Observation of Chiral Surface Excitons in a Topological Insulator Bi2se3 H.-H

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Observation of chiral surface excitons in a topological insulator Bi2Se3 H.-H. Kunga,1,2, A. P. Goyalb, D. L. Maslovb,1, X. Wanga,c, A. Leea, A. F. Kemperd, S.-W. Cheonga,c, and G. Blumberga,e,1

aDepartment of Physics & Astronomy, Rutgers University, Piscataway, NJ 08854; bDepartment of Physics, University of Florida, Gainesville, FL 32611; cRutgers Center for Emergent Materials, Rutgers University, Piscataway, NJ 08854; dDepartment of Physics, North Carolina State University, Raleigh, NC 27695; and eLaboratory of Chemical Physics, National Institute of Chemical Physics and Biophysics, 12618 Tallinn, Estonia

Edited by Angel Rubio, Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany, and approved January 22, 2019 (received for review August 5, 2018)

The protected electron states at the boundaries or on the surfaces troscopy to study the secondary emission from a 2D electronic of topological insulators (TIs) have been the subject of intense the- system of significant current interest: the surface state of a oretical and experimental investigations. Such states are enforced 3D topological insulator (TI), Bi2Se3. [Polarized PL was also by very strong spin–orbit interaction in solids composed of observed in Bi1.95In0.05Se3 crystals (SI Appendix, section 3)]. In heavy elements. Here, we study the composite particles—chiral 3D TIs, strong SOC and time-reversal symmetry collaborate to excitons—formed by the Coulomb attraction between electrons support topologically protected massless surface states (denoted and holes residing on the surface of an archetypical 3D TI, Bi2Se3. by SS1 in Fig. 1 A and B) with chiral spin-momentum texture Photoluminescence (PL) emission arising due to recombination of (19–23). Both angular-resolved photoemission (ARPES) data excitons in conventional semiconductors is usually unpolarized and first-principle calculations show that there are two more sur- because of scattering by phonons and other degrees of freedom face bands near the Brillouin zone center (Γ-point) in Bi2Se3 during exciton thermalization. On the contrary, we observe almost (17, 18, 24): (i) a high-energy unoccupied Dirac cone (SS2) and perfectly polarization-preserving PL emission from chiral excitons. (ii) fully occupied Rashba-like surface states (RSS). These bands We demonstrate that the chiral excitons can be optically oriented are depicted by red lines in Fig. 1A and enclosed in boxes in with circularly polarized light in a broad range of excitation ener- Fig. 1B. Due to strong SOC, all three surface bands exhibit gies, even when the latter deviate from the (apparent) optical band spin-momentum locking, which could lead to optical orienta- gap by hundreds of millielectronvolts, and that the orientation tion of single-electron spins and excitons (4). So far, most of remains preserved even at room temperature. Based on the depen- the research on TIs has been focused on spin dynamics and col- dences of the PL spectra on the energy and polarization of incident lective modes of Dirac fermions in SS1 (25–28), and far less is photons, we propose that chiral excitons are made from massive known about the properties of RSS and SS2. We excite interband holes and massless (Dirac) electrons, both with chiral spin textures transitions between surface states RSS and SS2 with circularly enforced by strong spin–orbit coupling. A theoretical model based polarized light and study polarization of PL emission in the on this proposal describes quantitatively the experimental obser- backscattering geometry, with light being incident normally on vations. The optical orientation of composite particles, the chiral the crystal surface. excitons, emerges as a general result of strong spin–orbit coupling in a 2D electron system. Our findings can potentially expand Significance applications of TIs in photonics and optoelectronics. We observe composite particles—chiral excitons—residing on exciton | topological insulator | photoluminescence spectroscopy | the surface of a topological insulator (TI), Bi2Se3. Unlike other optical orientation known excitons composed of massive quasiparticles, chiral excitons are the bound states of surface massless electrons pin–orbit coupling (SOC) plays a central role in spintronic and surface massive holes, both subject to strong spin–orbit Sand optoelectronic applications by allowing optical control coupling which locks their spins and momenta into chiral of spin excitations and detection with circularly polarized light, textures. Due to this unusual feature, chiral excitons emit in the absence of an external magnetic field (1–3). This effect circularly polarized secondary light (photoluminescence) that is also known as optical orientation, where nonequilibrium dis- conserves the polarization of incident light. This means that tribution of spin-polarized quasiparticles is optically created in the out-of-plane angular momentum of a chiral exciton is pre- semiconductors with strong SOC (4, 5). Detailed information served against scattering events during thermalization, thus on spin dynamics can be obtained by studying polarized pho- enabling optical orientation of carriers even at room temper- toluminescence (PL) (6–9). Typically, the degree of PL polar- ature. The discovery of chiral excitons adds to the potential of ization in semiconductors decreases rapidly as the excitation TIs as a platform for photonics and optoelectronics devices. photon energy deviates from the optical band gap or with heating (4, 5, 10). Elaborative layer and strain engineering is often Author contributions: G.B. designed research; H.-H.K. and G.B. designed the experiments; required to lift spin degeneracy of the bulk bands to achieve X.W. and S.-W.C. grew the single crystals; H.-H.K., A.L., and G.B. acquired and analyzed the optics data; A.P.G. and D.L.M. developed the theoretical model of chiral excitons; a higher degree of PL polarization (11). In contrast, nearly A.F.K. performed calculations of the band structure; and H.-H.K., A.P.G., D.L.M., X.W., complete PL polarization, observed recently in transition metal A.L., A.F.K., S.-W.C., and G.B. contributed discussions and wrote the paper.y dichalcogenide (TMD) monolayers up to room temperature, was The authors declare no conflict of interest.y attributed to spin–orbit-mediated coupling between the spin and This article is a PNAS Direct Submission.y valley degrees of freedom (12–14). The reduced dimensionality Published under the PNAS license. y suppresses dielectric screening and restricts the number of scat- 1 To whom correspondence may be addressed. Email: [email protected], sean. tering channels, resulting in long-lived coherent 2D excitons (15). [email protected], or [email protected]fl.edu.y These results have attracted significant interest due to possible 2 Present address: Quantum Matter Institute, University of British Columbia, Vancouver, applications and also as an insight into the nature of many-body BC V6T 1Z4, Canada.y interactions in 2D electronic and photonic systems (3, 13, 16). This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. In this paper, we discuss a class of excitons which produce 1073/pnas.1813514116/-/DCSupplemental.y helicity-preserving PL. We use polarization-resolved PL spec- Published online February 20, 2019.

