This is the submitted version of the article: Valbuena M.A., Chudzinski P., Pons S., Conejeros S., Alemany P., Canadell E., Berger H., Frantzeskakis E., Avila J., Asensio M.C., Giamarchi T., Grioni M.. Polarization dependence of angle-resolved photoemission with submicron spatial resolution reveals emerging one-dimensionality of electrons in NbSe3. Physical Review B, (2019). 99. 075118: - . 10.1103/PhysRevB.99.075118. Available at: https://dx.doi.org/10.1103/PhysRevB.99.075118 Polarization dependence of sub-micro ARPES reveals emerging one-dimensionality of electrons in NbSe3 M. A. Valbuena*,1, 2 P. Chudzinski*,3, 4 S. Pons,1, 5 S. Conejeros,6 P. Alemany,7 E. Canadell,8 H. Berger,1 E. Frantzeskakis,9 J. Avila,9 M. C. Asensio,9 T. Giamarchi,10 and M. Grioni1 1 Institute of Physics (IPHYS), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 2 IMDEA Nanociencia, C/ Faraday 9, Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain 3 Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 4 School of Mathematics and Physics, Queens University Belfast 5 Laboratoire de Physique et d’Études des Matériaux, École de Physique et Chimie Industrielles de la ville de Paris, ESPCI, PSL Research University; CNRS UMR8213; Sorbonne Universités, UPMC University, 10 rue Vauquelin, 75005 Paris, France 6 Departamento de Quı́mica, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta 124000, Chile 7 Departament de Ciència de Materials i Quı́mica Fı́sica and Institut de Quı́mica Teòrica i Computacional (IQTCUB), Universitat de Barcelona, Martı́ i Franqués 1, 08028 Barcelona, Spain 8 Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra (Barcelona), Spain 9 Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin-BP 48, 91192 Gif sur Yvette, France 10 Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland (Dated: June 4, 2020) In materials with nearly commensurate band filling the electron liquid may spontaneously sepa- rate into components with distinct properties, yielding complex intra- and inter-unit cell ordering patterns and a reduced dimensionality. Polarization-dependent angle-resolved photoemission data with sub-micron spatial resolution demonstrate such an electronic self-organization in NbSe3 , a compound considered to be a paradigm of charge order. The new data indicate the emergence of a novel order, and reveal the one-dimensional (1D) physics hidden in a material which naively could be considered the most 3D of all columnar chalcogenides. The 1D physics is evidenced by a new selection rule – in two polarization we observe two strikingly different dispersions each closely resembling apparently contradicting results of previous studies of this material. PACS numbers: I. INTRODUCTION of a novel kind of order. The resulting electrons’ self- organization reveals the one-dimensional (1D) physics Recent developments – e.g. in the field of cuprates1–6 hidden in a material which naively could be considered and pnictides7,8 – prompt to seriously consider whether, to be the most 3D of all columnar chalcogenides. in an ordered phase, the electrons can self-differentiate in The quasi-1D compound NbSe3 has a columnar struc- such a way that only some of them develop an order and ture, shown in Fig. 1(a)22 . It undergoes two succes- coexist9–11 with the remaining non-gapped carriers12 . sive CDW transitions, which affect electrons on dis- This scenario is of uttermost importance in materials tinct columns23–25 , at TCDW 1 = 145 K with wave vector with reduced dimensionality where interactions can par- q1 = (0, 0.241, 0), and at TCDW 2 =59 K, with wave vector ticularly easily lead to to new exotic instabilities13–15 , q2 =(0.5, 0.259, 0.5)26,27 . The resulting charge ordering but these systems are both fragile with respect to dimen- is not strong enough to remove the entire FS26,28 , and sional crossover16,17 , and prone to coupling to lattice vi- metallic conduction is observed even below TCDW 2 . The brations. This is unfortunate, because they completely instabilities are usually interpreted in terms of imperfect pre-empt all exciting non-Fermi liquid phenomena which nesting of the warped multi-sheet Fermi surface (FS). could take place at the lowest energies18–20 . However, the nature of the gapless states has proven quite Materials where distinct carriers independently gov- elusive, with several proposals for the location of residual erned by each of these perturbations can potentially co- pockets in the Brillouin zone28–30 . exist, give us an opportunity to overcome this longstand- Previous ARPES experiments have reported dispersive ing stalemate. Here we report Angle Resolved Photoe- bands and CDW-related gaps21,31 and, more recently, mission Spectroscopy measurements with sub-µm spatial possible signatures of 1D physics32 . In our measure- resolution (NanoARPES) of NbSe3 , a material that has ments an electronic self-organization, driven by inter- been considered a paradigm of charge order. Our new actions, is revealed by emergent polarization-dependent data question a canonical scenario for charge-density- selection rules, which reflect the underlying 1D nature wave (CDW) formation21 and suggests the emergence of the electronic states. The number of observed bands 2 FIG. 2. (a) Sample holder with several NbSe3 samples; (b) Optical microscope image of NbSe3 whiskers. The scale of the image is given in (c). fects. The sample is mounted on a high-precision scan- ning stage with five degrees of freedom comprising linear and angular motion. Piezoscanners and an interferomet- ric control of the sample and ZP positions guarantee zero FIG. 1. (a) The crystal structure of NbSe3 (monoclinic, a = backlash and enable sub-µm accuracy and reproducibil- 10.01Å, b = 3.48 Å, c = 15.63 Å, β = 109.5◦ ) is built out ity as different parts of the sample are scanned through of prismatic columns of three types, hereafter labeled column the beam. All these parts ensure spatial resolution of A-C, running along the b axis. (b) ac plane cross-section. the acquired data without sacrificing any of the qualities Green (orange) dashed lines indicate strong (weak) Se...Se of conventional ARPES. Images of the sample chemical intercolumnar contacts. composition and angle-resolved valence band states at a sub-µm scale may be acquired by the synchronized scan- ning of the sample with respect to the focused beam, is also doubled with respect to a single-particle picture. while the electron analyzer collects photoelectrons of a The ability to zoom in on a small selected, disorder-free selected kinetic energy. area