Electron Transfer Methods in Open Systems

Nicolas Bergmann; Michael Galperin

arXiv:1906.07827·cond-mat.mes-hall·September 5, 2019

Electron Transfer Methods in Open Systems

Nicolas Bergmann, Michael Galperin

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TL;DR

This paper introduces a nonequilibrium Hubbard Green's functions diagrammatic technique to improve the construction of electron transfer rates in open quantum systems, surpassing traditional second and fourth order methods.

Contribution

It presents a novel diagrammatic approach that generalizes rate calculations and incorporates additional baths or degrees of freedom naturally.

Findings

01

Previous rate calculations are special cases of the new diagrammatic series.

02

The Hubbard Green's function approach offers advantages over traditional methods.

03

Standard diagram dressing allows for inclusion of more complex system-bath interactions.

Abstract

Utilization of electron transfer methods for description of quantum transport is popular due to simplicity of the formulation and its ability to account for basic physics of electron exchange between system and baths. At the same time, necessity to go beyond simple golden rule-type expressions for rates was indicated in the literature and ad hoc formulations were proposed. Similarly, kinetic schemes for quantum transport beyond usual second order Lindblad/Redfield considerations were discussed. Here we utilize recently introduced by us nonequilibrium Hubbard Green's functions diagrammatic technique to analyze construction of rates in open systems. We show that previous considerations for rates of second and fourth order can be obtained as a particular case of zero and second order Green's function diagrammatic series with bare diagrams. We discuss limitations of previous considerations,…

Equations273

2 \mbox R e K \in {L, R} \sum \int_{- \infty}^{t} d t^{'}

2 \mbox R e K \in {L, R} \sum \int_{- \infty}^{t} d t^{'}

\displaystyle\bigg{(}\ \sigma^{K\,>}_{S_{f}S_{i},S_{f}S_{i}}(t-t^{\prime})\,g^{<}_{S_{f}S_{i}}(t^{\prime}-t)\,\langle\hat{F}_{S_{f}S_{i},S_{f}S_{i}}(t)\rangle

\displaystyle+\sigma^{K\,<}_{S_{i}S_{f},S_{i}S_{f}}(t-t^{\prime})\,g^{>}_{S_{i}S_{f}}(t^{\prime}-t)\,\langle\hat{F}_{S_{i}S_{f},S_{i}S_{f}}(t)\rangle\bigg{)}

W^{(2)}_{S_{f}\leftarrow S_{i}}=i\sum_{K\in\{L,R\}}\bigg{(}\sigma^{K\,>}_{S_{f}S_{i},S_{f}S_{i}}(E_{S_{i}}-E_{S_{f}})-\sigma^{K\,<}_{S_{i}S_{f},S_{i}S_{f}}(E_{S_{f}}-E_{S_{i}})\bigg{)}

W^{(2)}_{S_{f}\leftarrow S_{i}}=i\sum_{K\in\{L,R\}}\bigg{(}\sigma^{K\,>}_{S_{f}S_{i},S_{f}S_{i}}(E_{S_{i}}-E_{S_{f}})-\sigma^{K\,<}_{S_{i}S_{f},S_{i}S_{f}}(E_{S_{f}}-E_{S_{i}})\bigg{)}

- 2 \mbox R e S \sum K \in {L, R} \sum \int_{- \infty}^{t} d t^{'}

- 2 \mbox R e S \sum K \in {L, R} \sum \int_{- \infty}^{t} d t^{'}

\displaystyle\bigg{(}\ \sigma^{K\,>}_{SS_{i},SS_{i}}(t-t^{\prime})\,g^{<}_{SS_{i}}(t^{\prime}-t)\,\langle\hat{F}_{SS_{i},SS_{i}}(t)\rangle

\displaystyle+\sigma^{K\,<}_{S_{i}S,S_{i}S}(t-t^{\prime})\,g^{>}_{S_{i}S}(t^{\prime}-t)\,\langle\hat{F}_{S_{i}S,S_{i}S}(t)\rangle\bigg{)}

