一維多量子線系統的量子傳輸性質之理論研究

Abstract

我們利用了數值以及解析方法研究一維量子系統的傳輸性質，像是分子導線以及多量子導線等量子元件的電子傳輸行為，並考慮了電聲交互作用以及電子與電子間的交互作用。首先，我們考慮了分子導線內包含電子與聲子的交互作用。分子導線是由許多單位元組成，而每個單位元皆與聲子源有局域性交互作用。我們所使用的解析計算主要是透過非平衡格林函數法（NEGF）來研究分子導線的傳導行為。對於基數個單位元的分子線模型，我們發現電導會隨分子線的長度震盪並且震盪週期與外加偏壓有關。當我們考慮外加偏壓的大小非常接近聲子能量時，這種震盪的行為將會消失。對於不考慮電子與聲子交互作用的情況下，我們發現電子傳輸機制的變化，可利用增長分子線的長度使得傳輸機制由穿隧機制過渡到熱活化躍遷機制。接下來我們考慮分子線具備電聲交互作用的情況，我們發現電子傳輸機制主要與熱驅動有關。此時，藉由增加分子線的長度將使得傳輸機制由熱抑制傳導轉變為熱活化傳導機制。 第二，我們考慮另一個模型也就是多量子導線模型其本身據有電電交互作用，特別的是我們所考慮的量子導線皆屬於強交互作用系統。對於利用數值方法研究這種具備強交互作用的多量子導線之量子元件的電子傳輸行為可能遭遇困難。主要的困難之一是為了計算多量子導線間的電導，藉由Kubo 方程式我們了解到可透過計算時變電流與電流間關聯函數來求得電導，而時變問題對於數值模擬將是一大挑戰。其二，理論上我們所考慮的量子導線為無窮長，因此這種無窮大的系統在數值模擬上也是一大挑戰。此外，我們所考慮的系統屬於臨界系統，而此系統可被相應的臨界量子態所描述，而目前的數值模擬方法中，可有效模擬對於擁有尺度不變性的臨界量子態的數值方法為數不多。 有鑑於此，我們希望發展一個數值方法可用於模擬臨界系統之臨界量子態，並研究其傳導特性像是多量子導線的普式電導，而我們所利用的數值方法是架構在多尺度之糾纏度重整化法(MERA) 之上。透過電導以及電流與電流間關聯函數的關係式，多量子導線的普式電導將可利用數值方法求得。我們主要針對兩個量子導線系統，發展特定的數值方法，而此數值方法可用於計算電流與電流間關聯函數及在保角場理論(CFT) 中的primary field 所對應之scaling dimension。我們展示了兩個量子導線元件的泛行為，並利用所發展的數值方法研究量子導線在重整群(RG) 的固定點(fixed points) 的特性，並對其分類。最後，此數值方法有很大的潛力可被推廣應用至更多量子導線的量子元件之研究。

We utilize both analytic and numerical methods to study the electron transport properties with the presence of electron-phonon interactions and electron-electron interactions in low-dimensional systems such as molecular junctions and the arbitrary junctions of multiple quantum wires. First, the weak electron-phonon interaction is present inside a molecular junction, in which each unit is coupled to a local phonon bath, and the non-equilibrium Green's function method (NEGF) is employed. We observe that the conductance oscillates with the molecular chain length and that the oscillation period in odd-numbered chains depends strongly on the applied bias. This oscillatory behavior is smeared out at the bias voltage near the phonon energy. For the phonon-free case, we find a crossover from tunneling to thermally activated transport as the length of the molecule increases. In the presence of the electron-phonon interaction, the transport is thermally driven and a crossover from thermally suppressed conduction to assisted conduction is observed. Second, it is hard to numerically study the conductance for qunatum multi-wire junctions with strongly interacting leads, especially with critical leads. The difficulty lies on the fact that the conductance is related to the dynamical correlation functions, and it is the properties of an open system. In addition, the critical state with scale invariance is rarely implemented in most of numerical methods. We therefore develop a numerical method to study transport properties in critical systems and the universal conductance of quantum multi-wire junctions based on the so-called scale invariant boundary muti-scale entanglement renormalization ansatz (MERA). Utilizing the key relationship between the conductance tensor and the ground-state correlation function, the universal conductance can be evaluated within the framework of the boundary scale invariant MERA. In particular, we study the Kane and Fisher fixed point of two interacting wires with an impurity. We demonstrate how to construct the boundary MERA to estimate the current-current correlation function and scaling dimensions of primary operators. We identify the universal behavior of the junction. This shows the grand potential of using the boundary MERA to classify the fixed points of the general multi-wire junctions.