1. Introduction

In recent several years,with the rapid improvement in nanostructure technology,such as molecular-beam epitaxy (MBE) and metal organic chemical-vapor deposition (MOCVD),it is has become possible to fabricate low-dimensional semiconductor heterostructures such as quantum well (QW),quantum well wire (QWW) and quantum dot (QD) structures. In these systems,one can find various physical properties,for instance,the discrete energy structure as well as the discrete absorption spectrum (both intra- and inter-band) expected^{[1, 2]}. Therefore,many theoretical and experimental works have been devoted to the study of the electronic and optical properties in these quantum systems,taking into account the impurity-induced effect,various confining potential and external field^[3-9].

It is well known that the photon absorption process can be described as an optical transition (inter-subband transitions,intra-band transitions and inter-band transitions) that takes place from an initial state to a final one assisted by a photon. The optical transitions can appear also in the low-dimensional semiconductor quantum systems,and display very interesting properties. The optical properties related to such transitions have great potential for device applications in laser amplifiers^[10],photodetectors^[11],high-speed optical modulators^[12],and so on in optic communications. Consequently,the optical properties of low-dimensional semiconductor systems such as the linear,nonlinear and total refractive index changes and absorption coefficients have attracted much attention in condensed matter and applied physics^[11-16].

The hydrogenic impurity-induced effect,geometry confinement of quantum structure and the external fields can greatly affect the physical properties,including energy spectrum and optical transitions. For a fixed confining configuration and the basic parameters of low-dimensional semiconductor systems,the presence of an impurity in the quantum system changes its effective potential due to the Coulomb interaction between the electron and the impurity,which results in analytical calculation of the eigenstates of the systems that are complicated or insoluble even. That is why most theoretical works attempted to analyze the donor impurities-induced effect on the linear,nonlinear refractive index changes and absorption coefficients by use of various methods in recent years^[17-40]. For example,recently,Barseghyan et al.^[35] (using a variational procedure) for quantum ring,Cakir et al.^[36] (Quantum Genetic Algorithm (QGA) and Hartree-Fock-Roothaan (HFR) method) for spherical quantum dot and Duque et al.^[37] (perturbation method) for disc-shaped quantum dot investigated the linear,nonlinear refractive index changes and absorption coefficients. All of the above mentioned studies display qualitatively similar results that these optical properties depend highly on the impurity,geometry of quantum dot,external fields as well as basic parameters of the quantum system. Also,it is noticed that most theoretical works mentioned above focus on the impurity located at the center of the quantum dot and the transition from the ground state to excited one due to the limitations of the calculation method. But also one may find few recent studies dealing with the linear and nonlinear optics for off-central impurity^[35,38] and for the transitions between the higher energy states^[36,39,40] in QDs. In addition,to our knowledge,up to now the nonlinear optical absorption and refractive index changes in a rectangular quantum dot (RQD) have not been studied for the transitions between the higher energy states and impurity position-induced effect.

In this paper,using the quasi-one-dimensional effective potential replacing the three-dimensional Coulomb potential proposed by Reference ^[41],giving the analytical form of the quasi-one-dimensional effective potential in our model,we calculate the wavefunctions and energy eigenvalues of the ground (j= 1) and first 2 excited states (j= 2 and 3) in RQD,and investigate influence of the impurity position,incident optical intensity and electric field on linear and nonlinear optical absorption as well as refractive index changes for the transitions j= 1-2 and j= 2-3.

2. Theory and calculation

We consider an RQD heterostructure with a hydrogenic donor impurity and the presence of an electric field applied parallel to the z-axis as shown in Figure 1,where the RQD's cross section is square with equal sides of length L,and the height denotes H. Note we adopt the quasi-one-dimensional effective potential replacing three-dimensional Coulomb potential proposed by Reference ^[41] in this paper. Coulomb interaction is small perturbation for transverse-motion electron states. Therefore,we introduced impurity affects only on the z-component of the QD wave functions but not on the x-y components.