of the sample was crucial to observe these features, which escaped previous investigations. Spatially resolved Se 3d intensity scans were obtained The outline of the paper is as follows. After a descrip- at different lateral scales and shown in Fig. 4(a,b). The tion of the experimental methods in Sec.II we present integrated intensity of the Se 3d core level spectrum is the experimental ARPES results in Sec.III. We illustrate represented as a function of position in a color scale, our DFT+U study of the material in Sec.IV and derive in yielding a “real-space” mapping of NbSe3 micrometric Sec.V the ARPES selection rules that explain the striking fibers. Figure 4(c) shows the Se 3d core level spectrum polarization dependence of the ARPES data. Finally, in acquired in a high-intensity area in one of the best qual- Sec.VI we present a consistent interpretation of our both ity chains (red area). The solid line is the result of a fit experimental and theoretical results. of the spectrum with five spin-orbit doublets. The com- plex line-shape, which reflects inequivalent Se sites, is a direct indication of the quality of the photoemission data II. EXPERIMENTAL METHODS acquired with sub-µm lateral resolution. NbSe3 single crystals with typical dimensions of 0.01 x 10 x 0.05 mm3 grown using the chemical vapor trans- port method33 were mounted on the sample holder, as shown in Fig. 2. Samples were cleaved “in-situ” in an Ul- tra High Vacuum (UHV) system at room temperature to expose the bc plane. As detailed below, with the help of the spatial photoemission scanning system it was possi- ble to focus the beam on one of the best cleaved NbSe3 whiskers. We performed NanoARPES measurements with the k- microscope of the ANTARES beamline at the SOLEIL synchrotron34,35 , with spatial and energy resolution of 100 nm and, respectively, 20 meV. The microscopy capa- bilities of the experimental end-station are schematically FIG. 3. Sub-µm ARPES experimental setup scheme. The depicted in Fig. 3. A photon beam of variable energy incoming light is focused in a sub-µm region by a zone plate. and polarization is focused on a sub-µm spot by means The sample is mounted on a high-precision 5 degrees of free- of a Fresnel zone plate (ZP). An order sorting aperture dom manipulator. Linear horizontal and linear vertical po- (OSA) removes undesirable higher-order diffraction ef- larized light is available. 3 a b FIG. 4. (a) Spatially resolved Se 3d intensity scan on NbSe3 whiskers, and (b) Higher resolution scan of dashed area. (c) Se 3d core level and the result of a fit with five spin-orbit doublets. c d III. ARPES RESULTS: POLARIZATION 0.0 0 DEPENDENT BAND DISPERSION -0.2 -10 Energy (eV) -20 -0.4 ARPES measurements of NbSe3 are challenging. Crys- -30 -40 tals grow in the form of very thin whiskers, parallel to -0.6 -50 the b axis, that assemble in bundles. It is thus difficult to -0.8 ensure that standard ARPES results are not affected by -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 inhomogeneities or by averaging over several domain ori- e f 0.0 entations. In our experiment we took advantage of the 0 sub-µm photon spot size to select a flat homogeneous -0.2 -20 Energy (eV) area of a well-cleaved single crystal whisker. This is es- -40 -0.4 pecially important for the analysis of the dependence of -60 -80 the spectra on the light polarization, that will be pre- -0.6 -100 sented below. Data were collected at 100 eV photon en- -0.8 ergy with light polarized either within (p−polarization) -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 or perpendicular to (s−polarization) the horizontal scat- Wave vector (�/b) Wave vector (�/b) p p tering plane, which contained the b axis of the sample. Figure 5 shows ARPES intensity maps measured at FIG. 5. ARPES intensity maps (hν=100 eV; T=55 K) 55 K, below TCDW 2 , along the b∗ k b axis. We used a soft measured along the b∗ k b axis (π/b=0.89 Å) with (a) median filter to remove spikes, and then symmetrized s−polarization, and (b) p−polarization. The inset shows the the data around k = 0. The average incoherent back- experimental geometry. The s−polarization is perpendicular ground, estimated from the noise level at k > 0.6 Å, to the long axis (i.e. to the b axis) of the NbSe3 crystalline is monotonous and can be fitted with a Shirley func- whiskers. The p−polarization has components along the b tion, which was then subtracted from the ARPES in- axis and perpendicular to the bc cleavage plane. (c) and (e) tensity. Curvature plots36 extracted from (a) and (b) to are curvature plots obtained from (a) and, respectively (b). enhance the visibility of the bands are shown in panels Peak positions determined from EDCs and MDCs are shown (c) and (e) in a smaller region around Γ. Figure 6 (a,b) in (d) and (f). The lines are guides to the eye. shows the corresponding raw, non-symmetrized, data, and Fig. 6 (c,d) the Energy Distribution Curves (EDCs) extracted from those maps. For a quantitative analysis 0.37 π/b. A gap opens at EF in band C2 owing to the we fitted the line shapes of EDCs and Momentum Distri- change of periodicity induced by CDW1. Further anal- bution Curves (MDCs) extracted from ARPES intensity ysis of the data at kF 1 and at kF 3 reveals a reduction maps to determine the energy and wave vector of the dis- of spectral weight near EF (see, e.g., Fig. 9 and Fig. 10). persive features. The resulting peak positions are shown This suggests the incipient opening of gaps due to CDW2, with different symbols in panels (d) and (f) of Fig. 5 and which were estimated at ≈20 meV by low-temperature by colored tick marks on the EDCs panels (c) and (d) of ARPES measurements21,31 . Fig. 6. Data taken with p-polarization, in Figs.5 (b,e,f) and For s−polarization, in Fig. 5 (c,d), we observe 3 6 (b,d), present a quite different picture. We now ob- parabolic bands, labeled A1+ , C2 and A3+ , near the serve two linearly dispersing bands, labeled A1− and Fermi energy (EF ). Extrapolating their dispersions to A3− , which do not correspond to any of the spectral EF yields kF 1 = 0.14 π/b; kF 2 = 0.22 π/b and kF 3 = features observed with s-polarization for the same area 4 a b c d FIG. 7. Energy-dependent ARPES intensity in the k > 0 branch of the A1− band, obtained by integrating the cor- responding peak in momentum-distribution curves (MDCs) extracted from Fig. 6 (b). The inflection point determines the Fermi level position. FIG. 6. ARPES intensity maps (raw data; T=55 K; hν = 100 eV) for (a) s-polarization and (b) p-polarization. The collection time was 8 