Dissipative Tunneling of Electrons in Vertically Coupled Double Asymmetric InAs/GaAs(001) Quantum Dots

Abstract

We report the results of experimental studies of the photoelectric properties of a p–i–n GaAs photodiode with InAs/GaAs(001) double asymmetric quantum dots (DAQDs) grown by self-assembling in the metal–organic vapor-phase epitaxy process. Three peaks were observed in the dependence of the photocurrent on the reverse bias measured at monochromatic photoexcitation of the DAQDs at the wavelength corresponding to the energy of interband optical transitions between the ground hole and electron states in the larger quantum dots (QDs). These peaks were related to the tunneling of the photoexcited electrons between the QDs, including a dissipative one (with emission and absorption of optical phonons). The experimental results agree qualitatively with the theoretical field dependence of the probability of 1D dissipative tunneling between QDs.

APPENDIX

Let us consider (3) taking into account the interaction with two local phonon modes (${{\omega }_{{L1}}} = {{\omega }_{2}}$ and ${{\omega }_{{L2}}} = {{\omega }_{3}}$). For simplicity, we will assume this interaction to be sufficiently small; i.e., $\frac{C}{{\omega _{0}^{2}}}$ $\ll$ 1 and $\frac{C}{{\omega _{L}^{2}}}$ $\ll$ 1. In this case ζn = $\nu _{n}^{2}\sum\nolimits_{\alpha = 2}^N {\frac{{C_{\alpha }^{2}}}{{\omega _{\alpha }^{2}(\omega _{\alpha }^{2} + \nu _{n}^{2})}}}$, where νn = $\frac{{2\pi n}}{\beta }$ and β = $\frac{\hbar }{{kT}}$.

$${{\zeta }_{n}} = \nu _{n}^{2}\frac{{C_{2}^{2}}}{{\omega _{2}^{2}(\omega _{2}^{2} + \nu _{n}^{2})}} + \nu _{n}^{2}\frac{{C_{3}^{2}}}{{\omega _{3}^{2}(\omega _{3}^{2} + \nu _{n}^{2})}};$$

$${{\sin }^{2}}{{\nu }_{n}}{{\tau }_{0}} = \frac{1}{2}(1 - \cos 2{{\nu }_{n}}{{\tau }_{0}}).$$

As a result, the sum in the last term of expression (3) will be rewritten as U = U₁ – U₂, where

$$\begin{gathered} {{U}_{1}} \\ = \frac{1}{2}\sum\limits_{n = 1}^\infty \frac{1}{{\nu _{n}^{2}\left( {\nu _{n}^{2} + \omega _{0}^{2} + \nu _{n}^{2}\frac{{C_{2}^{2}}}{{\omega _{2}^{2}(\omega _{2}^{2} + \nu _{n}^{2})}}\nu _{n}^{2} + \frac{{C_{3}^{2}}}{{\omega _{3}^{2}(\omega _{3}^{2} + \nu _{n}^{2})}}} \right)}} \\ \\ {{U}_{2}} \\ = \frac{1}{2}\sum\limits_{n = 1}^\infty \frac{{{\text{cos}}2{{\nu }_{n}}{{\tau }_{0}}}}{{\nu _{n}^{2}\left( {\nu _{n}^{2} + \omega _{0}^{2} + \nu _{n}^{2}\frac{{C_{2}^{2}}}{{\omega _{2}^{2}(\omega _{2}^{2} + \nu _{n}^{2})}}\nu _{n}^{2} + \frac{{C_{3}^{2}}}{{\omega _{3}^{2}(\omega _{3}^{2} + \nu _{n}^{2})}}} \right)}}. \\ \end{gathered} $$

(A.1)

Denote $\nu _{2}^{2}$ = x and transform the expression in the denominator:

$$\begin{gathered} x[(x + \omega _{0}^{2})\omega _{2}^{2}\omega _{3}^{2}(x + \omega _{2}^{2})(x + \omega _{3}^{2}) \\ + \,xC_{2}^{2}\omega _{3}^{2}(x + \omega _{3}^{2}) + xC_{3}^{2}\omega _{2}^{2}(x + \omega _{2}^{2})] \\ \end{gathered} $$

$$\begin{gathered} = x[\omega _{2}^{4}\omega _{3}^{4}x + \omega _{2}^{2}\omega _{3}^{2}{{x}^{3}} + \omega _{3}^{2}\omega _{3}^{2}{{x}^{2}}(\omega _{2}^{2} + \omega _{3}^{2}) \\ + \,\,\omega _{2}^{4}\omega _{3}^{4}\omega _{0}^{2} + \omega _{2}^{2}\omega _{3}^{2}\omega _{0}^{2}{{x}^{2}} + \omega _{2}^{2}\omega _{3}^{2}\omega _{0}^{2}x(\omega _{2}^{2} + \omega _{3}^{2}) \\ + \,\,C_{2}^{2}\omega _{3}^{4}x + C_{2}^{2}\omega _{3}^{2}{{x}^{2}} + C_{3}^{2}\omega _{2}^{4}x + C_{3}^{2}\omega _{2}^{2}{{x}^{2}}] \\ \end{gathered} $$

