Stable topological insulators achieved using high energy electron beams

Topological insulators are potentially transformative quantum solids with metallic surface states which have Dirac band structure and are immune to disorder. Ubiquitous charged bulk defects, however, pull the Fermi energy into the bulk bands, denying access to surface charge transport. Here we demonstrate that irradiation with swift (∼2.5 MeV energy) electron beams allows to compensate these defects, bring the Fermi level back into the bulk gap and reach the charge neutrality point (CNP). Controlling the beam fluence, we tune bulk conductivity from p- (hole-like) to n-type (electron-like), crossing the Dirac point and back, while preserving the Dirac energy dispersion. The CNP conductance has a two-dimensional character on the order of ten conductance quanta and reveals, both in Bi2Te3 and Bi2Se3, the presence of only two quantum channels corresponding to two topological surfaces. The intrinsic quantum transport of the topological states is accessible disregarding the bulk size.

U nconventional quantum matter can be easily hidden within the rich existing library of condensed matter and to recognize it often novel concepts have to be invoked. It also takes a profound understanding of the real material constraints that can prevent the unconventional properties from being detected. Three-dimensional (3D) topological insulators are a spectacular example of this 1 -narrow-band semiconductors, well known for their high performance as thermoelectrics 2 , they were discovered to support unusual gapless robust two-dimensional (2D) surface states that are fully spin-polarized with Dirac-type linear electronic energy-momentum dispersion 3,4 , which makes them protected against backscattering by disorder 3-6 . These materials have narrow bulk gaps (B200-300 meV) and have charge carriers donated by intrinsic crystalline lattice defects such as vacancies and antisites 7 . As a result, the conduction through the bulk and its intermixing with surface channels is what largely denies direct access to surface charge transport required for the implementation in spin-based nanoelectronics 8 or fault-tolerant topological quantum computing 4 .
Extensive attempts to reduce the contribution of bulk carriers, involving techniques such as nanostructured synthesis/ growth 9,10 , chemical doping 11 or compositional tuning 12 , have relied on electrostatic gating of micro-or nanostructures comprising tens of nanometre thin films 13 or similarly thin exfoliated crystals 14 to gain less ambiguous access to the surface states. Here we demonstrate that bulk conductivity in topological insulators (TIs) can be decreased by orders of magnitude to charge neutrality point on a large (depth) scale by the controlled use of electron beams, which for energies below B3 MeV are known to produce a well-defined, stable and uniform spread of Frenkel (vacancy-interstitial) pairs 15 within their penetration range of hundreds of micrometres. The combined effect of these pairs is to compensate for the intrinsic charged defects responsible for the conductivity of the bulk while crystal lattice integrity is maintained and, as we demonstrate by angularly resolved photoemission spectroscopy (ARPES), the topological Dirac surface states remain robust. Stable surface conduction channels are achieved when a sufficient irradiation dose is followed by the optimally engineered annealing protocol, thereby resolving one of the key limitations of bulk TIs.

Results
Compensating bulk carriers using electron beams. Under exposure to light particles such as electrons, the production of vacancy-interstitial point defects 16 in solids known as Frenkel pairs is well established. The pair formation has an energy threshold which scales with atomic weight-it is high for heavy Bi and is lower for lighter Te and Se, and the effective cross-section s for pair creation on different sublattices in materials containing these elements depends on the projectile energy (Fig. 1a,b and Supplementary Figs 1,2). This defines a pair-production window: below these thresholds, the given sublattice becomes immune to irradiation, whereas at energies above B3 MeV, clustering and even nuclear reactions 17 may occur. The process of Frenkel pair formation is straightforward and can be well controlled. The charge is primarily delivered by introduced vacancies, since at room temperature interstitials do not contribute owing to their much lower (compared with vacancies) migration barriers. For electron beam energies above B1.5 MeV, the effective cross-section s on a Bi sublattice is the highest and we will show that donor type defects on this sublattice do prevail. For energies below Bi threshold (B1.2 MeV), the creation of Frenkel pairs will be mostly on Se or Te sublattices and, correspondingly, by tuning electron beam energy a different mix of donor/acceptor defects and a net acceptor charge doping can be expected.
