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Abstract
We theoretically demonstrated the energy levels and wave functions of Tunnel-Coupled Quantum Dots of undoped GaAs/AlxGa1-xAs with various Al contents. The influence of applied electric fields on the energy levels and wave functions was investigated by applying a single-band effective mass approximation to the Schrödinger equation. The Stark effect induced shifts of energies that led to optimal absorption coefficients. TCQDs exhibited no significant change in the energy difference and matrix element between the ground state and first excited state, therefore absorption coefficients were the same at different lattice temperatures. The discovered study will be used to develop room-temperature optoelectronic devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Although new technology dominates today's world, the quest for smaller and smaller devices continues. Understanding the fundamentals of quantum mechanics can help us design smaller, faster, and more sensitive devices. A wide range of nanoscale semiconductors can now be synthesized, thanks to the advancement of nanofabrication processes. Examples of these nanoscale semiconductors include quantum wells [1,2], quantum wires [3], quantum dots [4,5], and coupled quantum dots (CQDs) [6,7], which are key to optoelectronic devices operating in the infrared region. Due to their special characteristics, a wide range of IR sensors with different wavelength bands can be utilized for many applications. For example, type II quantum wells, enable the detection of gases that are important to the ecosystem, such as HCl, CO2, N2O [8]. And medical imaging innovations are made possible by infrared nanostructure photodetectors [9]. Additionally, in recent years, single-photon sources driven by optical and electrical pumping have also been developed using quantum dots [10].
The Stark effect is described by perturbation theory in the Schrödinger equation as an external electric field that perturbs the quantum system and then significantly changes the energy levels and wave functions in the system. These changes impact the optical absorption coefficients in a big way [11]. Thus, it is not surprising that researchers have used this phenomenon to deal with optoelectronic modulators. The quantum confinement Stark effect in a quantum dot is an outstanding candidate for modulators due to the large oscillator strength and the low energy transition. These confinement effect features are required for more sensitive and lower voltage optoelectronic devices. In an applied electric field, the optical transition energies of self-assembled InAs/GaAs quantum dots exhibits distinct redshifts. This theoretical result is in good agreement with the existing experimental data [12]. Changing the strength of electric and laser fields can influence not only the optical transition energy, but also the total absorption coefficient and refractive index. Blue or red shifts can be induced without causing design complications [13]. A study of a lens-shaped quantum dot [14] demonstrated that the resonant peak of optical absorption decreased as the electric field increased. The red-shift phenomenon was observed since increasing electric fields change the potential geometry and reduce confinement effects, leading to a lower transition energy between the ground and first excited states. And the implications for the design of prospective semiconductor nanostructures, such as asymmetrical quantum wells and quantum dots, are promising. The tuning of energy levels, especially under the influence of the Stark effect, could be facilitated. For example, the structure of asymmetrically coupled quantum wells has been studied theoretically, and the large Stark effect they exhibit enabled the design of a tunable photodetector that is perfect for use in the atmospheric window region [15]. More recently, asymmetrical coupled quantum dots were instrumental in extending the range of semiconductor laser in the THz radiation range [16].
In this paper, we theoretically studied the influence of an applied electric field on light absorption properties of Tunnel-Coupled Quantum Dots (TCQDs). TCQDs is a unique structure for electron confinement at different lattice temperatures. Its active region was designed in such a way that the first and second dot quantization levels were located at a given close energy distance and electrons could tunnel between the first and second dot. This was done in order to observe and study the phenomenon of charge carrier heating. We focused on the theoretical study of the Stark effect, which influences energy levels of electrons, wave functions and consequently, oscillator strengths at lattice temperatures of 77 K and 300 K. Moreover, we also studied the effect of incident light intensity on the total absorption coefficient, and in order to better understand this, we also provided a discussion of the saturation incident light intensity. The specific design of active regions and the Stark effect play important roles in the performance improvement of optoelectronics devices operating in the IR range, and the study of their properties is always a challenge and motivation for researchers.
