Polarizability of Exciton in Surface Quantum Well

Surface quantum wells are seeking considerable attention due to their asymmetrical nature of polarized interface and its consequences. Their results with and without external perturbations are expected to be remarkably different from their counterparts in symmetrical quantum wells. Effect of electric field on binding energies of light hole and heavy hole exciton in surface quantum well composed of vacuum/GaAs/Ga1xAlxAs are theoretically calculated as a function of well width and Al composition. Effect of image charges arising due to the mismatch of the dielectric constant at the vacuum/GaAs interface is considered. Stark shift and polarizability of exciton in this surface quantum well is also calculated for various strengths of electric field with different well width confinement as well as Al concentration. Our results show that: (1) exciton binding energy increases as the electric field applied along the growth axis increases; (2) stark shift in exciton energy decreases as electric field, Al composition and well width increase; and (3) polarizability of exciton decreases when the electric field increases, but increases when well width increases. Variation of our results with those for other symmetrical wells will provide a choice of the well for electric field applications.


INTRODUCTION
10][11][12][13] The response of exciton to the electric field in such confined systems has to be considered for the fabrication of spintronic and opto-electronic devices. 14,15Monier et al. determined the oscillator strength of excitons as a function of well width in In x Ga 1-x As/GaAs quantum wells under large electric field. 16The influence of uniform electric field on binding energy of excitons in inhomogeneous quantum dot is analysed by Khamkhami et al. 17 The Stark shift and ground state energy of exciton bound to ionised donor in Spherical quantum dots are determined as a function of electric field and dot radius. 18Wu and Xia studied the effect of electric field on exciton polarizability and binding energy in CdSe/ZnS nanocrystal quantum dots using the diagonalisation method. 19Feddi et al. have determined the effect of electric field on polarizability of an exciton bound to an ionised donor in a spherical quantum dot by perturbative-variational method. 20vestigations on surface quantum wells (SQW) composed of a quantum well with infinite barrier on one side and finite barrier on another side, are of special interest due to the asymmetric nature that leads to: (1) the presence of localised resonant states above the finite single quantum barrier; and (2) the formation of image charges for electron and hole at the single infinite barrier/well interface with large dielectric constant mismatch, unlike the symmetrical quantum wells, where the image charge formation is at both interfaces. 21 our previous work, we have estimated the exciton binding energy in SQW formed by vacuum/GaAs/Ga x Al 1-x As considering with and without the image charge potentials for isotropic and anisotropic masses. 22We observed that SQW shows certain differences in the behaviour of binding energy of excitons.It is of interest to extend our studies on excitons in such SQW under external perturbations especially the quantum confined Stark effect.In the present work, we have studied the effect of electric field in such a SQW composed of vacuum/GaAs/Ga x Al 1- x As with image charge potentials.The binding energies and polarizabilities of the excitons are calculated variationally as a function of well width as well as Al concentration for different strengths of electric field.

MODEL AND FORMULATION
SQW is considered as a nanostructure of well material GaAs with barrier Ga 1-x Al x As on one side and vacuum on other side shown in Figure 1. 21cuum

Well State and Ground State of Excitons
The electron states (without conduction band non-parabolicity), light hole states, heavy hole states and exciton states (with image charges and isotropic masses) are obtained using the theory of our previous work. 22

Effect of Electric Field
The Hamiltonian of exciton in SQW with electric field along the growth axis is given by: where ℋ 0 is unperturbed Hamiltonian dealt with in Section 2.2 of our previous work. 22e perturbation term for electric field along the growth axis is expressed as: 15 where z e and z h are the z-coordinates of the electron and hole, h is the dimensionless measure of the electric field defined as R The trial wavefunction of excitons in the SQW with electric field is taken to be of the form: 20 where ψ o is the wavefunction for unperturbed Hamiltonian H o used in our previous work. 22Whereas, a in ψ o , b and c are the variational parameters.
Exciton energy levels in the presence of electric field are calculated as: For an applied electric field F, the Stark shift on the exciton energy can be calculated as: The polarizability of the exciton is then determined from quadratic Stark effect as: 24 Well state energies of electron, light hole and heavy hole are also affected by the presence of electric field, which are calculated as follows: where j = e or ih (i = l for light hole and i = h for heavy hole) and ψ j is the wavefunction used in Section 2.1 of our previous work. 22e binding energy of the exciton in the presence of electric field is calculated as:

