Advances in Condensed Matter Physics
Volume 2019 (2019), Article ID 2138264, 14 pages
https://doi.org/10.1155/2019/2138264

Review Article

Survey of the Thermodynamic Properties of the Charge Density Wave Systems

M. Saint-Paul1,2 and P. Monceau1,2

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

Correspondence should be addressed to M. Saint-Paul; michel.saint-paul@neel.cnrs.fr

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 TCDW increases with increasing TCDW 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 temperature TCDW. 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 TCDW [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 Q₁ of the prototypal rare earth tritelluride ErTe₃ at the upper CDW phase transition TCDW1 = 265 K is parallel to the axis, whereas the incommensurate ordering parameter Q₂ observed at the lower CDW phase transition TCDW2 = 150 K with ErTe₃ is parallel to the 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 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 TCDW 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 around TCDW 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 transition TC, 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 F₀ 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 by

The 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 TC given by [47]

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 is related to the specific heat jump ∆CP and to the stress dependence components, , at the phase transition TCDW:

where i=1, 2, and 3 correspond to the , , and crystallographic axes directions and Vm is the molar volume.

(a)

(b)

(c)

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].

The elastic constant component is related to the elastic velocity by , being the mass density. The discontinuities of the elastic constants (or velocity ) at a second-order phase transition are related to the stress dependence by

The term, , proportional to the entropy variation and multiplied by the second derivative [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)

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 at TCDW 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 transition TCDW is proposed in [3]. A second approach was developed by Rhewald [51] based on the Landau phenomenological theory including the interaction between the strain components ei and the square of the order parameter Q [51, 52]. The expansion of the free energy density in power of Q² and ei 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

where g and h are the coupling constants.

The interacting terms linear in ei and quadratic in Q as are responsible for a decrease of the longitudinal elastic constant Cii (velocity Vii). The decrease of the longitudinal elastic constant Cii is proportional to the square of the coupling constant :

The coupling second terms in (7) show that several elastic constants (or velocities) follow the temperature dependence of the square of the static value of the order parameter in the ordered phase below TCDW [51, 52]:

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 transition TCDW 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 temperature TCDW in Figure 2. A linear dependence is expected . The experimental data are situated inside the area determined by the two linear dependence types and (Figure 2). The first line followed by TTF-TCNQ, 2H-TaS₂, and 2H-TaSe₂ has a larger coefficient A₁=4 ×10⁻² J/molK⁻². The second line 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 .

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 , and measured and calculated elastic constants in GPa and ratios.

Materials
Molar volume
cm³
TCDW 
K
Specific heat

J/molK
Thermal expansion
coefficient along the axis
∆αc (K⁻¹)
Elastic velocity
decrease
∆V/V
Elastic constant
Measured
GPa
calculated
ρV² 
GPa
Ratio
Calculated
/measured

TbTe₃
71
330
3
[41]
-1.2x10⁻⁵ [8]
along the or axis
-0.01 [41]
C₃₃~ 50
25
~0.5

HoTe₃
70.4
280
120

Along the axis
-0.025 [43]
-0.001
C₃₃~ 50

ErTe₃
69.7
260
150
1
[42]
0.5
-3.5x10⁻⁶ [8]
along the or axis
-0.015 [42]
-0.001
C₃₃~ 50
20
~0.4

Table 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 , 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 /experimental elastic constant.

Materials
Molar volume
cm³
TCDW 
K
Specific heat

J/molK
Thermal expansion
coefficient
∆α  
K⁻¹
Elastic velocity decrease
∆V/V
Elastic constant
GPa
calculated

GPa
Ratio

2H-NbSe₂
40
30
0.5 [19]
axis
-3x10⁻⁶ [19]
axis
-2x10⁻⁷ [18]
axis
-5x10⁻⁴ [16, 17]
C₁₁=108 [17]
C₃₃=460
35
~0.3

TiSe₂
39.6
200
1.15 [22]
0.5 [23]
axis
-2.5x10⁻⁶ [19]
along the axis
-0.05 [19]
axis
C₁₁~ 190 [20]
1800
800
~10
~4

2H-TaSe₂

120
4
[24]
axis
-8x10⁻⁶ [25]
axis
-0.0005
[16, 17]
Young modulus ( axis)
[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 , and measured and calculated elastic constants in GPa and ratios.

