Simultaneous Effects of Electric Field and Magnetic Field of a Confined Exciton in a Strained GaAs

Combined effects of electric field and magnetic field on exciton binding energy as a function of dot radius in a cylindrical GaAs0.9P0.1/GaAs0.6P0.4 strained quantum dot is investigated. The strain contribution includes the strong built-in electric field induced by the spontaneous and piezoelectric polarizations. Numerical calculations are performed using variational procedure within the single band effective mass approximation. The interband emission energy as functions of geometrical confinement, electric and magnetic field strengths in the quantum dot is discussed. The diamagnetic shift of exciton as a function of dot radius is found. The quantum confined Stark shifts of exciton for various dot radii are discussed. Optical rectification in the GaAs0.9P0.1/GaAs0.6P0.4 strained quantum dot is computed in the presence of electric and magnetic field strengths. Our results show that the optical rectification strongly depends on the spatial confinement, electric field and magnetic field strength. [DOI: 10.1380/ejssnt.2013.29]


I. INTRODUCTION
Quantum dots are given due attention because of their exotic optical behaviour and interesting confined systems in condensed matter physics for the past few decades.Quantum confinement of the charge carriers in quantum dots structures leads to the formation of discrete energy levels.Semiconductor quantum dots are considered to be an artificial atom exhibiting some interesting electronic and optical properties which are strongly affected by some external perturbations such as electric, magnetic fields, pressure and temperature.The structural parameters and properties of these structures have been investigated both experimentally and theoretically [1,2].Electric field induced excitons [3,4] and magnetic field induced excitons [5,6] in quantum dots have been studied earlier.The effect of an electric field on the exciton in the confined excitonic states is generally known as quantum confined Stark effect in low dimensional semiconductor systems.Combined effects of electric and magnetic fields applied parallel to each other in semiconductor quantum dots have been investigated [7,8] but the study of magnetoexciton in the quantum dots with the tilted angle with the external field is sparse [9].
Nonlinear optical properties in low dimensional semiconductor systems are quite interesting and properties are enhanced due to the geometrical confinement [10][11][12][13].Nonlinear optical effects in quantum dots have been discussed earlier [14,15].Among nonlinear optical properties, second order nonlinear optical rectification is given due attention nowadays because its coefficients are higher than the other nonlinear optical coefficients [16].
In the present work, the exciton binding energy as a function of dot radius in a cylindrical GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 strained quantum dot, in the presence of electric and magnetic fields applied in the z direction, is investigated.The strain contribution includes the strong built-in electric field induced by the spontaneous and piezoelectric polarizations.Combined effects of electric and magnetic fields on the interband emission energy as a function of geometrical confinement in the quantum dot are discussed.The diamagnetic shift and the quantum confined Stark shift of exciton as a function of dot radius are investigated.The dependence of optical rectification on the dot radius of GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 strained quantum dot in the presence of external perturbations is brought out.In Section 2, the method used in our calculation is briefly described and the results and discussion are presented in Section 3. A brief summary and results are presented in the last Section.

II. THEORY AND MODEL
Combined effects of electric and magnetic fields on the binding energy and the optical rectification of an exciton are studied in a cylindrical GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 strained quantum dot.The strain effect incorporates the effects of piezoelectric field.The inner quantum dot, GaAs 0.9 P 0.1 , is considered with the radius R surrounded by the barrier height H of a larger band gap of the material, GaAs 0.6 P 0.4 .The Hamiltonian of an exciton in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot consists of a single electron part (H e ), the single hole part (H h ) and the Coulomb interaction term between electronhole pair.The Hamiltonian of the electron (hole), in the presence of electric and magnetic fields, in the cylindrical co-ordinates, within the effective mass-approximation, is given by Volume 11 (2013) Vinolin and Peter where the subscript j = e, h refers the electron or hole, ε is the dielectric constant of material inside the quantum dot, ρ is the radial component of the electron (hole) coordinate, F is the effective electric field due to piezoelectric effect, G is the applied electric field, µ B is the Bohr magneton and σ is the spin taken as ±1/2 [17].The z-component of the spin has been taken as ±1/2 for simplicity for holes.The parameters used in our calculations are given in Table I.V (ρ j , z j ) is the confinement potential due to the band offset of GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 and the contribution by the strain effect (V c(v)strain ).
