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Abstract
A new method for regulating optical properties of a coaxial cylindrical quantum well using the electric field and intense laser field is investigated in the effective mass approximation. By means of the finite difference method and the correct dressing effect of the confinement potential, the results show that the enhancement and recovery of optical absorption and refractive index change strongly depend on the multifaceted-cooperative regulation of the laser parameter, the electric field strength, the angle between the electric field and polarization direction of laser, and the barrier width. This is promising for the design of a new generation of highly polarization sensitive devices, optical repair equipments and optical phase modulators by adopting the multistage combination of electric and intense laser fields.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The goal-oriented adjustment of intersubband transitions in conducting bands of semiconductor quantum wells have received considerable attention over the past few decades, owing to its practicability in acquiring novel and controllable optical effects, such as optical absorption and refractive index changes [1–3]. Recently, it has been found that since the degree localization of the electronic states strongly depends on the barrier width, the two-dimensional coaxial cylindrical quantum well wires (CCQW) exhibits multifaceted tunable optical properties compared with the traditional solid cylindrical quantum well (SCQW) or a hollow cylindrical quantum well (HCQW), and have been widely applied in newfashioned optical devices, such as nanowire photodetector [4], nanowire solar cells [5], and biomedicine [6].
The asymmetry of the confining potential rules the variation of electronic and optical properties. So far, plenty of practical measures to realize asymmetry have been well developed, for example, by compositionally grading the quantum well (QW) [7], by applying static electric [8], magnetic [2], and non-resonant intense laser field (ILF) [9] to a symmetric QW, etc. It has been confirmed theoretically by Karimi et al. that under the external magnetic field, the anticrossing effect of CCQW leads to a large increase of the resonant peak values of refractive index changes (RIC) and optical absorption coefficients (OAC) [2]. In 2018, Sangtawee et al. found that crossing and anticrossing behaviors of energy levels can be controlled by the thickness of a nonmagnetic semiconductor layer in CCQW [10]. To our knowledge, the influence of an external electric field on the RIC and OAC of a typical GaAs/AlxGa1-xAs CCQW has not been investigated. Recently, in the research of the effect of ILF on the optical properties of CCQW, we have proposed definitely, for the first time, a physical explanation for the anticrossing of energy levels, and found that the CCQW exhibits great potential for ultra-wide frequency tunability of the OAC and RIC by modulating the laser parameter and barrier thickness [11]. In addition to destroying the preexisting symmetry of nanostructures, ILF have also been reported to be capable of restoring the damaged symmetry [12]. However, the conditions required to restore the symmetry of confining potential are highly demanding. Even so, the recovery of optical properties is expected if multiple field controls can be combined. Ozturk has discussed the intense laser field and applied electric field effect on the linear and nonlinear intersubband absorption coefficient in graded quantum well [13]. Further, the influence of the static electric, magnetic, and non-resonant intense THz laser fields on nonlinear optical rectification and second harmonic generation in Woods-Saxon quantum well has been investigated by Ungan et al [14]. These studies manifest that the combination of multiple fields enables the desired multilevel adjustment of optical properties. Nevertheless, previous studies on multi-field interference are limited to one-dimensional nanostructures, and there are relatively few reports about the restoration of optical properties by multi-field combination.
In this paper, the effects of static electric field and non-resonant ILF on the electronic states, intersubband OAC and RIC in GaAs/AlGaAs CCQW are investigated in detail. To the best of our knowledge, with the frame of multi-field regulation, it is the first time that we not only discovery that the anticrossing of energy levels can be generated and disappeared, which leads to the abrupt change of optical properties, but also demonstrated that the optical properties of CCQW can be restored and enhanced. Besides, we pay particular attention to the dependence of OAC and RIC on the laser parameter, the electric field strength, the angle between electric field and polarization direction of ILF, and the barrier width, which reflects the multifaceted tunability of optical properties in CCQW.
