8 – The reduced 2D
scattering cross section for the potentials: a - Gaussian ring with radii of 30
and 50 nm; b - Gaussian point with radius of 40 nm; ñ - cylindrical ring with
radii of 30 and 50 nm.
The problem of scattering on a disk-shaped barrier was solved in the
paper [11]. The Fourier method given above is also suitable for such a problem,
which allows us to compare the results. Qualitatively and numerically the
solutions coincide, the squares of the modulus of wave functions and the
reduced 2D scattering cross sections were compared:
.
The program with the implementation of this method is written in R,
the system of linear equations for the expansion coefficients is solved using
the built-in function solve(). The visualization of the square of the modulus
of the wave function is done using the ggplot2 library: an image is created
from the given array with the values of the function.
Figure 9 – The motion of the
wave packet (the square of the wave function modulus) during scattering on a Gaussian
quantum ring.
The finite-difference method was developed for a time-dependent
problem. Examples of the results of the scattering problem are presented in
Figures 9-11 for the cylindrical and Gaussian quantum ring, Gaussian quantum
dot, respectively. It can be seen that for the quantum ring the wave packet is
divided into two identical parts, while for the quantum dot the scattering is
focused inside the nanostructure.
The program for this method is written in the Rust programming
language. As a result, we obtain images of the potential and position of the
wave packet at several points in time. To visualize the motion of the wave
packet, a custom function was implemented that translates the values of the
square of the modulus of the wave function on the grid into integers from 0 to
99 with known maximum and minimum values. They are correlated with color values
from a given palette of 100 colors.
In Figures 9-11 the size of the calculation area is 500 nm (the x
axis) by 300 nm (the y axis).
Figure 10 – The motion of the
wave packet (square of the modulus of the wave function) at scattering on a
Gaussian quantum dot.
Figure 11 – Motion of the wave
packet (square of the wave function modulus) when scattering on a hole in the
form of a ring with radii of 30 and 50 nm.
A
comparison of the results obtained by the Fourier method and the
finite-difference method is shown in Figure 12. The pictures are similar: the
main centers of oscillations and the position of parts of the wave packet,
their shape and direction coincide. The finite-difference method produces a
sequence of images with a scattering picture with some time step. Because of
this the states may not coincide.
Figure 12 – Comparison of the
Fourier method (left) and the finite-difference method (right): a - for a
Gaussian ring; b - for a Gaussian point; c - for a cylindrical ring.
The Fourier method and the finite-difference method both cope very
well with solving the problem of two-dimensional electron scattering on the
considered potentials. Visualization of the wave function and probability
current confirms the correctness of the developed programs by comparing the
visualized results with other existing works and with each other. The obtained
images of the quantities sought show that the results of both methods are
physically meaningful and consistent with each other.
The results of the Fourier method for another problem were compared
to the article [11] and found to be in agreement. A solution for centrally
symmetric potentials has been developed. But the method is not limited to this
condition. For an arbitrary form of potential, it is necessary to solve a large
system of equations. The only requirement for using this method is to divide
the potential into several piecewise constants.
The developed finite-difference method is suitable for any shapes
and positions of the potential. To solve the problem, it is necessary to
specify its values on the coordinate grid. For the developed method, the most
important thing is the ratio of intervals between neighboring grid points for
the three variables. Even though the Sauliev scheme for the Schrödinger
equation can be proven to be unconditionally stable, in practice a sufficiently
small time interval is required for the computational scheme to converge.
When using wave packets to model scattering, one needs to consider
the widening of the wave packet as it moves. The speed of the widening is
inversely proportional to the initial width. Thus, two free parameters appear -
the initial position of the wave packet and the initial width (dispersion),
which require careful selection.
The obtained solutions demonstrate that scattering on a quantum ring
devides the wave function into two equal parts. For the quantum dot we see the
focusing of the wave function inside the nanostructure.
The present work was supported by the Ministry of Science and Higher
Education within the framework of the Russian State Assignment under contract
No FSWU-2023-0075.
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To solve the non-stationary problem, the time grid is also
introduced:
As indicated in Figure 3 in the computational scheme, the motion
goes in four directions. The equation for the wave function of each grid point
for each direction is (A.1-A.4), where the index on top denotes the time step,
is the potential value on the coordinate grid. The time derivative
and the value of the wave function at the required point of the grid are
written out according to the Sauliev finite-difference scheme [14].
