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

Authors: M. Saint-Paul; P. Monceau
Journal: Advances in Condensed Matter Physics
 (2019)

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## 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 transitionTCDW increases with increasingTCDW 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 TbTe3, transition metal dichalcogenide compound 2H-NbSe2, and quasi-one-dimensional conductor K0.3MoO3. In contrast inconsistency exists in the Ehrenfest relationships with the transition metal dichalcogenide compounds 2H-TaSe2 and TiSe2 having a different thermodynamic behavior at the transition temperatureTCDW. 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.


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## Body

## 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 temperatureTCDW [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 parameterQexp⁡(iΦ) [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 vectorQ1 of the prototypal rare earth tritelluride ErTe3 at the upper CDW phase transitionTCDW1 = 265 K is parallel to the c→ axis, whereas the incommensurate ordering parameterQ2 observed at the lower CDW phase transitionTCDW2 = 150 K with ErTe3 is parallel to the a→ axis. In contrast the CDW order in the dichalcogenide compounds (for example, 2H-NbSe2) 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 RTe3 (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 ink 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 belowTCDW 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 RTe3 (TbTe3, ErTe3, and HoTe3) [8, 41–43], transition metal dichalcogenides MX2 compounds (2H-NbSe2 [17–19], 2H-TaSe2 and 2H-TaS2 [16, 24, 25], and TiSe2[20–23]), quasi-one-dimensional conductors (NbSe3 [25, 27], K0.3MoO3 [30–33], (TaSe4)2I [39, 40], and TTF-TCNQ [35–38]), and in the system (LaAgSb2) [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-TaSe2, 2H-TaS2) [24]. Small fluctuation effects are observed aroundTCDW in the rare earth tritellurides TbTe3 [41] and ErTe3 [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 K0.3MoO3 and NbSe3. 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 transitionTC, the order parameterQ increases continuously in the ordered phase at T<TC. The Landau free energy [47] can be written without knowing the microscopic states as (1)F=F0+aT-TCQ2+BQ4,whereF0 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 (∂F/∂Q=0) is given by(2)Q2=aTCDW2B1-TTCDW.The entropy (S) is derived from the free energy (F), S=-∂F/∂T, and the specific heat at constant pressure is given by CP=T∂S/∂TP. There is a jump in the specific heat (Figure 1(a)) at the second-order phase transitionTC given by [47](3)ΔCP=a2TC2B.Discontinuities in the thermal expansion coefficients and the elastic constants are also observed at a second-order phase transition. An example (TbTe3) 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 αi is related to the specific heat jump∆CP and to the stress dependence components, ∂TCDW/∂σi, at the phase transitionTCDW:(4)Δαi=-ΔCPVmTCDW∂TCDW∂σi,where i=1, 2, and 3 correspond to the a→, b→, and c→ crystallographic axes directions andVm is the molar volume.Figure 1
Color on line. Temperature dependence of the thermodynamic properties of TbTe3. (a) Specific heat measurements taken from [41]. (b) Relative change of the velocity (blue), ultrasonic attenuation (red), and fit (dashed black curve) taken from [41]. (c) The lattice parameter c measured by X-rays taken from [8].
(a) (b) (c)The elastic constant componentCii is related to the elastic velocity Vii by Cii=ρVii2, ρ being the mass density. The discontinuities of the elastic constants ∆Cii/Cii (or velocity ∆Vii/Vii=∆Cii/2Cii) at a second-order phase transition are related to the stress dependence ∂TCDW/∂σi by(5)ΔViiVii=-ρVii2ΔCP2VmTCDW∂TCDW∂σi2.The term, ρVii2ΔS∂2TCDW/∂σi2, proportional to the entropy variation and multiplied by the second derivative ∂2TCDW/∂σi2 [3, 31], is neglected in (5). Isothermal elastic constants must be used in (5). But the adiabatic elastic constants are measured in the MHz range and the adiabatic values are generally used in (5).From (4) and (5) (6)ΔViiVii=-ρVii2VmTCDW2ΔCPΔαi2.Thus the discontinuities in the elastic velocities are proportional to the square of the discontinuities in the expansion coefficients. Typical discontinuities of the specific heat, thermal expansion coefficient, and elastic velocity at the charge density wave transition are shown in Figure 1. The discontinuities of the elastic constants atTCDW are evaluated using the extrapolated linear temperature dependence of the high temperature background as shown in Figures 1(a) and 1(b).


### 2.2. Elastic Constants

CDW materials acquire lattice distortions that are incommensurate with the basic lattice. They form part of a wider field of interest developed in the incommensurate structures [49, 50]. Incommensurate structures may arise with insulators as K2SeO4 [51]. The structural changes are characterized by a distortion whose wave vector cannot be expressed by a rational fraction of the lattice vector. The resulting ordered phase is not strictly crystalline and is described by an incommensurate phase.The amplitude of the modulation increases continuously as the temperature is lowered. The relationship between the crystalline and the modulated phases can be formulated in the framework of the Landau theory [28]. In some materials, as 2H-TaSe2, the modulation periodicity is temperature dependent and may be lock-in at low temperatures to a value that is commensurate with the periodicity of the basis structure [28, 46]. The lock-in transition is a first-order phase transition and very different in nature from the incommensurate instability [46]. 2H-TaSe2 undergoes a normal to incommensurate transition (second-order) at 122 K and an incommensurate-commensurate transition (first-order) at 90 K [16]. The transition to the incommensurate structural phase is reflected in the elastic stiffness components analyzed in [51, 52].In order to explain the stiffening of the elastic constants (velocities) in the ordered phase below the incommensurate structural phase transition, a first approach based on the analysis of the entropy variation around the CDW phase transitionTCDW is proposed in [3]. A second approach was developed by Rhewald [51] based on the Landau phenomenological theory including the interaction between the strain componentsei and the square of the order parameterQ [51, 52]. The expansion of the free energy density in power ofQ2 andei is developed in agreement with the symmetry point group of the material [51, 52].In the orthorhombic symmetry, for example, the free interaction energy is given by(7)Fcei,Q=g1e1+g2e2+g3e3Q2+12∑i,j3hijeiej+∑i=46hiiei2Q2,where g and h are the coupling constants.The interacting terms linear inei and quadratic inQ as gieiQ2 are responsible for a decrease of the longitudinal elastic constantCii (velocityVii). The decrease of the longitudinal elastic constantCii is proportional to the square of the coupling constant gi.:(8)∆C11~-gi2∆C22~-g22∆C33~-g32.The coupling second terms ei2Q2 in (7) show that several elastic constants Cij (or velocities) follow the temperature dependence of the square of the static value of the order parameter Q in the ordered phase belowTCDW [51, 52]:(9)∆Cij~hijQ2forij=11,22,33,44,55,66.The temperature dependence of the sound velocity and the amplitude of the superlattice reflections gives directly the temperature dependence of the order parameter [1].This general behavior has been observed at the CDW phase transitionTCDW in different materials [16, 17, 20, 36, 37, 41–43]. The hardening observed in the ordered phase with several compounds is analyzed in Section 3.5.


