A Raman study of the Charge-Density-Wave State in A$_{0.3}$MoO$_3$ (A = K,Rb)

We report a comparative Raman spectroscopic study of the quasi-one-dimensional charge-density-wave systems \ab (A = K, Rb). The temperature and polarization dependent experiments reveal charge-coupled vibrational Raman features. The strongly temperature-dependent collective amplitudon mode in both materials differ by about 3 cm, thus revealing the role of alkali atom. We discus the observed vibrational features in terms of charge-density-wave ground state accompanied by change in the crystal symmetry. A frequency-kink in some modes seen in \bb between T = 80 K and 100 K supports the first-order lock-in transition, unlike \rb. The unusually sharp Raman lines(limited by the instrumental response) at very low temperatures and their temperature evolution suggests that the decay of the low energy phonons is strongly influenced by the presence of the temperature dependent charge density wave gap.


I. INTRODUCTION
Quasi-one dimensional (1D) metals are interesting materials because of their tendency to undergo a phase-transition inevitably due to the inherent instability called the Peierls instability [1]. This instability, originating from the electron-phonon interaction leads to a metal-insulator transition at reduced temperature, where the insulating state may be characterized by a lattice deformation in conjunction with a charge modulation and the opening of a single particle excitation gap in the electronic spectrum. [2] One of the striking features of these materials are their unusual non-linear electrical conduction properties [3][4][5], which arises due to the collective motion of the charge-density wave (CDW). Along with the conventional single particle excitations, these materials also have collective excitations, viz., amplitudons and phasons, which dominate the low energy physical properties [3]. The Raman active collective amplitudon mode, i.e. the transverse oscillation of the amplitude of the coupled charge-lattice wave, has been observed by various researchers [6,7].
Blue bronze, A 0.3 MoO 3 (A = alkali metal) is one of the best known quasi-1D materials which undergoes a CDW transition [3,6,8] via the Peierls channel. The first evidence of the CDW transition in K 0.3 MoO 3 by Raman measurements came from the experiments by Travaglini and Wachter [6]. In this paper a mode around 50 cm −1 was observed at temperatures below T CDW = 180 K that showed a strong temperature dependence of, in particular, its frequency when the temperature approached TCDW from below. As it was also found that this is a fully symmetric (A g -mode) breathing mode it was assigned to the amplitude oscillation of the charge density wave. Further evidence came from the observation that the line width tends to diverge as the T CDW was approached from below. This anomalous behaviour of the line width could be due to the fluctuation effects that become pronounced in the vicinity of the phase transition. Mean-field theory of a generalized second-order phase transition predicts that, due to a temperature dependent order parameter, the frequency (and the amplitude) of the soft mode tends to zero as the temperature is approaches T CDW . [2] However, in the the study by Travaglini et al. [6] the softening of the amplitudon mode with increasing temperature was found to be incomplete. The observed softening of only ∼13 % was ascribed to 3D-correlations. The 3D-correlations are nothing but the long range Coulomb interaction between density-modulated electrons, which might effectively screen the electron-phonon interaction [9,10], thereby reducing the expected complete mode-softening. The effects of strong local electron-electron interaction leads to both dynamical and static screening processes dressing the electron-phonon coupling. These screening effects are thought to be a generic feature of strongly correlated electron systems.
Recently a Raman study of the anomalous profile of the amplitudon mode in K 0.3 MoO 3 was reported [11] in which for temperatures below 100 K the amplitudon mode showed an anomalous "fin"-like profile toward the Stoke's side of it's central frequency. This was interpreted as splitting of the amplitudon mode into two or more closely spaced harmonic oscillator type modes. The splitting was claimed to be due to a strong perturbation of At the charge density wave transition temperature the structure transforms in an incommensurately modulated structure with a temperature dependent incommensurate wave vector q ic close to (but not equal to) 0.25b * + 1 2 c * . [13,14] At low temperature (T < 100 K) the incommensurate wave vector becomes temperature independent, even though this does not seem to correspond to a lock in transition, i.e. the modulation remains incommensurate. [14] K 0.3 MoO 3 and Rb 0.3 MoO 3 single crystals with typical size 5 × 3 × 0.5 mm 3 were prepared by the temperature gradient flux method [15]. After polishing, samples were mounted in an optical flow cryostat (temperature stabilization better than 0.1 K) and backscattering Raman spectra were recorded (frequency resolution ∼2 cm −1 ) using a triple grating monochromator in subtractive mode equipped with a liquid nitrogen cooled CCD detector.
A solid-state diode-pumped, frequency-doubled Nd : Y V O 4 laser system is used as excitation source (wavelength 532 nm, spot size 10 µm). The power density was kept below 100 W/cm 2 in order to minimize heating effects. The polarization was controlled on both the incoming and outgoing beam, giving access to all the polarizations allowed for by the back-scattering configuration. In general both parallel ( (xx), where x is in the [20-1] direction, and (bb) in Porto notation) and perpendicular polarizations have been measured. The perpendicular spectra did not yield appreciable scattering intensity and will therefore not be discussed here.  [16,17], and is consistent with the interpretation of the Raman spectra of pure MoO 3 [18]. There are quite a few different modes originating from the MoO 6 units, due to the large distortion of the MoO 6 "octahedra"; the metal-oxygen distances in blue bronzes typically range from 1.7 to 2.3Å [13]. The Mo-O stretching modes appear at higher frequencies (between 900 to 1000 cm −1 ) [18] as reported by Shigeru et al., [19] and Massa [20]. In general, the frequencies for the modes observed in Rb 0.3 MoO 3 are somewhat lower than the corresponding modes in K 0.3 MoO 3 . This is consistent with the fact that Rb 0.3 MoO 3 has slightly larger lattice constants in the a and c directions [13,21] resulting from the larger ionic radius of the rubidium ion. A small variance in the various Mo-O distances between K 0.3 MoO 3 and Rb 0.3 MoO 3 has also been observed in the study of Schutte et al., [13]. Finally, note that group theory predicts 71 A g modes for the high temperature phase of the bronzes. The experimental spectra, however, reveal only 35 modes. The "missing" modes most likely result from accidental degeneracies and from weak intensity and/or screening, in particular in the metallic direction. For the predicted 55 B g modes ((xb) geometry) the intensity is to weak to be detected in the present experiments.
The low temperature phase shows a much richer spectrum than the high temperature  compound, which is exemplified by the right inset of Fig. 3 for the 480 cm −1 mode. As a general rule, splitting of Raman modes occurs mainly due to two mechanisms. The first one occurs when a static distortion transforms equivalent atomic positions to inequivalent ones [23]. A second mechanism is the correlation field or Davydov [23][24][25] splitting that occurs due to coupling of vibrations of molecular units at different equivalent sites in the unit cell, and is sensitive to transitions involving multiplication of the unit cell size (like for instance the Peierls distortion). The incommensurate nature of the blue bronzes may also activate vibrational modes in Raman spectra [26]. Modes with wave vector k= nq become active in the incommensurate phase, where q is the incommensurate modulation vector, and their scattering strength depends strongly on the modulation amplitude.