4006–4011 | PNAS | March 5, 2019 | vol. 116 | no. 10 www.pnas.org/cgi/doi/10.1073/pnas.1813514116 Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 iclryplrzdP inl xie yrgtcrual polar- right-circularly by light, excited ized signals PL polarized circularly PL. Polarized Results states Experimental bound excitonic the pairs, interaction, electron–hole ﬁnite transition the With only gray. of eV. The in momenta 2.5 transfer. shown of angular energy momentum are total others zero the blue; with by in SS2, denoted shown to are is and RSS study minimum from this sity, band transitions in the used possible below energy for form excitation pairs the electron–hole with noninteracting consistent of relation dispersion etectto yrgt rlf iclryplrzdphotons. al. et circularly-polarized Kung left or right- indepen- by to and states excitation amenable light-emitting 2 A) components dent the with (Fig. that degenerate, line doubly suggests right- same are This with the intensity. show signals and 2B) shape (Fig. PL excitation polarized polarized geom- left-circularly the “time-reversed” the i.e., in reproducible etry; geome- are polarization results different The in tries. measured spectra the compare decay the can from non- state gapped bound a PL. are observed electron–hole in SS2 in the resulting energy and radiatively, thus RSS Fermi and case, the level, our Fermi to electron– In rapidly produced way. relax optically radiative be would the exciton gap, to pairs the the if likely hole Even within is states. crossing remains state band level level exciton conduction the Fermi the with the because hybridized band, with conduction semiconductors, the doped and in als not but channel RR excita- the the in one. RL occurs as the emission polarization i.e., is same eV photon; the 2.3 tion with at polarized peak fully the in contrast, almost observed In pairs. signal optically electron–hole of PL thermalization generated during lost ordinary inci- is the polarization an photon about dent memory as the behaves where The semiconductors, eV intensity. conventional same 1.5 of has at behave emission light i.e., peak range polarized unpolarized; left-circularly is visible polarized eV and the circularly 1.5 right- at in by peak excited eV the when Namely, 2.3 light. ways and different eV strikingly 1.5 in about at and peaks photons emitted of energy by designated are signals background, PL PL polarized unpolarized left-circularly and shows Right- line by K. black denoted the 15 orientations The along at spin excitation structure in-plane 2.7-eV band the polarized Calculated for (E with (B) layer lines, energy gray. red ple Fermi in thick the shown by near depicted are (SS1) are states (SS2) states surface surface topological unoccupied 1. Fig. I I RR RL BCD AB (ω ofrhreuiaetentr fplrzdP in PL polarized of nature the elucidate further To semimet- in observed usually not are excitons that note We (ω ,T ,T R 2.0 )−f )−f 2.5 h lcrncbn tutr erthe near structure band electronic The (A) (ω (ω eV ,T ,T [I ) I ) RR RR J spotda ucino (ω of function a as plotted is (ω z i.1C Fig. = (ω , T .(C 2). section Appendix, (SI bands surface three the highlight squares blue The 1/2. , ) − T I ) RL and (ω eit h neste frgt n left- and right- of intensities the depicts , T )]d I RL ω (ω sona h hddbu rain area blue shaded the as (shown , T T stmeaue w emission Two temperature. is epciey where respectively, ), 0 ,with ), F r eitdb hnrdlns h ukbns(hc ontcnrbt ocrual oaie PL) polarized circularly to contribute not do (which bands bulk The lines. red thin by depicted are ) T f ≈ Γ (ω 5K h e iei ieretaoainto extrapolation linear a is line red The K. 15 on sifre rmAPSmaueet 1,1) h ahasraesae RS n the and (RSS) states surface Rashba The 18). (17, measurements ARPES from inferred as point , T The (D) clarity. for 3 of factor by multiplied is range 2.6-eV to 1.8- the in intensity The ). Bi Kcti h rloi oeo h eaoa atc,poetdot h o quintu- top the onto projected lattice, hexagonal the of zone Brillouin the in cut Γ–K 2 ω Se sthe is 3 we , C ,v.ectto nrymaue t1 .(F K. 15 at measured energy excitation vs. ), i.3dpcsteitniyo oaie Lmaue iha with measured PL polarized intensity. of peak intensity the the of polarized depicts dependence for excitation 3 responsible the Fig. bands study electron we the PL, identify polarized circularly To with spins light. of sur- orientation optical three to all bulk lead could However, 1 the Fig. (29). from in degenerate bands originate face spin cannot are PL which for bands, polarization circular Photons. of Incident of Energy the on quantum Dependence circular valley both to due where preserved 14). monolayers, of (13, are coherence TMD property polarizations in linear This and observed polarization. pre- PL which circular ized emitters, not from independent but emission two linear the as of serve result, act relaxation a states the during excitonic As preserved pairs. not electron–hole is coherence quantum with coincides both right- excite into independently the decomposed can 2 ones, being polarized (Fig. photon, left-circularly polarized and polarized Y exci- linearly both or the a the in X that that whether shape is of ﬁnd line photon and regardless We tation intensity channels, incidence. same polarization the of circular almost paral- plane has polarization the signal linear PL to denoting (orthogonal) (Y) X lel with light, polarized as states these will denote that same reasons we For on, the |J one. later with excitation angular clear the emitted their become of are preserve that photons as states polarization secondary these the of momenta, processes relaxation If o rsre nteP rcs Fg 2F (Fig. process PL the in preserved not z nFg 2 Fig. In = |J −1i z 1 = Fg 1D). (Fig. i C–F and [I Bi RL eso nest fP xie ihlinearly with excited PL of intensity show we |J 2 r Se A (ω PNAS (T z I RR = ) and 3 , (ω = h Lsetamaue ihright-circularly with measured spectra PL The ) T −1i ufc ttsi ncnrs opolar- to contrast in is states surface , ,sgetn iiu xiainthreshold excitation minimum a suggesting 0, T + ) | B nrdand red in ) ac ,2019 5, March J xii pnmmnu okn and locking spin-momentum exhibit z I tts htlna oaiainis polarization linear That states. (E . LL (ω h nertdplrzdP inten- PL polarized integrated The ) , h eoaiainratio depolarization The ) T )]/2 I RL (ω | | J

