Abstract
In recent years, a class of solidstate materials, called threedimensional topological insulators, has emerged. In the bulk, a topological insulator behaves like an ordinary insulator with a band gap. At the surface, conducting gapless states exist showing remarkable properties such as helical Dirac dispersion and suppression of backscattering of spinpolarized charge carriers. The characterization and control of the surface states via transport experiments is often hindered by residual bulk contributions. Here we show that surface currents in Bi_{2}Se_{3} can be controlled by circularly polarized light on a picosecond timescale with a fidelity near unity even at room temperature. We reveal the temporal separation of such ultrafast helicitydependent surface currents from photoinduced thermoelectric and drift currents in the bulk. Our results uncover the functionality of ultrafast optoelectronic devices based on surface currents in topological insulators.
Introduction
Layered materials, such as Bi_{2}Te_{3} (refs 1, 2), Sb (ref. 3), Bi_{2}Se_{3} (refs 2, 4, 5, 6), Bi_{2}Te_{2}Se (ref. 7) and (Bi_{1x}Sb_{x})_{2}Te_{3} (ref. 8), are important narrowbandgap semiconductors for tunable, highperformance infrared detectors and thermoelectric applications^{9}. They have been demonstrated to be reference threedimensional topological insulators^{10,11,12} exhibiting exceptional transport mobilities^{1,2,3,4,5,6}. The latter suggests a reduced energy consumption that is very attractive for semiconductor devices in highspeed communication applications. In this respect, it is very advantageous that the helical surface states in topological insulators can be addressed by polarized light^{2,9,13,14,15,16}. Particularly, the circular photogalvanic effect results from a helicitydependent asymmetric optical excitation of spinsplit surface states in momentum space, allowing the generation and control of spinpolarized surface currents in topological insulators by circularly polarized light^{17,18,19}.
Here we demonstrate that such surface currents can be accessed and readout independently of the bulk photocurrents on a picosecond timescale with nearunity fidelity. Therefore, our results open the avenue for a highspeed transmission of information based on topological insulators. Furthermore, our experiments address the connection between the nonequilibrium currents of hot electrons directly after the laser excitation and the timeaveraged thermoelectric currents in these materials. A timeofflight analysis of the photogenerated hot electrons yields a speed that is consistent with the group velocity at the Fermienergy of the Bi_{2}Se_{3}.
Results
Timeresolved and timeintegrated photocurrent spectroscopy
The investigated ntype Bi_{2}Se_{3} films are embedded in a metaltopological insulatormetal photodetector geometry (Fig. 1a). We characterize the ultrafast photocurrents by an onchip, timedomain THz photocurrent spectroscopy with a picosecond timeresolution (Fig. 1b and Methods)^{20,21,22}. We excite the Bi_{2}Se_{3} films with a circularly polarized pump pulse under an oblique angle θ (Fig. 1b). Such helical photons excite spins asymmetrically in kspace owing to angular momentum selection rules in Bi_{2}Se_{3} (refs 9, 23). In turn, a net outofequilibrium spin polarization is acquired in the Dirac cone. After photoexcitation, spin and charge degrees of freedom relax on different timescales in the bulk and surface states of Bi_{2}Se_{3} (ref. 24). For the surface states, spin depolarization, intraband cooling via surface electronphonon scattering and interband electronhole recombination occur on a subpicosecond to picosecond timescale^{24,25,26}.
Figure 1c shows an image of a Bi_{2}Se_{3} film with two gold striplines serving as electronic contacts. We scan the pump laser across such a Bi_{2}Se_{3} film and record the timeintegrated photocurrent I_{photo} as a function of the laser position (Fig. 2a). At the metal interfaces (triangles), a photothermoelectric current is generated because of a laserinduced heat gradient and the large thermoelectric power of Bi_{2}Se_{3}. In between the two striplines, the timeintegrated photocurrent averages to be close to zero (circles in Fig. 2a,b). Changing the polarization of the exciting laser from circularly lefthanded to horizontally linear and then to circularly righthanded (Fig. 2c), we observe the sinusoidal fingerprint^{15} of a circular photogalvanic current with an amplitude C (Methods and Supplementary Fig. 1). The sinusoidal fits consider the circular and linear photogalvanic currents depending on the oblique angle θ=+17° (Supplementary Note 1 and Supplementary Fig. 1). The different background amplitudes of I_{photo} are highlighted as dashed lines in Fig. 2c. They represent the spatially varying, dominant photothermoelectric current I_{thermo}. We introduce a fidelity f=C/(C+I_{thermo}), which describes the ability to resolve the circular photogalvanic current with respect to I_{thermo}. For all samples, we find f to be smaller than 40% in timeintegrated measurements. The maximum value is achieved for the laser being positioned at the centre of the Bi_{2}Se_{3} film (for example, circles in Fig. 2), where the overall I_{thermo} generated within the laser spot averages out towards the noise amplitude (I_{thermo}→A_{noise}~40 pA). As we show below, highspeed, timeresolved measurements provide a fidelity close to unity. This high fidelity can be achieved because photocurrent contributions on different timescales with different directions do not average out.
