Evidence for topological surface states in amorphous Bi 2 Se 3
Paul Corbae, Samuel Ciocys, Daniel Varjas, Steven Zeltmann, Conrad H. Stansbury, Manel Molina-Ruiz, Sinead Griffin, Chris Jozwiak, Zhanghui Chen, Lin-Wang Wang, Andrew M. Minor, Adolfo G. Grushin, Alessandra Lanzara, Frances Hellman
EEvidence for topological surface states in amorphous Bi Se Paul Corbae, , ∗ Samuel Ciocys, , ∗ Daniel Varjas, Steven Zeltmann, , Conrad H. Stansbury, , Manel Molina-Ruiz, Sin´ead M. Griffin, , Chris Jozwiak, Zhanghui Chen, Lin-Wang Wang, Andrew M. Minor, , Adolfo G. Grushin, Alessandra Lanzara, , and Frances Hellman , Department of Materials Science, University of California,Berkeley, California, 94720, USA Department of Physics, University of California,Berkeley, California, 94720, USA Materials Science Division, Lawrence Berkeley National Laboratory,Berkeley, California, 94720, USA Advanced Light Source, Lawrence Berkeley National Laboratory,Berkeley, California, 94720, USA National Center for Electron Microscopy,Molecular Foundry, Lawrence Berkeley National Laboratory,Berkeley, California, 94720, USA Molecular Foundry, Lawrence Berkeley National Laboratory,Berkeley, California, 94720, USA QuTech and Kavli Institute of NanoScience, Delft University of Technology,2600 GA Delft, The Netherlands Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel,38000 Grenoble, France ∗ To whom correspondence should be addressed; E-mail: [email protected];[email protected]; [email protected]@berkeley.edu1 a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b rystalline symmetries have played a central role in the identification of topo-logical materials. The use of symmetry indicators and band representationshave enabled a classification scheme for crystalline topological materials, lead-ing to large scale topological materials discovery. In this work we addresswhether amorphous topological materials, which lie beyond this classificationdue to the lack of long-range structural order, exist in the solid state. Westudy amorphous Bi Se thin films, which show a metallic behavior and anincreased bulk resistance. The observed low field magnetoresistance due toweak antilocalization demonstrates a significant number of two dimensionalsurface conduction channels. Our angle-resolved photoemission spectroscopydata is consistent with a dispersive two-dimensional surface state that crossesthe bulk gap. Spin resolved photoemission spectroscopy shows this state hasan anti-symmetric spin-texture resembling that of the surface state of crys-talline Bi Se . These experimental results are consistent with theoretical pho-toemission spectra obtained with an amorphous tight-binding model that uti-lizes a realistic amorphous structure. This discovery of amorphous materialswith topological properties uncovers an overlooked subset of topological mat-ter outside the current classification scheme, enabling a new route to discovermaterials that can enhance the development of scalable topological devices. Introduction
Much of condensed matter physics has focused on exploiting crystal symmetries to understandphysical properties, headlined by topological phases and spontaneously broken symmetries inquantum materials. The unusual properties of topological materials, such as the robustness todisorder and their quantized electromagnetic responses, have prompted extensive efforts to clas-2ify crystalline topological matter. Non-magnetic crystalline topological insulators and metalswith topological bands close to the Fermi level are relatively abundant, representing ∼ % ofall materials ( ), a number that may increase by including magnetic space groups ( ). Toidentify topological crystals one asks if the band representations of a particular space group ad-mit a trivial insulator limit compatible with the crystal symmetries; if not, the material is labeledtopological. The absence of a crystal lattice places amorphous matter outside this classification,even though it is a subset of materials of comparable size to their crystalline counterpart. Thisraises the question we have set to answer in this work: is there an amorphous topological insu-lator in the solid state?Theoretically, amorphous matter can be topological since there are non-spatial symmetries,such as time-reversal symmetry, that protect topological phases. Topological insulator crystalsare robust against disorder; topological states do not rely on a periodic crystal lattice at all.Therefore, amorphous materials, which lack translational symmetry and cannot be understoodin the context of Bloch states, can still present topological properties. Specifically, electrons in alattice of randomly distributed atoms with strongly disordered electron hoppings—so strong thatno memory of a lattice can be used to label the sites—can present topologically protected edgestates and quantized Hall conductivity, hallmarks of topological insulators ( ). As a proofof principle, a random array of coupled gyroscopes ( ) was designed to act as a mechanicalanalogue of an amorphous topological state with protected edge oscillating modes, but therehas so far been no realization in a solid state system.In this work, we have grown and characterized amorphous Bi Se . In its crystalline formBi Se is a textbook three-dimensional topological insulator ( ). We find that amorphousBi Se , despite being strongly disordered and lacking translational invariance, hosts two di-mensional surface states with features common to topological surface states. The temperatureand field dependent resistance reveals the existence of low dimensional carriers with a reduced3ulk contribution. Angle-resolved photoemission (ARPES) and spin-resolved ARPES showsa two dimensional surface state with strong spin-momentum locking, strongly resembling thatexpected for a topological Dirac cone. This interpretation is supported by numerical simulationsof the surface spectrum using an amorphous topological tight-binding model. Main