Generally, crystal growth of tetradymite crystals such as Bi 2 Te 3 and Bi 2 Se 3 results in equilibrium defect configurations comprising vacancies and antisites on both sublattices 7,18 . The net free charge balance that delivers carriers to bulk conduction or valence bands from these defects can be varied by growth conditions or doping 11 . In undoped Bi 2 Se 3 , where Se vacancies are presumed to dominate, the net conduction is by electron carriers or n-type. In Bi 2 Te 3 , where antisites are prevalent 19 the conductivity is usually p-type, namely by hole carriers, although by varying stoichiometry it has been grown of either conductivity type. Without a priori knowledge of the net donor or acceptor flavour of the pairs on different sublattices, we have chosen first to irradiate p-type topological materials Bi 2 Te 3 and Ca-doped 11 Bi 2 Se 3 with 2.5 MeV electron beams that create Frenkel pairs on all sublattices according to s(E), shown in Fig. 1b and Supplementary Fig. 2. Our results demonstrate that in the above-mentioned Bi-based TIs at this energy, the net flavour is of donor type. It is also important to note that even for relatively high (\1 C cm À 2 ) electron irradiation doses, the lattice parameters remain unchanged (see Supplementary Fig. 3).
Ambipolar conduction tuned by irradiation dose. Longitudinal resistivity r xx of the initially p-type Bi 2 Te 3 measured at 20 K in situ in the irradiation chamber (see 'Methods' section) as a function of irradiation dose is shown in Fig. 1e. Most immediately notable features in the figure are a nearly three orders of magnitude resistivity increase to a maximum r max xx and the observed ambipolar conduction as a function of irradiation dose with well-distinguished p (hole) and n (electron) conduction regions. The resistivity maximum r max xx (the conductivity minimum s min xx ) is at the charge neutrality point where conduction is converted from p-to n-type, as determined from Hall resistivity (see Fig. 2). The same type conversion is observed in Ca-doped Bi 2 Se 3 (Supplementary Fig. 4a and Supplementary Note 1), which is also p-type 11 . As long as the terminal irradiation dose f is relatively low, below B0.1 C cm À 2 , r xx (f) traces its shape upon temperature cycling to room temperature and back to 20 K, with r max xx reproduced by the next irradiation cycle. Here it is apparent that the value of f max at the charge neutrality point is not universal; it depends on the starting free carrier concentration n b but can be straightforwardly scaled using universal slope ' @n b @f of quasi-linear variation of n b versus dose ( Fig. 2a and Supplementary Figs 4b and 5).
Longitudinal and Hall resistivities across CNP. Ex situ measured resistivities of Bi 2 Te 3 crystals irradiated to different terminal doses and taken to room temperature before the chill-down to 4.2 K are in full correspondence with the in situ results. Figure 2a shows three orders of magnitude increase in r xx to the maximum, r max xx , at f max E90 mC cm À 2 higher than the in situ f max (Fig. 1e) likely owing in part to some defect migration (mostly interstitials 16 ) above 100 K. The p-to n-type conversion is clearly seen in Hall resistance R xy (Fig. 2b,f-h) flipping its slope dR xy /dH and Hall coefficient R H ¼ À 1 n b e changing sign in the conversion region. Near CNP, the net residual bulk carrier density n b ¼ n D À n A is very low (n D and n A are concentrations of donors and acceptors, respectively). In this region, local charge fluctuations can be very large creating inhomogeneity akin to a network of puddle-like p-n junctions that nonlinearly screen random potential on long length scales 20 and a simple estimate of n b is no longer appropriate 14 . Mobilities in our crystals are significantly higher (mD7,000-11,000 cm 2 V À 1 s À 1 , see Supplementary Table 1) than a commonly observed t1,000 cm 2 V À 1 s À 1 range 14 near the CNP region.  