Lead position and lead-ring coupling effects on the spin-dependent transport properties in a two-dimensional network of quantum nanorings in the presence of Rashba spin–orbit interaction

Abstract

The effects of lead positions and lead-ring coupling regimes are investigated for spin-related transport through a two-dimensional network of quantum nanorings (2DNQRs) considering Rashba spin–orbit interaction (RSOI) and magnetic flux. A matrix representation of the transmission and reflection coefficients through a single ring connected to the arbitrary number of leads has been introduced. As a specific example of 2DNQRs, the conductance, spin polarization and system efficiency are obtained via a triangular network of quantum rings (TNQRs). TNQRs are completely opaque or transparent versus RSOI strength and wave vector (k) of the incident electron. The periodicity of these functions along the k axis depends on the lead positions and is independent of lead-ring coupling. Also, the symmetric geometry and strong lead-ring coupling regime significantly improve the performance of the system as a multipurpose spintronic device (i.e., a perfect spin filter, spin splitter, spin switching, Stern–Gerlach device and an electronic switching device).

Appendix

Here we calculate the details of $M_{i}^{\sigma }$ matrix elements. Using Eqs. (3)–(7) in Sect. 2, it can be read;

$$\left( {\begin{array}{*{20}c} {r_{2i}^{1\sigma } } \\ {a_{1i}^{1;2\sigma } } \\ {a_{2i}^{3;1\sigma } } \\ \end{array} } \right) = S_{i}^{1} \left( {\begin{array}{*{20}c} {r_{1i}^{1\sigma } } \\ {\tau_{2i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{2i}^{1;2\sigma } } \\ {\tau_{1i}^{3;1\sigma } (\gamma_{i}^{3;1} )a_{1i}^{3;1\sigma } } \\ \end{array} } \right)$$

(23)

$$\left( {\begin{array}{*{20}c} {t_{2i}^{2\sigma } } \\ {a_{2i}^{1;2\sigma } } \\ {a_{1i}^{3;2\sigma } } \\ \end{array} } \right) = S_{i}^{2} \left( {\begin{array}{*{20}c} {t_{1i}^{2\sigma } } \\ {\tau_{1i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{1i}^{1;2\sigma } } \\ {\tau_{2i}^{3;1\sigma } (\gamma_{i}^{3;1} )a_{2i}^{3;1\sigma } } \\ \end{array} } \right)$$

(24)

and

$$\left( {\begin{array}{*{20}c} {t_{2i}^{3\sigma } } \\ {a_{2i}^{2;3\sigma } } \\ {a_{1i}^{1;3\sigma } } \\ \end{array} } \right) = S_{i}^{3} \left( {\begin{array}{*{20}c} {t_{1i}^{3\sigma } } \\ {\tau_{1i}^{2;3\sigma } (\gamma_{i}^{2;3} )a_{1i}^{2;3\sigma } } \\ {\tau_{2i}^{1;3\sigma } (\gamma_{i}^{1;3} )a_{2i}^{1;3\sigma } } \\ \end{array} } \right)$$

(25)

where

$$r_{2i}^{1\sigma } = c_{i} r_{1i}^{1\sigma } + \, \sqrt {\epsilon_{i} } \tau_{2i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{2i}^{1;2\sigma } + \, \sqrt {\epsilon_{i} } \tau_{1i}^{3;1\sigma } (\gamma_{i}^{3;1} )a_{1i}^{3;1\sigma }$$

(26)

$$a_{1i}^{1;2\sigma } = \sqrt {\epsilon_{i} } r_{1i}^{1\sigma } + a_{i} \tau_{2i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{2i}^{1;2\sigma } + b_{i} \tau_{1i}^{3;1\sigma } (\gamma_{i}^{3;1} )a_{1i}^{3;1\sigma }$$

(27)

$$a_{2i}^{3;1\sigma } = \sqrt {\epsilon_{i} } r_{1i}^{1\sigma } + b_{i} \tau_{2i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{2i}^{1;2\sigma } + a_{i} \tau_{1i}^{3;1\sigma } (\gamma_{i}^{3;1} )a_{1i}^{3;1\sigma }$$

