Electron Transport Through a Molecular Junction Using a Multi-configurational Description

Abstract

The importance of the electronic description of the junction electronic structure is investigated in quantum transport through molecular devices. Using an accurate wave-function-based description of the low-energy spectroscopy, transport through a 2-electron/2-molecular orbital prototype is evaluated. The contributions arising from the presence of competing singlet and triplet states in magnetic systems are analyzed. It is shown that the electronic conductivity provides a signature of the full multiplet energy spectrum, as well as of the multideterminant structure of wave-functions. We then inspected the current-potential characteristics as a function of the differential magnetization of the electrodes. From the wave-function description, a modulation of the magnetoresistance ratio is anticipated and both direct and inverse regimes are observed depending on the electronic structure of the junction.

Appendices

A. Coefficients

$$\begin{aligned} w_{{\rm{r}}I^-,\sigma }= &\;x_{\rm{r},\sigma }\frac{2\uppi }{\hbar Z} \sum _{I}\rho _{\rm{r}}(E_{N}-E_{N+1}) \hat{D}(E_{N}-E_{N+1})\\&\quad \quad \quad \quad\times \sum _{f}\left| \gamma _{r,f}\langle {I^-}|f^{\dag }_{\sigma }r_{\sigma }|{I} \otimes r \rangle \right| ^2 \exp \left( -\frac{E_{N+1}}{k_{B}T}\right) \end{aligned}$$

$$\begin{aligned} w_{I^-{\rm{l}},\sigma }=&\;x_{\rm{l},\sigma }\frac{2\uppi }{\hbar Z}\sum _{I}\left( 1- \rho _{{\text{ l }}}(E_{N}-E_{N+1})\right) \hat{D}(E_{N}-E_{N+1})\\&\quad \quad \quad \quad \quad \quad \times \sum _{f}\left| \gamma _{l,f} \langle {I} \otimes l|l^{\dag }_{\sigma }f_{\sigma }| {I^-}\rangle \right| ^2 \exp \left( -\frac{E_{N+1}}{k_{B}T}\right) \end{aligned}$$

$$\begin{aligned} w_{I^-{\rm{r}},\sigma }=&\;x_{\rm{r},\sigma }\frac{2\uppi }{\hbar Z}\sum _{I}\left( 1- \rho _{\rm{r}}(E_{N}-E_{N+1})\right) \hat{D}(E_{N}-E_{N+1})\\&\quad \quad \quad \quad \quad \quad \times \sum _{f}\left| \gamma _{r,f} \langle {I} \otimes r|r^{\dag }_{\sigma }f_{\sigma }| {I^-}\rangle \right| ^2 \exp \left( -\frac{E_{N+1}}{k_{B}T}\right) \end{aligned}$$

B. Cancellation of Coefficients in $\mathbf{i }_{\text{ tot }}$ and $\mathbf{P }^{{\text{ i }}}$ for a Parallel Magnetization

All coefficients can be factored depending on the corresponding spin and the electrode involved. All the $w_{\uparrow }$ coefficients can be factorized as $w= \bar{w} x_{1}$. While all $w_{\downarrow }$ coefficients can be factorized as: $w= \bar{w} \left( 1- x_{1}\right)$.

As the difference between $w_{I^-\text{ r,l } ,\sigma }$ and $w_{r,1 I^-,\sigma }$ is only the proportionality to $1-\rho _{1,\rm{r}}$ or $\rho _{1,\rm{r}}$. We have:

$$\begin{aligned} A_1+A_2=&\;B_1+B_2\\ =&\;\frac{2\uppi }{\hbar Z}\sum _{I} \hat{D}(E_{N}-E_{N+1})\sum _{f}\left| \gamma _{l,f} \langle {I} \otimes l|l^{\dag }_{\sigma }f_{\sigma }| {I^-}\rangle \right| ^2 \exp \left( -\frac{E_{N+1}}{k_{B}T}\right) \end{aligned}$$

and $C_1+C_2=D_1+D_2$

Table 2 Simplified expression of the different coupling coefficients for the parallel situation

Using equations (3) and (5) we can show that the polarization $P^{i}$ is equal to $2x-1$ and the total current is constant.

C. Origin of the Quadratic Term in the Total Current for an Antiparallel Situation

For the antiparallel situation, only the proportionality to $x$ or $1-x$ is changed for the coefficients involving the right electrode.

Table 3 Simplified expression of the different coupling coefficients for the antiparallel situation

Using Eqs. (3) and (5) we can show that the quadratic term in the total current is the following:

$$\begin{aligned} \frac{1}{A_1+A_2+C_1+C_2}((A_1-C_1)(D_1-B_1)+(A_2-C_2)(B_2-D_2)) \end{aligned}$$

As a consequence everything is controlled by the difference between $\uparrow$ and $\downarrow$ coefficients deprived of their dependence upon $x$.