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APPLIED PHYSICAL SCIENCES with intensity deﬁned as LR(ω, T ) = IRR(ω, T ) − f (ω, T ). We ADnote that f (ω, T ) and LR(ω, T ) have distinct lineshapes and therefore are likely to have different origins. We henceforth focus on the polarized PL signal, LR(ω, T ). A small fraction of LR(ω, T ) is also present in the orthogonal polarization emission, LL(ω, T ) = IRL(ω, T ) − f (ω, T ) = r(T )LR(ω, T ), where r(T ) ≡ IRL(ω,T)−f (ω,T) is the depolarization ratio (Material and IRR(ω,T)−f (ω,T) Methods). In Fig. 4, we plot the temperature dependence of LR(ω, T ), BEexcited with 2.6-eV right-circularly polarized light. While PL is much stronger at 15 K, emission remains polarized even at 300 K with the same r(T ) ≈ 0.1 for all temperatures (SI Appendix, Fig. S6). This demonstrates that while heating shortens the exciton lifetime, it has little impact on polarization of the exciton emission. Assuming that the depolarization process occurs mostly during the energy relaxation of the electron and hole within the corre- CFsponding bands, we expect r(T ) → 0 as ω0 approaches the direct surface band gap. Fig. 1F shows that r(T ) linearly extrapolates to zero at ω0 ≈ 2.5 eV, which suggests that the band gap should be close to this value. This is consistent with a direct transition between the top branches of RSS and SS2, shown by the blue arrow in Fig. 1A.

15 IRL(ω,T) 532 nm

IRR(ω,T) (2.33 eV) Fig. 2. Polarization dependence of PL measured with 2.6 eV excitation 10 f(ω,T) ω at 26 K. (A and B) The thick and thin lines show the right- and left- 5 0 handed PL spectra under right- and left-circularly polarized excitation. The black dashed line shows the unpolarized background. C–E compare circu- 0 I (ω,T) larly polarized PL spectra excited with linearly polarized light, where X 15 RL 521 nm (Y) denotes linear polarization parallel (orthogonal) to the plane of inci- I (ω,T) (2.38 eV) dence. (F) Comparison of the spectra with excitation polarization parallel 10 RR and orthogonal to the PL polarization. f(ω,T) 5 0 I (ω,T) right-circular excitation with six different energies, denoted by 15 RL 482 nm the arrows in each panel. As one can see from Fig. 3, the polar- I (ω,T) (2.57 eV) 10 RR ized PL peak, whose position is marked by the dashed line, is f(ω,T) absent for excitation energies below 2.4 eV. This implies that 5 the electron and hole bands involved in forming the exciton are separated by at least 2.4 eV. By comparing the PL spectra in 0 I (ω,T) the top two panels of Fig. 3, we note, however, that the spec- 15 RL 476 nm I (ω,T) (2.60 eV) trum for the excitation energy of 2.38 eV does not exhibit any 10 RR visible features at the exciton energy, i.e., at 2.3 eV, whereas f(ω,T) weak PL for the excitation energy of 2.33 eV is enhanced at 5 2.3 eV. Also, the enhancement occurs primarily in the polarized 0 (RR) channel, whereas PL at the excitation energy of 2.38 eV 15 IRL(ω,T) 468 nm is not polarized. We argue that this reentrant behavior is an Photoluminescence intensity (arb. unit) I (ω,T) (2.65 eV) indication of the resonant excitation of dipole-allowed exciton 10 RR f(ω,T) states (30). 5 Besides the overall intensity, the difference between IRL(ω, T ) and IRR(ω, T ), depicted in Fig. 3 by the light and dark col- 0 I (ω,T) ored lines, respectively, also changes with the excitation energy. 15 RL 447 nm

