Mesoscopic electron focusing in topological insulators

The particle wave duality sets a fundamental correspondence between optics and quantum mechanics. Within this framework, the propagation of quasiparticles can give rise to superposition phenomena which, like for electromagnetic waves, can be described by the Huygens principle. However, the utilization of this principle by means of propagation and manipulation of quantum information is limited by the required coherence in time and space. Here we show that in topological insulators, which in their pristine form are characterized by opposite propagation directions for the two quasiparticles spin channels, mesoscopic focusing of coherent charge density oscillations can be obtained at large nested segments of constant energy contours by magnetic surface doping. Our findings provide evidence of strongly anisotropic Dirac fermion-mediated interactions. Even more remarkably, the validity of our findings goes beyond topological insulators but applies for systems with spin orbit lifted degeneracy in general. It demonstrates how spin information can be transmitted over long distances, allowing the design of experiments and devices based on coherent quantum effects in this fascinating class of materials.

theoretical studies showed that iso-energy cuts progressively evolve from an almost circular (and convex) shape very close to the Dirac energy to a more concave, snowflake-like shape at higher energies [10,11]. Once this transition takes place, large parallel sections face each other, a scenario supporting good nesting vectors which can strongly enhance the susceptibility of the system to external perturbations.
In pristine TIs, an active role of these vectors is strongly suppressed by time reversal symmetry [7,8]. Here, we demonstrate that once this protection is lifted by the introduction of magnetic surface dopants, the nesting leads to strongly focused interference patterns in the charge density, resulting in coherent quantum oscillations which can be observed by quasiparticle interference (QPI) imaging with the scanning tunneling microscope (STM) over distances of tens of nanometers without any significant intensity loss. Theoretical calculations rationalize these findings in terms of the combined action of the iso-energy contour shape and the local magnetic moments present on the surface, thereby providing guidelines to control this effect. In particular, for a long coherence length two conditions need to be fulfilled: (i) the Fermi energy contour must exhibit large nearly parallel segments (nesting) with pairs of initial and final k-points which can be connected by the same scattering vector, and (ii) the magnetic dopants must couple to generate a high-spin state with a magnetic moment well beyond a single atom. As revealed by x-ray magnetic circular dichroism measurements (XMCD) this is fulfilled already at very dilute Mn concentrations on Bi 2 Te 3 . Our observations provide evidence that the emergence of superparamagnetic order on magnetically doped TIs with nested constant-energy contours can trigger Dirac fermion-mediated highly anisotropic indirect interactions. These results suggest that-by appropriate band engineering-spin-dependent quantum coherent transport can be achieved over mesoscopic distances and pave the way to design novel device concepts that rely on quantum coherent effects in this fascinating class of materials.  the Dirac point is schematically represented in Fig. 1(b). Selected CECs relevant for the following discussion are sketched in Fig. 1(c)-(e). Just above the Dirac point [ Fig. 1(e)] the CEC is circular (black line) and the spin is perpendicularly locked to the momentum, thereby leading to a helical spin structure indicated by colorized arrows. As we move further away from the Dirac point the warping increases. This first leads to a hexagonal [ Fig. 1(d)] and eventually to a snowflake-like shape of the CEC [ Fig. 1(c)]. This deformation goes along with the development of an alternating out-of-plane component of the spin polarization along ΓK directions of the surface Brillouin zone [11], as indicated by symbols (⊗, ) in Fig. 1(c) and (d). Fig. 1 show quasiparticle interference (dI/dU ) maps measured at three energies, E − E F = +70 meV, −10 meV, and −70 meV, respectively. These energies are chosen such that they mark the transition from convex to concave CEC, as consistently shown by theoretical calculations [11] and photoemission experiments [10]. As indicated by green and red arrows in Fig. 1(c)-(e), the scattering vectors q 1 and q 2 specific for each energy can be obtained by using the stationary phase approximation [13]. At E − E F = −70 meV The scattering channel along ΓM directions (q 2 ) is routinely found on TI materials [8,14]. Its appearance is related to the strength of the warping term which allows to effectively scatter between next-nearest neighbors segments of a CEC because of their parallel spin polarization. This channel is rather weak at the energy discussed here, E − E F = −10 meV, since nesting along q 2 is still rather poor [see scheme in Fig. 1(d)]. In contrast, high-intensity spots are visible along the ΓK directions, which cannot be found on pristine TI surfaces. Their appearance implies an active role of the time-reversal symmetry breaking perturbations, i.e. the Mn adatoms, which, following theoretical predictions [15], we recently suggested to be magnetically coupled by the Dirac fermions present on the TI surface [12] (see XMCD discussion below).
Further raising the energy to E − E F = +70 meV leads to a weaker modulation of the dI/dU signal detected in QPI images (see scale bars of Fig. 1). This finding is confirmed by the FT-QPI map shown in the top panel of Fig. 1(i) which only reveals six weak spots along the ΓM direction corresponding to q 2 .
Interestingly, closer inspection of the real space images reveals that, once backscattering channels are opened, coherent waves propagate without any significant intensity loss over distances larger than 30 nm, as can clearly been seen in the line section of Fig. 1(k). Furthermore, although point-shape scattering centers should result in spherical waves emanating in all radial directions, our experimental data presented in Fig. 1(g) indicate that the charge density oscillations remain highly focused over mesoscopic distances well beyond the atomic scale. These observations provide compelling evidence that Dirac fermion-mediated interactions in TIs are highly anisotropic, where some crystallographic directions are preferred over others, as predicted theoretically [16]. Even more remarkably they demonstrate that, by appropriate band engineering, spin coherence can be achieved over mesoscopic distances.