Photothermoelectric currents at the contacts
We measure the timeresolved photocurrent I_{sampling} for different excitation positions on a Bi_{2}Se_{3} film from the right to the left contact (right and leftward triangles in Fig. 3a). For all positions, we observe that I_{sampling} changes sign and therefore direction at a certain Δt. This means that for short (long) times, a current with a direction towards the closer (farther) contact dominates. The origin of the opposing currents can be understood as follows. Within the first picosecond after excitation, the electrons thermalize to form a hot carrier ensemble^{27,28}. They propagate away from the laser spot due to sampleinternal potentials such as the thermopower and the density gradient. Close to the metal interfaces, the thermopower generated between Bi_{2}Se_{3} and Au gives rise to the photothermoelectric current.
Space and voltage dependence of the timeresolved photocurrents
In a timeofflight analysis, we determine the ultrafast transport currents with opposing directions for each laser position (Supplementary Figs 2 and 3). The analysis allows us to estimate the fastest timeofflight velocity of photogenerated hot electrons to be v_{e}=(5.7±1.5) × 10^{5} ms^{−1} at room temperature (Supplementary Note 2), which agrees remarkably well with the group velocity of the Bi_{2}Se_{3} at the given Fermienergy^{29,30}, as discussed below. In addition, we determine the decay time of the photocurrent signals to be (3.0±0.5) ps at room temperature, which is in agreement with the picosecond relaxation dynamics of hot electrons^{26,27}.
In a next step, we focus the laser onto the centre of the Bi_{2}Se_{3} film (Fig. 3b) where the timeintegrated photocurrent is close to zero (for example, circle in Fig. 2a). At high bias, the data mimic the curves measured at zero bias at the right and left contacts (right and leftward triangles in Fig. 3a). This means that at the centre in between the striplines (Fig. 3b), an externally applied V_{sd} gives rise to an additional drift of the photogenerated hot electrons. The polarity of the applied bias allows us to confirm that the dominating contribution stems from electrons and not holes (Supplementary Note 2 and Supplementary Fig. 4). The similarity of the top and bottom traces in Fig. 3a,b suggests that for certain values of V_{sd} at the centre, the corresponding electrostatic potential has an equivalent impact on the hot electron dynamics as the thermopower at the metal interfaces. In turn, at the centre of the sample and for V_{sd}=0 V, the net field due to sampleinternal potentials is zero. Particularly, the temporal tails of I_{sampling} are close to zero for Δt≥10 ps at zero bias. We note that there is always an ultrafast photocurrent within the first picoseconds at this position (Fig. 3b).
Helicitydependent ultrafast photocurrents
Peculiarly, the sign and amplitude of the prevailing ultrafast contribution are completely controlled by the polarization of the exciting photons and their oblique angle of incidence. Figure 4a shows I_{sampling}(Δt) measured close to the centre of another Bi_{2}Se_{3} film for a varying photon polarization. The fitting curves consider ultrafast transport currents with changing directions depending on the photon polarization (Supplementary Fig. 2). Particularly, for Δt≈4 ps, the current amplitude depends only on the photon polarization. Microscopically, the helicitydependent contribution results from an asymmetric excitation in kspace of the spinpolarized surface states caused by the peculiar helical symmetry of the surface states (Supplementary Fig. 1). The panels in Fig. 4a sketch the direction of the helicitydependent current for circularly lefthanded and righthanded polarized light (red and blue). The corresponding, measured peak amplitude follows the sinusoidal fingerprint of the circular and linear photogalvanic effects (Fig. 4b) with a fidelity f exceeding 95% for all temperatures. At the same excitation position, when the polarization dependence is measured in a timeintegrated manner (Fig. 4c), the timeaveraged background reduces f to be 32.6%, because the photocurrent contributions with different directions compete on long timescales.
