Due to the dimensions being reduced down to nanometric scale, low-dimensional semiconductor structures, such as quantum wells (QWs), quantum well wires (QWWs), and quantum dots (QDs), have been investigated from both theoretical and experimental point of view.^[1–6] One of the most interesting fields of the nanostructures is the multi-layer quantum dot (MLQD). This interest is because these structures play an important role in optoelectronic devices such as photodetectors,^[7,8] and lasers.^[9,10] It is possible to produce spherical MLQD with the development of the lithographic techniques and wet chemical synthesizing methods.^[11,12] MLQD structures have many advantages compared to single-layer QD. For example, physical properties are tunable by changing the core diameter and shell thickness. Also, they have higher absorption cross-sections and lower Auger recombination coefficients.^[13–15] The MLQD structure has greater active volume than that of a single layer system and so the gain and sensitivity of the electronic device can be increased.

An impurity effect plays a fundamental role in the thermal, optical and electrical properties of the nano-scaled structures since it can alter the characteristic of any quantum device impressively. The first attempt to calculate the binding energy of a hydrogenic impurity of QW was made by Bastard^[16] who used a variational approach. After this pioneering study, a number of theoretical investigations have been published concerning the impurity problem and its various aspects in nanostructures.^[17–19] In addition, many studies have been carried out on the effects of external perturbations, such as temperature,^[20–24] and hydrostatic pressure,^[25–28] on the impurity properties in a QD. These studies have shown that external factors play an important role in the electronic properties of the nanostructures.

In the literature, various studies^[29–36] related to the electronic properties of MLQD structures have been reported. Salini et al.^[37] studied the electronic and optical properties of a single exciton in MLQD by using effective mass approximation with parabolic confinement. The change of the electronic properties of spherical MLQD in the presence and absence of a donor impurity is discussed numerically by Akgul et al.^[38] Their results show that the electronic properties and impurity binding energies are strongly dependent on the layer thicknesses.

Until now, to the best of our knowledge, there have been no reports on the hydrostatic pressure and temperature effects on the electronic and optical properties of spherical MLQD with parabolic confinement potential. In this paper, we intend to study the effects of hydrostatic pressure and temperature on the donor binding energy and optical absorption of spherical core/shell/well/shell (CSWS) structure. In theoretical calculations, the shooting technique within the effective mass approximation has been used to determine the energy levels and their wave functions. The rest of the paper is organized as follows: In the next section the theory and the details of the calculation are presented. Section 3 contains results and discussion. Finally, a brief conclusion is given in Section 4.

We consider GaAs/GaAlAs spherical CSWS nanostructure with electron confinement in the conduction band. The schematic representation of a spherical CSWS structure and its potential profile are given in Fig. 1. As seen from Fig. 1, GaAs core is a sphere of radius r₁. The well region is that another GaAs spherical shell of inner radius r₂ and outer radius r₃ is concentric with the core region. In the structure, W₀ = r₁, W_s = r₂ − r₁, and W_w = r₃ − r₂ are defined as the core radius, the shell thickness, and the well width, respectively.

Fig. 1.

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	Fig. 1. Schematic representation of a spherical CSWS structure and its potential profile Vr.

Within a framework of the effective mass approximation the Hamiltonian associated with an electron bound to an on-center donor impurity can be written as

Table 1.

Table 1.

Table 1. The recommended values of material parameters for GaAs. . Parameters Recommended values 1.519a Δso/meV 341b 7.51a α/10−4 (eV/K2) 5.405c β/K 204c b/(meV/GPa) 107.3d S11/ × 10−2 GPa−1 1.16e S12/ × 10−2 GPa−1 0.37e a Ref. [51], b Ref. [52], c Refs. [41] and [53], d Ref. [54], e Ref. [40]

Table 1. The recommended values of material parameters for GaAs. .