$$\begin{gathered} = x[\omega _{2}^{2}\omega _{3}^{2}{{x}^{3}} + {{x}^{2}}\{ \omega _{2}^{2}\omega _{3}^{2}(\omega _{2}^{2} + \omega _{3}^{2}) + \omega _{0}^{2}\omega _{2}^{2}\omega _{3}^{2} \\ \, + C_{2}^{2}\omega _{3}^{2} + C_{3}^{2}\omega _{2}^{2}\} + x\{ \omega _{2}^{4}\omega _{3}^{4} + \omega _{0}^{2}\omega _{2}^{2}\omega _{3}^{2}(\omega _{2}^{2} + \omega _{3}^{2}) \\ \, + C_{2}^{2}\omega _{3}^{4} + C_{3}^{2}\omega _{2}^{4}\} + \omega _{0}^{2}\omega _{2}^{4}\omega _{3}^{4}] \\ \end{gathered} $$

$$\begin{gathered} = x\omega _{2}^{2}\omega _{3}^{2}\left[ {{{x}^{3}} + {{x}^{2}}\left\{ {\omega _{2}^{2} + \omega _{3}^{2} + \omega _{0}^{2} + \frac{{C_{2}^{2}}}{{\omega _{2}^{2}}} + \frac{{C_{3}^{2}}}{{\omega _{3}^{2}}}} \right\}} \right. \\ \left. { + \,x\left\{ {\omega _{2}^{2}\omega _{3}^{3} + \omega _{0}^{2}(\omega _{2}^{2} + \omega _{3}^{2}) + \frac{{C_{2}^{2}\omega _{3}^{2}}}{{\omega _{2}^{2}}} + \frac{{C_{3}^{2}\omega _{2}^{2}}}{{\omega _{3}^{2}}}} \right\} + \omega _{0}^{2}\omega _{2}^{2}\omega _{3}^{2}} \right]. \\ \end{gathered} $$

We introduce the following notation: A = $\omega _{2}^{2}$ + $\omega _{3}^{2}$ + $\omega _{0}^{2}$ + $\frac{{C_{2}^{2}}}{{\omega _{2}^{2}}}$ + $\frac{{C_{3}^{2}}}{{\omega _{3}^{2}}}$,

$${{B}_{\omega }} = \omega _{2}^{2}\omega _{3}^{2} + \omega _{0}^{2}(\omega _{2}^{2} + \omega _{3}^{2}) + \frac{{C_{2}^{2}\omega _{3}^{2}}}{{\omega _{2}^{2}}} + \frac{{C_{3}^{2}\omega _{2}^{2}}}{{\omega _{3}^{2}}},$$

$$C = \omega _{0}^{2}\omega _{2}^{2}\omega _{3}^{2}.$$

The expression in the denominator of the first term in (A.1) then takes the form

$$\begin{gathered} x\omega _{2}^{2}\omega _{3}^{2}[\underbrace {{{x}^{3}} + A{{x}^{2}} + {{B}_{\omega }}x + C}_{ = 0}] \\ = x\omega _{0}^{2}\omega _{3}^{2}(x - {{x}_{1}})(x - {{x}_{2}})(x - {{x}_{3}}). \\ \end{gathered} $$

Let us designate Q = $\frac{{{{A}^{2}} - 3{{B}_{\omega }}}}{9}$; R = $\frac{{2{{A}^{3}} - 9A{{B}_{\omega }} + 27C}}{{54}}$, S = Q³ – R²; Φ = $\frac{1}{3}\arccos \left( {\frac{R}{{\sqrt {{{Q}^{3}}} }}} \right)$. If S > 0, then

$$\begin{gathered} {{x}_{1}} = - 2\sqrt Q \cos (\Phi ) - \frac{A}{3}, \\ {{x}_{2}} = - 2\sqrt Q \cos \left( {\Phi + \frac{2}{3}\pi } \right) - \frac{A}{3}, \\ {{x}_{3}} = - 2\sqrt Q \cos \left( {\Phi - \frac{2}{3}\pi } \right) - \frac{A}{3}. \\ \end{gathered} $$

(A.2)

The first sum in (A.1) is converted to the form

$$U_{1}^{{}} = \frac{1}{2}\sum\limits_{n = 1}^\infty \frac{{\omega _{2}^{2}\omega _{3}^{2}(\omega _{2}^{2} + \nu _{n}^{2})(\omega _{3}^{2} + \nu _{n}^{2})}}{{\nu _{n}^{2}\omega _{2}^{2}\omega _{3}^{2}(\nu _{n}^{2} - {{x}_{1}})(\nu _{n}^{2} - {{x}_{2}})(\nu _{n}^{2} - {{x}_{3}})}}.$$

(A.3)

The last expression in (A.1) is divided into simple fractions:

$$\begin{gathered} \frac{{{{\beta }_{0}}}}{x} + \frac{\gamma }{{x - {{x}_{1}}}} + \frac{\varphi }{{x - {{x}_{2}}}} + \frac{\Delta }{{x - {{x}_{3}}}} \\ = \frac{{{{x}^{2}} + x(\omega _{2}^{2} + \omega _{3}^{2}) + \omega _{2}^{2}\omega _{3}^{2}}}{{x(x - {{x}_{1}})(x - {{x}_{2}})(x - {{x}_{3}})}}, \\ \end{gathered} $$

where

$${{\beta }_{0}} = - \frac{{\omega _{2}^{2}\omega _{3}^{2}}}{{{{x}_{1}}{{x}_{2}}{{x}_{3}}}},$$