Amorphous Bi Se thin films were thermally evaporated in a UHV chamber with base pressureof − Torr. The films were grown at room temperature from high purity ( . ) elementalBi and Se single sources (for further growth and composition details see the SupplementaryMaterials). High resolution transmission electron microscopy (HRTEM), Fig. 1(A), shows nosigns of crystalline order. The diffraction pattern in Fig. 1(A) inset shows a pattern typicalof amorphous materials with well defined coordination and inter-atomic distance. There is adiffuse but well defined inner ring corresponding to the short-range ordering of nearest neigh-bors. The nearest neighbor spacing set by the inner ring ranges from . ˚A to . ˚A, comparedto . ˚A in crystalline Bi Se ( ). A 1D cut plot of the electron diffraction pattern intensitiesis shown in Fig. 1(B) as well as the expected reflections for the single crystal Bi Se system.The ED pattern cut plot shows a well-defined diffuse diffraction ring proving the lack of long-range ordering. The Raman spectrum, Fig. 1(C), shows one broad peak between
135 cm − and
174 cm − . As the laser power increases two peaks are defined, which correspond to the bulkE g and A g vibrational modes, respectively. The A g van der Waals mode at ∼
72 cm − , whichcorresponds to the layered structure of the crystal, is absent in our samples. Instead, we observea peak at
238 cm − not present in crystalline Bi Se , which we attribute to amorphous Se-Sebonding seen in Se films ( ). The Raman peaks broaden compared to the crystalline system;the full width half maximum of the E g mode is . − compared to . − ( ). Theresults in Fig. 1 show that our samples are amorphous and, while lacking a layered structure4ith a van der Waals gap, have a local bonding environment similar to the crystalline phase.Figure 1: Structural and spectral evidence for the amorphous atomic structure of Bi Se (A) An abberation corrected HRTEM image at
300 kV . Inset: Diffraction pattern for the amor-phous Bi Se films. There is a dominant spacing due to strong local ordering at . ˚A. (B)A 1D intensity cut along the diffraction pattern presented in the inset of (A) with crystallinediffraction peaks presented. The amorphous samples show no intensity peaks corresponding tonanocrystallites. (C) Raman spectra for
50 nm amorphous Bi Se films using a
488 nm laser.The peaks are labeled with their respective Raman mode. Different curves (blue and purple)correspond to different laser powers, indicating bulk Raman modes become more well-definedand do not shift as the films become more ordered. All samples presented in this work showsimilar spectra to the lower power spectra shown in this figure. Crystalline data ( ) (greencurve) is overlayed to show the lack of a Van der Waal mode at ∼
72 cm − in the amorphousfilms. 5igure 2 shows the temperature dependent transport data for different thicknesses in amor-phous Bi Se . The resistivity, ρ ( T ) , is shown in Fig. 2(A). Quantitatively, the ρ ( T ) values( − m Ω · cm ) are larger than the crystalline system ( ∼ − sm Ω · cm ( )). The amorphoussystem demonstrates a weakened T dependence compared to the crystalline counterpart ( ).As in all amorphous metals, the carrier mean free path is determined more by disorder-drivenlocalization than phonon interactions, leading to a largely temperature independent resistiv-ity ( ). Moreover the high ρ and the weak temperature dependence is inconsistent with eithera purely metallic or purely insulating material, and suggestive of a metallic surface on an insu-lating bulk state with potential topological origins.Due to the potential metallic surface, we consider the sheet resistance in Fig. 2(B), R S ( T ) = ρ ( T ) /t , where t is the film thickness. Again, the R S values are increased compared to thecrystal ( ∼ − for similar thicknesses ( )). The small variations in sheet resistancebetween sample thicknesses results from < at % deviation in composition. The R S valueslie within a factor of two, which is consistent with a thickness independent two-dimensionaltransport channel as in Ref ( ).In Fig. 2(B), R S ( T ) increases below
30 K with decreasing temperature for all thicknesses.The calculated activation energy for the thermally activated behavior below 30 K gives values < . This energy scale is very small (compared to ∼
37 meV ( ) reported in the crystallinephase) and indicates that the low temperature increase in resistance cannot be explained byactivated behavior from impurity levels ( ). This suggests that the upturn in R S ( T ) is a resultof bulk carrier freeze-out and dominant surface state transport (
18, 23 ).The R S ( T ) data was fit using a two channel conductance model ( ) represented by thedashed curves in Fig. 2(B). The model includes parallel contributions from both the bulk andsurface states, and provides an overall good fit to the data. This allows us to extract the bulk andsurface state contributions with high accuracy as presented in the supplemental materials . The6igure 2: Transport signatures of putative topological electronic states in amorphousBi Se (A) ρ ( T ) for
20 nm ,
50 nm ,
75 nm and
155 nm films. All films show an increasedresistivity with a lack of apparent temperature dependence. Inset: Schematic of thin film sys-tem studied. (B) The sheet resistance for
20 nm ,
50 nm ,
75 nm and
155 nm films. All filmsshow metallic behavior in R S with decreasing temperature and a low temperature upturn. Two-channel conductance fits the data reasonably well indicating a metallic surface and insulatingbulk behavior. (C) Symmetrized resistance change as a function of the magnetic field for a
50 nm film, measured at ,
50 K and
150 K , where ∆ R = R ( B ) − R (0) . The deep cuspin low field regime is characteristic of the WAL effect. (D) Magnetoconductance showing α values extracted from the HLN equation indicating decoupled surface surface states at anda single conduction channel at
50 K . 7bserved T − / scaling at low temperatures in our system is evidence for strong localizationof the bulk states and variable-range hopping transport, implying an Anderson insulating bulk.At low temperature the surface state contribution increases for all films and in thinner filmsthe contribution is > near room temperature (see supplemental materials). The modelindicates that the surface state contribution to conduction is metallic and dominant over a largetemperature range for the amorphous samples, suggesting the existence of a topological surfacestate.The magnetoresistance (MR) provides another means to probe the topological properties.Fig. 2(C) shows MR data for a
