現今半導體已進入奈米等級製成，量子效應也變得不可忽視。因此，近年來量子傳輸的研究也盛行。這項子學門主要藉由電性傳輸揭露了量子干涉、量子化、疊加、拓樸、旋性等種種性質。而在未來應用方面，因為量子性質，可更加對於度量衡學的精準化、促進未來電晶體的發展，甚至於量子電腦的實現。

在我的博士研究中，我主要在研究一維以及二維系統中的量子傳輸。在低溫、低維度系統中，量子現象的顯著得以利用外加電性量測來加以證實，並且進一步的去操控、應用。在一維系統中，因為外加電場導致著電子傳輸的限制，電子-電子交互作用變得極為顯著，影響著傳輸性質讓運動中電子展現出液態特性甚至是固態特性。而另一方面，曲率在二維系統中扮演著很重要的角色，影響著電子的傳輸也產生出類似磁場的行為，被稱之為擬磁場。因此，在本論文中，我主要在研究這兩種特殊的效應在不同維度中的貢獻。

第一個題目，是關於研究一維系統中的鋸齒形線條(zigzag)的Wigner晶格。Wigner晶格自1934被理論提出後就不斷的被研究，但由於本身的不穩定結構的特性，使得大部分的探針偵測系統都難以描繪本身晶格結構。在我的研究題目中，利用了可調式的量子線，在其中形成一維的Wigner晶格，並且可隨著量子線的寬度調整，讓線性晶格轉換成鋸齒形排列晶格。我們在此量子線旁邊設置一偵測器，利用磁聚焦的技術可以直接量測電子密度的分布。當施加一小磁場時，由量子線所射出的電子會因為迴旋運動的關係，注入到偵測器。因此在實驗結果中，當晶格結構轉換時，會因為射出電子束從一束變成兩束，而可以利用偵測器量測到電子束的變化。進一步，磁聚焦技術甚至可以量測電子自旋的極化方向藉由量子線本身的自旋極化特性。此兩項研究數據證明了Wigner晶格的轉換，也驗證了我們確實觀測到鋸齒狀排列的晶格結構。最後，在實驗中也量測了Wigner晶格的溶解溫度，並且與先前的研究做比較。我們的數據藉由全電性的操控，揭示了一維的Wigner晶格的相位轉換以及自旋相位轉換，提供了一項可能的途徑來深入研究Wigner晶格的相位變化以及未來可能應用。

第二項題目，我們展現了兩種新形態的霍爾效應。這兩種霍爾效應分別稱為線性和非線性異常霍爾效應，而且是存在於時間返演對稱之中，這也不同於以往的認知:霍爾電導(電阻)的出現必須要破壞時間返演對稱。我們將雙層石墨烯放置在一周期性波浪紋路的氮化硼上，使石墨烯產生應力。因為這應力的關係，影響了雙層石墨烯之間的層間交互作用(Interlayer coupling)產生了時空間以及動量空間的擬磁場，也扭曲石墨烯的費米分布而最後導致了傾斜的狄拉克能量錐伴隨著類Rashba的自旋軌道耦合效應以及貝利曲率偶極(Berry curvature dipole)。實驗中，我們觀測到了在零磁場下，霍爾電阻隨著閘極的改變呈現非零的特殊分布，此數據也與理論非常符合。而進一步的，也在此樣品中量測到二次諧振的電阻訊號，即為非線性異常霍爾效應。在這項研究中，我們利用應力工程，可直接利用奈米製程技術來調控石墨烯的能帶結構，甚至可以創造出特殊的貝利曲率，提供了一個方法來操控層間交互作用的途徑。

Current scaling technology in the field of semiconductor engineering has advanced to include nano-size architecture, a context in which it is necessary to consider quantum effects, which, in turn, provide a basis for research into quantum transport. This subject can be apprehended through the lens of electronic transport in regard to interference, quantization, superposition, topology, and chirality on semi-conducting materials. Simultaneously, scaling technology also has practical implications given its applicability as a criterion of metrology, as a means of improving transistor performance, and its potential to realize prospective quantum computers. In this thesis, I focus on 1D and 2D quantum transport in low-temperature environments in order to investigate both the impact of electronelectron interactions when electrons are constricted in quantum wires and the manifestation of pseudo-magnetic fields in engineered 2D materials. These experiments produced interesting and intuitive results via electronic measurement and thereby demonstrate the evolution of electronic crystal and topological consequences. Two projects are described herein: imaging the zigzag Wigner phase transition in confinement-tunable quantum wires and the anomalous Hall effect in the strained bilayer graphene without a magnetic field. In the first project, I present the observation of the zigzag Wigner lattice in a 1D system. The Wigner crystal can be formed in confinement-tunable quantum wires, which can be controlled further in its phase by changing the effective width. The operation of an on-chip detector is introduced including in reference to spatial mapping and spin analysis. The experimental results show the Wigner phase transition via the use of the detector, which relies on electron focusing to determine the electron density distribution and probe the spin properties. The melting temperature for Wigner crystallization and the accompanying spin depolarization are also reported. On this basis, the results describe the formation of the 1D Wigner crystal. In addition, the results account for the crystals spin physics, which can be controlled electrically in semiconductor systems. Thus, a feasible route is proposed for experimental research into Wigner crystals and their technological applications in quantum information science. In the second project, I demonstrate two new types of Hall effects, referred to as the linear and nonlinear anomalous Hall effects under time-reversal symmetry, in an artificial corrugated bilayer graphene system designed by myself in collaboration with a research team. The interplay between the pseudo-magnetic field and the interlayer coupling is induced by the distortion of the bilayer graphene, which leads to tilted Dirac cones with Rashba-like valley-orbit coupling and inhomogeneous Berry curvature. Simultaneously, numerical calculations of the band dispersion and Hall contribution support the results consistent with the theoretical prediction. This new class of condensed-matter systems makes it possible to modify the band structure and more remarkably to access a new exotic phase of matter via strain engineering. The ability to artificially create a nontrivial band structure and Berry curvatures from conventional two-dimensional materials, such as graphene, via patterned lattice deformation has the potential to open new avenues for the exploration of a new category of matter in condensed physics.

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