Temperature Dependent Optical Properties in a Strained Diluted Magnetic Quantum Well

Temperature dependence of the exciton in a Cd1−xoutMnxoutTe/Cd1−xinMnxinTe/Cd1−xoutMnxoutTe strained quantum well is investigated for various Mn content. Spin polaronic shifts are estimated using mean field theory for different Mn concentration and the well sizes. Calculations are performed for various Mn ion content in a Cd1−xMnxTe material in which the strong built-in electric field due to piezoelectric polarizations are included. A theoretical study of diluted magnetic semiconductors treating local sp–d exchange interaction J between the itinerant carriers and the Mn electrons are treated within a realistic band structure. The temperature dependent optical absorption and the refractive index changes as a function of normalized photon energy with various Mn ion content are analyzed. Our results show that the occurred red shifts of the absorption resonant peak due to the effect of temperature give the information about the variation of two energy levels in the quantum well. The optical absorption coefficients and the refractive index changes strongly depend on the incident optical intensity, the temperature effect and the Mn content. [DOI: 10.1380/ejssnt.2012.388]


I. INTRODUCTION
Spintronics, an emerging area of recent research work, is attracted much attention due to the potential application for fabricating new devices.It is a new field of spin based electronics having a potential for technological advancements nowadays.The semiconductors with incorporation of spin degree of freedom can drastically enhance the performance and their functions are found to be more than the conventional charge based electronic devices.Spin polarized electrons and spin transport current play important roles in designing these spin based materials into semiconductors at room temperature [1].Semimagnetic semiconductors are the most promising candidates for developing such novel devices where the electronic and optical properties are controlled by Mn impurities and the external perturbations.The knowledge of basic band parameter values are required for the development of new devices based on semimagnetic semiconductor nano-heterostructures [2].Cadmium manganese telluride is one of the best-known semimagnetic semiconductors, for high efficiency and high-resolution roomtemperature radiation detector, when the Mn magnetic impurities are introduced with a proper proportion.It is considered to be a room temperature mid IR radiation detector having high quantum efficiency.
The main part of taking advantages of carrier spin of the semimagnetic semiconductor heterostructures based electronic devices is to develop techniques to control and manipulate the spin alignment of magnetic ions impurities in the devices [3].Further, the spin degrees of freedom of the electrons can be utilized for the transferring the information.The remarkable properties observed in these semimagnetic semiconductor compounds in the presence of Mn, bring out the strong ion-ion and ion-carrier exchange interactions through the hybridization between the sp band and the Mn 3d states [4].In addition to these interactions, energy transfer can occur between an extended electron and a local d-electron [5].These exchange interactions of electrons and holes with the Mn impurities show some unusual magnetooptical properties and phase transitions.Various effects such as magnetic field-induced type I-type II transitions, spinmagnetic polaron and the variation of band offsets with temperature will come into play a role in those devices when the magnetic ions are introduced [6][7][8][9][10].Investigations on temperature-dependent changes in the trion and exciton photoluminescence spectra in modulation-doped semimagnetic semiconductor quantum wells in high magnetic fields have been carried out [11].The temperature dependence for the Mn 2+ photoluminescence peak energy in CdMnTe has been reported [12].Cregus et al. [13] have reported the large shift in Mn 2+ PL peak energy by photoluminescence measurements on zinc blende Cd 1−x Mn x Te thin films.
Optical band gap of semimagnetic semiconductors strongly depends on the manganese impurities and it shows some interesting nonlinear optical properties such as thermally induced optical bistability.These bistable elements will have some interesting potential application for device fabrications (optical modulators) in which the temperature dependent absorption coefficients will play a vital role [14].The effect of ferromagnetic order on the electrical properties of the diluted magnetic semiconductors has been studied by analyzing the temperature dependence of the dielectric constant and the resistivity by Lopez-Sancho and Brey [15].The optical properties of Cd 1−x Mn x Te quantum wells across the Mott transition have studied very recently by Teran et al. [16].The energy spectrum of the correlated particles in Cd 1−x Mn x Te material was described within a hydrogenic like energy model [17] in which excitonic features characterize the optical properties of insulating systems.Some optical properties of Mg based II-VI ternary and quaternary materi-als have been investigated by Eunsoon Oh [18] who have observed the large redshift of the band-gap photoluminescence and the Raman electron paramagnetic resonance of Mn 2+ .
In the present work, the temperature effect on the 1slhh, exciton states and their related optical properties are analyzed in a Cd 1−xout Mn xout Te/ Cd 1−xin Mn xin Te/ Cd 1−xout Mn xout Te strained quantum well for various Mn ion content and the well width.Numerical calculations are performed within a single band effective mass approximation using the variational technique.The interband emission energy as a function of well width is calculated for various Mn ion content with and without the temperature effect.The spin polaron energy is included in the Hamiltonian.The optical properties such as the total optical absorption and the refractive index changes as a function of normalized photon energy in the presence of temperature and the Mn ion concentration are investigated.In Sec.II, we briefly describe the method and the model used in our calculations of obtained eigenfunctions and eigenenergies of electron states, oscillator strengths and the linear and non-linear optical absorption coefficients.The results and discussion are presented in Sec.III.A brief summary and results are presented in the last Section.