50 nm film, revealing a sharp increase in the low field MR ( < Se ( ). Due to strong spin-orbit coupling (SOC), backscatteringis suppressed when a field is absent and time reversal symmetry is present. When time-reversalsymmetry is broken with the application of a magnetic field, backscattering increases leading toa positive MR. The magnetoconductance can be fit with the standard Hikami-Larkin-Nagaoka(HLN) formula for WAL ( ): ∆ G ( B ) = α e πh (cid:34) Ψ (cid:32) ¯ h eBl φ + 12 (cid:33) − ln (cid:32) ¯ h eBl φ (cid:33)(cid:35) (1)where Ψ is the Digamma function, B is the out-of-plane field, l φ (the phase coherence length)and α are used as fitting parameters. According to this model, each conductance channel witha π Berry phase should contribute an α = − / factor to ∆ G ( ). In an ideal TI the twosurfaces should each contribute an α = − / factor, while the bulk does not contribute, givinga total of α = − . Fitting our low field data, Fig. 2(D), at gives a value of α = − . ,suggesting we have two decoupled surface states corresponding to the top and bottom surfaces.At
50 K , α = − . suggesting the two putative topological surface states are coupled to a bulkstate, causing the entire film to act effectively as one channel, as seen in crystalline Bi Se for a8ide range of thicknesses ( ). As the temperature increases to
150 K the WAL contributionis diminished and the MR becomes quadratic in field as seen in Fig. 2(C), and is dominatedby the Lorentz force ( ). The MR at high fields for both low and high temperatures trendslinearly, which has been ascribed to spatially distorted current paths in disordered topologicalinsulators ( ). The observed behavior in the sheet resistance and MR is consistent with ametallic topological surface state that dominates over a wide range of temperatures.If amorphous Bi Se is a time-reversal invariant three-dimensional topological state, itshould present topologically protected Dirac surface states. It is important to note that an amor-phous system lacks translational symmetry; unlike crystalline systems, none of the momentumcomponents, k x , k y , and k z , are good quantum numbers. However, since the nearest neighbordistance is still well defined (inset Fig. 1(A)), there exists a good reciprocal length scale. Thespherical coordinate | k | is always a good quantum number due to its relationship to energy( E ∝ k ). Furthermore, the lack of a preferential direction in the bulk of an amorphous systemimplies spherical symmetry in momentum space. This allows for a spherically symmetric bandstructure and the existence of topologically protected surface states. The amorphous wavefunc-tions can be now parameterized in spherical coordinates by quantum numbers ( k , θ, φ ) , where θ is the azimuthal angle and φ is the polar angle. The spherical coordinates θ and φ are experi-mentally measured and refer to the respective angles of photoemission from the sample surface(see supplemental materials for details).The interpretation of a topological surface state in our amorphous Bi Se films is furthersupported by our numerical model that realizes a Dirac-like state regardless of the absence ofcrystalline symmetry (see supplemental materials for details). Motivated by the similarity oflocal environments between the crystalline and amorphous Bi Se found in Fig. 1, we usean amorphous variant of the three dimensional 4 band (spin-1/2 x 2-orbital) BHZ model ( ). The Hamiltonian features direction-dependent spin-orbit hoppings set by the normalized9opping vector ˆ d and is the sum of onsite and hopping terms H onsite = mσ τ z , (2) H hop (ˆ d ) = it (ˆ d · σ ) τ x + t σ τ z (3)where σ i and τ i are the spin and orbital Pauli matrices respectively, m sets the splitting be-tween the local s and p -like orbitals, t is the spin-orbit hopping, and t is the normal hoppingamplitude. In the crystalline case this Hamiltonian correctly reproduces key features of thetopologically nontrivial bands closest to the Fermi level ( ). We implement this tight-bindingmodel on large systems of short-range correlated amorphous structures (see supplemental ma-terials) and investigate the topological surface states by calculating spectral functions using theKernel Polynomial Method (
31, 32 ).From the model we numerically obtain a spin resolved spectral function, shown in Fig. 3(A).The inclusion of a strong surface potential that breaks inversion symmetry results in spin-splittrivial surface states at E F and a Dirac point that has been pushed to higher binding energies.This surface potential depends on the details of the surface termination (such as dangling bonds,Se vacancies, or surface reconstruction ( )) and can tune the Dirac point to arbitrary bindingenergies ( ). However, it does not affect the bulk topological properties, and preserves theDirac cone pinned to φ = 0 , spanning the bulk gap (see Fig. 3(A)).Fig. 3(B) displays the raw ARPES spectrum as a function of energy and emission angle φ at a specific θ , a momentum space slice that intersects Γ ( φ = 0 ◦ ) . The spectrum shows twovertical features at the Fermi level crossing the bulk gap which we attribute to the putative topo-logical surface state. The raw spectrum reveals an intensity peak near E F starting at − . and a sharp rise in intensity below − . . The increased intensity of the surface states near E F may be due to photoemission enhancement from the less-visible bulk conduction band. Theincreased intensity below − . coincides with a flat band which is most likely the bulk va-10igure 3: ARPES spectra of putative topological electronic states in amorphous Bi Se (A) A calculated spin-resolved surface spectral function as a function of φ . The Dirac point islow in binding energies and Rashba spin-split states develop near the Fermi level. (B) ARPESspectrum E vs. φ taken at normal emission at hν = 117 . . The spectrum reveals verticalstates that cross the bulk gap and meet at − . near the bulk valence states. (C) Proposedspin-resolved electronic structure in amorphous Bi Se corresponding to the ARPES in (B).The topological surface state traverses the electronic gap and meets near the bulk valence states,while trivial Rashba states develop near the bulk conduction states. (D) The ring-like in-planeFermi surface. φ are the angles simultaneously collected by the detector referenced to normalincidence at a given sample tilt φ (cid:48) . (E) hν vs. φ with binding energy integrated from − . tothe Fermi level and normalized by photon energy. The hν vs. φ plot displays no photon energydependence of the photoemission angle. Red dotted lines are fit to intensity peaks in the hν vs. φ spectrum. 11ence band. The exact bottom of the conduction band and top of the valence band is obscuredin the ARPES spectra due to intrinsic broadening. However, using angle-integrated photoemis-sion, we can roughly estimate the band gap to be ∼