Since the system has spherical symmetry, equation (1) can be separated in different independent equations when it is expressed in spherical coordinates. Using separation of variables in the form

In the above equations, n, ℓ, m are the principle, angular, and magnetic quantum numbers, respectively. Y_ℓm(θ,ϕ) are the spherical harmonics, R_nℓ(r) is the radial wave functions, and E_nℓ is the energy eigenvalues. In order to obtain the energy eigenvalues and the corresponding wave functions, equation (10) is solved numerically by using the shooting method. Note that we have taken into consideration of the lowest subbands which are 0s (n = 0, ℓ = 0) and 1p (n = 1, ℓ = 0). 0s and 1p states correspond to the ground state and the first excited state, respectively.

The solution to Eq. (10) allows for the impurity binding energies to be obtained taking into account both the temperature and the pressure effect. The donor binding energy, E_b, is defined as the difference of energy for that state in the absence of donor impurity (Z = 0) and in its presence (Z = 1):

The linear α⁽¹⁾(ω) and the third-order nonlinear α⁽³⁾(ω,I) optical absorption coefficients (ACs) are given by^[45–49]

The linear and nonlinear ACs that are induced by an optical transition from the ground state (0s) to the first excited state (1p) are evaluated. In highly symmetrical QDs, transitions from a lower state to an upper state are forbidden according to the selection rules. In spherical QDs, dipole transitions are allowed only between states satisfying the selection rules △ℓ = ±1.^[50]

In this section, we will discuss the effects of the pressure, temperature, spatial confinement and the presence of impurity on the electronic and optical properties of spherical MLQD with parabolic confinement. Throughout the numerical calculations, we used atomic units so that ħ = m₀ = e = 1, (m₀ is the mass of a free electron). Table 2 presents the variation of GaAs/Ga_0.7Al_0.3As effective Bohr radius a*(P, T) and pressure-dependent function C(P) = [1−(S₁₁ + 2S₁₂)P].

Table 2.

Table 2.

Table 2. The pressure- and temperature-dependent effective Bohr radius and function C(P) = [1 − (S11 + 2S12)P] for GaAs. . P/GPa (T =100 K) a*(P, T)/nm C(P) 0 10.19 1 2.5 8.34 0.9525 5.0 7.14 0.9250

Table 2. The pressure- and temperature-dependent effective Bohr radius and function C(P) = [1 − (S11 + 2S12)P] for GaAs. .

In Fig. 2, we compare the impurity binding energy of the ground state as a function of the core radius for cases with and without the hydrostatic pressure. At given temperature 100 K, the fixed QD shell and well sizes are selected as nm and nm for P = 0 GPa, respectively. In the nm unit, pressure-dependent sizes are W_s (0 GPa) = 3.67 nm, W_w (0 GPa) = 6.11 nm, W_s (2.5 GPa) = 2.86 nm, W_w (2.5 GPa) = 4.77 nm. Firstly, we focus on the behavior of the binding energy without pressure. When the core radius increases, the ground state binding energy decreases until a critical core radius value (W₀ = 7.2 nm). The physical origin of this result is that the probability distribution in the well region is larger than that in the core region. So the impurity and electron distributions are in different regions, the Coulomb interaction decreases due to the weak overlap and the binding energy decreases. After a certain value, the binding energy is rapidly increasing due to the probability of electron tunneling to the core region rises with increasing the core radius. With further increasing of the core radius, the binding energy decreases and is similar to that of single core-shell QD. With increasing the hydrostatic pressure, the dot radius and dielectric constant decrease and the effective mass of electron increases. Thus, the donor impurity becomes more confined and the binding energy increases. Moreover, it is seen from Fig. 2 that critical core radius in the presence of pressure moves to smaller values.

Fig. 2.

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	Fig. 2. The variation of ground state binding energy as a function of core radius W0 for different values of pressure at given temperature 100 K.