(28)

$$t_{2i}^{2\sigma } = c_{i} t_{1i}^{2\sigma } + \, \sqrt {\varepsilon_{i} } \tau_{1i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{1i}^{1;2\sigma } + \, \sqrt {\varepsilon_{i} } \tau_{2i}^{3;2\sigma } (\gamma_{i}^{3;2} )a_{2i}^{3;2\sigma }$$

(29)

$$a_{2i}^{1;2\sigma } = \sqrt {\varepsilon_{i} } t_{1i}^{2\sigma } + a_{i} \tau_{1i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{1i}^{1;2\sigma } + b_{i} \tau_{2i}^{3;2\sigma } (\gamma_{i}^{3;2} )a_{2i}^{3;2\sigma }$$

(30)

$$a_{1i}^{3;2\sigma } = \sqrt {\varepsilon_{i} } t_{1i}^{2\sigma } + b_{i} \tau_{1i}^{1;2\sigma } (\gamma_{i}^{1;2} )a_{1i}^{1;2\sigma } + a_{i} \tau_{2i}^{3;2\sigma } (\gamma_{i}^{3;2} )a_{2i}^{3;2\sigma }$$

(31)

$$t_{2i}^{3\sigma } = c_{i} t_{1i}^{3\sigma } + \sqrt {\varepsilon_{i} } \tau_{1i}^{2;3\sigma } (\gamma_{i}^{2;3} )a_{1i}^{2;3\sigma } + \sqrt {\varepsilon_{i} } \tau_{2i}^{1;3\sigma } (\gamma_{i}^{1;3} )a_{2i}^{1;3\sigma }$$

(32)

$$a_{2i}^{2;3\sigma } = \sqrt {\varepsilon_{i} } t_{1i}^{3\sigma } + a_{i} \tau_{1i}^{2;3\sigma } (\gamma_{i}^{2;3} )a_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;3\sigma } (\gamma_{i}^{1;3} )a_{2i}^{1;3\sigma }$$

(33)

$$a_{1i}^{1;3\sigma } = \sqrt {\varepsilon_{i} } t_{1i}^{3\sigma } + b_{i} \tau_{1i}^{2;3\sigma } (\gamma_{i}^{2;3} )a_{1i}^{2;3\sigma } + a_{i} \tau_{2i}^{1;3\sigma } (\gamma_{i}^{1;3} )a_{2i}^{1;3\sigma }$$

(34)

One can write the above equations in matrix form as;

$$A_{i}^{\sigma } \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {r_{2i}^{1\sigma } } \\ {t_{2i}^{2\sigma } } \\ {t_{2i}^{3\sigma } } \\ {a_{1i}^{1;2\sigma } } \\ \end{array} } \\ {a_{2i}^{1;3\sigma } } \\ {a_{2i}^{1;2\sigma } } \\ {\begin{array}{*{20}c} {a_{1i}^{2;3\sigma } } \\ {\begin{array}{*{20}c} {a_{2i}^{2;3\sigma } } \\ {a_{1i}^{1;3\sigma } } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {c_{i} r_{1i}^{1\sigma } } \\ {c_{i} t_{1i}^{2\sigma } } \\ {c_{i} t_{1i}^{3\sigma } } \\ {\sqrt {\varepsilon_{i} } r_{1i}^{1\sigma } } \\ \end{array} } \\ {\sqrt {\varepsilon_{i} } r_{1i}^{1\sigma } } \\ {\sqrt {\varepsilon_{i} } t_{1i}^{2\sigma } } \\ {\begin{array}{*{20}c} {\sqrt {\varepsilon_{i} } t_{1i}^{2\sigma } } \\ {\begin{array}{*{20}c} {\sqrt {\varepsilon_{i} } t_{1i}^{3\sigma } } \\ {\sqrt {\varepsilon_{i} } t_{1i}^{3\sigma } } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right)$$