300 meV , consistent with our analysis of theactivated behavior from high-T resistivity (
293 meV ) and with calculated DOS from amorphousstructures using ab-initio molecular dynamics (
299 meV ) (see supplemental materials). In Fig.3(C), we present our interpretation of the electronic structure in amorphous Bi Se drawn fromour experimental data. The topological surface states cross the bulk gap, forming a Dirac pointat the top of the valence band, similar to the surface states observed in related topological insu-lator crystals ( ). We expect trivial spin-split states of Rashba type, with opposite polarizationfrom the topological surface state, to develop at the Fermi level due to broken inversion sym-metry as observed in Ref. ( ) and is consistent with our spin-ARPES data later on.Fig. 3(D) presents the experimental in-plane Fermi surface in amorphous Bi Se . The annu-lar Fermi surface is consistent with crystalline Bi Se where the conical dispersion associatedwith the topological surface states produces a ring at the Fermi surface. To confirm these statesare localized to the surface, in Fig. 3(E) we show the photon energy plotted versus emissionangle φ . Due to conservation of energy, photon energy ( hν ) and k of the photoemitted electronare proportional, and related by ¯ h k / m = hν − Φ − E b where Φ is the work function of thematerial and E b is the binding energy. In the plane ( k , φ ) the states are near vertical features(red lines in Fig. 3(E)), which cannot be attributed to spherically symmetric bulk states that areindependent of φ , revealing their 2D character. These observations motivate us to interpret thesestates as two-dimensional topological surface states. It is important to note that there exist sig-nificant density of states at the Fermi energy associated with the 2D surface states, confirminga two-dimensional transport channel as determined by our magnetoresistance measurements.If the structure in Fig. 3 is truly a topological surface state, then the contributing stateswill be spin-polarized as seen in our calculations Fig. 3(A). Fig. 4 shows spin-resolved angle-12igure 4: Spin-resolved ARPES spectra of electronic states in amorphous Bi Se (A) S-ARPES experimental geometry. (B) Spin-resolved EDC’s taken at φ = − ◦ , Γ , and φ = 6 ◦ ,respectively. The spin contributions at each binding energy vary with respect to φ = 0 ◦ . (C)Spin-resolved EDC map of E vs φ with SME background subtraction taken from φ = − ◦ to φ = 9 ◦ . The spin polarization switches from red to blue (or vice versa on the other side of Γ ) at − . and from blue to red at − .
55 eV . 13esolved photoemission spectroscopy (S-ARPES) taken with
11 eV photons. Fig. 4(A) illus-trates the experimental geometry where the measured spin polarization is P y in which y pointsalong the axis of rotation used in the experiment, parallel to the sample surface.The spin-polarized energy distribution curves (EDCs) with p -polarized light are shown inFig. 4(B) at φ = − ◦ , ◦ and ◦ . The spin-polarization is measured by the relative differencebetween spin-up and spin-down photoelectrons weighted by the Sherman function ( S ) of thedetector in the form P y = S ∗ ( I ↑ − I ↓ ) / ( I ↑ + I ↓ ) . The most evident feature from the threespin-polarized EDCs is the large positive polarization between − . and . that reachesa maximum of ∼ % . This large polarization offset is due to spin-dependent photoemissionmatrix elements (SMEs) in which SOC leads to selective emission of electrons with a partic-ular spin-state. This is observed in crystalline Bi Se near the upper Dirac cone with similarintensity ( ). Our observed spectra are qualitatively similar to other time-reversal symmetrictopological insulators that lack inversion symmetry (see Fig. 2 and 3 in Ref. ( )). In conjunc-tion with the observed gapless surface spectrum (Fig. 3(B)), the anti-symmetric spin textureis a strong evidence for the topological character of the surface state. In order to uncover theintrinsic spin texture within the SME background we follow a similar background subtractionto Ref. ( ) (see supplemental materials).Fig. 4(C) presents the spin polarization as a function of binding energy and φ after per-forming the background subtraction. From this spin polarized map, three ranges of bindingenergy exhibit distinct anti-symmetric spin polarization with respect to φ . Between E F and − .
20 eV (region I) we observe a spin polarization that corresponds to the trivial spin-splitstates of Rashba form that are seen in our TB model. In the range − .
20 eV to − .
55 eV (re-gion II) an opposite spin polarization is seen from the trivial spin-split states, this correponds tothe upper part of the topological surface state. From − .
55 eV to − .
75 eV (region III) the spinpolarization switches sign, according to the helical nature of the topological surface state. The14pin polarization has a magnitude of ± % and flips sign between these ranges as a function ofbinding energy. Furthermore, the spin texture (i.e. the sign of the polarization) in the amorphoussample matches that of our tight-binding model in the presence of a surface potential as shownin Fig. 3(A). Based on the spin polarization observed in Fig. 4(C) we conclude the topologicalsurface state forms a node around − .
55 eV , and the anti-symmetric spin-resolved spectrumaround Γ at E F is reminiscent of trivial states localized at the surface stemming from Rashbatype spin-splitting in our system and similarly seen in crystalline Bi Se ( ). The most im-portant details uncovered by photoemission are that ARPES reveals that these vertical surfacestates cross the bulk band gap and SARPES shows they are spin-polarized. Discussion and Conclusion
Our data is consistent with a spin-polarized two-dimensional surface state resulting from a topo-logical bulk state in amorphous Bi Se and poses a rich set of questions for further investigation.These findings suggest that amorphous phases of known crystalline topological insulators canbe topological as well, potentially resulting in a route to find a large pool of topological materi-als for applications.Spin resolved ARPES showed an anti-symmetric spin-polarization spectra close to the nodalregion observed at − .