Our studied structure was GaAs/AlxGa1-xAs TCQDs structure which was varied Al concentrations and the magnitude of electric field application in the z–direction. To provide an alternative way to modulate optoelectronic devices based on theoretical calculations, the energy levels and wave functions of the structure were determined by using a single-band effective mass approximation for the Schrödinger equation. And we decided to focus on the investigation of the GaAs/Al0.36Ga0.64As TCQDs absorption coefficients as the next stage. Our results showed that the total absorption coefficient of our studied structure did not change at different lattice temperatures. Therefore, this unique structure can be considered a prospective design for THz radiation sources operating at room temperature, and perhaps, eventually, for a single-photon source.
2. Research materials and methods
The studied structure was an undoped GaAs/AlxGa1-xAs Tunnel Coupled Quantum Dots (TCQDs). The period of the TCQDs comprised GaAs QDs arranged in pairs. The first dot was 6.8 nm wide, and the second dot was 4.8 nm wide, The GaAs/Al0.36Ga0.64As layer separating the QDs of each pair was 1.5 nm wide and each period was separated by an Al0.36 Ga0.64As barrier 12 nm wide. This purposed structure is the arbitrary design structure for Tunnel Coupled Quantum Wells [17]. This time we want to focus on the light absorption property under the influence of electric field application on TCQDs of the same designed structure. The studied structure is presented in Fig. 1.
Fig. 1. The structure of the GaAs/AlxGa1-xAs TCQDs: the widths (z) of the first and the second dot are 6.8 and 4.8 nm, respectively.
The advantage of the tunnel-coupled structure is the possibility of free electron gas occupying the first excited state (${E_2}$) at a lattice temperature of 300 K. In the studied TCQDs structure, the difference between the quantum-confinement energy levels of the ground state and the first excited state (${E_1}$ and ${E_2}$) was about 47 meV. This energy difference is not much more than thermal energy at room temperature (26 meV). Also, the probability density of electrons at the ground state (${|{{\psi_1}} |^2}$) was located at the first QD and the probability density of electron at the first excited state (${|{{\psi_2}} |^2}$) was located at the second QD. Therefore, at room temperature, under sufficiently high intersubband optical pumping, electrons can occupy ${E_1}$ and ${E_2}$ at the same time. The conduction band profiles and the probability density of the studied structure at lattice temperatures of 77 K and 300 K without an applied electric field are presented in Fig. 2.
Fig. 2. The conduction band profiles of the undoped GaAs/Al0.36Ga0.64As TCQDs at lattice temperature of 77 K and 300 K. The calculated quantum-confinement energy levels and the probability density of electrons at ${E_1}$ and ${E_2}$ are also presented.
We calculated the energy levels $({E(x,y,z)} )$ and the wave functions $({\psi (x,y,z)} )$ of the TCQDs as functions of the percentage of Al contents $(x)$ and the magnitude of electric field applied in the growth direction (z–direction). The results were obtained by using a single-band effective mass approximation for the Schrödinger equation [18], thus, the Hamiltonian, for this case, was written as:
In Eq. (1), the first term of the right-hand side is the kinetic energy, $\hbar $ is Planck's constant, ${m^\ast }$ is the effective mass of electrons in the system. The second term is the potential energy of x and y dimensions, and $V(z)$ is the confining potential well in the z direction. The last term is the external applied electric field along the z direction [19].