IV. TEMPERATURE DEPENDENT RAMAN SPECTRA
The temperature dependence of the low frequency modes of both compounds is depicted in Fig. 4 for a few selected temperatures. As mentioned before, in K 0. an abrupt change in their energy in the between 80 K and 100 K. This is exemplified in Fig. 5. Clearly some of the modes show a discontinuous shift to higher energy of a few wave numbers indicating a phase transition. It should be noted that not all modes exhibit the frequency-'kink', as can for instance be seen for the 75 cm −1 mode in Fig.5. In general, a temperature-dependent frequency-jump is a signature of a first-order transition. In the present case, it could be due to a lock-in transition to a commensurate state. However, the low temperature phase of K 0.3 MoO 3 has been studied in quite some detail and temperature dependent neutron scattering experiments [14] failed to show a lock-in transition in K 0.3 MoO 3 . It was found that the incommensurate wave vector is strongly temperature dependent down to T = 100 K. Below this temperature the wave vector becomes temperature independent, but remains incommensurate. Still, the present study strongly indicates that there is indeed a first order phase transition in K 0.3 MoO 3 below 100 K. The most salient feature in the low frequency spectra is the appearance of the amplitudon in both compounds which shows a softening and broadening upon approaching the phase transition (see Fig.4). The low temperature frequencies of these modes are 56 cm −1 in the K 0.3 MoO 3 compound, and 53 cm −1 in Rb 0.3 MoO 3 . The interpretation of these modes as amplitudon modes is consistent with the previous neutron study by Pouget et al., [22], and Raman scattering studies by Travaglini et al., [6] and Massa [20]. Recently this mode has also been observed in K 0.3 MoO 3 as coherent excitations in pump-probe transient reflectivity experiments [27,28], in addition to coherent phonon excitations [28] at 74 cm −1 , and 85 cm −1 also observed in the present data. The amplitudon modes disappear from the spectra already around 150 K. This is due to the strongly over damped nature of the amplitudon in the vicinity of the phase transition to the metallic state originating from strong fluctuations of the charge density wave. These fluctuations lead to a diverging line width (see Fig. 6, right panel), which in turn makes it impossible to observe complete amplitude softening.
The observable amplitudon softening upon approaching the phase transition amounts only about 10 %, as is shown in Fig. 6 (left panel).
Finally, turn back to the phonon modes observed in the CDW phase. The phonon modes observed at low temperatures are rather narrow, and often resolution limited. This is in particular true for the phonons below 400 cm −1 .  The observed anomalous line broadening hints to a coupling of the phonons to electronic excitations with an energy above ∼350 cm −1 , which, given the temperature dependence, scale with the CDW gap. Since the CDW energy gap in K 0.3 MoO 3 is about 900 cm −1 it can not be a coupling to the usual quasi-particle excitations. It has been argued in the literature that a common feature of incommensurate CDW materials is the presence of so called midgap states [29]. Therefore the anomalous broadening of the phonons above ∼350 cm −1 is tentatively assigned to a coupling of the phonons to midgap state excitations.

V. SUMMARY
In summary, we have studied two representatives of the prototypical quasi-onedimensional charge density wave system, blue bronze, using temperature and polarization to perform additional studies to confirm the present interpretation. This also holds for the observed changes in the K 0.3 MoO 3 spectra between 80 and 100 K, which indicate a first order phase transition. This is in line with the recently discussion of the maximum in the threshold field for CDW conduction which was interpreted as due to a incommensuratecommensurate phase transition (lock-in transition). [30] This transition is not related to the recently reported observation of a glass transition in K 0.3 MoO 3 by Staresinic et al. [31] at T g ∼ 10 K, for which no further evidence was found in the present study. Finally, evidence is found for a coupling of the high frequency phonons to mid-gap excitations.