Ultrafast electron dynamics in a topological surface state observed in two-dimensional momentum space

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Introduction
The most remarkable properties of the topological surface state (TSS) of three-dimensional (3D) topological insulators (TIs) are its Dirac-like, quasi-relativistic energy dispersion and its chiral spin texture in momentum (k-)space 1,2 .The latter locks momentum and spin of electrons in the TSS and suggests that momentum scattering within the Dirac cone is strongly suppressed with even complete absence of direct backscattering 3 .This implies that surface currents are not only automatically spin-polarized, but also flow ballistically over large distances.Together with the robustness of the TSS against nonmagnetic pertubations due its topological protection, this makes these surface states very promising for use in ultrafast low-loss electronics and spintronics.It turned out, however, that not only the application of 3D TIs in real devices, but already the confirmation of these unique properties by transport measurements is typically impeded by the dominant role of bulk carriers which are induced by intrinsic electron-or hole-doping 4,5 in combination with the rather small band gap of even the prototype 3D TIs such as the binary chalcogenides Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 .
Direct evidence for long-lasting ballistic surface currents has been provided by ultrafast pump-probe experiments that combine surface current generation and its time-resolved detection by angle-resolved photoelectron spectroscopy (ARPES).The most direct approach resolves on a subcycle time scale the momentum distribution of Dirac fermions close to the Fermi level E F of Bi 2 Te 3 as they are accelerated by the carrier wave of a THz pulse 6 .Its dynamics have been described by the semi-classical Boltzmann equation which accounts for inelastic scattering as well as for elastic momentum scattering.Both corresponding scattering times have been found to be longer than 1 ps which shows that THz-accelerated Dirac fermions may propagate coherently over several hundred nanometres before under-going scattering 6 .In a preceding work, we have shown that such long momentum scattering times are also inhere in electrons that are optically excited into the initially unoccupied upper branch of the Dirac cone in intrinsically p-doped Sb 2 Te 3 7 .This has been accomplished in a time-and angle-resolved two-photon photoemission (2PPE) experiment by using linear polarized mid-IR laser pump pulses.These pulses do not only permit a direct optical excitation between the occupied lower and initially unoccupied upper branch of the TSS, but also generate a strong asymmetry along a given direction in k-space parallel to the surface concentrated at an energy of a few hundred meV above E F 7 .In both experiments, the momentum distribution of the electrons in the TSS has been recorded in one direction of the two-dimensional (2D) momentum space of the surface and the current has been deduced from its asymmetry for opposite parallel momenta along the direction of current flow.The extracted momentum scattering times therefore represent effective phenomenological times for backscattering (180 • scattering), although multiple scattering processes with smaller scattering angles are in fact involved, because direct backscattering should be completely suppressed in the TSS 8 .
Here, we present the investigation of the ultrafast population dynamics in the TSS of Sb 2 Te 3 after direct optical excitation by mid-IR pulses in the full 2D momentum space.We will show that the excitation is strongly enhanced along three of the six Γ-M directions, which allows to monitor the subsequent redistribution of the electrons within the whole Dirac cone in unprecedented detail.We find that the enhancement significantly differs along these three directions, if a mirror plane ( Γ-M direction) of the sample surface is oriented perpendicular to the plane of light incidence.This confirms that the excitation by linear polarized mid-IR pulses can in fact generate a macroscopic photocurrent which is automatically spin polarized due to the spin texture of the TSS.
The time-resolved observation of the decay and the redistribution of the initially inhomogeneous TSS population in 2D momentum space allows to disentangle inelastic decay out of the TSS and elastic momentum scattering within the TSS in great detail.We demonstrate this capability for pristine and vanadium doped Sb 2 Te 3 and show how the vanadium atoms affect these two scattering processes in a distinctly different way.The dominating mechanism for inelastic decay in p-doped samples such as intrinsicially p-doped stoichiometric Sb 2 Te 3 has been identified by conventional 2PPE experiments to be electron-hole pair creation in the partially filled valence band [9][10][11] .The corresponding population lifetime therefore strongly depends on the position of E F and it could be shown that it can be strongly enhanced by Fermi level tuning through doping 12,13 .Consequently, the emergence of additional states in the bulk band gap can be expected to introduce further decay channels and we have in fact shown that doping of Sb 2 Te 3 by vanadium strongly reduces the population lifetime even for small concentrations of a few percent.This has been attributed to V-induced impurity states which have been spectroscopically identified by scanning tunneling spectroscopy (STS) 14,15 .We show here that the momentum scattering is on the other hand surprisingly almost unaffected by the presence of vanadium atoms, which demonstrates the robustness of momentum scattering in the TSS against defects even for magnetic scattering centers.