The optical properties of the studied structure can be calculated if the matrix element is determined, since intersubband transitions between states can only occur in accordance with the selection rules. In this case, since the electric field is applied in the z direction, the transition matrix element between the initial state $({i^{th}})$ and final state $({f^{th}})$ is given by [20]:
The dimensionless quantity known as oscillator strength, which explains the probability of optical absorption in intersubband transitions between ${E_1}$ and ${E_2}$, can be written by [21]
where $\Delta {E_{21}} = {E_2} - {E_1}$ is the energy difference between the first excited state and the ground state.The optical properties of the studied structure can be obtained by using the above equations. These equations provide the energies, the related wave functions, and the intersubband matrix element. To calculate the linear ${\alpha ^{(1)}}(\omega )$ and the nonlinear ${\alpha ^{(3)}}(\omega ,I)$ optical absorption coefficients, we apply the following equations [21–23]:
The total absorption coefficient can be written as:
Here, $\mu = 1/{\varepsilon _0}{c^2}$ is the permeability of the system, ${\varepsilon _0} = 1/{c^2}{\mu _0}$ is the electrical permittivity of a vacuum, ${\mu _0}$ is the permeability of the vacuum, ${\varepsilon _r}$ is relative permittivity, e is electron charge, c is the speed of the light, ${n_r} = \sqrt {{\varepsilon _r}} $ is the refraction index of the semiconductor, $\sigma $ is the electron density in the QD, $\hbar \omega $ is the incident photon energy, $\Gamma = 1/\tau $ is the relation rate, $\tau $ is the relaxation time of carriers and, I is the incident optical intensity.
3. Results and discussions
3.1 Part I. The effect of Al concentration on energy levels and wave functions
In order to demonstrate the light absorption of the studied structure, the energy difference between electron energy levels ($\Delta {E_{21}}$) has to be studied, since $\Delta {E_{21}}$ is the most important parameter of oscillator strength that has to be clarified. Using Eqs. (1) to (4), we obtained ${E_1}$, ${E_2}$, $\Delta {E_{21}}$ and $({|{{\psi_1}} |^2})$ as functions of the percentage of Al content. The results are presented in Fig. 3(a) and (b).
Fig. 3. (a) ${E_1}$ and ${E_2}$ as functions of the percentage of Al contents, $\Delta {E_{21}}$ is presented in the inset. (b) The probability density of electrons at ${E_1}$ as a function the percentage of Al content.
Figure 3(a, inset) shows that $\Delta {E_{21}}$ decreases as Al content increases because the linear increase in ${E_2}$ as a function of Al contens less steep than the linear increase in ${E_1}$. Consequently, $({|{{\psi_1}} |^2})$ of the first QD, whose beginning and ending positions were varied from 12 nm to 18.8 nm, increases as the Al content increases (Fig. 3(b)), while ${|{{\psi_1}} |^2}$ of the second QD decreases (z = 21.3–25.1 nm) since the confining potential height has risen. Another influence on the energy difference is the change in the energy gap of the structure in accordance with Vegard’s law: ${E_{gap}}(x) = 1.519 + x \times 1.249\,\textrm{eV}$ [24]. Since the energy band gap is influenced by the percentage of alloy content ($x$), its variation can affect the energy of the system, including ${E_1}$, ${E_2}$ and $\Delta {E_{21}}$.
3.2 Part II. The influence of the applied electric field on optical properties
To investigate the effect of the electric field on the optical properties of undoped TCQDs of GaAs/Al0.36Ga0.64As, we set the lattice temperature of our simulation model at 77 and 300 K. We focused on x = 0.36 because this percentage of Al content produced a redshift in energy, which lowered the excitation photon energy necessary to excite electrons from the ground state to the first excited state in the absorption coefficient. We studied the influence of the applied electric field on optical properties by varying the magnitude of the electric field applied in the z direction of the studied structure. Under the influence of a sufficiently strong electric field, the lattice temperature of TCQDs will rise. In an electric field, the lattice temperature of the structure increases and this affects the band gap energy of GaAs following the well-known equation [25], ${E_g}({T_L}) = 1.519 - [5.405 \times {10^{ - 4}}T_L^2/({T_L} + 204)]$, where ${T_L}$ is lattice temperature (here varied from 77 K to 300 K). It is clear that ${E_g}$ decreases when ${T_L}$ increases. As a result, at a lattice temperature of 300 K, the band gap energy of GaAs was narrower than it was at 77 K. Therefore, ${E_1}$ and ${E_2}$ are smaller at T = 300 K than at T = 77 K, while $\Delta {E_{21}}$ remains the same at both lattice temperatures, as shown in Fig. 4(a) and (b), respectively.
