Polarization dependence of angle-resolved photoemission with submicron spatial resolution reveals emerging one-dimensionality of electrons in ${\mathrm{NbSe}}_3$

In materials with nearly commensurate band filling the electron liquid may spontaneously separate into components with distinct properties, yielding complex intra- and interunit cell ordering patterns and a reduced dimensionality. Polarization-dependent angle-resolved photoemission data with submicron spatial resolution demonstrate such an electronic self-organization in ${\mathrm{NbSe}}_3$, 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 three dimensional of all columnar chalcogenides. The 1D physics is evidenced by a new selection rule—in two polarizations we observe two strikingly different dispersions each closely resembling apparently contradicting results of previous studies of this material.

Figure 1

(a) The crystal structure of ${\mathrm{NbSe}}_3$ (monoclinic, $a=10.01\AA , b=3.48\AA , c=15.63\AA , \beta =109.5^{\circ }$) is built out of prismatic columns of three types, hereafter labeled columns A–C, running along the $b$ axis. (b) $ac$ plane cross section. Green (orange) dashed lines indicate strong (weak) $\mathrm{Se}\cdots \mathrm{Se}$ intercolumnar contacts.

Figure 2

(a) Sample holder with several ${\mathrm{NbSe}}_3$ samples. (b) Optical microscope image of ${\mathrm{NbSe}}_3$ whiskers. The scale of the image is given in (c).

Figure 3

$\mathrm{Sub}-\mu \mathrm{m}$ ARPES experimental setup scheme. The incoming light is focused in a $\mathrm{sub}-\mu \mathrm{m}$ region by a zone plate. The sample is mounted on a high-precision 5 degrees of freedom manipulator. Linear horizontal and linear vertical polarized light is available.

Figure 4

(a) Spatially resolved Se $3d$ intensity scan on ${\mathrm{NbSe}}_3$ 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.

Figure 5

ARPES intensity maps ($h\nu =100$ eV; $T=55$ K) measured along the $b^{*}\parallel b$ axis ($\pi /b=0.89\AA$) with (a) $s$ polarization and (b) $p$ polarization. The inset shows the experimental geometry. The $s$ polarization is perpendicular to the long axis (i.e., to the $b$ axis) of the ${\mathrm{NbSe}}_3$ crystalline whiskers. The $p$ polarization has components along the $b$ axis and perpendicular to the $bc$ cleavage plane. (c) and (e) Curvature plots obtained from (a) and (b), respectively. Peak positions determined from EDCs and MDCs are shown in (d) and (f). The lines are guides to the eye.

Figure 6

ARPES intensity maps (raw data; $T=55$ K; $h\nu =100$ eV) for (a) $s$ polarization and (b) $p$ polarization. The collection time was 8 h for each map. (c) and (d) Energy distribution curves (EDCs) extracted from (a) and (b), respectively. Tick marks indicate peak positions of the various spectral features.

Figure 7

Energy-dependent ARPES intensity in the $k>0$ branch of the $\text{A}{1}^{-}$ band, obtained by integrating the corresponding peak in momentum-distribution curves (MDCs) extracted from Fig. 6. The inflection point determines the Fermi level position.

Figure 8

(a) Comparison of curvature plots obtained from data measured with $s$ polarization (green) and $p$ polarization (red). (b) A closeup of (a) near the Fermi level.

Figure 9

(a) Second derivative (with respect to energy) of the ARPES intensity measured with $s$ polarization. (b) Corresponding raw data measured in $s$ polarization. Both (a) and (b) evidence the backfolding resulting from the hybridization of bands $\mathrm{A}{1}^{+}, \mathrm{A}{2}^{+}$, and C2. The gap amplitudes (arrows) can be estimated from the distance between the $E_F$ and the edge profile. (c) Second derivative with respect to energy of the ARPES intensity measured with $p$ polarization. The spectral intensity exhibits a gapless behavior and a linear dispersion with no backfolding evidences. The dashed wriggling line shown on each panel follows the inflection point of the spectral intensity at the vicinity of $E_F$ in the $s$ polarization.

Figure 10

(Left) EDCs measured at $k_{F1}$ with $s$ (blue symbols) and $p$ polarization (red symbols), extracted from the intensity maps of Fig. 6. (Right) Same for $k_{F3}$. The solid lines are obtained by symmetrizing the raw spectra around $E_F$.

Figure 11

MDCs extracted 30 meV below $E_F$ for the two polarizations.

Figure 12

(a) Band dispersion obtained from MDCs fits for $p$ polarization. (b) Results of the MDCs fit superimposed on the ARPES intensity image. The dashed lines indicates the energy profiles for MDCs shown in (c). Dark and light orange features display the double structure composing band $\mathrm{A}{3}^{-}$ related to possible spinon and holon contributions.

Figure 13

Theoretical DFT GGA band calculations including finite $U=4$ eV. The bands can be associated with their experimental counterparts in Fig. 5 by comparing their respective Fermi wave vectors. The energy zero corresponds to the Fermi level. $\mathrm{\Gamma }=(0,0,0), Y=(1/2,0,0), Z=(0,1/2,0), B=(0,0,1/2)$, and $D=(0,1/2,1/2)$ in units of the monoclinic reciprocal lattice vectors.

Figure 14

Band structure for ${\mathrm{NbSe}}_3$ 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.

Figure 15

(a) Calculated Fermi surface for ${\mathrm{NbSe}}_3$ showing the four pairs of warped sheets. (b) Particular sections of the Fermi surface of ${\mathrm{NbSe}}_3$. $\mathrm{\Gamma }=(0,0,0), Y=(1/2,0,0), Z=(0,1/2,0), C=(1/2,1/2,0), B=(0,0,1/2), D=(0,1/2,1/2), A=(1/2,0,1/2)$, and $E=(1/2,1/2,1/2)$ in units of the monoclinic reciprocal lattice vectors.

Figure 16

Mixing of Nb $d$ orbitals on column C yields composite orbitals visible in both polarizations.

Figure 17

(a) $ac$ plane cross section showing schematically the perpendicular dimerization and emergent self-organization. Yellow dots represent col.C where CDW1 takes place. Pairs of red-blue rectangles linked by a thick black line are doublets of A+B columns. The perpendicular dimerization is indicated by the dashed lines. (b) Model band structure of rehybridized $\gamma +$ states showing the proximity of an interband $\text{A}{3}^{+}\to \text{A}{1}^{+}$ backscattering (red arrows) to the CDW1 wave vector ${\mathbf{q}}_1$ (blue arrow). The color scale identifies the “hot spot” (c) (schematic) bonding ${\gamma }_{+}$ and antibonding ${\gamma }_{-}$ orbitals have a different spatial extension. Their amplitude is indicated by the strength of turquoise shading. Red and blue color indicates the phase of the involved $p$ orbitals. The orientation of a $p$ orbital determines the direction in which the resulting ${\psi }_{\gamma }$ orbital will be extended. ${\gamma }_{+}$ (${\gamma }_{-}$) exhibits odd (even) character vs the ${\sigma }_l$ plane indicated by the dashed line.