Figure 3 shows the probability distributions for different values of the core radius for P = 0 (the left panel) and P = 2.5 GPa (the right panel) with Z = 0 and Z = 1 at T = 100 K, nm and nm. In small values of the core radius, the probability distributions in the well region are larger than those in the core region for both of the Z = 0 and Z = 1 cases as seen from Fig. 3(a). By further increase in the W₀, the electron starts tunneling to the the core region (Fig. 3(b)) due to the Coulomb interaction and it is confined completely in the core region (Fig. 3(c)) for the Z = 1 case at W₀(0 GPa) = 9.0 nm. However, for the Z = 0 case, the electron starts tunneling to the core region after W₀(0, GPa) = 9.0 nm (Fig. 3(d)). In Fig. 3(e), the probability distributions of the electron are completely localized for both the Z = 0 and Z = 1 cases and the multi-layer quantum dot structure exhibits the properties of a single quantum dot. It is observed that the behavior of the probability distributions for P = 2.5, GPa case (the right panel in Fig. 3) is similar to the P = 0 case. However, the wave function is more strongly confined inside the multi-layer quantum dot. We should also mention that the magnitudes of the probability distributions with pressure (the right panel) are smaller than those without pressure (the left panel).

Fig. 3.

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	Fig. 3. The ground state probability distributions for different values of core radius for P = 0 GPa (left panel) and P = 2.5 GPa (right panel) with Z = 0 and Z = 1.

The variation of the binding energy of the donor impurity as a function of the core radius is plotted in Fig. 4 for two different temperature values (T = 25 K and T = 300 K) at nm, nm, nm, nm and P = 2.5 GPa. It is clear from Fig. 4 that the donor binding energy decreases as temperature increases. With the increase of temperature, the effective mass of electron decreases and the dielectric constant increases. Therefore, the binding energy decreases for all values of temperature and the critical core radius moves to larger values with increasing temperature. Moreover, it is clear that the hydrostatic pressure effect on the binding energy is more apparent than that of the temperature.

Fig. 4.

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	Fig. 4. The variation of ground state binding energy as a function of core radius W0 for different values of temperature at given pressure 2.5 GPa.

In Fig. 5, the binding energy of the ground state is plotted as a function of shell size W_s for nm, nm, and T = 100 K, in the absence and presence of the hydrostatic pressure. The binding energy decreases with the increase in shell thickness. This is due to the fact that the electron is less affected by the impurity in the center of MLQD with the increase in the shell thickness. When the pressure is increased, the binding energy increases due to the hydrogenic impurity becoming more confined. To understand clearly this behavior, we have plotted in Fig. 6 the probability distributions for different values of the shell thickness in the presence and absence of the hydrostatic pressure.

Fig. 5.

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	Fig. 5. The variation of ground state binding energy as a function of shell width Ws for different values of pressure at given temperature 100 K.

Fig. 6.

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	Fig. 6. The ground state probability distributions for different values of shell width for P = 0 GPa and P = 2.5 GPa with Z = 0 and Z = 1.

In Fig. 6(a), the electron is localized in the well region as shell thickness increases. Figure 6(b) shows that the presence of impurity causes the increase in tunneling between the core and well regions. The effect of hydrostatic pressure on the probability distributions is plotted in Figs. 6(c) and 6(d). The character of the probability distributions under pressure is similar to that without pressure. However, the probability distribution in the presence of pressure is narrower due to the additional spatial confinement by increasing pressure.

In the following Fig. 7, the binding energies of the 0s and 1p states are plotted as a function of the well size without and with pressures for nm, nm, and T = 100 K. In the P = 0 case, the binding energy remains almost constant until certain W_w value and after this value it decreases rapidly. For very large W_w values, the binding energy approaches the free electron energy values.

Fig. 7.

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	Fig. 7. The variation of 0s and 1p state binding energies as a function of well width Ww for different values of pressure at given temperature 100 K.

In order to understand this behavior, we plot the probability distributions for different well sizes in Fig. 8. We plot Fig. 8 for nm, nm, and T = 100 K. In small W_w values, the electron is localized in the core region. As W_w increases, the probability distribution in the well region increases and it decreases in the core region and the binding energy decreases. Finally, for large W_w values, the electron is localized in the well region and the electron behaves like a free particle. The behavior of the binding energies without pressure (P = 0) is similar to the previous results found in Ref. [32]. It should also be noted that the hydrostatic pressure changes the behavior of the donor binding energy. With the presence of the pressure, the binding energy monotonously decreases as a function of the well size.