$$\begin{gathered} \Delta = \frac{{x_{3}^{2}}}{{({{x}_{3}} - {{x}_{2}})({{x}_{1}} - {{x}_{3}})}} \\ \times \,\,\left\{ {\frac{{\omega _{2}^{2}\omega _{3}^{2}}}{{{{x}_{1}}{{x}_{2}}{{x}_{3}}}}\left( {\frac{{{{x}_{1}}{{x}_{2}} + {{x}_{1}}{{x}_{3}} + {{x}_{2}}{{x}_{3}}}}{{{{x}_{2}}{{x}_{3}}}} - 1} \right)} \right. \\ + \,\,\frac{{\omega _{2}^{2} + \omega _{3}^{2}}}{{{{x}_{2}}{{x}_{3}}}} - \frac{1}{{{{x}_{3}}}}\left( {1 + \frac{{\omega _{2}^{2}\omega _{3}^{2}}}{{{{x}_{1}}{{x}_{2}}{{x}_{3}}}}\left[ {\frac{{{{x}_{1}}{{x}_{2}} + {{x}_{1}}{{x}_{3}} + {{x}_{2}}{{x}_{3}}}}{{{{x}_{2}}{{x}_{3}}}}} \right.} \right. \\ \left. {\left. {\left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}} + \,\,({{x}_{2}} + {{x}_{3}} - {{x}_{1}})} \right]} \right) + \frac{{(\omega _{2}^{2} + \omega _{3}^{2})({{x}_{2}} + {{x}_{3}})}}{{{{x}_{2}}{{x}_{3}}}}} \right\}, \\ \end{gathered} $$

$$\begin{gathered} \varphi = \frac{{{{x}_{2}}}}{{{{x}_{3}}({{x}_{2}} - {{x}_{1}})}}\left\{ {\Delta \frac{{{{x}_{2}}}}{{{{x}_{3}}}}({{x}_{1}} - {{x}_{3}}) - 1} \right. \\ - \,\,\frac{{\omega _{2}^{2}\omega _{3}^{2}}}{{{{x}_{1}}{{x}_{2}}{{x}_{3}}}}({{x}_{2}} + {{x}_{3}} - {{x}_{1}}) - \frac{{{{x}_{2}} + {{x}_{3}}}}{{{{x}_{2}}{{x}_{3}}}} \\ \left. { \times \,\,\left\{ {\omega _{2}^{2} + \omega _{3}^{2} + \frac{{\omega _{2}^{2}\omega _{3}^{2}}}{{{{x}_{1}}{{x}_{2}}{{x}_{3}}}}({{x}_{1}}{{x}_{2}} + {{x}_{1}}{{x}_{3}} + {{x}_{2}}{{x}_{3}})} \right\}} \right\}, \\ \end{gathered} $$

(A.4)

$$\begin{gathered} \gamma = \frac{1}{{{{x}_{2}}{{x}_{3}}}}\{ \omega _{2}^{2} + \omega _{3}^{2} - \Delta {{x}_{1}}{{x}_{2}} - {{\varphi }_{1}}{{x}_{3}} \\ - \,\,{{\beta }_{0}}({{x}_{2}}{{x}_{3}} + {{x}_{1}}({{x}_{2}} + {{x}_{3}}))\} ,\quad {{\nu }_{n}} = \frac{{2\pi n}}{\beta }. \\ \end{gathered} $$

Eventually, U₁ is converted to the form

$${{U}_{1}} = \frac{1}{2}\sum\limits_{n = 1}^\infty {\left( {\frac{{{{\beta }_{0}}}}{{\nu _{n}^{2}}} + \frac{\gamma }{{\nu _{n}^{2} - {{x}_{1}}}} + \frac{\varphi }{{\nu _{n}^{2} - {{x}_{2}}}} + \frac{\Delta }{{\nu _{n}^{2} - {{x}_{3}}}}} \right).} $$

$$\begin{gathered} \sum\limits_{n = 1}^\infty {\frac{{{{\beta }_{0}}}}{{\nu _{n}^{2}}} = {{\beta }_{0}}\sum\limits_{n = 1}^\infty {\frac{{{{\beta }^{2}}}}{{4{{\pi }^{2}}{{n}^{2}}}}} } \\ = {{\beta }_{0}}\frac{{{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\sum\limits_{n = 1}^\infty {\frac{1}{{{{n}^{2}}}} = {{\beta }_{0}}\frac{{{{\beta }^{2}}}}{{24}}} , \\ \end{gathered} $$

$${{x}_{1}} = - 2\sqrt Q \cos \phi - \frac{A}{3} = - {{x}_{{10}}} = - \left( {2\sqrt Q \cos \phi + \frac{A}{3}} \right).$$

If x₁ < 0;

$$\begin{gathered} \sum\limits_{n = 1}^\infty {\frac{\gamma }{{\nu _{n}^{2} + {{x}_{{10}}}}}} = \sum\limits_{n = 1}^\infty {\frac{\gamma }{{\frac{{4{{\pi }^{2}}{{n}^{2}}}}{{{{\beta }^{2}}}} + {{x}_{{10}}}}}} \\ = \frac{{\gamma {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\sum\limits_{n = 1}^\infty {\frac{1}{{{{n}^{2}} + \frac{{{{x}_{{10}}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}}}} \\ \end{gathered} $$

$$\begin{gathered} = \frac{{\gamma {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{1}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}{\text{cot}}\left( {\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right)} \right]; \\ \tilde {x}_{{10}}^{2} = \frac{{{{x}_{{10}}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}, \\ \end{gathered} $$

$$\begin{gathered} {{x}_{2}} = - 2\sqrt Q \cos \left( {\Phi + \frac{2}{3}\pi } \right) - \frac{A}{3} = - {{x}_{{20}}}, \\ \tilde {x}_{{20}}^{2} = \frac{{{{x}_{{20}}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}, \\ \end{gathered} $$

$$\begin{gathered} {{x}_{3}} = - 2\sqrt Q \cos \left( {\Phi - \frac{2}{3}\pi } \right) - \frac{A}{3} = - {{x}_{{30}}}, \\ \tilde {x}_{{30}}^{2} = \frac{{{{x}_{{30}}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}. \\ \end{gathered} $$

