Survey of the Thermodynamic Properties of the Charge Density Wave Systems

We reexamine the thermodynamic properties such as specific heat, thermal expansion, and elastic constants at the charge density wave (CDW) phase transition in several oneand 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 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.


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 .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  exp(Φ) [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.
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].

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 T < T C .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 (/ = 0) 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 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 ûC P and to the stress dependence components,   /  , at the phase transition T CDW : ( The term,  2  Δ( 2   / 2  ), proportional to the entropy variation and multiplied by the second derivative  2   /  2  [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 T CDW are evaluated using the extrapolated linear temperature dependence of the high temperature background as shown in Figures 1(a) and 1(b).

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 K 2 SeO 4 [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-TaSe 2 , 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-TaSe 2 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 T CDW 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 e i and the square of the order parameter Q [51,52].The expansion of the free energy density in power of Q 2 and e i 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 e i and quadratic in Q as      2 are responsible for a decrease of the longitudinal elastic constant C ii (velocity V ii ).The decrease of the longitudinal elastic constant C ii is proportional to the square of the coupling constant  .: The coupling second terms  2   2 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 T CDW [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 T CDW in different materials [16,17,20,36,37,[41][42][43].The hardening observed in the ordered phase with several compounds is analyzed in Section 3.5.

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 û of the thermal expansion coefficient in the basal plane at the upper phase transition T CDW1 = 330 K of TbTe 3 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.
Similar discontinuities are observed along the  →  and  →  axes.
Only the values of û along the  →  axis are reported in Table 1.
(d) Finally the thermal expansion discontinuities along the  →  axis observed in LaAgSb 2 at the upper (T CDW1 = 210 K) and lower CDW phase transition (T CDW2 = 185 K) [44] are reported in Table 4. Thermal expansion discontinuity along the  →  axis observed at the spin density wave transition T SDW = 310 K for chromium [54] is also reported in Table 4.
The stress dependence deduced using (4) from the thermal expansion coefficient discontinuities û measured at T CDW along one crystallographic direction is given by The stress dependence   / values deduced at T CDW from the values given in Tables 1-4 are reported versus the transition temperature T CDW in Figure 3.It seems that   / increases with increasing T CDW .The high values of the stress dependence   / are found with the rare earth tritellurides and 2H-NbSe 2 .Such a high value   / ∼ 100 K/GPa obtained for TbTe 3 is in agreement with the value dT CDW /dp = 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.
Two different values û/ ∼ 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 (  = 310 K ) of Chromium [54] is also reported in Table 4.All the absolute values û/ are shown in Figure 4.
Very small values are reported for 2H-TaSe 4 and (TaSe 4 ) 2 I.A general tendency is observed: the amplitude of the sound velocity discontinuities increases with T CDW .We mention that large discrepancies exist among the experimental Young modulus values.

Consistency.
The consistency of Ehrenfest relations ( 1) and ( 2) may be checked by evaluating the value  2 , equivalent to an effective elastic constant, from the discontinuities ûV/V, û, and ûC p measured at the CDW phase transition from different experiments following (6) which is rewritten as The values  calculated =  2 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 TbTe 3 and ErTe 3 .An unrealistic value of about 5000 GPa is evaluated with the very small thermal expansion jump value, û ∼ 2 × 10 −7 K −1 , measured with 2H-NbSe 2 in [18].In contrast the thermal expansion results, û ∼ 3 × 10 −6 K −1 , reported in [19] give a value ∼35 GPa.A realistic value 250 GPa is evaluated for K 0.3 MoO 3 in [31].A smaller value of 37 GPa is obtained for the one-dimensional conductor (TaSe 4 ) 2 I.In contrast large values 1800 GPa and 800 GPa are obtained for TiSe 2 .A small value of about 16 GPa is evaluated for 2H-TaSe 2 .No discontinuity, û ∼0, is observed for TTF-TCNQ and C calculated given by ( 11) cannot be evaluated for this material (Table 3).Finally a realistic value C calculated is evaluated (see (11)) at the SDW phase transition in chromium which has been previously discussed in [53][54][55].The ratio values between C calculated and the measured elastic constant are shown in Figure 5.   1-4), C calculated / C measured (column 10 in Tables 1-4).C calculated / C measured ∼ 1 indicates that the Ehrenfest relations are satisfied.
In conclusion the Ehrenfest equations are approximately satisfied by several materials: the rare earth tritellurides TbTe 3 and ErTe 3 , the transition metal dichalcogenide 2H-NbSe 2 , and the one-dimensional conductors K 0.3 MoO 3 and (TaSe 4 ) 2 I.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-TaSe 2 and TiSe 2 compound do not satisfy the Ehrenfest equations.

