1. Introduction
Recently, in the field of condensed matter physics, because of the continuous development of nanofabrication technology, people are more and more interested in the low dimensional system like quantum dots or quantum wells (QWs) [1 –7]. QW is a kind of micro structure with two-dimensional degrees of freedom in a plane parallel to the well wall, which has a quantum confinement effect in the other direction [8]. Its some basic characteristics are its unique optical and electronic transport properties. Zhao et al [9] studied the cyclotron mass and frequency of the magnetopolaron in the wurtzite ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1-x}{\rm{N}}/\mathrm{GaN}$ QW. Miller et al [10], for the first time, determined a set of QW parameters to describe the observed exciton transition energy, which are applicable to parabolic and square GaAs QWs grown with the $\mathrm{GaAs}-{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1-x}\mathrm{As}$ system. Ryczko et al [11] utilized the 10-band k-p modeling spectroscopic experiment in order to verify the chemical conduction band offset and research the electron effective mass in GaAsN/GaAs QWs. Meanwhile, in presence of electromagnetic field, temperature of the system and Coulomb impurity potential (CIP) field, numerous researchers have deliberated the nature of the polaron in QWs. Guo et al [12] studied the variation of refractive index and optical absorption coefficients in asymmetrical Gaussian potential quantum wells (AGPQWs), considering the impact of external electric field, by employing the approach of the compact density matrix and the effective mass approximation. Sarengaowa et al [13] derived the ground state energy (GSE) of the strong-coupled polaron in AGPQWs by utilizing the second unitary transformation and the linear combination operator approaches. Miao et al [14] analyzed the impact of hydrogen-like impurities on the GSE, vibration frequency and other characters of the weak-coupling bound polaron which is under an asymmetric Gaussian confinement potential QW by the same method. Ma et al [15] derived a strong-coupled polaron’s excitation energy and GSE in an AGPQW with applied electric field by the Pekar-type variational method (PTVM). Xiao [16] studied the characters of a strong-coupling impurity bound polaron in the AGPQW such as its energy levels and different states’ transition frequency. In particular, the combination of the Lee–Low–Pines unitary transformation method (LLPUTM) and the PTVM is a common research method which is used to research the properties of a polaron. The LLPUTM was firstly cited by Lee et al [17], and Landau and Pekar [18, 19] firstly used the PTVM to think over the nature of the strong-coupling polaron. In recent years, much attention has been paid to these two methods. For instance, Khordad et al [20], who considered a system under the condition of the asymmetrical Gaussian QW, explored the influence of the temperature on a bound polaron’s GSE and lifetime by the LLPUTM and the PTVM. Chen [21, 22], one of the authors of this paper, employed the combination of these two methods to study Rashba effects on a bound polaron’s first excited state and influences of the Rashba spin–orbit interaction on the period of bound polarons’ oscillation in the quantum pseudodot, considering the temperature effect. For more information about this author’s works on polarons, please refer to [23 –29].
However, we find that few researchers studied the impact of temperature on the bound polaron’s GSE within an AGPQW. For the purpose of this article, we investigate it in presence of electromagnetic field by using the LLPUTM and the PTVM, considering the influence of the temperature. In general, the harmonic oscillator potential or parabolic potential is considered to describe the confinement potential in nano-materials, because they are similar to the molecular vibration potential in the system. However, in some real or experimental conditions, using them to describe the phenomena or experimental results is unreasonable [30 –32]. Many works show that the anharmonic oscillator or Gaussian potential is more reasonable in different physical systems [30, 33]. Moreover, with the rapid development of nanofabrication techniques [31, 32], it has become feasible to design and fabricate QWs with different geometrical sizes, shapes and confinement potentials. Therefore, it is necessary to study the optical properties of various constraint potentials in detail. We believe that the results are of theoretical significance to the low-dimensional nano-science in condensed matter physics.