Table 1). The Fermi vector in pristine crystals k F E0.025 Å À 1 is consistently larger than after irradiation and, for example, the dose fD89 mC cm À 2 that tunes the crystal close to CNP results in k F E0.014 Å À 1 . The corresponding net carrier densities are n b ¼ 5.06 Â 10 17 cm À 3 and n b C1.08 Â 10 17 cm À 3 , respectively. At higher irradiation dose fD99 mC cm À 2 , with k F E0.022 Å À 1 Fermi surface size becomes comparable to the pristine material. A comparison of these numbers with carrier densities obtained independently from Hall data (Supplementary Table 1) reveals a good agreement between the two techniques near CNP but about an order of magnitude higher estimate of n b by the latter 21 well outside the CNP region. Berry phase j B ¼ 2pb can be estimated from SdH oscillations using a semiclassical description 22 where A SdH is the oscillation amplitude, H F is the frequency in 1/H and b is the Berry factor. Near CNP, the obtained Berry factor is b ¼ 0.5 ± 0.06 as expected for the topological Dirac particles, while outside the CNP region, a trivial value of bC0 is obtained (Fig. 2i).
Nonequilibrium defect compensation at low irradiation dose. The stability of net carrier density crucially depends on the terminal irradiation dose. First, we illustrate that for low terminal electron doses (ft0.1 C cm À 2 ), the resistivity is not stable as it evolves from the n-region after irradiation through CNP back into the p-region. The experiment was to simply change the dwelling time at room temperature to allow for vacancies to diffuse. The result is the reverse conversion ( Fig. 3a) from metallic-like resistivity on the n-side just after irradiation, through insulating resistivity at CNP (after nearly 250 h at room temperature) and back to a weakly semiconducting-like resistivity on the p-side. As the system evolves across CNP, a weak antilocalization quantum interference correction 23 to classical magnetoresistance (MR) emerges in the nonequilibrated conversion region (Fig. 3b), with a complex MR structure and nonlinear Hall resistivity (see Supplementary  Fig. 6), likely reflecting a presence of two carrier types. From isochronal annealing experiments (Supplementary Fig. 7 and Supplementary Note 2), we estimate energy barriers controlling defect migration to be B0.8 eV corresponding to B9,000 K, in line with literature values for vacancy migration in solids 16 . We conclude that at low terminal doses while defect migration is sluggish, the equilibrium is not attained.
Achieving stability at CNP with high irradiation dose. Next we demonstrate that stability can be controlled and optimized in a reverse conversion process by the annealing protocol when the terminal electron dose is sufficiently high. This is the key part of the material modification process by electron irradiation; without it p-type n-type . Cycling to room temperature reverses the process, which can be recovered by further irradiation (blue circles) and stabilized. The conversion from n-to p-type is obtained in a material such as Pb-based TI, where vacancies on the Pb sites are known to be of acceptor type.
surface transport via electron irradiation could not be obtained 24 . We demonstrate this for the samples exposed to the dose 10 times higher, f ¼ 1 C cm À 2 . Figure 4a shows temperature dependence of sheet resistance R xx and conductance G xx of Bi 2 Te 3 tuned back to CNP through a thermal protocol shown in Fig. 4b, where it remained for months of testing (Supplementary Figs 8,9 and Supplementary Note 3). As clearly seen in ARPES (Fig. 4c,d and Supplementary Fig. 10), under this annealing protocol Dirac dispersion remains preserved. The effect of tuning conduction from the irradiation-induced n-type (Fig. 4c) back towards CNP (Fig. 4d) is a relative shift of the Fermi level toward the Dirac point, consistent with transport data and with the calculated band structure 5 of Bi 2 Te 3 (see cartoons in Fig. 2f-h).