A. Exciton binding energy
We consider the Hamiltonian of an exciton in a strained Cd where is the Hamiltonian operator.In Eq. ( 2), e is the absolute value of the electronic charge, m * e(h) (x, T ) is the Mn and temperature dependent effective mass of electron (hole) of Cd 1−x Mn x Te, z e and z h are the electron and hole co-ordinates along the growth direction of the structure, V e(h) (x, T ) is the Mn and temperature dependent strain induced confined potential for electrons (holes) [19,20].F is the built-in electric field due to piezoelectric polarization, H SP (x, T ) refers the interaction between electron spin and Mn 2+ ions respectively (Sec.II.C).
The Luttinger parameters which are related to the band edge effective masses through the relations given by [21]  where γ 1 , γ 2 , α and E p are the Luttinger parameters and the Kane energy of CdTe.E g and ∆ are Mn and Temperature dependent band gap and spin orbit gap respectively.The variation of Mn and temperature dependent band gap differences of CdTe with Mn concentration is given by where β = −3.5 × 10 −4 eV/K, θ has been taken as 60 K [22], δ e(h) N 0 ≈ 220(880) meV with N 0 = 2.94 × 10 22 cm −3 , the bowing parameter has been taken form Ref. [23] with ε xx (x) = ε yy (x) = (a 0 (x) − a(x))/a 0 (x) where a 0 (x) and a(x) are the Mn-dependent lattice parameters of bulk CdTe and MnTe respectively and , where C ij are the values of elastic constants and e ij are the piezoelectric stress constants as given in Table I.The piezo electric polarization is given by The strain induced confinement potential can be written as a sum of energy band offsets and the static electric potential induced by the built-in electric field.The straininduced potential for the conduction band in the influence of Mn-incorporation is given by where a c is the deformation potential constant of conduction band, where a v and b are the deformation potential constants of valence band.
We have chosen the trial wave function for the exciton ground state (1s-1hh), within the variational scheme.
We take the problem of an exciton in a Cd 1−xout Mn xout Te/Cd 1−xin Mn xin Te/Cd 1−xout Mn xout Te quantum well within the single band effective mass approximation.Since the inclusion of impurity potential leads to a nonseparable differential equation which cannot be solved analytically, it is necessary to use a variational approach to calculate the eigen function and eigen value of the Hamiltonian and to calculate the bound exciton ground state energy.Considering the correlation of the electron-hole relative motion, the trial wave function can be chosen as where N is the normalization constant, f e and f h are ground state solution of the Schrödinger equation for the electrons and holes in the absence of the Coulomb interaction.The above equation describes the correlation of the electron-hole relative motion.δ and β are variational parameters responsible for the in-plane correlation and the correlation of the relative motion in the z-direction respectively.By matching the wave functions and the effective mass and their derivatives at boundaries of the quantum well and along with the normalization, we fix all the constants except the variational parameters.So the wave function Eq. ( 10) completely describes the correlation of the electron-hole relative motion.The Schrödinger equation is solved variationally by finding ⟨H⟩ min and the binding energy of the exciton in the quantum well is given by the difference between the energy with and without Coulomb term.First, we concentrate on the calculation of the electronic structure of the CdMnTe quantum well system by calculating its subband energy (E) and subsequently the exciton binding energy.Then, by using the density matrix approach, within a two-level system approach, the explicit expressions for the nonlinear optical properties such as the nonlinear optical absorption and the changes of refractive index are computed in saturation limit.We assume the non-linear optical process through the third order harmonic generation.
The binding energy of the excitonic system in the presence of electric field strength is defined as where E e,h is the sum of the free electron and the free hole self-energies in the same quantum well.