Fig. 4. (a) ${E_1}$ and ${E_2}$ at lattice temperatures of 77 K and 300 K as functions of the applied electric field. (b) $\Delta {E_{21}}$ as a function of the electric field at $T$= 77 K and $T$= 300 K.
Due to the Stark effect (which was dominant in the conduction band profiles and the electron sub-band levels of the studied structure), when the magnitude of the electric field was varied from 0 to 50 kV/cm, ${E_1}$ increased at the same time as ${E_2}$ decreased at both lattice temperatures (Fig. 4(a)). Consequently, the energy difference between electron energy levels ($\Delta {E_{21}}$) shifted. Therefore, the wave functions of electrons at ${E_1}$ and ${E_2}$ changed their forms of overlap. ${E_1}$ and ${E_2}$ behaved in the opposite way as the electric field was varied from 60 to 200 kV/cm. These magnitudes of electric field are sufficiently high to change the conduction band profiles of the structure, which induced increases in ($\Delta {E_{21}}$) (Fig. 4 (a) and (b)).
Figure 5(a) – (d) presents the probability densities of electrons at the ground state and first excited state in conduction band profiles of the TCQDs at T = 77 and T = 300 K in applied electric fields of F = 0 kV/cm, F = 50 kV/cm, and F = 100 kV/cm. For F = 0 kV/cm at both lattice temperatures, the probability density of electrons at the ground state overlaps the probability density of electrons at the first excited state in both the first and second QD (Fig. 5(a) and 5(b)). As mentioned in the Introduction, this overlap property of tunneling wave functions is special for this unique active region. When electrons were heated by an electric field of F = 50 and F = 100 kV/cm (Fig. 5(c)-(d)), the wave functions of electrons at the ground state in the first QD were lower, because the conduction band profiles were changed by the influence of the Stark effect and the tunnel effect between quantum dots of electrons. Therefore, the wave functions of electrons at ${E_1}$ overlapped the wave functions of electrons at ${E_2}$. Correspondingly, the matrix element ($|{{M_{fi}}} |$) strongly responded to the shifting behavior of wave functions.
Fig. 5. The conduction band profiles of GaAs/Al0.36Ga0.64As TCQDs as functions of applied electric fields at two lattice temperatures. (a) F = 0 kV/cm at 77 K, (b) F = 0 kV/cm at 300 K, (c) F = 50 kV/cm at 77 K, and (d) F = 100 kV/cm at 77 K.
The optical absorption probability of transitions between the ground state and the first excited state, or oscillator strength (${P_{21}}$), and the transition matrix element ($|{{M_{21}}} |$) are presented in Fig. 6(a) and 6(b), respectively. At both lattice temperatures, the oscillator strength is proportional to the applied electric field from 0 to 50 kV/cm, since $|{{M_{21}}} |$ increases at these magnitudes of electric field (Fig. 6(b)), even if the energy difference decreases in accordance with Eq. (3). As the electric field increases from 60 to 200 kV/cm, oscillator strength decreases until it reaches zero at 200 kV/cm. This behavior is caused by the Stark effect, which changes conduction band profiles in a strong electric field. As a result, the electrons in ${E_2}$ are unable to reach the confinement potential, and correspondingly, no overlap of probability occurs between the ground state and first excited state, resulting in Eq. (2). Moreover, due to this overlap in the matrix element term, the maximum oscillator strength occurs at 50 kV/cm. Consequently, electrons have the best chance of transition from the ground state to the first excited state at 50 kV/cm. Finally, when the electric field is applied to the lattice at temperatures of 77 K and 300 K, there is no noticeable difference in oscillator strength, due to the very close energy difference shown in Fig. 4(b). The transition matrix element $|{{M_{21}}} |$ is shown in Fig. 6(b).