hours for each map. (c, d) Energy Distribution Curves (EDCs) extracted from (a) and, respec- FIG. 8. (a) Comparison of curvature plots obtained from tively, (b). Tick marks indicate peak positions of the various data measured with s-polarization (green) and p-polarization spectral features. The inset in (c) is an intensity map near (red). (b) A close-up of panel (a) near the Fermi level. the Fermi level for s-polarization, obtained by applying the curvature procedure to the raw data. Figure 8 presents combined images of intensity plots obtained from data recorded with s- and p-polarizations. of the single crystal, and cross EF with no indications of The images support the idea of distinct dispersions for CDW-related gaps. This dramatic polarization depen- the bands close to the EF measured with the two po- dence suggests that two different types of carriers, sepa- larizations. This is clear in particular for the A1± rable by some selection rule, are simultaneously present bands. We also notice that A3+ exhibits a large gap in a homogenous crystal. at k ∼ ± 0.3 π/b associated with the CDW1 superlat- The accurate determination of the Fermi energy is cru- tice potential. By contrast, A3− exhibits a continuous, cial to address the behavior of the electronic states af- unaffected dispersion. fected by CDW instabilities in low dimensional materials. Figure 9 shows the second derivative of the spectral The Fermi level position in Fig. 6 (c,d) was determined intensity with respect to the energy near EF . This proce- from the Fermi cutoff of a polycrystalline gold sample. dure enhances the visibility of the band edges and there- We have verified that it coincides with an internal energy fore of the opening of energy gaps, same was used in reference extracted from the raw data. To this end we Ref.31,32 . It is based on the fact that a maximum of a have quantitatively analyzed the energy-momentum dis- curve always implies a presence in a vicinity of an inflec- persion of the linear band A1− , which provides a clear tion point of the curve (to be precise, mathematically, and distinct ARPES signal. We have extracted from there are two inflection points flanking the maximum). the intensity map of Fig. 6 (b) MDCs at various energies From panel c) one sees that it is a good method to fol- for the k > 0 branch and integrated the intensity under low a dispersion as the dashed lines follows reasonably the corresponding peak. The resulting energy-dependent well the green/yellow area of the raw data. One has to momentum-integrated intensity (Fig. 7) exhibits a clear keep in mind that the value of the gap extracted in panel cutoff which coincides with the Fermi energy determined a), although our best attempt, is only an estimate on from the gold sample. This MDC-based EF determi- which one needs to put 20meV error bar (due to all res- nation, besides being intrinsic, is largely unaffected by olution effects). Nevertheless, our estimated value falls possible pitfalls affecting the more standard EDC-based close to CDW2 gap extracted from a finite temperature procedure (see e.g. Ref.37 ). tunneling spectroscopy38 which gives ≈ 15meV in our 5 a 0.05 0.00 � 2 ∼ 20 meV D ~ Energy (eV) -0.05 0.0 � 1 ∼ 55 meV 0.2 d I/dE (a.u.) 2 -0.10 D ~ 0.4 2 -0.15 0.6 0.8 -0.20 -3 1.0x10 -0.25 -0.4 -0.2 0.0 0.2 0.4 b 0.05 0.00 0.00 Energy (eV) -0.05 -0.05 -0.10 d I/dE (a.u.) 2 -0.10 -0.15 2 -0.20 -0.15 -0.25 -0.20 -0.30 -0.35 -0.25 -0.4 -0.2 0.0 0.2 0.4 0.05 c 0.00 Energy (eV) -0.05 400 FIG. 10. (left) EDCs measured at kF 1 with s- (blue symbols) Intensity (a.u.) -0.10 300 and p-polarization (red symbols), extracted from the intensity 200 -0.15 maps of Fig. 6. (right) Same for kF 3 . The solid lines are 100 -0.20 obtained by symmetrizing the raw spectra around EF -0.25 -0.4 -0.2 0.0 0.2 0.4 0.05 d 0.00 0 sults of this procedure applied to band A1± at kF 1 and Energy (eV) -0.05 -10 d I/dE (a.u.) 2 -0.10 -20 to band A3± at kF 2 are illustrated in Fig. 10. Sym- 2 -0.15 -30 -0.20 -40 metrization of the spectra measured with p-polarization -0.25 yields in both cases prominent peaks centered at EF , con- -0.4 -0.2 0.0 0.2 0.4 Wave vector (π /b) sistent with gapless metallic behavior. The situation is p qualitatively different for the spectra measured with s- FIG. 9. (a,b) 2nd derivative with respect to energy of the polarization where, even after symmetrization, one ob- ARPES intensity measured with s-polarization shown in (c). serves a nearly constant intensity at kF 3 , and even a dip The contrast of image (a) was chosen to plot the line profile at kF 1 . There is no direct evidence of CDW2 gaps, which following the inflection point of the spectral intensity at the have been estimated at ≈ 20 meV from low-temperature vicinity of EF . The wriggling line is reported in images (b) ARPES data (following the same procedure as Ref.31 ). and (c) in order to evidence the backfolding resulting from the However one must consider that our data have been mea- hybridization of bands A1+ , A2+ and C2.The gap amplitude sured close to TCDW 2 , where the gap is smaller than its can be estimated from the distance between the EF and the edge profile. (d) 2nd derivative with respect to energy of the zero-temperature limit38 . It would not be possible to ARPES intensity measured with p-polarization. The spectral observe a direct signature of such a reduced gap open- intensity exhibits a gapless behavior and a linear dispersion ing with our experimental resolution. Nevertheless, the with no backfolding evidences. symmetrized spectra of Fig. 10 suggest an underlying in- cipient gap, consistent with results shown on Fig. 9 and previous tunneling transport38 , optical spectroscopy28,39 temperature range (with ±10meV error bar for uncer- and spectral31,32 properties of NbSe3 . Theoretical pre- tainty of temperature and their uncertainty of zero en- diction for the spectral function (without finite resolu- ergy gap). Leaving the exact value of the CDW2 gap tion broadening and in-gap states included) have been aside, it is well known that bands’ dispersion backfold- proposed in Ref. 40 where (in Fig.2 therein) we observe ing is an equally good hallmark of the presence of CDW. that even very moderate temperature produces a broad With s-polarization the wriggling of the spectral intensity EDCs with rather soft gap. underlines the hybridization of the A1+ -C2 and C2-A3+ The differences illustrated in Fig. 10 support the claim bands and reveals the backfolding produced by their anti- that the spectral features observed with different