**1. Introduction**

The quantum interaction of charge carriers with external and internal forces in different materials remains an open question for the condensed matter community. The enigma began with Edwin Hall's discovery of the classical Hall Effect in 1879 [1]. The Hall effect shows a voltage difference formed by injecting steady current and magnetic fields across a conductor or semiconductor. Due to the current and external magnetic fields interplay, voltage generation occurs. This voltage difference is due to charge confinement. In the 1980s, Von Klitzing discovered the quantum hall effect (QHE) in a two-dimensional system [2]. Due to the intense magnetic fields, charge carriers are constrained into two dimensions and exhibit topologically ordered states. The edge states at the surface cause current to flow at the superiorities, and these states are formed as a result of the high external magnetic field. These geometrical

states give birth to a new phase of matter known as topological insulators (TI) [3]. Charles Kane and Eugene Mele anticipated the development of TI in 2005 [3–8]. It is very similar to QHE, except that no external magnetic field is necessary since the inherent characteristics of materials generate the magnetic field. The spin-orbit interaction generates this magnetic field. TI is unusual because it is insulating in bulk yet conducts at the surface. The electronic wave function of a charge carrier is dependent on its shape, which varies from bulk to surface. As a result, it is referred to as a topological insulator. The SS renders the system impervious to non-magnetic doping due to protected SS. These SS are protected by time-reversal symmetry (TRS). This small property enables many unique applications due to the impervious topological states to non-magnetic disruption and their dissipation-free spin current transit. Spintronic, thermoelectric, magnetic memory storage, magnetoelectric devices, next-generation batteries, THz generators, transistors, photodetectors, and sensor applications are only a few of these applications, which is possible in these TI [3–13].

This article is focused on topological insulators, providing an overview of the topological phases and states found in TIs. Additionally, the dynamics of the carrier and phonon scattering are also discussed. TI's surface and bulk states are probed using various optical methods. The ultrafast laser pulse is employed in particular to characterize the functional characteristics of Fermions in TI. These pulses have also been used to examine the phonon vibration mode. Finally, it establishes the existence of the coherent optical phonon (COP) and coherent acoustic phonon (CAP) modes. The temperaturedependent evolution of these modes has also been examined as these phonon vibrations progress with the charge carrier dynamics. Transient Reflectance Ultrafast Spectroscopy (TRUS) was utilized to explore the non-conventional conducting surface charge carriers. A femtosecond pump beam was employed to stimulate the sample. The material was probed with a wide beam ranging from visible to near-infrared. TRUS measurements aid in determining the charge carrier dynamics and the capacity of terahertz production. Additionally, investigating carrier and phonon dynamics in a temperature-dependent manner aids in the understanding of crucial transitions associated with surface states.
