Millimetre-long transport of photogenerated carriers in topological insulators

Excitons are spin integer particles that are predicted to condense into a coherent quantum state at sufficiently low temperature. Here by using photocurrent imaging we report experimental evidence of formation and efficient transport of non-equilibrium excitons in Bi2-xSbxSe3 nanoribbons. The photocurrent distributions are independent of electric field, indicating that photoexcited electrons and holes form excitons. Remarkably, these excitons can transport over hundreds of micrometers along the topological insulator (TI) nanoribbons before recombination at up to 40 K. The macroscopic transport distance, combined with short carrier lifetime obtained from transient photocurrent measurements, indicates an exciton diffusion coefficient at least 36 m2 s−1, which corresponds to a mobility of 6 × 104 m2 V−1 s−1 at 7 K and is four order of magnitude higher than the value reported for free carriers in TIs. The observation of highly dissipationless exciton transport implies the formation of superfluid-like exciton condensate at the surface of TIs.


Supplementary Note 1: Extraction of photocurrent decay length.
A hyperbolic function must be used instead of exponential when L is comparable or shorter than L d . We have fitted the photocurrent distributions by an exponential function or a hyperbolic function , where L is the channel length and L D is the photocurrent decay length. The two fittings are similar when L d << L, but the hyperbolic function fits more accurately when L d is comparable or longer than L (Supplementary Figure 4). We justify the hyperbolic fitting below. The steady state continuity equation describing exciton concentration is, where D is the diffusion coefficient,  is the lifetime, G is the generation rate proportional to laser power, and the device geometry is shown in Supplementary Figure 5. The local laser generation is considered as a delta function. For boundary conditions, we assume all excitons are separated at contact, so n drops to zero at x = 0. The excitons cannot flow out of the nanoribbon, so drops to zero at x = L. The exciton concentration is continuous at x = x 0 and its derivative follows a relation that can be found out by integrating equation (1). The solution of exciton concentration then can be found out, from which we can calculate current distribution as, where . The above derivation considers a diffusive process. For a ballistic process, if we assume that the injected excitons have equal probability of moving left or right and bounce back at the end of the nanoribbon without loss, we can derive a similar photocurrent distribution, where I L is the current contributed by the excitons moving left from the injection point to contact and I R is the current contributed by the excitons moving right from the injection point and then bounced back at the end of the nanoribbon.
is the average distance that an exciton condensate can travel ballistically before recombination. The exciton velocity v is expected to be close to the Fermi velocity of the massless electrons at the TI surface. Therefore, we showed the current follows a hyperbolic function with the excitation position, in both diffusive and ballistic cases.

Supplementary Note 2: Error analysis.
The main error in L d originates from error in photocurrent and laser power. The photocurrent error is about 1%. Because of the large scan areas, the incident power is not uniform and the photocurrent presented has already been corrected with the power at different injection position. The uncertainty in power measurement then also affects the corrected photocurrent and is estimated to be 0.5%. From equation (2) or (3), it is easy to find , where . We then propagate these error sources to estimate the error in L d , where . Both currents are corrected by power, i.e., where J is the uncorrected current and P is laser power. The factor of 2 is because z is the ratio of two currents at and . The error is large when L d is long at low temperature since z is close to 1. The error of L d at different temperature is shown in Fig. 1e.

Supplementary Note 3: Exclusion of other photocurrent mechanisms.
Mechanisms that can generate photocurrent include thermo-electric effects, lateral Photo-Dember effects 1,2 , doping level gradient 3,4 , photo-recycling 5 , and in the case of TIs, spin-polarized current 28 , chiral spin mode 6 . The temperature gradient caused by local laser heating drops exponentially over a thermo decay length of , where κ is the thermal conductivity of the material, and is the heat dissipation into the environment 7 . In our case, is dominated by the heat transfer through the 300 nm-thick SiO 2 . An estimate yields a thermal decay length shorter than 1 m in Bi 2 Se 3 . Hence thermoelectric effects cannot be used to explain the observed mm-long photocurrent decay length. Both lateral Photo-Dember effects and doping level gradient effects only generate photocurrent when the laser is in the source-drain channel (Supplementary Figure  12) and thus cannot explain the photocurrent outside the channel. Photo-recycling has recently been used to understand the high power conversion efficiency in halide perovskite materials 5 , where emitted photons can travel inside the material and are re-absorbed, leading to an efficient energy transfer in a distance over 50 m. However, this mechanism does not explain the mm-long photocurrent decay length in Sb-doped Bi 2 Se 3 , which has weak photoluminescence. Due to spin-momentum locking of the TI surface states, oblique angle circular polarized photon injection creates a spin imbalance which results in an electric current. The photocurrent we observed is not from this mechanism because linear polarized light and normal incidence are used. Magnons in reference 6 is a result of excitation to the second Dirac cone located 1.5 eV above the conduction band edge. In our study, photocurrent is observed down to 0.7 eV photon energy, which indicates that second Dirac cone is not involved. In addition, the observed chiral spin mode in reference 6 sensitively depends on the circular and linear polarization of the light. On the other hand, our photocurrent distribution under normal incidence configuration does not depend on the polarization of the laser.

