Mobilitat en transport públic

As discussed in section 2.2, FEFs can be induced into the TI either through the proximity effect, by capping with a ferromagnetic insulator, or via doping with transition metal atoms.

This can lead to different spatial inhomogeneities. For the beginning, however, the FEFs are considered homogeneous within magnetic domains. Spatial inhomogeneities of a proximity field will be studied in section 4.5.5.

In section 4.1, it has been shown that an FEF of sufficient strength perpendicular to the surface (in z-direction) of a 2D TI in contact with vacuum removes only edge states with one spin direction, leaving the other edge states basically untouched. At an interface with a TI, either without FEF or with FEF of opposite polarization, additional edge states emerge.

The pseudo-spin of the edge states thereby depends on the interface type. Considering for example an FEF in positive z-direction, edge states with spin-up exist at the interface with vacuum or a trivial insulator and edge states with spin-down at the interface with a TI. If the FEF is now restricted to a small area of the TI device where it is in contact with the device boundary (vacuum) on one side and with a pure TI on all other sides, it is expected that a spin-down wave packet moves around this area, while a spin-up wave packet should pass straight through. Quantum transport calculations largely confirm these assumptions. According to section 4.1, an FEF of|V_z| = 0.34eV is needed to achieve these effects for the whole bulk gap and is therefore chosen for the following simulations. By looking at LDOS, it can be clearly seen that a spin-down wave packet (Ψ⁺_↓) encountering such a positive FTI leaves the edge

4.3 2D transport calculations

Figure 4.8: Scattering probabilities of the wave packetΨ⁺_↓ in a TI strip without scattering region after an evolution time ofτ = 1000_eV¹ . The shown energy range is that covered by the edge states contributing to the wave packets. The range of the bulk gap is indicated by vertical black lines. Even though there are no perturbations present,

|S_↓+,↓+|²deviates from unity at the edges of the energy range and has even values bigger than one. This effect, called Gibbs phenomenon, arises due to the Fourier transformation at the energy cut-off of the wave packet and can therefore not be completely removed.

Figure 4.9: LDOS at a local FEF (orange) of strengthV_z = 0.34eV in positivez-direction (in-dicated by black arrows) located at the lower edge of a TI stripe (blue). (a) A spin-down wave packet (Ψ⁺_↓) takes a detour around the FTI as no spin-down channel exists at the FTI-vacuum interface. (b) A spin-up wave packet (Ψ⁻_↑) coming from the opposite direction at the same edge moves straight through the FTI. Arrows in the color of the wave packet show the propagation direction. The Fermi energy is chosen asE_F = 0eV, about in the middle of the bulk gap. Figures taken from Ref. [1].

and moves around the FTI (Fig. 4.9a). On the other hand, a spin-down wave packet (Ψ⁻_↑) coming from the opposite direction moves straight through the same FTI (Fig. 4.9b), just as expected. However, LDOS show only the path for one specified energy, which is chosen as E_F= 0, approximately in the middle of the bulk gap. To get a quantitative, energy dependent measure of the device efficiency, one has to look at the scattering probabilities. Inside the bulk gap, the spin-down wave packet is perfectly transported from left to right (see Fig. 4.10).

Outside the bulk gap, the transmission probability|S_↓+,↓+|² now drops instantly to very low values, meaning that scattering into bulk states is strongly enhanced compared to Fig. 4.8.

The reason for this is that the momentum of the wave packet is changed at every corner of the FTI, making scattering into the much more numerous bulk states more likely. As the density of states is higher below the bulk gap, the effect is more prominent. Any part of the wave packet that is scattered into bulk states remains basically undetected as the scattering matrix is insensitive to bulk states. As a consequence, no notable scattering probability is measured for the other exit channels. Scattering into spin-up channels is even completely forbidden because spin-up and spin-down states are completely decoupled in the 2D Hamiltonian as long as there is no in-plane FEF. In the following, therefore, only spin-up or spin-down channels will be shown as long as the Hamiltonian forbids spin-flip scattering.

The transition probability of the spin-up wave packet Ψ⁻_↑ through the FTI is one for the biggest part of the bulk gap as well (see Fig. 4.11). Only at the lower edge of the bulk gap, it drops rapidly. As has been seen in Fig. 4.4, a positive FEF gives rise to additional edge states with pseudo-spin up at the FTI-TI interface. So, at the two points where the wave packet enters and leaves the FTI, it is scattered into this additional channel going around the FTI. A very low group velocity with momentum dependent sign results in long dwell times in this channel. Only for very long evolution times (at least ten times longer than shown), the

4.3 2D transport calculations

Figure 4.10: Scattering probabilities corresponding to the setup shown in Fig. 4.9a after an evolution time ofτ = 2000_eV¹ . Inside the bulk gap, the wave packet is perfectly transferred around the FTI into the Φ⁺_↓ exit channel. Outside of the gap, the probability drops instantly to nearly zero due to increased scattering into bulk states at all corners of the FTI. These parts remain basically undetected since the scattering matrix is insensitive to bulk states. The steep drop in the scattering probability causes Gibbs oscillations inside the bulk gap. No notable scattering into other exit channels is measured.