The electron effective mass m * e is given by m * I denotes the electron effective mass of GaAs 0.9 P 0.1 and m * II denotes the electron effective mass of the outer material, GaAs 0.6 P 0.4 .Since it has a large influence on the electron energy levels in a semiconductor quantum dot, especially for the narrow dots the material dependent effective mass is employed in this calculation [18].
The energy dependent effective mass is given by, (3) where E denotes the electron energy in the conduction band, m(0) is the conduction band effective mass, E g and ∆ are the main band gap and spin-orbit band splitting respectively.And is the effective Lande factor of the electron [19].In Eq. ( 4), m 0 denotes free electron mass.The strain effects will induce an extra potential field, V strain in the z-direction.For the strained quantum dot nanostructure, the confinement potential can be written as a sum of energy offsets of the conduction band (or valence band) and the strain-induced potential.The V (z j ) and V (ρ j ) are the z-direction and in-plane confinement potential due to the conductor band offset in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot respectively.The electron (hole) confinement potential V (ρ j ) due to the band offset in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot is given by and where is the barrier height and Q c is the conduction band offset parameter taken as 80:20 [20].∆E Γ g , the band gap difference between quantum dot and the barrier at Γ-point, is given by [21,22], with [23] ∆E Γ g (unstrained) = 1.424 + 1.174x + 0.186x 2 , (9) and where a c(v) is the deformation potential constants of conduction (valence) band, b is the uniaxial strain, the strain in the layer is given by ε = (a 0 − a)/a where a 0 and a are the lattice parameters of the dot and the barrier respectively.The strain causes the shift in the conduction band resulting the increase in the band gap of the dot and decrease in the band gap of barrier.The uniaxial strain removes the degeneracy between the heavy hole and light hole subband energy in the valence band.
The conduction and valence band offsets, V c and V h are calculated using the expressions as where E GaAs0.6P0.4 c and E GaAs0.6P0.4 hh are the energies of the conduction and heavy hole bands in the barrier dot, E GaAs0.9P0.1 c and E GaAs0.6P0.4 v are the energies of the conduction and heavy hole bands in the inner dot.
The exciton binding energy is reduced when the piezoelectric effect is included and the expression of piezoelectric field is given by where d 31 is the piezoelectric constant of GaAs 0.9 P 0.1 , ε is the dielectric constant of material inside the quantum dot, C 11 , C 12 , C 13 and C 33 are the elastic stiffness constants and ε xx is the in-plane component of the strain tensor, given by  where a in and a out are the lattice constants of GaAg 0.9 P 0.1 and GaAs 0.6 P 0.1 material respectively.The heavy-hole and light-hole masses in terms of Luttinger parameters, γ 1 and γ 2 , (Table I) are given by [24] where m 0 is the free electron effective mass.
We have chosen the trial wave function for the exciton ground state, within the variational scheme.An exciton in a GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot is considered within the single band effective mass approximation, it is necessary to use a variational approach to calculate the eigen function and eigen value of the Hamiltonian and thereby to calculate the bound exciton ground state energy.
A two-parameter variational Gaussian wave function is used to calculate the energy eigen values considering the correlation of the electron-hole relative motion.The trial wave function is chosen as where N is the normalization constant, the wave functions f (ρ j ) and h(z j ) express the motions of the electron (hole) in in-plane and z-direction.They are the ground state solutions of Schrödinger equation for the electrons and holes in the absence of Coulomb interaction.α (responsible for in-plane correlation) and β (responsible for the relative motion in the z-direction) are the variational parameters.The Eq. ( 18) describes the correlation of the electron-hole relative motion.α and β are variational parameters responsible for the in-plane correlation and the correlation of the relative motion in the z-direction respectively [25].By matching the wave functions and the effective mass and their derivatives at boundaries of the quantum dot and along with the normalization, we fix all the constants in the above equations except the variational parameters.These constants are obtained by the interface conditions between the dot and the barrier.So the wave function Eq. ( 18) completely describes the correlation of the electron-hole relative motion.These two variational parameters are the responsible of anisotropy of cylindrical nature of the quantum dots.And hence, it is believed that this type of Gaussian wave function is suitable approximation in a quantum limit region [26].The application of electric field in the z-direction has been followed from the Ref. [27].