55 eV , that we attributed to a spin-polarized two-dimensional topologicalsurface state. The spectra are largely broadened by the inherent atomic disorder as well as thepresence of vacancies and dangling bonds at the surface, which are a significant source of finalstate scattering. While the lack of inversion symmetry could give rise to trivial surface Rashbastates and hence explain the observed spin asymmetry ( ), it cannot account for the followingobservations. First, the trivial surface state can contribute a value of − / ≤ α ≤ to the HLNfit depending on if the states are weakly antilocalized or weakly localized, respectively. Theobserved values of α ≈ − at and α = − / at
50 K (Fig. 2(D)) is suggestive that the15eak antilocalization originates from topological surface states rather than trivial Rashba statessince the magnitude of α from trivial Rashba states would likely be smaller at both tempera-tures. Secondly, the observed spin asymmetry due to Rashba does not agree with our theoreticalcalculations which show the asymmetry originates from a nontrivial topological state. Finally,we observe states spin-polarized states that cross the putative band gap, which is a requirementfor topological surface states but not for trivial Rashba surface states.TEM and Raman data suggests that the typical length scales of the amorphous structure arecomparable to the crystalline case, suggesting that a possible condition to preserve the topolog-ical nature of the bulk in the amorphous state is to retain a local ordering similar to that of thecrystal. The impact of coupling strengths and local field environments can be assessed theoret-ically via ab-initio calculations to refine the Hamiltonian modeling of amorphous topologicalmaterials ( ). This approach can be extended to amorphous material systems without topolog-ical crystalline counterparts, where local ordering coupled with disorder and strong SOC canmix energy levels to produce a topologically nontrivial electronic structure.In conclusion, we have found that amorphous Bi Se hosts a two-dimensional metallic sur-face state with a spin-texture consistent with helical spin-polarization, indicating that the bulkis a three-dimensional amorphous topological insulator. This first experimental evidence con-sistent with topological states in an amorphous solid state system highlights that topologicalphysics is not restricted to crystalline systems. Our results represent the first step towards a ma-terials realization of recently proposed non-crystalline topological phases ( ) that lie outsidethe known classification schemes for topological crystalline matter ( ). We expectour work to motivate an effort to understand topological amorphous matter, enabling materialsdiscovery that can provide a path towards implementation into modern thin film processes.16 eferences
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P. C. and S. C. would like to thank E. Parsonnet and Dylan Rees for their understanding anddiscussions. A. G. G. is grateful to J. H. Bardarson, S. Ciuchi, S. Fratini, and Q. Marsal fordiscussions. D. V. thanks A. Akhmerov, A. Lau and P. Perez Piskunow for discussions.
Fund-ing
The growth and transport work was primarily funded by the U.S. Department of Energy,Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Di-vision under Contract No. DE-AC02-05-CH11231 within the Nonequilibrium Magnetic Ma-terials Program (MSMAG). The ARPES and SARPES work was supported by Berkeley lab’sprogram on ultrafast materials sciences, funded by the U.S. Department of Energy, Office ofScience, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under20ontract No. DE-AC02-05-CH11231 and the Quantum Materials Program at Lawrence Berke-ley National Laboratory, funded by the U.S. Department of Energy, Office of Science, Officeof Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-05CH11231. TEM at the Molecular Foundry was supported by the Office of Science,Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Computational resources were provided by the National Energy ResearchScientific Computing Center and the Molecular Foundry, DoE Office of Science User Facilitiessupported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The work performed at the Molecular Foundry was supported by the Officeof Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under the samecontract. P. C. is supported by the National Science Foundation Graduate Research Fellowshipunder Grant No. 1752814. S.C. was supported by the National Science Foundation GraduateResearch Fellowship under Grant No. DGE1852814 and DGE1106400. A. G. G. is supportedby the ANR under the grant ANR-18-CE30-0001-01 and the European Unions Horizon 2020research and innovation programme under grant agreement No 829044. D. V. is supported byNWO VIDI grant 680-47-53. S. E. Z. was supported by the National Science Foundation underSTROBE Grant No. DMR 1548924.
Author contributions
P. C. and S. C. contributed equallyto this work. The project was initiated and oversaw by P.C., S.C., A. G. G., A.L., and F. H..P. C. grew the films. S.C. performed synchrotron ARPES, S. C. and P. C. performed SARPESmeasurements, and S.C. performed the data analysis. P. C. performed transport measurements.S. Z., S. C., and P. C. performed TEM. M. M. R. and P. C. performed Raman measurements. A.G. G. and D. V. constructed the tight-binding model, and D. V. performed numerical calcula-tions. Z.C. performed the molecular dynamics and S.G. performed the DOS calculations. P. C.,S. C., A. G. G. and D. V. took part in interpreting the results. All authors contributed to writingthe manuscript. 21 upplementary Materials
Materials and Methods
Thin film growth and characterization
The Bi Se films used for this study were grown on amorphous Si N substrates (10x10x0.5mm ) using a standard thermal evaporation technique. A custom built UHV chamber with basepressure of − Torr was used to grow the films. The films were co-deposited from high purity( . ) elemental Bi and Se single sources. The films were grown at room temperature withfluxes to match the stoichiometric ratio.Stoichiometry of the films was confirmed using XPS (X-ray photoelectron spectroscopy),EDS (Energy dispersive X-ray spectroscopy), and RBS (Rutherford backscattering spectroscopy).RBS, Fig. S1(A), shows our films are within ± of the exact stoichiometry of 40/60 % . Ad-ditional XPS measurements, Fig. S1(B), of the Bi f and Se d core levels prove we arein fact measuring Bi Se films. EDS spectra were taken via STEM HAADF showing there isno clustering or composition gradients in the films (Fig. S1(C)). The EDS compositional mapcharacterization, with a lateral resolution of ˚ A , reports that the highest Se concentration in theSe-rich regions is ∼ , and accounts for the . of the total measured area, while the sys-tem nominal Se concentration is ∼ . The amorphous nature of the films and the isotropicdistribution of the atoms achieved during simultaneous evaporation of Se and Bi atoms argueagainst the presence of highly energetic structures, such as layers of columns, leaving spheresas the most favorable shape of possible clusters. The EDS results constrain the size of possibleSe clusters to be ≤ ˚ A . Summarizing, if Se clusters exist, they are so small that cannot bequantified with the techniques used in this work.TEM was performed on