polar- crossing. By contrast, the A1− and A3− bands measured izations can be attributed to different bands with distinct with p-polarization exhibit linear dispersion and metallic near-EF properties. This conclusion is strengthened by behavior. Undoubtedly, the two (on panels b) and d) ) differences in the MDCs line shapes for the two polar- represent different behaviour. izations. The MDCs of Fig. 11 were extracted for k > 0 For a detailed assessment of bands dispersion near EF 30 meV below EF , i.e. close to EF , and yet sufficiently it is convenient to adopt a symmetrization procedure that far to prevent artifacts due to the finite temperature and removes the perturbing effect of the Fermi-Dirac distri- experimental resolution. Both line shapes exhibit peaks bution function. Specifically, to an EDC measured at corresponding to bands A1± and A3± . Band C2 is vis- kF is added a mirror image of the spectrum with re- ible as a distinct feature with s-polarized light but, be- spect to EF . This yields the underlying spectral func- cause of matrix elements, only as a shoulder to A1− for tion A(kF , E) free from the Fermi-Dirac cutoff, with the p-polarization. A comparison of the two MDCs demon- usual mild assumption A(kF , E) = A(kF , −E) The re- strates that the momenta of the A1 and A3 ARPES fea- 6 FIG. 11. MDCs extracted 30 meV below EF for the two po- larizations. a c MDCs fits 0.0 A1- p-Pol C2 80 meV Energy (eV) -0.2 -0.4 A3s- A3h- Intensity (arb. units) -0.6 138 meV FIG. 13. Theoretical DFT GGA band calculations including -0.4 -0.2 0.0 0.2 0.4 Wave vector (�/b) finite U = 4 eV. The bands can be associated with their exper- p b imental counterparts in Fig. 2 by comparing their respective MDC profile 0 Fit MDC A3h - Fermi wave vectors. - Energy (eV) -0.2 A3s - A1 -0.4 C2 313 meV results are shown in Fig. 13. The present calculations -0.6 -0.4 -0.2 0.0 0.2 0.4 -0.5 0.0 0.5 Wave vector (�/b) Wave vector (�/b) p p FIG. 12. (a) Band dispersion obtained from MDC fits for p-polarization. (b) Results of the MDC fits superimposed on the ARPES intensity image. The dashed lines indicates the energy profiles for MDC shown in panel (c). Dark and light orange features display the double structure composing band A3− related to possible spinon and holon contributions. tures are not the same in the two polarizations, and that therefore they must be attributed to different bands. A further analysis of the p-polarization ARPES data reveals an even more complex picture. Figure 12 presents FIG. 14. Calculated band structure for NbSe3 at 298 K. The fits of MDCs extracted at selected binding energies. Best energy zero corresponds to the Fermi level. Γ = (0, 0, 0), Y fits are obtained when two peaks, with k-dependent sep- = (1/2, 0, 0), Z = (0, 1/2, 0), B = (0, 0, 1/2) and D = (0, aration, are used to fit the A3− band. They are labelled 1/2, 1/2) in units of the monoclinic reciprocal lattice vectors. A3− − h (dark orange) and A3s (light orange). A possible interpretation can be made in terms of 1D signatures of were carried out using a numerical atomic orbitals den- spin-charge separation and will be discussed elsewhere41 . sity functional theory (DFT) approach,42,43 which was The MDC fits confirm the presence of the C2 band, vis- developed for efficient calculations in large systems and ible as a satellite of the more intense A1− band. implemented in the SIESTA code.44,45 We have used the generalized gradient approximation (GGA) to DFT and, in particular, the functional of Perdew, Burke and Ernz- IV. DFT STUDY erhof46 . Only the valence electrons are considered in the calculation, with the core being replaced by norm- As a first step to interpret our data we performed conserving scalar relativistic pseudopotentials47 factor- (GGA+U) DFT calculations. Compared to previous ized in the Kleinman-Bylander form48 . We have used a DFT calculations on this material21 the inclusion of a lo- split-valence triple-ζ basis set including polarization or- cal interaction term U allows us to better differentiate Nb bitals with an energy shift of 50 meV for all atoms49 . atoms with formally different oxidation states. The main The energy cutoff of the real space integration mesh was 7 250 Ry. An on-site Coulomb energy (U) of 4 eV has separately for each column (Sec.V D and Sec.V E). We been taken into account within the rotationally invariant conclude that the experimental data imply that, in con- DFT+U approach of Dudarev et al50 . The Brillouin zone trast to the DFT results, the coherent motion of the elec- was sampled using grids of (8×8×8) and (30×30×30) k- trons within the a − c plane is inhibited, endowing NbSe3 points51 for determination of the density matrix and the with a stronger 1D character than anticipated. Fermi surface, respectively. The x-ray crystal structure at 298 K was used in the calculations22 . The calculated band structure at 298 K as well as its fat A. Definitions band analysis are reported in Figs.13 and 15, respectively. The calculated Fermi surface as well as several particular ARPES is a probe where a high frequency electromag- sections are shown in Figs. 16 and 17, respectively. netic wave suddenly removes an electron from a given They confirm that the A1+ , A3+ and C2 bands ob- electronic orbital. We probe the propagation (and sub- served with s-polarization are associated with crystal sequent recombination) of an excited photo-hole. The columns A and C. The calculated kF values are in good ARPES intensity is approximately given by (see e.g.54 ) agreement with the measured data. The two bands la- beled C2 are degenerate near EF as seen in the ARPES ~ ~q, ω) = |Mf i (A)| I(A, ~ 2 A(~q, ω)f (ω) (1) experiment where they appear as a single band, C2, im- plying a very small hybridization within a pair of C where ω and q are respectively the energy and the columns. The bands crossing EF mainly derive from d or- in-plane momentum deposited by the photon, and A ~ bitals of a given Nb atom, but the d weight is never more is the vector potential of the photon field. A(~q, ω) is than 50%, a rather different situation from e.g. transi- the one-particle spectral function, f the Fermi factor55 . tion metal oxides. This is a manifestation of the covalent ~ 2 is the matrix element we will focus on in this |Mf i (A)| character of the Nb-Se bonds and implies that the elec- section. It depends on the shape of the initial and final tron clouds are not locked on the Nb d orbitals, and can states and the photon polarization. be deformed by the CDW potential. The band disper- The matrix elements