## 2.1. Ehrenfest Relations

At a second-order phase transitionTC, the order parameterQ increases continuously in the ordered phase at T<TC. The Landau free energy [47] can be written without knowing the microscopic states as (1)F=F0+aT-TCQ2+BQ4,whereF0 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 (∂F/∂Q=0) is given by(2)Q2=aTCDW2B1-TTCDW.The entropy (S) is derived from the free energy (F), S=-∂F/∂T, and the specific heat at constant pressure is given by CP=T∂S/∂TP. There is a jump in the specific heat (Figure 1(a)) at the second-order phase transitionTC given by [47](3)ΔCP=a2TC2B.Discontinuities in the thermal expansion coefficients and the elastic constants are also observed at a second-order phase transition. An example (TbTe3) 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 αi is related to the specific heat jump∆CP and to the stress dependence components, ∂TCDW/∂σi, at the phase transitionTCDW:(4)Δαi=-ΔCPVmTCDW∂TCDW∂σi,where i=1, 2, and 3 correspond to the a→, b→, and c→ crystallographic axes directions andVm is the molar volume.Figure 1
Color on line. Temperature dependence of the thermodynamic properties of TbTe3. (a) Specific heat measurements taken from [41]. (b) Relative change of the velocity (blue), ultrasonic attenuation (red), and fit (dashed black curve) taken from [41]. (c) The lattice parameter c measured by X-rays taken from [8].
(a) (b) (c)The elastic constant componentCii is related to the elastic velocity Vii by Cii=ρVii2, ρ being the mass density. The discontinuities of the elastic constants ∆Cii/Cii (or velocity ∆Vii/Vii=∆Cii/2Cii) at a second-order phase transition are related to the stress dependence ∂TCDW/∂σi by(5)ΔViiVii=-ρVii2ΔCP2VmTCDW∂TCDW∂σi2.The term, ρVii2ΔS∂2TCDW/∂σi2, proportional to the entropy variation and multiplied by the second derivative ∂2TCDW/∂σi2 [3, 31], is neglected in (5). Isothermal elastic constants must be used in (5). But the adiabatic elastic constants are measured in the MHz range and the adiabatic values are generally used in (5).From (4) and (5) (6)ΔViiVii=-ρVii2VmTCDW2ΔCPΔαi2.Thus the discontinuities in the elastic velocities are proportional to the square of the discontinuities in the expansion coefficients. Typical discontinuities of the specific heat, thermal expansion coefficient, and elastic velocity at the charge density wave transition are shown in Figure 1. The discontinuities of the elastic constants atTCDW are evaluated using the extrapolated linear temperature dependence of the high temperature background as shown in Figures 1(a) and 1(b).


## 2.2. Elastic Constants

CDW materials acquire lattice distortions that are incommensurate with the basic lattice. They form part of a wider field of interest developed in the incommensurate structures [49, 50]. Incommensurate structures may arise with insulators as K2SeO4 [51]. The structural changes are characterized by a distortion whose wave vector cannot be expressed by a rational fraction of the lattice vector. The resulting ordered phase is not strictly crystalline and is described by an incommensurate phase.The amplitude of the modulation increases continuously as the temperature is lowered. The relationship between the crystalline and the modulated phases can be formulated in the framework of the Landau theory [28]. In some materials, as 2H-TaSe2, the modulation periodicity is temperature dependent and may be lock-in at low temperatures to a value that is commensurate with the periodicity of the basis structure [28, 46]. The lock-in transition is a first-order phase transition and very different in nature from the incommensurate instability [46]. 2H-TaSe2 undergoes a normal to incommensurate transition (second-order) at 122 K and an incommensurate-commensurate transition (first-order) at 90 K [16]. The transition to the incommensurate structural phase is reflected in the elastic stiffness components analyzed in [51, 52].In order to explain the stiffening of the elastic constants (velocities) in the ordered phase below the incommensurate structural phase transition, a first approach based on the analysis of the entropy variation around the CDW phase transitionTCDW is proposed in [3]. A second approach was developed by Rhewald [51] based on the Landau phenomenological theory including the interaction between the strain componentsei and the square of the order parameterQ [51, 52]. The expansion of the free energy density in power ofQ2 andei is developed in agreement with the symmetry point group of the material [51, 52].In the orthorhombic symmetry, for example, the free interaction energy is given by(7)Fcei,Q=g1e1+g2e2+g3e3Q2+12∑i,j3hijeiej+∑i=46hiiei2Q2,where g and h are the coupling constants.The interacting terms linear inei and quadratic inQ as gieiQ2 are responsible for a decrease of the longitudinal elastic constantCii (velocityVii). The decrease of the longitudinal elastic constantCii is proportional to the square of the coupling constant gi.:(8)∆C11~-gi2∆C22~-g22∆C33~-g32.The coupling second terms ei2Q2 in (7) show that several elastic constants Cij (or velocities) follow the temperature dependence of the square of the static value of the order parameter Q in the ordered phase belowTCDW [51, 52]:(9)∆Cij~hijQ2forij=11,22,33,44,55,66.The temperature dependence of the sound velocity and the amplitude of the superlattice reflections gives directly the temperature dependence of the order parameter [1].This general behavior has been observed at the CDW phase transitionTCDW in different materials [16, 17, 20, 36, 37, 41–43]. The hardening observed in the ordered phase with several compounds is analyzed in Section 3.5.