If x₁ > 0, x₂ > 0, x₃ > 0:

$$\begin{gathered} {{U}_{1}} = \frac{1}{2}\left\{ {{{\beta }_{0}}\frac{{{{\beta }^{2}}}}{{24}} + \frac{{\gamma {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{1}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cot \left( {\frac{{\sqrt {{{x}_{1}}} \beta }}{2}} \right)} \right]} \right. \\ + \frac{{\varphi {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{2}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cot \left( {\frac{{\sqrt {{{x}_{2}}} \beta }}{2}} \right)} \right] \\ \left. { + \frac{{\Delta {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{3}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{3}}} \beta }}\cot \left( {\frac{{\sqrt {{{x}_{3}}} \beta }}{2}} \right)} \right]} \right\}. \\ \end{gathered} $$

(A.5)

Let us move on to calculation of U₂:

$$\begin{gathered} {{U}_{2}} = \frac{1}{2}\sum\limits_{n = 1}^\infty {\left( {\frac{{{{\beta }_{0}}\cos 2{{\nu }_{n}}{{T}_{0}}}}{{\nu _{n}^{2}}} + \frac{{\gamma \cos 2{{\nu }_{n}}{{T}_{0}}}}{{\nu _{n}^{2} - {{x}_{1}}}}} \right.} \\ \left. { + \frac{{\varphi \cos 2{{\nu }_{n}}{{T}_{0}}}}{{\nu _{n}^{2} - {{x}_{2}}}} + \frac{{\Delta \cos 2{{\nu }_{n}}{{T}_{0}}}}{{\nu _{n}^{2} - {{x}_{3}}}}} \right), \\ \end{gathered} $$

$$\begin{gathered} \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{{{{\beta }_{0}}{{{\cos }}^{2}}\frac{{2\pi {{T}_{0}}n}}{\beta }}}{{\frac{{4{{\pi }^{2}}{{n}^{2}}}}{{{{\beta }^{2}}}}}} = \frac{1}{2}\left[ {\frac{{{{\beta }^{2}}{{\beta }_{0}}}}{{4{{\pi }^{2}}}}\sum\limits_{n = 1}^\infty {\frac{{{{{\cos }}^{2}}\frac{{2\pi {{T}_{0}}}}{\beta }n}}{{{{n}^{2}}}}} } \right]} \\ = \frac{1}{2}\left[ {\frac{{{{\beta }^{2}}{{\beta }_{0}}}}{{4{{\pi }^{2}}}}\frac{1}{{12}}\left( {3\frac{{{{{(4\pi {{T}_{0}})}}^{2}}}}{\beta } - 6\pi \frac{{4\pi {{T}_{0}}}}{\beta } + 2{{\pi }^{2}}} \right)} \right] \\ + \,\,\frac{1}{2}\gamma \sum\limits_{n = 1}^\infty {\frac{{\cos \frac{{4\pi {{T}_{0}}n}}{\beta }}}{{\frac{{4{{\pi }^{2}}{{n}^{2}}}}{{{{\beta }^{2}}}} - {{x}_{1}}}}} = \frac{1}{2}\left[ {\frac{{{{\beta }^{2}}\gamma }}{{4{{\pi }^{2}}}}\sum\limits_{n = 1}^\infty {\frac{{\cos \frac{{4\pi {{T}_{0}}}}{\beta }n}}{{{{n}^{2}} - \frac{{{{x}_{1}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}}}} } \right] \\ = \frac{1}{2}\left[ {\frac{{{{\beta }^{2}}\gamma }}{{4{{\pi }^{2}}}}\left\{ {\frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cos \left[ {\left( {\pi - \frac{{4\pi {{T}_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right]} \right.} \right. \\ \left. {\left. { \times \,{\text{cosec}}\frac{{\sqrt {{{x}_{1}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}}} \right\}} \right]. \\ \end{gathered} $$

At ${{x}_{1}},{{x}_{2}},{{x}_{3}} > 0$,

$$\begin{gathered} {{U}_{2}} = \frac{1}{2}\left\{ {\frac{{{{\beta }_{0}}{{\beta }^{2}}}}{{48}}\left( {3\frac{{{{{(4\pi {{T}_{0}})}}^{2}}}}{\beta } - \frac{{24{{\pi }^{2}}{{T}_{0}}}}{\beta } + 2{{\pi }^{2}}} \right)} \right. \\ + \,\,\frac{{\gamma {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left\{ {\frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cos \left[ {\left( {\pi - \frac{{4\pi {{T}_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right]{\text{cosec}}\frac{{\sqrt {{{x}_{1}}} \beta }}{2}} \right\} \\ \end{gathered} $$

$$\begin{gathered} + \frac{{\varphi {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left\{ {\frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{2}}} \beta }}\cos \left[ {\left( {\pi - \frac{{4\pi {{T}_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{2}}} \beta }}{{2\pi }}} \right]} \right. \\ \left. { \times \,{\text{cosec}}\frac{{\sqrt {{{x}_{2}}} \beta }}{2} - \frac{{2{{\pi }^{2}}}}{{{{x}_{2}}{{\beta }^{2}}}}} \right\}. \\ \end{gathered} $$

