Configuration Resonance and Generation Rate of Surface Plasmon Polaritons Excited by Semiconductor Quantum Dots near a Metal Surface

Abstract

We discuss particular features of generation of surface plasmon polaritons in a metal–dielectric planar interface that is coupled to semiconductor quantum dots by near-field interactions. As a model of working medium for performing numerical experiment, we use a gold metal surface onto which a polyethylene terephthalate film containing CdSe semiconductor spherical quantum dot is deposited. The problem of optimizing the radius of a quantum dot and its distance to a metal surface is solved for achieving the maximum transfer efficiency of the quantum dot energy for the generation of surface plasmon polaritons. Dispersion effects of the surface wave generation rate associated with deviations of the radius of quantum dots and their distance to the metal surface from the corresponding average values are taken into account.

Appendices

DERIVATION OF RELAXATION RATES b AND FREQUENCY ω OF DIPOLE OSCILLATIONS

Let us reduce Eq. (1) to a homogeneous differential equation and obtain relations for new relaxation rates and frequency shift. By substituting solutions (3) into Eq. (1), we obtain

$$\begin{gathered} {{p}_{0}}{{e}^{{\left( { - b/2 - i\omega } \right)t}}}{{\left( { - \frac{b}{2} - i\omega } \right)}^{2}} + \omega _{0}^{2}{{p}_{0}}e{}^{{\left( { - b/2 - i\omega } \right)t}} \\ = \frac{{{{e}^{2}}}}{m}{{E}_{0}}{{e}^{{\left( { - b/2 - i\omega } \right)t}}} - {{b}_{0}}{{p}_{0}}{{e}^{{\left( { - b/2 - i\omega } \right)t}}}\left( { - \frac{b}{2} - i\omega } \right). \\ \end{gathered} $$

After dividing this expression into ${{e}^{{\left( { - b/2 - i\omega } \right)t}}}$ and representing complex amplitude E₀ in the algebraic form E₀ = Re(E₀) + iIm(E₀), we obtain

$$\begin{gathered} {{p}_{0}}{{\left( { - \frac{b}{2} - i\omega } \right)}^{2}} + \omega _{0}^{2}{{p}_{0}} \\ = \frac{{{{e}^{2}}}}{m}\left( {\operatorname{Re} \left( {{{E}_{0}}} \right) + i\operatorname{Im} \left( {{{E}_{0}}} \right)} \right) - {{b}_{0}}{{p}_{0}}\left( { - \frac{b}{2} - i\omega } \right). \\ \end{gathered} $$

We will separate the imaginary and real parts of the expression as follows:

$$\begin{gathered} - \frac{{{{e}^{2}}\operatorname{Re} \left( {{{E}_{0}}} \right)}}{m} + \frac{{{{p}_{0}}{{b}^{2}}}}{4} - \frac{{b{{b}_{0}}{{p}_{0}}}}{2} - {{p}_{0}}{{\omega }^{2}} + \omega _{0}^{2}{{p}_{0}} \\ - \;i\left( { - \frac{{{{e}^{2}}\operatorname{Im} \left( {{{E}_{0}}} \right)}}{m} + b{{p}_{0}}\omega - {{b}_{0}}{{p}_{0}}\omega } \right) = 0 \\ \end{gathered} $$

and obtain the systems of the equations

$$ - i\left( { - \frac{{{{e}^{2}}\operatorname{Im} \left( {{{E}_{0}}} \right)}}{m} + b{{p}_{0}}\omega - {{b}_{0}}{{p}_{0}}\omega } \right) = 0,$$

$$ - \frac{{{{e}^{2}}\operatorname{Re} \left( {{{E}_{0}}} \right)}}{m} + \frac{{{{p}_{0}}{{b}^{2}}}}{4} - \frac{{b{{b}_{0}}{{p}_{0}}}}{2} - {{p}_{0}}{{\omega }^{2}} + \omega _{0}^{2}{{p}_{0}} = 0.$$

