The polaronic effect on the linear, third-order nonlinear and total optical absorption coefficien... more The polaronic effect on the linear, third-order nonlinear and total optical absorption coefficients have been calculated in the case of GaAs/AlAs core/shell quantum dot, with the impurity is positioned at the central radial position of the GaAs shell. The calculations are realized in the framework of the effective mass approximation and the numerical results are obtained by using a variational method and an infinite confining potential. The results show that the polaronic effect has a great influence on optical properties of core/ shell quantum dot, he causes a red-shift of the nonlinear optical coefficients associated to light absorption. Also, the polaronic effect is enhanced with the decreasing of quantum dot radius.
A quantum ring is a physical object similar to a real life ring that we find in our daily life, t... more A quantum ring is a physical object similar to a real life ring that we find in our daily life, the only difference is in its very small nanometer dimensions (0.000000001 m). Furthermore, the ring is made of a semiconductor material called gallium arsenide (GaAs), as opposed to ordinary rings generally made of gold or silver. Another difference with the real ring is that it only has two dimensions, this means that it looks like a hole coin with negligible thickness. To the naked eye, this system could appear too ideal and not useful; but, these structures can be produced experimentally with very small thicknesses close to the system studied here and they have large enough important applications for the actual world.
Abstract The problem of exciton states in spherical semiconductor quantum dots is revisited, empl... more Abstract The problem of exciton states in spherical semiconductor quantum dots is revisited, employing the finite element method to numerically solve the system of differential equations for the center of mass and relative motion of the interacting electron-hole pair. This process is performed within the effective mass and parabolic bands approximations. The use of a finite confinement together with a parabolic description of the conduction and valence band profiles prevents the two equations from uncoupling. The allowed energies are reported as functions of the quantum dot radius. A comparison of theoretically determined fundamental photoluminescence peak energies with available experimental reports in the cases of CdS, CdSe and CdTe is presented and discussed, showing a good agreement between calculated and measured results. Graphical abstract
The effects of external applied fields on the electron-related optical nonlinear properties in a ... more The effects of external applied fields on the electron-related optical nonlinear properties in a short-range bottomless exponential potential quantum well are investigated. The potential profile behaves as inverse square root in the vicinity of the origin and vanishes exponentially toward infinity, supporting a finite number of bound states whose energies and wave functions are determined within the effective-mass approximation. The linear, third-order nonlinear and total optical absorption coefficients (TOACs) as well as the coefficient of relative refractive index changes (RRICs) of the system are calculated from the expressions derived in the framework of the compact density matrix approach. The results obtained after numerical calculations show that with the effect of the static electric and magnetic fields (high-frequency THz laser field), the resonant peak position of the optical response shifts toward higher (lower) energies and the magnitude of the peak increases.
The electronic states in GaAs-AlxGa1−xAs elliptically-shaped quantum rings are theoretically inve... more The electronic states in GaAs-AlxGa1−xAs elliptically-shaped quantum rings are theoretically investigated through the numerical solution of the effective mass band equation via the finite element method. The results are obtained for different sizes and geometries, including the possibility of a number of hill-shaped deformations that play the role of either connected or isolated quantum dots (hills), depending on the configuration chosen. The quantum ring transversal section is assumed to exhibit three different geometrical symmetries - squared, triangular and parabolic. The behavior of the allowed confined states as functions of the cross-section shape, the ring dimensions, and the number of hills-like structures are discussed in detail. The effective energy bandgap (photoluminescence peak with electron-hole correlation) is reported as well, as a function of the Al molar fraction.
The conduction band and electron-donor impurity states in elliptic-shaped GaAs quantum dots under... more The conduction band and electron-donor impurity states in elliptic-shaped GaAs quantum dots under the effect of an externally applied electric field are calculated within the effective mass and adiabatic approximations using two different numerical approaches: a spectral scheme and the finite element method. The resulting energies and wave functions become the basic information needed to evaluate the interstate optical absorption in the system, which is reported as a function of the geometry, the electric field strength, and the temperature.
Abstract The donor-impurity-related optical absorption, relative refractive index changes, and Ra... more Abstract The donor-impurity-related optical absorption, relative refractive index changes, and Raman scattering in GaAs cone-like quantum dots are theoretically investigated. Calculations are performed within the effective mass and parabolic band approximations, using the variational procedure to include the electron-impurity correlation effects. The study involves 1 s -like, 2px-like, and 2pz-like states. The conical structure is chosen in such a way that the cone height is large enough in comparison with the base radius thus allowing the use a quasi-analytic solution of the uncorrelated Schrodinger-like electron states.