are defined between one-body sion has quasi-1D character, but the anisotropy is mod- wave functions in the initial and final state: erate: the calculated bandwidth is Wb = 2.5 eV along the b axis and up to 250 meV along perpendicular direc- Mf i (A) ~ · P̂~ |ψi i ~ = hψf | A (2) tions. Previously proposed band structures21,29,30 hold also when local correlations are included, but U shifts where P̂ is the momentum operator. Typically |ψf i will band B above EF . This is consistent with scanning tun- have the symmetry of a plane wave whose wave vector neling microscopy measurements25 where a significant ~kf defines, together with the incident photon wave vec- spectral weight appears just above EF , and also with tor, the scattering plane. In the following we denote it σl . our experiment, where no extra band is detected close to The polarization of the incident light can thus be perpen- the Γ point. By contrast, the sample of Ref. 32 appears dicular to this plane (s-polarization; A ~ ⊥ σl ) or within to be slightly electron doped, possibly owing to a small the plane σl (p-polarization). Note that if we choose a off-stoichiometry. gauge with zero scalar potential we can rewrite the in- ~ ~ · R̂, teraction term as E where R̂ is the position opera- tor. This makes the symmetries of (2) more transparent, V. EMERGENT SYMMETRY RULES especially if one re-writes the states as real space wave- functions |ψi i ≡ ψi (~r) and integrates over space to obtain From the DFT results one could infer that, at tempera- ~ Mf i (A). tures smaller than the band dispersion, electrons should In the following we use a shorthand bra-ket notation move coherently in the crystal in all directions, and in |e, oi to indicate an even/odd state with respect to the particular perpendicular to the chains. Then the symme- plane σl . The final state |ψf i is always even, thus |ei, try of the electronic system would be that of the P 21 /m because it can be expressed as an even parity plane-wave monoclinic space group of NbSe3 , with only two mirror state on the detector’s surface 52 . Therefore, we have symmetry planes, both parallel to the ac plane. Since the two possibilities depending on the polarization. For s- scattering plane in our experiment is perpendicular to polarization the initial state |ψi i must be odd with re- these symmetry planes, no definite polarization selection spect to the plane σl to make the overall space integral rule52 should apply and one should thus observe identical Mif ∼ he|o|oi = 6 0. On the contrary, for p-polarization dispersions for both polarizations53 , albeit with very dif- the initial state |ψi i must be even. ferent intensity. This is in stark contrast with the results In most systems, electron hopping can be considered of Fig. 5. In this section we introduce the concept of se- reasonably isotropic, and the electronic states are well lection rules in ARPES (Sec.V A), then briefly discuss the represented by Wannier functions, which are coherent su- result that would have been observed if the system were perpositions in the three directions of the local orbitals. an usual 3D material (Sec.V B), then develop symmetry |ψi i is a quantum coherent sum of these local entities, rules for the 1D case (Sec.V C) and analyze the outcome which we need to sum to get the Mif . Actually, this 8 FIG. 15. Band structure for NbSe3 at 298 K where the size of the blue (a), green (b) and red (c) circles is proportional to the contribution of the niobium atoms in chains C, B and A, respectively. summation allows us to pick a state with a well defined B. Rough selection rules quasi-momentum inside a crystal. We see that, in order to have a well defined symmetry, the plane σl must coin- Note that in addition to the symmetry selection rules cide with a true symmetry plane Σ of the 3D crystal lat- above, the strength of interaction with light does depend tice52 . Then the bands observed in s-and p-polarization on the shape of the orbitals which are at the root of the can be different. If this is not the case, then there is Wannier functions. From (2), we see that the ampli- no symmetry protection and there will be in general a tude is expected to be larger, the electric dipole moment mixing of the two signals. In the NbSe3 crystal lattice, %j bigger, when the transition involves a j−orbital that where the space group is P 21 /m, there are two possible ~ The orbitals Σ planes, both parallel to the ac plane (see Fig. 1 (a)), is extended in the direction given by E. thus perpendicular to the σl plane. We conclude that, spread along the c-axis are likely to be lead to larger vis- if the system could be reasonably described as a three ibility with the s-polarized light, while those which have dimensional material, the spectra should be essentially a large cross-section within the σl plane will be more vis- independent of light polarization. ible in the p-polarized light. From this we deduce that the px component of Se occupancy (and a minor dxy com- FIG. 17. Particular sections of the Fermi surface of NbSe3 . Γ = (0, 0, 0), Y = (1/2, 0, 0), Z = (0, 1/2, 0), C = (1/2, 1/2, 0), B = (0, 0, 1/2), D = (0, 1/2, 1/2), A = (1/2, 0, 1/2) FIG. 16. Calculated Fermi surface for NbSe3 showing the four and E = (1/2, 1/2, 1/2) in units of the monoclinic reciprocal pairs of warped sheets. lattice vectors. 9 ponent of Nb occupancy as well) will favour interaction in the direction perpendicular to the chains is essentially with s-polarized light while all other components with incoherent 16,17,56 . In this high-energy limit the electron p-polarized light. liquid is then in a 1D regime. At low energy the gaps Such “rough selection rules” are based on a single elec- ∆1,2 that appear in the spectrum (see 57 ) also prevent tron 3D description. Both tight binding and DFT result single particle hopping between chains. Thus, in both yield the atomic content for each electronic band near the limits the electronic waves perpendicular to the b-axis Fermi level. It is therefore possible to predict their vis- are incoherent. In that case, the matrix element in (2) ibility according to this rough selection rule. There are must be computed for a single entity only, and thus the two tests that we can run against our experimental data: symmetry is based on a wave function for a single entity. one qualitative and one quantitative. Quantitatively, we One could think naively that the entities to be considered notice that the px component is about half of the total are the chains themselves, in which case any plane that Se (for