Georeferenciació de cartografies històriques

Figure 4.49: Scattering spectrum of the wave packet Ψ⁺_↓ encountering an FEF of strength V_z = −66meV that is tilted byθdegrees intox-direction. Only a relatively small fraction of about5 − 20%of the wave packet is scattered into exit channels with opposite spin orientation.

other exit channel with the same spin. For a tilt intoy-direction, the effect is much smaller with less than5%off-scattering at a20° tilt. Changes of the scattering rates are approximately the same for positive and negativez-fields. Random impurity potentials and rough edges show no big influence on the scattering spectra.

4.6 Summary

By numerical time-evolution of wave packets in 2D TIs with local FEFs, it has been shown that these structures can be used to steer spin polarized edge currents. In the absence of in-plane fields, the edge state pseudo spin is conserved and a100% steering efficiency can be obtained. Using only out-of-plane fields, which are naturally stable in the Bi2Se3 class of TIs, devices were conceived, which are capable of creating, switching and detecting pure spin currents with high efficiency. Because charge currents are directly converted into spin currents, these devices can be more efficient than other state of the art concepts. The large bulk gap of the 2D TI model promises even operation at room-temperature, provided that the induced FEFs are just as strong. As the size of device structures is only dictated by the spatial extent of edge states, they can be reduced to a very small scale in large gap TIs.

More realistic calculations were performed using thin films of 3D TIs. These calculations showed smaller bulk gaps but a QAH effect already for arbitrary small FEFs, in consistence with experimental observations. Even though some realistic effects can change the spin of

propagating electrons, the pure spin current devices were shown to be very stable because scattering events in different electron channels largely compensate each other. Sizable devi-ations of10%-20%arise only due to a breaking of structural inversion symmetry or a global in-plane field component. These effects may, however, be avoidable by careful device engi-neering. If they are present, they can cause a small residual charge current at the right side of the spin current transistor in the “off”-state because the number of scattering sites for spin-up and spin-down current is unequal in that case.

Due to the rather small size of the hybridization gaps and currently achievable magnetic excitation energies, measurements of the proposed devices will require relatively low temper-atures. To achieve room temperature applicability, other materials with larger 2D gaps have to be considered and the excitation gaps have to be enhanced.

5 Meservey Tedrow method

Due to their intrinsic locking of momentum and spin, TIs make interesting candidates for spin-tronic applications. In many cases, e.g. when combined with ferromagnets [2, 64, 65] or in the spin current devices discussed in the previous chapter, the efficiency of TI based de-vices strongly depends on the surface state spin polarization. Thus, to give a quantitative measure of the efficiency of TI based spintronic devices, it is crucial to get a precise pic-ture of the surface state spin texpic-ture of TI materials. Currently the most prominent method for measuring the surface state spin is spin- and angle-resolved photoemission spectroscopy ((S)ARPES). In ARPES measurements, photoelectrons emitted by a crystal due to irradiation with a strong monochrome light source, e.g. a laser, are analyzed in terms of their energy and emission angle, which are in direct connection to the vacuum momentum. By applying energy and momentum conservation laws, the dispersion of the initial electrons in the crystal can be determined from this information [103]. SARPES additionally determines the spin of photoelectrons, which is assumed to be conserved in the photoemission process [104].

For 3D TIs of the Bi2Se3 class, it was theoretically predicted that the spin lies mainly in the surface plane and is orthogonal to the momentum. Due to a hexagonal deformation of the Fermi surface, some of these materials also feature a small out-of-plane polariza-tion, which increases with momentum. SARPES measurements confirm these predictions.

However, where theoretical calculations predict in-plane spin polarizations of about50%-65%

[42, 87, 88], values reported by SARPES measurements show no uniform result but vary be-tween about45%and up to100%[38, 39, 42, 43]. The reason for this discrepancy is that the spin polarization of emitted photoelectrons in SARPES measurements can be manipulated by the photon energy and the geometry of the setup. Hence, it can deviate from the spin of the original electrons in topological surface states, i.e. for TIs, the spin is not necessar-ily conserved in the photoemission process [39, 42, 104, 105]. An alternative approach for measuring the spin polarization would therefore be useful to verify SARPES measurements and theoretical predictions. This alternative could be based on spin polarized tunneling. Liu et al. [24] investigated topological surface states in a spin Hall effect like fashion, using spin polarized electrons tunneling from a ferromagnet. However, they did not determine the sur-face state spin polarization, only the charge spin conversion efficiency. A closer look at such a device is taken in chapter 6. Here, another method, developed by Meservey and Tedrow [21, 22], based on quantum tunneling from a thin superconducting film, will be investigated regarding its applicability to TIs.

It was shown by Meservey, Tedrow and Fulde [106] that a strong parallel magnetic field splits the quasiparticle states of spin-up and spin-down electrons in thin (approximately50Å) superconducting aluminum (Al) films. In a magnetic field of strength B, the BCS energy spectra of opposite spin are shifted by±µ_BB with respect to the spectrum without magnetic field. So, the total spectrum, i.e. the sum of both, shows four peaks (see Fig. 5.1). µ_B is the

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

-3 -2 -1 0 1 2 3

N(E)

(eU-µ)/Δ antiparallel

parallel sum

Figure 5.1: In a strong parallel magnetic field, the BCS density of states of a thin layer of superconducting aluminum splits. Electron states are shifted either in positive or negative direction with respect to the chemical potentialµ, depending on whether their spin is parallel or antiparallel to the applied field. The sum of both spectra features four peaks whose relative hight can be used to measure spin polarizations in tunneling experiments.

Bohr magneton. As the spin of electrons tunneling from this superconducting layer, through an insulating layer (I), into the surface of a third material is conserved in the tunneling process [21], such a device structure can be used to measure the polarization of this material.

If the material is ferromagnetic, electron spins in the ferromagnet and the superconduct-ing aluminum film are aligned either parallel or antiparallel because of the strong magnetic field. So, when the conductance of this tunnel junction is measured, the polarization of the ferromagnet can be calculated from how the different densities of states of up and spin-down electrons in the ferromagnet change the relative hight of the four peaks in the split BCS spectrum. When the ferromagnet is unpolarized, i.e. a normal metal, the conductance of the junction simply reproduces the split BCS spectrum shown in Fig. 5.1 because the densities of states of spin-up and spin-down electrons are equal at the Fermi level. Different densities of state of spin-up and spin-down states result in an asymmetric tunnel spectrum, where the BCS density of states of spin-up states is reduced and that of spin-down states increased or vice versa, depending on the sign of the polarization. Consequently, the spectrum of a fully polarized ferromagnet, which has only states of one spin type at the Fermi level, would show only one of the two shifted spectra.

In a TI, the spin of surface electrons does not align with the magnetic field. Instead, it rotates with momentum around the Fermi surface so that all spin orientations are equally populated. So, when the above scheme would be applied to a TI, the measured polarization would always be zero. In fact, the TI in total is unpolarized in the absence of an electric field.

The solution to this issue is to make use of the fact that electrons with opposite spin move in opposite directions. When only electrons moving, e.g., in positivex-direction are measured,