Experimental method
Details of the optical setup are described in Refs. 7,14 Electrons were excited above the Fermi level E F by pump laser pulses with a photon energy in the mid-IR (hν pump = 0.33 eV, 100 fs) or in the visible (VIS) (hν pump = 2.52 eV, 80 fs) and subsequently photoemitted by time-delayed ultraviolet (UV) probe laser pulses (hν probe = 5.04 eV, 100 fs) at a repetition rate of 200 kHz.The chosen UV photon energy just suppresses direct photoemission of the occupied states but provides full access of the unoccupied part of the TSS in 2PPE with high dynamic range.Pump and probe pulses were p-polarized and focused on the sample into a spot with a diameter of ∼ 100 µm at angles of incidence of both beams close to 45 • .The experiments were carried out in a µ-metal shielded UHV chamber at a base pressure of 4 × 10 −11 mbar with the samples cooled to 110 K after in-situ cleaving by the Scotch tape method.Characterization of the samples, which were grown by the modified vertical Bridgman method with a rotating heat field 16 , is described in detail in Ref. 14 .Photoelectrons were collected by a hemispherical analyzer (Scienta DA30) equipped with deflection plates in the electron lens which makes it possible to aquire energy-momentum (E-k x ) maps with the electron momentum k x oriented along the orientation of the entrance slit of the hemisphere for varying momentum k y perpendicular to k x .In this way, the ultrast dynamics of the optically excited population in the initially unoccupied band structure can be sequentially mapped in the full two-dimensional momentum space of the sample surface without moving the sample.

Homogenous population of the TSS by visible excitation
Before turning to the discussion of the dynamics of the momentum distribution of photocurrents excited by mid-IR pump pulses, we first discuss data taken with visible pump pulses in order to disentangle the impact of the pump and probe pulses on the energy and momentum distribution of the detected final states.The latter is governed by both the transient population of the intermediate state excited by the pump pulses and the sequential photoemission by the probe pulses into the detected final states.Visible pump pulses have been shown to initially excite electrons far above the TSS and result in an indirect population of the Dirac cone 9,17 .This involves many scattering events which completely homogenize the electron distribution in momentum space 7 .Therefore, any momentum dependence of the specral weight of the 2PPE data can be in this case related to the photoemission probe process and can be used to correct the momentum-dependent 2PPE data for the optical matrix element of the probe process as we have shown in Ref. 18 .Figure 1(a) and (f) show two selected cuts through 2PPE data of the initially unoccupied part of the TSS taken at 1 ps after temporal overlap with visible pump pulses of photon energy hν = 2.52 eV.At this delay the indirect population of the TSS has reached its maximum 9 .The E-k x map for k y = 0 depicted in Fig. 1(a) is symmetric in k x and shows the linear dispersion of the TSS along Γ-K with the Dirac point (DP) located 200 meV above E F .The TSS is homogeneously populated in energy up to the bulk conduction band minimum at E − E F ≈ 330 meV which feeds the population of the TSS for several ps 9 .The k x -k y map centered at E − E D = 140 meV depicted in Fig. 1(f) shows the expected warping of the Dirac cone at this energy with a slight flattening of the linear dispersion along Γ-M19 , but most notably a strong asymmetry of the 2PPE data with respect to k y with a half-moon shaped intensity distribution.This asymmetry does not result from an inhomogeneous population of the TSS, but from the oblique incidendence of the p-polarized UV probe pulses.This momentum distribution is independent of the sample orientation as has been tested by azimuthal rotation of the sample and measurements with two different orientations of the plane-of-incidence.For p-polarized probe pulses incident along the k y -direction as presented in Fig. 1, a half moon shaped intensity distribution indicates that the TSS is dominated by out-of-plane sp z orbitals with neglible in-plane contributions which would show a threefold symmetric pattern 20 .The k x -k y maps can be therefore corrected for the photoemission probe process by dividing the intensity by (1 − sin φ ) where φ is the azimuthal angle counting anticlockwise with respect to the +k x direction.This is, however, not applicable for φ close to 90 • and we apply this correction only for k y < 0 and mirror the data with respect to the k x axis.The corrected and symmetrized data depicted in Fig. 1(k) shows a homogeneous intensity distribution around the Dirac cone and in particular no kink at the mirror axis which demonstrates that this correction accounts well for the matrix element of the UV probe pulses.This makes it now possible to correct the 2PPE data to reveal the actual population in the intermediate state also for other pump photon energies, as long as the same photon energy and polarization of the photemission probe is kept.Mirroring of the data, however, can be only applied if either the population is homogeneous in k-space or if the plane of probe incidence is oriented perpendicular to a mirror axis of the sample surface such as the Γ-M direction of the threefold symmetric surface of Sb 2 Te 3 (0001).

