Fermi-surface-induced lattice distortion in $\mathrm{Nb}{\mathrm{Te}}_2$

The origin of the monoclinic distortion and domain formation in the quasi-two-dimensional layer compound $\mathrm{Nb}{\mathrm{Te}}_2$ is investigated. Angle-resolved photoemission shows that the Fermi surface is pseudogapped over large portions of the Brillouin zone. Ab initio calculation of the electron and phonon band structure as well as the static RPA susceptibility lead us to conclude that Fermi surface nesting and electron-phonon coupling play a key role in the lowering of the crystal symmetry and in the formation of the charge density wave phase.

Figure 1

Side view of the monoclinic structure of $\mathrm{Nb}{\mathrm{Te}}_2$ [projection onto the (010) plane within the monoclinic space group]. Each layer of $\mathrm{Nb}{\mathrm{Te}}_2$ consists of a Te-Nb-Te sandwich. The Nb atoms (filled circles) are displaced within the plane and form “trimers,” whereas the Te atoms (empty circles) exhibit an out-of-plane buckling (a,b,c). Successive Te-Nb-Te sandwiches are shifted within the plane. The stacking sequence is repeated after three layers.

Figure 2

(a) LEED pattern $\left(75\mathrm{eV}\right)$ of $\mathrm{Nb}{\mathrm{Te}}_2$ at room temperature. (b) Schematic illustration of a single domain $(3\times 1)$ superstructure as observed in reciprocal space. The fat dots indicate the $\Gamma$ points of the undistorted trigonal lattice. Due to the $(3\times 1)$ distortion in real space, two additional spots appear in between these main reflections represented by the small dots. Their spacing is $\frac{1}{3}{a}^{*}$. The bold hexagons outline the surface Brillouin zone of the trigonal $(1\times 1)$ structure. The thin compressed hexagons show the surface Brillouin zone of the $(3\times 1)$ superlattice. (c) Projection onto the basal plane of the first three Te-Nb-Te sandwiches of the monoclinic structure. The size of the circles is proportional to the $z$-coordinate. The lattice vectors of the trigonal $(1\times 1)$ and $(3\times 1)$ cells are indicated. (d) Bulk and surface Brillouin zone for $1T\text{−}\mathrm{Nb}{\mathrm{Te}}_2$ with high symmetry points.

Figure 3

(a) Angular distribution of electrons from $E_F$ mapped as a function of ${\mathbf{k}}_{\Vert }$ at room temperature, white and black correspond to high and low intensity, respectively. (b) Symmetrized and flattened data of (a). (c) Theoretical $\mathrm{APW}+\mathrm{lo}$ Fermi surface contours for the undistorted trigonal $\mathrm{Nb}{\mathrm{Te}}_2$ in the free electron final state approximation (Ref. 16) with photon energy $h\nu =21.2\mathrm{eV}$, workfunction $\varphi =4\mathrm{eV}$ and an assumed inner potential $V_0=13\mathrm{eV}$. Brillouin zones of the $(1\times 1)$ and one particular orientation of the $(3\times 1)$ lattice are superimposed.

Figure 4

Comparison between ARPES and STS data. (a) Room temperature ARPES spectra along $\overline{\Gamma }\overline{\mathrm{M}}$. (b) STM topography of the $(3\times 1)$ phase of $\mathrm{Nb}{\mathrm{Te}}_2$ at $77\mathrm{K}$ ($V=3\mathrm{mV}$, $I=2.9\mathrm{nA}$). (c) STS spectrum at $77\mathrm{K}$ measured with a lock-in amplifier, modulation frequency $1\mathrm{kHz}$, modulation amplitude $30\mathrm{mV}$. The sample-tip distance was chosen in order to have a tunneling current of $0.8\mathrm{nA}$ for a bias voltage of $-1.5\mathrm{V}$.

Figure 5

(a) $\mathrm{APW}+\mathrm{lo}$ band structure along high symmetry directions for $1T\text{−}\mathrm{Nb}{\mathrm{Te}}_2$. (b) Cuts through the Fermi surface at $k_z=0$, $k_z=c^{*}∕4$, $k_z=c^{*}∕2$. Nesting vectors of length $q=1∕3a^{*}$ are indicated.

Figure 6

RPA susceptibility in the first Brillouin zone at $q_z=0$ ($\Gamma \mathrm{M}\mathrm{K}$ plane), $\frac{1}{4}{c}^{*}$ and $\frac{1}{2}{c}^{*}$ (ALH plane) obtained by integration of the $\mathrm{APW}+\mathrm{lo}$ Fermi surface of Fig. 5b. The peaks marked 1a and 1b occur exactly at $q_x=\frac{1}{3}{a}^{*}$. Feature 2 is close to the nesting vector of the $\sqrt{19}\times \sqrt{19}$ CDW phase.

Figure 7

Phonon dispersion of $1T\text{−}\mathrm{Nb}{\mathrm{Te}}_2$ along high symmetry lines obtained by the response function capabilities of ABINIT. Frequencies below $0\mathrm{meV}$ are imaginary. All free degrees within the trigonal $P\overline{3}m1$ space group were relaxed using the Broyden-Fletcher-Goldfarb-Shanno minimization scheme as implemented in the ABINIT code. The optimized LDA lattice parameters underestimate the values derived from Brown’s experimental values (Ref. 3) for the trigonal structure by less then 1%, following the general trend observed for LDA results (Ref. 55). The LDA equilibrium value for $z_{\mathrm{red}}=0.274$ is also in good agreement with the derived averaged value $z_{\mathrm{red}}=0.277$ (see the Appendix).