Materials
Molar volume
cm³
TCDW 
(K)
Specific heat

J/molK
Thermal expansion
coefficient
(K⁻¹)
Elastic velocity decrease

Elastic constant
GPa
calculated

GPa
Ratio
calculated
/measured

K0.3MoO₃
36.3
180
3.6 [30]
2
[34]
axis
7x10⁻⁶ [31]
axis
-0.02 [31]
axis
Young modulus
250 [31]
60 [33]
250 [31]
~1
~4

(TaSe₄)₂I
180
260
0.83 [40]
axis
-1x10⁻⁶ [38]
axis
-0.001 [37]
C₃₃=115 [37]
37
~0.3

NbSe₃
40.8
145
58
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]
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 , and measured and calculated elastic constants in GPa and ratios.

Materials
Molar volume
cm³
TCDW,TN 
K
Specific heat
J/molK
Thermal expansion coefficient

K⁻¹
Elastic velocity decrease
Elastic Constant
GPa
calculated
ρV² 
GPa
Ratio
calculated
/measured

LaAgSb2

62.8
TCDW1= 210
TCDW2= 185
0.5
[44]
0.1
[44]
axis
2.2x10⁻⁶  
-5x10⁻⁷[44]
No measurement

Cr
7.2
TN= 310
3 [53]
along  
-7x10⁻⁶ [54]
along  
-0.03 [54]
C₁₁=400GPa
[54]
770
~2

Figure 2: Color on line. Heat specific discontinuities at the CDW phase transition (Table 1) as a function of . TbTe₃ (blue circle) [41]; ErTe₃ (black circles TCDW1 = 260 K, TCDW2 = 155 K) [42]; TiSe₂ (violet circles, TCDW=200 K with two different results given in [22, 23]); LaAgSb₂ (red square TCDW1 = 210 K) [44]; NbSe₃ (green circle TCDW1 = 145 K, TCDW2 = 48 K) [26]; 2H-NbSe₂ (blue squares TCDW = 30 K) [19]; 2H-TaSe₂ (red down triangle TCDW =120 K) and 2H-TaS₂ (red up triangles TCDW = 80 K) mean field contribution given by [24]; TTF-TCNQ (black diamond symbol mean field contribution given in [35]); K0.3MoO₃ (blue up triangles TCDW = 180 K) two different results given by [30]; (TaSe₄)₂I (brown diamond TCDW = 260 K [40]); Cr (light blue square [53]). Lines and 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 of the thermal expansion coefficient in the basal plane at the upper phase transition TCDW1 = 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 axis. Large anisotropic behavior is observed for the thermal expansion along the and axes. The largest discontinuity is observed along the axis [8] and is only reported in Table 1. The discontinuities along the and axes at the lower CDW phase transition TCDW2 = 150 K of ErTe₃ were obtained from the thermal expansion measurements using X-rays technique by Ru [8]. Similar discontinuities are observed along the and axes. Only the values of along the axis are reported in Table 1.(b)Thermal expansion coefficients discontinuities in the basal plane along the and 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 determined along the 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 the 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 axis observed at the spin density wave transition TSDW = 310 K for chromium [54] is also reported in Table 4.

The stress dependence deduced using (4) from the thermal expansion coefficient discontinuities measured at TCDW along one crystallographic direction is given by

The stress dependence values deduced at TCDW from the values given in Tables 1–4 are reported versus the transition temperature TCDW in Figure 3. It seems that increases with increasing TCDW. The high values of the stress dependence are found with the rare earth tritellurides and 2H-NbSe₂. Such a high value K/GPa obtained for TbTe₃ is in agreement with the value = 85K/GPa obtained in the hydrostatic measurements [15]. Smaller (one order of magnitude smaller) values of the stress dependence are found with the transition metal dichalcogenide compounds and the quasi-one-dimensional conductors.