atoms on short bond) occupancy while the prob- would contain the columns (see Fig.1a)) could be a good ability of finding one electron from the A1/3 band on Se symmetry plane. However one should not neglect the atom is approximately 1/3 (note that apical Se atoms col.A(1)-col.A(2) hybridization, which manifests itself as have less px content). Hence the ratio of intensities of s- a splitting of the b1/b3 bands. The symmetry plane Σ vs p-polarization should be (at most) [(0.5 ∗ 0.3(3) + 0.1 ∗ must accommodate these strong links, and a plane per- 0.6(6))2 /(1 − (0.5 ∗ 0.3(3) + 0.1 ∗ 0.6(6))2 )] = 0.058 (and pendicular to the bc plane must be chosen. This plane here we added an extra 10% dxy component of Nb occu- coincides with the scattering plane, σl . We thus find that pancy ). We conclude that, if we were observing the same in the 1D regime Σ = σl and that the selection rules are 3D state in both polarizations, then such state would be strictly obeyed. Let us now examine the consequences visible above the noise level only in one polarization, in for each column or group of columns. clear contrast with the result of our experiment. This prediction clearly contrasts with the experimentally ob- served ratio ≈ 0.5. Actually, since the beamline through- D. Column C put is larger for p-polarization, the intrinsic ratio is even closer to 1.0 (i.e. equal intensities for both polarizations). The prisms out of which col.C is built are strongly dis- Qualitatively, one observes a substantial drop in the torted. This means that the high symmetry of an ideal intensity of the A3 band at binding energy 0.25 eV, but D3h chain, out of which the double prismatic columns only in s-polarization. The orbital components do not ex- are made, is lowered to C2h . The dz02 orbital, which is hibit any anomaly for the corresponding momenta. For decoupled in a D3h environment, for C2h crystal field kc , ka ≈ 0 we see only a very weak dependence of or- contains an admixture of dx02 −y02 , because now dz02 and bital content on momentum and nearly identical orbital dx02 −y02 belong to the same, rank-1 symmetry class a1 . content for the A1 and A3 bands (in accordance with The two orbitals N b(C) − dz02 and N b(C) − dx02 −y02 re- an intuitive two-leg ladder model proposed below). Ac- main split by the crystal field potential, but its effect can tually, no sensible model for lattice hybridization could be effectively reduced owing to a relatively large value explain a sudden variation of A3 content without any of the Hund coupling (one peculiarity of N b58 ), which visible change of A1 content. in a mean field treatment favours the symmetry-allowed Finally, one could point out that the different sensitiv- orbital mixing. ities of different atoms to light at a given energy should So far, we have used the primed system of coordinates, be taken into account. Indeed in our case there is a larger related to a local Nb atom, because the local crystal field photo-ionization cross-section for the Se atomic sites, but is associated with a given Nb site, see Fig. 18. However, at the same time the hybridization along the b-axis is while the admixed orbital is the dx02 −y02 in the local sys- much stronger than along the c-axis (note, that for the tem of coordinates, in the system of coordinates linked moment, for the sake of argument we are considering 3D with the scattering plane σl the spatial shape of the ad- system, but it is an anisotropic 3D system) hence the mixed Nb d orbital is very close to a dxy orbital. This is elongation is larger along the former direction. It is safe due to the tilting of the col. C prisms with respect to the to assume that these two secondary effects compensate bc plane (the tilt angle is α = 43◦ ). This explains why each other. C2 is visible in s-polarization. Actually, the amplitude of this s-polarization signal can be stronger than one could expect solely from a rather weak Nb−dxy component. C. New (1D) selection rules The Nb-Se bond has mostly a covalent character, with nearly 50% Se p spectral weight. The admixed dxy has a The summation that defines |ψi i is done over a quan- larger overlap (than dz2 ) with the p-Se orbitals (including tum coherent part of the electron liquid. Thus new pos- the p-Se(B)) and thus it plays a bigger role in p−d overlap sibilities for Σ arise in a quasi-one dimensional material. integrals. These in turn determine the symmetry proper- If the hopping between one-dimensional entities, which ties of the outer part of the “C2-molecular orbital”, the will be discussed below, is small compared to the inverse ψiC2 in our ARPES description. This expectation can be time-scale of the ARPES process t⊥  t < ~ω, hopping put on more solid ground. Fom our DFT calculation, we 10 obtain that the p-Se part of the “C2-molecular orbital” is An issue is why the even component of C2 is not well located on a short side of col.C prisms (as expected), the visible in p-polarization, since the new 1D-selection rules Se(C< ) atoms, and the Se-Se bond has mostly a π char- and the DFT results imply that for p-polarization Mf i is acter, turned away from Nb-site, thus with larger Se py close to that of the odd part61 . It should be noted that component (in a system of coordinates of Fig. 18). Then previous experiments, performed with different photon indeed the p-Se part has stronger odd character vs. σl . energies, revealed the doubly degenerated band C2 as The local part of the initial wave function corresponding the strongest, in accordance with the expectation from to band C2, this must be convoluted with Bloch wave the DFT and the sudden electron approximation. Thus along b-axis and anti-symmetrized to get the full ψiC2 (~r), spectral weight is missing at the particular condition we is a composite orbital: probe. We need to search for a cause that depends on X energy and on the fact that electrons are ordered. A pos- ψiC2 (~r) = adz2 χdz2 (~r − RN b(C) )+ sible explanation lies beyond the sudden electron approx- imation. For the chosen photon energy the wavelength adxy χdxy (~r − RN b(C) ) + apy χpy (~r − RSe(C < ) ) + ... (3) of the excited photoelectrons matches well the distance between the nearest Nb(C) sites. This enhances pho- where χj (~r − R ~ 0 ) is an atomic orbital j localized around toelectrons back-scattering, which reduces the ARPES a given site R ~ 0 and aj are coefficients of the expan- signal. Mathematically, these processes would produce sion, the Se(C< ) indicates atoms on the shorter edge strong vertex corrections to w̄dz2 , a coefficient that en- of