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Effects of an anisotropic parabolic potential and Coulomb【-逻*辑*与-】apos;s impurity potential on the e

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Chun-Yu Cai, Wei Qiu, Yong Sun, Cui-Lan Zhao, Jing-Lin Xiao,Institute of Condensed Matter Physics, Inner Mongolia University for Nationalities, Tongliao 028043, China

Received:2020-08-2Revised:2020-09-25Accepted:2020-10-26Online:2020-12-18


Abstract
Because of its unique optoelectronic properties, people have studied the characteristics of polarons in various quantum well (QW) models. Among them, the asymmetrical semi-exponential QW (ASEQW) is a new model for studying the structure of QWs in recent years. It is of great significance to study the influences of the impurity and anisotropic parabolic confinement potential (APCP) on the crystal’s properties, because some of the impurities, usually regarded as Coulomb’s impurity potential (CIP), will exist in the crystal more or less, and the APCP has flexible adjustment parameters. However, the energy characteristics of the ASEQW under the combined actions of impurities and APCP have not been studied, which is the motivation of this paper. Using the linear combination operation and Lee–Low–Pines unitary transformation methods, we investigate the vibrational frequency and the ground state energy of the strong coupling polaron in an ASEQW with the influences of the CIP at the origin of coordinates and APCP, and make a comparison between our results and previous literature’s. Our numerical results about the energy properties in the ASEQW influenced by the CIP and APCP may have important significances for experimental design and device preparation.
Keywords: asymmetrical semi-exponential quantum well;anisotropic parabolic confinement potential;vibrational frequency;Coulomb's impurity potential;ground state energy


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Chun-Yu Cai, Wei Qiu, Yong Sun, Cui-Lan Zhao, Jing-Lin Xiao. Effects of an anisotropic parabolic potential and Coulomb's impurity potential on the energy characteristics of asymmetrical semi-exponential CsI quantum wells. Communications in Theoretical Physics, 2021, 73(1): 015701- doi:10.1088/1572-9494/abc470

1. Introduction

The development and progress of experimental technologies has made the manufactures of various low-dimensional structures possible. Because nano-low-dimensional systems have important scientific research value and their unique photoelectric properties and transport characteristics, they have extremely broad application prospects and potential economic values. Compared with other low-dimensional structures, quantum wells (QWs) have higher quantum efficiency, so they are used to make light-emitting devices and photodetector devices. Due to the broad development prospects of QWs in the field of optoelectronics, the study of polarons in QWs of various structures has made considerable progress [16]. The optical properties of single QW [7], multiple QW [8] and doped QW [9] were studied respectively. Ungan et al shown the optical properties of QWs under the triple influence of electric field, magnetic field and laser field [10].

Among many types of QWs, the asymmetrical semi-exponential QW (ASEQW) is a new model to study the structure and performance of QWs in recent years. Guo Kangxian’s team [1116] and Wang et al [17] have studied the polaron’s optical properties and the temperature effect of the ground state energy (GSE) in ASEQW, respectively. Impurities are inevitably introduced during the material manufacturing process. Therefore, people had paid attention to the influences of impurities on the properties of crystals for a long time [18, 19]. Xiao studied the influences of the Coulomb’s impurity potential (CIP) on the energy characteristics [20] and coherent properties [21] in the asymmetric Gaussian confinement QW, and found that the CIP can improve the stability of the AGCPQW system, however, the CIP can seriously damage the coherent properties of the polaron. So far, according to our knowledge, only [22, 23] have studied the influences of the CIP and temperature on the ground state binding energy and GSE in the ASEQW.

However, none of the above references considering the properties of the ASEQW under the combined actions of the anisotropic parabolic confinement potential (APCP) and CIP. Inspired by the [10], we consider and investigate the impacts of these factors on the energy level structure of the ASEQW. Numerical calculations are performed on the actual material CsI, and the obtained results are compared with the previous literatures.