(A.6)

$$\begin{gathered} + \frac{{\Delta {{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left\{ {\frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{3}}} \beta }}} \right.\cos \left[ {\left( {\pi - \frac{{4\pi {{T}_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{3}}} \beta }}{{2\pi }}} \right] \\ \left. {\left. { \times \,{\text{cosec}}\frac{{\sqrt {{{x}_{3}}} \beta }}{2} - \frac{{2{{\pi }^{2}}}}{{{{x}_{3}}{{\beta }^{2}}}}} \right\}} \right\}. \\ \end{gathered} $$

The semiclassical action, with allowance for two promoting modes, reduces to an expression of the form

$$\begin{gathered} {{S}_{B}}2\omega _{0}^{2}(a + b)a{{\tau }_{0}} - \frac{2}{\beta }\omega _{0}^{2}{{(a + b)}^{2}}\tau _{0}^{2} \\ - \frac{4}{\beta }\omega _{0}^{4}{{(a + b)}^{2}}\{ {{U}_{1}} + {{U}_{2}}\} , \\ \end{gathered} $$

where

$$\begin{gathered} {{\tau }_{0}} = \frac{1}{{2{{\omega }_{0}}}}{\text{arcsinh}}\left[ {\frac{{b - a}}{{b + a}}\sinh \frac{{{{\omega }_{0}}\beta }}{4}} \right] + \frac{\beta }{4} \\ = \frac{1}{{2{{\omega }_{0}}}}{\text{arcsinh}}\left[ {\frac{{\frac{b}{a} - 1}}{{\frac{b}{a} + 1}}\sinh \frac{{{{\omega }_{0}}\beta }}{4}} \right] + \frac{\beta }{4} \\ \end{gathered} $$

or

$$\tau _{0}^{*} = {{\tau }_{0}}{{\omega }_{0}} = \frac{1}{2}{\text{arcsinh}}\left[ {\frac{{b{\text{*}} - 1}}{{b{\text{*}} + 1}}\sinh \beta {\text{*}}} \right] + \beta {\text{*}};$$

$$\tau _{0}^{*} = \tau {{\omega }_{0}};\quad \beta \text{*} = \frac{{{{\omega }_{0}}\beta }}{4}.$$

Finally, the renormalized expression for the 1D semiclassical instanton action, taking into account two local modes of the medium thermostat, takes the form

$$\begin{gathered} {{{\tilde {S}}}_{{10}}} = \frac{{{{S}_{{10}}}}}{{{{\omega }_{0}}{{a}^{2}}}} = 2(b{\text{*}} + 1)\tau _{0}^{*} - \frac{1}{{2\beta {\text{*}}}}{{(b{\text{*}} + 1)}^{2}}\tau _{0}^{{*2}} \\ - \,\,\frac{{{{{(b{\text{*}} + 1)}}^{2}}}}{{\beta {\text{*}}}}\left\{ {\frac{1}{2}\left[ {{{\beta }_{0}}\omega _{0}^{2}{{{\left( {\frac{{\beta {{\omega }_{0}}}}{4}} \right)}}^{2}}\frac{2}{3} + 4\frac{{\gamma \omega _{0}^{2}{{{\left( {\frac{{\beta {{\omega }_{0}}}}{4}} \right)}}^{2}}}}{{{{\pi }^{2}}}}} \right.} \right. \\ \times \,\,\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{1}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cot \left( {\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right)} \right] + 4\frac{{\varphi \omega _{0}^{2}\beta {{{\text{*}}}^{2}}}}{{{{\pi }^{2}}}} \\ \times \,\,\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{2}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{2}}} \beta }}\cot \left( {\frac{{\sqrt {{{x}_{2}}} \beta }}{{2\pi }}} \right)} \right] \\ \end{gathered} $$

$$\begin{gathered} \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} + \,\,4\frac{{\Delta \omega _{0}^{2}\beta {{{\text{*}}}^{2}}}}{{{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{2{{x}_{3}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{3}}} \beta }}\cot \left( {\frac{{\sqrt {{{x}_{3}}} \beta }}{{2\pi }}} \right)} \right]} \right] \\ - \,\,\frac{1}{2}\left[ {{{\beta }_{0}}\omega _{0}^{2}{{{\left( {\frac{{\beta {{\omega }_{0}}}}{4}} \right)}}^{2}}\frac{1}{3}\left( {3{{{\left( {\frac{{4\pi {{\tau }_{0}}{{\omega }_{0}}}}{{\beta {{\omega }_{0}}}}} \right)}}^{2}} - \frac{{6{{\pi }^{2}}{{\tau }_{0}}{{\omega }_{0}}4}}{{\beta {{\omega }_{0}}}} + 2{{\pi }^{2}}} \right)} \right. \\ + \,\,\frac{{4\gamma \omega _{0}^{2}{{{\left( {\frac{{\beta {{\omega }_{0}}}}{4}} \right)}}^{2}}}}{{{{\pi }^{2}}}}\left\{ {\frac{{{{\omega }_{0}}{{\pi }^{2}}4}}{{4\sqrt {{{x}_{1}}} \beta {{\omega }_{0}}}}\cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}{{\omega }_{0}}}}{{\beta {{\omega }_{0}}}}} \right)\frac{{\sqrt {{{x}_{1}}} 2\beta {{\omega }_{0}}}}{{{{\omega }_{0}}\pi 4}}} \right]} \right. \\ \left. { \times \,\,{\text{cosec}}\frac{{2\sqrt {x1} }}{{{{\omega }_{0}}}}\frac{{{{\beta }_{0}}{{\omega }_{0}}}}{4} + \frac{{\omega _{0}^{2}{{\pi }^{2}}4}}{{8{{x}_{1}}\beta {{\omega }_{0}}}}} \right\} + \frac{{4\phi \omega _{0}^{2}\beta {{{\text{*}}}^{2}}}}{{{{\pi }^{2}}}} \\ \end{gathered} $$