W^{(2)}_{S_{i}\leftarrow S_{i}}=-i\sum_{S}\sum_{K\in\{L,R\}}\bigg{(}\sigma^{K\,>}_{SS_{i},SS_{i}}(E_{S_{i}}-E_{S})-\sigma^{K\,<}_{S_{i}S,S_{i}S}(E_{S}-E_{S_{i}})\bigg{)}

W^{(2)}_{S_{i}\leftarrow S_{i}}=-i\sum_{S}\sum_{K\in\{L,R\}}\bigg{(}\sigma^{K\,>}_{SS_{i},SS_{i}}(E_{S_{i}}-E_{S})-\sigma^{K\,<}_{S_{i}S,S_{i}S}(E_{S}-E_{S_{i}})\bigg{)}

S_{f} \sum W_{S_{f} \leftarrow S_{i}}^{(2)} = 0

S_{f} \sum W_{S_{f} \leftarrow S_{i}}^{(2)} = 0

B_{S S^{'}} (τ, τ^{'})

B_{S S^{'}} (τ, τ^{'})

D_{S S^{'}} (τ, τ^{'})

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\ \sigma^{K_{1}\,<}_{S_{i}S_{1},S_{3}S_{2}}(t-t^{\prime})\,g_{S_{3}S_{2}}^{>}(t^{\prime}-t_{1})\,g_{S_{3}S_{i}}^{<}(t_{1}-t_{2})\,\sigma^{K_{2}\,>}_{S_{3}S_{i},S_{2}S_{1}}(t_{2}-t_{1})\,g_{S_{i}S_{1}}^{>}(t_{1}-t)\langle\hat{F}_{S_{i}S_{1},S_{i}S_{1}}(t)\rangle

- σ_{S_{1} S_{i}, S_{2} S_{3}}^{K_{1} >} (t - t^{'}) g_{S_{2} S_{3}}^{<} (t^{'} - t_{1}) g_{S_{i} S_{3}}^{>} (t_{1} - t_{2}) σ_{S_{i} S_{3}, S_{1} S_{2}}^{K_{2} <} (t_{2} - t_{1}) g_{S_{1} S_{i}}^{<} (t_{1} - t) ⟨ \hat{F}_{S_{1} S_{i}, S_{1} S_{i}} (t)⟩

+ σ_{S_{1} S_{i}, S_{2} S_{3}}^{K_{1} >} (t - t^{'}) g_{S_{2} S_{3}}^{>} (t^{'} - t_{2}) g_{S_{2} S_{1}}^{<} (t_{2} - t_{1}) σ_{S_{2} S_{1}, S_{3} S_{i}}^{K_{2} >} (t_{1} - t_{2}) g_{S_{1} S_{i}}^{<} (t_{2} - t) ⟨ \hat{F}_{S_{1} S_{i}, S_{1} S_{i}} (t)⟩

\displaystyle-\sigma^{K_{1}\,<}_{S_{i}S_{1},S_{3}S_{2}}(t-t^{\prime})\,g^{<}_{S_{3}S_{2}}(t^{\prime}-t_{2})\,g^{>}_{S_{1}S_{2}}(t_{2}-t_{1})\,\sigma^{K_{2}\,<}_{S_{1}S_{2},S_{i}S_{3}}(t_{1}-t_{2})\,g^{>}_{S_{i}S_{1}}(t_{2}-t)\langle\hat{F}_{S_{i}S_{1},S_{i}S_{1}}(t)\rangle\bigg{)}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\ \sigma^{K_{1}\,>}_{S_{1}S_{i},S_{2}S_{3}}(t-t^{\prime})\,g^{>}_{S_{2}S_{3}}(t^{\prime}-t_{2})\,\sigma^{K_{2}\,<}_{S_{i}S_{3},S_{1}S_{2}}(t_{2}-t_{1})\,g^{>}_{S_{1}S_{2}}(t_{1}-t_{2})\,g^{<}_{S_{1}S_{i}}(t_{2}-t)\langle\hat{F}_{S_{1}S_{i},S_{1}S_{i}}(t)\rangle