2. Model description and calculation
At present, we take such a system into account, in which the electron moves in a QW with an asymmetric Gaussian potential and interacts with bulk LO phonons in presence of an external electromagnetic field. Based on the theoretical framework of the effective mass approximation, the system Hamiltonian with the electron–phonon interaction can be indicated by the following form: $ \begin{eqnarray}\begin{array}{c}\begin{array}{rcl}H & = & \displaystyle \frac{{\left({P}_{x}-\tfrac{1}{2}{eBy}\right)}^{2}}{2{m}^{\ast }}+\displaystyle \frac{{\left({P}_{y}+\tfrac{1}{2}{eBx}\right)}^{2}}{2{m}^{\ast }}+\displaystyle \frac{{({P}_{z})}^{2}}{2{m}^{\ast }}+V(z)\\ & & +\,\displaystyle \sum _{{\boldsymbol{q}}}\hslash {\omega }_{{\rm{LO}}}{a}_{{\boldsymbol{q}}}^{\dagger }{a}_{{\boldsymbol{q}}}+\displaystyle \sum _{{\boldsymbol{q}}}[{V}_{q}{a}_{{\boldsymbol{q}}}{{\rm{e}}}^{{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}}+{\rm{h.c.}}]\\ & & -\,\displaystyle \frac{\beta }{r}-{e}^{\ast }{zF},\end{array}\end{array}\end{eqnarray}$ where $ \begin{eqnarray}V(z)=\left\{\begin{array}{ll}-{V}_{0}\exp \left(-\tfrac{{z}^{2}}{2{L}^{2}}\right), & Z\geqslant 0\\ \infty, & Z\lt 0\end{array}\right..\end{eqnarray}$ In equation (1 ), the physical quantities are identical with those in [34]. $-\tfrac{\beta }{r}$ expresses the CIP about the hydrogen-like impurity and the electron. Furthermore, the CIP strength β can be denoted as $\beta =\tfrac{{e}^{2}}{{\varepsilon }_{0}}$ . V (z) indicates the potential of z -direction, where this direction means the QW’s increasing direction [12, 35]. Besides, L is the scope of the confinement potential and V 0 is the height of the AGPQW. The expressions of V q and α in equation (1 ) are as follows: $ \begin{eqnarray}{V}_{q}={\rm{i}}\left(\displaystyle \frac{{\hslash }{\omega }_{{\rm{LO}}}}{q}\right){\left(\displaystyle \frac{{\hslash }}{2m{\omega }_{{\rm{LO}}}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{4\pi \alpha }{V}\right)}^{\tfrac{1}{2}},\end{eqnarray}$ $ \begin{eqnarray}\alpha =\left(\displaystyle \frac{{e}^{2}}{2{\hslash }{\omega }_{\mathrm{LO}}}\right){\left(\displaystyle \frac{2m{\omega }_{\mathrm{LO}}}{{\hslash }}\right)}^{\tfrac{1}{2}}\left(\displaystyle \frac{1}{{\varepsilon }_{\infty }}-\displaystyle \frac{1}{{\varepsilon }_{0}}\right),\end{eqnarray}$ here, the two quantities severally represent the electron-LO-phonon coupling amplitude and strength.
To obtain the GSE of the polaron, we should derived the lowest eigenvalue of the Hamiltonian. Enlighten by [17], i.e. according to the LLPUTM, we diagonalize approximately the phonon part of Hamiltonian (1 ) with the canonical transformation: $ \begin{eqnarray}U=\exp \left[\displaystyle \sum _{{\bf{q}}}({a}_{{\bf{q}}}^{+}{f}_{q}-{a}_{{\bf{q}}}{f}_{q}^{* })\right],\end{eqnarray}$ here f q and ${f}_{q}^{* }$ are the conjugate variational functions. After the self-consistent transformation, the Hamiltonian becomes as follows: $ \begin{eqnarray}H^{\prime} ={U}^{-1}{HU},\end{eqnarray}$ which makes it easier to solve the GSE of the polaron.