2. Theoretical models

The electron interacts with bulk longitudinal optical phonons and moves in the ASEQW influenced by the CIP at the origin of coordinates and APCP. Inspired by the [13, 22, 24], the Hamiltonian of the system in the ASEQW under the combined actions of the CIP and APCP reads as follows:$\begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \frac{{p}^{2}}{2m}+\displaystyle \sum _{{\boldsymbol{q}}}{\hslash }{\omega }_{\mathrm{LO}}{a}_{{\boldsymbol{q}}}^{\dagger }{a}_{{\boldsymbol{q}}}+\displaystyle \sum _{{\boldsymbol{q}}}({V}_{q}{a}_{{\boldsymbol{q}}}\exp ({\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}})+{\rm{h}}.{\rm{c}}.)\\ & & +\displaystyle \frac{1}{2}m{\omega }_{x}^{2}{x}^{2}+\displaystyle \frac{1}{2}m{\omega }_{y}^{2}{y}^{2}+U\left(z\right)-\displaystyle \frac{\beta }{r},\end{array}\end{eqnarray}$where$\begin{eqnarray}U\left(z\right)=\left\{\begin{array}{cc}{U}_{0}\left({{\rm{e}}}^{z/\sigma }-1\right) & z\geqslant 0\\ \infty & z\lt 0\end{array}\right.,\end{eqnarray}$$\begin{eqnarray}\left\{\begin{array}{l}{V}_{q}={\rm{i}}({\hslash }{\omega }_{\mathrm{LO}}/q){\left({\hslash }/2m{\omega }_{\mathrm{LO}}\right)}^{1/4}{\left(4\pi \alpha /V\right)}^{1/2}\\ \alpha =({{e}}^{2}/2{\hslash }{\omega }_{\mathrm{LO}}){\left(2m{\omega }_{\mathrm{LO}}/{\hslash }\right)}^{1/2}(1/{\varepsilon }_{\infty }-1/{\varepsilon }_{0})\end{array}\right..\end{eqnarray}$

$U\left(z\right)$ denotes the asymmetric semi-exponential confinement potential (ASECP), and both U0 and σ are positive parameters. β represents the CIP’s strength (CIPS) and satisfies $\beta ={e}^{2}/{\varepsilon }_{0}$. The fourth and the fifth terms in the equation (1) describe the two-dimensional xy plane APCP.

Giving Fourier expansion to the CIP [25],$\begin{eqnarray}-\displaystyle \frac{\beta }{r}=-\displaystyle \frac{4\pi \beta }{v}\displaystyle \sum _{{\boldsymbol{q}}}\displaystyle \frac{1}{{q}^{2}}\exp \left(-{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}\right).\end{eqnarray}$

For electron–phonon strong coupling, the equation (1) is performed on the second Lee–Low–Pines unitary transformation (LLPUT) [26] and employed by the linear combination operation (LCO) [27]:$\begin{eqnarray}{U}_{2}=\exp \left[\displaystyle \sum _{{\boldsymbol{q}}}({a}_{{\boldsymbol{q}}}^{\dagger }{f}_{q}-{a}_{{\boldsymbol{q}}}{f}_{q}^{* })\right],\end{eqnarray}$$\begin{eqnarray}\left\{\begin{array}{l}{p}_{j}={\left(m{\hslash }\lambda /2\right)}^{1/2}({b}_{j}+{b}_{j}^{+})\ \\ {r}_{j}={\rm{i}}{\left({\hslash }/2m\lambda \right)}^{1/2}({b}_{j}-{b}_{j}^{+})\ \end{array}\right.,\ \left(j=x,y,z\right).\end{eqnarray}$

The system’s ground state wave function is chosen as$\begin{eqnarray}\left|{\psi }_{0}\right\rangle =| 0{\rangle }_{a}| 0{\rangle }_{b},\end{eqnarray}$where ${\left|0\right\rangle }_{a}$ and ${\left|0\right\rangle }_{b}$ satisfy ${b}_{j}{\left|0\right\rangle }_{b}={a}_{{\boldsymbol{q}}}{\left|0\right\rangle }_{a}=0$.