## 3. Results

### 3.1. Specific Heat Anomaly at the CDW Phase Transition

We reexamine the specific heat discontinuities∆CP measured at the CDW phase transitions in the following materials:(a)
Rare earth tritellurides TbTe3 [41] and ErTe3 [42](b)
Transition metal dichalcogenides 2H-NbSe2 [19], TiSe2 [22, 23], 2H-TaSe2, and 2H -TaS2 [24]: The mean field contribution for 2H-TaSe2 and 2H-TaS2 was estimated in [23]. These two compounds are characterized by large fluctuations(c)
Quasi-one-dimensional conductors NbSe3 [26], K0.3MoO3 [30, 31], (TaSe4)2I [40], and TTF-TCNQ [35](d)
Three-dimensional material LaAgSb2 [44] and Cr [53]The specific heat discontinuities∆CP are reported in Tables 1–4 and they are shown as a function of the CDW phase transition temperatureTCDW in Figure 2. A linear dependence is expected ∆CP=ATCDW. The experimental data are situated inside the area determined by the two linear dependence types (1) and (2) (Figure 2). The first line (1) followed by TTF-TCNQ, 2H-TaS2, and 2H-TaSe2 has a larger coefficient A1=4 ×10−2 J/molK−2. The second line (2) has a coefficient A2=3 ×10−3 J/molK−2, 10 times smaller than A1. This second line is followed approximately by the rare earth tritelluride compounds ErTe3 and TbTe3 at the upper and lower CDW phase transitions (blue circle and black circles). The specific heat discontinuity found at the upper CDW phase transition with LaAgSb2 (red square symbol) is also situated on line (2).Table 1
Materials (2D CDW) TbTe3, HoTe3, and ErTe3 rare earth tritellurides. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa and ratios.
Materials Molar volumecm3 T CDW K Specific heat∆Cp  J/molK Thermal expansioncoefficient along the c→ axis∆αc (K−1) Elastic velocitydecrease∆V/V Elastic constantMeasuredGPa calculatedρV2 GPa RatioCalculated/measured TbTe3 71 330 3[41] -1.2x10−5 [8] along thea→ or c→ axis-0.01 [41] C33~ 50 25 ~0.5  HoTe3 70.4 280120   Along thec→ axis-0.025 [43]-0.001 C33~ 50  ErTe3 69.7 260150 1[42]0.5 -3.5x10−6 [8] along thea→ or c→ axis-0.015 [42]-0.001 C33~ 50 20 ~0.4Table 2
Materials (2D CDW) transition metal dichalcogenides 2H-NbSe2, TiSe2, 2H-TaSe2, and 2H-TaS2. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa. Discrepancies exist among the experimental values of the Young modulus; the largest values are only indicated. Ratio = calculated ρV2/experimental elastic constant.
Materials Molar volumecm3 T CDW K Specific heat∆Cp  J/molK Thermal expansioncoefficient ∆α K−1 Elastic velocity decrease∆V/V Elastic constantGPa calculatedρV2 GPa Ratio 2H-NbSe2 40 30 0.5 [19] c → axis -3x10−6 [19]c→ axis-2x10−7 [18] a → axis-5x10−4 [16, 17] C11=108 [17]C33=460 35 ~0.3  TiSe2 39.6 200 1.15 [22]0.5 [23] a → axis-2.5x10−6 [19] along thea→ axis-0.05 [19] a → axisC11~ 190 [20] 1800800 ~10~4  2H-TaSe2  120 4[24] a → axis-8x10−6 [25] a → axis-0.0005[16, 17] Young modulus (a→ axis)Ea~120 [16] 16 ~0.1  2H-TaS2 35.7 78 2.8 [24]  no discountin.Table 3
Materials (1D CDW), quasi-one-dimensional conductors K0.3MoO3, NbSe3, (TaSe4)2I, and TTF-TCNQ. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa and ratios.
Materials Molar volumecm3 T CDW (K) Specific heat∆Cp J/molK Thermal expansioncoefficient ∆α (K−1) Elastic velocity decrease∆V/V Elastic constantGPa calculatedρV2 GPa Ratiocalculated/measured K0.3MoO3 36.3 180 3.6 [30]2  [34] [ 102 ] axis7x10−6 [31] [ 102 ] axis-0.02 [31] [ 102 ] axis Young modulus250 [31]60 [33] 250 [31] ~1~4  (TaSe4)2I 180 260 0.83 [40] c → axis-1x10−6 [38] c → axis-0.001 [37] C33=115 [37] 37 ~0.3  NbSe3 40.8 14558 3 [26]0.4  [26] in plane-2x10−6 [25] -0.0003[27] 550[27] 70 ~0.1  TTF-TCNQ 250 55 2.5 [35] No discontinuity [36] c →axis-0.01 [37]~0 [36] Young modulus E~60 [37]E~15 [36]Table 4
Three-dimensional materials. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa and ratios.
Materials Molar volumecm3 T CDW,TN K Specific heatJ/molK Thermal expansion coefficient∆α  K−1 Elastic velocity decrease Elastic ConstantGPa calculatedρV2 GPa Ratiocalculated/measured LaAgSb2 62.8 T CDW1= 210TCDW2= 185 0.5[44]0.1 [44] c → axis2.2x10−6  -5x10−7[44] No measurement  Cr 7.2 T N= 310 3 [53] along[100]  -7x10−6 [54] along[100]  -0.03 [54] C11=400GPa[54] 770 ~2Figure 2
Color on line. Heat specific discontinuities ∆CP at the CDW phase transition TCDW (Table 1) as a function of TCDW. TbTe3 (blue circle) [41]; ErTe3 (black circlesTCDW1 = 260 K,TCDW2 = 155 K) [42]; TiSe2 (violet circles,TCDW=200 K with two different results given in [22, 23]); LaAgSb2 (red squareTCDW1 = 210 K) [44]; NbSe3 (green circleTCDW1 = 145 K,TCDW2 = 48 K) [26]; 2H-NbSe2 (blue squaresTCDW = 30 K) [19]; 2H-TaSe2 (red down triangleTCDW =120 K) and 2H-TaS2 (red up trianglesTCDW = 80 K) mean field contribution given by [24]; TTF-TCNQ (black diamond symbol mean field contribution given in [35]); K0.3MoO3 (blue up trianglesTCDW = 180 K) two different results given by [30]; (TaSe4)2I (brown diamondTCDW = 260 K [40]); Cr (light blue square [53]). Lines (1) and (2) are lineal temperature dependence.However it should be noted that substantial differences exist between the experimental specific heat results obtained from different groups.


