Light absorption in coated ellipsoidal quantum lenses

Abstract

In the framework of adiabatic approximation the energy states of electron and direct light absorption in coated ellipsoidal quantum lens is investigated. Analytical expressions for particle energy spectrum is obtained taking into account that electron effective masses are different in the coat and in quantum lens. Obtained results were applied for the case of rectangular quantum well of finite height. The investigation of energy level dependence on geometrical parameters of quantum lens was performed. In particular, it has been shown, that the particle energy is equidistant for both ellipsoidal and spherical segment cases, and dependence of energy on geometrical parameters has root character. Influence of the presence of a coating at a quantum lens on direct light absorption is investigated. Dependence of edge of absorption on geometrical parameters of a quantum lens and coating is obtained. The selection rules as in case of presence so and at absence of a coating for both ellipsoidal and spherical segment cases were derived.

Appendices

Appendix 1

1.1 Estimation of relative energy error at adiabatic approximation

Let us define relative error for one dimensional energy as ratio ${{\left( {\varepsilon _{1}^{{num}} \left( r \right) - \varepsilon _{1}^{int} \left( r \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\varepsilon _{1}^{{num}} \left( r \right) - \varepsilon _{1}^{int} \left( r \right)} \right)} {\varepsilon _{1}^{{num}} \left( r \right)}}} \right. \kern-\nulldelimiterspace} {\varepsilon _{1}^{{num}} \left( r \right)}}$, where $\varepsilon _{1} \left( r \right)$ is exact numerically calculated energy of CC in one dimensional step-like quantum well, $\varepsilon _{1}^{int} \left( r \right) \approx \alpha _{1} + \beta _{1}^{2} {\kern 1pt} r^{2} /4$ is interpolated Taylor series of adiabatic approximated energy of CC. This estimation approach can be used as a useful tool for designing objects for practical applications from theoretically modeled samples.

According to Fig. 8 at utilization of the adiabatic approximation the magnitude of the error comprises 10⁻⁴, which demonstrates the high accuracy which is obtainable via implementation of this approximation.

Fig. 8

Curve of relative energy error at adiabatic approximation

Appendix 2

$$I_{{n,n^{\prime}}} = \left| \begin{gathered} \frac{2}{{h_{1} \left(r \right)}}\int\limits_{0}^{{h_{1} \left( r \right)}} {\sin\left({\sqrt {\varepsilon _{1}^{e} \left( r \right)} z + \pi n}\right)\sin\left( {\sqrt {\varepsilon _{1}^{h} \left( r \right)} z +\pi n^{\prime}} \right)dz} \hfill \\ + \left( \begin{gathered}\frac{{e^{{2\sqrt {\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0} -\varepsilon _{1}^{e} \left( r \right)} \right)} h_{1} \left( r\right)}} - e^{{2\sqrt {\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0}- \varepsilon _{1}^{e} \left( r \right)} \right)} h_{1} \left( r\right)}} }}{{2\sqrt {\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0} -\varepsilon _{1}^{e} \left( r \right)} \right)} }} - \frac{{e^{{ -2\sqrt {\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0} - \varepsilon_{1}^{e} \left(r \right)} \right)} h_{1} \left( r \right)}} - e^{{- 2\sqrt {\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0} - \varepsilon_{1}^{e} \left( r \right)} \right)} h_{1} \left( r \right)}}}}{{2\sqrt {\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0} -\varepsilon _{1}^{e} \left( r \right)} \right)} }} \hfill \\ +2\left( {h_{2} \left( r \right) - h_{1} \left( r \right)} \right)\hfill \\ \end{gathered} \right)^{{ - {1 \mathord{\left/ {\vphantom{1 2}} \right. \kern-\nulldelimiterspace} 2}}} \hfill \\\times \left( \begin{gathered} \frac{{e^{{2\sqrt {\frac{{\mu _{2}}}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} h_{1} \left( r \right)}} - e^{{2\sqrt {\frac{{\mu_{2} }}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} h_{1} \left( r \right)}} }}{{2\sqrt {\frac{{\mu_{2} }}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} }} - \frac{{e^{{ - 2\sqrt {\frac{{\mu _{2} }}{{\mu_{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r \right)}\right)} h_{1} \left( r \right)}} - e^{{ - 2\sqrt {\frac{{\mu _{2}}}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} h_{1} \left( r \right)}} }}{{2\sqrt {\frac{{\mu_{2} }}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} }} \hfill \\ + 2\left( {h_{2} \left( r \right) -h_{1} \left( r \right)} \right) \hfill \\ \end{gathered} \right)^{{- {1 \mathord{\left/ {\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} \hfill \\ \times \int\limits_{{h_{1}\left( r \right)}}^{{h_{2} \left( r \right)}} {\left( {e^{{\sqrt{\frac{{\mu _{2} }}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{e}\left( r \right)} \right)} z}} + e^{{ - \sqrt {\frac{{\mu _{2}}}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} z}} } \right)\left( {e^{{\sqrt {\frac{{\mu _{2}}}{{\mu _{1} }}\left( {U_{0} - \varepsilon _{1}^{h} \left( r\right)} \right)} z}} + e^{{ - \sqrt {\frac{{\mu _{2} }}{{\mu _{1}}}\left( {U_{0} - \varepsilon _{1}^{h} \left( r \right)} \right)}z}} } \right)dz} \hfill \\ \end{gathered} \right|^{2} $$

$$J_{{N,N^{\prime}}}^{{n,n^{\prime}}} = \left|\begin{array}{l}\frac{{\sqrt {\beta _{n} } \sqrt {\left( {\frac{{N - \left| m\right|}}{2}} \right)!} \Upgamma \left( {\left| m \right| + 1}\right)}}{{\Upgamma ^{{{3 \mathord{\left/ {\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}} \left( {\left| m \right| + 1 +\frac{{N - \left| m \right|}}{2}} \right)}}\frac{{\sqrt {\beta_{{n^{\prime}}} } \sqrt {\left( {\frac{{N^{\prime} - \left|{m^{\prime}} \right|}}{2}} \right)!} \Upgamma \left( {\left|{m^{\prime}} \right| + 1} \right)}}{{\Upgamma ^{{{3 \mathord{\left/{\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} \left({\left| {m^{\prime}} \right| + 1 + \frac{{N^{\prime} - \left|{m^{\prime}} \right|}}{2}} \right)}} \\\times \int\limits_{0}^{{R_{0} }} e^{{ - \frac{{\beta _{n}r^{2} }}{4}}} e^{{ - \frac{{\beta _{{n^{\prime}}} r^{2} }}{4}}}\left( {\frac{{\beta _{n} r^{2} }}{2}} \right)^{{\frac{{\left| m\right|}}{2}}} \left( {\frac{{\beta _{{n^{\prime}}} r^{2} }}{2}}\right)^{{\frac{{\left| {m^{\prime}} \right|}}{2}}} {}_{1}F_{1}\left\{ { - \left( {\frac{{N - \left| m \right|}}{2}} \right),\left|m \right| + 1;\frac{{\beta _{n} r^{2} }}{2}} \right\} \\\quad \times {}_{1}F_{1} \left\{ { - \left( {\frac{{N^{\prime} - \left|{m^{\prime}} \right|}}{2}} \right),\left| {m^{\prime}} \right| +1;\frac{{\beta _{{n^{\prime}}} r^{2} }}{2}} \right\}rdr \end{array}\right|^{2}. $$

Here $I_{{n,n^{\prime}}}$ is vertical direction integral, and $J_{{N,N^{\prime}}}^{{n,n^{\prime}}}$ is radial integral.