Photocurrent generation by mid-IR excitation
The four rightmost columns of panels in Fig. 1 show 2PPE data for mid-IR pump pulses which drive a resonant excitation across the Dirac point 7 and result in a strongly enhanced population centered at E − E D = 140 meV as can be most clearly seen in Fig. 1(b).Moreover, the mid-IR excitation induces a strong asymmetry with respect to k x when the sample is oriented with the Γ-M direction aligned perpendicular to the plane of probe incidence.As can be already seen in the raw and uncorrected k x -k y map (Fig. 1(g)), the population is not only enhanced in one direction, but in three of the six Γ-M directions.Even if the photoemission probe is much less efficient in the direction of the upper right M point, the enhancement is still faintly visible in the uncorrected data.The threefold pattern indicates that the optical excitation is associated with the Sb-Te bonds which have a threefold arrangement in the unit cell 21 .The degree of the population enhancement, however, differs in the three directions.It is much stronger in direction of the left as compared to that of the lower right M point.This difference is even further enhanced when the data is corrected for the photoemission probe as shown in Fig. 1(l).This clearly demonstrates that the direct excitation by the mid-IR pulses in fact generates a macroscopic photocurrent along the k x direction while an asymmetry in a E-k x cut as shown Fig. 1(b) could also result from a threefold symmetric excitation with equal weight along the Γ-M directions as noticed in Ref. 22 .However, even in such a case it is possible to gain information about elastic momentum scattering of the electrons within the Dirac cone by the investigation of the population balancing along a line in k-space that shows a population asymmetry 7,23 .