In our theoretical approach,we adopt the effective Bohr radius ${{a}^{*}}={{\hbar }^{2}}\varepsilon /{{m}_{\text{e}}}{{e}^{2}}$ as the unit of length,the effective electron Rydberg $R\text{y}={{e}^{2}}/2\varepsilon {{a}^{*}}$ as the unit of energies and ${{F}_{0}}=e/\varepsilon {{a}^{*2}}$ as the unit of electric field,where e and ${{m}_{\text{e}}}$ denote the absolute value of electronic charge and the electron effective mass respectively,ε is GaAs static dielectric constant. The effective-mass Hamiltonian of an electron in an RQD can be described as in the Cartesian coordinates

$H=-\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} \right)+2Fz+{{V}_{\text{c}}}(x,y,z)-\frac{2}{{{r}_{\text{i}}}},$

(1)

where $r_{\rm i} =\sqrt {(x-x_{\rm i} )^{2}+(y-y_{\rm i} )^{2}+(z-z_{\rm i} )^{2}}$ is the electron-impurity distance with x _i,y _i and z _i coordinates of the impurity in the RQD,F>0 is the electric field,and V_c (x,y,z) is the confining 3D potential which takes the value zero in the interior of the RQD and it is infinity otherwise.

In our previous work^[42],the Schr\"{o}dinger equation

$H{{\psi }_{nmk}}(x,y,z)={{E}_{nmk}}{{\psi }_{nmk}}(x,y,z),$

(2)

was obtained by use of the quasi-one-dimensional effective potential model. Without the impurity potential,the eigenfunction and eigenvalue were calculated as

$\begin{align} & split\psi _{nmk}^{(0)}(x,y,z)=\frac{2}{L}\sin \left[ m\pi \left( \frac{x}{L}-\frac{1}{2} \right) \right] × \\ & \sin \left[ n\pi \left( \frac{y}{L}-\frac{1}{2} \right) \right] × [CAi(\tilde{z})+DBi(\tilde{z})], \\ \end{align}$

(3)

and

$E_{mnk}^{(0)}=\frac{{{\pi }^{2}}}{{{L}^{2}}}({{m}^{2}}+{{n}^{2}})+E_{k}^{(0)},$

(4)

where $Ai(\tilde{z})$ and $Bi(\tilde{z})$ are the two independent solutions of the Airy equation, $\tilde{z}={{F}^{1/3}}(z-{{E}_{k}}/2F)$ . The energy levels $E_{k}^{(0)}$ were obtained by solving the transcendental equation as follows

$\begin{align} & Ai\left[ {{F}^{1/3}}\left( -\frac{E_{k}^{(0)}}{2F}+\frac{H}{2} \right) \right] × Bi\left[ {{F}^{1/3}}\left( -\frac{E_{k}^{(0)}}{2F}-\frac{H}{2} \right) \right]= \\ & Ai\left[ {{F}^{1/3}}\left( -\frac{E_{k}^{(0)}}{2F}-\frac{H}{2} \right) \right] × Bi\left[ {{F}^{1/3}}\left( -\frac{E_{k}^{(0)}}{2F}+\frac{H}{2} \right) \right], \\ \end{align}$

(5)

where m,n and k are quantum numbers in the x,y and z directions respectively.

In the presence of the impurity,the eigenfunction and the eigenvalue were obtained as follows,

$\begin{align} & {{\psi }_{nmk}}(x,y,z)=\psi _{nm}^{(0)}(x,y){{\psi }_{k}}(z) \\ & =\frac{2}{L}\sin \left[ m\pi \left( \frac{x}{L}-\frac{1}{2} \right) \right] \\ & × \sin \left[ n\pi \left( \frac{y}{L}-\frac{1}{2} \right) \right] × {{\psi }_{k}}(z), \\ \end{align}$

(6)

and

${{E}_{mnk}}=E_{nm}^{(0)}+{{E}_{k}}=\frac{{{\pi }^{2}}}{{{L}^{2}}}({{m}^{2}}+{{n}^{2}})+{{E}_{k}},$

(7)

the longitudinal eigenfunction $\psi_{k} (z)$ and eigenvalue $E_{k}$ were obtained by solving numerically the following longitudinal Hamiltonian operator (8) with the finite-difference method on a one-dimensional mesh

${{H}_{z}}=-{{\nabla }^{2}}-{{V}_{\text{eff}}}(z)+2Fz,$

(8)

where $V_{\rm eff} (z)$ is the analytical form of the effective interaction potential,and expressed as the integral in cylindrical coordinates,

