In magnetic-doped topological insulators such as Cr-doped (Bi,Sb)₂Te₃, the interplay between topology and magnetism gives rise to a unique electromagnetic response governed by axion electrodynamics. This framework extends Maxwell's equations to include a topological magnetoelectric term, leading to phenomena like the quantized magnetoelectric effect. The underlying mechanism hinges on the breaking of time-reversal symmetry by magnetic doping, which opens a gap in the surface states of the topological insulator while preserving the bulk insulating state. The resulting electromagnetic response is characterized by a topological theta term in the action, where theta is quantized to π in units of the fine structure constant. This term couples electric and magnetic fields in a manner distinct from conventional materials, leading to exotic effects such as the topological magnetoelectric effect.

The quantized magnetoelectric effect manifests as a direct relationship between an applied electric field and an induced magnetic polarization, or vice versa, with a quantized coefficient. In Cr-doped (Bi,Sb)₂Te₃, the surface states acquire a mass gap due to magnetic ordering, and the bulk responds to electromagnetic fields through the axion term. The effective Lagrangian for the electromagnetic field in such a system includes an additional term proportional to theta times the dot product of the electric and magnetic fields, E·B. Theta is quantized to an odd multiple of π, reflecting the topological nature of the material. This quantization is robust against perturbations as long as the bulk gap remains intact and time-reversal symmetry is broken.

The implications of this effect are profound for exotic electromagnetism. One consequence is the prediction of a half-integer quantum Hall effect on the surface of the topological insulator when a magnetic field is applied. The surface Hall conductivity is quantized to e²/2h, a signature of the axion electrodynamics at play. Additionally, the magnetoelectric coupling leads to unusual electromagnetic wave propagation, including non-reciprocal effects where the transmission of light depends on the direction of propagation. This non-reciprocity is a direct result of the topological magnetoelectric polarization and has potential applications in designing optical isolators and circulators.

Another key implication is the emergence of image magnetic monopoles. When an electric charge is brought near the surface of a magnetic-doped topological insulator, the axion field induces a mirror image magnetic monopole in addition to the conventional image charge. This effect arises because the theta term modifies the boundary conditions for the electromagnetic fields at the surface. While true magnetic monopoles do not exist in isolation, the image monopole provides a tangible signature of the topological magnetoelectric effect. Experimental verification of this phenomenon would constitute strong evidence for the axion electrodynamics framework in these materials.

The quantized magnetoelectric effect also has consequences for the dynamical properties of the system. For instance, the low-frequency electromagnetic response includes a term proportional to the time derivative of theta, which can lead to anomalous Hall-like effects even in the absence of an external magnetic field. This dynamical axion field couples to the electromagnetic field and can influence the dispersion relations of collective modes such as plasmons and magnons. The resulting hybrid modes exhibit unique properties, including gap openings and modified propagation characteristics, which could be exploited in novel optoelectronic devices.

In terms of material realizations, Cr-doped (Bi,Sb)₂Te₃ has been extensively studied due to its relatively high Curie temperature and well-defined magnetic ordering. The doping level of Cr is critical, as it must be sufficient to induce long-range magnetic order without disrupting the topological insulating phase. Typical doping concentrations range from 5% to 20%, with the exact value depending on the desired balance between magnetic and electronic properties. The magnetic transition temperature in these systems is typically below 100 K, but efforts are ongoing to identify materials with higher ordering temperatures to facilitate practical applications.

The experimental observation of the quantized magnetoelectric effect requires precise measurements of the magnetoelectric susceptibility. Techniques such as torque magnetometry and polarized neutron reflectometry have been employed to probe the coupling between electric and magnetic fields. These measurements confirm that the magnetoelectric coefficient is indeed quantized in units of the fine structure constant, as predicted by theory. However, challenges remain in isolating the topological contribution from other magnetoelectric effects that may arise due to material imperfections or extrinsic factors.

Beyond Cr-doped (Bi,Sb)₂Te₃, other magnetic topological insulators such as Mn-doped Bi₂Te₃ and V-doped Sb₂Te₃ have been explored for their axion electrodynamic properties. Each system offers unique advantages and challenges, depending on the specific magnetic interactions and band structure details. For example, Mn-doped systems tend to exhibit stronger magnetic anisotropy, which can enhance the stability of the quantized magnetoelectric effect but may also complicate the interpretation of experimental results. The choice of material system thus depends on the intended application and the specific phenomena under investigation.

The broader implications of axion electrodynamics in topological insulators extend to fundamental physics and potential technological applications. On the fundamental side, these materials provide a condensed matter analog of the axion, a hypothetical particle originally proposed in high-energy physics to solve the strong CP problem. The realization of axion-like electrodynamics in topological insulators offers a platform to study related phenomena in a controlled laboratory setting. On the technological side, the unique electromagnetic properties could enable new types of sensors, memory devices, and quantum computing components. For instance, the non-reciprocal optical effects could be harnessed for on-chip optical isolation, while the quantized magnetoelectric response might be used in precision metrology.

In summary, magnetic-doped topological insulators exhibit a rich interplay between topology and magnetism that gives rise to axion electrodynamics and the quantized magnetoelectric effect. This framework leads to exotic electromagnetic phenomena, including non-reciprocal wave propagation, image magnetic monopoles, and hybrid collective modes. Experimental realizations in materials like Cr-doped (Bi,Sb)₂Te₃ have confirmed key predictions of the theory, though challenges remain in optimizing material properties and isolating topological effects. The implications span both fundamental physics and potential applications, making this a vibrant area of research in condensed matter physics and materials science.