Review Article  Open Access
M. SaintPaul, P. Monceau, "Survey of the Thermodynamic Properties of the Charge Density Wave Systems", Advances in Condensed Matter Physics, vol. 2019, Article ID 2138264, 14 pages, 2019. https://doi.org/10.1155/2019/2138264
Survey of the Thermodynamic Properties of the Charge Density Wave Systems
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 twodimensional materials. The amplitude of the specific heat anomaly at the CDW phase transition T_{CDW} increases with increasing T_{CDW} and a tendency to a lineal temperature dependence is verified. The Ehrenfest mean field theory relationships are approximately satisfied by several compounds such as the rare earth tritelluride compound TbTe_{3}, transition metal dichalcogenide compound 2HNbSe_{2}, and quasionedimensional conductor K_{0.3}MoO_{3}. In contrast inconsistency exists in the Ehrenfest relationships with the transition metal dichalcogenide compounds 2HTaSe_{2} and TiSe_{2} having a different thermodynamic behavior at the transition temperature T_{CDW}. It seems that elastic properties in the ordered phase of most of the compounds are related to the temperature dependence of the order parameter which follows a BCS behavior.
1. Introduction
The electron density of a low dimensional (onedimensional (1D) or twodimensional (2D)) compound may develop a wavelike periodic variation, a charge density wave (CDW), accompanied by a lattice distortion when temperature drops below a critical temperature T_{CDW} [1–46]. CDW ordering is driven by an electron phonon coupling. The concept of charge density wave is related to the initial work of Peierls [1], followed by Fröhlich [2] when it was demonstrated that a onedimensional metal is instable with respect to a phase transition in the presence of electron phonon coupling.
A charge density wave is characterized by a spatial periodic modulation of the electronic density concomitant with a lattice distortion having the same periodicity. The properties of the CDW state can be described by an order parameter [1]. The fluctuations of the lattice distortions can be described by amplitude and phase modes [1]. This variation, charge density wave, in the electron density is receiving intense study because it often competes with another ground state (superconductivity). A CDW order can be formed with one fixed wave vector or multiple wave vectors. For example, the incommensurate ordering vector Q_{1} of the prototypal rare earth tritelluride ErTe_{3} at the upper CDW phase transition T_{CDW1} = 265 K is parallel to the axis, whereas the incommensurate ordering parameter Q_{2} observed at the lower CDW phase transition T_{CDW2} = 150 K with ErTe_{3} is parallel to the axis. In contrast the CDW order in the dichalcogenide compounds (for example, 2HNbSe_{2}) is formed by three superposed charge density waves.
The origin of the CDW phase transition observed in the twodimensional 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 Xray 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 twodimensional transition metal dichalcogenide compounds [16–25] and in quasionedimensional conductors [26–37] has generated a considerable interest over the past 30 years. The onset of the CDW order has remarkable effects on the thermodynamic properties since below T_{CDW} a gap opens up in the density of the electronic states. A microscopic model is given by McMillan [28]. The elastic properties of quasi low dimensional conductors undergoing charge and spin density phase transitions are reviewed by Brill [3]. Several reviews discuss the properties of the charge density wave systems [4–6, 45].
We reexamine the thermodynamic experimental data such as specific heat, thermal expansion, and elastic constants of several CDW compounds. We give a survey of the Ehrenfest relations using the experimental data obtained at the CDW phase transition in different materials: rare earth tritellurides RTe_{3} (TbTe_{3}, ErTe_{3}, and HoTe_{3}) [8, 41–43], transition metal dichalcogenides MX_{2} compounds (2HNbSe_{2} [17–19], 2HTaSe_{2} and 2HTaS_{2} [16, 24, 25], and TiSe_{2}[20–23]), quasionedimensional conductors (NbSe_{3} [25, 27], K_{0.3}MoO_{3} [30–33], (TaSe_{4})_{2}I [39, 40], and TTFTCNQ [35–38]), and in the system (LaAgSb_{2}) [44, 45].
Departures from the mean field behavior of the thermodynamic properties are generally attributed to fluctuations which belong to the 3D XY criticality class [27–34]. The contribution of the fluctuations is important in the quasionedimensional conductors [28] and in the transition metal dichalcogenides (2HTaSe_{2}, 2HTaS_{2}) [24]. Small fluctuation effects are observed around T_{CDW} in the rare earth tritellurides TbTe_{3} [41] and ErTe_{3} [42].
The amplitude of the lattice distortion is governed by the electron phonon coupling strength [46]. A moderately strong electron phonon coupling is reported for the rare earth tritellurides (ARPES experiments [10, 11]), similar to that observed in quasi1D CDW systems such as K_{0.3}MoO_{3} and NbSe_{3}. In a weak coupling CDW, the specific heat behavior at the CDW phase transition is driven by the electronic entropy [28, 46]. In a strong coupling CDW the transition is also governed by the entropy of the lattice [28, 46].
2. Thermodynamic Properties
2.1. Ehrenfest Relations
At a secondorder phase transition T_{C}, the order parameter Q increases continuously in the ordered phase at . The Landau free energy [47] can be written without knowing the microscopic states as where F_{0} describes the temperature dependence of the high temperature phase and the constant parameters a and B are positive. The order parameter that minimizes the free energy () is given byThe entropy (S) is derived from the free energy (F), , and the specific heat at constant pressure is given by . There is a jump in the specific heat (Figure 1(a)) at the secondorder phase transition T_{C} given by [47]Discontinuities in the thermal expansion coefficients and the elastic constants are also observed at a secondorder phase transition. An example (TbTe_{3}) is shown in Figures 1(b) and 1(c). The thermodynamic quantities at a secondorder phase transition such as a charge density wave phase transition are generally discussed with the Ehrenfest relations reformulated by Testardi [48]. The discontinuity in the thermal expansion coefficients is related to the specific heat jump ∆C_{P} and to the stress dependence components, , at the phase transition T_{CDW}:where i=1, 2, and 3 correspond to the , , and crystallographic axes directions and V_{m} is the molar volume.