- σ_{S_{i} S_{1}, S_{3} S_{2}}^{K_{1} <} (t - t^{'}) g_{S_{3} S_{2}}^{<} (t^{'} - t_{2}) σ_{S_{3} S_{i}, S_{2} S_{1}}^{K_{2} >} (t_{2} - t_{1}) g_{S_{2} S_{1}}^{<} (t_{1} - t_{2}) g_{S_{i} S_{1}}^{>} (t_{2} - t) ⟨ \hat{F}_{S_{i} S_{1}, S_{i} S_{1}} (t)⟩

+ σ_{S_{i} S_{1}, S_{3} S_{2}}^{K_{1} <} (t - t^{'}) g_{S_{3} S_{2}}^{>} (t^{'} - t_{1}) σ_{S_{1} S_{2}, S_{i} S_{3}}^{K_{2} <} (t_{1} - t_{2}) g_{S_{i} S_{3}}^{>} (t_{2} - t_{1}) g_{S_{i} S_{1}}^{>} (t_{1} - t) ⟨ \hat{F}_{S_{i} S_{1}, S_{i} S_{1}} (t)⟩

\displaystyle-\sigma^{K_{1}\,>}_{S_{1}S_{i},S_{2}S_{3}}(t-t^{\prime})\,g^{<}_{S_{2}S_{3}}(t^{\prime}-t_{1})\,\sigma^{K_{2}\,>}_{S_{2}S_{1},S_{3}S_{i}}(t_{1}-t_{2})\,g^{<}_{S_{3}S_{i}}(t_{2}-t_{1})\,g^{<}_{S_{1}S_{i}}(t_{1}-t)\langle\hat{F}_{S_{1}S_{i},S_{1}S_{i}}(t)\rangle\bigg{)}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\sigma^{K_{1}\,<}_{S_{i}S_{1},S_{3}S_{2}}(t-t^{\prime})\,g^{<}_{S_{3}S_{2}}(t^{\prime}-t_{2})\,\sigma^{K_{2}\,>}_{S_{3}S_{i},S_{2}S_{1}}(t_{2}-t_{1})\,g^{>}_{S_{2}S_{1}}(t_{1}-t)\,B^{<}_{S_{2}S_{i}}(t-t_{2})

\displaystyle+\sigma^{K_{1}\,>}_{S_{1}S_{i},S_{2}S_{3}}(t-t^{\prime})\,g^{>}_{S_{2}S_{3}}(t^{\prime}-t_{2})\,\sigma^{K_{2}\,<}_{S_{i}S_{3},S_{1}S_{2}}(t_{2}-t_{1})\,g^{<}_{S_{1}S_{2}}(t_{1}-t)\,B^{>}_{S_{i}S_{2}}(t-t_{2})\bigg{)}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

2 \mbox I m S_{1}, S_{2}, S_{3} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\sigma^{K_{1}\,<}_{S_{i}S_{1},S_{3}S_{2}}(t-t^{\prime})\,g^{<}_{S_{3}S_{2}}(t^{\prime}-t_{2})\,D_{S_{i}S_{2}}^{>}(t_{2}-t)\,g^{<}_{S_{1}S_{2}}(t-t_{1})\,\sigma^{K_{2}\,<}_{S_{1}S_{2},S_{i}S_{3}}(t_{1}-t_{2})

\displaystyle+\sigma^{K_{1}\,>}_{S_{1}S_{i},S_{2}S_{3}}(t-t^{\prime})\,g^{>}_{S_{2}S_{3}}(t^{\prime}-t_{2})\,D^{<}_{S_{2}S_{i}}(t_{2}-t)\,g^{>}_{S_{2}S_{1}}(t-t_{1})\,\sigma^{K_{2}\,>}_{S_{2}S_{1},S_{3}S_{i}}(t_{1}-t_{2})\bigg{)}