(A.7)

$$\begin{gathered} \times \,\,\left\{ {\frac{{{{\omega }_{0}}{{\pi }^{2}}4}}{{4\sqrt {{{x}_{2}}} \beta {\text{*}}}}} \right.\cos \left[ {\left( {\pi - \frac{{4\pi \tau _{0}^{*}{{\omega }_{0}}}}{{\beta {\text{*}}}}} \right)\frac{{\sqrt {{{x}_{2}}} 2\beta {\text{*}}}}{{{{\omega }_{0}}\pi }}} \right] \\ \,\left. { \times \,\,{\text{cosec}}\frac{{2\sqrt {{{x}_{2}}} }}{{{{\omega }_{0}}}}\beta {\text{*}} + \frac{{\omega _{0}^{2}{{\pi }^{2}}}}{{8{{x}_{{20}}}\beta {{{\text{*}}}^{2}}}}} \right\} + \frac{{4\Delta \omega _{0}^{2}\beta {{{\text{*}}}^{2}}}}{{{{\pi }^{2}}}} \\ \times \,\,\left\{ {\frac{{{{\omega }_{0}}{{\pi }^{2}}4}}{{4\sqrt {{{x}_{3}}} \beta {\text{*}}}}\cos \left[ {\left( {\pi - \frac{{4\pi \tau _{0}^{*}{{\omega }_{0}}}}{{\beta {\text{*}}}}} \right)\frac{{\sqrt {{{x}_{3}}} 2\beta {\text{*}}}}{{{{\omega }_{0}}\pi }}} \right]} \right. \\ \left. {\,\left. { \times \,\,{\text{cosec}}\frac{{2\sqrt {{{x}_{3}}} }}{{{{\omega }_{0}}}}\beta {\text{*}} + \frac{{\omega _{0}^{2}{{\pi }^{2}}}}{{8{{x}_{3}}\beta {{{\text{*}}}^{2}}}}} \right\}} \right]. \\ \end{gathered} $$

Let us proceed to the calculation of the pre-exponential factor taking into account two promoting phonon modes:

$$B = \frac{{2\omega _{0}^{2}{{{(a + b)}}^{2}}}}{{{{{(2\pi \beta )}}^{{\frac{1}{2}}}}}}\frac{{\sum\limits_{n = - \infty }^\infty {\frac{{{{{\sin }}^{2}}{{\nu }_{n}}{{\tau }_{0}}}}{{{{\lambda }_{{0n}}}}}} }}{{{{{\left[ {\sum\limits_{n = - \infty }^\infty {\frac{{\cos 2{{\nu }_{n}}{{\tau }_{0}}}}{{{{\lambda }_{{0n}}}}}} } \right]}}^{{\frac{1}{2}}}}}},$$

(A.8)

where

$${{\lambda }_{{0n}}} = \nu _{n}^{2} + \omega _{0}^{2} + {{\zeta }_{n}},$$

$$\begin{gathered} \sum\limits_{ - \infty }^\infty {\frac{{{{{\sin }}^{2}}{{\nu }_{n}}{{\tau }_{0}} = \frac{1}{2}(1 - \cos 2{{\nu }_{n}}{{\tau }_{0}})}}{{\nu _{n}^{2} + \omega _{0}^{2} + \frac{{\nu _{n}^{2}C_{2}^{2}}}{{\omega _{2}^{2}(\omega _{2}^{2} + \nu _{n}^{2})}} + \frac{{\nu _{n}^{2}C_{3}^{2}}}{{\omega _{3}^{2}(\omega _{3}^{2} + \nu _{n}^{2})}}}}} \\ = \frac{1}{2}\sum\limits_{n = - \infty }^\infty {\frac{{(1 - \cos 2{{\nu }_{n}}{{\tau }_{0}})\omega _{2}^{2}\omega _{3}^{2}(\omega _{2}^{2} + \nu _{n}^{2})(\omega _{3}^{2} + \nu _{n}^{2})}}{{(\omega _{0}^{2} + \nu _{n}^{2})\omega _{2}^{2}\omega _{3}^{2}(\omega _{2}^{2} + \nu _{n}^{2})(\omega _{3}^{2} + \nu _{n}^{2}) + \nu _{n}^{2}C_{2}^{2}\omega _{3}^{2}(\omega _{3}^{2} + \nu _{n}^{2}) + \nu _{2}^{2}C_{3}^{2}\omega _{3}^{2}(\omega _{3}^{2} + \nu _{n}^{2})}}} , \\ \end{gathered} $$