B. Interband Emission energy, Eph
The Hamiltonian of an exciton consisting of a single electron part (H e ), the single hole part (H h ) and the Coulomb interaction term between electron-hole pair is given by, where all the terms are defined as earlier.The Mndepended band gap of Cd 1−x Mn x Te is explained as earlier.It is inferred the direct band gap of CdMnTe increases with Mn concentration linearly.Thus the incorporation of Mn into CdTe will significantly change its band gap.The ground state energy of the exciton in the Cd 1−x Mn x Te quantum well is calculated by using the following equation The temperature and Mn-dependent exciton binding energy E b and the interband emission energy E ph associated with the exciton is calculated using the following equations: where E e (x, T ) and E h (x, T ) are the Mn and temperature dependent confinement energies of the electron and hole respectively.

C. Spin Polaronic effect
When a single confined carrier interacts with magnetic impurities (Mn), it forms a new stable state so called a magnetic polaron.The incorporation of magnetic impurities in non-magneitc semiconductors extends the possibility of analyzing the interaction between carriers and magnetic impurities under strong geometrical and external confinement.The antiferromagnetic exchange interaction arising between the spin of a conduction electron and the Mn 2+ spins is described by the Hamiltonian H m as Here S j is the spin of the Mn 2+ ion at position R j and s is the spin of the conduction electron (hole) centered at r.Here J s(p)−d (r, R j ) is the electron (hole)-Mn exchange coupling constant which depends on the overlap between the orbital of the conduction electron (hole) and of the 3d electrons.
The mean field approximation incorporating the exchange interaction between the carrier and the magnetic impurity, yields the spin polaronic shift, E sp , with the modified Brillouin function [27] ] dτ, (17) where δ e(h) is the exchange coupling parameter, S is the Mn 2+ spin, and xN 0 is the Mn ion concentration.The integration is on spatial coordinates.Also g ≈ 2 , S 0 (x), the effective spin, and T 0 (x), the effective temperature are the semi-phenomenological parameters, which describe the paramagnetic response of the Mn 2+ ions in the bulk Cd 1−x Mn x Te.In Eq. ( 17), ϕ(r e , e h ) is the envelope function as given in Eq. (10) with the appropriate values of the variational parameters, k B is the Boltzmann constant and B s η is the modified Brillouin function.
The parameters used in our calculations are N 0 = 2.94×10 22 cm −3 , δ e(h) N 0 ≈ 220 (880) meV, and the semiphenomenological parameters S 0 (X in = 0.02) = 1.97, S 0 (X out = 0.1) = 1.08,T 0 (X in = 0.02) = 0.94, and T 0 (X out = 0.1) = 3.84.Using the envelop function given in Eq. ( 10) with the proper variational parameters, we obtain where .Choosing the cell dimension in CdTe to be 6.48 Å, taking 8 atoms per unit cell, we have calculated the total number of ions present in a well for different concentration.J sp−d is the coupling strength due to the spin-spin exchange interaction between the d electrons of the Mn +2 cations and the p-band holes, and it is positive for valence band holes.The value of J sp−d is taken to be 15×10 −3 eVnm −3 , the spin of Mn +2 cation is 5/2, and the above approach of Brillouin function is quite general especially when dealing with the two dimensional problems.