Supplementary Note 4: Theoretical calculation of critical temperature.
We assume a quasi-equilibrium state where the excited electrons and holes stay in the upper and lower Dirac cones respectively with well-defined chemical potentials and . The noninteracting Hamiltonian can be written as, (5) where and are annihilation operators for electrons in the upper and lower Dirac cones respectively. In the picture of electrons, we consider a Coulomb interaction, (6) where the screening effect has been taken into account with the screening wave number , where and is the wavenumber at the Fermi surfaces of the electron and hole pockets. The Coulomb repulsive interaction between electrons, equivalent to the attractive interaction between electrons and holes, can generate excitons and gap out the Dirac state. Similar to reference 10, we perform a mean-field calculation of the possible Cooper instability, analogous to the BCS theory but in the particle-hole channel. Specifically, we introduce mean-field order parameter in the particle-hole channel with , where The self-consistent equation can be obtained by minimizing the energy of the mean-field ground state as, where , , and the integral is restricted to a momentum cutoff set by . Since the photons generate equal number of electrons and holes and the original chemical potential of the sample in the dark is close to the Dirac point, we expect that the electron and hole Fermi surfaces enjoy a perfect nesting with chemical potential (therefore , ). Indeed, the self-consistent solution of the equation with a -resolved order parameter indicates that the ground state can develop a full excitonic gap due to BCS condensation. Furthermore, if one considers a spatially-resolved complex order parameter, the thermal fluctuation is found to generate vortices at finite temperature that are more stable than the real order parameter solutions. Meanwhile, the KT transition takes place characterizing the phase transition. The KT temperature is the energy scale for the proliferation of vortices and antivortices. To calculate the KT temperature, we first evaluate the exciton current with a small momentum of the excitons. The superfluid density can then be obtained from . The superfluid density directly determines the transition temperature through . Since is dependent on which again relies on , is a function of , and therefore dependent on the laser intensity. When lowering the chemical potential , the obtained first increases because of the weaker screening and then decreases due to the lower density of states at small . Since is proportional to the laser intensity, these results reveal the trend of the growth of with lowering intensity, and are qualitatively in agreement with the experimental observation (Supplementary Figure 10). However, the evolution of T KT at very low laser intensity is not experimentally observable due to low signal. The obtained mean-field phase diagram as well as the KT temperature are shown in Supplementary Figure 11, where an excitonic condensate supporting the long-range photocurrent is stabilized from the electron-hole gas after crossing KT transition curve ( as a function of ). For  = 0.4, the maximum T KT is calculated to be 37 K, comparable to the experimental observation. This value is expected to be in the range of 0.1 to 0.5, as at low frequency at the TI surface is about one half of the bulk value and is expected to vary from 50 to 10 in the previous reports 8,9 .
Finally, we estimate the electric field required to split the excitons. We first calculate the condensation energy E cond of the excitonic phase, i.e., the energy reduction of the excitonic order compared to the original Dirac surface state. E cond = -3.0 meV for the optimal  2 (0.1eV) with  = 0.4. The Bohr radius can then be estimated by , which gives around 110 nm. This leads to an electric field of E cond / ea B ~ 2.7 × 10 4 V/m, much higher than the maximum electric field (600 V/m) used in Fig. 2. . The two diodes D 1 and D 2 correspond to the metal-TI junctions at the drain and source contacts, respectively. R is the TI nanoribbon resistance. The current is measured by a preamp as I 2 . The IQE is I 2 h / eP, where P is the absorbed power and h is the photon energy. There are three major loss mechanisms: (1) recombination of exciton condensates, (2) recombination of normal excitons, and (3) loss at the junction. (1) is much less than (2) because of the ballistic transport of the condensate. The last mechanism is expected to be small because of the efficient charge transfer at TI and metal junction. Therefore, IQE is an evaluation of the fraction of condensed excitons out of the total excitons. . The experimental data is in good agreement with the theory near T KT but deviates at higher temperature. b, T c as a function of laser power. T KT is extracted from the turning point in a. At higher power, the T KT extracted from the L d vs T curves decreases rapidly. The red curve at 100 W does not reach condensate in the experimental temperature range. Here L d is measured as a function of temperature at different excitation power in a device different from the one shown in Fig. 4c and f. Note that because of the limited length of the nanoribbon, only the lower limit of the L d can be accurately determined, which we use to represent the L d value after saturation. The T KT value in this device appears to be higher than that in Fig. 1e. Figure 11: Phase diagram from mean-field calculations. The electron-hole gas state excited by photons is stable at high temperature, while the system starts to enter into the BCS excitonic condensation phase for temperature lower than the mean-field transition temperature . The calculated KT transition temperature is denoted by the brown data curve . For temperature lower than , the vortices become closely bound and the system displays long-range transport of excitons. The parameters used are  = 0.4, v F = 5 × 10 5 m/s, the energy cutoff of the surface state 0.3 eV, and we assume a perfect nesting with .

Supplementary
Supplementary Figure 12: Exclusion of photo-Dember effects and doping inhomogeneity effects as photocurrent generation mechanisms. a, Lateral photo-Dember effects: difference between electron and hole mobilities results in a mismatch between the electron and hole centers. b, Doping level gradient effects: a gradient in doping level causes tilting in the valence band (VB) and conduction band (CB) and results in a build-in electric field. c, Schematic of the equivalent electrical circuit, showing that neither mechanism can generate photocurrent when laser is outside the source-drain channel.