In Fig. 1E we show the integrated PL intensity difference, IRR(ω,T) (2.77 eV) R 2.5 eV 10 2.0 [IRR(ω, T ) − IRL(ω, T )]dω, vs. the excitation energy ω0. f(ω,T) The polarized PL peak is observed only for excitation energies 5 between 2.6 eV and 3.0 eV. Comparing the excitation proﬁle with 0 the known band structure of Bi2Se3 (SI Appendix, section 2), we 2.0 2.2 2.4 2.6 conclude that the only possible interband transition is from RSS to SS2 bands. Photon energy (eV) Fig. 3. Low-temperature PL intensity is plotted against photon energy for Depolarization. To analyze polarization-preserving PL quantita- six different excitations (shown by arrows in each panel): ω0= 2.33 eV, tively, we decompose IRR(ω, T ) and IRL(ω, T ) into two spectral 2.38 eV, 2.57 eV, 2.60 eV, 2.65 eV, and 2.77 eV. The light and dark colored contributions (Fig. 3): (i) a broad unpolarized emission band, lines denote IRL(ω, T) and IRR(ω, T), respectively. The smooth background f (ω, T ), and (ii) a narrower peak that is almost fully polarized, f(ω, T) is plotted by black lines.

4008 | www.pnas.org/cgi/doi/10.1073/pnas.1813514116 Kung et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 .With 2). section Hamiltonians Appendix, the (SI use we neglected term assumptions, be corresponding these can the than thus smaller and 4 RSS of in factor a than more by is the on by momentum described to angular be correspond total states the of RSS tions and SS2 both elec- that the bands, 1B of (Fig. RSS structure calculation and band first-principle tronic SS2 the consider- in on by of holes Based and observations, all respectively. electrons explains experimental excited that the optically model of ing minimal aspects a key build the we bands. following, surface to the due entirely In be must PL polarized observed that S ad,isbn aaeesaesilepce ob universal and be SS1 to expected the still to are contrast parameters band in its protected, bands, SS2 topologically not is band at and spin respectively, subspaces, spin linear-in-p RSS and SS2 1 the in surface, matrices the to normal vector mass, hole effective SS2, the and is RSS of points touching Dirac RSS the the touching near Here, holes SS2 respectively. Rashba (32), the point massive near the electrons and (31) Dirac point massless the describe to bulk a Since break- by symmetries. lifted inversion is or states Bi electron time-reversal of either degeneracy ing spin the possible is if PL only polarization-preserving general, In results. mental States. Surface Model Theoretical ( and maximum half signal, in colors at PL line the width polarized to corresponds full the coding of the intensity (B) integrated of dependence ature ftepoo nryfrvrostmeaue.(B temperatures. various for energy photon the of 4. Fig. uge al. et Kung σ 2 e Se and 3 h nest fteplrzdP signal, PL polarized the of intensity The (A) AB 90 rsa snnantcadcnrsmerc eargue we centrosymmetric, and nonmagnetic is crystal 1 σ ◦ h em ecieteefc fSCwihlcselectron locks which SOC of effect the describe terms oismmnu.W oeta lhuhteRSS the although that note We momentum. its to r h dniymtie ntesm usae.The subspaces. same the in matrices identity the are H H o etr oteitrrtto fteexperi- the of interpretation the to turn we Now RSS SS2 2 × = (p) ∆ = (p) 2 al arcs lo h astr nSS2 in term mass the Also, matrices. Pauli − α 1 2m p σ steRsb coefficient, Rashba the is 2 e and h σ ∆ + 1 e steeeg ifrnebetween difference energy the is v σ and ,w assert we S4 ), Fig. Appendix, SI h (σ v A. − steDrcvelocity, Dirac the is e σ α(σ × h p) r h etr fPauli of vectors the are h R · × z ˆ L L z , p) J R R and (ω xsadtu can thus and axis z (ω · = C , z ˆ , T T C ±1/2 ) ,a function a as ), h temper- The ) d ω z ˆ h color The . saunit a is m projec- C h the ) > [1] 0 ie tmclysot n rsl lae ufcs hc are which surfaces, cleaved in freshly realized and smooth atomically given oyHamiltonian body ietotcltastossuidi hsrpr.If pair electron–hole report. an (k this the momentum in symmetric, total