The exotic surface state of TIs protected by TRS can be broken by certain perturbations, such as the application of a magnetic field out of plane of the surface or elemental doping that leads to magnetic ordering out of plane. As shown in Figure 1.6, the breaking of TRS in the system results in opening a surface energy gap at the Dirac point, separating the energy states of spin-up carriers from spin-down upon the surface.
This phenomenon can be understood by considering a simple Hamiltonian for the surface states without turbulence from magnetic field as 𝐻⃑⃑ 0 = 𝑣 𝐹(kxσy – kyσx) where 𝑣 𝐹 is the Fermi velocity, 𝑘⃑ = (kx; ky) is the wave vector and σx and σy are Pauli matrices. The energy of electrons are E = ±vF k. When magnetic elements are doped in topological insulators, with inducing the ferromagnetic phases when placing the TIs at low temperature environment. With magnetization direction normal to the surface, the Hamiltonian of the system changes to 𝐻⃑⃑ =𝐻⃑⃑ 0+JMσz/2 [16], where the magnetization is 𝑀⃑⃑ =(0, 0, M) and J the dimensionless exchange coefficient. To solve the eigenfunction, energy eigenvalues of electrons are given by Ek = ±[(vF
Figure 1.6 Ideal 3D topological insulator band structure and ferromagnetic topological insulators band structure.
k)2 + (JM/2)2]1/2, where k2=kx2+ky2, which describes surface bands separated by an energy gap of JM. This can be compared directly with the well-known expression for the dispersion of a free relativistic particle from Dirac equation, Ep = ±[(cp)2 + (mc2)2] 1/2 where c is the speed of light, m is the particle mass and p is the momentum. Through comparison of the two dispersion relations, the dispersion characteristics of surface carriers in TIs are very similar to free relativistic particles, with the magnetization and exchange coupling accounting for the energy gap instead of the particle mass. It is in this situation, with a surface energy gap, that topological insulators find many of their interesting properties such as the anomalous quantum Hall effect (QAHE) and its quantized conductivity [17-20]. The anomalous Hall effect in a ferromagnet can be induced by spontaneous magnetization [21].
Figure 1.7 Schematic picture of quantum Hall effect and quantum anomalous Hall effect [22] As introduced above, the discover of quantum spin Hall effect leads to the development of TIs. Researchers also recognized that QSHE in nonmagnetic systems is funmanetally related to the anomalous Hall effect in ferromagnetic systems [21]. Through suppressing one of the
spin channels in the QSH system by inducing ferromagnetism, it will naturally leads to the QAHE [23]. Different from QHE, QAHE can happen without a external magnetic field. The combination of spontaneous magnetization in ferromagnetic TIs and strong spin-orbital coupling could take over the role of external magnetic field in QHE. QAHE was first observed in Cr-doped Bi2Te3 in 2013 [24] where the Hall resistance shows h/e2 as magnetic field drops to zero (Figure 1.7). One can observed that Hall resistance jumps from -1ℎ/𝑒2 to 1ℎ/𝑒2 directly when applied magnetic field and stay at 1ℎ/𝑒2 when remove the field in QAHE. This is different from the QHE in which Hall resistance reach to 1ℎ/𝑒2 step by step as magnetic field increasing.
Figure 1.8 Quantum Hall trio [23].
The discovery of QAHE completes the Hall effect family. Figure 1.8 shows the Hall effect family since the first Hall effect been discovered by Edwin Hall in 1897. The year of each discovery lists in the parentheses. For all three quantum Hall effect, electrons flow on the edges whereas the bulk of the systems keep insulating. When there is a net flow of electrons for Hall resistance measurements, the extra electrons only occupy the left edge channels regardless of spin direction in QHE, spin with opposite directions occupy the opposite sides
of edges in QSHE, and only spin-down electrons flow through the left edge in QAHE [23]. Besides, QHE requires the external magnetic field in 2D semiconductor or conductor. QSHE happens in nonmagnetic thin films. QAHE happens in ferromagnetic TIs.
The lossless edge channel and the exact quantization of QAHE not only spur the research of TIs but also could be applied in many fields including spintronic devices and quantized resistance can be used as a resistance standard [23]. In this work, we explore the sensor applications of TIs for ultrahigh sensivity sensors.
1.3.3 3D topological insulators
Although Bi1-xSbx alloys were the first 3D TI material discovered by Fu and Kane [25] but it is not suitable for detailed study of topological surface state and the further application due to its complicated surface state [26, 27]. Some theoretical prediction and first-principle calculations [28, 29] of Bi1-xSbx surface are not consistent with the experimental results. The understanding is still in an incomplete state for this system [7]. Therefore, Zhang et al. theoretically predicted that Bi2Se3, Bi2Te3 and Sb2Te3 should be 3D TIs using a low-energy effective model. Soon after the prediction, the experimental observation of surface states of Bi2Se3 [30], Bi2Te3 [31] and Sb2Te3 [32] were reported. These materials are all tetradymite structure which can be defined as five layer of atoms stacked in X-Y-X-Y-X form in covalence bond [7]. These five atomic layers form one quintuple layer which is about 1nm thick. The interaction between quintuple layers is via van der Waals force. This provides a change to deposite TI thin films on any substrates by van der Waals epitaxy which requires less on lattice matching. This will be discussed in next chapter. Crystal structure of 3D TI systems and quintuple layer are shown in Figure 1.9.
Figure 1.9 Crystal structure and quintuple layer [33].