\displaystyle-2\,\mbox{Im}\sum_{S_{1},S_{2},S_{3}}\sum_{K_{1},K_{2}\in\{L,R\}}\int\frac{d\omega_{1}}{2\pi}\int\frac{d\omega_{2}}{2\pi}\,\bigg{(}\frac{P_{S_{i}}\,P_{S_{3}}}{P_{S_{i}}+P_{S_{3}}}+P_{S_{2}}\bigg{)}\frac{P_{S_{i}}}{P_{S_{2}}+P_{S_{3}}}

\displaystyle-2\,\mbox{Im}\sum_{S_{1},S_{2},S_{3}}\sum_{K_{1},K_{2}\in\{L,R\}}\int\frac{d\omega_{1}}{2\pi}\int\frac{d\omega_{2}}{2\pi}\,\bigg{(}\frac{P_{S_{i}}\,P_{S_{3}}}{P_{S_{i}}+P_{S_{3}}}+P_{S_{2}}\bigg{)}\frac{P_{S_{i}}}{P_{S_{2}}+P_{S_{3}}}

\displaystyle\bigg{[}\ \frac{\Gamma^{K_{1}}_{S_{i}S_{1},S_{3}S_{2}}(\omega_{1})f_{K_{1}}(\omega_{1})\Gamma^{K_{2}}_{S_{3}S_{i},S_{2}S_{1}}(\omega_{2})[1-f_{K_{2}}(\omega_{2})]}{(E_{S_{3}}-E_{S_{i}}+\omega_{2}+i0)(E_{S_{2}}-E_{S_{i}}-\omega_{1}+\omega_{2}+i0)(E_{S_{1}}-E_{S_{i}}-\omega_{1}+i0)}

+ \frac{Γ _{S_{i} S_{1}, S_{3} S_{2}}^{K_{1}} ( ω _{1} ) f _{K_{1}} ( ω _{1} ) Γ _{S_{1} S_{2}, S_{i} S_{3}}^{K_{2}} ( ω _{2} ) f _{K_{2}} ( ω _{2} )}{( E _{S_{3}} - E _{S_{i}} - ω _{2} + i 0 ) ( E _{S_{2}} - E _{S_{i}} - ω _{1} - ω _{2} + i 0 ) ( E _{S_{1}} - E _{S_{i}} - ω _{1} + i 0 )}

+ \frac{Γ _{S_{1} S_{i}, S_{2} S_{3}}^{K_{1}} ( ω _{1} ) [ 1 - f _{K_{1}} ( ω _{1} )] Γ _{S_{2} S_{1}, S_{3} S_{i}}^{K_{2}} ( ω _{2} ) [ 1 - f _{K_{2}} ( ω _{2} )]}{( E _{S_{i}} - E _{S_{3}} - ω _{2} + i 0 ) ( E _{S_{i}} - E _{S_{2}} - ω _{1} - ω _{2} + i 0 ) ( E _{S_{i}} - E _{S_{1}} - ω _{1} + i 0 )}

\displaystyle+\frac{\Gamma^{K_{1}}_{S_{1}S_{i},S_{2}S_{3}}(\omega_{1})[1-f_{K_{1}}(\omega_{1})]\Gamma^{K_{2}}_{S_{i}S_{3},S_{1}S_{2}}(\omega_{2})f_{K_{2}}(\omega_{2})}{(E_{S_{i}}-E_{S_{3}}+\omega_{2}+i0)(E_{S_{i}}-E_{S_{2}}-\omega_{1}+\omega_{2}+i0)(E_{S_{i}}-E_{S_{1}}-\omega_{1}+i0)}\bigg{]}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\ \sigma^{K_{1}\,>}_{S_{f}S_{i},S_{2}S_{1}}(t-t^{\prime})\,g^{>}_{S_{2}S_{1}}(t^{\prime}-t_{2})\,g^{<}_{S_{2}S_{f}}(t_{2}-t_{1})\,\sigma^{K_{2}\,>}_{S_{2}S_{f},S_{1}S_{i}}(t_{1}-t_{2})\,g^{<}_{S_{f}S_{i}}(t_{2}-t)\langle\hat{F}_{S_{f}S_{i},S_{f}S_{i}}(t)\rangle