Discussion
It is insightful to highlight the dynamics of the photocurrents in Fig. 4a again. After the spin depolarization and intraband cooling (Δt≥5 ps), the spin information and helicity protection are predominantly lost. Then, the expansion dynamics of the same (hot) electrons dominates the transport dynamics. These currents follow the combined influence of the thermopower and the electrostatic potentials. In the following, we will derive further insights into the interplay of the two potentials. Starting point is the fact that the optoelectronic expansion dynamics are dominated by photogenerated hot electrons that propagate at v_{e}=(5.7±1.5) × 10^{5} ms^{−1} (Supplementary Note 2). Generally, the dispersion of the surface states can be written as^{31}
with ℏk the inplane crystal momentum, v_{Dirac} is group velocity close to the Dirac point, D=−12.4 eV Å^{2} a quadratic term resulting from the broken particlehole symmetry^{32}, B=0 eV Å^{2} describing massive states^{32} and Δ the energy gap for the intersurface coupling, which is zero for the investigated samples. The value for the binding energy of the Dirac point E_{0}=(0.55±0.05) eV is determined from Hallmeasurements (assuming an effective mass of 0.13·m_{e})^{33}. Recent photoemission experiments on ntype Bi_{2}Se_{3} and related topological materials demonstrate that, only a few 100 fs after photoexcitation, the electron distribution is already centred at the Fermienergy^{25,27,28}. From equation (1), we derive the group velocity to be (5.5±0.2) × 10^{5} ms^{−1} at the Fermienergy E_{Fermi}. We note that for bulk states, the principal order of the group velocity is in agreement with this value at the given Fermienergy^{32}. Importantly, the experimental value of v_{e} agrees remarkably well with the derived value and with the reported one determined from photoemission experiments^{29,30}. All values exceed the saturation drift velocity as measured in Bi_{2}Se_{3}based transistor devices^{34}. In other words, the group velocity of photogenerated charge carriers determines the ultimate speed of the optoelectronic response in topological insulators.
Experimentally, this manifests itself in a linear timeofflight diagram with v_{e}=(5.7±1.5) × 10^{5} ms^{−1} (Supplementary Figs 2 and 3). To further discuss this surprising result, we use a twotemperature model T_{e} and T_{phonon} for the electron and phonon baths^{35,36}. The electron heat capacity is well approximated by a linear temperature dependence C_{e}~C′_{e}T_{e}, if T_{e}≪T_{Fermi} with T_{Fermi}=3,320 K the Fermitemperature of the measured ntype Bi_{2}Se_{3} films^{37}. Within this model, we calculate the maximum electron temperature T_{e}^{max}=1,500 K for the experimental parameters as in Fig. 3a (ref. 35). The electron–electron collision time can be estimated to be τ_{ee}~ℏE_{Fermi}/(k_{B}T_{e})^{2}=9 fs, with k_{B} the Boltzmann constant^{37}, and the electronphonon scattering time is on the order of τ_{ephonon}~ℏ/(k_{B}T_{bath})=94 fs. The latter was recently determined to be ~0.7 ps for optical phonons and ~2.3 ps for acoustic phonons^{24,27}. In this regime (τ_{ee}≪τ_{ephonon}), there exists a strong electronlattice nonequilibrium, and the electron relaxation is governed by the electron–electron collisions^{35}. It was reported that, hereby, the electronic heat transport occurs at the Fermi velocity after an ultrafast optical excitation of the electron bath^{36}. The dynamics can be analytically described by a twotemperature model in the limit of a strong electronlattice nonequilibrium in combination with an electron heat conductance κ_{e}∝(T_{e})^{−1} (ref. 35). In the present experiment on Bi_{2}Se_{3} films, the ultrafast current of hot electrons is measured. In our interpretation, this current carries the electronic heat to the contacts, and the corresponding heat transport at the Fermi velocity explains our data with linear timeofflight diagrams (Supplementary Figs 2 and 3).