By computations, the strong coupling system’s vibrational frequency (VF) in the ASEQW under the combined actions of the CIP and APCP can be obtained and written as$\begin{eqnarray*}F(\lambda ,{f}_{q})=\left\langle {\psi }_{0}\right|{U}_{2}^{-1}{{HU}}_{2}\left|{\psi }_{0}\right\rangle \end{eqnarray*}$$\begin{eqnarray}\begin{array}{l}{\lambda }^{2}-\left(\displaystyle \frac{4\beta }{3{\hslash }}{\left(\displaystyle \frac{m}{\pi {\hslash }}\right)}^{1/2}+\displaystyle \frac{2\alpha }{3}\sqrt{\displaystyle \frac{{\omega }_{\mathrm{LO}}}{\pi }}\right){\lambda }^{3/2}\\ \quad -\,\left(\displaystyle \frac{{U}_{0}}{3m{\sigma }^{2}}+\displaystyle \frac{{{\hslash }}^{2}}{3{m}^{2}}\left(\displaystyle \frac{1}{{l}_{x}^{4}}+\displaystyle \frac{1}{{l}_{y}^{4}}\right)\right)=0.\end{array}\end{eqnarray}$

Assuming the root of equation (8) is λ0, the system’s GSE can be gotten variationally$\begin{eqnarray}\begin{array}{rcl}{E}_{0} & = & \displaystyle \frac{3}{4}{\hslash }\lambda \ +\ \displaystyle \frac{{\hslash }{U}_{0}}{4m\lambda {\sigma }^{2}}-\displaystyle \frac{1}{\sqrt{\pi }}\alpha {\hslash }{\omega }_{\mathrm{LO}}{\left(\displaystyle \frac{\lambda }{{\omega }_{\mathrm{LO}}}\right)}^{1/2}\\ & & +\displaystyle \frac{1}{4\lambda }\displaystyle \frac{{{\hslash }}^{3}}{{m}^{2}{l}_{x}^{4}}+\displaystyle \frac{1}{4\lambda }\displaystyle \frac{{{\hslash }}^{3}}{{m}^{2}{l}_{y}^{4}}-2\beta \sqrt{\displaystyle \frac{m\lambda }{\pi {\hslash }}},\end{array}\end{eqnarray}$where lx and ly are the x and y directions effective confinement lengths, respectively.

3. Results and discussion

In order to fully explore the energy characteristics of the polaron in the ASEQW under the combined actions of the CIP and APCP, numerical results about its GSE and the VF have been displayed in figures 14. Choosing the CsI crystal, adopted experimental parameters are α=3.67, ℏωLO=11.616 meV, m=0.42m0 and m0=9.1×10−31 kg [28].

Figure 1.

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Figure 1.The GSE E0 versus the CIPS β and parameter U0.


Figures 1 and 2 depict the GSE E0 and the VF λ of the strong coupling impurity polaron in the CsI ASEQW versus the CIPS and the ASECP’s positive parameter U0 when the x and y directions effective confinement lengths satisfy lx=ly=1.0 nm for other three different positive parameters σ=0.2, 0.3, 0.4 nm. The numerical results of figure 1 indicate that the GSE of the strong coupling impurity polaron in the ASEQW is an enhancing (decreasing) function of the positive parameter U0 $\left(\sigma \right)$. The reason is that the main property of the GSE is due to ASECP added in the growth direction of the ASEQW. The ASECP is an elevating (decreasing) function of the parameter U0 (σ) seen from the equation (2). Therefore, the GSE of impurity polaron increases (decreases) with increasing the parameter U0 $\left(\sigma \right)$. In this figure, We also can observe that the GSE E0 is enlarged by decreasing the CIPS β. As can be seen from the expression of the GSE that the last term of the equation (9), which value is negative, representing the contribution of the CIP to the GSE. Therefore, the GSE will decrease with increasing the CIPS. The changing laws obtained here are consistent with the laws in the asymmetric Gaussian confinement QW [20] and the ASEQW [23]. Simultaneously, comparing the results of this paper with the conclutions in [20, 23], which consider the temperature effects, we indirectly found that the introductions of the CIP will cause the increase of the energy level in the ASEQW, which will destroy the stability of the system and then weaken the luminescence characteristics of devices based on the ASEQW.