- σ_{S_{i} S_{f}, S_{1} S_{2}}^{K_{1} <} (t - t^{'}) g_{S_{1} S_{2}}^{<} (t^{'} - t_{2}) g_{S_{f} S_{2}}^{>} (t_{2} - t_{1}) σ_{S_{f} S_{2}, S_{i} S_{1}}^{K_{2} <} (t_{1} - t_{2}) g_{S_{i} S_{f}}^{>} (t_{2} - t) ⟨ \hat{F}_{S_{i} S_{f}, S_{i} S_{f}} (t)⟩

+ σ_{S_{i} S_{f}, S_{1} S_{2}}^{K_{1} <} (t - t^{'}) g_{S_{1} S_{2}}^{>} (t^{'} - t_{1}) g_{S_{1} S_{i}}^{<} (t_{1} - t_{2}) σ_{S_{1} S_{i}, S_{2} S_{f}}^{K_{2} >} (t_{2} - t_{1}) g_{S_{i} S_{f}}^{>} (t_{1} - t) ⟨ \hat{F}_{S_{i} S_{f}, S_{i} S_{f}} (t)⟩

\displaystyle-\sigma^{K_{1}\,>}_{S_{f}S_{i},S_{2}S_{1}}(t-t^{\prime})\,g^{<}_{S_{2}S_{1}}(t^{\prime}-t_{1})\,g^{>}_{S_{i}S_{1}}(t_{1}-t_{2})\,\sigma^{K_{2}\,<}_{S_{i}S_{1},S_{f}S_{2}}(t_{2}-t_{1})\,g^{<}_{S_{f}S_{i}}(t_{1}-t)\langle\hat{F}_{S_{f}S_{i},S_{f}S_{i}}(t)\rangle\bigg{)}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\ \sigma^{K_{1}\,<}_{S_{i}S_{f},S_{1}S_{2}}(t-t^{\prime})\,g^{>}_{S_{1}S_{2}}(t^{\prime}-t_{1})\,\sigma^{K_{2}\,<}_{S_{f}S_{2},S_{i}S_{1}}(t_{1}-t_{2})\,g^{>}_{S_{i}S_{1}}(t_{2}-t_{1})\,g^{>}_{S_{i}S_{f}}(t_{1}-t)\langle\hat{F}_{S_{i}S_{f},S_{i}S_{f}}(t)\rangle

- σ_{S_{f} S_{i}, S_{2} S_{1}}^{K_{1} >} (t - t^{'}) g_{S_{2} S_{1}}^{<} (t^{'} - t_{1}) σ_{S_{2} S_{f}, S_{1} S_{i}}^{K_{2} >} (t_{1} - t_{2}) g_{S_{1} S_{i}}^{<} (t_{2} - t_{1}) g_{S_{f} S_{i}}^{<} (t_{1} - t) ⟨ \hat{F}_{S_{f} S_{i}, S_{f}, S_{i}} (t)⟩

+ σ_{S_{f} S_{i}, S_{2} S_{1}}^{K_{1} >} (t - t^{'}) g_{S_{2} S_{1}}^{>} (t^{'} - t_{2}) σ_{S_{i} S_{1}, S_{f} S_{2}}^{K_{2} <} (t_{2} - t_{1}) g_{S_{f} S_{2}}^{>} (t_{1} - t_{2}) g_{S_{f} S_{i}}^{<} (t_{2} - t) ⟨ \hat{F}_{S_{f} S_{i}, S_{f} S_{i}} (t)⟩