10 nm thick Bi Se films deposited on a
10 nm thick SiN window,using an image-corrected FEI ThemIS operated at
300 kV . The amorphous structure of the22lm was confirmed with XRD, Raman spectroscopy, and TEM. A large-scale real space imageis shown in Fig. S2(A) to show the lack of incipient ordered structures or crystalline precur-sors, such as grain boundaries, nanocrystallites, etc. In order to check that decap process wasnot crystallizing our films we performed TEM on samples undergoing the same decap proce-dures. The diffraction patterns showed the same ring pattern that was seen in non-decappedsamples as well as in the FFT, Fig. S2(B). XRD, Fig. S2(C), shows no peaks in the spectrumthat correspond to crystalline Bi Se interatomic planes and at low θ values, where θ is theincident angle, there is an amorphous hump. Data from Ref. ( ) is overlaid to show the dif-ference between our amorphous Bi Se films and disordered Bi Se films in X-ray diffraction.Quantitative phase analysis using Rietveld method (red line) and the diffraction pattern of anamorphous Bi Se film (black circles) is shown in Fig. S2(D). We used the Bi Se crystal struc-ture (space group 166) to quantify the crystalline fraction, and an unconstrained and isotropicstructure based on the same unit cell to squantify the amorphous fraction. It is not clear whetherthe second diffuse ring is real or a background artifact, for that reason we did not include it inthe quantitative phase analysis. Raman was performed with a Renishaw inVia
488 nm
Ar/Nelaser using linearly polarized light and operating at µW - mW with a spot size of . µm . The amorphous Bi Se samples resistivity was measured in a Van der Pauw (VdP) geometryusing Indium contacts. The VdP samples were grown and transferred to vacuum within 30minutes to minimize atmospheric effects. Additional measurements were made on films witha ∼ Se cap layer. These films showed the same behavior as well as values of R sheet ( T ) ,Fig. S6(F). 23 RPES/SARPES
The experimental geometry used to obtain the ARPES spectra is shown in Fig. S3. φ is thedetector angle away from the z-axis, φ (cid:48) is the sample tilt angle, and θ is the azimuthal rotationangle. We performed ARPES at the Advanced Light Source MAESTRO (7.0.2) and MERLIN(4.0.3) beamlines with photon energies in the range of
65 eV to
125 eV . The spin-resolvedspectra were acquired from a high-efficiency and high-resolution spin-resolved time-of-flight(TOF) spectrometer that utilizes the spin-dependent reflection from a magnetic thin-film due tothe exchange interaction ( ). The light source for the spin measurements was a Lumeras
11 eV
Xenon gas-cell laser with 1MHz repetition rate ( ). Synchrotron ARPES measurements andspin-resolved measurements were taken at ∼
20 K and ∼
75 K , respectively.
Theoretical tight-binding calculations
To construct an amorphous system, we randomly add atomic sites in a fixed volume from anuncorrelated uniform distribution. Treating atoms as hard spheres, we reject atoms closer thandistance one to existing atoms and this procedure is performed until the goal density is reached.This procedure minimizes density fluctuations and produces a peak in the radial distributionfunction at the typical nearest neighboring distance, matching the distribution function of anamorphous system more closely than independent uniformly distributed points, see Fig. S4.We then connect each neighbor to its 6 other nearest neighbors with hoppings of fixed mag-nitude, preserving the key features of octahedral coordination and fixed bond lengths, as incrystalline Bi Se . We do not distinguish Bi and Se atoms in this simplified effective model.The Hamiltonian of the system contains an onsite and a hopping term, H = H onsite + H hop (ˆ d ) (S4) H onsite = mσ τ z , (S5)24 hop (ˆ d ) = it (ˆ d · σ ) τ x + t σ τ z . (S6)where ˆd is the normalized nearest neighbor vector.To calculate the ground state (or finite temperature) expectation values of observables inlarge systems we use the Kernel Polynomial Method (KPM) ( ). The observable relevant tophotoemission spectroscopy discussed in the main text, sensitive to the topological nature ofthe system, is the energy E and momentum k dependent surface spectral function, A ( k , E ) = (cid:88) l (cid:104) k , l | δ ( H − E ) | k , l (cid:105) (S7)where | k , l (cid:105) is a plane wave state of wavevector k nonzero only in the local orbital time l .When we restrict the local part of the plane wave vectors to a certain spin or orbital polarizationthe spectral function is spin-resolved. We use the Kwant ( ) software package to generatethe tight-binding Hamiltonians and calculate spectral functions using the code published inRef. ( ). The results shown in the main text were obtained using a sample of dimensions50x50x20 and 0.4 average density, × sites in total. The bulk parameter values are (inarbitrary units) t = 1 , t = 2 , M = 4 , and we use a constant electric field in the surface layer ofthickness d=3 with electric field strengths E z = 0 , − . , and − . To obtain the surface spectralfunction we used plane waves that decay exponentially in the z direction with k z = 0 . i . Density of states calculation
Density functional theory (DFT) calculations were performed using the projector augmentedwave (PAW) formalism in the Vienna ab initio Simulation Package (VASP) (
