高溫銅氧化物是準二維材料，當材料被電洞摻雜時可以觀察到許多新現象。在本論文中，我們重點研究了低摻雜區域中高溫銅氧化物的穿隧能譜。我們首先分析了高溫銅氧化物的穿隧能譜，在緊束縛極限中具有均勻的 d 波配對對稱性。通過在緊束縛極限中使用 BTK 公式，我們計算了普通金屬 -（110）方向上高溫銅氧化物接面的穿隧能譜，並驗證了零偏壓電導峰值出現在能譜中。我們接下來推廣 BTK 公式來研究當高溫銅氧化物處於贗能隙相時的穿隧能譜。贗能隙相由具有 d 波對稱性的配對密度波 (PDW) 態建模，並且有序波向量在 x 方向或 y 方向。我們發現與來自 PDW 態的 Andreev 反射相關的穿隧能譜表現出幾個重要特徵，在微分電導曲線的特定電壓處具有峰值或階躍。這些特徵是穿隧實驗中 PDW 態的重要特徵。

The high-Tc cuprates are quasi-two dimensional materials with many novel phenomena observed when the materials are hole-doped. In this thesis, we focus on the tunneling spectrum of high-Tc cuprates in the underdoped region. We first analyze the tunneling spectrum for high Tc cuprates with a uniform d-wave pairing symmetry in tight-binding limit. By using the BTK formalism in the tight-binding limit, we compute the tunneling spectrum for the junction of normal metal - high Tc cuprates in the (110) direction and verify that a zero-bias-conductance-peak (ZBCP) emerges in the spectrum. We next generalize the BTK formalism to investigate the tunneling spectrum when the high-Tc cuprates are in the pseudo-gap phase. The pseudo-gap phase is modeled by the pair density wave (PDW) state with d-wave symmetry and the ordering wavevector being either in x direction or y direction. We find that the tunneling spectrum that is associated with the Andreev reflections from the PDW state exhibit several important features with either peaks or steps at specific voltages in the differential conductance curve.
These features serve as important signatures for the PDW state in tunneling experiments.

摘要 I
Abstract II
Acknowledgements III
Contents V
List of Tables VI
List of Figures IX
1 Introduction 1
2 Tunneling Spectroscopy of d-wave Superconductors 5
2.1 Local density of states . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Zero Energy Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Non-Zero Momentum Gap Function . . . . . . . . . . . . . . . . . . . 10
3 BTK method for NS Junction 11
3.1 N-Isotropic Superconductors in the continuum limit . . . . . . . . . . . 11
3.2 N-Anisotropic Superconductors in the continuum limit . . . . . . . . . 16
4 N-dSC Junction in the tight-binding limit 18
4.1 N(100)-I-dSC Junction . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.1 N-I-d(100) junction . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 N-I-d(110) junction . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.3 Boundary conditions for two find of junctions . . . . . . . . . . 23
IV
4.2 N-I-N junction for normalized conductance . . . . . . . . . . . . . . . 26
5 Pseudo-gap Phase of Cuprate Superconductors and N/dPDW junction 30
5.1 Coupled to three-band PDW . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 dPDW state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 N-I-dPDW(100) Junction . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Conclusions and Future Works 40
Bibliography 41

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