Theoretical Studies on Novel Electronic Transport Properties of Graphene Systems

石墨烯是具有二維蜂窩狀晶格的單原子層石墨。它的價帶跟導帶在六角形布里淵區的六個頂點相接在一起。這六個頂角即被稱為Dirac 點。而在每個Dirac點附近，石墨烯擁有線性能帶跟手徵性本徵態。因此，石墨烯裡的電荷載子即使處於低能量還是能夠展現相對論力學。好幾個理論跟實驗工作已經顯示這種Dirac類型的費米子可能導致許多有趣的傳輸現象,例如奇異量子霍爾效應跟Klein 穿隧效應。 在本論文，我們將報告我們關於石墨烯電子傳輸方面的研究。我們的研究可以分成三類: (i)超導傳輸， (ii)量子霍爾效應， 跟(iii)近藤效應。 在工作(i)中，我們基於Blonder-Tinkham-Klapwijk 法則，透過解自旋極化的Dirac-Bogoliubov-de-Gennes方程式，估計了石墨烯鐵磁-絕緣體-超導體接面的穿隧電導。在薄屏障限制，如同正常金屬-絕緣體-超導體接面，電導會隨著阻礙強度震盪。有趣地，對於電導對阻礙強度的圖，我們發現了交換能量大於費米能量 跟小於費米能量的曲線之間，普遍存在一個pi/2 的相位差。這項研究已經被發表在Phys. Rev. B 81, 045412 (2010)。 在工作(ii)中，我們研究AA 堆疊雙層石墨烯的量子霍爾效應。使用線性響應理論裡的Kubo 公式，我們同時估算了霍爾電導跟縱向電導。有趣地，我們發現AA 堆疊雙層石墨烯的量子霍爾效應具有三個奇異特徵: 電導為零的霍爾平台，周期性的8e^2/h 台階，跟強烈受磁場、化學位勢影響的性質。這項研究已經被發表在Phys. Rev. B 82, 165404 (2010)。 在工作(iii)中，我們使用數值重整群方法解石墨烯的Anderson 模型，藉此來研究石墨烯的近藤效應。 不同於傳統材料，對於石墨烯近藤效應的研究，必須考慮石墨烯的兩個特徵，手徵性跟非常數的能態密度。石墨烯的近藤效應性質被顯示會視雜質位置而定。因此，我們研究石墨烯兩個不同結構的近藤物理: (i) 雜質位於碳原子上，(ii) 雜質位於六角晶格中心。目前，我們已經發展了可應用於石 墨烯的數值重整群法則。我們解析地切割哈密爾頓函數並透過數值運算求得了Lanczos 係數。未來，我們將更進一步將我們所發展的數值重整群方法運用到石墨烯參雜鐵雜質系統上。現今，石墨烯內磁矩的起源還有導致多重通道近藤效應的原因是具有爭議性的。我們預期我們的工作會有助於釐清這兩個議題。Graphene is a single atomic layer of graphite with a two-dimensional honeycomb lattice structure. The valence and conduction bands of graphene touch each other at six corner points of the hexagonal Brillouin zone. These six corner points are known as Dirac points. Graphene possesses a linear energy dispersion and chiral eigenstates near each Dirac point. Hence, charge carriers in graphene would exhibit relativistic dynamics even at low energy. Several theoretical as well as experimental works have shown that the Dirac-type fermions may result in many intriguing transport phenomena such as anomalous quantum Hall effect and Klein tunneling.In this thesis, we will report our works on the electronic transport properties in graphene, which could be categorized into three: (i) superconducting transport,(ii) quantum Hall effect (QHE), and (iii) Kondo effect. In work (i), we calculate the tunnelling conductance of graphene ferromagnet-insulatorsuperconductor (FIS) juctions within the Blonder-Tinkham-Klapwijk formalism by solving spin polarized Dirac-Bogoliubov-de-Gennes equation. In the thin-barrier limit, the conductance G of a graphene FIS junction oscillates as a function of barrier strength chi, the same as graphene normal metal-insulator-superconductor junctions. Interestingly, we find a universal phase difference of pi/2 exists between the G − chi curves for exchange energy E_{ex} > E_F (Fermi energy) and E_{ex} < E_F. This research has been published in Phys. Rev. B 81, 045412 (2010). In work (ii), we investigate the quantum Hall effect of AA-stacked bilayer graphene. We calculate the Hall conductivity as well as longitudinal conductivity within linear response Kubo formalism. Interestingly, we find that QHE in AA-stacked bilayer graphene possesses three unique characteristics: the filling factor ar{ u} = 0 Hall plateau, the periodic 8e^2/h-steps, and the strong dependence on magnetic field and chemical potential. This research has been published in Phys. Rev. B 82, 165404 (2010). In work (iii), we study Kondo effect in graphene by solving the graphene Anderson model, using numerical renormalization group (NRG) method. Unlike conventional host metal, for the study on kondo physics in graphene, the two characteristics of graphene, namely chirality and non-constant density of states have to been taken into account. Graphene was shown to exhibit impurity-position-dependent Kondo effcet. Therefore, we investigate the Kondo physics of two diffenent configurations in graphene: (i) an adatom impurity on the top of one carbon atom, and (ii) an adatom impurity at the center of the honeycomb. So far, we have develop a NRG formalism applicable to graphene. We have analytically discretized Hamiltonian and numerically obtained Lanczos coefficiences. In the furture, we will use our NRG formalism to investigate the Kondo effect of iron-impurities in graphene. Nowadays, the origin of magnetic moment and multi-channel Kondo effect in graphene are controversial. We expect our work would contribute to clarify the two issues