${{V}_{\text{eff}}}(z)=\frac{1}{\pi }\int_{0}^{2\pi }{\text{d}\theta }\int_{0}^{\infty }{{{I}_{n}}(q,\theta ){{I}_{m}}(q,\theta ){{\text{e}}^{-q\left| z-{{z}_{\text{i}}} \right|}}}\text{d}q,$

(9)

where

$\begin{align} & {{I}_{n}}(q,\theta )=\text{sinc}\left( \frac{Lq\sin \theta }{2\pi } \right)\cos ({{y}_{\text{i}}}q\sin \theta )- \\ & \text{sinc}\left( n-\frac{Lq\sin \theta }{2\pi } \right)\cos ({{y}_{\text{i}}}q\sin \theta -n\pi ), \\ \end{align}$

(10)

$\begin{align} & {{I}_{m}}(q,\theta )=\text{sinc}\left( \frac{Lq\cos \theta }{2\pi } \right)\cos ({{x}_{\text{i}}}q\cos \theta )- \\ & \text{sinc}\left( m-\frac{Lq\cos \theta }{2\pi } \right)\cos ({{x}_{\text{i}}}q\cos \theta -m\pi ). \\ \end{align}$

(11)

In the electric dipole approximation,if the polarization of the electromagnetic radiation is chosen in the z-direction,we may write the electric dipole transition matrix elements between ith state (lower) and jth state (upper) as

${{M}_{ji}}=\left| \left\langle {{\psi }_{j}}\left| z \right|{{\psi }_{i}} \right\rangle \right|,$

(12)

where ψ_j and ψ_i are the wavefunctions of the upper and the lower states.

The photon absorption process can be described as an optical transition that takes place from an initial state to a final one assisted by a photon. The linear and third-order nonlinear optical absorption coefficient and refractive index change due to intersubband transitions in an RQD being derived by the compact density matrix approach and the iterative procedure. This assumes the interaction of electromagnetic radiation (frequency ω ) with the RQD. The incident optical field is

$E(t)={{E}_{0}}\text{cos}(\omega t)=\tilde{E}\text{exp}(-\text{i}\omega t)+\tilde{E}\text{exp}(\text{i}\omega t),$

(13)

due to the symmetric structure and nonlinear polarization of medium,we do not consider the second-order and higher-order nonlinear susceptibilities^[43]. The electronic polarization $P(t)$ caused by the incident field is written by

$P\left( t \right)={{\varepsilon }_{0}}{{\chi }^{\left( 1 \right)}}(\omega )\tilde{E}{{\text{e}}^{-\text{i}\omega t}}+{{\varepsilon }_{0}}{{\chi }^{\left( 3 \right)}}(\omega )\tilde{E}{{\text{e}}^{-\text{i}\omega t}}+c.c.,$

(14)

where ε₀ denotes vacuum permittivity, ${{\chi }^{(1)}}(\omega )$ and ${{\chi }^{(3)}}(\omega )$ are the linear and third-order nonlinear optical susceptibilities,respectively.

The absorption coefficient is related to the susceptibility $\chi (\omega )$ as

$\alpha (\omega )=\omega \sqrt{\frac{\mu }{{{\varepsilon }_{\text{r}}}{{\varepsilon }_{0}}}}\text{Im}({{\varepsilon }_{0}}\chi (\omega )).$

(15)

The analytical form of the linear and third-order nonlinear optical absorption coefficients is given by^{[29, 36, 37]}

${{\alpha }^{\left( 1 \right)}}\left( \omega \right)=\omega \sqrt{\frac{\mu }{{{\varepsilon }_{\text{r}}}{{\varepsilon }_{0}}}}\frac{\rho \hbar \Gamma M_{ji}^{2}}{{{\left( {{E}_{ji}}-\hbar \omega \right)}^{2}}+{{\left( \hbar \Gamma \right)}^{2}}},$