Figure 2.

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Figure 2.The VF λ versus the CIPS β and parameter U0.


The results of figure 2 also imply that the VF λ is an increasing (decreasing) function of the positive parameter U0 $\left(\sigma \right)$. This phenomenon is still due to the fact that the confinement of the ASECP becomes strong (weak) with enlarging U0 $\left(\sigma \right)$. Thereby, the VF of impurity polaron lifts (reduces) with increasing the positive parameter U0 $\left(\sigma \right)$. Furthermore, the figure 2 also reveals that as the CIPS β enlarges, the VF of the impurity polaron increases. The reason is that when the CIPS is applied to a system, an additional energy will be added to the system. It is clear that the presence of the CIPS is equivalent to introduce another new confinement to the system. It will lead to greater electron wavefunction overlapping, then the polaron’s internal interactions will be raised. Therefore, the VF elevates. This numerical result is consistent with [22].

In figures 3 and 4, we respectively plot the GSE E0 and the VF λ versus the effective confinement lengths lx and ly for the ASECP’s two positive parameters and the CIPS satisfy U0=10 meV, σ=0.2 nm and β=0.1 meV · nm. The numerical results in figures 3 and 4 show that the GSE and the VF of the polaron reduces rapidly with elevating the effective confinement lengths. Their changing reasons are same. Because the motion of the electrons is confined by the APCP in the x and y directions. With the deduce of the effective confinement lengths, the electronic thermal motion energy and the polaron’s internal interactions are enlarged rapidly. Therefore, the GSE and the VF are increased. The changing laws are consistent with the laws of the energy level and the VF versus the effective confinement lengths in the quantum dot [29, 30], which were influenced by the APCP. It shows that the introduction of the APCP makes the polaron’s characteristics in the ASEQW have a more flexible adjustment methods.

Figure 3.

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Figure 3.The GSE E0 versus the x and y directions effective confinement lengths lx and ly.


Figure 4.

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Figure 4.The VF λ versus the x and y directions effective confinement lengths lx and ly.


4. Conclusions

Utilizing the LCO and LLPUT methods, this paper theoretically explores the influences of the CIP and APCP’s combined actions on the energy level characteristics of the polaron in the ASEQW. The numerical results show that: ①We have found five ways for adjusting the energy level structures of polarons in ASEQW: changing the CIPS, the x and y directions effective confinement lengths of the APCP, and two positive parameters of ASECP; ②Not only the introduction of the APCP makes the polaron characteristics in ASEQW more flexible to adjust; but also comparing the results in the previous references, which consider the temperature effects, we indirectly find that the introductions of the CIP will cause the increase of the energy level in the ASEQW, which will damage the stability of the system and the luminescence characteristics of devices based on the ASEQW.

Acknowledgments

This project was supported by the National Natural Science Foundation of China under Grant No. 11 464 034, the National Science Foundation of Inner Mongolia Autonomous Region under Grant Nos. 2016MS0119 and 2016BS0107, Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY16183, and the Scientific Research Fund of Inner Mongolia University for Nationalities under Grant Nos. NMDYB1756 and NMDYB18024.


