The Charge Density Wave (CDW) phase is a broken symmetry state widely occurring in condensed matter and coexisting with several other phases, including superconductivity[1]. The electronic instability involved in this transition is induced by electron-phonon coupling which leads to two consequences. First a gap opening appears at the Fermi wave vector with a periodic modulation of the electronic density. From a structural point of view, the metal-insulator transition is associated with a Periodic Lattice Distortion (PLD) of the crystal lattice at the same Fermi wave vector 2k\(_F\).

The CDW phase is highly sensitive to many external excitations such as temperature[2], ultra-short laser pulses[3]–[9] or pressure[10]. CDW materials are also strongly sensitive to chemical pressures[11] or tensile stress [12], [13]. Another property of incommensurate CDW is its ability to slide. Under an external applied field, CDW systems may display an excess of current. This charge transport is highly specific because it is a pulsed and collective mode which occurs at the macroscopic scale and leads to electronic noise oscillations, commonly referred to as Narrow-Band Noise[14], [15].

The sliding phenomenon is well observed by transport measurements but it is also observed by diffraction since the transport of charges is associated to deformations of the PLD. Two types of deformations were observed by diffraction, along and tranverse to the applied field. A change of the \(2k_F\) norm was observed at the threshold current close to electrical contacts [16], [17], while shear was observed through a change in the transverse width of the satellite peak[18], [19]. All these measurements were performed in the NbSe\(_3\) model system. In the TbTe\(_3\) system studied here, the situation is more complex. TbTe\(_3\) stabilizes also an incommensurate CDW which is able to slide[20] and the slding state is associated with a PLD deformation[21]. However, deformations in TbTe\(_3\), although very small in amplitude, include the three directions of the lattice and mainly the tranverse direction along b\(^*\).

These deformations and the excess of current are well described by a time and space dependent phase \(\phi(\vec{r},t)\) of the periodic modulation \(\cos(2 k_F x+\phi(\vec{r},t))\). The additional current \(j\) is related to temporal derivative \(\frac{\delta\phi(t)}{\delta t}\), while CDW deformations are related to the spatial one, including the longitudinal (\(\delta\phi/\delta x\)) and the transverse (\(\delta \phi/\delta y\)) deformation. In both cases, PLD deformations and the excess current alter the band structure. However, the band deformation is profoundly different in the two cases: the former changes the k\(_F\) value while preserving k-symmetry, whereas the latter breaks k-space symmetry (see Fig.1). In the case of a longitudinal deformation \(\delta\phi/\delta x\), corresponding to an expansion or a contraction of the CDW period with a change in \(2k_F\) and therefore to a change of the band filling (see Fig. 1b-c). In that case, the modification of the dispersion curve preserves the symmetry in k-space since any spatial deformation do not favour one direction over another. On the other hand, the existence of an additional current \(j\) across the crystal essentially breaks the band symmetry by favouring electrons along the direction of the applied field (see Fig. 1d). The band structure is shifted of \(\delta k\) due to the additional momentum \(\delta p=\hbar \delta k=m^* v_s\), where \(v_s\) is the sliding velocity,

\[v_s=\frac{1}{2k_F}\frac{\delta\phi}{\delta t}\]

Furthermore, the band structure is also shifted in energy. In the simple case of free like electron dispersion curve (\(\epsilon=\hbar^2k^2/2m^*\)), the shift in energy is [22]: \[\delta \epsilon=\hbar k v_s.\] In fact, the shift in energy depends of the Fermi velocity \(v_F\). The case where the curvature of the band clsoe to the gap openning is taken into account is treated in [23].

In any case, if the signature of the siding state through the band structure should be present, it should be weak due to the orders of magnitude involved, making observation difficult by ARPES. However, in an attempt to observe it, the TbTe\(_3\) system has been chosen because it displays a large gap (\(\delta\approx\) 230 meV)[24] as well as sliding properties[20]. In this work, we will present experimental measurements compatible with the onset of sliding CDW.