Fig. 6. (a) Oscillator strength at lattice temperatures of 77 K and 300 K. (b) The transition matrix element $|{{M_{21}}} |$ as a function of an applied electric field at both lattice temperatures.
Based on the previous discussions, it is evident that the energy difference and the transition matrix element are the key factors affecting the absorption coefficient, and the importance of absorption coefficients in device applications must be underlined. In order to present the values of the absorption coefficient of the studied structure, we chose the following parameters [23] to be applied in Eqs. (4) and (5):${\mu _0} = \mathrm{4\pi \times 1}{\textrm{0}^{\textrm{ - 7}}}\textrm{H}{\textrm{m}^{\textrm{ - 1}}}$, ${\varepsilon _r} = 12.53$, $c = \mathrm{3\ \times 1}{\textrm{0}^\textrm{8}}\textrm{m}{\textrm{s}^{ - 1}}$, $\sigma = \mathrm{3\ \times 1}{\textrm{0}^{\textrm{22}}}{\textrm{m}^{\textrm{ - 3}}}$ and $\tau = \textrm{0}\textrm{.14 ps}$. Figure 7(a) presents the influence of the electric field on the linear $({\alpha ^{(1)}}(\omega ))$ and nonlinear $({\alpha ^{(3)}}(\omega ,I))$ absorption coefficients, plotted as a dashed line and dotted line respectively. Figure 7(b) then presents the total absorption coefficient $({\alpha _{tot}}(\omega ,I))$, plotted as a solid line, as functions of photon energy $(\hbar \omega )$, for electric fields of six different strengths (F = 0, 20, 50, 70,100, and 150 kV/cm) at T = 77 K for an incident light intensity of 0.2 MW/cm2. The linear absorption coefficient peak takes place at $\hbar \omega \approx \Delta {E_{21}}\,$, due to $d{\alpha ^{(1)}}(\omega )/d\omega = 0$, and it increases as the strength of the electric field increases (F = 0, 20, 50 kV/cm). The highest peak occurs when $F = $50 kV/cm, then the coefficient tends to decrease when $F > $50 kV/cm ($F = $70, 100, 150 kV/cm), and finally the lowest peak occurs at $F = $150 kV/cm.
In the previous paragraph we discussed oscillator strength. Figure 7(a) shows that when electrons are excited by incident photon energy ($I$), they have the chance to move from the ground state to the first excited state and produce an absorption coefficient peak in accordance with oscillator strength values. From Eq. (6), the total absorption coefficient is the sum of the linear and nonlinear absorption coefficients. Since the nonlinear absorption coefficient is negative, the total absorption coefficient is smaller than the linear absorption coefficient. Both the total and nonlinear absorption coefficient peaks tend to be the same as the linear coefficient peak when the magnitude of the electric field is varied. As the electric field strength is varied from 0 to 100 kV/cm, the absorption coefficient peaks red shift. However, at F = 150 kV/cm, the absorption coefficient peak blue shifts under the influence of the Stark effect in the stronger electric field. The total absorption coefficients for $F = $ 0, 50 and 100 kV/cm at $T$ = 77 K and $T$= 300 K are presented in Fig. 7(b). The total absorption peaks at both temperatures are similar due to the similar energy difference and matrix element, which correspond to the oscillator strength.
Fig. 7. (a) The linear and nonlinear absorption coefficients of a GaAs/Al0.36Ga0.64As TCQDs under the influence of an applied electric field are shown as a dashed line above the zero level and a dotted line below the zero level, respectively. The total absorption coefficient is shown as a solid line (b) the total absorption coefficient of the studied structure at $T$= 77 K and $T$= 300 K under the influence of an applied electric field.