Reference By original order
By published year
By cited within times
By Impact factor

Wu Y Z et al. 2020 Synthesis and properties of InGaN/GaN multiple quantum well nanowires on Si (111) by molecular beam epitaxy
Phys. Status Solidi a 217 1900729

DOI:10.1002/pssa.201900729 [Cited within: 1]

Jawhar N N Soheyli E Yazici A F Mutlugun E Sahraei R 2020 Preparation of highly emissive and reproducible Cu-In-S/ZnS core/shell quantum dots with a mid-gap emission character
J. Alloys Compd. 824 153906

DOI:10.1016/j.jallcom.2020.153906

Guo Y M Li J W 2020 MoS2 quantum dots: synthesis, properties and biological applications
Mater. Sci. Eng. C 109 110511

DOI:10.1016/j.msec.2019.110511

Wang Q W Uddin R Du X Z Li J Lin J Y Jiang H X 2019 Synthesis and photoluminescence properties of hexagonal BGaN alloys and quantum wells
Appl. Phys. Express 12 011002

DOI:10.7567/1882-0786/aaee8d

Jang T Sohn S H 2018 Study of quantum well shell structures grown using one-pot synthesis
Mol. Cryst. Liq. Cryst. 663 34 39

DOI:10.1080/15421406.2018.1467646

Surana K Salisu I T Mehra R M Bhattacharya B 2018 A simple synthesis route of low temperature cdse-cds core–shell quantum dots and its application in solar cell
Opt. Mater. 82 135 140

DOI:10.1016/j.optmat.2018.05.060 [Cited within: 1]

Sailai M Aierken A Qiqi L Heini M Zhao X Mo J Jie G Hao R Yu Z Qi G 2020 Effects of 1 MeV electron irradiation on the photoluminescence of GaInNAs∣GaAs single quantum well structure
Semiconductors 54 554 557

DOI:10.1134/S1063782620050103 [Cited within: 1]

Shi K J Li H B Ma W Wang C X Li C F Wei Y H Ji Z W Xu X G 2020 Effect of InGaN growth interruption on photoluminescence properties of an InGaN-based multiple quantum well structure
Physica E 119 113982

DOI:10.1016/j.physe.2020.113982 [Cited within: 1]

Piotrowski P Pacuski W 2020 Photoluminescence of CdTe quantum wells doped with cobalt and iron
J. Lumin. 221 117047

DOI:10.1016/j.jlumin.2020.117047 [Cited within: 1]

Ungan F Bahar M K Mora-Ramos M E 2020 Optical properties of n-type asymmetric triple delta-doped quantum well under external fields
Phys. Scr. 95 055808

DOI:10.1088/1402-4896/ab7a37 [Cited within: 2]

Liu G G Guo K X Wu Q J 2012 Linear and nonlinear intersubband optical absorption and refractive index change in asymmetrical semi-exponential quantum wells
Superlattices Microstruct. 52 183 192

DOI:10.1016/j.spmi.2012.04.023 [Cited within: 1]

Mou S Guo K X Liu G H Xiao B 2014 Second-harmonic generation coefficients in asymmetrical semi-exponential quantum wells
Physica B 434 84 88

DOI:10.1016/j.physb.2013.10.029

Mou S Guo K X Xiao B 2014 Studies on the third-harmonic generation coefficients in asymmetrical semi-exponential quantum wells
Superlattices Microstruct. 65 309 318

DOI:10.1016/j.spmi.2013.11.016 [Cited within: 1]

Xiao B Guo K X Mou S Zhang Z M 2014 Polaron effects on the optical rectification in asymmetrical semi-exponential quantum wells
Superlattices Microstruct. 69 122 128

DOI:10.1016/j.spmi.2014.01.016

Mou S Guo K X Xiao B 2014 Polaron effects on the linear and nonlinear intersubband optical absorption coefficients in quantum wells with asymmetrical semi-exponential potential
Superlattices Microstruct. 72 72 82