\displaystyle-\sigma^{K_{1}\,<}_{S_{i}S_{f},S_{1}S_{2}}(t-t^{\prime})\,g^{<}_{S_{1}S_{2}}(t^{\prime}-t_{2})\,\sigma^{K_{2}\,>}_{S_{1}S_{i},S_{2}S_{f}}(t_{2}-t_{1})\,g^{<}_{S_{2}S_{f}}(t_{1}-t_{2})\,g^{>}_{S_{i}S_{f}}(t_{2}-t)\langle\hat{F}_{S_{i}S_{f},S_{i}S_{f}}(t)\rangle\bigg{)}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

\displaystyle\bigg{(}\ \sigma^{K_{1}\,>}_{S_{f}S_{i},S_{2}S_{1}}(t-t^{\prime})\,g^{>}_{S_{2}S_{1}}(t^{\prime}-t_{2})\,\sigma^{K_{2}\,<}_{S_{i}S_{1},S_{f}S_{2}}(t_{2}-t_{1})\,g^{<}_{S_{f}S_{2}}(t_{1}-t)\,B^{>}_{S_{i}S_{2}}(t-t_{2})

\displaystyle+\sigma^{K_{1}\,<}_{S_{i}S_{f},S_{1}S_{2}}(t-t^{\prime})\,g^{<}_{S_{1}S_{2}}(t^{\prime}-t_{2})\,\sigma^{K_{2}\,>}_{S_{1}S_{i},S_{2}S_{f}}(t_{2}-t_{1})\,g^{>}_{S_{2}S_{f}}(t_{1}-t)\,B^{<}_{S_{2}S_{i}}(t-t_{2})\bigg{)}

- 2 \mbox I m S_{1}, S_{2} \sum K_{1}, K_{2} \in {L, R} \sum \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t_{1}} d t^{'} \int_{- \infty}^{t^{'}} d t_{2}

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Supporting Information:

Electron Transfer Methods in Open Systems

Nicolas Bergmann

Department of Chemistry, Technical University of Munich, D-85748 Garching, Germany

Michael Galperin

[email protected]

Department of Chemistry & Biochemistry, University of California San Diego, La Jolla, CA 92093, USA

keywords:

electron transfer, kinetic schemes, quantum transport, Hubbard nonequilibrium Green’s functions

Here we give explicit expressions for second and fourth order transfer rates: first in terms of locators, spectral weights and correlation functions; after that, result of substitution of zero order expressions for the latter. results of Refs. 1, 2 for $W^{(n)}_{S_{f}\leftarrow S_{i}}$ rate is obtained by setting $P_{S_{i}}=1$ and all other probabilities to [math].

1 Second Order Rates

The only diagram contributing to second order rate is shown in Fig. 1a of the main text.

First and fourth terms in the right side of Eq. (11) in the main text

[TABLE]

contribute to $W^{(2)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$ . Explicit expression for the rate is

[TABLE]

Second and third terms in the right side of Eq. (11) in the main text

[TABLE]

contribute to $W^{(2)}_{S_{i}\leftarrow S_{i}}\,P_{S_{i}}$ . Explicit expression for the rate is

[TABLE]

It is easy to check that

[TABLE]

2 Fourth Order Rates

Contributions are classified by projection in Fig. 3 and diagram in Fig. 1 of the main text. In the expressions below

[TABLE]

2.1 A.(0).(s) projections

This contribution is of the type $W^{(4)}_{S_{i}\leftarrow S_{i}}\,P_{S_{i}}$

2.1.1 Diagram (c)

[TABLE]

2.1.2 Diagram (d)

[TABLE]

2.1.3 Diagram (g)

[TABLE]

2.1.4 Diagram (h)

[TABLE]