Figure 3: Color on line. Stress dependence derived from Ehrenfest eq. (4) with the thermal expansion coefficient discontinuities (Tables 1 and 2) measured at the CDW phase transitions as a function of TCDW. along the axis in TbTe₃ at TCDW1 = 330 K (blue circle) and ErTe₃ at TCDW2 = 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 circle TCDW = 145 K) [25]; 2H-NbSe₂ (blue square) [18, 19]; K0.3MoO₃ (blue up triangle TCDW = 180 K) [31, 32]; (TaSe₄)₂I (brown diamond) [39]; LaAgSb₂ (red square TCDW1 = 210 K ) [44]; and Cr (light blue square [54]). Lines and are lineal temperature dependence.

It results in the fact that a high lattice anharmonicity is responsible for such a large stress dependence of 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 along the 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 ) were measured in the a-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 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 ( = 310 K ) of Chromium [54] is also reported in Table 4. All the absolute values are shown in Figure 4.

Figure 4: Color on line. Experimental discontinuities (absolute values) of the velocity of the longitudinal C₁₁ and C₃₃ shown as a function of TCDW (Tables 1 and 2). Results obtained with TbTe₃ (blue circle TCDW = 330 K) [41], ErTe₃ (black circles TCDW1 = 260 K and TCDW2 = 155 K) [42], and HoTe₃ (brown circles TCDW1 = 280 K and TCDW2 = 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 with TCDW. 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 , equivalent to an effective elastic constant, from the discontinuities ∆V/V, , and ∆Cp measured at the CDW phase transition from different experiments following (6) which is rewritten as

The values 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, 0⁻⁷ K⁻¹, measured with 2H-NbSe₂ in [18]. In contrast the thermal expansion results, 0⁻⁶ 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 given by (11) cannot be evaluated for this material (Table 3). Finally a realistic value is evaluated (see (11)) at the SDW phase transition in chromium which has been previously discussed in [53–55]. The ratio values between and the measured elastic constant are shown in Figure 5.

Figure 5: Color on line. Ratio between the calculated elastic constants () using (11) and the measured values of the elastic constants (Tables 1–4), / (column 10 in Tables 1–4). / ~ 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 below TCDW shown by the dotted black line in Figure 1(b) is related to the square of the order parameter Q (T) (see (9)). is analyzed with the following relation:

where is the value of the order parameter at T= 0K and is the maximum value of the relative velocity at T=0K and at TCDW. For simplicity all the data are normalized at T=0 where . It results in the fact that (12) is changed by

The temperature dependence of the velocity 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 [1]:

The blue dashed curve is calculated with for 2H-NbSe₂ with TCDW = 32 K [16, 17]. The pink dashed curve is calculated with for 2H-TaS₂ with TCDW=75 K [16, 17]. The black dashed curve is calculated with for TTF-TCNQ with TCDW = 50 K [36] and the violet dashed curve with for TiSe₂ with TCDW = 200 K [20]. The black circles are values for ErTe₃ with TCDW = 260 K [42].

Figure 6: Color on line. Temperature dependence of the measured relative velocity ∆V/V as a function of the reduced temperature . 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 transition TCDW = 260 K [42]. The dashed curves are calculated with the BCS theory (see (14)).

A remarkable feature is the increase of the amplitude with , , in Figure 7. It yields the fact that the order parameter Q(0) proportional to increases with the charge density wave transition temperature TCDW in agreement with BCS theory.

Figure 7: Color on line. Amplitudes of the square of the order parameter at T = 0K, which is proportional to the values deduced from Figure 6, are reported as a function of . The solid line increases as .