DOI:10.1016/j.spmi.2014.03.048

Guo K X Xiao B Zhou Y C Zhang Z M 2015 Polaron effects on the third-harmonic generation in asymmetrical semi-exponential quantum wells
J. Opt. 17 035505

DOI:10.1088/2040-8978/17/3/035505 [Cited within: 1]

Wang X Q Xiao J L 2018 Effects of temperature on the ground state energy of the strong coupling polaron in a rbcl asymmetrical semi-exponential quantum well
Int. J. Theor. Phys. 57 3436 3442

DOI:10.1007/s10773-018-3857-5 [Cited within: 1]

Bastard G 1981 Hydrogenic impurity states in a quantum well: a simple model
Phys. Rev. B 24 4714 4722

DOI:10.1103/PhysRevB.24.4714 [Cited within: 1]

Greene R L Bajaj K K 1983 Energy levels of hydrogenic impurity states in gaas-ga1 xalxas quantum well structures
Solid State Commun. 45 825 829

DOI:10.1016/0038-1098(83)90809-8 [Cited within: 1]

Xiao J L 2018 The effects of hydrogen-like impurity and temperature on state energies and transition frequency of strong-coupling bound polaron in an asymmetric Gaussian potential quantum well
J. Low Temp. Phys. 192 41 47

DOI:10.1007/s10909-018-1873-8 [Cited within: 3]

Xiao J L 2019 Effects of hydrogen-like impurity on the coherence time of asymmetric Gaussian confinement potential quantum well qubit
Superlattices Microstruct. 135 106279

DOI:10.1016/j.spmi.2019.106279 [Cited within: 1]

Xiao J L 2018 Temperature and impurity effects on the vibrational frequency of the strongly-coupled polaron in asymmetrical semi-exponential rbcl quantum wells
Superlattices Microstruct. 120 459 462

DOI:10.1016/j.spmi.2018.06.014 [Cited within: 3]

Sun Y Ding Z H Xiao J L 2019 Temperature effect on the ground state energy and the longitudinal optical-phonon mean number of the impurity polaron in asymmetrical 2D RbCl semi-exponential quantum wells
Mater. Express 9 371 375

DOI:10.1166/mex.2019.1496 [Cited within: 3]

Zhao C Cai C Xiao J 2013 Influence of an anisotropic parabolic potential on the quantum dot qubit
J. Semicond. 34 112002

DOI:10.1088/1674-4926/34/11/112002 [Cited within: 1]

Zhu K-D Gu S-W 1991 Polaronic states that incorporate effects of phonon confinement in a rectangular quantum well wire
Solid State Commun. 80 307 310

DOI:10.1016/0038-1098(91)90135-I [Cited within: 1]

Lee T D Low F E Pines D 1953 The motion of slow electrons in a polar crystal
Phys. Rev. 90 297 302

DOI:10.1103/PhysRev.90.297 [Cited within: 1]

Huybrechts W J 1976 Note on the ground-state energy of the Feynman polaron model
J. Phys. C: Solid State Phys. 9 L211 L212

DOI:10.1088/0022-3719/9/8/006 [Cited within: 1]

Devreese J T 1972 Polarons in Ionic Crystals and Polar Semiconductors Amsterdam North-Holland Publishing Co.
[Cited within: 1]

En-Hui X Cui-Lan Z Chun-Yu C Jing-Lin X 2013 Effects of an anisotropic parabolic potential on the excited state energy of a strong-coupling polaron in a quantum dot
J. Inner Mongolia Univ. Nationalities 28 4 6 + 10

DOI:10.14045/j.cnki.15-1220.2013.01.038 [Cited within: 1]

Chun-Yu C Cui-Lan Z Jing-Lin X 2012 Influences of an anisotropic parabolic potential on the ground state binding energy of polaron in quantum dot
J. Inner Mongolia Univ. Nationalities 27 504 507

DOI:10.14045/j.cnki.15-1220.2012.05.038 [Cited within: 1]

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