2. Theory
In this section, we discuss the system in which the electrons are tightly bound in two GaAs cylindrical wires regions (A), one within the other, and each encircled by AlxGa1-xAs layers (B) with a wider bandgap (Fig. 1(a)). The confinement potential, composed of three jump discontinuities at the interfaces between A and B regions, can be expressed as [15]:
Fig. 1. The laser-dressed potential profiles of the CCQW for TB = 7 nm and: (a) F = 0, a0 = 0; (b) F = 60 kV/cm, a0 = 0, θ = 0°; (c) F = 60 kV/cm, a0 = 3 nm, θ = 0°; (d) F = 60 kV/cm, a0 = 3 nm, θ = 30°.
We assume that the structure, with the growth direction along the x-axis, is in the presence of a static electric field $\vec{F}$ and a non-resonant long-wavelength ILF (such as THz) represented by a monochromatic plane wave with linearly polarized parallel to the growth-axis and photon energy less than the field-free binding energy. The one-electron Schrödinger equation in the effective-mass approximation in this system can be described by [16]:
Performing Floquet method and Fourier transform, Eq. (5) can be further reduced to [17–20]:
A 2D finite difference method will be adopted to numerically calculate the Schrödinger equation as thoroughly explained in several previous works [22–25]. Performing the central difference approximation to the derivatives, the discrete Schrödinger equation can be derived as:
To avoid the electron not to be too fast to see the laser-dressed potential well, ωL needs to satisfy the high-frequency regime ωLτ ≫ 1, in which τ represents the transit time of the electron in the well region. In this case, the angular frequency ωL of ILF in our GaAs CCQW system has an inferior limit of ∼1014 s−1, which can be provided by CO2 and Nd-YAG lasers. Moreover, For the dipole approximation to be effective, the laser power has a upper limit of ∼4×10−11 ωL2 W/cm2, for example, the ILF intensity of Nd-YAG laser is on the order of 104 W/cm2 [26].
By virtue of the compact density matrix approach and the iterative method, the linear and third-order nonlinear OAC can be respectively given by [27]:
The total OAC is obtained as [27]:
Similarly, the linear and the third-order nonlinear RIC can be writed as [27]:
The total RIC is given by [27]:
Here, σν expresses the electron density, µ denotes the permeability, nr is the refractive index, ɛr is the real part of the permittivity, ω expresses the incident photon angular frequency, and $I = 2{\varepsilon _0}{n_r}{|{\hat{E}} |^2}$ is the optical intensity of incident electromagnetic wave that excites the structure and leads to the intersubband optical transition. ${M_{ij}} = \left\langle {{\varPsi _i}} \right|e\vec{r}|{{\varPsi _j}} \rangle $ is the dipole matrix element, and ${E_{ij}} = \hbar \omega = {E_i} - {E_j}$ is the energy interval with i, j = 1, 2.
3. Results and discussions
In this section, the parameters used in our calculations are as follows: V0 = 228 meV, m* = 0.067 m0 (m0 is the free electron mass), x = 0.3, Ri = 4 nm, R0 = 15 nm, σν = 5.0×1022 m−3, nr = 3.2, ɛ0 = 8.85×10−12 Fm−1, Γ0 = 1/0.14 ps, and ɛr = 10.89 ɛ0 in the high-frequency laser field [26,27].
3.1 Electric field and ILF effects on the electronic state
Figure 1 shows the effect of the electric field and ILF radiation on the CCQW with TB = 7 nm. Electric field and ILF have different dressed-effects on the confinement potential: (i) The electric field can excite a symmetric confinement potential into a linear change along the x axis, see in Fig. 1(b). The triangular potential is created in the original three quantum wells along the x axis. (ii) The ILF is able to effectively elevate the confinement in CQWW along the polarization direction, see in Fig. 1(c). The three quantum wells along the x axis evolve into different V-shaped potential wells with wider width. (iii) By controlling the laser parameter a0, the strength of electric field F, and the angle θ, the confinement condition of CCQW can be regulated as expected in two-dimensional plane, see in Fig. 1(d).