By solving it simultaneously, we obtain

$$b = {{b}_{0}} + {{b}_{0}}\frac{{{{e}^{2}}\operatorname{Im} \left( {{{E}_{0}}} \right)}}{{m{{p}_{0}}\omega {{b}_{0}}}},$$

$$\Delta \omega \approx - \frac{{{{e}^{2}}\operatorname{Re} \left( {{{E}_{0}}} \right)}}{{2{{\omega }_{0}}m{{p}_{0}}}} + \frac{{{{b}^{2}}}}{{8{{\omega }_{0}}}} - \frac{{b{{b}_{0}}}}{{4{{\omega }_{0}}}},$$

assuming that ${{\omega }^{2}} - \omega _{0}^{2}$ = $(\omega - {{\omega }_{0}})(\omega + {{\omega }_{0}})$ ≈ $2{{\omega }_{0}}\Delta \omega$, where $\Delta \omega = \omega - {{\omega }_{0}}$. Then, we obtain the following expression for frequency ω of dipole oscillations:

$$\omega \approx {{\omega }_{0}} - \frac{{{{e}^{2}}\operatorname{Re} \left( {{{E}_{0}}} \right)}}{{2{{\omega }_{0}}m{{p}_{0}}}} + \frac{{{{b}^{2}}}}{{8{{\omega }_{0}}}} - \frac{{b{{b}_{0}}}}{{4{{\omega }_{0}}}}.$$

CALCULATION OF THE INTEGRAL FOR THE RATE OF SPP GENERATION BY THE CHANGE OF VARIABLES TECHNIQUE

Let us divide the improper integral of the second kind in (4) by a sum of integrals, taking into account that the integrand has an infinite discontinuity at the point u = 1:

$${{b}_{ \bot }} = 1 + {\text{3/2}}q{\text{Im}}\left[ {\mathop {\lim }\limits_{{{\delta }_{1}} \to - 0} \int\limits_0^{1 + {{\delta }_{1}}} {{{r}_{p}}\exp ( - 4\pi \sqrt {{{\varepsilon }_{1}}} {{l}_{1}}d{\text{/}}\lambda {\text{)}}{{u}^{3}}{\text{/}}{{l}_{1}}{\text{d}}u} } \right]$$

$$ + \;{\text{3/2}}q\operatorname{Im} \left[ {\mathop {\lim }\limits_{{{\delta }_{2}} \to + 0} \int\limits_{1 + {{\delta }_{2}}}^\infty {{{r}_{p}}\exp ( - 4\pi \sqrt {{{\varepsilon }_{1}}} {{l}_{1}}d{\text{/}}\lambda ){{u}^{3}}{\text{/}}{{l}_{1}}{\text{d}}u} } \right].$$

Let us introduce u into the differential:

$$\begin{gathered} {{b}_{ \bot }}\, = \,1\, + \,{\text{3/2}}q{\text{Im}}\left[ {\mathop {\lim }\limits_{{{\delta }_{1}} \to - 0} \int\limits_0^{1 + {{\delta }_{1}}} {\frac{{{{r}_{p}}}}{{4{{l}_{1}}}}{\text{exp}}( - 4\pi \sqrt {{{\varepsilon }_{1}}} {{l}_{1}}d{\text{/}}\lambda {\text{)d}}{{u}^{4}}} } \right] \\ + \;{\text{3/2}}q{\text{Im}}\left[ {\mathop {\lim }\limits_{{{\delta }_{2}} \to + 0} \int\limits_{1 + {{\delta }_{2}}}^\infty {\frac{{{{r}_{p}}}}{{4{{l}_{1}}}}{\text{exp}}( - 4\pi \sqrt {{{\varepsilon }_{1}}} {{l}_{1}}d{\text{/}}\lambda ){\text{d}}{{u}^{4}}} } \right]. \\ \end{gathered} $$

((B1))

By making the change of variables u⁴ = x₁ in (B1), we obtain

$${{b}_{ \bot }} = 1 + {\text{3/2}}q\operatorname{Im} \left[ {\mathop {\lim }\limits_{{{\delta }_{1}} \to - 0} \int\limits_0^{1 + {{\delta }_{1}}} {\frac{{{{r}_{p}}({{x}_{1}})}}{{4{{l}_{1}}({{x}_{1}})}}} } \right.$$

$$\left. {^{{^{{^{{^{{^{{}}}}}}}}}} \times \;{\text{exp}}( - 4\pi \sqrt {{{\varepsilon }_{1}}} {{l}_{1}}({{x}_{1}})d{\text{/}}\lambda ){\text{d}}{{x}_{1}}} \right]$$