(A.9)

$$x = \nu _{n}^{2} = \frac{1}{2}\sum\limits_{n = - \infty }^\infty {\frac{{(1 - \cos 2{{\nu }_{n}}{{\tau }_{0}})(\omega _{2}^{2} + \nu _{n}^{2})(\omega _{3}^{2} + \nu _{n}^{2})}}{{{{x}^{3}} + A{{x}^{2}} + {{B}_{\omega }}x + C}},} $$

where the notation A = $\omega _{2}^{2}$ + $\omega _{3}^{2}$ + $\omega _{0}^{2}$ + $\frac{{C_{2}^{2}}}{{\omega _{2}^{2}}}$ + $\frac{{C_{3}^{2}}}{{\omega _{3}^{2}}}$,

$${{B}_{\omega }} = \omega _{2}^{2}\omega _{3}^{2} + \omega _{0}^{2}(\omega _{2}^{2} + \omega _{3}^{2})\frac{{C_{2}^{2}\omega _{3}^{2}}}{{\omega _{2}^{2}}} + \frac{{C_{3}^{2}\omega _{2}^{2}}}{{\omega _{3}^{2}}},$$

$$C = \omega _{0}^{2}\omega _{2}^{2}\omega _{3}^{2},$$

we also denote

$$Q = \frac{{{{A}^{3}} - 3{{B}_{\omega }}}}{9};\quad R = \frac{{2{{A}^{3}} - 9A{{B}_{\omega }} + 27C}}{{54}};$$

$$S = {{Q}^{3}} - {{R}^{2}};\quad \Phi = \frac{1}{3}\arccos \left( {\frac{R}{{\sqrt {{{Q}^{3}}} }}} \right).$$

At S > 0,

$${{x}_{1}} = - 2\sqrt Q \cos (\Phi ) - \frac{A}{3},$$

$${{x}_{2}} = - 2\sqrt Q \cos \left( {\Phi + \frac{2}{3}\pi } \right) - \frac{A}{3},$$

$${{x}_{3}} = - 2\sqrt Q \cos \left( {\Phi - \frac{2}{3}\pi } \right) - \frac{A}{3}.$$

Let us expand the denominator of relation (A.9):

$$\begin{gathered} \frac{1}{2}\sum\limits_{n = - \infty }^\infty {\frac{{(\omega _{2}^{2} + \nu _{n}^{2})(\omega _{3}^{2} + \nu _{n}^{2})}}{{(\nu _{n}^{2} - {{x}_{1}})(\nu _{n}^{2} - {{x}_{2}})(\nu _{n}^{2} - {{x}_{3}})}}} \\ = \frac{D}{{\nu _{n}^{2} - {{x}_{1}}}} + \frac{E}{{\nu _{n}^{2} - {{x}_{2}}}} + \frac{F}{{\nu _{n}^{2} - {{x}_{3}}}}, \\ \end{gathered} $$

$$F = \frac{{(\omega _{2}^{2} + \omega _{3}^{2} + {{x}_{2}} + {{x}_{3}})[{{x}_{2}}{{x}_{3}}({{x}_{1}} + {{x}_{3}}) - {{x}_{1}}{{x}_{3}}({{x}_{2}} + {{x}_{3}})] + ({{x}_{2}} - {{x}_{1}})[({{x}_{2}} + {{x}_{2}})\omega _{2}^{2}\omega _{3}^{2} + {{x}_{2}}{{x}_{3}}(\omega _{2}^{2} + \omega _{3}^{2})]}}{{({{x}_{2}} - {{x}_{1}})[{{x}_{1}}{{x}_{2}}({{x}_{2}} + {{x}_{3}}) - {{x}_{2}}{{x}_{3}}({{x}_{1}} + {{x}_{2}})] - ({{x}_{1}} - {{x}_{3}})[{{x}_{2}}{{x}_{3}}({{x}_{1}} + {{x}_{3}}) - {{x}_{1}}{{x}_{3}}({{x}_{2}} + {{x}_{3}})]}},$$

$$E = \frac{{\omega _{2}^{2} + \omega _{3}^{2} + {{x}_{2}} + {{x}_{3}} + F({{x}_{1}} - {{x}_{3}})}}{{{{x}_{2}} - {{x}_{1}}}};$$

$$D = - \frac{{\omega _{2}^{2} + \omega _{3}^{2} + E({{x}_{1}} + {{x}_{3}}) + F({{x}_{1}} + {{x}_{2}})}}{{{{x}_{2}} + {{x}_{3}}}},$$

$$\begin{gathered} \frac{1}{2}\sum\limits_{n = - \infty }^\infty {\frac{D}{{\nu _{n}^{2} - {{x}_{1}}}}} = \frac{D}{2}\sum\limits_{n = - \infty }^\infty {\frac{1}{{\frac{{4{{\pi }^{2}}{{n}^{2}}}}{{{{\beta }^{2}}}} - {{x}_{1}}}}} = \frac{1}{2}\frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\sum\limits_{n = - \infty }^\infty {\frac{1}{{{{n}^{2}} - \frac{{{{x}_{1}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}}}} \\ = \frac{1}{2}\frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}} + 2\sum\limits_{n = - \infty }^\infty {\frac{1}{{{{n}^{2}} - \frac{{{{x}_{1}}{{\beta }^{2}}}}{{4{{\pi }^{2}}}}}}} } \right] \\ \end{gathered} $$