The concurrent term is the nonlinear absorption coefficient term. It obviously reduces the total absorption coefficient, as shown in Fig. 7 and Eq. (5). Therefore, the effect of incident light intensity (I) on the nonlinear absorption coefficient must be taken into account. The total absorption coefficient at I = 0.3 MW/cm2 and I = 0.4 MW/cm2 are presented in Fig. 8(a) and (b), respectively. This parameter should be selected with caution.This limit should prevent the occurrence of the saturation incident light intensity (Is), defined as the intensity that accounts for half of the linear absorption coefficient, thus, $\alpha ({\omega ,{I_s}} )= {\alpha ^{(1)}}(\omega )/2.$Therefore, as a function of the saturation incident light intensity, the nonlinear absorption coefficient is defined as ${\alpha ^{(3)}}({\omega ,{I_s}} )={-} {\alpha ^{(1)}}(\omega )/2$ [26].
Fig. 8. The absorption coefficient of GaAs/Al0.36Ga0.64As TCQDs under the influence of an applied electric field with (a) $I = $0.3 MW/cm2 and (b) $I = $ 0.4 MW/cm2
If the saturation incident light intensity is reached, the nonlinear absorption coefficient will be larger than the linear absorption coefficient. This phenomenon is known as the bleaching phenomena in total optical absorption coefficient and is considered a fallibility of the density-matrix perturbative equation [27,28]. As shown in Fig. 8, when the value of I increases, the magnitude of the nonlinear absorption coefficient decreases due to the negative sign following Eq. (5). Moreover, for I = 0.4 MW/cm2, the highest total absorption peak is no longer at F = 50 kV/cm. Consequently, the incident light intensity is another key component that has a significant impact on the total absorption coefficient.
Figure 9(a) shows the influence of the bleaching phenomenon on the total optical absorption coefficient, which affected the height as well as the position of the intensity peaks. The positions of the peaks depended on the strength of the electric field applied. The dependence of saturation light intensity (${I_s}$) on the electric field was also studied (Fig. 9 (b)). ${I_s}$ is the factor which determines the half of the linear absorption coefficient. The Stark effect from the applied electric field influences the electron sub-band energies (${E_1}$ and ${E_2}$) and wave functions of the structure, and hence the absorption coefficient. As a result, the highest total absorption coefficient (Fig. 9(a)) produced the lowest ${I_s}$ in an electric field of 50 kV/cm (Fig. 9(b)). Finally, in order to conclude the results of our investigation, we presented the marking results of our investigation in Table 1.
Table 1. Parameters dependent on temperature, electric field strength, and the dependence of absorption coefficient on incident light intensity and electric field strength
Fig. 9. (a) Total absorption coefficient of TCQDs with different incident light intensities. (b) Saturation incident light intensity as a function of applied electric field at T = 77 K.
4. Conclusion
Our investigations revealed that increasing the Al concentration in the structure led to increased energy of electron sub-band levels due to the higher potential confinement. Because of the tunnel-couple active region, the wave function of the ground state was always located at the first QD, while the first excited state was found at the second QD (F = 0 kV/cm). The influence of the Stark effect from the applied electric field induced shifts in both the energy levels and the wave functions. Despite the fact that the energy of the ground state and the first excited state decreased as temperature increased, the magnitude of the energy difference remained the same, as did the wave function used to calculate the matrix element. As a result, the absorption coefficient was not significantly different at lattice temperatures of 77 K and 300 K. Therefore, it can be inferred that temperature does not affect the optical properties of this structure. Further, the incident light intensity is one of the most significant parameters to consider in the nonlinear absorption coefficient term. Our results offer an option to lower the voltage caused by the redshift of energy from an electric field. The major discovery of this study is that the total absorption coefficient of our studied structure did not change at different lattice temperatures. Therefore, this unique structure can be considered a prospective design for THz radiation sources such as Quantum Cascade Lasers operating at room temperature and modulators operating in the IR region.
Funding
National Science, Research and Innovation Fund (NSRF) (B05F650024); Development and Promotion of Science and Technology Talents Project (DPST).
Acknowledgments
This research has received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation [grant number B05F650024], and the Development and Promotion of Science and Technology Talents Project (DPST). Mr.Thomas Duncan Coyne helped with the English grammatical revision.
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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