The Schrödinger equation is solved variationally by finding ⟨H⟩ min and the binding energy of the exciton in the quantum dot is given by the difference between the energy with and without Coulomb term.First, we concentrate on the calculation of the electronic structure of GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot system by calculating its lowest binding energy with the inclusion of magnetic field and subsequently the electric field are applied in the z-direction in order to get the corresponding exciton binding energy.And then, by using the density matrix approach, within a two-level system approach, the explicit expressions for the nonlinear optical rectification coefficients are computed in saturation limit.
The ground state energy of the exciton in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot in the presence of external magnetic field, E exc , is obtained by minimizing the expectation value of H exc with respect to the variational parameters using Eq. ( 18).The ground state energy of the exciton in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot is calculated by using the following equation The binding energy of the excitonic system is defined as where E e,h is the sum of the free electron and the free hole self-energies in the same quantum dot and γ the measure of magnetic field with γ = ℏω c /2Ry * where Ry * is the effective Rydberg (6.937 meV) and ω c is the cyclotron frequency.
The stark shifts of the exciton binding energy is obtained as The oscillator strength of the exciton ground state is given by [28] where P is the matrix element, m 0 is the bare electron mass, E exc is the exciton binding energy and E is the ground state energy of electron and hole.The value of P 2 /2m 0 is assumed to 1 eV.The interband emission energy E ph associated with the exciton are calculated by where E Γ g is band gap energy of GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot.The second order nonlinear optical rectification coefficient is given by [29,30] Volume 11 (2013) Vinolin and Peter where σ s is the electron density in the quantum dot, ε 0 is the vacuum permittivity, Γ 0 = 1/τ is the relaxation rate for states 1 and 2 and ℏω is the photon energy.The Matrix element, µ 01 = ⟨ψ 0 |z|ψ 1 ⟩, is defined as the electric dipole moment of the transition from the ground state (ψ 0 ) to the first excited state ψ 1 with δ 01 = ⟨ψ 1 |z|ψ 1 ⟩−⟨ψ 0 |z|ψ 0 ⟩.∆E (transition energy) is the absorption energy from ψ 0 to ψ 1 .We have taken the relaxation rate as 1 ps and the electron density is taken as 1×10 24 m −3 .

III. RESULTS AND DISCUSSION
All the numerical calculations are carried out to investigate the binding energy, the interband emission energy, the oscillator strength and the optical rectification of an exciton in a strained GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot in the presence of combined effects of magnetic field and electric field.The geometrical confinement and the confinement due to magnetic field and electric field are employed.The atomic units are followed in the computations in which the electronic charge and the Planck's constant are assumed to be unity.In all our calculations of exciton binding energy and the optical properties, the heavy hole mass is used since the heavy excitons are more common in experimental results.The used material parameters are given in Table I.