(at x₁ > 0):

$$ = \frac{1}{2}\frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{2{{\pi }^{2}}}}{{{{x}_{{10}}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cot \frac{{\sqrt {{{x}_{1}}} \beta }}{2}} \right\}} \right].$$

The amount containing cos2ν_nτ₀ yields in this case

$$\begin{gathered} = - \frac{1}{2}\frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)} \right.} \right.} \right. \\ \left. {\left. {\left. { \times \,\,\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right]{\text{cosec}}\frac{{\sqrt {{{x}_{1}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{1}}\beta }}} \right\}} \right]. \\ \end{gathered} $$

(A.10)

Eventually, the dimensionless preexponential factor is determined by sums of two types:

$$\tilde {B} = \frac{B}{{{{a}^{2}}{{\omega }^{{\frac{2}{3}}}}}} = \frac{{2\omega _{0}^{2}{{{\left( {\frac{b}{a} + 1} \right)}}^{2}}}}{{{{{(2\pi \beta )}}^{{\frac{1}{2}}}}}}\frac{{{{V}_{1}}}}{{{{{({{V}_{2}})}}^{{\frac{1}{2}}}}}},$$

$$\begin{gathered} {{V}_{1}} = \sum\limits_{n = - \infty }^\infty \frac{{{\text{si}}{{{\text{n}}}^{2}}{{\nu }_{n}}{{\tau }_{0}}}}{{{{\lambda }_{{0n}}}}} \\ = \frac{1}{2}\frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{2{{\pi }^{2}}}}{{{{x}_{{10}}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}{\text{cot}}\frac{{\sqrt {{{x}_{1}}\beta } }}{2}} \right\}} \right] \\ \end{gathered} $$

$$\begin{gathered} + \,\,\frac{1}{2}\frac{{E{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{2}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{2{{\pi }^{2}}}}{{{{x}_{{20}}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{2}}} \beta }}\cot \frac{{\sqrt {{{x}_{2}}} \beta }}{2}} \right\}} \right] \\ + \,\,\frac{1}{2}\frac{{F{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{3}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{2{{\pi }^{2}}}}{{{{x}_{{30}}}{{\beta }^{2}}}} - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{3}}} \beta }}\cot \frac{{\sqrt {{{x}_{3}}} \beta }}{2}} \right\}} \right] \\ - \,\,\frac{1}{2}\frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}\cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right]} \right.} \right. \\ \left. {\left. { \times \,\,{\text{cosec}}\frac{{\sqrt {{{x}_{1}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}}} \right\}} \right] \\ \end{gathered} $$

$$\begin{gathered} - \,\,\frac{1}{2}\frac{{E{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{2}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{2}}} \beta }}\cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{2}}} \beta }}{{2\pi }}} \right]} \right.} \right. \\ \left. {\left. { \times \,\,{\text{cosec}}\frac{{\sqrt {{{x}_{2}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{2}}{{\beta }^{2}}}}} \right\}} \right] - \frac{1}{2}\frac{{F{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{3}}{{\beta }^{2}}}}} \right. \\ + \,\,2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{3}}} \beta }}} \right.\cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{3}}} \beta }}{{2\pi }}} \right] \\ \left. {\left. { \times \,\,{\text{cosec}}\frac{{\sqrt {{{x}_{3}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{3}}{{\beta }^{2}}}}} \right\}} \right]; \\ \end{gathered} $$

$$\begin{gathered} {{V}_{2}} = \sum\limits_{n = - \infty }^\infty \frac{{{\text{cos}}2{{\nu }_{n}}{{\tau }_{0}}}}{{{{\lambda }_{{0n}}}}} \\ = \frac{{D{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{1}}} \beta }}{\text{cos}}\left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{1}}} \beta }}{{2\pi }}} \right]} \right.} \right. \\ \left. {\left. { \times \,{\text{cosec}}\frac{{\sqrt {{{x}_{1}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{1}}{{\beta }^{2}}}}} \right\}} \right] + \frac{{E{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{2}}{{\beta }^{2}}}}} \right. \\ \end{gathered} $$

$$\begin{gathered} + 2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{2}}} \beta }}} \right.\cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{2}}} \beta }}{{2\pi }}} \right] \\ \left. {\left. { \times \,{\text{cosec}}\frac{{\sqrt {{{x}_{2}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{2}}{{\beta }^{2}}}}} \right\}} \right] \\ + \frac{{F{{\beta }^{2}}}}{{4{{\pi }^{2}}}}\left[ { - \frac{{4{{\pi }^{2}}}}{{{{x}_{3}}{{\beta }^{2}}}} + 2\left\{ { - \frac{{{{\pi }^{2}}}}{{\sqrt {{{x}_{3}}} \beta }}} \right.} \right. \\ \left. {\left. { \times \cos \left[ {\left( {\pi - \frac{{4\pi {{\tau }_{0}}}}{\beta }} \right)\frac{{\sqrt {{{x}_{3}}} \beta }}{{2\pi }}} \right]{\text{cosec}}\frac{{\sqrt {{{x}_{3}}} \beta }}{2} + \frac{{2{{\pi }^{2}}}}{{{{x}_{3}}{{\beta }^{2}}}}} \right\}} \right]. \\ \end{gathered} $$

(A.11)

As a result, the expressions for the probability of 1D tunneling transfer:

$$\Gamma = B\exp ( - S).$$

(A.12)