Edge states of topological insulators

Topological insulators are materials which are described by a topological number characterizing their phase.

Within the boundaries of the topological insulator, this integer number is constant and thus constitutes a global property of the insulator. This is in stark contrast to more conventional materials (e.g. a magnet) which can be described by a local order parameter (e.g. the magnetization) which might vary within the material.

Topological insulators are protected by the energy gap in their spectrum. This means that the gap has to close for the topological number to change. At the boundaries of topological insulators, the phase changes from topologically non-trivial in the insulator to topologically trivial in the vacuum. As a consequence, the gap has to close leading to edge states localized at the boundary of a topological insulator. We now briefly introduce two kinds of two-dimensional topological insulators which exhibit one-dimensional, current-carrying edge states. For further information on the quickly expanding field of topological insulators, we refer the reader to the book by Bernevig and Hughes [Bernevig13] and the reviews by Qi and Zhang [Qi11]

and by Hasan and Kane [Hasan10]. For more information on quantum Hall effects, we recommend the review by Goerbig [Goerbig09].

Quantum Hall insulators

The most prominent two-dimensional topological insulator is provided by a two-dimensional electron gas brought into the quantum Hall regime by a strong perpendicular magnetic field. We illustrate this by considering free electrons in two-dimensions. The single-particle Hamiltonian in real space governing such a system reads

Hˆ = 1

2m(−i∇ −eA)²+eV(y) +gµ_BBˆσ_z, (2.40) where∇= (∂_x, ∂_y) andA= (A_x, A_y),V(y) is a confinement potential,B =∂_xA_y−∂_yA_xthe magnetic field which points in the z-direction, g is the g-factor, µ_B the Bohr magneton, and the Pauli matrix ˆσ_z acts on the spin degree of freedom. In the Landau gauge (Ax=−ByandAy= 0) the Hamiltonian is translationally invariant in the x-direction implying that the momentum in the x-direction is a good quantum number. In the absence of a confinement potential, the eigenstates of the last Hamiltonian can be expressed in terms of the Hermite polynomialsHn(x)

Ψn,k_x,σ(x, y) =e^ikxxHn

y−y0

lB

e^{−(y−y0)/2l2B}|σi, (2.41)

with the eigenenergies

E_n,σ=nω_C+σgµ_BB (2.42)

Here we introduced the magnetic length lB = 1/p

|eB|, y0 =kxl²_B, the cyclotron frequency ωC =|eB|/m and σ= (−)1 for spin up (down) electrons.

We note that the eigenstates are localized aroundy0which is proportional tokx, while their eigenenergies are independent of kx. The spectrum of a two-dimensional electron gas in the quantum Hall regime thus consists of flat bands which are called Landau levels [see Fig. 2.2 (c)]. Whenever the chemical potential is located in between two Landau levels, the system is insulating. Furthermore, it is described by a topological invariant [Thouless82], the so-called Chern number denotedν. For the simple example of free electrons, the Chern number is given by the number of filled Landau levels below the chemical potential and can thus take on any positive integer number.

At the boundary of a quantum Hall insulator, edge states appear due to the topological nature of the system. To see this, we introduce a confinement potential which limits the size of the system. We consider the strip geometry sketched in Fig. 2.2(a) and thus take the potential to be independent of x. Assuming furthermore that V(y) varies slowly as a function of y compared to the magnetic length, we can replace the argument of the potential by y0 due to the localization of the wavefunction [cf. Eq. (2.41)]. Then the only effect of the potential is to add the term eV(y0) to the energy [Goerbig09]. The Landau levels thus bend upwards for states that are localized at the edges following the confinement potential. Because the localization in they-direction depends linearly onkx, this affects the spectrum as illustrated in Fig.2.2(c).

We note that the effect of a quickly varying potential is similar [Goerbig09]. Due to this upward bending, all Landau levels which are below the chemical potential in the bulk cross the chemical potential somewhere close to the edge. Therefore the Chern number indicates how many edge states there are. Furthermore, all states at a given edge propagate in the same direction [cf. Fig.2.2(a) and (c)] while the states at the other edge propagate in the other direction motivating the namechiral edge states. This large spatial separation between counter propagating states suppresses backscattering and results in the quantized Hall resistance which can be measured with a relative uncertainty as small as 3·10⁻¹¹[Schopfer13] and currently serves as the resistance standard

Finally, we note that the behavior discussed above can be interpreted in a semi-classical picture where the electrons in the bulk are confined to closed cyclotron orbits due to the Lorentz force [Goerbig09]. The electrons close to the edge can not complete their orbits but get scattered of the edge. This leads to propagating “skipping orbits”, as illustrated in Fig. 2.2 (a), which constitute the current carrying edge states.

Quantum spin Hall insulator

The second type of topological insulators we discuss are two-dimensional quantum spin Hall insulators. The basic idea is to combine a quantum Hall effect for spin up electrons with a quantum Hall effect for spin down electrons corresponding to an inverted magnetic field. The resulting system exhibits counter propagating edge states with opposite spin polarization.