(35)

where

$$A_{i}^{\sigma } = \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & { - \sqrt {\varepsilon_{i} } \tau_{2i}^{1;2\sigma } } & 0 & 0 & { - \sqrt {\varepsilon_{i} } \tau_{1i}^{3;1\sigma } } \\ 0 & 1 & 0 & { - \sqrt {\varepsilon_{i} } \tau_{1i}^{1;2\sigma } } & 0 & 0 & 0 & { - \sqrt {\varepsilon_{i} } \tau_{2i}^{2;3\sigma } } & 0 \\ 0 & 0 & 1 & 0 & { - \sqrt {\varepsilon_{i} } \tau_{2i}^{3;1\sigma } } & 0 & { - \sqrt {\varepsilon_{i} } \tau_{1i}^{2;3\sigma } } & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & { - a_{i} \tau_{2i}^{1;2\sigma } } & 0 & 0 & { - b_{i} \tau_{1i}^{3;1\sigma } } \\ 0 & 0 & 0 & 0 & 1 & { - b_{i} \tau_{2i}^{1;2\sigma } } & 0 & 0 & { - a_{i} \tau_{1i}^{3;1\sigma } } \\ 0 & 0 & 0 & { - a_{i} \tau_{1i}^{1;2\sigma } } & 0 & 1 & 0 & { - b_{i} \tau_{2i}^{2;3\sigma } } & 0 \\ 0 & 0 & 0 & { - b_{i} \tau_{1i}^{1;2\sigma } } & 0 & 0 & 1 & { - a_{i} \tau_{2i}^{2;3\sigma } } & 0 \\ 0 & 0 & 0 & 0 & { - b_{i} \tau_{2i}^{3;1\sigma } } & 0 & { - a_{i} \tau_{1i}^{2;3\sigma } } & 1 & 0 \\ 0 & 0 & 0 & 0 & { - a_{i} \tau_{2i}^{3;1\sigma } } & 0 & { - b_{i} \tau_{1i}^{2;3\sigma } } & 0 & 1 \\ \end{array} } \right)$$

(36)

To calculate the elements of the matrix $M_{i}^{\sigma }$ which relates the coefficients $r_{1i}^{1\sigma } , \, r_{2i}^{1\sigma } , \, t_{1i}^{2\sigma } , \, t_{2i}^{2\sigma } , \, t_{1i}^{3\sigma }$ and $t_{2i}^{3\sigma }$ to each other, we require to find the three first rows of the inverse of the matrix $A_{i}^{\sigma }$. For the convenience of calculations, we first define some auxiliary variables, namely $f_{1i}^{\sigma }$, $f_{2i}^{\sigma }$, $f_{3i}^{\sigma }$, $f_{4i}^{\sigma }$, $f_{5i}^{\sigma }$, $f_{6i}^{\sigma }$, $f_{7i}^{\sigma }$, $f_{8i}^{\sigma }$, $f_{9i}^{\sigma }$, $f_{10i}^{\sigma }$, $f_{11i}^{\sigma }$, $f_{12i}^{\sigma }$, $f_{13i}^{\sigma }$, $f_{14i}^{\sigma }$, $f_{15i}^{\sigma }$, $f_{16i}^{\sigma }$, $f_{17i}^{\sigma }$, $f_{18i}^{\sigma }$, $f_{19i}^{\sigma }$, $f_{20i}^{\sigma }$, $f_{21i}^{\sigma }$, $f_{22i}^{\sigma }$, $f_{23i}^{\sigma }$, $f_{24i}^{\sigma }$, $f_{25i}^{\sigma }$, $f_{26i}^{\sigma }$, $f_{27i}^{\sigma }$, $f_{28i}^{\sigma }$, $f_{29i}^{\sigma }$ and $f_{30i}^{\sigma }$.

$$f_{1i}^{\sigma } = 1 - \tau_{2i}^{3;2\sigma } (a_{i}^{2} \tau_{1i}^{3;2\sigma } + b_{i}^{3} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )$$

(37)

$$f_{2i}^{\sigma } = - \sqrt {\varepsilon_{i} } \tau_{2i}^{2;3\sigma } (a_{i} \tau_{1i}^{2;3\sigma } - b_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )$$