A priori, however, the exact mechanisms that lead to the emergence of highly focused and anisotropic QPI patterns are not evident. Good nesting is known to be necessary to trigger the focusing effect [5,17]. In the present case, this condition is fulfilled by large portions of the FS especially for the hexagonal case. This configuration supports additional scattering vectors with v k ≈ −v k (v k is the group velocity), such as q 1 or q 1 , which-in contrast to  The quantity that we examine is the extended joint density of states (exJDOS), an extension of the widely used joint density of states approach [7,18] that constitutes an approxi-mation to Fourier-transformed dI/dU images. We define the exJDOS as the convolution at energy E, involving the spectral density A surf k (E) integrated in the spatial region between tip and sample surface, and the matrix element M kk (E) = P kk (E) γ kk (E). Here, P kk (E) is the scattering rate (including the suppression of time-reverse scattering) and γ kk (E) = 1 − cos(v k , v k ) accounts for the fact that the STM probes standing waves, i.e., states with opposite group velocities are favored, in the spirit of the stationary phase approximation.
The quantities A surf k (E) and γ kk (E) are derived from the band structure, while P kk (E) is calculated by means of the Golden Rule from the T -matrix of the impurity obtained by a calculation of the impurity Green function.
Our density-functional calculations are based on the local density approximation [19].
We employ the Korringa-Kohn-Rostoker Green-function (KKR) method for the calculation of the electronic structure and scattering properties (T -matrix and surface state scattering rate) of the impurity. The Bi 2 Te 3 surface is modeled by a film of 6 quintuple layers. Te vacancies show that the ferromagnetic state has the lowest total energy as compared to the other configurations discussed above. We speculate that this may be the result of indirect RKKY-type coupling mediated by the Dirac fermions which exhibit a Fermi wavelength of approximately 7 nm, i.e. well above the average Mn-Mn spacing of about 3 nm, making the interaction always ferromagnetic as predicted in Ref. [15].
In parallel we performed model-calculations based on the Hamiltonian of Lee et al. [20].
As described in the supplementary information, the model was extended by exchangeinteraction terms to represent (i) the scattering by Mn moments, as well as (ii) the possibility of a uniform magnetization of the surface state caused by ferromagnetic coupling among the Mn defects. These model calculations, in agreement with the ab-initio results, provide further evidence that the experimentally observed standing wave patterns arise from the combined action of ferromagnetically coupled Mn atoms and the hexagonally shaped CEC supporting the focusing effect.
Achieving such a high magnetic moment in a Mn-doped system is possible only if Mn atoms couple ferromagnetically since both bulk Mn [21] and Mn nanostructures [22], are known to exhibit an antiferromagnetic ground-state. In order to directly determine the magnetic moment and configuration of Mn on Bi 2 Te 3 , we have performed XMCD mea- The XMCD, calculated as I − − I + and shown in the bottom panel of Fig. 3(a), highlights the considerable magnetic polarization of the Mn surface dopants. Fig. 3(b) displays the magnetization cycles, recorded on the same sample at T = 2.5 K, at normal (green line) and grazing incidence (yellow line), by following the XMCD magnitude at the Mn L 3 -edge (full circles) as a function of the applied magnetic field. Both, the larger slope in M (H) at small fields and the saturation at large fields in normal incidence geometry indicate an out-of-plane magnetic anisotropy for Mn on Bi 2 Te 3 . The saturation magnetization (in Bohr magnetons per Mn atom) can be evaluated by applying the XMCD sum rules [24,25] to the data in Fig. 3(a). Independent information on the magnitude of the fluctuating total moments is contained in the shape of the magnetization isotherms, strongly determining their slopes near µ 0 H = 0. An analysis in terms of classical Langevin paramagnetism augmented with a uniaxial magnetic anisotropy term [26], yields an effective value of the (total ) saturation moment M sat and the associated magnetic anisotropy energy. We envision that exchange coupling between Mn adatoms may be mediated by the surface electron gas of Bi 2 Te 3 . We speculate that-although ferromagnetic correlations are present-thermal fluctuations and disorder are too strong to establish stable ferromagnetic order. Nevertheless, assemblies of Mn atoms with small enough Mn-Mn separation will exhibit sufficiently strong magnetic interactions to couple their individual moments to a macro-spin, resulting in the enhanced susceptibility characteristic for superparamagnets which is experimentally observed in Fig 3(b). Since XMCD spatially averages over a macro-scopic sample area, a value of 7 µ B implies the existence of assemblies with considerably larger magnetic moments. This result is consistent with the theoretical finding that units composed of at least three ferromagnetically interacting Mn atoms are required to activate the scattering channel along the ΓK direction.
Our findings provide evidence that, through electron focusing, quantum coherent information can be transferred in topological insulators over distances of more than 30 nm, making it compatible with device dimensions and thereby paving the way to design experiments and devices based on spin quantum coherent phenomena in this fascinating class of materials. More generally, they provide evidence that, through appropriate band engineering in materials with spin-split states, many interesting phenomena may appear, with important implications for spintronic and quantum computation.