RESULTS AND DISCUSSION
The parameters of the material used in the present work such as effective masses of light hole, heavy hole and electron; reduced masses of light hole and heavy hole exciton; dielectric constant for GaAs are given in Table 1.The energy gap difference ∆E g between GaAs and Ga 1-x Al x As is determined by the following expression: 23 ∆E g = 1.155x 0.37x 2 eV (9)  The barrier height V oe or the conduction band discontinuity is taken to be 0.65∆E g , hence the potential well height of the valence band V oh is 0.35∆E g . 25e have taken x = 0.3 for all calculations in the present work.In symmetrical quantum wells, the applied electric field pulls the electron against the direction of electric field and hole along the direction of electric field.Hence, the exciton becomes more polarized and the exciton binding energy is reduced. 15ut in SQW, due to the asymmetric nature and large dielectric constant mismatch at the vacuum/GaAs interface, image charges are formed for the electron and the hole.A dead layer, free of excitons, is formed near the interface, due to the repulsion of charge carriers and their images.Hence unlike the symmetrical quantum wells, application of electric field cannot pull apart the electron and hole of the exciton since they will be repelled by their image charges at the interface.This contrary behaviour leads to the increased confinement of exciton and increase in the binding energy with increased applied electric field as shown in Figures 2 and 3.But this result is contrary to the results reported by other authors. 14,15,19ason for this significant dissimilarity may also be due to: (1) the large reduction in the electron-hole kinetic energy as applied electric field increases; and (2) the competing effects of the confining potential, image potential and the potential resulting from the applied electric field that changes the optical properties significantly. 26A detailed study of this mechanism can be undertaken as a future work for quantitative understanding.It is also observed that the binding energy of hh-exciton is greater than that of lh-exciton for a given applied electric field.We show the variation of polarizability of lh-exciton and hh-exciton as a function of electric field for different well widths in Figures 10 and 11, respectively.The polarizability of both lh and hh-exciton strongly increases with the well width and gradually decreases with electric field increases up to F = 150 kV cm -1 , beyond which the polarizability remains unchanged. 19,24This is consistent with our result shown in Figures 2 and 3 that the applied electric field reduces the kinetic energy of electron and hole, hence binding energy of the exciton increases with the electric field.Polarizability of the exciton is sensitive to the well width for lower strength of electric field and insensitive for higher strength of electric field.These observations may be of use in the choice of electric field and confinement for potential applications.

CONCLUSION
We have performed a theoretical investigation of the effect of electric field on binding energy of exciton, Stark shift in exciton energy and polarizability of exciton in SQW formed by vacuum/GaAs/Ga x Al 1-x Al as a function of well width and Al concentration considering the effect of image charges.It is found that lh and hh-exciton binding energy increase with increase in electric field.Stark shift in excitonic energy shift towards higher energy as the electric field, well width and Al concentration decrease.The result also shows that the polarizability of the exciton decreases when the electric field increases, but increases with well width.
Our attempt to study the quantum confined Stark effect in SQW showed that these behaviours in SQW vary with those in other symmetrical quantum wells due to their asymmetrical nature of polarized interface.Hence our results may provide an understanding on the choice of the well for electric field applications.The role of band bending due to the application of electric field, which is also expected to be of different nature in SQW, is not exclusively studied at present in our calculations and may be considered for future work.

5.
and R* as the effective Bohr radius and effective Rydberg respectively.

Figures 2 23 F = 0 kV cm − 1 F = 50 kV cm − 1 F = 200 kV cm − 1 Well
Figures 2 and 3 display the variation of binding energy of lh and hh exciton as a function of well width for different electric fields.For a given electric field F, the exciton binding energy increases as the well width decreases, reaches a maximum at a certain well width (L = 8 nm for lh-exciton and L = 6 nm for hh-exciton) and starts to decrease for further reduction of the well width.The Coulomb attraction energy is roughly proportional to 1/r, and the square well quantisation energy to 1/r 2 .Therefore, in most simplistic approximation, at sufficiently small values of r, one may ignore the Coulomb term which can lead to a maxima of excitonic binding energy with respect to reducing well widths.Moreover, when the well width L is reduced, the wave function of the exciton compressed in the well, leads to more binding, however the exciton wave function spread out into the surrounding interface after certain value of L. This causes the binding energy to be decreased as L is further reduced.22,23

Figure 2 :
Figure 2: lh-exciton binding energy as a function of well width for different electric fields.

Figure 3 :
Figure 3: hh-exciton binding energy as a function of well width for different electric fields.

Figure 4 : 1 F 1 F
Figure 4: lh-exciton binding energy as a function of Al concentration for different electric fields.

Figure 5 :
Figure 5: hh-exciton binding energy as a function of Al concentration for different electric fields.

Figures 6 and 7 Figure 6 :Figure 7 :
Figures 6 and 7 illustrate the Stark shift in lh-exciton and hh-exciton energy respectively as a function of electric field for different well widths.It is observed that the Stark shift decreases as the electric field increases, since the exciton energy shifts monotonically towards lower energy as the electric field increases due to the quantum-confined Stark effect.This behaviour is in good agreement with the results reported by Wu et al. and Dujardin et al.19,24It is also noted that the Stark effect on exciton energy with large well width is less than that with small well width and it shows nearly linear behaviour with electric field.The negative values of Stark shift indicate a red shift in the exciton energy in the presence of electric field.

Figures 8 and 9 5 F = 150 kV cm − 1 F
Figures 8 and 9 show the Stark shift in lh and hh exciton energy as a function of Al concentration for different electric fields.Stark shift decreases monotonically with Al concentration up to x = 0.4, after which it remains constant for a given electric field.The reason for these variations may be due to the raise of the barrier height as the increase of the Al concentration.

Figure 9 :Figure 10 :
Figure 9: Stark shift in hh-exciton energy for various Al concentrations.

Figure 11 :
Figure 11: Polarizability of hh-exciton as a function of electric field for different well widths.

Table 1 :
Material parameters used in the calculations.