studied ally transitions zero optical of direct case the Transitions. on Optical and Eigenstates bound excitonic to lead holes states. and electrons between attraction n ol xetta cteigb phonons by scattering that expect min- could Mexican-hat One the 1D. to Fig. close of energies imum with photons by excited the 1C). (Fig. of data recombination the with from consistent is arising which peak exciton, PL single a with between excitations cross-relaxation only polarized produce circularly can experiment, with light our states of two geometry the backscattering whereas degenerate, doubly J are 1D, Fig. in with states bound The amplitudes 1A). section for equations algebraic small by case, screened this ψ interaction In Coulomb carriers. free for approximation sonable 1B), section Appendix, the (SI in representation wavefunction momentum-space spinor four-component Eq. the for in ansatz Hamiltonian lowing the by own. defined its on Eq. number of angu- quantum states orbital good a the is neither (Here, spin although nor number momentum quantum lar good a is originat- bands two-body and the However, from possible. ing be not would sisrciet elc h neato oeta yamodel a it by structure, potential spin interaction their the attraction particular, replace short-range in to and, instructive is states discrete ing where utlyrMxcnht(opr i.1 Fig. S2 (compare hat Mexican multilayer of values to conjugate momentum V where † npaeeeti dipole. electric in-plane degenerate the both of to operators belong symmetry therefore all to respect with symmetric Eq. fully in functions wave exciton the Inspecting 2. is which titysekn,ectnsae aet ecasfidwti h ufc ymtygroup, symmetry surface the within classified be to have states exciton speaking, Strictly z 1 (r) nitrcigeeto–oepi sdsrbdb a by described is pair electron–hole interacting An h bv rueti utbefrepann oaie PL polarized explaining for suitable is argument above The .I oheeto n oebnswr ases on state bound a massless, were bands hole and electron both If ). (p 0 = H ψ α ), eh = (p) ecie h olm interaction, Coulomb the describes u nyfrsae with states for only but λ, aee as labeled , p ythe by r C . . . (p, J ˆ φ 6v = oudrtn h eea rpriso h result- the of properties general the understand To |p. ≡ z H o culBi actual for |J , = steaiuhlageof angle azimuthal the is H r = k) Bi z eh ψ e e eh = 1 − iJ 4 2 0) (p, + p 2 (p ±1i σ (p, z Se 2 a ecasfidby classified be can r e φ H 1 h ) ⊗ emi h S ad hrfr,teCoulomb the Therefore, band. RSS the in term 3 σ SS2 sterltv oiino h lcrnadhole, and electron the of position relative the is k) ttsbln othe to belong states aennrva ouin o ninfinitesimally an for solutions nontrivial have . |J e ψ 2 1 A Se r one rmblwfrayvle of values any for below from bounded are ⊗ PNAS 1 1 z z σ aeaWsae iprinrsmln a resembling dispersion W-shaped a have (p 3 h reuil ersnain nteohrhn,tedoubly the hand, other the On representation. irreducible 0 = 1 p opnn fteaglrmmnu of momentum angular the of component 3)or (33) (−i )e σ V + h |J aedfeetenergies different have i, | V −i = (r) k 2 ∂ )(r (1/2) z C ac ,2019 5, March (r), φ φ ∞v = , + ) ⊗ ψ ±1i o oainlyivratHmloini Eq. in Hamiltonian invariant rotationally a for J −λδ 2 1 z (p 1 2 E σ 0 = 1 efidta h two the that find we 4, J e 1 h J ), and z + ersnain hc rnfr san as transform which representation, σ hc rvdsarea- a provides which (r), − z nwa olw,w focus we follows, what In 2 h Schr The . ψ e = )Teeoe h eigen- the Therefore, p.) r and 1 ⊗ a esle ytefol- the by solved be can 3 h |J ∆J (p aee as labeled ±1, σ ) | D For . σ e z p ), o.116 vol. h z J ⊗ and 0 = z 0 = = z + ψ = H = −i 4 C prpit for appropriate ) 1 2 i RSS suigno Assuming ±1. (p k dne equation odinger Fig. Appendix, SI ¨ 6v ±1 tts eexpect we states, σ ∇ 0 = )e and e z | † r ⊗ IAppendix, (SI |J i and , −p o 10 no. φ V h eigen- the , ihnthe Within . z C 1 ∞v = 4 (r) | | σ T + J J 0i × h rusand groups z z , k ttsare states k 2 = = 4 saxi- is | sthe is two- ±1i ±1i 4009 [3] [4] [2] v