Figure 4.11: Scattering probabilities corresponding to the setup shown in Fig. 4.9b after an evolution time ofτ = 2000_eV¹ . As spin-flip scattering is forbidden by the Hamil-tonian, only spin-up channels are shown. Over the vast majority of the bulk gap, the wave packet is perfectly transferred through the FTI into the Φ⁻_↑ exit chan-nel. Only at the lower end of the bulk gap, the transmission probability drops rapidly due to scattering into the additional edge states at the FTI-TI interface (see Fig. 4.4).

transmission probability approaches one for the whole bulk gap. Scattering into bulk states for energies above the bulk gap is less prominent compared to Fig. 4.10 because the propagation direction of the wave packet is conserved at the FTI area.

When the FTI is extended towards the opposite edge of the TI strip, the incoming spin-down wave packetΨ⁺_↓ is no longer transferred into theΦ⁺_↓ exit channel because the spin-down path is now removed from the other edge of the FTI as well. Instead, it moves along the edge of the FTI towards the opposite edge and then in negative x-direction, leaving the device through theΦ⁻_↓ exit channel (Fig. 4.12a). Reflection into the counter propagating channel is now logically perfect inside the bulk gap as there is no spin-down channel inside the FTI.

Inverting the polarization of the FEF closes the main path at the FTI-TI interface and opens that at the FTI-vacuum interface. Hence, the spin-down wave packet can now pass through the FTI and leaves through theΦ⁺_↓ exit channel (Fig. 4.12b). Transmission through the FTI is again perfect for the biggest part of the bulk gap (see Fig. 4.13). Only at the lower bulk gap edge, the wave packet is still reflected into the counter propagating channel because of the additional edge states at the FTI-TI interface. The low group velocity in these states requires again long propagation times (aboutτ = 10000_eV¹ ) in order to detect the majority of the reflected wave packet.

The dispersion at the interface of two FTIs is similar to that of an FTI-TI interface. As a result, at such an interface, spin-up and spin-down wave packets propagate in the same direction (Fig. 4.14). Propagation into the opposite direction happens at the FTI-vacuum interfaces, like in Fig. 4.12b. Interchanging the magnetization directions of the two domains changes the paths, i.e. propagation from left to right happens along the domain wall and from right to left at the vacuum interfaces. Scattering probabilities are similar to Fig. 4.13, i.e. reflection happens only at the lower edge of the bulk gap.

The observation of chiral fermion modes at such a domain wall is consistent with the ex-istence of similar states at a domain wall on the surface of a 3D TI [17]. Here, an instant

4.3 2D transport calculations

Figure 4.12: LDOS at E_F = 0eV of the spin-down wave packet Ψ⁺_↓ encountering an FEF of strengthV_z = ±0.34eV that is spanning the whole width of the TI strip. (a) If the FEF is directed in positive z-direction, the wave packet is reflected into the counter propagatingΦ⁻_↓ exit channel. (b) An FEF in negative direction can be traversed by the spin-down wave packet, just like the positive FEF can be traversed by a spin-up wave packet (Fig. 4.11). Thus, the incoming wave packet remains at the same TI edge and leaves through theΦ⁺_↓ exit channel. Figure (a) taken from Ref. [1].

Figure 4.13: Full reflection- (left) and transmission-spectrum (right) of the spin-down wave packetΨ⁺_↓ at a negative FEF that spans the full width of the TI strip, as shown in Fig. 4.12b. At the lower edge of the bulk gap, the wave packet is reflected by the FTI into the counter propagating spin-down channel at the opposite edge, while the rest of the bulk gap shows perfect transmission. The scattering matrix was calculated after a propagation time ofτ = 10000_eV¹. At lower propagation times, only a small part of the reflected wave packet is detected due to the low group velocity in edge states at the FTI-TI interface.

Figure 4.14: At a domain boundary, both spin-down (a) and spin-up (b) wave packets can only propagate in one direction along the boundary. Propagation into the opposite direction happens along the FTI-vacuum interface, analog to Fig. 4.12b. Figure (b) taken from Ref. [1].

inversion of the magnetization at the domain wall was assumed. The effect of a rotation via an in-plane magnetization is discussed in section 4.5.1.