A quantum ring is a physical object similar to a real life ring that we find in our daily life, t... more A quantum ring is a physical object similar to a real life ring that we find in our daily life, the only difference is in its very small nanometer dimensions (0.000000001 m). Furthermore, the ring is made of a semiconductor material called gallium arsenide (GaAs), as opposed to ordinary rings generally made of gold or silver. Another difference with the real ring is that it only has two dimensions, this means that it looks like a hole coin with negligible thickness. To the naked eye, this system could appear too ideal and not useful; but, these structures can be produced experimentally with very small thicknesses close to the system studied here and they have large enough important applications for the actual world.
The electronic states of two‐dimensional (2D) semiconductor quantum wells and quantum wires of di... more The electronic states of two‐dimensional (2D) semiconductor quantum wells and quantum wires of disk‐ and ring‐like geometries, under the application of lateral static electric fields, are investigated. The effects of a shallow hydrogenic impurity on the energy levels of both the electron and hole states are reported and the respective binding energies are calculated. The eigenfunctions and eigenvalues of the Hamiltonian are retrieved under the effective mass approximation applying a direct matrix diagonalization scheme in conjunction with a 2D finite element method. A finite confinement potential is chosen for the structures, which are embedded in a 2D‐box bounded by a hard‐wall potential. It is shown that the first order optical absorption coefficient strongly depends on the geometrical parameters and external perturbations of the systems allowing for an optimization of the opto‐electronic properties through control of the latter.
This paper presents the results of the theoretical study of the effects of non-resonant intense l... more This paper presents the results of the theoretical study of the effects of non-resonant intense laser field and electric and magnetic fields on the optical properties (the linear and third-order nonlinear refractive index and absorption coefficients) in an asymmetric quantum well. The electric field and intense laser field are applied along the growth direction of the asymmetric quantum well and the magnetic field is oriented perpendicularly. To calculate the energy and the wave functions of the electron in the asymmetric quantum well, the effective mass approximation and the method of envelope wave function are used. The asymmetric quantum well is constructed by using different aluminium concentrations in both right and left barriers. The confinement in the quantum well is changed drastically by either the effect of electric and magnetic fields or by the application of intense laser field. The optical properties are calculated using the compact density matrix approach. The results show that the effect of the intense laser field competes with the effects of the electric and magnetic fields. Consequently, peak position shifts to lower photon energies due to the effect of the intense laser field and it shifts to higher photon energies by the effects of electric and magnetic fields. In general, it is found that the concentration of aluminum, electric and magnetic fields and intense laser field are external agents that modify the optical responses in the asymmetric quantum well.
We investigate the properties of the vortex singularities in two-component exciton-polariton cond... more We investigate the properties of the vortex singularities in two-component exciton-polariton condensates in semiconductor microcavities in the presence of transverse-electric-transverse-magnetic (TE-TM) splitting of the lower polariton branch. This splitting does not change qualitatively the basic (lemon and star) geometry of half-quantum vortices (HQVs), but results in warping of both the polarization field and the supercurrent streamlines around these entities. The TE-TM splitting has a pronounced effect on the HQV energies and interactions, as well as on the properties of integer vortices, especially on the energy of the hedgehog polarization vortex. The energy of this vortex can become smaller than the energies of HQVs. This leads to modification of the Berezinskii-Kosterlitz-Thouless transition from the proliferation of half-vortices to the proliferation of hedgehog-based vortex molecules.
Using a variational procedure within the effective mass approximation, the mixing between the Γ a... more Using a variational procedure within the effective mass approximation, the mixing between the Γ and X conduction band valleys in GaAs-Ga1-xAlxAs quantum wells is investigated by taking into account the effect of applied hydrostatic pressure. Some optical properties such as donor and/or acceptor binding energy and impurity-related transition energies are calculated and comparisons with available experimental data are presented.