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 transition TCDW are found in all the CDW materials under review. The amplitude of the specific heat anomaly at the CDW phase transition TCDW is sample dependent but the amplitude increases (roughly) linearly with increasing TCDW 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 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

G. Grüner, “Density Waves in Solids,” Frontiers in Physics Ed D. Pines (Addison-Wesley) 1994 and Rev. Mod. 60, 1129, 1988.

H. Frölich, Proc. R. Soc. London A223, 296, 1954.

J. W. Brill, “Elastic properties of low dimensional materials,” in Chap. 10 in Handbook of Elastic Properties of Solids (Volume II) Liquids and Gases, Levy, Bass, and Stern, Eds., Academic Press, 2001.

P. Monceau, “Electronic crystals: An experimental overview,” Advances in Physics, vol. 61, no. 4, pp. 325–581, 2012.

X. Zhu, J. Guo, J. Zang et al., “Misconceptions associated with the origin of charge density waves,” Advances in Physics, vol. 2, pp. 325–581, 2017.

X. Zhu, Y. Cao, J. Zhang et al., “Classification of charge density waves based on their nature,” Proceedings of the National Academy of Sciences, vol. 24, pp. 2367–2371, 2015.

H.-M. Eiter, M. Lavagni, R. Hackl et al., “Alternative route to charge density wave formation in multiband systems,” Proceedings of the National Academy of Sciences, vol. 110, pp. 64–96, 2013.

N. Ru, C. L. Condron, G. Y. Margulis et al., “Effect of chemical pressure on the charge density wave transition in rare earth tritellurides RTe₃,” Physical Review B, vol. 77, Article ID 035114, 2008.

N. Ru and I. R. Fisher, “Thermodynamic and transport properties of YTe₃, LaTe₃ and CeTe₃,” Physical Review B, vol. 73, Article ID 033101, 2006.

V. Brouet, W. L. Yang, X. Zhous et al., “Angle-resolved photoemission study of the evolution of band structure and charge density properties in RTe₃ (R=Y, La,Ce,Sm, Gd, Tb, and Dy),” Physical Review B, vol. 77, Article ID 235104, 2008.

R. G. Moore, V. Brouet, R. He et al., “Fermi surface evolution across multiple charge density wave transition in ErTe₃,” Physical Review B, vol. 81, Article ID 073102, 2010.

R. G. Moore, W. S. Lee, P. S. Kirchman et al., “Ultrafast resonant soft x-ray diffraction dynamics of the charge density wave in TbTe₃,” Physical Review B, vol. 93, Article ID 024304, 2016.

M. Maschek, S. Rosenbranz, R. Heid et al., “Wave-vector-dependent electron-phonon coupling and the charge density wave transition in TbTe₃,” Physical Review B, vol. 91, Article ID 235147, 2015.

N. Lazarevic, Z. V. Popovic, R. Hu et al., “Evidence of coupling between phonons and charge density waves in ErTe₃,” Physical Review B, vol. 83, Article ID 024302, 2011.

D. A. Zocco, J. J. Hamlin, K. Grube et al., “Pressure dependence of the charge density wave and superconducting states in GdTe₃,” Physical Review B, vol. 91, Article ID 205114, 2015.

M. Barmatz, L. R. Testardi, and F. J. Di Salvo, “Elasticity measurements in the layered dichalcogenides TaSe₂ and NbSe₂,” Physical Review B, vol. 12, no. 10, pp. 4367–4376, 1975.

M. H. Jericho, A. M. Simpson, and R. F. Frindt, “Velocity of ultrasonic waves in 2H-NbSe₂, 2H-TaS₂ and 1T-TaS₂,” Physical Review B, vol. 22, no. 10, pp. 4907–4914, 1980.

O. Sezerman, A. M. Simpson, and M. H. Jericho, “Thermal expansion of 1T-TaS2 and 2H-NbSe₂,” Solid State Communications, vol. 36, pp. 737–740, 1980.

V. Eremenko, V. Sirenko, V. Ibulaev, J. Bartolomé, A. Arauzo, and G. Reményi, “Heat capacity, thermal expansion and pressure derivative of critical temperature at the superconducting and charge density wave (CDW) in NbSe₂,” Physica, vol. 469, no. 7-8, pp. 259–264, 2009.