D. Absorption coefficients and refraction index changes
For any electronic system transitions, these calculations are imperative to compute the different optical properties.However, the dipole transition transitions are allowed using the selection rules ∆l = ±1 where l is the angular momentum quantum number.In addition to that the oscillator strength which is related to the dipole transition, expressed as where ∆E f i = E f − E i refers the difference of the energy between the lower and upper states.M f i = 2⟨f |R|i⟩ is the electric dipole moment of the transition from i state to f state in the quantum well width.Here, we have considered the selection rule, ∆l = ±1 which determines the fine state of the electron after absorption.Hence the state l = 0 is taken as the ground state and the state l = 1 is taken as the final state.The observation of oscillator strength is imperative especially in the study of optical properties and they are related to the electronic dipole allowed absorptions.Moreover, the outcome of the results will viewed on the fine structure of the optical absorption.
The optical absorption calculations are based on the Fermi Golden rule from which the total absorption coefficient is given by [28,29] where µ 0 is the permeability of the material, ε r is the real part of the permittivity and I is the incident light intensity.χ 1 (ω) and χ 3 (ω) describe the linear and nonlinear contribution to the polarization with the same frequency of the incident field.The optical absorption coefficient, for the linear term, is given by and for the third order nonlinear term, Thus the total absorption coefficient is given by The susceptibilities are related to the changes of refractive index as where n r is the refractive index of the material.The expressions of the linear and nonlinear changes in the refractive index are given by ∆n Hence the total refractive index change is given by where n r is the refractive index of the semiconductor, ε 0 is the vacuum permittivity, ω the angular frequency of the incident photon energy, N = 10 18 cm −3 , E i and E f denote the confinement energy levels for the ground and the first excited state, respectively, Γ is the line width of the exciton and in our calculation we use ℏΓ = 6 meV [30].
The above two equations are linear and third order nonlinear optical absorption coefficients.