APPLIED PHYSICAL SCIENCES couples the |Jz = +1i and |Jz = −1i states for energies above the the bound-state energy arises because massive holes with ener- minimum, which would cause an increase of r(T ) with ω0. How- gies close to the minimum of the Rashba spectrum exhibit an ever, we see only a moderate increase of r(T ) even if ω0 is about effectively 1D motion (38). In 1D, the bound-state energy in a 300 meV above the Mexican-hat minimum (Fig. 3). To explain weak potential U (x) is proportional to R dxU (x)2 (40) (hence the preservation of optical orientation during energy relaxation, the ln2 factor for the 1/x potential). A more accurate result can we note that a transition between the |Jz = ±1i states that do be obtained by numerical solution of the Schrodinger¨ equation not conserve the z component of the angular momentum Jˆz (39) which gives 0.22 eV for the bound-state energy, whereas the would require scattering by nonsymmetric bosons or by a mag- observed value is 0.2 eV. We thus conclude that our theoretical netic impurity. It is known that nonsymmetric surface phonons model is in quantitative agreement with the data. in Bi2Se3 are weak (34, 35), hence Jz remains approximately We note that while α and v may vary slightly from sample to conserved during the energy relaxation. It would be interesting sample, the chiral exciton energy depends only logarithmically on to study in the future the interaction between the chiral exci- the band structure parameters, at least as long as the Coulomb ton and other more exotic collective modes, such as the Dirac attraction between electron and hole is sufficiently weak. Fur- plasmons and chiral spin modes (27, 36, 37). Importantly, the thermore, it is known from ARPES measurements, nonlinear |Jz = ±1i exciton states can also be resonantly populated with optics, and first-principle calculations that, while the position of circularly polarized 2.3-eV excitation (Fig. 3), which suggests the Fermi level is very sensitive to surface preparation, the sur- that the exciton states are dipole allowed and thus confirms the face states are rather robust against nonmagnetic dopants (21, proposed model. 41, 42). In our case, the surface states composing the chiral exciton are far away from the Fermi level and thus should be even Bound-State Energies. The theoretical model described above less sensitive to surface contamination. This naturally explains allows one to extract quantitative characteristics of the exciton the reproducibility of the observed features between samples. spectra. The exciton energies for a short-range interaction as 2 functions of the dimensionless coupling constant u = mh λ/2π~ Conclusions are shown in Fig. 5. With the band parameters extracted from We used polarization-resolved PL spectroscopy to study the ARPES data (17, 18), one finds for the absorption edge in secondary emission from the surface states of an archetypical ±1 2 the Jz = ±1 channel Eg = ∆ − mh (α + v) /2 ≈ 1.8 eV, which topological insulator Bi2Se3. When the crystal is excited with is somewhat smaller but comparable to the observed value of 2.5–2.8 eV circularly polarized light, we detect emission of the 2.48 eV. For a more realistic case of the Coulomb interac- same polarization at 2.3 eV. Polarization of emitted light is tion (which is assumed to be weak), the binding energy can be preserved even if the excitation energy is hundreds of millielec- ±1 ∗ 2 2 estimated as ±1 − Eg = −4Ry ln 2mh (α + v)aB /e ~ (38, tronvolts above the emission threshold energy. We assign such ∗ 4 2 2 39), where Ry = mh e /2~ εeff ≈ 0.02 eV is the effective Ryd- emission as resulting from recombination of exciton states: chiral berg, εeff ≈ 13 is the effective dielectric constant of semiinfinite excitons. We propose that chiral excitons are made of (topo- Bi2Se3 for frequencies above the topmost phonon mode, aB = logically protected) massless electrons and massive holes, both 2 2 ~ εeff/mh e is the effective Bohr radius, and e = 2.718 ... is the residing on the surface of Bi2Se3 and characterized by chiral spin base of the natural logarithm. The (large) logarithmic factor in textures. The exciton states can be characterized by the eigen- values of the out-of-plane total angular momentum, Jz . Based on the results of our theoretical model, we identify the dou- blet of degenerate states with Jz = ±1 as being responsible for observed polarization-preserving PL. The most surprising find- ing is that polarization of chiral exciton PL is preserved up to room temperature and robust with respect to chemical substitu- tion, which we attribute to the weakness of spin-flip scattering between surface states with opposite helicity. In this way, chiral excitons are fundamentally different from other known excitons that also preserve helicity (4, 13). Controlled optical orientation of chiral surface excitons may facilitate new photonics and optoelectronics applications of topological insulators.

Materials and Methods Material Growth. All data presented in the main text are collected from bulk single crystals grown by a modiﬁed Bridgman method. Mixtures of high-purity bismuth (99.999%) and selenium (99.999%) with the mole ratio Bi : Se = 2 : 3 were heated to 870 ◦C in sealed vacuum quartz tubes for 10 h and then slowly cooled to 200 ◦C at the rate of 3 ◦C/h, followed by furnace cooling to room temperature.

Experimental Setup. The crystals were cleaved before cooldown in a glove bag filled with nitrogen gas and were transferred into a continuous-flow liquid helium optical cryostat without exposure to atmosphere. A solid-state laser was used for 2.33-eV (532 nm) excitation, a diode laser was used for 2.77-eV (447 nm) excitation, and a Kr+ ion laser was used for all other excitations, with laser spot size roughly 50 × 50 µm2. The power density on the sample is kept below 0.7 kW/cm2, and all temperatures shown were corrected for laser heating with 1 K/mW. The polarized secondary emission was Fig. 5. Bound-state energies of excitons with Jz = ±1 (blue) and Jz = 0 (red analyzed and collected by a custom triple-grating spectrometer with a liquid and black) obtained by numerically diagonalizing Eq. 2 with V(r) = −λδ(r). nitrogen-cooled CCD detector. The band structure parameters are taken from fitting the ARPES data of The intensity Iµν (ω, T) was corrected for the laser power and spectral refs. 17 and 18 into the spectrum of Eq. 1: ∆ = 2.7 eV, v = 2.0 eVA,˚ α = 5.2 response of the spectrometer and CCD, where µ (ν) denotes the direction of −1 −2 2 eVA,˚ mh = 0.036 eV · A˚ (SI Appendix, section 2), and u = mhλ/2π~ . incident (collected) photon polarization, ω is energy, and T is temperature.

