=Paper=
{{Paper
|id=Vol-3628/short13
|storemode=property
|title=Mathematical model of coherent the electron transfer for nanostructures with an applied magnetic field of constant strength
|pdfUrl=https://ceur-ws.org/Vol-3628/short13.pdf
|volume=Vol-3628
|authors=Igor Boyko,Julia Nestor
|dblpUrl=https://dblp.org/rec/conf/ittap/BoykoN23
}}
==Mathematical model of coherent the electron transfer for nanostructures with an applied magnetic field of constant strength==
https://ceur-ws.org/Vol-3628/short13.pdf
Mathematical model of coherent the electron transfer for
nanostructures with an applied magnetic field of constant
strength
Igor Boyko and Julia Nestor
Ternopil Ivan Puluj National Technical University, 56, Ruska Street, Ternopil, 46001, Ukraine
Abstract
A mathematical model, which describes the processes of coherent electron transfer through a
plane semiconductor nanosystem with a constant magnetic field applied longitudinally to it is
proposed. A finite difference scheme has been constructed that provides software
implementation of time-dependent solutions to the complete Schrödinger equation. The direct
implementation of the mathematical model was carried out using the Wolfram Mathematica
system. The mathematical model was verified for the parameters of an experimentally
realized nanosystem with typical geometric and physical parameters.
Keywords 1
Nanostructure, electron transfer, finite difference method, quantum transitions
1. Introduction
The transport properties of the nanostructures that make up these devices play a decisive role in the
practical functioning of nanoscale devices. Consequently, theoretical studies are usually carried out in
mathematical models of two-three and multilayer open nanosystems without constant external fields,
and mainly only taking into account the interaction of electrons with the electromagnetic field [1-3].
Later, the influence of constant electric and magnetic fields on electron tunneling through
nanostructures was studied [4-8]. It turned out that a constant electric field directed along the electron
flow through the nanostructures plays an important positive role in the operation of quantum cascade
lasers, since it coordinates the operation of all cascades [4-6]. As for the constant magnetic field, its
role in the operation of nanodevices turned out to be quite complex. As has been established
experimentally and theoretically [9, 10], a magnetic field with a strength parallel to the current
through the nanostructure does not affect the peak current value for a resonant tunnel diode. This is
due to the fact that a longitudinal magnetic field does not affect the movement of charge carriers
along the current, but only causes a change in the density of states.
It was also investigated that the inclusion of a longitudinal magnetic field causes the appearance of
steps in the current-voltage characteristic, the number of which is related to the number of operating
Landau levels. With the advent of optoelectronic nanodevices, experimental studies of the behavior of
electrons in quantum superlattices in a constant magnetic field have intensified. Thus, in experimental
papers [11, 12], quantum cascade nanodevices were first implemented, the operation of which is
based on quantum transitions between Landau energy levels in a magnetic field with a strength
parallel to the current through the nanosystem. Theoretical papers [11-13] were mainly concerned
with the calculation of the quasi-stationary energy spectrum of an electron in such superlattices, and
they also studied the dependence of the electron transparency coefficient of multilayer nanosystems
on the magnitude of the applied magnetic field. Thus, we can conclude that the most complete correct
Proceedings ITTAP’2023: 3rd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24,
2023, Ternopil, Ukraine, Opole, Poland
EMAIL: boyko.i.v.theory@mail.com (A. 1); nazarko.julia26@gmail.com (A. 2);
ORCID: 0000-0003-2787-1845 (A. 1); 0000-0003-0737-8965 (A. 2)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
model of the electronic states of a nanosystem with a constant magnetic field, which would
correspond to most practical problems, is missing
Inn proposed paper, we consider the general case that corresponds to the mathematical model of
electron tunnel transport with the presence of a constant longitudinal magnetic field and satisfies
experimentally realized quantum cascade lasers and detectors.
2. Mathematical model of electronic ballistic transport in a longitudinal
magnetic field of constant strength
The mathematical model of a ballistic electron flow through flat nanosystems in a magnetic field
with a strength parallel to the direction of the current in the proposed paper will be developed in an
approach as described below. We consider a plane semiconductor nanostructure geometric and the
energy scheme of which, such as those shown in Fig. 1.
To begin with, we choose the vector potential of the magnetic field in the following form:
1
A [B r ] . (1)
2
where the magnetic field is such that it is directed perpendicular to the planes of the nanosystem, i.e.
along the direction of electron tunneling (Oz axis) (Fig. 1).
The electron Hamiltonian using a model of various effective masses in the wells and barriers of a
multilayer nanosystem looks is obtained as follows:
1 1 1
px eAx p y eAy pz
2 2
H p U ( z)
2 ( z ) 2 ( z) z
. (2)
1 eB eB 1 1
2 2
p x y py x pz pz U ( z ).
2 ( z ) 2 2 2 ( z )
where px , p y are the components of the electron momentum in the direction perpendicular to the
tunneling direction, pz is the component along the tunneling direction, U ( z ) and ( z ) are the
potential energy and effective mass of the electron in the layers of the nanosystem, respectively.