III. RESULTS AND DISCUSSIONS
The atomic units have been followed in the determination of electronic charges and the wave functions in which the electronic charge and the Planck's constant have been assumed as unity.In all our calculations we have used the heavy hole mass as the heavy excitons are more common in experimental results.
All the values of parameters pertaining to Cd 1−x Mn x Te which are interpolated used in our calculation are given in Table I. Figure 1  the increase in Mn content, we observe the nonlinearity behaviour.This non-linear effect is more pronounced for greater Mn content.It shows that the non-parabolicity effect should be included for higher Mn concentration in CdMnTe materials.∆E c is found to be greater than ∆E v so as to satisfy the electron confinement.X out has been taken as 0.3 throughout our calculation.We present the variation of exciton binding energy as a function of Cd 1−xout Mn xout Te/Cd 1−xin Mn xin Te/Cd 1−xout Mn xout Te strained well width for various Mn content in Fig. 2 and the insert figure shows the variation of binding energy with a Mn content for two different well width.It is observed that the binding energy increases first with decreasing well width and it reaches the maximum value for a critical well width and then rapidly decreases when the well width is still reduced for all the cases of Mn incorporation.It is because the contribution of confinement is dominant for smaller well width making the electron unbound and ultimately tunnels through the barrier.The electron and hole wave functions penetrate into the barrier for narrow wells.These results are well known [31].It is clear from the figure that the exciton binding energy depends on both well width and the potential barrier height.The binding energy of 1s-1hh free exciton remains constant for the well width beyond 500 Å for the Mn content.It implies that the barrier height has no significant effect for the larger well sizes.The insert figure shows that the variation of exciton binding energy is drawn with the Mn content for two different well width and it is found that it increases linearly as the Mn concentration increases taking into account of spin polaron energy in the Hamiltonian for all the well width.Further, we notice that the binding energy is more for smaller well width than the larger size due to the additional spatial confinement.
Figure 3 displays the temperature dependent exciton binding energy as a function of well width for constant Mn content (x = 0.1) and the insert figure shows the variation of exciton binding energy with a temperature for http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology x = 0.1 FIG.3: Temperature dependence on the exciton binding energy as a function of well width for constant Mn content (x = 0.1) and the insert figure shows the variation of exciton binding energy with a temperature for three different well width.
Well width (Å) x = 0.2 three different well width.The behavior of the variation of binding energy is the same as that of Fig. 2.However, we notice that the temperature has a stronger influence on the exciton binding energy which decreases linearly with the temperature.Further, we find that decrease of binding energy with the temperature pertaining to narrow well width is found to be steeper than the wider well sizes.It is because there occurs a competition between the effect of temperature and spatial confinement while reducing the well width.Our results are in good agreement with the earlier literature [32] in which the temperature dependent binding energy of a donor was calculated.In Fig. 4, we present the variation of interband emis-  It is noted that the interband emission energy decreases monotonically as the well width is increased for all the temperature.This is due to the confinement of electron-hole with respect to z-plane when the well width is increased.Moreover, it is clearly shown that the effect of bound exciton has influence on the interband emission energy.This representation clearly brings out the quantum size effect.Moreover, we observe the nonlinear behaviour of the interband emission energy with the temperature as shown in the insert figure.It implies the reduction of band gap with increasing in temperature [33].Figure 5 displays the variation of total absorption coefficient of an exciton in a Cd 1−xout Mn xout Te/ Cd 1−xin Mn xin Te/Cd 1−xout Mn xout Te strained quantum well for two different well width (30 Å) and (100 Å) as a function of photon energy for different values of Mn content for I = 2 × 10 8 W/m 2 with hω = 150 meV.Here, we observe that the variation of magnitude of absorption coefficient becomes more when the Mn-incorporation is taken into account.We notice that the increase in linear variation of resonant absorption coefficient is observed with the Mn incorporation for two different well widths.It is because the exciton binding energy increases with the Mn content due to the addition of spin polaron effect.This result is in good agreement with the other investigator [13,34].It is observed that the magnitude of the absorption coefficient is more for narrow wells due to the geometrical confinement.Moreover, we notice that the binding energy is more for when the Mn content is included for all the well width due to the enhancement of exciton binding energy when the Hamiltonian is included with the spin polaronic effect.Hence it is concluded that intensity dependent nonlinear absorption coefficients near the resonant frequencies are important and it should be taken into account in studying the optical properties of exciton in the quantum well.We display the variation of total absorption coefficient of an exciton in a Cd 1−xout Mn xout Te/Cd 1−xin Mn xin Te/ Cd 1−xout Mn xout Te quantum well for two different well width (30 Å) and (100 Å) as a function of photon energy for different values of temperature with a constant Mn concentration in Fig. 6.It is observed that the absorption coefficient peak moves towards the lower photon energy as the temperature is increased.As the effect of temperature increases, the total absorption coefficient shifts toward lower energies and the magnitude also decreases.It implies that the temperature red shifts the absorption resonance coefficients in a quantum well.It is because that the spacing between the energy levels decreases due to the increase in temperature.The reason for the redshift is due to the smaller transition energy (E 2 − E 1 ) and with the reduction of overlap integral due to the decrease in dipole matrix when the temperature effect is increased.It is because there occurs a competition between the energy interval and the dipole matrix element which determines these features.Thus by increasing the temperature a remarkable red-shift of the absorption resonant peak is induced, leading to a smaller energy interval.
Figure 7 shows the variation of total refractive index changes of an exciton confined in a Cd 1−xout Mn xout Te/ Cd 1−xin Mn xin Te/ Cd 1−xout Mn xout Te quantum well (100 Å) as a function of photon energy in the influence of Mn concentration with a constant value of incident optical intensity (2 × 10 8 W/m 2 ).This figure has been drawn with the combining effects of two components of refractive indices, namely, ∆n (1) (ω)/n r and ∆n (3) (ω)/n r as a function of incident energy for different values of Mn content with the constant incident optical intensity.It is observed that as the Mn incorporation increases, the total refractive index changes with the higher  magnitude of total refractive index.This is because the increase in exciton binding energy occurs with the Mncomposition.
We display the variation of total refractive index changes of an exciton confined in a Cd 1−xout Mn xout Te/ Cd 1−xin Mn xin Te/ Cd 1−xout Mn xout Te quantum well as a function of photon energy for various temperature with a constant Mn concentration (x = 0.3) for a constant well width, 100 Å in Fig. 8.We notice that as the temperature increases the magnitude of the refractive index decreases as well as there occurs a red shift due to the decrease of the energy difference of the first excited state and the ground state of the system and the decrease of dipole matrix.This causes the magnitude of the refractive index shifts towards the lower energy.It shows the weaker http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology Volume 10 (2012) confinement effect due to the application of temperature.Also, it is noticed that the linear and nonlinear changes in refractive index depend on photon intensity and the nonliner term varies quadratically with the matrix element of the electric dipole moment of the transition.Thus, the nonlinear term must be considered when calculating the refractive index changes of quantum well systems in which the incident light propagates along the z-axis [35].Thus, the nonlinear term must be considered when calculating the refractive index changes in low dimensional semiconductor systems.