Our onchip, timedomain THz spectroscopy allows us to reveal the impact of an electrostatic potential on such nonequilibrium expansion dynamics (Fig. 3b). Along this line, we estimate the maximum gradient of the local electron temperature profile to be ∇T_{e}^{max}~T_{e}^{max}/(0.5d_{stripline})=1,500 K/(0.5 × 15 μm)=200 K μm^{−1}, with d_{stripline} the distance between the two striplines. In first approximation, the measured expansion dynamics of the hot electrons are identical for exciting the Bi_{2}Se_{3} films at the contacts (top and bottom traces in Fig. 3a) and for an excitation spot in the centre of the sample at finite bias (top and bottom traces in Fig. 3b). For the results in Fig. 3b, the laser position is carefully chosen such that influence of the thermopower induced at the contacts is close to zero. Then for a finite bias V_{sd}, the electrostatic potential gradient ∇V_{electrostatic} at the laser spot has an equivalent impact on the hot electron dynamics as the thermopower at the metal interfaces. We describe this thermopower at very short timescales as the product of S_{Bi2Se3}·∇ T_{e}, with S_{Bi2Se3} a nonequilibrium, effective Seebeck coefficient. Comparing the amplitudes of the top and bottom traces of Fig. 3a,b, we extract ∇V_{electrostatic}=(820±440) V m^{−1} (Methods). In turn, we can estimate the effective Seebeck coefficient to be in the order of S_{Bi2Se3}^{minimum}~∇V_{electrostatic}/∇ T_{e}^{max}=−(4.1±2.2) μV K^{−1}. The extracted value phenomenologically describes the nonequilibrium thermopower at the Bi_{2}Se_{3}metal contacts. To the best of our knowledge, the above derivation is the first estimate of an effective Seebeck coefficient of nonequilibrium hot electron ensembles after a pulsed laser excitation. The derived nonequilibrium Seebeck coefficient is smaller than the typical quasiequilibrium value S~−50 μV K^{−1} (ref. 38). This can be understood as follows: for the experimental parameters in Fig. 3, we concurrently find a timeaveraged amplitude I_{photo}=370 pA at the Bi_{2}Se_{3}metal contact and at an acquisition time of ~ms. At this long timescale, the heat transport is governed by phonons. Accordingly, we numerically calculate a temperature increase at a Bi_{2}Se_{3}metal contact of ΔT=47 mK (for both the phonon and electron baths). In turn, we derive a quasistatic Seebeck coefficient by the following expression ΔV~S_{Bi2Se3}^{quasistatic} ·ΔT, with S_{Bi2Se3}^{quasistatic}=−23 μV K^{−1}. For the sample and experimental conditions as in Fig. 2, we calculate S_{Bi2Se3}^{quasistatic}=−62 μV K^{−1} at room temperature. Both values are consistent with the above quasiequilibrium value of the Seebeck coefficient. In other words, at long timescales, the heat transport is dominated by phonons. However, at ultrashort timescales, it is governed by a highly nonequilibrium expansion of hot electrons. In our interpretation, this nonequilibrium current of hot electrons carries the electronic heat to the contacts as long as there exists a strong electronlattice nonequilibrium as discussed above.
Intriguingly, despite of such highspeed dynamics of the hot electrons, the data at Δt~4 ps in Fig. 4a demonstrate that the helicitydependent currents can be still addressed and readout. The underlying physical reason is the protection of the helical states within the spin depolarization time. Only for Δt≥5 ps in Fig. 4a, all currents have the same negative sign. That means that, for this particular experiment, still a tiny thermopower potential drives these currents to the left contact, which is consistent with the negative offset observed in the timeintegrated measurement (Fig. 4c). Again, for Δt≥5 ps, there is no polarization control for these currents anymore, since the pumplaser pulse is off and the spin depolarization has occurred (Fig. 4a).
The circular and linear photogalvanic effects are induced via surface states and not bulk states because of the broken inversion symmetry at the surface of Bi_{2}Se_{3} (refs 15, 17). A normal incidence geometry (θ=0°) suppresses the photogalvanic effects (Supplementary Fig. 1), indicating an inplane spin distribution and more fundamentally, an inplane rotational symmetry of the involved electron states in Bi_{2}Se_{3}. Generally, there may be a small contribution from a helicityindependent transverse photon drag effect^{17}. Contributions from Rashbasplit bulk states at a surface inversion layer can be assumed to be negligible for the examined range of E_{photon} (Supplementary Note 3 and Supplementary Fig. 5). Overall, the ultrafast polarizationcontrolled photocurrent is limited by the spin lifetime in Bi_{2}Se_{3}.