51, 52 ). Theexchange-correlation potentials were treated in the framework of generalized gradient approx-imation (GGA) of Perdew-Burke-Ernzerbof (PBE) ( ). Bi (6s, 6p) and Se(4s, 4p) electronswere treated as valence, and their wavefunctions expanded in plane waves to an energy cutoffof 500 eV. A k-point grid of 3x3x1 with Gamma sampling was used. Spin-orbit coupling was25dded self-consistently for all density of states calculations. Amorphous structures were gen-erated with ab initio molecular dynamics using VASP. Ten representative amorphous structureswere set up by constructing a box with 54 Bi atoms and 81 Se atoms which underwent a meltingstep (at 2000 K), a quenching step, and an annealing step (at 200 K) in a canonical ensemble,followed by a further relaxation step. Supplementary Text
Electronic Structure
To calculate the band gap we performed Density Functional Theory calculations of 10 represen-tative amorphous structures as described in the methods section. The results of our calculationsare shown in Fig. S5. Several of the structures have mid-gap defect states (the upper five),which are often present in the electronic structure of amorphous materials ( ). Therefore, toestimate the bandgap, we neglected any structures with clear mid-gap states as their appearanceis not uniform across the series. Averaging the bandgaps of the five remaining structures givesa bandgap of 299 meV. We note that many of these also contain extended Urbach tails and sothis value is a lower bound. Finally, since GGA typically underestimates the bandgap we alsocompare previous calculations of crystalline Bi Se for different exchange correlation function-als and experimentally determined values. Surprisingly, Park et al. ( ) found that the PBE gapof crystalline Bi Se is 336 meV while HSE06 underestimates the gap at 37 meV compared tothe experimental gap of 300 meV. We conclude, therefore, that PBE is an appropriate choice foraccurate gap calculations. Transport
The R S ( T ) data was fit using a two-channel conductance model. The total conductance is theparallel sum of bulk conductance and a metallic surface conductance. The bulk conductanceconsists of thermally activated behavior and variable range hopping, while the surface conduc-26ance is metallic, G T = 1 /R S ( T ) = G Bulk + G Surface (S8) G Bulk = ( Ce ∆ /k B T + De ( T /T ) / ) − (S9) G Surface = ( A + BT ) − . (S10)This model gives good fits to the resistance data and allows us to extract the surface state con-tribution, G surface /G total , as seen in Fig. S6(E).The bulk conductance behaves as expected for different thicknesses, especially at low tem-perature where extrinsic effects have been frozen out, as seen in Fig. S6(A). As thickness goesfrom
155 nm to the bulk conductance drops (bulk resistance increases), as expected forfilms with less carriers. Anderson localization implies that there should be no conduction as T approaches 0, meaning the bulk conductance should have no finite intercept at low temperatures.We see in our G bulk vs. T plots that G bulk trends towards 0 as seen in Fig. S6(A). Additionally,the requirement of a T − / scaling at low temperature (VRH) for our fits implies we are in ornear the Anderson insulating regime. Taking values for bulk conductivity extracted from ourfits at low temperature, we obtain a dimensionless conductivity, g , of 1 implying the bulk is atthe crossover of diffusive transport and strong localization. Therefore, when we are at low T ,such as in our magnetotransport, we can expect the bulk has entered the Anderson insulat-ing regime. The surface conductance falls on top of each other for
20 nm ,
50 nm , and
75 nm films expected of surface state transport. The G surface for the
155 nm film is ∼ × smaller, thisis due to small compositional differences among films likely increasing surface-bulk scattering.We find for the
20 nm ,
50 nm , and
75 nm films the D (electron-phonon coupling) parameter tobe 4.4, 3.6, and 4.5 Ω /K , respectively. These values are smaller than previously reported valueof ∼ /K in the bulk insulating TI BiSbTeSe ( ). Considering the bulk resistivity, which isan intrinsic quantity, we see all the resistivity values fall within the range of ∼ . − . · cm27nd all films exhibit the same temperature dependence.Fig. S6(D) shows the Hall resistance as a function of magnetic field at various temperaturesin amorphous Bi Se . Hall resistance is linear with magnetic field and is nearly temperatureindependent. This indicates that transport is determined by a single carrier (electron type) andthe lack of temperature dependence implies that carrier density is unaffected by temperature.By taking the slope of the Hall resistance we can obtain n sheet at different temperatures. Thebulk carrier density is related to the sheet carrier density by N bulk = n sheet /t where t is the filmthickness. This yields N bulk ∼ . × cm − at ,
50 K , and
150 K .An activated fit on the low temperature data does not fit our data well giving unrealisticactivation energies ( .
004 meV ). At low temperatures ( < ), resistivity values begin to de-crease in slope and level off indicating at lower temperatures the values saturate. The resistivitydata exhibits metallic and low temperature behavior that is consistent with multiple channels ofconduction. ARPES
In order to estimate the size of the bulk electronic energy gap, we consider the angle-integratedphotoemission spectrum seen in Fig. S7. Different curves in Fig. S7(A,B) correspond todifferent photon energies, ranging from 55-83 eV (light to dark lines) spaced by 2 eV. Fig.S7(A) shows the spectral intensity at different photon energies. At photon energies near 83eV, the surface state intensity drops and two separate regions with spectral intensity becomeapparent. The first is directly below the Fermi level and the second around -0.4 eV.In Fig. S7(B) the dI/dE curve demonstrates two humps with an upturn near -0.125 eVassociated with the bulk conduction band bottom and a downturn near -0.425 mV associatedwith the bulk valence band top. This allows us to estimate the bulk band gap at 0.3 eV. TheFermi energy appears as a sharp dip centered at 0 eV.28ig. S8(A) displays the ARPES spectrum E vs. φ taken at normal emission at hν = 86 . with enhanced contrast and normalized by binding energy. The spectrum reveals a possibleDirac cone at higher binding energy. The putative Dirac cone is marked by the dotted lines.Constant energy cuts, Fig. S8(B), are radially symmetrized across θ . The ring near the Fermilevel gradually shrinks, becomes nodal, and then opens again at higher binding energies. Thenode is a local maximum in the spectral intensity. However, the spectral weight in the valencegreatly exceeds the spectral weight near the Fermi level and can only be uncovered via bindingenergy normalization. It is possible that this node is related to the true Dirac node and is shiftedto more negative binding energies due to the convolution of the surface states with the largespectral gradient in the valence.Momentum distributions curves (MDC) are presented in Fig S9. The MDCs show two peaksin spectral intensity in the bulk gap corresponding to the surface state dispersion. The large peakin intensity at lower binding energies can be attributed to the valence states. To get a better viewof the surface states, Fig. S10(A) presents an ARPES spectrum normalized by binding energy inthe gap. These states traverse the bulk gap and show constant width in φ at all binding energies,Fig. S10(B), providing evidence that these surface states are not gapped. SARPES: Spin Matrix Element and Background Subtraction in Spin-Polarized Map