Here we focus on systems that can be described by the Hamiltonian proposed by Bernevig, Hughes, and Zhang (BHZ) [Bernevig06] which includes quantum wells of HgTe/CdTe [Bernevig06], InAs/GaSb/AlSb [Liu08] and Ge/GaAs [Zhang13]. The Hamiltonian reads

Hˆ =

ˆh 0 0 ˆh_{T R}

, ˆh=

C+M −(B+D) ˆp² Apˆ+

Apˆ₋ C − M+ (B − D) ˆp²

, (2.43)

where A, B, C, D, and M are material specific parameters, ˆp² = ˆp²_x+ ˆp²_y, and ˆp_± = ˆp_x±iˆp_y and ˆh_{T R} is obtained from by ˆh by time reversal (i.e. complex conjugation and ˆpj → −pˆj). Here ˆh describes states with positive spin (or total angular momentum) polarization while ˆhT R describes the time-reversed states with negative spin (or total angular momentum) polarization. The full Hamiltonian is thus invariant under time-reversal. We follow the common trend in the literature in denoting the states described by ˆh (ˆh_{T R}) as spin up (down) states even when these states do not correspond to eigenstates of the spin operator. In Sec.2.3.4, we discuss the basis states of Eq. (2.43) and their spin polarization in more detail.

It can be shown that the above Hamiltonian supports counter-propagating, spin-polarized edge states forM/B>0 with their propagation direction determined by the sign ofA/B[Qi11]. Since the propagation direction is coupled to the spin of the particles, these states are called helical edge states. Figure 2.2 (a)

illustrates these edge states while Fig.2.2(d) shows the spectrum of a quantum spin Hall insulator in a strip geometry.

x

Figure 2.2: Edge states of topological insulators and their corresponding spectra. (a) Illustration of a quantum Hall insulator in a strip geometry. The insulator is characterized by the Chern number ν = 1 while the surrounding vacuum is topologically trivial (ν = 0). In a semiclassical picture, the bulk states are confined to closed cyclotron orbits while at the edge, skipping orbits provide chiral one-dimensional edge states as illustrated at the lower edge. A top gate (gray shading) can be used to guide the edge states. (b) Illustration of a quantum spin Hall insulator in a strip geometry. The insulator is characterized by the Z2

topological numberZ2 = 1 and exhibits helical edge states. Due to time-reversal symmetry, backscattering within a pair of edge states is prohibited. A top gate (gray shading) only changes the penetration depth and can not be used to guide the edge states. (c) Spectrum of a quantum Hall insulator. The momentum in x-direction is directly related to the localization iny-direction. Within the bulk, the spectrum exhibits flat bands called Landau levels which are spin split by the Zeeman energy ∆EZ = 2gµBB. The Chern number is given by the number of Landau levels that lie below the chemical potential µ. At the edges, the bands bend upwards due to the confinement potential. The same effect can be exploited to guide the edge states using a top gate (gray shading). (d) Spectrum of a quantum spin Hall insulator. The band gap of the bulk states is crossed by the spin polarized edge states. Here only the edge states of one edge are shown (adding the other edge states would result in spin degeneracy at each k_x). The crossing atk_x = 0 is protected by time-reversal symmetry guaranteeing that the edge states cross the chemical potentialµ.

In Eq. (2.43), there are no terms coupling the two spin blocks and thus backscattering within a pair of helical edge states is obviously absent. Interestingly, coupling the two spin blocks does not introduce backscattering as long as the Hamiltonian remains time-reversal invariant. This can be motivated from the spectrum shown in Fig.2.2(d). Time-reversal invariance dictates that the spectrum remains invariant when invertingkx→ −kxand exchanging spin up and down. This implies that there exists a degeneracy of the edge states atkx= 0. Since this degeneracy can only be lifted by breaking time-reversal invariance or by closing the bulk gap (then the edge states can hybridize with the bulk bands), the edge states necessarily cross the chemical potential (which lies within the band gap) even if the two spin blocks are coupled [Hasan10;Qi11].

Unlike in the quantum Hall regime, where multiple topologically protected edge states can coexist, only

a single pair of edge states is protected in quantum spin Hall insulators. Indeed if there is an even number of edge states, they can hybridize and a gap can open even for couplings which respect time-reversal symmetry.

Only if there is an odd number of edge states is one pair guaranteed to cross the full gap. Thus, quantum spin Hall insulators are characterized by aZ² topological invariant denotedZ2 equal to 0 (1) if there is an even (odd) number of edge states [Hasan10;Qi11].

Top gates

For many setups investigating transport through edge channels, it is essential to control the location of the edge states. For instance, a quantum-point contact (QPC) is formed by bringing the edge states of opposite edges in close proximity. In this section, we briefly discuss how this is possible in quantum Hall insulators using top gates and why the same method does not work in quantum spin Hall insulators.

A local top gate [as illustrated in Figs.2.2(a), (b) and (c)] adds the termeV_tg(x, y) to the Hamiltonian.

In principle, one would have to solve the Schr¨odinger equation in the presence of this potential. Here we only give a handwaving motivation for the qualitative influence of the top gate.

For a quantum Hall insulator, the effect of the top gate is equivalent to the effect of the confinement potential. For a given x0, the bands will be shifted by the amount eVtg(x0, y0 =kxl²_B). As illustrated in Figs. 2.2(a) and (c), this can change the Chern number underneath the top gate. If this happens, the edge state will propagate around the top gate (at the interface of two distinct topological phases) allowing for a controlled manipulation of the edge state localization.

In quantum spin Hall insulators, there is no correspondence betweenkxand the localization iny-direction.

Furthermore, there is no insulating state with a different topological number available by shifting the chemical potential. Therefore, as long as the chemical potential remains away from the bulk bands, the top gate only locally shifts the bands corresponding to the edge states. This might change their penetration depth [Krueckl11] but it does not move the states away from the edge of the sample. As a consequence, controlling the localization of helical edge states is extremely challenging and to date no QPCs have been experimentally implemented.