the prism. Ellipsis accounts for further p-Se con- p ters in Mij . Only the p-polarization part of C2 is strongly tributions present on all surrounding atoms. Assum- affected, because this is the one that originates mostly ing that there is no interaction between the outgoing from an on-site Nb(C)-dz2 . photoelectron and other carriers (this is at the core of the sudden approximation), then the symmetry charac- ter of the composite orbital ψiC2 (~r) is a weighted sum E. Column A of the symmetry characters of its atomic constituents. The MP f i becomes a weighted sum of aj coefficients: Contrary to the wave function ψiC2 , which is local- Mf i = j wj aj , where only aj with an appropriate sym- ized around a given col.C, the “molecular orbitals” cor- metry contribute. Since we have linked our system of co- responding to well split bonding/anti-bonding bands A1 ordinates to σl it is straightforward to establish whether and A3, the ψiA1 and ψiA3 , must be delocalized over an a given χj contributes or not for a given polarization. In entire quartet col.A+col.B. The single column starting the case of ψiC2 (~r) we have χdz2 ≡ |ei, χdxy ≡ |oi and point cannot be taken. On the other hand, here the ar- χpy ≡ |oi thus one expects: Mfpi = j wdz2 adz2 + ... and P rangement of most Se p bonds is perpendicular to the σl Mfsi = j wdxy adxy + wpy apy + .... P plane. Then, also in accordance with w̄N b < w̄Se , it is To evaluate Mf i we need to know the weights wj = more convenient to start with the Se p part of the ψiA1 %j w̄j , where ρj is the already discussed geometric factor, and ψiA3 . In particular we focus on Se p-(A< ), the or- which is certainly largest for a well delocalized orbital bitals around a short link between col.A-col.B, shown as pz −Se(C). w̄j is the bare photoionization cross-section a green link in Fig. 1. tabulated in Ref. 59. For ~ω=100 eV the Nb 4d cross- Focusing on Se p-(A< ) one can construct two states: section is close to the Cooper minimum59,60 , while the Se 4p orbitals have a relatively large cross-section (around X eight times larger). Then, surprisingly, the Se p part ψiγ+ (~r) = aγ+ r − RSe(A< ) )+ py χpy (~ enters with larger weight. As a result the band C2 has a Ri significant odd component with respect to σl and thus it 0 apγ+ χpy (~r − RSe(A,B) ) + aγ+ r − RN b(A) ) + ... (4) y d 2 χdz2 (~ can be well visible in s-polarization. z where the second term is implied by the σ character of the Se-Se bonds, as determined by the py orientation, and: X γ− ψiγ− (~r) = ad 2 χdz2 (~r − RN b(A) )+ z Ri 0 aγ− r − RSe(A< ) ) + apγ− px,z χpx,z (~ χ x,z px,z (~r − RSe(A,B) ) + ... (5) where we take a sum over all atomic orbitals within a double unit cell that constitute a given eigenstate in real space. By reasoning along the same lines like as for FIG. 18. Mixing of Nb d orbitals on column C yields compos- ψiC2 , taking χdz2 ≡ |ei, χpy ≡ |oi and χpx,z ≡ |ei, we de- 0 ite orbitals visible in both polarizations. duce for the γ+ state: Mfsγ+ = j wpy (aγ+ γ+ P i py + apy ) + ..., 11 0 Mfpγ+ wdz2 aγ+ γ+ γ+ γ+ P i = j d 2 . Since (apy + apy ) > ad 2 and a b z z wpy > wdz2 we obtain Mfsγ+ i  Mfpγ+i . For the γ− c state the selection rule is strictly obeyed: Mfpγ− i = γ− 0 sγ− γ− γ− a* P j wdz2 adz2 + wpx,z (apx,z + apx,z ) + ..., Mf i = 0. One can also invert the reasoning to realize that we have one initial state that interacts strongly with s-polarized light, while the other only with p-polarized light. It is then quite reasonable to expect that a photohole created by s- polarized light |ψhs i will have a large overlap with only one c eigenstate, the γ+ shown in Fig. 4, that is hψhs |ψiγ+ i ≈ 1 and by analogy hψhp |ψiγ− i ≈ 1. Then one observes a dou- ψγ+ ble dispersion of bands A1, A3 because the two different initial states are probed with the two different polariza- tions. σl VI. THEORETICAL INTERPRETATION In the previous section we showed that ARPES selec- ψγ- tion rules can emerge due to the one dimensional charac- ter of the material. We now need to check if the peculiar physics of NbSe3 can be captured in this situation. Obvi- ously, the standard nesting scenario for the lower CDW transition, based on a warped 2D Fermi surface, must be abandoned. Note that such nesting scenario was al- ready hardly compatible with results from X-ray diffuse FIG. 19. (a) ac plane cross section showing schematically the scattering62 . It has been proposed already long ago27 perpendicular dimerization and emergent self-organization. that phase transitions in NbSe3 must be considered from Yellow dots represent col. C where CDW1 takes place. Pairs the perspective of strong interactions, but so far experi- of red-blue rectangles linked by a thick black line are dou- ments managed to avoid, albeit sometimes narrowly21 , blets of A+B columns. The perpendicular dimerization is to tackle this admittedly very complex theoretical is- indicated by the dashed lines. (b) Model band structure of sue. However, any nesting-based weak coupling anal- re-hybridized γ+ states showing the proximity of an inter- band A3+ → A1+ back-scattering (red-arrows) to the CDW1 ysis could hardly explain the observed doubling of the wave vector q1 (blue arrow). The colour scale identifies the bands. Therefore, an interpretation of our results can- “hot spot” (c) (Schematic) bonding γ+ and anti-bonding γ− not avoid correlations (due to interactions) that are only orbitals have a different spatial extension. Their amplitude partly taken into account in DFT calculations. is indicated by the strength of turquoise shading. Red and In order to see how interactions and low dimensional- blue color indicates the phase of the involved p-orbitals. The ity could affect the DFT results, we first consider bands orientation of a p-orbital determines the direction in which C2, which are degenerate and nearly commensurate, and the resulting ψγ orbital will be extended. γ+ (γ− ) exhibits hence act as a doublet of 1D chains, prone to a Peierls odd (even) character vs. the σl plane indicated by the dashed distortion. Indeed the C2 dispersion is clearly back- line). folded, an effect of the CDW1 with wave vector q1 , for which a charge distortion is known to be located mostly on C columns23,25,27 . The 1D character of this mental data. Let us denote by ψAi+ (r) the wave func- transition manifests itself in an anomalously large ra- tion corresponding to a DFT band Ai+ . Let us assume a tio (∆1 /kB TCDW 1 ) = 6.757 . The CDW1 transition thus spontaneous dimerization in the ac plane, such transition partitions the rest of the