A. Caillé, Y. Lepine, M. H. Jericho et al., “Thermal expansion, ultrasonic velocity, and attenuation measurements in TiS₂,TiSe₂ and TiS0.5Se1.5,” Physical Review, vol. 26, no. 10, p. 5454, 1983.

W. G. Stirling, B. Dorner, J. D. N. Cheeke et al., “Acoustic phonons in the transition metal dichalcogenide layer compound TiSe₂,” Solid State Communications, vol. 18, no. 7, pp. 931–933, 1976.

R. A. Craven, F. J. Di Salvo, F. S. L. Hsu et al., “Mechanisms for the 200 K transition in TiSe₂ : A measurement of the specific heat,” Solid State Communication, vol. 25, pp. 39–42, 1978.

J. P. Castellan, S. Rosenkranz, R. Osborn et al., “Chiral phase transition in charge ordered 1T-TiSe₂,” Physical Review Letters, vol. 110, no. 19, Article ID 196404, 2013.

R. A. Craven and S. F. Meyer, “Specific heat and resisitivity near the charge density wave phase transitions in 2H-TaSe₂ and 2H-TaS₂,” Physical Review B, vol. 16, no. 10, p. 4583, 1977.

D. Maclean and M. H. Jericho, “Effect of the charge density wave transition on the thermal expansion of 2H-TaSe₂, NbSe₃ and o-TaS₃,” Physical Review B, vol. 47, no. 24, pp. 16169–16177, 1993.

S. Tomic, K. Biljakovic, D. Djurek et al., “Calorimetric study of the phase transition in Niobium triselenide NbSe₃,” Solid State Communications, vol. 38, pp. 109–112, 1981.

J. W. Brill and N. P. Ong, “Young's modulus of NbSe₃,” Solid State Communications, vol. 25, no. 12, pp. 1075–1078, 1978.

W. L. McMillan, “Microscopic model of charge density waves in 2H-TaSe₂,” Physical Review B, vol. 16, no. 2, pp. 643–650, 1977.

J. A. Aronovitz, P. Goldbart, and G. Mozurkewich, “Elastic singularities at the Peierls transition,” Physical Review Letters, vol. 64, no. 23, pp. 2799–2802, 1990.

R. S. Kwok, G. Gruner, and S. E. Brown, “Fluctuations and Thermodynamics of the charge density wave phase transition,” Physical Review Letters, vol. 65, no. 3, pp. 365–368, 1990.

J. W. Brill, M. Chung, Y.-K. Kuo et al., “Thermodynamics of the charge density wave transition in blue bronze,” Physical Review Letters, vol. 74, no. 7, pp. 1182–1185, 1995.

M. R. Hauser, B. B. Platt, and G. Mozurkewich, “Thermal expansion associated with the charge density wave in K0.3MoO₃,” Physical Review B, vol. 43, no. 10, p. 8105, 1991.

L. C. Bourne and A. Zettl, “Elastic anomalies in the charge density wave conductor K0.3MoO₃,” Solid State Communications, vol. 60, no. 10, pp. 789–792, 1986.

D. C. Johnston, “Thermodynamics of charge density waves in quasi one dimensional conductors,” Physical Review Letters, vol. 52, no. 23, p. 2049, 1984.

R. A. Craven, M. B. Salamon, G. DePasquali et al., “Specific heat of Tetrathiofulvalinium-Tetracyanoquinodimethane (TTF-TCNQ) in he vicinity of the metal-insulator transition,” Physical Review Letters, vol. 32, no. 14, pp. 769–772, 1974.

T. Tiedje, R. R. Haering, M. H. Jericho et al., “Temperature dependence of sound velocities in TTF-TCNQ,” Solid State Communications, vol. 23, no. 10, pp. 713–718, 1977.