((B2))

$$ + \;{\text{3/2}}q\operatorname{Im} \left[ {\mathop {\lim }\limits_{{{\delta }_{2}} \to + 0} \int\limits_{1 + {{\delta }_{2}}}^\infty {\frac{{{{r}_{p}}({{x}_{1}})}}{{4{{l}_{1}}({{x}_{1}})}}{\text{exp}}( - 4\pi \sqrt {{{\varepsilon }_{1}}} {{l}_{1}}({{x}_{1}})d{\text{/}}\lambda ){\text{d}}{{x}_{1}}} } \right],$$

where

$${{r}_{p}}({{x}_{1}}) = \frac{{{{\varepsilon }_{2}}{{l}_{1}}({{x}_{1}}) - {{\varepsilon }_{1}}{{l}_{2}}({{x}_{1}})}}{{{{\varepsilon }_{2}}{{l}_{1}}({{x}_{1}}) + {{\varepsilon }_{1}}{{l}_{2}}({{x}_{1}})}},$$

$${{l}_{1}}({{x}_{1}}) = - i\sqrt {1 - \sqrt {{{x}_{1}}} } ,\quad {{l}_{2}}({{x}_{1}}) = - i\sqrt {{{\varepsilon }_{2}}{\text{/}}{{\varepsilon }_{1}} - \sqrt {{{x}_{1}}} } .$$

Then, we will make the following changes of variables in (B2): ${{t}_{1}}$ = $\sqrt {1 - \sqrt {{{x}_{1}}} }$ (${{x}_{1}}$ = ${{(1 - t_{1}^{2})}^{2}}$) in the second term and $t_{1}^{'}$ = $\sqrt {\sqrt {{{x}_{1}}} - 1}$ (${{x}_{1}}$ = ${{(t{{_{1}^{'}}^{2}} + 1)}^{2}}$) in the third term. As a result, we obtain the following expressions:

$${{b}_{ \bot }} = 1 + {\text{3/2}}q\operatorname{Im} \left[ {\mathop {\lim }\limits_{{{\delta }_{1}} \to - 0} \int\limits_1^{0 - {{\delta }_{1}}} { - i{{r}_{p}}({{t}_{1}})} } \right.$$

$$\left. {^{{^{{^{{^{{^{{}}}}}}}}}} \times \;\exp (4\pi \sqrt {{{\varepsilon }_{1}}} i{{t}_{1}}d{\text{/}}\lambda )(1 - t_{1}^{2}){\text{d}}{{t}_{1}}} \right]$$

$$ + \,{\text{3/2}}q{\text{Im}}\left[ {\mathop {\lim }\limits_{{{\delta }_{2}} \to + 0} \int\limits_{0 + {{\delta }_{2}}}^\infty {{{r}_{p}}{\text{(}}t_{1}^{'}{\text{)}}\exp ( - 4\pi \sqrt {{{\varepsilon }_{1}}} t_{1}^{'}d{\text{/}}\lambda )(t{{{_{1}^{'}}}^{2}}\, + \,1){\text{d}}t_{1}^{'}} } \right],$$

where

$${{r}_{p}}({{t}_{1}}) = \frac{{{{\varepsilon }_{2}}{{l}_{1}}({{t}_{1}}) - {{\varepsilon }_{1}}{{l}_{2}}({{t}_{1}})}}{{{{\varepsilon }_{2}}{{l}_{1}}({{t}_{1}}) + {{\varepsilon }_{1}}{{l}_{2}}({{t}_{1}})}},\quad {{l}_{1}}({{t}_{1}}) = - i{{t}_{1}},$$

$${{l}_{2}}({{t}_{1}}) = - i\sqrt {{{\varepsilon }_{2}}{\text{/}}{{\varepsilon }_{1}} - 1 + t_{1}^{2}} ,$$

$${{r}_{p}}(t_{1}^{'}) = \frac{{{{\varepsilon }_{2}}{{l}_{1}}(t_{1}^{'}) - {{\varepsilon }_{1}}{{l}_{2}}(t_{1}^{'})}}{{{{\varepsilon }_{2}}{{l}_{1}}(t_{1}^{'}) + {{\varepsilon }_{1}}{{l}_{2}}(t_{1}^{'})}},\quad {{l}_{1}}(t_{1}^{'}) = t_{1}^{'},$$

$${{l}_{2}}(t_{1}^{'}) = - i\sqrt {{{\varepsilon }_{2}}{\text{/}}{{\varepsilon }_{1}} - t{{{_{1}^{'}}}^{2}} - 1} .$$