We pay particular attention to the effect of double fields on the electronic state of CCQW with special barrier thickness TB, such as TB = 6, 7, 8, and 11 nm, due to the particularity of electronic distribution, as shown in Fig. 2. Figure 2 depicts the electron probability density for TB = 6, 7, 8, and 11 nm in the absence of electric field and ILF, which has been analyzed in detail in our previous work [11]. Notably, when TB is raised to 11 nm, the CCQW becomes HCQW, which is conducive to compare the characteristic of the double quantum wells and the single quantum well.
Fig. 2. Electron probability density for different barrier thicknesses TB in the absence of electric field and ILF.
First of all, we study the effect of electric field on electron probability density of the ground and second subbands with a0 = 0 and θ = 0°, as presented in Fig. 3. It can be clearly seen that the ground and the second subbands are pushed up to the bottom of the outer well for CCQW with the increasing F, and the electrons mainly move to the right-side of the outer well, which inevitably leads to the decrease of the subband energy levels and the enhancement of the overlap between different electron states. As shown in Fig. 4, the energy levels of CCQW decrease with the increase of F. More importantly, it is found that the crossings (accidental degeneracies) and anticrossings (repulsion) of energy levels occur with the augment of F, which has also been found in the work of the influence of ILF on CCQW [11]. According to the explanation we proposed before [11], the anticrossings of energy levels arise from the exchange of energy level roles and electron states at the intersection (see green circles in Fig. 4). As is confirmed by Fig. 3 and the insets of Fig. 4, Fig. 3(b5) and Fig. 4(e6) retain the same wave-function form of Fig. 4(e5) and Fig. 3(b6), respectively. Similarly, Fig. 3(c2) and Fig. 3(c6) correspond to Fig. 3(c7) and Fig. 3(c3), respectively. Figure 3(c7) and Fig. 4(e8) correspond to Fig. 4(e7) and Fig. 3(c8), respectively. The exchange of electron states greatly affects the intersubband transitions, because only transitions between opposite parity states are allowed [12,11]. As depicted in Fig. 5(a), the allowable polarization direction of incident wave changes when the non-vanishing matrix elements locate at the position of anticrossings. Besides, as F continues to increase, the M21 with different TB tend to be stable and remain consistent, because when F is large enough, electronic distributions in different quantum wells tend to be stable, and the transition probabilities from the ground state to the second state are basically consistent, as seen in Fig. 3.
Fig. 3. Electron probability density of the ground and second subbands with different values of F for a0 = 0 and θ = 0°.
Fig. 4. Variation of the first three subband energies with F and the insets show squared modulus of the probability amplitudes $\varPsi _2^2$ and $\varPsi _3^2$ for a0 = 0 and θ = 0°.
Fig. 5. Non-vanishing matrix elements: (a) as functions of F for a0 = 0 and θ = 0°; (b) as functions of a0 for F = 60 kV/cm and θ = 0°.
When the electric field F is fixed at 60 kV/cm and θ = 0°, $\mathit{\Psi}_{1}^{2}$ and $\mathit{\Psi}_{2}^{2}$ changing with the laser parameters a0, a0 = 1, 5, and 9 nm, is plotted in Fig. 6. By increasing a0, the effective dressed-well width decreases and the subbands tend to extend to the upper part of the effective dressed-well with wider width. More interestingly, when the angle θ is taken into account, one can find from Fig. 7 that the electron distribution varies along the periphery of the effective dressed-well with θ, which means that it is possible to control the electronic state using double fields. It is worth mentioning that θ has little effect on the electron probability density of HCQW. Due to the existence of multi-well coupling and barrier thickness, the probability of tunneling of electron in CCQW has a non-monotonic change with a0 in the presence of electric field F, which leads to the corresponding change of the energy intervals E21. Figure 8(a) reflects that E21 decreases first and then increases with the enlargement of a0 with F = 60 kV/cm and θ = 0° for TB = 6, 7, 8 nm, while it decreases monotonously in the same case for TB = 10 nm. On the contrary, E21 can be further improved as the increasing θ, as seen in Fig. 8(b). It is worth noting that the anticrossings of energy levels will disappear when the electric field and ILF are applied simultaneously, which is quite different from the case with the ILF alone [11]. To our knowledge, this is the first time that a two-field regulation is proposed to eliminate the anticrossings of energy levels. Therefore, the abrupt change of the polarization direction of the incident wave does not occur when the non-vanishing matrix element M21 varies with a0 in the present of electric field, as shown in Fig. 5(b).
