High-temperature superconductors have been extensively studied over the past four decades. Much progress has been made in terms of understanding the basic fundamental mechanisms that lead to exceptionally high transition temperatures as compared to conventional superconductors. However, many questions still remain unanswered. In this thesis, we use the scanning tunneling microscopy (STM) technique to investigate the electronic properties of Bi₂Sr₂CaCu₂O_8+δ (Bi-2212) high-temperature cuprate superconductor. This material has a complex phase diagram encompassing many exotic electronic phases.

In this work, we mainly focus on three major problems. First, we study the doping and temperature evolution of the periodic local density of states (LDOS) modulations in the superconducting and the pseudogap state of Bi-2212. We use different STM acquisition modes to investigate the nature of different electronic orders, particularly q₁ ≈ 4a_o and q₅ ≈ (4/3)a_o modulations, where a_o is the Bi-Bi (or Cu-Cu) lattice constant. Our findings indicate that the LDOS features observed in conductance measurements are consistent with an impurity-induced, quasi-particle scattering scenario. However, we do not observe any STM signatures of the charge density wave (CDW) state, which is widely believed to exist in this compound. Furthermore, the energy range where these features are observed in conductance maps reveals a striking doping dependence.

The second problem we address is the vortex core electronic structure as a function of the magnetic field in a broad doping range. Vortices are key to answering some of the fundamental questions related to the properties of d-wave superconductors. Our results clearly show that the vortex core spectral features reveal a doping dependence. We demonstrate that to investigate the doping evolution of Bi-2212 vortex cores, it is essential to use a local parameter such as the local gap instead of a macroscopic sample property like the bulk critical temperature. We observe a potential link between the vortex core electronic structure, the pseudogap, the Fermi surface topology, and possibly the quantum critical point (QCP).

The third problem we discuss is a spatial variation of local gap scales (superconducting gap and the pseudogap) from the underdoped to the overdoped regime of Bi-2212. Our extremely high-resolution conductance measurements reveal that both gap scales modulate at the atomic length scale, confirming the very small size of the Cooper pairs. Furthermore, we observe local symmetry breaking of the gap within the unit cell, which could be a potential signature of the recently discovered nematic orbital-ordered state in Bi-2212. Additionally, our results indicate that the pseudogap also modulates at a long-range order correlated with the structural supermodulation. In contrast, the superconducting gap lacks striking supermodulation-induced variations.