4010 | www.pnas.org/cgi/doi/10.1073/pnas.1813514116 Kung et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 1 hnY,e l 20)Eprmna elzto fatredmninltopological three-dimensional a of realization Experimental (2009) al. et YL, Chen 21. 0 hn ,e l 20)Tplgclisltr nBi in topological insulators Topological (2009) a al. transport et in Dirac H, helical Zhang spin photocurrent the 20. in of insulator topological tunable imaging A (2009) resolved al. et D, Band Hsieh (2017) 19. al. state surface et Dirac unoccupied H, an Soifer to coupling 18. optical Direct (2013) al. et JA, platform Sobota A dichalcogenides: 17. transition-metal on Graphene (2015) J Fabian M, Gmitra 16. 5 i Y aJraaF,LueS 21)Otclsetu fMoS of spectrum Optical (2013) SG Louie FH, Jornada da monolayer DY, in Qiu coherence valley 15. excitonic of generation Optical (2013) al. metal et transition layered AM, monolayer in Jones pseudospins in and 14. Spin polarization (2014) valley TF Heinz of D, Xiao Control W, Yao (2012) X, TF Xu Heinz 13. J, Shan K, p He heavy KF, of Mak Effects 12. (1999) V Kalevich Y, Yashin B, Oskotskii Y, Mamaev A, Subashiev 11. 0 ontA lnlR Benoit R, Planel A, Bonnot 10. uge al. et Kung rud easm neeg needn eoaiainratio depolarization independent energy an assume We ground. and respectively, where xiain h esrdP neste a edcmoe notoparts, two into decomposed be can intensities PL measured the excitation, Subtraction. Background Photoluminescence photon. the of helicity the the to in than measured the rather momentum angular is, frame, the laboratory That to light. refer “spectroscopy polarizations polarized the left circularly and follow right of we “handedness” Here, the incidence. for of parallel convention” plane polarization the linear to denotes (orthogonal) (Y) X where respectively, polarizations, L X n X R YX. and XX, RL, h cteiggoere sdi hseprmn r eoe as denoted are experiment this in used geometries scattering The .GosE emgrvS ooek ,KalmvB(93 o-xio luminescence Hot-exciton (1973) B Kharlamov Y, Morozenko S, Permogorov E, Effect Gross (1971) 9. IB Rusanov YP, Veshchunov RI, Dzhioev VG, Fleisher in BI, transitions Zakharchenya interband in 8. carriers of orientation Optical (1970) VI Safarov AI, Ekimov 7. .ProsR 16)Bn-obn pia upn nsld n oaie photolumi- polarized and solids in pumping optical Band-to-band (1969) RR Parsons 6. (2017) MI D’yakonov 5. (1984) C Guillaume transition R, semiconductor 2D Planel spin- of optoelectronics 4. and and Photonics resonance (2016) J spin Shan KF, Chiral Mak (2005) 3. AM Finkel’stein M, Khodas A, Shekhter 2. 1. nuao,Bi insulator, surface. the on cone Dirac regime. 2018. 19, March posted Preprint, arXiv:1712.08694. insulator. Bi insulator topological the in optospintronics. and physics spin-orbit proximity for states. exciton of diversity and WSe dichalcogenides. MoS spectra structures. luminescence strained-layer low-temperature GaAs/GaAsP and of spectra emission polarized the on doping CdS. nCS crystals. CdSe in crystal. GaAs a in spins electron 137–139. of orientation optical of semiconductors. nescence. 16–32. pp Switzerland), 353–380. pp dichalcogenides. metal interactions. electron-electron the of presence 71:165329. the in conductivity Hall Phys Mod Zuti ˇ ,Fba ,DsSraS(04 pnrnc:Fnaetl n applications. and Fundamentals Spintronics: (2004) S Sarma Das J, Fabian I, c ´ 2 2 hsRvB Rev Phys L . yotclhelicity. optical by a Nanotechnol Nat R (ω Nature hsRvLett Rev Phys 76:323–410. , 2 T Te and ) 3 460:1101–1105. 9:690–702. . hsSau Solidi Status Phys x ho hsLett Phys Theor Exp J a Phys Nat = f Science (ω X L , pnPyisi Semiconductors in Physics Spin + T L a Photon Nat I 23:1152–1154. (ω I eoe h etrls noaie ra back- broad unpolarized featureless the denotes ) 8:634–638. RR RL i a Nanotechnol Nat 325:178–181. n L and Y 10:343–350. a Phys Nat , (ω (ω aGilueC(94 pia retto fectn in excitons of orientation Optical (1974) C Guillaume la a ` T eoergtadlf iclryplrzdPL, polarized circularly left and right denote ) , , 2 pia retto fExcitons of Orientation Optical T T Se hsRvLett Rev Phys ) ) 3 = = 59:551–560. . = hsRvLett Rev Phys 10:216–226. 5:438–442. L L X 12:198–201. L R (ω − (ω Semiconductors , i , 7:494–498. eoetergt n left-circular and right- the denote Y T T 111:216805. ) ) + + 2 ihrgtcrual polarized right-circularly With Se f f 111:136802. (ω (ω 3 Bi , hsRvB Rev Phys , , Srne nentoa,Cham, International, (Springer T T 2 ), ) Te 33:1182–1187. 