R xx (T) increases exponentially as the temperature decreases, turning below B200 K into variable range hopping (Supplementary Fig. 11 and Supplementary Note 4). This bulk behaviour is cut off at low temperatures when the contribution from the surface becomes comparable and a temperature-independent surface transport with minimum conductance 14 G min xx ffi 20 e 2 /h reveals its quantum nature. The 2D character of this region is witnessed by MR that depends only on the transverse component of magnetic field H > ¼ Hcosy (Fig. 4e-h). Unlike the quadratic-in-field MR of a conventional metal, the large (\10%) magnetoresistance at CNP (Fig. 4a) is found to be linear in field (Fig. 4f,h). The topological protection of the surface states near CNP can be tested in at least three ways. One, by breaking time reversal symmetry (TRS) which should gap out Dirac states. Another, by the appearance of topological Berry phase 4 of p (see Fig. 2i). And yet another, by detecting 2D weak antilocalization (WAL) of Dirac particles travelling through time-reversed paths 23 associated with this Berry phase. Figure 4a shows that applying TRS-breaking magnetic field indeed causes R xx (T) at the lowest temperatures to upturn, showing localizing behaviour consistent with opening of the Dirac gap (see, for example, ref. 25). From this, we conclude that near CNP, even with the complex Bi 2 Te 3 band structure, topological conduction channels dominate.
Stable 2D quantum transport at CNP. As we approach CNP, WAL quantum correction to classical conductivity emerges at low magnetic fields as a characteristic positive magnetoresistance cusp 23 (inset in Fig. 4b). In close proximity to CNP, the corresponding cusp in negative magnetoconductance scales with the transverse field H > ¼ Hcosy (Fig. 4e,g), confirming its 2D character. In this case, the number n Q of quantum conduction channels contributing to WAL can be estimated from 2D localization theory 26 where DG(B) is the low-field quantum correction to 2D magnetoconductance, coefficient a ¼ n Q /2 equals to 1/2 for a single 2D channel, f x ð Þ lnx À c 1=2 þ x ð Þ , c is the digamma function, and field B f ¼ ' is related to the dephasing length l f of interfering electron paths. In Bi 2 Te 3 at CNP, the fit (in Fig. 4e) yields aC1.26±0.1 corresponding to n Q B2, smaller than G min xx . The obtained value of B2 is quite remarkable since 'universality' of n Q has been questioned 27 given a likely formation of subsurface 2D electron gas states of bulk origin. In Bi 2 Se 3 , where the Dirac point is expected to coincide with CNP, the fit similarly yields aC1.12±0.1 (Fig. 4g), corresponding to two 2D quantum channels we associate with two independent topological surfaces.
Finally, we remark that using thermal protocol illustrated in Fig. 4b, we found that once CNP is reached, the system remains there for months on cycling, which was the duration of our experiments. This robustness suggests that at higher electron doses, vacancies are likely correlated; for example, di-vacancies can form 16,28 and it is known that point-defect complexes can be stabilized 29 in systems with multiple sublattices. With the choice of electron beam energy and terminal electron dose controlling the stability of pairs, and with a suitably designed thermal tuning to charge neutrality, the high-energy electron irradiation offers a path to large scale access to topological states.

Methods
Crystal growth and structural characterization. The standard Bridgman-Stockbarger technique employing a vertical pull through the temperature gradient was used to grow single crystals of Bi 2 Te 3 and Ca(0.09%)-doped Bi 2 Se 3 . X-ray diffraction of crystals was performed in a Panalytical diffractometer using Cu Ka (l ¼ 1.5405 Å) line from Philips high intensity ceramic sealed tube (3 kW) X-ray source with a Soller slit (0.04 rad) incident and diffracted beam optics. Transmission electron microscopy was performed at Evans Analytical Group.
Angularly resolved photoemission spectroscopy. ARPES measurements were performed on the FemtoARPES setup 30 using a Ti:sapphire laser operated at 0.25 MHz repetition rate. The specimens were probed with the fourth harmonic (at 6.26 eV), with an overall spectral resolution better than 30 meV. The data were taken along G-M and G-K directions. We found no substantial differences for measurements performed on several cleaved surfaces.