The raw spin-polarized spectrum collected from the S-ARPES contains a large spin-polarizedbackground associated with spin matrix elements (SME) from −
600 meV to that reachesapproximately at maximum. The magnitude of the SME is an interesting phenomenon onits own, needing further investigation. Moreover, over the course of the entire measurement (thespectra at each angle took approximately 3 hours), the background maximum increased slightly.We captured each spectra in non-sequential order with respect to angle in order to determinethat the increase in background was due to time-related changes to the sample in vacuum and29as not dependent on emission angle. Since the underlying spin signal is a fraction of theSME background, normalizing by the SME peak value provides a more accurate spin-map afterbackground subtraction.Similar to Fig. 4(C) in the main text, Fig. S11 displays four spin-polarized maps on amor-phous Bi Se . Fig. S11(A) is the total spin polarization including the SME after normalizingto the SME peak intensity. Only the spin polarization of the bands near −
700 meV are dis-tinguishable. To remove the SME to determine the underlying spin-polarization, we subtractthe spin-polarized spectra at φ = 0 since time-reversal symmetry requires spin-degeneracy atzero-momentum. Fig. S11(B) and Fig. S11(D) show the SME subtracted spin-maps withoutand with SME peak normalization, respectively. The difference between these two maps is sub-tle and suggests that the normalization process is not drastically affecting the results. Lastly,if, instead of subtracting the normalized map with the φ = 0 spectrum, we subtract off theaveraged spectrum with respect to φ , then we get Fig. S11(C). The difference between Fig.S11(C) and Fig. S11(D) are even less noticeable, suggesting that the final spin-maps are robustto background subtraction methods. 30igure S1: Compositional characterization.
Compositional characterization. (A) RBS spec-trum fit indicating near stoichiometric films. (B) XPS spectra of the Bi f and Se d core levelsindicating near stoichiometric films. (C) EDS compositional mapping performed from STEMHAADF showing no composition gradients. 31igure S2: Structural characterization.
Structural characterization. (A) Larger scale HRTEMimage showing an absence of any long-range order. (B) FFT of HRTEM image. Diffuse ringsattributed to the amorphous structure are observed. (C) XRD patterns showing and amorphoushump compared to disordered Bi Se films from Ref. ( ). (D) FFT of HRTEM image. Diffuserings attributed to the amorphous structure are observed.32igure S3: ARPES experimental geometry. φ is the detector angle away from the z-axis inwhich the z-axis is defined as the normal vector from the sample surface. φ (cid:48) is the sample tiltangle and θ is the azimuthal angle. 33 Figure S4:
Tight Binding Structure
Histogram of the relative positions of atoms in the xyplane for a one-unit thick slice of the amorphous lattice structure used in the numerics. Thecorrelation hole for distances under one and an annular peak corresponding to close packing arevisible. 34igure S5:
Amorphous density of states calculation
Density of states for ten amorphousstructures. The mean bulk gap is ∼ meV. 35igure S6: Resistivity Analysis (A) The variation bulk conductance as a function of temper-ature for various thicknesses of amorphous Bi Se . The conductance trends towards 0 as thetemperature decreases, indicating a localized bulk. (B) The intrinsic bulk resistivity as a func-tion of temperature. (C) G surface versus temperature. The
20 nm ,
50 nm , and
75 nm films alldisplay the same surface state transport. (D) The variation of Hall resistance as a function ofmagnetic field at various temperatures in amorphous Bi Se showing a linear behavior. (E)Surface state conductance contributions taken from the ratio of G surface / G total . All films showincreased surface state contribution with decreasing temperature. (F) The sheet resistance fora
50 nm Bi Se film with of Se cap. The film displays similar behavior and R sheet ( T ) values to the films presented in the main text. 36igure S7: Angle-integrated photoemission spectra . (A) Spectral intensity vs. binding energyfor photon energies between 55-83 eV (light to dark curves), two distinct regions of increasedspectral weight are observed at higher photon energies. (B) dI/dE vs. binding energy allowsus to extract estimates for the bulk E g . The shaded regions correspond to the approximated bulkand valence band locations. The bottom of the conduction band is located at ∼ − . eV andthe top of the valence band is near ∼ − . eV.37igure S8: Dirac-like ARPES spectra (A) ARPES spectrum E vs. φ taken at normal emissionat hν = 86 . with enhanced contrast and normalized by binding energy. The spectrumreveals a possible Dirac cone at higher binding energy. The putative Dirac cone is marked bythe dotted lines. (B) Constant energy cuts from (A) radially symmetrized across θ .Figure S9: Momentum distribution curves
MDCs of the surface state dispersion taken at hν = 117 . . Increasing spectral intensity at higher binding energies is associated with thebulk valence state. 38igure S10: Mid-gap surface states. (A) ARPES spectrum taken at hν = 117 . normalizedby binding energy to show states in the gap clearly, the surface states remain in the gap. (B) Fitsto the peak locations and widths across binding energy to the spectral intensity in (A).39igure S11: Spin map background subtraction (A) spin-map before SME subtraction inwhich the spectra at each φ are normalized by peak SME value. (B) SME subtracted spin-mapusing φ = 0 spectrum for subtraction, no SME peak normalization. (C) SME subtracted spin-map using φ -integrated spectrum as background, SME peak normalized. (D) SME subtractedspin-map using φ = 0= 0