Physica E: Low-dimensional Systems and Nanostructures, 2008
The mixing between G and X conduction band valleys in GaAs-Ga 1Àx Al x As quantum wells is invest... more The mixing between G and X conduction band valleys in GaAs-Ga 1Àx Al x As quantum wells is investigated along the lines of a variational model. Trial wavefunctions are depending on a weighting variational parameter that accounts for the mixing by acting as a coefficient in the combination of both uncorrelated G and X states in the system. The dependencies of the calculated binding energy of a donor impurity and the correlated electron-hole photoluminescence peak energy upon hydrostatic pressure and quantum well width are presented.
The exciton binding energy of an asymmetrical GaAs-Ga1−x Al x As cylindrical quantum dot is studi... more The exciton binding energy of an asymmetrical GaAs-Ga1−x Al x As cylindrical quantum dot is studied with the use of the effective mass approximation and a variational calculation procedure. The influence on this quantity of the application of a direct-current electric field along the growth direction of the cylinder, together with that of an intense laser field, is particularly considered. The resulting states are used to calculate the exciton-related nonlinear optical absorption and optical rectification, whose corresponding resonant peaks are reported as functions of the external probes, the quantum dot dimensions, and the aluminum molar fraction in the potential barrier regions.
In this work, we study the exciton states in a zincblende InGaN/GaN quantum well using a variatio... more In this work, we study the exciton states in a zincblende InGaN/GaN quantum well using a variational technique. The system is considered under the action of intense laser fields with the incorporation of a direct current electric field as an additional external probe. The effects of these external influences as well as of the changes in the geometry of the heterostructure on the exciton binding energy are discussed in detail.
ABSTRACT The effects of intense laser radiation on the exciton states in GaAs-Ga1−x Alx As quantu... more ABSTRACT The effects of intense laser radiation on the exciton states in GaAs-Ga1−x Alx As quantum dots are studied with the inclusion of applied dc electric fields oriented along the growth direction of the system. The calculations are made within the effective mass and parabolic band approximations. The intense laser effects have been included along the lines of the Floquet method, modifying the confinement potential associated to the heterostructure. The laser field modifies the Coulomb potential via the generation of two interaction centers. The exciton binding energy behaves as a decreasing function of the laser field strength, as well as of the size of the quantum dot. The normalized photoluminescence peak energy increases with the laser field strength and behaves as a decreasing function of the dot’s dimensions for fixed laser field intensity.
We report the theoretical calculation of the electronic states in a AlxGa1−xAs-based quantum well... more We report the theoretical calculation of the electronic states in a AlxGa1−xAs-based quantum well with inverse parabolic connement under the combined eects of the intense laser eld and hydrostatic pressure. Calculations are in the eective mass and parabolic band approximations and using a variational procedure and the so-called Floquet method in order to obtain the energies and wave functions for the conduction band states. We use the obtained information to investigate the intersubband-related nonlinear optical absorption and optical rectication coecients.
We theoretically investigate the electron and hole states in a semiconductor quantum dot-quantum ... more We theoretically investigate the electron and hole states in a semiconductor quantum dot-quantum ring coupled structure, inspired by the recent experimental report by Elborg and collaborators (2017). The finite element method constitutes the numerical technique used to solve the three-dimensional effective mass equation within the parabolic band approximation, including the effects of externally applied electric and magnetic fields. Initially, the features of conduction electron states in the proposed system appear discussed in detail, under different geometrical configurations and values of the intensity of the aforementioned electromagnetic probes. In the second part, the properties of an electron-hole pair confined within the very kind of structure reported in the reference above are investigated via a model that tries to reproduce as close as possible the developed profile. In accordance, we report on the energies of confined electron and hole, affected by the influence of an ex...
Este artículo ofrece un esquema para el cálculo de la movilidad de electrones en una estructura d... more Este artículo ofrece un esquema para el cálculo de la movilidad de electrones en una estructura de pozo delta dopado, a bajas temperaturas y para campos eléctricos aplicados no muy intensos. El análisis se hace para un sistema dopado tipo n, considerando una lámina de impurezas de silicio en una muestra de GaAs. Se tiene en cuanta explícitamente el carácter tridimensional de los estados electrónicos y de las magnitudes que se miden –en lugar de realizar aproximaciones en que se reduzcan las dimensiones del sistema–, lo que facilita la ejecución del cálculo. Se exponen los resultados de otros esquemas utilizados y se dan las fórmulas que se emplearán para comprobar y comparar nuestros cálculos con los reportados previamente. En la actualidad, se está en el proceso de implementación del cálculo numérico, del que se expone el algoritmo que se tiene en ejecución.