(38)

$$f_{3i}^{\sigma } = - a_{i} b_{i} \tau_{2i}^{3;2\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )$$

(39)

$$f_{4i}^{\sigma } = f_{3i}^{\sigma } a_{i} b_{i} \tau_{1i}^{1;3\sigma } \tau_{2i}^{1;3\sigma } + f_{1i}^{\sigma } (1 - a_{i}^{2} \tau_{1i}^{1;3\sigma } \tau_{2i}^{1;3\sigma } )$$

(40)

$$f_{5i}^{\sigma } = - \sqrt {\varepsilon_{i} } a_{i} b_{i} \tau_{1i}^{1;3\sigma } \tau_{2i}^{1;3\sigma } \tau_{2i}^{2;3\sigma } )$$

(41)

$$f_{6i}^{\sigma } = a_{i} \tau_{1i}^{1;3\sigma } \tau_{2i}^{1;3\sigma } (b_{i} f_{2i}^{\sigma } - \sqrt {\varepsilon_{i} } f_{1i}^{\sigma } )$$

(42)

$$f_{7i}^{\sigma } = - \sqrt {\varepsilon_{i} } \tau_{1i}^{1;3\sigma } (f_{1i}^{\sigma } + a_{i} b_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } )$$

(43)

$$f_{8i}^{\sigma } = - b_{i} \tau_{1i}^{1;3\sigma } (f_{1i}^{\sigma } + b_{i} a_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } )$$

(44)

$$\begin{aligned} f_{9i}^{\sigma } = & - f_{8i}^{\sigma } b_{i} \tau_{1i}^{1;2\sigma } (f_{3i}^{\sigma } a_{i} (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } ) + f_{1i}^{\sigma } (b_{i} \tau_{1i}^{2;3\sigma } + a_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ & - f_{4i}^{\sigma } (f_{1i}^{\sigma } ( - 1 + a_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } ) + a_{i}^{2} b_{i}^{2} \tau_{1i}^{1;2\sigma } \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ \end{aligned}$$

(45)

$$\begin{aligned} f_{10i}^{\sigma } = & - f_{5i}^{\sigma } b_{i} \tau_{1i}^{1;2\sigma } (a_{i} f_{3i}^{\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } ) - f_{1i}^{\sigma } (b_{i} \tau_{1i}^{2;3\sigma } + a_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ & - \sqrt {\varepsilon_{i} } \tau_{1i}^{1;2\sigma } f_{4i}^{\sigma } (f_{1i}^{\sigma } + a_{i} b_{i} \tau_{2i}^{2;3\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ \end{aligned}$$

(46)

$$\begin{aligned} f_{11i}^{\sigma } = & - b_{i} \tau_{1i}^{1;2\sigma } f_{6i}^{\sigma } (a_{i} f_{3i}^{\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } ) - f_{1i}^{\sigma } (b_{i} \tau_{1i}^{2;3\sigma } + a_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ & - b_{i} \tau_{1i}^{1;2\sigma } f_{4i}^{\sigma } (\sqrt {\varepsilon_{i} } f_{1i}^{\sigma } (\tau_{1i}^{2;3\sigma } + a_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } ) - a_{i} f_{2i}^{\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ \end{aligned}$$

(47)

$$\begin{aligned} f_{12i}^{\sigma } = & - b_{i} \tau_{1i}^{1;2\sigma } f_{7i}^{\sigma } (a_{i} f_{4i}^{\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } ) - f_{1i}^{\sigma } (b_{i} \tau_{1i}^{2;3\sigma } + a_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ & - \sqrt {\varepsilon_{i} } a_{i} \tau_{1i}^{1;2\sigma } \tau_{2i}^{1;2\sigma } f_{4i}^{\sigma } (f_{1i}^{\sigma } + b_{i}^{2} \tau_{2i}^{2;3\sigma } (\tau_{1i}^{2;3\sigma } + b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } )) \\ \end{aligned}$$