Figure 1: Schematic energy and geometric diagram of the studied multilayer nanosystem
By introducing the creation a and annihilation a operators by analogy with a harmonic
oscillator, the electron Hamiltonian is presented in the following convenient form:
1 1 1
H n p z pz U ( z ) . (3)
2 2 ( z)
where is the cyclotron frequency; n a a – filling number operator. Next, we take into account
the fact that n ( x, y ) these are the known eigenfunctions of the Hamiltonian of a two-dimensional
harmonic oscillator:
nn ( x, y ) nn ( x, y ), n 0, 1, 2, ... (4)
then from the form of Hamiltonian (3) it immediately follows that the Landau quantum energy levels
are now determined by the formula:
1
En n ; n 0, 1, 2, ... (5)
2
Consequently, the energy of longitudinal motion of an electron in a nanosystem is determined as:
Ez ( z ) E En ( z ). , (6)
The spatial wave function of electron motion is now found from the stationary Schrödinger equation
(expressions (5) and (6) was used in the equation (3)):
1 1 1
2 n i 2 pz ( z ) pz U ( z ) (r ) E (r ) . (7)
The wave function in equation (7) in order to separate the longitudinal motion of the electron from the
transverse one is sought in the following form:
( r ) n ( x , y ) ( z ) (8)
Now the Schrödinger equation (7) takes the following appearance:
2 d 1 d
( z ) U eff ( z )( z ) E ( z ) . (9)
2 dz ( z ) dz
where
1
E z E n . (10)
2
is the energy of the longitudinal motion of the electron, and U eff is the effective potential of the
electron, which has the form:
0, ( wells )
U eff mw 1
. (11)
U b ( n, B ) U 0 1 n , (barriers )
m b 2
mw , mb are the effective electron masses in the wells and barriers of the nanosystem, respectively. As
can be seen, the effective potential in which the electron moves along the Oz axis depends on the
quantum number n and on the magnetic field induction B .
Boundary conditions for solutions of the Schrödinger equation (9), describing their continuity and the
continuity of probability flows at the boundaries of the sth layer of the nanosystem:
1 d (s) ( z) 1 d ( s 1) ( z )
( s ) ( z ) (s) ( z) ; . (11)
z z s 0 z z s 0 s dz z z s 0
s 1 dz z zs 0
Now solutions to the Schrödinger equation (9) taking into account are sought on a one-dimensional
grid:
m z : zm mzm , m 0,1, 2,3...M . (13)
As a result, we obtain the following finite difference scheme:
Ф0 Ф1 0;
ФM 1 ФM 0;
m1 m 1 m m m1 0; (14)
m 1 m1
m1 2m E h h2 U 0 1 mw 1 n 2 m m1 0.
2
2
m b 2
Here we used well-known approximations for the first and second derivatives:
d m d 2 2 m m 1
m 1 ; 2
m 1 ; h zm .
dz h dz h2
Outside the nanostructure, it is advisable to present solutions to the Schrödinger equation in the
following simple analytical form:
( z ) z eikz Be ikz ;
. (15)
( z ) z Aeikz ; k 2m0 E .
The difference scheme, presented in the form (14), can already be directly implemented using applied
software. As a result, the eigenvalues of the difference scheme (14) determine the electronic spectrum
En . This allows us to directly determine the wave function inside the nanostructure, satisfying the
normalization condition:
( E , z ) dz 1 .
2
n (16)
As a result, this makes it possible to calculate the transparency coefficient of the nanosystem based on
the following expression:
2
D( E ) A( E ) . (17)
3. Results and discussion
Direct calculations using a developed mathematical model of resonant electron transport were
performed for an experimentally studied nanosystem [5], containing ten AlAs potential barriers, each
2 nm thick, and nine GaAs potential wells, each 3 nm thick. The height of the potential barrier was
taken to be 520 meV, the effective mass of the electron in the potential barriers and wells was 0.72me
and 0.67me, respectively (me is the mass of a free electron).
1,0
a 0T b
400
1T
0,8 3T
5T
En=1, 2,3, (meV)
300
0,6
E3
D
200
0,4
E2
0,2 100
E1
0,0 0
12,90 12,92 12,94 12,96 12,98 13,00 13,02 0 5 10 15 20 25
E, meV B, (T)
Figure 2: Dependence of the nanosystem transparency coefficient on electronic energy at different
values of magnetic field induction (B) (a) and dependence of electron energy levels on the induction
value (b).
In Fig. 2a are shown the dependence of the transparency coefficient of the nanosystem on the
electronic energy scale. Calculations were performed depending on the magnitude of the applied
magnetic field induction B. As can be seen from the figure, the magnetic field leads to a decrease in
the maximum value of the transparency coefficient, while the levels of the electron energy spectrum
also shift to the high-energy region. This can also be seen more clearly from Fig. 2b, which shows the
dependences of the first three electronic levels on the magnetic field induction values. It should also
be noted that as the magnetic field induction increases, the electronic levels form a “bottleneck”
effect, which usually occurs when the geometric parameters of the nanosystems under study are
varied.
The directly established effects will have a significant impact on the electron tunneling process. In
particular, a decrease in the maximum value of the transparency coefficient will lead to a significant
decrease in the intensity of quantum transitions between electronic levels. In turn, a shift in energy
levels leads to a change in the working part of the devices due to the fact that electromagnetic waves
will be generated with different frequencies different from the operating frequency.
4. Conclusions
A mathematical model of electron transport in nanosystems is proposed, taking into account the
influence of an applied constant longitudinal magnetic field. Calculations based on the developed
model using the parameters of an experimentally created nanosystem were used to study the influence
of the magnetic field on the transparency coefficient of the nanosystem and the electronic spectrum. It
has been established that the magnetic field reduces the transparency coefficient of the nanosystem
and shifts the electron energy levels to a high-energy region.
The presented mathematical model can be the immediate basis for further research into problems
associated with electron transfer in nanosystems.
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