Electron irradiation. Electron irradiations were carried out in NEC Pelletron-type electrostatic accelerator in Laboratoire de Physique des Solides at École Polytechnique, Palaiseau, configured with a low-temperature target maintained at B20 K in a chamber filled with liquid hydrogen fed from a close-cycle refrigerator. All irradiations were performed with samples kept at 20 K, below the mobility threshold of the interstitials which tend to be more mobile than vacancies 16 -this ensured the stability of all charges introduced by the irradiation process. It also allowed us to evaluate thermal migration of interstitials during the annealing process. The size of the 2.5 MeV electron beam spot was reduced to 5 mm by a circular diaphragm aperture, with uniformity of the beam current ensured with a constant beam sweep in x and y-directions at two non-commensurate frequencies.
Careful calibration of the beam current density, typically of 2 mA on a 0.2 cm 2 surface, was performed by periodic introduction of a Faraday cage between the diaphragm and the sample chamber and the measurement of current collected on the control metallic sample. Beam current densities, limited to 10 mC cm À 2 s À 1 by the cooling rate, allowed modifications of carrier concentration on the order of 10 20 cm À 3 .
Transport measurements. The system was set up for in situ transport measurements that could be monitored as a function of electron dose in real time. Crystals used in the irradiation experiments were typically 15 mm thick with spark-weld electrical contacts placed in van der Pauw contact configuration. Ex situ transport and Shubnikov-de Haas oscillations measurements were performed in Quantum Design Physical Property Measurement System equipped with 14 T magnet. In ex situ measurements, when samples were exfoliated to thicknesses of 200 nm and below, lithographically designed contacts with Ti/Au metallurgy were used (see Fig. 4a). Annealing experiments up to 200°C (such as shown in Fig. 4b and We note that in Bi 2 Te 3 , CNP and Dirac point do not coincide, with the Dirac point situated within a valley in the bulk valence bands 5 , and G min xx is expected to reflect that 23 . Magnetic field breaks time reversal symmetry and gaps Dirac bands, resulting in localizing behaviour shown as dash. Inset: optical image of the crystal showing van der Pauw contact configuration used. (b) Annealing protocol with time steps Dt ¼ 30 min implemented to tune Bi 2 Te 3 crystal with dose 1 C cm À 2 back to stable CNP. Inset: magnetoresistance at 1.9 K after each annealing step, with colours matched to indicate different annealing temperatures. (c,d) ARPES spectra of a Bi 2 Te 3 crystal irradiated with electron dose of 1.7 C cm À 2 taken along G À K direction in the Brillouin zone. (c) Before annealing, the irradiated sample is n-type, with Dirac point at E DP B À 290(10) meV relative to the Fermi level E F . (d) After annealing at 120°C, Dirac point upshifts to a binding energy of E DP B À 160(10) meV. The same shift is seen for the scans along G À M. (e) WAL low-field quantum interference correction to the linear-in-field magnetotransport ((f), also see text) at CNP in a Bi 2 Te 3 crystal at 1.9 K with its characteristic low-field cusp. The 2D character of WAL is evident in its scaling with transverse field H > ¼ Hcosy, where y is the tilt angle of the field measured from sample's c axis. A fit to 2D localization (HLN) theory 26 (solid line) confirms that the contribution is only from two surfaces and yields a dephasing field B f ¼ h 4el 2 f $ 0:01 T. (f) Linear magnetoresistance at CNP shows 2D scaling with H > . (g) WAL contribution at CNP in a Bi 2 Se 3 :Ca(0.09%) crystal at 1.9 K also scales with H > . At high fields outside the cusp, the scaling is seen to fail for y\60°. A fit to HLN theory (solid line) again confirms the contribution only from two surfaces and yields a smaller dephasing field B f B0.004 T (corresponding to dephasing length l f B220 nm). (h) Linear magnetoresistance at CNP also scales with H > .