Temperature Dependence of the CDW Order Parameter.
The increase of the elastic velocity û below T CDW shown by the dotted black line in Figure 1(b) is related to the square of the order parameter Q (T) (see ( 9)).û/ is analyzed with the following relation: where (0) is the value of the order parameter at T= 0K and [û/] 0 is the maximum value of the relative velocity at T=0K and û/ = 0 at T CDW .For simplicity all the data are normalized at T=0 where û/ = 0.It results in the fact that ( 12) is changed by  The temperature dependence of the velocity û/ 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 [()/(0)] 2  [1]: The blue dashed curve is calculated with 0.0017{[()/ (0)] 2  − 1]} for 2H-NbSe 2 with T CDW = 32 K [16,17].The pink dashed curve is calculated with 0.0086{[()/ (0)] 2  − 1} for 2H-TaS 2 with T CDW =75 K [16,17].The black dashed curve is calculated with 0.014{[()/(0)] 2  − 1} for TTF-TCNQ with T CDW = 50 K [36] and the violet dashed curve with 0.06{[()/(0)] 2  − 1} for TiSe 2 with T CDW = 200 K [20].The black circles are values for ErTe 3 with T CDW = 260 K [42].
A remarkable feature is the increase of the amplitude [û/] 0 with T 2 CDW , [û/] 0 ∼ 10 −6 × T 2 CDW , in Figure 7.It yields the fact that the order parameter Q(0) proportional to {[û/] 0 } 0.5 increases with the charge density wave transition temperature T CDW in agreement with BCS theory.
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].

Conclusions
Similar features in the thermodynamic properties at the CDW phase transition T CDW are found in all the CDW materials under review.The amplitude of the specific heat anomaly at the CDW phase transition T CDW is sample dependent but the amplitude increases (roughly) linearly with increasing T CDW in agreement with a second-order phase transition.The (mean field theory) Ehrenfest equations are approximately satisfied by several compounds: the rare earth tritellurides TbTe 3 , ErTe 3 compounds, the transition metal dichalcogenide 2H-NbSe 2 compound, and several quasi-one-dimensional conductors.In contrast large inconsistency in the Ehrenfest relationships is found with the transition metal dichalcogenide compounds 2H-TaSe 2 and TiSe 2 .Lattice anharmonicity acting through the stress dependence of the phase transition temperature T CDW 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 LaAgSb 2 has been classified as a 3D CDW system.The Ehrenfest relationships should be verified in this material.

Figure 1 :
Figure 1: Color on line.Temperature dependence of the thermodynamic properties of TbTe 3 .(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].

Figure 3 :
Figure 3: Color on line.Stress dependence   / derived from Ehrenfest eq.(4) with the thermal expansion coefficient discontinuities û (Tables1 and 2) measured at the CDW phase transitions as a function of T CDW .û along the  →  axis in TbTe 3 at T CDW1 = 330 K (blue

Figure 5 :
Figure 5: Color on line.Ratio between the calculated elastic constants (C calculated ) using (11) and the measured values of the elastic constants (Tables1-4), C calculated / C measured (column 10 in Tables1-4).C calculated / C measured ∼ 1 indicates that the Ehrenfest relations are satisfied.

2 Figure 7 :
Figure 7: Color on line.Amplitudes of the square of the order parameter Q 2 0 at T = 0K, which is proportional to the values [ûV/V] 0 deduced from Figure 6, are reported as a function of T CDW .The solid line increases as  2  .
is related to the elastic velocity   by   =  2  ,  being the mass density.The discontinuities of the elastic constants û  /  (or velocity û  /  = û  /2  ) at a second-order phase transition are related to the stress dependence   /  by 2.

Table 1 :
Materials (2D CDW) TbTe3 , HoTe 3 , and ErTe 3 rare earth tritellurides.Molar volumes are given in cm 3 , specific heat discontinuities in J/molK, thermal expansion discontinuities in K −1 , elastic velocity discontinuities in ûV/V, and measured and calculated elastic constants in GPa and ratios.

Table 4 :
Three-dimensional materials.Molar volumes are given in cm 3 , specific heat discontinuities in J/molK, thermal expansion discontinuities in K−1