The energy-level and vibrational frequency properties of a polaron weak-coupled in a quantum well with asymmetrical Gaussian confinement potential

Wei Xiao; Jinglin Xiao

doi:10.1088/1674-4926/40/4/042901

Journals >Journal of Semiconductors >Volume 40 >Issue 4 >Page 042901 > Article

Journal of Semiconductors
Vol. 40, Issue 4, 042901 (2019)

The energy-level and vibrational frequency properties of a polaron weak-coupled in a quantum well with asymmetrical Gaussian confinement potential

Wei Xiao¹ and Jinglin Xiao²

Author Affiliations

¹Department of Basic Sciences, University of Informational Science and Technology of Beijing, Beijing 100101, China

²Institute of Condensed Matter Physics, Inner Mongolia University for the Nationalities, Tongliao 028043, China

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DOI: 10.1088/1674-4926/40/4/042901 Cite this Article

Wei Xiao, Jinglin Xiao. The energy-level and vibrational frequency properties of a polaron weak-coupled in a quantum well with asymmetrical Gaussian confinement potential[J]. Journal of Semiconductors, 2019, 40(4): 042901 Copy Citation Text

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Abstract

The vibrational frequency (VF), the ground state (GS) energy and the GS binding energy of the weak electron-phonon coupling polaron in a quantum well (QW) with asymmetrical Gaussian confinement potential are calculated. First we introduce the linear combination operator to express the momentum and coordinates in the Hamilton and then operate the system Hamilton using unitary transformation. The results indicate the relations of the quantities (the VF, the absolute value of GS energy and the GS binding energy) and the parameters (the QW barrier height and the range of Gaussian confinement potential in the growth direction of the QW).

$\begin{split} H =\;& \frac{{{p^2}}}{{2m}} + V\left( z \right) + \sum\limits_{ q} {\hbar {\omega _{{\rm{LO}}}}a_{ q}^ + } {a_{ q}} \\ &+ \sum\limits_{ q} {\left( {{V_{q}}{a_{ q}}\exp \left( {i{ q} \cdot { r}} \right) + h\cdot c} \right)} , \end{split}$

(1)

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$V\left( z \right) = \left\{ {\begin{aligned} &{ - {V_0}\exp \left( { - \frac{{{z^2}}}{{2{R^2}}}} \right),\quad z \geqslant 0},\\ &{\infty, \quad z < 0}. \end{aligned}} \right.$

(2)

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$\begin{split} &{V_{{q}}} = i\left( {\frac{{\hbar {\omega _{{\rm{LO}}}}}}{q}} \right){\left( {\frac{\hbar }{{2m{\omega _{{\rm{LO}}}}}}} \right)^{\frac{1}{4}}}{\left( {\frac{{4\pi \alpha }}{v}} \right)^{\frac{1}{2}}},\\ &\quad \alpha = \left( {\frac{{{e^2}}}{{2\hbar {\omega _{{\rm{LO}}}}}}} \right){\left( {\frac{{2m{\omega _{{\rm{LO}}}}}}{\hbar }} \right)^{\frac{1}{2}}}\left( {\frac{1}{{{\varepsilon _\infty }}} - \frac{1}{{{\varepsilon _0}}}} \right). \end{split}$

(3)

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${U_1} = \exp \left( { - i\sum\limits_{ q} {{ q} \cdot { r}a_{ q}^ + {a_{ q}}} } \right),\tag{4a}$

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${U_2} = \exp \left[ {\sum\limits_{ q} {\left( {a_{ q}^ + {f_{ q}} - {a_{ q}}f_{ q}^ * } \right)} } \right],\tag{4b}$

(4)

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$ \begin{split} &{p_{{j}}} = {\left( {\displaystyle \frac{{m\hbar \lambda }}{2}} \right)^{\frac{1}{2}}}\left( {{b_{{j}}} + b_{{j}}^ + } \right),\\ &\quad {r_{{j}}} = i{\left( {\displaystyle \frac{\hbar }{{2m\lambda }}} \right)^{\frac{1}{2}}}\left( {{b_{{j}}} - b_{{j}}^ + } \right), \end{split} $

(5)

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