Fig. 6. Electron probability density of the ground and second subbands with different values of a0 for F = 60 kV/cm and θ = 0°.
Fig. 7. Electron probability density of the ground and second subbands with different values of θ for a0 = 3 nm and F = 60 kV/cm.
Fig. 8. The first three subband energies: (a) as a function of a0 with F = 60 kV/cm θ = 0°; (b) as a function of θ with a0 = 3 nm and F = 60 kV/cm.
It is well known that the intersubband transitions determine the optical properties, while the polarization direction of incident wave is critical to the amplitude variation of matrix element M21. Figure 9 depicts that matrix element M21 as functions of φ (where φ is the angle between the polarization direction of incident wave and x axis) with different θ for a0 = 3 nm and F = 60 kV/cm. It can be clearly seen that the peak of M21 decreases with the enhancement of θ for CCQW, and φ corresponding to the peak of M21 has an opposite trend to θ. Specially, θ has little effect on the peak of M21 and φ for HCQW.
3.2 Influences of CCQW with the electric field and ILF on the OAC
The linear α(1), third-order α(3) and total absorption coefficients α(ω, I) as functions of incident photon energy, for (1–2) intersubband transition, are drawn in Fig. 10, with different values of TB and F for I = 0.1 MW/cm2 and a0 = 0. A lot of interesting phenomena can be observed. Firstly, the electric field is capable of exciting the previously forbidden (1–2) intersubband transition with TB = 7 and 8 nm. Secondly, with the increasing F, the resonant peak of α(ω, I) exhibits a blue shift with TB = 6 and 7 nm, while it shows a red shift at first and then blue one with TB = 8 nm, and a red shift with TB = 11 nm, respectively. The reason for these traits can be explained as follows. As confirmed in previous studies, the resonant peaks will appear at the energy difference E21 [25].The decrease rate of E1 is greater than that of E2, resulting in the increase of E21 with TB = 6 and 7 nm, but it shows the opposite change with TB = 11 nm. Differently, anticrossings of energy levels causes E1 and E2 to come together and then repulse, leaving E21 decrease and then increase. Thirdly, the resonant peak of α(ω, I) enhances with the increasing F for CCQW, but that is the opposite for HCQW. The reason of these traits lie in the fact that the resonant peak of α(ω, I) changes mainly originate from the linear term α(1) and the third-order nonlinear term α(3), which is mainly decided by ${E_{21}}{|{{M_{21}}} |^2}$ and ${E_{21}}{|{{M_{21}}} |^4}$, respectively. The increase of E21 makes larger contribution to α(1) than ${|{{M_{21}}} |^2}$, while α(3) is up to ${|{{M_{21}}} |^4}$. Hence, the resonant peak of α(1) increases and the resonant peak of α(3) declines with strengthening of F, which contributes to the enhancement of α(ω, I) for CCQW. Quantitatively, if F < 1kV/cm, the absorption coefficient curve will be stable in the gray area, which is equivalent to the “saturation area” of the absorption coefficients at the lower limit in the electric field. More interestingly, when F is less than 1 kV/cm, the photon energy corresponding to the resonant peak of α(ω, I) is the same as that without the electric field (about 127 meV), while the peak amplitude is twice than that of F = 0. This means that applying a weak electric field to the HCQW can enhance the original absorption properties.
Fig. 10. The absorption coefficients as a function of incident photon energy with different values of TB and F for I = 0.1 MW/cm2 and a0 = 0.