We follow the LLPUTM and write the ground state wave function of the polaron no regard of the confinement potential before and after transformation as $ \begin{eqnarray}| {\phi }_{{\rm{p}}}\rangle =U| 0{\rangle }_{\mathrm{ph}}\exp ({\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}),\end{eqnarray}$ $| 0{\rangle }_{\mathrm{ph}}$ represents a phonon’s vacuum state, which can be satisfied ${a}_{{\bf{q}}}| 0{\rangle }_{\mathrm{ph}}=0$ , and $U| 0{\rangle }_{\mathrm{ph}}$ is stated by the phonon coherent state. Then we follow the PTVM and the ground state trial wave function under the confinement potential can be selected: $ \begin{eqnarray}| {\varphi }_{0}\rangle =U| {\phi }_{0}\rangle ={\pi }^{-\tfrac{3}{4}}{\lambda }_{0}^{\tfrac{3}{2}}\exp \left(-\displaystyle \frac{{\lambda }_{0}^{2}{r}^{2}}{2}\right)| {\phi }_{{\rm{p}}}\rangle ,\end{eqnarray}$ in which ${\lambda }_{0}$ is one variational parameter. In above equation, the part except for $| {\phi }_{{\rm{p}}}\rangle $ is the ground state wave function of the system in which the limited potential is considered, and it is orthogonally normalized.
We minimize the Hamiltonian expectation value, namely ${E}_{0}=\langle {\varphi }_{0}| H| {\varphi }_{0}\rangle $ , in order to obtain the bound polaron GSE. After a complicated calculation, we can get the expression of E 0 as follows by selecting the commonly used polaron units ℏ = 2m = ω LO = 1 as well as letting ${R}_{{\rm{b}}}=\sqrt{\tfrac{{\hslash }}{{eB}}}$ , then $ \begin{eqnarray}\begin{array}{rcl}{E}_{0} & = & \displaystyle \frac{3}{2}{{\lambda }_{0}}^{2}+\displaystyle \frac{1}{4{R}_{{\rm{b}}}^{4}{{\lambda }_{0}}^{2}}-{V}_{0}{\left(1+\displaystyle \frac{1}{2{L}^{2}{{\lambda }_{0}}^{2}}\right)}^{-\tfrac{1}{2}}\\ & & -\,\displaystyle \frac{2\beta }{\sqrt{\pi }}{\lambda }_{0}-\displaystyle \frac{F}{\sqrt{\pi }{\lambda }_{0}}-\sqrt{\displaystyle \frac{2}{\pi }}\alpha {\lambda }_{0}.\end{array}\end{eqnarray}$
3. Temperature effects
Within the theoretical model of quantum statistics, the bulk LO phonons’ statistical mean number can be selected as: $ \begin{eqnarray}\overline{N}={\left[\exp \left(\displaystyle \frac{{\hslash }{\omega }_{\mathrm{LO}}}{{k}_{{\rm{B}}}T}\right)-1\right]}^{-1},\end{eqnarray}$ in which T is this system’s temperature, and k B is the Boltzmann constant. In addition, the LO phonon’s average number around the electron can take the form as follows: $ \begin{eqnarray}\overline{{N}_{0}}=\langle {\phi }_{0}| {U}^{-1}\displaystyle \sum _{{\boldsymbol{q}}}| {a}_{{\boldsymbol{q}}}^{\dagger }{a}_{{\boldsymbol{q}}}U| {\phi }_{0}\rangle =\sqrt{\displaystyle \frac{2}{\pi }}\alpha {\lambda }_{0}.\end{eqnarray}$ After the corresponding calculation of equations (10 ) and (11 ), we gain the relation between the temperature and the variational parameter λ 0 . Fundamentally, we can conclude that the GSE depends on the external electromagnetic field, the electron-LO-phonon coupling, the CIP, the asymmetrical Gaussian confinement potential and the temperature.
4. Results and discussions
At this part, we carry out the numerical calculations to determine the influence of independent variables R b (the external magnetic field constant), V 0 (the height of the AGPQW), L (the range of the confinement potential), β (the strength of the CIP), F (the external electric field constant), α (the electron-LO-phonon coupling strength) and T (the temperature) on the GSE in the AGPQW. The results are presented in figures 1 –6 . Meanwhile, we discuss each relationship in two cases: (a) lower temperature range $T\in (-150\,{\rm{K}}\sim 0\,{\rm{K}})$ and (b) higher temperature range $T\in (0\,{\rm{K}}\sim 400\,{\rm{K}})$ . It should be noted that the GSE changes from positive to negative value when the temperature rises from low to high temperature region. This shows that the effects of each term of equation (9 ) are different in different temperature regions. The positive value means that the first two terms of equation (9 ) play a greater role, while the negative value indicates that the last four terms of the equation will play a major role. And only its absolute value has the physical meaning.