### 3.2. Thermal Expansion Anomaly at the CDW Phase Transition

Anisotropic anomalies of the elastic velocities and thermal expansion coefficients are observed at the CDW phase transition of the compounds under review.(a)
Discontinuities∆α of the thermal expansion coefficient in the basal plane at the upper phase transitionTCDW1 = 330 K of TbTe3 were obtained from thermal expansion measurements using X-rays technique by Ru et al. [8]. At the upper phase transition, the incommensurate wave vector is along the a→ axis. Large anisotropic behavior is observed for the thermal expansion along the c→ and a→ axes. The largest discontinuity is observed along the c→ axis [8] and is only reported in Table 1.The discontinuities∆α along the a→ and c→ axes at the lower CDW phase transitionTCDW2 = 150 K of ErTe3 were obtained from the thermal expansion measurements using X-rays technique by Ru [8]. Similar discontinuities are observed along the a→ and c→ axes.Only the values of∆α along the c→ axis are reported in Table 1.(b)
Thermal expansion coefficients discontinuities in the basal plane along thea→ and c→ axes obtained at the CDW transitions on transition metal dichalcogenides 2H-NbSe2 [18, 19], TiSe2 [20], and 2H-TaSe2 [25] are reported in Table 2. Very different experimental results were found for 2H-NbSe2 [18, 19].(c)
Thermal expansion discontinuities∆α determined along the [102] directions in NbSe3 [25], K0.3MoO3 [31, 32], (TaSe4)2I [39], and TTF-TCNQ [38] are reported in Table 3.(d)
Finally the thermal expansion discontinuities along thec→ axis observed in LaAgSb2 at the upper (TCDW1= 210 K) and lower CDW phase transition (TCDW2 = 185 K) [44] are reported in Table 4. Thermal expansion discontinuity along the a→ axis observed at the spin density wave transitionTSDW = 310 K for chromium [54] is also reported in Table 4.The stress dependence deduced using (4) from the thermal expansion coefficient discontinuities ∆α measured atTCDW along one crystallographic direction is given by(10)∂TCDW∂σ=-ΔαVmTCDWΔCp.The stress dependence ∂TCDW/∂σ values deduced atTCDW from the values given in Tables 1–4 are reported versus the transition temperatureTCDW in Figure 3. It seems that ∂TCDW/∂σ increases with increasingTCDW. The high values of the stress dependence ∂TCDW/∂σ are found with the rare earth tritellurides and 2H-NbSe2. Such a high value ∂TCDW/∂σ~100 K/GPa obtained for TbTe3 is in agreement with the value dTCDW/dp = 85K/GPa obtained in the hydrostatic measurements [15]. Smaller (one order of magnitude smaller) values of the stress dependence ∂TCDW/∂σ are found with the transition metal dichalcogenide compounds and the quasi-one-dimensional conductors.Figure 3
Color on line. Stress dependence ∂TCDW/∂σ derived from Ehrenfest eq. (4) with the thermal expansion coefficient discontinuities ∆α (Tables 1 and 2) measured at the CDW phase transitions as a function ofTCDW. ∆α along the c→ axis in TbTe3 atTCDW1 = 330 K (blue circle) and ErTe3 atTCDW2 = 150 K (black circle) deduced from the thermal expansion measurements using X-rays technique reported by Ru et al. [8]. Results with TiSe2 (violet circle)TCDW = 200 K [20]; 2H-TaSe2 (pink down triangle) [25]; NbSe3 (green circleTCDW = 145 K) [25]; 2H-NbSe2 (blue square) [18, 19]; K0.3MoO3 (blue up triangleTCDW = 180 K) [31, 32]; (TaSe4)2I (brown diamond) [39]; LaAgSb2 (red squareTCDW1 = 210 K ) [44]; and Cr (light blue square [54]). Lines (1) and (2) are lineal temperature dependence.It results in the fact that a high lattice anharmonicity is responsible for such a large stress dependence ofTCDW observed in the rare earth tritelluride materials.


### 3.3. Elastic Constant (Velocity) Anomaly at the CDW Phase Transition

The steplike decrease of the longitudinal elastic velocity∆V33/V33=∆C33/2C33 along the c→ axis measured at the upper and lower CDW phase transitions in the rare earth tritelluride TbTe3 [41], ErTe3 [42], and HoTe3 [43] compounds is reported in Table 1.The sound velocity and the Young modulus E discontinuities (velocity discontinuity deduced from E is given by∆V/V=∆E/2E) were measured in thea-b plane at the CDW phase transition in dichalcogenides 2H-NbSe2 [16, 17], TiSe2 [20], 2H-TaSe2 [16, 17], and 2H-TaS2 [16, 17], in quasi-one-dimensional conductors K0.3MoO3 [31, 33] and (TaSe4)2I [39] (Tables 2 and 3).Two different values∆V/V~0 and ~0.01 (dotted black line in Figure 4) are reported for the organic conductor TTF-TCNQ [36, 37]. The discontinuity measured at the SDW phase transition (TN = 310 K ) of Chromium [54] is also reported in Table 4. All the absolute values ∆V/V are shown in Figure 4.Figure 4
Color on line. Experimental discontinuities (absolute values) ∆V/V of the velocity of the longitudinal C11 and C33 shown as a function ofTCDW (Tables 1 and 2). Results obtained with TbTe3 (blue circleTCDW = 330 K) [41], ErTe3 (black circlesTCDW1 = 260 K andTCDW2 = 155 K) [42], and HoTe3 (brown circlesTCDW1 = 280 K andTCDW2 = 120 K) [43]. Results obtained by Young and velocity measurements NbSe3 (green circle) [27], 2H-NbSe2 (blue squares) [16, 17], TiSe2 (violet circle) [20], 2H-TaSe2 (red down triangle) [16], K0.3MoO3 (blue up triangle [31]), (TaSe4)2I (brown diamond) [39], TTF-TCNQ (black diamonds) with a large discrepancy (~0.0001, 0.01 shown by the dotted black line [36, 37]), and Cr (light blue [54]).Very small values are reported for 2H-TaSe4 and (TaSe4)2I. A general tendency is observed: the amplitude of the sound velocity discontinuities increases withTCDW. We mention that large discrepancies exist among the experimental Young modulus values.