IV. CONCLUSION
The effects of temperature on excitons in a strained Cd 1−xout Mn xout Te/ Cd 1−xin Mn xin Te/ Cd 1−xout Mn xout Te quantum well have been investigated for various Mn incorporation and thereby the effects of some optical prop-erties have been analyzed.The interband emission energy including the effects of temperature and the Mn concentration as function of well width has been studied.The linear and nonlinear absorption coefficients and the refractive index changes as a function of photon energy in the presence of temperature effect have been discussed for different Mn content.The occurred high magnitude shift of the peak absorption coefficient due to the increase in Mn content and the effect of temperature will give the information about the variation of two energy levels in the quantum well width.Thus the effect of temperature as well as the manganese impurities can play an important role in the optical properties of the CdMnTe semiconductors which are considered to be promising diluted magnetic materials for a room temperature radiation detector materials, solar cells and visible and near-IR lasers.However, our results will pave the way for IR transmission technique which would give more complete knowledge in II-VI materials.
1−xout Mn xout Te/Cd 1−xin Mn xin Te/Cd 1−xout Mn xout Te semiconductor quantum well, within the single band effective mass approximation.The Schrödinger equation of the exciton is given by ĤΨ(r e , r h ) = EΨ(r e , r h ),

with η 1 =
Sδ e(h) |ϕin(ee,r h )| 2 2kB[T +T0(xin)] and η 2 = Sδ e(h) |ϕout(ee,r h )| 2 2kB[T +T0(xout)] FIG.1: Variation of conduction and valence band offset and band gap of a strained Cd1−x out Mnx out Te/Cd1−x in Mnx in Te/Cd1−x out Mnx out Te quantum well as a function of well width for Mn content and the insert figure shows the variation of band gap energy with the temperature.

FIG. 2 :
FIG.2: Variation of exciton binding energy as a function of well width for various Mn content and the insert figure shows the variation of binding energy with a Mn content two different well width.

FIG. 4 :
FIG. 4: Temperature dependent interband emission energy of the ground state exciton of a strained Cd1−x out Mnx out Te/ Cd1−x in Mnx in Te/ Cd1−x out Mnx out Te quantum well width for various Mn concentration and the insert figure shows the variation of interband emission energy as a function of temperature for various well width with a constant Mn content (x = 0.2).

( 1 )100ÅFIG. 5 :
FIG. 5: Variation of total absorption coefficient of an exciton in a Cd1−x out Mnx out Te/Cd1−x in Mnx in Te/Cd1−x out Mnx out Te quantum well (30 Å) and (100 Å) as a function of photon energy for different values of Mn content

xout Mn xout Te/ Cd 1−xin Mn xin Te/ Cd 1−xout Mn xout Te strained quantum well expressed as
[26]in Te and Cd 1−xout Mn xout Te respectively.The strain-induced potential for the valence band in the influence of Mn incorporation in CdTe can be written as[26] a where a 0 and a http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) are the lattice parameters of bulk Cd 1−xin