Critical Mechanism of Magnetic Doped Cr$_{x}$Bi$_{2-x}$Te$_{3}$ Topological Insulator Thin Films*

Introducing magnetic dopants into topological insulators can lead to the opening of the surface band gap which can induce interesting phenomena such as the quantized anomalous Hall effect (QAH) and magnetoelectric effect. However, the critical properties of ferromagnetism in magnetic doped TIs are not well studied. In this work, we investigated the effect of magnetic doping on magnetic and transport response in Bi$_{2}$Te$_{3}$ thin films. Cr$_{x}$Bi$_{2-x}$Te$_{3}$ thin films with x $=$ 0.03, 0.14, and 0.29 were grown with low surface roughness ($\sim$ 0.4 nm). It is found that Cr is an n-type doping element, which reduces the carrier density of p-type Bi$_{2}$Te$_{3}$. Moreover, doping Cr induces long range ferromagnetism when x$=$0.14 and 0.29, where anomalous Hall effect and weak localization of magnetoconductance were observed. The Arrott-Noakes plot for Cr$_{x}$Bi$_{2-x}$Te$_{3}$ demonstrates that the critical mechanism of the ferromagnetism can be described well with 3D-Heisenberg model. Our work may benefit for the practical applications of ferromagnetic TIs with opened surface band gap in spintronics and magnetoelectric devices.