(48)

$$\begin{aligned} f_{13i}^{\sigma } = & \frac{{b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } (f_{10i}^{\sigma } f_{8i}^{\sigma } - f_{5i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - a_{i} \tau_{2i}^{1;2\sigma } f_{10i}^{\sigma } f_{4i}^{\sigma } (f_{1i}^{\sigma } + b_{i}^{3} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }} \\ & + \frac{{\sqrt {\varepsilon_{i} } b_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } f_{9i}^{\sigma } f_{4i}^{\sigma } }}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }} \\ \end{aligned}$$

(49)

$$\begin{aligned} f_{14i}^{\sigma } = & \frac{{b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } (f_{11i}^{\sigma } f_{8i}^{\sigma } - f_{6i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - a_{i} \tau_{2i}^{1;2\sigma } f_{11i}^{\sigma } f_{4i}^{\sigma } (f_{1i}^{\sigma } + b_{i}^{3} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }} \\ & + \frac{{b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } (\sqrt {\varepsilon_{i} } f_{1i}^{\sigma } - b_{i} f_{2i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }} \\ \end{aligned}$$

(50)

$$f_{15i}^{\sigma } = \frac{{b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{1;3\sigma } (f_{12i}^{\sigma } f_{8i}^{\sigma } - f_{7i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - \tau_{2i}^{1;2\sigma } f_{4i}^{\sigma } (f_{1i}^{\sigma } + b_{i}^{3} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } )(a_{i} f_{12i}^{\sigma } - \sqrt {\varepsilon_{i} } f_{9i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(51)

$$f_{16i}^{\sigma } = \frac{{\tau_{1i}^{2;3\sigma } (f_{10i}^{\sigma } f_{8i}^{\sigma } - f_{5i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - a_{i} \tau_{1i}^{2;3\sigma } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } (a_{i} b_{i} \tau_{2i}^{1;2\sigma } f_{10i}^{\sigma } - \sqrt {\varepsilon_{i} } f_{9i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(52)

$$f_{17i}^{\sigma } = \frac{{\tau_{1i}^{2;3\sigma } (f_{11i}^{\sigma } f_{8i}^{\sigma } - f_{6i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - \tau_{1i}^{2;3\sigma } f_{4i}^{\sigma } (a_{i}^{2} b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } f_{11i}^{\sigma } + f_{9i}^{\sigma } (\sqrt {\varepsilon_{i} } f_{1i}^{\sigma } - a_{i} f_{2i}^{\sigma } ))}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(53)

$$f_{18i}^{\sigma } = \frac{{\tau_{1i}^{2;3\sigma } (f_{12i}^{\sigma } f_{8i}^{\sigma } - f_{7i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - a_{i} b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{1i}^{2;3\sigma } f_{4i}^{\sigma } (a_{i} f_{12i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{9i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(54)

$$f_{19i}^{\sigma } = \frac{{\tau_{2i}^{1;3\sigma } (f_{10i}^{\sigma } f_{8i}^{\sigma } - f_{5i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - b_{i} \tau_{1i}^{2;3\sigma } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } (a_{i} b_{i} \tau_{2i}^{1;2\sigma } f_{10i}^{\sigma } - \sqrt {\varepsilon_{i} } f_{9i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(55)

$$f_{20i}^{\sigma } = \frac{{\tau_{2i}^{1;3\sigma } (f_{11i}^{\sigma } f_{8i}^{\sigma } - f_{6i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - \tau_{2i}^{1;3\sigma } f_{4i}^{\sigma } (b_{i}^{2} a_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } f_{11i}^{\sigma } + f_{9i}^{\sigma } (\sqrt {\varepsilon_{i} } f_{1i}^{\sigma } - b_{i} f_{2i}^{\sigma } ))}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(56)

$$f_{21i}^{\sigma } = \frac{{\tau_{2i}^{1;3\sigma } (f_{12i}^{\sigma } f_{8i}^{\sigma } - f_{7i}^{\sigma } f_{9i}^{\sigma } )(a_{i} f_{1i}^{\sigma } - b_{i} f_{3i}^{\sigma } ) - b_{i}^{2} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } \tau_{2i}^{1;3\sigma } f_{4i}^{\sigma } (a_{i} f_{12i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{9i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(57)