(16)

and

${{\alpha }^{\left( 3 \right)}}\left( \omega ,I \right)=-\omega \sqrt{\frac{\mu }{{{\varepsilon }_{\text{r}}}{{\varepsilon }_{0}}}}\frac{2I\rho \hbar \Gamma M_{ji}^{4}}{{{\varepsilon }_{0}}nc{{\left[ {{\left( {{E}_{ji}}-\hbar \omega \right)}^{2}}+{{\left( \hbar \Gamma \right)}^{2}} \right]}^{2}}},$

(17)

where $E_{ji} =E_{j} -E_{i}$ is the energy difference between i and j electronic states,ρ denotes the electron charge density in RQD,μ is the magnetic permeability of the material,c is the speed of light in vacuum, $\varepsilon_{\rm r}$ is the real part of relative electric permittivity of medium, $I=2{{\varepsilon }_{0}}nc{{\left| {\tilde{E}} \right|}^{2}}$ is the incident optical intensity, $n=\sqrt{{{\varepsilon }_{\text{r}}}}$ is the refractive index, $\hbar \omega$ is the incident optical intensity and $\Gamma =1/\tau$ (τ is relaxation time) is the decay rate from j to i states.

The total optical absorption coefficients can be written as

$\alpha \left( \omega ,I \right)={{\alpha }^{\left( 1 \right)}}\left( \omega \right)+{{\alpha }^{\left( 3 \right)}}\left( \omega ,I \right).$

(18)

The refractive index change is associated with the susceptibility χ(ω) as

$\frac{\Delta n}{n}=\text{Re}\frac{\chi (\omega )}{2{{n}^{2}}}.$

(19)

The linear and third-order nonlinear refractive index change expressions are defined as follows^{[29, 36, 37]},respectively,by

$\frac{\Delta {{n}^{\left( 1 \right)}}\left( \omega \right)}{n}=\frac{{{e}^{2}}\rho }{2{{n}^{2}}{{\varepsilon }_{0}}}\frac{{{E}_{ji}}-\hbar \omega }{{{\left( {{E}_{ji}}-\hbar \omega \right)}^{2}}+{{\left( \hbar \Gamma \right)}^{2}}}M_{ji}^{2}\quad ,$

(20)

and

$\frac{\Delta {{n}^{\left( 3 \right)}}\left( \omega ,I \right)}{n}=-\frac{{{e}^{4}}\rho \mu cI}{{{n}^{3}}{{\varepsilon }_{0}}}\frac{{{E}_{ji}}-\hbar \omega }{{{\left[ {{\left( {{E}_{ji}}-\hbar \omega \right)}^{2}}+{{\left( \hbar \Gamma \right)}^{2}} \right]}^{2}}}M_{ji}^{4}.$

(21)

The total refractive index change is obtained as

$\frac{\Delta n(\omega ,I)}{n}=\frac{\Delta {{n}^{(1)}}(\omega )}{n}+\frac{\Delta {{n}^{(3)}}(\omega ,I)}{n}.$

(22)

3. Results and discussion

In the present study,we take the following basic parameters in GaAs^[37]: ε=13,ρ =5 × 10²²m^-3,Γ = 1/(0.14 ps),ε₀ =8.85 × 10₁₂ F/m,μ =1.256 × 10_-6Tm/A,n=3.2 and m_e=0.067 m₀ is the electron effective mass,where m₀ is the free electron mass. With this set of parameters,the effective units correspond to ${{a}^{*}}\approx 100{\AA},Ry\approx 5.7$ meV and F₀ ≈ 11.5 KV/cm. In the following figures,the side of square cross section of RQD is L=a^* and the height is _H=2_a^* in the atomic units we are using. The states in the figures are labeled with the index j corresponding to the states (m,n,k),where j= 1 → (1,1,1); j= 2 → (1,1,2); j= 3 → (1,1,3). We consider the linear and nonlinear optical absorption as well as refractive index changes for the transitions j= 1-2 and j= 2-3.