(a)
(b)
(c)
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 secondorder phase transition are related to the stress dependence byThe 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 T_{CDW} 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 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 2HTaSe_{2}, the modulation periodicity is temperature dependent and may be lockin at low temperatures to a value that is commensurate with the periodicity of the basis structure [28, 46]. The lockin transition is a firstorder phase transition and very different in nature from the incommensurate instability [46]. 2HTaSe_{2} undergoes a normal to incommensurate transition (secondorder) at 122 K and an incommensuratecommensurate transition (firstorder) 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 bywhere g and h are the coupling constants.
The interacting terms linear in e_{i} and quadratic in Q as 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 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, 41–43]. 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 ∆C_{P} measured at the CDW phase transitions in the following materials:(a)Rare earth tritellurides TbTe_{3} [41] and ErTe_{3} [42](b)Transition metal dichalcogenides 2HNbSe_{2} [19], TiSe_{2} [22, 23], 2HTaSe_{2}, and 2H TaS_{2} [24]: The mean field contribution for 2HTaSe_{2} and 2HTaS_{2} was estimated in [23]. These two compounds are characterized by large fluctuations(c)Quasionedimensional conductors NbSe_{3} [26], K_{0.3}MoO_{3} [30, 31], (TaSe_{4})_{2}I [40], and TTFTCNQ [35](d)Threedimensional material LaAgSb_{2} [44] and Cr [53]
The specific heat discontinuities ∆C_{P} are reported in Tables 1–4 and they are shown as a function of the CDW phase transition temperature T_{CDW} 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 TTFTCNQ, 2HTaS_{2}, and 2HTaSe_{2} has a larger coefficient A_{1}=4 ×10^{−2} J/molK^{−2}. The second line has a coefficient A_{2}=3 ×10^{−3} J/molK^{−2}, 10 times smaller than A_{1}. This second line is followed approximately by the rare earth tritelluride compounds ErTe_{3} and TbTe_{3} 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 LaAgSb_{2} (red square symbol) is also situated on line .




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 T_{CDW1} = 330 K of TbTe_{3} were obtained from thermal expansion measurements using Xrays 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 T_{CDW2} = 150 K of ErTe_{3} were obtained from the thermal expansion measurements using Xrays 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 2HNbSe_{2} [18, 19], TiSe_{2} [20], and 2HTaSe_{2} [25] are reported in Table 2. Very different experimental results were found for 2HNbSe_{2} [18, 19].(c)Thermal expansion discontinuities determined along the directions in NbSe_{3} [25], K_{0.3}MoO_{3} [31, 32], (TaSe_{4})_{2}I [39], and TTFTCNQ [38] are reported in Table 3.(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 measured at T_{CDW} along one crystallographic direction is given byThe 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 2HNbSe_{2}. Such a high value K/GPa obtained for TbTe_{3} 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 quasionedimensional conductors.
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 TbTe_{3} [41], ErTe_{3} [42], and HoTe_{3} [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 ab plane at the CDW phase transition in dichalcogenides 2HNbSe_{2} [16, 17], TiSe_{2} [20], 2HTaSe_{2} [16, 17], and 2HTaS_{2} [16, 17], in quasionedimensional conductors K_{0.3}MoO_{3} [31, 33] and (TaSe_{4})_{2}I [39] (Tables 2 and 3).
Two different values and ~0.01 (dotted black line in Figure 4) are reported for the organic conductor TTFTCNQ [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.
Very small values are reported for 2HTaSe_{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.
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 ∆C_{p} measured at the CDW phase transition from different experiments following (6) which is rewritten asThe 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 TbTe_{3} and ErTe_{3}. An unrealistic value of about 5000 GPa is evaluated with the very small thermal expansion jump value, 0^{−7} K^{−1}, measured with 2HNbSe_{2} 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 K_{0.3}MoO_{3} in [31]. A smaller value of 37 GPa is obtained for the onedimensional 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 2HTaSe_{2}. No discontinuity, ∆α ~0, is observed for TTFTCNQ 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.
In conclusion the Ehrenfest equations are approximately satisfied by several materials: the rare earth tritellurides TbTe_{3} and ErTe_{3}, the transition metal dichalcogenide 2HNbSe_{2}, and the onedimensional 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 2HTaSe_{2} and TiSe_{2} compound do not satisfy the Ehrenfest equations.
3.5. Temperature Dependence of the CDW Order Parameter
The increase of the elastic velocity 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)). 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 T_{CDW}. For simplicity all the data are normalized at T=0 where . It results in the fact that (12) is changed byThe 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 2HNbSe_{2} with T_{CDW} = 32 K [16, 17]. The pink dashed curve is calculated with for 2HTaS_{2} with T_{CDW}=75 K [16, 17]. The black dashed curve is calculated with for TTFTCNQ with T_{CDW} = 50 K [36] and the violet dashed curve with 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 with , , in Figure 7. It yields the fact that the order parameter Q(0) proportional to 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].
4. 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 secondorder 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 2HNbSe_{2} compound, and several quasionedimensional conductors. In contrast large inconsistency in the Ehrenfest relationships is found with the transition metal dichalcogenide compounds 2HTaSe_{2} and TiSe_{2}. 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 quasionedimensional 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright © 2019 M. SaintPaul 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.