The polaronic effect on the linear, third-order nonlinear and total optical absorption coefficien... more The polaronic effect on the linear, third-order nonlinear and total optical absorption coefficients have been calculated in the case of GaAs/AlAs core/shell quantum dot, with the impurity is positioned at the central radial position of the GaAs shell. The calculations are realized in the framework of the effective mass approximation and the numerical results are obtained by using a variational method and an infinite confining potential. The results show that the polaronic effect has a great influence on optical properties of core/ shell quantum dot, he causes a red-shift of the nonlinear optical coefficients associated to light absorption. Also, the polaronic effect is enhanced with the decreasing of quantum dot radius.
A quantum ring is a physical object similar to a real life ring that we find in our daily life, t... more A quantum ring is a physical object similar to a real life ring that we find in our daily life, the only difference is in its very small nanometer dimensions (0.000000001 m). Furthermore, the ring is made of a semiconductor material called gallium arsenide (GaAs), as opposed to ordinary rings generally made of gold or silver. Another difference with the real ring is that it only has two dimensions, this means that it looks like a hole coin with negligible thickness. To the naked eye, this system could appear too ideal and not useful; but, these structures can be produced experimentally with very small thicknesses close to the system studied here and they have large enough important applications for the actual world.
Abstract The problem of exciton states in spherical semiconductor quantum dots is revisited, empl... more Abstract The problem of exciton states in spherical semiconductor quantum dots is revisited, employing the finite element method to numerically solve the system of differential equations for the center of mass and relative motion of the interacting electron-hole pair. This process is performed within the effective mass and parabolic bands approximations. The use of a finite confinement together with a parabolic description of the conduction and valence band profiles prevents the two equations from uncoupling. The allowed energies are reported as functions of the quantum dot radius. A comparison of theoretically determined fundamental photoluminescence peak energies with available experimental reports in the cases of CdS, CdSe and CdTe is presented and discussed, showing a good agreement between calculated and measured results. Graphical abstract
The effects of external applied fields on the electron-related optical nonlinear properties in a ... more The effects of external applied fields on the electron-related optical nonlinear properties in a short-range bottomless exponential potential quantum well are investigated. The potential profile behaves as inverse square root in the vicinity of the origin and vanishes exponentially toward infinity, supporting a finite number of bound states whose energies and wave functions are determined within the effective-mass approximation. The linear, third-order nonlinear and total optical absorption coefficients (TOACs) as well as the coefficient of relative refractive index changes (RRICs) of the system are calculated from the expressions derived in the framework of the compact density matrix approach. The results obtained after numerical calculations show that with the effect of the static electric and magnetic fields (high-frequency THz laser field), the resonant peak position of the optical response shifts toward higher (lower) energies and the magnitude of the peak increases.
The electronic states in GaAs-AlxGa1−xAs elliptically-shaped quantum rings are theoretically inve... more The electronic states in GaAs-AlxGa1−xAs elliptically-shaped quantum rings are theoretically investigated through the numerical solution of the effective mass band equation via the finite element method. The results are obtained for different sizes and geometries, including the possibility of a number of hill-shaped deformations that play the role of either connected or isolated quantum dots (hills), depending on the configuration chosen. The quantum ring transversal section is assumed to exhibit three different geometrical symmetries - squared, triangular and parabolic. The behavior of the allowed confined states as functions of the cross-section shape, the ring dimensions, and the number of hills-like structures are discussed in detail. The effective energy bandgap (photoluminescence peak with electron-hole correlation) is reported as well, as a function of the Al molar fraction.
The conduction band and electron-donor impurity states in elliptic-shaped GaAs quantum dots under... more The conduction band and electron-donor impurity states in elliptic-shaped GaAs quantum dots under the effect of an externally applied electric field are calculated within the effective mass and adiabatic approximations using two different numerical approaches: a spectral scheme and the finite element method. The resulting energies and wave functions become the basic information needed to evaluate the interstate optical absorption in the system, which is reported as a function of the geometry, the electric field strength, and the temperature.
Abstract The donor-impurity-related optical absorption, relative refractive index changes, and Ra... more Abstract The donor-impurity-related optical absorption, relative refractive index changes, and Raman scattering in GaAs cone-like quantum dots are theoretically investigated. Calculations are performed within the effective mass and parabolic band approximations, using the variational procedure to include the electron-impurity correlation effects. The study involves 1 s -like, 2px-like, and 2pz-like states. The conical structure is chosen in such a way that the cone height is large enough in comparison with the base radius thus allowing the use a quasi-analytic solution of the uncorrelated Schrodinger-like electron states.
