單分子量子輸送中的電流分佈與電流遲滯現象之理論研究

Abstract

本論文共包含三個研究主題。首先，在第一章我們分析單分子接點 (single molecular junction) 中電流分佈 (current distributions) 與分子軌域的關係。藉由休克爾模型 (Hückel model) 搭配非平衡格林函數方法 (non-equilibrium Green's function technique) 及一階微擾理論，我們由局域電流公式 (local current formula) 出發，在考慮分子有對稱性的情況下推導出一個簡化的局域穿透度公式；此簡化的局域穿透度公式顯示電子的局域穿透度與躍遷密度矩陣非對角元 (off-diagonal elements of transition density matrices) 有密切關係。文中以單環分子接點 (single-ring molecular junctions) 和多環分子接點 (multi-ring molecular junctions) 為例，說明躍遷密度矩陣非對角元與局域穿透度之間的關聯。在單環及多環分子接點的分析中，我們發現只需考慮簡併或近簡併分子軌域的躍遷密度矩陣就能描述分子內環電流 (circular current) 的分佈。 其次，在第二章我們研究於電流-電壓特性曲線所呈現的電流遲滯現象。基於Longuet-Higgins-Salem (LHS) 模型，我們發展了一個簡單的絕熱理論 (adiabatic theory)，並且用此絕熱理論探究由一個可形變π-電子雙原子分子 (deformable π-electron diatomic molecule) 所組成的分子接點的電荷輸送性質 (charge transport properties)。文中以絕熱位能曲線 (adiabatic potential energy curve) 分析電極-分子耦合強度與偏壓電位對分子結構的影響。在弱電極-分子耦合的情況，我們發現在特定偏壓範圍中分子的絕熱位能曲線具有雙位能阱 (double potential wells)，並且在這特定偏壓範圍分子可以有兩種穩態結構 (steady-state geometry) 分別對應到雙位能阱的兩個最小值處；這會導致電流-電壓特性曲線顯示奇特的電流遲滯現象。然而，隨著電極-分子耦合強度增強，雙位能阱之間的能量障礙會逐漸減小，並且電流遲滯現象也會逐漸消失。 最後，在第三章我們藉由 LHS 模型搭配非平衡格林函數方法，在晶格弛豫理論的框架下分析不同鏈長 (chain length) 聚甲炔分子接點 (polymethine molecular junctions) 及聚乙炔分子接點 (polyacetylene molecular junctions) 的電荷輸送性質 (charge transport properties)。在不考慮外加電場的情況下，對於相似鏈長的分子，電流-電壓特性曲線顯示聚甲炔分子接點所呈現的電流平台 (current plateaus) 相較於聚乙炔分子呈現的電流平台有較大的電流值；這歸因於聚甲炔分子有孤子 (soliton)，而偶數碳鏈本身沒有孤子所導致。另外，聚甲炔分子接點以及聚乙炔分子接點的電流-電壓特性曲線皆呈現電流遲滯現象。隨著鏈長變長，聚甲炔分子接點以及聚乙炔分子接點其電流遲滯區間的寬度皆會逐漸變大。這是由於長鏈聚甲炔分子以及長鏈聚乙炔分子的前沿軌域 (frontier orbitals) 於鏈末端原子 (chain end atoms) 的係數大小 (coefficient magnitude) 會隨鏈長變長而變小，使得長鏈共軛分子接點於前沿軌域能量 (frontier orbital energy) 附近的電子態密度分佈 (density of state distribution) 被寬化 (broadened) 的程度較不明顯，因而電流遲滯現象於長鏈共軛分子有較明顯的表現。在考慮外加電場的情況下，聚乙炔分子接點其電流遲滯區間的寬度，相對於沒考慮外加電場的情況有明顯變小；然而，聚甲炔分子接點其電流遲滯區間的寬度與沒考慮外加電場下遲滯區間的寬度相類似，此奇偶效應是因為在聚甲炔分子接點中有孤子所導致。

Three topics are included in this thesis. In the chapter 1, we study the connections between the current distribution within a single molecular junction and molecular orbitals. By using the first order perturbation theory and non-equilibrium Green’s function techniques in the framework of Hückel theory, we derive a simplified local transmittance formula from the local current formula. The simplified local transmittance formula shows that the local transmittances are closely related to the off-diagonal elements of transition density matrices. Moreover, a few simple molecular junctions consisting of single- and multi-ring conjugated systems illustrate the connections between the off-diagonal elements of transition density matrices and the local transmittances. We find that the distribution of local currents inside the molecular junctions can be faithfully described through consideration of only the transition density matrices of degenerate or nearly-degenerate molecular orbitals. In the second chapter, we study the current hysteresis phenomenon in the current-voltage characteristics. A simple adiabatic theory based on the Longuet-Higgins-Salem (LHS) model for investigating charge transport through a molecular junction consisting of a deformable conjugated two-site molecule bridging in between two electrodes is developed. Influences of the electrode-molecule coupling strengths and bias voltages on the geometry of a deformable two-site molecular bridge are also investigated. In the case of the weak electrode-molecule coupling, we found that the molecule has two steady-state geometries corresponding to the double minima of the potential energy curves at a specific bias regime. In this regime, the current-voltage characteristics can thus exhibit a peculiar hysteresis phenomenon. However, as the electrode-molecule coupling strength increases, the energy barrier of the double minima is gradually reduced and the hysteresis phenomenon is diminished. In the final chapter, by using the LHS model and the non-equilibrium Green’s function techniques in the framework of lattice relaxation theory, we study the electron transport properties of polymethine molecular junctions and polyacetylene molecular junctions with different chain lengths. In the case without considering the influences of the external electric field, for the molecular junctions with similar chain lengths, the current plateaus in current-voltage characteristics of polymethine molecular junctions show larger current values than that of polyacetylene molecular junctions. The analysis reveals that this is because polymethines have solitons but polyacetylenes have not. In addition, both of the current-voltage characteristics of polymethine and polyacetylene molecular junctions exhibit current hysteresis phenomenon. The widths of the hysteresis regions of both polymethine and polyacetylene molecular junctions increase as the chain length increases. Our analysis shows that the magnitudes of the coefficients on the chain end atoms in the frontier orbitals decrease when the chain length increases. This causes the density of state associated with the frontier orbitals less broadened for the long-chain conjugated molecular junctions. Therefore, the current hysteresis phenomenon in the long-chain conjugated molecular junctions is more apparent. On the other hand, the widths of the hysteresis regions of the polyacetylene molecular junctions with considering the influences of the external electric field are relatively small compared to the case without considering the influences of the external electric field. However, the widths of the hysteresis regions of polymethine molecular junctions in the two cases, with and without considering the influences of the external electric field, are similar. Further investigation reveals that this even-odd effect is due to the solitons inside the polymethine molecular junctions.