Dynamics of momentum scattering
The time-resolved observation of the redistribution in the full two-dimensional k-space of the surface studied here, however, provides a much more detailed insight into the initial excitation and the sequential scattering processes.This is demonstrated by the data of Fig. 1(b-e), (g-j) and (k-o) which shows a time series of cuts through the 2PPE data for selected delays ∆t between mid-IR pump and UV probe.These data show that the population at the resonant excitation energy of E − E D = 140 meV has almost reached its maximum already at ∆t = 40 fs because it is created by a direct optical transition from the lower to the upper part of the Dirac cone 7 .For increasing delay, the whole population at the resonant excitation energy gradually decreases due to inelastic scattering of the electrons.Simultaneously, the population around the Dirac cone becomes more uniform as can be seen in the k x -k y maps.Electrons are elastically scattered in direction within the cone that were initially less populated.However, even at ∆t = 1.04 ps, an enhancement of the population along the Γ-M directions is still clearly visible that shows that elastic scattering, which randomize the population around the Dirac cone, is weak as compared to inelastic scattering.This results from the suppression of momentum scattering due to the chiral spin structure of the TSS and is in strong contrast to the electron dynamics in topological trivial materials such as GaAs where the electrons in the Γ valley have been shown to quasiequilibrate in momentum space within 100 fs 24 .The E-k x maps show that the filling of the lower part of the Dirac cone is delayed with respect to the population at the resonant excitation energy.This is caused by sequential inelastic scattering of the directly excited electrons to lower energies.The intensity at all energies below the resonant excitation energy still increases from ∆t = 0.04 ps to ∆t = 0.24 ps and decays only for longer delays whereas the resonantly populated part at negative k x is already reduced at ∆t = 0.24 ps.
The redistribution of the electrons due to elastic scattering within the TSS is analyzed in more detail in Figure 2, where we present the time-evolution of the 2PPE intensity at the resonantly excited energy along the different Γ-M directions.For this purpose, the 2PPE intensity, which is proportional to the TSS population, has been integrated over four quadratic regions shown in the inset of Fig. 2. The data show that the populations in these regions all decay after their differently strong initial excitation, but merge with each other at different delays.At first, the populations in regions 2 and 3 merge at ∆t > 0.5 ps followed by merging with the initially largest population in region 1 at ∆t > 1 ps.The population in region 4 also approaches the others, but even at the largest observed delay of 2.3 ps it is still the smallest.The merging shows that the decay of the population in the different regions is not independent and not only governed by inelastic scattering, but also accompied by exchange between the regions due to elastic momentum scattering.The latter can be characterized by a weighted distribution of scattering angles whereas the spin structure of the TSS suggests an enhanced probability for small angle scattering and a complete suppression of direct backscattering 8 .
We analyze our data by a simplified rate-equation model for the transient populations n i in the four regions i = 1 . . . 4 by considering population loss of all regions with a common rate Γ d = 1/τ d and population exchange between neighboring regions with a common rate Γ e 60 • = 1/τ e 60 • as indicated by the white arrows in the inset of Fig. 2. Here, τ d and τ e 60 • are the corresponding mean scattering times for inelastic scattering to lower energies within the TSS or into the bulk and for 60 • momentum scattering, respectively.Considering that the regions 2 and 3 centered at k y − 0.06 Å −1 in the inset of Fig. 2   identical to those centered at k y + 0.06 Å −1 due to the mirroring of the data, the rate equations are given by Here, δ (t) is the temporal intensity profile of the Gaussian shaped mid-IR laser pulse, and the A i indicate the different excitation probabilities in the four different regions.This model represents an extension of the rate-equation model used in Ref. 7 where elastic scattering was characterized by an effective scattering time for 180 • scattering which is in fact composed of sequential scattering events with smaller scattering angles.The solid lines in Fig. 2 show the best fit of the data within our model assuming that the 2PPE intensity is proportional to the TSS population.This fit yields an elastic scattering time of τ e 60 • = 1.21(15) ps for 60 • scattering and an inelastic scattering time of τ d = 0.44 (10)ps.The number for τ e 60 • roughly fits to the previously determined effective scattering time of 2.5 ps for 180 • scattering 7 .The latter would correspond to a population exchange only between regions 1 an 4. The inelastic scattering time is about 30% smaller as reported in Ref. 7 , but within the variation range observed for different cleaves of the sample, which has been attributed to variations of the defect density and of the position of E F with respect to E D and the valence band maximum 9 .