TbTe\(_3\) is a layered quasi-2D crystal stacked along the \(b\)-crystal axis which undergoes a CDW transition at 335 K with 2k\(_F\)=5/7 [\(c^*\)] along the \(c\)-axis. The TbTe\(_3\) sample was studied by ARPES at the ANTARES beamline of SOLEIL synchrotron at a photon energy of 30 eV, at room temperature and low pressure \(\sim5 \times10^{-10}\) mbar. The sample was cleaved inside the ultra-high vacuum (UHV) chamber. The analyser slit was parallel to the \(c\)-axis, first determined from inhouse X-ray diffraction measurements. The beam size was 100\(\mu m^2\) and located far from the electrodes to avoid any perturbation. The threshold current (I\(_s\)=30mA) has been checked before and after the experiment and the Fermi surface and the electronic band dispersions has been measured versus applied currents, below and above the threshold current. Fermi Surfaces (FS) are determined by integrating the photoemission signal in a 20 meV integration window across E\(_F\). A Shirley background has been subtracted from the line profiles drawn on the Energy Distribution Curves (EDCs) for quantitative analysis. The energy reference was selected the Fermi level determined from the inflection point of the Fermi step.

The Fermi Surface is presented in Fig. 2a without current. The investigations to observe tiny band structure modifications have been studied at \(k_x = -0.192\) Å\(^{-1}\) (red dotted line in Fig. 2a) located far enough from intense bands. The corresponding surface sheet is shown without current in Fig. 2 (b). A certain asymmetry of the bands is observed between k\(>\)​0 and k\(<\)​0 because of photoemission matrix elements and a slight sample misorientation during measurements. Insets in Fig. 2(b) are shown for a better visibility of the CDW-induced band gaps.

The Fermi surface at different applied currents below and threshold current I\(_s\), negative and positive, are shown in Fig.3. As expected, the overall FS exhibits only minor changes. Fig. 4 thus shows close-ups of the gapped regions for 0 and \(\pm 70\) mA currents in order to appreciate if there is any displacement of the band onset with respect to E_F. The energy band offset shift can be appreciated at the intensity contour plots in panel \(b\) and the k-shift in panel \(a\) by comparing to the red line highlighting the band edge at zero current. It can be appreciated that (i) positive currents promote an upward energy shift at the left band (k\(<\)​0) and an opposite downward shift in the right band (k\(>\)​0), (ii) For negative currents, the opposite behavior is observed and (iii) when the right band shifts to higher binding energies at positive current, it exhibits a concomitant \(\Delta k <0\) shift. All these changes are small and close to our experimental resolution but seem to be compatible with expected results.

In order to quantify the energy shift under applied current, Fig. 5 (a-b) shows the EDCs indicated by the lines in Fig. 2 (b) at \(k_z = \pm0.264\) Å\(^{-1}\). The fitted peak energies as a function of applied current are shown in Figures 5 (c) and 5 (d). At \(k<0\) (left side of the \(\Gamma\) point), the two visible bands in the gap region make difficult a very precise determination. At k\(>\)​0, the intensity of one of the two bands is significantly reduced due to the photoemission matrix elements. However, a small but systematic energy shift is observed, where the band shifts to higher binding energies for positive current to lower binding energies for negative current. The estimated energy shift between the \(+70\) mA and \(-70\) mA is \(\sim\)​12 meV.

A global energy shift is induced by the applied electric field. For a comparison between currents, the EDCs have been rescaled from the inflection point of the Fermi step. For this, we have analyzed the energy shift of a reference band not affected by a CDW gap (Fig. 4 (e-f) corresponding to the dashed lines in fig. 2b). Fig. 4e-f shows the energy shift of the reference bands (k\(<\)​0 and k\(>\)​0) as a function of the applied current. The energy shift is \(\sim 2\) meV and much smaller than the energy shift of the CDW gapped band. This shows that the electric field is not affecting all the bands but preferable the bands involved in the CDW formation.

In conclusion, we have investigated the deformation of the band structure under electrical current predicted by the Fröhlich theory for a Charge Density Wave (CDW) phase through an analysis of the Fermi surface and the electronic band dispersion of TbTe₃ measured by ARPES. A slight energy shifts of the CDW band gaps as a function of applied current were observed, which reverse when the current is reversed. This effect has to be confirmed but seems to be in agreement with expected behavior of band structure in the slding CDW regime.

Acknowledgements

The authors would like to thank Véronique Brouet and Gilles Abramovici for useful discussions.

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