A quantum ring is a physical object similar to a real life ring that we find in our daily life, t... more A quantum ring is a physical object similar to a real life ring that we find in our daily life, the only difference is in its very small nanometer dimensions (0.000000001 m). Furthermore, the ring is made of a semiconductor material called gallium arsenide (GaAs), as opposed to ordinary rings generally made of gold or silver. Another difference with the real ring is that it only has two dimensions, this means that it looks like a hole coin with negligible thickness. To the naked eye, this system could appear too ideal and not useful; but, these structures can be produced experimentally with very small thicknesses close to the system studied here and they have large enough important applications for the actual world.
The electronic states of two‐dimensional (2D) semiconductor quantum wells and quantum wires of di... more The electronic states of two‐dimensional (2D) semiconductor quantum wells and quantum wires of disk‐ and ring‐like geometries, under the application of lateral static electric fields, are investigated. The effects of a shallow hydrogenic impurity on the energy levels of both the electron and hole states are reported and the respective binding energies are calculated. The eigenfunctions and eigenvalues of the Hamiltonian are retrieved under the effective mass approximation applying a direct matrix diagonalization scheme in conjunction with a 2D finite element method. A finite confinement potential is chosen for the structures, which are embedded in a 2D‐box bounded by a hard‐wall potential. It is shown that the first order optical absorption coefficient strongly depends on the geometrical parameters and external perturbations of the systems allowing for an optimization of the opto‐electronic properties through control of the latter.
This paper presents the results of the theoretical study of the effects of non-resonant intense l... more This paper presents the results of the theoretical study of the effects of non-resonant intense laser field and electric and magnetic fields on the optical properties (the linear and third-order nonlinear refractive index and absorption coefficients) in an asymmetric quantum well. The electric field and intense laser field are applied along the growth direction of the asymmetric quantum well and the magnetic field is oriented perpendicularly. To calculate the energy and the wave functions of the electron in the asymmetric quantum well, the effective mass approximation and the method of envelope wave function are used. The asymmetric quantum well is constructed by using different aluminium concentrations in both right and left barriers. The confinement in the quantum well is changed drastically by either the effect of electric and magnetic fields or by the application of intense laser field. The optical properties are calculated using the compact density matrix approach. The results show that the effect of the intense laser field competes with the effects of the electric and magnetic fields. Consequently, peak position shifts to lower photon energies due to the effect of the intense laser field and it shifts to higher photon energies by the effects of electric and magnetic fields. In general, it is found that the concentration of aluminum, electric and magnetic fields and intense laser field are external agents that modify the optical responses in the asymmetric quantum well.
We investigate the properties of the vortex singularities in two-component exciton-polariton cond... more We investigate the properties of the vortex singularities in two-component exciton-polariton condensates in semiconductor microcavities in the presence of transverse-electric-transverse-magnetic (TE-TM) splitting of the lower polariton branch. This splitting does not change qualitatively the basic (lemon and star) geometry of half-quantum vortices (HQVs), but results in warping of both the polarization field and the supercurrent streamlines around these entities. The TE-TM splitting has a pronounced effect on the HQV energies and interactions, as well as on the properties of integer vortices, especially on the energy of the hedgehog polarization vortex. The energy of this vortex can become smaller than the energies of HQVs. This leads to modification of the Berezinskii-Kosterlitz-Thouless transition from the proliferation of half-vortices to the proliferation of hedgehog-based vortex molecules.
Using a variational procedure within the effective mass approximation, the mixing between the Γ a... more Using a variational procedure within the effective mass approximation, the mixing between the Γ and X conduction band valleys in GaAs-Ga1-xAlxAs quantum wells is investigated by taking into account the effect of applied hydrostatic pressure. Some optical properties such as donor and/or acceptor binding energy and impurity-related transition energies are calculated and comparisons with available experimental data are presented.