$$f_{22i}^{\sigma } = \frac{{ - f_{10i}^{\sigma } (f_{8i}^{\sigma } f_{3i}^{\sigma } + a_{i} b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } ) + f_{9i}^{\sigma } (f_{3i}^{\sigma } f_{5i}^{\sigma } + \sqrt {\varepsilon_{i} } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(58)

$$f_{23i}^{\sigma } = \frac{{ - f_{11i}^{\sigma } (f_{8i}^{\sigma } f_{3i}^{\sigma } + a_{i} b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } ) + f_{9i}^{\sigma } (f_{3i}^{\sigma } f_{6i}^{\sigma } - f_{2i}^{\sigma } f_{4i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(59)

$$f_{24i}^{\sigma } = \frac{{ - f_{12i}^{\sigma } (f_{8i}^{\sigma } f_{3i}^{\sigma } + a_{i} b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } ) + f_{9i}^{\sigma } (f_{3i}^{\sigma } f_{7i}^{\sigma } + \sqrt {\varepsilon_{i} } b_{i} \tau_{2i}^{1;2\sigma } \tau_{2i}^{2;3\sigma } f_{4i}^{\sigma } )}}{{f_{1i}^{\sigma } f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(60)

$$f_{25i}^{\sigma } = \frac{{f_{10i}^{\sigma } f_{8i}^{\sigma } - f_{9i}^{\sigma } f_{5i}^{\sigma } }}{{f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(61)

$$f_{26i}^{\sigma } = \frac{{f_{11i}^{\sigma } f_{8i}^{\sigma } - f_{9i}^{\sigma } f_{7i}^{\sigma } }}{{f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(62)

$$f_{27i}^{\sigma } = \frac{{f_{12i}^{\sigma } f_{8i}^{\sigma } - f_{9i}^{\sigma } f_{7i}^{\sigma } }}{{f_{4i}^{\sigma } f_{9i}^{\sigma } }}$$

(63)

$$f_{28i}^{\sigma } = \frac{{ - f_{10i}^{\sigma } }}{{f_{9i}^{\sigma } }}$$

(64)

$$f_{29i}^{\sigma } = \frac{{ - f_{11i}^{\sigma } }}{{f_{9i}^{\sigma } }}$$

(65)

$$f_{30i}^{\sigma } = \frac{{ - f_{12i}^{\sigma } }}{{f_{9i}^{\sigma } }}$$

(66)

Finally, using these auxiliary variables, the elements of the $3 \times 3$ matrix $M_{i}^{\sigma }$ can be obtained as:

$$M_{11i}^{\sigma } = c_{i} + \sqrt {\varepsilon_{i} } f_{13i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{16i}^{\sigma } ,$$

(67)

$$M_{12i}^{\sigma } = \sqrt {\varepsilon_{i} } f_{22i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{25i}^{\sigma } ,$$

(68)

$$M_{13i}^{\sigma } = \sqrt {\varepsilon_{i} } f_{19i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{28i}^{\sigma } ,$$

(69)

$$M_{21i}^{\sigma } = \sqrt {\varepsilon_{i} } f_{14i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{17i}^{\sigma } ,$$

(70)

$$M_{22i}^{\sigma } = c_{i} + \sqrt {\varepsilon_{i} } f_{23i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{26i}^{\sigma } ,$$

(71)

$$M_{23i}^{\sigma } = \sqrt {\varepsilon_{i} } f_{20i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{29i}^{\sigma } ,$$

(72)

$$M_{31i}^{\sigma } = \sqrt {\varepsilon_{i} } f_{15i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{18i}^{\sigma } ,$$

(73)

$$M_{32i}^{\sigma } = \sqrt {\varepsilon_{i} } f_{24i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{27i}^{\sigma } ,$$

(74)

$$M_{33i}^{\sigma } = c_{i} + \sqrt {\varepsilon_{i} } f_{21i}^{\sigma } + \sqrt {\varepsilon_{i} } f_{30i}^{\sigma } ,$$

(75)