Figure 1. The GSE E 0 of bound polaron versus the temperature T and the external magnetic field constant R b . |
Figure 1 indicates the bound polaron’s GSE as a function of the external magnetic field constant R b and the temperature T for V 0 = 5, L = 2, β = 30, F = 20 and α = 3. From figure 1 (a), we can survey that in the range of lower temperature, the GSE E 0 is a decreasing function of the external magnetic field constant R b, i.e., E 0 increases with the increase of the external magnetic field B because R b is inversely proportional to B ’s square root. The variational trend is that E 0 decreases sharply to the minimum then gradually decreases in the process of the R b increasing. Similarly, figure 1 (b) shows that in the higher temperature, the absolute value of E 0 booms near R b = 0.5, and then remains unchanged with the increase of R b . This result indicates that the GSE will enhance with decaying magnetic field in this temperature range. These phenomena indicate that the effect of the external magnetic field on the GSE is different in the high and low temperature range, and its effect is just opposite. In addition, one can see that there are differences of the temperature’s influence on E 0 in the high and low temperature range. Comparing figure 1 to 6, it can be seen that this difference still exists when other factors such as the electric field or the asymmetrical Gaussian potential changes.
Figure 2 describes the relationship of the GSE E 0 with the height of the AGPQW V 0 and the temperature T for R b = 2, L = 2, β = 30, F = 20 and α = 3. Figure 3 shows the GSE E 0 changing with the range of the temperature T and the confinement potential L for R b = 2, V 0 = 5, β = 30, F = 20 and α = 3. It is shown in figures 2 (a) and 3 (a) that in lower temperature, the GSE is a minus function of the range of the confinement potential and the height of the AGPQW. Nevertheless, from figures 2 (b) and 3 (b), we find that the absolute of E 0 increases with the increase of V 0 and L in the higher temperature. These results reveal that the interaction between phonon and electron is enhanced and the GSE is increased with the increase of the asymmetrical Gaussian potential’s restriction on the polaron.
Figure 2. The GSE E 0 of bound polaron versus the temperature T and the height of the AGPQW V 0 . |
Figure 3. The GSE E 0 of bound polaron versus the temperature T and the range of the confinement potential L . |
In figure 4, we illustrate the change in the GSE E 0 with the constant of the strength CIP β and temperature T for R b = 2, V 0 = 5, L = 2, F = 20 and α = 3. In figure 5, the GSE E 0 varying with temperature T and the constant of external electric field F is plotted for R b = 2, V 0 = 5, L = 2, β = 30 and α = 3. Figure 6 presents the GSE E 0 versus the temperature T and the electron-LO-phonon coupling strength α for R b = 2, V 0 = 5, L = 2, β = 30 and F = 20. From figures 4 –6, we can see that the GSE’s absolute value increases with the increasing strength of β, F and α in the range of lower and higher temperature. The reasons are as follows: when the temperature belongs to the category of low or high temperature, the fourth, the fifth and the sixth terms in equation (9 ), respectively representing the contribution from the CIP strength, the external electric field and the electron-LO-phonon coupling strength, have positive values. All of them show that the Coulomb interaction between charges, the external electric field and the interaction between electrons and phonons can promote the interaction between the ground state electrons and phonons, thus increasing the GSE. It is also shown that the contribution of the electricity and the coupling of electrons and phonons are stronger than that of magnetic field in both the low and high temperature regions.
Figure 4. The GSE E 0 of bound polaron versus the temperature T and the strength of the CIP β . |
Figure 5. The GSE E 0 of bound polaron versus the temperature T and the external electric field constant F . |
Figure 6. The GSE E 0 of bound polaron versus the temperature T and the electron-LO-phonon coupling strength α . |
5. Conclusion
In the current work, we have used the LLPUTM and PTVM to study the GSE of the bound polaron in an AGPQW considering the temperature and electromagnetic field. The impact of the temperature and asymmetrical Gaussian potential, electromagnetic field, CIP and electron–phonon coupling on the GSE are obtained. According to our researched results, the GSE of the bound polaron, on the one hand, oscillates as the temperature changes regardless of electron–phonon coupling, CIP, asymmetrical Gaussian potential and the electromagnetic field, on the other hand, has different rules with the electron–phonon coupling, CIP, asymmetrical Gaussian potential and electromagnetic field at different temperature zones.