In Figure 2,the linear,third-order nonlinear and total absorption coefficient for the transitions j= 1-2 and j= 2-3 are shown as a function of photon energy without the impurity in Figure 2(a) and with consideration of a donor impurity located at the center of the RQD in Figure 2(b),where the incident optical intensity I= 0.2 MW/cm². Dotted lines represent the results obtained with zero electric field,while the solid lines denote the calculation with electric field F=10F₀. From Figure 2,the energy and intensity of resonance peak for the transition j= 1-2 is less than that of the transitions j= 2-3 for a given electric field. The reason is that the energy difference E₂₁ for j= 1-2 is less than the energy difference E₃₂ for j= 2-3,while the absolute value of the electric dipole matrix element M₂₁ for j= 1-2 is less than M₃₂ for j= 2-3. By comparing Figure 2(a) and Figure 2(b),when the electric field F= 0,we can see that one of the effects of impurity is to induce a blueshift on the resonant peaks of the coefficients for the transitions j= 1-2,while redshift for transitions j= 2-3.This is because the presence of impurity leads to the decrease of the energy of the system. The interaction potential between the impurity and electron depends on the average distance between electrons and impurities due to inverse square Coulomb potential. Without impurity,it is known from the probability distribution of the electrons in the rectangular quantum dot that the position of the maximum probability density is in the center z= 0 for the ground state j= 1,are located at z=-0.5a^* and z=0.5a^* for j= 2 and are at z= 0,and z=-0.5a^* and z=0.5a^* for j= 3. When the impurity is at the center of the quantum dot,the average distance between electron and impurity is minimum for j=1,followed by j= 3,and the farthest for j= 2. It results in the energies E₂₁> E₂₁₀,E₃₂<E₃₂₀. Also,the presence of impurity results in a slight decrease of intensity of resonance peak for j= 1-2,but an increase for j= 2-3. The intensity of resonance peak depends mainly on the absolute magnitude of the transition matrix element M_ij,which is related to the overlapping of the involved wavefunctions. The larger the energy level is,the smaller the two level interval is,and the more the overlapping of the wavefunctions is. Therefore,the optical intensity of transitions from j= 1 to j= 2 is much weaker than that from j= 2 to j= 3. The similar results are also found in Reference ^[36]. When the electric field F=10F₀ is applied along the z axis,it can be seen from this figure that the resonance peaks of the transitions j= 1-2 and j= 2-3 move toward higher energy region and the peak intensities are decreasing. Because the differences between adjacent energies increase,and the overlaps of the wave function decrease,this results in a decrease of the absolute value of electric dipole moments with the electric field. At the same time,although the impurity effect induces the blueshift for j= 1-2 and redshift for j= 3-2 in the absence of the electric field,it leads to a redshift for j= 1-2 and blueshift for j= 3-2 in the existence of the field. The reason is that the electric field effect can interfere with the Coulomb potential induced by the impurity. In the RQD,the electron moves along negative direction of the applied electric field. Based on distribution of the electron probability density in RQD,the electrons located at the center of RQD for the ground state j= 1 are far away from the impurity due to the field applied along the z axis,which results in reduction of Coulomb potential. Similarly,the electron located at z=0.5a^* for the states j= 2 move toward the center of the impurity,which induces the increase of Coulomb potential. Also,the electrons at the center of RQD for the state$j=$ 3 are away from impurities,which also lead to a decrease of Coulomb potential. But for the state j= 3,the electrons at the z=0.5a^* are simultaneously close to the impurity center with the electric field^[42]. The result is that reduction of Coulomb potential for j= 3 is less than that for j= 2.

These features mentioned the impact of impurity effect,the electric field on absorption coefficient above can be also seen in Figure 3 which depicts the linear,the third-order nonlinear and the total refractive index change for the transitions j= 1-2 and j= 2-3 as a function of photon energy. The refractive index changes are also important parameters in the study of optical properties. We can see from Figure 3 that the curves of refractive index changes between the energy values corresponding to the max and min values decrease drastically as the photon energy increases. It is a phenomenon of anomalous dispersion defined as absorption band because the photon is very strongly absorbed. The photon energy corresponding resonant peak of the coefficient in Figure 2 is corresponding to a point where the refractive index change curve intersects to the horizontal axis. So the frequency of the energy is the resonance frequency of the system. Moreover,the refractive index changes increase smoothly with the increasing photon energy in the outside region of the anomalous dispersion region,which is called a normal dispersion region.