3 x ho hsLett Phys Theor Exp J n Sb and Esve,Amsterdam), (Elsevier, 92:155403. 2 aybd effects Many-body : 2 Te 3 ihasingle a with hsRvB Rev Phys µν r = (T RR, Rev ) [5] 13: = 6 uL ta.(06 unie aaa n errtto n xo electrodynamics axion and rotation Kerr and Faraday Quantized (2016) al. et L, Wu 26. unoccupied the in dynamics electron spin-polarized Ultrafast (2017) insulators. al. et topological D, Three-dimensional Bugini (2011) 24. JE Moore MZ, Hasan 23. 1 se ,e l 21)Nnierotclpoeo ual ufc lcrn na on Bi electrons insulator topological the surface on doping tunable Gas (2013) of B Yan probe T, Frauenheim optical M, Koleini Nonlinear 42. (2011) al. et D, Hsieh short (1977) 41. EM two-dimensional Lifshitz and a spin-orbit LD, with Landau atom in hydrogen 40. two-dimensional a states of levels Energy Bound (2008) C Grimaldi topological (2006) 39. in LI plasmons Dirac Magarill and surface AV, of Interplay Chaplik (2015) 38. liquid. al. helical et a of A, modes Collective Politano (2010) SC 37. Zhang XL, Qi SB, Chung S, coupling Raghu electron-phonon 36. surface and bulk Distinguishing topological (2014) 3D al. Bi et insulator in topological JA, the transitions Sobota of modes predicted 35. vibrational Surface Symmetry (2017) al. (2013) et HH, JL Kung Birman 34. JJ, topological Tu the J, of Li states 33. surface the in (2003) R effects Winkler warping 32. resonant Hexagonal under wells (2009) quantum L GaAs in Fu dynamics 31. Exciton (1994) al. et A, Vinattieri 30. insulator. topological a of surface the on mode spin Chiral (2017) al. et HH, Kung 27. Vald 25. superconductors. and insulators Topological (2011) SC Zhang XL, Qi 22. 9 i ,e l 21)Sac n eino omgei etoymti layered centrosymmetric nonmagnetic of design and Search (2015) al. et Q, insulator topological the Liu in states 29. surface to due rotation Faraday (2017) al. et Y, Shao 28. .-VP ek h iclryplrzdP a ecluae knowing calculated be can PL polarized circularly The peak. PL 2.3-eV Then, as emission unpolarized the .X hn .Sinv .A ooa .Sie,adD adrit We Rashba, Vanderbilt. D. I. and E. (to Soifer, DMR-1629059 Lu, Grant and H. NSF Z. H.-H.K., and X.W.). Sobota, D.L.M.), G.B., and Levitov, (to S.-W.C. (to A. DMR-1720816 S. DMR-1104884 Grant J. Grant NSF L. A.L.), NSF Smirnov, from with D. support discussions acknowledge Shen, and X. Rustagi Z. A. and ACKNOWLEDGMENTS. L L R L (ω odn atrPhys Matter Condens 83:1057–1110. fa3 oooia insulator. topological 3D Bi a of insulator topological the of states 087403. surface the from Bi of state surface topological surface. insulator. topological York). New (Pergamon, interaction. spin-orbit interaction. Rashba the by induced 126402. potential range Bi of case The insulators: Lett Rev Lett Rev Phys Bi insulator topological the in spectroscopy. Raman by observed insulators. Systems Bi insulator excitation. rsaswt ag oa pnpolarization. spin local large with crystals (Bi Lett Rev (ω ,T ,T 1−x ) r ) sAulrR ta.(02 eaet epneadclsa errotation Kerr colossal and response Terahertz (2012) al. et R, Aguilar es ´ (T Inserting . Sb sdtrie ymnmzn hr etrsin features sharp minimizing by determined is ) Srne,Berlin). (Springer, hsRvLett Rev Phys 104:116401. 119:136802. x ) oi tt Commun State Solid 2 hsRvB Rev Phys 2 Te Te 113:157401. 3 3 . . aoLett Nano hsRvLett Rev Phys pnObtCuln fet nToDmninlEeto n Hole and Electron Two-Dimensional in Effects Coupling Spin–Orbit r (T hsRvB Rev Phys f noteaoeepeso of expression above the into ) (ω 2:55–78. hsRvLett Rev Phys 110:016403. 50:10868–10879. L PNAS eaegaeu o ehia upr rmB .Dennis S. B. from support technical for grateful are We , R 2 T (ω Se ) 17:980–984. 3 , = . T Science 2 2 hsRvLett Rev Phys 103:266801. Se 77:113308. ) Se | I 163:11–14. RL = 3 3 hsRvB Rev Phys ac ,2019 5, March (ω . sn iersle hteiso spectroscopy. photoemission time-resolved using 106:057401. I hsCnesMatter Condens Phys J RR , unu ehnc NnRltvsi Theory) (Non-Relativistic Mechanics Quantum 354:1124–1127. T (ω ) 1 − , hsRvB Rev Phys 1 T − r − 115:216802. ) (T 95:245406. − r r (T ) (T I · RL ) I ) RR (ω | 91:235204. (ω , o.116 vol. T , ) T . I 2 ) 29:30LT01. RL Se . (ω 3 . f hsRvLett Rev Phys , (ω hsRvLett Rev Phys | T ,w a write can we ), , o 10 no. T rudthe around ) e o Phys Mod Rev nuRev Annu | r (T 4011 2 2 Phys Phys 108: Se Se [6] [7] 96: ), 3 3

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