Impact of V-doping
In the following, we will show that the capability to observe the electron dynamics in the full two-dimensional momentum space makes it possible to disentangle elastic and inelastic electron scattering, even if no macroscopic photocurrent is generated, as long as the initial excitation is inhomogeneous in momentum space.For this purpose, we compare the electron dynamics of pristine and vanandium doped Sb 2 Te 3 for a different sample orientation.Recently, we have shown that even small vanadium concentrations of a few percent drastically reduce the inelastic scattering time in Sb 2 Te 3 , which was attributed to impurity states 14,15 .This has been deduced from the observation of the dynamics after mid-IR excitation in one-dimensional cuts through the TSS band structure along one surface direction.
In figure 3, we present here a comparison of time-resolved 2PPE data in the two-dimensional momentum space obtained for pristine Sb 2 Te 3 and a Sb 2−x V x Te 3 sample with x = 0.015.The measurements were done at a temperature of 110 K, which is well below the Debye temperature of Sb 2 Te 3 (Θ D = 162 K 25 ), but at the same time well above the Curie temperature of V-doped Sb 2 Te 3 14 .Both samples were equally oriented with the Γ-M direction aligned along the plane of incidence.This results in a pure threefold symmetric pattern of the excited population in the TSS and no macroscopic photocurrent is generated, as can be most clearly seen in Fig. 3  however, can still be extracted from these data because of the initially strong selective enhancement of the excited population in three of the Γ-M whereby only the lower two are clearly observed in the raw data because of the matrix element of the probe transition.This is demonstrated in Fig. 4 where we show an analysis of the transient TSS population at the resonantly excited energy for both data after correction of the 2PPE intensity for the probe matrix element.Symmetrization by mirroring is not possible for this threefold symmetric data because the Γ-K direction is no mirror plane.Similar to the analysis shown in Fig. 2 the 2PPE intensity has been evaluated at the three visible M-points.The blue and red solid dots show these data for two of the initially differently populated M-points as indicated by the blue and red rectangles in the k x -k y map of the inset, respectively.The solid lines show again fits of the corresponding rate-equation model considering inelastic decay and 60 • elastic momentum scattering.In order to well describe the initial rise of the transients, we additionally consider a small contribution of the third image-potential state to the 2PPE intensity, which is only visible at negative delays (not shown).It appears at similar final state energies as the TSS but with pump and probe pulses exchanged 18,23 .Its rise and decay therefore appears towards negative delays here.
From the data of the pristine sample shown in Fig. 4a, we obtain in this way an inelastic decay time of τ d = 0.68(1)ps which is very close to the result of Ref. 7 , but longer by ∼ 50% if compared to the one determined from the data in Fig. 2 due to the variations for different cleaves as discussed above.In contrast, the elastic momentum scattering time τ e 60 • = 1.40( 5) is comparable to the one obtained from the data of Fig. 2 which already indicates that elastic scattering is rather robust against defects and coupling to the bulk.Comparing Fig. 4(a) and (b) makes it apparent that V-doping strongly reduce the inelastic scattering time and we determine a four times smaller value of τ d = 0.16(1)ps as compared to pristine Sb 2 Te 3 .This can be most probably attributed to the V-induced impurity states that have been identified by scanning tunneling spectroscopy (STS) 14 .Such impurity states can introduce additional decay channels for intraband scattering to lower energies and were shown to persist even in the presence of magnetic order 26 .Surprisingly, however, the momentum scattering time is not much affected by V-doping as can be already seen by comparing the 2D maps of the pristine and V-doped sample in the insets of Fig. 4 taken at 0.33 ps.Although the overall intensity for the V-doped sample is more strongly reduced as compared to the pristine sample due to the stronger inelastic scattering, the inhomogeneity of the intensity distribution around the Dirac cone is comparable for both samples.This can be even more clearly seen by comparing the time dependence of the intensity in region 2 with those of region 1 and 3 shown in Fig. 4b.It does not still take considerable time for these intensities to merge.The still long momentum-scattering time is also reflected by the fact that the maximum difference of these intensities, which is reached around 0.12 ps, is comparable to the one of the pristine sample.An accelerated momentum-scattering would lead to a faster and stronger merging of theses intensities.A fit of the data indeed gives only a slightly reduced momentum-scattering time of τ e 60 • = 1.0(5) ps, but with an enhanced uncertainty due to the stronger inelastic scattering.Such still weak momentum scattering is surprising because it might be expected that impurities also enhance momentum scattering, in particular if the impurities carry a magnetic moment.On the other hand, time-resolved quantum-beat photoelectron spectroscopy of image-potential  states on a Cu(001) surface has shown that single scattering centers can affect inelastic decay and momentum scattering in a very different way depending on the details of the scattering potential 27 .While CO adatoms mainly lead to the decay of the quantum beats between different image-potential states due to momentum scattering 28 , an even much smaller concentration of Cu adatoms has a stronger impact on the inelastic decay 29 .Ab initio calculations suggest that for scattering of electrons in a TSS at single magnetic impurities, the non-magnetic part of the scattering potential with resonant scattering into defect states dominates 30 .This is in line with our observation that the magnetic impurities strongly reduce the inelastic scattering time, but have only a small impact on elastic momentum scattering.