Fig. 8.

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	Fig. 8. The ground state probability distributions for different values of well width for P = 0 GPa and P = 2.5 GPa, T = 100 K with Z = 0 and Z = 1.

The effect of the core radius on the ACs is plotted in Fig. 9. Figure 9 shows the linear α⁽¹⁾(ω), the nonlinear α⁽³⁾(ω,I) and the total α (ω,I) ACs as a function of the photon energy for three different core radius values at P = 2.0 GPa, T = 300 K, nm and nm. It can be seen that peak positions of ACs shift to higher energies with increasing the core radius as the energy difference between the electronic states increases. This behavior can be explained with more localized wave function in the core region. The separation between the neighboring energy levels becomes wider with increasing the core radius. Moreover, it is obvious that the magnitude of ACs will increase when the W₀ increases.

Fig. 9.

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	Fig. 9. The absorption coefficients as a function of the incident photon energies for different core radius.

In Fig. 10, we report the linear α⁽¹⁾(ω), the nonlinear α⁽³⁾(ω,I), and the total α (ω,I) ACs for several values of the hydrostatic pressure at T = 300 K, nm, nm, and nm. From Fig. 10, we can see that the resonant peak value of the AC is highly related to the hydrostatic pressure values. This figure shows that decreasing the pressure causes the resonant peaks of ACs to experience a red-shift. An explanation for this behavior is the variations of the energy difference between ground and first excited state by varying the pressure. Also, it is obvious that the magnitude of the optical absorption augments when the hydrostatic pressure increases.

Fig. 10.

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	Fig. 10. The absorption coefficients as a function of the incident photon energies for different hydrostatic pressure.

We plot the ACs as a function of the incident photon energy for T = 25 K and T = 300 K with P = 2.0 GPa, nm, nm, and nm in Fig. 11. As seen from Fig. 11, the peak of AC shifts to the lower energy regions with increasing temperature. The main reason for this behavior is the variation in energy difference of two different electronic states, the ground and first excited states, with increasing temperature. As can be seen from Fig. 9 to Fig. 11, the hydrostatic pressure, temperature and width of the layers permit us to adjust the resonance conditions and hence many optical properties related to it in any optical system using these materials.

Fig. 11.

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	Fig. 11. The absorption coefficients as a function of the incident photon energies for different temperature.

We have investigated the effects of pressure, temperature, confinement and presence of an impurity on the electronic properties in a spherical MLQD with parabolic confinement. As quantum dot materials, we have used GaAs in the core and well layers and AlGaAs in the barrier (shell) layers. The energies of structure are obtained numerically by using the shooting method and the effective mass approximation. This study provides an understanding of how the electronic and optical properties of an MLQD structure are affected by pressure, temperature and width of the layers.

The obtained results reveal that the electronic properties of MLQD can be adjustable with pressure, temperature, core radius, shell thickness and well width. The results show that the donor binding energy increases (decreases) by increasing the pressure (the temperature).

Furthermore, we have found that for the parabolic QD, the presence of impurity provides important effects on the probability distributions. The probability distributions in the presence of impurity exhibits a different behavior as compared with the absence of impurity.

It can be said that the pressure, the temperature and core radius play an important role on the optical absorption in a spherical MLQD. The results in this paper are shown with increasing of pressure, resonant peak of absorption exhibits blue shift and the magnitude of absorption increases. The magnitude of absorption reduces while the temperature and core radius increase and the variation of the temperature and core radius causes the peak position to move to lower photon energies (red shift). The peak position and magnitude of absorption could be adjusted with a good choice of these parameters and a red- or/and a blue-shift could be obtained.

To our knowledge, the effects of the pressure and temperature on optical properties of a spherical MLQD with parabolic confinement have been investigated for the first time. The theoretical investigation of the optical ACs in a spherical MLQD will lead to a better understanding of the physical properties of a new kind of quantum dot structure. We expect that the results of this theoretical investigation could stimulate and guide more experimental studies of the MLQD.