### 3.4. Consistency

The consistency of Ehrenfest relations (1) and (2) may be checked by evaluating the value ρV2, equivalent to an effective elastic constant, from the discontinuities∆V/V, ∆α, and∆Cp measured at the CDW phase transition from different experiments following (6) which is rewritten as(11)ρV2calcultaed=21VmΔα2-ΔVVΔCPTCDWThe values Ccalculated=ρV2 evaluated using (11) with different materials are indicated in Tables 1–4.A realistic value of about 20 GPa is found for the rare earth tritelluride compounds TbTe3 and ErTe3. An unrealistic value of about 5000 GPa is evaluated with the very small thermal expansion jump value, ∆α~2×10−7 K−1, measured with 2H-NbSe2 in [18]. In contrast the thermal expansion results, ∆α~3×10−6 K−1, reported in [19] give a value ~35 GPa. A realistic value 250 GPa is evaluated for K0.3MoO3 in [31]. A smaller value of 37 GPa is obtained for the one-dimensional conductor (TaSe4)2I. In contrast large values 1800 GPa and 800 GPa are obtained for TiSe2. A small value of about 16 GPa is evaluated for 2H-TaSe2. No discontinuity, ∆α ~0, is observed for TTF-TCNQ and Ccalculated given by (11) cannot be evaluated for this material (Table 3). Finally a realistic value Ccalculated is evaluated (see (11)) at the SDW phase transition in chromium which has been previously discussed in [53–55]. The ratio values between Ccalculated and the measured elastic constant are shown in Figure 5.Figure 5
Color on line. Ratio between the calculated elastic constants (Ccalculated) using (11) and the measured values of the elastic constants (Tables 1–4), Ccalculated/ Cmeasured (column 10 in Tables 1–4). Ccalculated / Cmeasured~ 1 indicates that the Ehrenfest relations are satisfied.In conclusion the Ehrenfest equations are approximately satisfied by several materials: the rare earth tritellurides TbTe3 and ErTe3, the transition metal dichalcogenide 2H-NbSe2, and the one-dimensional conductors K0.3MoO3 and (TaSe4)2I. In the same manner the Ehrenfest equations are quantitatively satisfied at the SDW phase transition temperature (Néel antiferromagnetic phase transition) in chromium as discussed in [55]. In contrast the metal transition dichalcogenide 2H-TaSe2 and TiSe2 compound do not satisfy the Ehrenfest equations.


### 3.5. Temperature Dependence of the CDW Order Parameter

The increase of the elastic velocity∆V belowTCDW shown by the dotted black line in Figure 1(b) is related to the square of the order parameterQ (T) (see (9)). ∆V/V is analyzed with the following relation:(12)∆VV=∆VV0QTQ02where Q(0) is the value of the order parameter at T= 0K and ∆V/V0 is the maximum value of the relative velocity at T=0K and ∆V/V=0 atTCDW. For simplicity all the data are normalized at T=0 where ∆V/V=0. It results in the fact that (12) is changed by(13)∆VV=∆VV0QTQ02-1.The temperature dependence of the velocity ∆V/V of the longitudinal modes measured in the different materials is reported in Figure 6. All the experimental data follow the temperature dependence of the square of the BCS order parameter QT/Q0BCS2 [1]:(14)∆VV=∆VV0QTQ0BCS2-1.The blue dashed curve is calculated with 0.0017{QT/Q0BCS2-1]} for 2H-NbSe2 withTCDW= 32 K [16, 17]. The pink dashed curve is calculated with 0.0086{QT/Q0BCS2-1} for 2H-TaS2 withTCDW=75 K [16, 17]. The black dashed curve is calculated with 0.014{QT/Q0BCS2-1} for TTF-TCNQ withTCDW= 50 K [36] and the violet dashed curve with 0.06{QT/Q0BCS2-1} for TiSe2 withTCDW= 200 K [20]. The black circles are values for ErTe3 withTCDW= 260 K [42].Figure 6
Color on line. Temperature dependence of the measured relative velocity ∆V/V as a function of the reduced temperature T/TCDW. Velocity of the longitudinal mode in the(a b) plane in TiSe2 [20], 2H-TaS2 [16, 17], 2H-NbSe2 [16, 17], and TTF-TCNQ [36]. Velocity of the mode C33 measured with ErTe3 at the upper CDW phase transitionTCDW = 260 K [42]. The dashed curves are calculated with the BCS theory (see (14)).A remarkable feature is the increase of the amplitude∆V/V0 with TCDW2, ∆V/V0~10-6×TCDW2, in Figure 7. It yields the fact that the order parameterQ(0) proportional to ∆V/V00.5 increases with the charge density wave transition temperatureTCDW in agreement with BCS theory.Figure 7
Color on line. Amplitudes of the square of the order parameter Q02 at T = 0K, which is proportional to the values ∆V/V0 deduced from Figure 6, are reported as a function of TCDW. The solid line increases as TCDW2.In conclusion the temperature dependence of the elastic velocity is compatible with the BCS behavior in agreement with the temperature dependence of the amplitude of the superlattice reflections and of the intensities of the Raman modes [1, 8, 14].