Conclusions
We have experimentally investigated the ultrafast population dynamics of electrons in the TSS of pristine and V-doped Sb 2 Te 3 in two-dimensional momentum space after direct optical excitation by linearly polarized mid-IR pulses using time-and angle-resolved two-photon photoemission.We have shown that the population at the resonant excitation energy in the upper Dirac cone is not uniform in momentum space, but enhanced along three of the six Γ-M directions.If the the plane of incidence is aligned along one of the Γ-K directions, this enhancement is not threefold symmetric and corresponds to a macroscopic photocurrent along a Γ-M direction.The inhomgeneous excitation, together with the detection in the full two-dimensional momentum space of the surface, makes it possible to observe the redistribution of the population around the Dirac cone by elastic momentum scattering and to lower energies by inelastic scattering.We have found that momentum scattering is much less efficient as compared to inelastic scattering with strong supression of large-angle scattering, as is expected from the chiral spin structure of the TSS.V-doping has been shown to strongly enhance inelastic scattering, while momentum scattering is almost unaffected, although the dopants carry a magnetic moment.This is in agreement with ab initio calculations which suggest that the scattering potential of single magnetic impurities is dominated by its non-magnetic part.

Figure 1 .
Figure 1.Angle-resolved 2PPE data of Sb 2 Te 3 excited by visible (hν = 2.52 eV) and mid-IR (hν = 0.33 eV) pump pulses.a) E-k x map for 2.52-eV excitation and k x along Γ-K at a pump-probe delay of ∆t = 1 ps.b-e) For 0.33-eV excitation and k x along Γ-K at different pump-probe delays as indicated.f-j) Corresponding k x -k y maps integrated over energy intervals centered at E − E D = 140 meV as is depicted by the cyan dashed lines in (a-e).k-o)The same cuts as (f-j) but with the intensity corrected for the matrix element of the probe transition and symmetrized by mirroring the data at the k x -axis.For all data, the plane of light incidence is oriented along the k y -axis.

Figure 2 .
Figure 2. Time-evolution of the 2PPE intensity at different points of the k x -k y maps (c.f.Fig. 1(l-o)) as is indicated by the square integration areas in the inset (data points).The solid lines show fits of a rate-equation model for the population of the Dirac cone at these points considering inelastic electron scattering to lower energies and elastic scattering between adjacent areas as indicated by the white arrows in the inset.
(a) and (e) for both samples, respectively.Information about elastic momentum scattering,

Figure 3 .
Figure 3.Comparison of the population dynamics between pristine Sb 2 Te 3 (a-d) and Sb 2−x V x Te 3 with x = 0.015 (e-h).The panels show uncorrected k x -k y maps centered at E − E D = 141 meV for different time delays ∆t.The sample orientation is indicated in panel (h).The plane of light incidence is along the k y -direction.

Figure 4 .
Figure 4. Comparison of the population dynamics between pristine (a) Sb 2 Te 3 and (b) Sb 2−x V x Te 3 with x = 0.015 with the Γ-M direction aligned along the plane of incidence.The insets show k x -k y maps centered at E − E D = 140 meV for two selected time delays ∆t.Data points show the intensity integrated over the blue and red square areas depicted in the k x -k y maps.Solid lines show fits to the data.