Advances in Condensed Matter Physics

Volume 2019, Article ID 2138264, 14 pages

https://doi.org/10.1155/2019/2138264

## Survey of the Thermodynamic Properties of the Charge Density Wave Systems

^{1}Université Grenoble Alpes, Institut Néel, F-38042 Grenoble, France^{2}CNRS, Institut Néel, F-38042 Grenoble, France

Correspondence should be addressed to M. Saint-Paul; rf.srnc.leen@luap-tnias.lehcim

Received 23 October 2018; Revised 19 December 2018; Accepted 20 January 2019; Published 3 March 2019

Academic Editor: Sergio E. Ulloa

Copyright © 2019 M. Saint-Paul and P. Monceau. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We reexamine the thermodynamic properties such as specific heat, thermal expansion, and elastic constants at the charge density wave (CDW) phase transition in several one- and two-dimensional materials. The amplitude of the specific heat anomaly at the CDW phase transition* T*_{CDW} increases with increasing* T*_{CDW} and a tendency to a lineal temperature dependence is verified. The Ehrenfest mean field theory relationships are approximately satisfied by several compounds such as the rare earth tritelluride compound TbTe_{3}, transition metal dichalcogenide compound 2H-NbSe_{2}, and quasi-one-dimensional conductor K_{0.3}MoO_{3}. In contrast inconsistency exists in the Ehrenfest relationships with the transition metal dichalcogenide compounds 2H-TaSe_{2} and TiSe_{2} having a different thermodynamic behavior at the transition temperature* T*_{CDW}. It seems that elastic properties in the ordered phase of most of the compounds are related to the temperature dependence of the order parameter which follows a BCS behavior.

#### 1. Introduction

The electron density of a low dimensional (one-dimensional (1D) or two-dimensional (2D)) compound may develop a wavelike periodic variation, a charge density wave (CDW), accompanied by a lattice distortion when temperature drops below a critical temperature* T*_{CDW} [1–46]. CDW ordering is driven by an electron phonon coupling. The concept of charge density wave is related to the initial work of Peierls [1], followed by Fröhlich [2] when it was demonstrated that a one-dimensional metal is instable with respect to a phase transition in the presence of electron phonon coupling.

A charge density wave is characterized by a spatial periodic modulation of the electronic density concomitant with a lattice distortion having the same periodicity. The properties of the CDW state can be described by an order parameter [1]. The fluctuations of the lattice distortions can be described by amplitude and phase modes [1]. This variation, charge density wave, in the electron density is receiving intense study because it often competes with another ground state (superconductivity). A CDW order can be formed with one fixed wave vector or multiple wave vectors. For example, the incommensurate ordering vector* Q*_{1} of the prototypal rare earth tritelluride ErTe_{3} at the upper CDW phase transition* T*_{CDW1} = 265 K is parallel to the axis, whereas the incommensurate ordering parameter* Q*_{2} observed at the lower CDW phase transition* T*_{CDW2} = 150 K with ErTe_{3} is parallel to the axis. In contrast the CDW order in the dichalcogenide compounds (for example, 2H-NbSe_{2}) is formed by three superposed charge density waves.

The origin of the CDW phase transition observed in the two-dimensional materials is still not completely settled [5]. Two alternatives have been proposed for describing the nature of the CDW in the family of rare earth tritelluride RTe_{3} (R=rare earth element) which represents a charge density model. Based on ARPES measurements [10, 11], one describes it in terms of Fermi surface nesting following the electron Peierls scheme. The other one emphasizes the role of the strongly momentum dependent electron phonon coupling as evidenced from inelastic X-ray scattering [13] and Raman [7, 14] experiments. As the electron phonon coupling is increased the importance of the electronic structure in* k* space is reduced.

Study of the thermodynamic properties of the charge density wave phase transition in two-dimensional transition metal dichalcogenide compounds [16–25] and in quasi-one-dimensional conductors [26–37] has generated a considerable interest over the past 30 years. The onset of the CDW order has remarkable effects on the thermodynamic properties since below* T*_{CDW} a gap opens up in the density of the electronic states. A microscopic model is given by McMillan [28]. The elastic properties of quasi low dimensional conductors undergoing charge and spin density phase transitions are reviewed by Brill [3]. Several reviews discuss the properties of the charge density wave systems [4–6, 45].

We reexamine the thermodynamic experimental data such as specific heat, thermal expansion, and elastic constants of several CDW compounds. We give a survey of the Ehrenfest relations using the experimental data obtained at the CDW phase transition in different materials: rare earth tritellurides RTe_{3} (TbTe_{3}, ErTe_{3}, and HoTe_{3}) [8, 41–43], transition metal dichalcogenides MX_{2} compounds (2H-NbSe_{2} [17–19], 2H-TaSe_{2} and 2H-TaS_{2} [16, 24, 25], and TiSe_{2}[20–23]), quasi-one-dimensional conductors (NbSe_{3} [25, 27], K_{0.3}MoO_{3} [30–33], (TaSe_{4})_{2}I [39, 40], and TTF-TCNQ [35–38]), and in the system (LaAgSb_{2}) [44, 45].

Departures from the mean field behavior of the thermodynamic properties are generally attributed to fluctuations which belong to the 3D XY criticality class [27–34]. The contribution of the fluctuations is important in the quasi-one-dimensional conductors [28] and in the transition metal dichalcogenides (2H-TaSe_{2}, 2H-TaS_{2}) [24]. Small fluctuation effects are observed around* T*_{CDW} in the rare earth tritellurides TbTe_{3} [41] and ErTe_{3} [42].

The amplitude of the lattice distortion is governed by the electron phonon coupling strength [46]. A moderately strong electron phonon coupling is reported for the rare earth tritellurides (ARPES experiments [10, 11]), similar to that observed in quasi-1D CDW systems such as K_{0.3}MoO_{3} and NbSe_{3}. In a weak coupling CDW, the specific heat behavior at the CDW phase transition is driven by the electronic entropy [28, 46]. In a strong coupling CDW the transition is also governed by the entropy of the lattice [28, 46].

#### 2. Thermodynamic Properties

##### 2.1. Ehrenfest Relations

At a second-order phase transition* T*_{C}, the order parameter* Q* increases continuously in the ordered phase at . The Landau free energy [47] can be written without knowing the microscopic states as where* F*_{0} describes the temperature dependence of the high temperature phase and the constant parameters a and B are positive. The order parameter that minimizes the free energy () is given byThe entropy (S) is derived from the free energy (F), , and the specific heat at constant pressure is given by . There is a jump in the specific heat (Figure 1(a)) at the second-order phase transition* T*_{C} given by [47]Discontinuities in the thermal expansion coefficients and the elastic constants are also observed at a second-order phase transition. An example (TbTe_{3}) is shown in Figures 1(b) and 1(c). The thermodynamic quantities at a second-order phase transition such as a charge density wave phase transition are generally discussed with the Ehrenfest relations reformulated by Testardi [48]. The discontinuity in the thermal expansion coefficients is related to the specific heat jump* ∆C*_{P} and to the stress dependence components, , at the phase transition* T*_{CDW}:where i=1, 2, and 3 correspond to the , , and crystallographic axes directions and* V*_{m} is the molar volume.