We present the variation of magnetic field as a function of measure of magnetic field strength for various concentration of Phosphorous alloy content in a GaAs 1−x P x quantum dot in order to calculate the lowest subband en- (3) -GaAs 0.7 P 0.3 (2) -GaAs 0.8 P 0.2 Magnetic field (T) Measure of magnetic field (g) (1) -GaAs 0.9 P 0.1 ergy.A linear variation of magnetic field with a measure of magnetic field is found for all the Phosphorous contents and this variation is more dominant for the larger Phosphorous alloy contents.The following magnetic field strength is found for different concentration. 1 gamma is 7.875 T for x = 0.1, 10.75 T for x = 0.2, 14.34 T for x = 0.3 and 18.43 T for x = 0.4.A constant Phosphorous alloy content (x = 0.1) for the inner quantum dot is taken throughout our paper.Figure 2 displays the variation of exciton binding energy as a function of dot radius in a GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot in the presence of magnetic field and electric field.The measure of magnetic field, one γ equals 7.875 T. The increase in exciton binding energy with the decrease in dot radius is observed for all the magnetic field and electric field.The binding energy is found to reach a maximum value and then decreases for the strong geometrical confinement.This is because the Coulomb interaction between the electron and hole is increased which ultimately causes the decrease in binding energy when the dot radius decreases.The binding energy decreases further as the dot radius approaches zero since the confinement becomes negligibly small, and in the finite barrier problem the tunneling becomes huge.Also, the contribution of geometrical confinement is dominant for smaller dot radii making the electron unbound and ultimately tunnels through the barrier [31].
The same behavior is observed for all the values of applied magnetic field and electric field.However, we notice that the binding energy increases with the magnetic field for all the dot radii.It is because an additional confinement due to the magnetic field is raised when the wave functions are squeezed also it is observed that the effect of  magnetic field has more influence on the bigger dot radii than the smaller ones.Since the cyclotron radii of the electron and hole are inversely proportional to the square root of the magnetic field strength and hence the strong magnetic field results in a squeezing of the exciton [32].Thus, we observe that the binding energy is more for smaller dot radii than the larger size due to the additional spatial confinement.On contrary, the exciton binding energy is decreased for all the dot radii when the electric field is applied.This is because as the electric field is increased the electron is pulled towards one side of the quantum dot which results overall decrease of the binding energies.However, the decrease of exciton binding energy is more for case of absence of magnetic field than in the cases which include the effects of magnetic field.This is because the domination of geometrical confinement in the case of absence of magnetic field whereas in the other cases, not only the geometrical confinement involves but also the additional confinement due to magnetic field is included.However, the application of electric field overcomes both the confinement effects.(1) ---γ=0.127,E=0 kV/cm (4) ---γ=0, E=50 kV/cm Dot radius (Å) Interband emission energy (eV) ( (2) ---γ=0.127,E=50 kV/cm (3) ---γ=0, E=0 kV/cm P GaAs FIG.5: Variation of interband emission E ph as a function of dot radius in the presence of magnetic field and electric field in a GaAs0.9P0.1/GaAs0.6P0.4 quantum dot.
We present the variation of diamagnetic shift for heavy hole exciton as a function of magnetic field strength for different dot radius in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot in the absence of electric field in Fig. 3.It is noticed that the exciton diamagnetic shift increases monotonically with the magnetic field strength for all the dot radii due to the effect of magnetic confinement.The diamagnetic shift is more pronounced for wider dots.Further, it is observed that the diamagnetic shift is less sensitive to the applied magnetic field strengths for smaller dot radii because the main contribution comes from the effect of geometrical confinement but the effect of magnetic field on the electron and hole dominates for the larger dot radii whereas the combined effects of magnetic confinement and the geometrical confinement take place in the intermediate region.Figure 4 shows the quantum confined Stark shifts for heavy hole exciton as a function of electric field for different dot radii.Electron-hole pair interaction on the Stark effect is brought out here.It is observed that the exciton energy becomes more red-shifted with increasing the strength of the electric field.The quantum confined Stark effects on narrow dots have less influence than the wider dots.It is because the polarizability increases with the dot radius and the effects of exciton binding energy are more for narrow dots in the stronger confinement region than in the weak confinements.