## 3.1. Specific Heat Anomaly at the CDW Phase Transition

We reexamine the specific heat discontinuities∆CP measured at the CDW phase transitions in the following materials:(a)
Rare earth tritellurides TbTe3 [41] and ErTe3 [42](b)
Transition metal dichalcogenides 2H-NbSe2 [19], TiSe2 [22, 23], 2H-TaSe2, and 2H -TaS2 [24]: The mean field contribution for 2H-TaSe2 and 2H-TaS2 was estimated in [23]. These two compounds are characterized by large fluctuations(c)
Quasi-one-dimensional conductors NbSe3 [26], K0.3MoO3 [30, 31], (TaSe4)2I [40], and TTF-TCNQ [35](d)
Three-dimensional material LaAgSb2 [44] and Cr [53]The specific heat discontinuities∆CP are reported in Tables 1–4 and they are shown as a function of the CDW phase transition temperatureTCDW in Figure 2. A linear dependence is expected ∆CP=ATCDW. The experimental data are situated inside the area determined by the two linear dependence types (1) and (2) (Figure 2). The first line (1) followed by TTF-TCNQ, 2H-TaS2, and 2H-TaSe2 has a larger coefficient A1=4 ×10−2 J/molK−2. The second line (2) has a coefficient A2=3 ×10−3 J/molK−2, 10 times smaller than A1. This second line is followed approximately by the rare earth tritelluride compounds ErTe3 and TbTe3 at the upper and lower CDW phase transitions (blue circle and black circles). The specific heat discontinuity found at the upper CDW phase transition with LaAgSb2 (red square symbol) is also situated on line (2).Table 1
Materials (2D CDW) TbTe3, HoTe3, and ErTe3 rare earth tritellurides. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa and ratios.
Materials Molar volumecm3 T CDW K Specific heat∆Cp  J/molK Thermal expansioncoefficient along the c→ axis∆αc (K−1) Elastic velocitydecrease∆V/V Elastic constantMeasuredGPa calculatedρV2 GPa RatioCalculated/measured TbTe3 71 330 3[41] -1.2x10−5 [8] along thea→ or c→ axis-0.01 [41] C33~ 50 25 ~0.5  HoTe3 70.4 280120   Along thec→ axis-0.025 [43]-0.001 C33~ 50  ErTe3 69.7 260150 1[42]0.5 -3.5x10−6 [8] along thea→ or c→ axis-0.015 [42]-0.001 C33~ 50 20 ~0.4Table 2
Materials (2D CDW) transition metal dichalcogenides 2H-NbSe2, TiSe2, 2H-TaSe2, and 2H-TaS2. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa. Discrepancies exist among the experimental values of the Young modulus; the largest values are only indicated. Ratio = calculated ρV2/experimental elastic constant.
Materials Molar volumecm3 T CDW K Specific heat∆Cp  J/molK Thermal expansioncoefficient ∆α K−1 Elastic velocity decrease∆V/V Elastic constantGPa calculatedρV2 GPa Ratio 2H-NbSe2 40 30 0.5 [19] c → axis -3x10−6 [19]c→ axis-2x10−7 [18] a → axis-5x10−4 [16, 17] C11=108 [17]C33=460 35 ~0.3  TiSe2 39.6 200 1.15 [22]0.5 [23] a → axis-2.5x10−6 [19] along thea→ axis-0.05 [19] a → axisC11~ 190 [20] 1800800 ~10~4  2H-TaSe2  120 4[24] a → axis-8x10−6 [25] a → axis-0.0005[16, 17] Young modulus (a→ axis)Ea~120 [16] 16 ~0.1  2H-TaS2 35.7 78 2.8 [24]  no discountin.Table 3
Materials (1D CDW), quasi-one-dimensional conductors K0.3MoO3, NbSe3, (TaSe4)2I, and TTF-TCNQ. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa and ratios.
Materials Molar volumecm3 T CDW (K) Specific heat∆Cp J/molK Thermal expansioncoefficient ∆α (K−1) Elastic velocity decrease∆V/V Elastic constantGPa calculatedρV2 GPa Ratiocalculated/measured K0.3MoO3 36.3 180 3.6 [30]2  [34] [ 102 ] axis7x10−6 [31] [ 102 ] axis-0.02 [31] [ 102 ] axis Young modulus250 [31]60 [33] 250 [31] ~1~4  (TaSe4)2I 180 260 0.83 [40] c → axis-1x10−6 [38] c → axis-0.001 [37] C33=115 [37] 37 ~0.3  NbSe3 40.8 14558 3 [26]0.4  [26] in plane-2x10−6 [25] -0.0003[27] 550[27] 70 ~0.1  TTF-TCNQ 250 55 2.5 [35] No discontinuity [36] c →axis-0.01 [37]~0 [36] Young modulus E~60 [37]E~15 [36]Table 4
Three-dimensional materials. Molar volumes are given in cm3, specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1, elastic velocity discontinuities in ∆V/V, and measured and calculated elastic constants in GPa and ratios.
Materials Molar volumecm3 T CDW,TN K Specific heatJ/molK Thermal expansion coefficient∆α  K−1 Elastic velocity decrease Elastic ConstantGPa calculatedρV2 GPa Ratiocalculated/measured LaAgSb2 62.8 T CDW1= 210TCDW2= 185 0.5[44]0.1 [44] c → axis2.2x10−6  -5x10−7[44] No measurement  Cr 7.2 T N= 310 3 [53] along[100]  -7x10−6 [54] along[100]  -0.03 [54] C11=400GPa[54] 770 ~2Figure 2
Color on line. Heat specific discontinuities ∆CP at the CDW phase transition TCDW (Table 1) as a function of TCDW. TbTe3 (blue circle) [41]; ErTe3 (black circlesTCDW1 = 260 K,TCDW2 = 155 K) [42]; TiSe2 (violet circles,TCDW=200 K with two different results given in [22, 23]); LaAgSb2 (red squareTCDW1 = 210 K) [44]; NbSe3 (green circleTCDW1 = 145 K,TCDW2 = 48 K) [26]; 2H-NbSe2 (blue squaresTCDW = 30 K) [19]; 2H-TaSe2 (red down triangleTCDW =120 K) and 2H-TaS2 (red up trianglesTCDW = 80 K) mean field contribution given by [24]; TTF-TCNQ (black diamond symbol mean field contribution given in [35]); K0.3MoO3 (blue up trianglesTCDW = 180 K) two different results given by [30]; (TaSe4)2I (brown diamondTCDW = 260 K [40]); Cr (light blue square [53]). Lines (1) and (2) are lineal temperature dependence.However it should be noted that substantial differences exist between the experimental specific heat results obtained from different groups.