From Equations (15),(16),(19),and (20),we can know that the incident light intensity I is irrelevant for the first-order terms,however,the amplitude of the third-order nonlinear contributions is directly proportional to I in the optical properties studied in this work. Therefore as I increases,the third-order optical intensity increases,and it is possible that the third-order terms are comparable to the first-order terms. In order to investigate the effect of the incident light intensity I,we have shown the optical absorption coefficient in Figure 4 and Δ _n/n in Figure 5 as a function of photon energy for the different values of the incident optical intensities,where the electric field F=0 with a donor impurity at the center of the RQD. As seen from Figure 4,the third-order nonlinear absorption coefficient has large negative increase while the linear absorption coefficient keeps a constant magnitude as the incident intensity I increases. In that case,the total coefficients decrease. Note that the third-order term is negative,while the first-order term is positive. It results in the total optical intensity less than the first-order one. For greater incident intensity,it appears to the optical absorption saturation phenomenon,and linewidths of absorption spectrum increase with the increasing I. When the incident light intensity increases to a certain value (e.g. I= 0.6 MW/cm²,the peak of the total coefficient will split.

In Figure 5,the linear and the third-order nonlinear refractive index changes are described for four different values of the incident optical intensity I= 0.2,0.3,0.4 and 0.5 MW/cm². From this Figure,the third-order nonlinear refractive index change decreases while the linear refractive index change remains a constant magnitude as the intensity of I increases. The results are that the amplitudes of the total refractive index changes diminish.

Finally,we focus on discussing the effect of the impurity positions on the optical absorption and reflection coefficient. It is known that the presence of impurities will greatly alter the electronic and optical properties of quantum dots,which strongly affect the optical and transport properties of devices made from these materials. So it is necessary to study hydrogenic impurity-related optical properties in the RQD.

Figure 6 displays the total absorption coefficient for the transitions j=1-2 and j=2-3 as a function of photon energy for the different impurity positions at points o= (0,0,0),a=(0,0,a^*),b=(0.5a^*,0,0),c=(0.5a^*,0,a^*),d=(0.5a^*,0.5a^*,0),e=(0.5a^*,0.5a^*,a^*) as shown in the inset,where F=0,I= 0.2 MW/m². From this Figure,the total absorption coefficient for the higher energy transitions j= 2-3 is insensitive to impurity positions,while that for the low energy transition j= 1-2 depend significantly on the positions of impurity. As impurity positions are away from the center of the RQD,the absorption peak moves toward the lower energy region,and the peak intensities decrease slightly for j= 1-2. Similar features are seen in Figure 7 which shows the total refractive index changes for the transitions j= 1-2 and j= 2-3 as a function of photon energies for the different impurity positions.

4. Conclusions

In this paper,adopting the quasi-one-dimensional effective potential model,we have calculated the linear and nonlinear optical absorption and relative refractive index change in a GaAs RQD with a hydrogenic donor impurity in the presence of electric field,and investigated in detail the dependence of the optical absorption and relative refractive index change for the transitions j= 1-2 and j= 2-3 on the electric fields,incident optical intensities and impurity positions. The results show the energy and intensity of resonance peak for the transition j= 1-2 is less than that of the transitions j= 2-3 for a given electric field. An applied electric field pushes the resonance peaks to move toward higher energy region and the peak intensities are decreasing due to the applied electric field. The effect of impurity on the optical coefficients appears to be associated with the electric field. When the field F= 0,the influence of the on-center impurity is to induce a blueshift on the resonant peaks of the coefficients for the transitions j= 1-2 (this result qualitatively agrees with those of Reference ^[37]),while redshift for transitions j= 2-3. But when an electric field is applied (F=10F₀),it will appear to have the reverse shift effect on the resonant peaks (a redshift for j= 1-2 and blueshift for j= 2-3). The third-order optical coefficients depend highly on the incident optical intensity,which results in a decrease of the total optical coefficients with the increasing optical intensity. It is interesting that the peak of the total coefficient shows firstly a saturation phenomenon,then a split as the optical intensity increases. The impurity positions have a significant influence on the optical coefficients of the low energy transition j= 1-2,while less effect for the high energy transition j= 2-3. When the impurity is located at the center,the energy of peak is largest. Although to our knowledge,there are no available experimental data to compare with our theoretical results,we believe they provide an indication for practical application of some photoelectric devices constructed based on rectangular GaAs quantum dot structures.