To conclude, our experiments demonstrate the temporal separation of helicitydependent surface photocurrents from polarizationindependent bulk currents even at room temperature, and they reveal the onset of photothermoelectric currents at ultrafast timescales. We elaborate the connection between the nonequilibrium currents of hot electrons directly after the pulsed laser excitation and the timeaveraged thermoelectric currents, which are typically described in the framework of a Seebeck coefficient. A timeofflight analysis yields a speed of the photogenerated electrons that is consistent with the group velocity at the Fermienergy of the Bi_{2}Se_{3}. Our results significantly advance the understanding of the optical excitation scheme and the ultrafast photocurrent dynamics in topological insulators. The timedelayed photothermoelectric currents in Bi_{2}Se_{3} are partly caused by the ndoping of the material. They can be reduced by utilizing topological insulators with a Fermienergy in the Dirac cone, such as Bi_{2}Te_{2}Se or (Bi_{1x}Sb_{x})_{2}Te_{3} (refs 7, 8). The picosecond response time of the surface currents proves the anticipated potential of topological insulators as promising materials for highspeed optoelectronic applications from the THz to the infrared range.
Methods
Fabrication of the Bi_{2}Se_{3} films
The investigated Bi_{2}Se_{3} films have a thickness of 50 nm≤250 nm and a lateral dimension of >15 μm. We produce the Bi_{2}Se_{3} films from 99.999% pure crystalline Bi_{2}Se_{3} granulate via exfoliation. We analyse the homogeneity and the dimensions of the films using optical microscopy, atomic force microscopy and white light interferometry. The presented results have been reproduced on three independent samples, which have a thickness of 75, 90 and 150 nm. Hallmeasurements on a 65nmthin Bi_{2}Se_{3} film from the same batch yield an electron density of ~4·10^{19} cm^{−3} at room temperature.
Design of the stripline circuit
Starting point is a 430μmthick sapphire substrate covered with a 300nmthin layer of ionimplanted silicon (Si). In a first optical lithography step, the Austonswitch geometry is formed via HF etching. The remaining silicon strip serves as a field probe (Austonswitch). Bi_{2}Se_{3} films are mechanically transferred onto the substrate by exfoliation. In a second optical lithography step, we evaporate 10 nm titanium (Ti) and 110–200 nm gold (Au) to form the waveguide circuits contacting the Bi_{2}Se_{3} films and the readout of the Austonswitch. The distance between two parallel striplines is 15 μm. Each stripline itself has an individual width of 5 μm. The Bi_{2}Se_{3} films are placed at a typical distance of 100 μm≤800 μm from the Austonswitch. The bias voltage V_{sd} is applied between striplines (see Fig. 1). The striplines including the Bi_{2}Se_{3} films have a twoterminal resistance of 2.9 kΩ, while the Bi_{2}Se_{3} films have a resistance of ~55 Ω. Hereby, the gradient ∇V_{electrostatic} can be estimated from the bias voltage via ∇V_{electrostatic}=V_{sd}/15 μm · 55 Ω/2.9 kΩ.
Timeintegrated photocurrent spectroscopy
The timeaveraged I_{photo} is measured in between the two striplines (Fig. 1b) with a currentvoltage amplifier.
