耗散奈米線中接近量子相變點的非平衡電子傳輸行為

Abstract

本論文目的在探討耗散環境中電子越過位能障礙的傳輸行為，主要涉及平衡(eV≪k_B T)與非平衡(eV≳k_B T)導電度的差異，及量子相變點附近的標度行為。杜克大學團隊最近的實驗(H. Mebrahtu et al., Nature Physics 9, 732–737 (2013))成功的在一個耗散系統中實現了Luttinger液體裡的共振穿隧現象。實驗中以一條奈米碳管上之兩點連接源電極與汲電極形成一個量子點的結構，在碳管與電極的接點處形成位能障礙。此外再於奈米碳管上外加一個閘電極控制量子點上的能階，並於兩個接點處各加上一個旁電極，以控制電子分別從源電極與汲電極到量子點的穿隧強度。在此一系統中，當兩個電極到量子點的穿隧強度相等時，調控閘極電壓可使源極中電子通過兩個接點處的位障到汲極的導電度達到共振值(e^2/h)。實驗的結果顯示導電度隨著閘電極變化的趨勢，在平衡態與非平衡態的情形之下分別在接近共振點的區域和微小導電度的區域滿足相同的指數律，但在由共振區到微小導電度區之間的過渡區域，兩種情形下的導電度存在明顯差異，這是我們想嘗試解釋的第一個現象。我們利用非平衡態格林函數的技巧，計算出在有限電壓之下此一系統的非平衡態電流公式，並結合重整化群方法所得出來的平衡態導電度，推導出非平衡狀態之下的導電度。將理論計算的結果與實驗值做定性與定量上的比對，我們所得的結果在一定程度上能有效說明實驗的結果。論文的第二個主題，在探討系統的導電度於接近共振點附近時與外界溫度或偏壓之間的標度關係。實驗顯示，同樣在兩端電極到量子點上的穿隧強度相同之下調控閘極電壓使系統處於共振點，此時電導隨著外界溫度或偏壓變化的標度律會與系統偏離共振態時有所不同。我們將電路中的耗散納入考慮，得出描述系統的有效作用量，並藉由重整化群的理論計算出與實驗結果一致的標度律。

This thesis investigates transport of electrons through a symmetric double-barrier in a dissipative environment and its critical behavior near the resonant tunneling point. One recent experiment (H. Mebrahtu et al., Nature Physics 9, 732–737 (2013)), exhibits the existence of resonant tunneling in a dissipative system equivalent to a Luttinger liquid. In the experiment, such a system is realized by employing a carbon nanotube onto which a source lead and a drain lead are attached at two different points to form a quantum dot structure. A gate voltage is then applied on the dot to control the energy level, while two additional side gates placed near where the dot and the leads meet are used to tune the coupling from the leads to the dot. When the system is in “symmetric coupling”, conductance through the double-barrier could be tuned to resonance (G=e^2/h) by manipulating the gate voltage. It is found in the experiment that the conductances in equilibrium (eV≪k_B T) and non-equilibrium (eV≳k_B T) both have the same power law dependence of the gate voltage in the resonant tunneling region and weak tunneling region but are remarkably different in the crossover region. This is the first phenomenon in the experiment that we attempt to explain. Combining the non-equilibrium Green’s function technique and the renormalization group theory to derive the current through a double-barrier quantum dot under a finite bias voltage, we are able to find a considerably persuasive analytical solution for describing the non-equilibrium conductance. The second concern of this thesis is to study the critical behavior of the conductance near the resonance point. The experiment shows that there exists a power law dependence of the system’s temperature or bias voltage for conductance through the double-barrier at the resonance, which becomes different when conductance is off-resonance. We include the dissipation in the circuit to find the effective action describing the system. In addition, via renormalization group method, we find the power law of conductance which is consistent with the experiment result.