M. Barmatz, L. R. Testardi, A. F. Garito et al., “Elastic properties of one dimensional compounds,” Solid State Communications, vol. 15, no. 8, pp. 1299–1302, 1974.

D. E. Schafer, G. A. Thomas, and F. Wudl, “High resolution thermal expansion measurements of tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ),” Physical Review, vol. 12, no. 12, p. 5532, 1975.

M. Saint-Paul, S. Holtmeier, R. Britel et al., “An ultrasonic and thermal expansion study of the quasi one dimensionbal compound (TaSe₄)₂I,” Journal of Physics: Condensed Matter, vol. 8, no. 12, pp. 2021–2041, 1996.

D. Staresinic, A. Kiss, K. Biljakovic et al., “Specific heat of the charge density wave compounds o-TaS₃ and (TaSe₄)₂I,” The European Physical Journal B, vol. 29, pp. 71–77, 2002.

M. Saint-Paul, C. Guttin, P. Lejay et al., “Elastic anomalies at the charge density wave transition in TbTe₃,” Solid State Communications, vol. 233, pp. 24–29, 2016.

M. Saint-Paul, G. Remenyi, C. Guttin et al., “Thermodynamic and critical properties of the charge density wave system ErTe₃,” Physica B, vol. 504, pp. 39–46, 2017.

M. Saint-Paul, C. Guttin, P. Lejay et al., “Elastic properties of the charge density wave system HoTe₃,” Internatiuonal Journal of Modern Physics B, vol. 32, no. 23, Article ID 1850249, 2018.

S. L. Bud’ko, S. A. Law, P. C. Canfield, and et al, “Thermal expansion and magnetostriction of pure and doped RAgSb₂ (R=Y, Sm, La) single crystals,” Journal of Physics: Condensed Matter, vol. 20, Article ID 115210, 2008.

C. S. Lue, Y. F. Tao, K. M. Sivakumar et al., “Weak charge density wave transition in LaAgSb₂ investigated by transport, thermal and NMR studies,” Journal of Physics: Condensed Matter, vol. 19, no. 40, Article ID 406230, 2007.

K. Rossnagel, “On the origin of charge density waves in select layered transition metal dichalcogenides,” Journal of Physics: Condensed Matter, vol. 23, Article ID 213001, 2011.

L. Landau and E. Lifshitz, Statistical Physics, Pergamon Press, 1968.

L. R. Testardi, “Elastic modulus, thermal expansion,and specific heat at a phase transition,” Physical Review B, vol. 12, no. 9, pp. 3849–3854, 1975.

R. A. Cowley, “Structural phase transitions,” Advances in Physics, vol. 39, no. 1, pp. 1–110, 1980.

R. Blink and A. P. Levanyk, Incommensurate Phases in Dielectrics: Fundamentals vol. 1 modern Problems in Condensed Matter Sciences, Elsevier Ltd, 1986.

W. Rehwald, A. Vonlanthen, J. K. Krüger et al., “Study of the elastic properties of K₂SeO₄ at the phase transition to incommensurate and the commensurate phase by ultrasonic techniques and Brillouin spectroscopy,” Journal of Physics C: Solid State Physics, vol. 13, no. 20, pp. 3823–3834, 1980.

W. Rehwald, “The study of structural phase transitions by means of ultrasonic experiments,” Advances in Physics, vol. 22, pp. 721–755, 1973, B. Luthi, "Physical Acoustic in the solid state" Springer-Verlag Berlin 2005.

R. Weber and R. Street, “The heat capacity anomaly of chromium at 311 K (antiferromagnetic to paramagnetic transition),” Journal of Physics F: Metal Physics, vol. 2, no. 5, pp. 873–877, 1972.

D. I. Bolef and J. De Klerk, “Anomalies in the elastic constants and thermal expansion of chromium single crystals,” Physical Review, vol. 129, no. 3, pp. 1063–1067, 1963.

M. B. Walker, “Phenomenological theory of the spin density wave state of chromium,” Physical Review, vol. 22, no. 3, p. 1338, 1980.