Physica E: Low-dimensional Systems and Nanostructures, 2008
The mixing between G and X conduction band valleys in GaAs-Ga 1Àx Al x As quantum wells is invest... more The mixing between G and X conduction band valleys in GaAs-Ga 1Àx Al x As quantum wells is investigated along the lines of a variational model. Trial wavefunctions are depending on a weighting variational parameter that accounts for the mixing by acting as a coefficient in the combination of both uncorrelated G and X states in the system. The dependencies of the calculated binding energy of a donor impurity and the correlated electron-hole photoluminescence peak energy upon hydrostatic pressure and quantum well width are presented.
The exciton binding energy of an asymmetrical GaAs-Ga1−x Al x As cylindrical quantum dot is studi... more The exciton binding energy of an asymmetrical GaAs-Ga1−x Al x As cylindrical quantum dot is studied with the use of the effective mass approximation and a variational calculation procedure. The influence on this quantity of the application of a direct-current electric field along the growth direction of the cylinder, together with that of an intense laser field, is particularly considered. The resulting states are used to calculate the exciton-related nonlinear optical absorption and optical rectification, whose corresponding resonant peaks are reported as functions of the external probes, the quantum dot dimensions, and the aluminum molar fraction in the potential barrier regions.
In this work, we study the exciton states in a zincblende InGaN/GaN quantum well using a variatio... more In this work, we study the exciton states in a zincblende InGaN/GaN quantum well using a variational technique. The system is considered under the action of intense laser fields with the incorporation of a direct current electric field as an additional external probe. The effects of these external influences as well as of the changes in the geometry of the heterostructure on the exciton binding energy are discussed in detail.
ABSTRACT The effects of intense laser radiation on the exciton states in GaAs-Ga1−x Alx As quantu... more ABSTRACT The effects of intense laser radiation on the exciton states in GaAs-Ga1−x Alx As quantum dots are studied with the inclusion of applied dc electric fields oriented along the growth direction of the system. The calculations are made within the effective mass and parabolic band approximations. The intense laser effects have been included along the lines of the Floquet method, modifying the confinement potential associated to the heterostructure. The laser field modifies the Coulomb potential via the generation of two interaction centers. The exciton binding energy behaves as a decreasing function of the laser field strength, as well as of the size of the quantum dot. The normalized photoluminescence peak energy increases with the laser field strength and behaves as a decreasing function of the dot’s dimensions for fixed laser field intensity.
We report the theoretical calculation of the electronic states in a AlxGa1−xAs-based quantum well... more We report the theoretical calculation of the electronic states in a AlxGa1−xAs-based quantum well with inverse parabolic connement under the combined eects of the intense laser eld and hydrostatic pressure. Calculations are in the eective mass and parabolic band approximations and using a variational procedure and the so-called Floquet method in order to obtain the energies and wave functions for the conduction band states. We use the obtained information to investigate the intersubband-related nonlinear optical absorption and optical rectication coecients.
We theoretically investigate the electron and hole states in a semiconductor quantum dot-quantum ... more We theoretically investigate the electron and hole states in a semiconductor quantum dot-quantum ring coupled structure, inspired by the recent experimental report by Elborg and collaborators (2017). The finite element method constitutes the numerical technique used to solve the three-dimensional effective mass equation within the parabolic band approximation, including the effects of externally applied electric and magnetic fields. Initially, the features of conduction electron states in the proposed system appear discussed in detail, under different geometrical configurations and values of the intensity of the aforementioned electromagnetic probes. In the second part, the properties of an electron-hole pair confined within the very kind of structure reported in the reference above are investigated via a model that tries to reproduce as close as possible the developed profile. In accordance, we report on the energies of confined electron and hole, affected by the influence of an ex...
Este artículo ofrece un esquema para el cálculo de la movilidad de electrones en una estructura d... more Este artículo ofrece un esquema para el cálculo de la movilidad de electrones en una estructura de pozo delta dopado, a bajas temperaturas y para campos eléctricos aplicados no muy intensos. El análisis se hace para un sistema dopado tipo n, considerando una lámina de impurezas de silicio en una muestra de GaAs. Se tiene en cuanta explícitamente el carácter tridimensional de los estados electrónicos y de las magnitudes que se miden –en lugar de realizar aproximaciones en que se reduzcan las dimensiones del sistema–, lo que facilita la ejecución del cálculo. Se exponen los resultados de otros esquemas utilizados y se dan las fórmulas que se emplearán para comprobar y comparar nuestros cálculos con los reportados previamente. En la actualidad, se está en el proceso de implementación del cálculo numérico, del que se expone el algoritmo que se tiene en ejecución.
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