## 3.2. Thermal Expansion Anomaly at the CDW Phase Transition

Anisotropic anomalies of the elastic velocities and thermal expansion coefficients are observed at the CDW phase transition of the compounds under review.(a)
Discontinuities∆α of the thermal expansion coefficient in the basal plane at the upper phase transitionTCDW1 = 330 K of TbTe3 were obtained from thermal expansion measurements using X-rays technique by Ru et al. [8]. At the upper phase transition, the incommensurate wave vector is along the a→ axis. Large anisotropic behavior is observed for the thermal expansion along the c→ and a→ axes. The largest discontinuity is observed along the c→ axis [8] and is only reported in Table 1.The discontinuities∆α along the a→ and c→ axes at the lower CDW phase transitionTCDW2 = 150 K of ErTe3 were obtained from the thermal expansion measurements using X-rays technique by Ru [8]. Similar discontinuities are observed along the a→ and c→ axes.Only the values of∆α along the c→ axis are reported in Table 1.(b)
Thermal expansion coefficients discontinuities in the basal plane along thea→ and c→ axes obtained at the CDW transitions on transition metal dichalcogenides 2H-NbSe2 [18, 19], TiSe2 [20], and 2H-TaSe2 [25] are reported in Table 2. Very different experimental results were found for 2H-NbSe2 [18, 19].(c)
Thermal expansion discontinuities∆α determined along the [102] directions in NbSe3 [25], K0.3MoO3 [31, 32], (TaSe4)2I [39], and TTF-TCNQ [38] are reported in Table 3.(d)
Finally the thermal expansion discontinuities along thec→ axis observed in LaAgSb2 at the upper (TCDW1= 210 K) and lower CDW phase transition (TCDW2 = 185 K) [44] are reported in Table 4. Thermal expansion discontinuity along the a→ axis observed at the spin density wave transitionTSDW = 310 K for chromium [54] is also reported in Table 4.The stress dependence deduced using (4) from the thermal expansion coefficient discontinuities ∆α measured atTCDW along one crystallographic direction is given by(10)∂TCDW∂σ=-ΔαVmTCDWΔCp.The stress dependence ∂TCDW/∂σ values deduced atTCDW from the values given in Tables 1–4 are reported versus the transition temperatureTCDW in Figure 3. It seems that ∂TCDW/∂σ increases with increasingTCDW. The high values of the stress dependence ∂TCDW/∂σ are found with the rare earth tritellurides and 2H-NbSe2. Such a high value ∂TCDW/∂σ~100 K/GPa obtained for TbTe3 is in agreement with the value dTCDW/dp = 85K/GPa obtained in the hydrostatic measurements [15]. Smaller (one order of magnitude smaller) values of the stress dependence ∂TCDW/∂σ are found with the transition metal dichalcogenide compounds and the quasi-one-dimensional conductors.Figure 3
Color on line. Stress dependence ∂TCDW/∂σ derived from Ehrenfest eq. (4) with the thermal expansion coefficient discontinuities ∆α (Tables 1 and 2) measured at the CDW phase transitions as a function ofTCDW. ∆α along the c→ axis in TbTe3 atTCDW1 = 330 K (blue circle) and ErTe3 atTCDW2 = 150 K (black circle) deduced from the thermal expansion measurements using X-rays technique reported by Ru et al. [8]. Results with TiSe2 (violet circle)TCDW = 200 K [20]; 2H-TaSe2 (pink down triangle) [25]; NbSe3 (green circleTCDW = 145 K) [25]; 2H-NbSe2 (blue square) [18, 19]; K0.3MoO3 (blue up triangleTCDW = 180 K) [31, 32]; (TaSe4)2I (brown diamond) [39]; LaAgSb2 (red squareTCDW1 = 210 K ) [44]; and Cr (light blue square [54]). Lines (1) and (2) are lineal temperature dependence.It results in the fact that a high lattice anharmonicity is responsible for such a large stress dependence ofTCDW observed in the rare earth tritelluride materials.


## 3.3. Elastic Constant (Velocity) Anomaly at the CDW Phase Transition

The steplike decrease of the longitudinal elastic velocity∆V33/V33=∆C33/2C33 along the c→ axis measured at the upper and lower CDW phase transitions in the rare earth tritelluride TbTe3 [41], ErTe3 [42], and HoTe3 [43] compounds is reported in Table 1.The sound velocity and the Young modulus E discontinuities (velocity discontinuity deduced from E is given by∆V/V=∆E/2E) were measured in thea-b plane at the CDW phase transition in dichalcogenides 2H-NbSe2 [16, 17], TiSe2 [20], 2H-TaSe2 [16, 17], and 2H-TaS2 [16, 17], in quasi-one-dimensional conductors K0.3MoO3 [31, 33] and (TaSe4)2I [39] (Tables 2 and 3).Two different values∆V/V~0 and ~0.01 (dotted black line in Figure 4) are reported for the organic conductor TTF-TCNQ [36, 37]. The discontinuity measured at the SDW phase transition (TN = 310 K ) of Chromium [54] is also reported in Table 4. All the absolute values ∆V/V are shown in Figure 4.Figure 4
Color on line. Experimental discontinuities (absolute values) ∆V/V of the velocity of the longitudinal C11 and C33 shown as a function ofTCDW (Tables 1 and 2). Results obtained with TbTe3 (blue circleTCDW = 330 K) [41], ErTe3 (black circlesTCDW1 = 260 K andTCDW2 = 155 K) [42], and HoTe3 (brown circlesTCDW1 = 280 K andTCDW2 = 120 K) [43]. Results obtained by Young and velocity measurements NbSe3 (green circle) [27], 2H-NbSe2 (blue squares) [16, 17], TiSe2 (violet circle) [20], 2H-TaSe2 (red down triangle) [16], K0.3MoO3 (blue up triangle [31]), (TaSe4)2I (brown diamond) [39], TTF-TCNQ (black diamonds) with a large discrepancy (~0.0001, 0.01 shown by the dotted black line [36, 37]), and Cr (light blue [54]).Very small values are reported for 2H-TaSe4 and (TaSe4)2I. A general tendency is observed: the amplitude of the sound velocity discontinuities increases withTCDW. We mention that large discrepancies exist among the experimental Young modulus values.