On chip timedomain THz photocurrent spectroscopy
The Bi_{2}Se_{3} films in the stripline circuit are optically excited by a pump pulse with ~200 fs pulse length generated by a titanium:sapphire laser at a repetition frequency of ~76 MHz with a photon energy of E_{photon}=1.53 eV and E_{photon}=1.59 eV. After excitation, an electromagnetic pulse starts to travel along the striplines. A field probe senses the transient electric field of the travelling pulse (Fig. 1b). Here we utilize an Austonswitch based on ionimplanted silicon. At a time delay Δt with respect to the pump pulse, the Austonswitch is shortcircuit by a probe pulse for the duration of the lifetime of the photogenerated charge carriers in the silicon (τ≤ 1 ps). During this time period, the transient electric field present at the field probe drives the current I_{sampling}. In turn, measuring I_{sampling}(Δt) yields information on the optoelectronic response of the Bi_{2}Se_{3} with a picosecond timeresolution. The time a THzpulse travels from the Bi_{2}Se_{3} film to the field probe can be estimated via t_{travel}=d· n_{sapphire}/c=0.5 mm · 3.07/c~5.1 ps for a distance of 0.5 mm between the Bi_{2}Se_{3} film and the Austonswitch. The striplines have a total length exceeding 48 mm. Thus, reflections at the end of the striplines are expected for Δt≥490 ps. These reflections are strongly reduced due to damping and radiation losses. Therefore, no reflections overlap with the electromagnetic signal coming directly from the Bi_{2}Se_{3} films in our data. The data of Figs 2 and 4 are presented for room temperature. The timeofflight data in Fig. 3 are depicted for 77 K, because the transfer characteristics of the striplines and therefore the signaltonoise ratio are enhanced at lower temperatures. A timeofflight analysis of data at room temperature is presented in the Supplementary Fig. 3. The position of the pumpspot is set with a spatial resolution of ~100 nm, while the position of the probespot is kept constant throughout the experiments. All measurements of I_{sampling} were carried out utilizing an optical chopper system, a currentvoltage converter connected to the field probe and a lockin amplifier. The spot size (FWHM) of the pumplaser is 3–4 μm for all timeresolved experiments and up to 9 μm for the timeintegrated experiments. The laser power of the pump (probe) pulse is chosen to be in the range of 0.1–20 mW (80–150 mW). All data are taken in vacuum (~10^{−5} mbar) to prevent the effect of photodesorption of oxygen on the surface of the Bi_{2}Se_{3}. The measurements have been reproduced for different temperatures between 4 and 295 K.
Polarization control and sinusoidal fitting function
The polarizationdependent photocurrents are characterized by measuring I_{photo} or I_{sampling}, while rotating a λ/4waveplate by an angle φ. The rotation changes the photon polarization with a period of 180° from linearly polarized (φ=0°) to lefthanded circular (φ=45°), to linearly (φ=90°), to righthanded circular (φ=135°) and to linearly (φ=180°). The fitting curves in Figs 2c and 4b,c describe the photocurrent in the ydirection of the Bi_{2}Se_{3} films (Fig. 1b) as j(φ)=C sin 2φ+L_{1} sin 4φ+L_{2} cos 4φ+D. As recently demonstrated^{15}, C describes the helicitydependent circular photogalvanic effect with a rotational inplane symmetry in Bi_{2}Se_{3}. L_{1} comprises the helicityindependent linear photogalvanic effect. L_{2} and D are bulk contributions (Supplementary Note 1).
Additional information
How to cite this article: Kastl, C. et al. Ultrafast helicity control of surface currents in topological insulators with nearunity fidelity. Nat. Commun. 6:6617 doi: 10.1038/ncomms7617 (2015).
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Acknowledgements
We thank Leonid Levitov and Michael Knap for insightful discussions, as well as Andreas Brenneis and Ursula Wurstbauer for technical assistance. This work was supported by the DFG via SPP 1666 (grant HO 3324/8), ERC Grant NanoREAL (n°306754), the DFG excellence cluster ‘Nanosystems initiative Munich (NIM)’, the ‘Center of NanoScience (CeNS)’ in Munich and the Munich Quantum Center (MQC).
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C. Kastl and C. Karnetzky performed the experiments and analysed the data together with A.W.H. A.W.H., C. Kastl and C. Karnetzky conceived the study and cowrote the paper with H.K.
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Supplementary Figures 15, Supplementary Notes 13, and Supplementary References (PDF 332 kb)
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Kastl, C., Karnetzky, C., Karl, H. et al. Ultrafast helicity control of surface currents in topological insulators with nearunity fidelity. Nat Commun 6, 6617 (2015). https://doi.org/10.1038/ncomms7617
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Further reading

Josephson current mediated by ballistic topological states in Bi2Te2.3Se0.7 single nanocrystals
Communications Materials (2020)

Deep tuning of photothermoelectricity in topological surface states
Scientific Reports (2020)

Scattering symmetrybreaking induced spin photocurrent from outofplane spin texture in a 3D topological insulator
Scientific Reports (2020)

Exploration of terahertz from timeresolved ultrafast spectroscopy in singlecrystal Bi2Se3 topological insulator
Journal of Materials Science: Materials in Electronics (2020)

A cascading nonlinear magnetooptical effect in topological insulators
Scientific Reports (2018)
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