be denoted by energy levels split by ~ω∗c and written as
ν∗= nh
eB∗, (2.20)
while fractional and integer filling factors can be related using the general form:
ν = ν∗
2pν∗± 1. (2.21)
2.4 The quantum anomalous Hall effect
Having introduced the IQHE and FQHE, we now touch on the concept of topology before turning to the quantum anomalous Hall effect (QAHE). Topology is a branch of mathematics concerned with the physical properties of systems that are preserved un-der continuous deformation, for example variations in the shape or size of an object (see Refs. [33, 34, 35]). While this field originated at the start of the 20th century1, the dis-covery of the quantum Hall effect in 1980 represented the first experimental realisation of topologically protected behaviour in nature [4]. As we saw in the previous sections, the conducting edges in the IQHE are quantised to remarkable precision in units of e2/h, and can only change in a discrete fashion. This quantisation is robust to physical perturbations, including the length of the edge or presence of disorder in a sample. In fact, it is the topologically non-trivial electronic band structure in the quantum Hall ef-fect that gives rise to this phenomena, and endows the edges withtopological protection.
To understand this, we first take a step back and consider 2D surfaces more gene-rally. Thegenus of an object is a global property that is related to the number of holes in a system. Objects with the same genus can be smoothly deformed into one another.
The canonical example is a sphere which has a genus of 0, and a doughnut which has a genus of 1. A doughnut can be pulled and elongated, but can not enter the same geometric class as a sphere without closing the gap in its centre. The two objects are topologically distinct. The Gauss-Bonnet and later Shiing-Shen Chern theorems [36]
convey this relationship between geometry and topology as the integral of a surface’s
1Although, it arguably has roots in the work of Euler in the 18th century.
2. THE QUANTUM HALL EFFECT
curvature over an area. This concept can be mapped onto condensed matter systems by considering particle wavefunctions in momentum space (see §1.1). Integrating the Berry curvature over the Bloch bands gives rise to an integer quantity known as the Chern number, n, which classifies the topological character of different systems (see Refs. [33, 35] for detailed discussion).
In the IQHE,n is unity for each Landau level within a sample, and zero outside the sample. Such a system can be considered as a topological insulator where a gap exists between the fully filled and empty energy levels. It follows then that in order for n to change from a finite value to zero, a gap in the system has to close. In this way, energy levels that give rise to non-zeron must depopulate by crossing EF at some point. This condition leads to the opening ofn gapless 1D modes at the sample boundary, forming conductive edge channels. In the quantum Hall effect, the Chern number is therefore equivalent to the number of QH edge states in a system, with the sum of all Chern numbers giving the total conductivity in units of e2/h [34].
While topological effects can be observed in the IQHE regime, realising this phe-nomenon requires a large magnetic field in order to break time reversal symmetry. In seminal work by Thouless [37] and later Haldane [38], a new type of ordering was predicted to occur in specific material systems in the absence of an external magnetic field. This topological phase of matter was later understood as the basis for the QAHE.
There are two key ingredients required to experimentally realise the QAHE: strong spin-orbit coupling, and the introduction of ferromagnetic doping (for reviews, see Refs.
[39, 40, 41, 42]). The spin-orbit (S-O) interaction is a relativistic effect experienced in the rest frame of an electron that couples its spin to its orbital moment. The effect of S-O coupling in the QAHE is to drive inversion between the conduction and valence bands, as shown in Fig. 2.5. This leads to a topologically non-trivial band structure.
The presence of ferromagnetic exchange then lifts the degeneracy at the Dirac point and opens a gap in 2D surface states ((c) in Fig. 2.5). This long-range coupling of mag-netic impurities is proposed to be mediated by two different mechanisms: the RKKY interaction or Van Vleck paramagnetism [43]. Experimentally, these properties can be realised by introducing chromium or vanadium dopants in thin films of materials with
2.4 The quantum anomalous Hall effect
(b)
EF
E k
(a) (b) (c)
Topological
insulator TI+ ferromagnetic exchange Normal
insulator
Figure 2.5: Band structure cartoon of topological insulators. (a) Energy band diagram for a normal insulator, with the fully filled valence band shown in red and the empty conduction band in blue. As the Fermi energy EF (dashed black line) lies within gap between the two bands, the flow of current is prohibited. (b) Illustrates a topolo-gical insulator, where there is bulk band inversion between the conduction and valence bands. Topologically protected surface states can exist within the gap. In (c), introducing ferromagnetic dopants like chromium or vanadium to a topological insulator leads to the breaking of time reversal symmetry, and opens a gap in the 2D surface states. Adapted from Ref. [34].
2. THE QUANTUM HALL EFFECT
Figure 2.6: 3D thin-film magnetic topological insulator. (a) Illustration of two chiral paths (dark blue) located around the top and bottom corners of a thin-film topolo-gical insulator with ferromagnetic doping. The sample has been uniformly magnetised in the z direction such that Mz = +1 (indicated by the red arrows). Each channel contribu-tes e2/2h to the total conductance, leading to a common fermionic, chiral path along the domain wall with σH= e2/h. The relevant energy band structure is shown in (b). The top and bottom surfaces have different topological characters (and therefore Chern numbers) owing to the net sample magnetisation and opposite pointing normal vectors. The energy gap between the bands is given by ±∆z. At the sidewall between the surfaces of different magnetisation, the gap closes and the Chern number is zero. Adapted from Refs. [42, 44].
a topologically non-trivial band structure. For example, the Bi2Se3/Bi2Te3/Sb2Te3 fa-milies.
Figure 2.6 illustrates the quantum anomalous Hall effect in a 3D magnetic thin film with strong S-O coupling. When polarised past the coercive field along z, all the elec-tron spins align in one direction leading to a net sample magnetisation Mz. However as the normal vector takes opposite signs for the top and bottom surfaces, the Chern number must transition between n = ±1. This leads to the formation of a chiral edge state along the domain boundary at the side wall. The total conductance of this edge is given by the sum of the top and bottom surfaces, each contributing e2/2h for a to-tal of σH = e2/h. Experimentally, the first realisation of the QAHE was in Cr-doped (Bi,Sb)2Te3 in 2013 [45].