electronic liquid into 1D sub- is permitted since a system of mutually exclusive weak- systems63 located on quartets of A+B columns, where strong bonds is in essence a 2D Ising model, so a discrete the charge is predominantly on column A. Each pair of symmetry is broken upon transition. The transition is A columns acts as a two-leg ladder system, as shown in bringing closer two A-B blocks as in Fig. 4 (a), then the Fig. 19. ψAi (r) orbital will deform, leading to bonding and anti- Such 1D substructure would be at the root of the ob- bonding combinations, denoted γ ± in Fig. 4 (c). These served polarization dependence. A complete calculation, two combinations have very different interactions with taking into account the full ladder structure and its pos- the inner C column, which carries the periodic charge sible instabilities15,64–66 , would be beyond the scope of modulation associated with CDW1. For the bonding the present paper and will be done elsewhere67 . We offer states ψγ + 1,3 (r) the local coupling, due to the emerg- here a plausible scenario, compatible with the experi- ing overlap between ψγ + 1,3 (r) and the electronic orbitals 12 on C columns, should thus enhance inter-band scattering sities in two polarizations (although we use a three-step with large momentum exchange (Fig. 4 (b)). The per- ARPES model, our confidence is strengthened by a recent pendicular deformation is energetically favorable because study [cite] which shows that within a one-step model, it allows to harvest electron-electron interaction energy final state’s effects cannot cause a substantial difference available thanks to the closeness (in momentum space) of at a chosen photon energy) which questions the standard a col.A inter-band instability and col.C induced hot-spot Fermi-liquid scenario and favors the 1D scenario. The in the A3+ dispersion, indicated by red and blue arrows emerging 1D scenario indeed captures all details of our in Fig. 4(b). In the experiment, this manifests itself as a experimental data and is in fact in agreement with previ- significant re-hybridization with the gapped band C2, an ous experimental findings. In particular, the new theory effect clearly visible in Fig. 2(c) and already noticed but allows to connect the re-hybridization of bands C2 and not directly addressed21 . A3+, clearly visible in s-pol (but not in p-pol) and de- One consequence of this scenario is that the 1D CDW2 tected already before21 (therein called ’mysterious’), with formation is accompanied by a dimerization along the c the CDW gap formation, which makes the entire picture axis. An organization in the ac plane with a chessboard consistent. pattern as in Fig. 4 (a) is a natural choice to minimze In summary, we exploited ARPES with sub-µm lat- Coulomb repulsion. It naturally yields the CDW2 pe- eral resolution and polarization control to study the elec- riodicity q2⊥ = ( 12 a∗ , 12 c∗ ), which is hard to justify in tronic states of NbSe3 . We observed a strong dependence a standard nesting scenario. Moreover this ansatz di- of the dispersion on the light polarization, incompatible rectly solves several experimental riddles: the band dou- with a simple anisotropic quasi-1D “Fermi-liquid” like de- bling into A1(3)+ and A1(3)− and their tiny splitting, scription of this material. Our results can be explained the number of electrons that remain ungapped, and the by a self-organization into orbitally distinct electronic presence of gapless carriers close to the Γ point. The systems. One set of electronic states is gapped as evi- hopping between 1D subunits remains incoherent down denced by back-folding of its electronic dispersion. The to very low temperatures. In this case the effective sym- other exhibits one-dimensional character and remains metry plane for the 1D system consisting of a quartet of metallic. The canonically accepted mechanism for the A+B columns must: i) be perpendicular to the c axis, ii) low-temperature CDW2 formation must be replaced by contain the b axis, and iii) cross the center of the quartet. a new scenario where part of the carries couples more The scattering plane σl fulfils these conditions and thus strongly with the preceding CDW1. Experimentally, our the dipole selection rules emerge. The bonding states γ + results demonstrate that pushing the lateral resolution of are by construction delocalized towards column C and ARPES to sub-micron scale can lead to qualitative break- involve p-Se σ-hybridization on the short link (green line throughs. This will be especially fruitful for studies of in Fig. 1 (b)). These states are anti-symmetric with re- novel and strongly correlated materials, that were so far spect to the σl plane (see Fig. 4 (c)) and hence are ex- hindered by by small crystal sizes and inhomogeneities. pected to appear in s−polarization, in agreement with Theoretically, the new scenario for the CDW formation, the experiment. A π-bond component on the short bond, based on self-differentiation leading to enhanced interac- required to keep the total density close to the DFT solu- tion with the preceding CDW, suggests a re-investigation tion, can be absorbed within the anti-bonding state γ − . of other materials where similar sequences of phase tran- These states are symmetric (vs. σl ) and hence visible in sitions are present. These results open the way to new p−polarization. Moreover, since they are mostly local- experiments searching for the emergence of hidden 1D ized in the core of the quartet and interact weakly with dimensional states in strongly correlated materials. CDW1, they are expected to be gapless. This is what we The work at Lausanne and Geneva is supported by the observe in our ARPES data. Swiss NSF. M.A.V. acknowledges financial support from Spanish MICIN Postdoctoral Mobility Program. SP ac- knowledges C’NANO Ile-de-France, DIM NanoK, for the VII. DISCUSSION AND CONCLUSION support of the Nanospecs project. Work in Spain was supported by a MINECO (Grants FIS2015-64886-C5-4-P We wish to emphasize that results of all four preced- and CTQ2015-64579-C3-3-P), Generalitat de Catalunya ing sections has to be taken altogether as a proof of the (2014SGR301) and XRQTC. E.C. and M.A.V acknowl- novel physics present in NbSe3 . In experimental part we edge support from MINECO through the Severo Ochoa pointed out the band back-folding in s-pol spectra (as it Centers of Excellence Program under Grants SEV-2015- facilitates data interpretation), but actually a minimal, 0496 and SEV-2016-0686. 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