The variation of interband emission, E ph , as a function of dot radius in the presence of magnetic field and electric field in a GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot is shown in Fig. 5.In all the cases, it is observed that the interband emission energy decreases monotonically as the radius of dot is increased.This is due to the confinement of electron-hole with respect to in-plane direction when the dot radius is increased.This representation clearly brings out the quantum size effect.Moreover, the interband emission energy increases with the magnetic field strength and decreases with the electric field strength.However, it is found that the interband transition energy is more influenced for both the electric field and magnetic fields.It is because the additional confinement due to the magnetic field strength. (2)

Dot radius (Å)
Oscillator strength (au) (1) ---γ=0.381 (2) ---γ=0.254 (3) ----γ=0.127(4) ----γ=0 all dotted lines ---E=50kV/cm P GaAs FIG.6: Variation of oscillator strength as a function of dot radius in the presence of magnetic field and electric field in a GaAs0.9P0.1/GaAs0.6P0.4 quantum dot.We present the variation of oscillator strength a function of dot radius in the presence of electric field and magnetic field in a GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot in Fig. 6.It is observed that the oscillator strength decreases with the increasing dot radius and increases with the magnetic field strength.It is also observed that the oscillator strength increases with magnetic field whereas it decreases with the application of electric field.The increase of oscillator strength with increasing magnetic field is due to the compression of exciton wave function.
Figure 7 displays the variation of second order non-linear coefficient as a function of incident energy for a confined exciton in the presence of electric and magnetic fields in a GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot.It is found that the resonant peak shifts towards blue region as the dot radius decreases due to the geometrical confinement.It is known that the exciton binding energy increases with the magnetic field strength.The magnitude of the resonant peak of nonlinear optical rectification is found to be around 10 −5 m/V [33].Here, we observe that the variation of magnitude of absorption coefficient decreases with the dot radius.The spacing between the energy levels increases due to decrease in dot radius.It is because the exciton binding energy decreases when the dot radius is increased and ∆E increases as the dot radius decreases according to Eq. ( 24).Thus, the resonant frequencies are important and it should be taken into account in studying the optical properties of exciton in the quantum nanostructures.The reason for the blue-shift is due to the higher transition energy when the dot radius is decreased.Further, we notice that the resonant absorption peak value is found to be linearly decreasing with the decrease in dot radius and the energy levels are separated largely with the reduction of overlap integral due to the increase in dipole matrix.It is because there occurs a competition between the energy interval and the dipole matrix element which determine these features.Thus by decreasing the dot radius a remarkable blue-shift of the absorption resonant peak is induced, leading to a higher energy interval.The application of magnetic field increases the nonlinear rectification coefficient due to the enhancement of exciton binding energy with the application of magnetic field whereas the application of electric field decreases the nonlinear rectification coefficient due to the decrease of exciton binding energy with the electric field.
In conclusion, the combined effects of electric and magnetic fields on exciton binding energy as a function of dot radius in a cylindrical GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 strained quantum dot have been discussed.The strong builtin electric field due to the spontaneous and piezoelectric polarizations has been included in the Hamiltonian.The interband emission energy as functions of geometrical confinement, electric field and magnetic field in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 quantum dot has been discussed.The diamagnetic shift and the quantum confined Stark shift of exciton as a function of dot radius are found.Electric field and magnetic field induced optical rectification in the GaAs 0.9 P 0.1 /GaAs 0.6 P 0.4 strained quantum dot has been investigated.Our results show that the optical rectification strongly depends on the spatial confinement, electric field and magnetic field.We hope that the present investigations would lead further experimental research works and open new openings of quantum size effects in optical devices.

FIG. 1 :
FIG. 1: Variation of magnetic field strength as a function of measure of magnetic field for various concentration of P in a GaAs1−xPx quantum dot. 1 gamma is 7.875 T for x = 0.1, 10.75 T for x = 0.2, 14.34 T for x = 0.3 and 18.43 T for x = 0.4.This figure is not referred in the text.Please check.

FIG. 2 :FIG. 3 :
FIG.2: Variation of exciton binding energy as a function of dot radius in a GaAs0.9P0.1/GaAs0.6P0.4 quantum dot in the presence of magnetic field and electric field.The measure of magnetic field, one γ equals 7.875 T.

FIG. 4 :
FIG. 4: Quantum confined Stark shifts for heavy hole exciton as a function of electric field for different dot radii.