2.1.5 Resulting zero-order expression

[TABLE]

2.2 A.(1).(s) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.2.1 Diagram (c)

[TABLE]

2.2.2 Diagram (d)

[TABLE]

2.2.3 Diagram (g)

[TABLE]

2.2.4 Diagram (h)

[TABLE]

2.2.5 Resulting zero-order expression

[TABLE]

2.3 A.(1).(t) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.3.1 Diagram (g)

[TABLE]

2.3.2 Diagram (h)

[TABLE]

2.3.3 Resulting zero-order expression

[TABLE]

2.4 A.(2).(t) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.4.1 Diagram (g)

[TABLE]

2.4.2 Diagram (h)

[TABLE]

2.4.3 Resulting zero-order expression

[TABLE]

2.5 B.(0).(s) projections

This contribution is of the type $W^{(4)}_{S_{i}\leftarrow S_{i}}\,P_{S_{i}}$

2.5.1 Diagram (b)

[TABLE]

2.5.2 Diagram (c)

[TABLE]

2.5.3 Diagram (d)

[TABLE]

2.5.4 Diagram (i)

[TABLE]

2.5.5 Resulting zero-order expression

[TABLE]

2.6 B.(1).(s) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.6.1 Diagram (b)

[TABLE]

2.6.2 Diagram (c)

[TABLE]

2.6.3 Diagram (d)

[TABLE]

2.6.4 Diagram (i)

[TABLE]

2.6.5 Resulting zero-order expression

[TABLE]

2.7 B.(1).(t) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.7.1 Diagram (b)

[TABLE]

2.7.2 Diagram (i)

[TABLE]

2.7.3 Resulting zero-order expression

[TABLE]

2.8 B.(2).(t) projections

This contribution is of the type $W^{(4)}_{S_{i}\leftarrow S_{i}}\,P_{S_{i}}$

2.8.1 Diagram (b)

[TABLE]

2.8.2 Diagram (i)

[TABLE]

2.8.3 Resulting zero-order expression

[TABLE]

2.9 C.(0).(s) projections

This contribution is of the type $W^{(4)}_{S_{i}\leftarrow S_{i}}\,P_{S_{i}}$

2.9.1 Diagram (c)

[TABLE]

2.9.2 Diagram (d)

[TABLE]

2.9.3 Diagram (g)

[TABLE]

2.9.4 Diagram (h)

[TABLE]

2.9.5 Resulting zero-order expression

[TABLE]

2.10 C.(1).(s) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.10.1 Diagram (c)

[TABLE]

2.10.2 Diagram (d)

[TABLE]

2.10.3 Diagram (g)

[TABLE]

2.10.4 Diagram (h)

[TABLE]

2.10.5 Resulting zero-order expression

[TABLE]

2.11 C.(1).(t) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.11.1 Diagram (b)

[TABLE]

2.11.2 Diagram (g)

[TABLE]

2.11.3 Diagram (h)

[TABLE]

2.11.4 Resulting zero-order expression

[TABLE]

2.12 C.(2).(t) projections

This contribution is of the type $W^{(4)}_{S_{f}\leftarrow S_{i}}\,P_{S_{i}}$

2.12.1 Diagram (b)

[TABLE]

2.12.2 Diagram (g)

[TABLE]

2.12.3 Diagram (h)

[TABLE]

2.12.4 Resulting zero-order expression

[TABLE]

Bibliography2

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1Leijnse and Wegewijs 2008 Leijnse, M.; Wegewijs, M. R. Kinetic Equations for Transport Through Single-Molecule Transistors. Phys. Rev. B 2008 , 78 , 235424
2Koller et al. 2010 Koller, S.; Grifoni, M.; Leijnse, M.; Wegewijs, M. R. Density-Operator Approaches to Transport Through Interacting Quantum Dots: Simplifications in Fourth-Order Perturbation Theory. Phys. Rev. B 2010 , 82 , 235307