To show clearly the electric field effect on the intersubband optical absorption in CCQW under the ILF, in Fig. 11, we plot the variation of the absorption coefficients as a function of the photon energy with different values of TB and a0 for I = 0.1 MW/cm2, F = 60 kV/cm and θ = 0. The trend of the resonant peaks are manifested by the orange lines with arrows. Different from the effect of single electric field, the resonant peaks of α(ω, I) move toward the lower energy regions at the beginning with the increase of a0, which is significant in restoring optical absorption. For instance, in the quantum structure grown with polar materials, polarized charges will be generated at the interface of the heterojunction, forming a polarized electric field. The existence of electric field makes the resonant peak of α(ω, I) appears blue shift and deviates from its original position. If the ILF is applied at the same time, the resonant peak of α(ω, I) is red-shifted to its original position, and the original optical characteristics can be restored. As shown in the insert of Fig. 11(a), if F =10 kV/cm and a0 = 2 nm, the photon energy corresponding to the resonant peak of α(ω, I) is the same as that without the electric field and ILF (about 7 meV), while the peak amplitude is enhanced. To our knowledge, this is the first time that two-field modulation has been proposed to restore and enhance optical properties, which is of great significance to the design of optical repair equipment.
Fig. 11. The absorption coefficients as a function of incident photon energy with different values of TB and a0 for I = 0.1 MW/cm2, F = 60 kV/cm and θ = 0.
In optical anisotropy, the variation of the absorption coefficient as a function of the photon energy is depicted in Fig. 12, with different values of TB and θ for I = 0.1 MW/cm2, F = 60 kV/cm and a0 = 3 nm. It can be clearly found that the resonant peaks of α(ω, I) enhance and move toward the higher energy regions as θ increases. The reason of this feature is same to the explanation for that of Fig. 10. These properties suggest that further control of optical properties can be achieved in two-dimensional plane by adjusting θ.
Fig. 12. The absorption coefficients as a function of incident photon energy with different values of TB and θ for I = 0.1 MW/cm2, F = 60 kV/cm and a0 = 3 nm.
3.3 Influences of CCQW with the electric field and ILF on the RIC
Similar to the study on the optical absorption coefficients, we also research the characteristic of refractive index changes Δn/nr in the above three cases, as shown in Fig. 13. As one may observe, the variation trend of total refractive index changes Δn/nr in different barrier thicknesses TB is similar to the case of total absorption coefficients α(ω, I), but maximum values of Δn/nr change in the opposite way. The reason lies in the fact that the maximum values of Δn/nr changes mainly come from the matrix element ${|{{M_{21}}} |^2}$, which is different from that of OAC. Therefore, by adopting the appropriate TB, a0, and θ, the RIC and the phase of the incident wave can be regulated at multiaspect as expected, which is of great significance for the manufacture of multifaceted-tunable optical phase modulator.
Fig. 13. The absorption coefficients as a function of incident photon energy with different values of TB for I = 0.1 MW/cm2: (a1-d1) a0 = 0; (a2-d2) F = 60 kV/cm and θ = 0; (a3-d3) F = 60 kV/cm and a0 = 3 nm.
4. Conclusion
In summary, we theoretically study the electronic state, OAC and RIC in a CCQW under ILF and electric field, and the results are compared with those of a single electric field or ILF. It is found that the optical properties of CCQW under double fields is completely different from that of single field. The effect of double fields can not only eliminate anticrossing of energy levels, but also restore the optical properties caused by the interfered distortion by regulating the laser parameter properly. The position and the magnitude of the OAC and RIC depend on the laser parameter, the electric field strength, the angle between electric field and polarization direction of ILF, and the barrier width. Such a restorable, enhanced, and multifaceted tunable optical properties under external fields in CCQW can be very useful for designing new optoelectronic devices with a performance tailored by the adjustment of the applied fields.
Funding
National Key Research and Development Program of China (2018YFB2201101); National Natural Science Foundation of China (51761135115, 61875199, 61935010, 61975208, 62175091); Foshan Science and Technology Program (1920001001680).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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