## 3.4. Consistency

The consistency of Ehrenfest relations (1) and (2) may be checked by evaluating the value ρV2, equivalent to an effective elastic constant, from the discontinuities∆V/V, ∆α, and∆Cp measured at the CDW phase transition from different experiments following (6) which is rewritten as(11)ρV2calcultaed=21VmΔα2-ΔVVΔCPTCDWThe values Ccalculated=ρV2 evaluated using (11) with different materials are indicated in Tables 1–4.A realistic value of about 20 GPa is found for the rare earth tritelluride compounds TbTe3 and ErTe3. An unrealistic value of about 5000 GPa is evaluated with the very small thermal expansion jump value, ∆α~2×10−7 K−1, measured with 2H-NbSe2 in [18]. In contrast the thermal expansion results, ∆α~3×10−6 K−1, reported in [19] give a value ~35 GPa. A realistic value 250 GPa is evaluated for K0.3MoO3 in [31]. A smaller value of 37 GPa is obtained for the one-dimensional conductor (TaSe4)2I. In contrast large values 1800 GPa and 800 GPa are obtained for TiSe2. A small value of about 16 GPa is evaluated for 2H-TaSe2. No discontinuity, ∆α ~0, is observed for TTF-TCNQ and Ccalculated given by (11) cannot be evaluated for this material (Table 3). Finally a realistic value Ccalculated is evaluated (see (11)) at the SDW phase transition in chromium which has been previously discussed in [53–55]. The ratio values between Ccalculated and the measured elastic constant are shown in Figure 5.Figure 5
Color on line. Ratio between the calculated elastic constants (Ccalculated) using (11) and the measured values of the elastic constants (Tables 1–4), Ccalculated/ Cmeasured (column 10 in Tables 1–4). Ccalculated / Cmeasured~ 1 indicates that the Ehrenfest relations are satisfied.In conclusion the Ehrenfest equations are approximately satisfied by several materials: the rare earth tritellurides TbTe3 and ErTe3, the transition metal dichalcogenide 2H-NbSe2, and the one-dimensional conductors K0.3MoO3 and (TaSe4)2I. In the same manner the Ehrenfest equations are quantitatively satisfied at the SDW phase transition temperature (Néel antiferromagnetic phase transition) in chromium as discussed in [55]. In contrast the metal transition dichalcogenide 2H-TaSe2 and TiSe2 compound do not satisfy the Ehrenfest equations.


## 3.5. Temperature Dependence of the CDW Order Parameter

The increase of the elastic velocity∆V belowTCDW shown by the dotted black line in Figure 1(b) is related to the square of the order parameterQ (T) (see (9)). ∆V/V is analyzed with the following relation:(12)∆VV=∆VV0QTQ02where Q(0) is the value of the order parameter at T= 0K and ∆V/V0 is the maximum value of the relative velocity at T=0K and ∆V/V=0 atTCDW. For simplicity all the data are normalized at T=0 where ∆V/V=0. It results in the fact that (12) is changed by(13)∆VV=∆VV0QTQ02-1.The temperature dependence of the velocity ∆V/V of the longitudinal modes measured in the different materials is reported in Figure 6. All the experimental data follow the temperature dependence of the square of the BCS order parameter QT/Q0BCS2 [1]:(14)∆VV=∆VV0QTQ0BCS2-1.The blue dashed curve is calculated with 0.0017{QT/Q0BCS2-1]} for 2H-NbSe2 withTCDW= 32 K [16, 17]. The pink dashed curve is calculated with 0.0086{QT/Q0BCS2-1} for 2H-TaS2 withTCDW=75 K [16, 17]. The black dashed curve is calculated with 0.014{QT/Q0BCS2-1} for TTF-TCNQ withTCDW= 50 K [36] and the violet dashed curve with 0.06{QT/Q0BCS2-1} for TiSe2 withTCDW= 200 K [20]. The black circles are values for ErTe3 withTCDW= 260 K [42].Figure 6
Color on line. Temperature dependence of the measured relative velocity ∆V/V as a function of the reduced temperature T/TCDW. Velocity of the longitudinal mode in the(a b) plane in TiSe2 [20], 2H-TaS2 [16, 17], 2H-NbSe2 [16, 17], and TTF-TCNQ [36]. Velocity of the mode C33 measured with ErTe3 at the upper CDW phase transitionTCDW = 260 K [42]. The dashed curves are calculated with the BCS theory (see (14)).A remarkable feature is the increase of the amplitude∆V/V0 with TCDW2, ∆V/V0~10-6×TCDW2, in Figure 7. It yields the fact that the order parameterQ(0) proportional to ∆V/V00.5 increases with the charge density wave transition temperatureTCDW in agreement with BCS theory.Figure 7
Color on line. Amplitudes of the square of the order parameter Q02 at T = 0K, which is proportional to the values ∆V/V0 deduced from Figure 6, are reported as a function of TCDW. The solid line increases as TCDW2.In conclusion the temperature dependence of the elastic velocity is compatible with the BCS behavior in agreement with the temperature dependence of the amplitude of the superlattice reflections and of the intensities of the Raman modes [1, 8, 14].


## 4. Conclusions

Similar features in the thermodynamic properties at the CDW phase transitionTCDW are found in all the CDW materials under review. The amplitude of the specific heat anomaly at the CDW phase transitionTCDW is sample dependent but the amplitude increases (roughly) linearly with increasingTCDW in agreement with a second-order phase transition. The (mean field theory) Ehrenfest equations are approximately satisfied by several compounds: the rare earth tritellurides TbTe3, ErTe3 compounds, the transition metal dichalcogenide 2H-NbSe2 compound, and several quasi-one-dimensional conductors. In contrast large inconsistency in the Ehrenfest relationships is found with the transition metal dichalcogenide compounds 2H-TaSe2 and TiSe2. Lattice anharmonicity acting through the stress dependence of the phase transition temperature TCDW in the rare earth tritelluride compounds is larger than that of the transition metal dichalcogenides and quasi-one-dimensional conductors.It seems that the elastic property in the CDW ordered phase is related to the temperature dependence of the order parameter which follows a BCS behavior. Finally LaAgSb2 has been classified as a 3D CDW system. The Ehrenfest relationships should be verified in this material.



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