This thesis is focused on the study of carrier transport behaviour in magnetically doped topological insulator (TI) materials. Magnetic TIs have been predicted to possess many exotic properties, such as the quantum anomalous Hall (QAH) effect, in which the edge states propagate adiabatically, similar to the quantum Hall effect, but without the need of an external magnetic field. In addition to the potential of being used for low-dissipation power transmission and for metrology as a resistance standard, the QAH effect also allows the exploration of a variety of new physics. However, it has been observed that dissipation mechanisms, which theoretically should be absent in the QAH effect, still exist in real measurements and impede the application of the QAH effect on further studies. Thus, this thesis presents the experiments designed to investigate the nature of dissipation in the QAH effect and discusses the possible ways to improve the QAH effect for future experiments.

The first part of the thesis introduces the study of the magnetic and electric properties of V-doped (Bi,Sb)$2$Te$3$ films. Transport measurements and various scanning probe microscopy (SPM) techniques are employed to probe the relation between magnetic domain structures and carrier transport behaviour in the samples during the QAH phase transition as well as in the QAH regime. Patch-like structures ranging from several hundred nanometers to 2 \textmu m with a stray magnetic field strength lower than 1 G are observed in the magnetic images, which may originate from the domains of the magnetic TI films. The activation energy extracted from Arrhenius plots of longitudinal resistance versus temperature is $\sim$ 4 \textmu eV (i.e. $\sim$ 47 mK). Such a small energy scale implies that the Fermi level is far away from the centre of the magnetic gap and the bulk conduction may dominate the transport. Thermal effects arising from the complicated interplay between the electronic and spin system in our samples are observed and the possible origins of them are discussed.

The second part of the thesis is dedicated to the study of transport properties of a Cr-doped (Bi,Sb)$2$Te$3$ sandwich heterostructure. The QAH effect with a non-vanishing longitudinal resistance $\sim$ 0.0667 $h/e2$ is observed, suggesting the presence of dissipative conducting channels in the sample. By multi-terminal measurements, we verify the existence of non-chiral edge states. A possible candidate of these non-chiral edge states is the quasi-helical states on the side surface of the sample that become dissipative due to the broken time-reversal symmetry. It is shown that the backscatterings of these quasi-helical states have a magnetic field dependence and contribute to the dissipation in parallel with the residual surface and bulk states. A theoretical model is employed to understand the interactions between different kinds of conducting channels. Based on this model, we speculate that a transition between different dominant dissipation mechanisms in the QAH effect occurs at a transition field $H\ast \sim$ 2.5 T at the base temperature, which suggests a mean free path $\sim$ 20 nm in this Cr-doped (Bi,Sb)$2$Te$3$ sample. The activation energy exhibits a magnetic field dependence as well, ranging from 35.0 \textmu eV at $\mu 0H=$ 0 T to 16.2 \textmu eV at $\mu 0H=\pm$ 5 T, which agrees with the transition behaviour observed at $H\ast$ and indicates that it may also result from the shift between different dominant dissipation mechanisms in the QAH regime. The magnetic gap of the sample is estimated to be at least larger than 43.7 \textmu eV. Moreover, $H\ast$ is found to have a temperature dependence, which is probably due to the increasing number of diffusive conducting states at higher temperatures according to our analysis. Finally, we demonstrate that the electron temperature can differ from the sample temperature due to Joule heating at millikelvin temperatures. The inferred electron temperature is found to be proportional to the source-drain current in our measurements. Because so far the QAH effect can only be observed at millikelvin temperatures and Joule heating should always play an important role in this regime, the relation obtained in our experiments will be useful for future studies on the QAH effect.