Dimensional Crossover in a Charge Density Wave Material Probed by Angle-Resolved Photoemission Spectroscopy

High-resolution angle-resolved photoemission spectroscopy data reveal evidence of a crossover from one-dimensional (1D) to three-dimensional (3D) behavior in the prototypical charge density wave (CDW) material ${\mathrm{NbSe}}_3$. In the low-temperature 3D regime, gaps in the electronic structure are observed due to two incommensurate CDWs, in agreement with x-ray diffraction and electronic-structure calculations. At higher temperatures we observe a spectral weight depletion that approaches the power-law behavior expected in one dimension. From the warping of the quasi-1D Fermi surface at low temperatures, we extract the energy scale of the dimensional crossover. This is corroborated by a detailed analysis of the density of states, which reveals a change in dimensional behavior dependent on binding energy. Our results offer an important insight into the dimensionality of excitations in quasi-1D materials.

Figure 1

(a) Schematic crystal structure of ${\mathrm{NbSe}}_3$ (adapted from Ref. [17]). $t_b$ and $t_c$ are the hopping amplitudes along the chains and along the $c$ axis, respectively, as used in the tight-binding model (see text). (b) Fermi surface obtained at 8 K with 31 eV photon energy in the $b^{*}{c}^{*}$ plane. The rectangle shows the first Brillouin zone as in (c). Dotted lines are guides to the eye. (c) DFT Fermi-surface contours in the $b^{*}{c}^{*}$ plane for various momenta covering the full Brillouin zone along $a^{*}$.

Figure 2

(a)–(c) Band dispersion along $k_{\parallel b}$ obtained at 6.5 K with 31 eV photon energy at the marked $k_{\parallel c}$ values. Black bars indicate the expected position of the band bottom based on a cosine dispersion along $k_{\parallel c}$ with bandwidth $4t_c=108\mathrm{meV}$ (see text). (d)–(f) Corresponding second-derivative plots. The position of the CDW gaps are marked in (d).

Figure 3

(a) Enlargement of the data from Fig. 2 including the $b^{*}$-axis component of the $\mathit{q}$ vectors for the two incommensurate CDWs: $q_{1,b^{*}}=0.435$; ${\mathit{q}}_{2,b^{*}}=0.468{\AA }^{-1}$ as measured by x-ray diffraction [18, 19]. (b) Simulation of the spectral function in a CDW system with ordering vectors $q_{1,b^{*}}$ and $q_{2,b^{*}}$ (see text). (c) EDCs of the raw ARPES data at $k_{\parallel b}=0.21$ (blue), $k_{\parallel b}=-0.20$ (green), $k_{\parallel b}=0.25$ (red), and $k_{\parallel b}=-0.24{\AA }^{-1}$ (black). The center of the CDW gaps is marked by dashed lines. All data are taken at $k_{\parallel c}=0$ and $T=6.5\mathrm{K}$. (d) Temperature dependence of the ${\mathit{q}}_1$ gap at $k_{\parallel b}=0.21{\AA }^{-1}$.

Figure 4

(a) Fermi surface obtained at $h\nu =22\mathrm{eV}$, overlaid with the tight-binding model described in the text. (b) $k_{\parallel b}$-integrated ARPES intensity at $k_{\parallel c}=0$ for various temperatures. White dashed lines indicate simple power laws with exponents close to 0.6; black dashed lines show best fits to a TLL model spectral function with $\alpha \approx 0.25$; vertical dashed lines indicate the energies of the CDW gaps seen in Fig. 2. (c) Energy-dependent exponent showing dimensional crossover at $E_C$ for $T>T_1$.